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

The first in the present series of two papers described a computer program for multidimensional scaling on the basis of essentially nonmetric data. This second paper reports the results of two kinds of test applications of that program. The first application is to artificial data generated by monotonically transforming the interpoint distances in a known spatial configuration. The purpose is to show that the recovery of the original metric configuration does not depend upon the particular transformation used. The second application is to measures of interstimulus similarity and confusability obtained from some actual psychological experiments.     

Internal multidimensional unfolding about a single-ideal—A probabilistic solution

A solution is presented for an internal multidimensional unfolding problem in which all the judgments of a rectangular proximity matrix are a function of a single-ideal object. The solution is obtained by showing that when real and ideal objects are represented by normal distributions in a multidimensional Euclidean space, a vector of distances among a single-ideal and multiple real objects follows a multivariate quadratic form in normal variables distribution. An approximation to the vector's probability density function (PDF) is developed which allows maximum likelihood (ML) solutions to be estimated. Under dependent sampling, the likelihood function contains information about the parametric distances among real object pairs, permitting the estimation of single-ideal solutions and leading to more robust multiple-ideal solutions. Tests for single- vs. multiple-ideal solutions and dependent vs. independent sampling are given. Properties of the proposed model and parameter recovery are explored. Empirical illustrations are also provided.     

The problem of the additive constant and eigenvalues in metric multidimensional scaling

This paper is concerned with the additive constant problem in metric multidimensional scaling. First the influence of the additive constant on eigenvalues of a scalar product matrix is discussed. The second part of this paper is devoted to the introduction of a new formulation of the additive constant problem. A solution is given for fixed dimensionality, by maximizing a normalized index of fit with a gradient method. An experimental computation has shown that the author's solution is accurate and easy to follow.     

A new solution to the additive constant problem in metric multidimensional scaling

A new solution to the additive constant problem in metric multidimensional scaling is developed. This solution determines, for a given dimensionality, the additive constant and the resulting stimulus projections on the dimensions of a Euclidean space which minimize the sum of squares of discrepancies between the formal model for metric multidimensional scaling and the original data. A modification of Fletcher-Powell style functional iteration is used to compute solutions. A scale free index of the goodness of fit is developed to aid in selecting solutions of adequate dimensionality from multiple candidates.This research is based in part on the author's Ph.D. dissertation at the University of Illinois at Urbana-Champaign. Computer time was provided by the Campus Computing Network of the University of California, Los Angeles.     

The Multidimensional Scaling Of A Set Of Artistic Drawings : Perceived Structure And Scale Correlates

Product-centered research on creativity approaches the criterion problem of what is to be the referent for creativity through the analysis of tangible products such as a r t objects, writing, or scientific achievements. The present research is concerned with the evaluation and study of artist drawings contributed by sophomore students a t the Rhode Island School of Design. Multi-dimensional scaling methods were applied to similarity judgments obtained from art experts on two separate sets of 26 drawings. Three similarity dimensions accounted for the interstimulus distances for each set of drawings. Although no statistical test was available, the dimensions from the two seta appeared to correspond. Scale values of 4 drawings common to the two sets were consistent, and the dimensions appeared to define very similar stimulus characteristics. It was concluded that multidimensional scaling procedures provided a means for differentiating among a set of complex, esthetic products. Scale values of drawings on the three dimensions also correlated differentially with cognitive and achievement measures available on the students, suggesting that product dimensions identified via similarity judgments were related to characteristics of individuals producing the products. Hypotheses were developed as to the psychological meaning of the three product dimensions.     

Beliefs,personality, and person perception: A theory of individual differences

The hypothesis tested was that we attend to aspects of our interpersonal environment that correspond to prominent features of our own personality. One hundred fifty-four men and women rated the way in which they typically communicate and interact with significant other people (e.g., mother, father, lover) on 11 bipolar adjective scales and the way in which those other people interact with them. An individual differences multidimensional scaling analysis of interstimulus distances among the significant other people based on the ratings of the subject's interaction with those other people indicated that subjects who weighted a particular dimension tended to think they possessed a personality trait corresponding to the dimension. Reasons for the individual difference hypothesis were given.     

Constant curvature Riemannian scaling

A multidimensional scaling algorithm is proposed for fitting distances to constant curvature Riemannian spaces. Examples are given and potential applications are discussed. Some general properties of Riemannian spaces are also discussed. It is argued that some restriction, such as that of constant curvature, is necessary to obtain simple unique solutions in Riemannian spaces.     

The integration of multidimensional scaling and multivariate analysis with optimal transformations

The recent history of multidimensional data analysis suggests two distinct traditions that have developed along quite different lines. In multidimensional scaling (MDS), the available data typically describe the relationships among a set of objects in terms of similarity/dissimilarity (or (pseudo-)distances). In multivariate analysis (MVA), data usually result from observation on a collection of variables over a common set of objects. This paper starts from a very general multidimensional scaling task, defined on distances between objects derived from one or more sets of multivariate data. Particular special cases of the general problem, following familiar notions from MVA, will be discussed that encompass a variety of analysis techniques, including the possible use of optimal variable transformation. Throughout, it will be noted how certain data analysis approaches are equivalent to familiar MVA solutions when particular problem specifications are combined with particular distance approximations.This research was supported by the Royal Netherlands Academy of Arts and Sciences (KNAW). An earlier version of this paper was written during a stay at McGill University in Montréal; this visit was supported by a travel grant from the Netherlands Organization for Scientific Research (NWO). I am grateful to Jim Ramsay and Willem Heiser for their encouragement and helpful suggestions, and to the Editor and referees for their constructive comments.     

Some characteristics of psychophysical data obtained with a group version of the up-and-down method

Effects of backgroun reflectance on the shapes of scales of lightness were investigated in two experiments. In Experiment I, 25 Ss compared similaritiesofpairs ofgray chips that were viewed against black, gray. or white backgrounds. In Experiment 2, 33 Ss positioned a series of gray chips, viewed against a white, gray. or black background, so that the distances between successive chips represented the perceived differences in lightness between them. The results of both studies indicated that scale shape was influenced by background. The nature of the effect was that interstimulus differences in the region of a background were enhanced relative to interstimulus differences far from a background.     

Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis

Multidimensional scaling is the problem of representingn objects geometrically byn points, so that the interpoint distances correspond in some sense to experimental dissimilarities between objects. In just what sense distances and dissimilarities should correspond has been left rather vague in most approaches, thus leaving these approaches logically incomplete. Our fundamental hypothesis is that dissimilarities and distances are monotonically related. We define a quantitative, intuitively satisfying measure of goodness of fit to this hypothesis. Our technique of multidimensional scaling is to compute that configuration of points which optimizes the goodness of fit. A practical computer program for doing the calculations is described in a companion paper.     

Expectancy and intermittency

When a human subject responds to the second of two closely succeeding stimuli, his reaction time to the second stimulus tends to increase sharply as the interstimulus interval decreases. Controversy has centred on the issue of whether this increase is mainly due to the effects of the first stimulus in producing some kind of block in the central analysing systems or whether it is mainly due to the temporal uncertainty of the second signal, as determined by the distribution of interstimulus intervals used.

By substituting for the first stimulus a spontaneous response on the part of the subject and holding the distribution of interstimulus intervals constant, it is shown that the delays in responding to the succeeding signal are eliminated, even at intervals as short as 50 millisec. This is interpreted as evidence in support of the intermittency hypothesis and as a clear indication that the increase in reaction times normally observed is not a result of the distribution of interstimulus intervals.     

The Earth is flat when personally significant experiences with the sphericity of the Earth are absent

Participants with personal and without personal experiences with the Earth as a sphere estimated large-scale distances between six cities located on different continents. Cognitive distances were submitted to a specific multidimensional scaling algorithm in the 3D Euclidean space with the constraint that all cities had to lie on the same sphere. A simulation was run that calculated respective 3D configurations of the city positions for a wide range of radii of the proposed sphere. People who had personally experienced the Earth as a sphere, at least once in their lifetime, showed a clear optimal solution of the multidimensional scaling (MDS) routine with a mean radius deviating only 8% from the actual radius of the Earth. In contrast, the calculated configurations for people without any personal experience with the Earth as a sphere were compatible with a cognitive concept of a flat Earth.     

A marginalization model for the multidimensional unfolding analysis of ranking data

A marginalization model for the multidimensional unfolding analysis of ranking data is presented. A subject samples one of a number of random points that are multivariate normally distributed. The subject perceives the distances from the point to all the stimulus points fixed in the same multidimensional space. The distances are error perturbed in this perception process. He/she produces a ranking dependent on these error-perturbed distances. The marginal probability of a ranking is obtained according to this ranking model and by integrating out the subject (ideal point) parameters, assuming the above distribution. One advantage of the model is that the individual differences are captured using the posterior probabilities of subject points. Three sets of ranking data are analyzed by the model.     

A general solution for a class of weakly constrained linear regression problems

This paper contains a globally optimal solution for a class of functions composed of a linear regression function and a penalty function for the sum of squared regression weights. Global optimality is obtained from inequalities rather than from partial derivatives of a Lagrangian function. Applications arise in multidimensional scaling of symmetric or rectangular matrices of squared distances, in Procrustes analysis, and in ridge regression analysis. The similarity of existing solutions for these applications is explained by considering them as special cases of the general class of functions addressed.The author is obliged to Henk Kiers and Willem Heiser for helpful comments.     

On the prediction of confusion matrices from similarity judgments

Three observers viewed visual representations of eight complex sounds in both a pairwise similarity-judgment task and an identification task. A multidimensional scaling procedure applied to the similarity judgments yielded a three-dimensional perceptual space and the relative positions of the stimuli in that space. A probabilistic decision model based on weighted interstimulus distances served to predict well the confusion matrices of the identification task. Three conditions of the identification task, calling for identification of different subsets of the eight stimuli, led the observers to vary the weights they placed on the dimensions; they apparently adjusted the weights to maximize the percent correct identification. An additional group of 14 subjects, participating only in the similarity-judgment task, manifested the same three dimensions as the observers (corresponding to the locus of low-frequency energy, the locus of midfrequency energy, and visual contrast), and also a fourth dimension (corresponding to the periodicity, or waxing and waning, of the sound). Although not evident in the scaling analysis for the three observers, our utilization of the additional dimension increased significantly the variance accounted for in their identification responses. The overall accuracy of the predictions from a perceptual space to identification responses supplies a substantial validation of the use of multidimensional scaling procedures to reveal perceptual structure in demonstrating the ability of that structure to account for behavior in an independent task. The empirical success of this approach, furthermore, suggests a relatively simple and practical means of predicting, and possibly enhancing, identification performance for a given set of visual or auditory stimuli.     

Spatial extent and figural factors in backward masking

Spatial and figural characteristics of backward masking were studied, with two collinear arcs presented end-to-end and serving as target and mask, respectively. Stimulus onset asynchrony was 50 ms while interstimulus interval was 0 ms. Mask exposure duration required for masking was determined as a function of target length with mask length as a parameter. The exposure duration of the mask required for complete masking varied directly with target length, but inversely with mask length. The fact that masking strength increased with mask duration while all other parameters were kept constant suggests that masking depended on stimulus termination asynchrony. Maximal masking occurred for target arcs as long as 5.0 deg of visual angle, exceeding previously reported distances. Misaligned or differently shaped stimuli produced less masking, suggesting that figural factors play a role in long-range backward masking.     

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.     

Perceptual organization of acoustic stimuli by budgerigars (Melopsittacus undulatus): I. Pure tones

A new combination of operant conditioning and psychophysical scaling procedures was used to study auditory perception in a small bird. In a same-different discrimination task, budgerigars learned to discriminate among pure tones that varied along one or more acoustic dimensions. Response latencies were used to generate a matrix of interstimulus similarities. Multidimensional scaling procedures were used to arrange these acoustic stimuli in a multidimensional space that supposedly reflects the bird's perceptual organization. For tones that varied in intensity, duration, and frequency simultaneously, budgerigars were much more sensitive to frequency changes. From a set of tones that varied only in intensity, it was possible to calculate the growth of loudness with intensity for the budgerigar. For tones that varied only in frequency, budgerigars showed evidence of an "acoustic fovea" for frequency change in the spectral region of 2-4 kHz. Budgerigars and humans also differed in their perceptual grouping of tone sequences that rise, fall, or remain constant in pitch. Surprisingly, budgerigars were much less responsive to pitch contour than were humans.     

The tunneling method for global optimization in multidimensional scaling

This paper focuses on the problem of local minima of the STRESS function. It turns out that unidimensional scaling is particularly prone to local minima, whereas full dimensional scaling with Euclidean distances has a local minimum that is global. For intermediate dimensionality with Euclidean distances it depends on the dissimilarities how severe the local minimum problem is. For city-block distances in any dimensionality many different local minima are found. A simulation experiment is presented that indicates under what conditions local minima can be expected. We introduce the tunneling method for global minimization, and adjust it for multidimensional scaling with general Minkowski distances. The tunneling method alternates a local search step, in which a local minimum is sought, with a tunneling step in which a different configuration is sought with the same STRESS as the previous local minimum. In this manner successively better local minima are obtained, and experimentation so far shows that the last one is often a global minimum.This paper is based on the 1994 Psychometric Society's outstanding thesis award of the first author. The authros would like to thank Robert Tijssen of the CWTS Leiden for kindly making available the co-citation data of the Psychometric literature. This paper is an extended version of the paper presented at the Annual Meeting of the Psychometric Society at Champaign-Urbana, Illin., June 1994.