A Ridge Restricted Maximum Likelihood Approach to Spatial Models

Abstract

Ridge restricted maximum likelihood (RREML) is a new method for regression analysis in linear models with dependent errors. Assume the linear model where the stochastic error terms are not independent, and the covariance structure is a function of some covariance parameter, in this case a spatial covariance parameter. Restricted maximum likelihood (REML) could be used to estimate this covariance parameter, but REML has no built-in methods for when multicollinearity exists in the design matrix. RREML takes the Bayesian analog of the ridge regression model, but modifies the context in order to incorporate the estimation of the variance parameter. The motivation behind such an approach is that by introducing a bit of bias in the estimator we will stabilize the variance of the estimates. By weighting the covariance of the prior distribution appropriately, the analysis should be able to both incorporate the information from the prior distribution and control the influence it has on the posterior estimates of the model. This work involves an approach that will be used in order to confront the inherent multi- collinearity of the design matrix obtained in inverse modeling as discussed in Kasibhatla et al. A Bayesian linear regression approach is commonly used in the atmospheric chemistry community in order to deal with the instability of the linear model, but it is found that these predetermined prior distributions can be too influential on the final results of the estimates. The goal of the proposed work is to control this sensitivity to the prior distribution while also incorporating a covariance structure on the error terms.