Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

3.9. Models for Principal Component Analysis 61the one non-trivial example considered by Martin (1988), the distributionsare identical for each observation and spherical, so that the underlyingcovariance matrix has the form Σ + σ 2 I p . Lynn and McCulloch (2000) usePCA to estimate latent fixed effects in a generalized linear model, and deFalguerolles (2000) notes that PCA can be viewed as a special case of thelarge family of generalized bilinear models.Although PCA is a largely descriptive tool, it can be argued that buildinga model gives a better understanding of what the technique does, helpsto define circumstances in which it would be inadvisable to use it, andsuggests generalizations that explore the structure of a data set in a moresophisticated way. We will see how either the fixed effects model or Tippingand Bishop’s (1999a) model can be used in deciding how many PCsto retain (Section 6.1.5); in examining mixtures of probability distributions(Section 9.2.3); in a robust version of PCA (Section 10.4); in analysing functionaldata (Section 12.3.4); in handling missing data (Section 13.6); and ingeneralizations of PCA (Section 14.1, 14.2). One application that belongsin the present chapter is described by Ferré (1995a). Here ˆµ 1 , ˆµ 2 ,...,ˆµ kare estimates, derived from k samples of sizes n 1 ,n 2 ,...,n k of vectors of pparameters µ 1 , µ 2 ,...,µ k .Ferré (1995a) proposes estimates that minimizean expression equivalent to (3.9.1) in which w i = ninwhere n = ∑ ki=1 n i;x i , z i are replaced by µ i , ˆµ i where ˆµ i is a projection onto an optimal q-dimensional space; and M is chosen to be S −1 where S is an estimate ofthe common covariance matrix for the data from which µ 1 , µ 2 ,...,µ k areestimated. The properties of such estimators are investigated in detail byFerré (1995a)