Exponential power mixture model for regression : estimation and variable selection

The mixture regression model is an important technique used in statistical modelling to investigate the relationship between variables. It has been applied in many fields such as genetics, finance and biology. In this research, we focus on its application to genetic data. As we know gene expression data normally contains unknown correlation structures even after normalization, hence it raises a great challenge for the existing clustering methods such as the Gaussian mixture(GM) model and k-mean. Here we use the exponential power distribution to robustly overcome the clustering of gene expression data by treating the data as a mixture of regression. The exponential power distribution (EPD) is a scale mixture of Gaussian distributions that has varying shape parameters. In this study we introduce and develop our method based on two different aspects of multiple regression with random errors distributed according to the exponential power distribution. The first aspect is estimation: we use both the ExpectationMaximisation algorithm (EM) and the Newton-Raphson method to estimate the parameters of the exponential power distribution mixture regression models. The second aspect is simultaneous variable selection and clustering: we develop a LASSO-type method to select only the related variables in a large dataset, especially for a high dimensional dataset. The novelty of this research regarding to the Expectation-Maximization algorithm is that we convert each penalised mixture regression estimation problem to a LASSO (Least absolute shrinkage and selection operator) problem. The performance of our method is assessed on both independent and dependent data. We also compared the EPD mixture regression with Gaussian mixture regressions by simulations and real data analyses. We also derive the model selection criteria such as AIC, BIC and EBIC for both EPD mixture and GM models.