The effects of temporal synchrony on the perceived organization of elements in spatially symmetric and asymmetric grids

Three experiments were conducted to investigate the influence of fast temporally segmented presentations of alternate rows or columns on the perception of grids of either symmetrically or asymmetrically spaced dot-like stimuli. Experiments 1a and 1b demonstrated that observers were able to discriminate between temporally segmented presentations of rows/columns stimuli, but showed a reduced performance in discrimination between non-temporally segmented stimuli and temporally segmented stimuli that were spatially unstructured. Experiment 2 confirmed the influence of temporally segmented presentations of rows and columns using elements whose mean luminance summed to the same as the display background. Experiment 3 demonstrated that the influence of temporal cues reduced when display elements were asymmetrically spaced. The results are discussed in terms of the influence of external spatio-temporal factors on the neural mechanisms of visual grouping.     

A Latent Block Distance-Association Model for Profile by Profile Cross-Classified Categorical Data

Abstract

Distance association models constitute a useful tool for the analysis and graphical representation of cross-classified data in which distances between points inversely describe the association between two categorical variables. When the number of cells is large and the data counts result in sparse tables, the combination of clustering and representation reduces the number of parameters to be estimated and facilitates interpretation. In this article, a latent block distance-association model is proposed to apply block clustering to the outcomes of two categorical variables while the cluster centers are represented in a low dimensional space in terms of a distance-association model. This model is particularly useful for contingency tables in which both the rows and the columns are characterized as profiles of sets of response variables. The parameters are estimated under a Poisson sampling scheme using a generalized EM algorithm. The performance of the model is tested in a Monte Carlo experiment, and an empirical data set is analyzed to illustrate the model.     

Inducing a blockmodel structure of two-mode binary data using seriation procedures

We show that seriation of the rows and columns of a two-mode, binary matrix can be an effective method for producing a reordering of the matrix that reveals a blockmodel structure of the data. The objective criterion of the seriation process is based on Robinson patterning of matrix elements. The key advantages of the proposed method are: (a) it can be used in conjunction with existing two-mode blockmodeling algorithms by facilitating selection of the number of classes for the rows and columns of the matrix and the appropriate types of ideal blocks; (b) the model uses a well-grounded index based on Robinson structure, (c) guaranteed optimal solutions can be obtained for problems of practical size, and (d) the seriation method is frequently capable of producing a solution that has a substantive interpretation with respect to the orderings of the row objects and column items.     

The conjunctive model of hierarchical classes

This paper describes the conjunctive counterpart of De Boeck and Rosenberg's hierarchical classes model. Both the original model and its conjunctive counterpart represent the set-theoretical structure of a two-way two-mode binary matrix. However, unlike the original model, the new model represents the row-column association as a conjunctive function of a set of hypothetical binary variables. The conjunctive nature of the new model further implies that it may represent some conjunctive higher order dependencies among rows and columns. The substantive significance of the conjunctive model is illustrated with empirical applications. Finally, it is shown how conjunctive and disjunctive hierarchical classes models relate to Galois lattices, and how hierarchical classes analysis can be useful to construct lattice models of empirical data.The research reported in this paper was supported by NATO (Grant CRG.921321 to Iven Van Mechelen and Seymour Rosenberg) and by the Research Fund of Katholieke Universiteit Leuven (Grants PDM92/19 and POR93/3 to Iven Van Mechelen; Grants OT89/9 and F91/56 to Paul De Boeck).     

Learning to Coordinate; a Recursion Theoretic Perspective

We consider two players each of whom attempts to predict the behavior of the other, using no more than the history of earlier predictions. Behaviors are limited to a pair of options, conventionally denoted by 0, 1. Such players face the problem of learning to coordinate choices. The present paper formulates their situation recursion theoretically, and investigates the prospects for success. A pair of players build up a matrix with two rows and infinitely many columns, and are said to “learn” each other if cofinitely many of the columns show the same number in both rows (either 0 or 1). Among other results we prove that there are two collections of players that force all other players to choose their camp. Each collection is composed of players that learn everyone else in the same collection, but no player that learns all members of one collection learns any member of the other. This revised version was published online in June 2006 with corrections to the Cover Date.     

The time course of perceptual grouping: the role of segregation and shape formation

The time course of perceptual grouping was examined in two experiments, using a primed matching task. In different conditions, elements were grouped into columns/rows by common lightness, into a shape (triangle/ arrow or square/cross) by common lightness, and into a shape without segregation of elements. The results showed an early and rapid grouping into columns/rows by common lightness and into a shape when no segregation from other elements was involved. Goodness of shape (i.e., triangle/arrow vs. square/cross) had no influence on how early grouping was evident, but the relatively poorer shapes appeared to consolidate with time. In contrast, grouping into a shape that involved segregation and required resolving figure-ground relations between segregated units, as grouping into a shape by common lightness, consumed time, regardless of shape goodness. These results suggest that the time course of grouping varies as a function of the processes involved in it (e.g., segregation and shape formation) and the conditions prevailing for each process.     

A latent class distance association model for cross‐classified data with a categorical response variable

In this paper we propose a latent class distance association model for clustering in the predictor space of large contingency tables with a categorical response variable. The rows of such a table are characterized as profiles of a set of explanatory variables, while the columns represent a single outcome variable. In many cases such tables are sparse, with many zero entries, which makes traditional models problematic. By clustering the row profiles into a few specific classes and representing these together with the categories of the response variable in a low‐dimensional Euclidean space using a distance association model, a parsimonious prediction model can be obtained. A generalized EM algorithm is proposed to estimate the model parameters and the adjusted Bayesian information criterion statistic is employed to test the number of mixture components and the dimensionality of the representation. An empirical example highlighting the advantages of the new approach and comparing it with traditional approaches is presented.     

Matrix reorganization and dynamic programming: Applications to paired comparisons and unidimensional seriation

A recursive dynamic programming strategy is discussed for optimally reorganizing the rows and simultaneously the columns of ann ×n proximity matrix when the objective function measuring the adequacy of a reorganization has a fairly simple additive structure. A number of possible objective functions are mentioned along with several numerical examples using Thurstone's paired comparison data on the relative seriousness of crime. Finally, the optimization tasks we propose to attack with dynamic programming are placed in a broader theoretical context of what is typically referred to as the quadratic assignment problem and its extension to cubic assignment.Partial support for this research was provided by NIJ Grant 80-IJ-CX-0061.     

A new scaling technique for absolute judgments

The results of an experiment using the method of absolute judgments can be viewed as a matrix of conditional probabilities in which the rows represent stimuli and the columns responses. The cosine of the angle between two row vectors is a measure of the similarity of the corresponding stimuli. This cosine provides the basis for a method of scaling the stimuli. Unlike the method of paired comparisons, this new technique does not require arbitrary fixing of a unit of measurement. A numerical example is given.     

Computing measures of simplicity of fit for loadings in factor-analytically derived scales

A very simple structure is sought when factor analysis is used to develop measurement scales. The SIMLOAD program computes measures of factorial simplicity for rows and columns of loading matrices (usually the factor pattern) as well as some overall measures. These include Kaiser’s (1974) index of factorial simplicity for variables (rows), the author’s scale fit index for factors (columns), Bentler’s (1977) scale-free matrix measure, and hyperplane counts. Routine use of these measures is recommended for multifactor scale development. The measures may also be useful in more general factor applications and in confirmatory as well as exploratory analyses. SIMLOAD also computes factor scale intercorrelations, scale alpha coefficients (including alpha when an item is removed), and sorted loadings for ease of interpretation.     

Positive Definiteness via Off-diagonal Scaling of a Symmetric Indefinite Matrix

Indefinite symmetric matrices that are estimates of positive-definite population matrices occur in a variety of contexts such as correlation matrices computed from pairwise present missing data and multinormal based methods for discretized variables. This note describes a methodology for scaling selected off-diagonal rows and columns of such a matrix to achieve positive definiteness. As a contrast to recently developed ridge procedures, the proposed method does not need variables to contain measurement errors. When minimum trace factor analysis is used to implement the theory, only correlations that are associated with Heywood cases are shrunk.     

A branch-and-bound algorithm for fitting anti-robinson structures to symmetric dissimilarity matrices

The seriation of proximity matrices is an important problem in combinatorial data analysis and can be conducted using a variety of objective criteria. Some of the most popular criteria for evaluating an ordering of objects are based on (anti-) Robinson forms, which reflect the pattern of elements within each row and/or column of the reordered matrix when moving away from the main diagonal. This paper presents a branch-and-bound algorithm that can be used to seriate a symmetric dissimilarity matrix by identifying a reordering of rows and columns of the matrix optimizing an anti-Robinson criterion. Computational results are provided for several proximity matrices from the literature using four different anti-Robinson criteria. The results suggest that with respect to computational efficiency, the branch-and-bound algorithm is generally competitive with dynamic programming. Further, because it requires much less storage than dynamic programming, the branch-and-bound algorithm can provide guaranteed optimal solutions for matrices that are too large for dynamic programming implementations.     

Evaluating the conformity of sociometric measurements

The problem of comparing two sociometric matrices, as originally discussed by Katz and Powell in the early 1950's, is reconsidered and generalized using a different inference model. In particular, the proposed indices of conformity are justified by a regression argument similar to the one used by Somers in presenting his well-known measures of asymmetric ordinal association. A permutation distribution and an associated significance test are developed for the specific hypothesis of no conformity reinterpreted as a random matching of the rows and (simultaneously) the columns of one sociometric matrix to the rows and columns of a second. The approximate significance tests that are presented and illustrated with a simple numerical example are based on the first two moments of the permutation distribution, or alternatively, on a random sample from the complete distribution.Partial support for the research of the first author was provided by the National Science Foundation through SOC 75-07860. Equal authorship is implied. The work was done when the first author was at the University of Wisconsin.     

Perceptual grouping and attention: Not all groupings are equal

We examined grouping under inattention using Driver, Davis, Russell, Turatto, & Freeman’s (2001) method. On each trial, two successive displays were briefly presented, each comprising a central target square surrounded by elements. The task was to judge whether the two targets were the same or different. The organization of the background elements stayed the same or changed, independently of the targets. In different conditions, background elements grouped into columns/rows by color similarity, a shape (a triangle/arrow, a square/cross, or a vertical/horizontal line) by color similarity, and a shape with no other elements in the background. We measured the influence of the background on the target same-different judgments. The results imply that background elements grouped into columns/rows by color similarity and into a shape when no segregation from other elements was involved and the shape was relatively “good.” In contrast, no background grouping was observed when resolving figure-ground relations for segregated units was required, as in grouping into a shape by color similarity. These results suggest that grouping is a multiplicity of processes that vary in their attentional demands. Regardless of attentional demands, the products of grouping are not available to awareness without attention.     

An algorithm for clustering relational data with applications to social network analysis and comparison with multidimensional scaling

A method of hierarchical clustering for relational data is presented, which begins by forming a new square matrix of product-moment correlations between the columns (or rows) of the original data (represented as an n × m matrix). Iterative application of this simple procedure will in general converge to a matrix that may be permuted into the blocked form [_?1¹₁^?1]. This convergence property may be used as the basis of an algorithm (CONCOR) for hierarchical clustering. The CONCOR procedure is applied to several illustrative sets of social network data and is found to give results that are highly compatible with analyses and interpretations of the same data using the blockmodel approach of White (White, Boorman & Breiger, 1976). The results using CONCOR are then compared with results obtained using alternative methods of clustering and scaling (MDSCAL, INDSCAL, HICLUS, ADCLUS) on the same data sets.     

The interaction of objective and subjective organizations in a localization search task

We investigated how both objective and subjective organizations affect perceptual organization and how this perceptual organization, in turn, influences observers’ performance in a localization search task. Two groups of observers viewing exactly the same stimuli (objective organization) performed in significantly different ways, depending on how they were induced to parse the display (subjective organization). In Experiments 1 and 2, the observers were asked to describe the location of a tilted target among a varying number of vertical or horizontal distractors. Subjective organization was induced by instructing observers to parse the display into either three horizontal regions (rows) or three vertical regions (columns). The position of the target was critical: location performance, as assessed by reaction time and errors, was consistently impaired at the locations adjacent to the boundaries defining the regions, producing what we refer to as thesubjective boundary effect. Furthermore, the extent of this effect depended on whether the stimulus-driven and conceptually driven information concurred or conflicted. This made location information more or less accessible. In Experiment 1, the strength of objective grouping was a function of the proximity of the items (near or far conditions) and their orientation in a 6×6 matrix. In Experiment 2, the strength of objective grouping was a function of similarity of color (items were color coded by rows or by columns) and the orientation of the items in a 9×9 matrix. The subjective boundary effect was more pronounced when the display promoted grouping in the direction orthogonal to that of the task (e.g., when observers parsed by rows but vertical distractors were closer together [Experiment 1] or color coded [Experiment 2] to induce global columns). In contrast, this effect decreased when the direction of both objective and subjective organizations was parallel (e.g., when observers parsed by rows and horizontal distractors were closer together [Experiment 1] or were color coded [Experiment 2] to induce global rows). A localization search task proved to be an ideal forum in which objective and subjective organizations interacted. We discuss how these results indicated that observers’ performance in a localization task was determined by the interaction of objective and subjective organizations, and that the resulting perceptual organization constrained coarse location information.     

Instability of hierarchical cluster analysis due to input order of the data: the PermuCLUSTER solution

Hierarchical agglomerative cluster analysis (HACA) may yield different solutions under permutations of the input order of the data. This instability is caused by ties, either in the initial proximity matrix or arising during agglomeration. The authors recommend to repeat the analysis on a large number of random permutations of the rows and columns of the proximity matrix and select a solution with the highest goodness-of-fit. This approach was implemented in an SPSS add-in, PermuCLUSTER, which can perform all HACA methods of SPSS. Analyses of 2 data sets show that (a) results are affected by input order, (b) instability in one method co-occurs with instability in other methods, and (c) some instability effects are more dramatic because they occur at higher agglomeration levels.     

Simple kinetic information for transparent depth

The present paper reports three investigations of new kinetic information for transparent depth using computer-generated dot patterns. An initial demonstration showed that separation in depth could be obtained by translating rectangular lattices of dots through one another like intersecting columns of marching soldiers. The first two experiments showed that diagonal interactions between lattices created significantly stronger separation than did horizontal or vertical interactions (horizontal was, in turn, stronger than vertical) and that patterns which translated through one another without any of the individual elements intersecting were better separated than those whose rows or columns intersected in register. The third experiment showed that random patterns interacting in any direction created the strongest separations of all the patterns observed. Results were taken to indicate that a unified theory of depth information, developed in the context of James Gibson’s ecological optics, must incorporate both spatial and kinetic structure in its specification of necessary and sufficient stimulus conditions.     

A simple procedure for rearranging matrices

Guttman's scalogram board technique for reordering the columns and rows of a matrix is described and its disadvantages are pointed out. A simple and inexpensive procedure for doing the same job without these disadvantages is outlined.I am grateful to Professor Jozef Cohen of the University of Illinois for a five-minute conversation which greatly simplified the procedure described here.     

Note on matrices of random numbers

The statistical and structural characteristics of 13 matrices of random numbers in which both the cells and the entries were randomly chosen are discussed. Each matrix was explored considering row means, standard deviations, and correlations as well as column means, standard deviations, and correlations. A study concerning the sequential arrangement of digits was performed by finding out in tables of random numbers how many times the values 0 to 9 are followed by any other digit. Analyses indicate clear factor structures when factor analyzing correlations of rows and of columns and when examining sequential arrangements, concluding that for a given set of digits it is possible to assert both randomness and nonrandomness depending on how the data are examined.