MATLAB Code Implementation for Multifractal Analysis Function

Multifractal analysis serves as a powerful tool for investigating local scaling properties of complex systems, with wide applications in turbulence, financial time series, and geographical morphology. Implementing multifractal functions in MATLAB typically requires integrating box-counting methods with statistical moment analysis techniques.

The core algorithm involves conducting multi-scale statistical analysis on input data (such as 1D signals or 2D images). The implementation begins by partitioning data into box grids of varying sizes using MATLAB's meshgrid or ndgrid functions, followed by calculating probability measure distributions at each scale. By constructing partition functions and analyzing their scaling behavior across different moment orders through linear regression, the function ultimately derives the singularity spectrum that characterizes the system's multifractal properties.

Critical implementation steps require attention: scale selection must employ logarithmically equidistant sampling (using logspace function) to ensure comparability between scales; probability measure computation should incorporate boundary effect handling through padding or circular indexing; partition function linear fitting necessitates robust regression methods like least absolute deviation (robustfit); singularity spectrum derivation involves numerical implementation of Legendre transformation using gradient or diff functions.

Compared to traditional monofractal dimension calculations, this multifractal analysis method provides more comprehensive characterization of nonlinear features, particularly for complex systems exhibiting multiple scaling behaviors. MATLAB implementation can leverage vectorized operations (arrayfun/bsxfun) for computational efficiency and utilize plotting functions (plot/semilogx) to visually display multifractal spectrum curves.

A distinctive feature of this function lies in its adaptive grid partitioning strategy, which dynamically adjusts analysis scales based on data characteristics using while-loops and conditional statements. This approach enables more accurate multifractal parameter estimation, particularly effective when analyzing data with local abrupt features through localized variance checks.