Matrices with a given number of violations of Regular Minimality

A row (or column) of an n×n matrix complies with Regular Minimality (RM) if it has a unique minimum entry which is also a unique minimum entry in its column (respectively, row). The number of violations of RM in a matrix is defined as the number of rows (equivalently, columns) that do not comply with RM. We derive a formula for the proportion of n×n matrices with a given number of violations of RM among all n×n matrices with no tied entries. The proportion of matrices with no more than a given number of violations can be treated as the p-value of a permutation test whose null hypothesis states that all permutations of the entries of a matrix without ties are equiprobable, and the alternative hypothesis states that RM violations occur with lower probability than predicted by the null hypothesis. A matrix with ties is treated as being represented by all matrices without ties that have the same set of strict inequalities among their entries.     

A variable neighborhood search method for generalized blockmodeling of two-mode binary matrices

The clustering of two-mode proximity matrices is a challenging combinatorial optimization problem that has important applications in the quantitative social sciences. We focus on one particular type of problem related to the clustering of a two-mode binary matrix, which is relevant to the establishment of generalized blockmodels for social networks. In this context, clusters for the rows of the two-mode matrix intersect with clusters of the columns to form blocks, which should ideally be either complete (all 1s) or null (all 0s). A new procedure based on variable neighborhood search is presented and compared to an existing two-mode K-means clustering algorithm. The new procedure generally provided slightly greater explained variation; however, both methods yielded exceptional recovery of cluster structure.     

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.     

A Note on the Linearly and Quadratically Weighted Kappa Coefficients

The linearly and quadratically weighted kappa coefficients are popular statistics in measuring inter-rater agreement on an ordinal scale. It has been recently demonstrated that the linearly weighted kappa is a weighted average of the kappa coefficients of the embedded 2 by 2 agreement matrices, while the quadratically weighted kappa is insensitive to the agreement matrices that are row or column reflection symmetric. A rank-one matrix decomposition approach to the weighting schemes is presented in this note such that these phenomena can be demonstrated in a concise manner.     

Constrained multidimensional unfolding of confusion matrices: Goal point and slide vector models

Two constrained multidimensional unfolding models, the goal point and slide vector models, are proposed for analyzing confusion matrices. In both models, the row and column stimuli are expressed as two sets of points in a low-dimensional space, where the difference vector connecting a column point to the corresponding row point indicates the change in the stimulus representation through a cognitive process. The difference vector is constrained by the hypothesis that the trend in the representational change is invariant across stimuli: the goal point model constrains all difference vectors to point toward a single point, and the slide vector model constrains all difference vectors to be parallel to each other. In both models the coordinates of points are estimated by the maximum-likelihood method. Examples illustrate that the two models allow us to examine hypotheses about invariant trends in representational changes and to grasp such trends from the resulting configurations.     

The approximation of two-mode proximity matrices by sums of order-constrained matrices

A least-squares strategy is proposed for representing a two-mode proximity matrix as an approximate sum of a small number of matrices that satisfy certain simple order constraints on their entries. The primary class of constraints considered define Q-forms (or anti-Q-forms) for a two-mode matrix, where after suitable and separate row and column reorderings, the entries within each row and within each column are nondecreasing (or nonincreasing) to a maximum (or minimum) and thereafter nonincreasing (or nondecreasing). Several other types of order constraints are also mentioned to show how alternative structures can be considered using the same computational strategy.     

The SIMCLAS Model: Simultaneous Analysis of Coupled Binary Data Matrices with Noise Heterogeneity Between and Within Data Blocks

In many research domains different pieces of information are collected regarding the same set of objects. Each piece of information constitutes a data block, and all these (coupled) blocks have the object mode in common. When analyzing such data, an important aim is to obtain an overall picture of the structure underlying the whole set of coupled data blocks. A further challenge consists of accounting for the differences in information value that exist between and within (i.e., between the objects of a single block) data blocks. To tackle these issues, analysis techniques may be useful in which all available pieces of information are integrated and in which at the same time noise heterogeneity is taken into account. For the case of binary coupled data, however, only methods exist that go for a simultaneous analysis of all data blocks but that do not account for noise heterogeneity. Therefore, in this paper, the SIMCLAS model, being a Hierarchical Classes model for the simultaneous analysis of coupled binary two-way matrices, is presented. In this model, noise heterogeneity between and within the data blocks is accounted for by downweighting entries from noisy blocks/objects within a block. In a simulation study it is shown that (1) the SIMCLAS technique recovers the underlying structure of coupled data to a very large extent, and (2) the SIMCLAS technique outperforms a Hierarchical Classes technique in which all entries contribute equally to the analysis (i.e., noise homogeneity within and between blocks). The latter is also demonstrated in an application of both techniques to empirical data on categorization of semantic concepts.     

Object relocation: a program for setting up, running, and analyzing experiments on memory for object locations

Object Relocation is a computer program for Windows 95, with which experiments on spatial memory for object locations can be designed, run, and analyzed. Because of its clear graphical user interface, no long and complex command syntax is needed. Basically, a stimulus consists of a frame that contains a chosen number of locations (i.e., the actual spatial layout) to which objects can be assigned. When the experiment is run, these stimuli are presented to the subject for a variable period of time. Subsequently (either with or without a delay), the objects are presented in a row above the frame and have to be relocated to the correct positions. Finally, the raw data can be analyzed efficiently, using various error scores, and an SPSS-ready output file can be produced. Object Relocation is a very flexible program: New objects and positions can easily be added, and various options for presentation and relocation are present.     

Optimal Least-Squares Unidimensional Scaling: Improved Branch-and-Bound Procedures and Comparison to Dynamic Programming

There are two well-known methods for obtaining a guaranteed globally optimal solution to the problem of least-squares unidimensional scaling of a symmetric dissimilarity matrix: (a) dynamic programming, and (b) branch-and-bound. Dynamic programming is generally more efficient than branch-and-bound, but the former is limited to matrices with approximately 26 or fewer objects because of computer memory limitations. We present some new branch-and-bound procedures that improve computational efficiency, and enable guaranteed globally optimal solutions to be obtained for matrices with up to 35 objects. Experimental tests were conducted to compare the relative performances of the new procedures, a previously published branch-and-bound algorithm, and a dynamic programming solution strategy. These experiments, which included both synthetic and empirical dissimilarity matrices, yielded the following findings: (a) the new branch-and-bound procedures were often drastically more efficient than the previously published branch-and-bound algorithm, (b) when computationally feasible, the dynamic programming approach was more efficient than each of the branch-and-bound procedures, and (c) the new branch-and-bound procedures require minimal computer memory and can provide optimal solutions for matrices that are too large for dynamic programming implementation.The authors gratefully acknowledge the helpful comments of three anonymous reviewers and the Editor. We especially thank Larry Hubert and one of the reviewers for providing us with the MATLAB files for optimal and heuristic least-squares unidimensional scaling methods.This revised article was published online in June 2005 with all corrections incorporated.     

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.     

Use of the recognition heuristic depends on the domain’s recognition validity,not on the recognition validity of selected sets of objects

According to the recognition-heuristic theory, decision makers solve paired comparisons in which one object is recognized and the other not by recognition alone, inferring that recognized objects have higher criterion values than unrecognized ones. However, success—and thus usefulness—of this heuristic depends on the validity of recognition as a cue, and adaptive decision making, in turn, requires that decision makers are sensitive to it. To this end, decision makers could base their evaluation of the recognition validity either on the selected set of objects (the set’s recognition validity), or on the underlying domain from which the objects were drawn (the domain’s recognition validity). In two experiments, we manipulated the recognition validity both in the selected set of objects and between domains from which the sets were drawn. The results clearly show that use of the recognition heuristic depends on the domain’s recognition validity, not on the set’s recognition validity. In other words, participants treat all sets as roughly representative of the underlying domain and adjust their decision strategy adaptively (only) with respect to the more general environment rather than the specific items they are faced with.     

Multiobjective Blockmodeling for Social Network Analysis

To date, most methods for direct blockmodeling of social network data have focused on the optimization of a single objective function. However, there are a variety of social network applications where it is advantageous to consider two or more objectives simultaneously. These applications can broadly be placed into two categories: (1) simultaneous optimization of multiple criteria for fitting a blockmodel based on a single network matrix and (2) simultaneous optimization of multiple criteria for fitting a blockmodel based on two or more network matrices, where the matrices being fit can take the form of multiple indicators for an underlying relationship, or multiple matrices for a set of objects measured at two or more different points in time. A multiobjective tabu search procedure is proposed for estimating the set of Pareto efficient blockmodels. This procedure is used in three examples that demonstrate possible applications of the multiobjective blockmodeling paradigm.     

Fluent, fast, and frugal? A formal model evaluation of the interplay between memory, fluency, and comparative judgments

A new process model of the interplay between memory and judgment processes was recently suggested, assuming that retrieval fluency-that is, the speed with which objects are recognized-will determine inferences concerning such objects in a single-cue fashion. This aspect of the fluency heuristic, an extension of the recognition heuristic, has remained largely untested due to methodological difficulties. To overcome the latter, we propose a measurement model from the class of multinomial processing tree models that can estimate true single-cue reliance on recognition and retrieval fluency. We applied this model to aggregate and individual data from a probabilistic inference experiment and considered both goodness of fit and model complexity to evaluate different hypotheses. The results were relatively clear-cut, revealing that the fluency heuristic is an unlikely candidate for describing comparative judgments concerning recognized objects. These findings are discussed in light of a broader theoretical view on the interplay of memory and judgment processes.     

Finding relations between systems of entities: a multivariate approach

A procedure to find out the relationship between systems of entities is described. The crux of the approach depends on projecting the systems to be compared on the same arbitrary hyperspace so that any two systems of entities can be related. The procedure described applies to data presented in square or rectangular non-singular matrices and provides a procedure to determine multiple regression coefficients, multiple correlations, canonical correlations and the relationships between the inner product of the row vectors and of the column vectors of the matrices involved. The operations described apply to raw values, deviation or standard scores, and may help in interdisciplinary research, though the risks involved in the blind application of a mathematical abstract formulation should not be ignored.     

Object segmentation by top-down processes

In cluttered scenes, some object boundaries may not be marked by image cues. In such cases, the boundaries must be defined top-down as a result of object recognition. Here we ask if observers can retain the boundaries of several recognized objects in order to segment an unfamiliar object. We generated scenes consisting of neatly stacked objects, and the objects themselves consisted of neatly stacked coloured blocks. Because the blocks were stacked the same way within and across objects, there were no visual cues indicating which blocks belonged to which objects. Observers were trained to recognize several objects and we tested whether they could segment a novel object when it was surrounded by these familiar, studied objects. The observer's task was to count the number of blocks comprising the target object. We found that observers were able to accurately count the target blocks when the target was surrounded by up to four familiar objects. These results indicate that observers can use the boundaries of recognized objects in order to accurately segment, top-down, a novel object.     

Procrustean analysis of matrices with different row order

Procrustean analysis is extended here to the case where matrices have different row order. Using a definition system in the sense of facet theory, fitting transformations are determined for the set of centroids that represent common substantive equivalence classes of all or some matrices. The resulting operators are used to transform the original matrices. An index, δ, which is consistent with this approach expresses the goodness of fit.     

Position effects in encoding briefly exposed item matrices: evidence for a reading bias or merely a matter of the selection criterion?

Position effects are frequently reported in experiments that investigate the recognition of items from briefly exposed stimulus matrices. A reliable finding is the ability to report items from the first row of the matrix more accurately than from the second row. The present experiments explore whether this position effect depends upon the selection criterion used to indicate the subgroup of items that has to be reported in a given trial. In Experiment 1, German and Chinese participants were presented with language-specific items which had to be selected by column. In Experiment 2, Germans were presented with Latin letters and the selection criterion was letter color. A strong row effect was evident in both experiments although the selection criteria did not prompt a line-by-line grouping of the items. The row effect is seen as a manifestation of top-down processing that is derived from reading habits.     

Some Properties Of Three Factor Contribution Matrices

Some properties of three matrices which have been proposed as quantitative bases for factor interpretation are discussed. The first of these contains as elements the covariances of the tests with weighted factor scores. Interpretations of the row sums and column sums of this matrix are given and its independence of the primary factor-reference vector distinction is demonstrated. Two matrices of part correlations are shown to be proportional by columns to the primary factor pattern (and thus to the reference vector structure). One of these matrices is shown to be equal to the reference vector structure.     

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.