Statistic Analysis Approach For Fuzzy Data

The statistical theory and approach is an essential research method in applied areas in nowadays. The research of the classical statistic theory has been already quite systematic and complete at present, however, the classical statistic theory also exits some limitation, and can not well solve all practical problems. Particularly, when observations of experiment can only be fuzzy data, How to deal with the statistic analysis becomes a problem we have to discuss.Developed along with the study of the f.r.v., fuzzy statistic theory is still at the beginning stage of its development. Though f.r.v. theory is not very complete, but with the past 20 years' development, some achievement has been make, and so has provided certain foundation for the research of fuzzy statistic theory. At present, there is relatively more study on fuzzy linear regression theory, but because of the complexity of fuzzy number space, each of the regression approach has certain limitation. And there is few discussion on the statistic properties of the estimator. As for the hypothesis-tests for the expectation of f.r.v., the approaches are too complicated to use in practice. As the development of fuzzy stochastic process, the study on fuzzy random dynamical system model appeared correspondingly, but the research on fuzzy stochastic system with a Gaussian initial state is remain very rare.This paper studies the statistic analysis approach for fuzzy data, and mainly in-eludes two parts: fuzzy linear regression theory and fuzzy valued Kalman filtering and its stability analysis. In the first part, we introduce a new estimation approach for the fuzzy linear regression model in which the observations are LR fuzzy data, discuss the statistic properties of each estimator, and give an interval estimation approach for the expectation and variance of f.r.v.. In the second part, we give the optimal estimation theorem for unknown fuzzy state X using the fuzzy observation data Yi, ? ? ? , Yr, study the optimal state estimate for linear discrete-time dynamic fuzzy system with a Gaussian initial state, discuss the stability of fuzzy valued Kalman optimal filtering, and give an example w.r.t. the stability of fuzzy valued Kalman filtering.In the discussion of fuzzy linear regression, we transform the fuzzy linear regression model to a family of crisp linear regression model, and discuss the estimation according to classical least squares theory. For the creativity of our approach, we get a more general result, which is the expand of Koner and Nather's result. We take advantage of Gaussian f.r.v.'s properties to discuss the statistic properties of the estimators, and discuss an interval estimate approach for fuzzy parameter, which are prepared some theory for the application of fuzzy linear regression in prediction and control. In the study of fuzzy valued Kalman filtering, we specify the estimation problem of two f.r.v.'s X and Y by constructing a mean-square performance index in the case of f.r.v.. And give the optimal estimation theorem w.r.t. fuzzy state firstly, and then, we use the same idea as before to transform the linear discrete-time dynamic fuzzy system into a family of crisp linear discrete-time dynamic system, and using the optimal estimation theorem, we transform the optimal state estimate into the combine of real valued non-random recurrence equation and ordinary discrete-time real-valued system finally.The approach that transforming fuzzy model into a family of real-valued models and then discussing the models with classical theory has universal significance. It provides a good idea for us to study each kind of fuzzy model.