Chebyshev approximation of log-determinants of spatial weight matrices

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.