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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/’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

Bel-20

Process with finite valued output: { Ç,È,= }

1. INTRODUCTION

Modeling— HMMs—Finite valued process—Open problems—Relation to LSM

60%

30%

50%

Bull Market

Bear Market

20%

10% BEL20 Ç60% BEL20 È30% BEL20 =

70% BEL20 Ç10% BEL20 È20% BEL20 =

40%

20%

10%

20%

StableMarket

30% BEL20 Ç30% BEL20 È40% 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

Modeling— HMMs—Finite valued process—Open problems—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, æ, a:, ai, ..., z}

{ Ç,È,= }

1. INTRODUCTION

Modeling— HMMs—Finite valued process—Open problems—Relation to LSM

Realization problem

Given: string prob’s

Find: HMM generating string prob’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

Modeling— HMMs—Finite valued process—Open problems—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

Modeling— HMMs—Finite valued process—Open problems—Relation to LSM

Hidden Markov model

Linear stochastic model

Realization

Identification

Realization

Identification

Estimation

Estimation

1. INTRODUCTION

Modeling— HMMs—Finite valued process—Open problems—Relation to LSM

Hidden Markov model

Linear stochastic model

Singular value decomposition

Realization

Identification

Realization

Identification

Estimation

Estimation

1. INTRODUCTION

Modeling— HMMs—Finite valued process—Open problems—Relation to LSM

Hidden Markov model

Linear stochastic model

Nonnegative matrix factorization

Singular value decomposition

Realization

Identification

Realization

Identification

Estimation

Estimation

1. INTRODUCTION

Modeling— HMMs—Finite valued process—Open problems—Relation to LSM

Matrix factorizations

Given: matrix

Find: low rank approximation of

2nd objective

Realization problem

Given: string prob’s

Find: HMM generating string prob’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 prob’s

Find: HMM generating string prob’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

Introduction— Existing factorizations—Structured NMF—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

Introduction— Existing factorizations—Structured NMF—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

Introduction— Existing factorizations—Structured NMF—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

Introduction— Existing factorizations—Structured NMF—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

Introduction— Existing factorizations—Structured NMF—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

Introduction— Existing factorizations—Structured NMF—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

Introduction— Existing factorizations—Structured NMF—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

Introduction— Existing factorizations—Structured NMF—NMF without nonneg. factors

Matrix factorizations

Given: matrix

Find: low rank approximation of

Realization problem

Given: string prob’s

Find: HMM generating string prob’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

Introduction— Realization—Quasi realization—Approx. realization—Modeling DNA

NONNEGATIVE

- String from
- String probabilities

- String probabilities generated by Mealy HMM

POSITIVE REALIZATION

such that

3. REALIZATION

Introduction— Realization—Quasi realization—Approx. realization—Modeling DNA

NONNEGATIVE

- Theoretical importance: transform ‘external’ model into ‘internal’ model
- Realization can be used to identify model from data

POSITIVE REALIZATION

3. REALIZATION

Introduction— Realization—Quasi realization—Approx. realization—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

Introduction— Realization—Quasi realization—Approx. realization—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

Introduction— Realization—Quasi realization—Approx. realization—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

Introduction— Realization—Quasi realization—Approx. realization—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

Introduction— Realization—Quasi realization—Approx. realization—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

Introduction— Realization—Quasi realization—Approx. realization—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

Introduction— Realization—Quasi realization—Approx. realization—Modeling DNA

- DNA

TGGAGCCAACGTGGAATGTCACTAGCTAGCTTAGATGGCTAAACGTAGGAATACCCT

ACGTGGAATATCGAATCGTTAGCTTAGCGCCTCGACCTAGATCGAGCCGATCGGTCT

ACTAGCTAGCTCGCTAGAAGCACCTAGAAGCTTAGACGTGGAAATTGCTTAATCTAG

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

Ù

SINGULAR VALUE

Ù

- Kullback-Leibler divergence

Ù

Ù

Ù

Ù

Ù

Ù

Ù

Ù

Ù

Ù

Ù

Ù

Ù

Ù

Ù

Ù

Ù

Ù

Ù

3. REALIZATION

Introduction— Realization—Quasi realization—Approx. realization—Modeling DNA

Matrix factorizations

Given: matrix

Find: low rank approximation of

Realization problem

Given: string prob’s

Find: HMM generating string prob’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

Introduction— Subspace inspired identification—HIV modeling

output sequence

system matrices

state sequence

state sequence

system matrices

Baum-Welch

Subspace inspired

4. IDENTIFICATION

Introduction— Subspace inspired identification—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

Introduction— Subspace inspired identification—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 – Subspace inspired identification

4. IDENTIFICATION

Introduction— Subspace inspired identification—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

Introduction— Subspace inspired identification—HIV modeling

Matrix factorizations

Given: matrix

Find: low rank approximation of

Realization problem

Given: string prob’s

Find: HMM generating string prob’s

Identification problem

Given: output sequence

Find: HMM that models the sequence

Estimation problem

Given: output sequence

Find: state distribution at time

- Filtering – smoothing – prediction

- State estimation – 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 HMMs— 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 HMMs— 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 HMMs— 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

Conclusions— Further research— 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

Conclusions— Further research— 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, 2006.
- 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, 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 991–1003, 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 826–831, 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. 1615–1626, 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

Conclusions— Further research— List of publications