Data-driven Transfer Operators

Previously, we have seen how to construct a transfer operator from a Markov chain. In this section, we will see how to construct a transfer operator from data. This is useful when we don't know the underlying transfer operator.

For example, suppose that we have the Markov chain

markov_chain = [1 1 2 2 2 1]

1×6 Matrix{Int64}:
 1  1  2  2  2  1

and we want to estimate the transfer operator from this data. We can do this by using the function perron_frobenius which is imported from the MarkovChainHammer.TransitionMatrix module. This function takes a Markov chain as input and returns the empirical transition matrix of the Markov chain.

using MarkovChainHammer.TransitionMatrix: perron_frobenius
P = perron_frobenius(markov_chain)
P

2×2 Matrix{Float64}:
 0.5  0.333333
 0.5  0.666667

This matrix is constructed by counting the number of times that the chain transitions from one state to another and dividing by the total number of transitions. For example, the entry P[1, 1] is the number of times that the chain transitions from state 1 to state 1 divided by the total number of transitions away from state 1. In this case, the chain transitions from state 1 to state 1, a total of 1 time. Furthermore, the chain transitions from state 1 to state 2 a total of 1 time. Thus the total number of transitions is 2 and the entries of the first column are P[1, 1] = 1/2 and P[2, 1] = 1/2.

The second column of the matrix is constructed in the same way. The chain transitions from state 2 to state 1 a total of 1 time and from state 2 to state 2 a total of 2 times. Thus the entries of the second column are P[1, 2] = 1/3 and P[2, 2] = 2/3.