Spatial,non-spatial and hybrid models for scaling

In this paper, hierarchical and non-hierarchical tree structures are proposed as models of similarity data. Trees are viewed as intermediate between multidimensional scaling and simple clustering. Procedures are discussed for fitting both types of trees to data. The concept of multiple tree structures shows great promise for analyzing more complex data. Hybrid models in which multiple trees and other discrete structures are combined with continuous dimensions are discussed. Examples of the use of multiple tree structures and hybrid models are given. Extensions to the analysis of individual differences are suggested.1976 Psychometric Society Presidential Address.While people too numerous to list here have contributed ideas, inspiration, and other help, I particularly wish to acknowledge the contributions of Sandra Pruzansky, without whom this paper could not have been written. I would also like to acknowledge the past contributions of my long-time colleague Jih-Jie Chang, without whose help I probably would not have beenasked to write it.     

An analysis and synthesis of multiple correspondence analysis,optimal scaling,dual scaling,homogeneity analysis and other methods for quantifying categorical multivariate data

We discuss a variety of methods for quantifying categorical multivariate data. These methods have been proposed in many different countries, by many different authors, under many different names. In the first major section of the paper we analyze the many different methods and show that they all lead to the same equations for analyzing the same data. In the second major section of the paper we introduce the notion of a duality diagram, and use this diagram to synthesize the many superficially different methods into a single method.The ideas in this paper were worked out by the first author, with some suggestions provided by the second. The current version of this paper has evolved from three previous versions, the first two written by the first author.     

Robust multidimensional scaling

A method for multidimensional scaling that is highly resistant to the effects of outliers is described. To illustrate the efficacy of the procedure, some Monte Carlo simulation results are presented. The method is shown to perform well when outliers are present, even in relatively large numbers, and also to perform comparably to other approaches when no outliers are present.This research was supported by Grant A8351 from the Natural Sciences and Engineering Research Council of Canada to Ian Spence.     

Context effects,reliability, and internal consistency of intermodal joint scaling

Context effects, intraindividual variability, and internal consistency of intermodal joint scaling with magnitude estimation (“magnitude matching”) were studied by instructing 12 subjects to judge the three pairs of odor intensity, loudness, and brightness on a common scale of perceived intensity as well as to judge odor intensity separately (unimodal magnitude estimation). Significant context effects were found by comparing odor intensity judgments obtained by separate versus intermodal joint scaling as well as across different modalities (loudness vs. brightness) in joint scaling. But no such effects were found for loudness or brightness when compared across modality of joint scaling. Intraindividual variability in the estimates imply about equal reliability in intermodal joint scaling and separate scaling. Good internal consistency was found, indicating that subjects are successful in expressing perceived intensities of different modalities on a common scale.     

Least squares metric,unidimensional scaling of multivariate linear models

The squared error loss function for the unidimensional metric scaling problem has a special geometry. It is possible to efficiently find the global minimum for every coordinate conditioned on every other coordinate being held fixed. This approach is generalized to the case in which the coordinates are polynomial functions of exogenous variables. The algorithms shown in the paper are linear in the number of parameters. They always descend and, at convergence, every coefficient of every polynomial is at its global minimum conditioned on every other parameter being held fixed. Convergence is very rapid and Monte Carlo tests show the basic procedure almost always converges to the overall global minimum.The author thanks Ivo Molenaar, three anonymous referees, and Howard Rosenthal for their many helpful comments.     

Image representation,scaling and cognitive model of object perception

Neuropsychological investigations of visual imagery and representations have led to a deeper understanding of the spatial perception, representation and memory. But how each individual perceives object’s geometrical properties and how they differ from person to person, both under event-related memory and normal recollecting memory in the presence or in the absence of direct sensory stimulation is still unclear. Spatial knowledge is diverse, complex, and multi-modal, as are the situations in which it is used. All seem to agree that a cognitive map is a mental representation of an external environment. The image scaling is important in understanding the psychological dysfunctions of patients suffering from spatial cognition problems. The scaling becomes self-evident in art forms, when people are asked to draw image of objects they see actively or from their short or long term memory. In this paper we develop a comprehensive model of this scaling factor and its implications in spatial image representation and memory. We also extend its notion in understanding the perception of objects whose representations are normally not possible (like the perception of universal scales, infinities and parallel lines) but are well comprehended by the human brains. Here we give a scaling factor which is variable depending on the situations for a person based on his visual memory and drawing capabilities. And then extend it to analyse his cognitive strengths, disorders and any imperfections. This model also helps in formalizing the architectural cognitive maps needed to change the scaling factor, depending on the types of visual works one performs.     

Nonmetric scaling of line lengths using latencies,similarity, and same-different judgments

An experiment investigating the relation between various ways of deriving psychophysical functions for line length is reported. A two-phase computer-controlled experiment in which the stimuli were pairs of lines was performed. In the first phase, the pair was presented for several seconds and the S was required to make a category judgment of the similarity of the lines. In the second phase, the pair was presented for a very brief period of time and the S decided whether the lines were the same or of different lengths. Three sets of data were analyzed: (1) the similarity judgments, (2) the proportion of correct different judgments, and (3) the latency of the correct different judgments. An individual differences analysis indicated that there were individual differences in the similarity and latency data. This analysis was not applicable to the proportion data. Nonmetric scaling revealed that the psychophysical functions based on all three sets of data were essentially identical     

Mixed-method mixed-modality psychophysical scaling

Four experiments are reported in which the direct psychophysical scaling methods of magnitude estimation, category judgment, and cross-modality matching were mixed in the same series of trials, both with a single stimulus modality and in a mixed-modality situation. The mixed-method scaling situation gave results consistent with those obtained when methods are used alone, and it has several advantages. Interactions between the methods were consistent with the idea that judgments made under all three are mediated by a primitive process of categorization that is influenced by heuristics used to achieve a single category identity for each stimulus.     

Restricted multidimensional scaling models

A class of multidimensional scaling models are developed wherein certain parameters may be fixed as known constants, or proportional to one another. Traditional multidimensional scaling can be obtained as a special case by fixing only the orientation and origin of a configuration. Methods of obtaining least-square estimates of the parameters via nonlinear programming are discussed, and an effective computer program is developed to implement application of the models to data. Several well-known data sets are reanalyzed under various restricted models, and the results demonstrate the possibility of achieving insight not attainable under the traditional approach. The potential distortion arising from inadequate model specification is discussed, and the importance of substantive theory to multidimensional scaling research is emphasized.     

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.     

Completely nonmetric multidimensional scaling

Most of the distance models underlying multidimensional scaling assume that if a stimulus y is between the stimuli x and z on each dimension, then x and z should be the farthest apart of the three stimuli. An iterative algorithm is described that uses only this betweenness prediction to infer the ordering of a set of stimuli on each of one or two dimensions. Applied to previously published semantic similarity data, this algorithm produced two-dimensional configurations that were similar in appearance to Euclidean configurations but generally involved fewer violations of the betweenness prediction.     

Pairwise nonmetric multidimensional scaling

A method of nonmetric multidimensional scaling is described which minimizes pairwise departures from monotonicity. The procedure is relatively simple, both conceptually and computationally. Experience to date suggests that it produces solutions comparable to those of other methods.     

Mental size scaling examined

Observers made same-different shape judgments of stimuli that were identical in shape and size different in shape but not in size, or differed in relative size along a number of steps for both same- and different- shaped forms. "Same" judgment RTs increased monotonically with increases in the magnitude of the relative size difference. In contrast, "different" judgment RTs were unaffected by changes in relative size. A second experiment in which stimulus presentation was successive rather than simultaneous yielded essentially the same results. Consideration was given to a dual- process model in which a time-consuming analog process normalizes stimuli that are size discrepant prior to a comparison stage that is operative for those structures responsible for "same" decisions but not for "different" judgments. Some evidence that seems inconsistent with a normalization process which operates prior to contact with memory was discussed, and an alternative explanation in which the early detection of a size difference causes changes in decision criteria was suggested.