Wei Zhao
email: zhaowei203@buaa.edu.cn

Piero Baraldi
email: piero.baraldi@polimi.it

Enrico Zio
email: enrico.zio@ecp.fr

Confidence in Signal Reconstruction by the Evolving Clustering Method

Keywords: confidence measure, signal reconstruction, condition monitoring, evolving clustering method I

Monitoring the health conditions of equipment allows supplying advanced warning of their incipient failures; this can provide evidence useful to maintenance and replacement practices. However, uncertainties in the signal measurements and incompleteness in the representativeness of the measured data can overshadow the conclusions drawn from condition monitoring, and possibly lead the decision-maker to take wrong actions. In order to reduce the risk of wrong actions, confidence measures on the condition monitoring indications of the state of a component must be provided, so that the decision-maker can know to what degree he or she should trust such indications. As condition monitoring is usually structured in two modules performed in succession, one of reconstruction of the signal values in normal operating conditions and a following one of equipment health state diagnosis, it is reasonable to define confidence measures for the two processes individually, and then integrate the two into a single criterion for the whole condition monitoring. The research presented in this paper focuses on the definition of confidence measures for the signal reconstruction part of condition monitoring. The Evolving Clustering Method (ECM) is adopted to build the empirical model of signal reconstruction. Requirements for the reconstruction confidence are originally defined, and a single confidence measure is proposed to meet all the requirements identified. The confidence measure is analyzed with respect to two-dimensional artificial datasets and a real dataset concerning the Reactor Coolant Pump of a French Pressurized Water Reactor. The results obtained show that the proposed confidence measure meets all requirements and is more informative than the reconstruction error.

INTRODUCTION

Systems for condition monitoring of the health state of an equipment are often based on empirical models of signal regression whose performances may vary depending on the density and information content of the example signal patterns available to train the models [START_REF] Baraldi | Signal Grouping for Condition Monitoring of Nuclear Power Plants Components[END_REF][START_REF] Hines | Development and Application of Fault Detectability Performance Metrics for Instrument Calibration Verification and Anomaly Detection[END_REF][START_REF] Chevalier | Assessment of Statistical and Classification Models For Monitoring EDF's Assets[END_REF][START_REF] Reifman | Survey of artificial intelligence methods for detection and identification of component faults in nuclear power plants[END_REF][START_REF] Roverso | Solutions for plant-wide on-line calibration monitoring[END_REF][START_REF] Gross | Sequential probability ratio test for nuclear power plant component surveillance[END_REF][START_REF] Baraldi | An ensemble approach to sensor fault detection and signal reconstruction for nuclear system control[END_REF]. Furthermore, the stochastic behavior of the processes and the signal measurements errors can overshadow the health-state conclusions drawn from condition monitoring, and possibly lead the decision-maker to take wrong actions. Given the criticality of these actions, it is important to provide the decision-maker with a measure of confidence on the condition monitoring system outcome [START_REF] Reifman | Survey of artificial intelligence methods for detection and identification of component faults in nuclear power plants[END_REF].

In this respect, the confidence measure should be useful to recognize the two potentially dangerous cases of missing and false alarms. In the first case, the condition monitoring indicates that an unhealthy equipment is operating in normal conditions; this may cause an unexpected failure of the equipment with associated long downtimes, high costs and possible safety and environmental problems. Contrarily, in case of false alarms, an healthy state of the equipment is erroneously recognized as unhealthy; this may lead to an unnecessary stop of the equipment operation with the associated loss of production, and may subtract resources to deal with other actual failures.

A typical scheme of condition monitoring can be described as follows. Historical data of equipment condition under normal operation are used to build a model (often empirical). The model is auto-associative in that it reconstructs the values of the signals measured to characterize the equipment condition. During operation, the actually observed signal values are compared with those reconstructed by the auto-associative model and the difference is computed. Based on this deviation, a decision about the health state of the equipment is made. The above two actions are often named fault detection [START_REF] Baraldi | A Method for Estimating the Confidence in the Identification of Nuclear Transients by a Bagged Ensemble of FCM Classifiers[END_REF][START_REF] Jolliffe | Principal Component Analysis[END_REF].

Because the condition monitoring is composed by the two phases of signal reconstruction and health state diagnosis, it is reasonable to define the confidence measures for these two phases individually and then to integrate them into a single confidence measure on the whole condition monitoring process.

In this paper, the research is focused on the development of a confidence measure for the signal reconstruction phase. The Evolving Clustering Method (ECM) [START_REF] Chevalier | Assessment of Statistical and Classification Models For Monitoring EDF's Assets[END_REF] is adopted to build the empirical reconstruction model of the equipment behavior. On the basis of ECM, the special requirements for the reconstruction confidence are analyzed and a confidence measure is proposed to meet all the requirements.

The remaining parts of the paper are structured as follows. Section II describes the process of condition monitoring in detail and points out the uncertainties existing in the process. Section III presents the ECM algorithm including the training phase and the procedure used for the signal reconstruction. In Section IV, the requirements for a confidence measure of signal reconstruction are discussed and a corresponding measure able to meet the requirement is proposed. In Section V, the proposed confidence measure is applied to both twodimensional artificial datasets and a real dataset concerning the condition monitoring of a Reactor Coolant Pump of a French Pressurized Water Reactor. Finally, Section VI states some conclusions and draws on potential future steps of the work.

II. CONDITION MONITORING

Fig. 1 shows the general framework used for condition monitoring of an equipment. This is typically based on:

1) a signal reconstruction module, 2) a diagnostic decision module.

The former receives in input the vector containing the q sensor measurements  

(1), ..., ( ) obs obs obs x x x q  and provides in output the signal values expected in case of normal condition

 

ˆˆ(1),..., ( )

nc nc nc x x x q 
. This module is usually based on an auto-associative model of the component behavior in normal conditions, obtained by techniques such as Principal Component Analysis (PCA) [START_REF] Jolliffe | Principal Component Analysis[END_REF], Auto-Associative Kernel Regression Method (AAKR) [START_REF] Baraldi | Signal Grouping for Condition Monitoring of Nuclear Power Plants Components[END_REF], Auto-associative Neural Networks [START_REF] Kramer | Autoassociative Neural Networks[END_REF], Evolving Cluster Method [START_REF] Chevalier | Assessment of Statistical and Classification Models For Monitoring EDF's Assets[END_REF], trained with data collected during operation in normal conditions. and observed values  

(1),..., ( )

obs obs

x x q to decide whether the system is in normal or abnormal conditions. In case of normal conditions, the measured values are expected to be very similar to the model reconstructed ones, i.e., the residuals are small; on the contrary, under abnormal conditions the measurements tend to deviate from the reconstruction allowing detection of the abnormality.

However, incompleteness of the training data, intrinsic stochasticity of the plant processes and measurement noises may lead to wrong diagnostic decisions on the system health state. For this reason, it is important to develop a confidence measure on the condition monitoring indications of the state of the equipment, so that the decision-maker can know to what degree he or she should trust such indications to take actions.

Proceeding systematically through the two successive phases of condition monitoring, the overall confidence on the condition monitoring,

III. SIGNAL RECONSTRUCTION BY EVOLVING CLUSTERING METHOD (ECM)

The algorithm considered in this work for reconstructing the equipment behavior in normal conditions is based on a clustering method called Evolving Clustering Method (ECM) [START_REF] Chevalier | Assessment of Statistical and Classification Models For Monitoring EDF's Assets[END_REF]. For completeness of the paper, Section III.A briefly describes the basics of ECM, whereas Section III.B is dedicated to the reconstruction procedure based on ECM.

A. The ECM algorithm

Given a training dataset

 

(1),..., ( )

obs nc bs nc bs nc i i i x x x q    
recorded during past operation of the equipment in normal conditions, the ECM algorithm provides a procedure to group the training patterns into clusters. The application of the algorithm requires to a priori fix the value of the parameter Dthr which defines the maximum allowed cluster radius. The clusters are then found by performing the following steps:

Step 0: Assume the center of the first cluster 1 v equal to the first pattern in the dataset 1 obs nc

x  , the corresponding cluster radius R1=0, and set the pattern counter i=1 and the cluster counter m=1.

Step 1: If all the patterns of the dataset have been processed (i=n), exit. Otherwise, i=i+1, consider the i-th pattern of the dataset obs nc i

x  and compute its distance Dik from the centers k v , 1,..., km  of all the already formed clusters:

m k v x D k nc obs i ik ,..., 1 , 2      (1)
Although the distance computation can be based on weighted distances

      q j k i k nc obs i j v j x j p v x 1 2 )) ( ) ( ( * ) (  (2)
with the weights p(j) related to the importance of signal j in the equipment monitoring, in this work the Euclidean distance is considered, i.e. all the signals are given equal weights

2 1 ( ( ) ( )) q obs nc i k i k j x v x j v j       (3)
Step 2: If, among the found distance values Dik, there are distances lower or equal to the corresponding cluster radii, i.e.

m k v x k k nc obs i k ,..., 1 , min arg 0       (4)
and go to step 1. Otherwise, if the i-th pattern does not belong to any cluster, perform steps 3, 4 and 5.

Step 3: For all m existing cluster centers, compute the values Sik=Dik+Rk, k = 1,… ,m and identify the cluster ka characterized by the minimum value of Sik:

m k S k k ik a ,..., 1 min arg   (5)
Step 4: If Notice that this procedure guarantees that the maximum possible distance between a pattern of the training dataset and the nearest cluster center is lower or equal to the threshold value Dthr.

B. Reconstruction Procedure

Once the m clusters have been identified, the reconstruction

 

ˆˆ(1),..., ( )

nc nc nc x x x q  of a test pattern

 

(1),..., ( )

obs obs obs x x
x q  is based on the following two steps:

1) identification of the cluster with the smallest distance from the test pattern. This is done by computing the distance ) , (

k test v x d  
of the test patterns with all the m cluster centers:

m k v x v x d k test k test ,..., 1 , ) , ( 2        (6)
and selecting as nearest cluster closest k the one with the minimum distance:

  m k v x d k k test k closest ,..., 1 , ) , ( min arg     (7) 2)
Reconstruction of the test pattern as the nearest cluster center:

closest k obs v x    ˆ (8)
Notice that contrary to other algorithms, such as AAKR which requires to perform the reconstruction of a test pattern to have the accessibility to all the training patterns, the ECM reconstruction is based only on the cluster centers and thus it does not require the continuous accessibility of the training set.

IV. CONFIDENCE MEASURE

In this Section, a measure of the degree of confidence of the signal reconstruction performed by using the ECM algorithm described in the previous Section III.A is proposed. Basically, the objective is to answer to the questions: how accurate is the reconstruction expected to be? To what degree can we trust the reconstruction?

To this purpose, notice that the metrics "accuracy" and "robustness" proposed in literature [START_REF] Baraldi | Signal Grouping for Condition Monitoring of Nuclear Power Plants Components[END_REF][START_REF] Hines | Development and Application of Fault Detectability Performance Metrics for Instrument Calibration Verification and Anomaly Detection[END_REF] to estimate the overall performance of the reconstruction algorithm on a set of test patterns different from those used to train the model are not satisfactory for our objective, since the reconstruction performance is expected to vary in different zones of the training space. Thus, the degree of confidence in the reconstruction should not be a fixed quantity independent from the location of the test pattern, but should vary according to the density and information content of the example patterns available to train the model.

A. Requirements for a confidence measure

The following four requirements which take into account the position of the test pattern with respect to the clusters found by the ECM algorithm should be considered in order to evaluate the confidence in the reconstruction:

1) smaller is the distance between the test pattern and the nearest cluster center, higher should be the reconstruction confidence;

2) if the test pattern has nearly the same distance from two or more cluster centers, the reconstruction confidence should be low;

3) higher is the number of training patterns in the cluster nearest to the test pattern, higher should be the reconstruction confidence;

4) higher is the density of the training patterns in the cluster nearest to the test pattern, higher should be the reconstruction confidence.

Requirement 1) is motivated by the fact that if a pattern does not belong to a cluster or is far away from its center, its reconstruction as the center of the nearest cluster is expected to be not reliable. Requirement 2) considers that we should not be confident in the reconstruction of ambiguous patterns, i.e. patterns whose position is such that they can belong with the same confidence to two or more clusters. Finally, requirements 3) and 4) are related to the density and information content of the training data: the number of patterns in the cluster is considered since we are less confident in a cluster formed by one or few training patterns which, for example, may correspond to an abnormal equipment condition and be erroneously introduced in the training dataset for the model of normal behavior.

B. Definition of the confidence measure

Given a test pattern  

(1),..., ( ) obs obs obs x x x q  , the degree of confidence of its reconstruction

 

ˆˆ(1),..., ( )

nc nc nc x x x q  is defined by: 2 2 2 2 ( , ) 2 1 0 ( , ) 2 ( 1)
( , ) 1 1 2 ( , ) ( 1)
d x v obs k k m obs k k k k r d x v obs k n m d x v e n d x v if n n Conf m d x e                              1 0 ) ( , ) nearest nearest k m obs k k k v d x v if n n                                  (9) ( 9)
where:  is related to the distance of the test pattern to all the cluster centers. This term may be rewritten as: ( can be interpreted as the mean distance between the test pattern and all the cluster centers except the nearest one, the term in [START_REF] Jolliffe | Principal Component Analysis[END_REF] results highest for those test patterns very close to the nearest cluster center and far away from all the other clusters, whereas it reaches its minimum value (1/m) in the case in which the test pattern has the same distance to all the cluster centers. These characteristics are in accordance to requirements 1) and 2). This term is derived from [START_REF] Duin | Classifier conditional posterior probabilities[END_REF] where it is used for the estimation of the classification confidence of a k-nearest neighbor algorithm. to take into account the density of the nearest cluster to the test pattern. It depends from the square of the distance between the cluster center and the cluster, and the inverse of the quantity k  which is a measure of the cluster dispersion. Thus, once fixed the distance of the test pattern to the nearest cluster center, the degree of confidence in the reconstruction of the test pattern is higher for denser clusters as requested by requirement 4).

 k nearest
Finally, the factor k n n has been introduced in the confidence measure in order to meet requirement 3); basically, if the number of patterns in the nearest cluster is lower than the threshold value the confidence of the reconstruction is decreased of the factor k n n .

V. VERIFICATION OF THE CONFIDENCE MEASURE

In Section V.A some case studies based on artificial bidimensional datasets are designed in order to verify if the proposed confidence measure effectively meets in practice the four requirements of Section IV.A. Then, in Section V.B the proposed confidence measure is applied to a real case study and the obtained results commented.

A. Application to artificial case studies

In the first case study, two clusters, each one formed by 100 training patterns, have been generated from two bi-dimensional Gaussian distributions centered on (30, 100) and (70, 100), respectively, both with variance equal to 10 (red dots in Fig. 2(a)). The application of the ECM procedure with threshold parameter Dthr = 0.15 has lead to the identification of three clusters whose center position along the horizontal axis is reported by a circle in Fig. 2(b), one containing the patterns on the left side of Fig. 2(a) and two containing the patterns on the right side. The obtained clusters have been used for the reconstruction of test patterns on the horizontal straight line passing for the centers of the Gaussian distributions (crosses in Fig. 2(a)). Fig. 2(b) reports the confidence in the reconstruction. Notice that in accordance with the first requirement, the confidence increases as the distance between the test pattern and the closest cluster center decreases. Although the data have been generated from two Gaussian symmetric distributions, a non symmetric behavior of the confidence measure in the two sides of Fig. 2(b) has been found due to the fact that the ECM algorithm results in one cluster for the right Gaussian distribution patterns and two for the left Gaussian distribution patterns.

The training data used for the second case study have been generated by using the same bi-dimensional Gaussian distributions considered in case study 1, whereas the test set is formed by patterns on a circle (crosses in Fig. 3(a)) with center (30, 70) equal to the center of the left Gaussian distribution. Fig. 3(b) shows the variation of the confidence measure as a function of the angle θ scanning the circle starting at the dark dotted line in Fig. 3(a) (angle values increase in the counter clockwise direction). The results show that the degree of confidence achieved in the reconstruction of patterns which have the same distance from the center of the Gaussian distribution but different angles θ, tends to be different. In particular, pattern A at θ=0 o (360 o ) is reconstructed with a degree of confidence equal to 0.33 whereas pattern B at θ = 180 o with a degree of confidence equal to 0.39. This is in accordance with the second criterion of Section IV.A: the confidence on the reconstruction of pattern A is lower than that of pattern B since A is closer to the cluster on the right and thus more ambiguous. The peak of the confidence measure at θ=270 o is due to an additional cluster center found by the ECM in proximity of the test pattern at θ=270 o .

In the third case study, 100 test patterns positioned on a vertical line in the middle between the two clusters (Fig. 4 (a), crosses) are reconstructed by the ECM with threshold Dthr equal to 0.30. Fig. 4 (b) shows that the degree of confidence in the reconstruction of this pattern is lower than 0.3. This result confirms the fulfillment of the second requirement according to which patterns with the same distance to two cluster centers should be reconstructed with a low degree of confidence. Furthermore, according to the first requirement, as the distance to the cluster center increases (values on the vertical axis tend to 50 or 150), the confidence tends to decrease.

The fourth case study is similar to the third one except that the ECM threshold Dthr is decreased to 0.15 in order to allow the generation of small clusters formed by few patterns. Notice that in the case in which the nearest cluster to the test pattern is formed by a single training pattern, the confidence in the reconstruction becomes 0. This is the case of the two test patterns indicated by the circles in Fig. 5 (b).

In the last case study, the training patterns are taken from two Gaussian distributions centered on (20, 100) and (80, 100) and with variance 10 and 100, respectively (Fig 6(a)). The degrees of confidence in the reconstruction of the test patterns belonging to the cluster on the left, characterized by higher density, are higher than those of the less dense cluster on the right. The ECM parameter Dthr has been set equal to the value of 0.05 in an attempt to find an optimal compromise between a low value which would generate very accurate but low robust reconstructions (several clusters formed by few patterns) and an high value which would lead to less accurate but more robust reconstructions (few clusters formed by several patterns). Fig. 7 reports the variation of the degree of confidence with the distance between the test pattern and the nearest cluster center. In accordance to the first requirement, the confidence tends to decrease as the distance between the test pattern and the nearest cluster center increases. Notice that this distance is equal to the reconstruction error since the reconstruction of the test pattern coincides with the nearest cluster center (Section III.B). Furthermore, Fig. 7 shows that there are some patterns reconstructed with confidence 0. These test patterns are assigned to a cluster formed by a single pattern far away from the other patterns; thus, their reconstructions are not believed reliable in accordance to the third criterion. These low degrees of confidence are justified from the practical dataset do not correspond to normal operation and for this reason should not be considered for the reconstruction of the equipment behavior in normal conditions. Given the composition of the training sets, the clusters obtained by applying the ECM method to training set 2 are expected to be less dense than those obtained based on training set 1. Table 1 reports the mean confidence achieved in the classification of the same test set. In accordance to requirement 4), when the denser training set 1 is used, the mean degree of confidence tends to become higher.

VI. CONCLUSIONS

Due to the criticality of condition monitoring in complex systems, it is important to provide decision makers with not only an estimation of the equipment health state but also a measure of the confidence in the condition monitoring model outcome. To this purpose, we have considered a condition monitoring scheme based on two modules performing (1) signal reconstruction and (2) diagnostics decision on the equipment health state. As reconstruction model, we have employed an auto-associative empirical algorithm based on the Evolving Clustering Method (ECM). We have, then, introduced a novel measure of confidence on the obtained signal reconstruction, verified it by application to artificial datasets and applied it to a real condition monitoring problem concerning an important component of a nuclear power plant. The results that we have obtained show that the proposed measure meets the requirements which the confidence is expected to depend on, related to the density and information content of the training space.

Future research activity should be devoted to the estimation of the confidence in the diagnostic decision on the equipment health state and to the proper combination of the developed confidence measures in order to obtain an overall measure of confidence in the condition monitoring process.

Figure 1

 1 Figure 1 The condition monitoring approach in this work. The latter module takes the difference between the reconstructed   ˆ(1),..., ( ) nc nc x x q

  CM Conf , is sought as a result of the confidence in the signal reconstruction, r Conf , and in the diagnostic decision, d Conf , properly aggregated.

  ik k DR  , assign the i-th pattern to the cluster k0 with the associated lowest distance Dik:

S

  is less than or equal to 2Dthr, the cluster ka is updated by moving its center

0 n

 0 represents the label of the cluster with the center nearest to the test pattern  training patterns belonging to the cluster  m the total number of clusters  k n the number of patterns belonging to the k -th cluster  n the total number of training patterns   an integer number representing a threshold for the number of patterns belonging to the cluster: when the number of patterns in the nearest cluster to the test pattern is below 0 n , the confidence in the reconstruction is decreased.

Figure 2

 2 Figure2Case study 1: variation of the degree of confidence in the reconstruction as a function of the distance of the test pattern to the cluster centers.

Figure 3

 3 Figure 3 Case study 2: variation of the degree of confidence in the reconstruction depending on the presence of other clusters.

Figure 4

 4 Figure 4 Case study 3: degree of confidence for patterns equally distant from two clusters (ECM with a large radius value).

Figure 5

 5 Figure[START_REF] Roverso | Solutions for plant-wide on-line calibration monitoring[END_REF] Case study 4: degree of confidence for patterns equally distant from two clusters (ECM with a small radius value).

Figure 6

 6 Figure 6 Case study 5: variation of the degree of confidence in function of the cluster density.These five case studies performed with properly designed bi-dimensional Gaussian data have shown that the proposed confidence measure meets the four requirements of Section IV.A.B. Confidence estimation in a real condition monitoringcase study A real case study concerning 48 signals used to monitor the Reactor Coolant Pump (RCP) of a French Pressurized Water Reactor (PWR) is considered in this Section. The signals values have been measured every hour for a period of 11 consecutive months and concern four RCPs, each one on a different line of the primary circuit. The 5768 48-dimensional available patterns have been divided into a training set of 3000 and a test set containing the remaining 2798 test pattern.

Figure 7

 7 Figure 7 Variation of the degree of confidence of the signal reconstruction with the distance between the test pattern and the nearest cluster center.. A second real case study in which two different training sets are considered has also been performed. Training set 1 is formed by 4000 patterns, training set 2 by 400 patterns randomly sampled from the training patterns of training set 1.Given the composition of the training sets, the clusters obtained by applying the ECM method to training set 2 are expected to be less dense than those obtained based on training set 1. Table1reports the mean confidence achieved in the classification of the same test set. In accordance to requirement 4), when the denser training set 1 is used, the mean degree of confidence tends to become higher.

TABLE I .

 I AVERAGE CONFIDENCE IN THE RECONSTRUCTION OF THE TEST PATTERNS IN FUNCTION OF THE CLUSTER DENSITY


INTRODUCTION

Uncertainty analysis is a fundamental part of the risk analysis of complex systems such as nuclear power plants, aerospace systems and others.

In risk analysis, uncertainty is conveniently considered of two different types: randomness due to inherent variability in the system behavior and imprecision due to lack of knowledge and information on the system. The former type of uncertainty is often referred to as objective, aleatory, stochastic whereas the latter is often referred to as subjective, epistemic, state-of-knowledge [START_REF] Helton | Alternative Representations of Epistemic Uncertainty[END_REF]).

In the context of risk analysis, the aleatory uncertainty is related to the occurrence of the events which define the various possible accident scenarios, whereas epistemic uncertainty arises from a lack of knowledge of fixed but poorly known parameter values entering the evaluation of the probabilities and consequences of the accident scenarios.

In the current risk assessment practice, both types of uncertainties are represented by means of probability distributions. However, resorting to a single probabilistic representation of epistemic uncertainty may not be possible when sufficient data is not available for statistical analysis, even if one adopts expert elicitation procedures to incorporate diffuse information into the corresponding probability distributions, within a subjective view of probability. Indeed, an expert may not have sufficiently refined knowledge or opinion to characterize the relevant epistemic uncertainty in terms of probability distributions [START_REF] Helton | Alternative Representations of Epistemic Uncertainty[END_REF].

As a result of the potential limitations associated to a probabilistic representation of epistemic uncertainty under limited information, a number of alternative representation frameworks have been pro-posed, e.g., fuzzy set theory, evidence theory, possibility theory and interval analysis [START_REF] Klir | Fuzzy Sets and Fuzzy Logic: Theory and Applications[END_REF][START_REF] Flage | Probabilistic and possibilistic treatment of epistemic uncertainties in fault tree analysis[END_REF].

Possibility theory is attractive for risk assessment, because of its representation power and its relative mathematical simplicity. It offers two measures of likelihood, namely possibility and necessity measures, that may be interpreted as lower and upper probabilities in the representation of imprecision in the experts' probability assignments.

The possibilistic representation of uncertainty can both be combined with and transformed into the traditional probabilistic representation. In this respect, an integrated ("hybrid") computational framework has been proposed for jointly propagating probabilistic and possibilistic representations through a model (Baudrit et al. 2006). This framework has been applied to propagate uncertainties in event trees [START_REF] Baraldi | A Combined Monte Carlo and Possibilistic Approach to Uncertainty Propagation in Event Tree Analysis[END_REF] and fault trees [START_REF] Flage | Possibilityprobability transformation in comparing different approaches to the treatment of epistemic uncertainties in a fault tree analysis[END_REF][START_REF] Flage | Probabilistic and possibilistic treatment of epistemic uncertainties in fault tree analysis[END_REF]).

In the present paper, the integrated framework of propagation is tested on a flood risk model [START_REF] Pasanisi | Some useful features of the Bayesian setting while dealing with uncertainties in industrial practice[END_REF][START_REF] Limbourg | Uncertainty analysis using evidence theory -confronting level-1 and level-2 approaches with data availability and computational constraints[END_REF] considered a realistic benchmark for uncertainty modeling.

The reminder of the paper is organized as follows. In Section 2, some basic concepts about possibility theory are summarized; in Section 3, the details about the integrated propagation framework are given; in Section 4, approaches for constructing possibility distributions are discussed; in Section 5, the flood model considered for the uncertainty propagation task is presented; in Section 6, the results of the joint propagation of aleatory and epistemic uncertainties through the flood model are reported and commented; finally, in Section 7, conclusions and direction for future work are provided.

Monte Carlo and fuzzy interval propagation of hybrid uncertainties on a risk model for the design of a flood protection dike P. Baraldi, N. Pedroni, E. Zio & E. Ferrario Politecnico di Milano, Milano, Italy A. Pasanisi, M. Couplet

Electricité de France, Chatou, France

ABSTRACT: A risk model may contain uncertainties that may be best represented by probability distributions and others by possibility distributions. In this paper, a computational framework that jointly propagates probabilistic and possibilistic uncertainties is compared with a pure probabilistic uncertainty propagation. The comparison is carried out with reference to a risk model concerning the design of a flood protection dike.

In possibility theory, uncertainty is represented by a possibility function ) ( y π

. For each y in a set Ω , ) ( y π expresses the degree of possibility of y . When 0 ) ( = y π for some y , it means that the outcome y is considered an impossible situation. When 1 ) ( = y π for some y , it means that the outcome y is possible, i.e., is just unsurprising, normal, usual [START_REF] Dubois | Possibility theory and statistical reasoning[END_REF]. This is a much weaker statement than when probability is 1.

The possibility function gives rise to probability bounds, upper and lower probabilities, referred to as necessity and possibility measures ( N , Π ). They are defined as follows.

The possibility of an event A , ( )

A Π , is defined by ( ) { } ) ( sup y A A y π Π ∈ = , (1)
and the necessity measure ( )

A N is defined by ( ) { } ) ( sup 1 ) ( 1 y notA A N A y π Π ∉ - = - = .
Let P P P P(π ) be a family of probability distributions such that for all events A , ( )

) ( ) ( A A P A N Π ≤ ≤ . Then, ( ) ) ( inf A P A N = and (
)

) ( sup A P A = Π (2)
where inf and sup are with respect to all probability measures in P P P P. Hence the necessity measure is interpreted as a lower level for the probability and the possibility measure is interpreted as an upper limit. Referring to subjective probabilities, the bounds reflect that the analyst is not able or willing to precisely assign his/her probability, and the bounds are the best he/she can do given the information available; in other words, he or she can only describe a subset of P P P P which contains his/her probability [START_REF] Dubois | Possibility theory and statistical reasoning[END_REF].

PROPAGATION OF EPISTEMIC AND ALEATORY UNCERTAINTIES THROUGH A MODEL

Let us consider a model whose output is a function

( ) n Y Y Y f Z ,..., , 2 1 = of n uncertain variables n j Y j , , 1 , L =
, ordered in such a way that the first k are random (aleatory uncertainty), with probability distributions )

n k Y Y π π , , 1 L + .
The propagation of this hybrid uncertainty information can be performed by combining the Monte Carlo technique with the extension principle of fuzzy set theory (Baudrit et al. 2006).

The operative steps of the procedure are:

1 set i = 0 2 set i = i + 1 3 sample the th i - realization ( ) i k i y y ,
i i f f α α , of ( ) n k i k i Y Y y y f , , , , , 1 1 L L + 7 if 1 < α then set α ∆ α α + = (e.
Y f Z = is constructed as the collection of its N α = 1/ α ∆ + 1 = 1/0.05 + 1 = 21 α-cut intervals [ ] i i f f α α ,
) 8 if i < m (e.g., m = 10000) then return to step 2. to generate a new realization of the random variables; otherwise, stop the algorithm At the end of the procedure an ensemble of m random realizations of fuzzy intervals is obtained, i.e., ( )

f m f π π , , 1 L
. Two considerations are in order with respect to the choices of N α and m. A small number N α of αcuts (e.g., N α = 5) leads to a rough and imprecise characterization of the fuzzy random realizations f i π of Z ; on the other hand, a large number N α of α-cuts (e.g., N α = 100) causes a remarkable increase in the computational time. Thus, the choice of the number N α of α-cuts is driven by the trade-off between estimation accuracy and computational cost. Similarly, the number m of realizations of the random variables has to be large enough to guarantee an accurate and precise propagation of the corresponding aleatory uncertainty: in practice, m is usually of the order of thousands.

For each set A contained in the universe of discourse Z U of the output variable Z , it is possible to obtain the possibility measure

) (A f i Π and the ne- cessity measure ) (A f i Ν from the corresponding possibility distribution ) (z f i π , by: { } ) ( max ) ( z A f i A z f i π Π ∈ = , Z U A ⊆ ∀ (3) { } ( ) A z A f i f i A z f i Π π Ν - = - = ∉ 1 ) ( 1 inf ) ( , Z U A ⊆ ∀ (4)
The m different realizations of possibility and necessity can then be combined to obtain the belief ) ( A Bel and the plausibility ) ( A Pl for any set A , respectively (Baudrit et al. 2006):

∑ = = m i f i i A N p A Bel 1 ) ( ) ( (5)
∑ = = m i f i i A p A Pl 1 ) ( ) ( Π (6)
where i p is the probability of sampling the th irealization ( )

i k i y y , , 1 L of the random variable vector ( ) k Y Y , , 1 L
. For each set A , this technique thus computes the probability-weighted average of the possibility measures associated with each output fuzzy interval.

The likelihood of the value ( )

Y f
passing a given threshold z can then be computed by considering the belief and the plausibility of the set Baudrit et al. 2006).

( ] z A , ∞ - = ; in this respect, ( ] ( ) z Y f Bel , ) ( ∞ - ∈ and ( ∈ ) (Y f Pl ( ]) z , ∞ - can be interpreted as bounding, average cumulative distributions ( ] ( ) z Y f Bel z F , ) ( ) ( ∞ - ∈ = , ( ] ( ) z Y f Pl z F , ) ( ) ( ∞ - ∈ = (

APPROACHES FOR CONSTRUCTING POSSIBILITY DISTRIBUTIONS

In this Section, a number of approaches for constructing possibility distributions of the variables subject to epistemic uncertainty are briefly described. In Section 4.1, triangular possibility distributions are considered; in Section 4.2, the use of Chebyshev inequality is illustrated; finally, in Section 4.3, two methods for transforming a probability distribution into a possibility distribution are described based on the principle of maximum specificity (Section 4.3.1) and on the normalization of the probability density function (Section 4.3.2).

Triangular function

Let us suppose that the analyst knows that an uncertain variable can take values in a given range

Chebyshev inequality

If the analyst knows the mean µ and the standard deviation σ of the uncertain variable of interest, then the Chebyshev inequality (Baudrit & Dubois 2006) can be used to construct a possibility distribution. Actually, the use of continuous possibility distributions for representing probability families heavily relies on probabilistic inequalities. Such inequalities provide probability bounds for intervals forming a continuous nested family around a typical value. This nestedness property leads to interpreting the corresponding family as being induced by a possibility measure. These bounds are usually used for proving convergence properties but, in this context, they can be used for representing knowledge. This is the case of the Chebyshev inequality, for instance.

As pointed, for instance by Baudrit & Dubois (2006), the classical Chebyshev inequality defines a bracketing approximation on the confidence intervals around the known mean µ of a random variable Y, knowing its standard deviation σ . The Chebyshev inequality can be written as follows:

( )

2 1 1 k k Y P - ≥ ≤ - σ µ for 1 ≥ k (7)
Chebyshev inequality defines a possibility distribution that dominates any density with given mean and variance: it allows to define a possibility distribution

π by considering intervals [ ] σ µ σ µ k k + -, as α- cuts of π and letting ( ) ( ) 2 / 1 k k k = + = - σ µ π σ µ π . Notice that from (7) P(Y ∈ [µ -kσ, µ + kσ]) ≥ 2 1 1 k - , for 1 ≥ k ; moreover, P(Y ∈ [µ -kσ, µ + kσ]) ≥ α - 1
by definition of α-cut interval. As a consequence, α = 1/k 2 by construction. Since in this paper α = 0.05, 0.1, …, 0.95, 1 (see Section 3), then k = 4.4721, 3.1623, …, 1.0260, 1. The resulting possibility distribution defines a probability family P P P P µ,σ (π ) which has been proven to contain all distributions with mean µ and standard deviation σ , whether the unknown probability distribution function is symmetric or not, unimodal or not (Baudrit & Dubois 2006).

Probability -possibility transformations

In this Section, we consider transformations from probability distributions to possibility distributions. It is worth noting that in the transformation procedure (i.e., going from probability to possibility) "some information is lost because there is a conversion from pointed-valued probabilities to intervalvalued ones" [START_REF] Dubois | On possibility/probability transformations[END_REF].

Given the interpretation of possibility and necessity measures as upper and lower probabilities, a possibility distribution π induces a family P P P P(π ) of probability measures. There is not a one-to-one relation between possibility and probability, and transformations from a probability measure P into a possibility distribution π can only ensure that -P P P P(π ) includes P; and -P P P P(π ) is selected according to some principle (rationale); e.g., "minimize loss of information", in some sense. The following should be basic principles for such transformations [START_REF] Dubois | On possibility/probability transformations[END_REF]): 1 The probability-possibility consistency principle.

The family P P P P(π ) is formally defined as

P P P P(π ) { } ) ( ) ( , : A A P Y A P Π ≤ ⊆ ∀ =
. It seems natural to require a transformation to select P from P P P P(π ) [START_REF] Dubois | On possibility/probability transformations[END_REF]. This is referred to as the probability-possibility consistency principle, formulated as

Y A A A P ⊆ ∀ ≤ ), ( ) (

The principle of maximum specificity

The most specific possibility distribution t π , or rather the minimum area under π , that dominates a given probability density p is given by:

= + = = ∫ ∫ +∞ ∞ - ) ( ) ( ) ( )) ( ( ) ( y h y t t dx x p dx x p y h y π π )) ( ( ) ( y h F y F + = (8) where ) ( 1 ) ( ⋅ - = ⋅ F F and { ≥ = ) ( : max ) ( x p x y h } ) ( y p
. It is interesting to observe that for this transformation:

[ ] ( ) [ ] ( ) [ ] ( ) 1 ) ( , ) ( , ) ( , = ≤ = y h y y h y P y h y N Π , i.e.
, the transformation prescribes equality between the necessity of a given α-cut and the probability of the same α-cut.

The transformation applies to unimodal, continuous and support bounded probability densities p. Moreover, this criterion is not necessarily adapted to the transformation of a subjective probability distribution reflecting an expert opinion [START_REF] Dubois | Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities[END_REF]).

Normalization of probability density

The possibility distribution resulting from the transformation is given by the normalization of probability density, i.e., ) ( sup / ) ( y p y p p = µ . Note that the distribution resulting from this normalization (when taken to be a possibility distribution) does not in general adhere to the probability-possibility consistency principle [START_REF] Dubois | Fuzzy Sets and Systems: Theory and Applications[END_REF].

CASE STUDY: FLOOD PROTECTION DESIGN

The case study deals with the design of a protection dike. The problem concerns a residential area that is closely located to a river; due to this location, there is a potential risk of flood. As prevention and mitigation measures, a dike has to be built to protect the area. Different design options must be considered taking into account that the construction of a dike involves high building costs and annual maintenance costs and the natural phenomenon of flooding is subject to a large amount of uncertainties. Thus, the analyst has to evaluate different design options while covering flood uncertainty.

The system model analyzed in this study is an analytical approximation for calculating the maximal water level of the river (i.e., the output variable of the model, c Z ) as a function of a number of parameters (i.e., the input variables of the model):

( ) 5 / 3 /         - * * + = L Z Z B K Q Z Z v m s v c (9)
where: Q is the yearly maximal water discharge (m3/s); m Z and v Z are the riverbed levels (m asl) at the upstream and downstream part of the river under investigation, respectively; s K is the Strickler friction coefficient; B and L are the width and length of the river part (m), respectively.

The

( )= β α, q Gum       -               - - = β α β α β q q exp exp exp 1 (10)
The Σ . An additional consideration is in order with respect to the fact that variable Z v appears twice in model function (9). Actually, the presence of repeated variables causes analytical difficulties in several uncertainty propagation methods, including interval analysis and fuzzy arithmetic [START_REF] Ferson | Construction Probability Boxes and Dempster-Shafer Structures[END_REF]: in particular, it typically leads to an overestimation of the uncertainty. However, this is not the case for MC simulation (here employed to propagate aleatory uncertainty) because it (automatically) escapes the problem of repeated variables by instantiating each occurrence to the same randomly sampled value.

-Epistemic variables:

• The Strickler friction coefficient s K is perhaps the most critical source of uncertainty. It is affected by epistemic uncertainty, since it is a simplification of a much more complex hydraulic model. In addition, assessing the uncertainty of s K is difficult because, in practice, even if this coefficient is strongly related to the morphology of the river, it cannot be measured. As a consequence, data may only be retrieved through indirect calibration noised by significant observational uncertain-ty: this is reflected in the availability of only a very small series of 5 data sets with ± 15% noise [START_REF] Limbourg | Uncertainty analysis using evidence theory -confronting level-1 and level-2 approaches with data availability and computational constraints[END_REF]. The absolute physical limits of s K are [a, b] = [5, 60], but the real value is expected to vary in a smaller range. In [START_REF] Pasanisi | Some useful features of the Bayesian setting while dealing with uncertainties in industrial practice[END_REF], this epistemic variable is treated within a probabilistic framework: it is considered that the probability distribution of s K is normal with mean µ and standard deviation σ equal to 30 and 7.5, respectively. In this paper, the epistemic uncertainty associated to s K is represented by means of possibility distributions; the four methods described in the previous Section 4 are used. For the method of Section 4.1 (i.e., triangular possibility distribution), the base of the triangle is [5, 60] (i.e., the absolute physical limits of s K ) and the most likely value is 30 (i.e., the mean µ of the normal probability density function of s K used in [START_REF] Pasanisi | Some useful features of the Bayesian setting while dealing with uncertainties in industrial practice[END_REF])); for the methods of Section 4.2, 4.3, 4.4 (i.e., Chebyshev inequality and probability-possibility transformations), the mean µ and the standard deviation σ used are 30 and 7.5 (i.e., the mean and the standard deviation of the probability density function of s K used in [START_REF] Pasanisi | Some useful features of the Bayesian setting while dealing with uncertainties in industrial practice[END_REF]). The possibility distributions for s K resulting from the application of the methods in Section 4.1-4.4 are shown in Figure 4. It is worth noting that the area underlying the possibility distribution is related to the imprecision in the knowledge of the possibilistic variable: the larger the area, the higher the imprecision. In Figure 4, a direct visual comparison of the areas underlying the four possibility distributions considered is shown. It can be noticed that the larger areas are those underlying the possibility distributions built using the triangular function and the Chebyshev inequality. In fact, the information available to the analyst for building these two possibility distributions is quite scarce: in the first case, only the physical limits and the most likely value of the variable are known; in the second case, only the mean value and the standard deviation are considered. On the contrary, the smaller areas underlying the possibility distributions constructed by the transformation methods are explained by the larger amount of information available to the analyst on the epistemic variable of interest, i.e., the probability distribution function itself.

For each of the four cases considered (i.e., for each of the possibility distributions built), the integrated uncertainty propagation procedure of Section 2 is run with m = 10000 realizations of the probabilistic variables; for each realization of the probabilistic variables, 21 values of α (0, 0.05, 0.1, …, 1) are considered to process the epistemic uncertainty associated to K s . The results are compared to those obtained with a pure probabilistic approach, as in [START_REF] Pasanisi | Some useful features of the Bayesian setting while dealing with uncertainties in industrial practice[END_REF], by sampling 10000 times a joint vector ) , , , (

s v m K Z Z Q by the joint pdf ) ( ) , ( ) ( s v m K p Z Z p Q p ⋅ ⋅
. With respect to the epistemic uncertain variable s K , this approach consists in marginalizing the joint pdf of ) , (

s c K Z : ) ( ) | (Z c s s K p K p ⋅
, over all possible values of s K .

RESULTS

Figure 5 (a-d) shows the comparison of the cumulative distribution functions of the maximal water level of the river (i.e., the output variable of the model, c Z ) obtained by the probabilistic uncertainty propagation approach (solid lines) with the belief (lower curves) and plausibility (upper curves) functions obtained by the integrated framework of uncertainty propagation where the possibility distributions for K s are constructed with the methods of Sections 4.1-4.3. It can be seen that: the integrated framework explicitly propagates the uncertainty by separating the contributions coming from the probabilistic and possibilistic variables; this separation is visible in the output distributions of the maximal water level of the river where the separation between the belief and plausibility functions reflects the imprecision in the knowledge of the possibilistic variable s K and the slope pictures the variability of the probabilistic variables

v m Z Z Q , ,
; the separation between the belief and plausibility functions is larger for the cases in Figures 5a and5b (where the possibility distributions are those of Figure 4, built using the triangular function and Chebyschev inequality, respectively) than for those in Figures 5c and5d (where the possibility distributions are those of Figure 4, built using the probability-possibility transformations); the larger gap between the belief and plausibility functions in Figures 5a and5b than in Figures 5c and5d is explained by the larger area contained under the corresponding possibility distribution functions (actually, the larger the area, the higher the imprecision in the knowledge of the possibilistic variable). the uncertainty in the output distribution of the pure probabilistic approach is given only by the slope of the cumulative distribution; as expected, the cumulative distribution of the maximal water level of the river obtained by the pure probabilistic method is within the belief and plausibility functions obtained by the hybrid approach Figure 5. Comparison of the cumulative distribution functions of the maximal water level of the river Z c obtained by the probabilistic uncertainty propagation approach (solid line) with the belief (lower dashed curve) and plausibility (upper dashed curve) functions obtained by the hybrid approach with the possibility distribution of K s built using a) a triangular function, b) the Chebyshev inequality, c) the principle of maximum specificity d) the normalization of probability density (see Section 4 and Figure 4).

The final goal of the risk model assessment is to determine i) the dike level necessary to guarantee a given flood return period or ii) the flood risk for a given dike level.

With respect to item i) above, a reasonable quantity of interest is the 99% quantile of c Z , i.e., 99 . 0 c Z , taken as the annual maximal flood level. This corresponds to the level of a "centennial" flood, the yearly maximal water level with a 100 year-return period. With respect to item ii) above, the quantity of interest that is mostly relevant to the decision maker is the probability that the maximal water level of the river c Z exceeds a given threshold * z , i.e., ( )

* z Z P c >
; in the present paper, * z = 55.5 m as in [START_REF] Limbourg | Uncertainty analysis using evidence theory -confronting level-1 and level-2 approaches with data availability and computational constraints[END_REF]. Table 1

( ) ( ) ( ) ( ) prob c c c z Z P z Z Bel z Z Pl W * / * * * > > - > = of the interval ( ) ( ) [ ] * , * z Z Pl z Z Bel c c > > have been reported.
The numerical results in Table 1 confirm the similarities between the cumulative distributions obtained by using the triangular function and the Chebyshev inequality for the possibilistic representation of the uncertainty on s K , and between the cumulative distributions obtained by the two different transformations from probability to possibility distributions.

CONCLUSIONS

We have applied to a risk flood model a computational framework for the joint propagation of probabilistic and possibilistic uncertainties. Aleatory and epistemic uncertainties have been kept separate in the model, i.e., some of the variables are purely probabilistic (aleatory uncertainty) and some are purely possibilistic (epistemic uncertainty).

The following analyses have been carried out: 1 A comparison between the "hybrid" and the "pure probabilistic" approach, highlighting that:

the "global" uncertainty of the output within the pure probabilistic approach is only given by the cumulative distribution: the contributions of aleatory and epistemic uncertainties are here merged; the hybrid approach explicitly propagates the uncertainty by separating the contributions coming from the probabilistic and possibilistic variables; the larger gap between the belief and plausibility functions is explained by the larger area contained under the corresponding possibility distribution functions; as expected, the cumulative distribution of the model output obtained by the pure probabilistic method is within the belief and plau-sibility functions obtained by the hybrid approach. 2 A comparison of four methods for constructing the possibility distributions of the variables subject to epistemic uncertainty, showing that:

the choice of the possibility distribution depends on the information available about the variable: when the physical limits and the most likely value are available, a triangular possibility distribution can be constructed; when the mean and the standard deviation can be computed, e.g., by means of empirical data, the Chebyshev inequality can be used; when a probability distribution is available, the methods for transforming probability into possibility distributions can be employed; there are similarities between the results obtained by using:

• the triangular function and the Chebyshev inequality; • the two transformations from probability to possibility distributions (i.e., those based on the principle of maximum specificity and on the normalization of the probability density function). These similarities are explained by the same "uncertainty content" borne by the corresponding possibility distributions (as demonstrated by the similar area limited by the possibility distribution functions). Future research will be devoted to:

the study and development of methods for the joint propagation of probabilistic and possibilistic uncertainty in cases when the uncertain variables are not purely aleatory or purely epistemic (for example, the uncertainty of a variable might be described by a probability distribution whose parameters are themselves poorly known and represented by a possibility distribution); the comparison of the integrated probabilistic -possibilistic approach to other computational frameworks for the joint propagation of aleatory and epistemic uncertainties, e.g., the double Monte Carlo method [START_REF] Baudrit | Representing parametric probabilistic models tainted with imprecision[END_REF]) and the Dempster -Shafer theory of evidence [START_REF] Ferson | Construction Probability Boxes and Dempster-Shafer Structures[END_REF]; the treatment of dependencies between probabilistic and possibilistic variables; the use of advanced integrated simulation methods within the framework of uncertainty propagation for reducing the associated computational cost.

  step 5. above; otherwise obtain the fuzzy random realization (fuzzy interval) α are considered in the procedure, i.e., N α = 21 α-cuts of the possibility distributions

  [a, b] and the most likely value is c. To represent this information a possibility distribution can be taken as a triangle with base determined by the range [a, b] (i.e., the absolute physical limits of the variable) and with vertex taken in correspondence of the most likely value c: in other words, the possibility distribution equals 0 in correspondence of the extreme values a and b of the physically allowable range and 1 in correspondence of the most likely value c. It has been shown that the family of probability distributions defined by a triangular possibility distribution with range [a,b] and vertex c contains all the probability distributions with support [a,b] and mode c(Baudrit & Dubois 2006).

  , two methods for transforming a probability distribution into a possibility distribution are considered: the first one is based on the principle of maximum specificity (Section 4.3.1); the second one on the normalization of the probability density function (Section 4.3.2).

  as the upstream and downstream sections are quite close it seems reasonable to model them as possibly dependent variables. A total of 29 pairs of data

Figure 4 .

 4 Figure 4. Comparison of the four different possibility distributions used for K s : triangular function (Section 4.1), Chebyshev inequality (Section 4.2), principle of maximum specificity (Section 4.3.1), normalization of the probability density (Section 4.3.2).

Table 1 .

 1 Lower and upper values of the Z c percentiles and the threshold exceedance probability, and calculation of the indicator W about the width of the confidence interval.

	Possibility distribution	Z c	0.99		P[Z c ≥ 55.5]
		(Pure probabilistic value = 56.10) [ [ [ [ ] ] ] ] 99 0 99 0 . , . , , upper c lower c Z Z [ [ [ [ ] ] ] ] % Zc W	(Pure probabilistic value = 0.0191) [ [ [ [ ] ] ] ] Pl Bel, [ [ [ [ ] ] ] ] % * W
	Triangular function	[54.57, 59.29]	8	[0.0015, 0.1682]	873
	Chebyshev inequality	[54.40, 60.00]	10	[0.0014, 0.1631]	847
	Transformation probability to	[54.83, 55.99]	2	[0.0043, 0.0344]	157
	possibility (principle of max-
	imum specificity)
	Transformation probability to	[54.60, 56.69]	4	[0.0028, 0.0705]	355
	possibility (normalization)