In the paper proposed we will make use of the gradient flow approach to consider a generalization of the well-known oblique Procrustes rotation problem, involving oblique simple structure rotation of both the core and component matrices resulting from three-mode factor analysis. The standard oblique Procrustes rotations to specified factor-structure and factor-pattern follw as special cases. The approach adopted leads to globally convergent algorithm and includes solving of initial value problem for certain matrix ordinary differential equation. Necessary conditions are established for the solution of the problem. The same approach is extended easily to the weighted oblique Procrustes rotation. Finally, some simulated numerical results are given and commented.
A study was undertaken to construct a descriptive model of both simultaneous contrast illusions and the figural after-effects in terms of a Riemannian space with Riemann-Christoffel curvature tensor. This differential-geometrical model was applied to the illusion of concentric circles (the Delboeuf illusion), the figural after-effect and the bent line illusion (the Orbison square illusion). The curvature tensor Rhijl, which can be calculated from the metric tensor specifying the perceived distortion in these illusions, implies another distortion effect which cannot be described within two-dimensional plane (depth effect). This effect occurs in any figural after-effects, while the simultaneous contrast and the figural after-effect only for small inspection time does not evoke this effect. Therefore, the later phenomenon can be formulated by Rhijl=0, and the former by Rhijl≠O. Assuming that an indicatrix is elliptic in the local perceptual field of parallel lines and solving R/hijl=0 uniquely determined the magnitudes of the illusions as a fractional function in which a parameter is involved. The sign of the parameter was decisive of whether the simultaneous illusion or the figural after-effect with small inspection time is evoked. Moreover, extending the present model to the summation of the effect of each circle led us to simulate a visually straight line in the Orbison square illusion.
The forced classification procedure of dual scaling can be regarded as discriminant analysis for categorical data. Although its procedure is simple, there are a number of interesting, but hidden, characteristics. This study employs dual scaling of the contingency table and that of projection to identify some of them, and a numerical example is used to bridge theory with practice. It is shown that this simple procedure provides comparatively comprehensive analysis of multiple-choice data, discriminant analysis and conditional analysis, a feature that is lacking in the analysis of the contingency table. It is hoped that the clarifications of the procedure will enhance its wider and appropriate use in data analysis.
This paper considers a method for measuring causal effects of factors in recursive causal systems described by hierarchical loglinear models. The total, direct and indirect effects of factors are defined by loglinear model parameters. A numerical example is also given to illustrate the present discussion.
A latent variable model for observed variables representing frequencies is proposed. The data type for the model is a subjects by variables two-way frequency table. The model has two groups of latent variables. The first group of latent variables represents the characteristics of subjects and corresponds to common factors in factor analysis. On the other hand, each of latent variables in the second group is related to one of the manifest variables and corresponds to a specific factor in factor analysis. The manifest variables in the model, when given the values of common latent variables, follow the negative binomial distributions. The latent variables in the first and second groups are integrated out of the model. The parameters in the model are estimated by the marginal maximum likelihood method, using a kind of the EM algorithm. The communality, specificity, and reliability for an observed variable are defined.