Computing theJacobian inGaussian Spatial Autoregressive Models: An Illustrated Comparison of Available Methods

Abstract

1When fitting spatial regression models by maximum likelihood us- Where maximum likelihood methods are chosen for fitting spatial regres-3 sion models, problems can arise when data sets become large because it is 4 necessary to compute the determinant of an n × n matrix when optimizing the 5 log-likelihood function, where n is the number of observations. As Bayesian 6 methods for spatial regression may also require the handling of the same ma-7 trix, they may face the same technical issues of memory management and 8 algorithm choice. We have chosen here to term the problem we are considering 9 the "Jacobian", although the expression of interest is ln |I − λW|, where | · | 10 here denotes the determinant of matrix ·, I is the identity matrix, λ is a spatial 11 coefficient, and W is an n × n matrix of fixed spatial weights, so the problem 12 perhaps ought to be termed finding the logarithm of the determinant of the 13 Jacobian. In order to optimize the log-likelihood function with respect to λ, 14successive new values of this calculation are required. 15The often sparse matrix of spatial weights W represents a graph of rela- Although it may seem that the computation of the Jacobian is an unimpor-34 tant technical detail in comparison with the substantive concerns of analysts, 35we feel that this review may provide helpful insight for practical research using 36 spatial regression with spatial weights matrices representing spatial processes. 26We continue by defining spatial regression models to be treated here, the 27 data sets to be used for this comparison, and how we, following Higham (2002),

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