A skew-normal factor model for the analysis of student satisfaction towards university courses

Classical factor analysis relies on the assumption of normally distributed factors that guarantees the model to be estimated via the maximum likelihood method. Even when the assumption of Gaussian factors is not explicitly formulated and estimation is performed via the iterated principal factors’ method, the interest is actually mainly focussed on the linear structure of the data, since only moments up to the second ones are involved. In many real situations, the factors could not be adequately described by the first two moments only. For example, skewness characterizing most latent variables in social analysis can be properly measured by the third moment: the factors are not normally distributed and covariance is no longer a sufficient statistic. In this work we propose a factor model characterized by skew-normally distributed factors. Skew-normal refers to a parametric class of probability distributions, that extends the normal distribution by an additional shape parameter regulating the skewness. The model estimation can be solved by the generalized EM algorithm, in which the iterative Newthon–Raphson procedure is needed in the M-step to estimate the factor shape parameter. The proposed skew-normal factor analysis is applied to the study of student satisfaction towards university courses, in order to identify the factors representing different aspects of the latent overall satisfaction.     

A robust biplot

This paper introduces a robust biplot which is related to multivariate M-estimates. The n × p data matrix is first considered as a sample of size n from some p-variate population, and robust M-estimates of the population location vector and scatter matrix are calculated. In the construction of the biplot, each row of the data matrix is assigned a weight determined in the preliminary robust estimation. In a robust biplot, one can plot the variables in order to represent characteristics of the robust variance-covariance matrix: the length of the vector representing a variable is proportional to its robust standard deviation, while the cosine of the angle between two variables is approximately equal to their robust correlation. The proposed biplot also permits a meaningful representation of the variables in a robust principal-component analysis. The discrepancies between least-squares and robust biplots are illustrated in an example.     

A permutation test for the spread of three-dimensional rotation data

The permutation test is a nonparametric test that can be used to compare measures of spread for two data sets, but is yet to be explored in the context of three-dimensional rotation data. A permutation test for such data is developed and the statistical power of this test is considered under various conditions. The test is then used in a brief application comparing movement around the calcaneocuboid joint for a human, chimpanzee, and baboon.     

Regression and correlation for 3 × 3 rotation matrices

This paper investigates a regression model for orthogonal matrices introduced by Prentice (1989). It focuses on the special case of 3 × 3 rotation matrices. The model under study expresses the dependent rotation matrix V as A₁UA^t₂ perturbed by experimental errors, where A₁ and A₂ are unknown 3 × 3 rotation matrices and U is an explanatory 3 × 3 rotation matrix. Several specifications for the errors in this regression model are proposed. The asymptotic distributions, as the sample size n becomes large or as the experimental errors become small, of the least squares estimators for A₁ and A₂ are derived. A new algorithm for calculating the least squares estimates of A₁ and A₂ is presented. The independence model is not a submodel of Prentice's regression model, thus the independence between the U and the V sample cannot be tested when fitting Prentice's model. To overcome this difficulty, permutation tests of independence are investigated. Examples dealing with postural variations of subjects performing a drilling task and with the calibration of a camera system for motion analysis using a magnetic tracking device illustrate the methodology of this paper.     

Numerically stable computations for general univariate linear models

We consider the general univariate linear model E(y) = Xb, V(y) = o² W, W symmetric nonnegative definite. A numerically stable method based on orthogonal rotations is given for computing the least squares estimate [bcirc] of b , as well as a representation of [/(b). It is shown how to extend the computations to update these results quickly and accurately when columns or rows of (y,X) are added or taken away. One of these techniques will handle the usual F-test for the general linear hypothesis, and the updating techniques can easily handle less than full rank X and W , while checking for consistency of the model. The first section describes some disadvantages of the original formulation of the problem and gives a general formulation which avoids these. The second section describes a numerically stable method for solution, while the third considers the statistical meaning of the computed quantities. Section 4 introduces the updating techniques as continuations of the original decomposition and Section 5 treats the special case of equality constraints     

基于CORDIC的无开方GIVENS旋转处理方法

自适应阵列处理能提高通信雷达等电子系统的抗干扰能力，因而获得了广泛的应用。许多重要的高速实时自适应阵列处理算法均需要采用一系列Ｇｉｖｅｎｓ旋转处理将输入数据矩阵变成三角阵。标准的Ｇｉｖｅｎｓ旋转包含开方运算，该运算是限制有关算法速度的一个重要因素。文中提出一种基于坐标旋转计算机技术的无开方Ｇｉｖｅｎｓ旋转处理方法，可以显著提高自适应算法的处理速度，并给出了这一方法的推导、Ｇｉｖｅｎｓ处理节点运算式和采用，这种方法的ＱＲ分解自适应阵并行处理算法的计算机模拟结果。