An evaluation of exact methods for the multiple subset maximum cardinality selection problem

The maximum cardinality subset selection problem requires finding the largest possible subset from a set of objects, such that one or more conditions are satisfied. An important extension of this problem is to extract multiple subsets, where the addition of one more object to a larger subset would always be preferred to increases in the size of one or more smaller subsets. We refer to this as the multiple subset maximum cardinality selection problem (MSMCSP). A recently published branch‐and‐bound algorithm solves the MSMCSP as a partitioning problem. Unfortunately, the computational requirement associated with the algorithm is often enormous, thus rendering the method infeasible from a practical standpoint. In this paper, we present an alternative approach that successively solves a series of binary integer linear programs to obtain a globally optimal solution to the MSMCSP. Computational comparisons of the methods using published similarity data for 45 food items reveal that the proposed sequential method is computationally far more efficient than the branch‐and‐bound approach.     

An enhanced branch‐and‐bound algorithm for a partitioning problem

This paper focuses on the problem of developing a partition of n objects based on the information in a symmetric, non‐negative dissimilarity matrix. The goal is to partition the objects into a set of non‐overlapping subsets with the objective of minimizing the sum of the within‐subset dissimilarities. Optimal solutions to this problem can be obtained using dynamic programming, branch‐and‐bound and other mathematical programming methods. An improved branch‐and‐bound algorithm is shown to be particularly efficient. The improvements include better upper bounds that are obtained via a fast exchange algorithm and, more importantly, sharper lower bounds obtained through sequential solution of submatrices. A modified version of the branch‐and‐bound algorithm for minimizing the diameter of a partition is also presented. Computational results for both synthetic and empirical dissimilarity matrices reveal the effectiveness of the branch‐and‐bound methodology.     

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.     

Multidimensional letter similarity derived from recognition errors

In order to provide a reliable measure of the similarity of uppercase English letters, a confusion matrix based on 1,200 presentations of each letter was established. To facilitate an analysis of the perceived structural characteristics, the confusion matrix was decomposed according to Luce’s choice model into a symmetrical similarity matrix and a response bias vector. The underlying structure of the similarity matrix was assessed with both a hierarchical clustering and a multidimensional scaling procedure. This data is offered to investigators of visual information processing as a valuable tool for controlling not only the overall similarity of the letters in a study, but also their similarity on individual feature dimensions.     

独体汉字的字形相似性研究

本文以速示呈现方式获得了一系列独体汉字和字符的混淆矩阵。用Luce（1963）的选择模型特混淆矩阵分解为对称的相似矩阵和反应倾向矢量。基于相似矩阵,以聚类分析和多维量表法探索了汉字和字符的视觉特征及其表征形式,为揭示汉字视觉特征的几何特性和潜在结构,在研究方法上作了有意义的偿试、此外,本研究的结果支持了作者（1995）关于在含“十”或“口”的独体汉字中,“十”与“口”是该类汉字的突出视觉特征的结论。     

Clustering, seriation, and subset extraction of confusion data

The study of confusion data is a well established practice in psychology. Although many types of analytical approaches for confusion data are available, among the most common methods are the extraction of 1 or more subsets of stimuli, the partitioning of the complete stimulus set into distinct groups, and the ordering of the stimulus set. Although standard commercial software packages can sometimes facilitate these types of analyses, they are not guaranteed to produce optimal solutions. The authors present a MATLAB *.m file for preprocessing confusion matrices, which includes fitting of the similarity-choice model. Two additional MATLAB programs are available for optimally clustering stimuli on the basis of confusion data. The authors also developed programs for optimally ordering stimuli and extracting subsets of stimuli using information from confusion matrices. Together, these programs provide several pragmatic alternatives for the applied researcher when analyzing confusion data. Although the programs are described within the context of confusion data, they are also amenable to other types of proximity data.     

S-multiplicativity of a stochastic matrix and applications to visual identification

A stochastic matrix is called S-multiplicative if the conditional probabilities in a specific subset S of its cells can be factorized into the product of two functions depending on the rows and columns, respectively. The concept of S-multiplicativity is related to Goodman's concept of quasi-independence and generalizes Falmagne's concept of a multiplicative confusion matrix, where S consists of the cells lying outside the main diagonal ($\text{D}$-multiplicativity). In the present study, the notion of S-multiplicativity is applied to the analysis of visual identification performance. The confusion matrices obtained from three identification experiments are tested for predictions of S-multiplicativity derived from the multicomponent theory of perception. If the stimuli do not overlap, i.e., different stimuli have no features in common, then the theory predicts a $\text{D}$-multiplicative confusion matrix. Alternatively, in the case of overlapping stimuli the theory predicts S-multiplicativity with respect to a more severely restricted subset S. These predictions are confirmed by the data.     

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.     

Sequential decision making: An aggregate scheduling methodology

Recent studies of sequential decision making performance have utilized optimal dynamic programming solutions as the criteria for evaluating the quality of human decisions. The difficulties incumbent in increasing the dimensionality of dynamic programming formulations effectively prohibit their extension beyond the case of a single decision variable. In the following article an alternative optimization formulation for a two-decision-variable case is presented, followed by the development of a corresponding one-decision-variable problem. In addition, related methodological problems limiting the comparability of results among previous studies are presented. It is suggested that the proposed problem formulation and its accompanying interpretation for the decision maker largely eliminate these previous methodological problems. Furthermore, the problem formulation permits quantification of the relative weights that should optimally be placed on future decision periods in the multiple-stage decision problem.     

A similarity-based range—frequency model for two-category rating data

This research examines whether an extension of range-frequency theory (Parducci, 1965) could successfully predict context effects on discrimination between stimuli for two-category rating data. The new model proposes that similarity comparisons between stimulus values and rating category prototypes take place on an implicit range—frequency scale, and thus it entails effects of context on both mean ratings and discrimination indices. Predictions of the similarity-based range—frequency model were tested in an experiment in which the frequency distribution was manipulated by varying the range and relative frequencies of squares, with participants rating square sizes on a two-category scale. The model provided a parsimonious account of how discriminability of common stimuli increases with a decrease in stimulus range or an increase in the density of stimuli within a subrange.     

Roles for Drosophila mushroom body neurons in olfactory learning and memory

Olfactory learning assays in Drosophila have revealed that distinct brain structures known as mushroom bodies (MBs) are critical for the associative learning and memory of olfactory stimuli. However, the precise roles of the different neurons comprising the MBs are still under debate. The confusion surrounding the roles of the different neurons may be due, in part, to the use of different odors as conditioned stimuli in previous studies. We investigated the requirements for the different MB neurons, specifically the alpha/beta versus the gamma neurons, and whether olfactory learning is supported by different subsets of MB neurons irrespective of the odors used as conditioned stimuli. We expressed the rutabaga (rut)-encoded adenylyl cyclase in either the gamma or alpha/beta neurons and examined the effects on restoring olfactory associative learning and memory of rut mutant flies. We also expressed a temperature-sensitive shibire (shi) transgene in these neuron sets and examined the effects of disrupting synaptic vesicle recycling on Drosophila olfactory learning. Our results indicate that although we did not detect odor-pair-specific learning using GAL4 drivers that primarily express in gamma neurons, expression of the transgenes in a subset of alpha/beta neurons resulted in both odor-pair-specific rescue of the rut defect as well as odor-pair-specific disruption of learning using shi(ts1).     

Acoustic confusions and order forgetting

This paper reports two experiments employing a PETERSON and PETERSON (1959) paradigm, aimed at showing order as opposed to item forgetting in acoustic confusion experiments. In the first experiment two subsets from the acoustically similar vocabulary C V B D G T P E, i.e. D V B and C G T, were selected. One group of Ss were presented with the items from subset D V B in different orders for 4 trials, another group with the subset C G T. Interpolated activity lasted twenty seconds and presentation was auditory. The second experiment was similar to the first, but presentation was visual and interpolated activity lasted only 5 sec. In both experiments the incidence of order errors significantly and considerably exceeded the incidence of intrusion errors. This leaves open the possibility that in acoustic confusion experiments, confusions can be accounted for by assuming order rather than item forgetting, contrary to CONRAD's (1965) hypothesis.     

Similarity ratings and confusability of lipread consonants compared with similarity ratings of auditory and orthographic stimuli.

Similarity ratings of pairs of lipread consonants were obtained using a 5-point scale. Matrices were constructed showing mean similarity ratings and confusions among stimuli. Both the similarity and the confusion data provide normative data useful for researchers in many areas. Lipread data collected here are compared with similarity ratings of orthographically and auditorily presented consonants collected by Manning (1977). These comparisons provide information about how stimulus similarity both within and between presentation formats may affect information processing of the three types of stimuli. These data are of special interest to researchers studying the visual processing of speech and the effect of format of presentation on recall.     

Impaired performance of right-brain-damaged patients on a sentence anagram task

A sentence anagram task was used to examine the right cerebral hemispheres's role in core grammatical functioning at the syntactic level. The test consisted of two subsets of stimuli involving empty categories: (a) those that required the empty category to be filled, and (b) those that allowed the category to remain empty. Three hypotheses were presented: (1) right-brain-damaged patients (RBDs) would be more sensitive to subset (a) than (b), (2) RBDs would perform similarly to left-brain-damaged subjects, and (3) non-brain-damaged controls would outerperform both brain-damaged groups. With all hypotheses supported, the discussion focuses on an explanation for RBDs' sensitivity to the first subset of stimuli.     

SIMILARITY IN THE CONTINUUM OF HEAVINESS WITH SOME METHODOLOGICAL AND THEORETICAL CONSIDERATIONS

An experiment in which the subjects estimated (a) the ratio and (b) the similarity of pairs of stimuli, consisting of weights, is reported. The report reveals that the relation obtained in earlier investigations between estimated similarity and the two subjective magnitudes of the pair, holds good for the continuum of heaviness too. In connection with this the intraindividual standard deviation of ratio estimates and the problem of internal consistency of a ratio matrix are treated. In a speculative discussion the concept of similarity is scrutinized, the similarity function is tentatively extended to multi-dimensional cases and an attempt is made to connect similarity and discrimination learning.     

Integer programs for one- and two-mode blockmodeling based on prespecified image matrices for structural and regular equivalence

Establishing blockmodels for one- and two-mode binary network matrices has typically been accomplished using multiple restarts of heuristic algorithms that minimize functions of inconsistency with an ideal block structure. Although these algorithms likely yield exceptional performance, they are not assured to provide blockmodels that optimize the functional indices. In this paper, we present integer programming models that, for a prespecified image matrix, can produce guaranteed optimal solutions for matrices of nontrivial size. Accordingly, analysts performing a confirmatory analysis of a prespecified blockmodel structure can apply our models directly to obtain an optimal solution. In exploratory cases where a blockmodel structure is not prespecified, we recommend a two-stage procedure, where a heuristic method is first used to identify an image matrix and the integer program is subsequently formulated and solved to identify the optimal solution for that image matrix. Although best suited for ideal block structures associated with structural equivalence, the integer programming models have the flexibility to accommodate functional indices pertaining to regular equivalence. Computational results are reported for a variety of one- and two-mode matrices from the blockmodeling literature.     

To what extent do Gestalt grouping principles influence tactile perception?

Since their formulation by the Gestalt movement more than a century ago, the principles of perceptual grouping have primarily been investigated in the visual modality and, to a lesser extent, in the auditory modality. The present review addresses the question of whether the same grouping principles also affect the perception of tactile stimuli. Although, to date, only a few studies have explicitly investigated the existence of Gestalt grouping principles in the tactile modality, we argue that many more studies have indirectly provided evidence relevant to this topic. Reviewing this body of research, we argue that similar principles to those reported previously in visual and auditory studies also govern the perceptual grouping of tactile stimuli. In particular, we highlight evidence showing that the principles of proximity, similarity, common fate, good continuation, and closure affect tactile perception in both unimodal and crossmodal settings. We also highlight that the grouping of tactile stimuli is often affected by visual and auditory information that happen to be presented simultaneously. Finally, we discuss the theoretical and applied benefits that might pertain to the further study of Gestalt principles operating in both unisensory and multisensory tactile perception.     

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.     

Finding the features that represent stimuli

We develop a model for finding the features that represent a set of stimuli, and apply it to the Leuven Concept Database. The model combines the feature generation and similarity judgment task data, inferring whether each of the generated features is important for explaining the patterns of similarity between stimuli. Across four datasets, we show that features range from being very important to very unimportant, and that a small subset of important features is adequate to describe the similarities. We also show that the features inferred to be more important are intuitively reasonable, and present analyses showing that important features tend to focus on narrow sets of stimuli, providing information about the category structures that organize the stimuli into groups.     

Comparison of perceptual dimensions uncovered from infant-cry signals using the method of pair-comparisons and the semantic differential

Since infant cry sounds may provide caregivers with information concerning an infant's needs, several studies have examined the perceptual features of cry signals. This tradition is followed in the present study, which utilises two techniques; the method of pair-comparisons (coupled to INDSCAL) and the Semantic Differential (SD). In each case 24 infant cries (six each of pain, hunger, pleasure and birth) are used as stimuli. Data were collected from mothers; 39 rated each cry on 50 SD scales, and 26 provided ratings of similarity for every possible pairwise combination of cries. The similarity of the perceptual spaces described by the two techniques is demonstrated by a canonical analysis which compared the SD factor matrix with the INDSCAL dimension-weight matrix. Two of three possible common dimensions were uncovered. This similarity is also indicated by a standard multiple regression analysis by which the INDSCAL dimensions were described in terms of the SD scales. However, it appears the SD emphasised emotional characteristics of the cry signals whereas INDSCAL emphasised physical attributes.