Diagnosis of transient states

Diagnosis of transient states A pattern recognition approach Nassim Boudaoud* — Mylène Masson** (*) Laboratoire CQP2, (**) Laboratoire HEUDIASYC Université de Technologie de Compiègne BP 529 - F-60205 Compiègne cedex Email : {boudaoud, masson}@utc.fr

RÉSUMÉ. Dans cet article nous présentons un système de diagnostic adaptatif en ligne basé sur la reconnaissance des formes. Ce système intrègre une boucle de supervision qui permet la détection et le diagnotic de dérives lentes et de nouveaux modes de fonctionnement en temps réel. Un mode de fonctionnement est supposé stationnaire, une dérive lente est considérée comme étant un état non stationnaire. La détection d’un changement de dynamique (stationnaire-non stationnaire) est basée sur l’évaluation d’un critère de perception par rapport à chaque mode de fonctionnement connu. Ces critères sont combinés pour engendrer le diagnostic final. Deux stratégies de fusion de décisions ont été testées : la fusion centralisée et la fusion distribuée, ces deux approches sont fondées sur un test d’hypothèses. Les résultats expérimentaux ont permis de mettre en évidence les performances du système de diagnostic en termes de détection précoce de changements de la qualité de l’eau d’une rivière et d’anticipation de son évolution.

In this paper an on-line adaptive diagnosis system based on a pattern recognition approach is presented. This system includes a supervision loop for on-line detection and diagnosis of slow evolutions and new operation states. We assume that an operation state (OS) is a stationary state and an evolution state is a transient state. The dynamical changes (stationarytransient) are detected using perception criteria. Decisions are generated according to the set of known OS, the final diagnosis is generated using an optimal combination decision strategy: a centralized or a distributed hypotheses test. The effectiveness of the diagnostic system is demonstrated on an environmental problem: water quality management. The experimental results demonstrate the abilities of the system to detect incipient water quality changes and to anticipate the evolution of water pollution.

ABSTRACT.

MOTS-CLÉS : diagnostic, supervision, reconnaissance

des formes, test d’hypothèses distribué, test

d’hypothèses centralisé KEYWORDS: fault diagnosis, supervision, pattern recogntion, distributed hypotheses test, centralized hypotheses test, fusion.

X. Volume X - nÆ X/2000, pages 1 à 19

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X. Volume X - nÆ X/2000

1. Introduction The diagnosis of a process is a key element in reducing the maintenance costs and improving the availability of production tools. The aim of a diagnosis system is, on the basis of multiple measurements, to detect the operating conditions of the process (normal/faulty), and if necessary, to localize (fault1/fault2) and quantify (low/medium/high) faulty behavior. Diagnosis techniques are classically divided into two main categories. In a model-based approach, decision-making is based on residual signals computed from analytical relationships between measured variables [BRU 90]. This approach requires the description of the dynamics of the system using a mathematical model. In a pattern-recognition approach, the knowledge about the system is limited to a learning set with recorded measurements and associated operating conditions. This knowledge is used to construct a statistical or a fuzzy model of the system operation. The design of a pattern recognition diagnosis system is made as follows: first, relevant features are selected from sensor measurements. A pattern is a vector gathering all these features. The second step is to partition the features space into decision regions corresponding to each operating conditions or rejection. Rejection is needed when the learning set is incomplete: for example, for costs or security reasons, only a small number of operating conditions have been recorded in the past. The reject option is a means of telling that a new observation does not look like previously seen patterns. The classifier is generally designed in such a way that the misclassification probability is minimised. In the exploitation step, each new observation is either assigned to one of the known class or rejected. However, this general scheme does not take into account the temporal behavior of the system. In fact, in some real applications, for example, tool wear monitoring in rotating machines, the system can gradually go from one state to another. It is of great importance to detect and to follow these slow evolutions to make the diagnosis system predictive. To deal with these considerations, in this paper we propose to include a supervison loop in the classical pattern recognition diagnosis system. The paper is organized as follows: in section 2, we present some background on the design of a pattern recogntion diagnosis systems, then we state the problem. In section 3, we describe the architecture of the supervision system: perception, detection and fusion module. Two strategies of decision fusion are presented and compared: a centralized and a distributed one. Finally, we present an application of the proposed system into an environmental problem: the monitoring of the water quality of a river.

2. Problem statement 2.1. Diagnosis and pattern recognition Pattern recognition provides a wide variety of methods for the design of a diagnosis system [DEN 97b]. Traditionally two main steps are required (see figure 1): analysis and exploitation.

Diagnosis of transient states

3

The analysis is an off-line step which leads to the design of the classification rule. It begins by first making a selection of features from measurements on the process in order to build a vector of d features (pattern vector) in a d-dimensional space. This step is carried out by using a data analysis, signal processing or a model based method. In this space, the past recorded patterns representative of different operation states of the system (nominal, fault 1, fault 2,...) occupy different regions in the feature space. Patterns of each operating state are separated into M classes or clusters !i ; i = 1 M . The set of classes is referred to as the learning set. To delimit the regions, one needs a classifier, that is, a discrimination rule between the different classes (see figure 1). Different frameworks can be used to elaborate the classification rule [FUK 90], [DUB 90] such as the k nn rule, artificial neural networks, and fuzzy classifiers.

Data

Pre-processig

Features selection

knowledge

Construction of Classes

Representation space Discrimination boundaries

Decision space

1 0 1 0

feature d

Class 1 (normal)

Class 2 (fault 1)

new observation

Analysis New observation

Decision rule

Classification feature 2

Class M (fault M)

feature 1

Exploitation

Figure 1. Diagnosis and pattern recognition

As the classification rule is available, new patterns gathered from the process are assigned on-line into one of the known classes !i . To take into account the fact that the learning set is often incomplete, the distance reject concept has been introduced by [DUB 93]: the reject option is a means to detect that a new observation does not look like previously seen patterns. A pattern which is lying far from all the classes in the learning set must be rejected. Once a number of patterns have been rejected, a new off-line learning procedure is carried out in order to detect the emergence of new classes. These new classes are in turn included into the classification rule.

2.2. Detection of transient states Traditionally, the adaptation of the classification rule is carried out off-line although the diagnostic procedure should sometimes react as soon as possible.To enable the real-time adaptation of the diagnosis system a supervision system should detect and update the classification rule on-line as soon as a new operation state appears. The main problem is to detect its occurence. This problem has been adressed in many contributions in the case of abrupt state changes [GRE 85], [GAN 87], [PEL 93] and [GOM 96] but few papers have treated the problem of on-line detection of incipient

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faults. In this paper, we propose to include a supervision loop in the classical recogniω2

ω2

ω1

ω1 Stabilisation in a new class Steady state in

ω1

(case 1)

(case 3) ω2

ω2

ω1

ω1 Going closer to Leaving ω 1

ω2

Leaving ω 1 Leaving ω 2

(case 2)

(case 4)

Figure 2. (case 1): Steady state. (case 2 and 4): Evolution. (case 3): Stabilisation in a new state

tion system to detect in real-time the departure (incipient) of the system from a known state, a stationary state, and to detect its stabilization in a known or unknown state. Four typical cases are considered (see figure 2): – case 1. The supervision system has to indicate that the system is in a stationary state !1 . – case 2 and 4. The supervision system has to indicate that the system is in a transient state: leaving the state !1 in the direction of !2 or in an unknown direction. – case 3. The supervision system has to indicate that the system is in stabilization in a new operation state. Fuzzy Membership Function Probability density ...

CLASSIFIER SUPERVISION

Pattern Vector Class characteristics

& FUSION

PERCEPTION

DETECTION

Figure 3. Architecture of the diagnosis system

Diagnosis


5

To take these constraints into consideration, three modules are added to the conventional classifier (see figure 3): a perception module generates indicator signals sensitive to an evolution or a stabilisation in the feature space. Indicators are computed from a measure of the dissimilarity between the current pattern to reference classes. A second module is used to detect if the system is stable or in a transient state. This detection is based on classical sequential hypothesis testing applied on the indicators signals. At last, a supervision module fuses the decisions to produce the final diagnosis.

3. The supervision system 3.1. Perception Let us consider a sequence of n pattern vectors fxk g1kn gathered from a process at each time step tk ; 1 k n. We define the perception criterion as a function which gives at each time tk the amount of similarity or dissimilarity between the current pattern xk and the reference class !i [BOU 96]. It can be, for example, a probability density, a fuzzy membership function or a distance measure. The choice of perception criterion is of great importance because it is at the first level of the supervision loop. It must be chosen in order to be sensitive to state changes (stationary and transient) in the process. In this context, we have selected a dissimilarity measure d(xk ; !i ), for example a distance (Euclidean, Mahanalobis,...) between xk and the center of gravity of !i (see figure 4). In the case of a multimodal distribution, we take the lower distance between xk and the different modes. Figure 4 illustrates the behaviour of different metrics: L1 (city block), L2 (Euclidean) and L1 (max). The distance measure is evaluated between the current pattern vector and the center of the class (+). The illustration (see figure 4) shows that for a stationary state (evolution inside class (+) or (o)) the distance changes are stationary and for a transient state the distance changes are not stationary. So to detect the state changes (stationary-transient) we propose to model the dynamical changes of the distance measure.

3.2. Indicator generation The detection of non-stationary (transient) behaviour can be performed by using parametric models in which one or several parameters may abruptly change. We consider M classes, and assume, for each class !i ; i = 1 M , that the dynamical changes of the perception function (i.e. d(xk ; !i )) are modelled by a polynomial function of order p:

dki = d(xk ; !i ) = yik + ki ; i = 1 M; with

yik =

p X l=0

al;i kl i = 1 M;

[1]

[2]

6

X. Volume X - nÆ X/2000 Linear Evolution

Perception criteria

14

30 ( : ) Cityblock (−) Euclidian (−.) Max

12 25 10 20

distance

x2

8

6

15

4 10 2 5 0

−2 −5

0

5 x1

10

0

15

0

50 time k

100

150

Figure 4. Perception criteria: (left) evolution in the feature space. (right) corresponding distances as a function of time (three metrics: city-block, Euclidean and max)

where k is the sampling time, the residuals ki are assumed i.i.d with a zero mean and a variance i2 . The parameters al;i carry an information about the magnitude of the lth derivative, for example, the first derivative is an indicator of the speed of the drift. In this paper, we assume that the non stationary behaviour is linear with respect to time. So, for each class !i , a first order polynomial model is assumed:

yik = a0;i + a1;i k;

[3]

where a0;i is the initial condition and a1;i is an indicator of the speed of the drift: – if a1;i

– if a1;i

=0

then the state is stationary.

6= 0 then the state is non-stationary (transient: increasing or decreasing).

As illustrated in figure 4, the behavioural changes of the simulated system correspond to an abrupt change of the slope in the perception criterion. In this case a1;i is a good indicator to detect the transition from a stationary state to a transient one (drift). The parameters akl;i ; l = 0; 1 are estimated using a recursive least squares method minimizing the following objective function:

J

min i =

ai

where ai

n X k=1

n

k

k yi (ai )

2 d(xk ; !i ) ;

[4]

a0;i ; a1;i ℄ and 2 [0 1℄ is a given weight.

=[

From a pattern recognition point of view, if a pattern vector is moving inside a class (i.e. steady state), then the expectation E [a1;i ℄ is equal to zero. If the pattern vector is leaving (slowly) a class, and thus in a transient state, the parameter ak1;i has a non zero mean value (> 0 or < 0). In this way, the transition from a steady state to


7

Estimation d( xk , ω i ) Perception Error

xk

Parametric model f( xk , a i )

y ik

^a k i Figure 5. Estimation procedure: abki is the estimated parameter vector

a transient one (and conversely) can be detected by tracking the slope variations (see figure 6). Initial state Steady Transient

Final state Transient Steady

Sign of E

k a

1;i

+

Figure 6. State transitions and corresponding sign changes of ak1;i This change detection problem can be solved by a statistical approach namely the likelihood ratio approach [BAS 93]. The following sections describe how to design k g . The detection statistical rules for the detection of changes in the mean of fad 1;i k strategy is based on a classical hypothesis testing procedure [BAS 93]. d k g slopes are independant. Theoretically, the test is applicable if the estimated fa 1;i k In the case of a sliding window, the independence of the estimated sequence is not ensured. In order to ensure this independence, the parameter a1;i could have been estimated on separated windows. However, this estimation strategy has the drawback of increasing the delay of detection dramatically. An altenative strategy could be to dik 1 but it is well known that the signalwork with a series of the differences dki to-noise ratio decreases. The sliding window method gives in practice satisfactory

8


performances with an acceptable delay of detection, as will be shown in the section dedicated to the experimental results. In the next section, single class and multi-class detection problems are treated separately.

3.3. Single class case: detection and diagnosis n g are estimated on different windows. They are assumed to be Thus, the fad 1;1 n independent.

We assume the following evolution model: n =+w ; ad n 1;1

where wn is a white noise with variance 2 , and

=

0 1

for n r 1: for r n:

The problem is simply to detect the change in mean and to estimate the change time

θ=θ0

1

2

nr

n

Figure 7. The detection problem

r . 0

and 1 are generally known, but if not they can be estimated. The hypothesis testing problem, as illustrated in figure 7, consists of making a decision between the two hypotheses: 8 < H0 :

H1

: :

= 0 = 0 = 1

for for for

1

1 k n: k r 1; r k n:


9

The likelihood ratio between H0 and H1 is:

Rn1 =

Qr

Qn 1 p ad d k k ) 1;1 k=r p1 (a1;1 ) ; Qn d k ) ( a p k=1 0 1;1

k=1 0 (

[5]

k and r is the unkwhere ps ; s = 0; 1 are the conditional probability densities of ad 1;1 nown change time. Thus, the log-likelihood ratio is: n

X Srn = ln Rn1 = sk ; k=r

where

sk = ln

[6]

p1 (ak1;1 ) : p0 (ak1;1 )

[7]

The unknown jump time r is estimated by its maximum likelihood estimate under H1 :

rb = arg

max

1rn

Srn (0 ; 1 ):

[8]

The detection rule is defined as follows:

u=

+1 1

H0 H1

if else

Srn (0 ; 1 ) < ; Srn (0 ; 1 ) ;

max1 r n max1 r n

[9]

where is the detection threshold. A recursive implementation of this rule is proposed in [BAS 93]: 1 : if gk h; u = +11 H [10] H0 : else; where gk = max (0; gk 1 + sk 1 ) : [11]

h is a detection threshold, its level is chosen in order to make a compromise between the probability of detection PD (or the delay of detection ) and the probability of false alarms PF (or the mean time between false alarms TF ) [BAS 93]. The detection time rd and the unknown jump time are given by the following rules: rbd = minfk : gk hg; rb = minfk : gk = 0g:

[12] [13]

If the noise sequence fwn gn is i.i.d and has a Gaussian distribution N (0; ) then the detection function is:

k gk = max 0; gk 1 + ad 1;1 0

2

;

[14]

where = j1 0 j is the magnitude of the jump. The detection function does not hold for a negative jump. To detect both positive and negative jumps requires the use of a two-sided detection function: – for a positive jump:

k gk+ = max 0; gk+ 1 + ad 1;1

0

2

;

[15]

10


– for negative jump:

gk

; gk 1

= max 0

k + ad 1;1 0

2

;

[16]

– and the respective decision rules are defined as follows:

u+ =

1

u

=

+1

+1 1

H1 H0

:

H1 H0

:

:

:

if else;

gk+ h;

[17]

h:

[18]

gk

if else:

The diagnosis is generated using the decisions rule:

u+ ; u

in the following decision

– if u+

= 1 and u = 1 then the diagnosis is: “Evolution inside class !1 ” + – if u = 1 then the diagnosis is “Leave class !1 ”

– else if u

=1

then the diagnosis is: “Stabilization in a new class”

3.4. Multiclass case: detection and diagnosis In this case, the design of the detection rule is more complex because we have to combine the information generated by each known class. The construction of the detection rule depends on the nature of the information. Two strategies are proposed for the combination of decisions: – a centralized hypothesis test – a distributed hypothesis test In the centralized case, the indicators fa1;i gi=1;M , where M is the number of known classes, are combined to make the final decision. In this case, only one decision rule is needed (see figure 8(a)). In a distributed strategy we consider M subordinate decisionmakers, one for each known class. Each detector generates a local decision ui then these decisions are combined by a primary decisionmaker to generate the final decision (see figure 8 (b)). In the following subsections we assume that d k g quences fa 1;i 1kn are estimated.

M

classes are known. So,

3.4.1. Centralized hypotheses test (CHT) – Detection: The hypothesis test is defined as follows: 8 < H0 :

H1

: :

= 0 = 0 = 1 or 1

for for for

1

1 k n: k r 1; r k n:

M

se-


11

(a)

ak

1,1

Estimation

d(x k,ω ) 1

ak 2

Decision

Fusion

1,2

d(x k,ω )

Estimation

u

Centralized test

ak

1,M

d(x k,ω )

Μ

Estimation

(b)

u1 ak

Detector 1

1,1

Fusion ak

Decision

u2

Detector 2

1,2

u of decisions

uM

ak

Detector M

1,M

Figure 8. (a) Centralized strategy. (b) Distributed strategy

This test includes positive and negative jumps as a same hypothesis. In this case, the likelihood ratio is defined as follows:

Rn =

Qn

k jH ) p a ad 1;M 1 : d d k k k=1 p(a1;1 a1;M jH0 )

d k k=1 ( 1;1

[19]

Qn

d k g If the sequences fa 1;i 1iM with respect to the different classes are i.i.d. then equation [19] is transformed as:

Rn =

M Y n Y

h

k )+p p1 (ad 1;i

i=1 k=r

k ) ad 1;i

1 (

k ) p0 (ad 1;i

i

:

[20]

Thus,

Srn = ln Rn =

M n X X k=r i=1

sik ;

[21]

12


where

h

sik = ln

k )+p p1 (ad 1;i

1 (

k ) ad 1;i

i

k ) p0 (ad 1;i

[22]

A recursive implementation of the detection rule is proposed:

u=

+1 1

H1 H0

: :

where

gk = max

gk h;

if else;

; gk 1 +

0

M X i=1

[23]

!

s 1 i k

[24]

Finally, the estimation of the jump time instant is given by:

rb = minfk : gk = 0g

[25]

– Diagnosis and action: The diagnosis generation procedure is based on the decision u and a boolean variable state. state is set to 0 if the operation state is stationary and set to 1 when an evolution is detected. The diagnosis is generated by means of the following decision rule: - if state = 0 and u =

1

then

- diagnosis: “Evolution within a class” - if state = 0 and u = 1 then - diagnosis: “Leave a class” - action: state is set to 1.

- else if state = 1 and u = 1 then - diagnosis: “Stabilization in a class” - action: state is set to 0. In the case of a centralized hypothesis test, the localization of the diagnosis is not possible i.e. the diagnosis system cannot deliver the actual operation state (class 1 or class 2 or ...class M), and is not able to give the direction of the evolution (evolution from class !i to class !j ). So, additional information must be agregated like the membership of the current pattern vector (crisp or fuzzy). To find the direction of the evolution, the problem is more difficult because no explicit information about the sign d k is available, an additional process is required, it makes the of the slope variations a 1;i decision system more complex, and so, less efficient. To deal with both problems, we propose to design a detection system based on a distributed test. In this case, more information is available before any fusion.


13

3.4.2. Distributed hypotheses tests (DHT) – Detection: In this case, we assume that, for each detector, individual decision rules have been designed. For a class !i the decision ui is defined as follows:

ui = max(u+i ; ui );

[26]

where u+ i and ui are respectively the decisions for the detection of positive and ned k g .u gative jumps in fa 1;i k i

= +1

f g

if any jump is detected and ui

=

1

in other cases.

d k If the sequences a 1;i k ; i = 1; M are assumed independant and distributed with a

Gaussian density probability then:

u+ =

+1

i

and,

ui where

1

+1

=

1

H1 H0

if else;

H1 H0

if else;

gki;+ hi ;

[27]

hi ;

[28]

gki;

gki;+ gki;

= =

i k ; gki 1 + ad 1;i 0 + 2i ; d i i k max 0; gk 1 + a1;i 0 + 2i ; max 0

[29]

i is the magnitude of the jump and hi is a decision threshold tuned thanks to the prior fixed probability of false alarms PFi = P (gki hi jH0 ). +

diagnosis i

ui

Diagnosis

u -i

by class ω i

state i

state i

Figure 9. Diagnosis generated by class !i The decisions ui are combined in order to generate the final diagnosis. Traditionally, the fusion rules are implemented as “k -out-of-M ” rule which means that the hypothesis H1 is chosen if k or more detectors decide hypothesis H1 :

u=

H1 H0

+1 1

PM

i=1

: :

ui k

M

(

k):

otherwise.

[30]

The Bayesian framework [CHA 86] gives a more general formulation of the fusion problem with individual detectors and proposes an optimal formulation:

u=

+1 1

H1 H0

: :

0 +

PM

i=1

i ui > 0;

otherwise;

[31]

14


where

0 i i

= = =

;

P log P1 1 P0M log P i F 1 PiF log P i Mi

if ui if ui

; 1:

= +1 =

PFi and PMi are respectively the probability of false alarm and non detection. P0 and P1 are the prior probability of H0 and H1 . If P0 = P1 , PFi = PF and PMi = PM for all the detectors then the combination rule is a particular case of the “k -out-of-M ” rule, with 2k > M . – Diagnosis and action: According to a given class, the diagnosis is generated by using the decisions and a boolean variable statei (see figure 9). Figure 10 summarizes the diagnosis generation logic.

u+i ; ui

u+i

1 1 +1 1j + 1 1 1 +1

ui

1 +1 1 1 +1 1j + 1 1

statei

0 0 0 +1 +1 1 1

diagnosisi steady state moving toward !i leave !i

diagnosisi

stabilization

action: statei 0

1 +1

statei 0

diagnosisi

statei

stabilization

0

Figure 10. Decision logic The final diagnosis is generated by the following decision rule:

1 be a pattern vector, u(k

1) the decision computed by equation [31] and a boolean variable set to 0 if the operation state is stationary and set to 1 if the operation state is transient. These variables are computed at a sampling time k 1:

Let xk

state(k

1)

- if the system is in a steady state: u(n 1) = 1 and state(n 1) = 0 then - Find the nearest class 1 !i - Diagnosis: D(n) =“steady state” !i .

- Action : state(n) is set to 0. - else if the system is in a transient state:

u(n

1) = +1

and state(n

1) = 0

then

- Find the nearest class !i

- Diagnosis: D(n) = diagnosisi (leave or move toward).

1. by using for example a fuzzy membership function or a distance measure


15

- Action: state(n) is set to 1. - else if the system still in the same transient state:

u(n

1) =

1

and state(n

1) = +1

- Diagnosis: D(n) = D(n

then

1)

- Action: state is not modified, state(n) = state(n - else if the system is in stabilization:

u(n

1) = +1

and state(n

1) = +1

- Find the nearest class new class is detected.

1).

then

!i , if it is far from all the known classes then a

- Diagnosis: D(n) =“stabilization in a class”(known or unknown) - Action: state(n) is set to 0. The global architecture of this decision strategy is represented in figure 11: D u1 u 2

Fusion of Decisions

uM

Global Diagnosis

diagnosis1 state1 diagnosis2 state2

state

Generator

diagnosisM stateM

Figure 11. A distributed diagnosis architecture

3.5. Comparison study between (CHT) and (DHT) In this section, we present a robustness study to compare the performances of a centralized (CT) and a distributed test (DT). The study consists in evaluating the effect of varying the Signal/Noise Ratio (SNR) on the performances of detection i.e. the delay of detection (DD) for a fixed rate of false alarm. We conduct the simulation by generating i.i.d Gaussian sequences fak1;i g, 1 i M; 1 k 200 (i.e 200 points) such as for 1 k 100 the sequences are normally distributed N (0; 1) and for 100 < k 200 the distributions are N (m; 1) with m > 1. In this case the SNR is defined as the jump magnitude m. For each Signal/Noise Ratio value the simulation is repeated 100 times. The performances are compared by computing the mean delay of detection (MDD) and the standard deviation of the delay of detection (SDD) over the repeated simulations. Figure 12 shows the effect of the SNR variations (in range [0:001 0:2℄) for M = 5.

16

X. Volume X - nÆ X/2000 100

Distriubted Test Centralized Test 80

Delay of detection

60

40

20

0

−20

0

0.02

0.04

Figure 12. Performances for M

0.06

= 5:

0.08

0.1 SNR

0.12

0.14

0.16

0.18

0.2

(solid line) MDD. (dashed line) SDD

For both DHT and CHT, the MDD is represented in solid line and the SDD limits in dashed lines. It is observed that the DHT is more efficient and robust than the CHT against noise. In this case both MDD and SDD are lower.

4. Application This diagnostic system has been developed within the EM2 S (Environmental Monitoring and Managing System) Esprit project. It has been applied to the problem of forecasting the water quality of a river. The monitoring of the water quality of the river is achieved by monitoring stations installed on strategic sites of the bank of the river. The stations provide on-line 7 physico-chemical parameter measurements: temperature of water, pH, conductivity, oxygen, turbidity, level of the river and the concentration in ammonium. The application concerns the supervision of the quality of the water. The main objective is to demonstrate the capability of the diagnosis system to anticipate the water quality. Four classes of water quality are defined by the French legislation. They depend on the level of different parameters such as pH, oxygen and ammonium. In this application, we have considered three classes of water quality which correspond to three different concentrations of ammonium (low-medium-high). The diagnosis system has been tested for water quality monitoring and prognosis. A learning set is composed of three classes of quality represented by four parameters: temperature, pH, conductivity and concentration of oxygen. The test set is a sequence of measures (sampling rate of 6 min) during three days. In order to evaluate the performances of the diagnosis of the system, fuzzy membership function described in [DEN 97a] were used as a reference label of the test set. It can be seen from


17

µ1

1

0.5

0

0

100

200

300

400

500

600

700

800

900

500

600

700

800

900

500

600

700

800

900

time

µ2

1

0.5

0

0

100

200

300

400 time

µ3

1

0.5

0

0

100

200

300

400 time

Figure 13. Fuzzy membership functions: (top) class 1. (middle) class 2. (bottom) class 3

figure 13 that the system goes alternatively from state 2 to state 3. Figures 14 and 15 illustrate respectively the evolution of the perception measure (distance) and the sequences of estimated slopes ak1;i ; i = 1 3. The following sequence of diagnosis is generated (between step 541 and 579): Example

541 Stabilization in lass 2 542 Stabilization in lass 2 543 Leaving lass 2 544 Leaving lass 2 ..... 549 Leaving lass 2 550 Leaving lass 2 551 Leaving lass 2 552 Moving toward lass 3 553 Moving toward lass 3 ....... 575 Moving toward lass 3 576 Moving toward lass 3 577 Moving toward lass 3 578 Stabilization in lass 3 579 Stabilization in lass 3

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Figure 14. Distance segmentations: (top) class 1. (middle) class 2. (bottom) class 3

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Figure 15. Evolution of fak1;i g : (top) class 1. (middle) class 2. (bottom) class 3

It demonstrates the ability of the system to detect a slow drift from the water quality class number 2 and to monitor and anticipate the future water quality class 3 with 26 steps ahead i.e. approximately 2 h 30 min. The prognosis makes predictive actions possible.

5. Conclusion In this paper, a system for monitoring and forecasting the operation state of an evolutive process is proposed. This diagnosis system is able to detect a slow evolutions and to give on-line a precise diagnosis for the current pattern. Two strategies of fusion have been tested. We have found that the performances of a distributed strategy are higher than a centralized fusion strategy. The decision given by the monitoring system


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can be used for the supervision of the clustering module to include a new state in the classification procedure as it is detected. This outcome will be included in future works.

6. Bibliography [BAS 93] BASSEVILLE M., N IKIFOROV I., Detection of abrupt changes : theory and application, Prentice Hall, New Jersey, 1993. [BOU 96] B OUDAOUD N., M ASSON M., D UBUISSON B., (( On-Line Diagnosis of a Technological System : a Fuzzy Pattern Recognition Approach )), International Federation of Automatic Control Congress, vol. N, San Fransisco, USA, 1996. [BRU 90] B RUNET J., Détection et diagnostic de pannes. Approches par modélisation, HERMES, Paris, 1990. [CHA 86] C HAIR Z., VARSHNEY P., (( Optimal data fusion in multiple sensor detection systems )), IEEE Transaction on Aerospace and Electronic Systems, vol. AES-22, p. 98–101, 1986. [DEN 97a] D ENŒUX T., B OUDAOUD N., C ANU S., ., DANG V.M. G OVAERT G., M ASSON M., P ETITRENAUD S., S OLTANI S., High level data fusion, Tech. Rep. CNRS/EM2 S, vol. /330/12-97, 1997. [DEN 97b] D ENOEUX T., M ASSON M., D UBUISSON B., (( Advanced Pattern Recognition Techniques for system monitoring and diagnosis : a survey )), RAIRO-APII-JESA, vol. 31, no 9, p. 1509–1539, 1997. [DUB 90] D UBUISSON B., Diagnostic et Reconnaissance des Formes. HERMES, Paris, 1990. [DUB 93] D UBUISSON B., M ASSON M., (( A statistical decision rule with incomplete knowledge about classes )), Pattern Recognition, vol. 26(1), p. 155–165, 1993. [FUK 90] F UKUNAGA K., Introduction to statistical pattern recognition, Academic Press, 2nd edition, New York, 1990. [GAN 87] G ANA K., Suivi d’évolution et aide au pronostic en maintenance de système industriel, PhD thesis, Université de Valenciennes et du Hainaut Cambrésis, 1987. [GOM 96] G OMM J., (( On-line Learning for Fault Classification using Adaptive Neuro-Fuzzy Network )), International Federation of Automatic Control Congress, p. 175–180, San Francisco, USA, 1996. [GRE 85] G RENIER D., D UBUISSON B., M. J., (( Une approche par apprentissage et reconnaissance des formes et du suivi de fonctionnement de systèmes technologiques )), AFCET Automatique, vol. 3, p. 683–697, 1985. [PEL 93] P ELTIER M., D UBUISSON B., (( A fuzzy algorithm based on the k-Nearest Neighbours rule for the detection of evolution )), IEEE Int. Conf. on Systems, Man and Cybernitics,Systems Engineering in the Service of Humans, vol. 4, p. 696–701, New York, 1993.

Nassim Boudaoud received an Engineering degree in Automatic Control from the National Polytechnic School of Algiers and a PhD from the Université de Technologie de Compiègne (UTC). He has been an assistant professor at the UTC and a member of the CQP2 Research

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team since 1998. His research interests include system diagnosis, statistical process control and data analysis. Mylène Masson received an Engineering degree in Computer Science and a PhD from the Université de Technologie de Compiègne (UTC). She has been an assistant professor at the Université de Picardie Jules Verne (IUT de l’Oise) and a member of the Heudiasyc laboratory (UMR CNRS 6599) since 1993. Her research interests include system diagnosis, statistical pattern recognition and data analysis.