{ "id": "ep6ecydna6", "slug": "multifractal-detrended-fluctuation-analysis", "title": "Multifractal detrended fluctuation analysis", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Non-geographic · 1946 to 2000: Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference" } ] } ], "excerpt": "Multifractal detrended fluctuation analysis (MF-DFA) is a statistical method that measures multifractality in nonstationary time series, reported by Jan W. Kantelhardt and colleagues in 2002.", "snippet": "Multifractal detrended fluctuation analysis (MF-DFA) is a statistical method that measures multifractality in nonstationary time series, reported by Jan W. Kantelhardt and colleagues in 2002.", "node": "physical.math.statistics.stats_statistical_inference", "markdown": "# Multifractal detrended fluctuation analysis\n\nMultifractal detrended fluctuation analysis (MF-DFA) is a statistical method that estimates how the scaling of fluctuations in a nonstationary time series depends on fluctuation magnitude, replacing the single [Hurst exponent](https://www.edgepedia.org/hurst-exponent) of ordinary [detrended fluctuation analysis](https://www.edgepedia.org/detrended-fluctuation-analysis) with a spectrum of generalized Hurst exponents h(q) and, derived from them, a singularity spectrum f(α).[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) The method was reported in 2002 in Physica A and has since been applied in physiology, finance, and geophysics.[2](https://www.scienceopen.com/document?vid=463b0b9c-985a-4474-b4b3-a6b5035233bf) • [3](https://juser.fz-juelich.de/record/909687/files/2104.10470.pdf) A 2025 review describes MF-DFA as the most widely used practical method for quantifying multifractality, due to its stability.[4](https://www.mdpi.com/2227-7390/13/2/205)\n\n| Key fact | Value |\n|---|---|\n| Output | Family of generalized Hurst exponents h(q), decreasing in q for multifractal signals and constant for monofractal ones[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) • [5](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.74.016103) |\n| Relation to DFA | For \\( q = 2 \\), MF-DFA reduces exactly to standard DFA[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) • [3](https://juser.fz-juelich.de/record/909687/files/2104.10470.pdf) |\n| Defining relations | τ(q) = qh(q) − 1; α = h(q) + qh′(q); f(α) = q[α − h(q)] + 1[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) • [3](https://juser.fz-juelich.de/record/909687/files/2104.10470.pdf) |\n| Typical q range | q-orders from −5 to 5; precision of \\( h(q) \\) decreases toward extreme q[6](https://www.frontiersin.org/journals/physiology/articles/10.3389/fphys.2012.00141/full) |\n| Segment sizes | Minimum segment above 10 samples and well above the polynomial order; maximum below N/10[6](https://www.frontiersin.org/journals/physiology/articles/10.3389/fphys.2012.00141/full) |\n| Short-record precision | About 5% error in \\( h(q) \\) at \\( |q| < 5 \\) for series of \\( 2^{10} \\) points[7](https://ar5iv.labs.arxiv.org/html/1311.2278) |\n| Introducing paper | Physica A 316, pp. 87–114, December 2002, DOI 10.1016/S0378-4371(02)01383-3[2](https://www.scienceopen.com/document?vid=463b0b9c-985a-4474-b4b3-a6b5035233bf) |\n\n## How it works\n\nMF-DFA rests on a q-dependent fluctuation function. After the series is segmented and locally detrended, the qth-order fluctuation function is\n\n\\[ F_{q}(s) = \\left\\{ \\frac{1}{2N_{s}} \\sum_{\\nu=1}^{2N_{s}} \\left[ F^{2}(\\nu,s) \\right]^{q/2} \\right\\}^{1/q}, \\]\n\nwhere the index q can take any real value except zero, and \\( F_{q}(s) \\) scales as \\( s^{h(q)} \\).[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) For \\( q = 2 \\) the standard DFA procedure is retrieved, and for stationary series h(2) equals the Hurst exponent H; h(q) is therefore called the generalized Hurst exponent.[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) A constant \\( h(q) \\) signals a monofractal series; a decreasing \\( h(q) \\) signals multifractality.[5](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.74.016103)\n\nThe exponent q weights segments differently: positive q emphasizes segments with large fluctuations and yields smaller h(q), while negative q emphasizes segments with small fluctuations and yields larger h(q).[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) At \\( q = 0 \\) the defining average diverges, so a logarithmic average is used instead: \\( F_{0}(s) = \\exp\\{(1/(4N_{s})) \\sum \\ln[F^{2}(\\nu,s)]\\} \\).[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) The fluctuation exponents connect to the standard multifractal formalism through the scaling exponent τ(q) = qh(q) − 1, and the singularity spectrum follows by a Legendre transform, with α = h(q) + qh′(q) and f(α) = q[α − h(q)] + 1.[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) • [3](https://juser.fz-juelich.de/record/909687/files/2104.10470.pdf) The spectrum width is defined as \\( \\Delta\\alpha = \\alpha_{\\max} - \\alpha_{\\min} \\).[8](https://www.actaphys.uj.edu.pl/R/46/10/1925/pdf)\n\n## How it is done\n\nThe protocol has five steps, the first three essentially identical to conventional DFA.[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf)\n\n1. Convert the series into a profile (the cumulative sum of the mean-subtracted values).\n2. Divide the profile into \\( N_{s} = \\mathrm{int}(N/s) \\) nonoverlapping segments of length s, counted from both ends of the record, giving \\( 2N_{s} \\) segments.\n3. Fit each segment with a least-squares polynomial of order \\(m\\) and compute the segment variance \\(F^{2}(\\nu,s)\\).\n4. Average over all segments with the q-order formula above to obtain \\( F_{q}(s) \\).\n5. Fit \\( \\log F_{q}(s) \\) versus \\( \\log s \\) to obtain \\( h(q) \\) for each q.\n\nIn MF-DFA of order m, trends of order m in the profile, equivalently order m − 1 in the original series, are eliminated, so comparing results across detrending orders helps identify the trend type present.[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) Practical parameter guidance from a Matlab tutorial: the minimum segment size should exceed 10 samples as a rule of thumb and must considerably exceed m; a maximum segment size below 1/10 of the series length leaves at least 10 segments for \\( F_{q} \\); and q-orders from −5 to 5 are typical.[6](https://www.frontiersin.org/journals/physiology/articles/10.3389/fphys.2012.00141/full) One persistent practical problem is the scaling range itself: published comparisons note there is no consensus on an objective determination of the fitting region, and an \\( R^{2} \\)-based, user-independent criterion for DFA and MF-DFA has been proposed to select it.[9](https://www.sciencedirect.com/science/article/abs/pii/S0378437113010960)\n\n## Origin\n\nMF-DFA was reported in 2002 by Jan W. Kantelhardt and colleagues, in Physica A: Statistical [Mechanics](https://www.edgepedia.org/mechanics) and its Applications, volume 316, pages 87–114.[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) • [2](https://www.scienceopen.com/document?vid=463b0b9c-985a-4474-b4b3-a6b5035233bf) The method generalizes detrended fluctuation analysis, an earlier technique for detecting long-range correlations in time series with nonstationarities; a precursor study extended DFA to higher detrending orders and analyzed how trends of order p cause artificial crossovers when the detrending order is not larger than p.[10](https://havlin.ph.biu.ac.il/wp-content/uploads/Publications/kkrhb416.pdf) The 2002 paper presents MF-DFA as a simpler alternative to the wavelet transform modulus maxima (WTMM) method, an improved multifractal formalism for nonstationary series developed in the early 1990s.[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf)\n\n## Variants\n\nSeveral named extensions adapt the detrending idea to other data structures.\n\n- **MF-DMA.** Multifractal detrending moving average algorithms generalize the DMA method; their authors report that the backward variant gives the most accurate scaling exponents and outperforms MF-DFA.[11](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.82.011136)\n- **Cross-correlation methods.** MF-DXA investigates multifractal behavior in power-law cross-correlations between two records in one or higher dimensions.[12](https://ideas.repec.org/p/arx/papers/0803.2773.html) A related multifractal detrending moving-average cross-correlation analysis was reported by Zhi-Qiang Jiang and Wei-Xing Zhou in [Physical Review E](https://www.edgepedia.org/physical-review-e) in 2011.[13](https://doi.org/10.1103/physreve.84.016106)\n- **Maxima MF-DFA.** Standard MF-DFA requires series of compact support; a modified version restricting the q-averaging sum to local maxima of \\( F^{2}(s,\\nu) \\), with \\( \\tau(q) \\) obtained from \\( \\Sigma [F^{2}(s,\\nu)]^{q/2} \\cdot s^{\\tau(q)} \\), handles fractal (non-compact) support.[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf)\n- **MFFDFA.** A flexibly detrended version lets the polynomial degree vary per segment and uses partly overlapping segments with step \\( \\lfloor s/k \\rfloor \\), giving roughly k times more intervals; on synthetic monofractal data its \\( \\Delta\\alpha \\) values are on average about 70% closer to the theoretical single point than standard MF-DFA.[8](https://www.actaphys.uj.edu.pl/R/46/10/1925/pdf)\n- **Generalized functions.** A 2024 Physica A paper by Suzielli M. Mendonça, Brenno C.T. Cabella, and Alexandre S. Martinez proposes an MF-DFA approach using generalized functions, validated by estimating generalized Hurst exponents on signals such as the fractional [Ornstein–Uhlenbeck process](https://www.edgepedia.org/ornstein-uhlenbeck-process).[14](https://doi.org/10.1016/j.physa.2024.129577)\n\n## Applications\n\nIn geophysics and hydrology, MF-DFA (orders 2 to 4, which give similar results) was applied to 99 daily precipitation and 42 daily river runoff records. Runoff records show stronger multifractality, with an average \\( \\Delta\\alpha = 0.49 \\pm 0.16 \\), than precipitation records, with \\( \\Delta\\alpha = 0.29 \\pm 0.14 \\); a generalized binomial cascade model fits the runoff records, and the multifractal exponents serve as fingerprints for individual stations and rivers.[15](https://www.rybski.de/diego/files/KantelhardtJ_jgpra_2006.pdf)\n\nIn finance, both MF-DFA and WTMM detect rich multifractality in American and German stock market data, though MF-DFA suggests the multifractality is poorer than WTMM does.[5](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.74.016103) Applied to 12 years of daily USD/Euro exchange rate data, MF-DFA returned results compatible with either monofractal behavior close to white noise or weak multifractality.[7](https://ar5iv.labs.arxiv.org/html/1311.2278) The software literature lists applications across heartbeat rate, EEG, precipitation, streamflow, finance, electricity prices, power-grid frequency, and epidemiology.[3](https://juser.fz-juelich.de/record/909687/files/2104.10470.pdf)\n\n## Limitations and alternatives\n\n**Finite-size effects.** Scales \\( s > N/4 \\) are excluded from fitting because too few segments remain for reliable averaging, and systematic deviations from scaling occur at very small scales around \\( s \\approx 10 \\); \\( F_{q}(s) \\) is only defined for \\( s > m + 2 \\).[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) Small-scale deviations grow stronger with higher detrending orders; a correction function constructed from shuffled data can remove these deviations and is useful for short records.[10](https://havlin.ph.biu.ac.il/wp-content/uploads/Publications/kkrhb416.pdf)\n\n**Crossovers.** Short-range correlations can produce notable crossovers in \\( F_{q}(s) \\), causing overestimation of \\( h(q) \\) for small q; crossovers were traditionally identified manually by experts, and CDV-A is an algorithm that detects them automatically, though it is limited to a single crossover.[16](https://link.springer.com/article/10.1007/s11071-024-10478-1)\n\n**Spurious multifractality.** For heavy-tailed (Lévy-type) data with \\( q > 5/3 \\), the variance-based MF-DFA spectrum is bifractal and broadened by finite-size effects; Δα → 0 as N → ∞ for uncorrelated heavy-tailed series, so such broadening should not be read as multifractality, since multifractality can arise from a broad probability distribution, temporal correlations, or both.[4](https://www.mdpi.com/2227-7390/13/2/205) • [17](https://arxiv.org/abs/physics/0202070) Comparing original with shuffled series distinguishes multifractality due to long-range correlations from multifractality due to a broad probability density.[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) For strongly anti-correlated signals with \\( h(q) \\) near zero, MF-DFA becomes inaccurate because it can only determine positive exponents; integrating the series first yields exponents h̃(q) = h(q) + 1.[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf)\n\n**Short records.** For monofractal signals, series as short as \\( 2^{10} \\) points can be analyzed with about 5% precision at \\( |q| < 5 \\); outside the reliable central q region, results can wrongly assign multifractal behavior to a monofractal signal, or reduce apparent multifractality in a genuinely multifractal one.[7](https://ar5iv.labs.arxiv.org/html/1311.2278)\n\n**Comparison with WTMM and MF-DMA.** Published comparisons disagree on the ranking. The 2002 paper reports equivalent detrending capability and accuracy for MF-DFA and WTMM on trended binomial multifractal series, with MF-DFA slightly better for short series and negative moments and its main advantage being simplicity.[1](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf) A 2006 comparison instead recommends MF-DFA in most situations where fractal properties are unknown a priori, because WTMM gives biased outcomes for fractional [Brownian motion](https://www.edgepedia.org/brownian-motion) and results depend on the chosen wavelet.[5](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.74.016103) On the MF-DMA side, its authors report that backward MF-DMA outperforms MF-DFA,[11](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.82.011136) a claim the MF-DFA literature does not concede; these discrepancies remain unresolved in the published literature.\n\n## References\n\n1. [Multifractal detrended fluctuation analysis of nonstationary time series (Kantelhardt et al., Physica A 316, 87–114, 2002, publisher-hosted published version)](https://www.uni-giessen.de/de/fbz/fb07/fachgebiete/physik/institute/theorie/institut-fur-theoretische-physik-iii/pub/paper/2002/papers2002/physica-1/@@download/file/Physica-1.pdf)\n2. [Multifractal detrended fluctuation analysis of nonstationary time series (ScienceOpen record)](https://www.scienceopen.com/document?vid=463b0b9c-985a-4474-b4b3-a6b5035233bf)\n3. [MFDFA: Efficient Multifractal Detrended Fluctuation Analysis in Python (Rydin Gorjão et al., arXiv 2104.10470, 2021)](https://juser.fz-juelich.de/record/909687/files/2104.10470.pdf)\n4. [Disentangling Sources of Multifractality in Time Series (Mathematics 13(2):205, 2025)](https://www.mdpi.com/2227-7390/13/2/205)\n5. [Wavelet versus detrended fluctuation analysis of multifractal structures (Oświęcimka et al., Phys. Rev. E 74, 016103, 2006)](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.74.016103)\n6. [Introduction to Multifractal Detrended Fluctuation Analysis in Matlab (Ihlen, Frontiers in Physiology 2012)](https://www.frontiersin.org/journals/physiology/articles/10.3389/fphys.2012.00141/full)\n7. [Performance of multifractal detrended fluctuation analysis on short time series (López & Contreras, Phys. Rev. E 87, 022918, 2013)](https://ar5iv.labs.arxiv.org/html/1311.2278)\n8. [A Multifractal Flexibly Detrended Fluctuation Analysis (MFFDFA), Acta Physica Polonica A](https://www.actaphys.uj.edu.pl/R/46/10/1925/pdf)\n9. [A criterion for the determination of optimal scaling ranges in DFA and MF-DFA (Physica A, 2014)](https://www.sciencedirect.com/science/article/abs/pii/S0378437113010960)\n10. [Detrended fluctuation analysis: DFA of higher orders and modified DFA (Kantelhardt, Koscielny-Bunde, Rego, Havlin, Bunde, Physica A 2001)](https://havlin.ph.biu.ac.il/wp-content/uploads/Publications/kkrhb416.pdf)\n11. [Detrending moving average algorithm for multifractals (Gu & Zhou, Phys. Rev. E 82, 011136, 2010)](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.82.011136)\n12. [Multifractal detrended cross-correlation analysis for two nonstationary signals (Zhou, 2008, arXiv:0803.2773)](https://ideas.repec.org/p/arx/papers/0803.2773.html)\n13. [Zhi-Qiang Jiang, Wei-Xing Zhou (2011). Multifractal detrending moving-average cross-correlation analysis. Physical Review E.](https://doi.org/10.1103/physreve.84.016106)\n14. [Suzielli M. Mendonça, Brenno C.T. Cabella, Alexandre S. Martinez (2024). A Multifractal Detrended Fluctuation Analysis approach using generalized functions. Physica A Statistical Mechanics and its Applications.](https://doi.org/10.1016/j.physa.2024.129577)\n15. [Long-term persistence and multifractality of precipitation and river runoff records (Kantelhardt et al., J. Geophys. Res. 2006)](https://www.rybski.de/diego/files/KantelhardtJ_jgpra_2006.pdf)\n16. [Crossover detection based on variances of slope differences for multi-fractal detrended fluctuation analysis (MF-DFA) (Nonlinear Dynamics, 2024)](https://link.springer.com/article/10.1007/s11071-024-10478-1)\n17. [Multifractal detrended fluctuation analysis of nonstationary time series](https://arxiv.org/abs/physics/0202070)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Statistical inference, estimation, sampling, and testing*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/multifractal-detrended-fluctuation-analysis", "markdown_url": "https://www.edgepedia.org/multifractal-detrended-fluctuation-analysis.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. 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