Uncertain Box-Cox And Logistic Regression Analysis

Expert's experimental data concerns the subjective judgement of different experts on the possibility of an uncertain event.Due to the characteristics of a small sample size and inaccu-racy,this judgement is different from the random sampling procedure in the study of classical statistics,rendering the direct application of classical statistical inference inappropriate.Pro-fessor Baoding Liu proposed the uncertainty theory to fix the inference problems for such kind of data.The theory describes uncertain phenomena by establishing a new axiomatic system and introducing the concepts of the uncertain measure,the uncertain distributions,the uncer-tain inverse distributions and their applications.This theoretical framework has opened up a new research direction,and has blossomed many important applications,such as uncertain programming,uncertain risk analysis,uncertain reliability analysis,uncertain finance,etcRegression analysis is a classical topic in statistical theory with enormous applications The uncertainty regression analysis,though,has been so far confined to linear models.In order to improve the theoretical system of uncertain regression analysis,this dissertation focuses on three perspectives of the uncertain regression analysis for two important uncertain non-linear models.Firstly,the analysis of the uncertain revised regression model is studied under three kinds of transformations respectively-namely the logarithmic,the square root and the recip-rocal transformation-of the response variable in the uncertain linear model.We propose the estimation of the regression coefficient and the variance of the error term by using the least squares principle,followed by the computation method for the prediction interval.These meth-ods are then applied to three kinds of uncertain revised regression models respectively,name-ly the uncertain revised linear regression model,the uncertain revised asymptotic regression model,and the uncertain revised Michaelis-Menten kinetics regression model,with numerical examples available to validate above methods.Secondly,the rescaled least squares method is used to study the regression analysis for the uncertain Box-Cox regression model,in estimation of the regression coefficient,the transformation coefficient and the variance of the error term.We also give the prediction interval.Moreover,we propose the residual analysis method to assess the validity of the fitted model.Compared with the least squares method,this method adds a new penalty term to the transformation coefficient,which could fix the theoretical prob-lem of the least squares method in estimating the transformation coefficient for the uncertain Box-Cox regression model.Finally,the uncertain maximum likelihood estimation of the uncer-tain Box-Cox and Logistic regression models are studied.We propose the uncertain maximum likelihood estimation for discrete uncertain models,parallel to the maximum likelihood esti-mation for continuous uncertain models already proposed,therefore extending the application scope of uncertain maximum likelihood estimation from linear models to non-linear models.Furthermore,We propose the cross-validation method to evaluate the fitted uncertainty model,together with the residual plots.In summary,this dissertation further enriches the theory of uncertain regression analysis,in a way that providing an effective analysis and evaluation method for uncertain non-linear models,especially the uncertain Box-Cox and Logistic regression models,thus has potential application values in the fields requiring the analysis of expert's experimental data.