Keywords: Seismic damage assessment in buildings, Machine learning, Efficiency and sufficiency of IM, Capacity curves Machine learning, 2015 Gorkha Nepal earthquake, post-earthquake building-damage data, Earthquake building-damage, DaDO building damage database, Machine learning, RISK-UE, Seismic vulnerability of buildings, Italy Machine learning, 2010 Haiti earthquake building damage dataset, host-to-target transfer, large-scale damage assessment Structural performance assessment, strong motion recordings in buildings, building frequency, efficiency experimental capacity curves, earthquake data recordings from buildings, building's coseismic capacity variation after damage, operational earthquake loss forecasting, structural health monitoring

Earthquakes are less frequent, but their consequences represent one-fifth of natural catastrophes.

Therefore, for efficient seismic risk management and mitigation, it is crucial to assess potential damage to a building (building-level) and its extent within an urban environment (large-scale) before the occurrence of an earthquake. Recently, data collected from buildings have become openly accessible, including post-earthquake damage survey databases and strong-motion data recordings. These datasets offer an opportunity to test the new models in seismic damage assessment: rapid loss assessment at large-scale and operational earthquake loss forecasting in building-level based on time-varying vulnerability assessment. Therefore, in this study, we utilized these earthquake data from buildings to develop data-driven-based insights into seismic damage assessment focusing in rapid damage assessment at large scale and building-level damage assessment. This thesis is divided into twofold.

First, focusing on rapid damage assessment at a large scale, we present our findings on testing machine learning models for damage prediction using post-earthquake building damage dataset. We evaluated different issues related to machine learning models for large-scale damage assessments, including the machine learning model selection, the relationship between building features and damage level, the effectiveness of readily available features (such as age, number of stories, floor area, height) in largescale damage assessment, the effectiveness of using machine learning model trained on past earthquake building damage data to predict potential damage during future earthquakes, the effectiveness of machine learning method compared to classical damage assessment methods, and the transferability of machine learning models developed in a region to assess damage in another region. Second, focusing on building-level damage assessment, we present our findings on evaluating the co-seismic building response during earthquake loading using the strong motion earthquake data recording from buildings.

More precisely, we examined the relationship between the co-seismic response of a building as a function of ground motion intensity measures and the associated variabilities related to the timevariation of structural state due to induced earthquake damage in a building, earthquake magnitude and distance, grouping buildings by class and region. In addition, we evaluated the lateral load-resisting capacity of a building and its variation due to the earthquake-induced damage and its accumulation over time in a specific building, among buildings with the same class, and the adjustments needed to account for the actual structural health required for time-varying damage modeling. Finally, some general conclusions are presented, together with some perspective work.

Résumé

Les tremblements de terre sont moins fréquents, mais leurs conséquences représentent un cinquième des catastrophes naturelles. Par conséquent, pour une gestion et une atténuation efficaces des risques sismiques, il est essentiel d'évaluer les dommages potentiels causés à un bâtiment (au niveau du bâtiment) et leur étendue dans un environnement urbain (à grande échelle) avant qu'un tremblement de terre ne se produise. Récemment, des données collectées sur les bâtiments sont devenues librement accessibles, notamment des bases de données d'enquêtes sur les dommages après les tremblements de terre et des enregistrements de données sur les mouvements violents. Ces ensembles de données permettent de tester les nouveaux modèles d'évaluation des dommages sismiques : évaluation rapide des pertes à grande échelle et prévision opérationnelle des pertes dues aux tremblements de terre au niveau des bâtiments sur la base d'une évaluation de la vulnérabilité variable dans le temps. Par conséquent, dans cette étude, nous avons utilisé ces données sur les tremblements de terre des bâtiments pour développer des idées basées sur les données dans l'évaluation des dommages sismiques en se concentrant sur l'évaluation rapide des dommages à grande échelle et l'évaluation des dommages au niveau des bâtiments. Cette thèse est divisée en deux parties. Premièrement, en se concentrant sur l'évaluation rapide des dommages à grande échelle, nous présentons nos conclusions sur les tests des modèles d'apprentissage automatique pour la prédiction des dommages à l'aide d'un ensemble de données de dommages aux bâtiments après le tremblement de terre. Nous avons évalué différentes questions liées aux modèles d'apprentissage automatique pour l'évaluation des dommages à grande échelle, y compris la sélection du modèle d'apprentissage automatique, la relation entre les caractéristiques des bâtiments et le niveau des dommages, l'efficacité des caractéristiques facilement disponibles (telles que l'âge, le nombre d'étages, la surface au sol, la hauteur) dans l'évaluation des dommages à grande échelle, l'efficacité de l'utilisation d'un modèle d'apprentissage automatique formé sur les données des dommages des bâtiments après un tremblement de terre pour prédire les dommages potentiels lors de futurs tremblements de terre, l'efficacité de la méthode d'apprentissage automatique par rapport aux méthodes classiques d'évaluation des dommages, et la transférabilité des modèles d'apprentissage automatique développés dans une région pour évaluer les dommages dans une autre région. Deuxièmement, en nous concentrant sur l'évaluation des dommages au niveau des bâtiments, nous présentons nos conclusions sur l'évaluation de la réponse cosismique des bâtiments pendant la charge sismique à l'aide de l'enregistrement des données sur les tremblements de terre à fort mouvement à partir des bâtiments. Plus précisément, nous avons examiné la relation entre la réponse cosismique d'un bâtiment en fonction des mesures d'intensité du mouvement du sol et les variabilités associées liées à la variation temporelle de l'état structurel due aux dommages induits par le tremblement de terre dans un bâtiment, à la magnitude du tremblement de terre et à la distance, en regroupant les bâtiments par classe et par région. En outre, nous avons évalué la capacité de résistance aux charges latérales d'un bâtiment et sa variation due aux dommages induits par le tremblement de terre et leur accumulation dans le temps dans un bâtiment spécifique, parmi les bâtiments de la même classe, et les ajustements nécessaires pour tenir compte de l'état structurel réel au cours de la modélisation des dommages variables dans le temps. Enfin, des conclusions générales sont présentées, ainsi que des perspectives de travail.

Mots-clés: Évaluation des dommages sismiques dans les bâtiments, apprentissage automatique, efficacité et suffisance de l'IM, courbes de capacité Acknowledgement I would like to express my sincere gratitude to my supervisors Philippe Guéguen and Danijel Schorlemmer. Philippe, thank you for providing me with valuable guidance, feedback, support, and care throughout this exciting journey. Danijel, your advice and support have been instrumental in helping me achieve my research goals.

I would like to express my gratitude to Jocelyn Chanussot, Sophie Giffard-Roisin, and Adrien Pothon for their invaluable advice during the annual thesis evaluation. Thank you for taking the time to provide me with constructive feedback, which has helped me to improve and refine my research.

I am also grateful to the URBASIS-EU project for their financial support and the invaluable experiences gained from my involvement organized during the project.

I extend my heartfelt thanks to the reviewers for their time and effort in reviewing my thesis.

To my family, thank you for your unwavering support and encouragement throughout my academic journey.

Lastly, I would like to thank all my friends for the memorable moments we shared. Your companionship has been a source of strength and inspiration for me.

Introduction

Earthquakes are less frequent among natural catastrophes; however, they contribute significantly to the physical and social consequences of natural hazards such as human casualties, economic losses, psychological trauma, social disruption, and environmental destruction.

The global population is increasing, and the proportion of the population living in urban areas is also remarkably increasing. For example, 55% of the global population was living in urban space at the end of 2020, which is anticipated to be 68% by 2050. As a consequence, urban infrastructures are developing at an accelerated rate (UN, 2018). Globally, 16.8% of urban areas are exposed to earthquake-prone areas. Out of 94 earthquake-prone countries, the United States, Iran, Turkey, Mexico, and Japan are the top 5 countries with earthquake-prone urban areas; and Pakistan, China, Iran, India, Afghanistan, Nepal, and Bangladesh are the top 7 countries expected with higher fatalities (He et al., 2021).

Earthquakes' consequences represent one-fifth of annual economic losses, with an annual average of approximately US$34.7 billion and more than 20 thousand deaths per year between 1900 and 2018 [START_REF] Daniell | The global role of earthquake fatalities in decisionmaking: earthquakes versus other causes of fatalities[END_REF]Silva et al., 2019). Out of all the earthquakes that occur worldwide, more than 56% of them result in casualties, and more than 91% of earthquake casualties are due to building collapses (He et al., 2021). Some of the deadliest earthquakes that occurred in the last 15 years include (Munich Re, IMF, World Bank, EM-DAT disaster database): the 2010 Haiti earthquake, which resulted in an estimated fatalities more than 250,000, left more than 2 million people injured and 1.5 million people homeless, with an estimated economic loss more than US$ 7.5 billion; the 2011 Tohoku earthquake, which resulted an estimated fatalities more than 20,000, left more than 6,000 people injured and 1.05 million people homeless, with an economic loss more than US$ 250 billion; the 2015 Nepal earthquake, which resulted in an estimated fatalities more than 8000, left more than 22,000 people injured and 3.0 million people homeless, with an economic loss more than US$ 1.9 billion; the 2021 Haiti earthquake, which caused an estimated fatalities more than 2500, left more than 12,000 people injured and 30,000 people homeless, with an economic loss more than US$ 1.6 billion; the latest 2023 Turkey-Syria earthquake, which resulted in an estimated fatalities more than 56,000, left more than 121,000 people injured and 2.7 million people homeless, with an economic loss more than US$ 109.1 billion, as of 01/04/2023. A significant number of buildings were damaged during these earthquakes. The building's poor performance against the seismic loading, observed in recent earthquakes, highlights the need to enhance our state-of-art knowledge of the complex response of buildings during seismic loading and to understand better the occurrence of damages, characterize their nature, and develop strategies to minimize them.

(i) Progress in collecting earthquake data from buildings

The availability of earthquake data, such as accelerometric recordings, parametric values, postearthquake damage surveys, collected from buildings is the key to understanding the earthquake's impact and necessary mitigation strategies [START_REF] So | Data and its role in reducing the risk of disasters in the built environment[END_REF]. In recent years, communities worldwide are collaborating to collect earthquake data from buildings in the form of (a) post-earthquake building damage datasets (Dolce et al., 2019;MINVU, 2021;MTPTC, 2010;NPC, 2015;[START_REF] Omoya | A relational database to support post-earthquake building damage and recovery assessment[END_REF][START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF], and (b) earthquake data recordings from buildings [START_REF] Astorga | /wooden buildings. Moment magnitude (Mw) varies from 3[END_REF], and share them with open access. This post-earthquake building damage dataset includes information on the damage sustained by buildings, collected through visual screening, and building parameters describing the structural and built-up environment. Damage surveys were carried out on many buildings covering a wide spatial range. For example, the post-earthquake building damage database of the 2015 Nepal earthquake (NPC, 2015) and 2010 Haiti earthquake (MTPTC, 2010) has more than 700,000 and 300,000 buildings collected from several districts of Nepal and Haiti, respectively. In addition, the database of observed damage in Italy (DaDO), a platform of the Civil Protection Department developed by the Eucentre Foundation, has more than 100,000 buildings collected after several earthquakes of different magnitudes from different regions (Dolce et al., 2019).

Similarly, earthquake recordings in buildings (NDE1.0) released by [START_REF] Astorga | /wooden buildings. Moment magnitude (Mw) varies from 3[END_REF] contains 8,250 records of strong motion data collected over the years from 2,737 events and 108 buildings from different designs and regions. The database includes structural engineering demand parameters (EDP) such as drift ratios (relative displacement between top and bottom floors normalized by their height), peak top values of acceleration, velocity, displacement, pre-and co-seismic fundamental frequencies, the ground motion intensity measures (IM) such as peak ground acceleration, velocity, displacement, spectral acceleration, velocity, displacement. In addition, this database contains information on buildings and earthquakes characteristics such as building design, location, magnitude, and epicentral distance.

(ii) Methods for assessing damage in buildings

Understanding potential damage to buildings and its extent within an urban environment during an earthquake event is crucial for decision-makers, emergency planners, insurers, and reinsurers for effective seismic risk management and mitigation in earthquake-prone areas (Bommer and Crowley, 2006;[START_REF] Earle | Prompt assessment of global earthquakes for response (pager): A system for rapidly determining the impact of earthquakes worldwide[END_REF]Riedel et al., 2015Riedel et al., , 2018)).

A building damage assessment usually involves the definition of the hazards affecting the region of interest (i.e., expected frequency and intensity of the ground shaking), the definition of the vulnerability of the assets exposed to the hazards (i.e., likelihood to suffer damage or loss), and a classification of the buildings in the region as exposure [START_REF] Rao | Earthquake Building Damage Detection based on Synthetic Aperture Radar Imagery and Machine Learning[END_REF]. The extent of damage to buildings is then calculated as a function of hazard, exposure, and vulnerability [START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF].

Several methods have been developed for assessing damage in buildings due to earthquakes depending on the level of information, methods, tools required for the assessment, and the objectives to be achieved [START_REF] Erdik | Rapid earthquake loss assessment after damaging earthquakes[END_REF][START_REF] Kassem | The seismic vulnerability assessment methodologies: A state-of-the-art review[END_REF]. Some of the most advanced methods include:

The machine learning method, first introduced by (Riedel et al., 2015), is an emerging robust method capable of handling a large amount of data, superior computational efficacy, ability to handle complex problems, and incorporation of uncertainties for building damage assessment [START_REF] Chi | Machine learning-based seismic capability evaluation for school buildings[END_REF][START_REF] Hegde | Applications of machine learning methods for engineering risk assessment -A review[END_REF][START_REF] Karmenova | Seismic Assessment of Urban Buildings Using Data Mining Methods[END_REF][START_REF] Rodriguez-Galiano | Classifying earthquake damage to buildings using machine learning[END_REF][START_REF] Sajedi | A data-driven framework for near real-time and robust damage diagnosis of building structures[END_REF][START_REF] Salehi | Emerging artificial intelligence methods in structural engineering[END_REF][START_REF] Sun | Machine learning applications for building structural design and performance assessment: State-of-the-art review[END_REF][START_REF] Zhang | A machine learning framework for assessing postearthquake structural safety[END_REF][START_REF] Zhao | Modelling and interpreting pre-evacuation decision-making using machine learning[END_REF]Mangalathu et al., 2020a;[START_REF] Harirchian | A synthesized study based on machine learning approaches for rapid classifying earthquake damage grades to rc buildings[END_REF][START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF][START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF]. In this method, the input features defining a building portfolio are initially mapped to the damage level (target) using supervised machine learning algorithms to create a damage prediction model. Subsequently, this model is utilized in other building portfolios with the same input features to assess the potential damage.

The Performance-based earthquake engineering (PBEE) method is a probabilistic framework designed to improve seismic risk decision-making [START_REF] Porter | An overview of PEER's performance-based earthquake engineering methodology[END_REF]. PBEE works in four stages: hazard analysis, structural analysis, damage analysis, and consequence analysis. In hazard analysis, intensity measures (IM) and their annual frequency of exceedance are defined by probabilistic seismic hazard assessment (PSHA). In structural analysis, the structure's response to a given IM is modeled and expressed in terms of engineering demand parameters (EDPs) such as structural drift, maximal top acceleration, etc. In damage analysis, damage measure (DM) is calculated based on EDP values.

Finally, the earthquake's consequences, in terms of repair costs, operability of the structure, and potential economic or human losses for a given DM, is calculated and expressed as decision variables (DVs) on which stakeholders base their decisions. In this framework, the relationship between the EDP and IM is utilized to assess damage in the building when compared with the reference EDP values at the damage-state threshold. Two natures of IM is defined: IM that results in the smallest variability in EDP|IM is defined as efficient IM, and IM that results in EDP|IM conditionally independent of earthquake magnitude and distance is defined as sufficient IM [START_REF] Luco | Probabilistic seismic demand analysis, SMRF connection fractures, and near-source effects[END_REF].

The vulnerability modeling method estimates the potential damage through the crossing of hazard, exposure, and vulnerability/fragility models [START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF]. The hazard model defines the ground-shaking potential in terms of expected frequency and intensity in a region of interest and can be calculated in three different ways: scenario-based, intensity-based, and frequency-based approach [START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF].

The exposure model defines the built environment in terms of vulnerability class, function, reconstruction cost, and spatial distribution. Exposure modeling involves grouping buildings into a class based on a set of attributes that characterize their vulnerability rather than classifying each building individually [START_REF] Rao | Earthquake Building Damage Detection based on Synthetic Aperture Radar Imagery and Machine Learning[END_REF].

The vulnerability and fragility model relates hazards with exposure to determine the likelihood of potential damage. The vulnerability model provides a more comprehensive overall estimate of damage using statistical functions. These functions are derived from post-earthquake damage observations from similar building classes experiencing a similar seismic load or based on experts' opinions [START_REF] Kassem | The seismic vulnerability assessment methodologies: A state-of-the-art review[END_REF][START_REF] Rossetto | Guidelines for the empirical vulnerability assessment[END_REF].

The fragility model estimates the probability of a building class reaching or exceeding a specific damage state as a function of ground motion intensity measures [START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF]. Fragility models are defined in three different ways: based-on expert opinion (heuristic), post-earthquake damage evaluation data (empirical), structural modeling and response simulation (analytical) (Lallemant et al., 2015).

(iii) Recent insights for assessing damage in buildings

In a recent European Project (RISE, [START_REF] Wiemer | Real-time earthquake rIsk reduction for a ReSilient Europe ( RISE )[END_REF], the dynamic damage assessment (DDA) framework was summarized as providing real-time assessment of seismic damage by addressing the two major limitations of the existing vulnerability modeling method: (a) the assumption of constant exposure and (b) the inability of structural vulnerability models to account for damage accumulation in buildings [START_REF] Wiemer | Real-time earthquake rIsk reduction for a ReSilient Europe ( RISE )[END_REF]. DDA method aims to consider the dynamic urban environment by using a time-varying exposure model and time-varying vulnerability models that can capture damage accumulation in buildings during different earthquakes over time [START_REF] Wiemer | Real-time earthquake rIsk reduction for a ReSilient Europe ( RISE )[END_REF]Nievas;[START_REF] Nievas | Deliverable 6.1 D6.1 Integration of RISE Innovations in the Fields of OELF, RLA and SHM Deliverable information[END_REF].

This framework was not new but compiled several ideas proposed by other authors for time-varying vulnerability and exposure.

First, the global dynamic exposure model is developing as a time-varying exposure model for DDA. It is a novel OpenStreetMap-based exposure model designed to capture the real-time spatial distribution of buildings, as well as their operational function, structural characteristics, occupancy, and valuation in real-time [START_REF] Schorlemmer | Global Dynamic Exposure and the OpenBuildingMap -A Big-Data and Crowd-Sourcing Approach to Exposure Modeling[END_REF]. The information related to building is collected explicitly (for example, location, number of floors, function), implicitly (for example, building shapes and positions), and semantically derived data based on expert judgment; and is growing at the rate of ~ 150,000 building data per day (consists of more than 375 million building footprints at the moment) [START_REF] Schorlemmer | Global Dynamic Exposure and the OpenBuildingMap -A Big-Data and Crowd-Sourcing Approach to Exposure Modeling[END_REF].

The time-varying vulnerability modeling both at large-scale (urban or regional) and building-level is achieved through the use of the rapid loss assessment (RLA) and Operational Earthquake Loss Forecasting (OELF) methods, respectively [START_REF] Wiemer | Real-time earthquake rIsk reduction for a ReSilient Europe ( RISE )[END_REF]Nievas;[START_REF] Nievas | Deliverable 6.1 D6.1 Integration of RISE Innovations in the Fields of OELF, RLA and SHM Deliverable information[END_REF].

The RLA method provides the first-order quantitative estimation of the damage induced in the buildings within a few hours after an earthquake at a larger scale through a time-varying exposure and vulnerability model (e.g., using machine learning methods as proposed initialy by Riedel et al. (2014Riedel et al. ( , 2015Riedel et al. ( , 2018)), updating existing vulnerability model with the Bayesian method).

The OELF method calculates the probable damage knowing the residual lateral load resisting capacity of a building after seismic loading. The residual capacity can be accessed through structural health monitoring, which involves continuous tracking of a building's structural state (for example, age, level of degradation, maintenance history, etc.) by measuring the dynamic parameters describing the building (e.g., structural period, (Trevlopoulos and Guéguen., 2018;[START_REF] Trevlopoulos | Earthquake risk in reinforced concrete buildings during aftershock sequences based on period elongation and operational earthquake forecasting[END_REF]).

The Host-to-target transfer method is an alternative approach for damage assessment when there is limited data available or when it is too costly due to time and resource-requirement to develop a regionspecific damage assessment method or for a specific building. In this case, the model developed in one condition (host) is transferred to another condition (target) by applying certain adjustments. The adjustment is applied from the generic host condition to the generic target conditions and/or targetspecific conditions: for example, transferring the damage assessment models from high-seismic areas to moderate-seismic areas with adjustments (e.g., [START_REF] Régnier | Contribution of ambient vibration recordings (free-field and buildings) for post-seismic analysis: The case of the Mw 7.3 Martinique (French Lesser Antilles) earthquake[END_REF][START_REF] Roca | A Simplified Method for Vulnerability Assessment of Dwelling Buildings and Estimation of Damage Scenarios in Catalonia[END_REF] or in regions with buildings of different seismic performance (e.g., Lagomarsino and Giovinazzi, 2006).

(iv) Key issues

During emergency situations, it may not be feasible to collect the necessary data for creating and updating the exposure model in relation to the vulnerability model for the RLA method due to time and resource constraints, as this information may not be readily available (Riedel et al., 2015). In this context, machine learning methods can offer a paradigm shift by reasonably assessing damage by relying on readily available data in a cost-effective way (Mangalathu et al., 2020a;[START_REF] Harirchian | A synthesized study based on machine learning approaches for rapid classifying earthquake damage grades to rc buildings[END_REF][START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF][START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF]. In addition, the machine learning method allows the development of relationships between readily available building features and damage level bypassing the vulnerability modeling step. It can handle a large amount of data so it could easily incorporate the global dynamic exposure model. However, the machine learning method is still in its early stages and requires rigorous testing across various issues such as method, data, effectiveness, and transferability before it can be implemented in the RLA framework for large-scale damage assessment (Mangalathu et al., 2020a;[START_REF] Harirchian | A synthesized study based on machine learning approaches for rapid classifying earthquake damage grades to rc buildings[END_REF][START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF][START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF].

Specifically, the following issues need to be addressed:

Which machine learning algorithm and damage assessment framework is robust for RLA application? What are the most crucial features, and how are they related to damage level? How about the host-to-target transfer, i.e., can machine models developed in one region be transferred between regions, and what parameters influence the model performance? The data are not clean and uniform, so how does that impact the performance of machine learning models? Do they outperform conventional damage assessment methods?

In PBEE and OELF methods, building response to ground motion intensity measures is typically developed through structural modeling and response simulation of a representative building in a given class, using simple (e.g., single-degree-of-freedom oscillators) to most advanced (e.g., finite elementbased methods) modeling methods [START_REF] Iervolino | Assessing uncertainty in estimation of seismic response for PBEE[END_REF][START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF]Nievas;[START_REF] Nievas | Deliverable 6.1 D6.1 Integration of RISE Innovations in the Fields of OELF, RLA and SHM Deliverable information[END_REF].

Structural modeling involves significant assumptions about the building's attributes (e.g., materials, design, geometry, foundation, period, damping, etc.). The response simulation uses different natural or synthetic accelerograms from different tectonic regions with or without scaling to obtain the desired building response values [START_REF] Iervolino | Assessing uncertainty in estimation of seismic response for PBEE[END_REF][START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF]). Such modeling assumptions can have a significant impact on the nonlinear seismic response of a building, potentially limiting our understanding of the true response of structures, and its variability from building to building and from earthquake to earthquake. This can also contribute to epistemic uncertainty in damage assessment (Astorga et al., 2018;[START_REF] Kalakonas | Exploring the impact of epistemic uncertainty on a regional probabilistic seismic risk assessment model[END_REF]Lestuzzi et al., 2016;Perrault et al., 2013Perrault et al., , 2020;;Perrault and Guéguen, 2015;[START_REF] Spence | Comparing loss estimation with observed damage: A study of the 1999 Kocaeli earthquake in Turkey[END_REF]. Mainly, it raises the following questions:

Which ground motion intensity measures (IM) can capture the best building response (EDP) in real-world problems? What about the experimental efficiency and sufficiency of IMs? How does the building response change over time after experiencing damage? Do all buildings with the same class respond similarly to a given ground motion? What features of the building contribute the most in response modeling uncertainties?

In addition, to advance the OELF method as a reliable tool for DDA, it is crucial to enhance our understanding of earthquake-induced damage and residual capacity of the building, including its variation from building to building and earthquake to earthquake [START_REF] Wiemer | Real-time earthquake rIsk reduction for a ReSilient Europe ( RISE )[END_REF]Nievas;[START_REF] Nievas | Deliverable 6.1 D6.1 Integration of RISE Innovations in the Fields of OELF, RLA and SHM Deliverable information[END_REF].

Specifically, the following questions need to be answered:

How does the building's co-seismic capacity change after experiencing damage? How do they vary from building to building? How can we consider the actual structural health required for the OELF method?

(v) Organization of the Thesis

This thesis is organized as follows:

Chapter One presents a comparative study on the efficacy of different machine learning models for predicting damage using the 2015 Nepal earthquake building damage dataset. This chapter also discusses crucial features for damage prediction, methods to address the imbalanced distribution of target features (damage grade) during model training, and the minimum number of data points required to develop a reasonable damage prediction model. The findings of this study have been published in the Earthquake Spectra journal [START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF] .

Chapter Two presents the findings on testing machine learning models for heuristic building damage assessment at a regional scale using the database of observed damage (DaDO) in Italy. We evaluate the efficacy of machine learning models trained on past earthquake building damage datasets to predict potential damage in future earthquakes. This chapter also explores the contribution of building features to the damage level and compares the damage prediction efficacy of the machine learning model with the Risk-UE method. The finding of this study is under review in the Natural Hazards and Earth System Science journal (Ghimire et al., 2023).

Chapter Three compares the damage prediction efficacy of machine learning models developed in three different regions (Nepal, Italy, and Haiti) to test their host-to-target adjustment for damage assessment. The chapter also discusses the impact of a building's characteristic features, such as typology, construction practice, seismic regulations, etc., on transferring machine learning models for damage prediction from host to target dataset.

Chapter Four presents the findings of the structural response (EDP) analysis to a given set of intensity measures (IM) using the NDE1.0. This chapter examines the efficiency and sufficiency of IMs, as well as the associated variability related to different factors such as structural health, typology, earthquake magnitude and distance. This chapter also discusses the variation of building frequency in response to earthquake loading. Moreover, a building damage prediction equation is provided for different building typologies. The finding from this study have been published in the Soil Dynamics and Earthquake

Engineering journal (Ghimire et al., 2021a).

Chapter Five presents the findings on the variability in the co-seismic capacity response of buildings under seismic loading using the NDE1.0. This chapter investigates the variation in the co-seismic capacity of a building before and after the first damage occurs, as well as the accumulation of damage over time and the variation between similar buildings. Additionally, this chapter provides the necessary adjustments to account for the actual structural health required for the OELF method. The finding of this study is currently under review at Bulletin of Earthquake Engineering for possible publication.

Finally, our conclusions are presented, along with perspectives for future work.

1 Testing machine learning models for seismic damage prediction at regional scale using building-damage dataset compiled after 2015

Gorkha Nepal earthquake

This chapter evaluates the effectiveness of machine learning models for a rapid damage assessment at a large scale using the 2015 Nepal earthquake building damage dataset. We analyze the efficacy of different machine learning models, methods for handling imbalanced target features, input feature's importance, the effectiveness of readily available building features for large-scale damage assessment, and minimum data points required for a reasonable prediction model. Our study's findings are published in Earthquake Spectra.

Ghimire, S., [START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF]. Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake. Earthquake Spectra, 38(4), 2970-2993.

Abstract

Assessing post-seismic damage on an urban/regional scale remains relatively difficult owing to the significant amount of time and resources required to acquire information and conduct a building-bybuilding seismic damage assessment. However, the application of new methods based on artificial intelligence, combined with the increasingly systematic availability of field surveys of post-seismic damage, has provided new perspectives for urban/regional seismic damage assessment. This study analyzes the effectiveness and relevance of a number of machine learning techniques for analyzing spatially distributed seismic damage after an earthquake at the regional scale. The basic structural parameters of a portfolio of buildings and the post-earthquake damage surveyed after the Nepal 2015 earthquake are analyzed and combined with macro-seismic intensity values provided by the United States Geological Survey ShakeMap tool. Among the methods considered, the random forest regression model provides the best damage predictions for specified ground motion intensity values and structural parameters. For traffic-light-based damage classification (three classes: green-, amber-, and red-tagged buildings based on post-earthquake damage grade), a mean accuracy of 0.68 is obtained. This study shows that restricting learning to basic features of buildings (i.e., number of stories, height, plinth area, and age), which could be readily available from authoritative databases (e.g., national census) or fieldsurveyed databases, yields a reliable prediction of building damage (4 features/3 damage grade accuracy accuracy: 0.64).

Introduction

Earthquakes may not occur frequently; however, they contribute significantly to the physical and social consequences of natural hazards. From 1990 to 2017, the consequences of earthquakes represent an annual average of approximately USD 34.7 billion (OECD, 2018). Information regarding the estimated extent and spatial distribution of potential seismic damage within a built environment is crucial for decision makers, emergency planners, insurers, and reinsurers (e.g., Bommer and Crowley 2006;[START_REF] Ranf | Post earthquake prioritization of bridge inspections[END_REF][START_REF] Earle | Prompt assessment of global earthquakes for response (pager): A system for rapidly determining the impact of earthquakes worldwide[END_REF]Riedel and Guéguen 2018). The level of detail, methods, and tools required for the assessment are conditioned by the objectives to be achieved [START_REF] Erdik | Rapid earthquake loss assessment after damaging earthquakes[END_REF]. The estimation of potential damage could be obtained through fragility or vulnerability modelling. Seismic fragility or vulnerability modelling is the crossing of hazard, exposure, and fragility/vulnerability component [START_REF] Crowley | The European Seismic Risk Model 2020[END_REF]. The exposure-related component provides information regarding the built environment, including vulnerability class, function, reconstruction cost, and spatial distribution. The hazard-related component defines the ground-shaking potential in a region of interest based on a probabilistic seismic hazard assessment or from a deterministic rupture scenario. Finally, the vulnerability/fragility-related component relates hazards with exposure to determine the likelihood of potential damage [START_REF] Silva | Developing a Global Earthquake Risk Model, 16th[END_REF]. Many advanced empirical methods have been developed for seismic vulnerability assessment (e.g., FEMA 2003;[START_REF] Milutinovic | Risk-UE An advanced approach to earthquake risk scenarios with applications to different european towns, Rep. to WP4 vulnerability[END_REF]Lagomarsino and Giovinazzi 2006;[START_REF] Régnier | Contribution of ambient vibration recordings (free-field and buildings) for post-seismic analysis: The case of the Mw 7.3 Martinique (French Lesser Antilles) earthquake[END_REF]Hancilar et al. 2010[START_REF] Hancilar | Empirical fragility assessment after the january 12, 2010 haiti earthquake[END_REF]Silva et al. 2014), based on field surveys of building parameters in relation to standardized building typologies, which are associated with the vulnerability or fragility functions for a specified seismic intensity measurement. The application of these damage assessment methods for rapid damage assessment in regional or urban scale is still challenging because the acquisition of building features and the application of classical methods is time-and resources-consuming on urban or regional scale.

Over the last decade, substantial progress has been realized in the field of machine learning tools and their applications in various domains. Within the scope of this study, Riedel et al. (2015) demonstrated the capacity of the support vector machine in providing seismic vulnerability assessments at regional and national scales. As the application of conventional vulnerability assessment methods on a large scale requires information that is not readily available, they proposed assessing the ability of available data from national census for a region to estimate building vulnerabilities and modeling seismic damage for specified seismic intensities. More recent studies have demonstrated the efficiency of using machine learning techniques in seismic-risk engineering to solve the aforementioned time and resource issues [START_REF] Chi | Machine learning-based seismic capability evaluation for school buildings[END_REF][START_REF] Hegde | Applications of machine learning methods for engineering risk assessment -A review[END_REF][START_REF] Karmenova | Seismic Assessment of Urban Buildings Using Data Mining Methods[END_REF]Mangalathu et al., 2020a;[START_REF] Sajedi | A data-driven framework for near real-time and robust damage diagnosis of building structures[END_REF][START_REF] Salehi | Emerging artificial intelligence methods in structural engineering[END_REF][START_REF] Sun | A Data-driven Building Seismic Response Prediction Framework: from Simulation and Recordings to Statistical Learning[END_REF][START_REF] Zhang | Pattern recognition approach to assess the residual structural capacity of damaged tall buildings[END_REF][START_REF] Zhao | Modelling and interpreting pre-evacuation decision-making using machine learning[END_REF]. [START_REF] Xie | The promise of implementing machine learning in earthquake engineering: A state-of-the-art review[END_REF] summarized the ongoing research on the application of machine learning methods in earthquake engineering, they concluded that the implementation of machine learning in earthquake engineering is still in its early stage and needs further investigations. Limited number of studies are available concerning the post-earthquake damage classification using machine learning (Mangalathu et al., 2020b;[START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF][START_REF] Harirchian | A synthesized study based on machine learning approaches for rapid classifying earthquake damage grades to rc buildings[END_REF][START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF]. In particular, Mangalathu et al., (2020b) tested the performance of machine learning techniques in a building damage survey performed after the 2014 South Napa earthquake and demonstrated the effectiveness of such tools in interpreting the patterns of seismic damage observed in buildings. [START_REF] Harirchian | A synthesized study based on machine learning approaches for rapid classifying earthquake damage grades to rc buildings[END_REF] studied the performance of different machine learning methods for damage prediction in RC buildings.

They observed reasonable damage prediction by machine learning models and suggested further investigation by considering a large number of buildings with more features to define the building vulnerability. [START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF] explored the performance of different machine learning methods using 2017 Puebla-Morelos earthquake building damage data and suggested further investigation using a large number of buildings in each damage typology.

In fact, the added value of post-earthquake studies based on building-damage observations has long been recognized for its contribution in improving our understanding and validating state-of-the-art seismic risk assessment models (Colombi et al., 2008;[START_REF] Spence | Estimation of vulnerability functions based on a global earthquake damage database[END_REF]Eleftheriadou andKarabinis, 2011, 2012;[START_REF] Guðmundsson | Comprehensive damage analysis of buildings affected by the 2008 South Iceland earthquake[END_REF]Karababa and Pomonis, 2011;Del Gaudio et al., 2017, 2020). In parallel to these machine learning developments, open access to a significant amount of information describing real-estate portfolios has improved (Crowley et al., 2020), and post-earthquake buildingdamage surveys are now available online (2015 Nepal Earthquake: Open Data Portal, 2021; Dolce et al., 2019;[START_REF] Loos | G-DIF: A geospatial data integration framework to rapidly estimate post-earthquake damage[END_REF]. For example, the National Planning Commission of Nepal shared a massive data survey of damaged buildings after the Mw 7.8 Nepal earthquake in 2015 (NPC, 2015a).

To prepare for the increased use of machine learning for earthquake and seismological engineering applications, combined with the increasing number of open-source data, independent studies on independent datasets and with different machine learning methods are essential to advance science in this emerging field. Unlike previous studies (e.g., Mangalathu et al., 2020b;[START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF], this study herein compares the damage prediction performance of various machine learning models to explain post-earthquake seismic damage, considering two different classifications of damage and two different sets of building features, applied to the 2015 Nepal earthquake building-damage portfolio (NBDP). This dataset is first described in the data section (1.2). In the method section (1.3), we present a summary of the selected machine learning tools applied to the dataset. In the results section (1.4), we present and discuss the results, considering damage assessment as the target value of this study. Finally, we provide our conclusions and perspectives in the last section (1.5) and (1.6).

Data

On April 25, 2015, a devastating MW 7.8 earthquake struck central Nepal, with an epicentral distance of approximately 80 km NW of Kathmandu, a hypocentral depth of 8.2 km, and a 120 km long rupture toward the east. Level VIII epicentral intensity was estimated in Nepal based on the observed damage (Martin et al., 2015). Thousands of residential buildings were damaged, resulting in 8,790 fatalities and 22,300 injuries (NPC, 2015). Additionally, 31 among 75 administrative districts of Nepal were affected, with 14 districts being severely affected and declared as crisis-hit areas, and an estimated loss of around seven billion US dollars was reported (NPC, 2015). The government of Nepal conducted a massive post-earthquake survey in the 11 most severely affected districts (Fig The color scale in Fig. 1.1 highlights the proportion of buildings tagged with DG5 in each district.

Additionally, a set of building parameters related to structure and environment, frequently used in empirical seismic vulnerability assessment, was assigned to each screened building for the NBDP database:

Numbers of stories: total numbers of floors above the ground surface.

Height of building: total height of the building above the ground surface measured in feet.

Building age: calculated from the date of construction to the date of the earthquake.

Plinth area: total area occupied by the building at ground floor level in square feet.

Ground slope condition: ground surface topography at the building location, considering three types of ground slope conditions, i.e., flat, mild slope, and steep slope.

Roof type: three types of roofs are considered based on the material used, i.e., light timber/bamboo roof, heavy timber/bamboo roof, and reinforced concrete roof.

Position of building: indications of the building's location relative to other buildings. Four types of positions are cited, i.e., stand-alone, attached on one side, attached on two sides, and attached on three sides.

Construction material: Eleven types of construction materials used in the superstructure are considered: adobe, stone flange, mud-mortar stone, mud-mortar brick, cement-mortar stone, cementmortar brick, reinforced concrete non-engineered, reinforced concrete engineered, timber, bamboo, and others.

Furthermore, in the NBDP database the geographic location of each building was assigned to a ward that belonged to a district. The ward was an elementary administrative cell. The total number of wards in 11 districts was 949. Considering building-by-building locations and the associated soil conditions, which may vary within a ward, can improve the damage prediction performance of a machine learning model (Mangalathu et al., 2020b;[START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF][START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF]. However, having as objective the large-scale classification of damages (and not the classification and representation per building), and considering the macro-seismic intensities, the attribution of each building within a ward is consistent with our approach.

In a first case, all the building features contained in the NBDP database are considered in the learning phase, without any preconceived idea on their significance or possible cross-correlation. Their importance score will then be evaluated in order to know their contribution in the learning phase. In a second case, a subset of input features assumed easy to collect without significant expertise or available via authoritative national or global open databases are considered to test their damage prediction performance, without taking into account their importance.

No specific data cleaning methods were applied to the NBDP database. In the NBDP database, 12 buildings had missing damage information so they were removed from the database and 5,731 buildings had missing age value so they were replaced by the average age value. Table 1.1 summarizes the distribution of the building parameters in the database. In the NBDP database, the distribution of the buildings per damage grade was as follows: 10.34% for DG1, 11.45% for DG2, 17.90% for DG3, 24.12% for DG4, and 36.19% for DG5. The number of stories ranged from one to nine floors, the building age ranged from 1 to 200 years, the building plinth area ranged from 70 to 5,000 square feet, and the building height was between 6 and 97 feet (approximately 2 to 30 m). The distribution of buildings based on the material used to construct the superstructure was as follows:

4.24% adobe, 80.02% mud-mortar stone, 3.51% stone flange, 1.58% cement mortar stone, 2.29% mudmortar brick, 7.15% cement-mortar brick, 25.87% timber, 8.04% bamboo, 3.97% reinforced concrete (RC) non-engineered, 1.63% RC engineered, and 1.20% other materials. Additional information provided by the ShakeMap tool from the United States Geological Survey (USGS) supplied the macro-seismic intensity (MSI) field and completed the database [START_REF] Wald | ShakeMap manual: technical manual, user's guide, and software guide[END_REF]. ShakeMap provides an estimate of the spatial distribution of earthquake ground shaking intensity by combining recorded ground motions, modeled ground motions, Did You Feel It? information, and slope-vs30-based site conditions. In recent studies requiring spatially distributed information as input ground motion, ShakeMap intensities have been successfully used to compensate for insufficient instrumental data [START_REF] Jaiswal | Development of a semi-empirical loss model within the USGS Prompt Assessment of Global Earthquakes for Response (PAGER) system[END_REF]Mak and Schorlemmer, 2016;[START_REF] Silva | Combining USGS ShakeMaps and the OpenQuake-engine for damage and loss assessment[END_REF]Del Gaudio et al., 2020;[START_REF] Pothon | Comparing Probabilistic Seismic Hazard Maps with ShakeMap Footprints for Indonesia[END_REF]. The USGS ShakeMap provides the MSI value in modified Mercalli intensity scale (Fig. 1.3a). Because the geo-localization of the buildings is available at the ward level, we computed the mean MSI value for each ward and assigned it to all the buildings located within the same ward. The MSI values ranged from 5.30 to 8.30 (Fig. 1.3b).

Method

In this study, damage prediction was considered first as a classification problem. Machine learning classification involves assigning a label (or class) to categorical response variables. Damage grades (from 1 to 5) are considered categorical variables; hence, the application of classification models is recommended. In this study, we focused on two typically used classification machine learning methods, i.e., random forest classification (RFC) [START_REF] Breiman | Random Forests[END_REF] and gradient boosting classification (GBC) [START_REF] Friedman | Greedy Function Approximation:A Gradient Boosting Machine[END_REF]. However, the damage grades were ordered and can be regarded as a continuous variable to minimize misclassification errors. In this study, regression models were considered simultaneously with the damage grade as a continuous variable ranging between 1 (DG1) and 5 (DG5).

Because the regression model outputs a real value between 1 and 5 and not an integer, we rounded the output (real number) to the nearest integer to plot the confusion matrix. However, the error matrices were computed without rounding the model outputs to the nearest integer. Four regression machine learning models were considered, i.e., linear regression (LR) (Hastie et al., 2009), support vector regression (SVR) [START_REF] Cortes | Support-vector networks[END_REF], gradient boosting regression (GBR) [START_REF] Friedman | Greedy Function Approximation:A Gradient Boosting Machine[END_REF], and random forest regression (RFR) [START_REF] Breiman | Random Forests[END_REF]. Using classification methods, every misclassification yields the same penalty loss, whereas regression methods penalize the relative penalty loss (i.e., difference error) by minimizing the squared loss between the true and predicted damage grades. In this study, we tested both approaches (regression and classification), and their strengths and weaknesses in terms of damage prediction were compared.

RFC/RFR and GBC/GBR are based on a set of several decision trees as base learners; this enables them to achieve greater efficiency while integrating the complexity and nonlinear interaction of the features in the dataset. LR has simple analytical and computational properties; additionally, it can easily illustrate the interpretable description of the effect of the input on the output, i.e., the contribution of each feature in the model. SVR is effective in high-dimensional spaces and is extremely versatile in terms of application (i.e., it can be used for linear and polynomial models).

Hereinafter, the damage grade is defined as the target variable, and the building parameters (i.e., number of stories, height, age, plinth area, ground-slope condition, building position, roof type, and construction material) and the ground-motion intensity (MSI) are defined as the input features. A one-hot encoding technique was used to transform categorical values (i.e., construction material, position, ground slope condition, and roof type) into nominal values [START_REF] Pedregosa | Scikit-learn[END_REF]. The one-hot encoding technique generates a unique binary value 1 or 0 for each categorical feature by creating a sparse matrix, which resulted in 26 input features for the model (Table 1.1).

The building damage dataset was randomly partitioned into three subsets In this study, we used the models provided in Python using the Scikit-learn package. The performance of each model was quantified by measuring the accuracy score (percentage of correctly predicted labels) [START_REF] Pedregosa | Scikit-learn[END_REF]. A high accuracy score (approximately 1) indicates the high efficacy (i.e., ability to perform a task to a satisfactory or expected degree) of the machine learning model in damage prediction. In addition, we computed the following statistical error indicators to compare the performance of each machine learning model: the mean of the absolute error (MAE) and the mean squared error (MSE). The scores of these indicators are referred to as the performance scores hereinafter. The smaller the MAE (approximately 0) and MSE (approximately 0), the higher is the efficacy of the models. We considered different metrics because we observed that one metric could not fully explain the predictive performance among classification and regression methods. For each machine learning model, the predictive performance is extremely sensitive to the model input parameters, which are known as hyperparameters. The hyperparameters of each machine learning model were tuned using a grid-search technique on the training set. Finally, computational time (time taken by each model in our personal computer, Macbook Pro 2019) is considered as one additional parameter to facilitate comparison among machine learning models.

The performance of each machine learning technique for damage prediction is presented graphically based on the distribution of errors as well as in a confusion matrix (see Fig. 1.4). A confusion matrix presents a comparison between the true and predicted damage grades. In the confusion matrix, the value of the predicted damage grades (cell value along the row of the confusion matrix) is normalized by the total number of true damage grades. The values along the main diagonal elements of the confusion matrix indicate the recall (i.e., the number of predicted values equal to the true values) of the machine learning model. A perfect model would have ones along the main diagonal of the confusion matrix, and zero everywhere else. In the confusion matrix, the accuracy score represents the number of correctly predicted instance (sum of the diagonal terms) over the total number of instances (sum of the full matrix). Because the regression model outputs a real value between 1 and 5 and not a label, we rounded the output (real number) to the nearest integer to plot the confusion matrix. But the error matrices were computed without rounding the model outputs to the nearest whole integer.

Result

Model selection

This section summarizes the performance of damage prediction provided by the machine learning models on the validation set. In conclusion, considering the damage-prediction task as a regression problem and using GBR/RFR machine learning models provides the best damage-prediction efficacy (see Fig. 1.4, 1.5 and Table 1.2).

For the GBR and RFR models, the damage prediction efficacy, and computational time were extremely sensitive to the hyperparameters. For example, the GBR model required a careful tuning of a significant number of hyperparameters, which increased the computational time compared with the RFR model.

In addition, the GBR model might be less generalizable to new data owing to its possible overfitting issue and is more difficult to implement compared with the RFR model. Hence, the RFR model was used in the remainder of this study to test the damage prediction.

Management of data imbalance

The distribution of labels in the training dataset affects the performance of machine learning models (e.g., [START_REF] Estabrooks | A mixture-of-experts framework for learning from imbalanced data sets[END_REF][START_REF] Japkowicz | The class imbalance problem A systematic study fulltext[END_REF][START_REF] Branco | SMOGN: a Pre-processing Approach for Imbalanced Regression[END_REF]). The NBDP dataset shows an unequal distribution of the damage grades, i.e., the highest damage grade represents the largest fraction of the dataset (Fig. 1.2a). This data imbalance may have affected the predictive performance of the RFR model (Fig. 1.4). In this study, we considered some extensively adopted techniques to manage an imbalanced dataset: over and undersampling the target features (e.g., [START_REF] Estabrooks | A mixture-of-experts framework for learning from imbalanced data sets[END_REF][START_REF] Japkowicz | The class imbalance problem A systematic study fulltext[END_REF]. Undersampling is achieved by selecting an equal amount of data through random selection of a minimum number of values in each damage grade from the training dataset, whereas oversampling is achieved by replicating the minority damage grades (i.e., DG1 and DG2 in our case).

The RFR model was trained using the resampled dataset to observe its performance on the validation dataset. By oversampling, the performance yielded was similar to that of the original method (1% increase in the MAE). However, undersampling increased the recall value for the lowest damage grades (Fig. 1.6a)

i.e., the recall value increased by 20%-25% for DG1 to DG3 and reduced by less than 15% for DG4 and DG5. Meanwhile, oversampling improved the recall value moderately for the lower grades without significantly affecting the recall value of the higher damage grades, e.g., the DG1/DG5 recall value changed from 0.33/0.54 (Fig. 1.4b) to 0.41/0.51 (Fig. 1.6b), respectively. Therefore, for further studies, we plan to use the oversampling technique to address data imbalance issues such that a better estimation of the lower grades can be obtained.

Feature importance

The importance of each input feature in the RFR damage-prediction model is analyzed in this section.

In decision trees, each node is a condition for dividing the values into single characteristics such that similar values of the dependent variable appear in the same set after the division. The condition is based on impurity, which is the variance in the case of regression trees. In other words, each feature's contribution can be measured by the average decrease in impurity using all the trees in the forest, without considering the linear separability of the data [START_REF] Pedregosa | Scikit-learn[END_REF]. The significance of each feature in the model is measured in terms of the feature importance scores. The feature importance score is a value assigned to each feature in the model while a model is developed. Note also that RFR uses the bagging algorithm (randomly splitting data into smaller subsets) while developing the trees, so the correlated features may or may not be used for a particular tree. Thus, correlated features may not affect the overall predictive performance, however, it may impact the importance ranking between two correlated features, removing one of the correlated features may lower the damage predictive performance of the RFR model. Buildings constructed using mud-mortar-stone constituted the highest proportion in our dataset (Fig.

1.2i), and this building class was damaged severely during the Nepal earthquake, independent of the MSI. In fact, this type of building is generally associated with the most vulnerable class in most

(b) (a)
RFR: under-sampling RFR: over-sampling vulnerability assessment methods. The fact that the RFR identified the mud-mortar-stone feature as the most important feature in damage prediction is consistent with previous studies [START_REF] Maheri | Performance of adobe residential buildings in the 2003 Bam, Iran, earthquake[END_REF][START_REF] Sayin | Failures of masonry and adobe buildings during the June 23, 2011 Maden-(Elaziǧ) earthquake in Turkey[END_REF][START_REF] Webster | Earthquake Damage To Historic and Older Adobe Buildings During the 1994 Northridge , California Earthquake[END_REF]. Additionally, Fig. 1.7 shows that, as expected, the MSIs that characterized the ground motion were one of the most significant input features for earthquake damage prediction (31%), i.e., damage was first conditioned by the MSI regardless of the building parameters. Other building-related input features, such as plinth area (8%), construction age (7%), and height (5%) contributed significantly to earthquake damage prediction model; meanwhile, the other features contributed only marginally to the damage prediction model (RFR).

As mentioned earlier (Tab. 1.2), a higher damage-prediction accuracy (0.49) was observed when all features were considered (MAE = 0.66). However, information regarding building construction materials is typically not easily accessible. The four basic building parameters (i.e., number of stories, age, height, and plinth area) could be easily accessible from the institutional databases (e.g., national census, national housing database) (Crowley et al., 2020;Riedel et al., 2015), from filed survey, and

partially available in open-source platform (e.g., OpenStreetMap (OSM) [START_REF] Bennett | OpenStreetMap[END_REF], OpenBuildingMap (OBM) [START_REF] Schorlemmer | Global Dynamic Exposure and the OpenBuildingMap -Communicating Risk and Involving Communities[END_REF]). Thus, the performance of RFR model considering only the basic parameters of the building (i.e., number of stories, age, height, and plinth area) in addition to the ground-motion intensities is explored. The RFR model provided a similar level of accuracy (0.46) in damage prediction (MAE = 0.72). This result suggests that basic building parameters enable the development of satisfactory RFR models for predicting damage, and therefore earthquake losses, for the given value of ground-motion intensity (as reported by Riedel et al. (2015) and Guettiche et al. (2017) for seismic vulnerability classification using a support vector machine).

Damage prediction using test dataset

This section presents the predictive performance of the RFR model for the test dataset. The RFR model was developed for two sets of features, i.e., all building parameters (i.e., full-feature setting) and the basic parameters (i.e., number of stories, age, height, plinth area, and ground-motion intensity) (basicfeature setting).

The full-feature/basic-feature settings yielded MSE scores of 0.81/0.93. Confusion matrices for the fulland basic-feature settings are shown in Figs. 1.8a and 1.8b, respectively. The main diagonals and their adjacent values are associated with a higher recall value, illustrating higher efficacy in damage prediction and a lower misclassification rate. As shown in Figs. 1.4b and 1.6b, the RFR model applied to the test dataset predicted the highest damage grades (DG4/DG5) with a higher recall value, corresponding to 60%/51% and 57%/46% in the full-and basic-feature settings, respectively. During field observations for damage classification, it is often difficult to assign a damage level to buildings falling between two damage grades. Moreover, it is more convenient to classify damage into three categories, i.e., in the same manner as using the traffic-light-system (TLS)-based classification system (green, yellow, and red) in post-earthquake emergency surveys (ATC, 2005;[START_REF] Bazzurro | GUIDELINES FOR SEISMIC ASSESSMENT OF DAMAGED BUILDINGS[END_REF]. Therefore, we classified the five damage grades into three categories by considering the severity of damage based on the damage grade definition (Section 1.2), as follows: S (slight: DG1+DG2), M (medium: DG3), and H (heavy: DG4+DG5). The MSE scores for the full-feature/basic-feature settings were 0.34/0.39. Fig. 1.9 shows the confusion matrices. The damage prediction accuracy increased significantly when the TLS-based approach was used. For example, the recall values for the S/M/H damage grades were 59%/60%/73% and 51%/61%/70% for the full-and basic-feature settings, respectively. Table 1.3 summarizes the MAE, and MSE scores observed with the full-and basic-feature settings. For the basic-feature setting, the values of these scores (e.g., MAE = 0.73/0.45), considering that the 5DG/TLS damage scales were similar to those of the full-feature setting (e.g., MAE = 0.69/0.41).

Section 4.3 shows that using the RFR model, a reasonable damage prediction was possible by merely considering the basic features of the buildings. acquisition for vulnerability assessments as well as earthquake damage and loss prediction on the urban or regional scale.

Testing the proportionating of the dataset

We explored the damage prediction efficacy of the RFR model as a function of the amount of data in the training dataset. Fig.

Discussion

In this study, we investigated the efficiency of various machine learning techniques for post-earthquake seismic damage assessment using the NBDP database compiled after the 2015 MW 7.8 earthquake in Nepal. Machine learning models were trained based on a number of building characteristics and some basic parameters that contributed significantly to damage prediction performance. We tested the predictive efficacies of LR, SVR, GBC/GBR, and RFR/RFC models. For classification methods significant misclassification is observed for intermediate damage grades whereas for regression methods the predicted values were similar to the ground truth (see. Fig 1.4, 1.5 andTable 1.2). This could be because for classification methods every mis-classification has the same penalty loss whereas regression methods penalize the relative penalty loss (i.e. difference error) by minimizing the squared loss between true and predicted damage grades. Among these models, RFR was the most relevant model for damage prediction when applied to our dataset; it provided the best cost/benefit ratio in terms of performance and computing time. For the baseline comparison, the result obtained from the RFR model is compared with a random uniform baseline (randomly assigning damage grades following an uniform distribution) and a random stratified baseline (following the distribution of damage grades in the training set) to every input. The RFR model resulted largely higher accuracy score (0.49) as compared to the random uniform baseline (0.20) and the random stratified baseline (0.25) to every input for 5DG classification. Moreover, in this case study, the RFR model demonstrated superior performance in predicting higher damage grades, which is ultimately the most pursued information for seismic damage assessment and earthquake loss reduction.

Additionally, we achieved a moderate improvement in damage prediction by addressing data imbalance issues via data resampling using the RFR model. In this study, we considered an unseen test dataset to ensure the overfitting issues of the model. The error values of validation and test datasets are very similar (same 0.49 accuracy on both cases), indicating that the model did not over-fit on the validation set.

However, the results herein show that considering a traffic-light damage classification and a limited number of building features, the results reach very satisfactory scores according to the objectives set:

damage prediction accuracy 0.64 for basic-features setting, to be compared to Mangalathu et al. (2020b), Roslin et al., (2020[START_REF] Harirchian | A synthesized study based on machine learning approaches for rapid classifying earthquake damage grades to rc buildings[END_REF] studies that reported accuracy values of 0.66, 0.67, and, 0.65, respectively also with the random forest method. The 3-class classification model lowers the number of targets and then lowers the confusion as is often observed during damage surveys for the intermediary damage grade (e.g., between DG3 and DG4). In addition, so-called basic building features (i.e., number of stories, age, height, and plinth area) associated with ground-motion intensity provided a relevant estimate of the damage grade with a significant level of accuracy. Similar observations have also been reported by Mangalathu et al. (2020b) and [START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF] in their studies using the 2014 South Napa and 2010 Kraljevo, Serbian earthquake building damage dataset, respectively. Moreover, the RFR model trained on a relatively small amount of dataset (5% -20% of the test dataset) resulted in a reasonable estimate of damage; similar observation has been reported by [START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF].

The input ground motion used was the USGS ShakeMap intensity of the mainshock, whereas the overall quality of the NBDP database results were based on the cumulative effects of the mainshock and aftershock events, which might have affected the prediction efficacy. In addition, missing building-bybuilding information in the NBDP database relative to their localization and their associated site condition reduce the damage prediction efficacy of the machine learning model (Mangalathu et al., 2020b;[START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF][START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF]. However, we also observed that by considering the geographic locations of buildings (ward-id in our case) slightly improved the damage prediction efficacy of the RFR model.

Finally, in Nepal, a majority of the building typology inventory collected after the 2015 earthquake holds a good match with the building typology inventory as observed in the 2011 national census (Chaulagain et al., 2015). Thus, the findings from this study seems promising to the Nepal's exposure context. Further investigations should be carried out to strengthen these findings: after the 2015 Gorkha Nepal earthquake, national and international communities are collaborating to develop an exposure model for Nepal [START_REF] Jordan | METEOR : Modelling Exposure Through Earth Observation Routines to Aid Su s tainable Development[END_REF].

Conclusion

To summarize, a framework for combining building parameters and earthquake building damage information collected after an earthquake event with machine learning techniques for seismic-damage assessment was presented herein. This study shows a possibility of using machine learning methods for immediate damage assessment once ground-motion information is published via operational tools, such as the USGS ShakeMap. This study shows that the building's features (number of stories, age, floor area, height) could result in reasonable damage prediction. These basic building parameters could be available in existing institutional databases (e.g., national census, national housing database), thereby resolving data acquisition issues associated with seismic-damage assessments at the urban or regional scale.

Without anticipating how a city planner can use these results, machine learning developed on a building portfolio makes it possible to explain post-earthquake damage. One may assume that the model herein for a specific portfolio might be used to predict and represent seismic damage in another region with the same portfolio (for example, Riedel et al., 2014) or for another characterization of the seismic hazard (Riedel et al., 2015), in an emergency situation (immediately after an earthquake if the exposure model is known) or in a planning process. This will of course have to be confirmed on another dataset, e.g.

consisting of a series of earthquakes affecting the same region, i.e. characterized by the same portfolio of buildings. In addition, the results herein show that using a traffic-light damage classification and a limited number of building features, the results reach very satisfactory scores according to the objectives set (damage prediction accuracy 0.64 for 4 basic features/3 damage grade settings, to be compared to the studies of Mangalathu et al. (2020b), Roslin et al. (2020[START_REF] Harirchian | A synthesized study based on machine learning approaches for rapid classifying earthquake damage grades to rc buildings[END_REF] that reported accuracy values of 0.66, 0.67, and, 0.65, respectively with the random forest method).

Additionally, the collection of the building exposure data is the key challenge faced by the damage assessment communities. The observed performance of machine learning considering only basic parameters, easy to collect and without requiring much technical expertise, may suggest the interest of exploring the potential of crowed-source databases (e.g. OSM/OBM) as input parameters. This approach will require validation before being able to assert it definitively, but it would solve the difficulty of considering the evolution of the building in the definition of the exposure models.

2 Testing machine learning models for heuristic building damage assessment applied to the Italian Database of Observed Damage (DaDO)

In chapter 1, we compared the damage prediction performance of various machine learning models to explain post-earthquake seismic damage, considering two different classifications of damage and two different sets of building features, applied to the 2015 Nepal earthquake building-damage portfolio (NBDP).

In this chapter, we evaluate machine learning models for heuristic building damage assessment at a regional scale using the database of observed damage (DaDO) in Italy. We analyzed the efficacy of machine learning models trained on past earthquake building damage datasets to predict potential damage in future earthquakes, the relation between building features and damage level, and methods to handle imbalance distribution of target features. We also compared the damage prediction efficacy of the machine learning model with the Risk-UE method. The content of this chapter is under review in the Natural Hazards and Earth System Science journal.

Ghimire, S., Guéguen, P., Pothon, A., and Schorlemmer, D. (2023). Testing machine learning models for heuristic building damage assessment applied to the Italian Database of Observed Damage (DaDO). Natural Hazards and Earth System Sciences Discussions, 1-29

Abstract

Assessing or forecasting seismic damage to buildings is an essential issue for earthquake disaster management. In this study, we explore the efficacy of several machine learning models for damage characterization, trained and tested on the database of damage observed after Italian earthquakes (DaDO). Six models were considered: regression-and classification-based machine learning models, each using random forest, gradient boosting and extreme gradient boosting. The structural features considered were divided into two groups: all structural features provided by DaDO or only those considered to be the most reliable and easiest to collect (age, number of storeys, floor area, building height). Macroseismic intensity was also included as an input feature. The seismic damage per building was determined according to the EMS-98 scale observed after seven significant earthquakes occurring in several Italian regions. The results showed that extreme gradient boosting classification is statistically the most efficient method, particularly when considering the basic structural features and grouping the damage according to the traffic-light based system used, for example, during the post-disaster period (green, yellow and red), 68% buildings were correctly classified. The results obtained by the machine

Introduction

Population growth worldwide increases exposure to natural hazards, increasing consequences in terms of global economic and human losses. For example, between 1985 and 2014, the world's population increased by 50% and average annual losses due to natural disasters increased from US$14 billion to over US$140 billion (Silva et al., 2019). Among other natural hazards, earthquakes represent one-fifth of total annual economic losses and cause more than 20 thousand deaths per year [START_REF] Daniell | The global role of earthquake fatalities in decisionmaking: earthquakes versus other causes of fatalities[END_REF]Silva et al., 2019). To develop effective seismic risk reduction policies, decision-makers and stakeholders rely on a representation of consequences when earthquakes affect the built environment.

Two main risk metrics generally considered at the global scale are associated with building damage: direct economic losses due to costs of repair/replacement and loss of life of inhabitants due to building damage. The damage is estimated by combining the seismic hazard, exposure models and vulnerability/fragility functions (Silva et al., 2019).

For scenario-based risk assessment, damage and related consequences are computed for a single earthquake defined in terms of magnitude, location, and other seismological features. Many methods have been developed to characterize the urban environment for exposure models. In particular, damage assessment requires vulnerability/fragility functions for all types of existing buildings, defined according to their design characteristics (shape, position, materials, height, etc.) and grouped in a building taxonomy (e.g. among other conventional methods FEMA, 2003;[START_REF] Grünthal | Escala Macro Sísmica Europea EMS -98[END_REF][START_REF] Régnier | Contribution of ambient vibration recordings (free-field and buildings) for post-seismic analysis: The case of the Mw 7.3 Martinique (French Lesser Antilles) earthquake[END_REF]Lagomarsino and Giovinazzi, 2006;[START_REF] Mouroux | Presentation of RISK-UE Project[END_REF]Silva et al., 2014). At the regional/country scale, damage assessment is therefore confronted with the difficulty of accurately characterizing exposure according to the required criteria and assigning appropriate vulnerability/fragility functions to building features. Unfortunately, the necessary information is often sparse and incomplete, and exposure model is suffering from economic and time constraints.

Over the past decade, there has been growing interest in artificial intelligence methods for seismic risk assessment, due to its superior computational efficiency, easy handling of complex problems, and the incorporation of uncertainties (e.g., Riedel et al., 2014Riedel et al., , 2015;;[START_REF] Azimi | Data-driven structural health monitoring and damage detection through deep learning: State-ofthe-art review[END_REF][START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF][START_REF] Hegde | Applications of machine learning methods for engineering risk assessment -A review[END_REF][START_REF] Kim | Pre-and post-earthquake regional loss assessment using deep learning[END_REF][START_REF] Mangalathu | Regional Seismic Risk Assessment of Infrastructure Systems through Machine Learning: Active Learning Approach[END_REF][START_REF] Morfidis | Approaches to the rapid seismic damage prediction of r/c buildings using artificial neural networks[END_REF][START_REF] Salehi | Emerging artificial intelligence methods in structural engineering[END_REF][START_REF] Seo | Metamodel-based regional vulnerability estimate of irregular steel moment-frame structures subjected to earthquake events[END_REF][START_REF] Sun | Machine learning applications for building structural design and performance assessment: State-of-the-art review[END_REF][START_REF] Wang | Machine learning-based regional scale intelligent modeling of building information for natural hazard risk management[END_REF][START_REF] Xie | The promise of implementing machine learning in earthquake engineering: A state-of-the-art review[END_REF]Y. Xu et al., 2020;Z. Xu et al., 2020). In particular, several studies have tested the effectiveness of machine learning methods in associating damage degrees with basic building features and spatiallydistributed seismic demand with acceptable accuracy compared with conventional methods or tested with post-earthquake observations (e.g., Riedel et al., 2014Riedel et al., , 2015;;Guettiche et al., 2017;[START_REF] Harirchian | A synthesized study based on machine learning approaches for rapid classifying earthquake damage grades to rc buildings[END_REF][START_REF] Mangalathu | Regional Seismic Risk Assessment of Infrastructure Systems through Machine Learning: Active Learning Approach[END_REF][START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF][START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF][START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF].

In parallel, significant efforts have been made to collect post-earthquake building damage observations after damaging earthquakes (Dolce et al., 2019;MINVU, 2010;MTPTC, 2010;NPC, 2015). With more than 10,000 samples compiled, the Database of Observed Damage (DaDO) in Italy, a platform of the Civil Protection Department, developed by the Eucentre Foundation (Dolce et al., 2019), allows exploration of the value of heuristic vulnerability functions calibrated on observations (Lagomarsino et al., 2021), as well as the training of heuristic functions using machine learning models [START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF] and considering sparse and incomplete building features.

The main objective of this study is to investigate the effectiveness of several machine learning models trained and tested on information from the DaDO to develop a heuristic model for damage assessment.

The model may be classified as heuristic because it applies a problem-solving approach in which a calculated guess based on previous experience is considered for damage assessment (as opposed to applying algorithms that effectively eliminate the approximation). The damage is thus estimated in a non-rigorous way defined during the training phase and the results must be validated and then tested against observed damage. By analogy with psychology, this procedure can reduce the cognitive load associated with uncertainties when making decisions based on damage assessment, by explicitly considering the uncertainties in the assessment, being aware about the incompleteness of the information and the accuracy level to make a decision. The dataset and methods are described in the data (2.2) and method (2.3) sections, respectively. The 2.4 section presents the results of damage prediction produced by machine learning models compared with conventional methods, followed by a conclusion section.

Data

The Database of Observed Damage (DaDO, Dolce et al., 2019) is accessible through a web-GIS platform and is designed to collect and share information about building features, seismic ground motions and observed damage following major earthquakes in Italy from 1976 to 2019. A framework was adopted to homogenize the different forms of information collected and to translate the damage information into the EMS-98 scale (Grunthal et al., 1998) using the method proposed by Dolce et al. (2019). For this study, we selected building damage data from seven earthquakes summarized in Table 2.1 and presented in Fig. 2.1. The converted EMS-98 damage grade (DG) ranges from damage grade DG0 (no damage) to DG5 (total collapse). The building features are available for each individual building and relate to the shape and design of the building and the built-up environment (Tab. 2.2, Fig. 2.2), as follows:

Building location -the location of each building is defined by its latitude and longitude, assigned using either the exact address of the building if available or the address of the local administrative centre (Dolce et al., 2019).

Numbers of storeys -total number of floors above the surface of the ground.

Age of building -time difference between the date of the earthquake and the date of building construction/renovation.

Height of building -total height of the building above the surface of the ground, in m.

Floor area -average of the storey surface area, in m 2 .

Ground slope condition -four types of ground slope conditions are defined (flat, mild slope, steep slope, and ridge).

Roof type -four types of roofs are defined (thrusting heavy roof, non-thrusting heavy roof, thrusting light roof, and non-thrusting light roof).

Position of building -indication of the building's position in the block: isolated, extreme, corner, and intermediate.

Regularity: building regularity in terms of plan and elevation, classified as either irregular or regular.

Construction material: vertical elements: good and poor-quality masonry, good and poor quality mixed frame masonry, reinforced concrete frame and wall, steel frame, and other.

For features defined as value ranges (e.g., date of construction/renovation, floor area, and building height), the average value was used. Furthermore, the Irpinia-1980 building damage portfolio (E1) was constructed using the specific Irpinia-1980 damage survey form, while the AeDES damage survey form was used for the others. The Irpinia-1980 dataset will therefore be analysed separately.

Building damage data from earthquake surveys other than the Irpinia-1980 earthquake damage survey primarily include damaged buildings. This is because the data was collected based on requests for damage assessments after the earthquake event (Dolce et al. 2019). The damage information in the DaDO database is still relevant for testing the machine learning models for heuristic damage assessment. Mixing these datasets to train machine learning models can lead to biased outcomes.

Therefore, the machine learning models were developed on the other earthquake dataset excluding the Irpinia dataset, and the Irpinia earthquake dataset was used only in the testing phase.

The distribution of the samples is very imbalanced (Fig. 2.2): for example, there is a small proportion of buildings in DG4+DG5 (7.59%), and a large majority of masonry (65.47%) compared to reinforced concrete frame (21.31%) buildings. This imbalance should be taken into account when defining the machine learning models. representing Irpinia-1980, Pollino-1998, Molise-Puglia-2002, Emilia-Romagna-2003, L'Aquila-2009, Emilia-Romagna-2012, and Garfagnana-Lunigiana-2013 Ground slope condition (GS1: ridge, GS2: plain, GS3: moderate slope, GS4: steep slope), (h) Regularity in plan and elevation (IRe: irregular, Re: Regular), (i) Roof type (RT1: heavy no thrust, RT2: heavy thrust, RT3: light no thrust, RT4: light thrust), (j) Construction material (CM1: poor-quality masonry, CM2: good-quality masonry, CM3: poor-quality mixed frame masonry, CM4: good-quality mixed frame masonry, CM5: reinforced concrete frame, CM6: reinforced concrete wall, CM7: steel frames, CM8: other), and (k) macro-seismic intensity.

Method

Machine learning models

E1 E2 E3 E4 E5 E6 E7 (a) (c) (b) (d) (e) (f) (g) (h) (i) (j) (k)
In chapter 1, we applied classification-and regression-based machine learning models to the damage observed after the 2015 Gorkha Nepal earthquake (NPC, 2015). The main concepts for method selection, the definition of the dataset for training and testing, and the representation of model performance are presented here.

To develop the heuristic damage assessment model, the damage grades are considered as the target feature. The damage grades are discrete labels, from DG0 to DG5. Three most advanced classification and regression machine learning algorithms were selected: random forest (RFC) and regression (RFR) [START_REF] Breiman | Random Forests[END_REF], gradient boosting classification (GBC) and regression (GBR) [START_REF] Friedman | Greedy Function Approximation:A Gradient Boosting Machine[END_REF], and extreme gradient boosting classification (XGBC) and regression (XGBR) [START_REF] Chen | XGBoost: A Scalable Tree Boosting System[END_REF].

A label (or class) was thus assigned to the categorical response variables (DG) for the classificationbased machine learning models. For the regression-based machine learning models, DG is converted into a continuous variable to minimize misclassifications [START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF].

Building features and macroseismic intensities were considered as input features. A one-hot encoding technique was used to convert the categorical features (i.e., ground slope condition, building position, roof type, construction material) into binary values (1 or 0), resulting in 28 input variables (Tab. 2.2).

No input features were removed from the dataset: some building features (e.g., number of storeys and height) may be correlated but we assumed that the presence of correlated features does not impact the overall performance of these machine learning methods [START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF]. No specific data cleaning methods were applied to the DaDO database.

The machine learning algorithms from the Scikit-learn package developed in Python [START_REF] Pedregosa | Scikit-learn[END_REF] were applied. The machine learning models were trained and tested on the randomly selected training (60% of the dataset) and testing (40% of the dataset) subsets of data, considering a single earthquake dataset or the whole DaDO dataset. The testing subset was kept hidden from the model during the training phase.

Machine learning model efficacy

The efficacy of the heuristic damage assessment model (i.e., its ability to predict damage to a satisfactory or expected degree) was analysed in three stages: comparison of the efficacy of the machine learning models using metrics; analysis of specific issues related to machine learning using the selected models; and application of the heuristic model to the whole DaDO dataset.

First stage: model selection

In the first stage, only the L'Aquila-2009 portfolio was considered for the training and testing phases. This is the largest dataset in terms of the number of buildings and was obtained using the AeDES survey format [START_REF] Baggio | Field Manual for post-earthquake damage and safety assessment and short term countermeasures (AeDES) Translation from Italian: Maria ROTA and Agostino GORETTI[END_REF]Dolce et al., 2019). Model efficacy was provided by the confusion matrix, which represents model prediction compared with the so-called "ground truth" value. Accuracy was then represented on the confusion matrix by the ratio of the number of correctly predicted DGs to the total number of observed values per DG (ADG).

Second stage: machine learning related issues

In the second stage, the best heuristic model for damage assessment was selected based on the highest efficacy, and used to analyse and test specific issues related to machine learning: (1) the imbalance distribution of DGs in the DaDO, (2) the performance of the selected model when only some basic, but accurately assessed, building features are considered (i.e., number of storeys, location, age, floor area), and (3) the simplification of the heuristic model, in the sense that DGs are grouped into a traffic-lightbased classification (i.e., green, yellow and red, corresponding to DG0+DG1, DG2+DG3 and DG4+DG5, respectively). In the second stage, the issues related to machine learning were first analysed using the L'Aquila-2009 portfolio. The whole DaDO dataset was then used.

Third stage: application to the whole DaDO portfolio and comparison with Risk-UE

In the third stage, several learning and testing sequences were considered, with the idea of moving to an operational configuration in which past information is used to predict damage from future earthquakes: either learning based on a portfolio of damage caused by one earthquake and tested on another portfolio, or learning based on a series of damage portfolios and tested on the portfolio of damage caused by an earthquake placed in the chronological continuity of the earthquake sequence considered. In this stage, the efficacy of the heuristic damage assessment model was analysed by comparing the prediction values with the so-called "ground truth" values through the error distribution, as follows:

ε ! (%) = & " ! # ' * 100 (2.1)
where n $ is the total number of buildings at a given error level (difference between observed and predicted DGs), N is the total number of buildings in the damage portfolio.

In this stage, the efficacy of the heuristic damage assessment model was compared with the conventional damage prediction framework proposed by the RISK-UE method [START_REF] Milutinovic | Risk-UE An advanced approach to earthquake risk scenarios with applications to different european towns, Rep. to WP4 vulnerability[END_REF]. The RISK-UE method assigns a vulnerability index (IV) to a building, based on its construction material and structural properties (e.g., height, building age, position, regularities, geographic location). For a given level of seismic demand (MSI), the mean damage (µd) and the probability, pk, of observing a given damage level k (k = 0 to 5) are given by:

µ ! = 2.5 11 + tanh & %&'().+,'-./0./ +.0 '7 (2.2) p 1 = ,! 1!(,.1)! & 5 " , ' , &1 - 5 " , '
,.1

(2.3)

Herein, comparing the heuristic model and the RISK-UE method amounts to considering the following steps, based on the equations given by RISK-UE:

Step 1 -The buildings in the training and testing datasets are grouped into different classes according to construction material.

Step 2 -For a given building class in the training dataset, computation of

Step 2.1 -mean damage (µ ! ) using the observed damage distribution at a given MSI value by:

µ ! = ∑ p 1 k , 167
(2.4)

Step 2.2 -vulnerability index (IV) with the µ ! obtained in step 2.1 by:

IV = / ).+, 113.1 -MSI + 2.3 Atanh ./ & 5 " +., -1'B7 (2.5)
Step 3 -For the same building class in the test dataset, calculation of

Step 3.1 -mean damage (µ ! ) Eq. 2.2 for a given MSI value with the value of IV obtained in step 2.2;

Step 3.2 -damage probability (p 1 ) Eq. 2.3 with the value of µ ! obtained in step 3.1;

Step 3.3 -distribution of buildings in each damage grade within a range of MSI values observed in the test dataset as follows:

N 89$!,1 = ∑ p 1 n ;<=,%&' %&'
(2.6)

where n ;<=,%&' is the total number of buildings observed in the test set for a given MSI value;

Step 3.4 -absolute error (ε 1 ) in each damage level k, given by:

ε 1 = C # #$%,' .# ()!",' # C (2.7)
where, N ;<=,1 is the total number of buildings observed in the given damage grade k.

Similarly, the heuristic damage assessment model was also compared with the mean damage relationship (Eq. 2.4) applied to the test set. Thus, for each building class in the test set, the error value (Eq. 2.7) for each DG was computed from the µ ! on the observed damage using Eq. (2.4), the probability p 1 of obtaining a given DG k (k= 0 to 5) using Eq. ( 2.3), and the distribution of buildings in each DG N 89$!,1 for a given MSI value using Eq. (2.6).

Result

First stage: model selection

The efficacy of the regression (RFR, GBR, XGBR) and classification (RFC, GBC, XGBC) machine learning models trained and tested on the randomly selected 60% (training set) and 40% (test set) of the 2009 -L'Aquila earthquake building damage portfolio is summarized in show that the efficacy of these models is higher for the lower DGs (around 60% for DG0 and 55% for DG1) and lower for the higher DGs (6% and 1% of the buildings are correctly classified in DG4 and DG5, respectively). and 39% for DG1) and lower efficacy for the higher DGs (5%, 23%, 12% and 17% buildings correctly classified in DG2, DG3, DG4 and DG5, respectively). The classification-based machine learning models thus yielded slightly better predictive efficacy, but still lower than recent studies applied to other datasets [START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF][START_REF] Harirchian | A synthesized study based on machine learning approaches for rapid classifying earthquake damage grades to rc buildings[END_REF][START_REF] Mangalathu | Regional Seismic Risk Assessment of Infrastructure Systems through Machine Learning: Active Learning Approach[END_REF][START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF][START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF]. The high classification error in the higher DGs could be related to the characteristics of the building portfolio and the imbalance of DG distribution. Among the classification methods, the XGBC model showed slightly higher classification efficacy; the XGBC model was therefore selected for the next stages 2.4.2 and 2.4.3.

Imbalance distribution of the DGs in the DaDO

The efficacy of the heuristic damage assessment model depends on the distribution of target features in the training dataset. This can lead to low prediction efficacy, especially for minority classes [START_REF] Estabrooks | A mixture-of-experts framework for learning from imbalanced data sets[END_REF][START_REF] Japkowicz | The class imbalance problem A systematic study fulltext[END_REF][START_REF] Branco | SMOGN: a Pre-processing Approach for Imbalanced Regression[END_REF][START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF]. The previous section reports significant misclassification associated with the highest DGs for all classification-and regression-based models (Fig. 2.3), i.e., for the DGs with the lowest number of buildings (Fig. (d) a combination of oversampling and undersampling methods: oversampling of the minority class using the SMOTE method, followed by the Edited Nearest Neighbours (ENN) undersampling method to eliminate data that is misclassified by its three nearest neighbours (SMOTE-ENN). The AT, MAE and MSE scores are given in Table 2.4 with the associated effects. In conclusion, the random oversampling method improves prediction in the minority class without significantly decreasing prediction in the majority class. The random oversampling method was therefore applied in this study.

Testing the XBGC model with basic features

This section begins by exploring the importance of each feature in the heuristic damage assessment model applied to the L'Aquila-2009 portfolio. We used the Shapely Additive Explanations (SHAP) method developed by [START_REF] Lundberg | A Unified Approach to Interpreting Model Predictions[END_REF]. The SHAP method compares the efficacy of the model with and without considering each input feature to measure its average impact, provided in terms of mean absolute SHAP values. 2021) observed a decrease in the vulnerability of structures as construction year increases, without distinguishing the DG considered, which is not the case herein. Note also that the importance score associated with the location feature can indirectly capture variations in local geological properties and the spatially distributed vulnerability associated with the built-up area of the L'Aquila-2009 portfolio (e.g., the distinction between the historic town and more modern urban areas). Furthermore, the average SHAP value obtained for poor quality masonry buildings for DG3/DG4/DG5 confirms the same high vulnerability of this typology as in the EMS-98 scale [START_REF] Grünthal | Escala Macro Sísmica Europea EMS -98[END_REF], regardless of DG. Some basic features of the building (e.g., location, age, floor area, number of storeys, height) are observed with a high mean SHAP value (Fig. 2.5a). Compared with others, these five basic features can be easily collected from the field or provided by national census databases, for example. Fig. 2.5b shows the efficacy of the heuristic damage assessment model using XGBC trained with a set of easily accessible building features (i.e., basic-features-setting: geographic location, floor area, number of stories, height, age, MSI), after addressing the class-imbalance issue using the random oversampling method. Compared with Fig. 2.4b (considering all features and named as the full-features-setting), the XGBC model with the basic-features-setting (Fig. 2.5b) gives almost the same efficacy with only a 6% average reduction in the accuracy scores.

Testing the XBGC model with the traffic-light system for damage grades

In this section, a simplified version of the DG scale was used, in the sense that the DGs are classified according to a traffic-light system (TLS) (i.e., green G, yellow Y and red R classes, corresponding to DG0+DG1, DG2+DG3 and DG4+DG5, respectively), as monitored during post-earthquake emergency situations (Mangalathu et al., 2020a;Riedel et al., 2015;ATC, 2005;[START_REF] Bazzurro | GUIDELINES FOR SEISMIC ASSESSMENT OF DAMAGED BUILDINGS[END_REF].

Testing the XGBC model with the whole dataset

The efficacy of the XGBC model was tested using a dataset with six building damage portfolios, excluding the 1980-Irpinia building damage portfolio. The XGBC model was trained and tested on the randomly selected 60% (training set) and 40% (test set) of the dataset for EMS-98/TLS damage classification, with two sets of features (full-features-setting and basic-features-setting), applying the random oversampling method to compensate for class-imbalance issues.

Third stage: application to the whole DaDO portfolio and comparison with Risk-UE

In this section, the efficacy of the heuristic damage assessment model was considered for building damage predictions, without respecting the time frame of the earthquakes. Two scenarios were considered: (1) a single building damage portfolio was used for training and the model was then tested on the others (named single-single), in situations using a single portfolio to predict future damage; and

(2) some building damage portfolios were used for training but testing was performed on a single portfolio (named aggregate-single), i.e. a larger number of damage portfolios were used as a training

Single-single scenario

First, a series of building damage portfolios, concerning earthquakes occurring in northern or southern

Italy and of different magnitudes, was used for training and testing:

Training set: E3 -test set: E1, E5, E7.

Training set: E5 -test set: E1, E3, E7.

Training set: E7 -test set: E1, E3, E5.

Figure 2.8 shows the distribution of correct DG classification (i.e., 1 -ε ! in % given by Eq. 2.1) observed for each building for the EMS-98 damage grade (2.8a) and the TLS (2.8b) systems. The xaxis represents the incremental error in the damage grade (e.g., 1 corresponds to the delta of damage grade between observation and prediction, regardless of the DG considered).

For the EMS-98 damage scale, correct classification (x-value centred on 0) in the range of 31% to 48% was found, depending on the training/test data sets. The error distribution is quite wide with incorrect predictions of +/-1 DG in the range of +/-13-35%. Remarkably, when considering the E1 portfolio (Irpinia-1980), for which the post-earthquake inventory was based on another form, as the test set, the error is larger. The predictions at +/-1 DG (i.e., the sum of the x-values Fig. 2.8a between -1 and +1)

were 70.5%, 69.9% and 72.8% with portfolios E3, E5 and E7 as the test set, respectively, for an average of 71%. For the other portfolios, the average of the predictions at +/-1 DG was 77%, 78% and 77%, respectively, for portfolios E5, E3 and E7 as the test set. This tendency was also observed for the TLS damage system (Fig. 2.8b). In this case, the classification of the E1 portfolio was correct on average (average of x-values centred on 0) at 63% and equal to 72%, 73%, and 70.5% for the test on portfolios E5, E3, and E7. For both damage scales, the distributions were skewed, with a larger number of predictions being underestimated (positive x-values), as certainly a consequence of the choice of machine learning models, their implementation (including imbalance issues), the distribution of input and target features considered, or all. The interest of machine learning model is also to have a relevant representation of the errors and limits of these methods.

Aggregate-single scenario

Secondly, several aggregated building damage portfolio scenarios were considered to predict a single earthquake, thus testing whether the prediction was improved by increasing the number of postearthquake damage observations. Three scenarios were tested. They are represented in Fig. 2.9 applying the EMS-98 damage grade (9a) and the TLS (9b): the quality and homogeneity of the input data (i.e., building features) affect the efficacy of the heuristic model and (2) this efficacy is limited and not improved by increasing the number of building damage observations, with a score (excluding E1) between 40% and 49% (x-value centred on 0), and up to 78%

(average of the two scenarios, Fig. 2.8a and Fig. 2.9a) at +/-1 DG. Considering the TLS damage scale (Fig. 2.9b), a damage prediction efficacy of about 72% was obtained (compared with 72% in Fig. 2.8b),

i.e., but no significant improvement was observed when the number of damaged buildings in the training portfolio was increased. For EMS-98 and TLS, the distributions were skewed, with a larger number of predictions being underestimated (positive x-values).

Finally, in conclusion, the heuristic damage assessment model based on the XGBC model gives a better score for TLS damage assessment than for the EMS-98 damage scale. The TLS system also allows for quick assessment of damage on a large scale such as a city or region from an operational point of view.

Comparing efficacy with the Risk-UE model

The efficacy of the heuristic damage assessment model was then compared with conventional damage prediction methods, i.e., RISK-UE and mean damage relationship (Eq. 2.2 to 2.7), considering the basic-features-settings. For RISK-UE, mean damage µ ! (Eq. 2.4) was computed using the training set and the vulnerability index IV for each building (Eq. 2.5). A vulnerability index was then attributed to all the buildings in each class defined according to building features. The vulnerability indexes were then attributed to every building in the test set, mean damage (µ ! ) was computed with Eq. 2.2 and then DG distribution with Eq. 2.3, before being compared with the damage portfolio used for testing. Finally, the distribution of the mean damage observed (Eq. 2.4) was compared with the distribution of damage directly on the test set, using Eq. 2.3. The x-axis is the damage grade and the y-axis is the percentage of absolute error (ε 1 in % given by Eq. 2.7). The blue bar corresponds to the mean damage relationship, the red bar corresponds to the RISK-UE method, the green and orange bars correspond to the heuristic model without (XGBC1) and with (XGBC2) compensation for the class-imbalance issues, respectively.

Discussion

Previous studies have aimed to test a machine learning framework for seismic building damage assessment (e.g., [START_REF] Mangalathu | Regional Seismic Risk Assessment of Infrastructure Systems through Machine Learning: Active Learning Approach[END_REF][START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF]Harirchan et al., 2021;[START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF]. They evaluated various machine learning and data balancing methods to classify earthquake damage to buildings. However, these studies [START_REF] Mangalathu | Regional Seismic Risk Assessment of Infrastructure Systems through Machine Learning: Active Learning Approach[END_REF][START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF], Harirchan 2022) also used a larger building damage database, but did not investigate the importance of input features as a function of damage levels and did not compare machine learning with conventional damage assessment methods.

Our study aims to go beyond previous studies by testing advanced machine learning methods and data resampling techniques using the unique DaDO dataset collected from several major earthquakes in Italy. This database covers a wide range of seismic damage and seismic demands of a specific region, including undamaged buildings. Most importantly, this study highlights the importance of input features according to the degrees of damage and finally compares the machine learning models with a classical damage prediction model (Risk-UE). The machine learning models achieved comparable accuracy to the Risk-UE method. In addition, TLS-based damage classification, using red for heavily damaged, yellow for moderate damage, and green for no to slight damage, could be appropriate when the information for undamaged buildings is unavailable during model training.

Indeed, it is worth noting that the importance of the input features used in the learning process changes with the degree of damage: this indicates that each feature may have a contribution to the damage that changes with the damage level. Thus, the weight of each feature does not depend linearly on the degree of damage, which is not considered in conventional vulnerability methods.

The prediction of seismic damage by machine learning remains until now tested on geographically limited data. The damage distribution is strongly influenced by region-specific factors such as construction quality and regional typologies, implementation of seismic regulations and hazard level.

Therefore, machine learning-based models can only work well in regions with comparable characteristics and a host-to-target transfer of these models should be studied. In addition, the distribution of damage is often imbalanced, impacting the performance of machine learning models by assigning higher weights to the features of the majority class. However, data balancing methods like random oversampling can reduce bias caused by imbalanced data during the training phase, but they may also introduce overfitting issues depending on the distribution of input and target features. Thus, integrating data from a wider range of input features and earthquake damage from different regions, relying on a host-to-target strategy, could help achieve a more natural balance of data sets and lead to less biased results. Moreover, the machine learning methods trained only on the data available in the learning phase, reflects the building portfolio in the study area. The importance of the features contributing to the damage could thus be modulated, and would require a host-to-target adjustment for the application of the model to another urban zone/seismic region.

However, the machine learning models trained and tested on the DaDO dataset resulted in similar damage prediction accuracy values reported in existing literature using different models and datasets with different combinations of input features. This might suggest that the uncertainty related to building vulnerability in damage classification may be smaller than the primary source of uncertainty related to the hazard component (such as ground motion, fault rupture, slip duration, etc.).

In recent years, there has been a proliferation of open building data, such as the OpenStreetMap-based dynamic global exposure model [START_REF] Schorlemmer | Global Dynamic Exposure and the OpenBuildingMap -A Big-Data and Crowd-Sourcing Approach to Exposure Modeling[END_REF] and building damage dataset after an earthquake (such as DaDO). We must therefore continue this paradigm shift initiated by Riedel et al. (2014Riedel et al. ( , 2015) ) which consisted in identifying the exposure data available and as certain as possible, and in finding the most effective relationships for estimating the damage, unlike conventional approaches which proposed established and robust methods but relying on data not available and therefore difficult to collect. The global dynamic exposure model will make it possible to meet the challenge of modelling exposure on a larger scale on available data, using a tool capable of integrating this large volume of data. Machine learning methods are one such rapidly growing tool that can aid in exposure classification and damage prediction by leveraging readily available information. It is therefore necessary to continue in this direction in order to evaluate the performance of the methods and their pros and cons for maximum efficacy of the prediction of damage.

Future works will therefore have to address several key issues that have been discussed here but that need to be further investigated. For example, the weight of the input features varies according to the level of damage, but one can question the systematization of this observation whatever the dataset and the feature considered. The efficiency of the selected models and the management of imbalance data remain to be explored, in particular by verifying regional independence. Taking advantage of the increasing abundance of exposure data and post-seismic observations, the imbalanced feature distribution and observed damage levels could be solved by aggregating datasets independent of the exposure and hazard contexts of the regions, once the host-to-target transfer of the models has been resolved. Finally, key input features (still not yet identified) describing hazard or vulnerability may be unexplored, and incorporating them into the models may improve the accuracy of damage classification.

Conclusion

In this study, we explored the efficacy of machine learning models trained using DaDO post-earthquake building damage portfolios. We compared six machine learning models: RFC, GBC, XGBC, RFR, GBR, and XGBR. These models were trained on numbers of building features (location, number of storeys, age, floor area, height, position, construction material, regularity, roof type, ground slope condition) and ground motion intensity defined in terms of macro-seismic intensity. The classification models performed slightly better than the regression methods and the XGBC model was ultimately found to be the most efficient model for this dataset. To solve the imbalance issue concerning observed damage, the random oversampling method was applied to the training dataset to improve the efficacy of the heuristic damage assessment model by rectifying the skewed distribution of the target features (DGs). Surprisingly, we found that the weight of the most important building feature evolves according to DG, i.e., the weight of the feature for damage prediction changes depending on the DG considered, which is not taken into account in conventional methods.

The basic-features-setting (i.e., considering number of storeys, age, floor area, height and macroseismic intensity, which are accurately evaluated for the existing building portfolio) gave the same accuracy (0.68) as the full-features-settings (0.72) with the TLS-based damage classification method. For training and testing, the homogeneity of the information in the portfolios is a key issue for the definition of a highly effective machine learning model, as shown by the data from the E1 earthquake (Irpinia-1990).

However, the efficacy of the model reaches a limit which is not improved by increasing the number of damaged buildings in the portfolio used as the training set, for example. For damage prediction, this type of heuristic model results in approximately 75% correct classification. Other authors (e.g., Riedel et al., 2014Riedel et al., , 2015;;[START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF]) have already reached this same conclusion by increasing the percentage of the training set compared with the test set.

Despite this limit threshold, the level of accuracy achieved remains similar to that attained by conventional methods, such as Risk-UE and the mean damage relationship, for the basic-featuressettings and TLS-based damage classification (error values less than 17 %). Machine learning models trained on post-earthquake building damage portfolios could provide a reasonable estimation of damage for a different region with similar building portfolios, after host-to-target adjustment. Some variability may have been introduced into the damage prediction model due to the framework defined to translate the original damage scale to the EMS-98 damage scale and because in the DaDO database, the year of construction and the floor area of each building are provided as interval values, and missing locations of buildings were replaced with the location of local administrative centres. The latter can lead to a smoothing of the macro-seismic intensities to be considered for each structure and also affect the distance to the earthquake. Similarly, the building damage surveys were carried out after the seismic sequence, which includes aftershocks as well as the mainshock, whereas the MSI input corresponds to the mainshock from the USGS ShakeMap. All these issues may reduce the efficacy of the heuristic model and its limit threshold. Addressing these issues could improve the damage prediction performance of machine learning models.

Introduction

The main objective of seismic risk mitigation is to ensure people's safety and protect their sources of sustenance by minimizing earthquake threats [START_REF] Bommer | Earthquake hazard and risk analysis for natural and induced seismicity: towards objective assessments in the face of uncertainty[END_REF]. To achieve this objective, accurate risk assessment requires a thorough examination of seismic damage as a crucial component of the process.

Advanced methods have been developed to estimate seismic damage (among other, FEMA, 2003;[START_REF] Milutinovic | Risk-UE An advanced approach to earthquake risk scenarios with applications to different european towns, Rep. to WP4 vulnerability[END_REF][START_REF] Porter | An overview of PEER's performance-based earthquake engineering methodology[END_REF][START_REF] Vamvatsikos | Applied incremental dynamic analysis[END_REF]Silva et al., 2014;[START_REF] Régnier | Contribution of ambient vibration recordings (free-field and buildings) for post-seismic analysis: The case of the Mw 7.3 Martinique (French Lesser Antilles) earthquake[END_REF]Hancilar et al., 2010;Lagomarsino and Giovinazzi, 2006). These methods provide damage as a function of seismic hazard, exposure models describing the portfolio of buildings, and the vulnerability/fragility functions describing the damage probability for building typologies. However, for large-scale damage assessment, the information necessary for these methods is often sparse, incomplete, or low resolution, making regional and national-scale damage assessment very challenging and time-consuming (Riedel et al., 2015).

Communities worldwide are collaborating to build exposure models and fragility/vulnerability functions for large-scale seismic damage assessment by aggregating available information from existing databases (Crowley et al. 2020). Similarly, efforts have been made to test novel methods that use satellite and ground-based remote sensing for exposure modeling and damage classification (Pittore and Wieland, 2013;Wieland et al., 2015;[START_REF] Stone | Earthquake damage data collection using omnidirectional imagery[END_REF][START_REF] Rao | Earthquake Building Damage Detection based on Synthetic Aperture Radar Imagery and Machine Learning[END_REF]. In parallel, platforms outlining basic building features suitable for damage assessment have increased in number (e.g., OpenBuildingMaps [START_REF] Schorlemmer | Global Dynamic Exposure and the OpenBuildingMap -Communicating Risk and Involving Communities[END_REF]; the National Institute of Statistics and Economic Studies in France (INSEE) (www.insee.fr); global exposure database [START_REF] Gamba | The GED4GEM project: development of a Global Exposure Database for the Global Earthquake Model Initiative[END_REF]; the World Housing Encyclopaedia (http://db.world-housing.net/); e-Stat Japan (https://www.e-stat.go.jp/)). These platforms offer access to the low-resolution, high-density building information, which opens an opportunity to be used for seismic damage assessment (Riedel et al., 2015). In this context, Riedel et al. (2014Riedel et al. ( , 2015) ) investigated machine learning methods to estimate seismic damage on a regional and national level using readily available data for many buildings in the INSEE database.

On the other hand, numerous studies have investigated the effectiveness of machine learning methods in accurately connecting building features and spatially-distributed ground motion to different levels of damage, with satisfactory levels of accuracy when compared to traditional methods or tested using postearthquake observations (e.g., Riedel et al. 2015;[START_REF] Harirchian | A synthesized study based on machine learning approaches for rapid classifying earthquake damage grades to rc buildings[END_REF][START_REF] Mangalathu | Regional Seismic Risk Assessment of Infrastructure Systems through Machine Learning: Active Learning Approach[END_REF][START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF][START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF][START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF]Ghimire et al., , 2023)). In parallel, significant efforts have been made to collect and share post-earthquake building damage observations after damaging earthquakes (MTPTC, 2010;NPA, 2021;MINUV, 2021;Dolce et al., 2019;[START_REF] Omoya | A relational database to support post-earthquake building damage and recovery assessment[END_REF][START_REF] Stojadinović | Rapid earthquake loss assessment based on machine learning and representative sampling[END_REF]. By taking advantage of the convergence of situations, i.e. rapidly developing learning methods and the increasing amount of available building open-data, it is time to evaluate the efficiency conditions of the damage classification methods, in particular for different post-earthquake data according to the building criteria and damage classification, but also for the construction of a building damage model through host-to-target region adjustments, i.e. developing a machine from one specific region in terms of seismicity and building design to be applied to another region.

The main objective of this study is to perform a comparative study on damage prediction across different regions using machine learning models trained on the readily available building features from the postearthquake damage database. Therefore, this study compares the damage prediction effectiveness of machine learning models trained on the age, number of storeys, and macroseismic intensity using building damage datasets from Haiti, Nepal, and Italy. The dataset and methods are described in the 3.2 and 3.3 sections. The 3.4 and 3.5 section presents the results of the comparative study of damage prediction, followed by a conclusion in 3.6.

Data

2010-Haiti earthquake building damage dataset (HBDP)

A Mw 7 earthquake struck Haiti on January 12, 2010, with the epicentre located 25 km SW of Port-au-Prince, with a hypocentre depth of 13 km and a 15-40 km long rupture, followed by three large aftershocks 6.0 Mw, 5.7 Mw, and 5.9 Mw, causing over 300,000 casualties, leaving over 1.3 million people homeless, and resulting an estimated losses of US$ 7 to US$ 14 billion, exceeding the Haiti gross domestic product [START_REF] Desroches | Overview of the 2010 Haiti Earthquake[END_REF]. The government of Haiti conducted a massive postearthquake damage survey with the help of more than 300 trained engineers, alongside a third-party structural engineers, and developed a database of observed damage (MTPTC, 2010).

During the field survey of damage, the ATC-20 damage classification methodology (ATC, 2005) was adopted for damage classification (MTPTC, 2010).

The damage was grouped into seven discrete classes based on visual observation.

Damage grades (DG): none for no damage (DG0) slight for 0-1% damage (DG1), light for 1-10% damage (DG2), moderate for 10-30% damage (DG3), heavy for 30-60% damage (DG4), major for 60-100% damage (DG5) and destroyed for 100% damage (DG6), respectively.

In the same survey, building features related to structural typology were also collected in the field: Number of storeys-total number of floors above the ground surface.

Age of building-time difference between the date of the earthquake and the date of building construction/renovation; buildings were grouped into four categories according to age: 0-10 years, 11-25 years, 26-50 years, and older than 50 years.

Floor plan-geometric shape defining the building plan; eight types of plans were defined: E-shape, Hshape, L-shape, O-shape, Rectangular-shape, T-shape, U-shape, and Other-shape.

Wall type-materials used in vertical wall; seven wall types were defined: block-masonry with reinforcement, block-masonry without reinforcement, brick-masonry, reinforced concrete, stonemasonry, wood-masonry, and others.

Structure type-material used in the vertical structure; four structure types were defined: reinforced concrete structures, load-bearing wall structures, steel sheet-metal structures, and wood sheet-metal structures.

Floor-type-materials used for flooring system in a building: three floor types were defined: reinforced concrete floor, concrete floor, and wooden floor.

For each building, the geographic location was provided in terms of latitude and longitude.

The 2010 Haiti earthquake building damage portfolio (HBDP) contained 351,819 buildings with complete information on the above-defined features. Fig. 3.1 shows their location, and Fig. 3.2 and Tab 3.1. show the distribution of building features in the HBDP database. The distribution of the samples was imbalanced: for example, there was a small portion of buildings with higher damage grades DG5+DG6 (16.01%) as compared with DG0+DG1 (42.23%), a large majority of buildings were 1storey (83.05%), reinforced concrete structure (82.11%), and unreinforced block masonry wall (87.81%).

For this study, the HBDP was completed with the spatially distributed ground motion information in terms of macroseismic intensities (MSI) values defined in terms of modified Mercalli intensities provided by the United States Geological Survey (USGS) ShakeMap tool [START_REF] Wald | ShakeMap manual: technical manual, user's guide, and software guide[END_REF]. The MSI values corresponding to the 2010 mainshock were added to each building information using their geographic location (Fig. 3.1).

(c) (d) (b) (a) (e) (f)

2015-Nepal earthquake building damage database (NBDP)

Following the 7.8 Mw earthquake on 25 April 2015 in Nepal, the government of Nepal conducted a post-earthquake damage survey in the 11 most affected districts and released a database of the observed damage. Tab. 3.2 summarizes the distribution of building features considered for this study and the full description of the database, including the building features, the damage classification and the seismic hazard classification according to USGS macroseismic shake-map is described in [START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF] (Chapter 1). In this database, information are given by districts (corresponding to an administrative unit).

Method

A detailed description on the development of machine learning models is presented in [START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF]Ghimire et al. ( , 2023)), and a short summary is presented here.

The damage grades were considered as target features. Damage grades were considered as categorical response variables from DG0 to DG6 to test the classification machine learning models. Two advanced classification-based machine learning methods were selected: random forest classification (RFC) [START_REF] Breiman | Random Forests[END_REF] and extreme gradient boosting classification (XGBC) [START_REF] Chen | XGBoost: A Scalable Tree Boosting System[END_REF].

Similarly, the damage grades were ordinal variables, and they were converted as continuous variables from 0 (DG0) to 6 (DG6). Two advanced regression models were selected: random forest regression (RFR) [START_REF] Breiman | Random Forests[END_REF] and extreme gradient boosting regression (XGBR) [START_REF] Chen | XGBoost: A Scalable Tree Boosting System[END_REF].

Building features and MSI were considered as input features. A one-hot encoding technique was used to convert the categorical features (i.e., wall type, structure type, floor plan, floor type) into binary values (1 or 0), resulting in 30 input variables (Tab. 3.2). No specific data cleaning methods were applied to the HBDP database.

The machine learning algorithms were applied from the Scikit-learn package developed in Python [START_REF] Pedregosa | Scikit-learn[END_REF]. The HBDP was randomly divided into training (60% of the whole HBDP) and test (40% of the whole HBDP) dataset with the identical distribution of target features in the training and the test set. The test dataset was kept hidden from the model during the training phase and used for model testing. The efficacy of the machine learning damage prediction model (i.e., its ability to predict damage to a satisfactory or expected degree) was analyzed in three different ways: at building level, at building-class level (defined according to the combination of three input features, i.e. number of storeys, age, and MSI), and at portfolio level.

Testing efficacy at building level

Model selection

At the building level, model efficacy was evaluated by the confusion matrix, representing model prediction compared with the observed (so-called "ground truth") value. Accuracy was then represented on the confusion matrix by the ratio of the number of correctly predicted DG to the total number of observed values per DG (ADG). Total accuracy (AT) was computed in a similar manner as the ratio of the number of correctly predicted DGs to the total number of observed buildings in the whole test dataset. AT and ADG values close to 1 indicate high efficacy. Moreover, the quantitative statistical error was also calculated as the average of the absolute value of errors (MAE): value close to 0 indicate high efficacy.

For classification-based machine learning models, the ordinal value of the DG was used to calculate the MAE score directly. For the regression-based machine learning models, the output DG values were rounded to the nearest integer for the accuracy scores plotted in the confusion matrix but not to calculate the MAE score.

Machine learning related issues

The best model for damage assessment was selected based on the highest efficacy and used to test

Testing efficacy by DG aggregation

DG aggregation in the whole test dataset

The buildings in the test dataset were aggregated according to the DG values and the error values was computed using Eq. 3.1:

ε 8 (%) = C D" *+ # C * 100 (3.1)
where Dn >? is the difference between the number of buildings observed and predicted at each DG level, and N is the total number of buildings in the test dataset.

DG aggregation in the building-class

The buildings in the test dataset were grouped into different building-class defined according to the combination of three input features (number of storeys, age, and MSI). The buildings in each buildingclass were aggregated according to the DG values and the error values were computed using Eq. 3.2:

ε @/ (%) = C D" *+/- # C * 100 (3.2)
where Dn >?/@ is the difference between the number of buildings observed and predicted at each DG level for a given building-class, and N is the total number of buildings in the test dataset.

Comparison between the datasets

The efficacy of the damage prediction model was explored in the HBDP, NBDP, and DaDO datasets.

These datasets were randomly divided into training (60%) and test (40%) datasets. The machine learning model was trained on the training set, and its efficacy was explored in the test dataset.

Efficacy at building-level between the datasets

The efficacy at the building level was compared through the error distribution computed using Eq. 3.3:

ε ! (%) = C " ! # $ C * 100 (3.3)
where n $ is the total number of buildings at a given error level (difference between observed and predicted DG), and N < is the total number of buildings in each test datasets.

Efficacy at building-class level between the datasets

The efficacy was compared at building-class level through the error distribution computed using Eq.

3.4:

ε @+ (%) = C D" *+/- #@ C * 100 (3.4)
where Dn >?/@ is the difference between the number of buildings observed and predicted at each DG in each building class, N @ is the total number of buildings in each building class.

Damage probabilities between the datasets

Finally, the damage probabilities were computed at the building-class level using Eq. 3.5 and compared between HBDP, NBDP, and DaDO datasets.

P !C (%) = C " *+/- #@ C * 100 (3.5)
where n >?/@ is the difference between the number of buildings observed and predicted at each DG in each building class, N @ is the total number of buildings in each building class.

Results

Testing efficacy at building level

Model selection

The efficacy of the regression (RFR, XGBR) and classification (RFC, XGBC) models trained and tested on the randomly selected 60% (training set) and 40% (test set) of the HBDP is summarized in Tab. 3.4.

For the ATC-20 damage classification, the regression-based machine learning models RFR and XGBR and Italian datasets in Chapter 1 and 2 [START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF](Ghimire et al. , 2023)).

In conclusion, the classification models showed slightly better efficacy as compared to the regression models and finally, the XGBC classification model is selected for this study.

Machine learning related issues

Handling of class-imbalance issue

The imbalance distribution of target features in the training dataset, known as class-imbalance issues, significantly impact the efficacy of the machine learning models [START_REF] Estabrooks | A mixture-of-experts framework for learning from imbalanced data sets[END_REF][START_REF] Japkowicz | The class imbalance problem A systematic study fulltext[END_REF][START_REF] Branco | SMOGN: a Pre-processing Approach for Imbalanced Regression[END_REF][START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF]). Resampling the training data using the random oversampling method effectively address the class-imbalance issues [START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF](Ghimire et al. , 2023)).

In this study, the random oversampling method was then applied in the training set before developing the XGBC model. The damage prediction efficacy on the test set is shown in Fig. 3.3(b). Compared with the result in Fig. 3.3(a), the XGBC model with random oversampling method yielded slightly better efficacy. For DG0/DG2/DG4/DG5/DG6, the ADG value increased by 30/5/15/11/12%; however, the ADG value for DG1 decreased by 42%, with a slightly higher MAE score (1.39) and lower AT score (0.33). The lower damage prediction efficacy could be originated from the confusion for the machine learning algorithms in multi-class classification depending on the resolution and distribution of input features [START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF][START_REF] Harirchian | A synthesized study based on machine learning approaches for rapid classifying earthquake damage grades to rc buildings[END_REF][START_REF] Roeslin | Damage assessment on buildings following the 19th september 2017 puebla, Mexico earthquake[END_REF]. This is also observed during damage surveys in the field, which sometimes find it hard to distinguish the intermediate damage grades, such as DG2 and DG3, or DG3 and DG4.

Testing traffic-light based damage classification

The ATC-20 damage classification was reframed into three classes according to the Traffic-Light based classification System (named TLS) (i.e., green G, yellow Y, red R classes, corresponding to DG0+DG1, DG2+DG3, DG4+DG5+DG6, respectively), as followed during post-earthquake emergencies [START_REF] Mangalathu | Regional Seismic Risk Assessment of Infrastructure Systems through Machine Learning: Active Learning Approach[END_REF]Riedel et al., 2015;ATC, 2005;[START_REF] Bazzurro | GUIDELINES FOR SEISMIC ASSESSMENT OF DAMAGED BUILDINGS[END_REF][START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF]Ghimire et al., , 2023)). The XGBC model is observed to have higher damage prediction efficacy for TLS-based damage classification, indicating that it is easier for the machine to classify damage into three-class than sevenclass (ATC-20 classification system) with the given level resolution of input features. This is consistent with previous studies by (Ghimire et al., 2023 and[START_REF] Nievas | Deliverable 6.1 D6.1 Integration of RISE Innovations in the Fields of OELF, RLA and SHM Deliverable information[END_REF][START_REF] Harirchian | A synthesized study based on machine learning approaches for rapid classifying earthquake damage grades to rc buildings[END_REF]Riedel et al., 2015;[START_REF] Roeslin | A machine learning damage prediction model for the 2017 Puebla-Morelos, Mexico, earthquake[END_REF].

Testing efficacy by DG aggregation

DG aggregation in the whole test dataset

Five XGBC models were developed using different combinations of input features and random oversampling method in HBDP dataset:

(a) XGBC1 with all input features and without handling class-imbalance issues;

(b) XGBC2 with all features and random oversampling;

(c) XGBC3 with the number of storeys, age, floor type, wall type, structure type, floor plan, MSI, and random oversampling;

(d) XGBC4 with location, number of storeys, age, MSI, and random oversampling;

(e) XGBC5 with the number of storeys, age, MSI, and random oversampling.

The damage prediction efficacy of these XGBC models was tested by DG aggregation in the HBDP test dataset. The distribution of error values (ε 8 from Eq. 3.1) is shown in Fig. 3.5.

The XGBC2 and XGBC4 models show a better damage prediction efficacy for ATC-20 classification (Fig. 3.5a), with error values less than 6% for all DGs except DG0 (9%) and DG1 (11%). Other models give error values of less than 10% for all DGs except DG0 (7-18%) and DG1(18-27%).

For TLS-based damage classification (Fig. 3.5b), XGBC1 shows higher error values (5-11%) among the DGs. Other models have smaller error values (less than 6%). Among them, XGBC4 and XGBC5 show higher damage prediction efficacy with the smallest error values (2-4% and 3-6%, respectively)

for the DGs prediction.

In conclusion, considering DGs aggregation through TLS, easily accessible building features (number of stories, age, location, and MSI) and random oversampling for class-imbalance issues provide reasonable damage estimates for ATC-20 and for TLS-based damage classification, highlighting their broad applicability for larger-scale damage classification. Therefore, the XGBC5 model for TLS-based damage classification is selected for further investigation in this study.

DG aggregation in the building-class using XGBC5 model

Building

Testing host-to-target machine learning models

The damage prediction efficacy was compared among three datasets: HBDP, NBDP, and DaDO. These datasets were randomly divided into training (60%) and test (40%) datasets. The XGBC5 model considering TLS-based damage classification was trained on the training set, and its efficacy was explored in the test dataset. The damage prediction efficacy was first compared at the building level and then at the building-class level.

Efficacy at building-level between the datasets

The distribution of error values in the DG (ε ! from Eq. 3.3) (e.g., 1 corresponds to the delta of 1 damage grade between observed and prediction, regardless of the DG considered) is shown in Fig. 3.7. The percentage of buildings correctly classified (x-value centred on 0 in Fig. 3.7a) in HBDP, DaDO, L'Aquila (a single earthquake of the DaDO dataset) and NBDP is 45%, 59%, 61% and 67%, respectively. The error values were concentrated in +/-1 DG in the range of 7-21% in NBDP, 5-23% in DaDO and L'Aquila, and 18-22% in HBDP. The error values in +/-2 DG was higher for HBDP (5-9%)

and DaDO (1-11%) as compared to NBDP (below 4%).

In the probability density function (Fig. 3.7b), skewed error distribution is observed, with a large number of predictions being underestimated in NBDP (positive x-value at peak: 0.21) and overestimated significantly in DaDO (negative x-value at the peak: -0.28) and slightly in HBDP (negative x-value at the peak: -0.08).

Efficacy at building-class level between the datasets

The HBDP, NBDP, and DaDO dataset have six common building classes according to the nomenclature considered for the Haiti dataset. VIII and H1-A3-VIII, the error are larger than 20% for H1-A2-VIII and H1-A3-VIII for DaDO and HBDP. Among the datasets, the lowest error values (below 10%) was observed in H1-A1-VIII, followed by H1-A2-VII (below 17%) and H1-A1-VII (below 18%) building classes. Interestingly, low-resolution features resulted in reasonable efficacy for large-scale damage classification at the building-class level or higher. The easily accessible building's features (age and the number of stories) and MSI can then provide a reasonable estimate of damage in TLS-based damage classification systems. This method is suitable for rapid damage classification at the regional and national scales, as Riedel et al. (2015) noted.

Damage probabilities between the datasets

We observed that the smallest error values for the H1-A1-VIII class, followed by H1-A1-VII, and higher error values for H1-A2-VIII and H1-A3-VIII class. Fig. 3.10 shows the damage probabilities (P !C ) for a building in these classes using Eq. 3.5.

For a building in the H1-A1-VIII class, the probability of being in no-to-slight (G), moderate (Y) and heavy (R) damage grade is 96%, 1% and 3% in Italy, 49%, 35% and 16% in Haiti, and 16%, 28% and 56% in Nepal, respectively. For the building class H1-A2-VII, the probability is 90%, 9% and 2% in Italy, 6%, 41% and 53% in Haiti, and 10%, 29% and 61% in Nepal, respectively. For H1-A1-VII class, the probability is 99%, 0.5% and 0.5% in Italy, 40%, 47% and 13% in Haiti, and 36%, 30% and 34%

in Nepal, respectively.

For H1-A3-VII class, the probability of G, Y and R is 28%, 45% and 27% for Italy, 7%, 28% and 65%

in Haiti, and 2%, 21% and 77% in Nepal, respectively.

Finally, for the H1-A3-VIII class, the probability of G, Y and R is 10%, 47% and 43% in Italy, 5%, 7% and 88% in Haiti, and 0.5%, 25.5% and 74% in Nepal, respectively.

Thus, a given building class with a given MSI value has different probability to be in each DG, as a consequence of the regional structural typologies and seismic regulations. For example, Nepal and Haiti provide very similar results whatever the class of building, compared to Italian dataset. Therefore, based on these three datasets, raw machine-learning based models for damage prediction are only valid for the hosting region. Thus, empirical adjustments for host-to-target region design differences using regional observation of damage are unlikely to be robust unless the buildings class are similar. In essence, these questions the relevance of generic vulnerability models developed for a host region and which are very tempting to apply to a target region. Testing empirical adjustments of machine learning model for regional characteristics is needed, isolating each host-to-target difference to analyse the contributions of the regional building design differences to the epistemic uncertainty of the model and the damage grades related to macroseismic intensity.

Conclusion

In this study, the damage classification efficacy of machine learning models was explored by using the 2010 Haiti earthquake building damage portfolio (HBDP), 2015-Nepal earthquake building damage portfolio (NBDP) and the observed damage for several earthquakes in Italy (DaDO). Compared to building-level damage classification, the XGBC model showed higher damage prediction efficacy when Higher-resolution features can result in higher efficacy in damage prediction. Interestingly, the lowresolution features (age, the number of stories, MSI) and ground truth damage information can also provide in reasonable efficacy for large-scale damage classification and enable a rapid large-scale damage classification, as also noted in previous studies (Riedel et al. 2014(Riedel et al. , 2015;;[START_REF] Ghimire | Testing machine learning models for seismic damage prediction at a regional scale using building-damage dataset compiled after the 2015 Gorkha Nepal earthquake[END_REF]Ghimire et al. , 2023)). This study opens an opportunity to test building data from the institutional database (e.g., national census, national housing database) for large-scale seismic risk assessment, as already discussed by Riedel et al. (2014Riedel et al. ( , 2015)). It can also benefit and encourage the exposure and vulnerability modeling communities (e.g., [START_REF] Jordan | METEOR : Modelling Exposure Through Earth Observation Routines to Aid Su s tainable Development[END_REF][START_REF] Schorlemmer | Global Dynamic Exposure and the OpenBuildingMap -Communicating Risk and Involving Communities[END_REF]. Further investigation should be carried out by collecting building damage portfolios from different regions and building information from different sources (e.g., OpenBuildingMap, European exposure model, INSEE database) to develop this emerging field.

Introduction

Performance-based earthquake engineering (PBEE) refers to the probabilistic framework in which earthquake consequences are expressed by a set of performance objectives, based on a comprehensive scientific foundation [START_REF] Porter | An overview of PEER's performance-based earthquake engineering methodology[END_REF]. Depending on the application, these performance objectives can help stakeholders to make decisions with regard to crisis management and structural capacity, such as immediate occupancy or near-collapse levels, by predicting human or economic losses. In the framework proposed by PEER [START_REF] Porter | An overview of PEER's performance-based earthquake engineering methodology[END_REF], PBEE works in four stages, starting with the hazard itself through to the consequence analysis. In hazard analysis, intensity measures (IM) and their annual frequency of exceedance (λIM) are defined by probabilistic seismic hazard assessment (PSHA). In structural analysis, the response of the structure to a given IM can be modeled and expressed in terms of engineering demand parameters (EDPs), such as structural drift, maximal top acceleration, etc. In damage analysis, damage measurement (DM) is calculated based on EDP values and models of structure capacity or fragility. Finally, the earthquake's consequences, in terms of repair costs, operability of the structure and potential economic or human losses for a given DM, can be calculated and expressed as decision variables (DVs) on which stakeholders can base their decisions in view of the expected performance levels.

The four steps of the underlying probabilistic framework of PBEE estimate the frequency of failure of a performance level over a given period of time; this involves uncertainties. For example, the annual frequency of exceeding a given EDP value (λEDP) is expressed by:

λ D>E = ∫ P[EDP/IM = im]|dλ im FG (4.1)
where P(EDP|IM=im) is the conditional probability of occurrence of each EDP value, taking into account the value of the IM, and dλim is the annual rate of exceeding an IM value, derived from the hazard curves. P(EDP|IM=im) is usually obtained by considering a series of nonlinear dynamic analyses of the structure. Baker and Cornell (2008a) provide a detailed description of approaches to characterize and propagate uncertainties at each step. Current research on PBEE is mainly focused on identifying the origins of uncertainties, distinguishing between epistemic and random uncertainties, in order to boost scientific efforts on the reducible elements that contribute most to performance uncertainty (e.g., [START_REF] Iervolino | Assessing uncertainty in estimation of seismic response for PBEE[END_REF]. In practice, P(EDP|IM=im) satisfies a chosen model of EDP distribution for a given IM and is obtained by regression of EDP values for IM values. [START_REF] Luco | Probabilistic seismic demand analysis, SMRF connection fractures, and near-source effects[END_REF], [START_REF] Luco | Structure-specific scalar intensity measures for near-source and ordinary earthquake ground motions[END_REF],

and Baker and Cornell (2008b) considered an IM to be sufficient if the prediction of EDP given IM is statistically independent of earthquake magnitude and epicentral distance values. Furthermore, the efficiency of IMs is assessed by measuring the variability of values of EDP (given IM) around the regression on the IM values. In general, Peak Ground Acceleration (PGA) or the acceleration spectral value at the resonance period T1 of the structure Sa(T1) (with 5% damping) are considered as outputs of the seismic hazard curves.

Structure response and the associated uncertainties are conditioned by time-history seismic excitation, considering the IM at which the EDP value is exceeded. The efficiency and sufficiency of Sa(T1) compared with PGA were investigated by [START_REF] Shome | Probabilistic Seismic Demand Analysis of Nonlinear Structures[END_REF] according to the type of building and the contribution of the higher modes to the total response was also considered [START_REF] Shome | Probabilistic Seismic Demand Analysis of Nonlinear Structures[END_REF]. Other ground motion parameters have also been investigated in terms of sufficiency and efficiency, such as peak values in velocity, duration or energy, or spectral values considering different resonant periods [START_REF] Buratti | A comparison of the performances of various ground-motion intensity measures[END_REF]Ebrahimian et al., 2015), spectral values in acceleration [START_REF] Luco | Structure-specific scalar intensity measures for near-source and ordinary earthquake ground motions[END_REF][START_REF] Shome | Probabilistic Seismic Demand Analysis of Nonlinear Structures[END_REF][START_REF] Bianchini | Prediction of Inelastic Structural Response Using an Average of Spectral Accelerations[END_REF][START_REF] Jayaram | A Computationally efficient ground-motion selection algorithm for matching a target response spectrum mean and variance[END_REF]Eads et al., 2015) or velocity [START_REF] Jayaram | A Computationally efficient ground-motion selection algorithm for matching a target response spectrum mean and variance[END_REF]Mollaioli et al., 2011), or by combining IMs based on vector-valued approach (Baker and Cornell, 2008b;[START_REF] Luco | Correlation of damage of steel moment-resisting frames to a vector-valued set of ground motion parameters[END_REF][START_REF] Vamvatsikos | Developing efficient scalar and vector intensity measures for IDA capacity estimation by incorporating elastic spectral shape information[END_REF]. All these studies are based on the numerical modeling of structures considering different ground motion datasets, mostly using the Incremental Dynamic Analysis approach (IDA). In structural analysis, the selection or generation of natural or synthetic accelerograms from different tectonic areas, the scaling applied to obtain the desired structural response values, the selection of physical modal parameters (e.g., structural period and damping) and their co-seismic variations, as well as other modeling assumptions related to component fragility functions, affect the overall uncertainty of the performance estimate. Furthermore, a typical assumption in the assessment of P(EDP|IM) is that the building response variability for a class of buildings is the same as the response variability for a given building in this class (this assumption is an ergodic assumption affecting fragility curves).

According to several authors (e.g., [START_REF] Guéguen | Nonlinear dynamics induced in a structure by seismic and environmental loading[END_REF]Trifunac et al., 2010), considering that data from full-scale observations in real buildings are much more representative than even the most sophisticated laboratory or numerical experiments, one way of improving engineering science to understand the physical behavior of structures is to use a complete database of earthquake recordings in real structures. For example, Perrault and Guéguen (2015) analyzed the variability of EDP versus IM using accelerometric data recorded in Californian buildings, taking structural drift as the EDP, and derived a single-building damage prediction equation (BDPE) with its associated uncertainties. Astorga et al. (2018[START_REF] Astorga | Recovery of the resonance frequency of buildings following strong seismic deformation as a proxy for structural health[END_REF][START_REF] Astorga | /wooden buildings. Moment magnitude (Mw) varies from 3[END_REF] completed the analysis, confirming the added value of physical data in understanding the seismic response of Japanese buildings in terms of co-seismic demand parameters related to modal (i.e. resonance frequency) parameter variations, especially during repetitive earthquake sequences.

In this study, the efficiency and sufficiency of several IMs for P(EDP|IM) from a large number of experimental datasets are analyzed using the regression model of EDP values for IM values. In the 4.2 section, the IMs and EDPs are described based on the study by [START_REF] Astorga | /wooden buildings. Moment magnitude (Mw) varies from 3[END_REF]. The 4.3 section describes the datasets and the methodology used. Then, the results in terms of efficiency and sufficiency are discussed in the 4.4 section, completed in the 4.5 by a specific analysis of the co-seismic frequency value versus EDP. Finally, the conclusion develops a simple empirical BDPE using available experimental data.

Description of IMs and EDPs

This study uses accelerometric data recorded in several sets of buildings and processed by [START_REF] Astorga | /wooden buildings. Moment magnitude (Mw) varies from 3[END_REF] for NDE1.0. Herein, we propose a brief description of the building information and earthquake data; more detailed information is available in NDE1.0 [START_REF] Astorga | /wooden buildings. Moment magnitude (Mw) varies from 3[END_REF].

Six ordinary IM values are considered in this study, computed from the data recorded at the bottom floor of each building:

-Peak ground acceleration (PGA) (Fig. 4.1a), velocity (PGV) (Fig. 4.1b), and displacement (PGD) (Fig. 4.1c), corresponding to the absolute values of maximum acceleration, velocity and displacement time histories, respectively.

-Arias intensity (AI), destructive potential (DP) and cumulative absolute velocity (CAV). Arias intensity [START_REF] Arias | MEASURE OF EARTHQUAKE INTENSITY[END_REF] includes both the amplitude and duration of seismic shaking, computed as follows:

AI = H +C ∫ a + (t)dt IJ 7 (4.2)
where g is the acceleration due to gravity, a(t) is the acceleration recorded at time t, and tf is the total duration of the recording. AI is an energy-based parameter that considers amplitude and duration of the ground motion, but it is unable to capture the frequency characteristics of ground motions. To overcome this, [START_REF] Araya | Earthquake accelerogram destructiveness potential factor[END_REF] define DP, as follows:

DP = K' L . / (4.3)
where v 7 + is the intensity of zero crossings, calculated over the entire duration of ground motion, as defined in its original version. [START_REF] Araya | Earthquake accelerogram destructiveness potential factor[END_REF] have shown a strong correlation between DP and observed real damage. Actually, v 7 + provides a measure of the dominant frequency content of the seismic ground motion.

Finally, Cumulative absolute velocity (EPRI, 1988) is computed as follows:

CAV = ∫ |a(t)|dt IJ 7 (4.4)
where |a(t)| is the absolute value of acceleration at time t.

Six spectral IM values are also considered: spectral acceleration (5% damping) (SAi and SAmin) (Fig. frequency values (i and min) impacting seismic demand. Index i corresponds to the spectral value computed at the elastic frequency of the system fi, i.e., the frequency obtained by Fourier analysis of the pre-event noise window before the earthquake. Index min corresponds to the minimal co-seismic value of the resonance frequency fmin during the strongest loading and obtained by applying a timefrequency distribution to the seismic recording (Fig. 4.1d), using the reassigned smoothed Wigner-Ville distribution [START_REF] Auger | Improving the Readability of T-F and T-S by Reassignment Method[END_REF] .The time-frequency distribution is applied to the top accelerometric time history (Fig. 4.1e). The time and frequency smoothing windows are Hamming windows, with N/10 and N/4 points, respectively. Then, a 3 rd order Savitzky-Golay filter is applied to the maximum values of energy window. Fig 4 .1(e) shows an example of the time-frequency process applied to the data. The fmin corresponds to the average value of ± 10 samples around the minimum value observed in the smoothing function (Astorga et al. 2018).

Finally, in this study, drift ratio (DR) is considered as an EDP to describe the building response [START_REF] Astorga | /wooden buildings. Moment magnitude (Mw) varies from 3[END_REF]. DR corresponds to the peak of the transient drift observed during the ground motion, obtained by computing the maximum relative displacement, as follows:

DR = ∆ 0#( . ∆ $#00#1 N (4.5)
where ∆top and ∆bottom are the horizontal displacements recorded at the top and bottom floors of each building, respectively, and h is the height between the top and bottom floors. ANX -One of the Japanese buildings, ANX, is a building that has been studied extensively by the BRI strong motion network. A detailed description of ANX is available in Astorga et al. (2018). ANX is an 8-story, steel-reinforced concrete building located approximately 60 km northwest of Tokyo, in Tsukuba (Japan) (Fig. 4.3a). ANX has one basement floor resting on spread foundations (8.2 m deep) lying on soft soil made up of alternating layers of clay and sandy-clay to a depth of 40 m. A description of the instrumentation is provided by [START_REF] Kashima | Dynamic Behavior of an Eight-Storey SRC Building Examined from Strong Motion Records[END_REF][START_REF] Kashima | Dynamic behaviour of SRC buildings damaged by the 2011 great east japan earthquake based on strong motion records[END_REF]. The ANX dataset is the largest of our One method of obtaining P[EDP|IM] is to perform a series of non-linear dynamic analyses for a given structure and for a given series of earthquakes [START_REF] Luco | Probabilistic seismic demand analysis, SMRF connection fractures, and near-source effects[END_REF]. Another method is to perform regression between EDP and IM, knowing the probability distribution [START_REF] Shome | Probabilistic Seismic Demand Analysis of Nonlinear Structures[END_REF][START_REF] Cornell | The Effect of Connection Fractures on Steel Moment Resisting Frame Seismic Demands and Safety[END_REF] for variability analysis. The degree of scattering around the fitted model represents the uncertainty of the EDP|IM model. To analyze EDP|IM uncertainty and testing efficiency and sufficiency of IMs, one-parameter log-log (log: natural logarithm) linear regression of EDP on IM [START_REF] Luco | Probabilistic seismic demand analysis, SMRF connection fractures, and near-source effects[END_REF] is used, defined as follows:

log(EDP) = a + b. log(IM) + ε (4.6)
where a and b are the estimated regression coefficients and ε is the standard error.

The variability associated with IMs and EDP is represented hereafter as σIM, and σEDP, respectively, i.e.

the standard deviation of the log of IM and EDP values, normalized by their mean value.

The efficiency of IMs is defined simply as the IM that results in a small variability of EDP given IM [START_REF] Luco | Structure-specific scalar intensity measures for near-source and ordinary earthquake ground motions[END_REF]. The variability associated with EDP for a given IM is measured by calculating the standard deviation of the residuals of the fitted regression model between EDP and IM (Eq. 4.6), represented hereafter as σEDP|IM. However, an efficient IM reduces the record-to-record variability between building responses. For practical purposes, this can then reduce the number of nonlinear time history analyses for IDA [START_REF] Shome | Probabilistic Seismic Demand Analysis of Nonlinear Structures[END_REF] with the necessary degree of precision.

The sufficiency of IMs is defined as the IM that makes EDP conditionally independent on earthquake parameters such as magnitude (M) and source-to-site distance (R). Sufficiency is estimated by computing the linear regression between EDP and IM regression residuals (ε|IM) of Eq. 4.6 and the corresponding value of M or log(R) [START_REF] Luco | Structure-specific scalar intensity measures for near-source and ordinary earthquake ground motions[END_REF]. As expected, σEDP|PGA varies significantly, reflecting several sources of uncertainties, which will be explored in the following section.

Results on EDP|IM

General trends -σ

For several candidate IMs, global and regional ground motion prediction equation (GMPE) models are continuously evolving [START_REF] Douglas | Ground Motion Prediction Equations 1964-2020[END_REF]. For e.g., [START_REF] Dhakal | Empirical analysis of path effects on prediction equations of pseudovelocity response spectra in northern Japan[END_REF], [START_REF] Podili | Ground motion prediction equations for higher order parameters[END_REF],

and Zhao et al. (2016) developed GMPE models for Japan, the value of sigma (σGMPE) reported in these models were 0.86 for PGA, 0.70 for PGV, 0.82 for PGD, 1.46 for AI, 0.69 for CAV, and 0.64-0.88 for acceleration response spectra, and 0.28-0.39 for velocity response spectra with a structural period ranging from 0.01-5 sec. Atkinson (2015) developed GMPE models for the US, the value of sigma reported for these models were 0.37 for PGA, 0.33 for PGV, and 0.31-0.41 for acceleration response spectra with a structural period ranging between 0.03-5 sec.

This section compares the efficiency of these candidates IMs in structural response prediction using our experimental data. The first observation is that the JPN dataset (σEDP=1.44), largest in terms of numbers, is predominant in our global dataset (1.48) compared with the US dataset (1.32). Fig. 4.5a shows that the value of σEDP|IM is oscillating between 0.8 and 1.1, with an average value of 0.9. The highest value of σEDP|IM corresponds to the spectral and peak ground acceleration values and DP intensity measures. Fig. 4.5b shows that PGA corresponds to the smallest value of σIM (1.15), whereas spectral acceleration corresponds to the highest value of σIM (in average around 1.60) after DP (3.33) and AI (2.29). Here we can see that the velocity related IMs (PGV, SVi and SVmin) corresponds to the relatively smaller value of σIM. However, for AI and DP, σEDP|IM remains similar to the other values. These IMs are also associated with the relatively higher value of σGMPE in the above mentioned GMPE models. This indicates that these parameters are not good indicators of the natural variability of ground motion and do not enable a high degree of certainty for predicting the response of structures for a given IM.

The velocity related IMs are observed with a smaller value of σEDP|IM (e.g., 0.80 for PGV, 0.79 for SVi, and 0.79 for SVmin). The efficiency of velocity IMs has already been reported for US data by Perrault and Guéguen (2015) and is confirmed herein, regardless of the dataset considered. It is interesting to note that these IMs are associated with the relatively smallest value of σGMPE in the above mentioned GMPE models, thus, the velocity related IMs are the most efficient IMs i.e. PGV, SVi, and SVmin.

In order to capture the origins of the uncertainties in building response prediction, several relationships are tested in the following sections, according to tectonic region, building typology and ageing effect.

The paucity of the data for specific analysis in some datasets makes it necessary to separate the studies;

sub-datasets are therefore presented. Atkinson and Morisson (2009) demonstrated that seismic ground-motion amplitudes in northern and southern California were significantly different for the same magnitude/distance pair of earthquakes, without identifying the origin of this difference, but related to different tectonic regions. In this study, the effect of the region on building response is examined for all building classes. Fig. 4.6 shows the effect of the tectonic region on the US dataset. The whole US dataset (US ALL) and both STS1 and STS2 subsets are tested, considering all of the previously mentioned US building typologies. Note that for STS1 and STS2, the σEDP values are the same (1.24 and 1.20 respectively, Appendix B), and σIM differs only marginally. In Fig. 4.6, the effect of considering the data by specific region barely minimizes the σEDP|IM values, for the same values of σIM (Fig. 4.6b). Some exceptions should be noted, the most remarkable being displacement and acceleration. Firstly, the figure shows that for velocity IMs (i.e. PGV, SVi and SVmin), the σEDP|IM values are similar, being around 1 for ALL, STS1 and STS2

Variability associated with the tectonic context -σR

(values in Appendix B). On the other hand, the σEDP|IM values for STS1 and STS2, respectively, correspond to 1.15 and 0.97 for SAi, 1.15 and 1.04 for SAmin, 1.03 and 1.10 for SDi and 1.03 and 1.10 for SDmin. Thus, a trend inversion (the smallest values for STS2 or STS1) is observed depending on whether acceleration or displacement IM values are considered. Although the origin of this inversion has not been confirmed, the class of the buildings concerned in these two geographical areas is likely to be cause, since some buildings are more sensitive to acceleration than others, depending on their period of resonance (i.e. stiff or flexible buildings). The following sections will therefore focus mainly on velocity IMs, testing the variability observed in relation to the class of structure in particular, and assuming an insignificant effect of the tectonic context.

Epistemic uncertainties related to building typology -σT

The seismic performance of buildings depends on their design and characteristics (construction material, height, plan area, regularity etc.). The uncertainties affecting vulnerability assessments are mainly epistemic because, according to [START_REF] Spence | Comparing loss estimation with observed damage: A study of the 1999 Kocaeli earthquake in Turkey[END_REF], they are due to the classification of buildings into typologies and the attribution of a single generic model to a whole class of buildings.

Furthermore, when evaluating seismic capacity, we suppose that many buildings of the same typology have the same ergodic epistemic uncertainties, implying that the values of the epistemic uncertainties do not change between buildings. In this section, the variability associated with different classes of buildings is explored using the US and JPN datasets. Only a basic description of the buildings, based on material, is available in our database. A more detailed classification according to international standards (e.g., HAZUS typology, GEM taxonomy) could be considered in a more comprehensive analysis.

Fig. 4.7(a) shows the variability observed for different classes of buildings in the US dataset. The trends are the same as those observed previously between the velocity IMs and the other IMs. Two typologies stand out: MA and WO. For these two typologies, the small amount of data in our dataset does not allow a more detailed analysis nor a definitive conclusion as to the effectiveness of certain IMs for EDP prediction. However, for MA buildings, the velocity IMs give higher values of σEDP|IM, and IMs in acceleration and displacement seem more and less efficient, respectively, than for the other typologies, due to the greater stiffness (i.e. smaller resonance period) generally observed in such buildings. For WO, all the σEDP|IM values are well below those of the other typologies, but the small number of buildings involved ultimately reduces the epistemic uncertainty related to structural differences within each building class. and ST, respectively, Appendix B). On the other hand, a notable difference exists between US ST and US RC buildings in particular, the latter having a lower σEDP|IM value for displacement IMs (i.e. 0.94 and 0.91 for PGD, 0.91 and 0.99 for SDi for RC and ST buildings, respectively).

There are significant differences between the JPN data (Fig. 4.7b) and the US data. First of all, the velocity IMs give different σEDP|IM values for different classes of buildings. For JPN ST, the σEDP|IM values are all lower, reflecting lower epistemic uncertainty related to the diversity of buildings within this class; this contrasts with JPN RC buildings (e.g. 0.86 and 0.76 for PGV, 0.86 and 0.70 for SVi and 0.82 and 0.50 for SVmin for JPN RC and JPN ST buildings, respectively). This epistemic uncertainty is confirmed in Fig. 4.7c, where the σEDP|IM values for one specific single building (ANX building) are compared with those of its building class. There is a significant contribution to the specific single building σEDP|IM values, with significantly reduced σEDP|IM values (e.g., 0.64 to 0.50 for PGV, 0.61 to 0.48 for SVi, and 0.60 to 0.40 for SVmin), particularly for parameters other than acceleration. It is also interesting to note an evident contribution of the response spectra calculated by taking into account the co-seismic response values to significantly reduce σEDP|IM for the SRC buildings (e.g., 0.78/0.48/0.48 for SAi/SVi/SDi and 0.46/0.40/0.39 for SAmin/SVmin/SDmin for the ANX building, Fig. 4.7c). Co-seismic resonance frequency, which modifies co-seismic demand, is known to vary for this building Astorga et al. (2018[START_REF] Astorga | Recovery of the resonance frequency of buildings following strong seismic deformation as a proxy for structural health[END_REF], as well as for a specific US ST building reported by [START_REF] Guéguen | Nonlinear dynamics induced in a structure by seismic and environmental loading[END_REF]. A similar variation is observed for JPN ST buildings (Fig. 4.7b), significantly reducing the σEDP|IM values (e.g., 0.80/0.70/0.78 for SAi/SVi/SDi and 0.59/0.50/0.48 for SAmin/SVmin/SDmin). This point will be analyzed more specifically in the last section of this manuscript.

Within-building variability associated with earthquake magnitude-distance -σMR

Fig. 4.8a shows the effect of M/R pairs on the variability of the ANX building response. The M/R criteria are described in Fig. 4.3 (section 4.3). Firstly, there is a significant effect on σEDP|IM values compared with the totality of the ANX data, regardless of the IMs considered, except for the displacement values of the IMs for the MR3 data subset (R = 250 ± 70% and M= 5.5 ± 0.5). These events generated longer periods of ground motion, to which the ANX building, with its resonance period of around 1Hz (Astorga et al., 2018) IM sufficiency is tested by considering the JPN building class dataset. The statistical significance of the coefficient obtained from the standard linear regression for M and log(R) is assessed based on the pvalue (i.e. the probability of obtaining an estimated value of the coefficient at least as large as the actual value, the actual value of the coefficient being zero) [START_REF] Benjamin | Solutions Manual to Accompany Probability, Statistics, and Decision for Civil Engineers[END_REF]. If the p-value observed is greater than or equal to 0.05, the estimated coefficient of M or log(R) is statistically insignificant and the IM is considered sufficient [START_REF] Luco | Structure-specific scalar intensity measures for near-source and ordinary earthquake ground motions[END_REF]. For the JPN building class, [START_REF] Karapetrou | Time-building specific" seismic vulnerability assessment of a hospital RC building using field monitoring data[END_REF] discussed the effect of aging over time on the seismic vulnerability of buildings.

Within-building variability associated with aging -σA

Moreover, Astorga et al. (2018[START_REF] Astorga | Recovery of the resonance frequency of buildings following strong seismic deformation as a proxy for structural health[END_REF] demonstrated the time-dependent response of the ANX building to cumulative events during a long sequence of moderate to strong earthquakes in experimental conditions. Incorporating the real state of a structure may therefore help to reduce variability, yielding more reliable results for PBEE analysis. Fig. 4.9 shows the σEDP|IM variations as a function of the age of the ANX building. Astorga et al. (2018[START_REF] Astorga | Recovery of the resonance frequency of buildings following strong seismic deformation as a proxy for structural health[END_REF] distinguished four specific periods (T1 to T4) during which the frequency of the building changed over time, depending on structural health related to the cumulative damage in the structure. Astorga et al. (2018[START_REF] Astorga | Recovery of the resonance frequency of buildings following strong seismic deformation as a proxy for structural health[END_REF] also showed that ANX's response between seismic events stabilized with its degradation, expressed as a function of the dispersion of structural drift values.

These four periods are considered here, focusing only on the M/R dataset corresponding to MR2 (i.e.

the dataset with the most data). A progressive reduction of σEDP|IM values is observed between T1 and T4, the last two periods being the most efficient (Appendix B), which confirms the results previously reported in Astorga et al. (2018) concerning the stability of the building response with degradation. It therefore appears that taking into account the ageing or actual state of a structure in performance analysis will help to modify the efficiency of the IMs, particularly as even moderate seismic shaking may change the building response (Perrault et al., 2020). For example, for PGV/SVi/SVmin, the σEDP|IM values correspond to 0.39/0.37/0.35 for the MR2-T2 ANX dataset and 0.37/0.37/0.26 for the MR2-T3

ANX dataset, compared with 0.41/0.41/0.37 for the MR2 ANX dataset as a whole. This results in a reduction of the performance prediction uncertainties as required during aftershock sequences for the short-time operative assessment of time-dependent building capacity assessment, based on resonance period shift [START_REF] Trevlopoulos | Earthquake risk in reinforced concrete buildings during aftershock sequences based on period elongation and operational earthquake forecasting[END_REF][START_REF] Trevlopoulos | Period elongation-based framework for operative assessment of the variation of seismic vulnerability of reinforced concrete buildings during aftershock sequences[END_REF].

Summary

Fig. 4.10 is a summary of the identification of building response prediction uncertainties for the different IMs considered, based on available data and metadata. In general, for the specific case of the ANX building, the σEDP|IM values are considerably lower than the model evaluated on the dataset to take into account structural characteristics such as construction type or ageing. For example, for PGV/SVi/SVmin, the σEDP|IM values decrease from 0.41/0.41/0.37 to 0.37/0.37/0.26.

Building frequency variation and the average response spectral value as an IM

Many previous studies (Astorga et al., 2018[START_REF] Astorga | Recovery of the resonance frequency of buildings following strong seismic deformation as a proxy for structural health[END_REF][START_REF] Calvi | State-of-the-knowledge on the period elongation of RC buildings during strong ground shaking[END_REF][START_REF] Clinton | The observed wander of the natural frequencies in a structure[END_REF]Guéguen et al., 2016;[START_REF] Kashima | Dynamic behaviour of SRC buildings damaged by the 2011 great east japan earthquake based on strong motion records[END_REF]Masi and Vona, 2010;[START_REF] Michel | Time-frequency analysis of small frequency variations in civil engineering structures under weak and strong motions using a reassignment method[END_REF][START_REF] Mucciarelli | Analysis of RC building dynamic response and soil-building resonance based on data recorded during a damaging earthquake (Molise, Italy[END_REF] have observed the co-seismic shifting of the resonance frequency for different building typologies.

Using a US dataset, Perrault and Guéguen (2015) showed that mean spectral values computed between the pre-and co-seismic periods provided the most effective IM for the EDP|IM model. On the other hand, several authors (Ebrahimian et al., 2015;[START_REF] Bianchini | Prediction of Inelastic Structural Response Using an Average of Spectral Accelerations[END_REF]Eads et al., 2015;Kazantzi and Vamvatsikos, 2015;[START_REF] Bommer | The influence of strong-motion duration on the seismic response of masonry structures[END_REF][START_REF] Adam | Optimal Spectral Acceleration-based Intensity Measure for Seismic Collapse Assessment of P-Delta Vulnerable Frame Structures[END_REF]Kohrangi et al., 2016[START_REF] Kohrangi | Conditional spectrum-based ground motion record selection using average spectral acceleration[END_REF] investigated the average spectral values computed between two periods to take into account the co-seismic nonlinear response of building in structural analysis. All these studies were carried using numerical modeling. Finally, [START_REF] Kohrangi | Conditional spectrum-based ground motion record selection using average spectral acceleration[END_REF]Kohrangi et al.( , 2016) ) observed that the average spectral acceleration value from 0.2*T1 to 1.5*T1 yields better prediction of structural response. We found no clear consensus on the values to be considered to reproduce co-seismic demand in structure analysis.

In this study, the frequency variation with respect to EDP was observed for the JPN dataset by building class. Fig. 4.11 summarizes the variation of the frequency ratio Rf =fmin/fi between pre-seismic frequency (fi) and co-seismic frequency fmin with respect to EDP for different JPN building classes and the whole US database, with EDP ranging from 5 10 -6 to 10 -2 . The variation of Rf confirms that, regardless of building class, the frequency shift between the pre-and the co-seismic period increases with EDP, which means large frequency drops occur for the strongest earthquakes. Significant variation of Rf is observed even at the lower end of the EDP range from 0.9 to 0.65 (below the slight damage threshold=0.0025). For EDP values between 10 -5 and 10 -3 , a relatively similar trend is observed regardless of building class, with Rf values decreasing from 0.78 to 0.65. There are no stronger earthquakes, which prevent us from extending this result to a higher level of drift, but this first experimental evaluation suggests the need to collect a large amount of earthquake data in buildings in order to refine our performance prediction models.

Conclusions

Experimental data is very useful in helping us to understand the complex physical processes at work in civil engineering structures to be able to integrate them into our models to reduce the epistemic uncertainty of these complex process. Earthquake data collected from buildings under long-term monitoring in Japan, the US and Romania were used to attempt to identify the components of the uncertainties associated with EDP|IM. Region-to-region, building-to-building and within-building uncertainties associated with earthquake magnitude-distance and ageing were explored.

Compared with the conventional IMs based on peak values or conventional spectral value (SAi, SVi or SDi), the ground motion intensity measure, denoted SAmin, SVmin and SDmin, which considers inelastic period lengthening, was found to be the most efficient IM for estimating EDP, taken as structural drift herein. In terms of sufficiency, generally speaking, it appears that no IMs are sufficient due to a significant conditional dependence of EDP on R (i.e. earthquake source-to-building distance) and M (i.e. magnitude). Some exceptions are pointed out in Fig. 4.8 for specific building classes and IMs. In fact, depending on the type of building and, in particular, its period, displacement and acceleration IMs might be more efficient or sufficient; this could be confirmed with additional data and a specific analysis of the building characteristics, which is not considered by this study. Nevertheless, all our results

indicate that velocity IMs are those that provide the lowest variability for predicting EDP given IM.

Based on the ANX building results, the components that make the largest contribution to overall uncertainties are building class and specific building associated with the M/R condition (Fig. 4.10, Tab.

4.2). When analyzing specific buildings using long-term monitoring data, the real structural state also appears to make a significant contribution to the uncertainties, reflecting the real co-seismic demand in EDP prediction. The underlying key issue is related to the variation of frequency, which is strongly dependent on EDP. Note that regardless of building class, this frequency variation follows the same trend for all the drift values in our dataset.

Several numerical studies have been carried out to investigate the efficiency and sufficiency of IMs considering one building or a group of buildings and using suits of input ground motions [START_REF] Buratti | A comparison of the performances of various ground-motion intensity measures[END_REF]Ebrahimian et al., 2015;[START_REF] Bianchini | Prediction of Inelastic Structural Response Using an Average of Spectral Accelerations[END_REF][START_REF] Jayaram | A Computationally efficient ground-motion selection algorithm for matching a target response spectrum mean and variance[END_REF]Eads et al., 2015;[START_REF] Jayaram | A Computationally efficient ground-motion selection algorithm for matching a target response spectrum mean and variance[END_REF]Mollaioli et al., 2011;[START_REF] Luco | Correlation of damage of steel moment-resisting frames to a vector-valued set of ground motion parameters[END_REF][START_REF] Vamvatsikos | Developing efficient scalar and vector intensity measures for IDA capacity estimation by incorporating elastic spectral shape information[END_REF]. In contrast, this study compares the efficiency and sufficiency of IMs and the variation of sigma in the EDP|IM relationship using real experimental data. We are able to capture the real structural response from different building typologies during several earthquakes, which is the main advantage of this study. This study shows that the most commonly used IMs i.e., PGA and SAi are not efficient and sufficient to predict the structural response within the range of our dataset. The velocity related IMs i.e., PGV, SVi, and SVmin are observed to be the most efficient in building response prediction. Our findings strongly agree with previous studies carried out on real data by Perrault and Guéguen (2015). It is very interesting note here that the IMs which are found efficient in our studies are also associated with relatively smaller variability in the available GMPE models.

This study highlights the importance of real experimental dataset. Having more information on the earthquakes and descriptions of the building characteristics would help to improve the prediction of structural response for analyzing seismic vulnerability or loss assessment. Although the amount of data contained in our dataset provides relevant results, the paucity of data concerning specific classes of buildings or components of uncertainties limits the strength of the conclusions that can be drawn. To resolve outstanding issues, we must continue our international collaborative efforts and motivate building owners to share their data, which would increase their interest in this type of study. In particular, having more specific data would enable verification of the aforementioned conclusions.

Moreover, building response prediction models can be developed considering several parameters related to earthquakes and buildings, such as ground motion IM, magnitude, distance, building typology, height, structural properties, etc. (Perrault and Guéguen, 2015;[START_REF] Fema | Case Studies-An Assessment of the NEHRP Guidelines for the Seismic Rehabilitation of Buildings[END_REF][START_REF] Hancock | Numbers of scaled and matched accelerograms required for inelastic dynamic analyses[END_REF]. In conclusion to this study, an empirical building damage prediction model is proposed (Tab.

4.3) based on the entire dataset (US, Japan and Romania) according to building class and considering the most efficient IMs (SVmin and PGV) using the functional form given in Eq. 4.6:

log(EDP) = a + b. log(IM) + ε

Introduction

Earthquake engineering is currently focused on improving the seismic resistance of structures through earthquake capacity design. Capacity design refers to the design of a building to ensure controlled ductile behavior to avoid collapse in a design-level earthquake specified by stakeholders. The main objective is to minimize the direct and indirect losses generally correlated with any level of structural damage that affect the assets and people exposed and, indirectly, to limit the downtime of vital functions and essential facilities in urban areas following an earthquake. Herein, damage refers to any undesirable change in structural properties that significantly affects the intended functions of a structure over its expected lifespan (Farrar & Worden, 2007). Anticipating seismic consequences requires prior information on the damage that structures might be expected to suffer in a given earthquake, i.e., capacity models for the structures exposed.

In the recent European Seismic Risk Model [START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF], three main approaches are considered to assess seismic risk: scenario-based, intensity-based and frequency-based. The main difference lies in the definition of seismic hazard, but once the latter has been defined, risk is calculated as the convolution of the hazard with the vulnerability or fragility functions. This results in a loss estimate for a given class of building, which is then aggregated to the geographical unit of the area concerned. The fragility functions are developed based on the capacity model of the building or building class, which describes the lateral strength and deformation capacity in acceleration-displacement-response-spectrum format (ADRS). Capacity curves are then developed, considering various structural attributes (e.g., materials, lateral load resistance system, number of storeys, etc.) to cover a wide range of building classes in the exposed area. In urban-specific seismic risk studies, it is common practice to start by assessing the exposure model at the site concerned and then to attribute class-capacity curves to each building [START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF]. This approach offers the potential advantage of taking into account a large set of buildings, but specific attributes within a class can cause significant deviations from the generic model.

For example, [START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF] released a databank of capacity curves that has been used to represent the vulnerability classes of current European buildings [START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF]. For each class, the backbone capacity curves were compiled from simulated design-based research studies, i.e., pushover analysis [START_REF] Fajfar | A Nonlinear Analysis Method for Performance-Based Seismic Design[END_REF] or incremental dynamic analysis [START_REF] Vamvatsikos | Incremental dynamic analysis[END_REF]. The method returned an average capacity curve for a generic building model without considering the effects of any specific building features. The application of generic backbone models to large sets of buildings in a single class introduces epistemic uncertainties into damage assessment [START_REF] Iaccarino | Earthquake Early Warning System for Structural Drift Prediction Using Machine Learning and Linear Regressors[END_REF]Perrault & Guéguen, 2015;[START_REF] Spence | Comparing loss estimation with observed damage: A study of the 1999 Kocaeli earthquake in Turkey[END_REF]. Among other attributes, specific features may control the nonlinear seismic response of a building (Lagomarsino & Giovinazzi, 2006;[START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF], thus bringing significant epistemic uncertainty to the damage estimation (Lestuzzi et al., 2016). The most advanced numerical models (e.g., finite element-based models, [START_REF] Mazzoni | OpenSees Command Language Manual, Pacific Earthq[END_REF] explicitly take into account all the specific attributes of the design and geometry of an individual building.

However, these more complex models may also have to cope with additional sources of uncertainties:

(i) the description of the design and materials, including foundation systems, and (ii) the assessment of the actual condition of the structures. These uncertainties are all the more critical in seismic prone regions because of the possible effects of cumulative seismic damage (e.g., Perrault et al., 2020).

To resolve the afore-mentioned issues, other approaches can be used to implement a so-called "host-totarget" adjustment to eliminate the adverse effects of considering average "generic" building class models. This adjustment can be based on a modal analysis of existing buildings to define their elastic properties related to design and actual condition, or the processing of earthquake data recorded in buildings (e.g., [START_REF] Mucciarelli | Analysis of RC building dynamic response and soil-building resonance based on data recorded during a damaging earthquake (Molise, Italy[END_REF][START_REF] Dunand | Utilisation du bruit de fond pour l'analyse des dommages des bâtiments de Boumerdès suite au séisme du 21 Mai 2003[END_REF][START_REF] Clinton | The observed wander of the natural frequencies in a structure[END_REF]Masi and Vona 2010;Michel et al. 2012;Vidal et al. 2014;Perrault et al., 2013;Astorga et al. 2018[START_REF] Astorga | Recovery of the resonance frequency of buildings following strong seismic deformation as a proxy for structural health[END_REF]Astorga and Guéguen 2020) to take into account the nonlinear processes activated in the buildings during earthquakes and estimate their residual capacity [START_REF] Dowgala | A Method for Extracting Building Empirical Capacity Curves from Earthquake Response Data[END_REF][START_REF] Freeman | Using strong motion recordings to construct pushover curves[END_REF][START_REF] Pan | Capacity-curve-based damage evaluation approach for reinforced concrete buildings using seismic response data[END_REF].

Using these data, within-building variability ϕ (i.e., the misfit between a single structure response for a given earthquake and the structure-specific median response model, which is defined as the median prediction of the model for a given structure plus the variability between earthquakes) and betweenbuilding variability τ (i.e., the average shift of the observed structure-specific median model from the class-specific median model of the same class) can be examined to quantify, understand, and process the variability of the seismic building response and to address the origins of uncertainties through residual analysis (Fig. 5. 1). Ultimately, the total variability σ for a single building is given by: σ = Yτ + + ϕ + (5.1)

Figure 5.1. Between-building and within-building components of building response variability modelled in acceleration-displacement-response-spectrum format.

The main objective of this study is to develop a chain of processing to obtain experimental capacity curves using strong motion data recorded in buildings and to quantify and process the variability of the building response, taking into account the building class adjustment to capacity curves provided by [START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF]. The dataset is described first, the selected buildings are presented in the second section, and the methods used to adjust the capacity curves from experimental data are discussed in the third section. The fourth section presents the results, and the paper ends with a discussion and conclusion.

Data

Seven Japanese buildings monitored by the Building Research Institute (BRI) with slight-to-extensive damage reported by [START_REF] Kashima | Dynamic behaviour of SRC buildings damaged by the 2011 great east japan earthquake based on strong motion records[END_REF] after the 2011 Tohoku Mw9.1 earthquake were selected from NDE1.0 (Astorga et al., 2020, Fig. 5.2). Accelerometric data were recorded at the top and bottom of each building in both horizontal directions and processed (double integration for displacement and timefrequency analysis for fundamental frequency assessment) by [START_REF] Astorga | /wooden buildings. Moment magnitude (Mw) varies from 3[END_REF].

(1) The Annex building (ANX) is an eight-storey steel reinforced concrete (SRC) structure. Since its completion in 1998, a seismic monitoring system has been installed and 1,630 earthquakes have been recorded over a 20-year period. During the 2011 Tohoku earthquake, slight-to-moderate damage was reported.

(2) The Tohoku University building (THU) is a nine-storey SRC structure constructed in 1969.

Monitoring started in 1994 and comprises 203 earthquake recordings, including a series of very strong ground motion events (e.g., 2005 Miyagi-ken earthquake, 2008 Iwate-Miyagi earthquake, 2008 Iwate earthquake). Moderate-to-extensive damage was reported after the Tohoku earthquake, during which multi-storey shear walls suffered serious bending failures. In addition to IM and EDP, resonance frequency variations were obtained by applying a Wiegner-Ville time-frequency distribution (see [START_REF] Astorga | /wooden buildings. Moment magnitude (Mw) varies from 3[END_REF] for details) to the top recordings. Analysis of the ante-, co-and post-seismic resonance frequency of buildings identifies the nonlinear processes activated during the earthquakes due to the opening/closing of cracks (Astorga et al., 2018[START_REF] Astorga | Recovery of the resonance frequency of buildings following strong seismic deformation as a proxy for structural health[END_REF][START_REF] Clinton | The observed wander of the natural frequencies in a structure[END_REF] and improves the efficiency analysis P(EDP|IM) [START_REF] Iaccarino | Earthquake Early Warning System for Structural Drift Prediction Using Machine Learning and Linear Regressors[END_REF]. In this study, fi corresponds to the frequency obtained by analyzing the noise window before the earthquake, and fmin is the minimal value of the co-seismic frequency observed during strong loading.

Method

Empirical capacity curves are typically represented in acceleration-displacement-response-spectrum (ADRS) format [START_REF] Freeman | Using strong motion recordings to construct pushover curves[END_REF]. This study uses the mass normalized capacity curve in the Sa-Sd spectrum proposed by [START_REF] Freeman | Using strong motion recordings to construct pushover curves[END_REF]. The following four-step procedure is used to obtain the empirical capacity curve.

Step 1. Calculation of the relative displacement time histories (δ_top) as follows:

δ I;8 = uT(t)-uB(t) (5.2)
where uT(t) and uB(t) are the top and bottom floor displacements (Fig. 5.4b) obtained from the accelerometric data (Fig. 5.4a).

Step 2. Calculation of the time history of the fundamental frequency from δ I;8 by applying a 4 th order Butterworth filter between fi and fmin (Fig. 5.4c).

Step 3. Calculation of Sa and Sd to develop the building response hysteresis (Fig. 5.4d) as follows: where Γ is the fundamental mode performance factor ranging from 1.3 to 1.4 for a multi-degree-offreedom system [START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF], and T corresponds to the fundamental period given by the time-frequency distribution (Fig. 5.4c).

S ! = O 0
Step 4. Finally, the empirical capacity curve is the envelope of the absolute value of the ADRS spectrum (Fig. 5.4e).

The mass-normalized co-seismic stiffness (i.e., mean ± standard deviation), measured in s -2 , is approximated by the slopes of the peaks in the ADRS map (Fig. 5.4d). In this study, a selection of data from NDE1.0 is used to analyze the variation of the experimentally derived capacity considering two different contexts (before and after the 2011 Tohoku earthquake) presented hereafter. In this manuscript, the ADRS curves are referenced according to the NDE1.0 format, i.e., date of the earthquake, building code, and direction of the component considered. First, we considered the within-building residual (Fig. 5.1) as the misfit between the capacity curves of a given structure for a single earthquake (i.e., single ADRS model) and the median response to several earthquakes (i.e., building-specific ADRS model). Within-building variability accounts for (1) the timevarying co-seismic dynamic properties related to the cumulative effect of earthquakes and the related structural ageing (Astorga et al. 2018[START_REF] Astorga | Recovery of the resonance frequency of buildings following strong seismic deformation as a proxy for structural health[END_REF][START_REF] Trevlopoulos | Earthquake risk in reinforced concrete buildings during aftershock sequences based on period elongation and operational earthquake forecasting[END_REF]) and ( 2) IM efficiency obtained by measuring the variability of the building response (i.e., EDP given IM) around the regression on the IM values. Post-seismic visual inspections of ANX, THU and IWK reported slight to moderate structural damage after the 2011 Tohoku earthquake. We explored the capacity variation of these buildings over time (within-building variation) using a set of seismic events recorded before and after the 2011 earthquake.

Secondly, we considered the between-building variability (Fig. 5.1), which refers to the average shift of the model of a single structure in a specific class of structures (i.e., building-specific ADRS model), from the median model defined for this class (i.e., class-specific ADRS model). This variability can be attributed to several components, including within-building components related to ground motion, variability of structural design, site conditions, and ageing of the building. Here, we explored the between-building variability of the capacity response for the two building classes that include the 8storey SRC buildings (i.e., IWK, TDS, and CHB) and the 6-storey RC buildings (i.e., NIT and HCN).

Finally, we compared the experimental co-seismic capacity curves of the ANX and THU buildings with the corresponding GEM-taxonomy capacity curves provided by [START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF]. The comparison is made by measuring the difference in the initial slope of the ADRS curves as follows:

D (%) = 23 2" 4!5!)6- . 23
2" !7(!)61!5038 23 2" 4!5!)6-x 100

(5.6) Figure 5.6 shows the within-building variation of the capacity curves before and after the Tohoku earthquake in the ANX building for three ranges of shaking amplitude: strong, corresponding to a drift ratio of 10 -3 or above (Fig. 5.6a), moderate for a drift ratio between 10 -4 and 1.1x10 -4 (Fig. 5.6b), and slight for a ratio between 1.25x10 -6 and 1.35x10 -6 (Fig. 5.6c). The impact of the 2011 Tohoku earthquake is clear even for moderate and slight amplitudes, as indicated by the change in slope of the before and after ADRS curves. The variation in the ADRS response is captured by the differences in the ϕ values before (0.31 for moderate and 0.007 for slight shaking) and after (0.32 for moderate and 0.005 for slight shaking) the Tohoku earthquake. The impact of the deterioration of structural health on response variability is more pronounced in the ADRS response to the earthquakes that caused strong shaking (Fig. 5.6a), also indicated by the higher ϕ value (2.50) after the Tohoku earthquake, compared with the events that caused only moderate or slight shaking.

Results

Within-building variability 𝜙

The different ϕ values for earthquakes causing strong, moderate and slight shaking illustrate the direct relationship between variability and degree of shaking, which has already been reported by other studies (e.g., Masi and Vona 2010;Trifunac et al. 2010;[START_REF] Michel | Quantification of fundamental frequency drop for unreinforced masonry buildings from dynamic tests[END_REF]Vidal et al. 2014;Astorga et al. 2018;Astorga and Guéguen 2020;[START_REF] Iaccarino | Earthquake Early Warning System for Structural Drift Prediction Using Machine Learning and Linear Regressors[END_REF].

Figure 5.6. Co-seismic ADRS curves for the ANX building along the lateral direction, illustrating the within-building variability during various earthquakes before (grey lines with square markers) and after (grey lines with circular markers) the 2011 Tohoku earthquake, considering events inducing (a) strong shaking (drift ratio ≥ 10 -3 ), (b) moderate shaking (10 -4 ≤ drift ratio ≤1.1x10 -4 ), and (c) slight shaking (1.25x10 -6 ≤ drift ratio ≤ 1.35x10 -6 ), respectively. The dashed-blue and dashed-red lines represent the curves of the mean ADRS model before and after 2011.

The same plots are given in Fig. 5.7b and 5.7c for the THU and IWK buildings, respectively, and Fig. the mass-normalised co-seismic stiffness value is 42.43 ± 15.65 (s -2 ) with a drift ratio of 2.7x10 -3 . The lowest co-seismic stiffness value (33.82 ± 7.54 s -2 ) is observed during the strongest aftershock (2011/03/11/15:15) with a drift ratio of 5.2x10 -3 . No further reduction of mass-normalised co-seismic stiffness is observed during the subsequent earthquakes with a similar loading amplitude. After 2011, the mass-normalised co-seismic stiffness of the ANX building dropped by an average of 45.35%.

THU and IWK exhibit a similar trend in mass-normalised co-seismic stiffness following the Tohoku earthquake. After this event, the mass-normalised co-seismic stiffness of THU and IWK dropped by 41.39% and 24.83%, respectively. These results show that the extent of the decline in co-seismic rigidity is proportional to the amplitude of loading and the level of residual damage in the building.

Furthermore, the initial slopes of the ADRS response (Fig. 5.7a) and the mass-normalised co-seismic stiffness (Fig. 5.7d) are similar for the ANX and THU buildings. The damage also reduces the variability of the structure response, as seen here with the rigidity variation (Fig. 5.7d). This reflects the decreasing variability of the within-building capacity curve based on experimental data, as previously reported by Astorga et al. (2018) for ANX. Astorga et al. (2018[START_REF] Astorga | Recovery of the resonance frequency of buildings following strong seismic deformation as a proxy for structural health[END_REF], Astorga and Guéguen (2020) Fig. 5.9 gives the between-event variability analysis for the 6-storey reinforced concrete buildings (NIT and HCN). As reported by [START_REF] Kashima | Dynamic behaviour of SRC buildings damaged by the 2011 great east japan earthquake based on strong motion records[END_REF], the NIT building suffered more damage than HCN in the 2011 Tohoku earthquake. As a consequence, for Sd < 0.4 cm (Fig. 5.9a), the ADRS curves are quite similar for both buildings, despite a minor slope difference due to the slight variation in the response of the buildings in the same class. For Sd > 0.4 cm, the NIT ADRS curve shows a broken slope, indicating damage. Also note that during the 2011 Tohoku earthquake, HCN suffered slight co-seismic degradation around Sd = 0.28cm. Furthermore, the difference in damage levels in both buildings is also illustrated by comparing the f value before (0.97 for NIT and 0.76 for HCN) and after (5.22 for NIT and 1.91 for HCN) the 2011 earthquake (Fig. 5.9b and Fig. 5.9c), confirming that the strongest 2011 earthquake caused damage to the NIT building. The before Tohoku values of [τ, 𝜎] are [6.45, 6.53] and [9.93, 9.96] for NIT and HCN,respectively,which become [3.46,6.26] and [17.96,18.06] after the earthquake.

Similarly, the total variability of the class before (5.67) and after (7.29) the Tohoku earthquake differs from the total variability for each specific building in the same class.

This shows that the variability [ϕ, τ, σ] associated with a building class is time-dependent, being related to the actual condition of the individual buildings in the class. Furthermore, the ergodic assumption, which supposes that class-specific variability is the same as that of a single building within that class, is not validated. This is confirmed by the total variability of the class-specific ADRS model compared with that of the building-specific ADRS model for that class, both before and after the Tohoku earthquake for 8-storey SRC and 6-storey RC buildings. This is a key observation for large-scale building damage assessment methods, for which a generic class-specific capacity curve is considered for all buildings in the same class. This can introduce significant epistemic uncertainty, as reported previously by other studies [START_REF] Iaccarino | Earthquake Early Warning System for Structural Drift Prediction Using Machine Learning and Linear Regressors[END_REF][START_REF] Spence | Comparing loss estimation with observed damage: A study of the 1999 Kocaeli earthquake in Turkey[END_REF]. Furthermore, an updated capacity curve based on experimental data enables the assessment of damage level and the residual capacity of monitored buildings after a mainshock. For the ANX building (Fig. 5.10a, zoomed in 5.10b), significant differences in the initial slope are observed between the generic and experimental ADRS curves (D calculated using Eq. 5.6). Over time, ANX experienced slight to moderate damage and the D values fall by 114% and 20% compared with the generic capacity model values for this 8-storey SRC building before and after the Tohoku earthquake, respectively. For THU (9-storey SRC class), the D values are -63% and +8% before and after the Tohoku earthquake, respectively.

Adjustments to the generic capacity curves

As reported by [START_REF] Kashima | Dynamic behaviour of SRC buildings damaged by the 2011 great east japan earthquake based on strong motion records[END_REF], the THU building (Fig. 5.10c, zoomed in 5.10d) was severely damaged by the Tohoku earthquake, and its D values are high. For example, the initial slope of the generic capacity model for the nine-storey building has a D value of -107% before the Tohoku earthquake and -4% after, while the generic model for the eight-storey building shows a D value of -63% before and +18% after. The difference observed between the experimental and generic capacity models in ANX and THU quantifies the adjustment needed to make the generic model more relevant to these buildings, and thus reduce epistemic uncertainty in the seismic damage assessment model. For example, for the ANX building, increasing the initial slope of the eight-storey model by 20% helps to account for the actual structural condition of ANX building. This significantly reduces the level of uncertainty in building vulnerability models applied to existing structures and contributes to the time-updating of seismic vulnerability, thus improving seismic risk assessment [START_REF] Iaccarino | Earthquake Early Warning System for Structural Drift Prediction Using Machine Learning and Linear Regressors[END_REF][START_REF] Guéguen | Testing Buildings Using Ambient Vibrations for Earthquake Engineering : a European Review[END_REF][START_REF] Hannewald | Development and validation of simplified mechanics-based capacity curves for scenario-based risk assessment of school buildings in Basel[END_REF][START_REF] Michel | Time-frequency analysis of small frequency variations in civil engineering structures under weak and strong motions using a reassignment method[END_REF]Michel et al., , 2012)). The generic capacity model curves are represented by the thick black lines, the thin, solid, and dashed lines represent the experimental capacity curves before, during, and after the Tohoku earthquake along the transversal and lateral directions.

Discussion

Previous studies have attempted to develop capacity models describing the lateral strength and deformation capacity of buildings or building classes [START_REF] Crowley | European Seismic Risk Model (ESRM20)[END_REF]. These capacity models are defined based on simulated design-based research studies that characterise building attributes (e.g., materials, lateral load resisting system, number of storeys, etc.) using simple single-or multi-degreeof-freedom systems. Such developments are based on significant modelling assumptions, which may prevent full understanding of the actual response of structures, as well as increasing epistemic uncertainty in risk assessment. Some studies have attempted to develop experimental capacity curves using strong motion data recorded in buildings [START_REF] Dowgala | A Method for Extracting Building Empirical Capacity Curves from Earthquake Response Data[END_REF][START_REF] Freeman | Using strong motion recordings to construct pushover curves[END_REF][START_REF] Pan | Capacity-curve-based damage evaluation approach for reinforced concrete buildings using seismic response data[END_REF]. Few of these studies have analysed the variability of the co-seismic response of buildings using experimental capacity curves.

In this study, we explore the variability of building response for specific buildings with time-varying conditions and different earthquakes, and between buildings with the same typology using experimental capacity curves developed from strong motion data recorded in buildings. Inspired by Al Atik et al.

(2010), who analysed the variability and variability components of empirical ground motion models, we examined the within-building variation of co-seismic capacity by analysing separately the ANX, THU and IWK buildings, which suffered slight to moderate structural damage during the 2011 Tohoku earthquake [START_REF] Kashima | Dynamic behaviour of SRC buildings damaged by the 2011 great east japan earthquake based on strong motion records[END_REF]. Our analysis of the experimental capacity curves revealed clear evidence of damage in these buildings resulting from the Tohoku earthquake, indicated by a shift in the initial slope and higher variability in the ADRS models. We also found that the degree of variability is linked to the degree of damage in the building. For instance, after the Tohoku earthquake, the ANX, THU, and IWK buildings show within-building variability of 1.72, 2.54, and 3.80, respectively. We also show that the variability of the ADRS curve depends on the amplitude of seismic loading. For example, for ANX, variability is 2.50, 0.32, and 0.01 for strong, moderate, and slight shaking, respectively, which we interpret as being due to the activation of existing cracks in the building (Astorga et al. 2018[START_REF] Astorga | Recovery of the resonance frequency of buildings following strong seismic deformation as a proxy for structural health[END_REF]Astorga and Guéguen, 2020;[START_REF] Abeele | Damage assessment in reinforced concrete using spectral and temporal nonlinear vibration techniques[END_REF]. Thus, the time-varying condition of the buildings, due to the cumulative effects of earthquakes, produces a change in the variability of the buildingspecific ADRS model for the same class. For example, IWK, TDS, and CHB (6-storey SRC class) show between-building variability of 0.53, 2.43, and 1.72 before the Tohoku earthquake and 4.66, 3.69, and 0.72 after the earthquake, respectively. The differences in response variability among buildings of the same class may be related to the degree of heterogeneity in the residual crack systems and the nature of the crack activation mechanisms during seismic activity.

Comparison of the experimental capacity curves of buildings with long-term monitoring (ANX and THU) and their GEM-taxonomy generic capacity models shows significant differences between the experimental (building-specific) and generic (class-specific) models. For example, following the Tohoku earthquake, a 20% difference was observed for the ANX building, indicating that the initial slope of the generic capacity model should be adjusted by 20% to account for the actual structural condition of the building. This updated ADRS model enables the use of time-varying co-seismic capacity for seismic risk assessment studies or operational loss forecasting, and at the same reducing epistemic uncertainty [START_REF] Trevlopoulos | Period elongation-based framework for operative assessment of the variation of seismic vulnerability of reinforced concrete buildings during aftershock sequences[END_REF][START_REF] Trevlopoulos | Earthquake risk in reinforced concrete buildings during aftershock sequences based on period elongation and operational earthquake forecasting[END_REF][START_REF] Iaccarino | Earthquake Early Warning System for Structural Drift Prediction Using Machine Learning and Linear Regressors[END_REF], Perrault and Gueguen 2015).

Our study found that, during the Tohoku earthquake, the ANX building experienced a 45.35% coseismic stiffness drop with slight-to-moderate damage, while the THU building experienced a 41.39% drop with moderate-to-extensive damage. Furthermore, when we analysed the THU building for a different range of shaking amplitudes, we observed different values of co-seismic stiffness degradation:

36.43% for strong shaking (drift ratio of 1x10 -3 or more), 21.71% for moderate shaking (drift ratio between 1x10 -4 and 1.4x10 -4 ), and 9.34% for slight shaking (drift ratio between 2x10 -5 and 3x10 -5 ).

These findings suggest that the change in co-seismic stiffness is proportional to the shaking amplitude and the level of residual damage in the building before the earthquake under consideration. Previous studies (e.g., [START_REF] Dunand | Utilisation du bruit de fond pour l'analyse des dommages des bâtiments de Boumerdès suite au séisme du 21 Mai 2003[END_REF]Masi and Vona 2010;Trifunac et al. 2010;[START_REF] Michel | Quantification of fundamental frequency drop for unreinforced masonry buildings from dynamic tests[END_REF]Vidal et al. 2014) have defined different thresholds of co-seismic stiffness/period elongation for detecting damage and estimating building condition. However, our results highlight the need for caution when implementing co-seismic stiffness-degradation-threshold-based approaches for structural health monitoring, as highlighted by Astorga et al. (2018).

Experimental capacity curves also reflect earthquake-induced damage in buildings, as illustrated by the broken slope of the ADRS model. The plateau of the break is proportional to the amount of damage in these buildings, as reported by [START_REF] Kashima | Dynamic behaviour of SRC buildings damaged by the 2011 great east japan earthquake based on strong motion records[END_REF]. These curves highlight the fact that actual structural condition is required to evaluate capacity immediately after an earthquake. The experimental capacity curves developed in this study only consider the fundamental mode, and additional parameters (high modes, damping, etc.) could be used to assess the impact of such parameters on total variability.

Conclusion

Experimental data from buildings are crucial for quantifying, understanding and processing the variability of the seismic building response and for identifying the origin of uncertainties through residual analysis. Experimental data offers the advantage of including the complex physical processes that affect buildings during seismic loading, and consideration of the time-varying capacity due to the actual condition of the building. In this study, we used strong motion data collected from seven Japanese buildings equipped with permanent instrumentation and post-processed in the NDE1.0 database [START_REF] Astorga | /wooden buildings. Moment magnitude (Mw) varies from 3[END_REF] to investigate variability in the co-seismic capacity curves. We developed an ADRS-based framework, considering the method proposed by [START_REF] Freeman | Using strong motion recordings to construct pushover curves[END_REF]. We explored the within-building variability related to the building-specific response for a given earthquake, including its change in condition, and the between-building variability related to the drift of building-specific and class-specific models, which are derived from experimental data or selected from the GEM-Taxonomy models.

Analysis of the ANX building's co-seismic capacity before and after the 2011 Tohoku earthquake shows that the damage helped to reduce ANX's co-seismic stiffness by 45% and increase the variability of the co-seismic capacity response by 39%. Furthermore, we observed that strong shaking earthquakes resulted in greater variability in the co-seismic capacity response compared with moderate to slight earthquakes. The IWK and THU buildings, which experienced none-to-slight and moderate-to-severe damage during the Tohoku earthquake, also showed the same trend. Similarly, the other buildings in the same class (IWK, CHB, TDS, and NIT, HCN) confirmed that cumulative seismic loading of varying amplitudes over time resulted in significant changes in the co-seismic capacity curves. This, in turn, led to changing levels of variability among the buildings in the same class, as indicated by different between-building variability values before and after the 2011 Tohoku earthquake. Consequently, the response variability of a given building in a given class differs from that of the class as a whole, which contradicts the ergodic assumption that the response variability for a class of buildings is the same as the response variability for any given building in that class. Time-varying experimental capacity models are useful for calculating time-varying seismic risk, especially after strong earthquakes or during a mainshock-aftershock sequence. Furthermore, the difference observed between experimental curves and the generic capacity curves obtained from the global databank of capacity curves can quantify the adjustments required to account for a building's actual structural condition.

Using the model developed in this study, we derived experimental capacity curves that took into account the damage-related condition and the necessary adjustment of the generic capacity models by reducing epistemic uncertainty. These experimental capacity curves can be helpful for characterising the actual seismic integrity of structures, for both time-variant seismic risk assessment and operational loss forecasting. These empirical capacity curves were developed by considering the fundamental mode period, and do not reflect the entire capacity of the building, i.e., considering the contribution of the higher modes in the building co-seismic response and the associated variability. Finally, there is a huge need for building testing to develop and calibrate actual models, which are superior to even the most sophisticated numerical models. The most important aspect is possibly not only related to the median values of engineering demand parameters, but their variability according to the physical processes in the structure that are activated by shaking.

General conclusion

Recently, earthquake data collected from buildings have become openly accessible, including postearthquake damage survey databases and strong-motion data recordings. These data represent the actual physical processes involved in buildings during earthquake loading more than even the most sophisticated numerical studies. In this study, we used the post-earthquake building damage survey database and strong-motion data recordings from buildings to develop new insights into the large-scale rapid damage assessment and building-level seismic damage assessment, respectively.

In the framework of the rapid damage assessment at a large scale, we evaluated the effectiveness of machine learning models by training and testing on the post-earthquake building damage dataset.

Machine learning models could reasonably estimate damage in a cost-effective way. Readily available building features such as the number of stories, age, floor area, and height can result in a reasonable assessment of damage at a large scale, particularly when using a traffic-light-based (green, yellow, and red) damage classification framework. Machine learning models trained on these readily available building features showed efficacy similar to conventional methods like Risk-UE. In addition, these parameters can be easily obtained from an institutional database, such as the national census or national housing database, thereby resolving data acquisition issues associated with seismic-damage assessments at the urban or regional scale.

We We also observed that the contribution of the building's feature changes depending on the damage level considered. This behaviour has not been discussed previously. The machine learning models trained on past earthquake building damage portfolios can reasonably estimate damage during the future earthquake for a different region with similar building portfolios. However, the damage distribution is strongly influenced by region-specific factors such as construction quality and regional typologies, implementation of seismic regulations, and hazard level. Thus, machine learning models trained with building damage datasets from one region may not directly apply to another region without first accounting for host-to-target adjustments.

Focusing on the building-level damage assessment, we evaluated the co-seismic response of a building during seismic loading using the earthquake data recordings from buildings. We found that using ground motion intensity measures that accurately capture a building's co-seismic response (for example, as spectral velocity at the co-seismic period) by considering the building's structural state, along with the earthquake's magnitude and distance, can significantly decrease the level of uncertainty in damage assessment.

We also observed that the damage caused by an earthquake and the damage accumulation over time could alter the co-seismic response of a building. The seismic loading of different amplitude can impact buildings with similar typologies differently. As a result, the co-seismic response of buildings with the same typologies can be significantly different over time. Assuming a generic co-seismic response for a building throughout its lifetime or grouping buildings into a class based on taxonomy and assuming a generic co-seismic response to all buildings during seismic loading (such as in fragility modeling or building response modeling in the performance-based earthquake engineering framework) can lead to significant uncertainty in damage assessment.

Co-seismic capacity curves can accurately capture a building's lateral load-resisting capacity during earthquake loading. The extent of damage caused by the first occurrence and the accumulation of damage over time can be easily reflected by a shift in the co-seismic capacity curves proportional to the degree of damage. The capacity curves help to quantify the necessary adjustments to generic capacity models and can help to reduce epistemic uncertainties in damage assessment. A time-varying capacity model is necessary for calculating time-varying seismic risk. The experimental capacity curves can be an efficient tool for characterizing a structure's actual seismic integrity for both time-variant seismic risk assessment and operational loss forecasting, particularly after strong earthquakes or during a mainshock aftershock sequence.

This study further highlighted the importance of real earthquake datasets to enhance our understanding of the seismic damage response of buildings and develop a tool for seismic damage assessment. More information on the earthquakes and descriptions of the building characteristics would help cover diverse aspects of the seismic damage or loss assessment framework. To further develop our insights into the seismic response of buildings and strengthen the conclusions mentioned above through data-driven studies, we must continue our global collaboration to collect and share earthquake data collected from buildings.

As a result, buildings may experience sudden demands for energy and displacement, leading to a larger concentration of drift demand (Mollaioli et al., 2011;Baker and Cornell, 2008b).

Additionally, higher modes may affect high-rise buildings near the fault area and the maximum displacement may not necessarily be at the top-story level (Mollaioli et al., 2011). In NDE1.0, we have earthquake data recordings in buildings located near to the epicentre. Further studies can be carried out to explore the building response for given intensity measures (EDP|IM) near the epicentre using the earthquake data recordings from the building available in NDE1.0.

(e) Chapter 5 presents co-seismic capacity curves, which enable real-time tracking of the residual load-resisting capacity of buildings that are equipped with strong motion sensors. In reality, not all buildings are currently being monitored by a strong motion sensors network. Various analytical methods have been developed to facilitate the real-time evaluation of potential damage to non-instrumented buildings based on structural health monitoring applications to assess the actual building's condition (e.g., by measuring certain dynamic parameters, such as the fundamental frequency) (e.g., [START_REF] Cremen | Quantifying the benefits of building instruments to FEMA P-58 rapid postearthquake damage and loss predictions[END_REF]Goulet, Michel and Kiureghian, 2015;[START_REF] Hwang | Nonmodel-based framework for rapid seismic risk and loss assessment of instrumented steel buildings[END_REF][START_REF] Porter | Near-real-time loss estimation for instrumented buildings[END_REF][START_REF] Reuland | Measurement-based support for post-earthquake assessment of buildings[END_REF][START_REF] Tubaldi | Examining the contribution of near real-time data for rapid seismic loss assessment of structures[END_REF][START_REF] Uma | Seismic Instrumentation of Buildings -A Promising Step for Performance Based Design in New Zealand[END_REF]. However, a paradigm shift from model-driven to data-driven methods is needed to test existing methods and to provide data-driven solutions improving our understanding in this domain [START_REF] Ozer | Vibration-Based and Near Real-Time Seismic Damage Assessment Adaptive to Building Knowledge Level[END_REF]. The NDE1.0 database contains a substantial amount of earthquake data that can be utilized to develop and test solutions for the structural health monitoring-based operational earthquake loss forecasting (OELF) framework required to facilitate real-time damage assessment for non-instrumented buildings. Further studies can be carried out in this domain.

(f) The earthquake data recordings in the NDE1.0 database can be used to test the feasibility of integrating machine learning-based solutions for the OELF framework (for e.g., [START_REF] Iaccarino | Earthquake Early Warning System for Structural Drift Prediction Using Machine Learning and Linear Regressors[END_REF]. Using earthquake data recordings from buildings available in NDE1.0, further studies can be carried out to develop a framework to integrate the machine learning methods with the OELF method for building level damage assessment.

  . 1.1), excluding the Kathmandu Valley. The survey comprised a visual screening of damaged buildings by experts to map the damage in each district and to develop the NBDP database. The NBDP database comprised information regarding 762,106 buildings, each characterized by socio-economic status, engineering properties, and damage. The damage grades were classified into five levels (5DG) that are generally used in postearthquake surveys following the EMS-98 damage classification system. Damage Grade 1 (DG1): Thin cracks in walls and falling of plaster or loose stones from the upper part of the building, few architectural repairs required. Damage Grade 2 (DG2): Cracks, falling of plaster or stones in many sections, damage to nonstructural parts such as chimneys, and projecting cornices, with no significant reduction in the loadbearing capacity of the building. Damage Grade 3 (DG3): Large, extensive cracks and collapse of a small portion of non-load-bearing walls. Detachment of roof tiles, tilting or falling of chimneys, failure of individual non-structural elements such as partitions/gable walls, and delamination of stone/adobe walls. Partial reduction in the load-bearing capacity of structural members, significant repairs required. Damage Grade 4 (DG4): Large gaps or collapse of walls and partial structural failure of floors/roofs, resulting in the building classified as dangerous. Damage Grade 5 (DG5): Complete or near collapse.

Figure 1 . 1

 11 Figure 1.1 Location of the studied area and the proportion of DG5 tagged buildings (percentage indicated and highlighted by the color scale) in the 11 districts surveyed by the Nepalese authorities (NPC 2015a).

Figure 1 . 2 .Figure 1

 121 Figure1.2. Distribution of building parameters in the whole NBDP dataset. The y-axis shows percentage and the x-axis is (a) damage grade, (b) number of stories (NS1: 0 -3, NS2: 3 -5, NS3: > 5), (c) age (Ag1: 0 -10, Ag2: 10 -50, Ag3: >50), (d) plinth area (Ar1: 0 -500, Ar2: 500 -1000, Ar3: > 1000) (e) building height (Ht1: 0 -10, Ht2: 10 -20, Ht3: > 20, in ft as contained in the original database), (f) ground slope condition at building location (FS: flat slope, MS: mild slope, SS: steep slope), (g) roof construction material (RT1: heavy bamboo/timber roof, RT2: light bamboo/timber roof, RT3: reinforced concrete), (h) position of building with respect to other buildings (A1: attached on one side, A2: attached on two sides, A3: attached on three sides, NA: stand-alone building), and (i) superstructure construction material (CM1: adobe, CM2: mud-mortar stone, CM3: stone-flange, CM4: cement-mortar stone, CM5: mud-mortar brick, CM6: cement-mortar-brick, CM7: timber, CM8: bamboo, CM9: RC non-engineered, CM10: RC engineered, CM11: other).

  . The training dataset constituted 60% of the entire dataset and was used to train the machine learning model. Meanwhile, the validation dataset constituted 20% of the entire dataset and was used to select the best model by comparing the strengths and weaknesses of the different machine learning models for damage prediction (Section 1.4.1). Model optimization was performed by investigating data imbalance arising from the unequal distribution of the target features in the training dataset (Section 1.4.2) and the importance of each feature in the model (Section 1.4.3). The test dataset constituted the remaining 20% of the entire dataset; it was kept hidden from the model until the final optimized model was developed. Once the optimized model was developed, the test dataset was used to test the performance of damage prediction of the optimized model (Section 1.4.4). The percentage distribution of each feature in the training, validation, and test dataset follows the same percentage distribution as in the entire NBDP database (Fig. 1.2).

Fig. 1 .

 1 Fig. 1.5d and SVR: Fig. 1.5e) that might be due to the complexity and nonlinear feature interactions in the dataset.

Figure 1 . 4 .

 14 Figure 1.4. Graphical representation of the predictive performance of the random forest regression (RFR) on the validation dataset. (a) Distribution of errors between the true and predicted damage grades (DG) and corresponding MAE value. (b) DG normalized confusion matrix for the RFR model. The model outputs were rounded to the nearest whole integer to plot the confusion matrix, which is not the case to compute the error matrices.

Figure 1 . 5 .

 15 Figure 1.5. Damage grade (DG) normalized confusion matrix for (a) gradient boosting classification (GBC) model, (b) random forest classification (RFC) model, (c) gradient boosting regression (GBR) model, (d) linear regression (LR) model, and (e) support vector regression (SVR) model on the validation dataset. A color bar shows the normalized cell value for each confusion matrix.

  Fig. 1.6 shows the performance of the RFR model after (a) undersampling and (b) oversampling. Compared with the previous model trained using an imbalanced dataset (Fig. 1.4b), the undersampling method degraded the efficacy of the RFR model with an increase in the MAE by 9%.

Figure 1 . 6 .

 16 Figure 1.6. Data imbalance testing applied to the RFR model. Damage grade (DG) normalized confusion matrix for (a) undersampling and (b) oversampling techniques.

Fig. 1 .

 1 Fig. 1.7 shows the importance score associated with each input feature considered in this study. The highest importance score (32% in Fig. 1.7) was associated with the mud-mortar stone material.

Figure 1 . 7 .

 17 Figure 1.7. Graphical representation of the importance scores associated with the different input features considered for the RFR model. Categorical features are transformed to binary features. The features (same as in Fig. 1.2) considered in this study are on the y-axis and the x-axis is the importance score. In the figure the building features are represented by a set of colors: no. of storey (blue), age (grey), plinth area (green), height (cyan), construction material (orange), MSI (red), ground slope (purple), roof type (magenta), and building's position (sky blue).

Figure 1 . 8 .

 18 Figure 1.8. Damage grade (DG) normalized confusion matrix for (a) the full-features setting and (b) the basic-features setting observed in the test dataset.

Figure 1 . 9 .

 19 Figure 1.9. Damage grade (DG) normalized confusion matrix for (a) the full-features setting and (b) the basic-features setting in the traffic-light (TLS) -based classification approach, grouping the five damage grades (DG) into three classes, small, medium, and heavy damage grades (S, M, H).

Figure 1 . 10 .

 110 Figure 1.10. Variation of the accuracy (a) and MSE (b) of the RFR model as a function of the percentage of the whole test dataset, NT, used for the training phase. The black-solid/red-dashed lines with the grey circle corresponds to TLS-based damage classification for the full-features/basic-features-setting, respectively. The black-solid/red-dash line with the grey square corresponds to 5DG-based damage classification for the full-features/basic-features-setting, respectively.

Figure 2 . 1 .

 21 Figure 2.1. Geographic location of the buildings considered in this study.

  Figure 2.2. Distribution of the different features in the database. E1, E2, E3, E4, E5, E6, and E7, representingIrpinia-1980, Pollino-1998, Molise-Puglia-2002, Emilia-Romagna-2003, L'Aquila-2009, Emilia-Romagna-2012, and Garfagnana-Lunigiana-2013 building damage portfolios, respectively. The y-axis is the percentage distribution and the x-axis is (a) Damage grade, (b) Number of storeys (NF1: 0-3, NF2: 3-5, NF3: >5), (c) Building age (AG1: 0-20, AG2: 21-40, AG3: 41-60, AG4: 61-80, AG5: >80), (d) Floor area (A1: 0-50, A2: 51-100, A3: 101-150, A4: 151-200, A5: >200), (e) Height (H1: 0-10, H2: 10-15, H3: >15), (f) Building position (P1: corner, P2: extreme, P3: internal, P4: isolated), (g) Ground slope condition (GS1: ridge, GS2: plain, GS3: moderate slope, GS4: steep slope), (h) Regularity in plan and elevation (IRe: irregular, Re: Regular), (i) Roof type (RT1: heavy no thrust, RT2: heavy thrust, RT3: light no thrust, RT4: light thrust), (j) Construction material (CM1: poor-quality masonry, CM2: good-quality masonry, CM3: poor-quality mixed frame masonry, CM4: good-quality mixed frame masonry, CM5: reinforced concrete frame, CM6: reinforced concrete wall, CM7: steel frames, CM8: other), and (k) macro-seismic intensity.

For

  the classification-based machine learning models, the XGBC model ([MSE, AT] = [1.78, 0.59]) was more effective than the RFC ([MSE, AT] = [1.86, 0.57]) and GBC ([MSE, AT] = [1.80, 0.58]) models, considering the EMS-98 scale. In the confusion matrix (Fig. 2.3d: RFC, Fig. 2.3e: GBC, and Fig. 2.3f: XGBC), the accuracy ADG values also show higher model efficacy for the lower DGs (86% for DG0

Figure 2 . 3 .

 23 Figure 2.3. Normalized confusion matrix between predicted and observed DGs. The values given in each main diagonal cell are the accuracy scores ADG. All values are also represented by the colour scale.

  2.2a). The efficacy of the XGBC model is analysed below, addressing the classimbalance issue with data resampling techniques applied to the training phase and considering the L'Aquila-2009 portfolio. Four strategies to solve the class imbalance issue were tested: (a) random undersampling: randomly selecting the number of data entries in each class equal to the number of data entries in the minority class (DG4 in our case); (b) random oversampling: randomly replacing the number of data entries in each class equal to the number of data entries in the majority class (DG0 in our case); (c) Synthetic Minority Oversampling Technique (SMOTE): creating an equal number of data entries in each class by generating synthetic samples by interpolating the neighbouring data in the minority class;

Fig. 2

 2 Fig. 2.4 shows the confusion matrices of the four strategies considered for the class imbalance issue.Compared with Fig.2.3f (i.e., XGBC), the effects of addressing the issue of imbalance were as follows:

Figure 2 . 4 .

 24 Figure 2.4. Confusion matrices for the four methods to solve the DG imbalance issue in the DaDO. The values given in each main diagonal cell are the accuracy scores ADG. All values are also represented by the colour scale.

Figure 2 .

 2 Figure2.5a shows the average SHAP value associated with each feature considered in this study as a function of DG. The most weighted features are building age, location (latitude and longitude), material (poor quality masonry, RC frame), MSI, roof type, floor area, and height. Interestingly, the mean SHAP values are dependent on the DG, i.e., the weight of the feature is not linear depending on the DG considered; this is never taken into account in vulnerability methods. For example,[START_REF] Scala | Influence of construction age on seismic vulnerability of masonry buildings damaged after 2009 L'Aquila earthquake[END_REF]

Figure 2

 2 Figure 2.5. (a) Graphic representation of the importance scores associated with the different input features considered for the XGBC model. The features (the same as in Fig. 2.2) considered in this study are on the y-axis, and the x-axis is the mean SHAP score according to DG. (b) Confusion matrices considering the basic-features-setting. The values given in each main diagonal cell are the accuracy scores ADG. All values are also represented by the colour scale.

  For the TLS-based damage classification, the XGBC model (after oversampling to compensate of the imbalance issue) with the basic-features-setting applied to the L'Aquila-2009 portfolio (Fig. 2.6a) gives almost the same efficacy compared to the full-features-setting (Fig. 2.6b). For example, accuracy values ADG using the basic-features-setting and the full-features-setting were 0.76/0.34/0.56 and 0.82/0.36/0.54 for G/Y/R classes, with the accuracy score AT of 0.68 and 0.72, respectively. Mangalatheu et al. (2020), Roslin et al., (2020), and Harirchian et al., (2021) reported similar damage grade classification accuracy values of 0.66, 0.67, and 0.65 respectively. The efficacy of the heuristic damage assessment model using TLS-based damage classification indicates that classifying damage into three classes is much easier for the machine learning model compared with the six-class classification system (EMS-98 damage classification). This is also observed during damage surveys in the field, which sometimes find it hard to distinguish the intermediate damage grades, such as DG2 and DG3, or DG3 and DG4. Similar observations have been reported in previous studies by Guettiche et al., (2017); Harirchian et al., (2021); Riedel et al., (2015); Roeslin et al., (2020) and Stojadinović et al., (2021).

Figure 2 . 6 .

 26 Figure 2.6. Confusion matrices for (a) the basic-features-setting and (b) the full-features-setting using the traffic-light (TLS)-based classification, grouping the EMS-98 damage grades (DG) into three classes (green for no or slight damage; yellow for moderate damage; and red for heavy damage). The values given in each main diagonal cell are the accuracy scores ADG. All values are also represented by the colour scale.

  (a) TLS: basic-features-setting (b) TLS: full-features-setting

  Fig. 2.7 shows the associated confusion matrix. The basic-features-setting resulted in a similar level of damage prediction compared with the fullfeatures setting for both EMS-98 and TLS-based damage classification systems. For EMS-98 damage classification (Fig. 2.7a, b), the accuracy ADG scores indicated in the confusion matrices are almost the same for the basic-features-setting and the full-features-setting. Furthermore, the accuracy AT and MAE scores are also almost the same (0.45 and 1.08 for the basic-features-setting and 0.48 and 0.95 for the full-features-setting). Likewise, for TLS-based damage classification (Fig. 2.7c, d), the accuracy values ADG for the basicfeatures-setting and the full-features-setting are almost the same, with similar accuracy AT/ MAE scores (0.63/0.45 and 0.67/0.39, respectively).

Figure 2 . 7 .

 27 Figure 2.7. Confusion matrices for EMS-98 (a, b) and TLS (c, d) damage classification systems using the basic-and full-features-settings (green for no or slight damage; yellow for moderate damage; red for heavy damage) with (c) the full-features-setting and (d) the basic-features-setting. The values given in each main diagonal cell are the accuracy scores ADG. All values are also represented by the colour scale.

  (c) TLS: full-features-setting (d) TLS: basic-features-setting (a) EMS-98: full-features-setting (b) EMS-98: basic-features-setting set to predict the damage caused by the next earthquake. The model XGBC was applied with the basicfeatures-setting (number of storeys, building age, floor area, height, MSI for EMS-98) and EMS-98and TLS-based damage classification.

Figure 2 . 8 .

 28 Figure 2.8. Distribution of the classification value (1 -ε ! in % given by Eq. 2.1) for (a) EMS-98-and (b) TLS-based damage classification using XGBC machine learning models and considering a single damage portfolio to predict a single portfolio (single-single scenario). The colour bar indicates the associated value in each cell. The x-values are the difference between the DG observed and the DG predicted, regardless of the DG considered.

  Figure 2.9. Distribution of the classification value (1 -ε ! in % given by Eq. 2.1) for (a) EMS-98-and (b) TLS-based damage classification using XGBC machine learning models and considering an aggregate damage portfolio to predict a single portfolio (aggregate-single scenario). The colour bar indicates the associated value in each cell. The x-values are the difference between the DG observed and the DG predicted, regardless of the DG considered.

  Fig.2.10 shows the distribution of absolute errors associated with the RISK-UE, mean damage relationship, and XGBC methods (with and without compensation for the class-imbalance issue) trained on earthquake building damage portfolio E5 and tested on E3. For EMS-98 damage classification (Fig.2.10a), the XGBC model (without compensation for class-imbalance issues) resulted in a level of absolute errors similar to that of the RISK-UE and/or mean damage relationship, except for DG0 (24%).Random oversampling to compensate for the class-imbalance issues improved the distribution of errors for the XGBC model (errors less than 8%, except for DG1: 13%).

Figure 2 .

 2 Figure2.11 shows the distribution of absolute errors trained using the E2456 portfolio and tested on the E3 portfolio. For EMS-98 damage classification (Fig.2.11a), the XGBC model (without compensation for class-imbalance issues) resulted in a level of errors similar to that of the RISK-UE and/or mean damage relationship; errors were highest for DG0 with 15.15%. With compensation for the classimbalance issues, the XGBC model achieved a slightly lower error distribution for DG0 (5%) and DG3 (4%); however, for other damage grades, the error value increased significantly (DG1: 11%, DG2: 12% DG4: 7%, DG5: 2%). For TLS-based damage classification, the distribution of absolute errors was similar for both the XGBC model and the mean damage relationship and/or RISK-UE methods (Fig.2.11b). The highest absolute error value was associated with the green (no or slight damage) class of buildings (16.40%). Compensation for the class-imbalance issues slightly increased the error distribution for the XGBC model with nearly 5% for buildings in the green (no or slight) and red (heavy) classes.

Figure 2 .

 2 Figure 2.10. Comparison of the efficacy of the heuristic model with the conventional model considering the DaDO portfolio (training set: E5; test set: E3) for (a) EMS-98-and (b) TLS-based damage classification.The x-axis is the damage grade and the y-axis is the percentage of absolute error (ε 1 in % given by Eq. 2.7). The blue bar corresponds to the mean damage relationship, the red bar corresponds to the RISK-UE method, the green and orange bars correspond to the heuristic model without (XGBC1) and with (XGBC2) compensation for the class-imbalance issues, respectively.

Figure 2 .

 2 Figure 2.11. Comparison of the efficacy of the heuristic model with the conventional model considering the DaDO portfolio (training set: E2456; test set: E3) for (a) EMS-98-and (b) TLS-based damage classification.The x-axis is the damage grade and the y-axis is the percentage of absolute error (ε 1 in % given by Eq. 2.7). The blue bar corresponds to the mean damage relationship, the red bar corresponds to the RISK-UE method, the green and orange bars correspond to the heuristic model without (XGBC1) and with (XGBC2) compensation for the class-imbalance issues, respectively.

  et al., 2021) had limitations such as limited data samples, damage classes, and building characteristics limited to a spatial coverage and range of seismic demand values.Ghimire et al. (

Figure 3 . 1 .

 31 Figure 3.1. The geographic location of buildings used in this study and the distribution of macroseismic intensity of mainshock obtained from the USGS ShakeMap.

Figure 3 . 2 .

 32 Figure 3.2. Distribution of different features in the Haiti building damaged database. The y-axis is the percentage distribution, and the x-axis is (a) Damage grades, (b) Number of storeys (NF1: 1-storey, NF2: 2-storeys, NF3: 3-10 storeys), (c) Building age (Ag1: 0-10 years; Ag2: 11-25 years; Ag3: 26-50 years; and Ag4: > 50 years), (d)Floor plan (FP1: E-shape; FP2: H-shape; FP3: L-shape; FP4: O-shape; FP5: other-shape; FP6: rectangle-shape; FP7: T-shape; FP8: U-shape), (e) Wall type (WT1: blockmasonry with reinforcement; WT2: block-masonry without reinforcement; WT3: brick-masonry; WT4: reinforced concrete; WT5: stone-masonry; WT6: other; WT7: wood-masonry), (f) Structure type (ST1: reinforced concrete; ST2: load-bearing wall; ST3: steel sheet-metal; ST4: wood sheet-metal), (g) Floor type (FT1: reinforced concrete floor; FT2: concrete floor; FT3: wooden floor), and (h) macroseismic intensity.

  specific issues related to (a) the imbalance distribution of DGs in the HBDP and (b) the simplification in the damage classification. In this case, DGs are grouped into a traffic-light based damage classification, i.e., green, yellow and red, corresponding to DG0+DG1, DG2+DG3, and DG4+DG5+DG6, respectively.

  yielded similar MAE scores (1.14, 1.18) and accuracy scores (AT = 0.30, 0.27). The classification-based machine learning models XGBC and RFC provide very similar scores ( [MAE, AT] = [1.21, 0.38] and [MAE, AT] = [1.21, 0.37], respectively). The efficacy of XGBC model is shown in Fig. 3.3a. In the confusion matrix (Fig. 3.3a), the accuracy ADG values show that the efficacy of the XGBC model is higher for DG1 (0.75) and DG6 (0.41) as compared to DG4 (0.03) and DG5 (0.18). The smaller ADG values in Fig. 3.3(a) may be due to the imbalance distribution of the target feature as reported for Nepal

Figure 3 . 3 .

 33 Figure 3.3. Normalized confusion matrix between predicted and observed damage grades (DG) (a) without and (b) with handling the DG imbalance issue in HBDP. The values given in each main diagonal cell are the accuracy scores ADG. The color scale represents all values.

  XGBC model was trained on the training dataset with the TLS-based damage classification, and its efficacy on the test dataset is shown in Fig. 3.4 without (Fig. 3.4a) and with (Fig. 3.4b) handling the class-imbalance issues using the random oversampling method. Without addressing the class-imbalance issue, the XGBC model resulted in ADG values of 0.76/0.42/0.45 for G/Y/R, respectively, with the MAE score of 0.51 and AT score of 0.57. After addressing the class imbalance issues, efficacy improved slightly, with ADG values of 0.62/0.46/0.60 for G/Y/R, respectively, and an MAE score of 0.52 and AT score of 0.57. The AT score obtained here (0.57) is slightly lower than those reported by other similar studies: 0.66 by Mangalathu et al. (2020), 0.67 by Roeslin et al. (2020), 0.65 by Harirchian et al. (2021), and 0.68 and 0.72 by Ghimire et al. (2022 and 2023) using Nepal and Italian datasets.

Figure 3 . 4 .

 34 Figure 3.4. Normalized confusion matrix between predicted and observed damage grades (DG) in HBDP using the traffic-light (TLS)-based classification, grouping the ATC-20 damage classification into three classes (green for no or slight damage; yellow for moderate damage; and red for heavy damage) (a) without and (b) with handling the DGs imbalance issue. The values given in each main diagonal cell are the accuracy scores ADG. The color scale represents all values.

Figure 3 . 5 .

 35 Figure 3.5. Distribution of error values (ε 8 from Eq. 3.1) for (a) ATC-20 and (b) TLS-based damage classification. The x-axis is the damage grades. The y-axis is the error values in percentage. The blue, orange, yellow, purple, and green bar corresponds to the XGBC1, XGBC2, XGBC3, XGBC4, and XGBC5 models, respectively.

  classes were defined based on the combination of number of stories, age, and MSI (H1= 1-3 stories, H2=4-6 stories, H3=7+stories. A1=0-20 years, A2=21-40 years, A3=41-60 years, A4=60+ years. MSI values between 4-9 as IV-XI). Only eight building classes had enough samples for analysis in the test dataset: (a) H1-A1-VII, (b) H1-A1-VIII, (c) H1-A2-VII, (d) H1-A2-VIII, (e)H1-A3-VII, (f) H1-A3-VIII, (g) H2-A1-VIII, and (h) H2-A2-VIII. The damage prediction efficacy of the XGBC5 model for TLS-based damage classification was tested by aggregating observed and predicted DG values in each building class. The distribution of buildings in each class and the associated error values (ε @/ from Eq. 3.2) is shown in Fig. 3.6. Most buildings in the test set were observed in H1-A1-VIII and H1-A2-VIII building classes (Fig. 3.6c and 6d), while other classes had smaller number of buildings. The XGBC5 model resulted higher error values (Fig. 3.6i) for H1-A2-VIII (3-7%) and H1-A1-VIII (0.5-3.5%) but smaller than 1% for other classes. The XGBC5 model showed reasonable damage prediction -based damage classification for large-scale classification at the building-class level considering age and number of storeys.

Figure 3 . 6 .

 36 Figure 3.6. Distribution of (a) -(h) the observed and predicted DG values and (i) the distribution of error values (ε @/ from Eq. 3.2) in each building class level for the TLS-based damage classification XGBC5 model. The x-axis is the damage grades. The y-axis for (a) -(h) is building density and (i) error values in percentage. In figure (a) -(g), the grey and blue bar corresponds to the observed and predicted DG.

Figure 3 . 7 .

 37 Figure 3.7. The distribution of (a) error values (ε ! from Eq. 3.3) and (b) probability distribution function for TLS-based damage classification using the XGBC5 model. The x-axis is the difference between the observed and predicted DG. The y-axis is the error values in percentage. The blue, orange, yellow, and purple colour corresponds to the NBDP, L'Aquila, DaDO, and the HBDP database, respectively.

Figure 3 . 8 .

 38 Figure3.8. Distribution of the error values (ε @/ from Eq. 3.2) for TLS-based damage classification using the XGBC5. The x-axis is the damage grades. The y-axis is the error values in percentage. The blue, red, and black lines with square, circle, and diamond markers correspond to the DaDO, NBDP, and HBDP databases.

Fig. 3 .

 3 Fig.3.9 shows the distribution of the error values (ε @+ from Eq. 3.4) by building classes for each dataset.For NBDP, error values were below 10%, except for class H1-A1-VII (4-18%). For DaDO, error values are below 9% for three building classes (H1-A1-VII, H1-A2-VII, and H1-A1-VIII), between 19-36% in H1-A3-VII, 1-31% in H1-A2-VIII, and 18-50% in H1-A3-VIII. For HBDP, error values are below 8% for H1-A1-VII and H1-A1-VIII, and between 1-19% for H1-A2-VII and H1-A3-VII. For H1-A2-

Figure 3 . 9 .

 39 Figure3.9. Distribution of the error values (ε @+ from Eq. 3.4) for TLS-based damage classification using the XGBC5. The x-axis is the damage grades. The y-axis is the error values in percentage. The blue, red, and black lines with square, circle, and diamond markers correspond to the DaDO, NBDP, and HBDP databases.

Figure 3 . 10 .

 310 Figure 3.10. Distribution of the damage probabilities values (P !C from Eq. 3.5) using the XGBC5 model. The x-axis is the damage grades. The y-axis is the probability values in percentage. The blue, red, and black lines with square, circle, and diamond markers correspond to the DaDO, NBDP, and HBDP databases, respectively.

  analysed by aggregating buildings according to DG values in building classes or in the whole test dataset.When tested with the readily available features (age, number of storeys, and MSI) with TLS-based damage classification, the XGBC model show smaller efficacy in the HBDP when compared with NBDP and DaDO. This could be due to the difference in the resolution of features in these datasets. For instance, HBDP and DaDO has interval age values whereas NBDP has discrete values, and damage information was collected adopting different damage survey form. XGBC model trained and tested on HBDP, NBDP, and DaDO showed different damage probabilities among the DGs for a given building class. It suggests that when transferring damage prediction models (e.g.,Ge et al., 2023) or vulnerability models from active seismic regions to moderate or less active regions, careful consideration on building typologies and their distribution, construction practices and seismic regulations are required.

4

  .1f), velocity (SVi and SVmin) (Fig. 4.1g) and displacement (SDi and SDmin) (Fig. 4.1h) for two specific

Figure 4 . 1 .

 41 Figure 4.1. Illustrations on the left side is the time history of (a) acceleration, (b) velocity, (c) displacement, recorded at the bottom floor sensor of the building. The red dots (a-c) correspond to the peak values. Similarly, the figures on the right are (d) the pseudo-Wigner-Ville time-frequency distribution, the red-line is the smoothing of the maximum of Wigner-Ville, (e) the acceleration recorded at the top floor sensors. Response spectrum of (f) acceleration, (g) velocity, and (h) displacement, the dashed vertical line represents the position of elastic frequency (fi) and co-seismic resonance frequency (fmin).

Figure 4 . 2 .

 42 Figure 4.2. View of the whole dataset used in this study. (a) Positions of epicenters (gray circles) and buildings (red squares) in the US (California), Romania and Japan. For the US dataset, the two red rectangular boxes define the area of the two specific regions discussed in the manuscript. (b) Magnitude versus epicentral distance distribution of the whole dataset including Japan (open circles), the US (gray squares) and Romania (solid diamonds). (c) Distribution of natural log(PGA) for American, Japanese and Romanian datasets, respectively. σ is the standard deviation of the distribution.

  datasets, comprising 1,630 recordings in both horizontal directions, made over a period of 20 years, starting immediately after building completion in March 1998 and including the main shock and aftershocks of the 2011 Tohuku earthquake. Magnitude varies from 2.6 to 9.1 and epicentral distance varies from 2.2 to 1,730 km.(Fig. 4.3b). A specific subset of data for ANX is considered, with a geographical boundary of 34 to 41.5 degrees (latitude) and 137 to 145 degrees (longitude). Furthermore, three data subsets are defined based on the distribution of magnitude-distance criteria considered to have an adequate number of data in each dataset (Fig.4.3b): MR1 corresponding to 166 entries with R = 20 ± 50% and M= 3.5 ± 0.5; MR2 corresponding to 575 entries with R =120 ± 60% and M= 4.5 ± 0.5 and MR3 corresponding to 274 entries with R = 250 ± 70% and M= 5.5 ± 0.5). The distribution of PGA for MR1/MR2/MR3 is shown in Fig.4.3(c).Astorga et al. ( 2018[START_REF] Astorga | Recovery of the resonance frequency of buildings following strong seismic deformation as a proxy for structural health[END_REF] analyzed the time variation of the resonance frequency of the ANX building since 1998. They defined four time periods corresponding to changes in its behavior. During the first period (T1), the fundamental frequency starts to decrease immediately after the completion of construction work, from 1998 to 2005. Frequency stabilizes during period T2 (2006-2011/02/30) until the Tohoku earthquake sequence in 2011. During period T3, the fundamental frequency drops significantly and a slow recovery of the resonance frequency is observed directly after the Tohoku earthquake during the immediate aftershock sequence between 2011/03/01 and 2011/09/30. Finally, T4 corresponds to the period between 2011/10/01 and 2018/05/15. T1/T2/T3/T4 comprise a total of 366/313/402/468 data, respectively. Four further subsets of data within the magnitude distance criteria MR2 are considered according to the period criteria: T1-MR2 (118 data), T2-MR2 (119 data), T3-MR2 (193 data), and T4-MR2 (121 data).

Figure 4 . 3 .

 43 Figure 4.3. Dataset for the specific Annex (ANX) building in Japan. (a) Location of the ANX building (red square) and related earthquake epicenters (gray circles). The black square represents the specific subset of data considered. (b) Magnitude versus epicentral distance distribution. The red rectangles define the boundaries of the three magnitude-distance criteria (MR1, MR2 and MR3) described herein. (c) Distribution of log(PGA) for specific magnitude-distance criteria MR1, MR2 and MR3. σ is the standard deviation of the distribution.

Fig. 4 .

 4 Fig. 4.4 shows the distribution of residual values EDP|PGA for the whole dataset, versus M or log(R).

Figure 4 . 4 .

 44 Figure 4.4. Distribution of residual values (EDP|PGA) as function of (a) M and (b) log(R), considering the whole dataset. The lines represent the fitted linear model between log(R)/M and the residuals.

  Fig. 4.5 shows the standard deviation of the residuals of the fitted standard log-linear regression model between EDP and IM (Eq. 4.6) for ALL (JPN+US+RO) datasets. For each IM parameter, σIM is given in Fig. 4.5b. All the σEDP, σIM and σEDP|IM values are provided in Appendix B. In the main body of the manuscript, only the most relevant results are discussed.

Figure 4

 4 Figure 4.5. (a) σEDP|IM values for the IMs concerned, computed for the whole dataset. (b) σIM values.

Figure 4 . 6 .

 46 Figure 4.6. (a) Values of σEDP|IM for the IMs concerned computed for US buildings by tectonic region (US STS1 and US STS2) and for all US buildings (US ALL). (b) σIM values associated with each IM.

Figure 4 . 7 .

 47 Figure 4.7. Variability of σEDP|IM values as a function of the class of buildings (a) US dataset, (b) JPN dataset, (c) ANX single building dataset.

Fig. 4 .

 4 Fig. 4.8b shows the p-value considering all the IM parameters, summarized in Tab. 4.1.

Figure 4 . 8 .

 48 Figure 4.8. (a) Variability of σEDP|IM values as a function of the IMs concerned for different magnitude and earthquake-to-building distance criteria, considering the ANX single building dataset. Magnitude/distance criteria are R = 20 ± 50% and M= 3.5 ± 0.5 for MR1; R =120 ± 60% and M= 4.5 ± 0.5 for MR2; R = 250 ± 70% and M= 5.5 ± 0.5 for MR3. (b) Sufficiency analysis with respect to magnitude (o) and distance (x) observed for the JPN building class datasets. The dashed line corresponds to a p-value of 0.05.

Figure 4 . 9 .

 49 Figure 4.9. Variation of σEDP|IM values as a function of the IMs for different periods, considering seismic cumulative damage in the ANX building. Magnitude/distance criteria (MR2) are R =120 ± 60% and M= 4.5 ± 0.5 and periods T1 to T4 are described in the manuscript.

  Boomer et al. (2004) considered the correlation of damage measures and average spectral IMs computed between T1 and F*T1, T1 being the elastic period of structure, for F values from 1.7 to 3. Bianchini et al. (2009) found the most efficient and sufficient spectral values computed from 0.2*T1 to 2*T1 and 0.2*T1 to 3*T1. Eads et al. (2015) concluded that average spectral acceleration computed between 0.2*T1 and 3*T1 yields lower variability in terms of structural response. Ebrahimian et al. (2015) also considered three average spectral values between 0.2*T1 and 1.5*T1, T1 and 2*T1, and 0.2*T1 and 2*T1. Adam et al. (2017) used average spectral acceleration values computed from T1 to 1.6*T1 and from 0.2*T1 to 1.6*T1 and observed a reduction of dispersion in the collapse capacity relationship.

Figure 4 .

 4 Figure 4.11 -Variation of the frequency ratio (Rf=fmin/fi) for different datasets.

  (3, 4, and 5) The Iwaki city hall building (IWK), Toda city hall building (TDS) and Chiba government office building (CHB) are eight-storey SRC buildings, with monitoring periods of 1993 to 2014, 1998 to 2013 and 2000 to 2012, respectively, corresponding to 415, 337 and 29 earthquakes.None-to-slight damage was reported in the IWK building.(6 and 7) The Hachinohe city hall building (HCN) and the Nippon Institute of Technology building (NIT) are six-storey reinforced concrete (RC) buildings, with recordings from 428 and 213 earthquakes over1998-2014 and 1994-2014, respectively.

Figure 5 . 2 .

 52 Figure 5.2. Building locations (open red squares) and earthquake epicentres (solid colour circles) for this study.

Figure 5 . 3 .

 53 Figure 5.3. Distribution of maximum acceleration at the bottom of the building versus the drift ratios (relative roof displacement normalized by height) for each building.

Figure 5 . 4 .

 54 Figure 5.4. Four-step procedure used to obtain the empirical capacity curve illustrated with the 2011 Tohoku earthquake recorded in the THU building. (a) Bottom and (b) Top floor recordings in acceleration and displacement time histories, (c) Time-frequency distribution of the fundamental frequency obtained from the Wigner-Ville distribution applied to the top recording with before (black) and after (red) Savitsky-Golay smoothing functions, (d) Fundamental mode building response hysteresis in the Sa-Sd spectrum, (e) The Sa-Sd spectrum in absolute value (black) with empirical ADRS curve (red dashes).

  linear regression on Sa and Sd is used to analyze ADRS curve variability, as follows: Sa=a+ b.Sd + e (5.5) where a and b are the estimated regression coefficients and e is the standard error. The variability associated with ADRS curves is measured by calculating the standard deviation of the residuals of the fitted regression model between Sa and Sd (Eq. 5.5), representing within-building variability ϕ.

Figure 5 .

 5 Figure 5.5 shows the variation of the co-seismic capacity of the ANX building along one horizontal direction (N270) during a selection of earthquakes from 2004 to 2016. Fig. 5.5a corresponds to the coseismic ADRS curve of the 2011 Tohoku earthquake (main shock file 201103111446-ANX and one immediate aftershock file 201103111515-ANX), which produced the largest displacement and acceleration values at the top of the building. Fig. 5.5b shows the ADRS curves for a selection of moderate earthquakes before and after the 2011 sequence, which clearly indicate the impact of theTohoku earthquake (and the damage induced) on the capacity curve. The seismic damage caused an increase in variability of the ADRS response (about 39%), as indicated by the higher ϕ value (1.72) after the Tohoku event compared with the ϕ value (1.24) from before.

Figure 5 . 5 .

 55 Figure 5.5. ADRS curve of the ANX building along the lateral direction (N270 component) before and after the Tohoku 2011 earthquake. (a) Co-seismic ADRS curves corresponding to the main shock (file 201103111446-ANX-270) and one immediate aftershock (file 201103111515-1NX-270). (b) Coseismic ADRS curves before (squares) and after (circles) the Tohoku event. The dashed-blue and dashed-red lines represent the mean ADRS models before and after 2011.

  5.7a includes ANX for the main shock. The three buildings belong to the same class of buildings (Steel Reinforced Concrete buildings) but have different numbers of storeys. During the Tohoku earthquake, THU, ANX, and IWK experienced severe, slight-to-moderate, and none-to-slight structural damage, respectively[START_REF] Kashima | Dynamic behaviour of SRC buildings damaged by the 2011 great east japan earthquake based on strong motion records[END_REF]. In Fig.5.7a, the slope break in the ADRS curve illustrates the co-seismic response of these buildings during the 2011 mainshock, with the threshold Sd values corresponding to the onset of damage: a significant increase in Sd values without a corresponding increase in Sa values indicates severe damage. In the building-specific ADRS model for THU (Fig.5.7b) and IWK (5.7c), the same trend as for ANX (Fig.5.5 and Fig.5.6) is observed, i.e., the slopes of the ADRS model change over time as a consequence of the damage caused by the 2011 earthquake. The THU and IWK models have ϕ values of 7.98 and 1.46 before Tohoku and 2.54 and 3.80 afterwards, respectively.

Fig. 5 .

 5 Fig. 5.7d shows the time variation of the ADRS slopes through mass-normalised co-seismic stiffness (mean ± standard deviation) for ANX, THU, and IWK. For ANX during Tohoku (2011/03/11/14:46),

  , and Abeele et al. (2000) related the variability of the building's response to the seismic activation of pre-existing cracks in the structure. These examples illustrate how long-term monitoring enables the development of building-specific SHM-based solutions for updating actual capacity and reducing the within-building variability of the vulnerability function.

Figure 5 . 7 .

 57 Figure 5.7. Co-seismic ADRS curves for the Steel Reinforced Concrete class of buildings. (a) ADRS curves for ANX, THU and IWK during the 2011 Tohoku earthquake. (b) ADRS curves before and after the 2011 earthquake for THU. (c) Same as (b) for IWK. (d) Mass-normalised co-seismic stiffness variation (mean ± standard deviation) for ANX, THU and IWK. The colour bar indicates the range of drift ratios observed.

Figure 5 . 8 .

 58 Figure 5.8. Co-seismic ADRS curves for 8-storey Steel Reinforced Concrete (SRC) buildings. (a) ADRS curves for TDS, CHB and IWK during the 2011 Tohoku earthquake. (b) ADRS curves before the 2011 Tohoku earthquake for IWK, TDS and CHB. (c) Same as (b) after the 2011 earthquake. The dashed-black, dashed-red, dashed-blue, and solid-red lines represent the mean ADRS curves for IWK, CHB, TDS, and for the whole class, respectively.

Figure 5 . 9 .

 59 Figure 5.9. Same as Fig. 5.8 for the 6-story Reinforced Concrete (RC) buildings, NIT and HCN.

Fig. 5 .

 5 Fig.5.10 compares the experimental co-seismic capacity curves for the ANX and THU buildings in both horizontal directions with the corresponding curves of the GEM-taxonomy generic class-specific ADRS model provided byMartin and Silva (2021). The GEM-taxonomy reference for these buildings is SRC-LDUAL-DUH-Hx, where x corresponds to the number of storeys. The experimental ADRS model curves are observed to be similar in both lateral and transversal directions, with only a slight variation observed during the 2011 Tohoku earthquake.

Figure 5 . 10 .

 510 Figure 5.10. Comparison between the building-specific ADRS model and the generic GEM-Taxonomy model for the THU and ANX class. (a) ANX building, zoomed in (b), (c) THU building, zoomed in (d).The generic capacity model curves are represented by the thick black lines, the thin, solid, and dashed lines represent the experimental capacity curves before, during, and after the Tohoku earthquake along the transversal and lateral directions.

  observed that the distribution of target features (damage grades) in the training set could significantly affect the damage prediction efficacy of the machine learning model. Random oversampling method or integrating naturally balanced datasets (selecting data from a wider range of input features and earthquake damage from different regions) while training the machine learning model can penalize the skewed distribution of the target features and improve the damage prediction efficacy of the machine learning model. The efficacy of a machine learning model is influenced by the quality and diversity of the training dataset, i.e., the efficacy of a machine learning model depends more on the quality of the dataset's resolution than on the quantity of dataset used in the training phase. Thus, increasing the number of damaged buildings in the training set may not improve the model's efficacy beyond a certain limit, but improving the dataset's resolution can enhance the prediction accuracy of the damage.

Table 1 .1

 1 Distribution of different building parameters present in the NBDP database.

	S.N. Parameters

Table 1 .2.

 1 Summary of the optimized input parameters and the performance scores of different machine learning models observed on the validation dataset. The values of the input parameters are the hyperparameters for the machine learning methods (the other hyperparameters not mentioned here are the default ones in the Scikit-learn documentation[START_REF] Pedregosa | Scikit-learn[END_REF]). The best model for each measure is indicated in bold.

	Machine learning model	Input parameters	Score Classification Accuracy	Statistical MAE	MSE	Computational time (sec)
	LR	Default	0.30	0.86	1.14	3
	SVR	C = 1 Kernel = rbf	0.43	0.77	1.09	80900
		learning rate = 0.1
	GBR	no_estimators = 500 max_depth = 10	0.51	0.64	0.72	875
	RFR	no_estimators =1000 max_depth = 20	0.49	0.66	0.76	563
		learning rate = 0.1
	GBC	no_estimators = 1000 max_depth = 10	0.57	0.60	1.05	8584
	RFC	no_estimators = 1000 max_depth = 30	0.53	0.67	1.25	450

Table 1 .3. Summary of the performance scores for RFR model considering the test datasets for the

 1

	full-

Table 2 .

 2 1. Building-damage data from the DaDO for the seven earthquakes considered in this study. 'Ref' is the reference to the earthquake. 'DL' is the number of the damage grade available in DaDO. 'NB' is the number of buildings considered in this study. AeDES is the post-earthquake damage survey form, first introduced in 1997 and become the official operational tool recognized by the Italian Civil Protection in 2002.

	Ref Earthquake	Event date Mag. Epicentre	Damage	DL NB
				Lat.	Long.	survey form
	E1	Irpinia-1980	23/11/1980 6.9	40.91 15.37 Irpinia-1980	8	37,828
	E2	Pollino-1998	09/09/1998 5.6	40.04 15.98 AeDES-1998	4	9,485
	E3	Molise-Puglia-2002	31/10/2002 5.9	41.79 14.87 AeDES-2000	4	6,396
	E4	Emilia-Romagna-2003	14/09/2003 5.3	44.33 11.45 AeDES-2000	4	239
	E5	L'Aquila-2009	06/04/2009 6.3	42.34 13.34 AeDES-2008	4	37,999
	E6	Emilia-Romagna-2012	20/05/2012 6.1	44.89 11.23 AeDES-2008	4	10,581
	E7	Garfagnana-Lunigiana-2013 21/06/2013 5.3	44.15 10.14 AeDES-2008	4	1,474

Table 2 .2. Distribution

 2

	No.	Parameters

of the different features used in this study.

Figure 2.2. Distribution of the different features in the database. E1, E2, E3, E4, E5, E6, and E7,

  Total accuracy (AT) was computed as the ratio of the number of correctly predicted DGs to the total number of observed values. AT and ADG values close to 1 indicate high efficacy. Moreover, the quantitative statistical error was also calculated as the mean of the absolute value of errors (MAE) and the mean squared error (MSE) (MAE and MSE values close to 0 indicate high efficacy). For classification-based machine learning models, the ordinal value of the DG was used to calculate the MAE and MSE scores directly. For the regression-based machine learning models, the output DG values were rounded to the nearest integer for the accuracy scores plotted for the confusion matrix, but not for the MAE and MSE value calculations.

Table 2

 2

.3. The hyperparameters indicated in Tab. 2.3 were choosen after tests performed by Ghimire et al. (2021). The regression-based machine learning models RFR, GBR and XGBR yielded similar MSE scores (1.22, 1.22 and 1.21) and accuracy scores (AT = 0.49, 0.50 and 0.50), considering the five DGs of the EMS-98 scale. In the confusion matrix (Fig. 2.3a: RFR, Fig. 2.3b: GBR, and Fig. 2.3c: XGBR), the accuracy ADG values

Table 2 .

 2 3. Summary of optimized hyperparameters parameters, accuracy AT and quantitative statistical error values for the regression-based and classification-based machine learning methods. The parameters are the hyperparameters chosen for the machine learning models (the other hyperparameters not mentioned here are the default parameters in the Scikit-learn documentation[START_REF] Pedregosa | Scikit-learn[END_REF]). The best accuracy and error values are indicated in bold.

	Method	Parameters	Accuracy AT	MSE	MAE
	RFR	n_estimators = 1000 max_depth = 25	0.49	1.22	0.77
		n_estimators = 1000
	GBR	max_depth = 10	0.50	1.22	0.77
		learning_rate = 0.01
		n_estimators = 1000
	XGBR	max_depth = 10	0.50	1.21	0.76
		learning_rate = 0.01
	RFC	no_estimators = 1000 max_depth = 25	0.57	1.86	0.77
		no_estimators = 1000
	GBC	max_depth = 10	0.58	1.80	0.77
		learning_rate = 0.01
		n_estimators = 1000
	XGBC	max_depth = 10	0.59	1.78	0.74
		learning_rate = 0.01

2 Second stage: issues related to machine learning

	2.4.
					(a) RFR							(b) GBR
		0	0.60	0.32	0.07	0.01	0.00	0.00		0	0.63	0.30	0.06	0.01	0.00	0.00
		1	0.19	0.57	0.20	0.04	0.01	0.00		1	0.22	0.55	0.18	0.05	0.00	0.00
	True DG	2 3	0.11 0.07	0.44 0.36	0.34 0.35	0.11 0.18	0.01 0.04	0.00 0.00	True DG	2 3	0.11 0.09	0.45 0.35	0.32 0.35	0.10 0.19	0.01 0.02	0.00 0.00
		4	0.04	0.23	0.37	0.28	0.08	0.00		4	0.04	0.24	0.40	0.26	0.06	0.00
		5	0.04	0.20	0.31	0.28	0.17	0.01		5	0.04	0.18	0.33	0.30	0.14	0.01
			0	1	2	3	4	5			0	1	2	3	4	5
					Predicted DG							Predicted DG
					(c) XGBR							(c) RFC
		0	0.63	0.30	0.06	0.01	0.00	0.00		0	0.85	0.12	0.01	0.02	0.00	0.00
		1	0.22	0.56	0.17	0.05	0.00	0.00		1	0.52	0.38	0.02	0.06	0.01	0.01
	True DG	2 3	0.12 0.08	0.45 0.36	0.31 0.35	0.12 0.18	0.01 0.02	0.00 0.00	True DG	2 3	0.43 0.38	0.33 0.30	0.06 0.04	0.14 0.20	0.02 0.04	0.02 0.03
		4	0.03	0.24	0.39	0.27	0.06	0.00		4	0.31	0.25	0.03	0.24	0.10	0.07
		5	0.04	0.18	0.34	0.31	0.13	0.01		5	0.33	0.18	0.03	0.25	0.06	0.15
			0	1	2	3	4	5			0	1	2	3	4	5
					Predicted DG							Predicted DG
					(e) GBC							(f) XGBC			1.0
			0.85	0.12	0.00	0.02	0.00	0.01		0	0.86	0.11	0.00	0.02	0.00	0.00
																	0.8
			0.49	0.40	0.02	0.07	0.01	0.01		1	0.52	0.38	0.01	0.07	0.01	0.01
			0.40 0.33	0.34 0.30	0.06 0.04	0.15 0.22	0.02 0.05	0.02 0.05	True DG	2 3	0.34 0.42	0.30 0.33	0.02 0.03	0.26 0.17	0.04 0.02	0.04 0.02	0.4 0.6
			0.25	0.26	0.02	0.24	0.14	0.09		4	0.28	0.26	0.03	0.26	0.10	0.08
																	0.2
			0.26	0.23	0.02	0.23	0.08	0.19		5	0.27	0.21	0.02	0.26	0.06	0.19
											0	1	2	3	4	5	0.0
													Predicted DG

Table 2 .

 2 4 -Scores of the accuracy AT, MSE and MAE metrics considering the imbalance issue and their variation Δ compared with values without consideration of the imbalance.

	Method	Accuracy AT	MSE		MAE
		Scores	Δ	Score	Δ	Score	Δ
	Undersampling	0.26	-0.33 1.24	-0.34	1.20	0.46
	Oversampling	0.53	-0.06 2.13	0.35	0.86	0.12
	SMOTE	0.57	-0.02 1.87	0.09	0.77	0.03
	SMOTE-ENN	0.49	-0.10 2.28	0.50	0.93	0.19

  as the training set, respectively, for an average of 64% (compared with the 70% score for the single portfolio scenario, Fig.2.8a). Other scenarios were also tested by aggregating the building damage portfolios differently (not presented herein), leading to the two main conclusions: (1)

	Training set: E2+E3+E4+E6 (shown as E2346) -test set: E1, E5 and E7.
	Training set: E2+E4+E5+E6 (shown as E2456) -test set: E1, E3 and E7.
	Training set: E2+E4+E6+E7 (shown as E2467) -test set: E1, E3 and E5.
	For the EMS-98 damage scale, correct classification (x-value centred on 0) in the range of 27% to 49%
	was found, depending on the training/test datasets. As in Fig. 2.8, using the E1 (Irpinia-1980)
	earthquake for testing score is lower regardless of the portfolio used for training (28.7%, 27.2% and
	27.4% prediction accuracy). With E1 as the test set, the predictions at +/-1 DG (i.e., the sum of the x-
	values on Fig. 2.9a between -1 and +1) were 65.7%, 63.8% and 62.4% considering the E2346, E2456
	and E2467 portfolios

Table 3 . 1 .

 31 Distribution of building features in the HBDP and their label reported in Fig.3.2.

	No.	Parameters

Table 3 . 2 .

 32 Distribution of building features in the NBDP.

	No.
		Parameters			Data type	Distribution (%)
				DG1		10.34
	1	Damage grades	DG2 + DG3	Categorical	29.35
				DG4 + DG5		60.31
				1-3		98.94
	2	Number storeys	of	4-6	Numerical	1.05
				7-10		0.01
				0-20		62.9
	3	Age (years)		21-40 41-60	Numerical	27.63 6.51
				> 60		2.95
				5		0.54
	4	MSI		6 7	Numerical	11.90 35.90
				8		51.66
	Italian earthquakes building damage database (DaDO)
	The Database of Observed Damaged (DaDO) provides information on damaged buildings surveyed
	after several earthquakes in Italy. The full description of the data used for this study is given in Ghimire
	et al. (2023) (Chapter 2). Tab. 3.3 summarizes the categories of the features from DaDO and the number

of building features used in this study, corresponding to seven Italian post-earthquake surveys.

Table 3 .

 3 3. Distribution of building features in the DaDO.

	No.	Parameters			Data type	Distribution (%)
				DG0+DG1		69.60
	1	Damage grades	DG2+DG3	Categorical	22.26
				DG4+DG5		8.15
				1-3		85.82
	2	Number storeys	of	4-6	Numerical	13.76
				7-10		0.43
				0-20		15.22
	3	Age (years)		21-40 41-60	Numerical	18.80 34.16
				> 60		31.82
				4		2.23
				5		21.75
	4	MSI		6 7	Numerical	31.78 33.57
				8		6.28
				9		4.39

Table 3 .

 3 4. Summary of optimized input parameters, accuracy AT, and quantitative statistical error values for the regression-based and classification-based machine learning methods for ATC-20 damage classification (seven-class classification). The parameters are the hyperparameters chosen for the machine learning models (the other modal parameters not mentioned here are the default parameters in the Scikit-learn documentation[START_REF] Pedregosa | Scikit-learn[END_REF]). The best accuracy and error values are indicated in bold.

	Method	Parameters	Accuracy AT	MAE
	RFR	n_estimators = 1000	0.30	1.14
		max_depth = 20
	XGBR	n_estimators = 1000	0.27	1.18
		max_depth = 10
		learning_rate = 0.01
	RFC	n_estimators = 1000	0.37	1.21
		max_depth = 25
	XGBC	n_estimators = 1000	0.38	1.21
		max_depth = 10
		learning_rate = 0.01

  , is more sensitive. For MR1 and MR2, the σEDP|IM values are lower than the values of the ANX dataset, particularly for the velocity IMs (values for PGV/SVi/SVmin are

	0.50/0.48/0.40 for all ANX data compared with 0.42/0.40/0.34 for MR1 and 0.41/0.41/0.37 for MR2)
	and for the displacement IMs (values for PGD/SDi/SDmin are 0.46/0.48/0.39 for all ANX data compared
	with 0.38/0.37/0.33 for MR1 and 0.35/0.42/0.29 for MR2).

  Table 4.2 summarizes the contribution of each component to the epistemic uncertainties of Eq. 4.6. While the Table 4.2 -Summary of the σEDP|IM values and their reduction (in %) applied to the specific ANX building. The Avg column is the mean value of all IMs.

		PGA PGV PGD AI	DP	CAV SAi SVi SDi SAmin SVmin SDmin Avg.
	σ	1.05 0.80 0.81 0.84 0.91 0.85 0.87 0.79 0.87 0.87 0.79 0.88 0.86
	σR	1.02 0.78 0.78 0.81 0.85 0.81 0.83 0.76 0.82 0.83 0.76 0.83 0.82
	σR/ σ	3 % 3 % 4 % 4 % 7 % 5 % 5 % 4 % 6 % 5 % 4 % 6 % 4 %
	σT	0.87 0.64 0.67 0.66 0.74 0.72 0.65 0.61 0.61 0.62 0.60 0.64 0.67
	σT/ σR	15 % 18 % 14 % 19 % 13 % 11 % 22 % 20 % 26 % 25 % 21 % 23 % 19 %
	σB	0.80 0.50 0.46 0.51 0.57 0.51 0.78 0.48 0.48 0.46 0.40 0.39 0.53
	σB/ σT	8 % 22 % 31 % 23 % 23 % 29 % -20 % 21 % 21 % 26 % 33 % 39 % 21 %
	σMR	0.55 0.41 0.35 0.41 0.41 0.48 0.61 0.41 0.42 0.37 0.37 0.29 0.42
	σMR/ σB 31 % 18 % 24 % 20 % 28 % 6 % 22 % 15 % 13 % 20 % 8 % 26 % 19 %
	σA	0.55 0.37 0.30 0.45 0.53 0.44 0.50 0.37 0.35 0.27 0.26 0.27 0.39
	σA/ σMR 0 % 10 % 14 % -10 % -29 % 8 % 18 % 10 % 17 % 27 % 30 % 7 % 8 %
	σA/ σ	48 % 54 % 63 % 46 % 42 % 48 % 43 % 53 % 60 % 69 % 67 % 69 % 55 %

Table 4 . 3 -

 43 Empirical building damage prediction model according to building class.

	IM	Parameter	BT-ALL	BT-RC	BT-SRC	BT-ST	BT-MA BT-WO
		a	-10.22	-10.65	-10.00	-9.81	-9.17	-9.26
	SVmin	b	0.87	0.83	0.89	1.08	0.55	0.58
		σ	0.79	0.84	0.60	0.60	1.10	0.68
		a	-9.41	-9.78	-9.21	-9.17	-8.80	-9.02
	PGV	b	0.94	0.86	0.95	1.16	0.60	0.55
		σ	0.80	0.87	0.64	0.80	1.07	0.71

Table B .

 B σIM, σEDP, and σEDP|IM values considering different sub datasets discussed in the manuscript PGA PGV PGD AI DP CAV SAi SVi SDi SAmin SVmin SDmin .24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 EDP|IM 1.14 1.00 0.93 1.00 1.07 0.93 1.15 1.03 1.03 1.15 1.01 1.03 STS2 IM 0.74 0.85 1.15 1.49 2.29 0.84 0.92 0.95 1.20 0.95 0.92 1.29 EDP 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 EDP|IM 1.12 1.01 0.99 1.00 1.06 0.98 0.97 1.01 1.10 1.04 1.01 1.10 RC IM 0.84 0.98 1.47 1.58 2.63 0.91 1.03 1.03 1.25 1.07 1.04 1.31 EDP 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 EDP|IM 1.21 0.94 0.94 0.99 1.03 0.93 1.20 1.01 0.91 1.22 0.99 0.90 ST IM 0.84 1.01 1.46 1.55 2.78 0.87 0.91 1.02 1.32 0.99 1.08 1.64 EDP 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.29 1.28 1.28 1.29 1.28 EDP|IM 1.20 0.93 0.91 0.97 1.01 0.89 1.13 1.02 0.99 1.19 0.98 0.99 MA IM 0.91 1.08 1.43 1.83 2.35 1.02 0.94 1.27 1.71 1.00 1.18 1.46 EDP 1.23 1.23 1.23 1.23 1.23 1.23 1.23 1.21 1.21 1.23 1.23 1.23 EDP|IM 1.11 1.07 1.01 1.02 1.10 1.00 1.11 1.12 1.15 1.14 1.12 1.14 .48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 EDP|IM 0.87 0.64 0.67 0.66 0.74 0.72 0.65 0.61 0.61 0.62 0.60 0.64 ST IM 1.11 1.24 1.61 2.18 3.24 1.21 1.72 1.37 1.56 1.65 1.43 1.70 EDP 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 EDP|IM 1.36 0.76 0.46 0.87 0.67 0.65 0.80 0.70 0.78 0.59 0.50 0.48 .80 0.90 1.30 2.10 0.65 1.06 0.85 0.91 0.89 0.88 0.97 EDP 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 EDP|IM 0.59 0.42 0.38 0.46 0.52 0.46 0.62 0.40 0.37 0.39 0.34 0.33 MR2 IM 0.71 0.75 0.78 1.16 1.82 0.57 1.21 0.79 0.89 0.89 0.79 0.86 EDP 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 EDP|IM 0.55 0.41 0.35 0.41 0.41 0.48 0.61 0.41 0.42 0.37 0.37 0.29 MR3 IM 0.79 0.86 0.96 1.47 2.17 0.73 1.10 0.85 0.93 0.88 0.92 1.03 EDP 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 EDP|IM 0.57 0.46 0.52 0.47 0.59 0.47 0.75 0.47 0.52 0.47 0.35 0.40 .71 0.80 1.14 1.98 0.57 0.91 0.78 0.82 0.79 0.80 0.90 EDP 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 EDP|IM 0.55 0.37 0.30 0.45 0.53 0.44 0.50 0.37 0.35 0.27 0.26 0.27 T4-MR2 IM 0.63 0.75 0.90 1.22 2.13 0.61 0.98 0.83 0.87 0.91 0.89 1.01 EDP 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 EDP|IM 0.61 0.39 0.30 0.43 0.42 0.41 0.46 0.37 0.34 0.25 0.25 0.25

	US+JPN +RO US data Japanese 1.48 1ANX ALL IM 1.15 1.33 1.63 2.29 3.35 1.21 1.53 1.31 1.43 1.63 1.44 1.61 EDP 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 EDP|IM 1.05 0.80 0.81 0.84 0.91 0.85 0.87 0.79 0.87 0.87 0.79 0.88 ALL IM 0.90 1.07 1.49 1.73 2.73 0.96 1.03 1.04 1.35 1.09 1.08 1.54 EDP 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 EDP|IM 1.19 1.00 0.98 1.01 1.07 0.97 1.14 1.03 1.08 1.18 1.01 1.08 STS1 IM 0.86 0.94 1.15 1.48 2.20 0.77 1.00 1.06 1.20 1.03 1.08 1.31 EDP 1.24 1WO IM 1.02 1.36 1.83 2.20 3.21 1.21 1.07 1.32 1.60 1.06 1.32 1.65 EDP 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 EDP|IM 0.66 0.71 0.81 0.71 0.77 0.75 0.67 0.69 0.72 0.70 0.68 0.70 ALL IM 1.11 1.31 1.61 2.26 3.34 1.22 1.50 1.30 1.42 1.61 1.43 1.61 EDP 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 EDP|IM 1.02 0.78 0.78 0.81 0.85 0.81 0.83 0.76 0.82 0.83 0.76 0.83 RC IM 0.98 1.13 1.50 2.02 2.98 1.12 1.25 1.09 1.22 1.37 1.20 1.35 EDP 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 EDP|IM 0.98 0.86 0.92 0.84 0.93 0.85 0.79 0.86 0.95 0.73 0.82 0.98 SRC IM 1.14 1.40 1.66 2.36 3.53 1.24 1.48 1.37 1.46 1.56 1.52 1.68 EDP building ALL IM 1.00 1.27 1.51 2.12 3.25 1.13 1.54 1.31 1.42 1.44 1.41 1.59 EDP 1.42 1.42 1.42 1.42 1.42 1.42 1.42 1.42 1.42 1.42 1.42 1.42 EDP|IM 0.80 0.50 0.46 0.51 0.57 0.51 0.78 0.48 0.48 0.46 0.40 0.39 STS IM 1.01 1.28 1.50 2.13 3.25 1.13 1.55 1.32 1.43 1.45 1.42 1.60 EDP 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 EDP|IM 0.80 0.50 0.45 0.50 0.56 0.50 0.78 0.48 0.47 0.45 0.39 0.39 IM MR2 IM 0.83 0.95 1.02 1.58 2.19 0.81 1.18 0.92 0.99 0.98 0.98 1.05 EDP 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94 EDP|IM 0.64 0.51 0.50 0.50 0.55 0.51 0.81 0.46 0.45 0.44 0.45 0.46 T2-MR2 IM 0.66 0.77 0.89 1.24 1.93 0.61 1.17 0.86 0.94 0.86 0.84 0.90 EDP 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 EDP|IM 0.59 0.39 0.37 0.43 0.57 0.42 0.62 0.37 0.33 0.33 0.35 0.31 MR2 0.69 0T1-T3-IM 0.60 0
	MR1

201103111515-NIT-288 201103191856-NIT-288 201104072332-NIT-288 201212071718-NIT-288 201311100737-NIT-288 ADRS-model-NIT 201103311615-HCN-254 201104072332-HCN-254 201106230650-HCN-254 201212071718-HCN-254 ADRS-model-HCN ADRS-model-all

Analysis of the efficiency of Intensity Measures from real earthquake

Based on Fig. 4.8b, it appears difficult to conclude on the sufficiency of the IMs tested on our dataset, i.e. EDP is not conditionally independent of magnitude and distance. For the ALL-JPN dataset, the most sufficient IM with respect to magnitude is ) and with respect to distance SDi (0.13) and SVmin (0.78), the latter value (the highest) allowing us to assume that the prediction of EDP knowing SVmin is statistically independent of distance. However, for individual building classes, the p-values differ between IMs. It seems that displacement IMs are the most sufficient in distance, such as PGD (p-value=0.57) and SDmin (0.41) for the ST class, as these buildings are the most slender -long-period buildings, i.e. more sensitive to ground displacements. For RC buildings, the most sufficient IMs (in distance) are ) and PGD (0.84) and, to a lesser extent, SDi (0.05) and SVmin (0.06).

For the same class of buildings, the sufficient IMs in magnitude are ) and, to a lesser extent, PGD (0.23) and DP (0.07). Finally, for SRC buildings, only two IMs (DP and CAV) are sufficient in magnitude, with p-values of 0.21 and 0.50, respectively.

Since sufficiency differs according to building class, these results suggest that particular attention should be paid when selecting the accelerometric time histories used to perform non-linear time history analysis or PBEE assessment. Furthermore, if insufficient IMs are considered, site-specific ground motion data must be provided to avoid inaccurate estimation of the damage levels or failure rates used in PBEE if the ground motion characteristics do not match the source and site requirements (Kazantzi and Vamvatsikos, 2015).

PGA PGV PGD AI

DP CAV SAi SVi SDi SAmin SVmin SDmin All JPN M 0.00 0.00 0.00 0.00 0.47 0.03 0.00 0.00 0.00 0.00 0.00 0.00 R 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.13 0.00 0.78 0.00 RC M 0.00 0.00 0.23 0.00 0.07 0.00 0.00 0.00 0.00 0.00 0.42 0.00 R 0.00 0.00 0.84 0.00 0.00 0.00 0.00 0.00 0.05 0.80 0.06 0.00 SRC M 0.00 0.00 0.01 0.00 0.21 0.50 0.00 0.00 0.00 0.00 0.00 0.02 R 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.24 ST M 0.00 0.00 0.57 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.41 R 0.00 0.00 0.03 0.00 0.74 0.00 0.00 0.00 0.44 0.00 0.00 0.00 regional distinction (σR) does not bring any significant gain (4% on average), distinction by type of construction (σT) and specific building (σB) contributes significantly (19% and 21% respectively).

Concerning the IMs that make EDP conditionally independent from magnitude M and source-to-site distance (σMR), the figure shows that all the IMs are globally non-sufficient, with a reduction of σEDP|IM values of approximately 19% for the specific case of the ANX building. When ageing (σA), i.e. the actual health of the structure, is taken into account, the σEDP|IM values are reduced by 8%.

In total, the spectral IMs benefit most from these successive components. For example, the SAmin/SVmin/SDmin value reductions are equal to 69%/67%/69% for σEDP|IM values corresponding to

Between-building variability τ

In this section, the between-building variation of co-seismic capacity is discussed by considering buildings from the same class. τ is computed as the standard deviation of the residual between the experimental building-specific ADRS model and the mean experimental class-specific ADRS model for this class. and [0.72, 1.44] after the earthquake. Finally, total variability for this class (corresponding to the mean class-specific ADRS model) before (1.50) and after (3.44) the Tohoku earthquake is observed to differ from the total variability for each building-specific ADRS model belonging to the same class.

Perspectives

Developing the dynamic damage assessment method requires rigorous testing of the damage response of buildings during seismic loading. Several issues need to be studied to fully develop the dynamic damage assessment method as a tool for seismic risk decision-making. Potential issues related to this study need to be addressed in further studies and are presented here as follows.

In the framework of the rapid damage assessment at a large scale:

(a) In Chapter 3, we observed that the accuracy of machine learning-based damage assessment at large-scale is heavily influenced by region-specific factors such as construction quality, regional typologies, implementation of seismic regulations, and hazard level; host-to-target adjustments are required to apply machine learning models trained in one region to another.

Further studies should focus on quantifying these adjustments based on the region-specific factors mentioned above. Similarly, the accuracy of damage assessment must also be compared between machine learning models developed in different regions with and without focusing on these specific features (e.g., typology and age). (c) Future studies could also focus on gathering earthquake loss data and evaluating machine learning models directly for loss assessment. These models will map building features with the loss value, bypassing vulnerability modeling and translating damage to loss.

In the framework of real-time building-level damage assessment:

(d) In Chapter 4, we observed that the engineering demand parameter (EDP) is not conditional independent with earthquake magnitude and distance. In reality, buildings near fault rupture were observed to have significant damage [START_REF] Bessason | Seismic vulnerability of low-rise residential buildings based on damage data from three earthquakes (Mw6.5, 6.5 and 6.3)[END_REF]. Previous study suggest that, the pulse-like signals of near-fault ground motions can induce fewer cycles in structures than far-field ground motions but with higher energy content (Mollaioli et al., 2011).

Appendix