MFDFA的计算原理及代码

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MFDFA的计算原理及代码

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MFDFA (Multifractal Detrended Fluctuation Analysis) is a powerful technique used to analyze the multifractal properties of time series data. This method extends the standard DFA (Detrended Fluctuation Analysis) by examining how the scaling behavior of fluctuations varies with different moments, providing a more comprehensive understanding of the underlying dynamics.

The computation of MFDFA involves several key steps. First, the time series is integrated to reduce noise and highlight underlying trends. Next, the series is divided into segments of varying lengths, and local trends within each segment are removed through polynomial fitting. The fluctuation function is then calculated for each segment, measuring the variance around the detrended data.

To capture multifractal characteristics, the fluctuation function is analyzed for different statistical moments (q-values), ranging from negative to positive values. These moments emphasize various aspects of the time series, with negative q-values focusing on small fluctuations and positive q-values highlighting large fluctuations. The scaling behavior is examined by plotting the fluctuation function against segment size on a log-log scale, with the slope indicating the generalized Hurst exponent.

Finally, the multifractal spectrum is derived by transforming the generalized Hurst exponents, revealing the distribution of fractal dimensions across different fluctuation magnitudes. This spectrum provides insights into the complexity and heterogeneity of the time series, making MFDFA a valuable tool in fields such as physiology, finance, and environmental science.

Visualization is crucial in MFDFA analysis, typically involving plots of the fluctuation function, scaling exponents, and the multifractal spectrum. These graphs help identify the range of scaling behaviors and the degree of multifractality present in the data. Proper implementation requires careful handling of parameters such as segment sizes, polynomial order for detrending, and the range of q-values to ensure accurate and meaningful results.