The Sigma-max System Induced from Randomness & Fuzziness and its Application in Time Series Prediction

TL;DR

This paper introduces the sigma-max system derived from randomness and fuzziness, providing a physical foundation for possibility theory and demonstrating its effectiveness in stock price prediction with up to 18.99% improvement.

Contribution

It establishes a rigorous connection between probability and possibility theories from intuitive definitions, and applies max inference to enhance time series prediction.

Findings

Max inference outperforms sigma inference by up to 18.99% in stock prediction.

Max is the only suitable disjunctive operator for fuzzy event spaces.

Provides a physical foundation for the axiomatic basis of possibility theory.

Abstract

This paper managed to induce probability theory (sigma system) and possibility theory (max system) respectively from the clearly-defined randomness and fuzziness, while focusing the question why the key axiom of "maxitivity" is adopted for possibility measure. Such an objective is achieved by following three steps: a) the establishment of mathematical definitions of randomness and fuzziness; b) the development of intuitive definition of possibility as measure of fuzziness based on compatibility interpretation; c) the abstraction of the axiomatic definitions of probability/ possibility from their intuitive definitions, by taking advantage of properties of the well-defined randomness and fuzziness. We derived the conclusion that "max" is the only but un-strict disjunctive operator that is applicable across the fuzzy event space, and is an exact operator for extracting the value from the fuzzy sample space that leads to the largest possibility of one. Then a demonstration example of stock price prediction is presented, which confirms that max inference indeed exhibits distinctive performance, with an improvement up to 18.99%, over sigma inference for the investigated application. Our work provides a physical foundation for the axiomatic definition of possibility for the measure of fuzziness, which hopefully would facilitate wider adoption of possibility theory in practice.