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201.
Some existing three-way factor analysis and MDS models incorporate Cattell's “Principle of Parallel Proportional Profiles”.
These models can—with appropriate data—empirically determine a unique best fitting axis orientation without the need for a
separate factor rotation stage, but they have not been general enough to deal with what Tucker has called “interactions” among
dimensions. This article presents a proof of unique axis orientation for a considerably more general parallel profiles model
which incorporates interacting dimensions. The model, Xk=AADk HBDk B', does not assume symmetry in the data or in the interactions among factors. A second proof is presented for the symmetrically
weighted case (i.e., whereADk=BDk). The generality of these models allows one to impose successive restrictions to obtain several useful special cases, including
PARAFAC2 and three-way DEDICOM.
We want to express appreciation for the contributions of several colleagues: Jos M. F. ten Berge and Henk A. L. Kiers carefully
went through more than one version of this article, found an important error, and contributed many improvements. J. Douglas
Carroll and Shizuhiko Nishisato acted with unusual editorial preserverance and flexibility, thereby making possible the successful
completion of a difficult assessment and revision process. Joseph B. Kruskal has long provided crucial mathematical insights
and inspiration to those working in this area, but this is particularly true for us. His contributions to this specific article
include early discussion of basic questions and careful examination of some earlier attempted proofs, finding them to be invalid.
The anonymous reviewers also made useful suggestions. Some portions of this work were supported in part by a grant from the
Natural Sciences and Engineering Research Council of Canada. 相似文献
202.
Jos M. F. ten Berge 《Psychometrika》1996,61(4):695-696
The solution of weakly constrained regression problems typically requires the iterative search, in a given interval, of a point where a certain function has a zero derivative. This note deals with improved bounds for the interval to be searched. 相似文献
203.
Joseph B. Kruskal 《Psychometrika》1976,41(3):281-293
Some methods that analyze three-way arrays of data (including INDSCAL and CANDECOMP/PARAFAC) provide solutions that are not subject to arbitrary rotation. This property is studied in this paper by means of the triple product [A, B, C] of three matrices. The question is how well the triple product determines the three factors. The answer: up to permutation of columns and multiplication of columns by scalars—under certain conditions. In this paper we greatly expand the conditions under which the result is known to hold. A surprising fact is that the nonrotatability characteristic can hold even when the number of factors extracted is greater thanevery dimension of the three-way array, namely, the number of subjects, the number of tests, and the number of treatments.This paper is being published in place of Dr. Kruskal's presidential address to the Psychometric Society, April, 1975. Further results like those in this paper, as well as a surprising connection with an area of mathematics called arithmetic complexity theory, will be found in a more recent paper [Kruskal, in press]. 相似文献
204.
A probabilistic model for the validation of behavioral hierarchies is presented. Estimation is by means of iterative convergence to maximum likelihood estimates, and two approaches to assessing the fit of the model to sample data are discussed. The relation of this general probabilistic model to other more restricted models which have been presented previously is explored and three cases of the general model are applied to exemplary data. 相似文献
205.
A modified version of a coordinate adjustment technique which permits the analysis of comparisons of psychological intervals for an unknown ordering of stimuli is described and compared to the original version and to TORSCA. For configurations with a large number of points, knowledge of the rank order of the stimuli does not improve the solution. For configurations with a small number of points, the performance of the new algorithm with an unknown ordering is equivalent to TORSCA.This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. 相似文献
206.
Principal component analysis of three-mode data by means of alternating least squares algorithms 总被引:4,自引:0,他引:4
A new method to estimate the parameters of Tucker's three-mode principal component model is discussed, and the convergence properties of the alternating least squares algorithm to solve the estimation problem are considered. A special case of the general Tucker model, in which the principal component analysis is only performed over two of the three modes is briefly outlined as well. The Miller & Nicely data on the confusion of English consonants are used to illustrate the programs TUCKALS3 and TUCKALS2 which incorporate the algorithms for the two models described. 相似文献
207.
A model and algorithm for multidimensional scaling with external constraints on the distances 总被引:1,自引:0,他引:1
A method for externally constraining certain distances in multidimensional scaling configurations is introduced and illustrated. The approach defines an objective function which is a linear composite of the loss function of the point configurationX relative to the proximity dataP and the loss ofX relative to a pseudo-data matrixR. The matrixR is set up such that the side constraints to be imposed onX's distances are expressed by the relations amongR's numerical elements. One then uses a double-phase procedure with relative penalties on the loss components to generate a constrained solutionX. Various possibilities for constructing actual MDS algorithms are conceivable: the major classes are defined by the specification of metric or nonmetric loss for data and/or constraints, and by the various possibilities for partitioning the matricesP andR. Further generalizations are introduced by substitutingR by a set ofR matrices,R
i
,i=1, ...r, which opens the way for formulating overlapping constraints as, e.g., in patterns that are both row- and column-conditional at the same time. 相似文献
208.
Bruce Bloxom 《Psychometrika》1978,43(3):397-408
A gradient method is used to obtain least squares estimates of parameters of them-dimensional euclidean model simultaneously inN spaces, given the observation of all pairwise distances ofn stimuli for each space. The procedure can estimate an additive constant as well as stimulus projections and the metric of the reference axes of the configuration in each space. Each parameter in the model can be fixed to equal some a priori value, constrained to be equal to any other parameter, or free to take on any value in the parameter space. Two applications of the procedure are described. 相似文献
209.
A monte carlo investigation of recovery of structure by alscal 总被引:1,自引:0,他引:1
A Monte Carlo study was carried out to investigate the ability of ALSCAL to recover true structure inherent in simulated proximity measures. The nature of the simulated data varied according to (a) number of stimuli, (b) number of individuals, (c) number of dimensions, and (d) level of random error. Four aspects of recovery were studied: (a) SSTRESS, (b) recovery of true distances, (c) recovery of stimulus dimensions, and (d) recovery of individual weights. Results indicated that all four measures were rather strongly affected by random error. Also, SSTRESS improved with fewer stimuli in more dimensions, but the other three indices behaved in the opposite fashion. Most importantly, it was found that the number of individuals, over the range studied, did not have a substantial effect on any of the four measures of recovery. Practical implications and suggestions for further research are discussed.The authors wish to thank Drs. Forrest W. Young, Paul D. Isaac and Thomas E. Nygren, who provided many helpful comments during this project. 相似文献
210.
Rolf Langeheine 《Psychometrika》1982,47(4):427-442
PINDIS, as recently presented by Lingoes and Borg [1978] not only marks the latest development within the scope of individual differences scaling, but, may be of benefit in some closely related topics, such as target analysis. Decisions on whether the various models available from PINDIS fit fallible data are relatively arbitrary, however, since a statistical theory of the fit measures is lacking. Using Monte Carlo simulation, expected fit measures as well as some related statistics were therefore obtained by scaling sets of 4(4)24 random configurations of 5(5)30 objects in 2, 3, and 4 dimensions (individual differences case) and by fitting one random configuration to a fixed random target for 5(5)30 objects in 2, 3, and 4 dimensions (target analysis case). Applications are presented. 相似文献