Detrended fluctuation analysis for fractals and multifractals in higher dimensions

Area of Science:

• Physics
• Data Analysis
• Complex Systems

Background:

• One-dimensional detrended fluctuation analysis (DFA) and multifractal DFA (MFDFA) are established methods for analyzing fractal and multifractal time series.
• These methods are valued for their accuracy and ease of implementation in scaling analysis.

Purpose of the Study:

• To generalize existing one-dimensional DFA and MFDFA techniques to higher-dimensional applications.
• To assess the efficacy of these generalized methods on synthetic and real-world data.

Main Methods:

• Development and application of generalized multi-dimensional detrended fluctuation analysis.
• Testing with synthetic fractional Brownian and multifractal surfaces.
• Application of two-dimensional MFDFA to analyze natural and experimental images.

Main Results:

• The generalized higher-dimensional DFA and MFDFA methods demonstrate successful application.
• Effective analysis of synthetic surfaces, revealing expected scaling behaviors.
• Identification of significant scaling laws in real-world images using 2D MFDFA.

Conclusions:

• The generalization of DFA and MFDFA to higher dimensions is a robust advancement.
• These extended methods provide powerful tools for analyzing complex multi-dimensional data, including surfaces and images.
• The approach successfully uncovers underlying scaling properties in diverse datasets.