Some expectation-maximization (EM) algorithm simplifications for spatial data

Some expectation-maximization (EM) algorithm simplifications for spatial data
Daniel A. Griffith

University of Texas at Dallas, School of Economic, Political, and Policy Sciences, 800 W. Campbell Rd., GR31, Richardson, TX 75080-3021, USA (

Abstract: The EM algorithm is a generic tool that offers maximum likelihood solutions when data sets are incomplete with data values missing at random or completely at random. At least for its simplest form, the algorithm can be rewritten in terms of an ANCOVA regression specification. This formulation allows several analytical results to be derived that permit the EM algorithm solution to be expressed in terms of new observation predictions and their variances. Implementations can be made with a linear regression, with a nonlinear regression, and with a generalized linear model routine, allowing missing value imputations, even when they must satisfy constraints or involve dependent observations. This paper extends to spatially correlated data findings already reported for non-spatial data, linking the EM algorithm solution with spatial autoregression, geostatistical kriging, and eigenvector spatial filtering. One theorem is proved, and two corollaries are derived that broadly contextualize imputation findings in terms of the theory, methodology, and practice of spatial statistical science.

Keywords: ANCOVA, eigenvector spatial filter, EM algorithm, kriging, spatial autoregression.

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