Accuracy Comparison of MFDFA and MFDMA Based on Existence Probability of Multifractal Spectrum in Finite‐Length Cascade Data

In this study, the estimation accuracy of multifractal detrended fluctuation analysis (MFDFA) and multifractal detrending moving average (MFDMA) was compared using finite‐length cascade data generated by log‐normal and log‐gamma cascades. For finite‐length data, not all regions of the theoretical spectrum derived under infinite‐length assumptions can be stably observed. Therefore, using the histogram method, the existence probability at each fractal dimension value under finite‐length conditions was quantified, and the comparable part of the spectrum was identified. Furthermore, the accuracy of MFDFA and MFDMA was compared using the error in the Hölder exponent at each fractal dimension value and the maximum distance error from the theoretical values. In addition, samples showing non‐arc‐shaped spectra were excluded, and the comparison used the same valid sample set. The results demonstrated that for finite‐length cascade data, the multifractal spectrum range that can be meaningfully compared depends on the time‐series length, whereas the existence probability decreases toward smaller fractal dimension values. MFDMA is more stable in preserving a properly arc‐shaped spectrum, whereas neither method exhibits uniform superiority in terms of estimation accuracy. The relative performance of MFDFA and MFDMA varies with the cascade model, degree of multifractality, and time‐series length.