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Bart Vanluyten

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Realization, identification and filtering for hidden Markov models using matrix factorization techniques

Bart Vanluyten

04/â€™06 06/â€™06 08/â€™06 10/â€™06 12/â€™06 02/â€™07 04/â€™07 06/â€™07 08/â€™07 10/â€™07 12/â€™07 02/â€™08 04/â€™08

Bel-20

Process with finite valued output: { Ã‡,Ãˆ,= }

1. INTRODUCTION

Modelingâ€” HMMsâ€”Finite valued processâ€”Open problemsâ€”Relation to LSM

60%

30%

50%

Bull Market

Bear Market

20%

10% BEL20 Ã‡60% BEL20 Ãˆ30% BEL20 =

70% BEL20 Ã‡10% BEL20 Ãˆ20% BEL20 =

40%

20%

10%

20%

StableMarket

30% BEL20 Ã‡30% BEL20 Ãˆ40% BEL20 =

50%

- Example: Bel-20
- Output process {up, down, unchanged}
- State process{bull market, bear market, stable market}

Andrey Markov (1856 - 1922)

- State process has Markov property and is hidden

1. INTRODUCTION

Modelingâ€” HMMsâ€”Finite valued processâ€”Open problemsâ€”Relation to LSM

BEL20

4-06 8-06 12-06 4-07 8-07 12-07 4-08

Bio-informatics

Coin flipping - dice-tossing (with memory)

TGGAGCCAACGTGGAATGTCACTAGCTAGCTTAGATGGCTAAACGTAGGAATAC

ACGTGGAATATCGAATCGTTAGCTTAGCGCCTCGACCTAGATCGAGCCGATCGGACTAGCTAGCTCGCTAGAAGCACCTAGAAGCTTAGACGTGGAAATTGCTTAATC

{ head, tail}

{ A, C, G, T}

{ 1, 2, ..., 6}

FINITE-VALUEDPROCESSES

Economics

Speech recognition

BEL20

4.800

4.600

4.400

4.200

4.000

3.800

3.600

{ i:, e, Ã¦, a:, ai, ..., z}

{ Ã‡,Ãˆ,= }

1. INTRODUCTION

Modelingâ€” HMMsâ€”Finite valued processâ€”Open problemsâ€”Relation to LSM

Realization problem

Given: string probâ€™s

Find: HMM generating string probâ€™s

Identification problem

Given: output sequence

Find: HMM that models the sequence

Obtain model from data

Estimation problem

Given: output sequence

Find: state distribution at time

Use model for estimation

1. INTRODUCTION

Modelingâ€” HMMsâ€”Finite valued processâ€”Open problemsâ€”Relation to LSM

- Mathematical model for stochastic processes
- Output process continuous range of values
- State processcontinuous range of values

NOISE

NOISE

STATE

OUTPUT

+

+

1. INTRODUCTION

Modelingâ€” HMMsâ€”Finite valued processâ€”Open problemsâ€”Relation to LSM

Hidden Markov model

Linear stochastic model

Realization

Identification

Realization

Identification

Estimation

Estimation

1. INTRODUCTION

Modelingâ€” HMMsâ€”Finite valued processâ€”Open problemsâ€”Relation to LSM

Hidden Markov model

Linear stochastic model

Singular value decomposition

Realization

Identification

Realization

Identification

Estimation

Estimation

1. INTRODUCTION

Modelingâ€” HMMsâ€”Finite valued processâ€”Open problemsâ€”Relation to LSM

Hidden Markov model

Linear stochastic model

Nonnegative matrix factorization

Singular value decomposition

Realization

Identification

Realization

Identification

Estimation

Estimation

1. INTRODUCTION

Modelingâ€” HMMsâ€”Finite valued processâ€”Open problemsâ€”Relation to LSM

Matrix factorizations

Given: matrix

Find: low rank approximation of

2nd objective

Realization problem

Given: string probâ€™s

Find: HMM generating string probâ€™s

Identification problem

Given: output sequence

Find: HMM that models the sequence

1st objective

Estimation problem

Given: output sequence

Find: state distribution at time

Matrix factorizations

Given: matrix

Find: low rank approximation of

Realization problem

Given: string probâ€™s

Find: HMM generating string probâ€™s

Identification problem

Given: output sequence

Find: HMM that models the sequence

Estimation problem

Given: output sequence

Find: state distribution at time

- Matrix

- Matrix decomposition

- Matrix rank
minimal inner dimension of exact decomposition

2. MATRIX FACTORIZATIONS

Introductionâ€” Existing factorizationsâ€”Structured NMFâ€”NMF without nonneg. factors

- Rank approximation of

James Sylvester

(1814 - 1897)

- Singular value decomposition (SVD)

orthogonal

- SVD yields (global) optimal low rank approximation in Frobenius distance

2. MATRIX FACTORIZATIONS

Introductionâ€” Existing factorizationsâ€”Structured NMFâ€”NMF without nonneg. factors

- In some applications is nonnegative and and need to be nonnegative too

- Nonnegative matrix factorization(NMF) of

NONNEGATIVE

NONNEGATIVE

NONNEGATIVE

- Algorithm (Kullback-Leibler divergence) [Lee, Seung]

- This thesis: 2 modifications to NMF

2. MATRIX FACTORIZATIONS

Introductionâ€” Existing factorizationsâ€”Structured NMFâ€”NMF without nonneg. factors

- Structured nonnegative matrix factorization of

NONNEGATIVE

NONNEGATIVE

NONNEGATIVE

NONNEGATIVE

- Algorithm (Kullback-Leibler divergence)

- Convergence to stationary point of divergence

2. MATRIX FACTORIZATIONS

Introductionâ€” Existing factorizationsâ€”Structured NMFâ€”NMF without nonneg. factors

PETAL

- Applications apart from HMMs: clustering data points

- petal width
- petal length
- sepal width
- sepal length

Given:

of 150 iris flowers

SEPAL

- Asked: Divide 150 flowers into clusters

Setosa

Versicolor

Virginica

2. MATRIX FACTORIZATIONS

Introductionâ€” Existing factorizationsâ€”Structured NMFâ€”NMF without nonneg. factors

cluster 1

cluster 2

cluster 3

- Computing distance matrix between points
- Applying structured nonnegative matrix factorization on distance matrix

- Clustering obtained by:

PETAL LENGTH

SEPAL WIDTH

PETAL WIDTH

PETAL LENGTH

SEPAL LENGTH

PETAL WIDTH

SEPAL WIDTH

SEPAL LENGTH

2. MATRIX FACTORIZATIONS

Introductionâ€” Existing factorizationsâ€”Structured NMFâ€”NMF without nonneg. factors

- NMF without nonnegativity constraints on the factors of

NONNEGATIVE

NO NONNEGATIVITY CONSTRAINTS

NONNEGATIVE

- Example

3

3

- We provide algorithm (Kullback-Leibler divergence)

- Problem allows to deal with upper bounds in an easy way

2. MATRIX FACTORIZATIONS

Introductionâ€” Existing factorizationsâ€”Structured NMFâ€”NMF without nonneg. factors

- 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

20

1000

361

. . .

NMF without

nonneg. factors

Upperbounded NMF

without nonneg. fact.

ORIGINAL

NMF

> 1

Kullback-Leibler divergence:

564

339

383

2. MATRIX FACTORIZATIONS

Introductionâ€” Existing factorizationsâ€”Structured NMFâ€”NMF without nonneg. factors

Matrix factorizations

Given: matrix

Find: low rank approximation of

Realization problem

Given: string probâ€™s

Find: HMM generating string probâ€™s

Identification problem

Given: output sequence

Find: HMM that models the sequence

Estimation problem

Given: output sequence

Find: state distribution at time

ORDER

- Moore HMM

=

NONNEGATIVE

- Mealy HMM

NONNEGATIVE

3. REALIZATION

Introductionâ€” Realizationâ€”Quasi realizationâ€”Approx. realizationâ€”Modeling DNA

NONNEGATIVE

- String from
- String probabilities

- String probabilities generated by Mealy HMM

POSITIVE REALIZATION

such that

3. REALIZATION

Introductionâ€” Realizationâ€”Quasi realizationâ€”Approx. realizationâ€”Modeling DNA

NONNEGATIVE

- Theoretical importance: transform â€˜externalâ€™ model into â€˜internalâ€™ model
- Realization can be used to identify model from data

POSITIVE REALIZATION

3. REALIZATION

Introductionâ€” Realizationâ€”Quasi realizationâ€”Approx. realizationâ€”Modeling DNA

- Generalized Hankel matrix

Hermann Hankel

(1839 - 1873)

- 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

3. REALIZATION

Introductionâ€” Realizationâ€”Quasi realizationâ€”Approx. realizationâ€”Modeling DNA

NO NONNEGATIVITY

CONSTRAINTS !

QUASI REALIZATION

such that

- 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

Introductionâ€” Realizationâ€”Quasi realizationâ€”Approx. realizationâ€”Modeling DNA

- 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

Introductionâ€” Realizationâ€”Quasi realizationâ€”Approx. realizationâ€”Modeling DNA

- 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

Upperbounded NMF without nonnegativity of the factorswith additional constraints

- Reconstruct Hankel matrix from largest block

We prove that rank does not increase in this step

- Apply partial quasi realization algorithm

3. REALIZATION

Introductionâ€” Realizationâ€”Quasi realizationâ€”Approx. realizationâ€”Modeling DNA

NONNEGATIVE

- Given: String probabilities of strings up to length t
- Asked: Positive HMM that approximately generates the string probabilities

APPROXIMATE POSITIVE REALIZATION

such that

3. REALIZATION

Introductionâ€” Realizationâ€”Quasi realizationâ€”Approx. realizationâ€”Modeling DNA

- Moore, t = 2
- Define

- If string probabilities are generated by Moore HMM

where

Structured nonnegative matrix factorization

- Mealy, general t
Generalize approachfor Moore, t = 2

3. REALIZATION

Introductionâ€” Realizationâ€”Quasi realizationâ€”Approx. realizationâ€”Modeling DNA

- DNA

TGGAGCCAACGTGGAATGTCACTAGCTAGCTTAGATGGCTAAACGTAGGAATACCCT

ACGTGGAATATCGAATCGTTAGCTTAGCGCCTCGACCTAGATCGAGCCGATCGGTCT

ACTAGCTAGCTCGCTAGAAGCACCTAGAAGCTTAGACGTGGAAATTGCTTAATCTAG

- 40 sequences of length 200
- String probabilities of strings up to length 4 stacked in Hankel matrix

Ã™

SINGULAR VALUE

Ã™

- Kullback-Leibler divergence

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

Ã™

3. REALIZATION

Introductionâ€” Realizationâ€”Quasi realizationâ€”Approx. realizationâ€”Modeling DNA

Matrix factorizations

Given: matrix

Find: low rank approximation of

Realization problem

Given: string probâ€™s

Find: HMM generating string probâ€™s

Identification problem

Given: output sequence

Find: HMM that models the sequence

Estimation problem

Given: output sequence

Find: state distribution at time

- Given: Output sequence of length T
- Asked: (Quasi) HMM that models the sequence

NO NONNEGATIVITY

CONSTRAINTS!

NONNEGATIVE

- Approach

Linear Stochastic Models

Prediction error identification

Subspace basedidentification

SVD

HiddenMarkovModels

Baum-Welch identification

Subspace inspiredidentification

NMF

4. IDENTIFICATION

Introductionâ€” Subspace inspired identificationâ€”HIV modeling

output sequence

system matrices

state sequence

state sequence

system matrices

Baum-Welch

Subspace inspired

4. IDENTIFICATION

Introductionâ€” Subspace inspired identificationâ€”HIV modeling

- 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

We have shown that:

. . .

. . .

. . .

. . .

. . .

. . .

. . .

- Compute the system matrices: least squares problem

Quasi HMM:

Positive HMM:

4. IDENTIFICATION

Introductionâ€” Subspace inspired identificationâ€”HIV modeling

A

- Mutation

- HIV virus

ENVELOPE

CORE

MATRIX

- 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 â€“ Subspace inspired identification

4. IDENTIFICATION

Introductionâ€” Subspace inspired identificationâ€”HIV modeling

- 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 topredict new viral strains and to distinguish between different HIV subtypes

4. IDENTIFICATION

Introductionâ€” Subspace inspired identificationâ€”HIV modeling

Matrix factorizations

Given: matrix

Find: low rank approximation of

Realization problem

Given: string probâ€™s

Find: HMM generating string probâ€™s

Identification problem

Given: output sequence

Find: HMM that models the sequence

Estimation problem

Given: output sequence

Find: state distribution at time

- Filtering â€“ smoothing â€“ prediction

- State estimation â€“ output estimation

HMM

HMM

= span of available measurements

FILTERING:

t

TIME

SMOOTHING:

t

TIME

PREDICTION:

t

TIME

- We derive recursive formulas to solve state and output filtering, prediction and smoothing problems

5. ESTIMATION

Estimation for HMMsâ€” Application

- 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 HMMsâ€” Application

Mef-2

Myf

Sp-1

SRF

TEF

- Findmotifsin 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.])

MOTIF PROBABILITY

MOTIF PROBABILITY

POSITION

POSITION

5. ESTIMATION

Estimation for HMMsâ€” Application

- 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

Conclusionsâ€” Further researchâ€” List of publications

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

. . .

6. CONCLUSIONS

Conclusionsâ€” Further researchâ€” 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, 367â€“374, 2006.
- B. Vanluyten, J.C. Willems and B. De Moor. Equivalence of State Representations for Hidden Markov Models. Systems and Control Letters, 57(5), 410â€“419, 2008.
- 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, 2008.
- 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 991â€“1003, 2004.
- 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 826â€“831, 2005.
- 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 4188-4193, 2006.
- 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. 1615â€“1626, 2007.
- 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, 2007.
- 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, 2007.

6. CONCLUSIONS

Conclusionsâ€” Further researchâ€” List of publications