UCLA

In this work, we explore a distance association method, Logistic Gifi, for categorical data which advances geometric data analysis techniques in much the same way that homogeneity analysis did with regard to correspondence analysis, multidimensional scaling, and general clustering methods. It uses the methods of multidimensional unfolding with a probability-based loss measure to create low-dimensional geometric representations of data in which distances correspond in a direct way to the probabilistic structure of the data. As with homogeneity analysis, a central feature of our method is the use of binary indicator matrices and, in some applications, fuzzy-coded (i.e., non-binary, row stochastic) indicator matrices to represent categorical data. This gives us a very versatile method with considerable flexibility in the types of data which can be analyzed. We create and study algorithms to use the method to compute low-dimensional geometric representations of various types of data. We analyze the convergence properties of this complex algorithm and show how minimal polynomial extrapolation can be used to accelerate it. We then study relationships between this logistic distance method and logit-based regressions. We present several applications of the method to visualizations of regression results as well as data types such as roll calls, social networks, and Markov chains. Finally, a version of the method with bias parameters is introduced and developed and used to emphasize features of data visualizations. We show how bias constraints can be used to represent certain types of model testing. Noting the similarities between the model with bias parameters and ideal point discriminant analysis, we examine these using bias constraints and different forms of indicator matrices. Last, we study the stability of the method.