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Unformatted text preview: CSE 6740 Lecture 12 How Do I Treat Temporal Data? II (Hidden Markov Models) Alexander Gray (Thanks to Nishant Mehta) agray@cc.gatech.edu Georgia Institute of Technology What Are Hidden Markov Models? Hidden Markov models (HMMs) are discrete Markov processes where every state generates an observation at each time step. A discrete process is a random variable that, at each time step, takes on a state value. A Markov discrete process is a memoryless process, where the transition probabilities into the next state are entirely dependent upon the current state. Data generated from a Markov discrete process is a Markov chain. Below is a Markov chain that emits symbols in the set { a,b,c } . t = 1 t = 2 t = 3 t = 4 t = 5 a b c b b 2 / 1 Discrete Markov Process An example discrete Markov process would be s 1 s 2 s 3 1 3 1 3 1 3 1 2 1 4 1 4 2 5 1 5 2 5 3 / 1 Discrete Markov Process Model What are the parameters of a discrete Markov process model? With what probability do we start in each state at t = 0? What is the probability that we transition from state s i to state s j at time t ? 4 / 1 Discrete Markov Process Model What are the parameters of a discrete Markov process model? With what probability do we start in each state at t = 0? The initial state probabilities i What is the probability that we transition from state s i to state s j at time t ? The state transition probabilities a ij , stored in a matrix A 5 / 1 Hidden Markov Models Since HMMs are discrete Markov processes where each state also emits an observation according to some probability distribution, we need to augment our model. What are the parameters of a hidden Markov model? The initial state probabilities i The state transition probabilities a ij , stored in a matrix A If we are in some state at time t , was is the density function from which we draw our observations? 6 / 1 Hidden Markov Models Since HMMs are discrete Markov processes where each state also emits an observation according to some probability distribution, we need to augment our model. What are the parameters of a hidden Markov model? The initial state probabilities i The state transition probabilities a ij , stored in a matrix A If we are in some state at time t , was is the density function from which we draw our observations? The probability distribution parameters B for each state. The corresponding probabilities are sometimes called the emission probabilities. Discrete b i ( k ) is the probability that state s i emits the k th symbol. Continuous b i ( x ) is s i s PDF evaluated at x 7 / 1 A Hidden Markov Model Example Bull Bear Neutral 1 3 1 3 1 3 1 2 1 4 1 4 2 5 1 5 2 5 Price P Up 0.8 Down 0.1 Flat 0.1 Price P Up 0.2 Down 0.8 Flat 0.0 Price P Up 0.1 Down Flat 0.9 8 / 1 Notation There are N states, s 1 . . . s N . The state label at time point is called q t ....
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 Fall '08
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