An essential characteristic of function transformation to reduce the class ratio dispersion and its application

The paper firstly points out the necessary and sufficient conditions of both monotone increasing and monotone decreasing function transformation which can reduce the class ratio dispersion respectively, and then proves that no matter what the function transformation is monotone increasing or monotone decreasing, and the effect object is the monotone increasing series or the monotone decreasing series, reducing the class ratio dispersion is equivalent to raising the smooth degree of data. Finally, according to these necessary and sufficient conditions, the paper proposes some examples to indicate whether or not the functions can reduce the class ratio dispersion and raise the smooth degree of data, and, consequently, obtains three compare criterions of the effect of two kinds of monotone function transformations which reduce the class ratio dispersion and raise the smooth degree of data.