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I do not come from a very mathematical background, but I am currently reading a paper on Cross-Entropy (http://en.wikipedia.org/wiki/Cross-entropy_method). This got me thinkging and led to my question. Given a matrix $M(t)$ whose elements $M_{ij}(t)$ vary with time, and given how the matrix has varied from time $t_0$ to $t_{\text{current}}$ is there some theory/branch of mathematics that can predict what will happen to the matrix at time $t_{\text{future}}$?

We basically record how the matrix has evolved over a period of time and we want to find out what the state of the matrix will be at some unspecified future time. Something like steady-state probabilities.

Is there any general area where I should start reading something about this?

Sujaan Kunalan
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hRob
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1 Answers1

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In general, even if the function from time $t_0$ to time $t_{current}$ is continuous which gives the matrix entries, you could extend the function to any continuous function you want from $t_{current}$ to $t_{future}$, as long as the values of the functions agreed at $t_{current}$. Even insisting on differentiability is not enough to guarantee uniqueness of extensions. In order to have unique extensions of functions you need to make a very strong assumption, e.g. the function is holomorphic.

user2566092
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  • Unfortunately, in my problem setting the function that generates the contents of the matrix from $t_0$ to $t_{current}$ is not continuous. It is an algorithm which uses the last $n$ instances of the matrix i.e. $M(0), ... , M(t_{current}-1)$ and a few other parameters (which are given constants) to generate $M(t_{current})$. My question would then be, is prediction even possible being agnostic to this generation function? And if it is, what would a reliable window size be for calculating steady-state probabilities? – hRob May 01 '14 at 17:03