Pith. sign in

REVIEW 3 major objections 4 minor 61 references

Disentangling sources of multifractality in time series

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Temporal correlations are the sole source of genuine multifractality; heavy tails only broaden the spectrum once correlations exist.

desk verdict A practically useful, mostly sound empirical study that gives MFDFA users a Gaussian-reference procedure for separating tail and correlation contributions, though the 'only scientifically valid' claim outruns the evidence. read the letter →

arxiv 2501.08898 v1 pith:FDPJBQLT submitted 2025-01-15 physics.data-an

Robert Kluszczyński , Stanis{l}aw Drożdż , Jaros{l}aw Kwapień , Tomasz Stanisz , Marcin Wątorek This is my paper · ORCID
classification physics.data-an
keywords multifractalitytimeseriesanalysistemporalcorrelationsheavytailsq-GaussiandistributionsMFDFAmultiplicativecascadessingularityspectrum
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 argues that the common practice of treating temporal correlations and heavy-tailed fluctuation distributions as two independent sources of multifractality is unfounded. Its central claim is that genuine multifractality arises only from temporal correlations, and that heavy tails can widen a multifractal spectrum only when such correlations are already present. To demonstrate this, the authors take four model cascades with built-in correlations, replace their fluctuation distributions with q-Gaussian distributions of varying tail thickness, and measure the singularity-spectrum width using multifractal detrended fluctuation analysis (MFDFA). The result is a quantitative prescription: compare a given series with a series that has the same temporal correlations but Gaussian fluctuations, and interpret the surplus in spectral width as the contribution of the heavier tails.

What carries the argument

The central mechanism is the rank-ordering probability-density transformation of Eq. (32), which takes a source time series, ranks its values, and replaces them with the corresponding ranks of a q-Gaussian sample, thereby keeping the temporal organization of the source while setting the fluctuation distribution to a q-Gaussian with tail thickness controlled by q. The q-Gaussian family serves as a dial for tail thickness: q = 1 gives the Gaussian, q > 1 gives increasingly heavy power-law tails, and q < 1 gives compact-support distributions thinner than Gaussian. MFDFA then converts each reshaped series into a singularity spectrum f(α), and the spectrum width Δα is compared across q. The Gaussian-reshaper at q = 1 provides the baseline against which tail-induced broadening is measured.

What would settle it

Generate a very long series of independent q-Gaussian increments with q = 2, so the increments have no temporal correlations, and compute its MFDFA singularity spectrum; if, as the series length grows, the spectrum retains a smooth concave shape whose width keeps growing with q instead of converging to the two-point bifractal structure, the paper's claim that tails alone cannot produce multifractality would be refuted.

Watch

Extended reading notes

Core claim

The central claim is that in the absence of temporal correlations only two outcomes are possible: monofractality when the fluctuation distribution lies in the Gaussian basin of attraction, and a two-point bifractal structure, broadened by finite-sample artifacts, when the distribution lies in the Lévy-Gnedenko basin. Genuine multifractality, meaning a concave singularity spectrum that persists in the ideal limit, requires temporal correlations, both linear and nonlinear. Using deterministic binomial and stochastic log-normal, log-gamma, and log-Poisson cascades, whose correlations are built in by construction, the authors show that even a Gaussian fluctuation distribution with cascade correlations produces a broad spectrum, and that replacing the PDF with heavier-tailed q-Gaussians while preserving the rank-ordered temporal organization broadens the spectrum further. They conclude that the Gaussian distribution, q = 1, is the correct reference point, and propose estimating the tail contribution as the surplus of the observed spectrum width over the width of the Gaussian-reshaper benchmark.

Load-bearing premise

The load-bearing premise is that replacing a cascade's fluctuation distribution by rank-ordering preserves enough of the original temporal correlations, across all moment orders q, that any change in spectrum width can be attributed to tail thickness rather than to q-dependent distortion of those correlations.

Editorial extensions

If this is right

  • If the paper is right, a detected multifractal spectrum in an empirical series is itself evidence of temporal correlations, even when the fluctuation distribution looks Gaussian.
  • The contribution of heavy tails can be quantified as the surplus of the observed spectrum width over the width of a series with the same temporal correlations but Gaussian fluctuations.
  • Studies that infer the source of multifractality by shuffling the original series are unreliable, because shuffled heavy-tailed series show spectral width from finite-size and bifractal artifacts rather than genuine multifractality.
  • Compact-support and bounded fluctuation distributions produce the same spectrum width as the Gaussian when correlations are present, so the attraction basin matters more than the exact distribution shape.
  • The disentangling procedure is expected to transfer to wavelet-based multifractal formalisms, since wavelet methods handle cascades effectively.

Reading between the lines

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

  • A stricter test of the paper's attribution would use a transformation that provably preserves all correlations, linear and nonlinear, while changing only the marginal distribution; the rank-ordering transform distorts correlations, and the paper does not quantify how this distortion varies with q.
  • For empirical data with measured long-range correlations, the proposed Gaussian-reshaper surplus can serve as a practical index separating correlation-induced from tail-induced multifractality, which the authors indicate they will apply to empirical series in future work.
  • For heavy-tailed uncorrelated series, the paper implies that existing MFDFA-based reports of multifractality should be re-examined as finite-size artifacts that dissolve into bifractality as series length grows.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the sources of multifractality in time series, arguing that temporal correlations are necessary for genuine multifractality and that heavy-tailed fluctuation distributions only broaden the multifractal spectrum when correlations are present. The authors use multiplicative cascades (deterministic binomial and stochastic log-normal, log-gamma, and log-Poisson) with known analytic singularity spectra, and reshape the cascade marginals to q-Gaussian distributions via a rank-ordering transformation that preserves the temporal ordering of the original series. They then apply MFDFA to quantify the multifractal spectrum width Δα as a function of the Tsallis parameter q. They find that in the absence of correlations, uncorrelated q-Gaussian noise is monofractal (or bifractal in the Lévy basin), while the correlated reshaped cascades are multifractal even for q=1, with Δα growing as q increases. The conclusion is that the Gaussian (q=1) reshaped series is the appropriate reference for estimating the tail contribution to multifractality, and that the same procedure can be used on empirical data.

Significance. If the central claim holds, the paper provides a practical recipe for disentangling correlation-induced and tail-induced contributions to multifractality, a question that is frequently asked in empirical multifractal analysis. The numerical work is careful: MFDFA results are validated against analytic cascade spectra, results are averaged over 10-100 realizations with error bars, and the cascade models are self-contained and not fitted to the conclusions. The paper also reports a new observation that compact-support PDFs with cascade-like correlations produce multifractal spectra similar to Gaussian ones. The main weakness is that the load-bearing assumption that the rank-ordering transformation preserves the correlation structure across q is asserted rather than demonstrated; the paper itself admits that reshaping a PDF distorts correlations, but does not quantify how this distortion varies with q. The 'only scientifically valid approach' phrasing in the Conclusions is an overstatement given the empirical nature of the q=1 reference. These issues are fixable with additional control analyses, so the central idea is worth pursuing.

major comments (3)
  1. [Section 5.2, Eq. (32), Figs. 12, 16, 20, 24] The rank-ordering transformation G~q = R^{-1}(R'(G_q)) preserves the empirical copula of the source series, but not its linear or nonlinear correlation functions, and the paper explicitly concedes in Section 5.2 that 'by reshaping a PDF, one distorts the correlations, but not remove them.' Because MFDFA's fluctuation functions for r>0 are dominated by large fluctuations and for r<0 by small ones, a q-dependent distortion of the autocorrelation or of higher-order correlations would shift h(r) asymmetrically, which is exactly the pattern reported in Figs. 9-24. The paper never quantifies this distortion or its dependence on q, so the surplus Δα(q)-Δα(1) cannot be unambiguously attributed to the tail shape alone. A concrete test is needed: for example, computing the autocorrelation function and the autocorrelation of squared values for the reshaped series across q, or comparing with an alternative reshuffling method that approximately preserves the full correlation structure (such as iterative amplitude-adjusted Fourier transform surrogates). Without such a control, the central quantitative claim that the Gaussian reference isolates the tail contribution remains unsupported.
  2. [Section 6, Conclusions] The claim that the only scientifically valid approach to estimate the tail contribution is to compare against a series with the same correlations but a Gaussian fluctuation distribution is too strong. The selection of q=1 as the reference is motivated by an observed plateau in Δα for q≲1.2 in Figs. 12, 16, 20, and 24, rather than by a theoretical derivation. The plateau is not exactly flat (e.g., a minor upward drift in Fig. 12), and the figures show that compact-support distributions (q<1) give nearly the same Δα, indicating that any distribution in the Gaussian basin of attraction could serve as a reference within numerical precision. The authors should either temper the 'only scientifically valid' statement or provide a theoretical argument why the Gaussian marginal, rather than, say, any thin-tailed distribution, is the unique baseline.
  3. [Section 5.2, Figs. 12, 16, 20, 24] For q>5/3 the q-Gaussian distribution has infinite variance, and Section 5.1 correctly warns that MFDFA results in this regime should be interpreted qualitatively because the variance calculation in Step 3 is not well defined in the population. However, the correlated-cascade results in Section 5.2 report quantitative Δα values up to q=2 without repeating or addressing this caveat. The paper should either restrict the quantitative tail-contribution claims to q≤5/3, or provide a sensitivity analysis showing that the reported Δα values for q>5/3 are robust to the choice of moment range r and series length, given that the infinite-variance regime is relevant to many empirical heavy-tailed series.
minor comments (4)
  1. [Section 5.2.3] There is a typo 'Fig, 18' where 'Fig. 18' is intended, and in Section 5.2 the sentence 'therefore, its is sufficient' should read 'therefore, it is sufficient.'
  2. [Section 5.2, Eq. (32)] The rank-ordering transformation is essentially a copula-based marginal transformation, but no reference to copula theory or to the existing surrogate-data literature is given; citing a standard reference would help readers understand the properties (and limitations) of the transformation.
  3. [Section 4, Eq. (24)] In the expression for C_q for q<1, there is a missing closing parenthesis in the printed formula; the intended expression is C_q = 2√π Γ(1/(1−q)) / ((3−q)√(1−q) Γ((3−q)/(2(1−q)))).
  4. [Section 5.2.1, Fig. 9] The text says 'the functions Fr(s) tend to spread out more and more' for increasing q, but it would be helpful to state explicitly over which range of r the spread is measured; the MFDFA r-range is given as -4 to 4, but the plotted h(r) in Fig. 9(b) extends to r=±20, and the relation between these ranges is not explained.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the q-Gaussian projection is an external transformation and the MFDFA outputs are simulation results, though the admitted correlation distortion weakens clean attribution.

full rationale

The paper's central claims are supported by newly generated simulations, not by a fitted parameter or by prior work alone. The rank-ordering transformation (Eq. 32) maps each cascade realization to a series with a prescribed q-Gaussian marginal while keeping the order statistics; the reported Delta-alpha values are then computed by the standard MFDFA routine (Eqs. 2-6) and checked against analytical cascade spectra (Eqs. 10, 14, 17, 20). No parameter is fitted to the quantity later presented as the main result: the identification of q=1 as the reference is an empirical observation that Delta-alpha is approximately constant for q below about 1.2, combined with the central limit theorem, and is not a consequence of the construction. The paper's own caveat in Section 5.2 that 'by reshaping a PDF, one distorts the correlations, but not remove them' is a genuine limitation for the clean attribution of the surplus, but it is a correctness/validity concern, not a circularity: the measured widening is an output of the algorithm, not an input redefined as output. The citations to the authors' prior work [26,28] provide background for the premise that genuine multifractality requires temporal correlations, but the present manuscript independently demonstrates that premise with new cascade simulations, so the self-citations are not load-bearing in the sense of forcing the conclusion. No circular step can be exhibited from the paper's equations.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central results are numerical; no new entities are postulated. The main extras beyond standard theory are the choice of the q-Gaussian family as a reshaping tool and the convention of using q=1 as reference, neither fitted to the target claim.

free parameters (2)
  • q (Tsallis parameter) = varies over (-infinity, 2]
    Independent control variable for tail thickness in the q-Gaussian family; not fitted to data.
  • beta (q-Gaussian width parameter) = 1/(3-q)
    Chosen to standardize the q-Gaussian family so the q-variance equals 1; a convention, not fitted.
assumptions (5)
  • domain assumption Multiplicative cascades are long-range autocorrelated by construction, with genuine multifractal properties.
    Stated in Section 3; the entire study uses cascades as ground truth for correlated multifractal time series.
  • domain assumption Rank-ordering density transformation preserves the temporal organization of the original series, though it distorts correlations.
    Section 5.2 states this; the method of comparing reshaped series across q relies on correlation structure being inherited rather than destroyed.
  • domain assumption MFDFA with detrending order m=2 and the chosen scale range reliably estimates singularity spectra for these series.
    Sections 2 and 5.1; the conclusions depend on the accuracy of MFDFA, validated here against analytic cascade spectra.
  • standard math Attraction basins determine limiting behavior: Gaussian basin gives monofractal, Levy-Gnedenko basin gives bifractal for uncorrelated series.
    Section 5.1 and refs [26,29]; used to interpret uncorrelated q-Gaussian noise results.
  • standard math The q-Gaussian family with beta=1/(3-q) is a standardized family with q-variance equal to 1.
    Section 4, Eq. (29); provides the distribution family used for reshaping.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Disentangling sources of multifractality in time series." pith.science (2026). https://pith.science/paper/FDPJBQLT

@misc{pith2026250108898,
  author       = {Pith},
  title        = {Pith review of: Disentangling sources of multifractality in time series},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FDPJBQLT}},
  note         = {Machine review of arXiv:2501.08898}
}
abstract

This contribution addresses the question commonly asked in scientific literature about the sources of multifractality in time series. Two primary sources are typically considered. These are temporal correlations and heavy tails in the distribution of fluctuations. Most often, they are treated as two independent components, while true multifractality cannot occur without temporal correlations. The distributions of fluctuations affect the span of the multifractal spectrum only when correlations are present. These issues are illustrated here using series generated by several model mathematical cascades, which by design build correlations into these series. The thickness of the tails of fluctuations in such series is then governed by an appropriate procedure of adjusting them to $q$-Gaussian distributions, and $q$ is treated as a variable parameter that, while preserving correlations, allows to tune these distributions to the desired functional form. Multifractal detrended fluctuation analysis (MFDFA), as the most commonly used practical method for quantifying multifractality, is then used to identify the influence of the thickness of the fluctuation tails in the presence of temporal correlations on the width of multifractal spectra. The obtained results point to the Gaussian distribution, so $q=1$, as the appropriate reference distribution to evaluate the contribution of fatter tails to the width of multifractal spectra. An appropriate procedure is presented to make such estimates.

Figures

Figures reproduced from arXiv: 2501.08898 by the authors.

Figure 1
Figure 1. (Top) Time series of a deterministic binomial multiplicative cascade with p = 0.3 and k = 17 iterations (N = 131, 072 data points). (Bottom left) Singularity spectrum f(α) for the cascade shown in the top panel calculated with MFDFA (blue symbols) and its theoretical form calculated from Eq. (10) (black solid line). (Middle right) The generalized Hurst exponents h(r) and (bottom right) the fluctuations functions Fr(… view at source ↗
Figure 2
Figure 2. (Top) Sample time series of a log-normal multiplicative cascade with k = 17 iterations (N = 131, 072 data points), µ = 1.1, and σ 2 = 1/5 ln 2. (Bottom left) Singularity spectrum for this cascade obtained by using MFDFA (blue symbols with error bars indicating standard deviation) and by using Eq. (14) (black solid line). (Middle right) the generalized Hurst exponents h(r) and (bottom right) the fluctuations function… view at source ↗
Figure 3
Figure 3. (Top) Sample time series of a log-gamma multiplicative cascade with k = 17 iterations (N = 131, 072 data points), and parameters γ = 2 and β = ln 2/( √ 2 − 1). (Bottom left) Singularity spectrum for this cascade obtained by using MFDFA (blue symbols with error bars indicating standard deviation) and by using Eq. (17) (black solid line). (Middle right) the generalized Hurst exponents h(r) and (bottom right) the fluct… view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: (Top) Sample time series of a log-Poisson multiplicative cascade with k = 17 iterations (N = 131, 072 data points), and parameter λ = 2 ln 2. (Bottom left) Singularity spectrum for this cascade obtained by using MFDFA (blue symbols with error bars indicating standard d…
Figure 5
Figure 5. Figure 5: Probability density functions of the q-Gaussian distributions pq(x) for selected values of the Tsallis parameter: q = −∞, −100, −10, 0.5, 1, 1.5, 2. For q = 1.5 and q = 2, one can observe increasingly fat tails. For q = 1, the normal distribution is restored. The finit…
Figure 6
Figure 6. Figure 6: Fluctuation functions Fr(s) with −4 ⩽ r ⩽ 4 obtained by using MFDFA for time series of uncorrelated q-Gaussian noise with sample values of q. All the time series have a length of N = 105 data points and the results have been averaged over 10 independent realizations of…
Figure 7
Figure 7. Figure 7: Singularity spectra f(α) for time series of uncorrelated q-Gaussian noise with sample values of q from the Lévy-Gnedenko basin of attraction: 5/3 < q ⩽ 2. The spectra have been calculated by using MFDFA with −4 ⩽ r ⩽ 4 and averaged over 10 independent realizations of a…
Figure 8
Figure 8. Figure 8: Width ∆α of the singularity spectra f(α) calculated for time series of uncorrelated q-Gaussian noise as a function of q in the range 1 ⩽ q ⩽ 2. Vertical line at q = 5/3 separates the Gaussian and Lévy-Gnedenko basins. In the latter basin, the orange dashed line denotes…
Figure 9
Figure 9. Figure 9: (a) Fluctuation functions Fr(s) and (b) the generalized Hurst exponents h(r) calculated by using MFDFA from q-Gaussian time series with 1 ⩽ q ⩽ 2 and their temporal organization inherited from a dyadic binomial cascade. Vertical lines denote the lower and upper boundar…
Figure 10
Figure 10. Figure 10: Singularity spectra f(α) for the same time series as in [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Singularity spectra f(α) for time series with q-Gaussian PDFs on a compact support, where −∞ < q < 1, and with temporal organization inherited from a dyadic binomial cascade. The results have been averaged over 10 independent realizations of the q-Gaussian time series…
Figure 12
Figure 12. Figure 12: Width ∆α of the singularity spectra f(α) calculated for q-Gaussian time series of length of N = 105 data points with 0 ⩽ q ⩽ 2 and their temporal organization inherited from a dyadic binomial cascade. Vertical line at q = 5/3 separates the Gaussian and Lévy-Gnedenko b…
Figure 13
Figure 13. Figure 13: (a) Fluctuation functions Fr(s) and (b) the generalized Hurst exponents h(r) calculated by using MFDFA from q-Gaussian time series with 1 ⩽ q ⩽ 2 and their temporal organization inherited from a dyadic log-normal cascade. Vertical lines denote the lower and upper boun…
Figure 14
Figure 14. Figure 14: Singularity spectra f(α) calculated from the same time series as the quantities shown in [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: Singularity spectra f(α) for time series with q-Gaussian PDFs on a compact support, where −∞ < q < 1, and with temporal organization inherited from a dyadic log-normal cascade. The results have been averaged over 10 independent realizations of the q-Gaussian time seri…
Figure 16
Figure 16. Figure 16: Width ∆α of the singularity spectra f(α) calculated for q-Gaussian time series of length of N = 105 data points with 0 ⩽ q ⩽ 2 and their temporal organization inherited from a dyadic log-normal cascade. Vertical line at q = 5/3 separates the Gaussian and Lévy-Gnedenko…
Figure 17
Figure 17. Figure 17: (a) Fluctuation functions Fr(s) and (b) the generalized Hurst exponents h(r) calculated by using MFDFA from q-Gaussian time series with 1 ⩽ q ⩽ 2 and their temporal organization inherited from a dyadic log-gamma cascade. Vertical lines denote the lower and upper bound…
Figure 18
Figure 18. Figure 18: Singularity spectra f(α) for the same time series as in [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]
Figure 19
Figure 19. Figure 19: Singularity spectra f(α) for time series with q-Gaussian PDFs on a compact support, where −∞ < q < 1, and with temporal organization inherited from a dyadic log-gamma cascade. The results have been averaged over 10 independent realizations of the q-Gaussian time serie…
Figure 20
Figure 20. Figure 20: Width ∆α of the singularity spectra f(α) calculated for q-Gaussian time series of length of N = 105 data points with 0 ⩽ q ⩽ 2 and their temporal organization inherited from a dyadic log-gamma cascade. Vertical line at q = 5/3 separates the Gaussian and Lévy-Gnedenko …
Figure 21
Figure 21. Figure 21: (a) Fluctuation functions Fr(s) and (b) the generalized Hurst exponents h(r) calculated by using MFDFA from q-Gaussian time series with 1 ⩽ q ⩽ 2 and their temporal organization inherited from a dyadic log-Poisson cascade. Vertical lines denote the lower and upper bou…
Figure 22
Figure 22. Figure 22: Singularity spectra f(α) for the same time series as in [PITH_FULL_IMAGE:figures/full_fig_p026_22.png]
Figure 23
Figure 23. Figure 23: Singularity spectra f(α) for time series with q-Gaussian PDFs on a compact support, where −∞ < q < 1, and with temporal organization inherited from a dyadic log-Poisson cascade. The results have been averaged over 10 independent realizations of the q-Gaussian time ser…
Figure 24
Figure 24. Figure 24: Width ∆α of the singularity spectra f(α) calculated for q-Gaussian time series of length of N = 105 data points with 0 ⩽ q ⩽ 2 and their temporal organization inherited from a dyadic log-Poisson cascade. Vertical line at q = 5/3 separates the Gaussian and Lévy-Gnedenk…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

61 extracted references · 49 canonical work pages

  1. [1]

    Multifractal phenomena in physics and chemistry

    Stanley, H.E.; Meakin, P . Multifractal phenomena in physics and chemistry. Nature 1988, 335, 405–409

    work page 1988

  2. [2]

    Multifractality of self-affine fractals

    Barabási, A.L.; Vicsek, T. Multifractality of self-affine fractals. Physical Review A 1991, 44

    work page 1991

  3. [3]

    Multifractal random walk

    Bacry, E.; Delour, J.; Muzy, J.F. Multifractal random walk. Physical Review E 2001, 64, 026103. https://doi.org/10.1103/PhysRevE.64.026103

    work page doi:10.1103/physreve.64.026103 2001

  4. [4]

    In Mathematics of Complexity and Dy- namical Systems; Meyers, R.A., Ed.; Springer New York: New York, NY, 2011; pp

    Kantelhardt, J.W., Fractal and Multifractal Time Series. In Mathematics of Complexity and Dy- namical Systems; Meyers, R.A., Ed.; Springer New York: New York, NY, 2011; pp. 463–487. https://doi.org/10.1007/978-1-4614-1806-1_30

    work page doi:10.1007/978-1-4614-1806-1_30 2011

  5. [5]

    Multifractal methodology

    Salat, H.; Murcio, R.; Arcaute, E. Multifractal methodology. Physica A 2017, 473, 467–487. https://doi.org/10.1016/j.physa.2017.01.041

    work page doi:10.1016/j.physa.2017.01.041 2017

  6. [6]

    Multifractal analysis of financial markets: A review

    Jiang, Z.Q.; Xie, W.J.; Zhou, W.X.; Sornette, D. Multifractal analysis of financial markets: A review. Reports on Progress in Physics 2019, 82, 125901. https://doi.org/10.1088/1361-6633/ab4 2fb

    work page doi:10.1088/1361-6633/ab4 2019

  7. [7]

    Multifractal nature of stock exchange prices

    Ausloos, M.; Ivanova, K. Multifractal nature of stock exchange prices. Computer Physics Com- munications 2002, 147, 582–585. Proceedings of the Europhysics Conference on Computational Physics Computational Modeling and Simulation of Complex Systems, https://doi.org/10.101 6/S0010-4655(02)00372-7

    work page 2002

  8. [8]

    Multifractal formalisms of human behavior

    Ihlen, E.A.; Vereijken, B. Multifractal formalisms of human behavior. Human Movement Science 2013, 32, 633–651. https://doi.org/10.1016/j.humov.2013.01.008

    work page doi:10.1016/j.humov.2013.01.008 2013

Show all 61 references
  1. [9]

    Quantifying origin and character of long-range correlations in narrative texts

    Dro˙zd ˙z, S.; O´ swi˛ ecimka, P .; Kulig, A.; Kwapie ´ n, J.; Bazarnik, K.; Grabska-Gradzi ´ nska, I.; Rybicki, J.; Stanuszek, M. Quantifying origin and character of long-range correlations in narrative texts. Information Sciences 2016, 331, 32–44. https://doi.org/10.1016/j.i...

    2016 doi

  2. [10]

    Statistical properties and multifractality of Bitcoin

    Takaishi, T. Statistical properties and multifractality of Bitcoin. Physica A 2018, 506, 507–519. https://doi.org/10.1016/j.physa.2018.04.046. Version January 16, 2025 submitted to Mathematics 31 of 33

    2018 doi

  3. [11]

    Multibranch multifractality and the phase transitions in time series of mean interevent times

    Klamut, J.; Kutner, R.; Gubiec, T.; Struzik, Z.R. Multibranch multifractality and the phase transitions in time series of mean interevent times. Physical Review E 2020, 101, 063303. https: //doi.org/10.1103/PhysRevE.101.063303

    2020 doi

  4. [12]

    Multifractal organization of EEG signals in multiple sclerosis

    W ˛ atorek, M.; Tomczyk, W.; Gawłowska, M.; Golonka-Afek, N.;˙Zyrkowska, A.; Marona, M.; Wnuk, M.; Słowik, A.; Ochab, J.K.; Fafrowicz, M.; et al. Multifractal organization of EEG signals in multiple sclerosis. Biomedical Signal Processing and Control 2024, 91, 105916. https: /...

    2024

  5. [13]

    Complex systems approach to natural language.Physics Reports 2024, 1053, 1–84

    Stanisz, T.; Dro˙zd˙z, S.; Kwapie ´ n, J. Complex systems approach to natural language.Physics Reports 2024, 1053, 1–84. https://doi.org/10.1016/j.physrep.2023.12.002

    2024 doi

  6. [14]

    Long-term persistence and multifractality of precipitation and river runoff records

    Kantelhardt, J.W.; Koscielny-Bunde, E.; Rybski, D.; Braun, P .; Bunde, A.; Havlin, S. Long-term persistence and multifractality of precipitation and river runoff records. Journal of Geophysical Research: Atmospheres 2006, 111. https://doi.org/10.1029/2005jd005881

    2006 doi

  7. [15]

    Direct determination of the f(α) singularity spectrum and its application to fuily developed turbulence

    Chhabra, A.B.; Meneveau, C.; Jensen, R.V .; Sreenivasan, K.R. Direct determination of the f(α) singularity spectrum and its application to fuily developed turbulence. Physical Review A 1989, 40, 5284–5294

    1989

  8. [16]

    The thermodynamics of fractals revisited with wavelets

    Arneodo, A.; Bacry, E.; Muzy, J.F. The thermodynamics of fractals revisited with wavelets. Physica A 1995, 213, 232–275

    1995

  9. [17]

    Multifractal detrended fluctuation analysis of nonstationary time series

    Kantelhardt, J.W.; Zschiegner, S.A.; Koscielny-Bunde, E.; Havlin, S.; Bunde, A.; Stanley, H.E. Multifractal detrended fluctuation analysis of nonstationary time series. Physica A 2002, 316, 87–

    2002

  10. [18]

    Fractal measures and their singularities: The characterization of strange sets

    Halsey, T.C.; Jensen, M.H.; Kadanoff, L.P .; Procaccia, I.; Shraiman, B.I. Fractal measures and their singularities: The characterization of strange sets. Physical Review A 1986, 33, 1141–1151

    1986

  11. [19]

    Multifractality in human heartbeat dynamics

    Ivanov, P .C.; Amaral, L.A.N.; Goldberger, A.L.; Havlin, S.; Rosenblum, M.G.; Struzik, Z.R.; Stanley, H.E. Multifractality in human heartbeat dynamics. Nature 1999, 399, 461–465

    1999

  12. [20]

    Multifractal properties of price fluctuations of stocks and commodities

    Matia, K.; Ashkenazy, Y.; Stanley, H.E. Multifractal properties of price fluctuations of stocks and commodities. Europhysics Letters 2003, 61, 422. https://doi.org/10.1209/epl/i2003-00194-y

    2003 doi

  13. [21]

    Components of multifractality in high-frequency stock returns

    Kwapie ´ n, J.; O´ swi˛ ecimka, P .; Dro˙zd ˙z, S. Components of multifractality in high-frequency stock returns. Physica A 2005, 350, 466–474. https://doi.org/10.1016/j.physa.2004.11.019

    2005 doi

  14. [22]

    Understanding the source of multifractality in financial markets

    Barunik, J.; Aste, T.; Matteo, T.D.; Liu, R. Understanding the source of multifractality in financial markets. Physica A 2012, 391, 4234–4251. https://doi.org/10.1016/j.physa.2012.03.037

    2012 doi

  15. [23]

    Origin of multifractality in solar wind turbulence: the role of current sheets

    Gomes, L.F.; Gomes, T.F.; Rempel, E.L.; Gama, S. Origin of multifractality in solar wind turbulence: the role of current sheets. Monthly Notices of the Royal Astronomical Society 2023, 519, 3623–3634. https://doi.org/10.1093/mnras/stac3577

    2023 doi

  16. [24]

    Improved Surrogate Data for Nonlinearity Tests

    Schreiber, T.; Schmitz, A. Improved Surrogate Data for Nonlinearity Tests. Physical Review Letters 1996, 77, 635–638

    1996

  17. [25]

    Constrained-realization Monte-Carlo method for hypothesis testing

    Theiler, J.; Prichard, D. Constrained-realization Monte-Carlo method for hypothesis testing. Physica D 1996, 94, 221–235

    1996

  18. [26]

    Quantitative features of multifractal subtleties in time series

    Dro˙zd ˙z, S.; Kwapie ´ n, J.; O´ swi˛ ecimka, P .; Rak, R. Quantitative features of multifractal subtleties in time series. EPL 2009, 88, 60003. https://doi.org/10.1209/0295-5075/88/60003

    2009 doi

  19. [27]

    Finite-size effect and the components of multifractality in financial volatility

    Zhou, W.X. Finite-size effect and the components of multifractality in financial volatility. Chaos, Solitons & Fractals 2012, 45, 147–155

    2012

  20. [28]

    Genuine multifractality in time series is due to temporal correlations

    Kwapie ´ n, J.; Blasiak, P .; Dro˙zd˙z, S.; O´ swi˛ ecimka, P . Genuine multifractality in time series is due to temporal correlations. Physical Review E 2023, 107, 034139. https://doi.org/10.1103/ PhysRevE.107.034139

    2023

  21. [29]

    Multi-scaling properties of truncated Lévy flights

    Nakao, H. Multi-scaling properties of truncated Lévy flights. Physics Letters A 2000, 266, 282–289. https://doi.org/10.1016/s0375-9601(00)00059-1

    2000 doi

  22. [30]

    Quantitative approach to multifractality induced by correlations and broad distribution of data

    Rak, R.; Grech, D. Quantitative approach to multifractality induced by correlations and broad distribution of data. Physica A 2018, 508, 48–66. https://doi.org/10.1016/j.physa.2018.05.059

    2018 doi

  23. [31]

    In Benoit Mandelbrot; 2015; chapter Chapter 5, pp

    Barral, J.; Peyriére, J., Mandelbrot’s cascades: a legendary destiny. In Benoit Mandelbrot; 2015; chapter Chapter 5, pp. 143–172. https://doi.org/10.1142/9789814366076_0005

    2015 doi

  24. [33]

    Effect of detrending on multifractal char- acteristics

    O´ swi˛ ecimka, P .; Dro˙zd ˙z, S.; Kwapie ´ n, J.; Górski, A.Z. Effect of detrending on multifractal char- acteristics. Acta Physica Polonica A 2013, 123, 597–603. https://doi.org/10.12693/APhysPolA.12 3.597

    2013 doi

  25. [34]

    Long-term storage capacity of reservoirs

    Hurst, H. Long-term storage capacity of reservoirs. T rans. Am. Soc. Civ. Eng. 1951, 116

    1951

  26. [35]

    Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes

    Heneghan, C.; McDarby, G. Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes. Physical Review E 2000, 62, 6103–6110. Version January 16, 2025 submitted to Mathematics 32 of 33

  27. [36]

    Asymmetrical singularities in real-world signals

    Ohashi, K.; Amaral, L.A.; Natelson, B.H.; Yamamoto, Y. Asymmetrical singularities in real-world signals. Physical Review E 2003, 68. https://doi.org/10.1103/PhysRevE.68.065204

    2003 doi

  28. [37]

    Asymmetric multifractal scaling behavior in the Chinese stock market: Based on asymmetric MF-DFA

    Cao, G.; Cao, J.; Xu, L. Asymmetric multifractal scaling behavior in the Chinese stock market: Based on asymmetric MF-DFA. Physica A 2013, 392, 797–807. https://doi.org/10.1016/j.physa. 2012.10.042

    2013 doi

  29. [38]

    Detecting and interpreting distortions in hierarchical organization of complex time series

    Dro˙zd ˙z, S.; O´ swi˛ ecimka, P . Detecting and interpreting distortions in hierarchical organization of complex time series. Physical Review E 2015, 91, 030902(R). https://doi.org/10.1103/PhysRevE. 91.030902

    2015 doi

  30. [39]

    Multifractal detrended fluctuation analysis of temperature in Spain (1960–2019)

    Gómez-Gómez, J.; Carmona-Cabezas, R.; Ariza-Villaverde, A.B.; Gutiérrez de Ravé, E.; Jiménez- Hornero, F.J. Multifractal detrended fluctuation analysis of temperature in Spain (1960–2019). Physica A 2021, 578, 126118. https://doi.org/10.1016/j.physa.2021.126118

    1960

  31. [40]

    Fractals, 4

    Feder, J. Fractals, 4. printing ed.; Physics of solids and liquids, Plenum Press: New York [u.a.],

  32. [41]

    Multifractal measures, especially for the geophysicist

    Mandelbrot, B.B. Multifractal measures, especially for the geophysicist. pure and applied geophysics 1989, 131, 5–42. https://doi.org/10.1007/bf00874478

    1989 doi

  33. [42]

    A Multifractal Model of Asset Returns

    Mandelbrot, B.; Fisher, A.; Calvet, L. A Multifractal Model of Asset Returns. Cowles Foundation Discussion Paper No. 1164. 1997

    1997

  34. [43]

    Multifractality in Asset Returns: Theory and Evidence

    Calvet, L.; Fisher, A. Multifractality in Asset Returns: Theory and Evidence. Review of Economics and Statistics 2002, 84, 381–406. https://doi.org/10.1162/003465302320259420

    2002 doi

  35. [44]

    Large Deviations and the Distribution of Price Changes

    Calvet, L.; Fisher, A.; Mandelbrot, B. Large Deviations and the Distribution of Price Changes. Cowles Foundation Discussion Paper 1997

    1997

  36. [45]

    Nonadditive entropy and nonextensive statistical mechanics -an overview after 20 years

    Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics -an overview after 20 years. Brazilian Journal of Physics 2009, 39, 337–356. https://doi.org/10.1590/s0103-9733200900 0400002

    2009 doi

  37. [46]

    Anomalous diffusion and Tsallis statistics in an optical lattice

    Lutz, E. Anomalous diffusion and Tsallis statistics in an optical lattice. Physical Review A 2003, 67, 051402. https://doi.org/10.1103/physreva.67.051402

    2003 doi

  38. [47]

    Introduction to Nonextensive Statistical Mechanics ; SpringerLink, Springer New York: New York, NY, 2009

    Tsallis, C., Ed. Introduction to Nonextensive Statistical Mechanics ; SpringerLink, Springer New York: New York, NY, 2009. Description based upon print version of record

    2009

  39. [48]

    Triangle for the entropic index q of non-extensive statistical mechanics observed by Voyager 1 in the distant heliosphere

    Burlaga, L.; Viñas, A. Triangle for the entropic index q of non-extensive statistical mechanics observed by Voyager 1 in the distant heliosphere. Physica A: Statistical Mechanics and its Applications 2005, 356, 375–384. https://doi.org/10.1016/j.physa.2005.06.065

    2005 doi

  40. [49]

    Tsallis Statistics of the Magnetic Field in the Heliosheath

    Burlaga, L.F.; Viñas, A.F.; Ness, N.F.; Acuña, M.H. Tsallis Statistics of the Magnetic Field in the Heliosheath. The Astrophysical Journal 2006, 644, L83–L86. https://doi.org/10.1086/505577

    2006 doi

  41. [50]

    Magnetic Fields in the Heliosheath and Distant He- liosphere:Voyager 1and2Observations During 2005 and 2006

    Burlaga, L.F.; Ness, N.F.; Acuna, M.H. Magnetic Fields in the Heliosheath and Distant He- liosphere:Voyager 1and2Observations During 2005 and 2006. The Astrophysical Journal 2007, 668, 1246–1258. https://doi.org/10.1086/521349

    2005 doi

  42. [51]

    Analysis of self-organized criticality in the Olami-Feder-Christensen model and in real earthquakes

    Caruso, F.; Pluchino, A.; Latora, V .; Vinciguerra, S.; Rapisarda, A. Analysis of self-organized criticality in the Olami-Feder-Christensen model and in real earthquakes. Physical Review E 2007, 75, 055101. https://doi.org/10.1103/physreve.75.055101

    2007 doi

  43. [52]

    Option Pricing Formulas Based on a Non-Gaussian Stock Price Model

    Borland, L. Option Pricing Formulas Based on a Non-Gaussian Stock Price Model. Physical Review Letters 2002, 89, 098701. https://doi.org/10.1103/physrevlett.89.098701

    2002 doi

  44. [53]

    Nonextensive statistical features of the Polish stock market fluctuations

    Rak, R.; Dro ˙zd˙z, S.; Kwapie ´ n, J. Nonextensive statistical features of the Polish stock market fluctuations. Physica A: Statistical Mechanics and its Applications 2007, 374, 315–324. https: //doi.org/10.1016/j.physa.2006.07.035

    2007 doi

  45. [54]

    The foreign exchange market: Return distribu- tions, multifractality, anomalous multifractality and the Epps effect

    Dro˙zd ˙z, S.; Kwapie ´ n, J.; O´ swi˛ ecimka, P .; Rak, R. The foreign exchange market: Return distribu- tions, multifractality, anomalous multifractality and the Epps effect. New Journal of Physics 2010, 12, 105003. https://doi.org/10.1088/1367-2630/12/10/105003

    2010 doi

  46. [55]

    Financial Return Distributions: Past, Present, and COVID-

    W ˛ atorek, M.; Kwapie ´ n, J.; Dro˙zd ˙z, S. Financial Return Distributions: Past, Present, and COVID-

  47. [56]

    On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics

    Umarov, S.; Tsallis, C.; Steinberg, S. On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics. Milan Journal of Mathematics 2008, 76, 307–328. https://doi.org/10.1007/ s00032-008-0087-y

    2008

  48. [57]

    Statistical-Mechanical Foundation of the Ubiquity of Lévy Distributions in Nature

    Tsallis, C.; Levy, S.V .F.; Souza, A.M.C.; Maynard, R. Statistical-Mechanical Foundation of the Ubiquity of Lévy Distributions in Nature. Physical Review Letters 1995, 75, 3589–3593. https://doi.org/10.1103/physrevlett.75.3589

    1995 doi

  49. [58]

    https://doi.org/10.3390/e23070884

    Entropy 2021, 23. https://doi.org/10.3390/e23070884

    2021 doi

  50. [59]

    Alternative way to characterize a q-Gaussian distribution by a robust heavy tail measurement

    de Santa Helena, E.; Nascimento, C.; Gerhardt, G. Alternative way to characterize a q-Gaussian distribution by a robust heavy tail measurement. Physica A: Statistical Mechanics and its Applica- tions 2015, 435, 44–50. https://doi.org/10.1016/j.physa.2015.04.032. Disclaimer/Pub...

    2015 doi

  51. [61]

    Nonextensive foundation of Lévy distributions

    Prato, D.; Tsallis, C. Nonextensive foundation of Lévy distributions. Physical Review E 1999, 60, 2398–2401. https://doi.org/10.1103/physreve.60.2398. Version January 16, 2025 submitted to Mathematics 33 of 33

    1999 doi

  52. [114]

    https://doi.org/10.1016/s0378-4371(02)01383-3

    doi

  53. [1989]

    Literaturverz. S. 244 - 257

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.