Visual image analysis by square wavelets: empirical evidence supporting a theoretical agreement between wavelet analysis and receptive field organization of visual cortical neurons

It was proposed that the human visual system analyzes images into square wavelets. To test this view, comparisons were made between the perceived similarity-dissimilarity of alphabet letters and the wavelet analyses of those same letters. For the proposal to be considered tenable, the coefficients of the wavelet analysis of similar letters must be similar, and the coefficients of the wavelet analysis of dissimilar letters must be dissimilar. From a selection of 12 letters, four pairs of letters had been reported by Van der Heijden, Mathas, and Van den Roovaart as very similar, and four other pairs of letters dissimilar. Each of the 12 letters was separately depicted in 8 x 8 matrices, and the signal represented by each of the matrices was analyzed into square wavelets using a new and original procedure which yielded a single set of coefficients for each matrix. Correlations between sets of coefficients were high (r ranged from .88 to .58) for those letter pairs judged high in similarity; correlations were low (r ranged from -.02 to .29) for those letter pairs judged low in similarity. When the correlations between the coefficients of wavelets of all eight-letter pairs were compared with the judged similarity-dissimilarity of all eight-letter pairs, the linear agreement was statistically significant. Agreement was found between the neurophysiological mapping of receptive fields of visual cortical neurons and the vectors or the pattern of pluses and minuses which characterized the wavelet analysis. Furthermore, regeneration of the visual image, or the pattern of neural activity representing the image, could be described by a tree-like flow of information among visual cortical neurons which received response data from visual receptive fields, the response data being wavelet coefficients. Results indicate the analysis accurately produces reliable transformations of visual patterns and may be a process used by the visual system.     

Separation of data as a principle in factor analysis

Two systems of factor analysis—factoring correlations with units in the diagonal cells and factoring correlations with communalities in the diagonal cells—are considered in relation to the commonly used statistical procedure of separating a set of data (scores) into two or more parts. It is shown that both systems of factor analysis imply the separation of the observed data into two orthogonal parts. The matrices used to achieve the separation differ for the two systems of factor analysis.     

Isolating Change And Invariance In Patterns Of Behavior

General approaches which may be taken in the study of change and in- variance in patterns of behavior are presented and compared. These are classified according to whether data are gathered by "ideal" means, in which the same entities are assessed with the same measuring devices on several occasions, or data are gathered by "less-than-ideal" means, in which change of occasion is confounded with change in the sample of entities or is confounded with change in the sample of measuring devices. Several estimation and hypothesis-testing procedures are presented as appropriate for use with data gathered in one or the other of these ways. These include Tucker's multi-mode factoring technique, his inter-battery factoring procedure, Meredith's methods for rotating to achieve factorial invariance and various techniques of use in evaluating the significance of difference of vectors and matrices. Newly developed least-squares techniques, based in part upon discriminant function principles, are presented in some detail. The various study-design and statistical-method possibilities are evaluated for the amount and kind of information they could provide about stable and dynamic patterns of behavior.     

The relation between sample and population characteristic vectors

Data are reported which show the statistical relation between the sample and population characteristic vectors of correlation matrices with squared multiple correlations as communality estimates. Sampling fluctuations were found to relate only to differences in the square roots of characteristic roots and to sample size. A principle for determining the number of factors to rotate and interpret after rotation is suggested.This study was supported by the National Science Foundation, Grant GB 4230. The author wishes to express his appreciation for the use of Western Data Processing Center and the Health Sciences Computing Facility, UCLA. He also thanks Dr. Roger Pennell for extremely valuable assistance in a number of phases of the study.     

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.     

A note on the analysis of subjective judgment matrices

Saaty (1977–1983) presents an eigenvector (EV) procedure for analyzing matrices of subjective estimates of the utility of one entity relative to another. The procedure is an especially effective tool for analyzing hierarchical problems where the dependence of the entities at one level on the entities in adjacent levels is estimated subjectively. Despite the absence of a formal proof that the procedure has desirable qualities as an estimator of the underlying relative utilities, the process has gained an active following. This paper derives a comparable estimate, the geometric mean (GM) vector (also known as the logarithmic least squares method or LLSM), that can be applied to hierarchical problems in exactly the same way but is developed from statistical considerations. It is shown to be optimal when the judge's errors are multiplicative with a lognormal distribution. The GM shares the desirable qualities of the EV and is preferable to it in several important respects.     

Some rao-guttman relationships

An examination of the determinantal equation associated with Rao's canonical factors suggests that Guttman's best lower bound for the number of common factors corresponds to the number of positive canonical correlations when squared multiple correlations are used as the initial estimates of communality. When these initial communality estimates are used, solving Rao's determinantal equation (at the first stage) permits expressing several matrices as functions of factors that differ only in the scale of their columns; these matrices include the correlation matrix with units in the diagonal, the correlation matrix with squared multiple correlations as communality estimates, Guttman's image covariance matrix, and Guttman's anti-image covariance matrix. Further, the factor scores associated with these factors can be shown to be either identical or simply related by a scale change. Implications for practice are discussed, and a computing scheme which would lead to an exhaustive analysis of the data with several optional outputs is outlined.     

Multivariate repeated measures designs

In the social, behavioral, and health researches it is a common strategy to collect data along time on more than one group of participants on multiple dependent variables. To analyse this kind of data is very complicated due to the correlations between the measures taken in different points of the time, and between the answers. Usually to analyse these data the multivariate mixed model, or the doubly multivariate model, are the most frequent approaches. Both of them require combined multivariate normality, equal covariance matrices, independence between the observations of different participants, complete measurements on all subjects, and time-independent covariates. When one ore more of these assumptions are not accomplished these approaches do not control in the correct way the Type I error, and this affects the validity and the accuracy of the inferences. In this paper some solutions that solve the problems with the error Type I will be shown. Several programs for a correct realization of the analyses through the SAS Proc Mixed procedure are presented.     

The simultaneous maximization of congruence for two or more matrices under orthogonal rotation

While a rotation procedure currently exists to maximize simultaneously Tucker's coefficient of congruence between corresponding factors of two factor matrices under orthogonal rotation of one factor matrix, only approximate solutions are known for the generalized case where two or more matrices are rotated. A generalization and modification of the existing rotation procedure to simultaneously maximize the congruence is described. An example using four data matrices, comparing the generalized congruence maximization procedure with alternative rotation procedures, is presented. The results show a marked improvement of the obtained congruence using the generalized congruence maximization procedure compared to other procedures, without a significant loss of success with respect to the least squares criterion. A computer program written by the author to perform the rotations is briefly discussed.     

The comparison of interdependent correlations between optimal linear composites

A general procedure is provided for comparing correlation coefficients between optimal linear composites. The procedure allows computationally efficient significance tests on independent or dependent multiple correlations, partial correlations, and canonical correlations, with or without the assumption of multivariate normality. Evidence from some Monte Carlo studies on the effectiveness of the methods is also provided.This research was supported in part by an operating grant (#67-4640) to the first author from the National Sciences and Engineering Research Council of Canada. The authors would also like to acknowledge the helpful comments and encouragement of Alexander Shapiro, Stanley Nash, and Ingram Olkin.     

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 design of ballantines

The shared variance between two variables can be expressed graphically by overlapping circles. A procedure is presented for locating the circles so that the graphical and statistical relations correspond exactly. The procedure does not generalize to the representation of multiple, part, and partial correlations between three or more variables. The reasons for the nonextension are explained, and some alternative graphical representations are suggested.     

Obtaining squared multiple correlations from a correlation matrix which may be singular

A theorem is presented relating the squared multiple correlation of each measure in a battery with the other measures to the unique generalized inverse of the correlation matrix. This theorem is independent of the rank of the correlation matrix and may be utilized for singular correlation matrices. A coefficient is presented which indicates whether the squared multiple correlation is unity or not. Note that not all measures necessarily have unit squared multiple correlations with the other measures when the correlation matrix is singular. Some suggestions for computations are given for simultaneous determination of squared multiple correlations for all measures.The research reported in this paper was supported by the Personnel and Training Branch of the Office of Naval Research under Contract Number 00014-67-A-0305-0003 with the University of Illinois.     

Comparing regressions when measurement error variances are known

In a multiple (or multivariate) regression model where the predictors are subject to errors of measurement with a known variance-covariance structure, two-sample hypotheses are formulated for (i) equality of regressions on true scores and (ii) equality of residual variances (or covariance matrices) after regression on true scores. The hypotheses are tested using a large-sample procedure based on maximum likelihood estimators. Formulas for the test statistic are presented; these may be avoided in practice by using a general purpose computer program. The procedure has been applied to a comparison of learning in high schools using achievement test data.     

A note on multiple and canonical correlation for a singular covariance matrix

A weaker generalized inverse (Rao's g-inverse; Graybill's c-inverse) can be used in place of the Moore-Penrose generalized inverse to obtain multiple and canonical correlations from singular covariance matrices.The author expresses his gratitude to a referee for his suggestions.     

The rowwise correlation between two proximity matrices and the partial rowwise correlation

This paper discusses rowwise matrix correlation, based on the weighted sum of correlations between all pairs of corresponding rows of two proximity matrices, which may both be square (symmetric or asymmetric) or rectangular. Using the correlation coefficients usually associated with Pearson, Spearman, and Kendall, three different rowwise test statistics and their normalized coefficients are discussed, and subsequently compared with their nonrowwise alternatives like Mantel'sZ. It is shown that the rowwise matrix correlation coefficient between two matricesX andY is the partial correlation between the entries ofX andY controlled for the nominal variable that has the row objects as categories. Given this fact, partial rowwise correlations (as well as multiple regression extensions in the case of Pearson's approach) can be easily developed.The author wishes to thank the Editor, two referees, Jan van Hooff, and Ruud Derix for their useful comments, and E. J. Dietz for a copy of the algorithm of the Mantel permutation test.     

A multiple group method of factoring the correlation matrix

There are a number of methods of factoring the correlation matrix which require the calculation of a table of residual correlations after each factor has been extracted. This is perhaps the most laborious part of factoring. The method to be described here avoids the computation of residuals after each factor has been computed. Since the method turns on the selection of a set of constellations or clusters of test vectors, it will be calleda multiple group method of factoring. The method can be used for extracting one factor at a time if that is desired but it will be considered here for the more interesting case in which a number of constellations are selected from the correlation matrix at the start. The result of this method of factoring is a factor matrixF which satisfies the fundamental relationFF'=R.This study is one of a series of investigations in the development of multiple factor analysis and application to the study of primary mental abilities. We wish to acknowledge the financial assistance from the Social Science Research Committee of The University of Chicago which has made possible the work of the Psychometric Laboratory.     

Developing stimulus control of young children's requests to teachers: classwide applications of multiple schedules

Children may recruit their teachers' attention at undesirably high rates or at inconvenient times. Tiger and Hanley (2004) described a multiple-schedule procedure to reduce ill-timed requests, which involved providing children with two distinct continuous signals that were correlated with periods in which teacher attention was either available or unavailable. The current study extended the application of multiple schedules by evaluating the effectiveness of the procedure when implemented by private-school teachers in 3 elementary classrooms. Following the introduction of the multiple schedules, student approaches toward their teacher were maintained during desirable periods but were minimized during undesirable periods.     

A heuristic method for the comparison of related structures

By extending a technique for testing the difference between two dependent correlations developed by Wolfe, a strategy is proposed in a more general matrix context for evaluating a variety of data analysis schemes that are supposed to clarify the structure underlying a set of proximity measures. In the applications considered, a data analysis scheme is assumed to reconstruct in matrix form the given data set (represented as a proximity matrix) based on some specific model or procedure. Thus, an evaluation of the adequacy of reconstruction can be developed by comparing matrices, one containing the original proximities and the second containing the reconstructed values. Possible applications in multidimensional scaling, clustering, and related contexts are emphasized using four broad categories: (a) Given two different reconstructions based on a single data set, does either represent the data significantly better than the other? (b) Given two reconstructions based on a single data set using two different procedures (or possibly, two distinct data sets and a common method), is either reconstruction significantly closer to a particular theoretical structure that is assumed to underlie the data (where the latter is also represented in matrix form)? (c) Given two theoretical structures and one reconstruction based on a single data set, does either represent the reconstruction better than the other? (d) Given a single reconstruction based on one data set, is the information present in the data accounted for satisfactorily by the reconstruction? In all cases, these tasks can be approached by a nonparametric procedure that assesses the similarity in pattern between two appropriately defined matrices. The latter are obtained from the original data, the reconstructions, and/or the theoretical structures. Finally, two numerical examples are given to illustrate the more general discussion.     

Factor analysis for clustered observations

Classical factor analysis assumes a random sample of vectors of observations. For clustered vectors of observations, such as data for students from colleges, or individuals within households, it may be necessary to consider different within-group and between-group factor structures. Such a two-level model for factor analysis is defined, and formulas for a scoring algorithm for estimation with this model are derived. A simple noniterative method based on a decomposition of the total sums of squares and crossproducts is discussed. This method provides a suitable starting solution for the iterative algorithm, but it is also a very good approximation to the maximum likelihood solution. Extensions for higher levels of nesting are indicated. With judicious application of quasi-Newton methods, the amount of computation involved in the scoring algorithm is moderate even for complex problems; in particular, no inversion of matrices with large dimensions is involved. The methods are illustrated on two examples.Suggestions and corrections of three anonymous referees and of an Associate Editor are acknowledged. Discussions with Bob Jennrich on computational aspects were very helpful. Most of research leading to this paper was carried out while the first author was a visiting associate professor at the University of California, Los Angeles.