Hidden Markov Models Sean Callen Joel Henningsen.

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Presentation transcript:

Hidden Markov Models Sean Callen Joel Henningsen

Example Discovering average yearly temperature at a particular location on Earth over a series of years using observed size of tree growth rings. Possible states (hidden) – Hot (H) and Cold (C) Possible observations – Small (S), Medium (M), and Large (L) HC H.7.3 C.4.6 SML H C.7.2.1

Notation T = length of the observation sequence N = number of states in the model M = number of observation symbols Q = {q 0, q 1, …, q N-1 } = distinct states of the Markov process V = {0, 1, …, M-1} = set of possible observations A = state transition probability matrix B = observation probability matrix π = initial state sequence O = (O 0, O 1, …, O T-1 ) = observation sequence

Example’s Notation

Probability Finding the probability of a state sequence given an observation sequence. X = {x 0, x 1, x 2, x 3 } O = (O 0, O 1, O 2, O 3 ) P(X) = π x0 b x0 (O 0 )a x0,x1 b x1 (O 1 )a x1,x2 b x2 (O 2 )a x2,x3 b x3 (O 3 ) Let O = (0, 1, 0, 2) P(HHCC) =.6(.1)(.7)(.4)(.3)(.7)(.6)(.1) =

Probability Optimal state sequence is CHCH.

The three problems  Given the model, find the probability of an observation sequence.  Given the model and an observation sequence, find the optimal state sequence.  Given an observation model, N, and M, determine a model to maximize the probability of O.