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I have been given a data set in which each sample represents a "quality indicator" (QI) of a signal. This QI ranges from 0 to 15 and therefore, there are 16 discrete values. Based on my knowledge of the system I can assume that the system's future state depends only on the present state, thus, it has the Markovian property.

Each signal sample in the data set has been taken at random times during a 6-month period, therefore the time interval between the samples is not regular.

I am trying to model the signal using a finite state Markov chain, and for this purpose, each QI value has been considered as a state in the chain.

I obtained the initial state probabilities based only on the frequency with which the signal has a given QI, or in other words, the frequency the chain is in a given state.

Then, I obtained the transition probabilities based on the frequency of the transitions that occur between all the states of the chain.

Finally, the Markov chain was iterated a number of times, and from the synthetic data generated by the chain, a PDF was obtained. This PDF was compared against the PDF from the original dataset which allowed me to observe that they have the same statistical properties.

I am not sure if the process to obtain a Markov chain and especially, the transition probabilities, is as simple as I described, or if I need to perform a more thorough analysis. Based on the material I've found, I think I need to answer the following questions:

  • ¿Does the time between samples need to be regular or can be random? If it does need to be regular, how can I consider the randomness in the analysis?
  • I am not interested in modeling the time whatsoever, does this mean that I can ignore the fact that the time between samples is not regular and just analyze the system as a function of transitions?
  • ¿Is there a more analytical way to validate the model?
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If the underlying model is a continuous-time Markov chain, then of course the time between samples needs to be considered. The transition matrix for a short time interval is typically much different than that for a long time interval.

You might look at my paper with Jeffrey Rosenthal and Jason Wei - however this considers a case where the empirical transition matrix is for a fixed time interval.

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