exponential of a matrix to determine transition probability matrix

I have a transition rate matrix for five states in period t as follows;

where, last state is an abosrbing state. Now using the MATLAB Command expm, I want to determine transition probabilty matrix. Q = [-(0.7725+1.73) 0.7725 0 00 1.732941;...

Q = [-2.5025 0.7725 0 00 1.732941;...
     0 -3.8651 0.8508 1.313 1.7013;...
     0 0.8265 -2.3376 0 1.5111;...
     0 0.4820 0 -2.1762 1.6942;...
     0 0 0 0 0]
     
     P(t) = expm(Q)

and the outcome is

P (t)= [0.08188 0.04309 0.0211 0.0345 0.820;

0 0.050 0.0519 0.087 0.8101;

0 0.05 0.120 0.039 0.789;

0 0.03225 0.0148 0.1376 0.815;

0 0 0 0 1]

My question is that I know that there will be some positive transition probability of state (e.g. 1 to state 3 in future) however, at initial period, there is no direct jump from state 1 to state 3 (as reflected in transition matrix, Q). Then why do I get a positive probability of say state 1 to state 3 using command expm(Q).

Since I have to use these probabilities to simulate a markov chain, therefore, please help me to understand if I am missing some point here?

Answer 1

[Animesh]

Hey @susman,

The function “expm” computes the matrix exponential. Matrix exponential is a standard mathematical operation on matrices, which is not element wise like other functions such as sine and cosine. Instead, the matrix exponential is defined through a power series. This means that even if an element “Q(i,j)” is zero, its corresponding element in “exmp(Q)” can be nonzero because it involves the sum of powers of the entire matrix Q, not just individual elements.

Even if there is no direct transition from state “1” to state “3” in the matrix “Q” (i.e., Q(1,3) = 0), the matrix exponential can result in a positive probability for transitioning from state “1” to state “3” over time “t”. This is because the matrix exponential considers the possibility of transitioning through intermediate states over time. In other words, it is possible to go from state “1” to state “3” by first transitioning to another state (like state “2”) and then to state “3”.

You can refer to the following MathWorks documentation for information on “expm” function:

https://www.mathworks.com/help/matlab/ref/expm.html

Hope this helps!