Analysis of individual differences in multidimensional scaling via an n-way generalization of 'eckart-young' decomposition
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.
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TL;DR: A new method for automatic indexing and retrieval to take advantage of implicit higher-order structure in the association of terms with documents (“semantic structure”) in order to improve the detection of relevant documents on the basis of terms found in queries.
Abstract: A new method for automatic indexing and retrieval is described. The approach is to take advantage of implicit higher-order structure in the association of terms with documents (“semantic structure”) in order to improve the detection of relevant documents on the basis of terms found in queries. The particular technique used is singular-value decomposition, in which a large term by document matrix is decomposed into a set of ca. 100 orthogonal factors from which the original matrix can be approximated by linear combination. Documents are represented by ca. 100 item vectors of factor weights. Queries are represented as pseudo-document vectors formed from weighted combinations of terms, and documents with supra-threshold cosine values are returned. initial tests find this completely automatic method for retrieval to be promising.
12,443 citations
Cites background from "Analysis of individual differences ..."
...Another is two-mode factor analysis [25] [26] [27] [28] , in which terms and documents would again be represented as points in a space, but similarity is given by the inner product between points....
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TL;DR: This survey provides an overview of higher-order tensor decompositions, their applications, and available software.
Abstract: This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.
9,227 citations
TL;DR: This paper presents a framework for organizational analysis that organizes the organizational effectiveness literature, indicates which concepts are most central to the construct of organizational effectiveness, makes clear the values in which the concepts are embedded, and provides an overarching framework to guide subsequent efforts at organizational assessment.
Abstract: This paper presents a framework for organizational analysis. The empirically derived approach does not emerge from the observation of actual organizations, but from the ordering, through multivariate techniques, of criteria that organizational theorists and researchers use to evaluate the performance of organizations.
In a two-stage study, organizational theorists and researchers were impaneled to make judgments about the similarity of commonly used effectiveness criteria. The model derived from the second group closely replicated the first, and in convergence suggested that three value dimensions control-flexibility, internal-external, and means-ends underlie conceptualizations of organizational effectiveness.
When these value dimensions are juxtaposed, a spatial model emerges. The model serves a number of important functions. It organizes the organizational effectiveness literature, indicates which concepts are most central to the construct of organizational effectiveness, makes clear the values in which the concepts are embedded, demonstrates that the effectiveness literature and the general literature on organizational analysis are analogues of one another, and provides an overarching framework to guide subsequent efforts at organizational assessment.
3,183 citations
01 Jan 1970
TL;DR: It is shown that an extension of Cattell's principle of rotation to Proportional Profiles (PP) offers a basis for determining explanatory factors for three-way or higher order multi-mode data.
Abstract: Simple structure and other common principles of factor rotation do not in general provide strong grounds for attributing explanatory significance to the factors which they select. In contrast, it is shown that an extension of Cattell's principle of rotation to Proportional Profiles (PP) offers a basis for determining explanatory factors for three-way or higher order multi-mode data. Conceptual models are developed for two basic patterns of multi-mode data variation, systemand object-variation, and PP analysis is found to apply in the system-variation case. Although PP was originally formulated as a principle of rotation to be used with classic two-way factor analysis, it is shown to embody a latent three-mode factor model, which is here made explicit and generalized frown two to N "parallel occasions". As originally formulated, PP rotation was restricted to orthogonal factors. The generalized PP model is demonstrated to give unique "correct" solutions with oblique, non-simple structure, and even non-linear factor structures. A series of tests, conducted with synthetic data of known factor composition, demonstrate the capabilities of linear and non-linear versions of the model, provide data on the minimal necessary conditions of uniqueness, and reveal the properties of the analysis procedures when these minimal conditions are not fulfilled. In addition, a mathematical proof is presented for the uniqueness of the solution given certain conditions on the data. Three-mode PP factor analysis is applied to a three-way set of real data consisting of the fundamental and first three formant frequencies of 11 persons saying 8 vowels. A unique solution is extracted, consisting of three factors which are highly meaningful and consistent with prior knowledge and theory concerning vowel quality. The relationships between the three-mode PP model and Tucker's multi-modal model, McDonald's non-linear model and Carroll and Chang's multi-dimensional scaling model are explored.
3,120 citations
TL;DR: The four Purposes of Multidimensional Scaling, Special Solutions, Degeneracies, and Local Minima, and Avoiding Trivial Solutions in Unfolding are explained.
Abstract: Fundamentals of MDS.- The Four Purposes of Multidimensional Scaling.- Constructing MDS Representations.- MDS Models and Measures of Fit.- Three Applications of MDS.- MDS and Facet Theory.- How to Obtain Proximities.- MDS Models and Solving MDS Problems.- Matrix Algebra for MDS.- A Majorization Algorithm for Solving MDS.- Metric and Nonmetric MDS.- Confirmatory MDS.- MDS Fit Measures, Their Relations, and Some Algorithms.- Classical Scaling.- Special Solutions, Degeneracies, and Local Minima.- Unfolding.- Unfolding.- Avoiding Trivial Solutions in Unfolding.- Special Unfolding Models.- MDS Geometry as a Substantive Model.- MDS as a Psychological Model.- Scalar Products and Euclidean Distances.- Euclidean Embeddings.- MDS and Related Methods.- Procrustes Procedures.- Three-Way Procrustean Models.- Three-Way MDS Models.- Modeling Asymmetric Data.- Methods Related to MDS.
3,096 citations
References
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TL;DR: The model for three-mode factor analysis is discussed in terms of newer applications of mathematical processes including a type of matrix process termed the Kronecker product and the definition of combination variables.
Abstract: The model for three-mode factor analysis is discussed in terms of newer applications of mathematical processes including a type of matrix process termed the Kronecker product and the definition of combination variables. Three methods of analysis to a type of extension of principal components analysis are discussed. Methods II and III are applicable to analysis of data collected for a large sample of individuals. An extension of the model is described in which allowance is made for unique variance for each combination variable when the data are collected for a large sample of individuals.
3,810 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
"Analysis of individual differences ..." refers background in this paper
...This argument has been developed in more detail by Horan [1969]. The model may not hold in every case, but if it does we gain a unique and hopefully psychologically meaningful orientation of axes, thus obviating the rotational problem and defining much stronger scales of measurement than is usual in multidimensional scaling....
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Book•
15 Jan 1958
3,060 citations
"Analysis of individual differences ..." refers methods in this paper
...To convert them into (ratio scale) distance estimates additive constants were est imated (separately for each subject) b y the "one dimensional subspace" method described in Torgerson [1958]. Scalar products matrices were computed from these est imated distances, normalized to unit sum of squares, and subjected to the analysis described above....
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...can be done by using one of the standard procedures described in Torgerson [1958]. We then use the equations also described in Torgerson [1958, pp....
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TL;DR: In this article, the authors used both mathematical and Monte Carlo results to establish and clarify the possibility of extracting metric information from purely ordinal data for two multidimensional cases: (a) analysis of proximities, in which one is given a single rank order of all n(n−1) 2 pairs of n objects with respect to psychological similarity or "proximity".
337 citations