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Understanding how to build a chain of higher-order stamps

Suppose I want to predict whether a person has class1=healthy or class2= fever . I have a dataset with the following domain: {normal,cold,dizzy}

The transition matrix will contain the probability of transition formed from our training data set, while the initial vector will contain the probability that a person will start (day1) with state x from the domain {normal,cold,dizzy} , again this is also generated from our training kit.

If I want to build a chain of first-order marks, I would generate a 3x3 transition matrix and an initial 1x3 vector for each class:

 > TransitionMatrix normal cold dizzy normal NA NA NA cold NA NA NA dizzy NA NA NA >Initial Vector normal cold dizzy [1,] NA NA NA

NA is filled with corresponding probabilities.

1-My question is about transition matrices in a higher order chain. For example, in a second-order MC, we would have a transition matrix of size domain²xdomain² like this:

  normal->normal normal->cold normal->dizzy cold->normal cold->cold cold->dizzy dizzy->normal dizzy->cold dizzy->dizzy normal->normal NA NA NA NA NA NA NA NA NA normal->cold NA NA NA NA NA NA NA NA NA normal->dizzy NA NA NA NA NA NA NA NA NA cold->normal NA NA NA NA NA NA NA NA NA cold->cold NA NA NA NA NA NA NA NA NA cold->dizzy NA NA NA NA NA NA NA NA NA dizzy->normal NA NA NA NA NA NA NA NA NA dizzy->cold NA NA NA NA NA NA NA NA NA dizzy->dizzy NA NA NA NA NA NA NA NA NA

here cell (1,1) represents the following sequence: normal->normal->normal->normal

or instead, there will simply be domain²xdomain as follows:

  normal cold dizzy normal->normal NA NA NA normal->cold NA NA NA normal->dizzy NA NA NA cold->normal NA NA NA cold->cold NA NA NA cold->dizzy NA NA NA dizzy->normal NA NA NA dizzy->cold NA NA NA dizzy->dizzy NA NA NA

here cell (1,1) represents normal->normal->normal , which is different from the previous view

2 - What about the initial vector for an MC of degree 2. Do we need two initial vectors of size 1xdomain like this:

  normal cold dizzy [1,] NA NA NA

leading to two initial vectors per class. the first gives the probability of occurrence {normal,cold,dizzy} on the first day for the class healthy/fever , and the second gives the probability of occurrence on the second day for healthy/fever . this will give 4 initial vectors.

OR we need only one initial vector of size 1xdomain² as follows:

  normal->normal normal->cold normal->dizzy cold->normal cold->cold cold->dizzy dizzy->normal dizzy->cold dizzy->dizzy [1,] NA NA NA NA NA NA NA NA NA

I see how the second way of representing the original vector would be problematic if we want to classify an observation with only one state.

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We say that the set of spaces is S. As a rule, in the nth order

  • The transition matrix has dimensions | S | n X | S |. This is because, given the current history of states, we need the probability of the next next state. It is true that this one following state induces another complex state of history n, but the transition itself refers to the only next state. See this example on Wikipedia , for example.

  • The initial distribution is the distribution of the elements | S | n (your second option).

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