Complexity of Financial Time Series: Multifractal and Multiscale Entropy Analyses

Farhad Shahbazi; Mohammad Sharifi; Oday Masoudi

REVIEW 5 major objections 6 minor 28 references

Complexity of Financial Time Series: Multifractal and Multiscale Entropy Analyses

T0 review · 5 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Bitcoin's daily log returns are the most complex of the four assets studied, under both RCMSE and MF-DFA, with the edge attributed to nonlinear correlations.

desk verdict Plausible directional claim, but the RCMSE complexity ranking is fragile because it leans on scales where the estimator is invalid; the MF-DFA finding is on firmer ground. read the letter →

arxiv 2507.23414 v1 pith:YHWEXJB4 submitted 2025-07-31 q-fin.ST physics.data-an

Oday Masoudi , Farhad Shahbazi , Mohammad Sharifi This is my paper · ORCID

classification q-fin.STphysics.data-an

keywords RCMSEMF-DFAmultifractalspectrumsampleentropyBitcoinfinancialtimeseriesnonlinearcorrelationsHurstexponent

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to show that Bitcoin's daily log-return series is more complex than the log returns of GBP/USD, gold, and natural gas, and that the extra complexity traces to nonlinear correlations rather than to simple volatility or linear autocorrelation. It applies two independent measures: refined composite multiscale sample entropy, which sums regularity across many coarse-grained time scales, and multifractal detrended fluctuation analysis, which measures how fluctuation scaling varies across moment orders. Both rank Bitcoin first: its RCMSE complexity score is 74.66 versus 51.48 to 67.88 for the others, and its multifractal spectrum width is 0.62 versus 0.21 to 0.50. Bitcoin shows the lowest entropy at short scales and the highest at long scales, so the claim is scale-dependent rather than a blanket statement of unpredictability. The result matters because complexity assessment bears on predictability and risk for investors and on which assets are adequately described by standard efficient-market models.

What carries the argument

Two named measures carry the argument. RCMSE, refined composite multiscale sample entropy, coarse-grains the series into multiple scale-factor windows, counts matching template vectors across all coarse-grained versions, and defines complexity as the sum of entropy over scales; it is chosen because ordinary multiscale entropy becomes undefined for short series at large scales. MF-DFA, multifractal detrended fluctuation analysis, builds a cumulative profile, splits it into segments, removes local polynomial trends, and computes q-th order fluctuation functions whose scaling exponents yield the singularity spectrum; its width, delta alpha, is the fractal complexity measure. The two methods are corroborated by shuffling tests, which remove temporal structure and show that Bitcoin's entropy and spectrum change far more than the other assets' do.

What would settle it

Run the same RCMSE and MF-DFA computations on the four assets over the identical calendar window from 2016 to 2024. If Bitcoin's complexity score and singularity-spectrum width drop into the range of the other assets, the claimed ranking is an artifact of the mismatched histories rather than a stable property of Bitcoin log returns.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that Bitcoin's log-return series combines short-scale regularity with long-scale irregularity to a degree the other assets do not show. At scale one its sample entropy is the lowest of the four, suggesting stronger short-horizon regularity, yet summed over 100 scales its RCMSE complexity reaches 74.66, well above 67.88 for gold, 67.24 for GBP/USD, and 51.48 for natural gas. The multifractal singularity spectrum is also widest for Bitcoin, with width 0.62 versus 0.50, 0.44, and 0.21, indicating a richer range of fluctuation strengths across time scales. Hurst exponents sit near 0.5 for all assets, so the authors interpret the entropy increase after shuffling and the spectrum narrowing after shuffling as evidence that Bitcoin's complexity comes from nonlinear autocorrelations rather than from linear memory.

Load-bearing premise

The load-bearing assumption is that the four series are directly comparable even though Bitcoin's data begin in 2016 while the others begin in 2010 and all are truncated to 3,730 points; if Bitcoin's higher complexity is a property of that later period rather than of Bitcoin itself, the ranking collapses.

Editorial extensions

If this is right

If Bitcoin's complexity ranking is right, short-horizon forecasts may be comparatively reliable for Bitcoin while long-horizon behavior is comparatively hard to predict.
Risk models for Bitcoin that assume simple random-walk or linear-autocorrelation dynamics would miss the nonlinear dependencies the shuffling tests expose.
The near-0.5 Hurst exponents across all assets mean standard detrended fluctuation analysis alone would not separate Bitcoin from the others; the multifractal and multiscale-entropy layers are what carry the ranking.
For GBP/USD, gold, and natural gas, the small changes after shuffling suggest their log-return series may be adequately described by random or linear processes, so complexity-based risk adjustments may matter less there.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: because Bitcoin's window starts in 2016 while the other assets start in 2010, the ranking should be re-tested on a common 2016-2024 window; the observed edge may reflect the crypto bull and bear regime rather than a stable asset-level property.
Editorial inference: the shuffling comparison could be turned into a formal significance test by generating many surrogate series and counting how often shuffled complexity exceeds the observed value, giving error bars for the Bitcoin-versus-others gap.
Editorial inference: the parameter choices of embedding dimension 3 and tolerance 0.15 times the standard deviation may affect the RCMSE ranking; a sensitivity sweep over embedding dimensions and tolerances would tell whether Bitcoin's lead is robust.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a textbook application of two standard complexity tools (RCMSE, MF-DFA) to four log-return series, and the headline result—Bitcoin is the most complex—is plausible but not actually established by the evidence as reported. The paper's real value is as a clean comparative snapshot; the main weakness is that the RCMSE complexity score is a sum over 100 scales, and the ranking is decided at scales where the estimator is operating outside its validity range.

The authors do some things well. They describe the algorithms accurately, use standard parameter choices (m=3, r=0.15*SD), and are transparent that 'complexity' is defined operationally as the sum of entropy over scales. The shuffled-series checks are a reasonable first pass, and the MF-DFA result—Bitcoin's wider singularity spectrum (Δα = 0.62 vs 0.21–0.50)—is consistent with existing literature on multifractality in crypto assets. So the directional claim is not a crank result.

The soft spots are significant, but proportionate. Tables 2 and 3 have no error bars, bootstrap intervals, or surrogate significance tests; the RCMSE margin (74.66 vs 67.88) is small and could easily be within finite-sample variability. More importantly, the RCMSE complexity integrates up to scale 100, where coarse-grained series are only 37 points long, far below the paper's own 10^m guideline (1,000 points for m=3). Bitcoin's lead appears only at those unreliable large scales; at small scales it actually has the lowest entropy. That makes the complexity ranking an artifact of the aggregation range. The window mismatch (Bitcoin starts 2016, others 2010) is a secondary confound, and the absence of code/data prevents independent checking. The MF-DFA half is more robust, but it too lacks uncertainty quantification, and the shuffle test reduces spectral width for all assets, not just Bitcoin, so the specificity of the nonlinear-correlation claim is weak.

This paper is for readers who want a quick, standard-method comparison of these four assets, especially the multifractal spectrum widths. It is not a methodological or conceptual advance. My recommendation: it is a marginal empirical note, but the flaws are fixable. A serious referee could push for matched sample periods, bootstrap or surrogate confidence intervals, and a scale cutoff consistent with the paper's own length guideline. As it stands, the headline ranking is not robust enough to stand on its own.

Referee Report

5 major / 6 minor

Summary. The manuscript applies Refined Composite Multiscale Sample Entropy (RCMSE) and Multifractal Detrended Fluctuation Analysis (MF-DFA) to daily log-return series of Bitcoin, GBP/USD, gold, and natural gas, each containing 3,730 points, and reports two complexity rankings. The RCMSE-based complexity, defined as the sum of entropy values over 100 coarse-graining scales, places Bitcoin highest (74.66) followed by gold (67.88), GBP/USD (67.24), and natural gas (51.48). The MF-DFA singularity-spectrum width also ranks Bitcoin highest (0.62) versus 0.50, 0.44, and 0.21 for the others. The authors interpret these results as evidence that Bitcoin displays the greatest complexity and that this complexity reflects nonlinear correlations in its log returns.

Significance. If the ranking were established with proper uncertainty quantification and comparable sampling windows, the finding would be a useful comparative stylized fact for cryptocurrency versus traditional financial markets, and the combination of RCMSE and MF-DFA is a reasonable and standard toolkit for the question. The paper clearly explains the algorithms and presents the results in an accessible way. However, the headline comparisons currently rest on entropy estimates at scales where the coarse-grained series are far shorter than the method's own length guideline, on unmatched historical windows, and on point estimates without confidence intervals. The manuscript also provides no code or data, so the central quantitative claims are not independently verifiable as presented.

major comments (5)

[Section 4.2, Eq. (6), Fig. 6, Table 2] The complexity score is the sum of RCMSE values over scales τ = 1 to 100. Section 2.1 states that meaningful sample entropy requires series lengths between 10^m and 30^m; for m = 3 this means 1,000 to 27,000 points. With N = 3,730, only τ ≤ 3 satisfies the lower bound. Scales 4 through 100 use coarse-grained series of length 932 down to 37 points, for which the estimates are subject to strong finite-sample bias and variance. RCMSE reduces the incidence of undefined entropies, but it does not eliminate this problem. The paper's own Fig. 6 shows that Bitcoin has the lowest entropy at small scales and the highest only at large scales, so its entire lead is generated by scales where the estimator is least reliable. The authors should report complexity computed over scales satisfying the stated guideline, or use an embedding dimension for which the guideline is met over a defensible scale range, and should accompany any summed score with bootstrap confidence intervals.
[Section 3, Fig. 3] The four series are not directly comparable as collected. Bitcoin begins in 2016, while GBP/USD, gold, and natural gas begin in 2010, and all series are truncated to 3,730 points. The Bitcoin sample therefore covers a different market regime, including the 2017 and 2021 cryptocurrency boom-bust episodes, whereas the other assets' samples include the 2010s sovereign-debt and commodity cycles. Any of the reported differences could be an artifact of this window mismatch. The authors should show that the ranking is robust to using a common sample period, for example 2016 onward for all assets, or at least provide a matched-window sensitivity analysis.
[Tables 2 and 3] No uncertainty quantification is given for the central point estimates. Bitcoin's RCMSE complexity (74.66) exceeds gold's (67.88) by about 10%, and its singularity-spectrum width (0.62) exceeds GBP/USD's (0.50) by about 24%, but without bootstrap intervals, surrogate-data tests, or parameter sensitivity analysis these differences cannot be distinguished from estimation error. This is especially important because Table 4 reports Hurst exponents with errors while Tables 2 and 3 do not. The authors should provide confidence intervals for the summed RCMSE values and the spectrum widths, and ideally a statistical test of the ranking.
[Section 4.2, Fig. 7 and Section 4.3, Fig. 9] The shuffled-data analysis does not support the claim that nonlinear correlations are distinctive to Bitcoin. Figure 9 shows that the singularity-spectrum width decreases after shuffling for all four assets, not only for Bitcoin, and the text acknowledges a 'notable reduction' across all assets. The claim that Bitcoin alone shows a significant entropy increase after shuffling is based on a bar chart with no error bars or statistical test, despite the statement that each series was shuffled 100 times. The authors should report the mean and standard deviation of the shuffled entropies and spectrum widths per asset, and test whether Bitcoin's shuffle-induced change is significantly larger than those of the other assets.
[Section 4.3, Eqs. (7)-(14)] The MF-DFA results are not reproducible because the parameter values are not reported. The manuscript does not state the range of q values used, the order of the detrending polynomial, or the range of segment lengths s over which the fluctuation functions were fitted. These choices materially affect the generalized Hurst exponents and the singularity-spectrum width, so without them Table 3 and Fig. 8 cannot be checked. The authors should list the exact parameters and ideally provide the code or a pseudocode description.

minor comments (6)

[Table 2 and Table 3 captions] The asterisks in the table captions are unexplained; either define them or remove them.
[Section 4.2, Table 2] The table header promises 'complexity values across 100 time scales, as well as the entropy at the first scale,' but the first-scale entropy is not shown in the table; either add these values or revise the heading.
[Section 4.2] The 'complexity' used in Table 2 is defined only in prose as a sum over scales; the authors should define it explicitly, for example as C = Σ_{τ=1}^{100} RCMSE(x, τ, m, r), to avoid ambiguity about whether scale 1 is included.
[Table 4 and Fig. 11] The Hurst exponent errors are given inconsistently: the text reports H = 0.51 ± 0.01 for Bitcoin, while the table reports 0.51 ± 0.1, and similar discrepancies appear for the other assets. These should be reconciled.
[Fig. 7] The figure shows entropy before and after shuffling but provides no error bars, even though the text states that shuffling was repeated 100 times; plotting means with standard deviations would make the claimed difference visible.
[Section 4.4] The statement that Hurst exponents near 0.5 indicate 'a lack of linear autocorrelation' is too strong; a Hurst exponent near 0.5 indicates an absence of long-range dependence, and short-range linear autocorrelation can still be present. The wording should be softened.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the complexity ranking follows from direct RCMSE and MF-DFA measurements; the only self-citation is minor and non-load-bearing.

full rationale

The paper's central claim is an empirical ranking, not a derivation whose conclusion is assumed in its inputs. The RCMSE complexity values in Table 2 are computed directly from Eq. (6) and summed over the 100 scales, while the multifractal spectrum widths in Table 3 follow from the standard MF-DFA procedure in Eqs. (7)-(14). Neither quantity is defined in terms of the claimed outcome, and no parameter is fitted to the reported ranking. The shuffled-series comparisons in Figs. 7, 9, and 10 provide independent empirical evidence about nonlinear correlations and do not presuppose the conclusion. The only self-referential element is reference [27], a prior paper by two of the present authors, cited for the standard Legendre-transform multifractal formalism; that formalism is a well-known general result, not a load-bearing uniqueness claim or an ansatz smuggled in through self-citation. A finite-sample validity concern exists — at scale factor 100 the coarse-grained series have only about 37 points, well below the 10^m to 30^m guideline quoted in Section 2.1 for m = 3 — but this is a methodological robustness issue, not circularity. The minor self-citation alone warrants a score of 2; the derivation chain itself is self-contained.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim does not depend on fitted parameters in the usual sense, but the ranking is sensitive to hand-chosen analysis constants such as m, r, the number of scales, and the unstated q-range. No new entities are introduced. The analysis relies on standard domain assumptions about what these entropy and multifractal measures actually capture.

free parameters (5)

embedding dimension m = 3
Chosen by convention for RCMSE; controls match counts and entropy values.
tolerance factor r = 0.15 times standard deviation
Standard choice for sample entropy; directly scales the distance threshold that defines matching vectors.
number of scales for RCMSE = 100
Arbitrary cutoff; complexity is defined as the sum of entropy values over these scales, and the ranking can depend on the chosen range.
q range for MF-DFA = not stated
The generalized moments used for the multifractal spectrum are not specified, but they determine the computed spectrum width.
detrending polynomial order = not stated
MF-DFA order (DFA1, DFA2, etc.) is not specified, yet it affects the fluctuation functions and resulting exponents.

assumptions (5)

domain assumption RCMSE and MF-DFA are valid measures of time-series complexity.
The entire analysis treats the outputs of these two algorithms as proxies for complexity and predictability without independent validation.
ad hoc to paper Summing RCMSE values over scales gives a meaningful complexity index.
The paper defines complexity as the sum of entropy values over 100 scales (Section 4.2, Table 2) without justifying this aggregate over other possible summaries.
domain assumption Shuffling removes temporal dependencies such that higher entropy in shuffled data implies nonlinear correlations in the original data.
The shuffling test is used as evidence for nonlinear correlations but no formal surrogate-data framework or statistical criterion is provided.
domain assumption Log-return transformation adequately represents market behavior and risk.
All complexity measures are applied to log-returns of closing prices, assuming this transformation captures the relevant dynamics.
standard math The multifractal formalism and Legendre transform relationships hold for the analyzed series.
Equations (12) to (14) are standard MF-DFA results, invoked from the literature without re-derivation.

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Cite this review

Pith. "Pith review of Complexity of Financial Time Series: Multifractal and Multiscale Entropy Analyses." pith.science (2026). https://pith.science/paper/YHWEXJB4

@misc{pith2026250723414,
  author       = {Pith},
  title        = {Pith review of: Complexity of Financial Time Series: Multifractal and Multiscale Entropy Analyses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YHWEXJB4}},
  note         = {Machine review of arXiv:2507.23414}
}

read the original abstract

We employed Multifractal Detrended Fluctuation Analysis (MF-DFA) and Refined Composite Multiscale Sample Entropy (RCMSE) to investigate the complexity of Bitcoin, GBP/USD, gold, and natural gas price log-return time series. This study provides a comparative analysis of these markets and offers insights into their predictability and associated risks. Each tool presents a unique method to quantify time series complexity. The RCMSE and MF-DFA methods demonstrate a higher complexity for the Bitcoin time series than others. It is discussed that the increased complexity of Bitcoin may be attributable to the presence of higher nonlinear correlations within its log-return time series.

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Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Chaos, Solitons & F ractals 139, 109962 (2020)

Zhao, X., Ji, M., Zhang, N., Shang, P.: Permutation transition entr opy: Measuring the dynamical complexity of ﬁnancial time series. Chaos, Solitons & F ractals 139, 109962 (2020)

work page 2020

[2] [2]

Springer, New Yor k (2011)

Gray, R.M.: Entropy and Information Theory. Springer, New Yor k (2011)

work page 2011

[3] [3]

Bell Sys t

Shannon, C.E.: A mathematical theory of communication. Bell Sys t. Tech. J. 27(3), 379–423 (1948)

work page 1948

[4] [4]

Richman, J.S., Moorman, J.R.: Physiological time-series analysis usin g approxi- mate entropy and sample entropy. Am. J. Physiol. Heart Circ. Phys iol. 278(6), 2039–2049 (2000)

work page 2000

[5] [5]

Bandt, C., Pompe, B.: Permutation entropy: A natural complexit y measure for time series. Phys. Rev. Lett. 88(17), 174102 (2002)

work page 2002

[6] [6]

Chaos Solitons Fractals 184, 115005 (2024)

Mastroeni, L., Mazzoccoli, A., Vellucci, P.: Wavelet entropy and com plexity– entropy curves approach for energy commodity price predictabilit y amid the transition to alternative energy sources. Chaos Solitons Fractals 184, 115005 (2024)

work page 2024

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Costa, M., Goldberger, A.L., Peng, C.-K.: Multiscale entropy analys is of complex physiologic time series. Phys. Rev. Lett. 89(6), 068102 (2002)

work page 2002

[8] [8]

Wang, J., Shang, P., Zhao, X., Xia, J.: Multiscale entropy analysis of traﬃc time series. Int. J. Mod. Phys. C 24(02), 1350006 (2013)

work page 2013