From empirical data to time-inhomogeneous continuous Markov processes

We present an approach for testing for the existence of continuous generators of discrete stochastic transition matrices. Typically, existing methods to ascertain the existence of continuous Markov processes are based on the assumption that only time-homogeneous generators exist. Here a systematic extension to time inhomogeneity is presented, based on new mathematical propositions incorporating necessary and sufficient conditions, which are then implemented computationally and applied to numerical data. A discussion concerning the bridging between rigorous mathematical results on the existence of generators to its computational implementation is presented. Our detection algorithm shows to be effective in more than $60\%$ of tested matrices, typically $80\%$ to $90\%$, and for those an estimate of the (nonhomogeneous) generator matrix follows. We also solve the embedding problem analytically for the particular case of three-dimensional circulant matrices. Finally, a discussion of possible applications of our framework to problems in different fields is briefly addressed.

Figure 1

(a) Region in parameter space of transition matrix ${\mathbf{T}}_C$, Eq. (8), for which a generator ${\mathbf{Q}}_C$ exists. The triangular region (line) is the one for which matrix ${\mathbf{T}}_C$ is a transition matrix, $a,b,c>0$ and $a+b+c=1$. The dark gray (blue) region indicates the region of parameter values for which only one generator exists, while the light gray (yellow) region indicates a region where two generators exist. (b) Zooming into this region shows a region where three generators exist (dark gray), and (c) continuing to zoom shows smaller and smaller regions, where a larger number of generators exist (see text).

Figure 2

Histogram of $\mathrm{\Delta }$ values, Eq. (41), from a sample matrices (see text).