P Baraldi

E Zio

G Gola

D Roverso

M Hoffmann

Signal reconstruction by a GA-optimized ensemble of PCA models

On-line sensor monitoring allows detecting anomalies in sensor operation and reconstructing the correct signals of failed sensors by exploiting the information coming from other measured signals. In field applications, the number of signals to be monitored is often too large to be handled effectively by a single reconstruction model. A more viable approach is that of decomposing the problem by constructing a number of reconstruction models, each one handling an individual group of signals. To apply this approach, two problems must be solved: 1) the optimal definition of the groups of signals and 2) the appropriate combination of the outcomes of the individual models. With respect to the first problem, in this work, Multi-Objective Genetic Algorithms (MOGAs) are devised for finding the optimal groups of signals used for building reconstruction models based on Principal Component Analysis (PCA). With respect to the second problem, a weighted scheme is adopted to combine appropriately the signal predictions of the individual models. The proposed approach is applied to a real case study concerning the reconstruction of 84 signals collected from a Swedish nuclear boiling water reactor.

Introduction

During plant operation, sensors may experience anomalies and convey inaccurate or misleading information on the actual plant state to the automated controls and the operators. Hence, it is important to develop accurate and robust systems capable of detecting such anomalies and correctly reconstructing the signals of the failed sensors.

On-line sensor monitoring allows detecting the degradation in the instrument performance; upon such detection, recalibration can be initiated. This way of proceeding based on the monitored health condition of the instrumentation bears important economic advantages, as the current quest for increased competitiveness by all industries requires streamlining all plant operations and reducing system downtime. In general, benefits of this procedure include the reduction of unnecessary maintenance and increased confidence in the actual values of the monitored parameters, with important consequences on system operation, production and accident management [START_REF] Hoffmann | On-line Monitoring for Calibration Reduction[END_REF][START_REF] Hoffmann | Signal Grouping Algorithm for an Improved On-line Calibration Monitoring System[END_REF].

The purpose of this work is to provide a robust technique for reconstructing faulty signals. In many field applications, the number of measured signals is too large to be handled effectively with one single reconstruction model [START_REF] Hoffmann | Signal Grouping Algorithm for an Improved On-line Calibration Monitoring System[END_REF][START_REF] Roverso | Solutions for plant-wide on-line calibration monitoring[END_REF][START_REF] Zio | Genetic Algoritms for Grouping of Signals for System Monitoring and Diagnostics[END_REF][START_REF] Baraldi | Genetic Algorithms for Signal Grouping in Sensor Validation: a Comparison of the Filter and Wrapper Approaches[END_REF]. A viable approach to address the issue of signal dimensionality amounts to subdividing the set of signals into smaller overlapping groups, developing a reconstruction model for each group of signals and then combining appropriately the outcomes of the models within an ensemble approach [START_REF] Perrone | When networks disagree: ensemble methods for hybrid neural networks[END_REF][START_REF] Krogh | Neural network ensembles, cross-validation and active learning[END_REF][START_REF] Sharkey | On combining artificial neural nets[END_REF] (Figure 1). Figure 1. The multi-group ensemble approach to signal reconstruction A major motivation behind the use of an ensemble of models is that reliance on different models increases the robustness of reconstruction. By assigning appropriate performance indicators to the individual models and basing the reconstruction on those models in the ensemble indicated as best performing for a robust reconstruction, failed or decalibrated signals can be effectively reconstructed.

Two issues are central to the ensemble approach for signal reconstruction: [START_REF] Hoffmann | On-line Monitoring for Calibration Reduction[END_REF] the partitioning of the signals into groups and (2) the combination of the outcomes of the individual models developed on the basis of the groups.

In this work, the first issue is tackled by resorting to Multi-Objective Genetic Algorithms (MOGAs) [START_REF] Holland | Adaptation in natural and artificial systems: an introductory analysis with application to biology[END_REF][START_REF] Goldberg | Genetic algorithms in search, optimization, and machine learning[END_REF][START_REF] Chambers | Practical handbook of genetic algorithms: applications[END_REF][START_REF] Sawaragy | Theory of multiobjective optimization[END_REF][START_REF] Raymer | Dimensionality reduction using genetic algorithms[END_REF][START_REF] Bozdogan | Statistical Data Mining with Informational Complexity and Genetic Algorithm[END_REF]. The groups thereby found are then used as bases for developing a corresponding number of reconstruction models based on Principal Component Analysis (PCA) [START_REF] Jolliffe | Principal Component Analysis[END_REF][START_REF] Diamantaras | Principal component neural networks: theory and applications[END_REF][START_REF] Scholkopf | Kernel principal component analysis[END_REF][START_REF] Moore | Principal component analysis in linear systems: Controllability, observability, and model reduction[END_REF][START_REF] Baldi | Neural networks and principal component analysis: learning from examples without local minima[END_REF]. Regarding the second issue, the outcomes of the models are combined by a weighted-combination scheme to provide the ensemble-reconstructed signal values.

The paper is organized as follows. Section 2 illustrates the MOGA-based method for signal grouping. The ensemble approach is then presented in Section 3 and applied in Section 4 to a real case study concerning the reconstruction of a data set of 84 signals measured at a Swedish nuclear Boiling Water Reactor located in Oskarshamn. Some conclusions on the advantages and limitations of the proposed methods are drawn in the last Section.

Multi-Objective Genetic Algorithms for Signal Grouping

The problem of signal grouping is here framed as a Multi-Objective Genetic Algorithm (MOGA) optimization search [10-14, 20, 21]. Formally, given 1

 n sensors' signals i f , i =1,2,…
, n , the aim of the MOGA search is to group them in K groups with some required characteristics and each one constituted by

k mn  signals,
k =1,2,…, K . The inclusion or not of a signal in a group can be encoded in terms of a binary variable which takes value 1 or 0, respectively.

The MOGA probabilistic search is thus performed on a population of K chromosomes, each one constituted by n bits representing all the signals: in the generic k -th chromosome, coding signal group k , the i -th bit, k i s , encodes the presence (1) or absence (0) of the i -th signal in the k -th group, 1,2,..., in  , 1, 2,..., kK  (Figure 2).

Figure 2. The structure of the generic k -th chromosome In general, in order to effectively group the signals, the objective functions of the genetic search should capture both the individual properties of the groups (i.e., the mutual information content of the group signals [START_REF] Zio | Genetic Algoritms for Grouping of Signals for System Monitoring and Diagnostics[END_REF][START_REF] Baraldi | Genetic Algorithms for Signal Grouping in Sensor Validation: a Comparison of the Filter and Wrapper Approaches[END_REF][START_REF] Gola | Signal Grouping for Sensor Validation: a Multi-Objective Genetic Algorithm Approach[END_REF]) and the global properties related to the ensemble of groups (i.e., the diversity between the groups, the redundancy of the signals and the inclusion of the majority of the signals in the groups [START_REF] Hoffmann | Signal Grouping Algorithm for an Improved On-line Calibration Monitoring System[END_REF][START_REF] Roverso | Solutions for plant-wide on-line calibration monitoring[END_REF][START_REF] Baraldi | Genetic Algorithms for Signal Grouping in Sensor Validation: a Comparison of the Filter and Wrapper Approaches[END_REF]).

The MOGA optimization devised in this work consists of a two-objective genetic search which at convergence leads to a final population constituted by groups which maximize the group signals correlation and the diversity between the groups in the population.

The first objective function is intuitively motivated by the fact that the signals in the groups are used to build models for their reconstruction and by the conjecture that strongly positively or negatively correlated signals are capable of regressing one another. In fact, the information content of strongly negatively correlated signals is also very high and comparable to the one derived from strongly positively correlated signals. The measure herein used to quantify these characteristics is the Pearson's correlation coefficient [START_REF] Lawrence | A Concordance Correlation Coefficient to Evaluate Reproducibility[END_REF][START_REF] Hunt | Percent agreement, Pearson's correlation, and kappa as measures of inter-examiner reliability[END_REF]. Considering N measurements of two signals

) (t f p and ) (t f q , 1,2,..., t N 
, the Pearson's coefficient is defined as: By definition, the value of q p corr , ranges from 1

, 1 ( ) ( ) 1 1 pq N pq pq pq ff t f t f f t f corr N S S                  (1)
 to 1 and assumes the value of 0 for statistically independent quantities. Signals that have the same trend (both increasing or both decreasing with respect to the mean) will have positive values of q p corr , , whereas those which vary in opposite ways will render q p corr , negative.

m p p q k q qp corr corr m      (2)
Finally, the group correlation k r , computed as the geometric mean of the average correlations of the k m signals in the group [START_REF] Kirschner | PEANO NNPLS: Advancements in 2002-03[END_REF], is taken as first objective function for the group optimization:

k k m m p p k corr r    1 (3)
This measure allows assigning a low correlation to those groups in which at least one signal has a low average correlation with respect to the others in the group [START_REF] Kirschner | PEANO NNPLS: Advancements in 2002-03[END_REF].

As for the second objective function, the measure used to compute the diversity between the groups is based on the concept of Hamming distance [START_REF] Hamming | Error-detecting and error-correcting codes[END_REF][START_REF] Forney | On the Hamming Distance Properties of Group Codes[END_REF]. According to the definition of the chromosome illustrated in Figure 2, to compute the pairwise diversity measure h j div , between the j -th and h -th group in the population, first the numbers of different, , jh diff , and common signals, , jh com , between the two groups are computed:

, 1 n jh j h i i i diff s s    (4)
,, 1 ()

n j h j h i com i     (5)
where , jh i s is 1 if the i -th signal is included in the j -th or h -th group, respectively, and 0 viceversa and , () jh i  equals 1 if the i -th bit of both the j -th and h -th group is equal to 1, i.e. if the two groups have the i -th signal in common, and 0 viceversa.

The pairwise diversity between the j -th and h -th groups is computed based only on the signals effectively included in the two groups, viz.: , , ,

jh jh j h j h diff div m m com   (6)
This measure is equal to 0 if two groups are constituted exactly by the same signals (i.e., , 0 jh diff  ) and to 1 if the groups have no signals in common.

The group diversity, k d , between the k -th group and the others 1, 2,..., jK  in the population is finally computed as the average pairwise diversity: , 1 [START_REF] Hoffmann | On-line Monitoring for Calibration Reduction[END_REF] 1

K k k j j jk d div K      (7)
At the end of the search, the final population is characterized by K highly correlated and diverse groups of signals; these groups are used as bases for developing a corresponding number of reconstruction PCA-based models; these models constitute the ensemble for reconstructing the signals, as illustrated in the following Section.

The Ensemble Approach to Signal Reconstruction

The aim of the ensemble approach is to ensure an accurate and robust reconstruction of the signals. The approach is sketched in Figure 3.

Figure 3. Sketch of the ensemble algorithm for signal reconstruction

To integrate the signal predictions provided by the K groups a weighted combination-scheme has been adopted. This way of combining the models outputs can be looked at as an extension to a regression problem of the Static Weighted Voting (SWV) technique adopted in classification problems to combine the class assignments of the single classifiers constituting the ensemble [START_REF] Zio | Feature-based classifier ensembles for diagnosing multiple faults in rotating machinery[END_REF].

In this approach, each group k is assigned N  patterns of the train-induce set. Thus, in the ensemble average more importance is given to the predictions of those groups whose corresponding models better reconstruct signal i .

Figure 4. Subdivision of the validation set

Operatively, the weights for the generic k -th group are obtained by first computing the absolute reconstruction error for each signal

i i K kk ii E k i K k i k f t w ft w       (10)
Finally, to evaluate the ensemble performance, first the absolute signal reconstruction error is computed according to Eq. ( 8) using the ensemble signal predictions:

1 1 ( ) ( ) TST N EE i i i TST t f t f t N     (11)
Then, the ensemble accuracy is retained as the average of the absolute signal reconstruction errors:

1 1 n EE i i n     (12)

Application

The ensemble approach has been applied to a real case study concerning n =84 signals collected at a nuclear Boiling Water Reactor (BWR) located in Oskarshamn, Sweden (Figure 5).

The MOGA code used for the calculations has been developed by the Laboratorio di Analisi di Segnale e di Analisi di Rischio (LASAR, Laboratory of Analysis of Signals and Analysis of Risk) of the Department of Nuclear Engineering of the Polytechnic of Milan (http://lasar.cesnef.polimi.it). The PCA code has been taken from http://lib.stat.cmu.edu/multi/pca and adapted to perform the signal reconstruction tasks.

MOGA signal grouping

The results of the two-objective MOGA search (Section 2) carried out to find the optimal set of groups are illustrated below. The MOGA settings here adopted are reported in Table 2, with a synthetic explanation of the terms used.

Selection FIT-FIT: the population, rank-ordered on the basis of the Pareto dominance criterion, is scanned and each individual is parent-paired with an individual of the next fittest Pareto rank class Replacement FITTEST: out of the four individuals (two parents and two chromosomes) involved in the crossover procedure, the fittest two replace the parents Mutation probability 10 -2 Population size 100 Number of generations 30000 Table 2. Main parameters used for the MOGA search. For further details on the FIT-FIT selection procedure and the FITTEST replacement procedure the interested reader may consult [START_REF] Holland | Adaptation in natural and artificial systems: an introductory analysis with application to biology[END_REF][START_REF] Goldberg | Genetic algorithms in search, optimization, and machine learning[END_REF].

At convergence of the search, 100 K  highly correlated and diverse groups of signals make up the final population. Figure 6 reports the main features (correlation, diversity and size) of the optimal groups of the final population. The groups contain from 2 to 70 signals and are characterized on average by a good degree of diversity. The value of diversity computed according to Eq. ( 7) is generally lower for small groups, which are naturally the most correlated [START_REF] Zio | Genetic Algoritms for Grouping of Signals for System Monitoring and Diagnostics[END_REF]. In general, the groups in the final population are characterized by different trade-offs between the two objectives. As mentioned in Section 2, an ensemble approach to signal reconstruction is effective only if the groups in the ensemble are sufficiently diverse, they ensure a good signal redundancy and include the majority of the signals [START_REF] Hoffmann | Signal Grouping Algorithm for an Improved On-line Calibration Monitoring System[END_REF][START_REF] Roverso | Solutions for plant-wide on-line calibration monitoring[END_REF]. Figure 7 shows that the maximization of diversity as objective function has led to fulfil both these global properties.

Ensemble signal reconstruction

Within the proposed ensemble weighted approach (Section 3), the signals of the K groups are used as bases for developing a corresponding number of PCA reconstruction models, as illustrated in Figure 1. For each group, the number of principal components retained to reconstruct the test patterns has been set equal to half the size of the group, for it represents a good compromise between the accuracy and the computational cost of the model [START_REF] Baraldi | Genetic Algorithms for Signal Grouping in Sensor Validation: a Comparison of the Filter and Wrapper Approaches[END_REF][START_REF] Jolliffe | Principal Component Analysis[END_REF][START_REF] Diamantaras | Principal component neural networks: theory and applications[END_REF][START_REF] Scholkopf | Kernel principal component analysis[END_REF][START_REF] Moore | Principal component analysis in linear systems: Controllability, observability, and model reduction[END_REF][START_REF] Baldi | Neural networks and principal component analysis: learning from examples without local minima[END_REF].

The validation set

The first row in Table 3 reports the results of the reconstruction in terms of the ensemble error E  defined as [START_REF] Sawaragy | Theory of multiobjective optimization[END_REF].

Compared with the error made by a single reconstruction model built on all 84 signals (last column of Table 3), the ensemble shows an improvement of 26.6%.

As stressed in the previous Sections, a robust ensemble of models must be able to reconstruct the signals when in presence of sensor failures, e.g. random noises, offsets or drifts. Within the proposed ensemble approach, a faulty sensor sends a faulty signal in input to the PCA models which include that signal; in this situation, the ensemble of models should still be capable of providing a good estimate of the true value of the signal by exploiting the information coming from the non-faulty signals in the groups of the ensemble.

i : * 1 ( ) ( ) () TST N ii i i t f t f t ft      (13)
where ()

n TST i i n           (14)
The magnitude of The second row in Table 3 reports the results obtained with disturbed patterns. Once again, the ensemble of models outperforms the single model, with an improvement of 20.7 %.

However, when applying the ensemble weighted approach on disturbed signals, the computation of the group weights becomes a crucial issue. In fact, calculating the group weights as before using undisturbed training patterns may lead to large weights for groups with indeed good reconstruction performances on the undisturbed signals, but not suitable for regressing correctly the disturbed signals.

To overcome this limitation, a robust training procedure has been embedded in the ensemble reconstruction scheme. In this respect, the training patterns are disturbed in the same manner as those of the test set, with a corresponding amount of disturb % TRN   defined as in Eq. [START_REF] Bozdogan | Statistical Data Mining with Informational Complexity and Genetic Algorithm[END_REF]. By so doing, the group weights are proportional to the group reconstruction performances on disturbed signals.

In the last two rows of Table 3, the accuracy and robustness of the multi-group ensemble approach with robust training are compared to those of the single reconstruction model built with all 84 signals. Robust training is seen not to improve the robustness of the model; on the other hand, if no robust training is performed (rows 1 and 2), the reconstruction error when the test set is disturbed is significantly increased (+1680% from row 1 to row 2 against +130% from row 3 to 4).

Reconstruction errors (× 10 The ensemble robustness has been then specifically tested on the reconstruction of a faulty signal when other sensor signals are damaged. Signal 7 has been chosen as the object of the analysis. A linear drift decreasing its value up to 25% of its real value has been introduced; at the same time, other signals have been either linearly drifted or forced in offset.

The different fault scenarios are reported in Table 4.

The validation set has been linearly divided into training and test (Figure 4) only once (no cross-validation), whereas a 10-fold cross-validation has been performed on the computation of the group weights. Robust training (last column) turns out to be effective when highly correlated signals are disturbed (cases A and D in Table 4, Figures 10a and10c

Conclusions

In this work, signal reconstruction has been carried out by resorting to a multi-group ensemble approach. The set of sensor signals, too large to be handled effectively with one single reconstruction model, is subdivided into many small, overlapping groups and a reconstruction model is developed for each group. The outcomes of the models are then combined to obtain the ensemble signal reconstruction.

Multi-Objective Genetic Algorithms have been used to generate groups of correlated signals; Principal Component Analysis has been used to build the corresponding reconstruction models; a weighted combination scheme has been used to combine the outcomes of the models.

The overall modelling scheme has been applied for the reconstruction of signals collected at a Swedish nuclear boiling water reactor. To enhance robustness, a robust training procedure has been introduced. The reconstruction performances obtained by the ensemble approach have been compared with those of a single model built using all the signals: the multi-group ensemble approach has proved to be more accurate and robust than the single model, independently of robust training.

  i -th weight being proportional to the accuracy of the model developed with group k in reconstructing the i -th signal. The set X VAL of VAL N patterns available to validate the ensemble performance is first partitioned into a training set X TRN (made of TRN N patterns) and a test set X TST (made of TST N patterns). The former is used to compute the sets of weights of the individual models, whereas the latter is used to verify the ensemble performance in the reconstruction task. To compute the weights, the training set must be further partitioned into a so-called train-induce set X TRN IND  (made of TRN IND N  patterns) and an induce set X IND (made of IND N patterns), as illustrated in Figure 4. The weight k i w is proportional to the error in reconstructing the IND N induce patterns of the i -th signal made by the model trained on the TRN IND

Figure 5 .

 5 Figure 5. The process diagram of the nuclear power plant in Oskarshamn. The 84 signals are identified by an alphanumeric code (e.g. 312 KA301, 423 KB509, etc…). The data available for the 84 signals have been sampled every 10 minutes from May 31, 2005 to January 5, 2006 from a corresponding number of sensors, providing a total amount of 30080 N  time samplings. Each recording instant t provides an 84-dimensional pattern ) ( ),..., ( ), (

  84

Figure 6 .

 6 Figure 6. Main features of the 100 K  groups of the final population in ascending order with respect to their size. Each group is identified by the value of its correlation k r (Eq. 3) (top), diversity

Figure 7 .

 7 Figure 7. Redundancy of each of the 84 n  signals (top) and corresponding distribution (bottom)

  one every ten. Note that preliminary studies have shown that reducing the number of samples has negligible effects on the values of the errors, while it considerably decreases the computational cost of validating the ensemble of models.The ensemble procedure has been 20 times cross-validated. At each run, 1200 constitute the training set X TRN needed for building the PCA models; the remaining 311 TST N  patterns form the test set X TST for evaluating the performance of the ensemble of models. Furthermore, a 10-fold cross-validation has been performed to compute the group weights. In this respect, at each cross-

  real and disturbed test patterns, respectively, the average percentage amount of disturb %

Figure 8

 8 Figure 8 shows that the effectiveness of robust training increases with

Figure 8 .

 8 Figure 8. Ensemble reconstruction error, E  , without and with robust training for different

Figure 9 .

 9 Figure 9. Absolute percentage gain (%) A g with robust training for different

  , dark dots); this is true, even if % TST   is small (case A). In fact, performing robust training helps reconstructing the highly correlated signals useful to reconstruct signal 7. On the contrary, when disturbing uncorrelated signals (cases B and E in Table 4), robust training is indeed useful for correctly reconstructing the disturbed signals, but it unavoidably introduces disturbs in the reconstruction of the correlated signals effectively exploited for reconstructing signal 7, thus worsening the reconstruction accuracy (Figure 10b, dark dots). In case C, highly and poorly correlated signals have been drifted and the effectiveness of robust training is due to the high value of % TST   .

Figure 10a .

 10a Figure 10a. Ensemble reconstruction of signal 7 (light line) when linearly drifted (dark line) in case A, with (dark dots) and without (light dots) robust training.

Figure 10b .

 10b Figure 10b. Ensemble reconstruction of signal 7 (light line) when linearly drifted (dark line) in case B, with (dark dots) and without (light dots) robust training.

Figure 10c .

 10c Figure 10c. Ensemble reconstruction of signal 7 (light line) when linearly drifted (dark line) in case D, with (dark dots) and without (light dots) robust training.

Finally, the reconstruction

  capabilities with and without robust training have been tested on one faulty signal in various multi-fault scenarios. The results have shown that robust training provides a more accurate reconstruction of the signal when highly correlated signals are affected by disturbs, whereas slightly lower performances are achieved when uncorrelated signals are drifted, for the introduced disturbs actually worsen the reconstruction of the signals correlated to the one of interest, upon which reconstruction is based.

  To associate a degree of correlation to the generic k -th group of

					m signals,	1, 2,..., kK 	, the average absolute
					k
	correlation of each	p -th signal,	1, 2,..., pm 	, is first computed as the mean of the absolute values of the correlation
					k
	between the	p -th and the remaining	1 m  signals, viz.
					k
			1	k
			1	1	,

  1, 2,...,

			im 	on the normalized induce data 1 , viz:
						k
			1 IND kk N 1 ( ) ( ) i i i IND t f t f t N   	(8)
	where, for	1, 2,..., tN 	, () ft is the real (normalized) value of the	i -th signal and ˆ() k ft is the reconstruction
						IND	i	i
	obtained by the PCA model built using the signals of the	k -th group and trained on the	TRN IND N 	patterns of the train-
	induce set.
	The corresponding weight k w ,	1, 2,..., im 	, is simply taken as the inverse of the signal reconstruction error:
							i	k
		k i w		1 k 		(9)
					i
	Obviously, this measure allows assigning large weights to those groups which provide the most accurate reconstruction
	of signal	i .
	Once the weights are computed, the generic PCA model built using the	m signals of group	k is trained on the	N
							k	TRN
	patterns of the training set to provide in output the predictions ˆ() k ft ,	1, 2,..., tN 	,	1, 2,..., im 	of the	N
							i	TST	k	TST
	patterns of the test set.
	The ensemble reconstruction of the	N	patterns of the generic	i -th signal, ˆ() E ft ,	1, 2,..., tN 	, is then obtained
							TST	i	TST
	by combining the predictions ˆ() k ft of the	K groups including signal	i with the corresponding weights k w ,
							i	i	i
	1, 2,..., kK 	:
			i
						ˆ()
		ˆ()	1
						1

Table 1 .

 1 The total number of available patterns is divided into a MOGA set X

						f	1	t	f	2	t	f	t	, t =1,…,
										MOGA of	N	MOGA		14966	samples used to perform
	the genetic optimization and a validation set	X VAL of	VAL N		15114	samples used to train and test the PCA models and
	to compute the ensemble reconstruction performances (Eqs. 11 and 12).
	Time samplings 	1, 2,..., tN 		Signals 	1, 2,..., in 	
										1	1 (1) f	f	2 (1)	…	84 (1) f
	N	MOGA		14966			…
										14966
										14967
	N		15114				…
		VAL
										30080	1 (30080) f	f	2 (30080)	… 84 (30080) f

N , identified by the values of the n =84 signals at the time instant t , as illustrated in

Table 1 .

 1 Partition of the total data set for the MOGA signal grouping and ensemble signal reconstruction

  To verify this, a disturb is introduced in the test set. More precisely, the signals of a test pattern are randomly affected either by a random noise (with probability 0.02

			RN p 	) or by setting their value equal to the offset value of the
	corresponding sensor (with probability	OF p 	0.01	); with probability 0.97 they are not affected at all.
	Defining

i  as the total disturb affecting signal

Table 3 .

 3 Ensemble approach compared to a single model, with and without robust training.

									-1 )
									Ensemble, E 	Single group
			  	% TRN		0	% TST %    TST    	0 0.61 	0.0094 ± 0.0096 0.1676 ± 0.0111	0.0128 ± 0.0151 0.2114 ± 0.0138
			  	% TRN		0.61	% TST %    TST    	0 0.61 	0.0756 ± 0.0247 0.1742 ± 0.0213	0.0933 ± 0.0259 0.2306 ± 0.0267
	To investigate further the effectiveness of robust training, larger amounts of	 	% TST	have been considered to verify
	whether the advantage brought by robust training in reconstructing disturbed signals overcomes the loss of accuracy in
	regressing undisturbed signals. As an indicator, let	g	(%)	be the absolute percentage gain obtained with robust
									A
	training:
	g	A	(%) 100 		%% 0 TRN TRN EE         0.61    	(15)

Table 4

 4

	reports the ensemble

Table 4 .

 4 Ensemble performance in the reconstruction of signal 7, when linearly drifted in different fault scenarios. When performing robust training, 20 differently disturbed training sets with

	 	% TRN	0.6	are generated and the mean

In the application which follows, each signal of the validation data set has been previously normalized in the range [0.2, 1].


P Baraldi

E Zio

G Gola

D Roverso

M Hoffmann

Two novel procedures for aggregating randomized model ensemble outcomes for robust signal reconstruction in nuclear power plants monitoring systems

Detecting anomalies in sensors and reconstructing the correct values of the measured signals is of paramount importance for the safe and reliable operation of nuclear power plants. Auto-associative regression models can be used for the signal reconstruction task but in real applications the number of sensors signals may be too large to be handled effectively by one single model. In these cases, one may resort to an ensemble of reconstruction models, each one handling a small group of sensor signals; the outcomes of the individual models are then combined to produce the final reconstruction. In this work, three methods for aggregating the outcomes of a feature-randomized ensemble of Principal Components Analysis (PCA)-based regression models are analyzed and applied to two case studies concerning the reconstruction of 215 signals monitored at a Finnish nuclear Pressurized Water Reactor (PWR) and 920 simulated signals of the Swedish Forsmark-3 Boiling Water Reactor (BWR). Based on the insights gained, two novel aggregation procedures are developed for optimal signal reconstruction.

Introduction

Accurate monitoring of the health state of systems, structures and components can contribute significantly to the safe and efficient functioning of nuclear power plants for it allows the timely detection of malfunctions and anomalies during operation.

The monitoring relies on the signals collected by a large number of sensors placed at various locations in the plant. The measured signals are transmitted to the control and protection systems and to the operators, to evaluate the plant health state and eventually take corrective or emergency actions for safely steering critical situations and preventing accidents.

For efficient monitoring, early detection of sensor failures and eventual correction of the delivered measurements is fundamental to avoid that misleading information be used for plant operation, possibly leading to unsafe and/or inefficient states [START_REF] Hoffmann | On-line monitoring for calibration reduction[END_REF][START_REF] Hoffmann | Signal grouping algorithm for an improved on-line calibration monitoring system[END_REF].

This work investigates the problem of reconstructing signals in realistic applications in which the number of measured signals is very large and cannot be handled effectively by a single reconstruction model [START_REF] Hoffmann | Signal grouping algorithm for an improved on-line calibration monitoring system[END_REF][START_REF] Roverso | Solutions for plant-wide on-line calibration monitoring[END_REF][START_REF] Zio | Genetic algorithms for grouping of signals for system monitoring and diagnostics[END_REF]Baraldi et al. 2008a). The problem is tackled by resorting to an ensemble-based signal reconstruction procedure. The ensemble approach is founded on the subdivision of the set of sensor signals into small overlapping groups; a reconstruction model is developed for each group of signals and the outcomes of the individual models are eventually aggregated to generate the reconstructed signal [START_REF] Perrone | When networks disagree: ensemble methods for hybrid neural networks[END_REF][START_REF] Krogh | Neural network ensembles, cross-validation and active learning[END_REF][START_REF] Sharkey | On combining artificial neural nets[END_REF][START_REF] Gola | Signal grouping for sensor validation: a multiobjective genetic algorithm approach[END_REF]Baraldi et al., 2008b;[START_REF] Gola | Reconstructing signals for sensor validation by a GA-optimized ensemble of PCA models[END_REF]) (Figure 1).

Figure 1. The multi-group ensemble approach to signal reconstruction Two issues are central to the ensemble approach to signal reconstruction: [START_REF] Hoffmann | On-line monitoring for calibration reduction[END_REF] how to define the signal groups and [START_REF] Hoffmann | Signal grouping algorithm for an improved on-line calibration monitoring system[END_REF] how to aggregate the outcomes of the models built on such groups.

In this work, the signals groups are randomly generated by the Random Feature Selection Ensemble (RFSE) technique [START_REF] Aly | Novel methods for the feature subset ensembles approach[END_REF][START_REF] Bryll | Attribute bagging: improving accuracy of classifiers ensembles by using random feature subsets[END_REF][START_REF] Baraldi | A procedure for the reconstruction of faulty signals by means of an ensemble of regression models based on principal components analysis[END_REF][START_REF] Polikar | Ensemble based systems in decision making[END_REF] which allows obtaining highly diverse signal groups and, correspondingly, diverse signal outcomes from the individual models. The regression models based on the signal groups are developed by Principal Component Analysis (PCA) [START_REF] Jolliffe | Principal component analysis[END_REF][START_REF] Diamantaras | Principal component neural networks: theory and applications[END_REF][START_REF] Scholkopf | Kernel principal component analysis[END_REF][START_REF] Moore | Principal component analysis in linear systems: controllability, observability, and model reduction[END_REF].

The issue of how to compute the ensemble-aggregated signal reconstruction output is the focus of this paper. The random composition of the signal groups is such that some of the individual models might provide largely incorrect signal reconstructions which negatively affect the ensemble aggregate. Thus, discarding the outcomes of some models can enhance the accuracy and robustness of the randomized ensemble approach. In this view, three different aggregation methods based on the distribution of the models' outcomes are analyzed: Simple Mean (SM), Median (MD) and Trimmed Mean (TM) [START_REF] Polikar | Ensemble based systems in decision making[END_REF]. The first amounts to using all the available outcomes of the models in a simple average; the second considers only the central outcome in the distribution of the outcomes of the individual models; the last one discards the outcomes falling in the tails of the distribution. Furthermore, two novel procedures for aggregating the models outcomes based on the combination of the Median and Trimmed Mean approaches are proposed to fully exploit the benefits of the two methods.

The paper is organized as follows. In Section 2, the randomized approach for generating diverse groups of signals is briefly summarized and the methods of model outcomes aggregation are described. In Section 3, the performance of the aggregation methods is analyzed with reference to two case studies: one concerns the reconstruction of a data set of 215 signals measured at a Finnish nuclear Pressurized Water Reactor (PWR) located in Loviisa; the other addresses the reconstruction of a data set of 920 simulated signals obtained with the HAMBO code [START_REF] Karlsson | The HAMBO BWR simulator of HAMMLAB[END_REF] which simulates the Forsmark-3 Boiling Water Reactor (BWR) located in Sweden. In Section 4, two novel aggregation procedures are introduced and verified on both case studies. Conclusions on the advantages and limitations of the proposed methods are drawn in the last Section.

Model ensemble construction and aggregation

Given a set of 1  n signals i f , n i ,..., 2 , 1 
, measured in the plant, the problem is to define a set of K groups of mn  signals, with characteristics favourable for their ensemble aggregation. In particular, diversity in the groups composition is important, balanced with a suitable redundancy of the signals in the groups (i.e. the same signal must be included in more than one group) so that their ensemble reconstruction can be based on several (diverse) models outcomes. In other words, groups must partially overlap in order to have some signals in common, while being sufficiently diverse among one another.

An effective method to pursue these characteristics is offered by the Random Feature Selection Ensemble (RFSE) technique [START_REF] Aly | Novel methods for the feature subset ensembles approach[END_REF][START_REF] Bryll | Attribute bagging: improving accuracy of classifiers ensembles by using random feature subsets[END_REF][START_REF] Baraldi | A procedure for the reconstruction of faulty signals by means of an ensemble of regression models based on principal components analysis[END_REF][START_REF] Polikar | Ensemble based systems in decision making[END_REF], which consists in randomly sampling, with replacement, from the n signals, K groups, each one constituted by m signals.

The procedure is simple and easily applicable in large scale applications. Furthermore, a reasonable choice of the ensemble parameters m and K allows the desired diversity among the groups and signal redundancy across the groups [START_REF] Baraldi | A procedure for the reconstruction of faulty signals by means of an ensemble of regression models based on principal components analysis[END_REF].

On the basis of the Simple Mean (SM) amounts to combining all the available model outcomes in a simple arithmetic average [START_REF] Baraldi | A procedure for the reconstruction of faulty signals by means of an ensemble of regression models based on principal components analysis[END_REF][START_REF] Polikar | Ensemble based systems in decision making[END_REF][START_REF] Kuncheva | Classifier ensembles for changing environments[END_REF][START_REF] Kittler | On combining classifiers[END_REF]:

, 1 1 ˆ( ) ( ) 1, 2,..., i K E SM k ii i k f t f t i n K    (1)
where , ˆ() E SM i ft is the ensemble aggregate of the reconstructions ˆ() k i ft of pattern t of signal i provided by the individual models 1, 2,...,

i kK  containing signal i .
Nevertheless, due to the random nature of the groups composition upon which the signal reconstruction models are built, some individual models might provide largely incorrect signal reconstructions, which compromise the simple average-aggregated ensemble outcome 1 . 1 In principle, if the outcomes of a very large number of (randomized) models were available for each signal (i.e., the signals are highly redundant), simple averaging would mitigate the effects of the group randomness and provide an accurate ensemble reconstruction. In practice, imposing a high redundancy ( i K ) for each signal 1,2,..., in  while, at the same time, requiring a small group size ( m ) for model effectiveness forces to increase the number of groups ( K ) in the ensemble [START_REF] Baraldi | A procedure for the reconstruction of faulty signals by means of an ensemble of regression models based on principal components analysis[END_REF], with consequent increase of the computational cost associated to the development of a corresponding number of models. ensemble aggregate. The methods are based on the randomness of the models outcomes, which, if unbiased, are expected to distribute around the correct (unknown) signal value. In this view, the outcomes lying around the centre of the distribution are conjectured to be close to the correct signal value, whereas those lying on the tails of the distribution are considered fairly incorrect.

The Median (MD) approach amounts to considering for the generic pattern t the single outcome ˆ() C k i ft lying in the centre of the distribution of the outcomes for that pattern, i.e.:

, ˆ( ) ( ) 1, 2,..., C k E MD ii f t f t i n  (2)
where C k denotes the index of the model whose outcome is central with respect to the reconstructed values of the i K models including signal i .

The Trimmed Mean (TM) approach amounts, instead, to discarding a fraction

TM  of the i K outcomes of signal i   ˆ( ), 1, 2,..., k ii f t k K 
at each tail of the distribution. The ensemble reconstruction of the generic pattern t , , ˆ()

E TM i ft ,
is then computed by simply averaging the

  12 TM i TM i KK   remaining outcomes: , 1 1 ˆ( ) ( ) 1, 2,..., TM i K E TM k ii TM i k f t f t i n K    (3)
In this sense, the TM represents a compromise between the SM and MD methods, for it allows discarding the tails of the distribution while still considering multiple (central) outcomes to reconstruct the signal 2 .

Finally, to evaluate the performance of the ensemble aggregates, first the absolute signal reconstruction error is

computed 3 : , ( , , ) , ( , , ) 1 1 ( ) ( )
TST N E SM MD TM E SM MD TM i i i TST t f t f t N     (4)
Then, the ensemble performance index is retained as the average of the absolute signal reconstruction errors (Eq. 4):

, , , ) , ( , , ) 1 1
n E SM MD TM E SM MD TM i i n     ( (5)

Application of the aggregation methods and analysis of their performance

2 Notice that 11 0, 1 2

TM i K         
. The lower limit corresponds to considering all the available outcomes (SM), whereas the upper limit represents the case of using the single central outcome (MD). 5 Once m and i K are set, the number of groups to generate with the RFSE procedure is obtained from the relation i mK nK  . For a more detailed explanation of the procedure, the interested reader may refer to [START_REF] Baraldi | A procedure for the reconstruction of faulty signals by means of an ensemble of regression models based on principal components analysis[END_REF].

Table 2 reports the results obtained by the three aggregation methods for the two case studies, in terms of ensemble reconstruction performance indexes (Eq. 5) on test patterns made of undisturbed ( ,

EU



) and disturbed ( , ED  ) signals.

In general, the results show the benefits of excluding some outcomes from the aggregation: overall, the MD ensemble provides the most accurate and robust signal reconstruction in both applications, with a slight improvement with respect to the TM technique. First of all, to show the effects of the three aggregation methods on the reconstruction, signal 163 of the Loviisa set is considered within the ensemble setting of Table 1 (i.e., 7

i K  , i  ). Figures 2 and3 illustrate the reconstruction errors obtained by the 163 7 K  groups including signal 163 and those achieved by the SM, MD and TM aggregation methods when the signal is undisturbed (Figure 2) and disturbed (Figure 3). Table 3 3).

In case of disturbed signals, resorting to the ensemble approach generally allows reducing the reconstruction errors of the individual groups, eventually resulting in a less noisy ensemble-aggregate signal reconstruction. Nevertheless, discarding the outlying predictions (i.e., in the MD and TM ensembles) enhances such effect and provides a more robust signal reconstruction (Table 3). The three aggregation methods have also been evaluated with an augmented signal redundancy

15 i K  , i  , i.e.
with more models' outcomes used in the ensemble to reconstruct the signals. With the increased signal redundancy and the optimal group size 38 m  , the number of groups to generate with the RFSE procedure is 85 K  6 . The fraction TM  used in the TM aggregation method has been kept equal to 0.2, which 6 See Note 5.

corresponds in this case to adopting 15, 1 and 9 models for the reconstruction of each signal by the SM, MD and TM methods, respectively. the performance in all cases. This effect is more evident when using the MD and TM methods on the disturbed signal set.

Furthermore, the reconstruction capabilities of the MD and TM approaches on each signal have been analyzed, with reference to the data of the Forsmark-3 case study, in terms of the distance between the TM i K PCA model outcomes used to produce the TM aggregate and the single PCA model outcome ˆ() C k i ft used as ensemble aggregate in the MD approach. In particular, the correlation between the distribution of the outcomes and the reconstruction performances of the two aggregation methods is analyzed.

To compute the average distance between the outcomes used to reconstruct the generic signal i by ensemble aggregation, first the absolute point-wise outcome distance tN  is computed:

1 1 ˆ( ) ( ) TM i C C K k tk i i i TM i k kk f t f t K      (6)

E TM E MD ii

  for MD outperforming TM or viceversa) and the corresponding average error reduction in using the best performing one. For the Forsmark-3 case study (Table 5a), approximately half of the are better reconstructed by MD (dots above the black line in the graphs) and half by TM (dots below the black line in the graphs); nevertheless, on disturbed signals, the average error reduction achieved by MD is larger, i.e. the improvements of using MD are more relevant than those obtained with TM, even though on a slightly smaller number of signals. This can also be visualized in the graphs of Figure 4

Two novel procedures of aggregation

The results of the above analyses show that neither MD, nor TM can clearly outperform the other; on the other hand, procedures for combining MD and TM can be sought, with the aim of exploiting the advantages of both methods. In particular, based on the signal reconstruction errors obtained on the training set, the idea is to define a reconstruction scheme finalized to discern automatically which method is the most effective for reconstructing the test patterns of a signal.

In this respect, two procedures for combining the MD and TM aggregation methods are proposed in this work. To tailor these procedures for robust signal reconstruction and since the error difference ,, E TM E MD ii   is quantitatively on undisturbed signals (so that using TM or MD makes really no difference, as shown for the Forsmark-3 case study by the dark dots in the left and right graphs of Figure 4) the procedures are devised with respect to the disturbed training signals (Figure 4, left) for application to the reconstruction of the test signals (be it undisturbed or disturbed).

Procedure 1

The first combination procedure, sketched in Figure 6, is based on the relationship between i d and

,, E TM E MD ii   . In fact, according to the results presented in Figure 4 for the Forsmark-3 case the value of the signals' average model outcome distances i d (Eq. 7) can provide some indications of the best method to adopt.

Operatively, the range of the average outcome distances ,,

:

il E TM E MD ii l id        (10)
In order to identify specific portions of the range of the average outcome distances in which one method outperforms the other, neighbour subsets characterized by the same sign of l  , e.g.   These clusters define the reconstruction scheme by which deciding the method to adopt for reconstructing the test signals (Figure 6). In this view, for the generic signal i , first the average outcome distance i d is computed using the (test) outcomes (Eq. 7); then, the (training) cluster j C to which i d belongs is retained for defining the most effective method for aggregating the outcomes.

Procedure 2

The proposed procedure (sketched in Figure 8) is based on the ensemble reconstruction errors obtained on the disturbed training patterns of each signal by the MD and TM methods. The conjecture is that if the (disturbed) training patterns of a signal are better reconstructed by one method (MD or TM), then the same method will better reconstruct also the (disturbed) signal's test patterns.

In this view, for each signal i , first the ensemble reconstruction errors 8).

iK

  and the outcomes of the corresponding models are aggregated into the final ensemble outcome.Three different methods are here considered to aggregate the individual model outcomes: Simple Mean (SM), Median (MD) and Trimmed Mean (TM).

4

  In the Forsmark-3 case study both TRN X and TST X include signal measurements related to start up, normal operation and shut down conditions, whereas in the Loviisa case study transients related to the first outage are included in TRN X , while those of the second outage are in TST X .

  reports the numerical values of the signal reconstruction errors (Eq. 4) on the undisturbed ( respectively, obtained by the SM, MD and TM techniques.

Figure 2 .Figure 3 .

 23 Figure 2. Groups and ensembles reconstruction errors for signal 163 (undisturbed)

   and therefore the difference in adopting MD or TM is negligible. This concerns the reconstruction of all the undisturbed training signals (black dots in the left graph) and most of the undisturbed test signals (black dots in the right graph). Nevertheless, the increase of i d (i.e. 0.02 i d  , a situation mostly occurring when signals are disturbedlight dots in left and right graphs) manifests the advantage of using either MD or TM to aggregate the models outcomes since the difference ,, E TM E MD ii   can reach values much larger than 0.005.

Figure 4 .

 4 Figure 4. Forsmark-3 case study: average PCA model outcomes distances versus difference between ensemble reconstruction errors by TM and MD, computed using the training (left) and test (right) sets, respectively, on undisturbed (dark dots) and disturbed (light dots) signals

  from the light dots representative of the values of the difference in ensemble reconstruction errors related to disturbed signals. On the other hand, in the Loviisa case study (

For a further understanding

  of the differences between the two methods, Figure5shows the results of a detailed analysis focused on six disturbed test signals, of which three are better reconstructed by MD (namely number 257, 501 and 132, left side of the Figure) and three by TM (namely number 508, 98 and 373, right side of the Figure). Each graph reports the values (in descending order) of the reconstruction errors k i outcomes of each PCA model 1, 2,...,7 k  are used to reconstruct the signal with MD and TM, respectively: of test patterns whose ensemble reconstruction is given by model k (thus to the central outcome), whereas in k TM  it represents the number of test patterns which include model k in the TM i K models used for their reconstruction. For the three signals on the left of the Figure (better reconstructed by MD), most of the models make low reconstruction errors and the central outcome is more likely to belong to one of these models, as proved by the high value of k MD  in correspondence of the accurate models; in such situations, using multiple outcomes might lead to include also those of the less accurate models, eventually compromising the pattern reconstruction. For the right-hand side signals (better reconstructed by TM), instead, fewer models make low reconstruction errors: in these cases, choosing one single outcome (as in the MD approach) entails the risk of using the one corresponding to a less accurate model, as shown by the peaks of k MD  in correspondence of those models; instead, including multiple outcomes increases the chances of considering also those provided by the (few) accurate models, as verified by the more evenly distributed values of k TM  .

Figure 5 .

 5 Figure 5. Forsmark-3 case study: model reconstruction errors ( k i  ) and model outcome inclusion frequency in MD

  disturbed training signals have the tendency to be better reconstructed by MD (

Figure 6 .

 6 Figure 6. Sketch of the first novel procedure for combining the MD and TM aggregation methods

Figure 7

 7 associated optimal method. Notice that 3 C (corresponding to MD, i.e.

   ) includes an exception, i.e. a subset whose average error difference is negative, but quantitatively negligible and in between two subsets with positive l  .

Figure 7 .

 7 Figure 7. Application of the first combination procedure to the Forsmark-3 case study: average error difference

Figure 8 .

 8 Figure 8. Sketch of the second novel procedure for combining the MD and TM aggregation methods

  This work addresses the problem of reconstructing the correct signal values measured by faulty sensors in nuclear power plants. One practical approach to effectively handle the dimensionality of the problem due to the large number of sensors typically involved is offered by resorting to an ensemble of auto-associative models for signal reconstruction.The set of signals is first subdivided into small, overlapping groups, made diverse by randomly sampling the signals according to the RFSE procedure. Then, one PCA-based reconstruction model is developed using the signals of each group. Finally, the outcomes of the models are aggregated to provide the ensemble reconstruction. Three aggregation methods have been investigated with respect to their performance in providing accurate and robust reconstructions: Simple Mean (averaging the outcomes of all individual models), Median (taking as ensemble aggregate the outcome located in the centre of the distribution of the outcomes of the individual models) and Trimmed Mean (discarding the outcomes lying on the tails of the distribution).The approach has been applied to two case studies concerning the reconstruction of 215 signals measured at a Finnish nuclear PWR and 920 simulated signals of a Swedish nuclear BWR. The accuracy and robustness of the three aggregation methods have been compared on the reconstruction of undisturbed and disturbed signals. In general, the analysis has proved the benefits of reconstructing signals by discarding the outlying outcomes, for they are likely to be fairly inaccurate and thus deteriorate the aggregate. In particular, the ensemble based on the Median aggregation has provided more accurate and robust performances in both applications, especially in reconstructing signals when in presence of disturbs.On the basis of the insights gained from the investigation, two novel procedures have been developed for combining the Median and Trimmed Mean aggregations in a way to fully exploit their benefits; the novel procedures have shown an increased robustness in reconstructing signals when in presence of disturbs.

Table 1

 1 concerns the reconstruction of 215 signals measured at a nuclear Pressurized Water Reactor (PWR) located in Loviisa, Finland; the second addresses the reconstruction of 920 simulated signals of the Swedish Forsmark-3 Boiling Water Reactor (BWR). The main characteristics and parameters adopted in the two cases studies are reported in Table1. The PCA models have been constructed with the code http://lib.stat.cmu.edu/multi/pca, adapted to perform the signal reconstruction task of interest here.

		Case study						Loviisa Pressurized Water Reactor (PWR)	Forsmark-3 Boiling Water Reactor (BWR)
		Number of signals, n			215	920
	Signals	Type of signals				Measured in situ	Simulated with the HAMBO code
		Number of patterns available,	N	12713	5463
		Number of training patterns,	TRN N	, in	X	TRN	8000	3600
		Number of test patterns,	TST N , in	X	TST	4713	1863
	Data set							Normal operation and	Start up, normal
		Type of measurements 4		transients related to	operation and shut
								two outages	down
		Sampling rate					1 hour	1 hour
	Wrapper-optimized	Signal redundancy,	K	i		7	7
	ensemble	Group size, m					38	70
	parameters	Ensemble size 5 , K				40	92
		TM fraction,	TM 				0.2	0.2
	Aggregation	Number of groups (i.e.					SM (	i K )	7	7
	parameters	models) used to reconstruct					MD	1	1
		each signal				TM ( TM K )	5	5
								i
	pattern are randomly affected either by noise (with probability	RN p 	0.12	) or by their value being set equal to the
	offset value of the corresponding sensor (with probability	OF p 	0.08	); with probability 0.8 the signals remain
	unmodified.

. Main characteristics and parameters of the two case studies An effective ensemble scheme of signal reconstruction must be capable of reconstructing the signals from faulty sensors, e.g. affected by random noises or offsets. Within the proposed ensemble approach, a faulty sensor sends a faulty signal in input to the PCA models which include that signal; in this situation, the ensemble must be capable of providing a good estimate of the true value of the signal by correlating the information coming from the non-faulty signals in the models of the ensemble.

To verify this, disturbs are randomly introduced in the patterns of the test set. More precisely, the signals of a test

Table 2 .

 2 Ensemble reconstruction performance indexes obtained with SM, MD and TM aggregations on undisturbed and disturbed signals for the Loviisa and Forsmark-3 case studies To further delve into the aggregation approaches, further analyses have been carried out on the Loviisa and Forsmark-3 data.

	2 10  	Loviisa Pressurized Water Reactor (PWR) Forsmark-3 Boiling Water Reactor (BWR) SM MD TM SM MD TM
		, EU	0.977	0.956	0.957	2.091	1.878	1.894
		, ED	6.221	5.169	5.261	8.172	7.607	7.638

Table 3 .

 3 Loviisa case study: ensemble performances on signal 163 (undisturbed and disturbed) obtained with SM, MD

	2 10  	SM	MD	TM
		, 163 EU	1.298	1.172	1.240
		, 163 ED	7.334	4.954	5.554
			and TM aggregations

Table 4

 4 reports the relative percentage improvement achieved using the higher signal redundancy (from 7

	i K  to

Table 4 .

 4 Loviisa case study: relative percentage improvements when increasing the signal redundancy obtained with SM, MD and TM aggregation on undisturbed and disturbed signals In general, increasing the number of outcomes upon which the ensemble reconstruction of the signal is based improves

  fed with the same undisturbed training patterns and with training patterns disturbed according to the procedure previously described, whereas in the right graph of Figure 4 the ensemble of models have been fed with undisturbed and disturbed test patterns.

	In general, if i d is small, i.e.	i d 	0.02	on both training (left graph) and test (right graph) patterns, the error difference
	,, E TM E MD	0.005
	ii
	Small values of t  are obtained when the TM K	PCA models considered in the ensemble for the aggregation provide
		i			i
	very similar outcomes, whereas if t  is large the TM K	PCA model outcomes for pattern t are more spread. Then, the
					i	i
	average model outcome distance associated to the PCA models reconstructions of signal	i is simply:
	() 1 TRN TST N ii () () 1 TRN TST t TRN TST t d N    	(7)
	Figure 4 plots the average model outcome distances	d ,	1,2,..., in 	, vs. the difference in the ensemble average
					i
	reconstruction errors	,, E TM E MD  	,	1,2,..., in 
			ii

, computed using the training (left) and test (right) sets, respectively.

In both cases, the ensemble models are built using the undisturbed training set. In the left graph of Figure

4

, the models have been

Table 5b )

 5b , results show that on disturbed signals MD performs better on a larger number of signals (more than two thirds) with an average error reduction considerably higher than TM.

	Aggregation	Number of signals for which the aggregation method is better / Average error reduction (	3 10  	)
	method	Training		Test
		Undisturbed	Disturbed	Undisturbed	Disturbed
	MD	356 / 0.14	451 / 2.92	474 / 1.58	420 / 2.97
	TM	564 / 0.13	469 / 1.60	446 / 1.34	500 / 1.92
	Table 5a. Forsmark-3 case study: comparison of MD and TM aggregation methods on undisturbed and disturbed
			training and test signals
	Aggregation	Number of signals for which the aggregation method is better / Average error reduction (	3 10  	)
	method	Training		Test
		Undisturbed	Disturbed	Undisturbed	Disturbed
	MD	78 / 0.79	155 / 5.68	113 / 4.55	163 / 6.12
	TM	137 / 0.73	60 / 2.88	102 / 4.89	52 / 2.91

Table 5b .

 5b Loviisa case study: comparison of MD and TM aggregation methods on undisturbed and disturbed training and test signals

Table 6

 6 reports the ensemble reconstruction performances on undisturbed and disturbed test signals obtained by MD, TM and the first proposed mixed approach (MX1). The MX1 approach, based on the reconstruction scheme built using disturbed training signals, provides a more robust reconstruction of the disturbed test set, at the expenses of a negligible loss of accuracy on undisturbed signals.

	2 10  	MD	TM	MX1
		, EU	1.878	1.894	1.881
		, ED	7.607	7.638	7.363

Table 6 .

 6 Application of the first combination procedure to the Forsmark-3 case study: ensemble reconstruction errors obtained with MD, TM and MX1 approaches on undisturbed and disturbed signals  , and thus no performance improvement with respect to using the pure MD approach to reconstruct the signals. This is in spite of the fact that Table5breports that in the Loviisa case study some signals are better reconstructed by TM. For this reason, a second combination procedure has been developed, aimed at capturing the signal reconstruction capabilities of the MD and TM methods on each individual signal.

	Notice however that, the application of this combination procedure (with	30 h  ) to the Loviisa case study has led to
	obtaining	0

l   , l

Table 7 .

 7 Application of the second combination procedure to the Loviisa and Forsmark-3 case studies: ensemble reconstruction errors obtained with MD, TM and MX2 approaches on undisturbed and disturbed signals

	2 10  	Loviisa Pressurized Water Reactor (PWR) Forsmark-3 Boiling Water Reactor (BWR)
			MD	TM	MX2	MD	TM	MX2
		, EU	0.956	0.957	0.958	1.878	1.894	1.890
		, ED	5.169	5.261	5.023	7.607	7.638	7.298

In the application that follows, each signal of the validation set has been previously normalized in the range [0.2, 1], for convenience.