A Bayesian Approach to FDD Combining Two Different Bayesian

Networks Modeling a Data-Driven Method and a Model-based Method

Mohamed Amine Atoui, Sylvain Verron and Abdessamad Kobi

LASQUO/ISTIA, L’UNAM University, 62 Avenue Notre Dame du Lac, 4900, Angers, France

Keywords:

FDD, Data-Driven Methods, Model-based Methods, Incidence Matrix, Bayesian Networks, Water Heater

System.

Abstract:

In this paper, we present an original FDD method. The interest of this method is her ability to coexist residuals

and measures, under a same and a single tool. Indeed, our proposal is to combine two different Bayesian

networks to FDD. A model-based method is associated to a data-driven method to enhance decision making

on the system operating state. This method is evaluated on a simulation of a water heater system in some

various circumstances.

1 INTRODUCTION

Nowadays, monitoring methods (also known as Fault

Detection and Diagnosis methods (FDD)) are more

and more used, due to the increasing systems com-

plexity. They contributeto the reduction of faults or in

the ideal case for their elimination by an entity (oper-

ator, engineer, automaton...) which will examine the

state of malfunction caused by these past and makes

a decision about the future of the system (adjusting

settings, maintenance, closure...). These methods are

used to describe and explain, at each instant, the situ-

ation in which the system is situated. They consist of

two phases usually associated: detection and diagno-

sis phases.

The detection phase seeks to conﬁrm if the sys-

tem is still in normal operating state (In control) or is

not (Out of control). The diagnosis phase is used in

order to designate the faults responsible for the devia-

tion of the system of his normal operating. This phase

can be deﬁned in different ways depending on the de-

sired description level. According to (Chiang et al.,

2001) we can distinguish three deﬁnitions: identiﬁca-

tion (determine the susceptible measures explaining

the occurred fault on the system), isolation (distin-

guishes the measure responsible for system abnormal

functioning), diagnosis (explains the faults occurred

in the system by expressing their type, location, am-

plitude and duration).

In the last years, many monitoring methods have

emerged (Chiang et al., 2001; Isermann, 2006; Qin,

2006; Ding, 2008). Among them, we can distin-

guish two classes of methods: model-based meth-

ods and data-driven methods. Model-based meth-

ods use a priori knowledge of the system for ex-

plaining its dynamic behavior. This knowledge cor-

responds to a speciﬁc set of mathematical equations

representing the dependencies that exist between the

variables of the system and contributing to the gener-

ation of residuals (differences between observed and

estimated measurements when the system is supposed

in normal operation). Once generated, their evalua-

tion contributes to the understanding of the operating

state. In contrast, data-driven methods are based only

on measures taken at different times, and analyzed in

relation to a historical of data regarding the system.

The ability of the data-driven methods to manage

a signiﬁcant number of data associated with the ca-

pacity of the model-based methods to describe ac-

curately the dynamic behavior of the system and

to provide a physical understanding, might improve

the monitoring, increase the number of scenarios

taken into account, beneﬁt from the advantages of

both methods and to struggle against the individual

shortcomings of each one when they are used sepa-

rately. However, despite many researchers (Chiang

et al., 2001; Venkatasubramanian et al., 2003; Ding

et al., 2009) suggesting that the creation of a common

framework using both classes of methods, would al-

low a better monitoring system, these research ﬁelds

remain unexplored. Nevertheless, we can ﬁnd some

recent works in the literature describing different as-

sociation of the two methods.

In (Schubert et al., 2011) a uniﬁed scheme is pro-

162

Amine Atoui M., Verron S. and Kobi A..

A Bayesian Approach to FDD Combining Two Different Bayesian Networks Modeling a Data-Driven Method and a Model-based Method.

DOI: 10.5220/0004432101620168

In Proceedings of the 10th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2013), pages 162-168

ISBN: 978-989-8565-71-6

Copyright

c

2013 SCITEPRESS (Science and Technology Publications, Lda.)

posed. The authors combine subspace approachesand

univariate and multivariate statistical control meth-

ods (data-driven methods) with inputs reconstruc-

tion method and banks of Unknown Input Observer

(model-based methods). Luo et al. (Luo et al., 2010)

for antilock braking system (ABS), propose a hybrid

approach using parity equations and a nonlinear ob-

server for residuals generation. These residuals are

used by statistical tests with the aid of SVM (sup-

port vector machine) to detect and isolate different

faults that may occur in the system. In (Ghosh et al.,

2011), for monitor a laboratory distillation column a

fusion of decisions of several monitoring methods is

proposed. The authors use four monitoring methods:

a model-based method: an extended Kalman ﬁlter,

and three data-driven methods: SOM (Self Organized

Map), artiﬁcial neural network and PCA (Principal

Component Analysis). The output of each method

corresponds to an assignment to one class of fault. A

fusion strategy is then applied using them to make the

right decision (Bayesian decision and other). Yew et

al. (Yew and Rajagopalan, 2010) propose collabora-

tion between different methods under a multi-agent

framework using some decision fusion methods.

The proposals mentioned above, although inter-

esting for the combination of data-driven and model-

based methods, does not seem to cover or address a

particular problem which is the lack of information

or approximations of the system (decrease in perfor-

mance of monitoring). We believe that the combina-

tion of the two methods is mainly interesting when the

two methods are able to complete their information

and to ﬁnally provide better oversight. For example,

a combination of a model-based method, without an

accurate model, and a data-driven method, with some

data are missing or insufﬁciently represented. In this

paper, we propose a new monitoring method based

on Bayesian networks. This method uses the comple-

mentarities that may have a data-driven and a model-

based method in a single and common tool. The ma-

jor interest of this combination is their ability to im-

prove the decision making when the two methods suf-

fer from a information lack or an approximations of

the system (decrease in performance of monitoring).

The paper is structured as follows: in section 2

we introduce Bayesian networks followed by a short

description of data-driven and model-based methods

in sections 4 and 3; section 5 describes the monitor-

ing methodology proposed; ﬁnally, the results of the

proposed method obtained in different conditions on

a simulation of a water heater system are outlined in

the last section.

2 BAYESIAN NETWORKS

A Bayesian network (Buntine, 1996; Jensen, 1996),

is a probabilistic directed acyclic graph. Each node in

the network represents a random variable that may be

discrete with n modalitees (multinomial) or continue

(univariate or multivariate). Each node has a condi-

tional probability table (marginal probability table for

root nodes). The oriented arcs show the conditional

dependencies/independencies that exist between dif-

ferent nodes of the graph. Each directed arc can link

only two nodes: among these nodes, one is called the

father and the other, the son. For updating the network

and calculate the different a posteriori probabilities

corresponding to each node, given the availability of

new information on the network (evidence), calcula-

tions (eg: Bayes rule) named inference is required. A

Bayesian network, in general, can be deﬁned formally

by:

• a directed acyclic graph G, G=(V,E), where V the

set of nodes of G, and E the set of arcs of G,

• E is a ﬁnite probabilistic space (Ω,Z, p), with Ω a

non-empty space, Z a set of subspace of Ω and p

a probability measure on Z with p(Ω) = 1,

• a set of random variables associated with to the

nodes of the graph G and deﬁned on (Ω,Z, p),

such that:

P(V

1

,V

2

,...,V

n

) =

n

∏

i=1

p(V

i

|C(V

i

)) (1)

where C(V

i

) is the set of parent nodes of V

i

in the

graph G.

Nowadays, several variants of Bayesian networks

exist. One of them is the Bayesian network calssiﬁer,

who is based on a discrete root node modelling the

fact of belonging to one class among others. Note that

under the assumption of dependence/independence of

variables X emitted, several types of structures are

proposed (Friedman et al., 1997). Among them, we

use two kind of Bayesian networks classiﬁers: one is

the Naive Bayes network, it’s making the strong as-

sumption that the variables are class conditionally in-

dependent and the second network is the semi-naive

condensed Bayesian network who provides a simple

structure that take into account correlation that may

exist under a group of variables.

3 BAYESIAN NETWORK AND

MODEL-BASED METHODS

The model-based methods, in the presence of an

analytical representation of the system, use resid-

ABayesianApproachtoFDDCombiningTwoDifferentBayesianNetworksModelingaData-DrivenMethodanda

Model-basedMethod

163

uals generators (Isermann, 2006) which as their

name suggests, contribute to the creation of residu-

als (r

1

,··· ,r

n

)

T

(the difference between the existing

measures on the system and their estimates). Once

generated a consistency test (evaluation of residuals)

is triggered to check if no residual is different from

zero. Indeed, during normal operating, the residu-

als are assumed to be equal to zero. However, some-

times they are not only sensitive to the faults but also

the noise measurements performed on the system, to

the disturbances and the modeling errors, makes them

different to zero even during normal operation, that’s

why generally each residual is considered to be sta-

tistically null with a given variation (eg: a residual

follows a normal distribution, with a standard mean

µ = 0 and variance σ

2

= 1). Thus, generally binary

statistical tests are used to making decision between

H

0

(corresponding to the distribution of residuals dur-

ing normal operating) and H

1

the alternative hypoth-

esis (corresponding to faults). This is achieved by at-

tempting to minimize the risk of ﬁrst and second kind,

respectively, α and β (see Figure 1).

Figure 1: Statistical test.

The result of residuals evaluation (u

t

1

,..., u

t

n

)

T

, in

the case of the diagnostic methods based on structured

residuals (constructed in order to be sensitive to cer-

tain faults and not to others), is compared (usually

a logic test) to another vector representing the char-

acteristics of each fault F

j

∈ {F

1

,F

2

,..., F

k

}. These

characteristics are generally assembled into a binary

array called incidence matrix (an example is shown in

Figure 1).

Table 1: Example of incidence matrix.

IC F

1

F

2

... F

k

u

1

0 b

1.2

b

1.3

... b

1.k

u

2

0 b

n.2

b

2.3

... b

2.k

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

u

n

0 b

n.2

b

n.3

... b

n.k

Furthermore, in (Verron et al., 2009), Verron

et al. model the last two phases of the model-

based monitoring (knowing that monitoring methods

based model usually consists of three complemen-

tary phases: generation of residuals, residual evalu-

ation (change detection) and decision making (diag-

nosis)). They propose a combination of two Bayesian

networks works (modeling a control chart T

2

(Verron

et al., 2007) and the modeling of the incidence ma-

trix (Weber et al., 2008)). To achieve this, a hybrid

Bayesian network, representing k naive Bayesian net-

work, is proposed. This network is made of discrete

nodes (representing the k faults) with two modalities

({presence (yes) and not presence (No) of F

j

}) and

continuous nodes (representing the n residuals) con-

sidered as a standard Gaussian variable (with a mean

µ and variance σ

2

).

Figure 2: Bayesian network for model-based monitoring.

In order to combine explicitly the probabilities of

belonging to one of the three faults (corresponding to

the states of the system when it is out of control) and

the probability that the system is always in normal

operating state IC, we propose to add to the network

a discrete parent node S

r

with k+1 modalitees (see 2)

linked to all the other nodes F

j

.

The conditional probability tables (CPT) of the

node S

r

and his son nodes F

j∈1,...,k

are given in tables

2 and 3.

Table 2: CPT of the node S

r

.

S

r

IC F

1

... F

k

1

k+1

1

k+1

1

k+1

1

k+1

4 BAYESIAN NETWORK AND

DATA-DRIVEN METHODS

Unlike methods that require an accurate model de-

signed from ﬁrst principles as a priori knowledge

of the system, this methods try to detect and ex-

plain a change in the normal operating system, rely-

ing solely on measurements collected on the system

ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics

164

Table 3: CPT of the nodes F

1

&F

2

&. ..F

k

.

S

r

F

1

&F

2

&...F

k

Yes No

IC 0 1

F

1

1

2

1

2

.

.

.

.

.

.

.

.

.

F

k

1 0

.

.

.

.

.

.

.

.

.

F

n

1

2

1

2

(temporary or not). Several data-driven methods for

monitoring purpose exist, they depends on the avail-

ability or unavailability of historical data on the sys-

tem. Among this methods, we can ﬁrstly mention the

subspace methods SMI (Subspace Model Identiﬁca-

tion)(Overschee and Moor, 1996), a linear identiﬁca-

tion algorithms, developed to address the problems of

building an accurate model for complex systems. The

principal component analysis (ACP) (Harkat et al.,

2006), is a statistical method that can be used to

modelise existing dependences between a set of sys-

tem variables or as a method of data reduction (used

when the number of variables is considerable). Men-

tion may also the control charts methods (MacGregor

and Kourti, 1995), statistics gathered over a period of

time t, which are also widely used in industry. They

(control chart T

2

of Hotelling, MEWMA (Multivari-

ate Exponentially Weighted Moving Average) are de-

signed to monitor the normal operation of the system.

Otherwise, when a history of faults is available,

supervised classiﬁcation methods, adapted to the di-

agnostic can be used. Indeed, the diagnosis prob-

lem can be formulated for a given observation, as

a problem of discrimination between several operat-

ing modes. Among this data-driven methods, we can

mention neural networks (Duda et al., 2001), SVM

(support vector machine) (Steinwart and Christmann,

2008), discriminant analysis (Fukunaga, 1990) (can

be modelled under a Bayesian network, where the

variables are assumed follow a multivariate normal

distribution).

In our work, one assuming our knowledge about

the faults covers almost all the space (assumed as

closed space) out of control (H

1

⊆ {F1,... , F

k

}), we

propose to use a Bayesian network classiﬁer to dis-

criminate between these faults and the state IC (where

the system is considered in normal operating condi-

tion H

0

). To achieve this, we use a Bayesian Network

(see ﬁgure 3) naive semi condensed (RBNSC) con-

sisting of a discrete node representing k+1 modalities

and continuous multivariate Gaussian node (with a

mean µ and variance Σ estimated on the fault database

by Maximum Likelihood Estimation (MLE) (Duda

et al., 2001)) combining all the m variables of the sys-

tem (x

1

,x

2

,..., x

m

).

Figure 3: Bayesian network for data-driven monitoring.

By making assumption about normality of each

modality, this network using a decision rule based on

the Bayes formula (2):

P(Y/X) =

P(Y)P(X/Y)

P(X)

(2)

corresponds to a quadratic discriminant analysis (5) :

δ : x ∈ M

∗

j

, if j

∗

= argmax

j=1,...,k+1

{P(M

j

/x)} (3)

= argmax

j=1,...,k+1

{

P(M

j

)P(x/M

j

)

P(x)

} (4)

= argmax

j=1,...,k+1

{P(M

j

)P(x/M

j

)} (5)

Where P(M

j

/x) is the probability a posteriori of

Y, P(x) is the density function of x, P(x/M

j

) is the

likelihood and P(M

j

) the prior probability of M

j

Thus, it making us able to decide to a given

instant, in which operating state, the system be-

longs among its various states separated quadratically

(IC, F

1

,F

2

,..., F

k

). The probability tables for each

node are shown in the tables 4, 5.

Table 4: CPT of the node S

m

.

S

m

IC F

1

... F

k

1

k+1

1

k+1

1

k+1

1

k+1

Table 5: CPT of the node X.

S

m

X

IC X ∼ N(µ

IC

,σ

2

IC

)

F

1

X ∼ N(µ

F

1

,σ

2

F

1

)

... ...

F

k

X ∼ N(µ

F

k

,σ

2

F

k

)

ABayesianApproachtoFDDCombiningTwoDifferentBayesianNetworksModelingaData-DrivenMethodanda

Model-basedMethod

165

5 DECISION FUSION

To improvethe decision making, in this work, we pro-

pose to combine the two monitoring methodology in a

same and a single tool. Indeed, our proposal consists

to combine the two methods discussed above under a

Bayesian network.

Thus to build our new network combining both

methods, a new discreet node S

r&m

is added (see ﬁg-

ure 4). This node represents like the root nodes S

r

and S

m

a variable with j + 1 modalities. One of these

corresponds to normal operating condition (IC) where

the other modalities j corresponds to the known faults

that may occur on the system. This new node S

r&m

is used to connect the two root nodes and so asso-

ciate the both methods by fusioning their decisions.

Thus, a conjunctivecombination (see (Xu et al., 1992;

Chen et al., 1997) for others decision fusion methods)

is made under the assumption that the two network

are conditionally independent to the node added. In-

deed, thanks to the use of the Bayes formula (2), the

Bayesian network offers a naturally probabilistic fu-

sion capacity.

Once the network is built, we enter the observa-

tions (evidences) in the network. They correspond to

the residuals obtained and the measures taken on the

system at a given instant. These evidences are then

transmitted to other unobserved nodes in the network.

Their marginal probabilities are then calculated us-

ing the inference method employed (we use junction

tree). After having carried out this inference, the node

S

r&m

indicates for each k+ 1 modalities, the probabil-

ity of its occurrence.

Figure 4: A two combination methods.

Regarding the decision, among others criterions,

we chose to use the maximum a posteriori like we

do for the others methods (model-based, data-driven),

Table 6: CPT of the node S

r&m

.

S

r&m

IC F

1

... F

k

1

k+1

1

k+1

1

k+1

1

k+1

where the modality with the higher probability a pos-

teriori is choose. The table of conditional probabili-

ties of node S

r&m

is shown in table 6.

6 APPLICATION

To illustrate our approach, we use a simulation of a

water heater. It consists of a tank equipped with two

resistors R

1

and R

2

. The inputs are the water ﬂow rate

Q

i

, the water temperature T

i

and the electric power for

heating P. The outputs are the rate of water ﬂow Q

0

and the temperature T regulated around an operating

point. The temperature of the incoming water T

i

is

assumed constant.

The objective of the system is to provide a water

ﬂow at a given temperature. Using hydraulic and ther-

mal equations, in this analysis, only sensor faults are

considered: water level sensor H, temperature sensor

T, sensor ﬂow of water from Q

0

. The detailed math-

ematical model of the system is presented in (Weber

et al., 2008).

A classic residuals generator is used: a Luen-

berger observer. The output vector is [H,T]

T

and

the input vector [Q

i

,P]. Structured residuals [r

1

,r

2

,r

3

]

are generated and evaluated to detect faults of water

level sensor H and temperature sensor T. Accord-

ing to the physical equations between the ﬂow rate

Q

0

and the liquid level H, other residual can be estab-

lished. The incidence matrix (the link between symp-

toms [u

1

,u

2

,u

3

] and faults [T,H,Q

0

]) deﬁned in table

7 will give the structure of our Bayesian network.

Table 7: Incidence Matrix of the heating water system.

IC T H Q

0

u

1

0 1 0 0

u

2

0 0 1 0

u

3

0 0 1 1

We simulated the system according to the sce-

narios described in table 8 in order to test the pro-

posed method under different assumptions. Indeed,

the value of combining the two methods is to be able

to beneﬁt of good results even when one or the other

method is not very efﬁcient (missing detection of ab-

normal operating state, faults misdiagnosis). Thus,

we propose to test the Bayesian network, taking into

account an accurate model (M+) or a less accurate

ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics

166

(M−), and a complete dataset of suitable size (D+)

or incomplete dataset (a lack faults data, a few data)

(D−). Thus, the scenarios previously presented will

be tested on four assumptions described in 9.

Table 8: Simulated scenarios.

period 1-30 31-60 61-90 91-120

case In Control faultT faultH faultQ

0

Table 9: Hypothesis matrix.

hypothesis M(Model) D(Data)

H

I

M

+

D

+

H

II

M

+

D

−

H

III

M

−

D

+

H

VI

M

−

D

−

Every simulation was performed using Mat-

lab/Simulink and BNT (BayesNet Toolbox). For each

observation, we attribute the fault to the modality with

the greater a posteriori probability. The different re-

sults of the simulation obtained by testing our meth-

ods under four different assumptions are presented in

ﬁgure 5.

Figure 5: Simulations results under the four hypothesis.

One can notice that for each hypothesis, the pro-

posed method can usually equalize the performance

of each method and even to improve the monitoring.

In hypothesis I, the three methods are good, they give

right answers. The proposed method is slightly more

efﬁcient. For the hypothesis II, we suppose that the

data of the fault H misses us. In this condition, we see

that the method proposed can have good results. The

same thing is happen, in the case of the third hypothe-

sis, where we have degraded the model. Finally, in the

hypothesis VI, the proposed approach allows a better

decision than the two other methods. This shows that

the proposition made is efﬁcient and can take advan-

tage of the two basic methods under a Bayesian net-

work.

7 CONCLUSIONS

The interest of this paper is to present a new method

for monitoring industrial systems. We have presented

a particular structure of Bayesian network which con-

sists of discrete and Gaussian nodes allowing to mod-

els and combines two Bayesian networks dedicated

to monitoring: one for data-driven monitoring and

one representing the incidence matrix and the eval-

uation of residuals for the model-based monitoring.

This original structure can enhance decision mak-

ing during monitoring using simultaneously data and

residuals. This method has been tested on a water

heater system, where an improvement of the decision

is made and this in the most cases (speciﬁc model,

model degraded and more or less data).

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