A GENERAL-MODEL FOR STATISTICAL-ANALYSIS USING FUZZY-SETS - SUFFICIENT CONDITIONS FOR IDENTIFIABILITY AND STATISTICAL PROPERTIES

Fuzzy sets and fuzzy state modeling require modifications of fundament al principles of statistical estimation and inference. These modificat ions trade increased computational effort for greater generality of da ta representation. For example, multivariate discrete response data of high (but finite) dimensionality present the problem of analyzing lar ge numbers of cells with low event counts due to finite sample size. I t would be useful to have a model based on an invariant metric to repr esent such data parsimoniously with a latent ''smoothed'' or low dimen sional parametric structure. Determining the parameterization of such a model is difficult since multivariate normality (i.e., that all sign ificant information is represented in the second order moments matrix) , an assumption often used in fitting the most common types of latent variable models, is not appropriate. We present a fuzzy set model to a nalyze high dimensional categorical data where a metric for grades of membership in fuzzy sets is determined by latent convex sets, within w hich moments up to order J of a discrete distribution can be represent ed. The model, based on a fuzzy set parameterization, can be shown, us ing theorems on convex polytopes [1], to be dependent on only the encl osing linear space of the convex set. It is otherwise measure invarian t. We discuss the geometry of the model's parameter space, the relatio n of the convex structure of model parameters to the dual nature of th e case and variable spaces, how that duality relates to describing fuz zy set spaces, and modified principles of estimation.