Suppose I have a collection of data $\{\boldsymbol x_t \in \mathbb S^d\}_{t = 1,\dots,T}$ where $\mathbb S^d$ is the $d$-dimensional unit simplex, i.e. the elements of $\boldsymbol x_t$ sum to $1$.

If I assume that the underlying process evolves according to a continuous time Markov chain, I end up with the following regression model $$\boldsymbol x_t = \exp(At)\boldsymbol x_0$$ where $A$ is the infinitesimal generator/transition rate matrix of the Markov chain.

Is there a well-established and specialised method for estimating the matrix $A$ and/or the initial proportions $\boldsymbol x_0$ from (possibly noisy) data?

The naive approach is just to do a constrained least squares optimisation making sure that $A$ meets the requirements to be a transition rate matrix, but I feel like there must be a better way to do it.


If my data are sampled at uniform timepoints I can write this out as $X_{1:T} = BX_{0:(T-1)}$ with $B = \exp(A\Delta t)$ and the matrix $X_{i:j}$ being the matrix containing the proportions data for all timepoints between $i$ and $j$ inclusive. Again, a naive way to estimate $B$ is just $X_{1:T}X_{0:(T-1)}^\dagger$ where $\dagger$ denotes the pseudoinverse, but this is unconstrained and can lead to matrices $B$ which are not actually transition matrices, and subsequently matrices $A$ which are not infinitesimal generators.

Any pointers to relevant literature appreciated.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.