# Interpretation of the fundamental matrix of Absorbing Markov Chain

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• Dec 11th 2017, 01:46 AM
Margodoy
Interpretation of the fundamental matrix of Absorbing Markov Chain
Hello!

I am studying the Absorbing Markov Chain theory, and I have a question about how to interpret the fundamental matrix (N). It is defined as the average number of times transient state j is encountered in the transition from transient state i to an absorbing state, but many times the values of N are not integer numbers. So, how should I interpret a N values of 0.2 or 1.8 for example? Should I just round it to the nearest integer? I have read many examples but in non of them this fraction number are explained.

Thanks for your help
• Dec 11th 2017, 10:12 AM
HallsofIvy
Re: Interpretation of the fundamental matrix of Absorbing Markov Chain
Can you give an example of a problem in which the entries of such a matrix are not non-negative integers? If it were not for that "1.8" I would suspect that the entry in the "i,j" position is the probability of transition from state i to state j rather than the "number of times" the transition occurs.
• Dec 21st 2017, 12:03 AM
Margodoy
Re: Interpretation of the fundamental matrix of Absorbing Markov Chain
Thanks for your answer, but no, this matrix gives the number of times a state is reached, and it is based on the transition matrix that gives the probability of the change of state.
• Dec 21st 2017, 12:40 AM
stans
Re: Interpretation of the fundamental matrix of Absorbing Markov Chain
Margodoy, you said yourself: "the average number of times". Averages (expectations) are rarely round numbers. If the expectation is 0.2, then the interpretation is: in most random scenarios there is 0 or 1 visit to transient state j before absorption, with the probability of 0-visit scenarios being roughly 80%. As uninspiring as this statement is, there is nothing more to say.
• Dec 21st 2017, 12:43 AM
Margodoy
Re: Interpretation of the fundamental matrix of Absorbing Markov Chain
Thanks a lot