A spatial variant of the Gaussian mixture of regressions model

Abstract

In this study the nite mixture of multivariate Gaussian distributions is discussed in detail including the derivation of maximum likelihood estimators, a discussion on identi ability of mixture components as well as a discussion on the singularities typically occurring during the estimation process. Examples demonstrate the application of the nite mixture of univariate and bivariate Gaussian distributions. The nite mixture of multivariate Gaussian regressions is discussed including the derivation of maximum likelihood estimators. An example is used to demonstrate the application of the mixture of regressions model. Two methods of calculating the coe cient of determination for measuring model performance are introduced. The application of nite mixtures of Gaussian distributions and regressions to image segmentation problems is examined. The traditional nite mixture models however, have a shortcoming in that commonality of location of observations (pixels) is not taken into account when clustering the data. In literature, this shortcoming is addressed by including a Markov random eld prior for the mixing probabilities and the present study discusses this theoretical development. The resulting nite spatial variant mixture of Gaussian regressions model is de ned and its application is demonstrated in a simulated example. It was found that the spatial variant mixture of Gaussian regressions delivered accurate spatial clustering results and simultaneously accurately estimated the component model parameters. This study contributes an application of the spatial variant mixture of Gaussian regressions model in the agricultural context: maize yields in the Free State are modelled as a function of precipitation, type of maize and season; GPS coordinates linked to the observations provide the location information. A simple linear regression and traditional mixture of Gaussian regressions model were tted for comparative purposes and the latter identi ed three distinct clusters without accounting for location information. It was found that the application of the spatial variant mixture of regressions model resulted in spatially distinct and informative clusters, especially with respect to the type of maize covariate. However, the estimated component regression models for this data set were quite similar. The investigated data set was not perfectly suited for the spatial variant mixture of regressions model application and possible solutions were proposed to improve the model results in future studies. A key learning from the present study is that the e ectiveness of the spatial variant mixture of regressions model is dependent on the clear and distinguishable spatial dependencies in the underlying data set when it is applied to map-type data.