Check memoryless property of a Markov chain

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

X=(ACDDBACBAACADABCADABE)

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?

EDIT

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

M=(0.18340.30770.07690.14790.28400.46970.11360.00760.25000.15910.18270.24040.22120.19230.16350.23780.18180.06290.33570.18180.24580.17880.11730.17880.2793)

Eigenvalues of M
E=(1.0000000000.2283000000.1344000000.11360.0430i000000.1136+0.0430i)

Eigenvectors of M
V=(0.44720.58520.42190.23430.0421i0.2343+0.0421i0.44720.78380.42110.44790.2723i0.4479+0.2723i0.44720.20060.37250.63230.63230.44720.00100.70890.21230.0908i0.2123+0.0908i0.44720.05400.05890.2546+0.3881i0.25460.3881i)

Answer

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:
    ˆpU,V=Prob[Xi+2=U|Xi=V]=W{A,B,C,D}Prob[Xi+2=U|Xi+1=W]Prob[Xi+1=W|Xi=V]
  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.

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

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