A $d$-D continuous-lag spatial Markov chain is probabilistic model which is developed by interpolation of the transition rate matrices computed for the main directions. It defines transition probabilities $\Pr(Z(s + h) = z_k | Z(s) = z_j)$ through
$$\mbox{expm} (\Vert h \Vert R),$$
where $h$ is the lag vector and the entries of $R$ are ellipsoidally interpolated.The ellipsoidal interpolation is given by
$$\vert r_{jk} \vert = \sqrt{\sum_{i = 1}^d \left( \frac{h_i}{\Vert h \Vert} r_{jk, \mathbf{e}_i} \right)^2},$$
where $\mathbf{e}_i$ is a standard basis for a $d$-D space.
If $h_i < 0$ the respective entries $r_{jk, \mathbf{e}_i}$ are replaced by $r_{jk, -\mathbf{e}_i}$, which is computed as
$$r_{jk, -\mathbf{e}_i} = \frac{p_k}{p_j} \, r_{kj, \mathbf{e}_i},$$
where $p_k$ and $p_j$ respectively denote the proportions for the $k$-th and $j$-th categories. In so doing, the model may describe the anisotropy of the process.