A Comparative Study Of Some Model Selection Methods For Principal Components Analysis

In recent decades,the study of signal processing has become a hot problem,where model selection methods are usually used to estimate the number of signals,such as AIC and BIC proposed by Wax and Kailath[18].With the development of science and technology,the dimension of the variables may be very high and in this case AIC may be not consistent.Thus,model selection methods for high dimensional case are proposed.We mainly consider the case p??.In the literature,the number of signals can be detected by two consistent methods,i.e.,KN and BFC proposed by Kritchman and Nadler[25]and Bai,Fujikoshi and Choi[33],respectively.For fixed p,Hu and Zhu[14]propesed ILP and demonstrated ILP is consistent,On the other hand,for p ??,Hu and Zhu[14]proposed EAIC and show that EAIC is a natural extension of AIC for fixed p.In the literature,the comparative study among the above model selection criteria mainly focuses on normal distribution.In this paper,we further compare them under non-normal distributions.In Chapter 1,we introduce the elementary knowledge of principal component analysis and some background of model selection methods.In Chapter 2,we introduce some random matrix theories.In Chapter 3,under non-normal distributions,we compare some model selection methods mentioned above by simulations.For fixed p,to compare ILP?AIC?MAIC and KN,we consider the t distribution with 5 degrees of freedom.By simulation,we find that AIC tends to overestimate the number of signals,while ILP?MAIC and KN are consistent.For high-dimensional case,we consider the following three cases:(1)n??,p?? p/n?c(<1);(2)n??,p?? p/n??(c = 1);(3)n??,p?? p/n?c(>1).To compare EAIC?ILP?AIC?BFC?KN and MAIC,we first consider the t distribution with 5 degrees of freedom.By simulation,we find that EAIC?BFC and KN are consistent.We also find that EAIC is in general better than other methods when the SNR is low.We also study the t distribution with 4 degrees of freedom,x2 distribution with 3 degrees of freedom,and ? distribution with parameters(2,3).In Chapter 4,we give the summary.