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.

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