Robust Bilinear Factor Analysis

Factor analysis(FA)is a dimensionality reduction method for probabilistic models.The observed values are vector data,and the matrix data must be first straightened into vectors for fitting,which not only destroys the potential correlation between the rows and columns of the matrix structure,but also makes the straightening data have a high dimension.In recent years,dimensionality reduction methods for matrix data have been proposed,such as bilinear probability principal component analysis(BPPCA).It directly reduces the dimension of the matrix data rather than straightening the vector,which reduces the computational complexity.Moreover,it can effectively update the analytical solution of the parameters whether the data contains hidden variables or not.Bilinear factor analysis model(BFA)also solves the problem of complexity and potential variables.However,the above model is built on the normal distribution density model of matrix variables and is very sensitive to irregular observation.When the data set has thicker tail than normal value or more outliers and outliers,the estimator obtained by using the normal distribution of matrix will be affected.To understand this problem,we extended the model from the normal distribution of matrix variables to the t-distribution,and proposed a bilinear factor analysis model(tBFA)based on the t-distribution of matrix variables.The matrix variable t-distribution fitting factor analysis,with a heavier tail and a more free parameter,is stable for irregular observations in practice.The matrix variable t distribution is used to fit the factor analysis instead of the normal distribution.It adjusts the thickness of the distributed tail,has a heavier tail,and contains a freer parameter.Therefore,it is stable to irregular observation values in practice,and can be used for robust estimation of mean value,regression coefficient,and variance-covariance matrix of multiple linear models,even in the absence of data.When they handle outliers in a natural way,the resulting new probabilistic model is more robust in practice.In order to obtain the maximum likelihood estimation of the robust bilinear factor analysis model,this paper proposes four algorithms,ECM,ECME,AECM1 and AECM2,to fit the model.These four algorithms are simple and stable.The difference between ECM(ECME)and AECM1(AECM2)is that ECM and ECME only introduce variables t but not potential variables Z.AECM1 and AECM2 introduce both variables t and potential variables Z.The difference between ECM and ECME algorithms lies in that ECM is the likelihood function of complete data with maximum expectation of each parameter respectively.ECME is the likelihood function of complete data with maximum expectation of row and column factor load matrix and noise variance matrix,but the likelihood function of observation data with maximum degree of freedom n.The difference between AECM1 and AECM2 is also that the maximum degree of freedom n of AECM2 is the likelihood function of observed data.Experiments verify the performance of the four algorithms.Since ECM and ECME do not contain missing data,ECME converges the fastest,while AECM2 enjoys the lowest computational complexity.The simulation results show that when the same likelihood value is convergent,the ECME algorithm needs the least number of iterations and the algorithm time,while the AECM2 algorithm has the most iterations.We verify that the four algorithms are insensitive to the initial values of the parameters,and obtain robust estimates of the degrees of freedom,mean matrix,factor loading matrix and variance-covariance matrix.In the case of simulated or actual grain data with outliers,the parameter estimation accuracy of the tBFA model is much higher than that of the BFA model,which verifies that the tBFA model is more stable to irregular data.