Penalized pseudolikelihood inference in spatial interaction models with covariates (Q2742760)

A new nonparametric method for estimating the interaction potentials of a Markov random field as functions of observed explanatory variables is proposed. Let \(\bar X = (X_1, \ldots, X_n)\) be a vector of random variables located on the \(n\) sites of an undirected finite graph \(G =(\Lambda, E)\). A Markov random field for \(X\) with the first-order pairwise interaction has the form NEWLINE\[NEWLINE P(\bar X = \bar x) = C^{-1} \exp \biggl( \sum_{(i,j) \in E} U_{ij}(x_i, x_j) + \sum_{i\in \Lambda} U_i(x_i)\biggr),NEWLINE\]NEWLINE where \(C\) is a normalizing constant. It is assumed that deterministic information is available at each spatial location \(\Lambda =\{ 1, 2,\ldots, n\}\) on the graph \(G.\) The explanatory variables are incorporated into the Markov random field by letting them modulate the potential functions locally as follows: NEWLINE\[NEWLINEP(\bar X = \bar x |\bar z) = C^{-1}(z) \exp \biggl( \sum_{(i,j) \in E} U_{ij}(x_i, x_j,\bar z) + \sum_{i\in \Lambda} U_i(x_i,\bar z)\biggr). NEWLINE\]NEWLINE A method is proposed to estimate the potentials as functions of \(z\) nonparametrically for a special case when \(U_i(x_i,\bar z) = \Phi(u_{ij}) g(x_i, x_j)\), \(U_i(x_i, z) =\Psi(z_i) h(x_i).\) The functions \(g(x_i, x_j)\) and \(h(x_i)\) are assumed to be known, \(u_{ij} =u_{ij}(\bar z).\) A pseudolikelihood approximation in a nonparametric setting to estimate \(\Phi\) and \(\Psi\) is proposed. The authors propose a nonparametric approach introducing a roughness penalty into the log-likelihood. They show that this approach is particularly convenient from a numerical point of view. B-splines are introduced to stabilize the numerical procedures that compute the estimators. A simulation experiment in the setting of Bayesian image reconstruction is presented.