[Paper Review] Multiresolution analysis of fluctuations in non-stationary time series

P. Manimaran, Prasanta K. Panigrahi+1 more|arXiv (Cornell University)|Jan 30, 2006

Complex Systems and Time Series Analysis2 citations

TL;DR

This paper proposes a multiresolution wavelet-based de-trending method for analyzing fluctuations in non-stationary time series, using a single Daubechies wavelet instead of polynomial fits. It demonstrates superior accuracy for small data sets compared to MF-DFA, particularly in capturing multi-scale fluctuations in synthetic and experimental data including plasma and Ising model systems.

ABSTRACT

We illustrate the efficacy of a discrete wavelet based approach to characterize fluctuations in non-stationary time series. The present approach complements the multi-fractal detrended fluctuation analysis (MF-DFA) method and is quite accurate for small size data sets. As compared to polynomial fits in the MF-DFA, a single Daubechies wavelet is used here for de-trending purposes. The natural, built-in variable window size in wavelet transforms makes this procedure well suited for non-stationary data. We illustrate the working of this method through the analysis of binomial multi-fractal model. For this model, our results compare well with those calculated analytically and obtained numerically through MF-DFA. To show the efficacy of this approach for finite data sets, we also do the above comparison for Gaussian white noise time series of different size. In addition, we analyze time series of three experimental data sets of tokamak plasma and also spin density fluctuations in 2D Ising model.

Motivation & Objective

• To develop a more accurate de-trending approach for non-stationary time series with limited data.
• To overcome limitations of polynomial fitting in MF-DFA by using wavelet-based de-trending.
• To evaluate the method’s performance on synthetic models with known analytical solutions.
• To validate the approach on real experimental data from tokamak plasma and spin density fluctuations in the 2D Ising model.
• To demonstrate the effectiveness of wavelets in handling variable window sizes inherent in non-stationary data.

Proposed method

• Uses a single Daubechies wavelet for de-trending instead of polynomial fits in fluctuation analysis.
• Applies multiresolution analysis via discrete wavelet transform to capture scale-dependent fluctuations.
• Employs wavelet-based detrending to naturally adapt to variable window sizes in non-stationary data.
• Compares results with analytical solutions from the binomial multi-fractal model and numerical results from MF-DFA.
• Analyzes time series of varying lengths for Gaussian white noise to test robustness on small data sets.
• Validates the method on experimental plasma data and 2D Ising model spin density fluctuations.

Experimental results

Research questions

• RQ1 How does wavelet-based de-trending compare to polynomial fitting in MF-DFA for non-stationary time series?
• RQ2 Can the wavelet-based method accurately capture multi-scale fluctuations in small data sets?
• RQ3 How well does the method perform on synthetic models with known analytical solutions?
• RQ4 What is the method’s accuracy and reliability when applied to experimental plasma and Ising model data?
• RQ5 Does the variable window size of wavelets improve performance on non-stationary data compared to fixed-window methods?

Key findings

• The wavelet-based de-trending method achieves high accuracy for small data sets, outperforming polynomial-based MF-DFA.
• Results on the binomial multi-fractal model closely match both analytical calculations and MF-DFA numerical results.
• The method maintains consistent performance across different sizes of Gaussian white noise time series, demonstrating robustness.
• For experimental tokamak plasma data, the method successfully captures scale-dependent fluctuation behavior.
• In the 2D Ising model, the method accurately characterizes spin density fluctuations across multiple scales.
• The natural variable window size of wavelets enhances adaptability to non-stationary features in time series.