Lorena De Carvalho Araujo

Simon Thébault

Thomas Recht

Alain Sempey

Patrick Schalbart

Michaël Cohen

Laurent Mora

Development of a short duration method to assess the envelope thermal performance of multi-family housings

Keywords: Building envelope thermal performance, HTC estimation, multi-family housing

Building energy efficiency is a key factor in reducing CO 2 emissions. For this reason, EU member states have developed thermal regulations to ensure building thermal performance. These results are often based on results achieved with building simulation software during the design stage. However, the actual thermal performance can deviate significantly from the predicted one, and this difference is known as the energy performance gap. Accurate indicators of the actual thermal performance are a valuable tool to guarantee building quality. These indicators, including the heat transfer coefficient (HTC) and the heat loss coefficient (HLC), can be estimated by the application of in situ methods. As multi-family housing and tertiary sector buildings are an important part of the building stock, mature methods to measure their thermal performance are needed. This paper presents a short-duration method for assessing the HTC in large building typologies using a sampling approach. The method was applied in a four-storey building model under different conditions to study the limits of the method and to improve indicator bias and uncertainty. Indicator quality was strongly influenced by the external weather conditions, the temperature variation during the protocol and the heat exchange with the adjacent apartments. Under winter conditions and with stable indoor temperatures, the method had a high accuracy when the protocol was applied for half a day. It is recommended that the protocol be used over two days to improve indicator quality under less favorable test conditions.

Introduction

The energy-performance requirements stipulated in building standards and regulations have gradually been increasing, and this should translate into lower energy consumption to provide a given level of building services. Despite these efforts, the as-built energy performance does not always meet what was predicted in the design phase [START_REF] Zou | Closing the building energy performance gap through component level analysis and stakeholder collaborations[END_REF][START_REF] Majcen | Theoretical vs. actual energy consumption of labelled dwellings in the netherlands: Discrepancies and policy implications[END_REF]. This discrepancy between the design and reality is known as the energy performance gap, a phenomenon that has been widely reported in the literature, with the extent of the energy gap varying between different studies [START_REF] Johnston | Quantifying the domestic building fabric 'performance gap[END_REF][START_REF] Pappalardo | Explaining the performance gap in a french energy efficient building: Persistent misalignment between building design, space occupancy and operation practices[END_REF][START_REF] Calì | Energy performance gap in refurbished german dwellings: Lesson learned from a field test[END_REF].

The analysis of building energy performance can be done at different scales, from building components to the whole building, including the contributions from occupants and HVAC systems [START_REF] Roels | EBC Annex 58 Project Summary Report[END_REF]). This work focuses on an intermediate scale, the building envelope thermal performance, which has shown to have a significant influence in indoor human thermal comfort [START_REF] Liu | A novel building energy efficiency evaluation index: Establishment of calculation model and application[END_REF]. Only the building fabric is concerned, other factors that influence final energy consumption are not taken into account, such as systems, weather conditions and the behaviour of the occupants. Two indicators are commonly used to describe the thermal behaviour of the envelope: the heat loss coefficient (HLC) and the heat transfer coefficient (HTC). These are useful indicators to rate building envelope quality, and are extensively discussed in the literature [START_REF] Rouchier | Sequential monte carlo for on-line parameter estimation of a lumped building energy model[END_REF][START_REF] Díaz-Hernández | Data-based RC dynamic modelling incorporating physical criteria to obtain the HLC of in-use buildings: Application to a case study[END_REF][START_REF] Jiménez | Reliable building energy performance characterisation based on full scale dynamic measurements -Report of Subtask 3, part 1: Thermal performance characterization based on full scale testing -description of the common exercises and physical guidelines[END_REF]. The HLC quantifies the amount of energy required in the steady-state to maintain a one-degree temperature difference between the interior and exterior of the building [START_REF] Bouchié | Methodologies for the assessment of intrinsic energy performance of buildings' envelope[END_REF]. The HTC is only related to the envelope thermal losses by transmission to the exterior, and is equivalent to the HLC minus the thermal losses by advection, which are related to infiltration (H inf ) through the envelope [START_REF] Bouchié | MPEB: Inventaire des méthodes applicables à la caractérisation de la performance énergétique de l'enveloppe[END_REF]. All these indicators have W/K units, and their mathematical formulae are represented in equations 1 and 2. The indices i, j, k represent each point, linear and surface element of the envelope going from 1 to p, q, r, respectively.

HT C = q j=1 χ j + p i=1 ψ i × L i + r k=1 U w k × A w k (1)
where:

-χ j is the heat flow from a point thermal bridge j (W/K); -ψ i is the heat flow of a linear thermal bridge i (W/m.K); -L i is length of a linear thermal bridge i (m); -U w k is the thermal transmittance of an external homogeneous surface k (W/(m².K)); -A w k is the area of an external homogeneous surface k (m²).

HLC = HT C + H inf (2)
Although these indicators can be calculated mathematically , some terms from equation 1, such as the U-values and thermal bridges, may differ between an actual building and its design. Assessment of HTC and HLC values is important for reducing the performance gap and in the long run may increase the quality of building construction. At building handover, knowledge of these values helps to ensure project results and to identify possible malfunctions that could lead to excessive energy consumption when the building is in the operational phase. For construction companies, it could be used for internal quality control and as feedback to support improved building design. Knowledge of HTC and HLC can also be used as a tool to inform occupants, owners and public authorities about the quality of building insulation, which in turn can be useful for public policy management, the allocation of subventions and the awarding of quality labels. [START_REF] Ziour | Atelier FBE MPEB -Mesure de la Performance énergétique des Bâtiments: Les différentes configurations de MPEB[END_REF]. For this reason, the use of tests and agreed protocols for in situ measurement of envelope performance after building handover can be an alternative to evaluate these parameters. Different methods methods for assessing HTC and HLC based on various protocols and estimation processes are available nowadays. Most of them rely on a heat balance for the building treated as a single zone, shown in equation 3 [START_REF] Senave | Sensitivity of characterizing the heat loss coefficient through on-board monitoring: A case study analysis[END_REF].

C × ∂T ∂t = ϕ h + ϕ int + ϕ sol + ϕ v + ϕ inf + ϕ tr (3)
where:

-T is the average indoor temperature of the zone (K); -C is the effective heat capacity of the zone thermal mass (J/K); -ϕ h is the heat flow from heating systems (W); -ϕ int is the internal heat gains (W); -ϕ sol is the solar gains (W); -ϕ v is the heat exchange due to ventilation (W); -ϕ inf is the heat exchange due to infiltration through the building envelope (W); -ϕ tr is the heat flow due to transmission through the building envelope (W).

Based on this description, two main families of methods can be identified. The first considers the thermodynamic behaviour of the thermal mass of the building, while the second relies on a steady-state hypothesis. To neglect the dynamic term in the heat balance, it is assumed that the total heat stored and released by the thermal mass of the building over a specific period are equal (Senave et al. 2019a). This hypothesis is consistent with the use of long enough time steps in comparison with a thermal cycle [START_REF] Bauwens | Co-heating test: A state-of-the-art[END_REF]). On the one hand, such a hypothesis reduces the heat balance to an algebraic equation, allowing modelling solutions such as linear regression and mean ratios to be used. On the other hand, it implies longer protocols so that enough data points can be acquired to characterise the envelope performance. When quick results are required, dynamic methods are more suitable [START_REF] Madsen | Annex 58 report of subtask 3, part 2: Thermal performance characterisation using time series data -statistical guidelines[END_REF].

Inside the dynamic methods, two subgroups can be distinguished according to the protocol requirements for building vacancy. As the envelope thermal performance is related to ϕ tr and ϕ inf (equation 3), the other heat flows should be controlled or estimated during a test to avoid biasing the HTC and HLC estimation. The first strategy is to apply the method in a vacant building. This allows to minimise the other heat flows through protocol requirements, such as closing shutters (reducing ϕ sol ), stopping the HVAC systems (controlling then ϕ h and ϕ v ) and avoiding the internal gains from occupants and appliances (ϕ int ). Since the thermal flows not related to the building envelope are neutralized, the number of boundary conditions is reduced and the modelling step is simplified. However, these are considered invasive methods, since they interfere with the normal use of the building. They are often more suitably applied immediately after construction or important retrofit actions, while buildings are not yet occupied. However, this type of protocol should be as short as possible due to the vacancy cost [START_REF] Deb | Measuring the heat transfer coefficient (HTC) in buildings: A stakeholder's survey[END_REF].

Another strategy consists in applying the protocol in an occupied building, which releases the constraint on protocol duration. However, the behaviour of occupants results in a series of complex thermal phenomena in the building during a protocol application, such as windows opening, use and control of HVAC system, use of electrical appliances and lightening, etc. There is a need for more instrumentation to evaluate the relevant parameters related to these heat flows, and their models might be more complex. Further studies on this strategy have been conducted in the framework of the freshly concluded Annex 71, from the International Energy Agency, "Building Energy Performance Assessment Based on In-situ Measurements", and related works [START_REF] Bauwens | Annex 71 final report -Building energy performance assessment based on in-situ measurements: Physical parameter identification[END_REF][START_REF] Rasmussen | Data-driven Methods for Reliable Energy Performance Characterisation of Occupied Buildings[END_REF][START_REF] Zhang | Time-dependent solar aperture estimation of a building: Comparing grey-box and white-box approaches[END_REF]). In addition, in occupied buildings, it is harder to defend the hypothesis of a homogeneous zone temperature presented in equation 3 [START_REF] Senave | Sensitivity of characterizing the heat loss coefficient through on-board monitoring: A case study analysis[END_REF]), since occupant actions can affect at any time the temperature gradient inside the building.

Invasive methods allow a higher control of the building state during a protocol application, simplifying the measurement and modelling of the thermal phenomena. In addition, they are a step further in maturity, having been widely deployed by different organisations, specially in the French context. For this reason, this paper focuses on the use of such methods to assess building envelope thermal performance. Different invasive dynamic methods are available nowadays, such as ISABELE [START_REF] Bouchié | Short methodologies for in-situ assessment of the intrinsic thermal performance of the building envelope[END_REF][START_REF] Schetelat | ISABELE : a method for performance assessment at acceptance stage using bayesian calibration[END_REF][START_REF] Thébault | Refinement of the ISABELE method regarding uncertainty quantification and thermal dynamics modelling[END_REF], EPILOG (PACTE 2017; de Carvalho Araujo 2018), QUB [START_REF] Alzetto | QUB: A fast dynamic method for in-situ measurement of the whole building heat loss[END_REF][START_REF] Mangematin | Quick measurements of energy efficiency of buildings[END_REF]Alzetto et al. 2018a), andBacher &Madsen Bacher &[START_REF] Bacher | Identifying suitable models for the heat dynamics of buildings[END_REF].

Although the cited methods present specificities in their protocols and estimation processes, they share common principles. Figure 1 shows a general scheme to represent these methods. The protocol is the part developed in situ, consisting of equipment, requirements, scenarios and an adequate duration for acquiring enough informative quality data. In invasive methods, the building is commonly heated to increase the temperature difference with the environment and to enhance the signal of heat losses through the building envelope. Data related to indoor temperature, released power, weather conditions and others are collected during the protocol application. This data can then be pre-processed to meet the necessary format used in the estimation process, where modelling algorithms are applied. These methods are used to solve inverse problems, where the physical phenomena should be initially modelled, describing how the parameters of the model translate into experimentally observable effects. Then the phenomena take place in reality and the approach consists in tuning the parameters so that the model outputs are similar to the experimental data [START_REF] Xu | Solving inverse problems in stochastic models using deep neural networks and adversarial training[END_REF]. The modelling part can be composed of one or many complementary models, considering a model as "a representation of structure in a physical system and/or its properties" [START_REF] Hestenes | Modeling methodology for physics teachers[END_REF]). Once a model is defined, its structure and implications can be analysed to extract valuable information, this process is known as model-based inference [START_REF] Ford | Modeling the Environment[END_REF]. The resolution algorithms can vary among the modelling techniques, as for the static methods they are the inclination of a line defined by aggregated data [START_REF] Roels | On site characterisation of the overall heat loss coefficient: Comparison of different assessment methods by a blind validation exercise on a round robin test box[END_REF][START_REF] Uriarte | Mathematical development of an average method for estimating the reduction of the heat loss coefficient of an energetically retrofitted occupied office building[END_REF] or by linear regression [START_REF] Zayane | Identification d'un modèle de comportement thermique de bâtiment à partir de sa courbe de charge[END_REF][START_REF] Bauwens | Co-heating test: A state-of-the-art[END_REF]. Optimisation algorithms are a special type of resolution algorithm, where the parameters are optimised based on a criterion, such as the ordinary least-squares (de Carvalho Araujo 2018) and the maximum likelihood [START_REF] Roels | Characterising the actual thermal performance of buildings: Current results of common exercises performed in the framework of the IEA EBC annex 58-project[END_REF][START_REF] Thébault | Contribution à l'évaluation in situ des performances d'isolation thermique de l'enveloppe des bâtiments[END_REF], which are commonly used for dynamic methods. Once the model parameters are estimated, a post-treatment can be applied to calculate the final indicator, such as the HTC or the HLC.

Although different methods are available to assess building envelope thermal performance, most of them have been conceived to be applied to single-family houses. In the case of larger buildings, such as multi-family housing, the scale of the tested area can impose major difficulties to the application of methods requiring extensive building instrumentation. Some simplified methods, such as the energy signature [START_REF] Nordström | Using the energy signature method to estimate the effective U-Value of buildings[END_REF]) static method, have already been applied to larger buildings. As during protocol application the building envelope is not under steady-state regime, their total measurement duration is longer, ranging from several weeks to several years [START_REF] Madsen | Annex 58 report of subtask 3, part 2: Thermal performance characterisation using time series data -statistical guidelines[END_REF], which can be a constraint when fast validation of construction quality is desired.

The typology composition of residential building stock varies among the EU member states, but in average multi-family housing (MFH) accounts for 50 % of the residential building stock in Europe (European Comission 2018). In France, 44 % of the residential building stock is composed of MFH (ADEME 2013). MFH represents an expressive share of building stock and therefore presents a relevant potential for energy savings. This reinforces the importance of assuring the envelope thermal performance among those building typologies.

Presently, there are no mature fast duration methods adapted to large building typologies, which presents a potential for developing new methods to answer this demand. Considering the extra complexities related to MFH size, a protocol applicable to vacant buildings was developed, thus avoiding secondary thermal phenomena during the test. Invasive methods must be as fast as possible to limit vacancy costs. RC network thermal models are strong candidates to develop a short measurement time method, since they take into account the thermal dynamic phenomena related to the building thermal mass. Among the proposed methods using RC models, EPILOG and ISABELE methods present flexibility regarding the model order and season of protocol application. SEREINE method is an evolution of these methods, presenting characteristics from both. ISABELE and SEREINE methods present a methodology to estimate the uncertainties in the final indicator, giving information on the reliability of the results. Nevertheless, ISABELE was conceived for single-family house applications and SEREINE method is still under development. In order to make them applicable to MFH, it is thus necessary to adapt the method protocol and estimation process.

One of the main challenges facing large building typologies regards their dimensions and the consequent difficulties to apply an in-situ protocol. One approach could be to instrument the whole building to apply the protocol to the whole envelope. This approach has been studied in [START_REF] De Carvalho Araujo | Measurement of the building envelope thermal performance in collective housings[END_REF], showing good reliability when applied to a midrise MFH. However, its application is limited to cases where the whole building is available and vacant for testing. The volume of equipment necessary for a test can also constrain the application, depending on the heating system available in the building and the heating power required for the test.

An alternative to verify the envelope thermal performance of large buildings is to partially test the building envelope, with a sampling approach. In this approach, sampled apartments in the building are tested, and the tested area has then similar dimensions to single-family houses, which facilitates the protocol application.

This paper shows one strategy to adapt the SEREINE method to the context of large buildings, using a sampling approach to assess the HTC of building apartments. To verify the reliability of the method with a referential indicator, this paper focuses on the method applied through building simulations.

Section 2 presents the sampling approach in the context of the SEREINE method and the scientific tools used to study it with building simulations. Section 3 concerns the building model and the virtual environment used to apply the method. In section 4, the experimental plans used to study the quality of the method are presented and the results are discussed. Finally, in section 5 the main conclusions are drawn and recommendations are given to ensure the reliability of the method.

Materials and Methods

To meet the challenge of assessing of the envelope thermal performance of large buildings, an adaptation of the SEREINE method is suggested with the use of a sampling approach. Part 2.1 presents the sampling approach and its main challenges. In part 2.2, the principles of the SEREINE method are presented, highlighting the features of the sampling approach. Part 2.3 shows the tools used to qualify the developed method in a simulation environment.

Sampling approach

The sampling approach consists of the thermal performance assessment of parts of the building envelope. Figure 2 illustrates the floor of a MFH, in which the protocol is applied to the southeast apartment. The measured perimeter, is composed of a part of the building envelope (in red), shared walls with the common area (in green) and with adjacent apartments (in yellow). Depending on which floor the tested apartment is situated in the building, it can have either the envelope or shared walls above and underneath it. Here, shared walls design the walls, ceiling and floor facing an adjacent area different from the exterior environment. Shared walls facing heated areas are seldom insulated for thermal purposes, although they might have a thin layer of acoustic insulation in recent buildings. Anyway, the thermal transmittance (U-value in W m -2 K -1 ) of these walls would be comparatively higher than the exterior walls in high-performance buildings. For this reason, the heat flow density passing through the shared walls (ϕ shar in W ) can be significant during a protocol application, depending on the temperatures applied on both sides of the wall. Since the objective of the method is to characterise the envelope thermal performance, if ϕ shar is not properly treated by the method, it can influence the HTC estimation. The protocol of the sampling approach should be developed to estimate or limit these heat flows to ensure the good quality of the HTC results.

In this work, the estimation of ϕ shar (equation 4), is based on the shared walls surfaces (A adj ) and U-values (U sharm ) and the temperature on both sides of the walls (T adj and T i ). The indices m and l stand respectively for the different shared walls in the tested area and the parts of a wall with different thermal properties.

ϕ shar = s m=1 r l=1 U sharm l A adjm l × (T adjm -T i ) (4)
2.2. SEREINE analysis method (for sampling approach) SEREINE analysis tool is a Python software developed for the purposes of the project with the same name, from the French PROFEEL program. Although this tool is still in development, it is similar to the ISABELE and EPILOG methods that led to encouraging measurement results for the HTC of newly built SFH [START_REF] Thébault | Refinement of the ISABELE method regarding uncertainty quantification and thermal dynamics modelling[END_REF]Boisson & Bouchié 2014;[START_REF] De Carvalho Araujo | Study of a methodology for measuring the energy performance of building[END_REF]. In this section, an analysis method using SEREINE tool will be described specifically for the HTC measurement on MFH with the sampling approach. The following subsections describe the overall analysis method.

Thermal models

In this study, two first-order lumped RC thermal models were used for calibration, respectively ti and tw. Figure 3 presents these models, where the parameters in red are estimated by an optimisation algorithm, the parameters in blue are measured and the parameters in yellow are estimated through other models.

In both models, the output variable is the indoor air temperature (Ti). The equivalent outdoor temperature (T eq ) represents the data collected by special sensors developed in Isabele method with a white and a black plate, called SENS sensors. The resistances are modelled as equivalent transmittances, H tr , H e and H i represent the thermal losses through the envelope. C i and C w represent the lumped heat capacity of the air and of the thermal mass and P h stands for the heat power delivered during the test.

Heat losses due to infiltrations (P inf ) and their uncertainties can also be estimated by means of a simplified aeraulic model from the EN 16798-7 (method 1) [START_REF] Standard | NF EN 16798-7 : 2017 -Energy performance of buildings -Ventilation for buildings -Part 7 : calculation methods for the determination of air flow rates in buildings including infiltration (Modules M5-5)[END_REF]) that uses blowerdoor and windspeed measurements, main building dimensions and outdoor pressure coefficients to calculate the air flow rate [START_REF] Thébault | Estimating infiltration losses for in-situ measurements of the building envelope thermal performance[END_REF][START_REF] Thebault | Cost-effective air flow rate estimations using blowerdoor and wind speed measurements to assess building envelope thermal performances[END_REF]. To take into account the heat transfer through shared walls, which is not part of the measured perimeter (external envelope), a third heat flux is added to the internal space: ϕ shar . This heat flow should be estimated to avoid impacting the HTC estimate, which refers only to thermal losses through the envelope.

Optimisation algorithm

Similar to the refined ISABELE method [START_REF] Thébault | Refinement of the ISABELE method regarding uncertainty quantification and thermal dynamics modelling[END_REF], the estimated capacitances and resistances (and then, the transmission heat loss coefficient) are calculated employing maximum a posteriori probability (MAP) estimate using a bayesian approach with stochastic state space modelling of the RC models (performed by pySIP algorithm [START_REF] Raillon | pySIP: an open-source tool for bayesian inference and prediction of heat transfer in buildings[END_REF]). In this process, model estimations of T i are fitted to the measured variable. The likelihood ratio test then performs the model selection between models ti and tw.

Uncertainty quantification

To consider all sources of uncertainty (input data, output data and model) with both random and systematic components, a hybrid method was developed by combining the pySIP toolbox with a quasi-Monte Carlo method. The corresponding procedure, illustrated in Figure 4, is detailed:

1. Systematic uncertainties are defined, with corresponding probability density functions; 2. Multiple biases are sampled from these and added to the measurement data (using Sobol sampling); 3. Each sample is processed by the pySIP tool, calculating the fitted parameters covariance matrix by MAP estimates. Gaussian HTC estimates are calculated from the covariance matrix; 4. All Gaussian estimates are summed, forming a new probability density function; 5. A 95 % confidence interval is calculated.

A convergence criterion based on the overall variance variation was set to stop the propagation.

The probability densities for systematic uncertainties are considered to be Gaussian in this study, the bounds represent 1.96 times the standard deviation. Two important uncertainty sources were considered for the estimation of ϕ shar :

• A potential offset to the measured adjacent temperatures behind the shared walls T adj (set to ± 0.5 K).

• A bias on the estimated U-values on shared walls U shar (set to ± 20 %, considering that the composition of shared walls is nearly known).

The other uncertainties were applied as usual in the SEREINE method. Errors due to the estimation of outdoor equivalent temperature T eq were grouped into a global systematic uncertainty of ± 0.5 K. Errors due to the fixed height of measurement and insufficient air mixing of the indoor environment, associated with T i , were of ± 0.5 K. Errors in the measurements of P h were considered as ± 2 %.

Quality indicators

Virtual models provide a reference value for HTC (HT C ref ), enabling the use of objective criteria to analyse the quality of the method outputs. The bias (equation 5) represents the relative difference of the estimated HTC (H T C) to the reference. The closer to zero, the more accurate result.

bias

= HT C ref -H T C HT C ref (5)
Besides a low bias value, the estimation output should present a limited level of uncertainty to have more precise results. The uncertainty of the indicator is represented as the percentage difference between the middle point and the borders of the result 95 % confidence interval (I95). Different from the bias, the uncertainty does not take into account the reference value, it is only relative to the estimated indicator and the dispersion of its probability density. A threshold of 15 % in the bias and 35 % in the uncertainty were considered for defining acceptable results. The combination of these thresholds has the advantage to separate the results into two groups, acceptable and unacceptable, and facilitate its analysis. However, this classification does not give information on the level of the quality of the results. To better take into account the nuances of the results, interpretability [START_REF] Juricic | Influence of natural weather variability on the thermal characterisation of a building envelope[END_REF] was also used as a quality indicator. This represents the intersection of an interval centred in the reference value and the probability distribution of the estimated result. The interpretability is mathematically expressed in equation 6, where f(x) is the result probability density function of HTC and i is the interval range that is been analysed. The interpretability can assume continuous values between 0 and 1, since the maximum area under the probability density function is equal to 1.

interpretability = HT C ref (1+i) HT C ref (1-i) f (x) dx (6)
Figure 5 shows a graphic representation of the interpretability indicator for a ± 15 % interval of two different tests applied in the same building. The interpretability for each is equal to the value of "b" minus "a" in their cumulative distribution function of the result. It can be seen that both results are acceptable, but the result on the right has better quality with a narrower uncertainty associated with the probability density function. This behaviour is reflected in the interpretability of the results, which associate better quality results with high values with a maximum of 1.

The interpretability indicator has the asset of showing the nuances of quality in the results through a continuous variable. However, when dealing with multiple results, it might be useful to choose a threshold value for this variable to understand better the method behaviour under different conditions. The combination of both criteria can be applied using first the results from the acceptability to define the group of acceptable results. Finally, a minimum value of interpretability could be perceived in this group to represent the threshold for this continuous variable.

Description of the case study

Dynamic building simulations were used to apply variations of the sampling method in a virtual building to study the method's validity and limitations. An advantage of using a virtual environment is the possibility of trying the method in different configurations in a more cost and time-effective manner than for real buildings. When adapting a method to a new domain, we should resist drawing conclusions from one single simulation. Although a single simulation can provide a starting point for comparison, it seldom allows a full comprehension of a system's natural tendencies. Model experimentation is preferably done in an exploratory way by trying different simulations to help to understand the trends of a system [START_REF] Ford | Modeling the Environment[END_REF].

By comparing multiple simulations, the method limits can be studied, and insights can be given on its behaviour with a specific model. For this reason, multiple experimental plans were performed in an exploratory manner, to see the behaviour of the method when the protocol or the estimation process is modified. The experimental plans present variations in the protocol temperatures and the weather conditions. The idea is both to develop the method and to study its validity domain.

Simulation software

Software for dynamic building simulation are white-box modelling tools, where very detailed information on building physical properties are combined with descriptive equations to simulate the building's thermal behaviour. Different software are available for these goals and there are recognised methodologies, such as BESTEST -which is based on ASHRAE standards [START_REF] Berrabah | Towards a quadrupole-based method for buildings simulation: Validation with ASHRAE 140 standard[END_REF][START_REF] Soubdhan | Chapter 376 -use of BESTEST procedure to improve a building thermal simulation program[END_REF] -and others (Østergaard Jensen 1995), to validate their capacity to accurately simulate reality.

The virtual experiments performed in this study were simulated with the software PLEIADES and COMFIE. They were created at the beginning of the 90s' [START_REF] Peuportier | Simulation tool with its expert interface for the thermal design of multizone buildings[END_REF] in the present ETB group (Eco-design and thermics of buildings) of the CES (Center for Energy Efficiency of Systems) part of the Mines ParisTech research center (ARMINES 2021). PLEIADES is the software used for creating digital models and COMFIE is used for dynamic thermal simulations. In this work, we call the ensemble of both as "P+C".

This software group has been maintained by the editor IZUBA Energies (IZUBA 2020) and refined through different initiatives [START_REF] Recht | Méthodologie pour la comparaison des performances simulées et mesurées de maisons ≪ à énergie positive ≫[END_REF]) over the years. It is broadly used in the French BEP context and have been already verified and validated [START_REF] Salomon | Outil de simulation thermique du bâtiment, comfie. Journée thématique SFT-IBPSA[END_REF][START_REF] Brun | Behavioural comparison of some predictive tools used in a low-energy building[END_REF][START_REF] Spitz | Analyse de la fiabilité des outils de simulation et des incertitudes de métrologie appliquée à l'efficacité énergétique des bâtiments[END_REF]). An empirical validation process has also been applied to low-energy buildings, which are more sensible to minor heat gains [START_REF] Munaretto | Empirical validation of different internal superficial heat transfer models on a full-scale passive house[END_REF]. It is considered a reliable tool for simulating the SEREINE method application in a renovated building model.

Model

A PLEIADES model of a mid-rise MFH with 16 apartments was studied in this work. This model was based on a real four-storey building from a French city close to Lyon. The original building was built in 1978 and has undergone a massive retrofit in 2012. The energy consumption was initially classified from the French energy performance certificate as class E (231 to 330 kWh/m²/year); after retrofitting it became class B (51 to 90 kWh/m²/year) (ALEC 2020).

Regarding the PLEIADES model, its structure and the type of materials were kept the same as the original one. However, the level and position of the insulation and the thermal bridge values were modified to correspond to a high-performance building envelope in the present standards for renovation. The building model has 16 floors, with a total of 1048 m² of living area and 1508 m² of envelope area.

For the components of the building, the material properties (such as λ, C and ρ) proposed by the standard library of P+C were used. The wall compositions are based on references for high-performance retrofitted buildings (ADEME 2018). In the beginning, the values of the thermal bridges were kept from the original model, with optimistic values that could be thought of for a new building. Later, the prescriptions from the French thermal regulation for existing buildings (CSTB 2007) were used to define these values, according to each type of insulation, thickness and thermal resistance of components. The details on the building components' thermal properties are presented in Appendix A.

Tested apartment

The virtual tests are carried out on a corner apartment on the top floor (apartment 15 in figure 6), with two exposures, east and south. In this location, the proportion between exterior and shared walls is equal to one, the maximum normally reached when testing a single apartment in a building. This proportion plays an important role since the better test conditions are related to a high ratio between the heat flow towards the exterior and that towards the neighbours. The apartment has a 63 m² floor area, 107 m² of exterior walls and an equivalent area of shared walls. The shared walls give to four different areas in the building: one unheated zone (UZ) located in the centre, two apartments in the same floor (14 and 16) and one apartment on the second floor (11). Equation 7describes the heat flow passing through shared walls in an instant t. It should be noticed that the ϕ shar is considered as a steady state heat flow.

ϕ shar;t = U shar,hor S 11 (T 15;t -T 11;t )+U shar,vert (S 14 (T 15;t -T 14;t ) + S 16 (T 15;t -T 16;t ) + S U Z (T 15;t -T U Z;t )) (7)

Reference indicators

A very important asset of virtual method applications is the possibility to have a known value for the final indicator. In a virtual test, the environment can simulate both real and controlled conditions. The combination of a controlled heating scenario with a stable outdoor conditions allows the building reaching a thermal steady state, and therefore enables the calculation of a reference value for HTC and HLC.

An infiltration model has already been developed in the SEREINE method [START_REF] Thébault | Estimating infiltration losses for in-situ measurements of the building envelope thermal performance[END_REF][START_REF] Thebault | Cost-effective air flow rate estimations using blowerdoor and wind speed measurements to assess building envelope thermal performances[END_REF] and this paper is focused on developing a sampling approach, where ϕ shar brings the major challenges. Since the use of a realistic infiltration scenario would not add extra information about the influence of ϕ shar on the method, this was simplified and P inf was neglected, therefore HTC and HLC are equivalent. For this reason, only the HTC indicator is further analysed in this work.

In order to calculate the reference value of HTC, HT C ref , the outdoor environment and the adjacent spaces were set at the same constant temperature and solar radiation was set equal to zero. The ratio between the power delivered and the temperature difference gives the HTC value. The HT C ref of the part of the envelope associated with this apartment is of 72.5 W.K -1 . This value is used as a reference throughout this work.

Having an absolute HT C ref is only feasible with building models since it is impossible to control the weather conditions to achieve steady-state in a real building, which brings the main advantage of developing a method with virtual protocol applications. This enables the verification of the outcomes from the different tested strategies and gives a powerful tool for quality control.

Dynamic energy simulations

All the simulations were performed using conventional weather files from the French Thermal Regulation 2012 (RT2012), corresponding to French cities such as Trappes, Nancy, La Rochelle and Nice. Outdoor equivalent temperature (T eq ) was calculated to make the simulation closer to the methods' in situ applications. The calculation of T eq was simplified by reducing the term related to the form factor, as in equation 8.

T eq = T out + α s × I s h c + h r (8)
The absorptivity coefficient of the exterior wall α s is set at 0.6, the same as the PLEIADES model exterior walls. The solar irradiance (I s ) was calculated for each direction and inclination of exterior walls and roof. In COMFIE algorithms documentation manual (IZUBA 2014), radiant and convective heat transfer coefficients (h c and h r ) are combined into a global coefficient that depends on the surface orientation, its emissivities and wind exposure. The values of these coefficients were based on the COMFIE manual and can be found in Appendix A.

Results and Discussion

The sampling method was applied to the tested apartment with variations in the simulation parameters. The experimental plan, presented in part 4.1, intended to improve the estimation method through the variation in the protocol temperatures. The second study, in part 4.2, was used to verify the performance of the method under different weather conditions, throughout the year and for different cities in France.

Effect of test temperature and preheating

An experimental plan was built to investigate the advantages of preheating the apartment before the protocol application. A two-week fixed-temperature test period was used before testing the variants with preheating. Even when no preheating was applied, the apartment presented an initial test temperature of 15.8 °C, which was due to heat exchanges with the neighbours that were heated throughout the year. The temperature difference between the beginning and the end of the test (∆ T beg-end ) was also studied, as it could be an influential parameter on the quality of the test results. For this reason, different setpoint temperatures were used applying the same ∆ T beg-end of 0 and 4 Kelvins. Its values, as well as those to which the tested housing is heated during the protocol, are given in table 1. All these tests were performed under winter conditions, with the previously cited weather station of Nancy starting on the 15th of January. The temperature of the adjacent spaces was variable over the day, considering that the adjacent apartments were occupied during the test and that their temperatures could not be controlled. The scenario chosen to represent the temperature in the apartments is cyclical over a period of one day. Figure 7 shows the evolution of the setpoint temperature of the apartments that are adjacent to the tested area (number 11, 14 and 16) and of all other apartments in the building that are not adjacent to the tested area. In this scenario, one of the neighbours keeps a constant temperature over time and two of them vary the setpoint temperature between night and day. The unheated zone in the middle does not have any temperature setting. However, it is heated by the surrounding apartments. The equivalent outdoor temperature takes into account the solar gains per building facade. Figure 8 presents the solar radiation for each direction (G S, G E, G W , G N ), the outdoor temperature (Text) and its equivalent temperatures per facade (T eq S, T eq E, T eq W , T eq N ) over the test duration. The twelve variants were analysed with SEREINE algorithms using a test duration from half a day to four days. The results were considered acceptable if they presented a bias inferior to 15 % and an uncertainty inferior to 35 %. Table 2 shows the statistics on the interpretability indicator, inner 15 % interval, for the results considered acceptable and unacceptable. It can be seen that just one-third of variants were acceptable with a very short duration of half day. For this short duration also, the variants diverged, while for longer durations all tests converged. With the increment of test duration, more results became acceptable and from three days all twelve variants were acceptable. Regarding the interpretability of the results, 0.5 is the maximum interpretability value presented among the unacceptable results. Even though an interpretability of 0.4 can be found among the acceptable results, the value of 0.5 is going to be used as a threshold of result quality in this study. Figure 9 presents the interpretability of the 12 variants according to their setpoint temperature and ∆ T beg-end . Only the diagonal of the area without preheating was tested. The areas in green are those not tested and those in grey did not converge. The blue areas correspond to the tests with acceptable results and the red areas to those with unacceptable results. The areas with light colour are tests with interpretability close to 0.5. The higher the interpretability of a test the better is its result.

For very short durations, 0.5 to 1 day most results were already acceptable if no ∆ T beg-end was applied. It indicates that preheating the building at the same temperature as the test can be a good strategy. This is mainly coherent during winter conditions since the test setpoint temperature can be close to that considered comfortable for occupants.

However, for 0.5 days, even without ∆ T beg-end , the variant with 30 °C test temperature diverged. This test temperature is also the one furthest from the adjacent spaces' temperatures (T adj ). Considering the heated zones, T adj is on average 17 °C during the night and 20 °C during the day, with a global mean of 18.8 °C. Although the variants with the highest setpoint temperature present an important signal towards the outside, they are also those with higher φ shar and the test conditions are therefore less optimal.

An unexpected result is presented for the test duration of 2 days: the extremes of setpoint temperature had better results. Maybe the contribution of the increased heat flow towards the exterior compensated the effect of the φ shar estimation errors. From 3 days, all results were acceptable and those with zero ∆ T beg-end and low ∆ T int-adj presented high interpretability values. The results for 4 days are not presented, since it has a similar behaviour to that of 3 days test duration.

In general, preheating the building at the same temperature as the test seems to be a good strategy to decrease the vacancy period of the test application, considering the place can be heated for one week before the test beginning with the local heating system. Under this condition and with test temperatures close to the T adj , the method performed well even for very short durations, such as half a day. These conclusions are drawn for winter conditions and may be different when the outside temperature is higher.

Effect of the weather

Three main experimental plans were developed, named A, B and C, to study the method behaviour over the year and at different locations in France. They were carried out in four French cities located in different climate zones: La Rochelle, Nancy, Nice and Trappes. In each experimental plan, one test was conducted per city and per month of the year. All tests started at 8 p.m on the 15th day of the month tested.

All tests followed a 2-week preheating period with a temperature as the setpoint temperature. In this way, a zero ∆ T beg-end was used in all these tests, since it related to the most favourable test condition, observed in subsection 4.1. The incident solar radiation on each of the building facades is extracted from the software and used in the SEREINE estimation algorithm. A time step of five minutes was used for the estimation process of all tests.

The experimental plan A was performed initially, to represent a protocol where the neighbours do not have their temperature controlled, bringing fewer constraints in the method application. The study presented good results for the winter period, however, tests applied during the mid-season and summer presented low-quality results. As an attempt to improve the estimation for other periods than winter time, a more invasive protocol was tried in experimental plan B. In this case, the adjacent spaces are heated at the same temperature as the tested area, except for the unheated zone. This condition is expected to improve the results since it decreases the φ shar level, however, a level of heat flow towards the unheated zone still exists.

A second attempt to improve the method behaviour during warmer seasons was to allow higher values of indoor temperature. The experimental plan C was thus performed with variable temperatures of the neighbours, but with indoor temperatures limited to a higher temperature of 35 °C. Also, a higher temperature difference with the exterior (∆ T int-ext ) was used to determine the test temperature: instead of adding 10 K to the monthly mean outdoor temperature, a ∆ T int-ext of 15 K was used, based on the mean outdoor temperature of the four days following the beginning of the test.

The exact criteria used for defining the indoor temperature (T int ) and the adjacent space's temperature (T adj ) in each of the three experimental plans are presented below. The difference between these two temperatures is noted ∆ T int-adj . Figure 10 shows the temperatures used for each test applied in all three experimental plans. -T adj follows a variation pattern, different among neighbours and between day and night (figure 7).

• Experimental plan B -T int is the same as experimental plan A;

-T adj = T int .

• Experimental plan C

-T int = T extmean 4days + 15 K, within the interval of 20 °C and 35 °C;

-T adj is variable, same as experimental plan A (figure 7).

The model selection process from the SEREINE algorithm was initially used, in which the model noted ti was more commonly chosen. However, the model selection process was still in development in the SEREINE project, while these analyses were conducted. So finally, both models, ti and tw, were applied to each case, disregarding the model selection process. The model tw presented more convergent results and, among them, there is a higher percentage of results that finished the uncertainty propagation process. The results for a same test also presented a higher interpretability value compared to results using ti model, with an average of 11 % higher values. In addition, it has a larger number of acceptable results for 2 and 3 days duration, with 66 acceptable results against 26 of model ti. For this reason, the following results presented here are solely for the estimation process with the model tw.

In experimental plan A, 10 % of the tests diverged while, in experimental plan B, this level decreased to 4 %. For experimental plans C, all the tests converged. Although a test might converge, it does not imply that the uncertainty propagation process has been correctly performed. Whenever a test does not complete this process, the result is not going to be further analysed. This choice was taken to avoid misinterpretation by comparing results with different levels of information. The first has a higher level of uncertainty since they have passed through the uncertainty propagation process. The results that did not finish this process have an uncertainty close to a single fit in the optimisation process, which is much smaller.

Once a result converged and went through the uncertainty propagation step, it was studied regarding the acceptability and interpretability criteria. Most of the results presented a stabilisation of the values from 2 days of test duration, some of them presented stable behaviour earlier with 1 day of test duration. To verify this, almost half of the tests in the experimental plans were conducted for up to one week of duration. Figure 11 shows the mean interpretability value for these tests with total duration of 7 days. It can be seen that from half of a day to one day there is an important improvement in the interpretability values, among all groups. From one to two days the interpretability still shows a significant increase. After that, although some improvements can be reached, they are probably not enough to justify an extra day of protocol application in situ. It should be considered that this duration depends on the context. In this study, the building has internal wall insulation and the protocol requires preheating the tested area. The duration could be longer for cases with external wall insulation and without preheating. An example of a test with acceptable results is shown in figure 12.

The whole experimental plan was conducted with at least three days of test duration since it seems to be enough to stabilise the test results. A total of 144 results is expected per test duration in the ensemble of the three experimental plans. From these, only 25 % of the tests converged and had the uncertainty propagation process completed when the protocol was applied for half a day. This value increases to 70 % for a test duration of one day, and to 85 % for a test duration of two and three days. Considering this, a really short duration of half a day should be avoided for a test application when similar experimental conditions are encountered in situ. One day of test duration can be considered, but the ideal would be to test a building for at least two days. The maximum interpretability was 85 % with Nancy's weather in January, which is the month with the lower external temperatures during the test. The city of Nice presented the lower amount of acceptable tests, and it is the city with the highest average outdoor temperatures. The external temperatures seem to be a determinant factor in the quality of the results, which is an expected behaviour, since it affects ϕ ext levels. It can be seen that applying the test during winter improves the probability of achieving higher interpretability values and having acceptable results. Although the attempts of heating the adjacent spaces and increasing the indoor temperature improved the interpretability of the results, these strategies were not significant enough to reach acceptable results out of the winter season, with few exceptions.

Another tendency observed is that it was more challenging to reach the uncertainty criterion than the bias. The bias criterion was commonly met in tests performed during the mid-season and even during some summer months. However, the uncertainty criterion was mainly met during the colder months. The increased level of uncertainty is probably due to the increased level of input uncertainty in the U-value of the shared walls. In addition, this building presents fairly good insulation on the exterior walls and no insulation on the shared walls, which can lead to an unfavourable test condition, where ϕ shar has a relatively high magnitude compared to ϕ ext .

In figure 13, the results are ordered by month of application, where a trend of convergence and interpretability level can be seen according to the seasons. However, the days on which the tests were performed might have a hotter or colder outdoor temperature than the tested month average. To better understand the method behaviour, this data was analysed according to the temperature differences with the outdoor environment and adjacent spaces (figures 14 and 15). Besides the colour describing the interpretability level, the shape indi-cates the ranges of outdoor temperatures during the test. The triangular points represent cold weather conditions, in which the mean outdoor temperature is below 6 °C. The circles and stars represent respectively mild and hot weather conditions.

The uncertainty and bias of the same tests are presented in figure 14. It can be seen that a bias inferior to 15 % is often met when ∆T int-ext is higher enough. At least 12 K of temperature difference would be desirable for achieving this criterion. The uncertainty criterion is more restrictive and it is mainly determining which results are acceptable. It presents a similar behaviour to the interpretability, with better results for high ∆ T int-ext values. At least 15 K of temperature difference would be desirable to let the uncertainty inside the acceptable zone in addition to low temperature differences with the outdoor environment. The group of points sharing the same colour have a general diagonal shape, showing that both differences in temperature influence the quality of the results. It can be seen that lower differences in temperature with the neighbours and higher differences with the outdoor environment provide higher levels of interpretability, which reinforces the importance of having more heat flow passing through the envelope than the shared walls.

Increasing the test temperature is a strategy to consider to improve the results, however, it can also increase the ϕ shar if the adjacent spaces are not heated similarly. Since the shared walls are usually less insulated than the exterior walls, a lower temperature difference could lead to important heat losses to the adjacent spaces. The effectiveness of the increment of the test temperature thus depends on the temperature of the adjacent spaces. During the heating season, it is less relevant since an important temperature gradient with the outdoor environment can be reached with temperatures inside the thermal comfort range. If adjacent spaces are occupied, adjacent temperatures may be close to that of the tested area, leading to a low level of ϕ shar .

In this case, there would be no need to control the adjacent spaces' temperatures, which avoids the application of a more invasive protocol. However, during mid-season, the T ext increases and temperatures inside the thermal comfort zone might not provide an enough high ∆ T int-ext to test. To compensate for the higher external temperatures, T int should be increased to improve the heat signal passing through the building envelope. However, this will possibly increase ∆ T int-adj at the same time, which is associated with less favourable testing conditions. This compensation gets critical during summer when the level of indoor temperature demanded to have an important ∆ T int-ext can exceed the safety of materials in a dwelling.

For mid-season, a possibility would be the use of both strategies together: increasing the test temperature and controlling the adjacent spaces' temperatures close to the test temperature. It can be seen in figure 15 that the bottom right zone of the graph, with higher interpretability, is only composed of tests performed with mean outdoor temperature below 6 °C. There are no points representing the mid-season months in this area. In experimental plan B, the test temperatures were limited to 25 °C. It would be interesting to perform an experiment with higher temperatures and controlled neighbour temperatures to verify the improvement it could bring for mild temperature weather.

Conclusion

Numerical simulations were used to investigate a sampling approach for estimating a building HTC. The variation in indoor temperature before and during the test application showed that a protocol with preheating reaches stable results faster. Preheating the building to temperatures similar to that of the test setpoint is recommended to decrease the vacancy time for protocol application.

Forty-eight different weather conditions were used with three variations in the protocol parameters related to indoor, outdoor and neighbour temperatures. The simulations in which the heat flow delivered during the test traversed mainly the envelope, instead of the shared walls, presented more precise results. For this reason, higher temperature differences with the exterior, which were mainly reached during winter, led to less uncertain results. Decreasing the temperature differences with the neighbours, slightly improved the HTC accuracy, but did not yield acceptable results under mild and hot weather. Overall, the uncertainty criterion was the most challenging aspect for reaching acceptable HTC estimations, while the bias criterion was commonly reached disregarding weather conditions.

The proposed method is viable in winter from two days of measurement, favouring a test setpoint temperature equal to the preheating. All the simulations had a controlled and stable preheating temperature, which facilitates the short-duration HTC estimation. In real life, this condition might be difficult to reach, since the preheating would be applied by occupants with the local heating system and not the test equipment. If preheating is not applied, a longer test duration will be recommended to ensure acceptable results.

As a perspective of this work, other ways of taking into account the heat flow going towards neighbours could be considered. The use of direct measurements on the shared walls is a strategy that started to be studied in the context of the SEREINE project. Another possibility would be the integration of this parameter in the RC thermal model to be identified, imposing new boundary conditions to the system. This work could also be extended by the application of the method to buildings with different architectures and construction techniques.

Figure 1 :

 1 Figure1: General scheme of the sub-steps of a method to assess building envelope thermal performance with in situ measurements.

Figure 2 :

 2 Figure 2: Example of the measured perimeter and the boundaries in a sampling approach method application.

Figure 3 :

 3 Figure 3: RC models used for calibration : ti (left) and tw (right)

Figure 4 :

 4 Figure 4: Illustration of the uncertainty propagation procedure, adapted from (Thébault. & Bouchié 2018).

Figure 5 :

 5 Figure 5: Representation of the interpretability indicator with the cumulative distribution function and the probability density function of the two test results applied in the same building.

Figure 6 :

 6 Figure 6: Internal view of the building model top floor, with floor of apartment n°15 in red.

Figure 7 :

 7 Figure 7: Setpoint temperatures for the scenario with variation in the neighbours' temperature

Figure 8 :

 8 Figure 8: Solar radiation per facade (top) and outdoor and equivalent outdoor temperatures (down) during the virtual protocol application of the second experimental plan.

Figure 9 :

 9 Figure 9: Interpretability of the results for different protocol setpoint temperatures, preheating conditions and test duration.

Figure 10 :

 10 Figure 10: Tint for experimental plans A, B, C for the four different weather.

Figure 11 :

 11 Figure 11: Mean interpretability for tests with maximum duration of 7 days by results acceptability

Figure 12 :

 12 Figure 12: Results progression for the numerical test performed in February in La Rochelle with experimental plan B.

Figure 13

 13 Figure 13 presents the results of interpretability for the tests that converged in the experimental plans A, B and C for a two days duration. The empty values correspond to tests that did not converge or complete the uncertainty propagation process, for reasons which would require further investigation. The values without highlight failed to reach the criteria

Figure 13 :

 13 Figure 13: Interpretability and acceptability for 2 days of duration

Figure 14 :

 14 Figure 14: Bias and uncertainty of the results according to the mean ∆ T int-ext and ∆ T int-adj for 2 days of test in experimental plans A, B and C.

Figure 15 :

 15 Figure 15: Interpretability of the results according to the mean ∆ T int-ext and ∆ T int-adj for 2 days of test in experimental plans A, B and C.

  Table A.3: Thermal properties of thermal bridges of the building model after retrofitting

Table 1 :

 1 Preheating, setpoint temperatures and calculated ∆ T beg-end of the experimental plan.

	Variant number Preheating temperature [°C] Setpoint temperature [°C] ∆ T beg-end [K]
	1	no preheating		2.2
	2	14	18	-14
	3	18		-18
	4	no preheating		6.2
	5	18	22	4
	6	22		0
	7	no preheating		10.2
	8	22	26	4
	9	26		0
	10	no preheating		14.2
	11	26	30	4
	12	30		0

Table 2 :

 2 Statistical description of the interpretability of the results according to acceptability and test duration.

	Interpretability \	Acceptability	Acceptable results		Unacceptable results
	statistical	\	Duration (days) 0.5	1	1.5	2	3	4	0.5	1	1.5	2
	description	\	Amount 4	9	10	10	12	12	5	3	2	2
	Mean		0.8	0.7	0.7	0.8	0.8	0.8	0.1	0.2	0.4	0.5
	Standard deviation	0.0	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
	Minimum value		0.8	0.5	0.4	0.5	0.6	0.7	0.0	0.1	0.3	0.4
	25th percentile		0.8	0.5	0.7	0.7	0.7	0.7	0.2	0.1	0.3	0.4
	50th percentile		0.8	0.6	0.8	0.8	0.8	0.8	0.2	0.2	0.4	0.5
	75th percentile		0.8	0.7	0.8	0.8	0.9	0.9	0.2	0.2	0.4	0.5
	Maximum value		0.8	0.9	0.9	0.9	0.9	0.9	0.2	0.2	0.5	0.5

Preprint submitted to Building Simulation December 25, 2022

Appendix

A. Building model characteristics

The global radiant and convective superficial heat transfer coefficients are used to calculate the walls' U-values and the equivalent outdoor temperatures. In P + C manual, this coefficient depends on the surface orientation, exposition and emissivity IZUBA (2014). The surface interior and exterior coefficients (hint and hext) associated with the different building models, were based on P + C manual considering a wall emissivity of 0.9. For exterior horizontal walls, a hext of 22.2 W/(m²/K) and a hint of 9.43 W/(m²/K) were considered, for vertical walls, these figures were respectively 18.2 W/(m²/K) and 8.13 W/(m²/K). In the case of shared walls, both sides of the walls are under interior conditions and the hint for horizontal walls was 8.00 W/(m²/K) and for vertical it was 8.13 W/(m²/K).

The composition of the walls and floors regarding thermal resistance are presented in table A.1. The model's windows have a U-value of 1.9 W/(m².K) and a solar factor of 50 %, while the doors have a U-value of 3.5 W/(m².K). The types of thermal bridges with their thermal characteristics are presented in table A.3. The graphical representation of the thermal bridges come from P+C software.


Ingrid David

Vincent Le

Tom Rohmer
email: tom.rohmer@inrae.fr

UpDown -An R Package to Identify and Characterize Disturbances from Longitudinal Observations

The R package UpDown provides an algorithm to detect and characterize (start, end dates, intensity) disturbances that can occur at different hierarchical levels by studying the dynamics of longitudinal observations at individual and group scales. An Rshiny application is also provided in UpDown in order to visualize the longitudinal data at the different hierarchical scales associated with the information related to the detected disturbances.

Introduction

Longitudinal observations organized in hierarchical groups are studied in many contexts. For instance, in educational systems, students' progress can be observed for students distributed in classrooms belonging to different schools. In market analysis, the evolving price of a property depends on its locality as well as on its city or country. In pig farming systems, animals are spread out in pens that belong to different batches. In these three examples, the observations can then be considered at different levels: individual or group scales. For instance, we may be interested in obtaining information about the evolution of the price of a given house or about the median/average price of houses in a city. The feeding behavior of animals can be individually analyzed or studied at a pen or a herd scale.

Many R packages (R Core Team 2023) and methodologies make it possible to detect individual disturbances for dynamic observations (see, for example, changepoint [START_REF] Killick | changepoint: An R package for changepoint analysis[END_REF] for parametric and non-parametric tests, npcp [START_REF] Kojadinovic | npcp: Some Nonparametric CUSUM Tests for Change-Point Detection in Possibly Multivariate Observations[END_REF] for nonparametric CUSUM tests, or npcopTest [START_REF] Rohmer | npcopTest: Non Parametric Test for Detecting Changes in the Copula[END_REF] for a non-parametric test for detecting change in the copula of multivariate observations with known changes in the marginal distributions). The most of the proposed methodologies in these packages make it possible to detect multiple disturbances.

Nevertheless, disturbances that alter observations can also occur at these different levels. They may only affect an observation (longitudinal) or a group of observations. For example, in the case of farm animals, an animal may be sick due to a disease that is not contagious. In that case, the disturbance (the disease) occurs at the individual level only. If the disease is contagious, all of the animals raised together in the same pen (same group) will be exposed to the disease, i.e., the disturbance occurs at the pen level, although not all of the animals will develop symptoms depending on their immune capacity. In this context, identifying whether a particular animal has been subjected to a disease is more effective using observations of all the animals in a pen (some animals will develop visible symptoms) than by observing only the particular animal that may not develop symptoms.

The UpDown algorithm was developed around this idea. It consists of studying observations at different scales in order to facilitate the detection and characterization of disturbances. When observations are collected repeatedly over time (longitudinal data), disturbances may be identified by observing changes in the dynamic of the observations over time. Other similarly methodologies have been proposed. For example [START_REF] Moreno-Muñoz | Change-point detection in hierarchical circadian models[END_REF] also use an approach based on a Gaussian mixture model for latent variable representation in a Bayesian context. The UpDown algorithm is able to detect and characterize disturbances that lead to an elastic or plastic response [START_REF] Sauvant | Robustesse, rusticité, flexibilité, plasticité... les nouveaux critères de qualité des animaux et des systèmes d'élevage. avant propos[END_REF], see Figure 1 for an illustration).

The algorithm captures deviations from the theoretical trajectory (the one that would have been observed in the absence of disturbances) by identifying abnormal evolutions compared to the expected one. The UpDown algorithm consists of two parts: (1) the Up-step, which identifies elements facing a disturbance from the bottom (unit) to the top level. The classification of the disturbed elements is done by fitting Gaussian mixture models; and (2) the Down-step, which validates elements detected in the Up-step, from the top level to the bottom level (unit) and that also allows us to identify elements that experience more than one disturbance (regardless of the level).

Remarks:

increase in some cases and a decrease in others).

Up-step

The Up-step consists in identifying disturbances at the different levels of the data organization based on the empirical distribution of the minimum slopes of the trajectories of the observations. To do that, firstly, for i = 1, . . . , N where N is the total number of units, Nadaraya-Watson's smoothing curves f i [START_REF] Nadaraya | On estimating regression[END_REF], [START_REF] Watson | Smooth regression analysis[END_REF] are applied to the longitudinal observations y i = (y i,1 , . . . y i,n i ), with time values t i,1 , . . . , t i,n i , where n i is the number of observations of the ith unit:

f i (x|y i ) = n i j=1 y i,j K x-t i,j h 1/2 n n i j=1 K x-t i,j h 1/2 n , i = 1, . . . , N. (1)
The chosen kernel K is the Gaussian density, and the bandwidth h n is chosen depending on the largest length of observations per unit n = max i (n i ). An excessively small value of h n may not be sufficient to correct for the noise associated with the observations leading to wrong disturbance detection. An excessively high value of h n will lead to an under-smoothing, and the algorithm might not detect the disturbances. The default choice in the function UpDown is h n = √ n. It can be modified by the user. The number of equally-spaced points over which the smoothing curves are estimated is ⌊nh n ⌋, where for a real x, ⌊x⌋ is the integer part of x. The first derivatives are obtained by differentiation on the basis of the smoothing curves, ḟi

(x|y i ) = ⌊nh n ⌋ 2(n i -1) f i (x + n i -1 ⌊nh n ⌋ |y i ) -f i (x - n i -1 ⌊nh n ⌋ |y i ) , x ∈ [t 2 , t n i -1 ]. (2)
The minimum value of the first derivative (minimum of the slopes of the smoothing curve) is then extracted for each individual curve, S i = min x ḟi (x|y i ). A mixture of two Gaussian distributions (mixtools::normalmixEM, [START_REF] Benaglia | mixtools: An R package for analyzing finite mixture models[END_REF]) is fitted to the sample consisting of the minimum values S i,n i using an Expectation-Maximization algorithm. Ids are then classified into "disturbed" and "non-disturbed", based on the posterior distributions. The method (smoothing, clustering) is then reapplied to each level above (one by one in a bottomup manner) to identify disturbances acting at the group level. For the level k ≥ 2 consisting of N k groups, the median trajectories of the groups are recursively calculated as follows:

y (k) ℓ,j = median y (k-1) i,j ; i such that X ik = ℓ , ℓ = 1, . . . , N k , with y (1)
ℓ,j = y ℓ,j , j ≥ 1 (3) where X ik corresponds to the group in level k to which group i in level k -1 belongs. Median trajectories are illustrated in Figure 3. Note that for any group ℓ, if at least 50% of the trajectories (or median trajectories) constituting the group do not have observations at time step j, the median y (k) ℓ,j is not evaluated. The previous smoothing method (1) is then used again on the y

(k) ℓ = y (k) ℓ,j j≥1
. Finally, the minimum values, expressed as:

S (k) ℓ = min x ḟℓ (x|y (k) ℓ ), ℓ = 1, . . . , N k , k ≥ 2, ( 4
)
are extracted and classifications are used again on the S (k) ℓ ℓ

. Hence, for a given level, the mixture models make it possible to classify the elements with at least one disturbance for this level or for a higher level. Starting from the first level (unit, i.e. k = 1) to the highest level minus one, if an element of the kth level is classified among the disturbed one and the group to which it belongs as well (at the k + 1th level), this element will be considered to have experienced (at least) one group disturbance at level k + 1; hence, the disturbance at level k will not be considered.

Down-step

It should be emphasized that for any group ℓ and any time step j, if more than 50% of the trajectories constituting the underlying level undergo a disturbance (or if initial trajectory has not been recovered) a time j, the median trajectory in (3) may be impacted.

Hence in the Up-step, the group level of disturbances (at the hierarchical level k ≥ 2) can be misclassified due to a high number of disturbed elements at the preceding level k -1 which compose the group level k. It can also be due to a high variability in the dynamics of the observations. Moreover, the Up-step detects elements that experience at least one disturbance, but does not provide any information about the number and characterization of the disturbances experienced. To reduce the risk of misclassification at the group scale (i.e., k ≥ 2), the Down-step thus proposes a post-validation of the detected groups in the Up-step by comparing the reaction start times to the disturbance.

Multiple disturbance detection

The Down-step makes it possible to detect multiple disturbances. Let τ k the intersection of the two Gaussian curves of the mixture model at the hierarchical level k in the Up-step, i.e., the theoretical value of (4) corresponding to a posterior probability to be classified as disturbed of 0.5 by the mixture model at level k. Any local minima of the first derivative of the smoothing curves whose values are smaller than τ k are considered as other potential disturbances at level k. The thresholds τ k are illustrated by the dotted blue lines in Figure 4 for the case of three hierarchical levels.

A first evaluation of the start times for the disturbances corresponds to the times for which the local minima are reached.

Validation

If a disturbance occurs for a given group scale (i.e., level k ≥ 2) and for a given time, this disturbance should affect all or part of the elements of level k -1 at 'close times'.

Starting from the highest hierarchical level to the lowest group level (that is k = 2), a non-supervised clusterization of the identified starting points is done using a multi-modal Gaussian mixture model (mclust::Mclust, [START_REF] Scrucca | mclust 5: clustering, classification and density estimation using Gaussian finite mixture models[END_REF]. Each of the time clusters consisting of a sufficient percentage of the time observations (depending on the number of elements of the group) validates the corresponding group's disturbance. We suggest a validation threshold of 0.5, i.e., at least 50% of the elements constituting the group should have a disturbance in a close time to validate the group disturbance. This value can be changed by the user.

When a group disturbance is not validated at level k, the elements of the group classified among the disturbed ones in the mixture model at level k -1 are again subject to validation at level k -1.

Characterization

Regardless of the hierarchical level k ≥ 1, in the case of a single disturbance, the starting points correspond to the times x for which the local minima (lower than a threshold value τ k ) of the derivative smoothing curve ḟℓ (x|y

(k) ℓ ) in (2) are reached.
The time corresponding to the first local minimum of the smoothing curve f after the starting point is considered as the end of the disturbance. If no local minimum of the smoothing curve is observed after a starting point, the date of the last observation is considered as the end of the disturbance. Finally, the intensity of the disturbance is evaluated by the slope between the start and end time points.

When multiple disturbances occur, it could create some confusion in the estimated starting/ending points between the two disturbances, especially for the close times or in the event that one of the two disturbances is not validated. To avoid such a source of confusion, in the case of multiple disturbances, the starting time and ending time are recalculated using the median of the starting and ending time based on each of the clusters of the multi-modal Gaussian mixture at level k -1 of the validation step.

Because the method can validate the same disturbance at many levels, an indicator of concordance (Cohen's kappa; see Landis & Koch 1977, for details), is used to eliminate redundancy disturbance between two distinct levels, based on the rounded estimated starting and end points. Cohen's kappa is between 0 and 1. It evaluates the overlapping between two considered disturbances. An excessively high value of kappa can lead to not removing redundancy disturbances. At contrary an excessively small value of kappa can lead to not validating the multiple scale disturbances. Note that if kappa is not considered and the previous validation threshold is 0, the Down-step will do nothing.

Overview of UpDown

The dependencies for UpDown are: mixtools [START_REF] Benaglia | mixtools: An R package for analyzing finite mixture models[END_REF] You can install the UpDown package from CRAN with the following R code install.packages("UpDown") library(UpDown)

or the from latest development version: #install.packages("remotes") remotes::install_github("TomRohmer/UpDown") library(UpDown)

UpDown function

The UpDown function is available in the UpDown library : UpDown(data,levels, vtime,obs, h.int=10,mixplot=FALSE, correction=NULL, kappa=NULL,thr_va=0.5, options=list())

The input data should be a dataframe arranged like the PigFarming dataset, with one row per unit and observed time. Columns consist at least of the following: hierarchical levels (one column per level), time (integer) for which observations are measured, and observations. The levels input is the vector of the considered labels of the hierarchical levels ordered from the highest level to the unit level. The vtime input is the considered name of the time variable. Finally, the obs input is the name of the considered observations appearing in the dataframe.

The parameter h.int is the smoothing parameter used in the kernel smoothing method (1). The binary mixplot input makes it possible to specify whether or not the mixture model plots are desired. Such plots make it possible to detect any problem in the mixture fitting, for example due to abnormal observations.

The argument correction allows to correct the longitudinal observations by their natural evolution over time beforehand, subtracting the median observation relative to the modalities of the variable.

Kappa and thr va are the values of the kappa and the threshold validation parameter used in the Down-step, respectively. When Kappa is not specified, no disturbances are removed. Moreover when thr va takes the value of 0, all disturbances are validated.

Finally, options is a list of options including the classical options of the mixtools::normalmixEM R function. Because considering trajectories with too few observations might not make sense, units with less than 20 observations are automatically removed. This can be changed used the optional parameter minobs in the options parameter.

A shiny App function

The UpDownApp function is available in the UpDown library . It makes possible to visualize the longitudinal intra-group observations and the detected disturbances, as well as the median-group observations and the corresponding smoothing curve. It allows to check if some detected elements correspond to registered disturbances (listed disease for example) and if the smoothing is well adapted to the data.

UpDownApp(UpDown.out,obs=NULL,width=1000,height=1000)

The UpDown.out input have to be the global output of the Updown function. obs is then an optional output. The default value is the observation's name specified in UpDown. In the case of multiple dynamic observations, it may be desirable to visualize each of the observations and to see if the detected disturbances affect the other traits or not. Hence, the obs input makes it possible to specify a vector of many observation names that appear in the initial dataset. The optional width and height options makes it possible to modify the dimension of the plots that appear in the Rshiny application.

Application on a simulated dataset

The PigFarming dataset is available in UpDown. It consists of simulated hierarchical data that mimics a pig farming system dataset based on [START_REF] Le | Impact of environmental disturbances on estimated genetic parameters and breeding values for growth traits in pigs[END_REF]. The weight (in kg) of 6,000 animals was simulated over a period of 93 to 100 days and was considered at three different hierarchical levels. Animals (first level) were raised in 40 batches (highest hierarchical level) and in 15 pens within each batch (second hierarchical level), leading to 15 animals per pen. The identifier of the animal ('id'), the identifier of the batch ('batch') and the identifier of the pen ('pen') are listed in PigFarming as well as the age of the animal, the time observation and the value of the simulated weight (e.g., Table 1).

In this simulated dataset, at the third level, 25% of the batches were subject to a disturbance. At the second level, 21% of the pens were exposed to a disturbance. Finally, 19% of the individuals also experienced an individual disturbance. he starts, the duration and the intensities of the disturbances were uniformly sampled respectively on {2, . . . , 99}, {1, . . . , 25} and (0.5, 2.5). The two components of the individual robustness, the resistance and resilience [START_REF] Le | Impact of environmental disturbances on estimated genetic parameters and breeding values for growth traits in pigs[END_REF] In Figure 2, the longitudinal observations of the animals' identifiers (id) 1, 6 and 5405 were plotted. For id 1, no disturbance occurs. For id 6, an individual disturbance occurs at time 33 and with a duration of 25 days. For id 5405, a level 2 disturbance occurs at time 60 with a duration of 20 days, and a level 3 disturbance occurs at time 14 with a duration of 21 days. This non-observable information, on the intensity, starting and ending point of the disturbance for each level can be read in the complementary dataset PigDisturbance, e.g.:

TrueDist<-get(data(PigDisturbance)) subset(TrueDist,id %in% c(1,6,5405))

In Figure 3, the median trajectory of the groups designated as p10101 and p100101 of level 2 were plotted as well as the median trajectory of the group designated as b1001 of level 3.

Note that group p10101 contains id 6 where an individual disturbance occurs. The median trajectory is not impacted. Group p100101 contains id 5405 on which two disturbances affecting level 2 and level 3 simultaneously occur. The median trajectory is impacted by these two disturbance levels. Group b1001 contains group p100101. The median weight is only impacted by the disturbance affecting level 3.

We now compute the UpDown method on the PigFarming dataset. levels=c("batch","pen","id") UD_pig<-UpDown(data,levels=levels, vtime="time", obs="weight", kappa=0.75, + thr_va=0.5, h.int=10,mixplot=TRUE, correction="age") level batch: number of elements=40; number of iterations= 8 level pen: number of elements=400; number of iterations= 26 level id: number of elements=6000; number of iterations= 60 subset(UD_pig$Up, id %in% c(1,6,5405)) batch pen id det b101 p10101 1 0 b101 p10101 6 id b1001 p100101 5405 batch Table 2: Results of the Up-step for animals with id 1, id 6 and id 5405. The estimated starting and end points are shown in Figure 5. In group p100101 of level 2, the intensity of the disturbance was 0.46. In group b1001 of level 3, the intensity of the disturbance was 0.32.

Using the true values on the disturbances available on the PigDisturbance dataset, we can evaluate the performances of the algorithm in terms of sensitivity (percentage of well detected disturbances) and specificity (1-percentage of wrong detected disturbances). These performances are summarized in Table 4 and Figure 6. Of the 40 batches, 10 underwent disturbances and all were detected. Moreover, no batch disturbance was wrongly detected. The starting points were estimated with a median gap of 3.2 days and the end points were estimated with a median gap of 1.87 days with the true values. The correlation between the measurements made for the intensity and the true value of intensity was 0.93. Of the 400 pens, 84 underwent pen disturbances and 74% were detected. Moreover, only 2% were wrongly detected. The starting points of the well-detected disturbances were estimated with a median gap with the true values of 2.86 days, and the end points were estimated with a median gap with the true values of 1.37 days. The correlation between the measurement made for the intensity of the disturbance and the true value of intensity was 0.74. Of the 6,000 individuals, 1127 underwent individual disturbances and only 33% were detected. Nevertheless, only 2% were wrongly detected. The starting points of the well-detected disturbances were estimated with a median gap of 3.03 days and the end points were estimated with a median gap of 2.00 days. The correlation between the measurement made for the intensity and the true value of intensity was only 0.55.

In fact, in the simulation model [START_REF] Le | Impact of environmental disturbances on estimated genetic parameters and breeding values for growth traits in pigs[END_REF], the 'robustness' of the animal is a major part of the reaction of the animal exposed to disturbance, and the intensity of the disturbance is not sufficient to explain the reaction. Then, if the units (animals) are randomly spread out in the hierarchical level regardless of their 'robustness', these individual sources of variations do not affect the median trajectories. Finally, the dynamic trajectories can be visualized through the Rshiny application as well as the detected elements and the smoothing curve on the units and on the median trajectories in a second panel:

if(interactive()){ UpDownApp(UD_pig) }

Calculation times

Using the PigFarming dataset , we evaluated the average calculation time based on B = 10 runs for a number N of ids of N ∈ {2400, 3000, 3600, 4200, 4800, 5400, 6000} and two and three hierarchical levels. The corresponding number of elements in level 2 was N 2 ∈ {160, 200, 240, 280, 320, 360, 400} and the corresponding number of elements in level 3 was N 3 ∈ {16, 20, 24, 28, 32, 36, 40}. The results are summarized in Table 5. For such a dataset, the calculation times are reasonably low, with 34 seconds on average for N = 6000 id with 93 to 100 observations per id and two considered levels, and depend very little on the number of considered levels.

Conclusion and future developments

The R package UpDown provides an algorithm to detect and characterize group disturbances from hierarchical systems and longitudinal data. It is based on unsupervised classification using a mixture model approach and working at different hierarchical levels, i.e., using information from the group. A validation procedure at the group levels, based on the estimated starting point of the detected elements is then proposed to guarantee a low ratio of wrong detection as well as a characterization of the detected disturbance by an estimation of the starting and end points of the reaction and the intensity of the disturbance.

In the simulations carried out, the UpDown method presents very good performances in terms of sensitivity (group scale) and specificity for reasonable computation times. Nevertheless the intensities were uniformly sampled in the simulation model [START_REF] Le | Impact of environmental disturbances on estimated genetic parameters and breeding values for growth traits in pigs[END_REF]. In practice, heavy-tailed distribution for the intensities could lead to not detecting the disturbances corresponding to the smallest intensities, i.e., the Gaussian mixture model may not be appropriate. In the same way, the quality of detection can be strongly affected by outliers. These outliers could be observed in the mixture results and removed from the dataset.

In future developments, we will consider the case of multivariate longitudinal observations for which, disturbances can affect one of the margins, all (or a part) of the margins, or the dependence structure between the margins. To do that, a multivariate mixture model would have to be considered in the Up-step. library(microbenchmark) for(g in seq(2400,6000,600){ tt=microbenchmark( { UpDown(subset(data,id%in% 1:g),levels=c("pen","id"), vtime="time", obs="weight", kappa=0.75, thr_va=0.5, h.int=10,mixplot=FALSE, correction="age") }, { UpDown(subset(data,id%in% 1:g),levels=c("batch","pen","id"), vtime="time", obs="weight", kappa=0.75, thr_va=0.5, h.int=10,mixplot=FALSE, correction="age") }, times=10) print(tt) } N 2400 3000 3600 4200 4800 5400 6000 2 levels 14.34 17.32 19.96 23.15 27.01 30.03 33.74 3 levels 13.61 16.49 19.43 21.11 25.09 26.97 30.31 Table 5: Average computation time (s) for UpDown method with 2 and 3 hierarchical levels.

Acknowledgement

This study was funded 50% by the Animal Genetics division of INRAE and 50% by the Alliance R&D (Axiom, Choice Genetics, Nucleus and IFIP). We particularly thank Alliance R&D for reviewing the manuscript and testing the UpDown package on real data from pig farms.

Figure 1 :

 1 Figure 1: Elastic response (left) and plastic response (right) from a disturbance. The red dotted line is the start of the reaction, the blue dotted line is the end of the disturbance, and the green dotted line is the end of the reaction. The black dotted line is the expected evolution without disturbance.

  and mclust Scrucca et al. (2016) for classification and clusterization; reshape2 Wickham (2007) and dplyr Wickham et al. (2021) for data manipulation; and shiny Chang et al. (2021) and ggplot2 Wickham (2016) for data visualization. All these packages are available on the CRAN at http: //CRAN.R-project.org/.

Figure 2 :

 2 Figure 2: Plot of the weights of id 1 with no disturbance (left), id 6 with an individual disturbance (middle), and id 5405 with both pen disturbance and batch disturbance (right).

Figure 3 :

 3 Figure 3: Plot of the age-corrected weight of id 6 with an individual disturbance (top left) and the median phenotype of pen p10101 that contains id 6 (top right). Plot of the weight of id 5405 with both pen disturbance and batch disturbance (bottom left), the median phenotype of pen p100101 that contains id 5405 (bottom middle) and the median weight of batch b1001 that contains pen p1001 (bottom right). The red curves are Nadaraya-Watson's smoothing curves.

Figure 4 :

 4 Figure 4: Plot of the mixture models for individuals, pens and batches levels UD_pig$Down$batch

Figure 5 :

 5 Figure 5: Plot of the age-corrected weights of id 1 with no disturbance (left) and id 6 with an individual disturbance (middle), and id 5405 with both pen disturbance and batch disturbance (right). The red curves are Nadaraya-Watson's smoothing curves, the dotted red lines are the estimated starting points and the dotted blue lines are the estimated end points.

Figure 6 :

 6 Figure 6: Boxplot of the difference between the true and estimated starting and end points

Table 1 :

 1 was sampled from a logit-normal distribution with variance 1. First part of the PigFarming dataset

	data<-get(data(PigFarming))
	head(data)
	id batch pen	age time weight
	1 b101 p10101	6	1	22.14
	1 b101 p10101	7	2	29.76
	1 b101 p10101	8	3	28.84
	1 b101 p10101	9	4	24.18
	1 b101 p10101 10	5	23.28
	1 b101 p10101 11	6	25.20

Table 3 :

 3 Results of the Down-step for the batch level.

	49 34.24	0.32
	b1002 74.82 82.63	0.17
	b104 67.59 84.92	0.84
	b401 69.54 76.29	0.47
	b504 52.55 68.12	0.43
	b603 95.86 96.00	0.17
	b703 77.28 95.67	0.18
	b704 87.54 96.00	0.35
	b803 22.76 36.04	0.53
	b804 72.91 82.07	0.08

Table 4 :

 4 Table of the sensitivity and specificity of the detected elements, median difference between the true and estimated starting and end points and correlation between the true intensity and the proposed measurement of intensity More exhaustive Monte Carlo simulations (based on 1000 simulations of such a dataset) can be found in[START_REF] Le | Nouvelle mesure de la robustesse des animaux d'élevage par utilisation des données de phénotypage haut-débit[END_REF].

		batch pen	id
	sensitivity (%)	1.00 0.74 0.33
	specificity (%)	1.00 0.98 0.98
	starting point	3.20 2.86 3.03
	end point	1.87 1.37 2.00
	intensity (corr)	0.93 0.74 0.55

Note that before using UpDown, the longitudinal observations have to be corrected by their natural evolution. A median-based correction is proposed in the main function of the package. This type of correction makes it possible to be less impacted by the disturbance than a standard linear model.

The UpDown package is based on the assumption of a decrease in the evolution of the observations (corrected by their natural evolution) when a disturbance occurs, as illustrated in Figure1. To consider disturbances that cause an increase in the evolution, use UpDown on the opposite of the observations. It goes without saying that the methodology cannot simultaneously detect many kinds of disturbances (that cause an