Robust Factor Analysis Based On Independent T Distribution

Factor analysis(FA)is a dimensionality reduction method by extracting common factors from groups of variables.In the FA model,its sampling model is based on Gauss assumption and is easily affected by outliers.In order to solve the robustness problem of factor analysis,some scholars replace multivariate normal distribution with multivariate t distribution,and propose a robust factor analysis model(TFA)based on multivariate t distribution,so as to obtain a simple robust extension of the factor analysis model.In the t distribution,the intervention of the parameter of freedom can adjust the weight of outliers adaptively,so as to accommodate the outliers in the data set and achieve a robust effect.However,in the TFA model,each sample shares a weight,so when some elements in the sample are abnormal,the whole sample will be treated as an outlier,resulting in the waste of other normal element information.Therefore,some scholars proposed to use independent t distribution for modeling of each data dimension of observation noise,which is called robust factor analysis based on independent t distribution(hereinafter referred to as it-FA).The research on it-FA only uses Bayesian method for parameter estimation,but it is complicated to calculate the posterior probability in the it-FA model.Therefore,it is difficult to use Bayesian method for parameter estimation.And it doesn’t explain how the independent t distribution solves the problem of local exceptions,which is our concern.Based on this,this paper does the following work: 1.In this paper,variational inference(VB)is used to approximate the posterior probability.In order to improve the convergence speed of VB to the approximate solution,VB and ECM(VBECM)algorithm are combined to solve the model parameters,and parameter extension VBECM(PXVBECM)algorithm is used to further accelerate the VBECM algorithm.The calculation speed of PX-VBECM algorithm is more efficient than that of VBECM algorithm.2.As for the robustness of the model we are concerned about,this paper explains how the it-FA model reduces the influence of local outliers from the perspective of element weight.A series of experiments show that it-FA can give a weight to each element in the sample.When abnormal elements appear in the sample,only the weight of abnormal elements is reduced,and the other normal elements can still provide accurate information and retain more effective element information,so as to improve the robustness of the model.At the same time,empirical studies show that when there are a few dimensional anomalies,the mean square error of model parameters is smaller,the test likelihood is larger,the abnormal elements can be accurately identified,and the factor structure is the same as FA.