# Estimation of infinitesimal generator/transition rate matrix from proportion data

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.

Note:

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.