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let´s say I estimated a Markov chain by splitting my data into traning and test data, and estimating the transition matrix for the traning data yields $$ P= \begin{bmatrix} p11 & p12 & p13 \\ p21 & p22 & p23 \\ p31 & p32 & p33 \\ \end{bmatrix} $$

i.e, the transitions between states 1, 2 and 3. It is, of course, possible to obtain the n-step ahead matrix from this.

Question 1: How do I evaluate the predictive capacity of this model on the test data? I know I should evaluate the MSE of this matrix on the transitions that actually occurred on the test, but how to do it? The caret package in R has some code, but only for other types of predictors (Knn, naive Bayes, etc, in any case, nothing whose output is a matrix). An easy approach would be to obtain the observed transitions matrix for the test data, divide by rows, and then get the element-wise difference with P, but thats probably not the best approach...

Question2 (Related to question 1): I am unable to deploy a model in my entire dataset, due to data size. Is it possible to evaluate several Markov models on samples and then take some kind of aggregation of these to better reflect the overall data? Kind of what bagging is about, but once again, no examples exist for Markov chains, only decision trees, etc.

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