SupposePi(i) (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σi=1m Σi=1n Δ2(Pj(i)Gi), whereGi is the centroid of them pointsPi(i) (i = 1, 2, ...,m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special casem = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.

Generalized procrustes analysis

It is often necessary in population biology to compare two sets of distance measures. These measures can be based on genetic markers, morphological traits, geographic separation, ecological divergence, and so on. The distance measures can take various forms and frequently have unknown distributional properties. Many different procedures have been developed to compare the correspondence of one set of distances with another set. Prominent among them are the: (1) matrix correlation techniques of Sokal and Rohlf (1962), and Sneath and Sokal (1973); (2) networkmatching techniques of Spielman (1973); (3) matrix dilation and rotation techniques of Gower (1971) and of Schonemann and Carrol (1970); and (4) smallest-space techniques of Lingoes (1965) and Guttman (1968). Each of these strategies has strong points, but all suffer from a difficulty in assessing the statistical significance of attained correspondence. The problem is that a set of all possible pairwise distances between k units (populations, taxa, habitats, etc.) cannot be independent. More recently, a test of matrix correspondence-originally developed by Mantel (1967) and widely applied in geography (Hubert and Golledge, 1982) and psychometrics (Hubert, 1979a, b)-has caught the attention of population biologists (Sokal, 1979; Sokal et al., 1980; Douglas and Endler, 1982; Dow and Cheverud, 1985; Schnell et al., 1985, 1986; Sokal et al., 1986, 1987; O'Brien, 1987; Smouse and Wood, 1987). Attractions of the Mantel procedure are its wide applicability and computational simplicity. Mantel (1967) presented a formal analysis of matrix correspondence based on the assumption of asymptotic normality for a particular test criterion. Later workers (Mielke et al., 1981) developed more general procedures for Mantel statistics that assume a Pearson Type III distributional form. The most widely used evaluation procedure, however, involves the construction of a null distribution by Monte Carlo randomization, whereby one of the matrices is held rigid and the other has its rows and corresponding columns randomly permuted (Cliff and Ord, 1981). Dietz (1983) evaluated the Mantel test as one of several permutational tests for association between distance matrices. When dimensions of two matrices are small (say K ' 7), it is customary to evaluate the Mantel test criterion for all K! equally likely permutations. If K is large, then a large number of random permutations are sampled with replacement. Although the test has been useful in its present form, there are some simple modifications and extensions that would encourage even wider deployment in population work. Our purpose here is to sketch these changes and to illustrate them with an example drawn from the Yanomama Indians of lowland South America.

http://www.pelagicos.net/MARS6300/readings/Smouse_et_al._1986.pdf

Multiple regression and correlation extensions of the mantel test of matrix correspondence

Simulated vegetation data were used to assess the relative robustness of ordination techniques to variations in the model of community variation in relation to environment. The methods compared were local non-metric multidimensional scaling (LNMDS), detrended correspondence analysis (DCA), Gaussian ordination (GO), principal components analysis (PCA) and principal co-ordinates analysis (PCoA). Both LNMDS and PCoA were applied to a matrix of Bray-Curtis coefficients. The results clearly demonstrated the ineffectiveness of the linear techniques (PCA, PCoA), due to curvilinear distortion. Gaussian ordination proved very sensitive to noise and was not robust to marked departures from a symmetric, unimodal response model. The currently popular method of DCA displayed a lack of robustness to variations in the response model and the sampling pattern. Furthermore, DCA ordinations of two-dimensional models often exhibited marked distortions, even when response surfaces were unimodal and symmetric. LNMDS is recommended as a robust technique for indirect gradient analysis, which deserves more widespread use by community ecologists.

An evaluation of the relative robustness of techniques for ecological ordination

Task complexity: Definition of the construct

Finding the relationship between two coordinate systems by using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. The solution has applications in stereophotogrammetry and in robotics. We present here a closed-form solution to the least-squares problem for three or more points. Currently, various empirical, graphical, and numerical iterative methods are in use. Derivation of a closed-form solution can be simplified by using unit quaternions to represent rotation, as was shown in an earlier paper [ J. Opt. Soc. Am. A4, 629 ( 1987)]. Since orthonormal matrices are used more widely to represent rotation, we now present a solution in which 3 × 3 matrices are used. Our method requires the computation of the square root of a symmetric matrix. We compare the new result with that obtained by an alternative method in which orthonormality is not directly enforced. In this other method a best-fit linear transformation is found, and then the nearest orthonormal matrix is chosen for the rotation. We note that the best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points.

/pdf/closed-form-solution-of-absolute-orientation-using-2j8f6jxfqj.pdf

Closed-form solution of absolute orientation using orthonormal matrices

A test of the progression-regression hypotheses in a cognitive inference task☆

A Monte Carlo comparison of nonmetric multidimensional scaling and factor analysis

A least squares method is presented for fitting a given matrixA to another given matrixB under choice of an unknown rotation, an unknown translation, and an unknown central dilation. The procedure may be useful to investigators who wish to compare results obtained with nonmetric scaling techniques across samples or who wish to compare such results with those obtained by conventional factor analytic techniques on the same sample.

Fitting one matrix to another under choice of a central dilation and a rigid motion

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A test of the progression-regression hypotheses in a cognitive inference task☆

A Monte Carlo comparison of nonmetric multidimensional scaling and factor analysis

Fitting one matrix to another under choice of a central dilation and a rigid motion