Discrepancy measures for transition matrices I'm doing some work on modelling transition matrices, and for this I need a measure of discrepancy or lack of fit: that is, if I have a matrix $T$ and a target matrix $T_0$, I want to be able to calculate how far $T$ is from $T_0$. Would anyone be able to provide pointers on what measure I should be using?
I've seen some references to using an elementwise squared-error measure, ie sum up the squared differences of the elements of $T$ and $T_0$, but this seems rather ad-hoc.
 A: Why does one want the measure of discrepancy to be a true metric? There is a huge literature on axiomatic characterizations of I-divergence as measure of distance. It is neither symmetric nor satisfies triangle inequality.
I hope by 'transition matrix' you mean 'probability transition matrix'. Never mind, as long as the entries are non negative, I-divergence is considered to be the "best" measure of discrimination. See for example http://www.mdpi.com/1099-4300/10/3/261/. In fact certain axioms which any one would feel desirable lead to measures which are nonsymmetric in general.
A: As long as your matrix represent conditional probability I think that using a general matrix norm is a bit artificial. Using some sort of geodesic distance on the set of transition matrix might be more relevant but I clearly prefer to come back to probabilities. 
I assume you want to compare $Q=(Q_{ij})$ and $P=(P_{ij})$ with $P_{ij}=P(X^P_{t}=j|X^P_{t-1}=j)$ and that for $P$ (resp. $Q$) there exists a unique stationnary measure $\pi_{P}$ (resp. $\pi_{Q}$).
Under these assumptions, I guess it is meaningfull to compare $\pi_{P}$ and $\pi_{Q}$ for example with a $L_{1}$ distance: $\sum_{j}|\pi_{P}[j]-\pi_{Q}[j]|$ or hellinger distance: $\sum_{j}|\pi^{1/2}_{P}[j]-\pi^{1/2}_{Q}[j]|^2$ or Kullback divergence: $\sum_{j}\pi_{P}[j] \log(\frac{\pi_{P}[j]}{\pi_{Q}[j]})$. 
