Generalized procrustes analysis
TL;DR: In this article, the authors investigated the problem of translating, rotating, reflecting and scaling configurations to minimize the goodness-of-fit criterion, where Gi is the centroid of the points in p-dimensional space.
Abstract: 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.
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TL;DR: This work describes a method for building models by learning patterns of variability from a training set of correctly annotated images that can be used for image search in an iterative refinement algorithm analogous to that employed by Active Contour Models (Snakes).
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TL;DR: It is shown that the protocol developed for these instruments successfully recaptures known biological results, and additionally that biological conclusions are consistent across sequencing platforms (the HiSeq2000 versus the MiSeq) and across the sequenced regions of amplicons.
Abstract: DNA sequencing continues to decrease in cost with the Illumina HiSeq2000 generating up to 600 Gb of paired-end 100 base reads in a ten-day run. Here we present a protocol for community amplicon sequencing on the HiSeq2000 and MiSeq Illumina platforms, and apply that protocol to sequence 24 microbial communities from host-associated and free-living environments. A critical question as more sequencing platforms become available is whether biological conclusions derived on one platform are consistent with what would be derived on a different platform. We show that the protocol developed for these instruments successfully recaptures known biological results, and additionally that biological conclusions are consistent across sequencing platforms (the HiSeq2000 versus the MiSeq) and across the sequenced regions of amplicons.
6,840 citations
Cites methods from "Generalized procrustes analysis"
...…driven by the results of principal coordinates analyses based on weighted UniFrac distances, we compared these plots using Procrustes analysis (Gower, 1975; Figure 1; Table 1) as implemented in QIIME and found that the observations were highly reproducible across lanes, read directions and…...
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TL;DR: FactoMineR an R package dedicated to multivariate data analysis with the possibility to take into account different types of variables (quantitative or categorical), different kinds of structure on the data, and finally supplementary information (supplementary individuals and variables).
Abstract: In this article, we present FactoMineR an R package dedicated to multivariate data analysis. The main features of this package is the possibility to take into account different types of variables (quantitative or categorical), different types of structure on the data (a partition on the variables, a hierarchy on the variables, a partition on the individuals) and finally supplementary information (supplementary individuals and variables). Moreover, the dimensions issued from the different exploratory data analyses can be automatically described by quantitative and/or categorical variables. Numerous graphics are also available with various options. Finally, a graphical user interface is implemented within the Rcmdr environment in order to propose an user friendly package.
6,472 citations
TL;DR: In this paper, a new method is presented that generalizes Siegel and Benson's (1982) resistant-fit theta-rho analysis so that more than two objects can be compared at the same time.
Abstract: Superimposition methods for comparing configurations of landmarks in two or more specimens are reviewed. These methods show differences in shape among specimens as residuals after rotation, translation, and scaling them so that they align as well as possible. A new method is presented that generalizes Siegel and Benson's (1982) resistant-fit theta-rho analysis so that more than two objects can be compared at the same time. Both least-squares and resistant-fit approaches are generalized to allow for affine transformations (uniform shape change). The methods are compared, using artificial data and data on 18 landmarks on the wings of 127 species of North American mosquitoes. Graphical techniques .are also presented to help sum- marize the patterns of differences in shape among the objects being compared. (Morphometrics; resistant-fit; least-squares; theta-rho analysis; rotational fit; affine transformations.) An important problem in morphomet- is now easy to display a transformation grid rics is that of comparing configurations of that maps the configuration of landmarks landmarks in two or more specimens. of one organism exactly into those of Thompson (1917) suggested an elegant ap- another. proach, using "transformation grids," that An alternative approach to fitting a mod- depicts the overall form of one organism el that completely describes the differences as a distortion in the shape of a reference between two organisms is to fit a very sim- organism. The basic idea was to place a ple model that only takes into consider- Cartesian coordinate grid over the refer- ation global parameters such as differences ence organism and then distort the image in rotation, translation, and scale. Geo- of the organism (including the grid) in var- metrically, this corresponds to superim- ious ways until the form of the second or- posing one organism on top of another so ganism was achieved. The differences in that its landmarks align as well as possible shapes of the two organisms are shown by (in some sense) with the positions of the the deviations of the fitted grid (usually corresponding landmarks on the second. bent and stretched in various ways) from Differences in shape are then shown by the original simple square grid. Thompson differences in positions of corresponding (1917) sketched the grids subjectively landmarks. Shape differences between two without an explicit specification of which organisms are found by studying these re- landmarks were used. Not all landmarks siduals. These methods are the subject of shown in pairs of drawings are located ex- the present paper. actly where the superimposed grids would Sneath (1967) investigated the problem imply they should be. This means that the of finding the optimal translation, rota- grids should be more complex than those tion, and size change of one object in order shown in Thompson (1917) in order to ac- for it to be superimposed on another. The curately show the differences between two two objects were represented as sets of x,y- organisms. Bookstein (1978) developed the coordinates of landmarks. A least-squares method of biorthogonal grid analysis criterion was used to measure the goodness which quantifies Thompson's approach and of fit of one object to another. Gower (1971) makes it objective. But it is complex and further developed Sneath's (1967) method has not been applied very often. A recent and expressed the operations in terms of b~ak-through (Bookstein, 1989) is the use matrix algebra. Siegel and Benson (1982) of methods based on thin-plate splines. It made the important observation that a
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Cites background or methods from "Generalized procrustes analysis"
...The purpose of this scaling is to ensure that 2i tr(X**X;*t) = n (Gower, 1975)....
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..., an Eckart-Young singular-value decomposition of X'tX' is performed (Gower, 1975)....
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TL;DR: This paper briefly update the discussion in that paper and summarize the advances in the ten years since the paper by Rohlf and Marcus (1993) and speculate on future directions in morphometric analysis.
Abstract: The analysis of shape is a fundamental part of much biological research. As the field of statistics developed, so have the sophistication of the analysis of these types of data. This lead to multivariate morphometrics in which suites of measurements were analyzed together using canonical variates analysis, principal components analysis, and related methods. In the 1980s, a fundamental change began in the nature of the data gathered and analyzed. This change focused on the coordinates of landmarks and the geometric information about their relative positions. As a by-product of such an approach, results of multivariate analyses could be visualized as configurations of landmarks back in the original space of the organism rather than only as statistical scatter plots. This new approach, called “geometric morphometrics”, had benefits that lead Rohlf and Marcus (1993) to proclaim a “revolution” in morphometrics. In this paper, we briefly update the discussion in that paper and summarize the advances in the ten years since the paper by Rohlf and Marcus. We also speculate on future directions in morphometric analysis.
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TL;DR: In this paper, an individual differences model for multidimensional scaling is outlined in which individuals are assumed differentially to weight the several dimensions of a common "psychological space" and a corresponding method of analyzing similarities data is proposed, involving a generalization of Eckart-Young analysis to decomposition of three-way (or higher-way) tables.
Abstract: An individual differences model for multidimensional scaling is outlined in which individuals are assumed differentially to weight the several dimensions of a common “psychological space”. A corresponding method of analyzing similarities data is proposed, involving a generalization of “Eckart-Young analysis” to decomposition of three-way (or higher-way) tables. In the present case this decomposition is applied to a derived three-way table of scalar products between stimuli for individuals. This analysis yields a stimulus by dimensions coordinate matrix and a subjects by dimensions matrix of weights. This method is illustrated with data on auditory stimuli and on perception of nations.
4,520 citations
TL;DR: In this paper, the problem of approximating one matrix by another of lower rank is formulated as a least-squares problem, and the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another.
Abstract: The mathematical problem of approximating one matrix by another of lower rank is closely related to the fundamental postulate of factor-theory. When formulated as a least-squares problem, the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another. The solution of the problem is simplified by first expressing the matrices in a canonic form. It is found that the problem always has a solution which is usually unique. Several conclusions can be drawn from the form of this solution. A hypothetical interpretation of the canonic components of a score matrix is discussed.
3,576 citations
TL;DR: In this paper, a least square 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.
Abstract: 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.
405 citations
01 Jan 1971
261 citations
TL;DR: A least squares method for approximating a given symmetric matrixB by another matrixB which is orthogonally similar to a second given matrixA is derived and then generalized to nonsymmetric (but square)A andB.
Abstract: A least squares method for approximating a given symmetric matrixB by another matrixB which is orthogonally similar to a second given matrixA is derived and then generalized to nonsymmetric (but square)A andB A possible application to ordering problems is discussed
49 citations