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

13.2. Analysis of Size and Shape 345components are orthogonal to the isometric vector, but the shape componentsthemselves are correlated with the isometric component. Cadima andJolliffe (1996) quote an example in which these correlations are as large as0.92.Ranatunga (1989) introduced a method for which the shape componentsare uncorrelated with the isometric component, but her technique sacrificesorthogonality of the vectors of coefficients. A similar problem, namelylosing either uncorrelatedness or orthogonality when searching for simplealternatives to PCA, was observed in Chapter 11. In the present context,however, Cadima and Jolliffe (1996) derived a procedure combining aspectsof double-centering and Ranatunga’s approach and gives shape componentsthat are both uncorrelated with the isometric component and have vectorsof coefficients orthogonal to a 0 . Unfortunately, introducing one desirableproperty leads to the loss of another. As pointed out by Mardia et al.(1996), if x h = cx i where x h , x i are two observations and c is a constant,then in Cadima and Jolliffe’s (1996) method the scores of the two observationsare different on the shape components. Most definitions of shapeconsider two observations related in this manner to have the same shape.Decomposition into size and shape of the variation in measurementsmade on organisms is a complex problem. None of the terms ‘size,’ ‘shape,’‘isometric’ or ‘allometry’ is uniquely defined, which leaves plenty of scopefor vigorous debate on the merits or otherwise of various procedures (see,for example, Bookstein (1989); Jungers et al. (1995)).One of the other approaches to the analysis of size and shape is to definea scalar measure of size, and then calculate a shape vector as the originalvector x of p measurements divided by the size. This is intuitivelyreasonable, but needs a definition of size. Darroch and Mosimann (1985)list a number of possibilities, but home in on g a (x) = ∏ pk=1 xa kk , wherea ′ =(a 1 ,a 2 ,...,a p )and ∑ pk=1 a k = 1. The size is thus a generalizationof the geometric mean. Darroch and Mosimann (1985) discuss a numberof properties of the shape vector x/g a (x) and its logarithm, and advocatethe use of PCA on the log-transformed shape vector, leading to shapecomponents. The log shape vector generalizes the vector v used by Aitchison(1983) in the analysis of compositional data (see Section 13.3), butthe PCs are invariant with respect to the choice of a. As with Aitchison’s(1983) analysis, the covariance matrix of the log shape data has the isometricvector a 0 as an eigenvector, with zero eigenvalue. Hence all the shapecomponents are contrasts between log-transformed variables. Darroch andMosimann (1985) give an example in which both the first and last shapecomponents are of interest.The analysis of shapes goes well beyond the size and shape of organisms(see, for example, Dryden and Mardia (1998) and Bookstein (1991)). Acompletely different approach to the analysis of shape is based on ‘landmarks.’These are well-defined points on an object whose coordinates definethe shape of the object, after the effects of location, scale and rotation have