Piero Baraldi

Roozbeh Razavi-Far

Enrico Zio

Classifier-Ensemble Incremental-Learning Procedure for Nuclear Transient Identification at Different Operational Conditions

Keywords: Classification, Fuzzy C-Means (FCM) Clustering, Bagging, Ensemble, Incremental Learning, BWR Nuclear Power Plant, Transient Identification

An important requirement for the practical implementation of empirical diagnostic systems is the capability of classifying transients in all plant operational conditions. The present paper proposes an approach based on an ensemble of classifiers for incrementally learning transients under different operational conditions. New classifiers are added to the ensemble where transients occurring in new operational conditions are not satisfactorily classified. The construction of the ensemble is made by bagging; the base classifier is a supervised Fuzzy C Means (FCM) classifier whose outcomes are combined by majority-voting. The incremental learning procedure is applied to the identification of simulated transients in the feedwater system of a Boiling Water Reactor (BWR) under different reactor power levels.

Introduction

Monitoring is a continuous real-time task of determining the conditions of a physical system, by recording information, recognizing and indication anomalies in the behavior [START_REF] Simani | Model-based fault diagnosis in dynamic systems using identification techniques[END_REF]. A fault diagnosis system is a monitoring system that is used to detect faults and diagnose their location and significance in a system [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF]). The diagnosis system performs the following tasks: fault detection-to indicate if a fault occurred or not in the system, fault isolation-to determine the kind, location and time of detection, and fault identification-to estimate the size and nature of the fault. The first two tasks of the system: fault detection and isolation are considered the most important. Fault diagnosis is then very often considered as fault detection and isolation [START_REF] Simani | Model-based fault diagnosis in dynamic systems using identification techniques[END_REF]. Here the term diagnosis indicates recognizing and indication transients and anomalies in the system behavior.

A number of diagnostic methods based on the advances of soft computing have been proposed for transient identification in nuclear systems [START_REF] Hines | A hybrid approach for detecting and isolating faults in nuclear power plant interacting systems[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] Embrechts | Hybrid identification of nuclear power plant transients with artificial neural networks[END_REF][START_REF] Na | Prediction of major transient scenarios for severe accidents of nuclear power plants[END_REF][START_REF] Evsukoff | Recurrent neuro-fuzzy system for fault detection and isolation in nuclear reactors[END_REF][START_REF] Zhao | Adaptive fuzzy inference causal graph approach to fault detection and isolation of field devices in nuclear power plants[END_REF][START_REF] Zio | Identification of nuclear transients via optimized fuzzy clustering[END_REF][START_REF] Razavi-Far | Model based fault detection and isolation of a steam generator using neuro-fuzzy networks[END_REF]. However, one factor that has limited their practical application is the difficulty of recognizing transients at different plant operational conditions, e.g. power levels [START_REF] Uhrig | Soft computing technologies in nuclear engineering applications[END_REF]. The objective of the present work is to overcome this limitation by proposing a procedure of empirical classification by incremental learning of transients at different plant conditions. The procedure is realistically applicable, with new examples of transients in new operational conditions becoming gradually available in time. Since the proposed classification procedure is based on the use of supervised learning algorithms, it requires a training phase in which some examples of patterns formed by the signal measurements observed during the transients (input) and the corresponding class (output) are available. Furthermore, we assume that the classes of the transients do not change at the different plant conditions, i.e. there can be a modification of the relationship between the inputs and the output, but the algorithm cannot be used to classify new classes of patterns.

One approach for learning new data (e.g. coming from new transients of new generated conditions) involves discarding the existing classifier and retraining a new one using all data (e.g. of all transients) that have been accumulated thus far. This approach, however, results in loss of all previously acquired information, a phenomenon known as catastrophic forgetting [START_REF] Polikar | Ensemble Based Systems in Decision Making[END_REF], and may be infeasible for real diagnostic systems due to the computational and financial efforts necessary for each model retraining. In order to avoid retraining a new model each time a new dataset becomes available, the classification algorithms must be able to learn the novel information content of the new data without forgetting the previously acquired knowledge. A further desiderata, important in those cases in which the datasets previously used for the model training may be lost, corrupted or otherwise unavailable, is the possibility of updating the model without requiring access to the previously seen datasets.

The ability of a classifier to learn under these circumstances is usually referred to as incremental learning [START_REF] Polikar | Learn++: An incremental learning algorithm for supervised neural networks[END_REF].

A further challenge comes from the fact that the input (transient data of the measured signals)output (corresponding fault class that originates the transients) relationships may change in different operational conditions. This problem is usually termed learning in a non-stationary environment. In particular, in the problem addressed in this work, the modification of the environment is assumed to be cyclic, since the plant usually returns to work in one operative condition previously left.

Recently, multiple classifier-based algorithms have been proposed for incremental learning in non-stationary environments. The proposed algorithms generate and then combine an ensemble of classifiers, where each classifier is trained on a different snapshot of the data. In particular, the following types of ensemble-based approaches have been distinguished [START_REF] Kuncheva | Classifier Ensembles for Changing Environments[END_REF]: (i) a fixed ensemble whose combination rules (weights) are changed based on the changing environment (dynamic combiners); (ii) an ensemble where new data are used to update some of the classifiers thanks to an on-line learning algorithm; (iii) a new ensemble structure obtained by altering the old ensemble structure.

In this work, the third approach is embraced within a procedure of modification of the ensemble structure when the classification in the current environment is not satisfactory. Firstly, an ensemble of classifiers is built using the datasets available. The ensemble is built according to the method proposed in [START_REF] Baraldi | Bagged Ensemble of FCM Classifier for Nuclear Transient Identification[END_REF]: the base classifier is a supervised Fuzzy C Means classifier [START_REF] Zio | Identification of nuclear transients via optimized fuzzy clustering[END_REF]; an ensemble of them is built on different bagging sets of the available data [START_REF] Breiman | Bagging predictors[END_REF]; the single classifiers outcomes are combined using a majority-voting scheme [START_REF] Parhami | Voting algorithms[END_REF]. When the plant starts working in a new operational condition and a corresponding new dataset of data becomes available, the classification performance of the ensemble previously built is verified and, in the case in which it is not satisfactory (i.e. if the fraction of patterns correctly classified is lower than a fixed threshold), the ensemble is updated by adding new classifiers. This is actually done by creating an additional ensemble of classifiers, each base classifier trained by using a different bagging set of the new dataset. Finally, the old and the new ensembles of classifiers are merged into a single ensemble of classifiers formed by all the classifiers of the new and old ensembles. The procedure is repeated each time a new dataset describing a new operational condition becomes available.

The capability of the overall ensemble system to identify faults that initiate from different plant operational conditions has been tested on an application regarding the identification of simulated transients occurring at different reactor power levels in the feedwater system of a Boiling Water Reactor (BWR) (Puska and Norman, 2002).

The remainder of the paper is organized as follows. Section 2 presents a brief description of the problem statement of incremental learning in a non-stationary environment. Section 3 illustrates an ensemble-based scheme for incremental learning, describing the method and algorithm. Section 4 describes how the ensemble-based scheme is used for transient identification in the feedwater system of a BWR at different power levels. Finally, conclusions are drawn in Section 5. For completeness: the procedure of ensemble construction and its algorithm are reported in Appendix A and the supervised, evolutionary-optimized FCM clustering algorithm used to train the base classifiers of the ensemble is briefly described in Appendix B.

Incremental learning in a non-stationary environment

Let us consider a plant which can work in several different operational conditions. We assume that at time j x is assumed to be fixed and equal in all the datasets j S . In general, the unknown mapping function between j k x and j k  may vary in different operational conditions, i.e. the class boundaries in the input space may be different in the different datasets j S , which contain transients occurring with the plant in different operational states.

The final objective of the present work is to develop a classification algorithm able to correctly classify transients of the signal measurement vector

x , independently from the plant operational conditions.

3. An ensemble-based procedure for incremental learning in a non-stationary environment

The idea underlying ensemble-based classification is to create many classifiers and combine their outputs in a way to improve the performance of a single classifier. This requires that individual classifiers perform well in different regions of the feature space and make errors on different patterns, which are balanced out in the combination. Intuitively, if each classifier makes different errors, then a strategic combination of these classifiers can reduce the total error. The overarching principle in ensembles is therefore to make each classifier as unique as possible, particularly with respect to misclassified instances. Specifically, we need classifiers whose decision boundaries are sufficiently different from those of others [START_REF] Polikar | Ensemble Based Systems in Decision Making[END_REF].

Various techniques have been suggested for obtaining diversity in the base models of an ensemble, e.g. using different training parameters [START_REF] Hansen | Neural network ensembles[END_REF], different training patterns [START_REF] Breiman | Bagging predictors[END_REF], different feature subsets [START_REF] Zio | Feature-based classifier ensembles for diagnosing multiple faults in rotating machinery[END_REF] and different learning methods for each classifier of the ensemble [START_REF] Xu | Methods of combining multiple classifiers and their applications to handwriting recognition[END_REF].

Here, the approach adopted in [START_REF] Baraldi | Bagged Ensemble of FCM Classifier for Nuclear Transient Identification[END_REF] where different training patterns are used to train individual classifiers is briefly described. The datasets are obtained through the resampling technique of bagging [START_REF] Breiman | Bagging predictors[END_REF] from a dataset containing all the available training patterns. Bagging, short for bootstrap aggregating, is one of the earliest ensemble-based algorithms [START_REF] Breiman | Bagging predictors[END_REF][START_REF] Breiman | Combining predictors[END_REF] and is based on the random sampling of the datasets, usually with replacement, from the entire training dataset. The main structure of this ensemble construction scheme is shown in Figure 1. With respect to the construction of the base classifier of the ensemble, the supervised FCM algorithm is considered [START_REF] Zio | Identification of nuclear transients via optimized fuzzy clustering[END_REF]. In this classification algorithm, the information regarding the known, physical class of the k-th pattern is used to supervise an evolutionary algorithm for finding c optimal Mahalanobis metrics which define c geometric clusters as close as possible to the a priori known physical classes [START_REF] Yuan | Intelligent Hybrid Systems Fuzzy Logic, Neural Network, and Genetic Algorithms[END_REF]. The Mahalanobis metrics are defined by the matrices , whose elements are identified by the supervised evolutionary algorithm so as to minimize the distances between the patterns belonging to class and the class prototype, i.e. the cluster center . The performance of the overall bagging ensemble approach has been verified by comparison with a single supervised, evolutionary-optimized FCM classifier with respect of the task of classifying artificial and nuclear transient datasets. The results obtained indicate that in the cases of datasets of large or very small sizes and/or complex decision boundaries, the proposed bagging ensemble improves the classification accuracy. However, the bagging approach does not allow incremental learning in a non-stationary environment since it requires that all the training patterns, which are used for training the ensemble base classifiers, be available in advance.

In order to overtake this limitation of the bagging algorithms such as [START_REF] Baraldi | Bagged Ensemble of FCM Classifier for Nuclear Transient Identification[END_REF], the basic idea of the procedure proposed in this work for adding the capability of incremental learning is to add new classifiers to an ensemble of classifiers whenever the current classification performance is not satisfactory due to the modification of the environment. This approach allows the ensemble to learn new information, without forgetting the previously acquired knowledge which is contained in the old classifiers which are kept in the ensemble. To control the proliferation of classifiers in the ensemble, new classifiers are added only if the transients occurring in the new operational condition are not satisfactorily classified. According to this procedure, diversity in the base models of the ensemble is obtained by using different training patterns. However, notice tha t the approach differs from the bagging approach in [START_REF] Baraldi | Bagged Ensemble of FCM Classifier for Nuclear Transient Identification[END_REF] since the different training datasets are not all obtained from the same dataset, but they come from different datasets corresponding to different operational conditions.

The diagnostic system is developed according to the following steps (Figure 2.b):

1) Fix the minimum classification performance * p which is always required to the diagnostic system and a fraction  indicating the maximum performance reduction which is acceptable when the diagnostic system is used to classify patterns corresponding to different operational conditions from those used to train the ensemble system. 2) At time 1 t when dataset 1 S becomes available, an ensemble system 1 E is built. The ensemble is formed by 1 T base . To ensure that there are adequate training samples in each subset, relatively large portions of the samples (F=0.75 -1.00) are drawn into each subset. This causes individual training subsets to overlap significantly, with many of the same instances appearing in most subsets, and some instances appearing multiple times in a given subset. The algorithm used to construct the bagging ensemble is briefly reported in Appendix A.

classifiers 1 l h , 1 ,..., 1 T l  , built

1.b)

Training: the building of a supervised, evolutionary-optimized FCM Classifier using the training data obtained in 1.a) and the procedure reported in Figure xyz.

3) The performance 1 p achieved by the first ensemble 1 E on the patterns of the dataset 1 test S not used for the training of the ensemble classifiers in step 1 is computed. In this respect, the test patterns are classified by the 1 T supervised FCM classifiers of the ensemble and the response of the single classifiers are aggregated according to the majority voting method (Appendix 1).

4) The achieved performance 1

p is compared to a minimum required classification performance * ) 1 ( * p p  . If the performance 1 p of 1 E is not satisfactory, i.e. 1 p p   , e.g. due to the presence of insufficient or poor quality data in 1 S , at time t 2 when dataset 2 S becomes available go to 7) in order to try to increase the classification performance by adding new classifiers to the ensemble, otherwise the diagnostic algorithm 6) The achieved performance j p is compared to

  ) 1 *, max( ) ( * 1     j p p j p  . The term   1 1    j p 
is considered in order to guarantee that the classification performance does not remarkably decrease when the diagnostic system is used for the classification of transient occurring at different operational conditions. If the classification performance j p is satisfactory, i.e. ) ( * j p p j  the ensemble system is left unmodified. In this case at time

1  j t
when the new dataset j S becomes available, go to 5). Otherwise, if ) ( * j p p j  , go to 7) in order to modify the ensemble. This usually occurs when the operational conditions verified in j S are significantly different from those previously experimented.

7) The ensemble system is updated by adding j T base classifiers j l h , j T l ,..., 1  , trained with bootstrapped replicas of j S according to the procedure in 1a) and 1b) applied to the data of j S . Thus, the obtained classification model E is an ensemble system formed by the union of the previous classifiers of E and the j T classifiers newly added, i.e. an ‗ensemble of ensembles'.

The main structure of the proposed incremental learning scheme is presented in Figure 2.a. Considering a generic time instant t at which an incremental learning ensemble system has been developed, the classification of an incoming new test pattern x is done by using the majority voting method, i.e. the class label which is supported by the majority of the individual classifiers is assigned to

Codice campo modificato

Subset 1 Subset i Subset Resampling F% of Training Dataset 1 j E  1 1 j C  1 j i C  1 1 j j T C    1 j T 

Training

x . In case the number of votes to different classes is equal, the class is assigned randomly among those classes with largest total votes.

Application to nuclear transient identification

In this Section, the capability of the proposed procedure is tested with respect to the classification of transients in the feedwater system of a BWR. The diagnosis considers three power operational levels, i.e. 50%, 80%, and 108% of full power. The corresponding transients have been simulated by the HAMBO simulator of the Forsmark 3 BWR plant in Sweden (Puska and Noemann, 2002).

The considered faults occur in the section of the feedwater system where the feedwater is preheated from 169°C to 214°C in two parallel lines of high-pressure preheaters while going from the feedwater tank to the reactor. Figure 3 shows a sketch of the system. A set of six faults, F1-F6, that are generally hard to detect for an operator have been chosen for this application (see [START_REF] Roverso | On-line early fault detection and diagnosis with the alladin transient classifier[END_REF]) for their description).

Figure 3: A sketch of the feedwater system of the BWR Among the 363 measured signals, only the 5 reported in Table 1 have been used for the fault classification in the two case studies here considered. These signals have been chosen considering the results of the application of a feature selection algorithm and some benchmark tests [START_REF] Zio | Selecting features for nuclear transients classification by means of genetic algorithms[END_REF]. have been considered, containing patterns taken from transients simulated with the plant working at 50%, 80% and 108% of full power, respectively. More specifically, each dataset is formed by 1800 patterns taken from three transients for each of the 6 faults, differing in the degrees of leakage and valve closure. The data relative to the selected 5 signals were recorded with a sampling frequency of 1 Hz. All transients start after 60 seconds of steady state operation. Given that the goal is early fault diagnosis, only the data from 70 seconds after the beginning of the transients have been considered for each transient.

Figure 4 shows the time evolution of the 5 features in transients of the 6 classes at the three power levels. Notice that signal variations are different at the different power levels, and more pronounced when the reactor is working at high power.

Case study 1: increasing power level

In this case study, the power level is increased firstly from 50% to 80% of the full power and then from 80% to 108%, at 50 80 108

t t t   .
The procedure for incremental learning in a non-stationary environment is applied as follows. At 50 t , an ensemble 1 E is constructed using the data in 50 train S . Although an overall investigation of the influence of the parameters used to build the ensemble on the classification performance is outside the scope of the present work, some considerations on the possible choices are here given. The two parameters of the ensemble   , FT , the fraction of the total number of training patterns in 50 train S randomly drawn to create the single classifier training set and the number of ensemble classifiers have been fixed following a trial and error procedure. The results of tests performed by the authors have shown that the key issue to guarantee high performance of the ensemble is the diversity between the ensemble classifiers. In particular, since a low value of F leads to training sets with few common patterns, high performances can be obtained by reducing F , and, at the same time, increasing the number of classifiers T in order to properly cover all the training space. Notice, however, that since the computational efforts necessary to develop the diagnostic system is directly proportional to T, the choice of the parameters  

, FT results from a compromise between high performance (low F, high T) and reduced computational effort (high F, low T).

The choice of the minimum classification performance, * p , is usually guided by requirements of the diagnostic system users. In this application, since the diagnostic system is devoted to the classification of faults which mainly produce efficiency losses if undetected, * p is set to 0.95. With respect to the parameter  indicating the maximum fraction of performance reduction which is acceptable when the diagnostic system is used to classify patterns corresponding to different operational conditions from those used to train the ensemble system, notice that a too low value of  will risk to cause the updating of the ensemble each time a new dataset becomes available with consequent high computational effort. In this respect, a value of  equal to 0.05 has been used.

Table 2 reports the basic parameters used in this work to build the ensemble of classifiers. The obtained performance in the classification of the test patterns of 50 S is 96.67% (Table 3).

Table 2: basic parameters used to build the ensembles of classifiers. At 80 t , the dataset 80 S becomes available and the performance of the previously developed ensemble 1 E is tested with the patterns of 80 test S : the fraction of patterns of 80 test S correctly classified is satisfactory (Table 3, first row, fifth column), so that it is not necessary to add classifiers to the ensemble structure i.e.

E E E   (Table 3, first row, sixth column).
Thus, in this case, it has not been necessary to update the first ensemble to learn the newly arriving information under different operational conditions: the ensemble constructed with data taken from transients occurring when the plant is working at 50% of full power is satisfactorily performing on transients at 80% and 108% of full power. The second row of Table 3 reports the classification results that would be obtained if the previously developed ensemble 1 E were discarded at 80 t and a new ensemble 1,2 E formed by 10 T 

classifiers built using all the patterns of 50 80 , train train S S is constructed: the performance of the latter ensemble is slightly better, but at the cost of high computational efforts since all the classifiers have to be retrained from scratch on an enlarged dataset (Table 3 and added to the previously constructed ensemble. The performance of the obtained new ensemble 3 E rises to 94.93% (Table 4, second row, sixth column).

Furthermore, if the power plant returns to work at 80% and 108% of full power, the performance of 3 E remains still satisfactory (Table 4, second row, fourth and fifth columns), this shows that the ensemble 3 E has incrementally learned the new information in 50 S without forgetting what it has learned before ( 80 S and 108 S ). The performance of 3 E is compared with those of the ensembles

Discussion

The above results show that the structure of the proposed ensemble is influenced by the order in which the datasets become available. In case study 1, the first ensemble of classifiers built on the information in 50 train S is able to classify the upcoming datasets with good accuracy, whereas in case study 2, the first ensemble of classifiers built using the information in 108 train S needs to be updated when the plant starts working at 50% of full power.

Conclusions

In this work, a realistic situation in which transient examples of plant behavior in different operational conditions become available in successive datasets has been considered. A practical procedure has been proposed based on the addition of classifiers to an ensemble, for incrementally learning new situations while keeping the computational efforts under control.

The approach used to construct the ensemble is bagging; the base classifier is a supervised Fuzzy C Means (FCM) classifier; the individual base classifiers outcomes are combined using a majority-voting scheme.

The novelty of the procedure is that it allows learning the new information contained in the data becoming available during the plant life without forgetting the previously acquired knowledge. This incremental learning capability is obtained by adding new classifiers to the ensemble if the transients occurring in the new operational conditions are not satisfactorily classified by the current ensemble model.

The procedure has been applied to the identification of simulated transients in the feedwater system of a Boiling Water Reactor (BWR) at different power levels. The proposed classification scheme has been compared with the classical approach which requires that the existing classification model is discarded when new data become available and a new one is retrained from scratch using all data that have been accumulated thus far. The obtained results show that the performance of the proposed procedure is comparable to that achieved by complete retraining of the models, but with the advantage of significant savings in computational efforts. Furthermore, the proposed procedure is suitable to be used in cases in which the datasets previously used for model training are lost, corrupted or otherwise unavailable.

One limitation of the proposed approach which will be object of future work is that the proposed diagnostic system cannot be used for the classification of new classes of faults for which transient examples are not available in the first dataset.

Acknowledgments :

The authors wish to thank Drs. Paolo Fantoni and Davide Roverso of the IFE, Halden Reactor Project for providing the transient simulation data. Also, many thanks go to the reviewers for their constructive comments which have allowed improving the paper.

  The iterative training scheme is summarized in Figure xyz. Once the classifier is constructed, a new test pattern x is classified, in fuzzy terms, by computing its value of membership to the c clusters, based on the Mahalanobis distances of matrices c i M  . Given the ordered corresponding between classes and clusters the fuzzy membership information is finally used for the crisp assignment of the pattern x to the class with the largest value of membership.

  using bootstrapped replicas of the training data 1 train S . In particular, 1 T bagging iterations are performed, each one based on: 1.a) Resampling: the creation of a new training dataset by randomly drawing, with replacement, a fraction F of the training patterns contained in 1 train S

Figure 2

 2 Figure 2.a. Main structure of the ‗ensemble of ensembles' scheme for incremental learning

Table 1 :

 1 Signals selected for the fault classification.

Figure 4 :

 4 Figure 4: Time evolution of the signals at different power levels in case of transients of classes F1 -F6.To test the incremental learning capability of the proposed algorithm under different operational conditions, it has been supposed that the datasets50 80 108 , , S S S become available at different time instants 50 80 108 , , t t t ; a fraction equal to 75% of the patterns of each dataset ( 50 80 108 , , train train train S S S ), has been used to train the models whereas the remaining 25% has been used to test the classification performance ( 50 80 108 , , test test test S S S ).

  of iterations of the supervised algorithm used to train the single base classifiers of the ensemble‖ 500

Figure 1

 1 Figure 1 shows the bagging algorithm used to train an ensemble E formed by T classifiers. A weak learning algorithm -WeakLearn" is used to train the individual base classifiers i h of the ensemble. The maximum number of iterations of the supervised algorithm used to train the single base classifiers of the ensemble is determined by rule of thumb. The flowchart of the training algorithm is shown in the Figure 5.

Figure 5 .

 5 Figure 5. Flowchart for ensemble bagging and training

Figure 6 .

 6 Figure 6. Flowchart of the majority voting algorithm

Table 3 :

 3 Performances and training computational time of the ensemble-based approaches in the classification of the test patterns: proposed approach for incremental learning in a non-stationary environment (first row), ensemble obtained by retraining all the base classifiers (second and third rows).

										Test Dataset
										Training time
	Approach				Train Dataset	50	80	108
										(min)
										S	S	S
										test	test	test
	Incremental Learning Ensemble -	1 E E E 2  	3	S	train 50				51	0.9666	0.9606	0.9659
	Retraining -	1,2 E			train 50 S	,	train 80 S		100	0.9694
	Retraining -	1,2,3 E			train 50 S	,	train 80 S	,	train 108 S	150	0.9715

  , third column). The training time on a Pentium IV 2.2 MHz PC is 51 minute and in case of retraining raises to 100 minute. A similar situation occurs at 108

				t if an ensemble
	1,2,3 E	of	10 T  classifiers is built using the patterns of 50 80 108 , , S S S .
			train	train	train
	4.2. Case study 2: decreasing power level
	In this case study the power level is decreased from 108% to 80% of full power and then from 80% to 50%, with datasets
	108 S S S becoming available at times 108 80 50 80 50 , , t t t   .
	The first developed ensemble 1 E formed by 1 10 T  classifiers trained at 108 t using only the patterns of dataset 108 train S	gives a
	satisfactory performance in the classification of the patterns of 108 S and 80 S i.e. 2 E E 1 	(Table 4, first row, fourth and fifth
				test	test
	column). However, when at 50 t the plant starts working at 50% of full power the performance of 1 E decreases to 71.52% of
	correctly classified patterns in 50 test S (Table 4, first row, sixth column). In the procedure proposed in this work, at 50 t , 3 10 T 
	new classifiers are trained with bagging of the dataset 50 S
				train

Table 4 :

 4 Performances and training computational time of the ensemble-based approaches in the classification of the test patterns: proposed approach for incremental learning in a non-stationary environment (first and second row), ensemble obtained by retraining all the base classifiers (third and fourth rows).

	1,2 E and 1,2,3 E	that would be obtained by discarding the


Piero Baraldi

Roozbeh Razavi-Far

Enrico Zio

Bagged Ensemble of Fuzzy C Means Classifiers for Nuclear Transient Identification

Keywords: Classification, Fuzzy C-Means (FCM) Clustering, Ensemble, Bagging, Transient Identification, BWR Nuclear Power Plant

This paper presents an ensemble-based scheme for nuclear transient identification. The approach adopted to construct the ensemble of classifiers is bagging; the novelty consists in using supervised Fuzzy C-Means (FCM) classifiers as base classifiers of the ensemble. The performance of the proposed classification scheme has been verified by comparison with a single supervised, evolutionary-optimized FCM classifier with respect of the task of classifying artificial datasets. The results obtained indicate that in the cases of datasets of large or very small sizes and/or complex decision boundaries, the bagging ensembles can improve classification accuracy. Then, the approach has been applied to the identification of simulated transients in the feedwater system of a Boiling Water Reactor (BWR).

Introduction

The increasing degree of automation and growing demand for performance, efficiency, reliability and safety of industrial systems, is motivating an increasing interest in fault diagnosis. Indeed, to maintain a high level of performance and reliability, system errors, component faults and abnormal operation must be diagnosed promptly. Faults in process equipment can result in offspecification production, increased operating costs, the chance of line shutdown and the possibility of detrimental environmental impact. The source and severity of each malfunction must be diagnosed so that corrective actions can be taken quickly. Therefore, prompt detection and diagnosis of process malfunctions are strategically important for companies to remain competitive in world markets. Furthermore, large-scale systems such as nuclear power plants (NPPs) are required to provide safe and reliable operation for long periods of time; nevertheless, system components are subject to manufacturing defects, interactions with the environment, wear and tear, and other causes of performance degradation [START_REF] Razavi-Far | Model based fault detection and isolation of a steam generator using neuro-fuzzy networks[END_REF]. For these safety-critical systems, the problem of detecting faults occurrence is of a high importance due to their disastrous consequences.

In this respect, a large number of diagnostic methods for nuclear power plant sensors and components have been proposed in the past decade [START_REF] Reifman | Survey of artificial intelligence methods for detection and identification of component faults in nuclear power plants[END_REF], based on the advances of soft computing techniques such as artificial neural networks [START_REF] Kim | Nuclear power plant fault diagnosis using neural networks with error estimation[END_REF][START_REF] Roverso | Soft computing tools for transient classification[END_REF][START_REF] Embrechts | Hybrid identification of nuclear power plant transients with artificial neural networks[END_REF], fuzzy logic [START_REF] Zio | Identification of nuclear transients via optimized fuzzy clustering[END_REF][START_REF] Zhao | Adaptive fuzzy inference causal graph approach to fault detection and isolation of field devices in nuclear power plants[END_REF] and neuro-fuzzy based techniques [START_REF] Evsukoff | Recurrent neuro-fuzzy system for fault detection and isolation in nuclear reactors[END_REF]Zio and Gola 2006;[START_REF] Razavi-Far | Model based fault detection and isolation of a steam generator using neuro-fuzzy networks[END_REF]. These methods differ for their diagnostic scope and for the type of computing approach. With regards to the former, the main tasks that these diagnostic systems tackle are fault detection (i.e., acknowledging the existence of a plant anomaly, based on observed trends of measured signals) and identification (i.e., establishing the cause of the detected anomaly). With regards to the latter, classification techniques are widely used.

Real applications are usually characterized by complex decision boundaries between the different classes which are difficult to be learned by linear or circular classifiers or lie outside the space of functions that can be implemented by the chosen classifiers. In this respect, an evolutionary optimized fuzzy clustering algorithm [START_REF] Zio | Identification of nuclear transients via optimized fuzzy clustering[END_REF] and local classification approaches such as Kernel methods [START_REF] Müller | An introduction to kernel based learning algorithms[END_REF], Radial Basis Networks [START_REF] Leonard | Radial basis function networks for classifying process faults[END_REF], Fuzzy rules [START_REF] Wang | Generating fuzzy rules by learning from examples[END_REF] Other two instances in which the performance of a single classifier in real applications may not be satisfactory are [START_REF] Polikar | Ensemble Based Systems in Decision Making[END_REF]):

(a) a large volume of data is available for training; (b) a small dataset, inadequate for satisfactory training, is available. In case of type (a), training a single classifier with such a huge amount of data can be not practical; on the contrary partitioning the data into smaller subsets, training different classifiers with different partitions of data, and combining their outputs using an intelligent combination rule often demonstrates to be a more efficient approach. In case of type (b), resampling techniques have been successfully used for drawing overlapping random subsets of the available data, each of which is used to train a different classifier, creating an ensemble. Furthermore, a classification scheme which follows a divide-and-conquer approach by dividing the dataset into smaller and easier-to-learn partitions, where each classifier learns only one of the simpler partitions allows to approximate the original complex decision boundary by a suitable combination of different classifiers. Figure 1, shows an example in which a complex decision boundary cannot be learned by a single linear or circular classifier (curve-shaped boundary), whereas ensemble of classifiers can span the decision space [START_REF] Polikar | Ensemble Based Systems in Decision Making[END_REF].

Figure 1. Ensemble of classifiers spanning the decision space [START_REF] Polikar | Ensemble Based Systems in Decision Making[END_REF] In an ensemble of classifiers, each base classifier is trained on different distributions of data. One of the most popular approaches for constructing different training set for the base classifiers of the ensemble is bagging [START_REF] Breiman | Bagging predictors[END_REF]. Bagging presents the classifiers with a training set that consists of training patterns randomly selected with replacement from the original dataset. Such a training set is called a bootstrap replicate of the original dataset. Empirical studies have established that bagging is a simple and robust method that generally increases the accuracy of a single learner [START_REF] Breiman | Bagging predictors[END_REF][START_REF] Bauer | An empirical comparison of voting classification algorithms: bagging, boosting and variants[END_REF]. The generalization performance of ensemble classifiers is dependent on the diversity and accuracy trade-off of the base classifiers. The combination of the single base classifiers outputs can be implemented using a variety of strategies, such as combination of multiple classifiers [START_REF] Woods | Combination of multiple classifiers using local accuracy estimates[END_REF], classifier fusion [START_REF] Keller | Advances in fuzzy integration for pattern recognition[END_REF], committees of neural networks [START_REF] Breiman | Bagging predictors[END_REF][START_REF] Schwenk | Boosting neural network[END_REF], decision tree ensembles [START_REF] Banfield | A comparison of decision tree ensemble creation techniques[END_REF][START_REF] Bauer | An empirical comparison of voting classification algorithms: bagging, boosting and variants[END_REF] and several combinations of these schemes [START_REF] Xu | Methods of combining multiple classifiers and their applications to handwriting recognition[END_REF].

Whereas neural network, decision tree and support vector machine ensembles have been largely investigated as base classifiers of ensemble systems, there is no comprehensive empirical research in fuzzy clustering ensembles. To fill this void, this paper considers a supervised, evolutionary-optimized Fuzzy C-Means (FCM) clustering procedure [START_REF] Zio | Identification of nuclear transients via optimized fuzzy clustering[END_REF] for building the base classifiers of the ensemble and proposes a bagging algorithm to generate the ensemble. The final classification decision is produced by using the majority-voting scheme [START_REF] Parhami | Voting algorithms[END_REF], one of the simplest and most common aggregation methods.

There are three versions of majority voting [START_REF] Polikar | Ensemble Based Systems in Decision Making[END_REF], where the ensemble chooses the class (i) on which all classifiers agree (unanimous voting); (ii) predicted by at least one more than half the number of classifiers (simple majority); or (iii) that receives the highest number of votes, whether or not the sum of those votes exceeds 50% (plurality voting or just majority voting). This latter is here used to determine the aggregated outcome of the base classifier decisions in the ensemble.

A comparison of the proposed bagging ensemble with a single supervised, evolutionary-optimized FCM classifier [START_REF] Zio | Identification of nuclear transients via optimized fuzzy clustering[END_REF] is firstly conducted on Gaussian datasets with different sizes and complexities. The results are then confirmed through an application to the classification of simulated transients in the feedwater system of a Boiling Water Reactor (BWR) (Puska and Norman, 2002).

In order to accurately estimate the performance of the considered classification methods on test data not previously seen, a crossvalidation (CV) procedure has been adopted [START_REF] Efron | Estimating the error rate of a prediction rule: Improvement on cross-validation[END_REF][START_REF] Efron | Improvements on cross-validation: The .632+ bootstrap method[END_REF][START_REF] Kohavi | A Study of Cross-Validation and Bootstrap for Accuracy Estimation and Model Selection[END_REF][START_REF] Polikar | Bootstrap-Inspired Techniques in Computational Intelligence[END_REF]. In particular, the so called "K-fold" cross-validation error estimate is used to compare the performance of the ensemble system with that of a single optimized classifier. A special case of the K-fold CV, called leave-one-out (LOO), is used in those cases in which the size of the available dataset is small. The remainder of the paper is organized as follows. Section 2 presents a brief overview of the ensemble techniques for classification, outlining the methods and their value to transient identification; further, an ensemble of classifiers based on the supervised, evolutionary-optimized FCM clustering is proposed. In Section 3, a comparison of the bagging ensemble and the single supervised, evolutionary-optimized FCM classifier is presented, using Gaussian datasets with different sizes and complexities. Section 4 describes how the ensemble-based model of the supervised, evolutionary-optimized FCM classifier is used for transient identification in the feedwater system of a BWR, the case studies and the experimental results are compared in this Section. Finally, conclusions are drawn in Section 5.

Ensemble-Based Classification for Transient Identification

The strategy underlying ensemble-based classification is to create many classifiers and combine their outputs so that the combination improves upon the performance of a single classifier. This requires that individual classifiers perform well in different regions of the feature space and make errors on different patterns, which are balanced out in the combination. The intuition is that if each classifier makes different errors, then a strategic combination of these classifiers can reduce the total error. The overarching principle in ensembles is therefore to make each classifier as unique as possible, particularly with respect to misclassified instances. Specifically, we need classifiers whose decision boundaries are sufficiently different from those of others.

Various techniques have been suggested for obtaining diversity in the base models of an ensemble, e.g. using different training parameters [START_REF] Hansen | Neural network ensembles[END_REF], different training patterns [START_REF] Breiman | Bagging predictors[END_REF], different feature subsets [START_REF] Zio | Feature-based classifier ensembles for diagnosing multiple faults in rotating machinery[END_REF] and different learning methods for each classifier of the ensemble [START_REF] Xu | Methods of combining multiple classifiers and their applications to handwriting recognition[END_REF].

In this work, the possibility of using different datasets to train individual classifiers is considered; such datasets are obtained through the resampling technique of bagging [START_REF] Breiman | Bagging predictors[END_REF], where training data subsets are drawn randomly, usually with replacement, from the entire training dataset. Bagging, short for bootstrap aggregating, is one of the earliest ensemble-based algorithms [START_REF] Breiman | Bagging predictors[END_REF][START_REF] Breiman | Combining predictors[END_REF]. It is also one of the most intuitive and simplest to implement, with a surprisingly good performance. In bagging, different training data subsets are used to train different classifiers of the same type. The individual classifiers are then combined by taking a majority vote of their decisions: for any given instance, the class chosen by most classifiers is the ensemble decision. To ensure that there are adequate training samples in each subset, relatively large portions of the samples (75% to 100%) are drawn into each subset. This causes individual training subsets to overlap significantly, with many of the same instances appearing in most subsets, and some instances appearing multiple times in a given subset. In order to ensure diversity under this scenario, a relatively unstable model is used so that sufficiently different decision boundaries can be obtained for small perturbations in different training datasets.

The main structure of our ensemble scheme is shown in Figure 2. In the next Section 2.1, the supervised, evolutionary-optimized FCM clustering algorithm used to build the base classifiers of the ensemble is briefly described [START_REF] Zio | Identification of nuclear transients via optimized fuzzy clustering[END_REF]. The procedure of ensemble construction and its algorithm is presented in Section 2.2.  of the k-th pattern is used to supervise an evolutionary algorithm for finding c optimal Mahalanobis metrics which define c geometric clusters as close as possible to the a priori known physical classes [START_REF] Yuan | Intelligent Hybrid Systems Fuzzy Logic, Neural Network, and Genetic Algorithms[END_REF]. The Mahalanobis metrics are defined by the matrices M i , i=1,…,c whose elements are identified by the supervised, evolutionary-optimized algorithm so as to minimize the distances

* * ( ) ( ) i T i ik k i i k i s x v M x v    between the patterns i k
x belonging to class i and the class prototype * i v  , i.e. the cluster center. The overall iterative training scheme can be summarized as follows [START_REF] Zio | Identification of nuclear transients via optimized fuzzy clustering[END_REF]:

1. At the first iteration (τ=1), initialize the metrics of all the c clusters to the Euclidean metrics.

2. At the generic iteration step τ, run the FCM clustering algorithm to partition the N training data into c clusters, based on the current metric M i () and on the "supervising" initial partition

  1 ,, t t t N     . Let   k  be the cluster to which the k-th pattern k x  is assigned. Then         1 ,..., N        is set
D + to D(Γ t ,Γ()), i M  to () i M  and D + i =D(Γ t i , Γ i ()).
5. Increment  by 1. Update each matrix i M  by exploiting its unique decomposition into Cholesky factors.

6. Return to step 2.

Once the classifier is constructed, a new test pattern x is classified, in fuzzy terms, by computing its value of membership to the c clusters, based on the Mahalanobis distances of matrices i M  . Given the order correspondence between classes and clusters, the fuzzy membership information is finally used for the crisp assignment of the pattern x to the class with the largest value of membership.

The Construction of the Bagging Ensemble

All ensemble models are built on two key components: first, a strategy is needed to build diverse classifiers; second, a strategy, is needed to combine the outputs of the individual classifiers that make up the ensemble in such a way that the correct decisions are amplified, and the incorrect ones are cancelled out. In this work, diversity is obtained by bagging and derives from using Integer T, number of iterations.

Integer F, fraction of the total number N of training patterns that constitute each bootstrapped replica.

Train: Bagging Do 1,...,T

 

1-Construct a set of bootstrap samples. Take a bootstraped replica S  by randomly drawing FN  patterns of the original dataset S , with replacement (Jiawei and Kamber, 2001).

2-Train a classifier

h  on S  .

3-Add

h  to the ensemble E . End

Aggregate: Majority Voting

Majority voting is one of the simplest and most intuitive methods to combine classification decisions. The majority voting method consists in assigning to

x the class label which is supported by the majority of individual classifiers. Given an unlabeled test pattern

x : Do 1,...,T

 

1-Call classifier h  from the ensemble E and classify the test pattern x .

2-Let   be the class assigned by classifier

h  to x . 3-Let , 1, 0, j if j v otherwise          
be the vote given to class j by classifier h  .

END 4-Obtain the total vote received by each class:

, 1 , 1,...,

T jj V v j c     
5-Choose as final assignment the class that receives the largest total vote. In case the number of votes to different classes is equal, assign randomly one among these classes with largest total vote.

In order to avoid over/under estimation of the ensemble performance, cross-validation of the results is performed according to the K-fold cross-validation scheme proposed in [START_REF] Polikar | Bootstrap-Inspired Techniques in Computational Intelligence[END_REF]. The original dataset is randomly partitioned into K blocks of equal size. One of these blocks is used as test data subset and the remaining K -1 blocks are combined together to constitute the training data subset. The cross-validation process is then repeated K times (the K-folds) using a different block as test set each time and consequently the bagging algorithm is repeated K times, each time with a different training set. The general structure of the bagging algorithm within the cross-validation scheme is shown in Figure 3. The K-fold cross-validation error estimate is then the average of the K individual error estimates.

In case of a small-size dataset, the so called "leave-one-out" (LOO) cross-validation method is used. As the name suggests, leaveone-out cross-validation (LOOCV) involves using a single pattern from the original dataset as the test data, and the remaining 1 N  patterns as the training data (Polikar, 2007). This is repeated N times so that each pattern of the dataset is used once as the test data. This is a special case of a K-fold cross-validation, with K being equal to N , the number of patterns in the original data. LOOCV is often computationally expensive because of the large number of times the training process is repeated, but it permits to obtain an unbiased estimation of the error [START_REF] Polikar | Bootstrap-Inspired Techniques in Computational Intelligence[END_REF].

Figure 3. The general structure of the ensemble scheme, with cross-validation; E-fold = Ensemble-fold

Experimental Results on Artificial Datasets

In this Section, the bagging ensemble is compared to a supervised, evolutionary-optimized FCM classifier with respect to the classification of Gaussian and artificial datasets of different size and complexity, including challenging artificial ("2-Spirals") and real-world datasets. A summary of the datasets characteristics is reported in Table 1.

Table 1: Description of the datasets used in the experiments.

The following datasets have been considered:

Small Gaussian dataset:

A Gaussian dataset, formed by 4 classes and 30 patterns is used to train, test and compare the performance of the bagging ensemble and the supervised, evolutionary-optimized FCM classifier.

Medium Gaussian datasets:

Two datasets constituted by 60 and 120 patterns, respectively, randomly sampled from the same Gaussian distributions of the Small Gaussian Dataset.

Large Gaussian dataset:

A dataset constituted by 1500 patterns randomly sampled from the same Gaussian distributions of the Small Gaussian Dataset. This dataset combines the difficulty of dealing with a large number of patterns with the advantages of having only two features.

Laryngeal Dataset:

It is a real medical dataset which contains 692 patterns of two classes representing normal/pathological voices. The Laryngeal dataset comes from the Bulgarian Academy of Sciences and is available at [START_REF] Kuncheva | Real medical datasets[END_REF]. The set was originally used for a computer decision support system, in order to aid diagnosing laryngeal pathology and especially in detecting its early stages. The main difficulty in dealing with the dataset is given by the large number of features of the patterns (16).

2-Spirals

Dataset: it contains 200 bi-dimensional patterns of 2 classes as shown in Figure 4, exhibiting complex cluster shapes [START_REF] Minaei-Bidgoli | Ensembles of Partitions via Data Resampling[END_REF]. This dataset has been considered due to its complex decision boundaries. Table 2: basic parameters used to build the ensemble of classifiers Table 3 reports the mean and the standard deviation of the performances, defined as the ratio between the number of test patterns correctly classified and the total number of test patters, of the bagging ensemble and optimized supervised FCM classifier, with a 10-fold cross-validation scheme.

Table 3: Mean and standard deviation of the performances i.e., the ratio between the number of test patterns correctly classified and the total number of test patters in a 10-fold cross-validation The bagging ensemble performs better than the optimized supervised FCM classifier in the case of the small Gaussian dataset considered. This is due to the fact that in absence of an adequate number of training patterns, the resampling method allows drawing overlapping random subsets, each of which is used to train a different classifier, creating a robust ensemble with more satisfactory performance. For the other datasets, the performances are similar although the bagging ensemble seems to perform slightly better than the optimized supervised FCM classifier, especially when the dimension and complexity of the dataset increase. In particular, in the cases of the Laryngeal and Spiral datasets, the bagging ensemble seems able to learn better the complexity of the decision boundaries that separate the data of the different classes.

Application to Nuclear Transient Identification

In this Section, the ensemble-based fuzzy FCM and the supervised, evolutionary-optimized FCM classifiers are applied to the classification of transients in the feedwater system of a BWR. The diagnostic problem of interest regards the early classification of a predefined set of transients in a BWR. The corresponding transients have been simulated by the HAMBO simulator of the Forsmark 3 BWR plant in Sweden (Puska and Noemann, 2002).

Process Description and Transient Simulation

The considered faults occur in the section of the feedwater system where the feedwater is preheated from 169°C to 214°C in two parallel lines of high-pressure preheaters while going from the feedwater tank to the reactor. Figure 5 shows a sketch of the system. A set of faults, that are generally hard to detect for an operator and that produce efficiency losses if undetected are chosen for this application (see [START_REF] Roverso | On-line early fault detection and diagnosis with the alladin transient classifier[END_REF] for their description). The proposed faults are as follows:

1. F1-F5 regard line 1 of the feedwater system.

2. F6, F7, F8 regard both lines.

Among the 363 measured signals, only the 20 reported in Table 4 have been used for the fault classification in the two case studies here considered. These signals have been chosen considering the results of the application of a feature selection algorithm [START_REF] Zio | Selecting Features for Nuclear Transients Classification by Means of Genetic Algorithms[END_REF] and some benchmark tests. Five transients were simulated for each of the 8 faults, considering different degrees of leakage and valve closures and with step and ramp changes at different leak sizes. The data relative to the selected 20 signals were recorded with a sampling frequency of 1 Hz. All transients start after 60 seconds of steady state operation. Given that the ramp changes cause variations of the parameters later than the step changes, only the three step changes for each fault were considered.

Transient identification in the case of a small dataset

In this case study, the two faults (F6 and F8) have been considered. The dataset is obtained by taking for each fault a transient caused by a step change of large magnitude. A pattern every 10 seconds has been considered in the interval [100,150] seconds.

Thus, the dataset contains 12 patterns and 20 signals. The dataset is classified by the bagging ensemble and the supervised, evolutionary-optimized FCM classifiers in a LOOCV scheme. Figure 6 shows the results of the classification of each single test pattern when the classifier is trained with the remaining 11 patterns, i.e. a single cross-validation in the LOOCV scheme. The bagging ensemble correctly classifies all the patterns (Performance = 1) whereas the supervised, evolutionary-optimized FCM classifier misclassifies pattern 5 (Performance = 0.917). membership values close to 0.5, indicating uncertainty in the assignment; however, the supervised, evolutionary-optimized FCM classifier assigns a slightly higher membership to class F8 and thus the pattern is assigned to class F8, whereas its true class is F6. On the other hand, 6 over 10 classifiers of the bagging ensemble assign pattern 5 with a higher membership degree to class F6, leading to a correct classification of the pattern. In this case, the diversity of the individual base classifiers obtained by using different training sets and aggregated by the majority voting rule allows the ensemble to correctly classify the test pattern.

Transient identification in the case of a large datasets

The objective of this case study is a prompt and correct classification of the seven faults F1,…, F7. For each fault, three transients Table 5 reports the mean performance obtained by the bagging ensemble and by the supervised, evolutionary-optimized FCM classifier in a 10-fold cross-validation scheme. The results indicate higher performance and robustness of the bagging ensemble in the classification of this high-dimensional dataset.

Table 5: Mean and standard deviation of the performance As seen, the performance of the bagging classifier is slightly higher than that of the supervised, evolutionary-optimized FCM classifier when the large dataset, with high dimensionality, is used.

In order to investigate the performance of the bagging classifier and its diversity, the results achieved testing a single fold during the cross-validation process are analyzed. Figure 9

Conclusions

The accurate and timely identification of transients in safety critical nuclear systems is important because it can decrease the probability of catastrophic failures, increase the life of the plant and reduce maintenance costs. In this paper, an ensemble-based scheme has been presented for nuclear transient identification. An ensemble is a group of learners which are combined together in order to obtain better generalization performance than that obtained by a single learner. The errors of the individual ensemble members cancel out to some degree when their predictions are combined. One of the most effective techniques for ensemble construction is the bagging method. This approach promises to give superior classification results with improved accuracy. The expectation is that each classifier will make a different error, and strategically combining these classifiers can reduce the total error. This is because the sets of data examples misclassified by the different individual classifiers would not necessarily overlap.

  and Locally Linear Neuro-Fuzzy system (Razavi-Far et al., 2009) have been proved to provide good performances due to their inherent ability of splitting the input space in local zones. However, given the difficulty in providing a test dataset sufficiently representative of the future field data, these classification algorithms may perform differently in the field. In order to overtake this limitation of single classifier systems, a solution may be of combining the outputs of several classifiers in order to reduce the risk of an unfortunate selection of a poorly performing classifier. The ensemble may or may not beat the performance of the best classifier in the ensemble, but it certainly reduces the overall risk of making a particularly poor selection.

Figure 2 .

 2 Figure 2. The main structure of the FCM-based ensemble scheme

  bootstrapped replicas of the training data: different training data subsets are randomly drawn, with replacement, from the entire training dataset; at each bagging iteration, a new training subset is created based on the original dataset. Each training data subset is used to train a different classifier of the FCM type. As for the combination of the outcomes of the individual base classifiers of the ensemble, this is, achieved by using a majority voting technique. Given the training dataset S of N patterns with known physical class labels 1

Figure 4 :

 4 Figure 4: "2-Spirals" Dataset with two classes, 200 patterns (100-100 per class)

Figure 5 :

 5 Figure 5: A sketch of the feedwater system of the BWR

Figure 6 :

 6 Figure 6: Performance comparison on each single test pattern -Small transient dataset (OFCM = Optimized FCM classifier)

Figure 7 :

 7 Figure 7: Values of membership to each class assigned to the 12 test patterns by the individual base classifiers of the bagging ensemble ( , ) and the supervised, evolutionary-optimized FCM classifier (OFCM , ).

  corresponding to step changes have been considered since they lead to an early variation of the measured variables. Given that the goal is early fault diagnosis, only the data of the first 220 seconds after the beginning of the transients have been considered. The obtained dataset contains 4641 patterns of 7 different classes with 20 features. Figure8shows the values of the 20 features in the patterns of the 7 classes.

Figure 8 :

 8 Figure 8: Behavior of the signals in the different fault classes (F1 -F7).

  shows the number of test patterns which have been correctly classified by 0, 1, 2, …, 10 individual base classifiers of the ensemble. Notice that all the ten individual base classifiers of the ensemble correctly classify 389 out of the 464 test patterns, indicating an overall satisfactory performance of the individual base classifiers. The remaining 75 patterns are assigned to the class which is supported by the majority of individual classifiers. Thus, if more than 5 of the 10 classifiers correctly classify a pattern, it is correctly classified by the ensemble. The remaining cases are analyzed in Figures 10 and 11 which show the distribution of the number of test patterns correctly classified and misclassified, respectively, by the ensemble. For example, Figure 10 shows that there are 3 patterns which are correctly classified by the ensemble although only 4 out of the 10 individual classifiers correctly classify them. This is due to the fact that the 6 classifiers misclassify the patterns assigning them to different classes, i,e, the diversity of the incorrect votes lead to a correct overall classification of the ensemble. Reversely, notice in Figure 11 that there are 2 cases in which all the 6 classifiers misclassifying the pattern agree in assigning them to the same incorrect class and thus the resulting classification of the pattern is incorrect. These analyses show the importance of obtaining an ensemble formed by single classifiers which make different errors. The diversity of the classification errors permits to correctly classify a pattern, even in those cases of Figure 10 in which the correct class is identified by a minority of the ensemble classifiers.

Figure 9 :

 9 Figure 9: Number of classifier responses coherent with true classes

Figure 11 .

 11 Figure 11. Analysis of the cases in which the ensemble final decision is wrong. The numbers [X,Y1,Y2,Y3] next to the column bars indicate the number of individual base classifiers with correct vote [X] and with incorrect votes to different classes [Y1,Y2,Y3].

  Thus, different classifier designs potentially offer complementary information about the data examples to be classified which could improve the performance of the selected classifiers. In this work, bagging is used to construct the ensemble of classifiers; the novelty is in the use of a supervised Fuzzy C-Means (FCM) classifier as individual base classifier of the ensemble. A comparative experimental evaluation of the bagging ensemble with a supervised, evolutionary-optimized FCM classifier has been performed with reference to the classification of artificial Gaussian datasets and simulated transients in the feedwater system of a Boiling Water Reactor. The experimental results show that the bagging ensemble performs as well or slightly better than the optimized FCM classifier on medium and large datasets and outperforms the optimized FCM classifier in the classification of small datasets. This is due to the fact that the single classifiers tend to overfit the data whereas the diversity of the ensemble base classifiers reduces the risk of overfitting. Notice that some issues have still to be addressed for the real application of automatic diagnostics models such as the one here proposed to operating NPPs. First of all, diagnostic systems based on empirical models have performance which may vary in different zones of the training space, according to the density and information content of the example patterns available to train the model. Then, for their application in practice, it is important to a-priori verify whether the model is working in a zone where its performance is expected to be satisfactory. Another factor that has limited the practical application of automatic diagnostic algorithms is the difficulty of obtaining the recognition of transients and faults at different plant operational conditions. Finally, diagnostic approaches able to learn novel information becoming available during the plant life without forgetting the previously acquired knowledge should be investigated in order to reduce the maintenance efforts and costs of the diagnostic system. In this respect, future developments of the present work include the investigation of ensemble techniques for estimating the degree of confidence in the fault diagnosis, for automatically taking into account changes in the plant operating state and updating the diagnostic model with new information.

  equal to the obtained optimal partition Γ * .

	3. Compute the distance D(Γ t ,Γ()) between the a priori known physical classes and the geometric clusters. At the first iteration
	(=1) initialize the best distance D + to D(Γ t ,Γ(1)), D i + to D(Γ i t ,Γ i (1)) and the best metric i M + to	M	i	( ) 1	and go to step 5.

4. If Γ() is close to Γ t or if the number of iterations  is greater than the predefined maximum allowed number of iterations  max , stop: Γ() is the optimal cluster partition Γ * ; otherwise, upgrade

Acknowledgments :

The authors wish to thank Drs. Paolo Fantoni and Davide Roverso of the IFE, Halden Reactor Project for providing the transient simulation data.