An alternative to working with geometrical configurations directly is to work with inter-landmark distances. Consider the squared Euclidean distance matrix D from the configuration X (k × m matrix) given by:
where (X)r are the coordinates of the rth point (r = 1, …, k). We consider methods for shape and size-and-shape analysis that involve working with the full collection of such distance matrices, and in some cases the estimates can be similar to Procrustes techniques. Traditional morphometrics studying lengths, ratios of lengths or angles usually considers just a subset of the inter-landmark distances, and was summarized in Section 2.3.
Multidimensional scaling (MDS) is concerned with constructing a configuration of k points in Euclidean space from information about the distances between the k points (see Mardia et al. 1979, pp. 394–398). Consider X to be a k × m configuration with k × k squared Euclidean distance matrix D, as in Equation (15.1). It can be shown that D is a squared Euclidean distance matrix if and only if
is positive semi-definite, where C is the k × k centring matrix of Equation (2.3). We can interpret B as the centred inner product matrix of the ...