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.     

A NEW MATRIX FOR THE ASSESSMENT OF FACTOR CONTRIBUTIONS

The matrix product of the initial factor loading matrix times the least- squares orthonormal approximation to the general oblique transformation matrix is suggested to be the most adequate matrix for the evaluation of the contributions of factors to variables. This factor contribution matrix can also re- place the primary factor pattern matrix in evaluating oblique simple structure.     

Information matrix estimation procedures for cognitive diagnostic models

Two new methods to estimate the asymptotic covariance matrix for marginal maximum likelihood estimation of cognitive diagnosis models (CDMs), the inverse of the observed information matrix and the sandwich-type estimator, are introduced. Unlike several previous covariance matrix estimators, the new methods take into account both the item and structural parameters. The relationships between the observed information matrix, the empirical cross-product information matrix, the sandwich-type covariance matrix and the two approaches proposed by de la Torre (2009, J. Educ. Behav. Stat., 34, 115) are discussed. Simulation results show that, for a correctly specified CDM and Q-matrix or with a slightly misspecified probability model, the observed information matrix and the sandwich-type covariance matrix exhibit good performance with respect to providing consistent standard errors of item parameter estimates. However, with substantial model misspecification only the sandwich-type covariance matrix exhibits robust performance.     

A non-graphical method for transforming an arbitrary factor matrix into a simple structure factor matrix

The most commonly used method of factoring a matrix of intercorrelations is the centroid method developed by L. L. Thurstone. It is, however, necessary to transform the centroid matrix of factor loadings into a simple structure matrix in order to facilitate the interpretation of the factor loadings. Current methods for effecting this transformation are chiefly graphical and require considerable experience and personal judgment. This paper presents a new method for transforming an arbitrary factor matrix into a simple structure matrix by methods almost completely objective. The theory underlying the method is developed and approximation procedures are derived. The method is applied to a matrix of factor loadings previously analyzed by Thurstone.     

Tatsuoka 矩阵理论的修正

K .K.Tatsuoka和她同事开发的规则空间模型（RSM）是一种在国内外有较大影响的认知诊断模型,但是Tatsuoka的RSM中 矩阵理论存在缺陷和错误,这些失误使得RSM中用布尔描述函数（BDF）计算被试理想项目反应模式（IRP）的方法缺乏理论依据。这里揭示了Tatsuoka的 矩阵理论的缺陷和错误并引进既不使用BDF又便于应用的计算IRP的方法;接着还介绍一种由可达阵计算简化 阵的方法,该方法显示了可达阵在构造认知诊断测验的重要性。这些结果对丰富 矩阵理论及正确使用RSM进行认知诊断有一定的意义     

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.     

Information Matrices and Standard Errors for MLEs of Item Parameters in IRT

The paper clarifies the relationship among several information matrices for the maximum likelihood estimates (MLEs) of item parameters. It shows that the process of calculating the observed information matrix also generates a related matrix that is the middle piece of a sandwich-type covariance matrix. Monte Carlo results indicate that standard errors (SEs) based on the observed information matrix are robust to many, but not all, conditions of model/distribution misspecifications. SEs based on the sandwich-type covariance matrix perform most consistently across conditions. Results also suggest that SEs based on other matrices are either not consistent or perform not as robust as those based on the sandwich-type covariance matrix or the observed information matrix.     

Testing pattern hypotheses for covariance matrices

Maximum likelihood estimates of the free parameters, and an asymptotic likelihood-ratio test, are given for the hypothesis that one or more elements of a covariance matrix are zero, and/or that two or more of its elements are equal. The theory applies immediately to a transformation of the covariance matrix by a known nonsingular matrix. Estimation is by Newton's method, starting conveniently from a closed-form least-squares solution.Numerical illustrations include a test for equality of diagonal blocks of a covariance matrix, and estimation of quasi-simplex structures.     

The orthogonal approximation of an oblique structure in factor analysis

A procedure is derived for obtaining an orthogonal transformation which most nearly transforms one given matrix into another given matrix, according to some least-squares criterion of fit. From this procedure, three analytic methods are derived for obtaining an orthogonal factor matrix which closely approximates a given oblique factor matrix. The case is considered of approximating a specified subset of oblique vectors by orthogonal vectors.Part of this research was carried out while the author was a psychometric fellow at the Educational Testing Service, Princeton, New Jersey.     

On the symmetric treatment of an asymmetric approach to factor analysis

On the assumption that a partitioning can be found such that three mutually exclusive test vector configurations span the same factor space, a procedure is derived whereby symmetric parts of the correlation matrix are estimated from functions of asymmetric parts treated symmetrically. This yields an explicit matrix formula for communality estimation which generalizes earlier work by Albert. Conventional factoring methods, with all their computational and fitting advantages, can be applied once the symmetric portions of the correlation matrix have been estimated. Extension to four subgroups of test vectors allows for a matrix generalization of the old tetrad difference criterion to the multiple-factor case.     

The relation between results obtainable with raw and corrected correlation coefficients in multiple factor analysis

This paper presents three theorems concerning the relation between results with obtained and corrected correlation coefficients in the Thurstone method of multiple factor analysis. (1) The rank of the correlational matrix, and thus the number of factors involved, is unaffected by correcting the obtained coefficients for attenuation. (2) The communality of a variable when the obtained coefficients have been corrected for attenuation is equal to the communality with obtained coefficients divided by the reliability coefficient of the variable. (3) The relationship is shown between the factorial matrix of a correlational matrix of raw correlation coefficients and the factorial matrix of a correlational matrix of corrected correlation coefficients, and a method of obtaining either of these factorial matrices from the other is indicated.     

A method of factor analysis by means of which all coordinates of the factor matrix are given simultaneously

In general, the methods of factor analysis developed during the past five years are based on the reduction of the correlational matrix by successive steps. The first factor loadings are determined and eliminated from the correlational matrix, giving a residual matrix. This process is continued for successive factor loadings until the elements of the last obtained residual matrix may be regarded as due to chance. The method outlined in this paper assumes the maximum number of factorsm in the correlational matrix. Them factor vectors are solved for simultaneously. Once them factor vectors are found, any vectors having only negligible factor loadings may be discarded.     

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.     

A discriminated similarity matrix construction based on sparse subspace clustering algorithm for hyperspectral imagery

The clustering of hyperspectral images is a challenging task because of the high dimensionality of the data. Sparse subspace clustering (SSC) algorithm is one of the popularly used clustering algorithm for high dimensionality data. However, SSC has not fully used the spectral and spatial information during similarity matrix construction based on single sparse representation coefficient for hyperspectral Imagery (HSI) clustering. In this paper, two novel similarity matrix construction methods named as Cosine-Euclidean similarity matrix (abbreviated as CE) and Cosine-Euclidean dynamic weighting similarity matrix (abbreviated as CEDW) are proposed for HSI clustering. They can combine the high spectral information and rich spatial information. Firstly, CE utilizes the cosine similarity of spectral information based on overall sparse representation vectors and classical Euclidean distance of spatial information to construct a novel similarity matrix. Secondly, inheriting CE merits, dynamic weighting adjustment method is introduced to CEDW for some external influence factors to the HSI information. Several experiments on HSI demonstrated that the proposed algorithms are effective for HSI clustering.     

认知诊断分类中心的确定

期望反应模式是认知诊断分类的类中心,确定了诊断范围中属性及其层级关系后,类中心的数目由测验Q矩阵决定.类中心应该是完备的,即理论上有多少个知识状态就应该有多少个类中心,这涉及评分方式、Q矩阵设计、Q矩阵评价与修改、认知诊断测验如何实施等问题的讨论.重点给出多级评分认知诊断测验不同属性层级对应的完备Q矩阵设计的例子和罗列了Wang等人（2013）的Q矩阵修改的方法.     

A Class of Population Covariance Matrices in the Bootstrap Approach to Covariance Structure Analysis

Model evaluation in covariance structure analysis is critical before the results can be trusted. Due to finite sample sizes and unknown distributions of real data, existing conclusions regarding a particular statistic may not be applicable in practice. The bootstrap procedure automatically takes care of the unknown distribution and, for a given sample size, also provides more accurate results than those based on standard asymptotics. But the procedure needs a matrix to play the role of the population covariance matrix. The closer the matrix is to the true population covariance matrix, the more valid the bootstrap inference is. The current paper proposes a class of covariance matrices by combining theory and data. Thus, a proper matrix from this class is closer to the true population covariance matrix than those constructed by any existing methods. Each of the covariance matrices is easy to generate and also satisfies several desired properties. An example with nine cognitive variables and a confirmatory factor model illustrates the details for creating population covariance matrices with different misspecifications. When evaluating the substantive model, bootstrap or simulation procedures based on these matrices will lead to more accurate conclusion than that based on artificial covariance matrices.     

Heuristic Implementation of Dynamic Programming for Matrix Permutation Problems in Combinatorial Data Analysis

Dynamic programming methods for matrix permutation problems in combinatorial data analysis can produce globally-optimal solutions for matrices up to size 30×30, but are computationally infeasible for larger matrices because of enormous computer memory requirements. Branch-and-bound methods also guarantee globally-optimal solutions, but computation time considerations generally limit their applicability to matrix sizes no greater than 35×35. Accordingly, a variety of heuristic methods have been proposed for larger matrices, including iterative quadratic assignment, tabu search, simulated annealing, and variable neighborhood search. Although these heuristics can produce exceptional results, they are prone to converge to local optima where the permutation is difficult to dislodge via traditional neighborhood moves (e.g., pairwise interchanges, object-block relocations, object-block reversals, etc.). We show that a heuristic implementation of dynamic programming yields an efficient procedure for escaping local optima. Specifically, we propose applying dynamic programming to reasonably-sized subsequences of consecutive objects in the locally-optimal permutation, identified by simulated annealing, to further improve the value of the objective function. Experimental results are provided for three classic matrix permutation problems in the combinatorial data analysis literature: (a) maximizing a dominance index for an asymmetric proximity matrix; (b) least-squares unidimensional scaling of a symmetric dissimilarity matrix; and (c) approximating an anti-Robinson structure for a symmetric dissimilarity matrix. We are extremely grateful to the Associate Editor and two anonymous reviewers for helpful suggestions and corrections.     

Scaling a conditional proximity matrix to symmetry

Two least squares procedures for symmetrization of a conditional proximity matrix are derived. The solutions provide multiplicative constants for scaling the rows or columns of the matrix to maximize symmetry. It is suggested that the symmetrization is applicable for the elimination of bias effects like response bias, or constraints on the marginal frequencies imposed by the experimental design, as in confusion matrices.The application of the scaling procedure to a matrix of conditional probabilities was suggested by one of the referees, whose helpful comments are gratefully acknowledged.     

Approximated Penalized Maximum Likelihood for Exploratory Factor Analysis: An Orthogonal Case

The problem of penalized maximum likelihood (PML) for an exploratory factor analysis (EFA) model is studied in this paper. An EFA model is typically estimated using maximum likelihood and then the estimated loading matrix is rotated to obtain a sparse representation. Penalized maximum likelihood simultaneously fits the EFA model and produces a sparse loading matrix. To overcome some of the computational drawbacks of PML, an approximation to PML is proposed in this paper. It is further applied to an empirical dataset for illustration. A simulation study shows that the approximation naturally produces a sparse loading matrix and more accurately estimates the factor loadings and the covariance matrix, in the sense of having a lower mean squared error than factor rotations, under various conditions.     

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.