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.     

A simplification of a result by zellini on the maximal rank of symmetric three-way arrays

Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix of ordern ×n as a linear combination of 1/2n(n+1) fixed rank one matrices, thus constructing an explicit tensor basis for the set of symmetricn ×n matrices. Zellini's decomposition is based on properties of persymmetric matrices. In the present paper, a simplified tensor basis is given, by showing that a symmetric matrix can also be decomposed in terms of 1/2n(n+1) fixed binary matrices of rank one. The decomposition implies that ann ×n ×p array consisting ofp symmetricn ×n slabs has maximal rank 1/2n(n+1). Likewise, an unconstrained INDSCAL (symmetric CANDECOMP/PARAFAC) decomposition of such an array will yield a perfect fit in 1/2n(n+1) dimensions. When the fitting only pertains to the off-diagonal elements of the symmetric matrices, as is the case in a version of PARAFAC where communalities are involved, the maximal number of dimensions can be further reduced to 1/2n(n–1). However, when the saliences in INDSCAL are constrained to be nonnegative, the tensor basis result does not apply. In fact, it is shown that in this case the number of dimensions needed can be as large asp, the number of matrices analyzed.     

Correspondence analysis and optimal structural representations

Many well-known measures for the comparison of distinct partitions of the same set ofn objects are based on the structure of class overlap presented in the form of a contingency table (e.g., Pearson's chi-square statistic, Rand's measure, or Goodman-Kruskal'sτ _b ), but they all can be rephrased through the use of a simple cross-product index defined between the corresponding entries from twon ×n proximity matrices that provide particular a priori (numerical) codings of the within- and between-class relationships for each of the partitions. We consider the task of optimally constructing the proximity matrices characterizing the partitions (under suitable restriction) so as to maximize the cross-product measure, or equivalently, the Pearson correlation between their entries. The major result presented states that within the broad classes of matrices that are either symmetric, skew-symmetric, or completely arbitrary, optimal representations are already derivable from what is given by a simple one-dimensional correspondence analysis solution. Besides severely limiting the type of structures that might be of interest to consider for representing the proximity matrices, this result also implies that correspondence analysis beyond one dimension must always be justified from logical bases other than the optimization of a single correlational relationship between the matrices representing the two partitions.     

Relations between a laboratory-based sustained attention task and traffic violations in an Iranian sample of drivers

The role of cognitive abilities regarding driver behavior is crucially important in the occurrence of traffic violations and preventing tragic motor vehicle collisions. Sustained attention is one cognitive ability that may contribute to safe driving behavior. The present study investigated the relation between sustained attention and traffic violations. One hundred and one Iranian male drivers (age: M = 37.17, SD = 8.37) with at least 2 years driving experience voluntarily participated in the study. Participants were categorized into two groups based on the number of driving rule violations they committed over the last 2 years (n = 48 clean record, n = 53 violators). Sustained attention was measured by performance on the Conjunctive Continuous Performance Task (CCPT). Four CCPT performance measures were computed: (1) Mean reaction time for correct responses (M-RT); (2) standard deviation of reaction times for correct responses (SD-RT); (3) percent of omission errors (failure to respond to target stimulus); (4) percent of commission errors (identification of a non-target stimulus as target). Results showed after controlling for age and education, there was a significant group difference on M-RT, indicating that individuals with no traffic violations had faster reaction times as compared to those who had 1 or more traffic violations. No effect of group on any of the other outcomes was present after correcting for alpha inflation. When assessing the effect of age and education, education was significantly related to average reaction time and percent of omission errors. No significant effect of age was apparent. Findings suggest cognitive function, specifically sustained attention measured with a laboratory-based measure, may be associated with safe driving behavior.     

Sources and influence of perceptual variance: Comment on Dzhafarov's Regular Minimality Principle

Dzhafarov [(2002). Multidimensional Fechnerian scaling: Pairwise comparisons, regular minimality, and nonconstant self-similarity. Journal of Mathematical Psychology, 46, 583-608] claims that Regular Minimality (RM) is a fundamental property of “same-different” discrimination probabilities and supports his claim with some empirical evidence. The key feature of RM is that the mapping, h, between two observation areas based on minimum discrimination probability is invertible. Dzhafarov [(2003a). Thurstonian-type representations for “same-different” discriminations: Deterministic decisions and independent images. Journal of Mathematical Psychology, 47, 184-204; (2003b). Thurstonian-type representations for “same-different” discriminations: Probabilistic decisions and interdependent images. Journal of Mathematical Psychology, 47, 229-243] also demonstrates that well-behaved Thurstonian models of “same-different” judgments are incompatible with RM and Nonconstant Self-Similarity (NCSS). There is extensive empirical support for the latter. Stimulus and neural sources of perceptual noise are discussed and two points are made:

Point 1: Models that require discrimination probabilities for noisy stimuli to possess the property that h is invertible would be too restrictive.

Point 2: In the absence of stimulus noise, violations of RM may be so subtle that their detection would be unlikely.

     

Identifying a Reordering of Rows and Columns for Multiple Proximity Matrices Using Multiobjective Programming

This paper is concerned with a problem where K (n×n) proximity matrices are available for a set of n objects. The goal is to identify a single permutation of the n objects that provides an adequate structural fit, as measured by an appropriate index, for each of the K matrices. A multiobjective programming approach for this problem, which seeks to optimize a weighted function of the K indices, is proposed, and illustrative examples are provided using a set of proximity matrices from the psychological literature. These examples show that, by solving the multiobjective programming model under different weighting schemes, the quantitative analyst can uncover information about the relationships among the matrices and often identify one or more permutations that provide good to excellent index values for all matrices.     

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.     

A nonmetric variety of linear factor analysis

The numbers in each column of ann ×m matrix of multivariate data are interpreted as giving the measured values of alln of the objects studied on one ofm different variables. Except for random error, the rank order of the numbers in such a column is assumed to be determined by a linear rule of combination of latent quantities characterizing each row object with respect to a small number of underlying factors. An approximation to the linear structure assumed to underlie the ordinal properties of the data is obtained by iterative adjustment to minimize an index of over-all departure from monotonicity. The method is “nonmetric” in that the obtained structure in invariant under monotone transformations of the data within each column. Except in certain degenerate cases, the structure is nevertheless determined essentially up to an affine transformation. Tests show (a) that, when the assumed monotone relationships are strictly linear, the recovered structure tends closely to approximate that obtained by standard (metric) factor analysis but (b) that, when these relationships are severely nonlinear, the nonmetric method avoids the inherent tendency of the metric method to yield additional, spurious factors. From the practical standpoint, however, the usefulness of the nonmetric method is limited by its greater computational cost, vulnerability to degeneracy, and sensitivity to error variance.     

Monotone spline transformations for dimension reduction

Consider a set of data consisting of measurements ofn objects with respect top variables displayed in ann ×p matrix. A monotone transformation of the values in each column, represented as a linear combination of integrated basis splines, is assumed determined by a linear combination of a new set of values characterizing each row object. Two different models are used: one, an Eckart-Young decomposition model, and the other, a multivariate normal model. Examples for artificial and real data are presented. The results indicate that both methods are helpful in choosing dimensionality and that the Eckart-Young model is also helpful in displaying the relationships among the objects and the variables. Also, results suggest that the resulting transformations are themselves illuminating.     

A Tabu-Search Heuristic for Deterministic Two-Mode Blockmodeling of Binary Network Matrices

Two-mode binary data matrices arise in a variety of social network contexts, such as the attendance or non-attendance of individuals at events, the participation or lack of participation of groups in projects, and the votes of judges on cases. A popular method for analyzing such data is two-mode blockmodeling based on structural equivalence, where the goal is to identify partitions for the row and column objects such that the clusters of the row and column objects form blocks that are either complete (all 1s) or null (all 0s) to the greatest extent possible. Multiple restarts of an object relocation heuristic that seeks to minimize the number of inconsistencies (i.e., 1s in null blocks and 0s in complete blocks) with ideal block structure is the predominant approach for tackling this problem. As an alternative, we propose a fast and effective implementation of tabu search. Computational comparisons across a set of 48 large network matrices revealed that the new tabu-search heuristic always provided objective function values that were better than those of the relocation heuristic when the two methods were constrained to the same amount of computation time.     

Kruskal's polynomial for 2×2×2 arrays and a generalization to 2×n×n arrays

A remarkable difference between the concept of rank for matrices and that for three-way arrays has to do with the occurrence of non-maximal rank. The set ofn×n matrices that have a rank less thann has zero volume. Kruskal pointed out that a 2×2×2 array has rank three or less, and that the subsets of those 2×2×2 arrays for which the rank is two or three both have positive volume. These subsets can be distinguished by the roots of a certain polynomial. The present paper generalizes Kruskal's results to 2×n×n arrays. Incidentally, it is shown that twon ×n matrices can be diagonalized simultaneously with positive probability.The author is obliged to Joe Kruskal and Henk Kiers for commenting on an earlier draft, and to Tom Wansbeek for raising stimulating questions.     

Geometric analogue of holographic reduced representation

Holographic reduced representations (HRRs) are distributed representations of cognitive structures based on superpositions of convolution-bound n-tuples. Restricting HRRs to n-tuples consisting of ±1, one reinterprets the variable binding as a representation of the additive group of binary n-tuples with addition modulo 2. Since convolutions are not defined for vectors, the HRRs cannot be directly associated with geometric structures. Geometric analogues of HRRs are obtained if one considers a projective representation of the same group in the space of blades (geometric products of basis vectors) associated with an arbitrary n-dimensional Euclidean (or pseudo-Euclidean) space. Switching to matrix representations of Clifford algebras, one can always turn a geometric analogue of an HRR into a form of matrix distributed representation. In typical applications the resulting matrices are sparse, so that the matrix representation is less efficient than the representation directly employing the rules of geometric algebra. A yet more efficient procedure is based on ‘projected products’, a hierarchy of geometrically meaningful n-tuple multiplication rules obtained by combining geometric products with projections on relevant multivector subspaces. In terms of dimensionality the geometric analogues of HRRs are in between holographic and tensor-product representations.     

The differences in hazard response time and driving styles of violation-involved and violation-free taxi drivers

The present study aimed to investigate the relationships between taxi drivers’ traffic violations in past driving and two domains: driving skill (hazard perception skill) and driving style. Five hundred and fifty taxi drivers aged 25 – 59 were recruited to finish a video-based hazard perception test and the Chinese version of the Multidimensional Driving Style Inventory (MDSI). The relationships between hazard response time, driving style and traffic violations were examined, and the differences in hazard response times and driving styles of violation-involved drivers (n = 220) and violation-free drivers (n = 330) were compared. The results showed that taxi drivers’ traffic violations are closely related to their driving styles and hazard response time. Violation-involved drivers scored significantly higher in hazard response time and maladaptive driving styles (i.e., anxious, risky and angry styles) and lower in careful driving style than violation-free drivers. More importantly, drivers’ hazard response time and driving styles can effectively predict their violation involvement in the last 12 months with an overall classification accuracy of 66.4%. The findings provide evidence for the usefulness of video-based hazard perception tests and the MDSI in taxi driver testing and training.     

Matrix differential calculus with applications to simple,hadamard, and kronecker products

Several definitions are in use for the derivative of an m × p matrix function F(X) with respect to its n × q matrix argument X. We argue that only one of these definitions is a viable one, and that to study smooth maps from the space of n × q matrices to the space of m × p matrices it is often more convenient to study the map from nq-space to mp-space. Also, several procedures exist for a calculus of functions of matrices. It is argued that the procedure based on differentials is superior to other methods of differentiation, and leads inter alia to a satisfactory chain rule for matrix functions.     

Graph-theoretic representations for proximity matrices through strongly-anti-Robinson or circular strongly-anti-Robinson matrices

There are various optimization strategies for approximating, through the minimization of a least-squares loss function, a given symmetric proximity matrix by a sum of matrices each subject to some collection of order constraints on its entries. We extend these approaches to include components in the approximating sum that satisfy what are called the strongly-anti-Robinson (SAR) or circular strongly-anti-Robinson (CSAR) restrictions. A matrix that is SAR or CSAR is representable by a particular graph-theoretic structure, where each matrix entry is reproducible from certain minimum path lengths in the graph. One published proximity matrix is used extensively to illustrate the types of approximation that ensue when the SAR or CSAR constraints are imposed.The authors are indebted to Boris Mirkin who first noted in a personal communication to one of us (LH, April 22, 1996) that the optimization method for fitting anti-Robinson matrices in Hubert and Arabie (1994) should be extendable to the fitting of strongly anti-Robinson matrices as well.     

Measuring and testing the agreement of matrices

The problem of comparing the agreement of two n × n matrices has a variety of applications in experimental psychology. A well-known index of agreement is based on the sum of the element-wise products of the matrices. Although less familiar to many researchers, measures of agreement based on within-row and/or within-column gradients can also be useful. We provide a suite of MATLAB programs for computing agreement indices and performing matrix permutation tests of those indices. Programs for computing exact p-values are available for small matrices, whereas resampling programs for approximate p-values are provided for larger matrices.     

Frequency and pattern of voluntary pedalling is influenced after one week of heavy strength training

Changes in voluntary rhythmic leg movement characteristics of freely chosen cadence (reflecting movement frequency) and tangential pedal force profile (reflecting movement pattern) were investigated during 4 weeks of (i) heavy hip extension strength training (HET, n = 9), (ii) heavy hip flexion strength training (HFT, n = 9), and (iii) no intervention (CON, n = 9). Training consisted of three 5RM–10RM sets per session, with two sessions/week. Submaximal ergometer cycling was performed before the training period (pretest) and after every week of training (test A1, A2, A3, and posttest). Strength increased by on average 25% in HET and 33% in HFT. Freely chosen cadence was only changed in HET, occurring already after 1 week of training. Thus, percentage reductions of cadence in HET at test A1, A2, A3, and posttest, with respect to the pretest value, amounted for maximally on average 17%, or 14 rpm, and were larger than the corresponding changes in CON (p = .037). Percentage increases in minimum tangential pedal force in HET at test A1, A2, A3, and posttest, with respect to the pretest value, were larger than the corresponding changes in CON (p = .024). Heavy hip flexion strength training did not cause such alterations.     

Age-related changes in cognitive processing of moral and social conventional violations

Moral and conventional violations are usually judged differently: Only moral violations are treated as independent of social rules. To investigate the cognitive processing involved in the development of this distinction, undergraduates (N = 34), adolescents (N = 34), and children (N = 14) read scenarios presented on a computer that had 1 of 3 endings: moral violations, conventional violations, or neutral acts. Participants judged whether the act was acceptable or unacceptable in a condition in which social rules were assumed, or in a condition in which they imagined the absence of rules (rule-removed condition). At all ages reaction times (RTs) were faster for moral than conventional violations when a rule was assumed. RTs in the rule-removed condition were longer than in the rule-assumed condition, only for adults’ moral judgments. In addition to this age difference, adolescents made more normative judgments than children. These findings extend previous work by showing different time courses of processing conventional versus moral violations and revealing age-related differences in the tendency to make normative judgments.     

An algebraic solution for the communalities

Factorial analysis begins with ann ×n correlation matrixR, whose principal diagonal entries are unknown. If the common test space of the battery is under investigation, the communality of each test is entered in the appropriate diagonal cell. This value is the portion of the test's variance shared with others in the battery. The communalities must be so estimated thatR will maintain the rank determined by its side entries, after the former have been inserted. Previous methods of estimating the communalities have involved a certain arbitrariness, since they depended on selecting test subgroups or parts of the data inR. A theory is presented showing that this difficulty can be avoided in principle. In its present form, the theory is not offered as a practical computing procedure. The basis of the new method lies in the Cayley-Hamilton theorem: Any square matrix satisfies its own characteristic equation.     

ConPar: a method for identifying groups of concordant subject proximity matrices for subsequent multidimensional scaling analyses

A common representation of data within the context of multidimensional scaling (MDS) is a collection of symmetric proximity (similarity or dissimilarity) matrices for each of M subjects. There are a number of possible alternatives for analyzing these data, which include: (a) conducting an MDS analysis on a single matrix obtained by pooling (averaging) the M subject matrices, (b) fitting a separate MDS structure for each of the M matrices, or (c) employing an individual differences MDS model. We discuss each of these approaches, and subsequently propose a straightforward new method (CONcordance PARtitioning—ConPar), which can be used to identify groups of individual-subject matrices with concordant proximity structures. This method collapses the three-way data into a subject×subject dissimilarity matrix, which is subsequently clustered using a branch-and-bound algorithm that minimizes partition diameter. Extensive Monte Carlo testing revealed that, when compared to K-means clustering of the proximity data, ConPar generally provided better recovery of the true subject cluster memberships. A demonstration using empirical three-way data is also provided to illustrate the efficacy of the proposed method.