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Math Help - MultiDimensional Markov Chain

  1. #1
    Junior Member
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    MultiDimensional Markov Chain

    Hello,
    I've done some stuff with Markov Chains before, and I was wondering if it is possible to have multi-dimensional Markov Chains. Using a simple example, lets say you have 3 rabbit fields (A, B, C) and 100 rabbits in total, and you 4 weight categories (1, 2, 3, 4).
    One the last day every month, each rabbit is weighed recorded, along with what field they were in. So basically from this set of values, you could essentially calculate the probabilities of moving from each weight category to each other weight category, depending on what field that rabbit was in - arriving at 3 transition matrices:

    For Field A:

    <br />
\begin{pmatrix}0.4 & 0.2 & 0.3 & 0.1 \\0.3 & 0.5 & 0.1 & 0.1\\0.4 & 0.3 & 0.2 & 0.1\end{pmatrix}<br />
    For Field B:

    <br />
\begin{pmatrix}0.4 & 0.2 & 0.4 & 0 \\0.3 & 0.2 & 0.3 & 0.2\\0.1 & 0.2 & 0.3 & 0.3 \\0.6 & 0.2 & 0.1 & 0.1 \end{pmatrix}<br />

    ect,


    I was wondering if its possible to combine these in some manner to predict what this population of rabbits will look like in 1 month, and in 12 months, given the total population in each Field in each Weight class (e.g. of the 100 rabbits, 30 are in field A and of those 30, 10 are in weight class 1, 5 in weight class 2, 10 in weight class 3, 5 in weight class 4.)

    I was thinking that the multidimensional transition matrix would look something like that attached image (which is really dodgy because I drew it in paint).

    If this is possible, how would that calculation go?

    Thanks in advance for any replies.
    Attached Thumbnails Attached Thumbnails MultiDimensional Markov Chain-matrix.bmp  
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  2. #2
    MHF Contributor

    Joined
    Aug 2008
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    Paris, France
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    A rabbit from one field can never go to another? Then you can study the Markov chains separately, since they don't interact. Or you bind them together artificial by considering the following matrix:

    \left(\begin{array}{c|c|c}\text{Field A}&0&0\\\hline 0&\text{Field B}&0\\\hline0&0&\text{Field C}\end{array}\right),


    where "Field A" represents the transition matrix of rabbits in field A, etc.. In other words, the state space is (rabbit sizes)x(rabbit fields).
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