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Procrustes analysis is concerned with matching two, or more, conﬁgurations of points. Imagine two maps of a region, each giving the positions of the same towns. The two maps are oriented diﬀerently, are of diﬀerent sizes, diﬀer in their accuracies, and use diﬀerent map projections. At a ﬁrst glance it may be diﬃcult to see much resemblance between the two maps. Procrustes analysis addresses such problems, usually taking one of the conﬁgurations as a target and ﬁtting the other to it, optimally, by using appropriate transformations. The ‘maps’ may be derived in many ways and may be in several dimensions but, for the concept of matching to be meaningful, it is important that the conﬁgurations refer to two representations of the same things: the same towns, the same cases (in multidimensional scaling), the same variables (in factor analysis), the same products (in marketing), the same landmarks (describing biological organisms).
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The name comes from Greek mythology where Procrustes was an innkeeper who had a bed (some-times two). Travelers were stretched or limbs chopped to make them ﬁt the bed; with two, adjustment was ensured. The pejorative implications are no worse in Procrustes analysis than in other forms of data-analysis where, misguidedly, data may be ﬁtted into the straightjacket of an unrealistic model.
1. Factor Analysis
The origins of Procrustes analysis (see Hurley and Catell 1962) are in factor analysis, where loadings X1 of k factors f occur in the form X1f and so may be replaced by (X1T)(T−1f) showing that they are determined only up to the arbitrary k × k matrix T. By choosing a suitable T, we may match X1 ‘the traveler’ to a previously determined, or hypothetical set of, loadings given in a matrix X2 ‘the bed’; T represents the ‘chopping/stretching transformation. Usually the degree of match is judged by least squares, minimizing ||X1T – X2|| the sum of the squared distances between like points of the target and transformed conﬁgurations. Without further constraint, this is the multivariate multiplenegression problem with well-known solution
Two of the most important constraints on T are:
(1) T must be orthogonal (denoted here by T = Q);
(2) T must be a direction cosine matrix (denoted here by T = C)
Both choices relate to the interpretation that f is expressed relative to an orthogonal system of coordinate axes and the columns of T give direction cosines of new axes relative to the original orthogonal axes. When T = Q, a new set of orthogonal axes is being sought that give a better match to X2; when T = C we imagine X2 as being referred to oblique axes. The orthogonal Procrustes least-squares solution is Q = VU´ where UΣV´ is the singular value decomposition of X2´X1. With C, minimization is intrinsically more diﬃcult and is less well deﬁned than in the orthogonal case. This is because there are at least two ways of expressing the position of a point relative to oblique axes and also because there are good arguments for replacing ordinary least squares by a weighted form which measures distance in the metric appropriate to the chosen system of oblique axes, giving four variants in which T is either C, C´, C−1 or (C´)−1. The simplest case, T = C, was ﬁrst discussed by Mosier (1939) but not ﬁnally solved until Browne (1967) gave a practical numerical method for solving the normal equations. Special algorithms have been proposed by Browne and Kristof (1969) and Gruvaeus (1970) for the case T = C−1. However, all problems may be tackled uniformly by proceeding as if the directions of all but one of the axes are known and then estimating the optimal direction of the remaining axis; cycling through all the axes iteratively until a stable solution is found for all the directions.
Rather than handling the indeterminacy of factor loadings as above, the modern tendency is to use an internal criterion of simple structure such as Varimax (Kaiser, 1958) that has no reference to an external target X2. Alternatively, X2 may be coded in binary form where xij = 1 indicates a desired high loading of the ith variable on the jth factor and xij = 0 indicates a low loading. This latter form remains a type of Procrustes problem but least squares is no longer appropriate and is replaced by choosing T to maximize the ratio of the sum of squares of the squared high loadings to the sum of squares of the squared low loadings (see Lawley and Maxwell 1971).
2. Variants And Later Developments
The problems outlined above are referred to as twosets Procrustes problems because two conﬁguration matrices X and X are involved. There are many variants:
(1) alternative constraints on T such as T is a rectangular column, orthonormal, or is a permutation matrix;
(2) two transformations: minimize ||X1T1 – X2T2|| ;
(3) isotropic scaling: minimize ||ρX1T – X2||;
(4) anisotropic scaling: minimize ||X1SRT – X2||, or ||X1TSR – X2||, or ||SLX1T – X2||, or ||SLX1SRT – X2||, or ||SLX1TSR – X2||, or ||SLX1 SR1 TSR2 – X2|| for unknown diagonal SR and SL;
(5) replace least squares by e.g., L1-norm, inner- product trace (X2´X1T), robust estimation method;
(6) two-sided Procrustes problems ||RX1T – X2|| where R and T are to be estimated, usually under speciﬁed constraints;
(7) weighted forms—minimize trace N(X1T1 – X2)´W(X1T1 – X2T2) where N and W are given matrices, one of which may be ignored;
(8) handling missing values.
Brief comments on the above list must suﬃce. It should be clear that X1 and X2 no longer represent factor loadings. Isotropic scaling allows for diﬀerences in size; the more esoteric versions of anisotropic scaling verge on the more pejorative interpretation of the Procrustes legend: real violence being done to the data. The product of two orthogonal transformation matrices is equivalent to one and the inner product and least-squares criteria then give the same result. Weights N are useful to cover replication or missing rows of X1 and need not be diagonal, though they usually are. Note that weights are given matrices whereas anisotropic scaling is expressed in terms of unknown diagonal matrices requiring estimation. However, sometimes, W may be a function of the unknown T, as with oblique Procrustes analysis in a metric where W is C´C or its inverse.
Changes in orientation are given by an orthogonal matrix Q, which includes rotations and reﬂections, leaving intact the essential distance and angular properties of the conﬁguration. A p × q orthonormal matrix T = P with q columns represents a projection from p to q dimensions. Projections may be interesting:
(1) because X2 has fewer dimensions than X1;
(2) a match may be required in some speciﬁed number of dimensions, say, q = 2.
When q < p we may still match in p dimensions by rotating in the higher dimensional space; e.g., we may rotate three points lying on a line to ﬁt three points lying in a plane. All we have to do is to augment X2 by p – q zero columns. That ||X1P – X2|| can be less than ||X1Q – X2|| (where X2 is augmented by zero columns) has been seen as a justiﬁcation for preferring projections to rotations but this is not comparing like with like. I believe that projection Procrustes analysis is rarely justiﬁed.
Nowadays, applications of Procrustes analysis are more often than not more to do with matching conﬁgurations other than factor loadings (Commandeur 1990). For example, X1 and X2 may be the two sets of points given by diﬀerent multidimensional scalings (MDS) of the same data, or they may refer to MDS solutions based on diﬀerent proximity/distance matrices measured between the same p objects, perhaps based on diﬀerent variables. Fruitful applications have been found in the food sciences, where X1 and X2 are derived from scores given by two judges on traits of several products (e.g., Dijksterhuis and Gower 1991/2). Applications in the biological sciences include the analysis of shape where the conﬁgurations refer to the directly measured coordinates of landmark features on organisms (see e.g., Bookstein 1991, Dryden and Mardia 1999). In shape analysis the matching of one conﬁguration to another is often termed registration.
Given k conﬁgurations X1, X2, X3,…, Xk we may match them in pairs by doing the corresponding two-sets Procrustes analyses, minimizing sij = ||X1T1 – X2T2|| for all pairs (i, j). Each sij may then be assigned to a matrix S and analyzed by some form of MDS (see Gower 1971 for an example from physical anthropology).
Alternatively, we may minimize Σki<j ||XiTiXjTj||, which for the least-squares criterion, rejecting the solution Ti = 0, is equivalent to minimizing Σki=1 ||XiTiG|| where G = 1/kΣi=jXiTi, often termed the group-average. Thus, generalized Procrustes analysis is a three-mode method of analysis. For some information on algorithms, see ten Berge (1977) To help interpretation, each dimension of the analysis may be summarized in an analysis of variance, partitioning the total into terms for the group average and for departures from the group average. Some dimensions may be identiﬁed with noise and some with signal, further subdivided into displayed and undisplayed parts (Dijksterhuis and Gower 1991 2).
The concept of a group average plays an important part in several psychometric methods as a useful summary of k data sets. Perhaps, for this reason, it has been suggested that maximizing the group average of Procrustes analysis should be a main objective of Procrustes analysis. For orthogonal T, both approaches are equivalent, but for projections, maximizing the group average is not the same as minimizing the criteria given above. The situation gets very convoluted, especially when the variants listed in Sect. 2 are admitted. This is because when focusing on the group average, the transformations, isotropic, and anisotropic scaling may be regarded as being attached to G rather than to the Xi, giving two versions of every variant.
The leading eigenvector of the matrix with (i, j)th element trace (T´iX´iXjTj) estimate isotropic scaling factors and also when Ti = I, this matrix is the basis of the STATIS method. This and similar links between Procrustes analysis and other methods that involve a group average, are likely to be an area for research in the immediate future. Research relating to biological landmark data and shape analysis is currently active and will continue (see Kendall et al. 1999.). Other research will focus on algorithms, the eﬀects of estimation criteria and the diﬃcult area of inference.
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