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.