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Part1 Markov Models for Pattern Recognition – Introduction

Part1 Markov Models for Pattern Recognition – Introduction. CSE717, SPRING 2008 CUBS, Univ at Buffalo. Textbook. Markov models for pattern recognition : f rom t heory to a pplications by Gernot A. Fink , 1st Edition, Springer, Nov 2007. Textbook. Foundation of Math Statistics

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Part1 Markov Models for Pattern Recognition – Introduction

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  1. Part1 Markov Models for Pattern Recognition – Introduction CSE717, SPRING 2008 CUBS, Univ at Buffalo

  2. Textbook • Markov models for pattern recognition: from theory to applications by Gernot A. Fink, 1st Edition, Springer, Nov 2007

  3. Textbook • Foundation of Math Statistics • Vector Quantization and Mixture Density Models • Markov Models • Hidden Markov Model (HMM) • Model formulation • Classic algorithms in the HMM • Application domain of the HMM • n-Gram • Systems • Character and handwriting recognition • Speech recognition • Analysis of biological sequences

  4. Preliminary Requirements • Familiar with Probability Theory and Statistics • Basic concepts in Stochastic Process

  5. Part 2 aFoundation of Probability Theory, Statistics & Stochastic Process CSE717 , SPRING 2008 CUBS, Univ at Buffalo

  6. Coin Toss Problem • Coin toss result: • X: random variable • head, tail: states • SX: set of states • Probabilities:

  7. Discrete Random Variable • A discrete random variable’s states are discrete: natural numbers, integers, etc • Described by probabilities of states PrX(s1), PrX(x=s2), … s1, s2, …: discrete states (possible values of x) • Probabilities over all the states add up to 1

  8. Continuous Random Variable • A continuous random variable’s states are continuous: real numbers, etc • Described by its probability density function (p.d.f.): pX(s) • The probability of a<X<b can be obtained by integral • Integral from to

  9. Joint Probability and Joint p.d.f. • Joint probability of discrete random variables • Joint p.d.f. of continuous random variables • Independence Condition

  10. Conditional Probability and p.d.f. • Conditional probability of discrete random variables • Joint p.d.f. for continuous random variables

  11. Statistics: Expected Value and Variance • For discrete random variable • For continuous random variable

  12. Normal Distribution of Single Random Variable • Notation • p.d.f • Expected value • Variance

  13. Stochastic Process • A stochastic process is a time series of random variables • : random variable • t: time stamp Stock market Audio signal

  14. Causal Process • A stochastic process is causal if it has a finite history A causal process can be represented by

  15. Stationary Process • A stochastic process is stationary if the probability at a fixed time t is the same for all other times, i.e., for any n, and , • A stationary process is sometimes referred to as strictly stationary, in contrast withweak or wide-sense stationarity

  16. Gaussian White Noise • White Noise: obeys independent identical distribution (i.i.d.) • Gaussian White Noise

  17. Gaussian White Noise is a Stationary Process Proof for any n, and ,

  18. Temperature Q1: Is the temperature within a day stationary?

  19. Markov Chains • A causal process is a Markov chain if for any x1, …, xt k is the order of the Markov chain • First order Markov chain • Second order Markov chain

  20. Homogeneous Markov Chains • A k-th order Markov chain is homogeneous if the state transition probability is the same over time, i.e., • Q2: Does homogeneous Markov chain imply stationary process?

  21. State Transition in Homogeneous Markov Chains • Suppose is a k-th order Markov chain and S is the set of all possible states (values) of xt, then for any k+1 states x0, x1, …, xk, the state transition probability can be abbreviated to

  22. Example of Markov Chain 0.4 0.6 Rain Dry 0.2 0.8 Two states : ‘Rain’ and ‘Dry’. Transition probabilities: Pr(‘Rain’|‘Rain’)=0.4 , Pr(‘Dry’|‘Rain’)=0.6 , Pr(‘Rain’|‘Dry’)=0.2, Pr(‘Dry’|‘Dry’)=0.8

  23. Short Term Forecast 0.4 0.6 Rain Dry 0.2 0.8 Initial (say, Wednesday) probabilities: PrWed(‘Rain’)=0.3, PrWed(‘Dry’)=0.7 What’s the probability of rain on Thursday? PThur(‘Rain’)= PrWed(‘Rain’)xPr(‘Rain’|‘Rain’)+PrWed(‘Dry’)xPr(‘Rain’|‘Dry’)= 0.3x0.4+0.7x0.2=0.26

  24. Condition of Stationary 0.4 0.6 Rain Dry 0.2 0.8 Pt(‘Rain’)= Prt-1(‘Rain’)xPr(‘Rain’|‘Rain’)+Prt-1(‘Dry’)xPr(‘Rain’|‘Dry’)= Prt-1(‘Rain’)x0.4+(1– Prt-1(‘Rain’)x0.2= 0.2+0.2xPrt(‘Rain’) Pt(‘Rain’)= Prt-1(‘Rain’) => Prt-1(‘Rain’)=0.25, Prt-1(‘Dry’)=1-0.25=0.75 steady state distribution

  25. Steady-State Analysis 0.4 0.6 Rain Dry 0.2 0.8 • Pt(‘Rain’)= 0.2+0.2xPrt-1(‘Rain’) • Pt(‘Rain’) – 0.25 = 0.2x(Prt-1(‘Rain’) – 0.25) • Pt(‘Rain’)= 0.2t-1x(Pr1(‘Rain’)-0.25)+0.25 • Pt(‘Rain’)= 0.25 (converges to steady state distribution)

  26. Periodic Markov Chain 0 1 Rain Dry 1 0 Periodic Markov chain never converges to steady states

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