KATHOLIEKE UNIVERSITEIT LEUVEN Promotoren:Prof.dr.ir. Bart De Moor, promotor Prof.dr.ir. Jan Willems, copromotor Juryleden: Prof.dr.ir. H. Van Brussel, voorzitter Prof.dr. A. Bultheel Prof.dr. V. Blondel (UCL, Louvain-la-Neuve) Prof.dr. P. Spreij (UVA, Amsterdam) Prof.dr.ir. L. Finesso (ISIB-CNR, Padova) Prof.dr.ir. K. Meerbergen Ph.D. defence Maandag 5 mei 2008 Realization, identification and filtering for hidden Markov models using matrix factorization techniques Bart Vanluyten

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 2 / 43 04/’06 06/’06 08/’06 10/’06 12/’06 02/’07 04/’07 06/’07 08/’07 10/’07 12/’07 02/’08 04/’08 Mathematical modeling Bel-20 Process with finite valued output: { , , = } 1. INTRODUCTION Modeling  —  HMMs  —  Finite valued process  —  Open problems  —  Relation to LSM

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 3 / 43 Hidden Markov model  Example: Bel-20 Output process {up, down, unchanged} State process{bull market, bear market, stable market}  State process has Markov property and is hidden Andrey Markov ( ) Bull Market Stable Market Bear Market 30% BEL20  30% BEL20  40% BEL20 = 50% 20% 60%30% 20% 10% 40% 20% 50% 70% BEL20  10% BEL20  20% BEL20 = 10% BEL20  60% BEL20  30% BEL20 = 1. INTRODUCTION Modeling  —  HMMs  —  Finite valued process  —  Open problems  —  Relation to LSM

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 4 / 43 BEL Finite-valued processes FINITE-VALUED PROCESSES { , , = } { A, C, G, T }{ 1, 2,..., 6 } { head, tail } { i:, e, æ, a:, ai,..., z } BEL20 Bio-informatics Speech recognitionEconomics Coin flipping - dice-tossing (with memory) TGGAGCCAACGTGGAATG TCACTAGCTAGCTTAGAT GGCTAAACGTAGGAATAC ACGTGGAATATCGAATCG TTAGCTTAGCGCCTCGAC CTAGATCGAGCCGATCGG ACTAGCTAGCTCGCTAGA AGCACCTAGAAGCTTAGA CGTGGAAATTGCTTAATC 1. INTRODUCTION Modeling  —  HMMs  —  Finite valued process  —  Open problems  —  Relation to LSM

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 5 / 43 Open problems for HMMs Obtain model from data Use model for estimation Estimation problem Given: output sequence Find: state distribution at time Identification problem Given: output sequence Find: HMM that models the sequence Realization problem Given: string prob’s Find: HMM generating string prob’s 1. INTRODUCTION Modeling  —  HMMs  —  Finite valued process  —  Open problems  —  Relation to LSM

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 6 / 43 Relation to linear stochastic model (LSM)  Mathematical model for stochastic processes Output process continuous range of values State processcontinuous range of values + NOISE STATEOUTPUT + 1. INTRODUCTION Modeling  —  HMMs  —  Finite valued process  —  Open problems  —  Relation to LSM

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 7 / 43 Relation to linear stochastic model RealizationIdentification Estimation Hidden Markov model RealizationIdentification Estimation Linear stochastic model 1. INTRODUCTION Modeling  —  HMMs  —  Finite valued process  —  Open problems  —  Relation to LSM

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 8 / 43 Relation to linear stochastic model RealizationIdentification Estimation Hidden Markov model RealizationIdentification Estimation Linear stochastic model Singular value decomposition 1. INTRODUCTION Modeling  —  HMMs  —  Finite valued process  —  Open problems  —  Relation to LSM

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 9 / 43 Relation to linear stochastic model RealizationIdentification Estimation Hidden Markov model RealizationIdentification Estimation Linear stochastic model Singular value decomposition Nonnegative matrix factorization 1. INTRODUCTION Modeling  —  HMMs  —  Finite valued process  —  Open problems  —  Relation to LSM

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 10 / 43 Outline Estimation problem Given: output sequence Find: state distribution at time Identification problem Given: output sequence Find: HMM that models the sequence Realization problem Given: string prob’s Find: HMM generating string prob’s Matrix factorizations Given: matrix Find: low rank approximation of 2 nd objective 1 st objective

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 11 / 43 Outline Estimation problem Given: output sequence Find: state distribution at time Identification problem Given: output sequence Find: HMM that models the sequence Realization problem Given: string prob’s Find: HMM generating string prob’s Matrix factorizations Given: matrix Find: low rank approximation of

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 12 / 43 Matrix – Decomposition – Rank : example  Matrix  Matrix decomposition  Matrix rank minimal inner dimension of exact decomposition 2. MATRIX FACTORIZATIONS Introduction  —  Existing factorizations  —  Structured NMF  —  NMF without nonneg. factors

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 13 / 43 Low rank matrix approximation  Rank approximation of James Sylvester ( ) orthogonal  SVD yields (global) optimal low rank approximation in Frobenius distance  Singular value decomposition (SVD) 2. MATRIX FACTORIZATIONS Introduction  —  Existing factorizations  —  Structured NMF  —  NMF without nonneg. factors

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 14 / 43 Nonnegative matrix factorization  In some applications is nonnegative and and need to be nonnegative too  Nonnegative matrix factorization(NMF) of  Algorithm (Kullback-Leibler divergence) [Lee, Seung]  This thesis: 2 modifications to NMF NONNEGATIVE 2. MATRIX FACTORIZATIONS Introduction  —  Existing factorizations  —  Structured NMF  —  NMF without nonneg. factors

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 15 / 43 Structured NMF  Structured nonnegative matrix factorization of  Algorithm (Kullback-Leibler divergence)  Convergence to stationary point of divergence NONNEGATIVE 2. MATRIX FACTORIZATIONS Introduction  —  Existing factorizations  —  Structured NMF  —  NMF without nonneg. factors

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 16 / 43 Structured NMF: application  Applications apart from HMMs: clustering data points SetosaVersicolorVirginica – petal width – petal length – sepal width – sepal length Asked: Divide 150 flowers into clusters Given: of 150 iris flowers PETAL SEPAL 2. MATRIX FACTORIZATIONS Introduction  —  Existing factorizations  —  Structured NMF  —  NMF without nonneg. factors

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 17 / 43 Structured NMF: application  Clustering obtained by: Computing distance matrix between points Applying structured nonnegative matrix factorization on distance matrix cluster 1 cluster 2 cluster 3 PETAL WIDTH SEPAL WIDTH SEPAL LENGTH PETAL LENGTH SEPAL LENGTH SEPAL WIDTH PETAL LENGTH PETAL WIDTH 2. MATRIX FACTORIZATIONS Introduction  —  Existing factorizations  —  Structured NMF  —  NMF without nonneg. factors

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 18 / 43 NMF without nonnegativity of the factors  NMF without nonnegativity constraints on the factors of  We provide algorithm (Kullback-Leibler divergence)  Problem allows to deal with upper bounds in an easy way NONNEGATIVENO NONNEGATIVITY CONSTRAINTS NONNEGATIVE  Example MATRIX FACTORIZATIONS Introduction  —  Existing factorizations  —  Structured NMF  —  NMF without nonneg. factors

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 19 / 43 NMF without nonnegativity of the factors ORIGINAL NMF without nonneg. factors Upperbounded NMF without nonneg. fact. NMF  Applications apart from HMMs: database compression Given: Database containing 1000 facial images of size 19 x 19 = 361 pixels Asked: Compression of database using matrix factorization techniques Kullback-Leibler divergence: > 1 2. MATRIX FACTORIZATIONS Introduction  —  Existing factorizations  —  Structured NMF  —  NMF without nonneg. factors

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 20 / 43 Outline Estimation problem Given: output sequence Find: state distribution at time Identification problem Given: output sequence Find: HMM that models the sequence Realization problem Given: string prob’s Find: HMM generating string prob’s Matrix factorizations Given: matrix Find: low rank approximation of

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 21 / 43 Hidden Markov models: Moore - Mealy  Moore HMM  Mealy HMM = NONNEGATIVE ORDER 3. REALIZATION Introduction  —  Realization  —  Quasi realization  —  Approx. realization  —  Modeling DNA

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 22 / 43 Realization problem  String from  String probabilities  String probabilities generated by Mealy HMM POSITIVE REALIZATION NONNEGATIVE such that 3. REALIZATION Introduction  —  Realization  —  Quasi realization  —  Approx. realization  —  Modeling DNA

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 23 / 43 Realization problem: importance  Theoretical importance: transform ‘external’ model into ‘internal’ model  Realization can be used to identify model from data POSITIVE REALIZATION NONNEGATIVE 3. REALIZATION Introduction  —  Realization  —  Quasi realization  —  Approx. realization  —  Modeling DNA

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 24 / 43 Realizability problem  Generalized Hankel matrix  Necessary condition for realizability: Hankel matrix has finite rank  No necessary and sufficient conditions for realizability are known  No procedure to compute minimal HMM from string probabilities  This thesis: two relaxations to positive realization problem Quasi realization problem Approximate positive realization problem Hermann Hankel ( ) 3. REALIZATION Introduction  —  Realization  —  Quasi realization  —  Approx. realization  —  Modeling DNA

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 25 / 43 Quasi realization problem QUASI REALIZATION such that NO NONNEGATIVITY CONSTRAINTS !  Finiteness of rank of Hankel matrix = N & S condition for quasi realizability  Rank of hankel matrix = minimal order of exact quasi realization  Quasi realization is more easy to compute than positive realization  Quasi realization typically has lower order than positive realization  Negative probabilities No disadvantage in several estimation applications 3. REALIZATION Introduction  —  Realization  —  Quasi realization  —  Approx. realization  —  Modeling DNA

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 26 / 43 Partial quasi realization problem  Given: String probabilities of strings up to length t  Asked: Quasi HMM that generates the string probabilities  This thesis: Partial quasi realization problem has always a solution Minimal partial quasi realization obtained with quasi realization algorithm if a rank condition on the Hankel matrix holds Minimal partial quasi realization problem has unique solution (up to similarity transform) if this rank condition holds 3. REALIZATION Introduction  —  Realization  —  Quasi realization  —  Approx. realization  —  Modeling DNA

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 27 / 43 Approximate quasi realization problem  Given: String probabilities of strings up to length t  Asked: Quasi HMM that approximately generates the string probabilities  This thesis: algorithm Compute low rank approximation of largest Hankel block subject to consistency and stationarity constraints Reconstruct Hankel matrix from largest block We prove that rank does not increase in this step Apply partial quasi realization algorithm Upperbounded NMF without nonnegativity of the factors with additional constraints 3. REALIZATION Introduction  —  Realization  —  Quasi realization  —  Approx. realization  —  Modeling DNA

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 28 / 43 Approximate positive realization problem APPROXIMATE POSITIVE REALIZATION NONNEGATIVE such that  Given: String probabilities of strings up to length t  Asked: Positive HMM that approximately generates the string probabilities 3. REALIZATION Introduction  —  Realization  —  Quasi realization  —  Approx. realization  —  Modeling DNA

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 29 / 43 Approximate positive realization problem  Moore, t = 2 Define If string probabilities are generated by Moore HMM Structured nonnegative matrix factorization  Mealy, general t Generalize approach for Moore, t = 2 where 3. REALIZATION Introduction  —  Realization  —  Quasi realization  —  Approx. realization  —  Modeling DNA

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 30 / 43 Modeling DNA sequences  DNA  40 sequences of length 200  String probabilities of strings up to length 4 stacked in Hankel matrix ORDER Quasi realization Positive realization  Kullback-Leibler divergence        SINGULAR VALUE TGGAGCCAACGTGGAATGTCACTAGCTAGCTTAGATGGCTAAACGTAGGAATACCCT ACGTGGAATATCGAATCGTTAGCTTAGCGCCTCGACCTAGATCGAGCCGATCGGTCT ACTAGCTAGCTCGCTAGAAGCACCTAGAAGCTTAGACGTGGAAATTGCTTAATCTAG   3. REALIZATION Introduction  —  Realization  —  Quasi realization  —  Approx. realization  —  Modeling DNA

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 31 / 43 Outline Estimation problem Given: output sequence Find: state distribution at time Identification problem Given: output sequence Find: HMM that models the sequence Realization problem Given: string prob’s Find: HMM generating string prob’s Matrix factorizations Given: matrix Find: low rank approximation of

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 32 / 43 Identification problem  Given: Output sequence of length T  Asked: (Quasi) HMM that models the sequence NONNEGATIVE NO NONNEGATIVITY CONSTRAINTS!  Approach Baum-Welch identification Linear Stochastic Models Hidden Markov Models Subspace based identification Subspace inspired identification Prediction error identification SVD NMF 4. IDENTIFICATION Introduction  —  Subspace inspired identification  —  HIV modeling

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 33 / 43 Identification problem output sequence system matrices state sequence Baum-WelchSubspace inspired 4. IDENTIFICATION Introduction  —  Subspace inspired identification  —  HIV modeling

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 34 / 43 Subspace inspired identification  Estimate the (quasi) state distribution quasi state predictor can be built from data using upperbounded NMF without nonnegativity of the factors state predictor can be built from data using NMF  Compute the system matrices: least squares problem Quasi HMM: Positive HMM: IDENTIFICATION Introduction  —  Subspace inspired identification  —  HIV modeling We have shown that:

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 35 / 43 Modeling sequences from HIV genome  Mutation  25 mutated sequences of length 222 from the part of the HIV1 genome that codes for the envelope protein [NCBI database] Training set Test set  HMM model using Baum-Welch – Subspace inspired identification A  HIV virus ENVELOPE MATRIX CORE 4. IDENTIFICATION Introduction  —  Subspace inspired identification  —  HIV modeling

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 36 / 43 Modeling sequences from HIV genome  Kullback-Leibler divergence (string probabilities of length-4 strings)  Mean likelihood of the given sequences  Likelihood of using third order subspace inspired model  Model can be used to predict new viral strains and to distinguish between different HIV subtypes ORDER Baum-Welch Subspace ORDER Baum-Welch Subspace TEST-SEQUENCE Likelihood IDENTIFICATION Introduction  —  Subspace inspired identification  —  HIV modeling

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 37 / 43 Outline Estimation problem Given: output sequence Find: state distribution at time Identification problem Given: output sequence Find: HMM that models the sequence Realization problem Given: string prob’s Find: HMM generating string prob’s Matrix factorizations Given: matrix Find: low rank approximation of

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 38 / 43 Estimation for HMMs  State estimation – output estimation  We derive recursive formulas to solve state and output filtering, prediction and smoothing problems HMM 5. ESTIMATION Estimation for HMMs  —  Application  Filtering – smoothing – prediction TIME t t t FILTERING: SMOOTHING: PREDICTION: = span of available measurements

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 39 / 43 Estimation for HMMs  Example: Recursive algorithm to compute  Recursive output estimation algorithms effective with quasi HMM  Finiteness of rank of Hankel matrix = N & S condition for quasi realizability  Rank of hankel matrix = minimal order of exact quasi realization  Quasi realization is easier to compute than positive realization  Quasi realization typically has lower order than positive realization  Negative probabilities No disadvantage in output estimation problems 5. ESTIMATION Estimation for HMMs  —  Application

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 40 / 43 Finding motifs in DNA sequences  Find motifs in muscle specific regulatory regions [Zhou, Wong] Make motif model Make quasi background model (see Section realization) Build joint HMM Perform output estimation  Results (compared to results from Motifscanner [Aerts et al.]) POSITION MOTIF PROBABILITY Mef-2 Myf Sp-1 SRF TEF 5. ESTIMATION Estimation for HMMs  —  Application

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 41 / 43 Conclusions  Two modification to the nonnegative matrix factorization Structured nonnegative matrix factorization Nonnegative matrix factorization without nonnegativity of the factors  Two relaxations to the positive realization problem for HMMs Quasi realization problem Approximate positive realization problem  Both methods were applied to modeling DNA sequences  We derive equivalence conditions for HMMs  We propose a new identification method for HMMs  Method was applied to modeling DNA sequences of HIV virus  Quasi realizations suffice for several estimation problems  Quasi estimation methods were applied to finding motifs in DNA sequences 6. CONCLUSIONS Conclusions  —  Further research  —  List of publications

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 42 / 43 Further research Matrix factorizations  Develop nonnegative matrix factorization with nesting property (cfr. SVD) Hidden Markov models  Investigate Markov models (special case of hidden Markov case)  Develop realization and identification methods that allow to incorporate prior-knowledge in the Markov chain  Method to estimate minimal order of positive HMM from string probabilities  Canonical forms of hidden Markov models  Model reduction for hidden Markov models  System theory for hidden Markov models with external inputs CONCLUSIONS Conclusions  —  Further research  —  List of publications

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 43 / 43 List of publications  Journal papers B. Vanluyten, J.C. Willems and B. De Moor. Recursive Filtering using Quasi-Realizations. Lecture Notes in Control and Information Sciences, 341, 367–374, B. Vanluyten, J.C. Willems and B. De Moor. Equivalence of State Representations for Hidden Markov Models. Systems and Control Letters, 57(5), 410–419, B. Vanluyten, J.C. Willems and B. De Moor. Structured Nonnegative Matrix Factorization with Applications to Hidden Markov Realization and Filtering. Accepted for publication in Linear Algebra and its Applications, B. Vanluyten, J.C. Willems and B. De Moor. Nonnegative Matrix Factorization without Nonnegativity Constraints on the Factors. Submitted for publication. B. Vanluyten, J.C. Willems and B. De Moor. Approximate Realization and Estimation for Quasi hidden Markov models. Submitted for publication.  International conference papers I. Goethals, B. Vanluyten, B. De Moor. Reliable spurious mode rejection using self learning algorithms. In Proc. of the International Conference on Modal Analysis Noise and Vibration Engineering (ISMA 2004), Leuven, Belgium, pages 991–1003, B. Vanluyten, J. C.Willems and B. De Moor. Model Reduction of Systems with Symmetries. In Proc. of the 44th IEEE Conference on Decision and Control (CDC 2005), Seville, Spain, pages 826–831, B. Vanluyten, J. C. Willems and B. De Moor. Matrix Factorization and Stochastic State Representations. In Proc. of the 45th IEEE Conference on Decision and Control (CDC 2006), San Diego, California, pages , I. Markovsky, J. Boets, B. Vanluyten, K. De Cock, B. De Moor. When is a pole spurious? In Proc. of the International Conference on Noise and Vibration Engineering (ISMA 2007), Leuven, Belgium, pp. 1615–1626, B. Vanluyten, J. C. Willems and B. De Moor. Equivalence of State Representations for Hidden Markov Models. In Proc. of the European Control Conference 2007 (ECC 2007), Kos, Greece, B. Vanluyten, J. C. Willems and B. De Moor. A new Approach for the Identification of Hidden Markov Models. In Proc. of the 46th IEEE Conference on Decision and Control (CDC 2006), New Orleans, Louisiana, CONCLUSIONS Conclusions  —  Further research  —  List of publications

1. INTRODUCTION2. MATRIX FACTORIZATIONS3. REALIZATION4. IDENTIFICATION5. ESTIMATION6. CONCLUSIONS SLIDE 44 / 43