Matrix derivatives with chain rule and rules for simple,Hadamard, and Kronecker products

The differential calculus for scalars is used to develop theorems for a calculus of functions of matrices. No appeal to scalar notation is necessary in the resulting calculus, so that the given chain and matrix product rules have wide applicability to matrix theory and models. The chain and product rules of scalar calculus become a special case of the matrix rules. The methods are illustrated by application to a generalized stochastic process model potentially relevant to long- and short-term memory.     

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.     

Stances and paradigms: a reflection

This paper compares and contrasts the concept of a stance with that of a paradigm qua disciplinary matrix, in an attempt to illuminate both notions. First, it considers to what extent it is appropriate to draw an analogy between stances (which operate at the level of the individual) and disciplinary matrices (which operate at the level of the community). It suggests that despite first appearances, a disciplinary matrix is not simply a stance writ large. Second, it examines how we might reinterpret disciplinary matrices in terms of stances, and shows how doing so can provide us with a better insight into non-revolutionary science. Finally, it identifies two directions for future research: “Can the rationality of scientific revolutions be understood in terms of the dynamic between stances and paradigms?” and “Do stances help us to understand incommensurability between disciplinary matrices?”     

"Enmeshment" and the Too Richly Cross-Joined System

This paper traces out a connection between Minuchin's concept of an "enmeshed" family and Ashby's discussion of the problems inherent in a system in which all the parts are tightly interlocked. Adaptiveness to change in such a system will depend on the possibility for the joins between parts to become temporarily inactive. This observation supports maneuvers in family therapy that emphasize the creation of boundaries between subsystems, differentiating individuals from one another, and blocking customary sequences of interaction.     

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.     

Reduced products of logical matrices

The class Matr(C) of all matrices for a prepositional logic (, C) is investigated. The paper contains general results with no special reference to particular logics. The main theorem (Th. (5.1)) which gives the algebraic characterization of the class Matr(C) states the following. Assume C to be the consequence operation on a prepositional language induced by a class K of matrices. Let m be a regular cardinal not less than the cardinality of C. Then Matr (C) is the least class of matrices containing K and closed under m-reduced products, submatrices, matrix homomorphisms, and matrix homomorphic counter-images.     

The Assumption of Proportional Components when Candecomp is Applied to Symmetric Matrices in the Context of Indscal

The use of Candecomp to fit scalar products in the context of INDSCAL is based on the assumption that the symmetry of the data matrices involved causes the component matrices to be equal when Candecomp converges. Ten Berge and Kiers gave examples where this assumption is violated for Gramian data matrices. These examples are believed to be local minima. It is now shown that, in the single-component case, the assumption can only be violated at saddle points. Chances of Candecomp converging to a saddle point are small but still nonzero.     

A generalization of Kristof's theorem on the trace of certain matrix products

Kristof has derived a theorem on the maximum and minimum of the trace of matrix products of the form \(X_1 \hat \Gamma _1 X_2 \hat \Gamma _2 \cdots X_n \hat \Gamma _n\) where the matrices \(\hat \Gamma _i\) are diagonal and fixed and theX _i vary unrestrictedly and independently over the set of orthonormal matrices. The theorem is a useful tool in deriving maxima and minima of matrix trace functions subject to orthogonality constraints. The present paper contains a generalization of Kristof's theorem to the case where theX _i are merely required to be submatrices of orthonormal matrices and to have a specified maximum rank. The generalized theorem contains the Schwarz inequality as a special case. Various examples from the psychometric literature, illustrating the practical use of the generalized theorem, are discussed.     

Some comments in connection with Rozeboom's linear correlation theory

It is shown that certain correlation measures suggested by information theory are simple functions of the determinants of the correlation matrices involved, if multivariate normality is assumed. This illuminates and in a sense strengthens Rozeboom's point in [3], that information theoretical statistics have classical correlational counterparts.     

A note on some modifications of latent roots and vectors

When an arbitrary positive scalar matrix is added to a correlation matrix the latent roots of the sum are equal to the corresponding roots of the correlation matrix plus an amount equal to the scalar number of the scalar matrix. The latent vectors of the sum are identical with those of the correlation matrix. An approximation to these relationships is suggested for the case in which the sum is of a correlation matrix and of a positive semidefinite diagonal matrix. The approximation is used to allow the solution of a characteristic problem for a correlation matrix with unities in the main diagonal to provide a family of solutions for the same correlation matrix.This research has been supported by a grant from the National Institute of Mental Health, MH 7864-01.     

Locational representation in imagery: a moving spot task

Subjects with eyes closed imagined a spot moving from cell to cell in a matrix in response to spoken instructions: Up, Down, Left, and Right. After 12 such moves from a given starting point, the subject indicated the final position of the spot. Accuracy was studied as a function of matrix order, which varied from 3 X 3 to 8 X 8. (Visual angles of the displayed matrices were also varied, from 4.6 degrees/cell in one experiment to 1 degree/cell in another, but with little or no effect on results.) The greatest decrement in accuracy occurred between 3 X 3 and 4 X 4 cells. Moreover, showing the subjects a systematic way of structuring matrices (i.e., of dividing them into parts that could be visualized more or less individually) improved performance markedly on all but the 3 X 3 matrix. These results, which agree fairly well with the verbal reports of the subjects, indicate that some capacity limitation of the image system is exceeded in going from a 3 X 3 to a 4 X 4 grid. Implications concerning the nature of imagery are discussed.     

An Exploratory Study of Neglect and Emotional Abuse in Adolescents: Classifications of Caregiver Risk Factors

Recent research has advanced exploration of developmental stage and age-related distinctions in understanding the perpetration of child maltreatment. Using longitudinal data from the National Survey of Child and Adolescent Well-being I (NSCAW-I), this exploratory study investigated heterogeneity in caregiver risks in cases of both neglect and emotional abuse of adolescents (N = 511). Using MPlus software, a person-centered data analytic strategy–latent class analysis was performed to identify distinct classes of caregiver risk factors that were associated with the reports of maltreatment among adolescents. Subsequently exploring the distinct characteristics of the classes, we examined the related demographic characteristics, child welfare system variables (e.g., type of placement status; change in placement type between waves), and youth factors such as the presence of clinical range problem behaviors (Child Behavior Checklist). Results demonstrate four distinct classes of caregiver risk factors, with a nearly even split between those who have an absence of risk factors and those who have a whole host of risk determinants. Differences between the classes regarding the duration of time in out-of-home placements between Wave 1 and Wave 3 also emerged significant.     

Recognition vs recall of visually vs acoustically confusable letter matrices

In an attempt to separate auditory and visual components in short-term memory, five subjects were exposed to letter matrices composed of six visually confusable letters, six acoustically confusable letters, or a mixture of the two, under two response conditions: recognition and recall. A 50-msec stimulus presentation was followed by a variable dark interval of 1, 250, 1,000, or 3,000 msec. In the recall condition, the interval was followed by a buzzer which signaled the subject to recall, in any order, as many letters as possible. In the recognition condition, the variable interval was followed by a second letter matrix which was either identical to the first matrix or differed from is by one letter. Subjects responded either "same" or "different." The results support the notion that the auditory component plays a major role in recall, whereas the visual component dominates in recognition.     

Gender differences in motivations for course selection: Academically talented students in an intensive summer program

This study examines the motivational factors contributing to course selection in a sample of students enrolled in an intensive three-week academic experience for exceptionally talented adolescents. Our sample included 947 gifted adolescents, of whom 3.7% were Black, 73.5% were White, 2.2% Hispanic, 0.1% Native American, 19.8% Asian, and 0.6% categorized themselves as other. Girls and boys selected different types of classes, with course participation falling along traditional gender-stereotyped lines. Boys and girls both performed exceptionally well, and said they chose their course because the subject was interesting, useful for future goals, and challenging. Females more often than males enrolled in classes because they perceived them as challenging, different from usual, not offered at school, and as making them more well-rounded. Males more often than females tended to select classes because they thought they would do well and/or because they viewed these classes as being useful for future schooling or career. However, these gender differences varied by the type of course the students took. The resulting complex picture of students' motivations is discussed in terms of the need to specify both the contexts in which males and females make their academic choices, and the interactive determinants of their motivations.An earlier version of this paper was presented at the Third Annual Esther Katz Rosen Symposium on the Psychological Development of Gifted Children, University of Kansas, Lawrence, Kansas, February 19–20, 1993.     

Joint Procrustes Analysis for Simultaneous Nonsingular Transformation of Component Score and Loading Matrices

In component analysis solutions, post-multiplying a component score matrix by a nonsingular matrix can be compensated by applying its inverse to the corresponding loading matrix. To eliminate this indeterminacy on nonsingular transformation, we propose Joint Procrustes Analysis (JPA) in which component score and loading matrices are simultaneously transformed so that the former matrix matches a target score matrix and the latter matches a target loading matrix. The loss function of JPA is a function of the nonsingular transformation matrix and its inverse, and is difficult to minimize straightforwardly. To deal with this difficulty, we reparameterize those matrices by their singular value decomposition, which reduces the minimization to alternately solving quartic equations and performing the existing multivariate procedures. This algorithm is assessed in a simulation study. We further extend JPA for cases where targets are linear functions of unknown parameters. We also discuss how the application of JPA can be extended in different fields.     

Divergence Behaviour of the Successive Geometric Mean Method of Pairwise Comparison Matrix Generation for a Multiple Stage,Multiple Objective Optimization Problem

This paper focuses on the divergence behaviour of the successive geometric mean (SGM) method used to generate pairwise comparison matrices while solving a multiple stage, multiple objective (MSMO) optimization problem. The SGM method can be used in the matrix generation phase of our three‐phase methodology to obtain pairwise comparison matrix at each stage of an MSMO optimization problem, which can be subsequently used to obtain the weight vector at the corresponding stage. The weight vectors across the stages can be used to convert an MSMO problem into a multiple stage, single objective (MSSO) problem, which can be solved using dynamic programming‐based approaches. To obtain a practical set of non‐dominated solutions (also referred to as Pareto optimal solutions) to the MSMO optimization problem, it is important to use a solution approach that has the potential to allow for a better exploration of the Pareto optimal solution space. To accomplish a more exhaustive exploration of the Pareto optimal solution space, the weight vectors that are used to scalarize the MSMO optimization problem into its corresponding MSSO optimization problem should vary across the stages. Distinct weight vectors across the stages are tied directly with distinct pairwise comparison matrices across the stages. A pairwise comparison matrix generation method is said to diverge if it can generate distinct pairwise comparison matrices across the stages of an MSMO optimization problem. In this paper, we demonstrate the SGM method's divergence behaviour when the three‐phase methodology is used in conjunction with an augmented high‐dimensional, continuous‐state stochastic dynamic programming method to solve a large‐scale MSMO optimization problem. Copyright © 2013 John Wiley & Sons, Ltd.     

The Scope and Limitations of Algebras: Some Historical and Philosophical Considerations

An important feature of mathematics, both pure and applied, during the nineteenth century was the widening from its common form to a proliferation, where the “objects” studied were not numbers or geometrical magnitudes but operations such as functions and differentiation and integration, abstract ones (as we now call them), linear algebras of vectors, matrices and determinants, and algebras in logic. In this article the author considers several of them, including the contributions of Hermann Grassmann and Benjamin Peirce. A notable feature of these developments was analogising from one algebra to another by adopting some of the same laws, such as associativity, commutativity and distributivity. In the final section we consider the normally secular character of these algebras.     

Financial product sensitivity predicts financial health

Recent research has aimed to understand how people consider financial decisions because they have important consequences for well-being. Yet existing research has largely failed to examine how attitudes and behaviors vary as a function of the specific financial product (e.g., debt type). We ask to what extent people differentiate between similarly categorized financial products (e.g., debt or investment) as a function of their terms (e.g., interest costs and expected returns) and whether such differentiation predicts financial health. Across four studies, we find not only that there are individual differences in attitudes toward similar financial products (e.g., two distinct loans), but also that the extent to which a consumer is averse to high-cost versus low-cost products predicts financial health. This relationship cannot be fully explained by financial literacy, numeracy, or intertemporal discounting. In addition, nudging people toward differentiating between financial products promotes decisions that are aligned with financial health.     

Dreamtelling as a request for containment: three uses of dreams in group therapy

This article is an attempt to develop a coherent, unified, and consistent conceptualization of dreaming and dreamtelling in the clinical setting. Dreams told in a therapeutic setting are challenging events: fantastically rich in content, but often overwhelming in their implications for peoples' relationships. When told in therapy groups, dreams provide additional challenges for all participants. Learning to work with dreams not only enhances understanding of unconscious intrapsychic and group processes, but may also have a strong impact on the therapeutic culture and working relationships in the group. After differentiating dreaming from dreamtelling, I briefly describe three uses of dreams in groups-the classical "informative" and more familiar "formative" uses, and a new perspective that focuses on the "transformative" aspects of a dream told. According to this perspective, a dream told has an interesting past, an important present, and a worthwhile future because of its interpersonal, intersubjective influence on the dreamer-audience relationship.     

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.