DIAGNOSIS METHOD FOR SPACECRAFT USING DYNAMIC BAYESIAN NETWORKS

Yoshinobu Kawahara1, Takehisa Yairi2, and Kazuo Machida2

1University of Tokyo, Department of Aeronautics and Astronautics2University of Tokyo, Research Center for Advanced Science and Technology

ABSTRACT

Development of sophisticated anomaly detection anddiagnosis methods for spacecraft is one of the impor-tant problems in space system operation. In thisstudy, we propose a diagnosis method for spacecraftusing probabilistic reasoning and statistical learn-ing with Dynamic Bayesian Networks (DBNs). Inthis method, the DBNs are initially from prior-knowledge, then modified or partly re-constructedby statistical learning with operation data, as a re-sult adaptable and in-depth diagnosis is performedby probabilistic reasoning using the DBNs. The pro-posed method was applied to the telemetry data thatsimulates the malfunction of thrusters in rendezvousmaneuver of spacecraft, and the effectiveness of themethod was confirmed.

Key words: Diagnosis, Fault Detection, DynamicBayesian Networks, Probabilistic Reasoning, Statis-tical Learning.

1. INTRODUCTION

A sophisticated diagnosis method automating a se-ries of tasks which contains anomaly detection, eval-uation of its dangerousness and appropriate corre-spondence is indispensable for the safety of space de-velopment. Such tasks are almost dealt with by man-ual operations in the present circumstances thoughspacecraft such as a satellite and a rocket is a highlyautomated system. However larger-scale and morecomplicated a space system is, more knowledge andmore tasks are required for its operation. So the es-tablished operation methods relying on manpowerbecome difficult to guarantee complete reliability.Acutually, the cases that a man slip over the signs ofsystems’ fault have increased. For this reason, it isindispensable to make men concentrate on the mostimportant decisions by automating these tasks withcomputers.

Several expert systems and model-based approachesincluding qualitative diagnosis have been developedas the intelligent diagnosis method using computers.Expert systems diagnose by classifying phenomenawith the experts’ empirical knowledge in the formof ”if-then” production rules (Tallo 92; Ciceri 94;Nishigori 01). And in model-based diagnosis, math-ematical or qualitative models are utilized to sim-ulate the system behavior and check the validity ofthe actual sensor values (Monsterman 99; Struss 03).An example model-based diagnosis system is Living-stone (Williams 96), which flew on the deep SpaceOne spacecraft as a part of the Remote Agent Ex-periment (Muscettola 98). And Recently the di-agnosis method using Particle Filter for a planetrover is also presented (Deaden 04). These ap-proaches using prior-knowledge for diagnosis havehigh cause-investigation power. However there isthe significant problem that the accuracy of the em-bedded knowledge representing the model or therules effects directly the possibility and the preci-sion of diagnosis for spacecraft because the past faultcases of spacecraft are absolutely few and space-craft are in the environment containing a lot ofuncertainty. On the other hand, in the field ofother than space, diagnosis methods putting em-phasis on data are proposed which detect the de-viances of systems’ behavior by using various sta-tistical techniqus or analysing the trends of data(Markou 03; Venkatasubramanian 03). These meth-ods, though don’t reach the approach putting em-phasis on knowledge in the point of accuracy or de-tails of diagnosis, need little knowledge beforehandand enable to diagnose in adaptable way.

In other words, the former approach enables ”nar-row and deep” diagnosis by using diagnosis modeldeductively and the latter ”wide and shallow” byaqcuiring one inductively. However, these two po-lar approaches have the difficulty for covering theother’s property and there are few researches fusingboth approaches. So we propose a diagnosis methodusing Dynamic Bayesian Networks for the purpose ofthe spacecraft diagnosis system which has the wideand deep diagnosis ability by using both knowledge

Proc. of 'The 8th International Symposium on Artifical Intelligence, Robotics and Automation in Space - iSAIRAS’, Munich, Germany.5-8 September 2005, (ESA SP-603, August 2005)

Z

Z

Z

ZZ

ZZ

Z

Z

Z

ZZ

ZZ

Figure 1. Abstract of the proposed diagnosis method

and data. DBNs have rich expressiveness and un-derstandability as containing Kalman Filter Mod-els (KFMs) and Hidden Markov Models (HMMs),and are the models suitable for representing dynam-ical systems such as spacecraft. In addition, thereare several effective algorithms for statistical learn-ing and probabilistic reasoning for DBNs, the diag-nosis method using knowledge and data in naturalway are possible to be formed.

The remainder of this paper is organized as follows.The diagnosis method using DBNs is suggested andexplanations are given for the each processes in thismethod. And the superiority of the proposed methodis ilustrated by applying to the telemetry data thatsimulates the malfunction of thrusters in rendezvousmaneuver of spacecraft. Finally, a conclusion isgiven.

2. APPROACH

The proposed diagnosis method uses DynamicBayesian Networks as diagnosis models and consistsof following three processes as shown in Figure 1.

1. Construct a diagnosis model with prior-knowledge

2. Learn the model with the stored data

3. Diagnose by probabilistic reasoning with themodel

First, in (1), the diagnosis model is constructed fromphysical knowledge, such as dynamics and design in-formation, and experts’ knowledge. Then, in (2), themodel is corrected or partly acquired by statisticallearning with operation or experiment data obtainedbeforehand. By this process, it becomes possible tore-construct the model of the parts which are dif-ficult to model with prior-knowledge or correct theerror according to modeling. And in (3), anomalydetection and investigation of its cause or positionare performed by probabilistic reasoning using themodel.

m

ku

x

Symbol Abstract Domain

X Position RealV Velocity 〃Y Observation 〃U Control Input 〃

XtXt-1

V V tt-1

Yt

v

X0

V0

B0 B

UtUt-1U0

Yt

x

Yt

v

Yt

x

Figure 2. The DBN representing a single-degree-of-freedom system

After this, we first give a summery of DBNs inSubsection 2.1. And in the following three subsec-tions, the explonations about each processes men-tioned above are given sequentially.

2.1. Dynamic Bayesian Networks

In this study, we use the probabilistic graphical mod-els called Dynamic Bayesian Networks (DBNs) asdiagnosis models1. A DBN is the model which ex-tend a Bayesian Network (BN) (alternatively Prob-abilistic Network, Belief Network or Causal Model)2

for applying to a dynamical system and is a generalstate-space-model containing Kalman Filter Model(KFM) or Hidden Markov Model (HMM). A BNis the model possible to deal with uncertain infor-mation which is relatively difficult to deal with forcomputers and studied actively in the field includingArtificial Intelligence(Haddawy 99). Its applicationsreach a wide area from user modeling (Horvitz 98),diagnosis (Jensen 01) to gene analysis (Segal 03).And a DBN, for which the research on various algo-rithms advances in recent years, is applied to voicerecognition etc. (Nefian 02).

A BN consists of the graphical structure G contain-ing the nodes which mean variables and the arcswhich represent the correlation between variablesand the conditional probability (and its parametersΘ) describing its correlation quantitatively. And aDBN consists of two BNs: BN0 which represents theprior-distribution Pr(X0) and BN→ the transitionprobability Pr(Xt|Xt−1). As a brief example, weconsider the DBN to estimate the states of a single-degree-of-freedom system represented as follows.

x = −k1 · x + u, u = kt

2y (1)

1(Murphy 02) is a famous doctoral thesis about DBNs.2We recommend (Pearl 88) as the most famous traditional

book, and (Jensen 01; Cowell 99; Pearl 00) as other famousones

The graphical structure of the DBN representing thissystem is shown in Figure 2. The observable vari-ables are represented by the gray nodes and the not-observable variables are the white nodes. And theconditional probabilities are given to each variables,for example about vt, the following conditional prob-ability can be given.

Pr (Vt|Xt−1, Vt−1, Ut−1)

= N (Vt−1 + ∆t (−k1Xt−1 + Ut−1) ,Σ)(2)

where Σ means the covariance matrix for the normaldistribution. In this case, the DBN contains only thecontinuous variables, however qualitative variables isalso possible to be incorporated.

2.2. Constructing DBNs with Prior-Knowledge

The first step is the construction of DBNs and needsthe following procedure.

1. Select variables in the model and knowledgeused in diagnosis

2. Initiallize the graphical structure and the pa-rameters of the DBN using diagnosis knowledge

First, the set of variables represented with the modelneeds to be selected from the purpose of diagnosisand the users’ understanding about spacecraft. Inthe case of the example described above, if a userneeds to consider the transition of the friction be-tween the mass point and floor, the variable rep-resenting the friction is added. Then the structureand parameters are setted in the following way fromthe differential equations representing the process ofspacecraft or the experts’ empirical rules.

• Physical knowledge represented as a differentialequation

x = f(x) ⇒ Pr(Xt|Xt−1) = N (xt−1+∆t·f(xt−1),Σ)

• Experts’ mpirical knowledge represented as aproduction rule

(if A = i then B = j) ⇒ Pr (A|B) = M(i, j)

where ∆t means the time-lag and M is the matrixrepresenting the probabilities of each cases. Herethough the parameters including variances need tobe given, users not necessarily set detailed these pa-rameters because the correctness by statistical learn-ing with data is possible later.

F F

X X

Y

t

t

tt-1

t-1

o o

Pr (Yt|Xt, Ft) =

N (xt, Σsmall) if Ft = 0(Normal)N (xt, Σlarge) if Ft = 1(Abnormal)

y

Figure 3. A DBN for diagnosis of sensors

2.3. Learning DBNs with Data

The DBN constructed from prior-knowledge is cor-rected or partly re-constructed by statistical learningwith the stored data as come close to the actual be-havior of spacecraft. The learning means the maxi-mization of log-likelihood L represented as followingequation.

L =τ∑

t=1

log Pr (yt|G, Θ) (3)

where yt means the observation at every time t, Gand Θ mean the graphical structure and the param-eters of the DBN respectively as described above. Alog-likelihood is the logarism of the conditional prob-ability of the observation when given the model, andis the measure how the model explanes the observa-tions obtained. For maximizing (3) of the model hav-ing some hidden variables, EM Algorithms (Expec-tation Maximization Algorithm)(Bilmes 98) are gen-erally applied. In our method, the graphical struc-ture and the parameters of a DBN are learnd to fitthe actual behavior of spacecraft using EM Algo-rithms with the stored data (Murphy 98; Geiger 94;Friedman 97).

2.4. Diagnosis by Probabilistic Reasoningwith DBNs

Our proposed method diagnoses by detecting statis-tical deviances every variables with on-line proba-bilistic reasoning using the DBN obtained.

A purpose of probabilistic inference targeting dy-namical systems is to calculate the marginal prob-ability described as the following equation at everytime.

Pr (Xt|y1:τ ) = cPr (yt+1:τ |Xt, y1:t) Pr (Xt, y1:t)

= cPr (yt+1:τ |Xt) Pr (Xt|y1:t)(4)

where Xt means the hidden variables at time t, y1:τ

the observation by now, and c the normalized term.

In our method, the mixture of the normal- andabnormal-distribution is attached to the objective

variable, and the transition of the each mixture pa-rameters are calculated for the purpose of detectingstatistical deviances every variables from the normalstates. For example in the case of a sensor, in ad-dition of the observed variable, the discrete variablewhose domain is ”normal” and ”abnormal” is addedas the condition of the observation variable. Thenthe distribution that has the mean equals to andthe variance larger than the normal one is attachedas the distribution of F = (abnormal). As a re-sult, the larger the deviance of the observatio fromthe normal behavior, the larger the probability ofF = (abnormal) becomes.

The calculation of probabilistic reasoning on DBNscan be generally performed by applying the junc-tion tree algorithm (Lauritzen 99). But in thecase that discrete variables have no continuous par-ent variables like our case, it is impossible to per-form the exact computation of probabilistic rea-soning because the number of situations increasesin exponential with the passage of time. So ourproposed method uses the approximate inferencealgorithm called Rao-Blackwellised Particle Filter(RBPF)(Doucet 00; Doucet 01).

3. RESULTS

In this section, our proposed diagnosis method isapplied to the telemetry data that simulates themalfunction of thrusters in rendezvous maneuver ofspacecraft3. The results are presented and the con-sideration is given.

3.1. Problem Set-up

The spacecraft in rendezvous maneuver process con-trols its attitude and transition using the fourteenthrusters. The data consists of the 26 series includingthe commands of 16 levels to these fourteen thrusters(14 series) and the states of the spacecraft (positionx, velocity v, euler angular θ and angular rate ω)(12 series). The sampling rate of the data is 125mili-seconds and the simulations are for 1000 sec-onds. And the mass properties (the weight and themoment of inertia of spacecraft) and the thrusters’properties (the normal thrusts and the location ofthrusters) are given beforehand, however the otherinformation such as the orbital charasteristics, thecontrol strategy, the sensors’ properties and so onare not given. So the nonlinear filters which need tobe modelled accurately beforehand (EKF or PF) aredifficult to be applied.

3The data is presented from Japan Aerospace ExplorationAgency (JAXA)

Table 1. Variables in the DBN

Symbol Description Domain

X Position RealV Velocity 〃W Angular Rate 〃Θ Euler Angular 〃T Thrust 〃Y Observation 〃C Command 1～16(Integer)F Presence of Fault 0(Normal),1(Abnormal)

XtXt-1

V V tt-1

X0

V0

B0 B

WtWt-1

tt-1

W0

0

Yt

v

Ft-2

1Ft-1

1

F 0

1

Tt-1

1C t-1

1

Yt

x

Yt

w

Yt

Ft-2

0Ft-1

0

F 0

14

Tt-1

14C t-1

14

Figure 4. Graphical Structure Used in the Experi-ment

3.2. Construction of Diagnosis Model

For this experiment, we constructed the DBN thathas the graphical structure shown in Figure 4 accord-ing to the procedures described below.

First, because of the purpose of the diagnosis de-scribed above, we added the continuous variablesthat mean the thrusts of each thrusters and the dis-crete variables that mean the presences of anomaliesand selected the set of variables shown in Table 1.Then the following linearized equations that meanthe attitude and transition dynamics on orbit areused as prior-knowledge.

θ = [ω1 + nθ3, ω2 + n, ω3 − nθ1]T

(5)

J1ω1 + n (J2 − J3) ω3 = M1

J2ω2 = M2 (6)

J3ω3 − n (J1 − J2) ω1 = M3

Table 2. Evaluated Diagnosis Models

prior-knowledgemodel structure parameter(initial value) learning

1 figure 4 from eq. (5)～(7) and prior no-learning2 figure 4 from eq. (5)～(7) and prior parameter learning3 figure 4 random parameter learning4 random random structure learning

x1 = 2nx3 + T1

x2 = −n2x2 + T2 (7)

x3 = 3n2x3 − 2nx1 + T3

where n means the mean motion of circular orbitand J means the moment of inertia of body, whoseaccurate values were not given beforehand. In thisexperiment, only physical knowledge above is used,however qualitative knowledge such as experts’ em-pirical rules which are represented by discrete vari-ables are posibble to be incorporated.

The procedure for constructing the model is followed.Here, the values of each variables at every discretetime t are represented as ∗[t], and its time-lag as ∆t.

1. Make each differential equations differences. Forexample about θ1, the following equation is ob-tained from Equation (5).

θ1[t] = θ1[t − 1] + ∆t · (ω1[t − 1] + nθ3[t − 1])

2. Set the graphical structure and the parameter ofthe DBN using the difference equations obtainedin Step 1 and the other prior-knowledge (for ex-ample, the thrust T = normal thrust × C/16in nominal phase). About Θ1[t], for example,Θ1[t− 1], Θ3[t− 1] and Ω1[t− 1] are selected asparent variables and the following conditionalprobability are given.

Pr (Θ1[t]|Θ1[t − 1],Θ3[t − 1],Ω1[t − 1])

= N (Θ1[t]; Θ1[t − 1] + ∆t (Ω1[t − 1] + nΘ3[t − 1]) ,Σ)

3. Using nominal operation data obtained before-hand, correct the graphical structure and theparameters of the DBN with statistical learningdescribed in Sub-section 2.3.

3.3. Learning Diagnosis Model

The transition of the log-likelihood in learning pro-cess in the case of modelling according to the proce-dure described in Sub-section 3.2 is shown in Figure5 (Model 2 in the figure), as well as the case of model-ing with no prior-knowledge (Model 4) and the caseof modeling with the graphical structure and ran-dom parameters. Also the Model1 in the figure isthe case that no-learning is done (Table 2). From

Figure 5. Transition of Log-Likelihood in Learning

Figure 5, we can make certain that the error in themodel and the uncertainty of the observation and soon are corrected to adapt to the actual behavior bylearning. And from the comparison of Model 2 andModel 3,4, because Model 3 and 4 fall into the localsolutions, we can make certain that prior-knowledgeis important to acquiring the better model.

3.4. Diagnosis Results and Discussion

In this sub-section, the results when applying ourproposed diagnosis method to the 3 fault cases us-ing the DBN obtained with the procedure describedabove (Model 2) are shown and the perspective aboutour method are given.

3.4.1. Simple Case

Figure 6. Thrust De-crease of Thruster 4 inthe Fault Case 1

In the first faultcase, the thrust ofThruster 4 decreaseslinearly for 60 sec-onds from 250 sec-onds. In this case,the result of theModel1 in Table 2,that is the modelnot learned, is alsoshown for compari-son.

First, Figure 7 shows the transition of each thrusters’abnormal probability, that is the probability thateach Fi in Figure 4 becomes 1. In these figures, thered lines show the case of the learned model and theblue ones the not learned model. The figure showsthat our method using the learned model suceededin detecting rapidly the anomaly of spacecraft as thevalues of abnormal probabilities of Thruster 4, 5, 7

Figure 7. The transition of each thrusters’ anomalyprobabilities (Fault Case 1)

and 9 change into 1 just after 250 seconds. How-ever the result also shows fail to specify Thruster 4in which the fault actually occurred, this is probablybecause of the limit of the states that the observationacquired can distinguish, that is because the observ-ability of the system is not complete. On the otherhand, the model with no-learning not succeeded indetecting appropriately as some of the fault proba-bility go up when the actual fault has not happened.

And Figure 8 and 9 show the transition of the thrustsestimated from the observation, that is the mean val-ues of each Ti in Figure 4 at every time. From Fig-ure 8, the situation of the abnormality of Thruster4 shown in Figure 6 is roughly reproduced when us-ing the learned model. However in the case of usingthe not learned model (Figure 9), the transitions arerandom and it is difficult to read some tendencies.

This result shows that the learning process functioswell in our proposed method. However at this time,the good results relevant to the structure learningare not obtained.

Figure 8. Transition of estimated thrusts (mean val-ues) of each thrusters (fault case1, Model2)

Figure 9. Transition of estimated thrusts (mean val-ues) of each thrusters (fault case1, Model1)

3.4.2. Complex Case

In the actual fault of a thruster, more complicatedchanges occur such as the gradual decrease of thrustor the repeat of temporal malfunction and recovery.So the results when applying to the fault cases inwhich the ratio (thrust / normal thrust) of Thruster9 repeats between 0 and 1 each 60 seconds from 250seconds as Figure 10 (fault case 2) and the thrust ofthruster 9 changes on the signature wave at 60 cyclesof the second from 250 seconds as Figure 11 (faultcase 3) are shown. For each fault cases, Figure 12 and14 show the transitions of the abnormal probabilitesand Figure 13 and 15 the transitions of the estimatedthrusts of each thrusters. About Figure 12 and 14,the graphs whose values keep around 0 almost allregion are removed. From Figure 12 and 14, ourproposed method, though failed to specify the actualfault thruster as well as the fault case 1, suceeded inrapidly detecting the changes of spacecraft in spiteof the complicated fault. However for Figure 13 and15, though the value of Thruster 9 which actually has

Figure 10. Thrust De-crease of Thruster 9 inthe Fault Case 2

Figure 11. Thrust De-crease of Thruster 9 inthe Fault Case 3

Figure 12. The transi-tion of each thrusters’anomaly probabilities(Fault Case 2)

Figure 13. Transition of estimated thrusts (mean val-ues) of each thrusters (fault case 2)

fault, those are considerably different from Figure 10and 11. Like this it is difficult to follow the largechanges of value from the limit of the observabilityand the accuracy of the model.

4. CONCLUSION

In this paper, we proposed the diagnosis method forspacecraft using DBNs which fuses and uses knowl-edge and data in the natural way. It has been shownthat this method can detect anomaly even in com-plicated fault cases and make a short list of the faultpositions. However we found the problems that itis difficult to specify the fault position completelyor reproduce the situation of faults because of in-

Figure 14. The transi-tion of each thrusters’anomaly probabilities(Fault Case 3)

Figure 15. Transition of estimated thrusts (mean val-ues) of each thrusters (fault case1 3)

completeness of the system’s observability and theinacuracy of the DBNs’ representation power. In thefuture, we are going to examine how to use the vari-ous diagnosis knowledge including experts’ empiricalrules and the representaion of the diagnosis model bywhom the application to the more practical situationis possible.

ACKNOWLEDGMENTS

We would like to thank Mr. Koji Yamanaka and Mr.Noriyasu Inaba for providing us with the simulationdata and a lot of insightful comments.

REFERENCES

[Tallo 92] Tallo, D. P., Durkin, J., and Petrik, E.J.: Intelligent Fault Isolation and Diagnosis forCommunication Satellite Systems, Telematics andInformatics, Vol. 9, No. 3-4, pp. 173-190 (1992)

[Ciceri 94] Ciceri, F. and Marradi, L.: Event Diag-nosis and Recovery in Real-time On-Board Au-tonomous Mission Control, in Proceedings of the1st Ada-Europe Symposium (1994)

[Nishigori 01] Nishigori, N., Hashimoto, M., Choki,A., and Mizutani, M.: Fully Automatic and

Operator-less Anomaly Detecting Ground Sup-port System for Mars Probe ’NOZOMI’, in Pro-ceeding of the 6th International Symposium on Ar-tificial Intelligence and Robotics and Automationin Space (I-SAIRAS) (2001)

[Monsterman 99] Monsterman, P. J. and Biswas, G.:Diagnosis of Continuous Valued Systems in Tran-sient Operating Regions, IEEE Transactions onSystems, Man and Cybernetics - Part A: Systemsand Humans, Vol. 29, No. 6, pp.554-565 (1999)

[Struss 03] Struss, P. and Price, C.: Model-BasedSystems in the Automotive Industry, AI Magazine(2003)

[Williams 96] Williams, B. C. and Nayak, P. P.:A Model-Based Approach to Reactive Self-Configuring Systems, in Proceedings of the Thir-teenth National Conference on Artificial Intelli-gence and Eighth Innovative Applications of Ar-tificial Intelligence Conference (1996)

[Muscettola 98] Muscettola, N., Nayak, P. P., Pell,B., and Williams, B. C.: Remote Agent : ToBoldly Go Where No AI System Has Gone Before,Artificial Intelligence, Vol.103, No.1-2 (1998)

[Deaden 04] Deaden, R., Willeke, T., and Hutter, F.:Real-time Fault Detection and Situational Aware-ness for Rovers: Report on the Mars TechnologyProgram Task, in Proceedings of IEEE AerospaceConference (2004)

[Markou 03] Markou, M. and Singh, S.: Novelty de-tection: a review - part 1,2, Signal Processing, Vol.83, No. 12, pp. 2481-2497 (2003)

[Venkatasubramanian 03] Venkatasubramanian, V.,Rengaswamy, R., Kavuri, S. N., and Yin, K.: A re-view of process fault detection and diagnosis PartIII: Process history based methods, Computersand Chemical Engineering, Vol. 27, pp. 327-346(2003)

[Murphy 02] Murphy, K. P.: Dynamic Bayesian Net-works: Representation, Inference and Learning,PhD thesis, University of California (2002)

[Pearl 88] Pearl, J.: Probabilistic Reasoning in In-telligent Systems, Morgan Kaufmann (1988)

[Jensen 01] Jensen, F. V.: Introduction to BayesianNetworks, University College London Press (1996)

[Cowell 99] Cowell, R. G., Dawid, A., Lauritzen, S.L., and Spiegelhalter, D. J.: Probabilistic Net-works and Expert Systems, Springe-Verlag (1999)

[Pearl 00] Pearl, J.: Causality: Models, Reasoningand Inference, Cambridge Unviersity Press (2000)

[Haddawy 99] Haddawy, P.: An Overview of SomeRecent Developments in Bayesian Problem Solv-ing Techniques, AI Magazine, Vol. 20, No. 2,pp.11-20 (1999)

[Jensen 01] Jensen, F. V., Kjaerulff, U., and Kris-tiansen, B.: The SACSO Methodology for Trou-bleshooting Complex Systems, Artificial Intelli-gence for Engineering Design, Analysis and Man-ufacturing (AIEDAM), Vol. 15, pp. 321-333 (2001)

[Horvitz 98] Horvitz, E., Breese, J., Heckerman, D.,Hovel, D., and Rommelse, D.: The LumiereProject: Bayesian User Modeling for Inferring theGoals and Needs of Software Users (1998)

[Segal 03] Segal, E., Shapira, M., Regev, A., Pe’er,D., Botstein, D., Koller, D., and Friedman, N.:Module Networks: Identifying Regulatory Mod-ules and their Condition-Specific Regulators fromGene Expression Data, Nature Genetics, Vol. 34,No. 2 pp. 166-176 (2003)

[Nefian 02] Nefian, A. V., Liang, L., Pi, X., Liu, X.,and Murphy, K.: Dynamic Bayesian Networks forAudio-Visual Speech Recognition, Journal of Ap-plied Signal Processing, Vol. 11, pp. 1-15 (2002)

[Bilmes 98] Bilmes, J. A.: An Gentle Tutorial ofthe EM Algorithm and its Application to Param-eter Estimation for Gaussian Mixture and HiddenMarkov Models, Technical report, InternationalComputer Science Institute (1998)

[Murphy 98] Murphy, K. P.: Fitting a conditionalgaussian distribution, Technical report, U. C.Berkeley, Dept. Comp. Sci (1998)

[Geiger 94] Geiger, D. and Heckerman, D.: Learn-ing Gaussian Networks, in Proceedings of the 16thAnnual Conference on Uncertainty in AI (UAI)(1994)

[Friedman 97] Friedman, N.: Learning the Structureof Dynamic Probabilistic Networks, in 14th Inter-nation conference on Machine Learning (ICML)(1997)

[Lauritzen 99] Lauritzen, S. L. and Jensen, F.: Sta-ble Local Computation with Conditional GaussianDistributions, Technical report, Aalborg Univer-sity (1999)

[Doucet 00] Doucet, A., Freitas, de N., Murphy, K.,and Russell, S.: Rao-Blackwellised Particle Filter-ing for Dynamic Bayesian Networks, in Proceed-ings of the 16th Annual Conference on Uncertaintyin AI (UAI) (2000)

[Doucet 01] Doucet, A., Freitas, de N., and Gor-don, N.: Sequential Monte Carlo Methods in Prac-tice, Statistics for Engineering and InformationScience, Springer Verlag (2001)