Maximum likelihood from spatial random effects models via the stochastic approximation expectation maximization algorithm

We introduce a class of spatial random effects models that have Markov random fields (MRF) as latent processes. Calculating the maximum likelihood estimates of unknown parameters in SREs is extremely difficult, because the normalizing factors of MRFs and additional integrations from unobserved random effects are computationally prohibitive. We propose a stochastic approximation expectation-maximization (SAEM) algorithm to maximize the likelihood functions of spatial random effects models. The SAEM algorithm integrates recent improvements in stochastic approximation algorithms; it also includes components of the Newton-Raphson algorithm and the expectation-maximization (EM) gradient algorithm. The convergence of the SAEM algorithm is guaranteed under some mild conditions. We apply the SAEM algorithm to three examples that are representative of real-world applications: a state space model, a noisy Ising model, and segmenting magnetic resonance images (MRI) of the human brain. The SAEM algorithm gives satisfactory results in finding the maximum likelihood estimate of spatial random effects models in each of these instances. H. Zhu ( ) Department of Biostatistics and Biomedical Research Imaging Center, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-7420, USA e-mail: [email protected] H. Zhu . B. Peterson Department of Psychiatry, Columbia University and New York State Psychiatric Institute, 1051 Riverside Drive, Unit 74, New York, New York 10032, USA M. Gu Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, P.R. China