Check memoryless property of a Markov chain

I suspect that a series of observed sequences are a Markov chain…


However how could I check that they indeed respect the memoryless property of P(Xi=xi|Xj=xj)?

Or at the very least prove that they are Markov in nature? Note these are empirically observed sequences. Any thoughts?


Just to add, the aim is to compare a predicted set of sequence from the observed ones. So we’d appreciate comments on as to how best to compare these.

First Order Transition matrix Mij=xijmxik where m=A..E states


Eigenvalues of M

Eigenvectors of M


I wonder if the following would give a valid Pearson χ2 test for proportions as follows.

  1. Estimate the one-step transition probabilities — you’ve done that.
  2. Obtain the two-step model probabilities:
  3. Obtain the two-step empirical probabilities ˜pU,V=i#Xi=V,Xi+2=Ui#Xi=V
  4. Form Pearson test statistic TV=#{Xi=V}U(ˆpU,V˜pU,V)2ˆpU,V,T=TA+TB+TC+TD

It is tempting for me to think that each TUχ23, so that the total Tχ212. However, I am not entirely sure of that, and would appreciate your thoughts on this. I am not likewise not co sertain about whether one needs to be paranoid about independence, and would want to split the sample in halves to estimate ˆp and ˉp.

Source : Link , Question Author : HCAI , Answer Author : StasK

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