Multidimensional rotation and scaling of configurations to optimal agreement

An integrated method for rotating and rescaling a set of configurations to optimal agreement in subspaces of varying dimensionalities is developed. The approach relates existing orthogonal rotation techniques as special cases within a general framework based on a partition of variation which provides convenient measures of agreement. In addition to the well-known Procrustes and inner product optimality criteria, a criterion which maximizes the consensus among subspaces of the configurations is suggested. Since agreement of subspaces of the configurations can be examined and compared, rotation and rescaling is extended from a data transformation technique to an analytical method.     

Multidimensional scaling in riemannian space

A metric which is a function of position is proposed for the analysis of the intrinsic geometry involved in preference or similarity judgments. Variation in the distance function or metric is characteristic of the Riemannian spaces and may be interpreted as curvature, stress or distortion in distance estimates and thus in the subjective perceptual space. It is possible to find the coefficients in the distance function at selected points by fitting a least-squares Riemannian surface to the Euclidean plane. The functional form of the distance can then be obtained by an application of the Laplace equation. Several examples are worked out for the two-dimensional solution but extension to higher spaces appears to be quite feasible.     

Multidimensional scaling and threshold graphs

A set of data has a Guttman scale if and only if a corresponding graph is a threshold graph. In this paper we relate the concepts of disjunctive and conjuctive Guttman scales, and biorder dimension to the threshold dimension of a graph. For those graphical properties that can be tested in polynomial time, the comparable Guttman scaling techniques can be performed in polynomial time. Fast algorithms are provided for computing a Guttman scale, and the conjunctive and disjunctive dimension of data with no 3-crowns. We define an extended Guttman scale to indicate strength of agreement, dominance, etc., and show that this, too, exists if and only if a particular graph is a threshold graph.     

Multidimensional scaling of pictorial informativeness.

The dimensions used in the judgment of the informativeness of picture sections were investigated by means of a recently proposed methodology, multidimensional similarity analysis. 10 judges (college students) rated the informational similarity of 32 areas within a single picture. The five extracted dimensions accounted for 86% of the judgmental variance and were all readily interpretable. These dimensions were discussed with respect to an earlier study in which eye movements of subjects viewing this picture were recorded.     

Multidimensional scaling of musical time estimations

The aim of this study was to identify the psycho-musical factors that govern time evaluation in Western music from baroque, classic, romantic, and modern repertoires. The excerpts were previously found to represent variability in musical properties and to induce four main categories of emotions. 48 participants (musicians and nonmusicians) freely listened to 16 musical excerpts (lasting 20 sec. each) and grouped those that seemed to have the same duration. Then, participants associated each group of excerpts to one of a set of sine wave tones varying in duration from 16 to 24 sec. Multidimensional scaling analysis generated a two-dimensional solution for these time judgments. Musical excerpts with high arousal produced an overestimation of time, and affective valence had little influence on time perception. The duration was also overestimated when tempo and loudness were higher, and to a lesser extent, timbre density. In contrast, musical tension had little influence.     

Motivating operations and terms to describe them: some further refinements

Over the past decade, behavior analysts have increasingly used the term establishing operation (EO) to refer to environmental events that influence the behavioral effects of operant consequences. Nonetheless, some elements of current terminology regarding EOs may interfere with applied behavior analysts' efforts to predict, control, describe, and understand behavior. The present paper (a) describes how the current conceptualization of the EO is in need of revision, (b) suggests alternative terms, including the generic term motivating operation (MO), and (c) provides examples of MOs and their behavioral effects using articles from the applied behavior analysis literature.     

Multidimensional scaling of hierarchical sorting data applied to facial expressions

A new algorithm for multidimensional scaling analysis of sorting data and hierarchical-sorting data is tested by applying it to facial expressions of emotion. We construct maps in “facial expression space” for two sets of still photographs: the I-FEEL series (expressions displayed spontaneously by infants and young children), and a subset of the Lightfoot series (posed expressions, all from one actress). The analysis avoids potential artefacts by fitting a map directly to the subject's judgments, rather than transforming the data into a matrix of estimated dissimilarities as an intermediate step. The results for both stimulus sets display an improvement in the extent to which they agree with existing maps. Some points emerge about the limitations of sorting data and the need for caution when interpreting MDS configurations derived from them.     

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.     

Multidimensional scaling of measures of distance between partitions

The techniques of multidimensional scaling were used to study the numerical behavior of twelve measures of distance between partitions, as applied to partition lattices of four different sizes. The results offer additional support for a system of classifying partition metrics, as proposed by Boorman (1970), and Boorman and Arabie (1972). While the scaling solutions illuminated differences between the measures, at the same time the particular data with which the measures were concerned offered a basis both for counterexamples to some common assumptions about multidimensional scaling and for some conjectures as to the nature of scaling solutions. The implications of the latter findings for selected examples from the literature are considered. In addition, the methods of partition data analysis discussed here are applied to an example using sociobiological data. Finally, an argument is made against undue emphasis upon interpreting dimensions in nonmetric scaling solutions.     

Multidimensional scaling of subjective colors by color-blind observers

Temporal coding theories of color vision suggest explanations of flicker-induced subjective colors such as those that appear on Benham’s disk. If color blindness were due simply to photopigment anomalies, then subjective colors might be elicited by central patterns of neural activity in color-blind observers that mimic those which the cones normally produce in colornormal observers. We had color-normal and color-blind observers scale subjective colors like those on Benham’s disk for similarity. The inferred color spaces for six normal observers resembled the familiar hue circle, but the spaces for five red-green-deficient observers were compressed along the red-green axis. This is consistent with the position that flicker colors are due to retinal processes, and suggests that color blindness may involve variations of the central nervous system in addition to photopigment anomalies.     

Multidimensional scaling of political parties A methodological study

Abstract.— With 7 Norwegian parties as stimuli, comparisons were made of various content and distance approaches to scaling. As expected, the former gave two factors, socialistic and nonsocialistic, and the latter gave one bipolar dimension, interpreted as socialistic-non-socialistic. Individual differences in structure were not found. Among several principles of similarity, Ekman's original (1961) gave the best fit. It is maintained, however, that the fundamental vector model of the content approach is inadequate, both for theoretical reasons and because subjects are unable to operate in accordance with it.     

Multidimensional scaling of color difference in the pigeon

Pigeons were required to discriminate between “identical” vs “different” pairs of lights in a yes/no signal-detection task with a symmetrical payoff matrix. If the two lights projected on the two halves of the bipartite field constituting the center response key in a three-key chamber were identical in wavelength composition, a single peck on the right key was reinforced with food. If the two lights differed in wavelength composition, then left-key pecks were reinforced. In Experiment 1, each of six pigeons experienced all possible pairs of 11 spectral lights and 1 purple light (66 pairs). In Experiment 2, the set of lights was expanded to 15 (105 pairs) and included a white light. The percentage of correct choices was taken as an index of the dissimilarity between the two lights constituting a pair. The rank-order information available in these dissimilarity measures was used to determine coordinate projections for each light in a two-dimensional Euclidean space. The configuration obtained in this manner was interpreted as a color circle for the pigeon.     

Multidimensional scaling reveals two dimensions of thermal pain

The Individual Differences Scaling (INDSCAL) model of multidimensional scaling was used to explore the dimensions of thermal pain. The observers made 66 similarity judgments to all pairs of 12 different thermal stimulus intensities ranging from zero to noxious. Analysis of the data revealed a two-dimensional group stimulus space. The major dimension ordered the stimuli with respect to their intensity. This quantitative, strength-of-sensation dimension may be interpreted as indicating how weak or strong a stimulus feels, apart from any secondary qualities of warmth or pain. The second dimension was related to the qualitative aspects of the stimuli. This bipolar dimension contained two attributes: a pain attribute ranging from just detectable warmth to painful, and a warm-hot attribute running from just detectable warmth to hot. This study demonstrates the utility of the INDSCAL approach to the understanding of pain and offers a new technique for answering the age-old question concerning the number and qualities of the dimensions underlying the pain experience.     

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.     

Multidimensional scaling: I. Theory and method

Multidimensional scaling can be considered as involving three basic steps. In the first step, a scale of comparative distances between all pairs of stimuli is obtained. This scale is analogous to the scale of stimuli obtained in the traditional paired comparisons methods. In this scale, however, instead of locating each stimulus-object on a given continuum, the distances between each pair of stimuli are located on a distance continuum. As in paired comparisons, the procedures for obtaining a scale of comparative distances leave the true zero point undetermined. Hence, a comparative distance is not a distance in the usual sense of the term, but is a distance minus an unknown constant. The second step involves estimating this unknown constant. When the unknown constant is obtained, the comparative distances can be converted into absolute distances. In the third step, the dimensionality of the psychological space necessary to account for these absolute distances is determined, and the projections of stimuli on axes of this space are obtained. A set of analytical procedures was developed for each of the three steps given above, including a least-squares solution for obtaining comparative distances by the complete method of triads, two practical methods for estimating the additive constant, and an extension of Young and Householder's Euclidean model to include procedures for obtaining the projections of stimuli on axes from fallible absolute distances.This study was carried out while the author was an Educational Testing Service Psychometric Fellow at Princeton University. The author expresses his appreciation to his thesis adviser, Dr. H. Gulliksen, for his guidance throughout the study and to Dr. B. F. Green, Jr., for valuable assistance on several of the derivations.