Solved – Regarding Transition Probability Matrix

markov-processMATLABprobability

I am calculating transition probability matrix (TPM) using hmmestimate command of MATALB.
The simple vector a=[ 1 1 1 1 3 3 3 2 1 3] and for this the TPM is

hmmestimate(a,a)

ans =
0.6000         0    0.4000
1.0000         0         0
     0    0.3333    0.6667

I believe it is calculated as : (1,3) element:

(# of 1->3 transitions)/(total # of 1) = 2/5 = 0.4

Same rule follows for all the elements. Then why not for element (3,2) and (3,3). For example: (3,2) element:

(# of 3->2 transitions)/(total # of 3) = 1/4 = 0.25

but the transition matrix calculated by MATLAB entry 0.33. For (3,3), according to my calculation the entry should have been (2/4)=0.5 but it is 0.6667

What am I missing?

Best Answer

You can't include the $3$ on the end in the count of $3$s because it has no successor. If you have $n$ values, there are only $n-1$ transitions.

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Solved – Creating a probability transition matrix

Modeling the system as a Markov chain, the maximum likelihood estimate for the $A \rightarrow B$ transition probability is simply the number of times you saw $B$ following $A$, divided by the number of times you saw any state following $A$. The transition matrix is a matrix $P$ where $P_{ij}$ contains the $i \rightarrow j$ transition probability. A simple way to calculate this is to construct the transition count matrix $N$, where $N_{ij}$ contains the number of times you observed an $i \rightarrow j$ transition. Then, calculate $P$ by dividing each row of $N$ by its sum.

Solved – How to calculate the probability of this claim

You might find it useful to write out a table of the bivariate distribution:

                Good driver number of claims
                 0       1       2       3   
             0  0.1    0.06 ...
 Bad driver  1
 number of   2
 claims      3

and then add the probabilities for the individual cells that have the same number of total claims. (e.g. so P(T=1) will be P(G=0,B=1)+P(G=1,B=0); you'll be adding along "diagonals" running this way $\it{/}$ )

This is not the fastest possible way to do it, but when you're first encountering these kinds of problems it's perhaps the way which makes it clearest what's going on.

Best Answer

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Solved – Creating a probability transition matrix

Solved – How to calculate the probability of this claim

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