An algebraic solution of maximum likelihood function in case of Gaussian mixture distribution

Abstract

Traditionally, the least-squares method has been employed as a standard technique for parameter estimation and regression fitting of models to measured points in data sets in many engineering disciplines, geoscience fields as well as in geodesy. If the model errors follow the Gaussian distribution with mean zero in linear models, the least-squares estimate is linear, unbiased and of minimum variance. However, this may not always be the case owing to contaminated data (i.e. the presence of outliers) or data from different sources with varying distributions.This study proposes an algebraic iterative method that approximates the error distribution model using a Gaussian mixture distribution, with the application of maximum likelihood estimation as a possible solution to the problem. The global maximisation of the likelihood function is carried out through the computation of the global solution of a multivariate polynomial system using numerical Groebner basis in order to considerably reduce the running time. The novelty of the proposed method is the application of the total least square (TLS) error model as opposed to ordinary least squares (OLS) and the maximisation of the likelihood function of the Gaussian mixture via an algebraic approach. Use of the TLS error model rather than OLS enables errors in all the three coordinates of the model of a 3D plane (i.e. [...]) to be considered. The proposed method is illustrated by fitting a plane to real laser-point cloud data containing outliers to test its robustness. Compared with the Random Sample Consensus and Danish robust estimation methods, the results of the proposed algebraic method indicate its efficiency in terms of computational time and its robustness in managing outliers. The proposed approach thus offers an alternative method for solving mixture distribution problems in geodesy.