Chebyshev approximation of log-determinants of spatial weight matrices

An O( n ) approximation to the log-determinant term required for likelihood-based estimation of spatial autoregressive models is introduced, which represents a tremendous increase in speed relative to exact computation that should allow researchers to explore much larger problems and facilitate spatial specification searches.

Abstract

To cope with the increased sample sizes stemming from geocoding and other technological innovations, this paper introduces an O(n) approximation to the log-determinant term required for likelihood-based estimation of spatial autoregressive models. It takes as a point of departure Martin's (1993) Taylor series approximation based on traces of powers of the spatial weight matrix. Using a Chebyshev approximation along with techniques to efficiently compute the initial matrix power traces results in an extremely fast approximation along with bounds on the true value of the log-determinant. Using this approach, it takes less than a second to compute the approximate log-determinant of an 890,091×890,091 matrix. This represents a tremendous increase in speed relative to exact computation that should allow researchers to explore much larger problems and facilitate spatial specification searches.