Research on the relationship between some important split measure functions for decision tree with purity law

This paper analysis some split measure functions of classical decision tree algorithm. After the research on the structure of these functions, we found out all of them are separable probability measure function, and their core functions are semi-purity functions. They achieve their minimums at row-column independent point, maximums at full-distinguish point, and accordance with purity law. Because chi-square does not support the symmetry, the purity law proposed has wider adaptability than the impurity theory. These can help us to analysis the theory of measure function and the relationship between measure functions and data, and it is important to find some more simple and effective split-measure functions in some special area.