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HIDDEN MARKOV MODELS Prof. Navneet Goyal Department of Computer Science BITS, Pilani Presentation based on: & on presentation on HMM by Jianfeng Tang Old Dominion University

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Topics Markov Models Hidden Markov Models HMM Problems

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Markov Analysis A technique that deals with the probabilities of future occurrences by analyzing presently known probabilities Founder of the concept was A.A. Markov whose 1905 studies of sequence of experiments conducted in a chain were used to describe the principle of Brownian motion

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Markov Analysis Applications: Market share analysis Bad debt prediction Speech recognition University enrollment prediction …

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Markov Analysis Two competing manufacturers might have 40% & 60% market share today. May be in two months time, their market shares would become 45% & 55% respectively Predicting these future states involve knowing the system’s probabilities of changing from one state to another Matrix of transition probabilities This is Markov Process

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Markov Analysis 1.A finite number of possible states. 2. Probability of change remains the same over time. 3. Future state predictable from current state. 4. Size of system remains the same. 5. States collectively exhaustive. 6. States mutually exclusive.

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The Markov Process Matrix of Transition New State Current State P

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Markov Process Equations P 11 P 12 P 13...P 1n P 21 P 22 P 23...P 2n P m1... P mn Matrix of transition probabilities = P = (i) = State probabilities = [ 1 2 3 … n ] (i+1) = (i)P

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Predicting Future States Market Share of Grocery Stores AMERICAN FOOD STORE: 40% FOOD MART: 30% ATLAS FOODS: 30% ∏(1)=[0.4,0.3,0.3]

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Predicting Future States

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Will this trend continue in the future? Is it an equilibrium state? WILL Atlas food lose all of its market share?

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Markov Analysis: Machine Operations P= 0.8 0.2 0.10.9 State1: machine functioning correctly State2: machine functioning incorrectly P 11 = 0.8 = probability that the machine will be correctly functioning given it was correctly functioning last month ∏[2]=∏[1]P=[1,0]P=[0.8,0.2] ∏[3]=∏[2]P=[0.8,0.2]P=[0.66,0.34]

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Machine Example: Periods to Reach Equilibrium Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 State 1 1.0.8.66.562.4934.44538.411766.388236.371765.360235.352165.346515.342560.339792.337854 0.0.2.34.438.5066.55462.588234.611763.628234.639754.647834.653484.657439.660207.662145 State 2

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Equilibrium Equations

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Markov System

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At regularly spaced discrete times, the system undergoes a change of state (possibly back to same state) Discrete first order Markov Chain P[q t =S j |q t-1 =S i,q t-2 =S k,…..]= P[q t =S j |q t-1 =S i ] Consider only those processes in which the RHS is independent of time State transition probabilities are given by a ij = P[q t =S j |q t-1 =S i ] 1<=i,j<=N

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Markov Models A model of sequences of events where the probability of an event occurring depends upon the fact that a preceding event occurred. Observable states: 1, 2, …, N Observed sequences: O 1, O 2, …, O l, …, O T P(O l =j|O 1 =a,…,O l-1 =b,O l+1 =c,…)=P(O l =j|O 1 =a,…,O l- 1 =b) Order n model A Markov process is a process which moves from state to state depending (only) on the previous n states.

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Markov Models First Order Model (n=1) P(O l =j|O l-1 =a,O l-2 =b,…)=P(O l =j|O l-1 =a) The state of model depends only on its previous state. Components: States, initial probabilities & state transition probabilities

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Markov Models Consider a simple 3-state Markov model of weather Assume that once a day (eg at noon), the weather is observed as one of the folowiing: State 1 Rain or (snow) State 2 Cloudy State 3 Sunny Transition Probabilities: 0.40.30.3 0.20.60.2 0.10.10.8 Given that on day 1 the weather is sunny What is the probability that the weather for the next 7 days will be S S R R S C S?

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Hidden Markov Model HMMs allow you to estimate probabilities of unobserved events E.g., in speech recognition, the observed data is the acoustic signal and the words are the hidden parameters

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HMMs and their Usage HMMs are very common in Computational Linguistics: Speech recognition (observed: acoustic signal, hidden: words) Handwriting recognition (observed: image, hidden: words) Machine translation (observed: foreign words, hidden: words in target language)

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Markov Model is used to predict what will come next based on previous observations. However, sometimes, what we want to predict is not what we observed. Example Someone trying to deduce the weather from a piece of seaweed For some reason, he can not access weather information (sun, cloud, rain) directly But he can know the dampness of a piece of seaweed (soggy, damp, dryish, dry) And the state of the seaweed is probabilistically related to the state of the weather Hidden Markov Models

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Hidden Markov Models are used to solve this kind of problems. Hidden Markov Model is an extension of First Order Markov Model The “true” states are not observable directly (Hidden) Observable states are probabilistic functions of the hidden states The hidden system is First Order Markov

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Hidden Markov Models A Hidden Markov Model is consist of two sets of states and three sets of probabilities: hidden states : the (TRUE) states of a system that may be described by a Markov process (e.g. weather states in our example). observable states : the states of the process that are `visible‘ (e.g. dampness of the seaweed). Initial probabilities for hidden states Transition probabilities for hidden states Confusion probabilities from hidden states to observable states

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Hidden Markov Models

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Initial matrix Transition matrix Confusion matrix

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The Trellis

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Coin Toss Problem Observed Sequence: HHTTTHTTH….H How do we build an HMM to explain (model) the observed sequence? What the states in the model correspond to? How many states should be there in the model? Single biased coin is tossed 2 state model Each state corresponds to a side of the coin ( H or T) Resultant Markov model is observable Only unknown in the value of the bias 2 biased coins are tossed 2 states in the model Each state corresponds to a different, biased coin being tossed Each state characterized by prob. dist. Of Hs & Ts 3 biased coins are tossed

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Coin Toss Problem Single biased coin is tossed 2 state model Each state corresponds to a side of the coin ( H or T) Resultant Markov model is observable Only unknown in the value of the bias

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Coin Toss Problem 2 biased coins are tossed 2 states in the model Each state corresponds to a different, biased coin being tossed Each state characterized by prob. dist. Of Hs & Ts

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Coin Toss Problem 3 biased coins are tossed

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Coin Toss Problem Which model best matches the actual observation? 1coin model has only one unknown parameter – the bias 2 coin model has 4 unknown parameters 3 coin model has 9 unknown parameters Degrees of freedom Larger HMMs more capable of modeling a series of coin tossing experiments?? Theoretically correct, but not practically Practical considerations impose limitations on the size of the HMM It might be the case that only 1 coin is being tossed

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Urn & Coloured Balls Model

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State corresponds to a specific URN, and for which a (ball) color probability is defined for each state Choice of URNS is dictated by the state transition matrix of the HMM

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Elements of an HMM N, number of states in the model, which are hidden Physical significance attached to the states Coin tossing experiment: Each state corresponds to a distinct biased coin Urn ball model State corresponds to urns Generally the states are interconnected Ergodic Model

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Elements of an HMM M, number of distinct observation symbols per state Coin tossing experiment: Hs or Ts Urn ball model Colors of the balls

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Elements of an HMM A, state transition probability distribution a ij = P[q t+1 =S j |q t =S i ] 1<=i,j<=N

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Elements of an HMM Observation symbol probability distribution in state j B={bj(k)}, where

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Elements of an HMM The initial state distribution

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HMM

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HMM problems HMMs are used to solve three kinds of problems Finding the probability of an observed sequence given a HMM (evaluation); Finding the sequence of hidden states that most probably generated an observed sequence (decoding). The third problem is generating a HMM given a sequence of observations (learning). –learning the probabilities from training data.

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HMM problems

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HMM Problems 1. Evaluation Problem: We have a number of HMMs and a sequence of observations. We may want to know which HMM most probably generated the given sequence. Solution: Computing the probability of the observed sequences for each HMM. Choose the one produced highest probability Can use Forward algorithm to reduce complexity.

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HMM problems Pr(dry,damp,soggy | HMM) = Pr(dry,damp,soggy | sunny,sunny,sunny) + Pr(dry,damp,soggy | sunny,sunny,cloudy) + Pr(dry,damp,soggy | sunny,sunny,rainy) +.... Pr(dry,damp,soggy | rainy,rainy,rainy)

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HMM problems 2. Decoding Problem: Given a particular HMM and an observation sequence, we want to know the most likely sequence of underlying hidden states that might have generated the observation sequence. Solution: Computing the probability of the observed sequences for each possible sequence of underlying hidden states. Choose the one produced highest probability Can use Viterbi algorithm to reduce the complexity.

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HMM Problems the most probable sequence of hidden states is the sequence that maximizes : Pr(dry,damp,soggy | sunny,sunny,sunny), Pr(dry,damp,soggy | sunny,sunny,cloudy), Pr(dry,damp,soggy | sunny,sunny,rainy),.... Pr(dry,damp,soggy | rainy,rainy,rainy)

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HMM problems (cont.) 3. Learning Problem: Estimate the probabilities of HMM from training data Solution: Training with labeled data Transition probability P(a,b)=(number of transitions from a to b)/ total number of transitions of a Confusion probability P(a, o)=(number of symbol o occurrences in state a)/(number of all symbol occurrences in state a) Training with unlabeled data Baum-Welch algorithm The basic idea Random generate HMM at the beginning Estimate new probability from the previous HMM until P(current HMM) – P( previous HMM) < e (a small number)

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Designing HMM for an Isolated Word Recognizer Vocabulary of V words Each word to be modeled by a distinct HMM For each word, we have a training data set of K occurrences of each word (spoken by one or more talkers) Each occurrence of the word constitutes an observation sequence Observations are some appropriate representations of the characteristics of the word

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Designing HMM for an Isolated Word Recognizer To do isolated word speech recognition:

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Designing HMM for an Isolated Word Recognizer

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HMM Application: Parsing a reference string into fields Problem Parsing a reference string into fields (author, journal, volume, page, year, etc.) Model as HMM Hidden states – fields (author, journal, volume, etc) and some special characters ( “,”, “and”, etc.) Observable states – words Probability matrixes --learning from training data Reference parsing Using Viterbi algorithm to find the most possible sequence of hidden states for an observation sequence

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Conclusions HMM is used to model What we want to predict is not what we observed The underlying system can be model as first order Markov HMM assumption The next state is independent of all states but its previous state The probability matrixes learned from samples are the actual probability matrixes. After learning, the probability matrixes will keep unchanged

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