Fault Diagnosis in an Industrial Process Using Bayesian Networks: Applicationof the Junction Tree Algorithm

Julio C. Ramírez1, Guillermina Muñoz2, Ludivina Gutierrez3

Instituto Tecnológico de Nogales, Ave. Tecnológico 911, Nogales, México1juliorv@itnogales.edu.mx, 2guillemunoz@itnogales.edu.mx, 3lgutierrez@itnogales.edu.mx

Abstract

In this paper we present a Bayesian Network for faultdiagnosis used in an industrial tanks system. We obtain theBayesian Network first and later based on this, we build adefined structure as Junction Tree. This tree is where wespread the probabilities with the algorithm known as LAZY-AR (also Junction Tree). Nowadays the state of the art ininference algorithms in Bayesian Networks is the JunctionTree algorithm. We prove empirically through a case studyas the Junction Tree algorithm has better performance withregard to the traditional algorithms as the Polytree.

1. Introduction

Bayesian Networks (BN) have become a paradigm veryused in medical diagnosis [1, 5] but in recent years theyhave also been preferred by researchers that study fault di-agnosis in engineering areas [8, 12]. Bayesian Networksadvantages with regard to other approaches is that they arebased on graph and probability theories. The graph theorymakes them more intuitive because it visualizes the causesand effects of each node and at this time the probability, isone of the sciences that better it represents the uncertainty.

A BN consists of a directed acyclic graph (DAG) anda corresponding set of conditional probability tables (CPT)[11]. The probabilistic conditional independences encodedin the DAG indicate the product of the conditionals is a jointprobability distribution. In many works that report the useof BN, one of the favorite inference algorithm is the Poly-tree [13, 3]. This algorithm is simple of applying but it hasas disadvantage that it only can be applied to a singly con-nected BN [11] and that it is less effective than others asthe Junction Tree or also known as LAZY-AR [2] (it can beused in a multiply connected BN). The Junction Tree (JT)algorithm effectiveness has been evaluated empirically byMadsen [7]. The present work also make an empiric evalu-ation of this algorithm, but in this occasion through a casestudy where we compare the performance of JT algorithm

with regard to the Polytree algorithm applied to the diagnos-tic system of interconnected deposits reported first in [9].Other works related with JT applications to fault diagnosisare in [10, 6].

The paper structure is the following one: in the section2, we explain the basic principles in the operations of the JTthat include marginalization and combination. In the sec-tion 3, by means of a case study we explain the JT construc-tion and the propagation of probabilities through the sameone. Section 4, consist in a comparative analysis among theresults produced by the JT and Polytree algorithms. In thesection 5, we provide the conclusions.

2. Basic definitions

In the Bayesian inference process using the JT algorithm,first we have to obtain the BN. This network is the descrip-tion of the probabilistic relationship that exists among thedifferent variables that represent any process. Each condi-tional probability p(Xi|pa(Xi)) will be stored and it willnecessarily associate to a tree node by means of potentials.A potential ψ defined on a set of variables XI ⊆ ν will bethe mapping ΩI → <+, where ν is the set of all the nodesand ΩI denotes all the configurations of the values taken bythe group of variables XI .

The groups or clusters in a JT also are called cliques. Aclique will be a group of variables that can form a completemaximum subgraph (triangular graph). The directly con-nected cliques in the tree shares a separator, which containcommon variables in both cliques. A separator is composedby Ci and Cj , these are two adjacent clusters in the JT [4].The separator for these clusters is Sij and it is defined bythe expression:

Sij = V |V ∈ Ci ∩ Cj (1)

The process of passing from a BN to a JT is called com-pilation and this formed by four basic steps: 1) Obtain themoral graph from the original DAG that represented the BN.2) Triangulate this moral graph. 3) Identify all the cliques.

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4) Connect these cliques in order to form a valid JT. Thesesteps will be illustrated in a case study in section 3.

Once formed the JT the separators mailboxes are repre-sented as potentials with regard to the variables set includedin the separator. If the separator Sij is located among cliqueCi and clique Cj , a mailbox will indicate a message in thedirection Ci → Cj and the other mailbox in the contrarydirection Cj → Ci. For clarity, the potentials inside a mail-box will be denotes as µ (e.g., µCi→Cj

). This is the in-formation about the probabilities that the cliques will con-tain and the separators will help to communicate amongthem. In order to understand how this communication willbe made, it is necessary to define two operations on po-tentials: marginalisation and combination [4]. These twooperations are basic in the probabilities propagation.

2.1. Marginalization

Suppose two variables setsXI andXJ in such a way thatXI ⊆ XJ . Let ψXJ

be a potential on XJ . We obtain themarginalization of ψXJ

to XI by means of the result of thefollowing summation:

ψ↓XI

XJ(xI) =

∑x↓XIJ =xI

ψXJ(xJ) (2)

2.2. Combination

LetXI andXJ be two sets of variables. And let ψXIand

ψXJbe two associated potentials. Then, the combination of

ψXIandψXJ

is a new potential defined overXI∪XJ whichis obtained by pointwise multiplication:

ψXI∪XJ (x) = ψXI (x↓XI )⊗ψXJ (x↓XJ ), ∀x ∈ ΩXI∪XJ . (3)

So, initially the probabilities given for every variable givenits parents (ψXI

) is assigned to every clique. The rest haveunitary potentials. The messages in the separator mailboxesare initially empty. Once a message is placed on one mail-box it is said to be full. A node Ci in a JT can send a mes-sage to its neighbour node Cj if and only if all Ci-incomingmessages are full except the one from Cj to Ci. Thus, ini-tially leaf nodes are the only capable of sending messages.The message Ci-outgoing (and Cj-incoming) is computedas:

µCi→Cj=

ψi.

∑Ck∈ne(Ci)−Cj

µCk→Ci

↓Ci∩Cj

(4)

where ψi is the initial probability potential on Ci, µCk→Ci

represent the messages from Ck to Ci and ne(Ci) are theneighbour clusters of Ci. With this scheme one messagecontains the information coming from one side of the treeand transmits it to the other side. It has been proved that it is

always possible to find, at least, one node to send a messageuntil all mailboxes are full. When the message passing endsit is said that the tree is consistent and the following holdsfor all the nodes in the tree:

ψmCi

= ψi.

∏Ck∈ne(Ci)

µCk→Ci

(5)

where ψmCi

is the potential (probability distribution) resultedfor the variables in Ci after this propagation. In equation(5) we see that for calculating the probability for a set ofvariables it is necessary to combine the initial potential ψi

with all the incoming messages. The desired probabilityfor a variable Xi can be calculated by marginalising ψm

Ci,

(where Xi ∈ Ci) over this variable and normalising theresult.

3. Case study

To illustrate the application of the algorithm JT com-pared with the algorithm Polytree, a BN structure will beused. This BN corresponds to diagnostic system of one in-dustrial process of two interconnected deposits (see figure1). It was used first in [9]. We model the system by means

h1 h

2

Vs

Rq deposit 1 deposit 2Va

ViR2

R1

q2q

1 C2

C1

Figure 1. System of interconnected deposits

of their differential equations. When varying the flow q ofeach valve, the different faults will be suitably simulatedin GNU Octave. The causes of the faults will be damagesin the valves V a, V i, V s, and the evidences that lead todiscover the faults will be the levels in q2, h1 y h2. The net-work structure and their a priori values, that were obtainedaccording to the techniques of parameters learning used in[9], are shown in figure 2. We are going to use this BN toillustrate the necessary steps for the construction of a JT.

3.1 Junction Tree construction

The first step is to obtain the moral graph that dependson the original GAD. This GAD is the BN depicted on thefigure 2.

262302302302

Vi

Vi 0.07298

¬ Vi 0.92702

Vs

Vs 0.10742

¬Vs 0.89258

h2Va ¬Va

Vs ¬Vs Vs ¬Vs

h2 0.1 1.0 1.0 0.01

¬h2 0.9 0.0 0.0 0.99

q2Va ¬Va

q2 0.99 0.01

¬q2 0.01 0.99

Va

Va 0.04592

¬Va 0.95408

h1Va ¬Va

Vi ¬Vi Vi ¬Vi

h1 0.1 1.0 1.0 0.01

¬h1 0.9 0.0 0.0 0.99

Figure 2. Bayesian Network for two depositdiagnosis system with their conditional prob-ability tables

3.1.1 Moralise the graph

So, we take the initial DAG that forms the graphical partof the network and makes it undirected following these tworules: join those nodes with common parents with a morallink and drop directions of the directed edges. The figure3 shows in what form our network would be, after beingmoralized.

VaVi Vs

q2 h

1 h

2

Figure 3. Obtaining the moral graph for twodeposits diagnosis system

3.1.2 Triangulate the moral graph

Once moralized the network, the triangulation process be-gins, that is part of the deletion sequence σ. In this se-quence, each node will have a position between 1 and |ν|in accordance with their deletion order. The rules for thedeletion sequence are: 1) remove from the graph the nodeνi and all its incident links, 2) if it is necessary, add a link totriangulate the nodes left by the deletion sequence (this pro-cess is different to the moralization, because in this instancethe links are undirected). In our case it becomes necessaryto add an edge among the nodes V i and V s, it is the dottededge of the figure 4.b, with this, our graph has been trian-gulated.

3.1.3 Identify the cliques

Once the graph is triangulated it is time to determine whichare the cliques in this triangulated graph. If G is an undi-rected graph, then all the maximal complete subgraphs inG are called cliques. After identifying the maximal com-plete subgraphs in the triangulated graph, we should givetree form to our structure, what leads us to the next stage.

3.1.4 Build the tree

In order to give the tree form to our structure, first we shoulddetermine σ, it is our deletion sequence of edges and nodes.Our deletion sequence is σ = q2, h1, h2, V a, V i, V s. Itcan be seen in the figure 4.

VaVi Vs

q2 h

1 h2

VaVi Vs

h1

h2

VaVi Vs Vi Vs

a) deletion of q2

b) deletion of h1 and h

2

c) deletion of Va d) deletion of Vi

Figure 4. Deletion sequence σ

Taking the descending order of σ, we define the cliquesas it is shown in the Table 1. Algorithms exist to transform anon binary tree in one binary, the JTs should be binary trees.In first instance, the JT that we obtained from our BN is not

Table 1. Descending order of σ in the JunctionTree definition

i νi Cliquesi Sepi parents6 V s5 V i4 V a V a, V i, V s φ -3 h2 h2, V a, V s [V a, V s] 42 h1 h1, V a, V i [V a, V i] 41 q2 q2, V a [V a] 4

binary, (it is depicted in the figure 5.a). If we designate tothe node 1 as the node father of the node 4, then the JTbecame binary tree, like one can observe in the figure 5.b.With the information of the Table 1, we can obtain the JTof the figure 5.b. This JT also has assigned the potential φcorresponding to each node.

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3.2 Probabilities propagation in a Junction Tree

For each potential φ it is necessary to find a group C,which may contain all the involved variables, like one ob-serves in the figure 5.b. We should include in the JT, eachpotential that one has of the BN in the group or appropri-ate clique, but we should not repeat the potentials inclu-sion. A JT is called consistent after propagating probabili-

1h 2h

2q

VsVi ,Va

Va, Vi, Vs 4

Va, Vi

h1, Va, Vi

2q

2, Va

1h

2, Va, Vs

3

Va Va, Vs

q2, Va

1

Va

Va, Vi, Vs 4

Va, Vi

h1, Va, Vi

2

Va, Vs

h2, Va, Vs

3a) JT with non binary structure b) JT with binary structure

Figure 5. Resulting Junction Tree

ties when ∀Ck, Xi ∈ Ck, normalization (ψmCk

)↓Xi returnsthe same probability distribution. This is, if we consult thesame probability but in different nodes that contain it, theresult should be the same one after the normalization of thisresult.

The propagation is usually divided in two stages: up-wards and downwards. They are also called collection anddistribution. For this scheme it is necessary to select a cliqueas the root one. In the first phase (upwards/collection),all the probabilities are sent recursively, starting from theleaves of below through each tree level toward the rootnode (superior node), it is the node 1 of the figure 5.b. Inthe second phase (downwards/distribution), the clique rootthat is the superior node sends recursively the informationcollected to its neighbouring until arriving to the inferiorleaves.

3.2.1 Collection

Once the general outline of probabilities propagation hasbeen presented in a JT, we will prove the inference algo-rithm taking as root node the node 1 of the JT. The poten-tials values can be observed in the BN depicted in the figure2. According to this figure, the nodes of input evidences(h1, h2, q2) possess information (conditional probability ta-bles) that can be updated by the information coming fromtheir respective sensors.

Now we will spread a present evidence in the sensor h1.We will propose that according to new readings, the prob-ability that h1 observes the manifestation of an abnormal

symptom it is of 100%. Later, using the combination oper-ation (3), we will obtain the updated potential for node 2 ofthe figure 5.b.

ψ2 = sensor h1 ⊗ φh1

ψ2 =sensorh1 1¬h1 0

⊗

V a ¬V aV i ¬V i V i ¬V i

h1 0.1 1 1 0.01¬h1 0.9 0 0 0.99

ψ2 =

V a ¬V aV i ¬V i V i ¬V i

h1 0.1 1 1 0.01¬h1 0 0 0 0

Then, we update the potential in node 3, where, according tonew readings in the sensor h2, the h2 level is 50% abnormal.

ψ3 = sensor h2 ⊗ φh2

ψ3 =

V a ¬V aV s ¬V s V s ¬V s

h2 0.05 0.5 0.5 0.005¬h2 0.45 0 0 0.495

As we are in the first phase, that is upward propagation orcollection, for not creating confusion, the last node of evi-dence input q2 will be updated when beginning the phase ofdescending propagation or distribution. Carrying on withthe probabilities collection, we will spread the probabilityfrom node 2 to node 4 in the following way.

µ2→4 = (ψ2)↓V a,V i =V a ¬V a

V i 0.1 1¬V i 1 0.01

In previous operation the expression (2) was applied, wherethe operator ↓ means marginalization of the updated poten-tial of node 2 with respect to V a and V i. The reason whywe must marginalize with respect to V a and V i, it is be-cause the separator between nodes 2 and 4, contains thosevariables (see figure 5.b). Marginalization can be seen asthe columns sum in ψ2. Now we will spread the probabilityfrom node 3 to node 4 as follows.

µ3→4 = (ψ3)↓V a,V s =V a ¬V a

V s 0.5 0.5¬V s 0.5 0.5

After this, we can continue collecting probabilities, spread-ing from node 4 to node 1 in the following way.

µ4→1 = (µ2→4 ⊗ µ3→4 ⊗ φV a ⊗ φV i ⊗ φV s)↓V a

The previous operation is the most complex that we willhave to solve in the probabilities collection, for what is rec-ommended to solve it by parts. As we can notice, it includesthe potentials that we have been collecting, besides the po-tentials belonging to node 4. The result of this operationshould be marginalized with respect to V a.

µ4→1 = V a 0.0215¬V a 0.0392

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3.2.2 Distribution

Once collected all the potentials, we begin the distributionphase. First, the potential in node 1 is actualized, whereaccording to new readings in sensor q2, the flow in q2 is 0%abnormal.

ψ1 = sensor q2 ⊗ φq2 =V a ¬V a

q2 0 0¬q2 0.01 0.99

Carrying on with the descending propagation, we spread thepotential from node 1 to node 4.

µ1→4 = (ψ1)↓V a = V a 0.01¬V a 0.99

To make the propagation from node 4 to node 2 we have totake into account the contributions of node 1 and also thoseof node 3, as well as the potentials belonging to node 4.

µ4→2 = (µ1→4 ⊗ µ3→4 ⊗ φV a ⊗ φV i ⊗ φV s)↓V a,V i

Due to the complex of the propagation operation from node4 to 2, it is recommended to solve by parts.

µ4→2 =V a ¬V a

V i 0 0.0345¬V i 0.0002 0.4378

To finish with the distribution we need to spread the poten-tials from node 4 to 3.

µ4→3 = (µ1→4 ⊗ µ2→4 ⊗ φV a ⊗ φV i ⊗ φV s)↓V a,V s

µ4→3 =V a ¬V a

V s 0 0.0083¬V s 0.0004 0.0693

3.3 Consistency test of the Junction Tree

Once spread all the potentials according to the JT algo-rithm, it is possible to compute the probabilities in V a, V iand V s. It should be done to see according to the new evi-dences introduced by the sensors, which are the causes thatoriginate the faults. According to the equation (5) consis-tency should exist in the JT, it means, that if we calculatep(V a) in the node 1, it can also be calculated in the node 4and the result should be the same one. To prove it, p(V a)will be computed in both nodes. First, we begin calculatingthe probability in the node 1.

p(V a) = (ψ1 ⊗ µ4→1)↓V a = V a 0.0002¬V a 0.0388

According to the JT algorithm the previous result should benormalized to obtain the probability that we are looking for,this means that p(V a) + p(¬V a) = 1. After normalizing,the result is the following one.

p(V a) = V a 0.0055¬V a 0.9945

Later, p(V a) is calculated in node 4. The computation inthis node is more costly than in the node 1.

p(V a) = (µ3→4 ⊗ µ2→4 ⊗ µ1→4 ⊗ φV a ⊗ φV i ⊗ φV s)↓V a

p(V a) = V a 0.0055¬V a 0.9945

We can see as p(V a) computed in the node 4 it is similarto the one calculated in the node 1, which confirms the JTconsistency. In the computation process of p(V a) in bothnodes, we observed: 1) the potentials of node where weare computing p(V a) should be used. 2) Also, all the po-tentials of the nodes related with the node used to comput-ing p(V a) should be collected. 3) After this, we computep(V a). The computation process actualize the potential toreflect the possible changes occurred in the JT. The prob-abilities computation of V i and V s, they are obtained re-spectively in the node 2 and 3, with the current evidencesthat we have on hand, in the following way.

p(V i) = (ψ2 ⊗ µ4→2)↓V i = V i 0.8825¬V i 0.1175

p(V s) = (ψ3 ⊗ µ4→3)↓V s = V s 0.1074¬V s 0.8926

In the previous operations we can notice that the marginal-ization ↓ is given with respect to the probability that iswanted to compute. It is clear that in accordance with theobserved results the fault is in the valve V i.

4. Results analysis

The Table 2 shows the results given for a comparativestudy among the algorithm Junction Tree and the algorithmPolytree, applied to diagnosis of a deposits system (see fig-ure 1). This Table shows, the tree possible single fault. Inthe first place, we can observe fault V a. It happens when thevalve V a is broken and let pass a flow q high that increasesall the levels. The fault of V i happens when the valve V i isclogged, and increases in abnormal form the h1 level. Thelast single fault, is the fault V s, it happens when the valveV s is clogged and as evidence the level in h2 is increased.

In the V a fault, the values of sensor h1, sensor h2

and sensor q2, rise gradually. It can be see in Table 2,in the rows where the fault is V a. These sensors actu-alize the values of the levels h1, h2 and q2, respectively.We can observe that when the probability in the sensors isp(sensor h1) = · · · = p(sensor q2) = 0.6, the behaviorof the two algorithms it is still similar. We notice that theybegin to diverge, when increasing at 0.8 the sensors proba-bilities. When being confirmed the symptoms in the sensors

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Table 2. Analysis of single fault with different algorithmsProbability in Junction Tree algorithm Polytree algorithm

the sensors S = sensors (h1, h2, q2) S = sensors (h1, h2, q2)Fault p(sensor h1) p(sensor h2) p(sensor q2) p(V a|S) p(V i|S) p(V s|S) p(V a|S) p(V i|S) p(V s|S)V a 0.01 0.01 0.01 0.95E-005 0.81E-003 0.0012 0.95E-005 0.0042 0.0065V a 0.2 0.2 0.2 0.0014 0.0197 0.0298 0.0014 0.0226 0.0341V a 0.4 0.4 0.4 0.0164 0.0509 0.0757 0.0164 0.0518 0.0770V a 0.6 0.6 0.6 0.1216 0.0987 0.1432 0.1216 0.1018 0.1477V a 0.7 0.7 0.7 0.2985 0.1186 0.1687 0.2985 0.1431 0.2034V a 0.8 0.8 0.8 0.6088 0.1068 0.1475 0.6088 0.2074 0.2857V a 0.9 0.9 0.9 0.8965 0.0549 0.0729 0.8965 0.3210 0.4195V a 1.0 1.0 1.0 0.9976 0.0099 0.0141 0.9976 0.5764 0.6754V i 1.0 0.0 0.0 6.04E-004 0.8868 6.04E-004 6.04E-004 0.5764 0.0052V s 0.0 1.0 0.0 2.70E-004 2.70E-004 0.9230 2.70E-004 0.0034 0.6754

at the 100%, the results difference of the two algorithms ismore significant.

When comparing the algorithms answers for the faultsof the valves V i and V s (last two rows of Table 2), wehave a behavior seemed in the performance of both algo-rithms. The two algorithms provide similar values for eachfault, until the symptoms are confirmed in a 100%, then itis when they diverge. Nevertheless the differences, we canobserve that the diagnosis given by the two algorithms it isvery similar, except that the algorithm Polytrees has lack ofcertainty. It can be seen in the single fault of V a, where thealgorithm for Polytree affirms with a certainty of 99% thatthe failure is V a, but at the same time, affirms that it may beV s with a severity of 67%. This undesirable effect doesn’texist in the JT algorithm, where V a is really a single fault.

5. Conclusions

In this paper, we have applied the Junction Tree algo-rithm used in the single faults diagnosis of interconnectedtanks. Although, it has a little higher computation cost, itoffers bigger certainty in the inference [2, 7], as we have ex-perienced it in our case study. Furthermore, it can be usedin a multiply connected BN, which, it is not possible withthe Polytree algorithm [11]. Another of the Polytrees dis-advantages, is that, for update of evidences, one has to adda node (as the sensors), for each variable that is wanted toactualize. This contributes in the complexity and little ver-satility, because with each extra node, the inference functionthat requires the Polytree algorithm has to be modified. Inthe Junction Tree this doesn’t happen, because the sensorsof update can be located in any node without affecting theinference, since they are not part of the Junction Tree.

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