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.
When some entries of the rates matrices are not identifiable, it is suggested to vary the tolerance coefficient and the rotation angles. This problem may be also avoided if the input argument mle is to set to be "mlk".