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 fundamental principles of statistical estimation and inference. These modifications trade increased computational effort for greater generality of data representation. For example, multivariate discrete response data of high (but finite) dimensionality present the problem of analyzing large numbers of cells with low event counts due to finite sample size. It would be useful to have a model based on an invariant metric to represent such data parsimoniously with a latent “smoothed” or low dimensional parametric structure. Determining the parameterization of such a model is difficult since multivariate normality (i.e., that all significant 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 analyze high dimensional categorical data where a metric for grades of membership in fuzzy sets is determined by latent convex sets, within which moments up to order J of a discrete distribution can be represented. The model, based on a fuzzy set parameterization, can be shown, using theorems on convex polytopes [1], to be dependent on only the enclosing linear space of the convex set. It is otherwise measure invariant. We discuss the geometry of the model's parameter space, the relation of the convex structure of model parameters to the dual nature of the case and variable spaces, how that duality relates to describing fuzzy set spaces, and modified principles of estimation.