Measure-Theoretic Probability of Complex Co-occurrence and E-Integral
Complex high-dimensional co-occurrence data are increasingly popular from a complex system of interacting physical, biological and social processes in discretely indexed modifiable areal units or continuously indexed locations of a study region for landscape-based mechanism. Modeling, predicting and interpreting complex co-occurrences are very general and fundamental problems of statistical and machine learning in a broad variety of real-world modern applications. Probability and conditional probability of co-occurrence are introduced by being defined in a general setting with set functions to develop a rigorous measure-theoretic foundation for the inherent challenge of data sparseness. The data sparseness is a main challenge inherent to probabilistic modeling and reasoning of co-occurrence in statistical inference. The behavior of a class of natural integrals called E-integrals is investigated based on the defined conditional probability of co-occurrence. The results on the properties of E-integral are presented. The paper offers a novel measure-theoretic framework where E-integral as a basic measure-theoretic concept can be the starting point for the expectation functional approach preferred by Whittle (1992) and Pollard (2001) to the development of probability theory for the inherent challenge of co-occurrences emerging in modern high-dimensional co-occurrence data problems and opens the doors to more sophisticated and interesting research in complex high-dimensional co-occurrence data science.