Some properties of Sugeno measure are further discussed, which is a kind of typical nonadditive measure. The definitions and properties of gλ random variable and its distribution function, expected value, and varianc...Some properties of Sugeno measure are further discussed, which is a kind of typical nonadditive measure. The definitions and properties of gλ random variable and its distribution function, expected value, and variance are then presented. Markov inequality, Chebyshev＇s inequality and the Khinchine＇s Law of Large Numbers on Sugeno measure space are also proven. Furthermore, the concepts of empirical risk functional, expected risk functional and the strict consistency of ERM principle on Sugeno measure space are proposed. According to these properties and concepts, the key theorem of learning theory, the bounds on the rate of convergence of learning process and the relations between these bounds and capacity of the set of functions on Sugeno measure space are given.展开更多
基金国家自然科学基金(the National Natural Science Foundation of China under Grant No.60773062,No.60673045)教育部科学技术研究重点项目(the Key Scientific and Technical Research Project of Ministry of Education of China under Grant No.206012)河北省教育厅科研计划重点项目(the Key Scientific Research Project of Department of Hebei of Education of China under Grant No.2005001D)
基金国家自然科学基金( the National Natural Science Foundation of China under Grant No.60773062) 教育部科学技术研究重点项目( the Key Scientific and Technical Research Project of Ministry of Education of China under Grant No.206012) +1 种基金河北省教育厅科研计划重点项目( the Key Scientific Research Project of Department of Hebei Education of China under Grant No.2005001D) 河北省自然科学基金资助项目( the Natural Science Foundation of Hebei Province of China under Grant No.2008000633)
基金supported by the National Natural Science Foundation of China(Grant No.60573069)the Natural Science Foundation of Hebei Province(Grant No.F2004000129)+1 种基金the Key Scientific Research Project of Hebei Education Department(Grant No.2005001D)the Key Scientific and Technical Research Project of the Ministry of Education of China(Grant No.20602).
文摘Some properties of Sugeno measure are further discussed, which is a kind of typical nonadditive measure. The definitions and properties of gλ random variable and its distribution function, expected value, and variance are then presented. Markov inequality, Chebyshev＇s inequality and the Khinchine＇s Law of Large Numbers on Sugeno measure space are also proven. Furthermore, the concepts of empirical risk functional, expected risk functional and the strict consistency of ERM principle on Sugeno measure space are proposed. According to these properties and concepts, the key theorem of learning theory, the bounds on the rate of convergence of learning process and the relations between these bounds and capacity of the set of functions on Sugeno measure space are given.
