Reasearch On Learning Theory And Method Based On Uncertain Mathematics

By means of machine learning technologies, human beings can learn thevaluable information underlying all kinds of data and finally apply the obtainedinformation to predict the results of the unknown cases. There is a precondition forthe traditional machine learning research, that is, the expression for each case mustbe clear, while the values assigned by the cases for every feature must be crisp. Theforesaid condition, to a great extent, restricts the application range of machinelearning. The data extracted from many real applications always possess uncertainty,which contradicts the precondition that the values need to be crisp. The caserepresentation is crisp, which indicates that the traditional machine learningalgorithms can only deal with the machine learning problems in certainenvironments. However, they are not fit for tackling the learning problems in theuncertain environments.In the past few years, with the developments of uncertain mathematics, e.g. theDempster-Shafer theory of evidence, fuzzy system theory and rough set theory,machine learning has become a research focus for dealing with uncertainty.Moreover, machine learning has already been generalized from the certainenvironment to the uncertain one. Nevertheless, machine learning for tacklinguncertainty is on the development stage, which needs to be further supplementedand perfected. For example, support vector machine can only deal with real crispdata. However, it lacks of theoretical basis and effective means for tacklinguncertain data. The core concept of fuzzy rough set, i.e., fuzzy similarity relationslack geometrical interpretation, which restricts the further development andapplication of fuzzy rough set theory. For the existing fuzzy extensions ofDempster-Shafer theory, the case of infinite fuzzy focal elements has not beenconsidered. Moreover, the probability of the fuzzy focal element is still real valuerather than fuzzy value. For solving the aforementioned problems, in this thesis, thelinear separabilities of samples in several different spaces are firstly investigated.Then, the geometrical interpretation and application of fuzzy similarity relation areanalyzed, while the attribute reduction theory of fuzzy decision system based onfuzzy rough set is introduced. Finally, the fuzzy generalization of theDempster-Shafer theory of evidence with respect to the T-fuzzy similarity relation isstudied. The main results are summarized as follows:The linear separability of two data sets in several spaces is investigated, whichincludes finite and infinite data sets. And necessary and sufficient conditions ofseparating two finite classes of samples by a hyper-plane are presented for Hilbert, Banach and fuzzy number spaces. Based on these results, machine learning modelcan be discussed in general spaces, and these conclusions are theoretical foundationof support vector machine for uncertainty data.The geometrical explanation of fuzzy similarity relations in Krein space isintroduced, and the geometrical explanation of the membership computed by lowerapproximations in fuzzy rough sets is presented. Based on this membership, wedevelop a new algorithm to compute reductions, and apply this membership tosupport vector machine, which two new fuzzy support vector machine models aredesigned.By means of general fuzzy rough sets, the attribute reduction method fordecision systems with fuzzy decision attributes is proposed. Firstly, inconsistentfuzzy decision systems and its relative reduction are introduced, and then algorithmsbased on discernibility matrix to compute all the reductions are proposed. Ourproposed method can deal with regression problems since every numerical decisionattribute of a regression problem can be transferred to a fuzzy decision attribute.The fuzzy generalization of the Dempster-Shafer theory of evidence isdiscussed. Two pairs of fuzzy valued belief and plausibility functions areconstructed by a fuzzy T-similarity relation. The relationship between thesefunctions and fuzzy rough sets is also examined. For given fuzzy valued functions,some axioms are developed to guarantee the existence of a fuzzy similarity relationwhich produces the same pairs of belief and plausibility functions.The foresaid research results enrich machine learning theories in uncertainenvironment, generalize the application range of machine learning, and improve theperformance of the traditional machine learning approaches.