Monotone spline transformations for dimension reduction

Consider a set of data consisting of measurements ofn objects with respect top variables displayed in ann ×p matrix. A monotone transformation of the values in each column, represented as a linear combination of integrated basis splines, is assumed determined by a linear combination of a new set of values characterizing each row object. Two different models are used: one, an Eckart-Young decomposition model, and the other, a multivariate normal model. Examples for artificial and real data are presented. The results indicate that both methods are helpful in choosing dimensionality and that the Eckart-Young model is also helpful in displaying the relationships among the objects and the variables. Also, results suggest that the resulting transformations are themselves illuminating.     

Some boundary conditions for a monotone analysis of symmetric matrices

This paper gives a rigorous and greatly simplified proof of Guttman's theorem for the least upper-bound dimensionality of arbitrary real symmetric matricesS, where the points embedded in a real Euclidean space subtend distances which are strictly monotone with the off-diagonal elements ofS. A comparable and more easily proven theorem for the vector model is also introduced. At mostn-2 dimensions are required to reproduce the order information for both the distance and vector models and this is true for any choice of real indices, whether they define a metric space or not. If ties exist in the matrices to be analyzed, then greatest lower bounds are specifiable when degenerate solutions are to be avoided. These theorems have relevance to current developments in nonmetric techniques for the monotone analysis of data matrices.This research in nonmetric methods is supported in part by a grant from the National Science Foundation (GS-929 & -2850).The very helpful comments and encouragement of Louis Guttman and J. Douglas Carroll are greatly appreciated. Finally, to that unknown, but not unsung, reviewer who helped in the clarification of the argument, I express my thanks.     

Nonmetric multidimensional scaling with clustering of subjects

A new nonmetric multidimensional scaling method is devised to analyze three-way data concerning inter-stimulus similarities obtained from many subjects. It is assumed that subjects are classified into a small number of clusters and that the stimulus configuration is specific to each cluster. Under this assumption, the classification of subjects and the scaling used to derive the configurations for clusters are simultaneously performed using an alternating least-squares algorithm. The monotone regression of ordinal similarity data, the scaling of stimuli and the K -means clustering of subjects are iterated in the algorithm. The method is assessed using a simulation and its practical use is illustrated with the analysis of real data. Finally, some extensions are considered.     

Binocular processing of brightness information: a vector-sum model

The relation between monocular and binocular brightness was examined. Clear evidence was found that the interaction between visual channels in binocular processing of brightness information implicates both an apparent averaging of monocular brightness when they are grossly different and a partial summation when they approach equality. A vector-sum model is shown to predict these properties. A nonmetric method was used to fit such a model to data from three experiments in each of which 15 subjects estimated brightness of binocularly fused targets. Magnitude estimation was used in two experiments, and cateogry ratings were obtained in the third experiment. When it was assumed only that subjects' responses were monotone with perceived brightness, estimates of the model's parameters from the data of the three experiments were almost identical, indicating that results from magnitude estimati;n and category rating can converge once nonlinear response functions are eliminated.     

Nonmetric grouping: Clusters and cliques

A class of related nonmetric (“monotone invariant”) hierarchical grouping methods is presented. The methods are defined in terms of generalized cliques, based on a systematically varying specification of the degree of indirectness of permitted relationships (i.e., degree of “chaining”). This approach to grouping is shown to provide a useful framework for grouping methods based on ana priori specification of the properties of the desired subsets, and includes a natural generalization for “complete linkage” and “single linkage” clustering, such as the methods of Johnson [1967]. The central feature of the class of methods is a simple iterative matrix operation on the original disparities (“inverse-proximities” or “dissimilarities”) matrix, and one of the methods also constitutes a very efficient single linkage clustering procedure.     

What are numbers?

This paper argues that (cardinal) numbers are originally given to us in the context ‘Fs exist n-wise’, and accordingly, numbers are certain manners or modes of existence, by addressing two objections both of which are due to Frege. First, the so-called Caesar objection will be answered by explaining exactly what kind of manner or mode numbers are. And then what we shall call the Functionality of Cardinality objection will be answered by establishing the fact that for any numbers m and n, if there are exactly m Fs and also there are exactly n Fs, then m = n.     

Failure of additivity in bisection of length

Subjects were intructed to select one rod to lie halfway in length between two given rods. These bisection instructions imply an additive model in the subjective metric. However, the data were inherently nonadditive; the length of the bisector could be an increasing or a decreasing function of the length of one given rod, depending on the length of the other given rod. A convexity analysis and a nonmetric analysis both showed that no monotone transformation could make the data additive. The bisection problem is used to contrast the axiomatic and functional approaches to measurement theory.     

The effectiveness of feedback in multiple-cue probability learning

How effective are different types of feedback in helping us to learn multiple contingencies? This article attempts to resolve a paradox whereby, in comparison to simple outcome feedback, additional feedback either fails to enhance or is actually detrimental to performance in nonmetric multiple-cue probability learning (MCPL), while in contrast the majority of studies of metric MCPL reveal improvements at least with some forms of feedback. In three experiments we demonstrate that if feedback assists participants to infer cue polarity then it can in fact be effective in nonmetric MCPL. Participants appeared to use cue polarity information to adopt a linear judgement strategy, even though the environment was nonlinear. The results reconcile the paradoxical contrast between metric and nonmetric MCPL and support previous findings of people's tendency to assume linearity and additivity in probabilistic cue learning.     

A Quasi-Metric Approach to Multidimensional Unfolding for Reducing the Occurrence of Degenerate Solutions

In multidimensional unfolding (MDU), one typically deals with two-way, two-mode dominance data in estimating a joint space representation of row and column objects in a derived Euclidean space. Unfortunately, most unfolding procedures, especially nonmetric ones, are prone to yielding degenerate solutions where the two sets of points (row and column objects) are disjointed or separated in the derived joint space, providing very little insight as to the structure of the input data. We present a new approach to multidimensional unfolding which reduces the occurrence of degenerate solutions. We first describe the technical details of the proposed method. We then conduct a Monte Carlo simulation to demonstrate the superior performance of the proposed model compared to two non-metric procedures, namely, ALSCAL and KYST. Finally, we evaluate the performance of alternative models in two applications. The first application deals with student rank-order preferences (nonmetric data) for attending various graduate business (MBA) programs. Here, we compare the performance of our model with those of KYST and ALSCAL. The second application concerns student preference ratings (metric data) for a number of popular brands of analgesics. Here, we compare the performance of the proposed model with those of two metric procedures, namely, SMACOF-3 and GENFOLD 3. Finally, we provide some directions for future research.     

A quasi-nonmetric method for multidimensional scaling VIA an extended euclidean model

An Extended Two-Way Euclidean Multidimensional Scaling (MDS) model which assumes both common and specific dimensions is described and contrasted with the standard (Two-Way) MDS model. In this Extended Two-Way Euclidean model then stimuli (or other objects) are assumed to be characterized by coordinates onR common dimensions. In addition each stimulus is assumed to have a dimension (or dimensions) specific to it alone. The overall distance between objecti and objectj then is defined as the square root of the ordinary squared Euclidean distance plus terms denoting the specificity of each object. The specificity,s _j , can be thought of as the sum of squares of coordinates on those dimensions specific to objecti, all of which have nonzero coordinatesonly for objecti. (In practice, we may think of there being just one such specific dimension for each object, as this situation is mathematically indistinguishable from the case in which there are more than one.)We further assume that _ij =F(d _ij ) +e _ij where _ij is the proximity value (e.g., similarity or dissimilarity) of objectsi andj,d _ij is the extended Euclidean distance defined above, whilee _ij is an error term assumed i.i.d.N(0, ²).F is assumed either a linear function (in the metric case) or a monotone spline of specified form (in the quasi-nonmetric case). A numerical procedure alternating a modified Newton-Raphson algorithm with an algorithm for fitting an optimal monotone spline (or linear function) is used to secure maximum likelihood estimates of the paramstatistics) can be used to test hypotheses about the number of common dimensions, and/or the existence of specific (in addition toR common) dimensions.This approach is illustrated with applications to both artificial data and real data on judged similarity of nations.     

The effectiveness of feedback in multiple-cue probability learning

How effective are different types of feedback in helping us to learn multiple contingencies? This article attempts to resolve a paradox whereby, in comparison to simple outcome feedback, additional feedback either fails to enhance or is actually detrimental to performance in nonmetric multiple-cue probability learning (MCPL), while in contrast the majority of studies of metric MCPL reveal improvements at least with some forms of feedback. In three experiments we demonstrate that if feedback assists participants to infer cue polarity then it can in fact be effective in nonmetric MCPL. Participants appeared to use cue polarity information to adopt a linear judgement strategy, even though the environment was nonlinear. The results reconcile the paradoxical contrast between metric and nonmetric MCPL and support previous findings of people's tendency to assume linearity and additivity in probabilistic cue learning.     

Constrained multidimensional scaling inN spaces

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.     

A technique for the nonmetric analysis of paired comparisons of psychological intervals

A nonmetric coordinate adjustment technique is developed which determines scale values for objects whose interobject intervals (differences in subjective value) have been directly compared. In Monte Carlo simulations, the degree of metric determinancy of the scale values is shown to be quite high even when the amount of error is relatively high. This robustness under high-error conditions permitted the analysis of individual subject data in experiments on the direct comparison of loudness differences and loudness ratios where only one judgment per interval comparison was obtained per subject.This research was supported by a grant from the National Research Council of Canada.     

The analysis of proximities: Multidimensional scaling with an unknown distance function. I.

A computer program is described that is designed to reconstruct the metric configuration of a set of points in Euclidean space on the basis of essentially nonmetric information about that configuration. A minimum set of Cartesian coordinates for the points is determined when the only available information specifies for each pair of those points—not the distance between them—but some unknown, fixed monotonic function of that distance. The program is proposed as a tool for reductively analyzing several types of psychological data, particularly measures of interstimulus similarity or confusability, by making explicit the multidimensional structure underlying such data.     

Nonmetric multidimensional scaling: Recovery of metric information

The degree of metric determinancy afforded by nonmetric multidimensional scaling was investigated as a function of the number of points being scaled, the true dimensionality of the data being scaled, and the amount of error contained in the data being scaled. It was found 1) that if the ratio of the degrees of freedom of the data to that of the coordinates is sufficiently large then metric information is recovered even when random error is present; and 2) when the number of points being scaled increases the stress of the solution increases even though the degree of metric determinacy increases.This report was supported in part by a PHS research grant No. M-10006 from the National Institute of Mental Health, Public Health Service, and in part by a Science Development grant No. GU-2059, from the National Science Foundation. The author is indebted to Charles R. Sherman for his assistance in gathering the data and for his critical re-writing of sections of this report. The assistance of Lyle V. Jones in his critical readings and comments is also deeply appreciated.     

A classification of all order-preserving homeomorphism groups of the reals that satisfy finite uniqueness

The only real relational structures of scale type (m, n) with 1 ≤ m ≤ n < ∞ are of scale type (1, 1), (1, 2), and (2, 2), and so are conjugate to structures whose automorphism groups are subgroups of the affines containing the group of translations. All real relational structures of scale type (0, n) with n < ∞ have automorphism groups abstractly isomorphic to; moreover, each contains subchain of order type theta, invariant under the action of the automorphism group, s.t. the action of the automorphisms on this subchain is conjugate to that of a sub-group of the affines.     

A locus problem involving the three-dimensional city-block metric

In connection with multidimensional scaling, representations have been considered of the form abDcd ← ?(f(a), f(b)) ≦ ?(f(c), f(d)), for all a, b, c, d ∈ A, where A is a nonvoid finite set, D is a four-place relation on A, f is a function from A into Euclidean n-space, $\text{R}{}^{\mathrm{n}}$, and ? is a metric in $\text{R}{}^{\mathrm{n}}$. For particular metrics there exist finite universal axiomatizations which are necessary and sufficient for the above representation. On the other hand, it is known that no such axiomatizations can be given for either the supremum metric or the ordinary Euclidean metric. Methods for showing this apply easily to the city-block metrics in $\text{R}{}^1$ and $\text{R}{}^2$. This article describes a computer-aided verification of a locus result which shows the impossibility of finite universal axiomatizability for the case of the city-block metric in $\text{R}{}^3$. The result was obtained by dealing with 21,780 cases, each of which involved a set of 10 equations in 12 unknowns along with a related set of inequalities.     

Multidimensional scaling of form: A psychophysical analysis

Both metric and nonmetric multidimensional scaling methods were used to analyze similarity estimates when random polygons were used as stimulus patterns. Three dimensions, dispersion, jaggedness, and elongation were obtained with both analyses and were related to physical measures of the patterns.     

Comparing the predictive powers of alternative multiple regression models

Mean squared error of prediction is used as the criterion for determining which of two multiple regression models (not necessarily nested) is more predictive. We show that an unrestricted (or true) model witht parameters should be chosen over a restricted (or misspecified) model withm parameters if (P _t ² ?P _m ² )>(1?P _t ² )(t?m)/n, whereP _t ² andP _m ² are the population coefficients of determination of the unrestricted and restricted models, respectively, andn is the sample size. The left-hand side of the above inequality represents the squared bias in prediction by using the restricted model, and the right-hand side gives the reduction in variance of prediction error by using the restricted model. Thus, model choice amounts to the classical statistical tradeoff of bias against variance. In practical applications, we recommend thatP ² be estimated by adjustedR ² . Our recommendation is equivalent to performing theF-test for model comparison, and using a critical value of 2?(m/n); that is, ifF>2?(m/n), the unrestricted model is recommended; otherwise, the restricted model is recommended.     

On the reciprocity of proximity relations

The degree of reciprocity of a proximity order is the proportion, P(1), of elements for which the closest neighbor relation is symmetric, and the R value of each element is its rank in the proximity order from its closest neighbor. Assuming a random sampling of points, we show that Euclidean n-spaces produce a very high degree of reciprocity, $\text{P(1) ≥}\text{1}\text{2}$, and correspondingly low R values, E(R) ≤ 2, for all n. The same bounds also apply to homogeneous graphs, in which the same number of edges meet at every node. Much less reciprocity and higher R values, however, can be attained in finite tree models and in the contrast model in which the “distance” between objects is a linear function of the numbers of their common and distinctive features.