Usage of the Mori-Zwanzig method in time series analysis

Area of Science:

• Statistical Physics
• Time Series Analysis
• Computational Mathematics

Background:

• The Mori-Zwanzig approach is a theoretical framework used in statistical physics to derive macroscopic equations from microscopic dynamics.
• Memory kernels are central to this approach, capturing the system's past influence on its present state.
• Interpreting these kernels is crucial for understanding the underlying physical processes in time series data.

Purpose of the Study:

• To investigate the interpretability of memory kernels derived from the Mori-Zwanzig formalism in time series analysis.
• To analyze the properties of a recently introduced discrete version of the memory kernel.
• To assess the applicability of the Mori-Zwanzig approach for analyzing complex time series.

Main Methods:

• Discussion of memory kernels derived from the Mori-Zwanzig approach.
• Analysis of a discrete approximation of the memory kernel.
• Rigorous mathematical analysis of the discrete kernel's properties.
• Comparison with analytically calculated memory kernels for simple time series processes.
• Examination of the relationship between finite Mori-Zwanzig kernel models and Autoregressive Moving Average (ARMA) models.

Main Results:

• Memory kernels from the Mori-Zwanzig approach are not easily interpretable, even for analytical models.
• Discretization of the memory kernel can lead to properties significantly different from its continuous form.
• Features of simple time series processes are often not detectable in their corresponding discrete memory kernels.
• Finite Mori-Zwanzig kernel models represent a specific subset of ARMA models.

Conclusions:

• The interpretability challenges of Mori-Zwanzig memory kernels, especially in their discrete forms, limit their practical application.
• The findings cast doubt on the reliability of the Mori-Zwanzig approach for analyzing complex time series data due to obscured properties.
• Further research is needed to develop more interpretable methods for time series analysis using Mori-Zwanzig formalism or alternative approaches.