MATLAB Implementation of MFDFA Toolbox

The MFDFA (Multifractal Detrended Fluctuation Analysis) Toolbox is an essential MATLAB tool for nonlinear time series analysis, primarily used to study multifractal characteristics and long-term correlations in temporal data.

Core Functions: Nonlinear Time Series Analysis: MFDFA eliminates local trends through detrending processing, suitable for analyzing non-stationary and nonlinear time series. Implementation typically involves dividing the series into segments and applying polynomial fitting to remove local trends. Hurst Exponent (H-index) Calculation: Quantifies long-range correlations in time series - H>0.5 indicates persistence, H<0.5 shows anti-persistence, and H=0.5 represents random series. The algorithm calculates root-mean-square fluctuations at different scales followed by log-log linear regression. Multifractal Feature Extraction: Analyzes multifractal properties through generalized Hurst exponents and scaling functions, revealing complex behaviors under different fluctuation scales. Code implementation involves multiple moment orders (q-values) to compute fluctuation functions. Prediction Applications: Combined with H-index and multifractal spectrum, it assesses system stability for predictive models in finance, physiological signals, and other fields.

Implementation Approach: Preprocessing: Segments original time series data and performs polynomial detrending using functions like polyfit and polyval. Fluctuation Calculation: Computes root-mean-square fluctuations of detrended data across multiple scales using window-based analysis. Scaling Analysis: Estimates Hurst exponent through double logarithmic linear regression, extended to multifractal analysis by computing fluctuation functions for different q-moments.

Extended Applications: MFDFA can be integrated with entropy measures or machine learning methods to enhance feature extraction capabilities for complex systems (such as EEG signals and stock price fluctuations), making it suitable for pattern recognition in high-noise environments. The toolbox can be extended with custom functions for specific domain applications.