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

346 13. Principal Component Analysis for Special Types of Databeen removed. Landmarks can be used to examine the shapes of animalsand plants, but they are also relevant to the analysis of shapes of manyother types of object. Here, too, PCA can be useful and it may be implementedin a variety of forms. Kent (1994) distinguishes four versionsdepending on the choice of coordinates and on the choice of whether to usereal or complex PCA (see also Section 13.8).Kent (1994) describes two coordinate systems due to Kendall (1984)and to Bookstein (1991). The two systems arise because of the differentpossible ways of removing location and scale. If there are p landmarks,then the end result in either system of coordinates is that each object isrepresented by a set of (p−1) two-dimensional vectors. There is now a choiceof whether to treat the data as measurements on 2(p − 1) real variables, oras measurements on (p − 1) complex variables where the two coordinatesat each landmark point give the real and imaginary parts. Kent (1994)discusses the properties of the four varieties of PCA thus produced, andcomments that complex PCA is rather uninformative. He gives an exampleof real PCA for both coordinate systems.Horgan (2000) describes an application of PCA to the comparison ofshapes of carrots. After bringing the carrots into the closest possible alignment,distances are calculated between each pair of carrots based on theamount of non-overlap of their shapes. A principal coordinate analysis(Section 5.2) is done on these distances, and Horgan (2000) notes thatthis is equivalent to a principal component analysis on binary variablesrepresenting the presence or absence of each carrot at a grid of points in twodimensionalspace. Horgan (2000) also notes the similarity between his techniqueand a PCA on the grey scale levels of aligned images, giving so-calledeigenimages. This latter procedure has been used to analyse faces (see, forexample Craw and Cameron, 1992) as well as carrots (Horgan, 2001).13.3 Principal Component Analysis forCompositional DataCompositional data consist of observations x 1 , x 2 ,...,x n for which eachelement of x i is a proportion, and the elements of x i are constrained tosum to unity. Such data occur, for example, when a number of chemicalcompounds or geological specimens or blood samples are analysed, and theproportion in each of a number of chemical elements is recorded. As notedin Section 13.1, Gower (1967) discusses some geometric implications thatfollow from the constraints on the elements of x, but the major reference forPCA on compositional data is Aitchison (1983). Because of the constraintson the elements of x, and also because compositional data apparently oftenexhibit non-linear rather than linear structure among their variables,Aitchison (1983) proposes that PCA be modified for such data.