Estimation of the Parameter a
In order to define the HMM, one has to estimate a from a observed data sequence.
Suppose that a sequence (in the steady state) of number of dots is observed as follows:
1, 2, 1, 2, 1, 2, 2, 4, 1, 2, 2, 1, 3, 3, 4, 1,
then the question is: what is the "best" a? To answer this question, we note that
P12 = |
æ ç ç ç ç ç è |
a |
1 - a |
 |
0 |
0 |
0 |
0 |
ö ÷ ÷ ÷ ÷ ÷ ø |
a |
1 - a |
0 |
0 |
0 |
0 |
 |
0 |
0 |
1/6+a/12 |
1/6+a/12 |
1/3-a/12 |
1/3-a/12 |
0 |
0 |
1/6+a/12 |
1/6+a/12 |
1/3-a/12 |
1/3-a/12 |
0 |
0 |
1/6+a/12 |
1/6+a/12 |
1/3-a/12 |
1/3-a/12 |
0 |
0 |
1/6+a/12 |
1/6+a/12 |
1/3-a/12 |
1/3-a/12 |
|
The structure of P12 can be explained by the fact that each state has a period of two. If
we ignore the hidden states (the first diagonal block of P12), then the observable states
follow the transition probability matrix given by
P12 |
= |
é ê ê ê ë |
1/6+a/12 |
1/6+a/12 |
1/3-a/12 |
1/3-a/12 |
ù ú ú ú û |
1/6+a/12 |
1/6+a/12 |
1/3-a/12 |
1/3-a/12 |
1/6+a/12 |
1/6+a/12 |
1/3-a/12 |
1/3-a/12 |
1/6+a/12 |
1/6+a/12 |
1/3-a/12 |
1/3-a/12 |
|
| = |
é ê ê ê ë |
a |
1-a |
ù ú ú ú û |
a |
1-a |
a |
1-a |
a |
1-a |
|
é ë |
1/4 |
1/4 |
1/4 |
1/4 |
ù û |
1/6 |
1/6 |
1/3 |
1/3 |
|
It is easy to see that the stationary probability distribution of is given by
p = ( 1/6 + a/12, 1/6 + a/12, 1/3 - a/12, 1/3 - a/12)
because 1 = (1, 1, 1, 1)Tp and therefore p 1 =
p. This probability distribution p should
be consistent with the observed distribution q of the observed sequence, i.e.
p = ( 1/6 + a/12, 1/6 + a/12, 1/3 - a/12, 1/3 - a/12)
» q = ( 6/16, 6/16, 2/16, 2/16)
This suggests a method to estimate a. The unknown transition probability a can be
obtained by solving
If we choose ||.|| to be the L2-norm then one may consider the following minimization
problem:
|
|
= |
min 0 £ a £ 1 |
4 S i=1 |
(p - q)2 |
|
= |
min 0 £ a £ 1 |
{a2/36 - 5a/36 + 25/144} |
|
This is a standard constrained least square problem.
The optimal value of a in this case is equal
to 1. For the problem of statistical inference of a HMM, we refer readers to [3,4].
|