Spatial data modeling by means of Gibbs–Markov random fields based on a generalized planar rotator model

We introduce a Gibbs Markov random field for spatial data on Cartesian grids which is based the generalized planar rotator (GPR) model. The GPR model generalizes recently proposed modified (MPR) by including in Hamiltonian additional terms that better capture realistic features of data, such as smoothness, non-Gaussianity, and geometric anisotropy. In particular, includes up to infinite number higher-order harmonics with exponentially vanishing interaction strength, directional dependence bilinear term between nearest grid neighbors, longer-distance neighbor interactions, two types an external bias field. Hence, contrast single-parameter MPR model, five parameters: $n$ parameter $\alpha$ controlling their decay rate, exchange anisotropy $J^{nn}$, further-neighbor coupling $J^{fn}$, (bias) parameters $K$ (or $K'$). present numerical tests various synthetic demonstrate effects respective model's prediction performance we discuss these results connection properties.