Chapman-Kolmogorov test for estimating memory length of two coupled processes

Area of Science:

• Complex Systems Analysis
• Time Series Modeling
• Statistical Mechanics

Background:

• Real-world processes often exhibit long-range memory, exceeding simple Markovian dependencies.
• System states can be influenced by the past states of multiple interdependent processes.

Purpose of the Study:

• To introduce a generalized Chapman-Kolmogorov equation (CKE) for estimating memory size in complex, dependent processes.
• To validate the generalized CKE's accuracy in measuring memory lengths.
• To apply the method to real-world coupled time series data, specifically cryptocurrencies.

Main Methods:

• Development of a generalized Chapman-Kolmogorov equation (CKE).
• Generation of synthetic coupled time series using autoregressive models to simulate memory.
• Application of the generalized CKE to analyze cryptocurrency market data.

Main Results:

• The generalized CKE demonstrated high accuracy in measuring predefined memory lengths in synthetic data.
• Analysis of cryptocurrency data revealed significant past influences on current states.
• Identified substantial interdependencies among different cryptocurrency time series.

Conclusions:

• The generalized CKE is a robust tool for quantifying memory in complex systems with multiple dependencies.
• The method confirms the interconnectedness and memory effects within cryptocurrency markets.
• The approach offers potential for forecasting coupled time series data in various domains.