Study Of The Regression And Classification Problems For Set-valued Data

Set-based classifications and regressions arise in a variety of applications,such as wind speed prediction,water quality assessment and video analysis.Currently,there is a lack of both theory and methods solving this kind of problems.Aiming to solve the set-based classifications and regressions,this thesis establishes a support vector regression(SVR)based on set-valued data and a support function machine(SFM),respectively,which extends the theory and applications of support vector machine(SVM)in essence.The contents are summarized as follows:1.In order to improve the prediction accuracy of the regression problems for setvalued data,firstly,a SVR based on the best representative points is established for the set-valued data with a small fluctuation degree.Secondly,for the set-valued data with a big fluctuation degree,a new ε-SVR based on prior information is proposed.At last,the applications of them in wind speed prediction are studied.The comparison experiments illustrate the validity and feasibility.2.In order to use the classification information provided by set-valued data comprehensively and reasonably,firstly,set-valued data are converted into the infinite dimensional Banach space C(S)via support functions,the classification hyperplane is defined,and the Hausdorff distance between two parallel hyperplanes is discussed.Secondly,two optimization problems are derived based on the maximum margin principle(MMP),the convexity and existence of optimal solutions are discussed,respectively,and then support function machines with hard and soft margin are established,respectively.Finally,experiments comparing with other methods illustrate the effectiveness and feasibility of the new SFM.3.In order to provide theoretical basis for support function machine,the consistency of structural risk minimization(SRM)and MMP are discussed firstly.Secondly,for the linearly separable set-valued data in R~d,we prove that they are still linearly separable in the infinite dimensional Banach space C(S).For the linearly inseparable set-valued data in R~d,a sufficient condition to judge the linear separability of them in C(S)is given.Finally,the existence of the hyperplane is proved,which provides a theoretical basis for improving the experiment and algorithm.4.In order to deal with the fuzzy classification of set-valued data more accurately,membership degree is introduced to describe the degree that input data belongs to a certain fuzzy class,and then a fuzzy classification of set-valued data is converted into the fuzzy classification of functional data,a fuzzy support function machine based on possibility measure(PMFSFM)is constructed.The PMFSFM can provide both the fuzzy class of input data and the membership degree that the input data belongs to the fuzzy class.The water quality assessment under fuzzy environment shows the effectiveness and superiority of the proposed PMFSFM.