Measure-Theoretic Probability of Complex Co-occurrence and E-Integral
- URL: http://arxiv.org/abs/2210.09913v1
- Date: Tue, 18 Oct 2022 14:52:23 GMT
- Title: Measure-Theoretic Probability of Complex Co-occurrence and E-Integral
- Authors: Jian-Yong Wang and Han Yu
- Abstract summary: The behavior of a class of natural integrals called E-integrals is investigated based on the defined conditional probability of co-occurrence.
The paper offers a novel measure-theoretic framework where E-integral as a basic measure-theoretic concept can be the starting point.
- Score: 15.263586201516159
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: 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.
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