Transforming Gaussian correlations. Applications to generating long-range power-law correlated time series with arbitrary distribution

Ana V. Coronado; Manuel G\'omez-Extremera; Pedro A. Bernaola-Galv\'an; Pedro Carpena

REVIEW 2 major objections 5 minor 53 references

Transforming Gaussian correlations. Applications to generating long-range power-law correlated time series with arbitrary distribution

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Gaussian-to-any-distribution transforms preserve power-law autocorrelations when the mapping is nearly linear.

desk verdict Honest, well-grounded treatment of how Gaussian autocorrelations transform under marginal changes, but the FFM spectrum in Eq. (36) is typo-level wrong and must be fixed before the main algorithm is reproducible. read the letter →

arxiv 1909.01725 v1 pith:N6EXPQJ5 submitted 2019-09-04 physics.data-an

Pedro Carpena , Pedro A. Bernaola-Galván , Manuel Gómez-Extremera , Ana V. Coronado This is my paper · ORCID

classification physics.data-an

keywords Gaussiantransformationautocorrelationfunctionpower-lawcorrelationsFourierFilteringMethodmarginaldistributionHurstexponentlong-rangedependencetimeseriessynthesis

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 establishes a transfer rule for correlations under monotone transformations: if a Gaussian time series with autocorrelation $C_G(\ell)$ is mapped to any new marginal distribution by $z_i=F^{-1}(\Phi(z_{G,i}))$, then the new autocorrelation is exactly $C(\ell)=C(C_G(\ell))$, where the function $C(C_G)$ is fixed by the destination distribution through a two-dimensional integral. It matters because standard generators such as the Fourier Filtering Method—a spectral method that imprints a prescribed power-law spectrum on Gaussian noise—produce Gaussian long-range correlated data, whereas real records are often non-Gaussian. The paper shows when and how the power-law exponent of the Gaussian series survives the transform: for symmetric destinations with bounded or fast-decaying tails, $C(C_G)$ is nearly linear, so $C(\ell)\simeq b_1 C_G(\ell)$ and the Hurst exponent (the exponent controlling the power-law decay) is preserved. Heavy-tailed or strongly skewed destinations compress and distort correlations, and negative correlations may become unreachable. The paper closes with a practical prescription for producing synthetic series of arbitrary distribution with a controlled power-law autocorrelation tail, demonstrated on absolute stock returns.

What carries the argument

The load-bearing object is the function $C(C_G)$, the Pearson correlation of two variables obtained by sending Gaussian variables with correlation $C_G$ through $F^{-1}(\Phi(\cdot))$, where $F$ is the destination cumulative distribution. It is computed from the double integral in Eq. (8); when no closed form exists, the paper expands it as $C(C_G)=\sum_{n=1}^\infty b_n C_G^n$, with $b_n=(1/(n!\, \sigma^2))\left(\int F^{-1}(\Phi(x)) H_n(x)\varphi(x)\,dx\right)^2$, where $H_n$ are Hermite polynomials and $\sigma^2$ is the destination variance. For symmetric marginals the even coefficients vanish, leaving an odd series; $b_1$ is about 0.90-0.99 for uniform, logistic, Laplace, and arcsine distributions and decreases systematically as distribution tails lengthen. The identity $C(\ell)=C(C_G(\ell))$ turns this function into a transfer map between Gaussian and final autocorrelations, which is what makes the generalized Fourier Filtering Method possible.

What would settle it

Generate a long Gaussian AR(1) series with negative coefficient, transform it with a heavy-tailed lognormal marginal ($s=2$), and compare the empirical negative-lag autocorrelation $C(\ell)$ with the analytic prediction $C(C_G(\ell))$ from Eq. (29); a systematic deviation beyond the $2/\sqrt{N-\ell}$ noise level at lags where both are significant would show that the bivariate-normal pair assumption fails and Eq. (31) is not universal. A direct check is to estimate the joint density of each lagged pair and test whether, after standardizing the marginals, it is bivariate Gaussian.

Watch

Extended reading notes

Core claim

The central discovery is that the transformation $z_i=F^{-1}(\Phi(z_{G,i}))$ converts Gaussian correlations through a deterministic, distribution-specific function: $C(\ell)=C(C_G(\ell))$, with $C(C_G)$ given by the double integral in Eq. (8) over the bivariate Gaussian density. The paper proves general properties: $C(0)=0$, $C(1)=1$, $C(C_G)$ is non-decreasing, and for symmetric destination marginals it is odd, so the feasible correlation range is $(-1,1)$; non-symmetric marginals have $C_{\min}$ in $(-1,0)$, and heavier tails push $C_{\min}$ toward $0$. It expands $C(C_G)=\sum_{n=1}^\infty b_n C_G^n$ with all coefficients nonnegative, and shows $b_1$ controls linearity: it is close to one for bounded, symmetric, and short-tailed distributions and shrinks toward zero as tails lengthen. Consequently, at sufficiently large lags, or whenever $b_1$ is large, $C(\ell)\simeq b_1 C_G(\ell)$, so a power-law $C_G(\ell)$ yields a power-law $C(\ell)$ with the same Hurst exponent. The paper verifies this numerically on AR(1) (first-order autoregressive) and Fourier-filtered Gaussian series, then uses it to synthesize a series matching both the empirical marginal distribution and the power-law autocorrelation tail of a real financial volatility series.

Load-bearing premise

The whole calculation assumes that every pair $(z_{G,i}, z_{G,i+\ell})$ of the Gaussian series is jointly bivariate normal with correlation $C_G(\ell)$; this is exact for fractional Gaussian noise (the Gaussian long-memory process) and AR(1) processes but only approximate for finite-length Fourier-filtered outputs, and if the underlying dependence between lagged values is not the Gaussian copula, $C(\ell)$ is not a function of $C_G(\ell)$ alone.

Editorial extensions

If this is right

The Fourier Filtering Method can be generalized: generate a Gaussian long-range correlated series, then apply $z_i=F^{-1}(\Phi(z_{G,i}))$ to obtain any target marginal; when $b_1$ is large the final series keeps the same Hurst exponent.
The observable lag range of power-law behavior is set by solving $b_1 C_G(\ell_{\max})=2/\sqrt{N-\ell_{\max}}$: long series or high $b_1$ give wide observable scaling, while short series with low $b_1$ may show no power-law at all.
For non-symmetric destination distributions, not every autocorrelation is realizable: the lower bound is $C_{\min}$, which approaches 0 as tails become heavier, so strongly skewed long-tailed marginals are effectively incompatible with anticorrelated time series.
Because symmetric marginals have only odd terms in the expansion of $C(C_G)$, the linear approximation holds to larger values of $|C_G|$, making symmetric destinations the safer choice for synthesis.
The recipe reproduces both the empirical marginal distribution and the power-law autocorrelation tail of observed absolute-return series, with discrepancies confined to small lags.

Reading between the lines

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

A diagnostic the authors leave implicit: for any observed non-Gaussian long-memory series, the empirical autocorrelation should collapse onto $C(C_G(\ell))$ for some Gaussian $C_G(\ell)$; checking this collapse would test whether the data are consistent with a monotone Gaussian-copula model.
Because the transfer function depends only on the destination marginal, the recipe extends beyond the named distributions to any empirical distribution with a numerically computed inverse CDF; the financial example shows the route, and biomedical or climate records would be natural further tests.
The $b_n$ expansion predicts a quantitative rule: two destination marginals with the same $b_1$ value should show the same observable scaling range regardless of their higher-order coefficients, a comparison that could be made with heavy-tailed marginals matched in $b_1$ but differing in skewness.
Since the transform is monotone, rank correlations and mutual information between lagged values are unchanged, so the framework could be paired with rank-based dependence measures to separate the effect of the marginal from the effect of the copula in long-memory data.

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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

Worth a look if you work with surrogate time series. The core result, C(l) = C(C_G(l)), is the standard Gaussian-copula/NORTA relation, and the authors properly cite Li-Hammond, Cario-Nelson, Chen, and Kugiumtzis rather than claiming it new. What is actually new is the systematic catalog of C(C_G) across symmetric and non-symmetric, bounded and unbounded distributions, plus the b1-based criterion in Eq. (39) for when power-law autocorrelation survives the transform and stays above the 2/sqrt(N) noise floor. That criterion is practical and, as far as I can check, correct. Fig. 7 is a nice validation that AR(1) and FFM series stay on the theoretical C(C_G) curve, and the uniform and lognormal cases are solved in closed form. The derivations in Eqs. (13)-(17) are clean.

The soft spots are real but manageable. Eq. (31) requires each pair of Gaussian variables to be jointly bivariate normal; this is exact for fGn and AR(1), only approximate for finite-N FFM output, and the paper does not push hard on that approximation. The IBM demonstration is a self-consistency check rather than an independent prediction, because the empirical CDF and the Hurst exponent are estimated from the same data whose autocorrelation is then matched. No code or data are shipped. None of these is fatal.

One thing is more serious. Eq. (36) states S(f_j) proportional to f_j^(2H-1). For H > 0.5 that gives a spectrum increasing with frequency, hence high-frequency-dominated noise, whereas fGn with H > 0.5 must have S(f) proportional to f^(1-2H), diverging at low frequency. The power-law tail in Eq. (35) and Wiener-Khinchin force the negative exponent. So a reader implementing the algorithm literally will not obtain the l^-0.3 decay shown in Fig. 10. The figures look correct, suggesting the authors used the right spectrum and the equation has a typo, but as written the central recipe is not reproducible. The stress-test note holds up. A referee should require the exponent be fixed, and ideally a short code snippet or pseudocode with the correct sign.

Bottom line: the theory and the b1 criterion are worth publishing, and the paper deserves serious peer review. I would send it to review, but with a required fix to Eq. (36) before acceptance.

Referee Report

2 major / 5 minor

Summary. The paper studies how the Pearson correlation between two Gaussian variables changes when both are transformed by the inverse-CDF map z = F^{-1}(Phi(z_G)) to a destination marginal distribution. It derives the pair-correlation function C(C_G) from the bivariate Gaussian integral (Eq. 8), obtains a power-series expansion with coefficients given by one-dimensional integrals (Eqs. 16-17), and establishes general properties such as oddness for symmetric marginals and the existence of a negative lower bound C_min for non-symmetric marginals. Closed forms are given for the uniform and lognormal cases, and numerical results are presented for logistic, Laplace, arcsine, symmetric Pareto, exponential, Weibull, and Pareto distributions. The paper then extends the pair result to time series via C(l) = C(C_G(l)), and uses this to propose a generalization of the Fourier Filtering Method (FFM) for generating power-law correlated time series with an arbitrary marginal distribution. The method is illustrated by synthesizing a series that mimics the marginal distribution and autocorrelation power-law tail of IBM daily absolute returns.

Significance. The central theoretical result is sound and useful: the relation C(l) = C(C_G(l)) is derived without fitting to the target autocorrelation, and the expansion in Hermite polynomials gives a principled, parameter-free way to quantify how different marginals attenuate or distort Gaussian correlations. The closed-form uniform and lognormal results are clean, and the numerical checks in Figs. 7 and 10 support the analytical formulas. The systematic comparison across symmetric, bounded, and heavy-tailed distributions is a useful practical guide. The proposed FFM generalization is potentially valuable, provided the algorithmic spectrum is stated correctly. The main weakness is that the FFM application rests on approximate finite-N Gaussianity, and, more seriously, the spectrum written in Eq. (36) has the wrong exponent and is not reproducible as stated.

major comments (2)

[IV.A, Eq. (36)] The exponent in the FFM spectrum is inverted. For a fractional Gaussian noise with autocorrelation (34)-(35), C_G(l) ~ H(2H-1) l^{2H-2}, so by the Wiener-Khinchin theorem the low-frequency spectrum must scale as S(f) ~ f^{1-2H}. Equation (36) instead prescribes S(f_j) proportional to f_j^{2H-1}. For H=0.85 this gives S(f) ~ f^{0.7}, which is high-frequency-dominated and would not produce the l^{-0.3} decay reported in Fig. 10; a reader implementing Eq. (36) literally would not obtain a long-range correlated series. The figures suggest the correct sign was used in the computations, but as written the algorithm is not reproducible. This is load-bearing because the generalized FFM is the central practical deliverable. Please correct the exponent and the sentence following Eq. (36), and re-verify the resulting spectra.
[IV, Eq. (31)] The identity C(l) = C(C_G(l)) requires that for every lag l the pair (z_G,i, z_G,i+l) is jointly bivariate normal with correlation C_G(l). This holds exactly for Gaussian processes such as AR(1) in stationarity and for true fGn, but FFM outputs are finite Fourier series with deterministic amplitudes and random phases, so they are only approximately Gaussian for finite N. The paper currently states without qualification that FFM outputs are Gaussian (Section IV.A, step 3). Please state this approximation explicitly and either add a quantitative check of the joint Gaussianity of FFM pairs or cite the standard large-N justification, since Eq. (31) is used to interpret all FFM-based examples, including Figs. 7 and 10.

minor comments (5)

[IV.A, Eq. (35)] The proportionality 'sign(H)' is a typo: H is positive throughout, so sign(H)=1. The intended sign is that of (2H-1), which is negative for H<1/2. Please replace sign(H) with sign(2H-1) or simply keep the explicit prefactor H(2H-1).
[IV.B, Application II] The Hurst exponent H=0.87 is estimated from the same IBM autocorrelation whose power-law tail is then reproduced by the synthetic series. The agreement in Fig. 12 is therefore a self-consistency check, not an out-of-sample validation. Please state this explicitly in the text so that readers do not overinterpret the match.
[II.A, Eq. (12)] The derivative identity in Eq. (12) is correct but is introduced as 'not difficult to check.' Since it is the key to the whole expansion, a one-line reference to the Mehler expansion of the bivariate Gaussian density would make the derivation more transparent.
[IV.A, Eqs. (38)-(39)] The noise level 2/sqrt(N-l) is a white-noise heuristic. For long-memory processes the fluctuations of autocorrelation estimates depend on H. The paper uses this only as a practical criterion, so this is not blocking, but acknowledging the heuristic nature would improve rigor.
[Abstract and Introduction] The abstract and introduction promise a study of 'arbitrary' distributions, but the actual analysis is confined to a selected family of parametric distributions plus one empirical CDF. This is fine, but the scope should be worded as 'a broad class of distributions' rather than literally arbitrary.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central C(C_G) relation is derived from a stated bivariate Gaussian integral, with no fitted parameter or self-citation chain forcing the predicted autocorrelations.

full rationale

The paper's central derivation is self-contained. Equation (8) defines C(C_G) as a double integral over the bivariate Gaussian density with the transformed variables inserted via the inverse-CDF mapping, and the expansion coefficients in Eq. (17) follow from the destination marginal alone. The time-series relation C(l) = C(C_G(l)) in Eq. (31) is obtained by substituting the lagged Gaussian variables directly into the pair result, not by fitting the final autocorrelation. The numerical checks in Fig. 7 compare independently generated AR(1) and FFM series with the theoretical curve and find agreement, so the relation is not imposed by construction. Application I uses the Fourier Filtering Method with a prescribed Hurst exponent and then checks that the transformed series has the predicted power-law autocorrelation; this is an application of the derived formula. Application II estimates H from the IBM autocorrelation tail and then generates a synthetic series that reproduces the same tail; this is explicitly a practical demonstration and self-consistency check, not a claim of independent predictive success. The self-citations to FFM implementations provide algorithmic context rather than load-bearing justification for the transformation theorem. The apparent reciprocal sign in Eq. (36) for the Fourier spectrum is a correctness/reproducibility concern, not a circularity: it does not reduce the derivation to its inputs. Therefore no circular step meeting the required evidentiary standard is present.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The theoretical core introduces no free parameters or invented entities. The only fitted number is H in the IBM demonstration, estimated from the same data whose autocorrelation is then reproduced, making that example a self-consistency check rather than a prediction.

free parameters (1)

Hurst exponent H for IBM example = 0.87
Estimated from the IBM absolute-return autocorrelation tail C(l) ~ l^-0.26 via 2-2H=0.26 in Section IV B. Used to generate the Gaussian seed; this is an input from the target data, not a parameter of the theoretical derivation.

assumptions (5)

domain assumption Bivariate normality of Gaussian series pairs
Equation (7) models (x_G,y_G) as bivariate Gaussian with correlation C_G, and Eq. (31) applies this to every lag of the Gaussian time series.
standard math Probability integral transform yields the desired marginal
Equation (3) defines z_i = F^{-1}(Phi(z_G,i)) and is the standard inverse-CDF transformation; it assumes F is continuous and strictly increasing.
standard math Hermite expansion converges
The power series in Eqs. (16)-(17) for C(C_G) assumes the Hermite coefficients are square-summable so the expansion converges on (-1,1).
domain assumption FFM output has the theoretical fGn autocorrelation
Section IV A assumes the FFM-generated series has C_G(l) as in Eq. (34) with a well-defined Hurst exponent H and random phases; in finite N this is approximate.
domain assumption Empirical CDF of IBM data is the true marginal
Application II uses the numerically estimated F and F^{-1} from the IBM absolute returns as if exact, then computes b_1 from them.

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Pith. "Pith review of Transforming Gaussian correlations. Applications to generating long-range power-law correlated time series with arbitrary distribution." pith.science (2026). https://pith.science/paper/N6EXPQJ5

@misc{pith2026190901725,
  author       = {Pith},
  title        = {Pith review of: Transforming Gaussian correlations. Applications to generating long-range power-law correlated time series with arbitrary distribution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N6EXPQJ5}},
  note         = {Machine review of arXiv:1909.01725}
}

read the original abstract

The observable outputs of many complex dynamical systems consist in time series exhibiting autocorrelation functions of great diversity of behaviors, including long-range power-law autocorrelation functions, as a signature of interactions operating at many temporal or spatial scales. Often, algorithms able to generate correlated noises reproducing the properties of real time series produce \textsl{Gaussian} outputs, while real, experimentally observed time series are often non-Gaussian, and may follow distributions with a diversity of behaviors concerning the support, the symmetry or the tail properties. Here, we study how the correlation of two Gaussian variables changes when they are transformed to follow a different destination distribution. Specifically, we consider bounded and unbounded distributions, symmetric and non-symmetric distributions, and distributions with different tail properties, from decays faster than exponential to heavy tail cases including power-laws, and we find how these properties affect the correlation of the final variables. We extend these results to Gaussian time series which are transformed to have a different marginal distribution, and show how the autocorrelation function of the final non-Gaussian time series depends on the Gaussian correlations and on the final marginal distribution. As an application of our results, we propose how to generalize standard algorithms producing Gaussian power-law correlated time series in order to create synthetic time series with arbitrary distribution and controlled power-law correlations. Finally, we show a practical example of this algorithm by generating time series mimicking the marginal distribution and the power-law tail of the autocorrelation function of a real time series: the absolute returns of stock prices.

Figures

Figures reproduced from arXiv: 1909.01725 by the authors.

**Figure 1.** b), c) and d) (symbols). As expected, since the three distributions are symmetric, C(CG) is and odd function in all cases, and therefore the corresponding Taylor expansion given by Eq. (20) includes only odd terms. Indeed, we also show in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 2.** FIG. 2. (Color online) a) The behavior of [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3. (Color online) a) Behavior of the correlation [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4. (Color online) Behavior of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5. (Color online) Behavior of the correlation [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6. Interval ( [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: FIG. 7. Top panel: the theoretical [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: FIG. 8. a) Gaussian time series [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: FIG. 9. Time series [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: FIG. 10. (Color online) a) Autocorrelation function [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

**Figure 11.** Figure 11: FIG. 11. a) Absolute returns time series [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]

**Figure 12.** Figure 12: FIG. 12. Autocorrelation function of the IBM absolute returns shown in Fig. 11a) (thick line). We also show in a thin line the [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]

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Finally, in Sec

A generalization of the FFM algorithm able to synthesize generic power-law correlated time series with arbitrary marginal distribution; and 2) a practical example where we generate a time series mimicking the distribution and the power-law tail of the autocorrelation function of a real-world power-law correlated time series: the absolute returns of the st...

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Uniform distribution The uniform distribution is one of the few cases for which the function C(CG) can be obtained analytically. Since the standardized distribution F (~x) is unique in this case (see Table II), the same happens with the function C(CG) and then it can be obtained using either F (~x) (Table II) or F (x) with any choice of the parameters a a...

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Logistic, Laplace and arcsine distributions The three distributions, apart from being symmetric, share also another property: they lack a shape parameter. As a consequence, the corresponding F (~x) does not have any parameter and is therefore unique in the three cases (Table II), so that each distribution presents a single C(CG) function. For the three di...

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Symmetric Pareto distribution For this distribution the function C(CG) has to be obtained numerically since there is no analytical solution of the integral in Eq. (8). The distribution is symmetric, so that C(CG) is odd. However, C(CG) is not unique since the distribution depends on three parameters (Table I): The location parameter µ and the scale parame...

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lognormal distribution We start with the case of lognormal distribution because it is the only one (together with the uniform distribution discussed above) for which the function C(CG) can be obtained analytically. In this case, noting that F−1(y) = exp ( m + √ 2s erf−1(2y− 1) ) (Table I), and that Φ(xG) is given in Eq. (2) we obtain that F−1(Φ(xG)) = exp...

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As can be seen in Table II, the standardized forms of the Weibull and Pareto distributions have shape parameters, δ and ε respectively, which control the tail behavior

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Finally, in Sec

A generalization of the FFM algorithm able to synthesize generic power-law correlated time series with arbitrary marginal distribution; and 2) a practical example where we generate a time series mimicking the distribution and the power-law tail of the autocorrelation function of a real-world power-law correlated time series: the absolute returns of the st...

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[2] [2]

Uniform distribution The uniform distribution is one of the few cases for which the function C(CG) can be obtained analytically. Since the standardized distribution F (~x) is unique in this case (see Table II), the same happens with the function C(CG) and then it can be obtained using either F (~x) (Table II) or F (x) with any choice of the parameters a a...

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[3] [3]

Logistic, Laplace and arcsine distributions The three distributions, apart from being symmetric, share also another property: they lack a shape parameter. As a consequence, the corresponding F (~x) does not have any parameter and is therefore unique in the three cases (Table II), so that each distribution presents a single C(CG) function. For the three di...

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[4] [4]

Symmetric Pareto distribution For this distribution the function C(CG) has to be obtained numerically since there is no analytical solution of the integral in Eq. (8). The distribution is symmetric, so that C(CG) is odd. However, C(CG) is not unique since the distribution depends on three parameters (Table I): The location parameter µ and the scale parame...

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[5] [5]

In this case, noting that F−1(y) = exp ( m + √ 2s erf−1(2y− 1) ) (Table I), and that Φ(xG) is given in Eq

lognormal distribution We start with the case of lognormal distribution because it is the only one (together with the uniform distribution discussed above) for which the function C(CG) can be obtained analytically. In this case, noting that F−1(y) = exp ( m + √ 2s erf−1(2y− 1) ) (Table I), and that Φ(xG) is given in Eq. (2) we obtain that F−1(Φ(xG)) = exp...

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[6] [6]

However, there is no analytical solution of the integral in Eq

Exponential distribution The exponential distribution lacks a shape parameter, and then there is a single standardized exponential distribu- tion so that the function C(CG) is unique (Table II). However, there is no analytical solution of the integral in Eq. (8) in this case, which has to be calculated numerically. The exact numerical result of the C(CG) ...

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[7] [7]

As can be seen in Table II, the standardized forms of the Weibull and Pareto distributions have shape parameters, δ and ε respectively, which control the tail behavior

Weibull and Pareto distributions We present these two distributions together because the corresponding C(CG) functions present similar properties. As can be seen in Table II, the standardized forms of the Weibull and Pareto distributions have shape parameters, δ and ε respectively, which control the tail behavior. For the Weibull case, the parameter δ mus...

work page

[8] [8]

C.-K. Peng, J. Mietus, J. M. Hausdorﬀ, S. Havlin, H. E. Stanley, and A. L. Goldberger. Long-range anticorrelations and non-Gaussian behavior of the heartbeat. Phys. Rev. Lett. 70, 1343 (1993)

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