Hidden Markov Autoregressive Models

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}

The sequence of states {X1, X2, 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).

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 Yt is normal with mean and variance

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]

The state means The state variances The state autoregressive parameters

Simulation of Autoregressive HMM’s HMM AR.xls

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:

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

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

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

Also and

Efficient Methods for computing Likelihood The Forward Method Let and Consider

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

Note: where

Then

where Finally

The Backward Procedure Let and Define

Also Note: