Weighted least squares fitting using ordinary least squares algorithms

A general approach for fitting a model to a data matrix by weighted least squares (WLS) is studied. This approach consists of iteratively performing (steps of) existing algorithms for ordinary least squares (OLS) fitting of the same model. The approach is based on minimizing a function that majorizes the WLS loss function. The generality of the approach implies that, for every model for which an OLS fitting algorithm is available, the present approach yields a WLS fitting algorithm. In the special case where the WLS weight matrix is binary, the approach reduces to missing data imputation.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author.     

Minimization of a class of matrix trace functions by means of refined majorization

A procedure is described for minimizing a class of matrix trace functions. The procedure is a refinement of an earlier procedure for minimizing the class of matrix trace functions using majorization. It contains a recently proposed algorithm by Koschat and Swayne for weighted Procrustes rotation as a special case. A number of trial analyses demonstrate that the refined majorization procedure is more efficient than the earlier majorization-based procedure. The research of H. A. L. Kiers has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences.     

Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: I: Algebraic lower bounds

Let Σ _x be the (population) dispersion matrix, assumed well-estimated, of a set of non-homogeneous item scores. Finding the greatest lower bound for the reliability of the total of these scores is shown to be equivalent to minimizing the trace of Σ _x by reducing the diagonal elements while keeping the matrix non-negative definite. Using this approach, Guttman's bounds are reviewed, a method is established to determine whether his λ₄ (maximum split-half coefficient alpha) is the greatest lower bound in any instance, and three new bounds are discussed. A geometric representation, which sheds light on many of the bounds, is described. Present affiliation of the second author: Department of Statistics, University of Nigeria (Nsukka Campus). Work on this paper was carried out while on study leave in Aberystwyth.     

A monotonically convergent algorithm for orthogonal congruence rotation

Brokken has proposed a method for orthogonal rotation of one matrix such that its columns have a maximal sum of congruences with the columns of a target matrix. This method employs an algorithm for which convergence from every starting point is not guaranteed. In the present paper, an iterative majorization algorithm is proposed which is guaranteed to converge from every starting point. Specifically, it is proven that the function value converges monotonically, and that the difference between subsequent iterates converges to zero. In addition to the better convergence properties, another advantage of the present algorithm over Brokken's one is that it is easier to program. The algorithms are compared on 80 simulated data sets, and it turned out that the new algorithm performed well in all cases, whereas Brokken's algorithm failed in almost half the cases. The derivation of the algorithm is given in full detail because it involves a series of inequalities that can be of use to derive similar algorithms in different contexts.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the first author. The authors are obliged to Willem J. Heiser and Jos M. F. ten Berge for useful comments on an earlier version of this paper.     

A theory of consistency of ordering generalizable to tailored testing

Measures of consistency and completeness of order relations derived from test-type data are proposed. The measures are generalized to apply to incomplete data such as from tailored testing. The measures are based on consideration of the items-plus-persons by items-plus-persons matrix as an adjacency matrix in which a 1 means that the row element, whether item or person, dominated the column element. From this the number of item-item and person-person dominance relations can be derived. Indices of consistency are developed which compare the number of dominance relations in a matrix to the number that would occur if it were perfectly consistent and to the number in a random matrix; relations to other indices are noted.The research reported here was supported by the Office of Naval Research, Contract N00014-75-C-0684, NR150-373. The author wishes to thank Dr. Thomas J. Reynolds for a number of helpful discussions of this material.     

Newton Algorithms for Analytic Rotation: an Implicit Function Approach

In this paper implicit function-based parameterizations for orthogonal and oblique rotation matrices are proposed. The parameterizations are used to construct Newton algorithms for minimizing differentiable rotation criteria applied to m factors and p variables. The speed of the new algorithms is compared to that of existing algorithms and to that of Newton algorithms based on alternative parameterizations. Several rotation criteria were examined and the algorithms were evaluated over a range of values for m. Initial guesses for Newton algorithms were improved by subconvergence iterations of the gradient projection algorithm. Simulation results suggest that no one algorithm is fastest for minimizing all criteria for all values of m. Among competing algorithms, the gradient projection algorithm alone was faster than the implicit function algorithm for minimizing a quartic criterion over oblique rotation matrices when m is large. In all other conditions, however, the implicit function algorithms were competitive with or faster than the fastest existing algorithms. The new algorithms showed the greatest advantage over other algorithms when minimizing a nonquartic component loss criterion.     

Simple structure in component analysis techniques for mixtures of qualitative and quantitative variables

Several methods have been developed for the analysis of a mixture of qualitative and quantitative variables, and one, called PCAMIX, includes ordinary principal component analysis (PCA) and multiple correspondence analysis (MCA) as special cases. The present paper proposes several techniques for simple structure rotation of a PCAMIX solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for PCA. In addition, a recently developed technique for the analysis of mixtures of qualitative and quantitative variables, called INDOMIX, is shown to construct component scores (without rotational freedom) maximizing the quartimax criterion over all possible sets of component scores. A numerical example is used to illustrate the implication that when used for qualitative variables, INDOMIX provides axes that discriminate between the observation units better than do those generated from MCA.The Netherlands organization for scientific research (NWO) is gratefully acknowledged for funding this project. This research was conducted while the author was supported by a PSYCHON-grant (560-267-011) from this organization. The author is obliged to Jos ten Berge for his comments on an earlier version.     

Probabilistic multidimensional scaling: Complete and incomplete data

Simple procedures are described for obtaining maximum likelihood estimates of the location and uncertainty parameters of the Hefner model. This model is a probabilistic, multidimensional scaling model, which assigns a multivariate normal distribution to each stimulus point. It is shown that for such a model, standard nonmetric and metric algorithms are not appropriate. A procedure is also described for constructing incomplete data sets, by taking into consideration the degree of familiarity the subject has for each stimulus. Maximum likelihood estimates are developed both for complete and incomplete data sets. This research was supported by National Science Grant No. SOC76-20517. The first author would especially like to express his gratitude to the Netherlands Institute for Advanced Study for its very substantial help with this research.     

Three-mode orthomax rotation

Factor analysis and principal components analysis (PCA) are often followed by an orthomax rotation to rotate a loading matrix to simple structure. The simple structure is usually defined in terms of the simplicity of the columns of the loading matrix. In Three-mode PCA, rotational freedom of the so called core (a three-way array relating components for the three different modes) can be used similarly to find a simple structure of the core. Simple structure of the core can be defined with respect to all three modes simultaneously, possibly with different emphases on the different modes. The present paper provides a fully flexible approach for orthomax rotation of the core to simple structure with respect to three modes simultaneously. Computationally, this approach relies on repeated (two-way) orthomax applied to supermatrices containing the frontal, lateral or horizontal slabs, respectively. The procedure is illustrated by means of a number of exemplary analyses. As a by-product, application of the Three-mode Orthomax procedures to two-way arrays is shown to reveal interesting relations with and interpretations of existing two-way simple structure rotation techniques.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author. The author is obliged to Jos ten Berge and two anonymous reviewers for useful comments on an earlier version of this paper.     

Action across emptiness

This paper examines how autism may affect the processing and containing of emotions and sensations. At the beginning of life we can hypothesise that the baby develops two means of processing experience: containment that involves a sensory and mimetic trace of relational experience (proto-containment) and containment which, through the α function and projective and introjective functions translates feelings into meaning (♀/♂). Both these means of processing emotional experience remain active throughout life but oscillate the one with the other. The main hypothesis of this work is that the autistic child is not able to oscillate between real containment (♀/♂) and a primitive form of proto-containment. The author aims to explore, through clinical material, the factors which may promote this oscillation. Autistic children often use non-verbal communication. The author examines how a particular type of action can cross the relational void in the therapeutic relationship through a process that can transform feelings of great distress in the analyst via acts that can be described as an Act of Faith, as described by Bion. This form of action that can cross the void is like an artistic performance.     

Average exponent from magnitude production

When magnitude production is used to obtain a psychophysical power function for a group ofSs, the group exponent is shown to be the harmonic mean of individual-S exponents under the conditions usually employed in empirical research. The conditions are (a) the same numbers are presented to eachS, (b) responses to each number are pooled overSs by taking geometric means, and (c) the function is fit by the method of least squares in which regression is of the logarithm ofSs' responses upon the logarithm of numbers presented byE.This paper was supported in part by grant no. APA-151 from the National Research Council of Canada. The author wishes to thank William W. Rozeboom for his helpful comments on this paper.     

Creating a safe place for growth in supervision

When people are afraid, learning is inhibited. In too many disturbed families, institutional settings, rigid organizations, and some supervision groups, a feeling of emotional safety is missing. The first step in teaching, supervising, or therapy is the necessity to provide a safe environment. The author suggests a model applicable to any supervision group, work group, organization, classroom, or family. A group model, specifically applied to the relationship between family therapy students or interns and supervisors, suggests ways to enhance growth by minimizing fear.     

Visual chimaeras obtained with the Riesz transform

Similar to an auditory chimaera (Smith et al, 2002 Nature 416 87-90), a visual chimaera can be defined as a synthetic image which has the fine spatial structure of one natural image and the envelope of another image in each spatial frequency band. Visual chimaeras constructed in this way could be useful to vision scientists interested in the study of interactions between first-order and second-order visual processing. Although it is almost trivial to generate 1-D chimaeras by means of the Hilbert transform and the analytic signal, problems arise in multidimensional signals like images given that the partial directional Hilbert transform and current 2-D demodulation algorithms are anisotropic or orientation-variant procedures. Here, we present a computational procedure to synthesise visual chimaeras by means of the Riesz transform--an isotropic generalisation of the Hilbert transform for multidimensional signals--and the associated monogenic signal--the vector-valued function counterpart of the analytic signal in which the Riesz transform replaces the Hilbert transform. Examples of visual chimaeras are shown for same/different category images.     

The information theoretic entropy function as a total expected participation index for communication network experiments

This paper shows how the concept of an incidence matrix of communications can be used to define the entropy of a finite scheme. The properties of the entropy function are examined and the function is found to be best interpreted as a total expected participation index. Data is presented showing the relationship between structural centrality and the new total expected participation index. In general, as the network becomes more centralized the smaller the value of the participation index and as the network becomes more structurally decentralized the greater the participation index.This research was supported in part by Ford Foundation Grant 1-40055 to the Graduate School of Industrial Administration at Carnegie Institute of Technology for Research in Organizational Behavior.The author wishes to acknowledge the aid of Terry B. Marbach in the preparation of data.     

Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis

One of the intriguing questions of factor analysis is the extent to which one can reduce the rank of a symmetric matrix by only changing its diagonal entries. We show in this paper that the set of matrices, which can be reduced to rankr, has positive (Lebesgue) measure if and only ifr is greater or equal to the Ledermann bound. In other words the Ledermann bound is shown to bealmost surely the greatest lower bound to a reduced rank of the sample covariance matrix. Afterwards an asymptotic sampling theory of so-called minimum trace factor analysis (MTFA) is proposed. The theory is based on continuous and differential properties of functions involved in the MTFA. Convex analysis techniques are utilized to obtain conditions for differentiability of these functions.     

Weighted minimum trace factor analysis

In the last decade several authors discussed the so-called minimum trace factor analysis (MTFA), which provides the greatest lower bound (g.l.b.) to reliability. However, the MTFA fails to be scale free. In this paper we propose to solve the scale problem by maximization of the g.l.b. as the function of weights. Closely related to the primal problem of the g.l.b. maximization is the dual problem. We investigate the primal and dual problems utilizing convex analysis techniques. The asymptotic distribution of the maximal g.l.b. is obtained provided the population covariance matrix satisfies sone uniqueness and regularity assumptions. Finally we outline computational algorithms and consider numerical examples.I wish to express my gratitude to Dr. A. Melkman for the idea of theorem 3.3.     

Consciousness without a cerebral cortex: a challenge for neuroscience and medicine

A broad range of evidence regarding the functional organization of the vertebrate brain - spanning from comparative neurology to experimental psychology and neurophysiology to clinical data - is reviewed for its bearing on conceptions of the neural organization of consciousness. A novel principle relating target selection, action selection, and motivation to one another, as a means to optimize integration for action in real time, is introduced. With its help, the principal macrosystems of the vertebrate brain can be seen to form a centralized functional design in which an upper brain stem system organized for conscious function performs a penultimate step in action control. This upper brain stem system retained a key role throughout the evolutionary process by which an expanding forebrain - culminating in the cerebral cortex of mammals - came to serve as a medium for the elaboration of conscious contents. This highly conserved upper brainstem system, which extends from the roof of the midbrain to the basal diencephalon, integrates the massively parallel and distributed information capacity of the cerebral hemispheres into the limited-capacity, sequential mode of operation required for coherent behavior. It maintains special connective relations with cortical territories implicated in attentional and conscious functions, but is not rendered nonfunctional in the absence of cortical input. This helps explain the purposive, goal-directed behavior exhibited by mammals after experimental decortication, as well as the evidence that children born without a cortex are conscious. Taken together these circumstances suggest that brainstem mechanisms are integral to the constitution of the conscious state, and that an adequate account of neural mechanisms of conscious function cannot be confined to the thalamocortical complex alone.     

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.