Analyzing a stochastic time series obeying a second-order differential equation.

Area of Science:

• Stochastic processes
• Nonlinear dynamics
• Time series analysis

Background:

• Markov analysis is a powerful tool for extracting stochastic properties from time series data.
• Analyzing processes governed by stochastically forced second-order differential equations requires advanced techniques like embedding.
• Discrete time series analysis introduces errors when approximating temporal derivatives, potentially biasing drift and diffusion function estimations.

Purpose of the Study:

• To analyze systematic errors in Markov analysis of discrete time series arising from derivative approximation.
• To propose a novel approach that accurately accounts for these differencing errors.
• To enable robust parameter estimation for Langevin-type Markov processes, even in the presence of weak measurement noise.

Main Methods:

• Utilizing a 2N-dimensional phase space embedding by augmenting the N-dimensional signal with its temporal derivative.
• Developing a method to correct for errors introduced by discrete differencing schemes in derivative calculation.
• Applying the corrected method to estimate drift and diffusion functions of stochastic processes.

Main Results:

• Identified and quantified systematic errors in drift and diffusion estimation due to uncorrected differencing errors.
• Developed and validated an approach that accurately corrects for these errors.
• Demonstrated the robustness of the proposed method in handling weak superimposed measurement noise.

Conclusions:

• Accurate Markov analysis of discrete time series requires explicit correction for differencing-induced errors.
• The proposed approach provides a reliable method for parameter estimation in Langevin-type Markov processes.
• This work enhances the applicability of Markov analysis to real-world, noisy time series data.