Kristina Yordanova

Samuel Whitehouse

Adeline Paiement

Majid Mirmehdi

Thomas Kirste

Ian Craddock

What's cooking and Why? Behaviour Recognition during Unscripted Cooking Tasks for Health Monitoring

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Nutrition affects our health and is an important factor for having a healthy lifespan [START_REF] Ohlhorst | Nutrition research to affect food and a healthy lifespan[END_REF]. Nutrition related diseases thus can impact our well-being and reduce the quality of life. This is particularly true for long term physical conditions, such as diabetes, eating disorders, or mental conditions such as depression, that affect the patient's willingness to prepare and consume healthy food, or people suffering from dementia disorders whose ability to prepare food is hampered by the disease's progression [START_REF] Serna | Modeling the progression of alzheimer's disease for cognitive assistance in smart homes[END_REF]. To reduce the costs associated with hospitalisation and treatment of these conditions, different works have attempted to provide automated home monitoring of the patient that besides reducing the hospitalisation costs, potentially improves the wellbeing of the patient as they can be monitored and treated in home settings [START_REF] Helal | Smart home-based health platform for behavioral monitoring and alteration of diabetes patients[END_REF].

To build such systems, one needs to recognise the user actions and goals as well as the causes behind the observed behaviour [START_REF] Krüger | Towards creating assistive software by employing human behavior models[END_REF]. To achieve that, different works make use of knowledge-based models [START_REF] Roy | A Possibilistic Approach for Activity Recognition in Smart Homes for Cognitive Assistance to Alzheimer's Patients[END_REF]. In contrast to data-driven approaches that rely on large amounts of sensor data and which can learn only situations that are present in the data, knowledge-based approaches have the advantage that they are able to reason beyond the sensor data. Thanks to the rules that define the possible behaviour, they can provide information about the person's situation, e.g. caused by the progression of the disease [START_REF] Serna | Modeling the progression of alzheimer's disease for cognitive assistance in smart homes[END_REF]. The main challenge for rule-based approaches is that they are usually unable to cope with the problems associated with real world scenarios: (a) the variability of the user behaviour results in complex models that are often computationally infeasible, and (b) the presence of imperfect sensors makes the purely symbolic models unable to cope with the ambiguity.

To address these problems, some works propose approaches that combine rules and probabilistic inference, such as [START_REF] Hiatt | Accommodating human variability in human-robot teams through theory of mind[END_REF][START_REF] Ramirez | Goal recognition over pomdps: Inferring the intention of a pomdp agent[END_REF]. This type of models is also known as computational state space models (CSSMs) [START_REF] Krüger | Computational state space models for activity and intention recognition. a feasibility study[END_REF]. CSSMs combine symbolic representation with probabilistic reasoning to cope with the combination of behaviour variability and sensor noise [START_REF] Hiatt | Accommodating human variability in human-robot teams through theory of mind[END_REF][START_REF] Krüger | Computational state space models for activity and intention recognition. a feasibility study[END_REF]. One challenge with CSSMs is that so far they have only been applied to scripted scenarios which implies simplified settings that do not address the challenges of complexity and behaviour variability present in real settings. Another challenge is the reconstruction of the user activities and goals from low level sensor data. As [START_REF] Sukthankar | Plan, Activity, and Intent Recognition: Theory and Practice, chapter Introduction[END_REF] point out, "bridging the gap between noisy, low-level data and high-level activity models is a core challenge". In this work, we address the above challenges by presenting first empirical results showing that CSSMs are able to reason about the user's actions and goals in unscripted kitchen scenarios based on low level sensor data.

Related Work

There are different rule-based approaches that can reason about the person's actions and goals based on context information. One such approach is where ontology-based behaviour libraries are explicitly provided by human experts [START_REF] Roy | A Possibilistic Approach for Activity Recognition in Smart Homes for Cognitive Assistance to Alzheimer's Patients[END_REF]. One problem with this type of approaches is that "library-based models are inherently unable to solve the problem of library completeness caused by the inability of a designer to model all possible execution sequences leading to the goal" [START_REF] Yordanova | A process for systematic development of symbolic models for activity recognition[END_REF].

A second option for arriving at a suitable model is to mine action sequences from observations of human behaviour. Such approaches manually define an initial library of behaviours. Later, behaviour variations are added or removed based on observations of the user activities [START_REF] Chen | An ontology-based hybrid approach to activity modeling for smart homes[END_REF]. Although this approach provides a solution to the problem of keeping behaviour libraries up-to-date, it still relies on initial manual definitions of behaviour variations.

To address the problem of designing models that represent the behaviour variability without relying on large amounts of sensor data, some works propose computational state space models [START_REF] Hiatt | Accommodating human variability in human-robot teams through theory of mind[END_REF][START_REF] Ramirez | Goal recognition over pomdps: Inferring the intention of a pomdp agent[END_REF][START_REF] Krüger | Computational state space models for activity and intention recognition. a feasibility study[END_REF][START_REF] Yordanova | A process for systematic development of symbolic models for activity recognition[END_REF]. CSSMs describe the actions in terms of preconditions and effects, and some of them allow probabilistic reasoning about the user state, goals and context. The manually defined model is very compact as it requires the definition of several action templates that are automatically expanded into different execution sequences based on the causal relations between them. This provides an alternative solution to the problem of manually defining all execution sequences, or relying on large amounts of annotated sensor data to learn them.

So far, CSSMs have been used only in controlled experiments with predefined execution sequences. Such experiments limit the behaviour variability typical for realworld problems, which in turn simplifies the CSSM model needed to recognise this behaviour. To our knowledge, so far there is no empirical evidence that CSSMs are able to cope with the behaviour complexity typical for a real world daily activities. What is more, so far CSSMs have been used for goal recognition only based on simulated data [START_REF] Hiatt | Accommodating human variability in human-robot teams through theory of mind[END_REF][START_REF] Ramirez | Goal recognition over pomdps: Inferring the intention of a pomdp agent[END_REF]. In a previous work we proposed a CSSM model that is able to recognise the protagonist's activities during unscripted kitchen tasks [START_REF] Whitehouse | Recognition of unscripted kitchen activities and eating behaviour for health monitoring[END_REF]. The model was tested on simulated sensor data. In this work, we extend this model for activity recognition based on real sensor data by building the appropriate observation model and actions' durations. Furthermore, we extend the model for goal recognition and we show first empirical evidence that it is possible to perform both activity and goal recognition based on noisy sensor data in real-world everyday scenarios.

Computational Causal Behaviour Models

The CSSM approach we chose for our problem is called Computational Causal Behaviour Models (CCBM) which has been shown to perform adequate activity recognition in problems with large state spaces and noisy sensor observations [START_REF] Krüger | Computational state space models for activity and intention recognition. a feasibility study[END_REF][START_REF] Yordanova | A process for systematic development of symbolic models for activity recognition[END_REF]. CCBM relies on the idea of Bayesian filtering to recognise the person's actions and goals based on observations. Figure 1 shows the dynamic Bayesian network (DBN) structure of a CCBM model. Informally, CCBM can be divided into two parts: observation model, which provides the probability of having an observation given a certain system state and system model, which provides the probability of the current system model state given the previous state. The observation model is defined through Y t = (W t , Z t ), the observation data for time step t, e.g sensor data collected from low level sensors or simulated data. So far CSSMs had either performed only activity recognition using real sensor data [START_REF] Krüger | Computational state space models for activity and intention recognition. a feasibility study[END_REF][START_REF] Yordanova | A process for systematic development of symbolic models for activity recognition[END_REF] or they have used simulated data to perform goal recognition [START_REF] Hiatt | Accommodating human variability in human-robot teams through theory of mind[END_REF][START_REF] Ramirez | Goal recognition over pomdps: Inferring the intention of a pomdp agent[END_REF]. In this work we use low level sensor data to perform both activity and goal recognition. The system model consists of causal model (expressed through G t , A t , S t ), duration model (expressed through D t and U t ), and action selection heuristics (which are not represented in Figure 1). G t is the current goal the person is pursuing. In difference to [START_REF] Krüger | Computational state space models for activity and intention recognition. a feasibility study[END_REF][START_REF] Yordanova | A process for systematic development of symbolic models for activity recognition[END_REF] who assume that the current goal is a constant, in this work we follow different goals and the goal can change dynamically over time. S t is the high-level model state (the state describing the person's behaviour). It is either the result of applying a new action A t or carrying over the old state S t-1 . Actions can last longer than a single time step, i.e. they have durations. U t denotes the starting time of an action, while the boolean random variable D t indicated the termination status of the previous action. V t is the time stamp associated to the DBN slice.

In CCBM the causal model is presented in terms of rules that describe the possible initial and goal states, the conditions that have to hold in order for an action to be executable and the changes to the world after the action is executed. To select a new action CCBM uses action selection heuristics such as goal distance, cognitive heuristic etc. The duration model is expressed through probability distribution indicating the t t -1 probability of terminating the action given the starting time of the action. For more details on CCBM see [START_REF] Krüger | Computational state space models for activity and intention recognition. a feasibility study[END_REF][START_REF] Yordanova | A process for systematic development of symbolic models for activity recognition[END_REF].

D t-1 V t-1 S t-1 A t-1 U t-1 W t-1 Z t-1 G t-1 D t V t S t A t U t W t Z t G t X t X t-1 Y t Y t-1

Experimental Setup

Data Collection A sensor dataset consisting of 15 runs of kitchen activities was recorded in the SPHERE House which is part of the SPHERE project (a Sensor Platform for HEalthcare in a Residential Environment) [START_REF] Zhu | Bridging e-health and the internet of things: The sphere project[END_REF]. The SPHERE House in Bristol (UK) is a 2-bedroom house equipped with a variety of environmental sensors. The sensor network in the kitchen of the house collects data on temperature, humidity, light levels, noise levels, dust levels, motion within the room, cupboard and room door state, and water and electricity usage. A head-mounted camera was used to record the actions of the participants to allow for annotation of the observations. The resulting dataset can be downloaded from [START_REF] Mirmehdi | Sphere unscripted kitchen activities[END_REF].

Data Processing

The original sensor data is in JSON format. The data was converted so that for each type of sensor there is a separate column. This conversion produced multiple rows with the same timestamp (in milliseconds). Rows with the same timestamp were then combined as long as per sensor type there was only one unique value. As this new format produces NAs for some sensors at a given time (due to the way in which the data is initially collected from the sensors), the NAs between two sensor readings were replaced with the first value. As the state of the most sensors is being read at a certain sampling rate but additionally there is a reading when a change in the state is detected, we believe that this simple replacement of NAs is sufficient. The resulting data contained identical observations for different action labels. To reduce the impact of this artefact on the model performance, a sliding window of 5 time steps with overlapping of 50% was used and the observations in this window were represented by the maximum value for each sensor in the window. [START_REF] Whitehouse | Recognition of unscripted kitchen activities and eating behaviour for health monitoring[END_REF] where they use it for activity recognition on simulated data. Here, we extend the model by adding probabilistic action durations and goal recognition and use it with real sensor data for following different goals.

CCBM Models In this work we use extended version of the model proposed in

Causal model: 15 specialised models, specifically fitted for the corresponding execution sequence, were developed. Furthermore, a general model was developed which can handle all sequences in the dataset. Each of the models can recognise the following action classes: clean, drink, eat, get, move, prepare, put, unknown. The model dimensions for the two model implementations can be seen in Table 1. Some additional discussion on the models can be found in [START_REF] Whitehouse | Recognition of unscripted kitchen activities and eating behaviour for health monitoring[END_REF]. Goals in the model: The model has three types of goals: 1) the type of meal the person is preparing (13 goals); 2) whether the meal / drink is healthy or not (4 goals); 3) whether the person is depressed or not (2 goals). For 3) we rely on the assumption that the person is depressed when they are preparing ready meals instead of cooking.

Duration model: The durations of the model were calculated based on the annotation. Empirical probability was assigned to each action class, indicating how long the model can stay in the same state before transitioning to another state.

Observation model: Two types of observation models were trained with a decision tree: 1) OM o : All data was used both for training the OM and for testing the CCBM model. 2) OM p : The first run was used for training the OM and the remaining runs were used for testing the CCBM. The first run was chosen because it is the only run where all action classes appear.

The decision tree for OM o without any additional underlying model achieved mean accuracy of .52. The decision tree for the OM p achieved mean accuracy of .39. This is to be expected as the tree was trained only on the first run and in each of the remaining experiments, a different meal was prepared usually by a different person.

Experiments For each of the observation models, the following experiments were conducted: 1) activity recognition of the action classes based on: the specific CCBM model (we call this model CCBM s ); the general CCBM model with one goal 1 (we call this model CCBM g ); the general CCBM model with multiple goals 2 (we call this model CCBM g1 ). 2) goal recognition on: the different meals and drinks that can be prepared (the goal recognition is done with the CCBM g1 model); whether the prepared meal is healthy (we call this model CCBM g2 ); whether the person is depressed (we call this model CCBM g3 ).

Results

Figure 2 shows the results from the activity recognition with the different observation and system models. It can be seen that for both OM, the CCBM models performed better than the classification with decision tree. For most models, Shapiro-Wilk normality test did not reject the null hypothesis that the samples come from normal distribution with the exception of CCBM s and CCBM g1 with OM p (p value ≤ 0.03). For that reason, to test whether the results significantly differ from each other we performed both signed t test and Wilcoxon test. Both tests showed that all CCBM models with OM o do not significantly differ from each other (p ≥ 0.84 for t test and p ≥ 0.74 for Wilcoxon test). This indicates that the general models do not significantly reduce the recognition rate in comparison to the specialised model. This however showed that all three CCBM models with OM o significantly differ from the decision tree tested on the training data (p ≤ 0.006 for t test and p ≤ 0.003 for Wilcoxon test). On the other hand, both tests illustrated that the CCBM models with OM p and the decision tree where a test and train dataset was used do not significantly differ (p ≥ 0.73 for t test and p ≥ 0.86 for Wilcoxon test). This stands to show that the very ambiguous observation model does not allow for the system model to improve the activity recognition performance. However, it also shows that despite the inaccurate OM, the system models do not reduce the recognition performance.

Figure 3 shows the results from the goal recognition for the different observation models. Here we measure the F score, as for some of the experiments more goals were recognised than were followed during the experiment. Surprisingly, OM p did not reduce recognition of the goal for healthy meal and the type of meal. On the contrary, despite the ambiguous OM and the low activity recognition results, the models were able to perform better than when using OM o . On the other hand, the recognition of depressed healthy meal whether the person is depressed or not was lower with OM p than with OM o . To test whether the results for the two OM significantly differ, we performed a Wilcoxon test (the Shapiro-Wilk normality test rejected the null hypothesis that the samples come from a normal distribution with a p ≤ 0.001). The Wilcoxon test showed that results for the two OMs for meal recognition and healthy meal do not significantly differ (p = 0.86 for CCBM g1 , and p = 0.40 for CCBM g2 ). This means that the models perform comparably and that the inaccurate observation model OM p does not reduce the accuracy of recognising the person's goal. The results however showed that the OM has influence on the recognition of whether the person is depressed or not (p = 0.004 for CCBM g3 ). In other words the more accurate OM o significantly improved the recognition of the cause for the prepared meal. Figure 4 shows an example of the probability of preparing a healthy meal / drink for one of the experiments. The true goal is "healthy meal" and it can be seen that after time step 125 the model converges to the real goal. Although in this example, the protagonist pursues only one goal, in 4 of the experiments the goal changes throughout the experiment as the protagonist prepares more than one meal or drink.

Conclusion and Future Work

This work investigates the applicability of CSSMs to real world everyday activities. We applied the approach to a sensor dataset containing 15 unscripted meal preparations. The results showed that the approach is able to perform activity recognition without reducing the recognition quality compared to the performance of decision tree. They also showed the approach is able to perform goal recognition and to accurately reason about the type of meal, whether it is healthy and whether the person preparing the meal is depressed even in the case of poor activity recognition results. These first results show that the approach has the potential to reason about the person's behaviour and the causes behind it that could hint at (the progression of) medical conditions.

In the future, we intend to compare the results from the goal recognition to state of the art approaches, such as HMMs. Furthermore, we intend to add the data from depth cameras to the observations and to investigate the influence of the type of sensor on the model performance.

Figure 1 :

 1 Figure 1: DBN structure of a CCBM model. Adapted from [4].

Figure 2 :

 2 Figure 2: Accuracy for the activity recognition.

Figure 3 :

 3 Figure 3: Accuracy for the goal recognition.

Figure 4 :

 4 Figure 4: Evolving of the goals' probability with the accumulation of new observations during the experiment execution.

Table 1 :

 1 Parameters for the different models.

	Parameters	General model Specialised model
	Action classes	8	8
	Ground actions	92	10 -28
	States	450 144	40 -1288
	Valid plans	21 889 393	162 -15 689

This means that all possible meals are described with "OR" statement.

This means that all possible meals are described as separate goals.

Acknowledgments

This work is partially funded by the UK Engineering and Physical Sciences Research Council (EPSRC) within the context of the SPHERE IRC, grant number EP/K031910/1.


Maksym Ivanov

Illia Thiele

Stefan Skupin

Danas Buozius

Virgilijus Vaicaitis

Terahertz Wave Generation in Air by Femtosecond Optical Vortex Pulses

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


C Chartin

S Salvador-Blanes

F Hinschberger

H Bourennane

J.-J Macaire

Soil and substrate morphology as witnesses of present and former agricultural landscape

-CONCLUSIONS

-Soil accumulations due to past and present field limits cannot be estimated by topographic attributes exclusively.

-Irregularity in the substrate morphology due to tillage operations should be taken into account.

-Surficial resistivity has to be further treated from background noise due to substrate heterogeneity to find a better link with soil accumulations. -Weak correlations, soil thickness and asymmetries interrogate about substrate morphology (Fig. 4). 3 -FIRST STUDY ON SURFICIAL RESISTIVITY (Fig. 5)

Figure 2 :Figure 1 :Figure 4 :

 214 Figure 2: Soil thickness perpendicularly to structures: variation relative to distance from field limit or road by transect and by mean +/-standard deviation. a) lynchet b) secondary structures

Figure 3 :

 3 Figure 3: examples of simple linear correlations: here between soil thickness and profile curvature. a) Lynchet. b) Secondary structures

Figure 5 :

 5 Figure 5: Resistivity at various depths


Eerik Mantere
email: emantere@u-bordeaux.fr

What Smartphones, Ethnomethodology, and Bystander Inaccessibility Can Teach Us About Better Design?

Keywords: Smartphones, Ethnomethodology, Collocated Interaction

Smartphones, the ubiquitous mobile screens now normal parts of everyday social situations, have created a kind of ongoing natural experiment for social scientists. According to Garfinkel's ethnomethodology social action gets its meaning not only from its content but also through its context. Mobility, small screen size, and the habitual way of using smartphones ensure that, while offering the biggest variety of activities for the user, in comparison to other everyday items, smartphones offer the least cues to bystanders on what the user is actually doing and how long it might take. This 'bystander inaccessibility' handicaps shared understanding of the social context that the user and collocated others find themselves in. Added considerations and interactive effort in managing the situation is therefore required. Future design needs to relate to this basic building block of collocated interaction to not be met with discontent.

Introduction and Background

In United States 81% owns a smartphone [START_REF] Taylor | Smartphone Ownership Is Growing Rapidly Around the World, but Not Always Equally[END_REF] and they are routinely used in the presence of others. How this impacts relationships with collocated others regularly hits the headlines [START_REF] Ducharme | Phubbing" Is Hurting Your Relationships. Here's What It Is[END_REF] [START_REF] Molina | Do smartphones keep us in or out of touch?: Devices often isolate, distract and disrupt acting with others[END_REF]. Previous research suggests various negative effects. Smartphone use can be distracting and undermine the benefits of social interactions [START_REF] Dwyer | Smartphone use undermines enjoyment of faceto-face social interactions[END_REF], which have previously found to be so crucial to psychological well-being [START_REF] Feeney | A New Look at Social Support: A Theoretical Perspective on Thriving Through Relationships[END_REF]. Although often aiming for connection with distant others, interactions online do not provide the same sense of social support as collocated interactions [START_REF] Kim | Smartphone-mediated communication vs. face-to-face interaction: Two routes to social support and problematic use of smartphone[END_REF]. Being distracted in collocated interactions due to smartphone use therefore seems like an ill-chosen trade-off. An Australian dictionary jumped on the idea by coming up with a new word for the phenomenon. "Phubbing" is defined at their marketing campaign's website site as "the act of snubbing someone in a social setting by looking at your phone instead of paying attention" [START_REF]Stop Phubbing Website[END_REF]. Researchers got on board with the term and phubbing has since been found to reduce communication quality and relationship satisfaction by reducing the feelings of belongingness and positive affect [START_REF] Chotpitayasunondh | The effects of "phubbing" on social interaction[END_REF], make both phubbers and the phubbed to be more likely to see phubbing as an inevitable social norm [START_REF] Chotpitayasunondh | How "phubbing" becomes the norm: The antecedents and consequences of snubbing via smartphone[END_REF], and be thought of as 'bad' by young people, even if they are doing it themselves [START_REF] Aagaard | Digital akrasia: a qualitative study of phubbing[END_REF]. "Partner phubbing" has further been found to reduce relationship satisfaction by creating conflicts over cell phone use [START_REF] Roberts | My life has become a major distraction from my cell phone: Partner phubbing and relationship satisfaction among romantic partners[END_REF] and cause depression in China for couples married more than seven years [START_REF] Wang | Partner phubbing and depression among married Chinese adults: The roles of relationship satisfaction and relationship length[END_REF]. A validated scale to measure phubbing has also been developed [START_REF] Chotpitayasunondh | Measuring phone snubbing behavior: Development and validation of the Generic Scale of Phubbing (GSP) and the Generic Scale of Being Phubbed (GSBP)[END_REF] and the capacity to predict phubbing risk has been pursued by forming a model constituting of communication disturbances and phone obsession [START_REF] Guazzini | An explorative model to assess individuals' phubbing risk[END_REF]. One should not then be surprised then that an article in the New York Times portrayed phubbing as if the term was developed by psychologists [START_REF] Roose | Do Not Disturb: How I Ditched My Phone and Unbroke My Brain[END_REF].

Not wanting to discredit the previous work, three points should be noted of their similar methodologies and the gap they fail to fill. First, though they study the social situation, they do not directly describe it, but rather produce second level constructs of it [START_REF] Garfinkel | Studies in ethnomethodology[END_REF]. Research participants have produced numeric or verbal accounts of imagined or previously lived situations. These are then used to make a scientific accounts-now two levels distanced from the phenomenon they aim to depict. Second, when directly observing social situations, they rely on a priori chosen qualities of interaction. Researchers observing social behavior then code it in regard to these qualities in order to use them as indicators in seeking relevance between them and general social categories like age or gender [START_REF] Goffman | The Neglected Situation[END_REF]. Third, none of them spring from a theoretical understanding of social action. Harold Garfinkel pointed out the problems of theories that rely on internalization of society's norms and found ethnomethodology (EM) to study how people themselves in everyday situations construct meaning and make and interpret social typifications as relevant. EM has quickly gained more and more ground as the theory of social action and has given birth to conversation analysis (CA), now considered the principal way to study verbal and non-verbal interaction alike [START_REF] Garfinkel | Studies in ethnomethodology[END_REF] [17] [START_REF] Heritage | Garfinkel and Ethnomethodology[END_REF] [19] [START_REF] Mondada | Multimodal resources for turn-taking: pointing and the emergence of possible next speakers[END_REF].

Though EM/CA literature covers a wide range of interactive contexts, research on spontaneous individual smartphone use in social situation is practically non-existent. One of the most closely related EM/CA studies looked at how smartphone use while driving is interleaved with traffic light stops. Users were looking for moments when the affordances of the phone's interface co-constructed transition relevant places with the activities of the user. In these moments a possible shift in orientation between smartphone use and other activities is sequentially made most available. The regularity in which the interface makes these moments possible was considered a central theme in organizing multiactivity with smartphone use and other concurrent activities [START_REF] Licoppe | Gaze Patterns and the Temporal Organization of Multiple Activities in Mobile Smartphone Uses[END_REF]. Another study of using public transport found gaze shifts away from the phone to be organized in relation to the sequential progression of the activity with the device. Beginning stages of phone use were suggested to be especially sequentially engaging but the methodology used and the level of granularity of the analyzes lacked the possibility to describe the interactive practices of in their sequential contexts [START_REF] Figeac | Video-ethnography of social media apps' connection cues in public settings[END_REF].

A study focusing on the use of a map applications found people sequentially organizing their phone use with actions like unilateral stopping, turning, and restarting, while walking together in public places. Again, phone use was found to have its own sequential progression which, then was interleaved with that of the concurrent social activities of the physical environment [START_REF] Laurier | Mediated Pedestrian Mobility: Walking and the Map App[END_REF]. The most relevant EM/CA work on smartphone use and collocated interaction addresses phone use in pubs [START_REF] Porcheron | Using Mobile Phones in Pub Talk[END_REF]. It does introduce and explore the topic but does not exhaust neither a single episode of interaction, nor describe any putative practice taking place in various interactions, to a satisfactory degree from the point of view of CA. Similarly, it does not make real use of the theoretical offer of EM. I encourage looking at smartphone use in social situation with a viewpoint rooted in EM, and adding in CA analysis, in order to understand how phone use may be constructed as unacceptable, and to find inspiration for more socially acceptable design.

Social Theory and Indexicality

Goffman [START_REF] Goffman | The Neglected Situation[END_REF] defined the social situation as an "environment of mutual monitoring possibilities, anywhere within which an individual will find himself accessible to the naked senses of all others who are ¨present,¨ and similarly find them accessible to him." All speaking and gesturing in face-to-face interaction takes place in the social situation and he emphasized the importance of the physical setting in any analysis of them. Even more than Goffman, Harold Garfinkel saw the context of interaction to be central in what the interaction itself means [16] [18]. Let us consider the following example:

I'm sorry
The phrase seems to clearly convey an apology. We might imagine that the person uttering the words feels regretful and elucidate how each of the words, I -am -sorry, convey something that together constitute an apology. We might reflect on how it differs from the more casual "sorry 'bout that", and we might even say that this apologizing seems humble and empathetic. But what if we added a context:

I got my diploma from the University of Honolulu

I'm sorry

Now the phrase "I'm sorry" doesn't seem so kind. This example shows how the same practice of "apologizing" can be used to do different things-one of them teasing. As the immediate social context changes, the meaning of the action changes too. Before Garfinkel, 'indexicality' was considered as a character only of words like "this", "here" or "now"-words that point, or index, a context in acquiring their meaning. Garfinkel planned a series of breaching experiments to claim that actually all human action is understood as indexing the context it takes place in. If people encounter behavior that is not designed in relation to the commonly shared situation, they feel awkward and severely challenged in knowing how to proceed. Whatever is done, through words or otherwise, always gets interpreted through what is seen as the shared understanding of the situation that the action takes place in [16] [18]. Garfinkel further proposed that this understanding was not only his, but people conducting their everyday lives actually orient to each other as accountable in entering social situations with the assumption that it is common knowledge [START_REF] Heritage | Garfinkel and Ethnomethodology[END_REF]. This knowledge is not rooted in detached reflection of the deep nature of social action. He does not suggest that all members of society passed sleepless nights in understanding the core concepts of ethnomethodology. Rather, in interacting with one another, a general thesis of interchangeability of perspectives is at work. To put it simply, people assume that what they see as relevant in a situation is seen relevant by others in the same situation. This is crucial for being able to trust to the shared understanding of the social situation as "good enough" for interaction to be meaningful. If we could not trust that we and another person have at least "good enough" match in understanding what is going on, we could not trust that anything we say or do in the situation would be understood as we would like it to be understood.

Bystander Inaccessibility

The participants of a social situation who start to use a smartphone, to a large extent stop giving hints of the goals of their actions to collocated others. Others can less often than is the case with other devices, infer from the posture and movements of the user, or from the shape and state of the smartphone, what the user is currently doing. The lack of visual and auditive cues to the bystander, the mobility of the device, bigger amount of variation in the types of actions possibilitie, than is the case with any other device, and the varied temporal organizations of the different smartphone activities are responsible of keeping some crucial aspects about the smartphone user's activity hidden to the person in their immediate vicinity:

I. Phase of action (e.g. preparatory phase, execution phase, or being already close to terminating the action) II. Category of action (e.g. entertainment, work, information seeking, or communication)

Not knowing what the activity of the smartphone user is, the other participants in the social situation are also in the dark about the "good enough" knowledge about nature of the situation as a whole. I call this bystander inaccessibility (BI). Imagine you want to ask something mundane of your partner, let's say, if she has gotten the mail. The mailbox is just outside, and you could easily check it yourself, but you would prefer not getting out of the house in vain. You see your partner sitting on the sofa, absorbed in their phone. Now if you would know that they are responding to an important work email, you might leave them alone and check the mail yourself. But if they were just scrolling a friend's Facebook feed, you might feel at ease to interrupt them. Being in the dark about the activity they are engaged in, you are also unable to know what your planned communicative action, "have you checked the mail?", would signify to them. It works the other way around as well. This is exemplified in the following data excerpt. Clo and Liz are eating out and exchanging funny stories together with a friend. Excerpt 1.

[overlapping speech] >faster speech< (0.9) silence in tenth of seconds (.) noticeable silence of maximum 0.3 seconds .mt smacking of the mouth @transformed speech, e.g. when quoting someone@ ºspoken silentlyº the-the production of the word is halted suddenly ((comments)) ((Clo is using her phone while talking)) 64 Clo: [>Nii nimeonomaa<] (.) ja sit vielä se ku tota noin ni toi [>Yeah exactly<] (.) and then also that when you know that 65 (0.9) .mt ((Clo stops typing and puts left hand to her face)) 66 (0.2) ((Clo continues to gaze at the phone))

67 Clo: öö iskä >oli sillee< [@↑nii joo mä muistan kun Niina umm dad >was like< [@oh yeah I remember when Niina 68 Liz:

[ºmä katon ton-º [ºI'll check th-º ((picks up her phone))

Clo is starting to tell a story that continues the theme of previous stories that night. While doing this she pauses (line 65, 66) and utilizes filler words (lines 64-67) before actually getting the story going (line 67). Before her turn she was using her phone. While beginning the telling at line 67, she is still looking at it. Liz is listening, gazing at Clo, and sees all this taking place. While Clo is struggling while visibly distributing her attention between two activities-telling a story and using her phone-she is also putting Liz in a difficult position. Clo has already prefaced her story and gained a silent "permission" from the group to occupy a speaker position for a longer duration than normal, i.e. until the story is finished. Therefore other participants are normatively restricted to the position of recipient. When regardless of this, Clo still does not put her full attention to the activity of telling the story, and is faltering in beginning the story, the next activities, being indexical, connect in their meaning also to this event.

When Liz begins to use her phone at line 68, BI makes Clo unable to automatically see the type and the goal of Liz's phone use. In this context it therefore risks being interpreted as motivated by dissatisfaction with the haphazard way Clo begun her responsibilities as a storyteller. Considering Goffman's [START_REF] Goffman | Interaction ritual: essays on face-to-face behavior[END_REF] face-work and the normative ways we protect the faces of ourselves, as well as other people from straightforward criticism, it is understandable that Liz chooses to counter this potentially face-threatening interpretation. She provides an account: "I'll check the-" at line 68. Interestingly, she does not actually specify the activity she will commence with the phone, but in providing the account, she nevertheless hints that there is something to be "checked" and the reason for her staring to use the phone could be in this "checking", rather than in the faltering conversational performance of Clo. To conclude, as BI hides Liz's activity from Clo, Liz has to produce an account to circumvent this lack. Providing this account in a sequentially appropriate manner encumbrances a very limited resource in the context of being a recipient to verbal storytelling: audible speech. BI -instigating technology (BI-tech) also makes it harder for collocated others to interpret responses, or lack of them, by a BI-tech user. Our study using role playing method and comic strips found most respondents more irritated when trying to unsuccessfully get the attention of a phone-using person, while no respondents evaluated the newspaper -condition as more irritating (p<0.001). Furthermore, the written responses often included descriptions on being bothered by not knowing what the phone user was actually doing [START_REF] Raudaskoski | Älypuhelin ja kasvokkaisen vuorovaikutuksen muuttuvat käytänteet[END_REF].

Conclusion

Designing socially acceptable technology should be informed by ethnomethodological study on the device's effect on social situation. What people do or do not accept is the way technology enters into the situation as part of the network of social activities. When engaged in technology use, a crucial aspect of it is that the activity is part of constituting the shared social reality that then gives meaning also to all the other activities of everyone else present in the situation. All their decisions to act or not to act are impacted by their understanding of what the technology use is about and whether they can trust that other participants see the situation similarly. There should be more work on design instigating affordances for collocated others to see, hear, or feel the nature of the technology use taking place in a social setting [START_REF] Ens | Candid Interaction: Revealing Hidden Mobile and Wearable Computing Activities[END_REF]. Crucially, I call for interdisciplinary work that benefits from EM/CA methodology to develop and test new prototypes. BItech handicaps participants in social encounters. While people find ways to circumvent it, the plethora of research reporting dislike of smartphone use in social situation suggests they would prefer to avoid these challenges. Interactional work and designing non face-threatening actions takes effort, and people do not like to be forced to make effort.

Fig. 1 .

 1 Fig. 1. Respondents identified with the person speaking and rated A and B in random order.


Mathieu Ribatet

Eric Sauquet

Jean-Michel Grésillon

Taha B M J Ouarda

A Regional Bayesian POT Model for Flood Frequency Analysis

Keywords:

published or not. The documents may come

A regional Bayesian POT model for flood frequency analysis

Introduction

Flood frequency analysis is essential in preliminary studies to define the design flood. Methods for estimating design flow usually consist of fitting one of the distributions given by the extreme value theory to a sample of flood events. If modelling exceedance over a threshold is of interest, a theoretical justification [START_REF] Fisher | Limiting forms of the frequency distribution of the largest or smallest member of a sample[END_REF][START_REF] Balkema | Residual life time at great age[END_REF][START_REF] Pickands | Statistical inference using extreme order statistics[END_REF] exists for the use of the Generalized Pareto distribution (GP).

F (x) = 1 -1 + ξ (x -µ) σ -1/ξ (1)
where 1 + ξ (x -µ) /σ > 0, σ > 0. µ, σ and ξ are the location, scale and shape parameters. This distribution is defined for ξ = 0, and can be derived by continuity in the case ξ = 0, corresponding to the Exponential case:

F (x) = 1 -exp - x -µ σ (2)
A comprehensive review of the Extreme Value Theory is given by [START_REF] Embrechts | Modelling Extremal Events for Insurance and Finance[END_REF] and [START_REF] Coles | An Introduction to Statistical Modelling of Extreme Values[END_REF].

However, frequency analysis can lead to unreliable flood quantiles when little data is available at the site of interest. A convenient way to improve estimates of flood statistics is to incorporate data from other gauged locations in the estimation procedures. This approach is widely applied in hydrology and is known as Regional Flood Frequency Analysis (RFFA). One of the most popular and simple approaches privileged by engineers is the Index Flood method [START_REF] Dalrymple | Flood frequency analysis[END_REF]. The standard procedure allows: a) the delineation of homogeneous regions, i.e. a set of sites which behave -hydrologically and/or statistically -in the same way; b) the derivation of a regional flood frequency distribution; c) and the estimation of the parameters and quantiles at the site of interest.

Regions are collections of gauged basins with similar site characteristics related to the flood magnitude. The pooled stations are not necessarily in the proximity of the site of interest. Forming homogeneous regions can be achieved in various ways. Regions have first been geographically established. More recent work promoted the use of geographically non-contiguous regions [START_REF] Burn | Evaluation of regional flood frequency analysis with a region of influence approach[END_REF][START_REF] Grehys | Inter-comparison of regional flood frequency procedures for canadian rivers[END_REF]. Recent research has defined the concept of "region of influence" [START_REF] Acreman | The regions are dead: Long live the regions[END_REF]. Other techniques can be used such as Artificial Neural Networks to identify groups of stations [START_REF] Hall | The application of data mining techniques for the regionalisation of hydrological variables[END_REF].

The Index Flood model assumes that flood distributions at all sites within a region are identical, up to a scale factor. The Index Flood approach is not exempt from critics as its application requires strong assumptions. One major implicit assumption, noticed by [START_REF] Gupta | Multiscaling theory of flood peaks: Regional quantile analysis[END_REF], is that the coefficient of variation of peak flows is to be constant across the region. This fundamental property seems not to be verified in practice [START_REF] Robinson | An investigation into the physical causes of scaling and heterogeneity of regional flood frequency[END_REF] and not physically justified [START_REF] Katz | Statistics of extremes in hydrology[END_REF].

The assumptions of the Index Flood model need often to be relaxed to suit the observations. For this purpose, [START_REF] Gabriele | A hierarchical approach to regional flood frequency analysis[END_REF] proposed a hierarchical approach to RFFA. The skewness is still supposed to be constant on the whole region, but the coefficient of variation and the mean annual flood can vary slowly from one subregion to another. However, the two authors underlined the practical difficulty to delineate these subregions.

In the Index Flood model, each observation from any site within the region have the same weight. However, it seems not optimal as, obviously, the most precious information come from the target site. Indeed target data -even short -are the only one which are "really" distributed as the target site.

We suggest here to carry out a Bayesian approach that encompasses the classical Index Flood model and uses the whole data in a more efficient manner. In summary, the proposed Bayesian approach differs from the Index Flood model as it: a) uses the at-site information in a more efficient way since this approach distinguishes the target site data and the regional data; b) does not impose a purely deterministic relationship between sites within the region.

The main goal of this article is to test the efficiency and robustness of the developed regional Bayesian model when dealing with short record length series. For this purpose, classical frequency analysis i.e. local and traditional RFFA will be compared to the suggested regional Bayesian approach. Section 2 presents a brief summary of the classical Index Flood model. Relevant theoretical aspects of Bayesian theory are introduced and applied to flood modelling in a RFFA context in Section 3. Section 4 describes the data set used to illustrate the method. Section 5 describes the procedure used to elicit the prior distribution. Section 6 outlines the weaknesses and strengths of each approach on a typical homogeneous region. Finally section 7 presents an analysis of the effect of homogeneity level on quantile estimation.

2 The Index Flood model

The Index Flood method states that flood frequency distributions within a particular region are supposed to be identical when divided by a scale factor -namely the Index Flood. Mathematically, this assumption is expressed as:

Q (S) = C (S) Q (R) (3)
where Q (S) is the quantile function at site S , C (S) the Index Flood at site S and Q (R) the regional quantile function i.e. the dimensionless quantile function valid across the homogeneous region.

Equation ( 3) is the core of the model and leads to strong constraints concerning at-site distribution parameters. Consequently, the shape parameter is the same throughout the homogeneous region, whereas the location and scale parameters have simple scaling behaviour -see Appendix A.

Equation ( 3) is supposed to be satisfied if all sites are hydrologically and/or statistically similar. Therefore, one of the main aspects of this approach is to identify a homogeneous region which includes the target site.

Similarity in basin characteristics is necessary but not sufficient to ensure the homogeneity of the region regarding the statistics of the flood peaks. Hosking andWallis (1993, 1997) suggested a heterogeneity measure H 1 to assess if a region is "acceptably homogeneous" (H 1 < 1), "probably heterogeneous"

(1 ≤ H 1 < 2) or "definitively heterogeneous" (H 1 ≥ 2). Note the case H 1 ≤ 0 seems to detect correlations between sites within the region.

Once the region satisfies the homogeneity test of Hosking andWallis (1993, 1997), the regional flood frequency distribution and the related at-site distribution is computed in a classical way. That is, by fitting the regional distribution to the weighted mean of sample L-moments. Details for computing heterogeneity statistics, regional flood frequency and at-site distribution can be found in (Hosking andWallis, 1993, 1997).

By definition of the Index Flood model, it can be seen that any realisations of each samples have the same weight. Giving equal weights to all site observations is debatable since the most relevant information is certainly the target site one. Relevance of the target site information is obvious as this is the only one which is "really" distributed as the target site. Thus, in this approach the available information is not efficiently used.

Regional Bayesian model

The Bayesian concepts have already been applied with success to the regional frequency analysis of extreme rainfalls [START_REF] Coles | Anticipating catastrophes through extreme value modelling[END_REF] and floods [START_REF] Madsen | Generalized least squares and empirical bayes estimation in regional partial duration series index-flood modeling[END_REF][START_REF] Northrop | Likelihood-based approaches to flood frequency estimation[END_REF]. Regional information is not used to build a regional distribution but to specify a kind of "suspicion" about the target site distribution. This is easily achieved in the Bayesian framework through the so called prior distribution.

The main goal of Bayesian inference is to compute the posterior distribution. The posterior distribution π(θ|x) is given by the Bayes Theorem:

π (θ|x) = π (θ) π (x; θ) Θ π (θ) π (x; θ) dθ ∝ π (θ) π (x; θ) (4)
where θ is the vector of parameters of the distribution to be fitted, Θ is the space parameter. π (x; θ) is the likelihood function, x is the vector of observation and π (θ) is the prior distribution.

In theory, the posterior distribution is entirely known but is often insolvable -because of the integral.

One of the solutions is to fix a prior model which leads to an analytical -or semi-analytical -posterior distribution and which allows the posterior distribution to be computed more easily [START_REF] Parent | Encoding prior experts judgments to improve risk analysis of extreme hydrological events via pot modeling[END_REF]. Nevertheless, the most convenient way is to implement Markov Chain Monte Carlo (MCMC) techniques to sample the posterior distribution. This approach avoids using a purely artificial prior model with no theoretical and/or physical justifications.

For our application, the likelihood function corresponds to the GP distribution as peaks over a threshold are of interest. From Eq. ( 4), if the prior distribution is known, posterior distribution can be computed -up to a constant. The next section describes how to define the prior distribution.

Prior Distribution

The prior model is usually a multivariate distribution which must represent beliefs about the distribution of the parameters i.e. µ, σ and ξ prior to having any information about the data.

As the proposed model is fully parametric, the prior distribution π(θ) is a multivariate distribution entirely defined by its hyper parameters. In our case study, the marginal prior distributions were supposed to be independent lognormal for both location and scale parameters and normal for the shape parameter. Thus,

π(θ) ∝ J exp (θ ′ -γ) T Σ -1 (θ ′ -γ) (5)
where γ, Σ are hyper parameters, θ ′ = (log µ, log σ, ξ) and J is the Jacobian of the transformation from θ ′ to θ, namely J = 1/µσ. γ = (γ 1 , γ 2 , γ 3 ) is the mean vector, Σ is the covariance matrix. As marginal priors are supposed to be independent, Σ is a 3-3 diagonal matrix with diagonal elements d 1 , d 2 , d 3 .

Estimation of the hyper parameters

Hyper parameters are defined through the Index Flood concept. Consider all sites of a region except the target site -say the j-th site. A set of pseudo target site parameters can be computed:

μ(i) = µ (i) * C (j) (6) σ(i) = σ (i) * C (j) (7) ξ(i) = ξ (i) * (8)
for all i = j, where C (i) is the at-site Index Flood and µ (i) * , σ (i) * , ξ (i) * are respectively the location, scale and shape at-site parameter estimates from rescaled sample.

Under the hypothesis of the Index Flood model, pseudo parameters μ(i) , σ(i) , ξ(i) for i = j are expected to be similar to the target site distribution parameters. Note that, information from the target site sample is not used to elicit the prior distribution. Thus, C (j) in equations ( 6) and ( 7) must be estimated without use of the j-th site sample.

From these pseudo parameters, hyper parameters can be computed:

γ 1 = 1 N -1 i =j log μ(i) , d 1 = 1 N -1 i =j V ar log μ(i) (9) γ 2 = 1 N -1 i =j log σ(i) , d 2 = 1 N -1 i =j V ar log σ(i) (10)
γ 3 = 1 N -1 i =j ξ(i) , d 3 = 1 N -2 i =j ξ(i) -γ 3 2 (11)
It is important at this step to incorporate the uncertainties on the elicitation of the prior distribution. Indeed, it may avoid problems related to misleading information resulting from a region not so homogeneous and moderating a "suspicion" that may be too true.

For this purpose, two types of uncertainties are taken into account: the one from parameter estimation, and the other one from target site Index Flood estimation. Thus, hyper parameters γ 1 and γ 2 are estimated differently than γ 3 as pseudo parameters for location and scale parameters depends on the target site Index Flood. Under the hypothesis of independence between C (j) and µ (i) * , σ (i) * the variance terms in Eq. ( 9) and ( 10) are computed according these two types of uncertainties:

V ar log μ(i) = V ar log C (j) + V ar log µ (i) * (12) V ar log σ(i) = V ar log C (j) + V ar log σ (i) * (13)
The independence assumption between C (j) and µ

(i) * , σ (i) * is not too restrictive as the target site Index Flood C (j) is estimated independently from µ (i) * , σ (i) * . Note that V ar log • (i) *
are estimated thanks to Fisher Information and the Delta method. Estimation of Var log C (j) is a special case and depends on the method for estimating the at-site Index Flood. Nevertheless, it is always possible to carry out an estimation of this variance, at least through standard errors.

Specificities of the proposed prior model

The construction of the prior distribution with the regional information was already suggested by [START_REF] Northrop | Likelihood-based approaches to flood frequency estimation[END_REF]. Nevertheless, the location parameter -or equivalently the threshold in the GP case -was supposed to be known. Yet, from a theoretical point of view, the location parameter can not be known prior to having any information from the target site sample. [START_REF] Northrop | Likelihood-based approaches to flood frequency estimation[END_REF] developed a similar approach based on the Index Flood but uncertainty associated with the scale factor prediction was not considered. The prior distribution was elicited directly from the distribution of the "pseudo target site" estimates (μ (i) , σ(i) , ξ(i) ). In this perspective, "pseudo target site" estimates are viewed as constant and not as random variables. When dealing with sites with a long record, uncertainties on parameter distributions are low. On the contrary, this have a much more impact for the Index Flood as uncertainties are as much larger as the target site Index Flood is estimated without use of at-site data -even with long record length sites. Note that if the prior distribution is overly accurate, estimation and credibility intervals are influenced. For these reasons and unlike the approach proposed by [START_REF] Northrop | Likelihood-based approaches to flood frequency estimation[END_REF], the target site Index Flood in the proposed methodology is considered to be a random variable and not a constant.

Thus, our prior distribution is not too falsely "tight fit". But it reflects "real" beliefs about target site behaviour without any use of target site sample. [START_REF] Madsen | Generalized least squares and empirical bayes estimation in regional partial duration series index-flood modeling[END_REF] and [START_REF] Fill | Using regional regression within index flood procedures and an empirical bayesian estimator[END_REF] both presented a regional empirical Bayesian estimator. Both models used conjugates families for prior distributions. However, even if conjugates families are convenient devices, they should not only be used just because computations are easier.

In their approaches, prior distributions are elicited with quantile regression on relevant physiographic characteristics.

Our approach differs differs from the two previous empirical Bayesian approaches (i.e target site sample is not used to elicit priors) and respects in that way absolutely the Bayesian theory. Moreover, conjugate priors are not considered, but priors are suited to the data. For example, the lognormal distribution for both location and shape parameters is justified by a physical and theoretical lower bound as: a) discharge data are naturally non negative; so the location parameter should also be non negative; b) the scale parameter is strictly positive by definition of the GP distribution.

This prior model is quite different from the one proposed by [START_REF] Coles | A bayesian analysis of extreme rainfall data[END_REF] who introduced a lognormal prior distribution only for the scale parameter. Note that it is possible to work with return levels [START_REF] Coles | A bayesian analysis of extreme rainfall data[END_REF] or return periods [START_REF] Crowder | Bayesian priors based on a parameter transformation using the distribution function[END_REF]) instead of working with distribution parameters. However, regional information is suited to work directly with distribution parameters. For other studies, such prior models could be of interest if "suspicion" is based on return levels or return periods.

Data description

Streamflow data were collected at 48 gauging stations in an area reaching from the 45 th to 47 th N and from the 3 rd to the 8 th E. The selection of the gauging sites was initially based on the 22 regions into which France is divided for the implementation of the Water Framework Directive [START_REF] Wasson | Hydro-ecoregions : a functional approach of river typology for the european water framework directive[END_REF]. Seven regions cover the area under study. These regions were manually delineated taking into account the spatial pattern of mean annual rainfall, elevation and underlying geology. All these variables might influence flood generation processes. Therefore this division is considered as a preliminary guide for pooling stations. According to [START_REF] Hosking | Regional Frequency Analysis[END_REF], pre-regions were slightly altered by identifying discordant sites while maximising the number of site within the region and meeting the heterogeneity test. Finally, a set of 14 stations was selected for this study. The heterogeneity statistic for this group is H 1 = 0.17 < 1. Consequently, the region be considered as "acceptably homogeneous".

The dataset includes seven tributaries to the Loire River and seven gauging stations located in the French part of the Rhône basin (Fig. 1, Tab. 1). The record length of the instantaneous discharge time series ranges from a minimum of 22 years to a maximum of 37 years, with a mean value of 32 years. The drainage areas vary from 32 to 792 km 2 . Moreover, most of the gauging stations monitored first-order stream catchments i.e. all but two pairs of catchments are unnested. The large majorities of flood in the region occur during autumn and winter and are caused by heavy liquid precipitation.

Partial duration flood series were extracted from the time series for each station. Fig. 2 illustrates time series for stations U4505010, U4635010 and V3015010 and their associated thresholds. Threshold levels were selected to extract in average around two events per year while meeting the criteria of independence between floods [START_REF] Lang | Towards operational guidelines for over-threshold modeling[END_REF].

Three stations U4505010, U4635010 and V3015010 were of particular interest because of their extended record length of 37 years. The time series of those three sites are displayed in Fig. 2.

In this case study, the scale factor was set to correspond to the 1-year return flood quantile -or equivalently the quantile associated with probability of non exceedance 0.5. Thus, our choice for the Index Flood 60 1978-2003 1975 1980 1985 1990 1995 2000 2005 Years is close to the sample median which was the reference in [START_REF] Robson | Flood Estimation Handbook[END_REF] but differs from [START_REF] Hosking | Regional Frequency Analysis[END_REF] where the sample mean was used. This particular choice for the Index Flood is not unintentional as estimating the quantile with probability of non exceedance 0.5 is more robust than estimating the sample mean. Analysing the influence of Index Flood selection is beyond the scope of this work. The main point is to keep the same Index Flood throughout the case study to compare approaches on the same basis.

Elicitation of the prior distribution

To estimate the target site Index Flood, the most popular way is to develop an empirical formula that relates the flow statistic to geomorphological, land-use and climatic descriptors. This relationship is usually established by multivariate regression procedures. In our case study, we consider a simple model for which only one explanatory variable is introduced in the regression analysis: the drainage area. The power form model is adopted:

C = aA b ( 14
)
where A is the area of the catchment. Parameters a and b are computed through ordinary least square procedures on logarithmically transformed data.

However, more sophisticated models could be carried out. Nevertheless, for our case study, observations demonstrate that Eq. ( 14) is a good parametrisation for estimating the Index Flood. Fig. 3 and4 illustrate the efficiency of the regressive and prior model for site U4505010 for which: Ĉ = 0.12A 1.01 R 2 = 0.94 (15)

Regional information was incorporated in the prior distribution through the Index Flood model. Moreover, uncertainties in the prior distribution were incorporated. Thus, the prior information is, on one

Performance of the Bayesian model on a homogeneous region

When making classical inference on small samples, the uncertainties may be too large. If an extremal event or too many "regular events" in this short record period are present, estimation can be impacted. It could lead to a dramatic overestimation or underestimation of quantiles corresponding to different return levels. A perfect model is expected : a) to perform well enough even with small samples; b) to be robust enough when an extreme event occurs in the sample; c) to be robust enough when too many "regular" events occur in the sample.

In this section, three different models will be applied. For this purpose, the three stations U4505010, U4635010 and V3015010 were selected to assess the robustness and efficiency of the local, regional and Bayesian regional models. These three different approaches correspond to : a) local: fit the GP distribution to the peaks over threshold data with the Maximum Likelihood Estimator (MLE), Unbiased Probability Weighted Moments (PWU) and the Biased Probability Weighted Moments (PWB); b) regional (REG): fit a regional GP distribution as described in section 2 and obtain the target site distribution; c) regional Bayesian (BAY): elicit the prior density from regional information, then compute the posterior density through MCMC techniques. As an illustration of MCMC output, Fig. 5 displays the prior and posterior marginal densities for the GP parameters of the proposed model. Marginal posterior distribution obtained from an uninformative prior model are also displayed. That is with the same prior model but with a large variance -i.e. d i = 1000, i = 1 . . . 3. Fig. 5 shows the relevance of regional information as the proposed prior model is clearly more accurate than an analysis directly from data. Moreover, for the proposed model and even with only 5 years record length, marginal posterior densities are more accurate than marginal prior densities -except for the shape parameter. Thus, combination of regional and target site information at two different stages is worthwhile, even when only few data are available. Location parameter is a special case as the modes of both marginal prior and posterior densities seem to be significantly dissimilar.

As the main goal of this work is to compare models on small samples, efficiency will be evaluated on sub-samples from the original data. Local Maximum Likelihood Estimation on the whole sample will be used as benchmark to assess the performance of each model. This particular case will be denoted THEO in the following sections. The choice of MLE estimate as a benchmark value is reasonable because of its theoretical motivation and asymptotic efficiency. Moreover, the MLE approach allows the calculation of profile confidence intervals. This is a key point as these profile confidence intervals are often more accurate than those based on the Delta Method and Fisher Information [START_REF] Coles | An Introduction to Statistical Modelling of Extreme Values[END_REF].

Furthermore, as interpretation on quantile estimates is more natural than for distribution parameter estimates, the analysis will focus on quantiles corresponding to return period 2, 5, 10 and 20 years. Benchmark values for these quantiles -and their associated 90% profile likelihood confidence intervals 10.8(10.1, 11.7) 15.3 (13.9, 17.4) 19.5(17.2, 23.4) 24.4 (20.6, 31.5) U4635010 33.0(30.0, 36.5) 52.2 (45.5, 62.5) 72.2(60.2, 95.4) 98.9(69.2, 200.5) V3015010 7.5 ( 6.9, 8.3) 11.7 (10.4, 13.7) 15.9(13.6, 19.9) 21.3 (17.3, 28. Moreover, for such return periods, benchmark values are quite equivalent to those obtained with PWM estimates -with a mean bias of 0.89%. So, performance of each model is not too much impacted by the choice of the MLE estimator for benchmark values.

Q 5 Q 10 Q 20 U4505010
Different frequency curves for site V3015010 with only the last 15 years recording are displayed in Fig. 6. Let us focus on the largest observation. Return period related to this event is very high for the REG approach. All other models lead to significantly lower return periods. This flood event is extreme at regional scale but not anymore in a local context. This underestimation is due to the misuse of the target site sample to establish the regional distribution. On the other side, the regional Bayesian model performs well for all return periods. Indeed, Fig. 6 indicates that the return level curve is very similar to benchmark one. This is quite logical as it adds up the advantage of using efficiently the target site sample and a good "suspicion" on the global behaviour of the flood peak distribution thanks to the so-called prior distribution. Local approaches suggest a very heavy tail as the extremal event of year 2004 (see Fig. 2, right panel) was in the last sequence of 15 years of records.

As one of the main goals of a RFFA procedure is to deal with small samples, the target site sample was truncated to obtain shorten periods of records of m years, m ∈ {5, 10, 15, 20, 25, 30, 37}. Robustness and efficiency of the methods to converge to the parameters of the target site distribution are measured. For this purpose, quantile estimates corresponding to return period 2, 5, 10, 20 years -corresponding to non-exceedance probabilities 0.75, 0.9, 0.95 and 0.975 respectively -are pointed. The evolution of Because of the extreme event observed in 1983 (see Fig. 2, middle panel), systematic underestimation of benchmark values for local and REG approaches can be noticed. This result shows that: on one hand, for small samples classical inference like MLE, PWB and PWU are too responsive if too many "regular" events occurred. On the other hand, for the Index Flood model, underestimation of quantiles is related to the underestimation of the scale factor C (S) in Eq. ( 3) because of these "regular" events. Only the Bayesian model performs well enough even with record lengths lower than 15 years.

A monotonic increase of the design flood estimates with the sample length can be noticed in Fig. 7. This behaviour is easily explained by Fig. 2, middle panel. Indeed, only the last part of the time series shows really extreme events. As the record length increases, much more extreme events occur leading to higher estimates. The Bayesian approach is the only one which does not really present this monotonic behaviour.

Moreover, the Bayesian approach is by far the most robust and accurate model as, on the whole range of record length, and for all benchmark values, estimation lies in the 90% profile likelihood confidence interval. This is not true with any other model. The advantage of incorporating regional information within a Bayesian framework is certainly to define a "restricted space" where distribution parameters belong to. Thus, the impact of a very extremal event -or conversely too many low-level events -should be regarded as an extreme event related to this "restricted space".

The gain of accuracy in the target site from using regional information is clearly established in section 6 (Fig. 6 and7). The Bayesian approach seems to be robust even with small samples while being accurate with larger sample. The poor performance of the REG model is related to a bad selection of sites within the "homogeneous" region being considered and estimates may be more accurate if "better" regions were considered. Unfortunately, building up such region is difficult because of the purely deterministic relation (3). As the Bayesian approach relaxes the REG model, the search for more homogeneous regions could 7 Effect of heterogeneity degree on quantile estimation

As indicated in the previous section, we focus now on the impact of the level of homogeneousness of the region. For this purpose, we consider four different regions -denoted He + , He, Ho and Ho + -which correspond to increasingly homogeneous regions according to the test of [START_REF] Hosking | Regional Frequency Analysis[END_REF]. The Ho region corresponds to the region analysed in the previous section and described in Tab. 1. All regions have 14 site except for the most homogeneous one Ho + which contains only 8 stations. He and He + regions are derived form Ho. One to five sites are withdrawn and replaced by other stations to obtain larger heterogeneity measure. The Ho + region is a sub-region of Ho. Heterogeneity statistics for these regions are summarised in Tab. 3.

To evaluate the influence of homogeneousness level of a region on quantile estimation, models are assessed using two performance criteria : the Normalised Bias (NBIAS) and the Normalised Root Mean Squared Error (NRMSE). These indices are defined as follows :

N BIAS = 1 k k i=1 Qi -Q Q (16) N RM SE = 1 k k i=1 Qi -Q Q 2 ( 17
)
where k is the number of estimates of Q and Qi is the i-th estimate of the benchmark value Q. To compute these two indices, we fit all models on all trimmed periods of size m years -m ∈ {5, 10, 15, 20, 25, 30}. Moreover, the overall performance of each model is evaluated using a rank score. This technique was already used to compare different models in [START_REF] Shu | Artificial neural network ensembles and their application in pooled flood frequency analysis[END_REF].

To calculate the rank score, the p models are ordered thanks to their performance indices -1 corresponding to the best model and p to the worst. For each model, the scores for the different criteria are summed to obtain the overall rank score R o for the model. For convenience, the overall rank score R o is standardised in such a way that in lies within the interval [0, 1]:

R s = pq -R o pq -q (18)
where p is the number of models being considered, and q the number of indices. A standardised rank score close to 1 -resp. 0 -is associated to a model with a goodresp. poor -performance.

Three quantiles are of particular interest Q 5 , Q 10 and Q 20i.e. associated to probability of nonexceedance 0.9, 0.95 and 0.975 respectively. N RM SE, N BIAS and the standardised rank score for station U4635010 and a record length of 5 years are illustrated in Tab. 4. Notations for different models in this table consist of one lowercase letter referring to the Bayesian approach b or Regional Index Flood r and the denomination of the homogeneity degree of the region. Only the M LE model does not use these notations as it is completely independent of the homogeneity level.

Results from Tab. 4 demonstrate that the Bayesian model performs quite well independently of the region being considered. However, this model seems to perform even better when applied to a "acceptably homogeneous" or "probably heterogeneous" region. For the Ho + region, the Bayesian approach performs poorly. This may be explained by the fact that the prior distribution is too informative and probably not consistent with the target site sample. This comment is yet not discrepant with the good overall performance of the REG model on this region. Indeed, as the prior distribution is elicited using equations ( 6)-( 8), and the scale factor C (j) is estimated without any use of the target site sample, this can lead to a misleading prior distribution while the REG model performs well. The bad estimation of the scale factor is less important with a more heterogeneous region as the prior information is less informative, thus the Bayesian model performance is not highly impacted.

On the other side, the overall rank score of the REG model increases with the homogeneity degree of the region. Yet, the overall rank score for the REG model never exceeds the value of 0.6 -reached for the Ho + region. This value remains much lower than the best rank score for the Bayesian modeli.e. 0.85. These results corroborate the superiority of the Bayesian approach.

From Tab. 4, two conclusions can be established. On one hand, for small samples, the Bayesian approach is the most competitive model. On the other hand, results seem to indicate that there is no need to keep increasing the homogeneousness of the region as it increases the risk of being too confident in the "homogeneous region" without increasing significantly the efficiency of the model.

These results are in line with similar results obtained for stations U4505010 and V3015010, except for the bad behaviour of the Bayesian model on the Ho + region. Indeed, for the other stations, the Bayesian model remains more efficient than the REG model within the Ho + region. However, its overall rank score remains stable through out the different region -He, Ho and Ho + . The "risk" to deal with too much homogeneous region -as Ho + -is also corroborated as the overall rank score for the Index Flood model for station U4505010 decreases dramatically until 0.06. Thus, the Index Flood for the Ho + region performs quite well for stations U4635010 and V3015010, while very surprisingly badly with U4505010.

In Fig. 8, the evolution of the overall rank score as a function of the record length is illustrated for station V3015010. The left panel corresponds to the REG part, while the right one stands for the Bayesian approach. In both panel, the M LE score is also presented. Fig. 8 indicates that the evolution of the overall rank score is more stable for regional models -that is REG and Bayesian models -than for the M LE. Furthermore, the benefit of increasing the homogeneity degree of the region is more relevant for the REG model than for the Bayesian model. Nevertheless, the worst Bayesian rank score is always quite close to the best REG rank score. This seems to indicate the superiority of the Bayesian approach. This last point is corroborated with the results corresponding to stations U4505010 and U4635010 except for the bHo + model for station U4635010 because of the bad estimation of the scale factor C (j) -as denoted earlier. The effect of bad estimation of the target site Index Flood on prior and thereby on posterior distributions is depicted in Fig. 9. From Fig. 9, it is overwhelming that the prior model is not appropriate -particularly for the location parameter. Prior for the shape parameter is not too false as it does not depend on the target site Index Flood estimate.

As the record length increases, the M LE model becomes more and more efficient. In particular, for record lengths greater than 15 years, it is more effective than rHe + , rHe and rHo models. On one hand, for record lengths smaller than 15 years, M LE is always less efficient than Bayesian approaches and even significantly for bHe, bHo and bHo + models. This is quite logical as Bayesian estimation can be looked at as a restrictive maximum likelihood estimator -restriction being defined by the prior distribution. So, under the hypothesis that the prior distribution is well-defined, the "restrictive estimator" is unbiased and has a smaller variance. On the other hand, for record lengths greater than 15 years, M LE, bHe and bHo seems to be equivalent.

Conclusion

A framework to perform a regional Bayesian frequency analysis for partially gauged stations is presented. The proposed model has the advantage of being less restrictive than the most widely used regional model, that is the Index Flood. Several case studies from French sites were analysed to illustrate the superiority of the Bayesian approach in comparison to the traditional Index Flood and to local approaches. The influence of the homogeneousness level of the pooling group on quantile estimates was also considered. Results demonstrate that working with quite large and homogeneous regions rather than small and strongly homogeneous regions is more efficient. Further work can focus on the regional estimation of other characteristics of the flood hydrograph. For instance, a regional Bayesian model can focus on Flood Duration Frequency.

All statistical analysis was carried out in the [START_REF] Development | R: A language and environment for statistical computing[END_REF] framework. For this purpose, two packages were contributed to this software under the framework of the present research work. These two packages integrate the tools that were developed to carry out the modelling effort presented in this paper. The first one POT performs statistical inference on Peaks Over Thresholds, while the second one, RFA, contains several tools to carry out a Regional Frequency Analysis. These two packages are available, free of charge, at the web site http://www.R-project.org, section CRAN, Packages.

Figure 1 :

 1 Figure 1: Location of the gauging stations within the studied area

Figure 2 :Figure 3 :Figure 4 :

 234 Figure 2: Times series for sites U4505010, U4635010 and V3015010 and thresholds associated

Figure 5 :

 5 Figure 5: Proposed prior (dashed line), proposed posterior (solid line) and posterior from an uninformative prior (dotted line) marginal densities for GP parameters. Site U4505010 with 5 years record length. Vertical lines denotes benchmark values.

Figure 6 :

 6 Figure 6: Comparison of frequency curves for site V3015010 with 15 last years recording

Figure 7 :

 7 Figure 7: Evolution of Q 2 , Q 5 , Q 10 , Q 20 estimates as the size increases for the site U4635010 and 90% profile likelihood confidence interval for the benchmark values -light blue area

Figure 8 :

 8 Figure 8: Score evolution as a function of record length for Station V3015010. (a) REG scores, (b) BAY scores

Figure 9 :

 9 Figure 9: Effect of bad estimation of target site Index Flood on marginal prior and posterior densities. Site U4635010 with 10 years record length.

Table 1 :

 1 Characteristics of the stations of the homogeneous region

	Code	Station	Area km 2	X (km) Y (km)	Record
	K0624510 The Bonson river at St Marcellin	104	744.72 2053.90 1971-2003
	K0663310 The Coise river at Larajasse	61	770.67 2072.11 1971-2003
	K0704510 The Toranche river at St Cyr	62.3	752.63 2076.68 1977-2003
	K0813020 The Aix river at St Germain Laval	193	729.48 2093.71 1973-2002
	K0943010 The Rhins river at Amplepuis	114	754.52 2111.10 1973-2003
	K0974010 The Gand river at Neaux	85	743.45 2107.75 1972-2003
	K1004510 The Rhodon river at Perreux	32	738.40 2116.64 1973-2003
	U4505010 The Ardieres river at Beaujeu	54.5	773.67 2130.75 1969-2003
	U4624010 The Azergues river at Chatillon	336	779.07 2099.72 1970-2003
	U4635010 The Brevenne river at Sain Bel	219	775.90 2092.57 1969-2003
	U4644010 The Azergues river at Lozanne	792	782.56 2098.09 1981-2003
	V3015010 The Yzeron river at Craponne	48	785.47 2084.50 1969-2003
	V3114010 The Gier river at Rive de Gier	319	780.54 2062.67 1981-2003
	V3315010 The Valencize river at Chavanay	36	786.54 2048.

Table 2 :

 2 Benchmark values for 2, 5, 10 and 20 years quantiles and the associated 90% profile likelihood confidence intervals in bracket Station Q 2

Table 3 :

 3 Heterogeneity statistics for the four region considered -statistics in bracket are obtained with the scale factor taken to be the 1-year quantile corresponding to non-exceedance probability 0.5

	Region	He +	He	Ho	Ho +
	H 1	7.11 (6.83) 1.35 (1.37) 0.17 ( 0.08) -0.60 (-0.67)
	H 2	3.46 (3.38) 1.00 (1.03) 0.41 ( 0.33) -1.28 (-1.31)
	H 3	1.40 (1.45) 0.30 (0.28) -0.09 (-0.14) -1.14 (-1.18)
	be ineffective. The goal of the next section is to measure the potential gain, for the Bayesian model,
	against homogeneity property.

Table 4 :

 4 Estimation of N RM SE and N BIAS for station U4635010 with a record length of 5 years

			Model	Q 5	N RM SE Q 10	Q 20	Q 5	N BIAS Q 10		Q 20	Rank Score
			M LE 0.33	0.34	0.39	0.01	-0.09 -0.18	0.26
			bHe + 0.16	0.13	0.18	0.09	-0.02 -0.13	0.65
			rHe + 0.27	0.30	0.37	-0.12	-0.22 -0.31	0.18
			bHe	0.10	0.07	0.11	0.08	0.00 -0.09	0.85
			rHe	0.27	0.26	0.28	-0.03	-0.10 -0.17	0.43
			bHo	0.14	0.09	0.08	0.12	0.05 -0.02	0.76
			rHo	0.27	0.26	0.27	0.01	-0.06 -0.12	0.58
			bHo + 0.29	0.28	0.25	0.29	0.27	0.25	0.19
			rHo + 0.28	0.27	0.26	0.02	-0.01 -0.04	0.60
			MLE rHe +	rHe rHo	rHo +				MLE bHe +	bHo bHe	bHo +
	Rank Score	0.0 0.2 0.4 0.6 0.8 1.0					Rank Score	0.0 0.2 0.4 0.6 0.8 1.0
		5	10	15 Record length (Years) 20	25	30		5	10	15 Record length (Years) 20	25	30
					(a)						(b)

Acknowledgements

The authors wish to thank the DIREN Rhne-Alpes for providing data. The authors are also very grateful to the two referees for their constructive remarks which improve the document.

Appendix A. Properties of the Index Flood on GP parameters

We provide in this appendix the proof for the following theorem:

Theorem 1. Let X be a random variable GP distributed. So X has the Cumulative Distribution Function defined by:

Then, Y is also GP distributed with parameters (Cµ, Cσ, ξ).

Proof. Let X be a r.v. GP distributed with parameters (µ, σ, ξ) and Y = CX where C ∈ R + * . Then:

So, Y is also GP distributed with parameters (µC, σC, ξ). The proof for the GEV case can be established in the same way.