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Hidden Markov Autoregressive Models

Hidden Markov Autoregressive Models. A Hidden Markov Model consists of. A sequence of states { X t |t  T } = { X 1 , X 2 , ... , X T } , and A sequence of observations { Y t |t  T } = { Y 1 , Y 2 , ... , Y T }.

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Hidden Markov Autoregressive Models

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  1. Hidden Markov AutoregressiveModels

  2. A Hidden Markov Model consists of • A sequence of states {Xt|t T} = {X1, X2, ... , XT} , and • A sequence of observations {Yt |tT} ={Y1, Y2, ... , YT}

  3. The sequence of states {X1, X2, ... , XT} form a Markov chain moving amongst the M states {1, 2, …, M}. • The observation Yt comes from a distribution that is determined by the current state of the process Xt. (or possibly past observations and past states). • The states, {X1, X2, ... , XT}, are unobserved (hence hidden).

  4. Given Xt = it, Yt-1 = yt-1, Xt-1 = it-1, Yt-2 = yt-2, Xt-2 = it-2, … , Yt-p = yt-p, Xt-p = it-p The distribution of Ytis normal with mean and variance

  5. Parameters of the Model • P = (pij) = the MM transition matrix where pij = P[Xt+1 = j|Xt = i] = the initial distribution over the states where = P[X1 = i]

  6. The state means • The state variances • The state autoregressive parameters

  7. Simulation of Autoregressive HMM’s HMM AR.xls

  8. Computing Likelihood Assuming that it is known that Y0 = y0, X0 = i0, Y-1 = y-1, X-1 = i-1, … , Y1-p = y1-p, X1-p = i1-p Let u1, u2, ... ,uT denote T independent N(0,1) random variables. Then the joint density of u1, u2, ... ,uT is:

  9. Given the sequence of states X1 = i1, X2= i2, X3 = i3, … , XT = iT we have: for t = 1, 2, 3, … , T:

  10. The jacobian of this transformation is: since for s > t and

  11. Hence the density of y1, y2, ... ,yT given X1= i1, X2 = i2, X3 = i3, … , XT = iT is where for t = 1, 2, 3, … , T:

  12. Also and

  13. Efficient Methods for computing Likelihood The Forward Method Let and Consider

  14. This will eventually be used to predict state (probability) from observations

  15. Note: where

  16. Then

  17. where Finally

  18. The Backward Procedure Let and Define

  19. Also Note:

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