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A time non-homogenous Markov Chain is one in which the transition probabilities are not constant over time. A non-linear Markov Chain is a model that is not linear in parameters and satisfies the Markov property (independence of future and distant past conditional on present/recent past).

(1) What is the relationship between time non-homogeneity and non-linearity of a Markov chain? (2) If time non-homogeneity and non-linearity are identical, please provide an intuitive, real world, example of such a Markov Chain process. If they are different types of Markov chains, please provide an intuitive, real world example of each.

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your current state can be expressed probabilistically $$S_i=\{Pr(X=1)=p_1,Pr(X=2)=p_2,...,Pr(X=n)=p_n\}^T$$

all the previous states in theory could affect the current state $$S_i=f_i(S_{i-1},S_{i-2},...,S_{0})$$

time homogeneity means f is always the same $$f_i \equiv f$$

linearity means the function $f()$ can be expressed as a linear transform, i.e. multiplying a matrix, which in the order 1 and homogeneous case is $$S_i=Q \cdot S_{i-1}$$ but may also be non-homogenous $$ S_i=Q_i \cdot S_{i-1}, \space S_{i-1}=Q_{i-1} \cdot S_{i-2} ...$$

These 2 things are different.

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  • $\begingroup$ I still don't fully understand the linearity bit. You say "linearity means the function f() can be expressed as a linear transform..". Can you expand on what a linear transform is? And a linear transform of what? $\endgroup$
    – ColorStatistics
    Commented Dec 11, 2018 at 17:54
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    $\begingroup$ function f() is a projection from vector S to another vector S. so a linear transform of the old state S to a new state S. Linear transform means taking the weighted sum of the S_{i-1} for each entry of the S_i, which can be written in a matrix form Q * S, where each row of Q is one set of weights. You can think that every state (e.g. X=i) has a probability to turn in to another sate (X=j), to describe the every possible transfer, you need a n-by-n matrix. $\endgroup$
    – Xiaoxiong Lin
    Commented Dec 13, 2018 at 9:53

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