# Spurious Results of Fluctuation Analysis Techniques in Magnitude and Sign Correlations

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## Abstract

**:**

## 1. Introduction

## 2. Fluctuation Analysis and Detrended Fluctuation Analysis

## 3. The Relationship of the Autocorrelation Function with FA and DFA

## 4. Magnitude and Sign Decomposition. Volatility

## 5. Magnitude and Sign Study Using FA and DFA

## 6. Exact Autocorrelation Function of Magnitude and Sign Series

#### 6.1. Spurious Results and Misinterpretations of FA and DFA on Sign and Magnitude Series

## 7. Analytical FA and DFA Scaling on Sign and Magnitude Series

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Hurst, H.E. Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng.
**1951**, 116, 770–799. [Google Scholar] - Peng, C.K.; Buldyrev, S.V.; Goldberger, A.L.; Havlin, S.; Sciortino, F.; Simons, M.; Stanley, H.E. Long-range correlations in nucleotide sequences. Nature
**1992**, 356, 168–170. [Google Scholar] [CrossRef] [PubMed] - Peng, C.K.; Buldyrev, S.V.; Havlin, S.; Simons, M.; Stanley, H.E.; Goldberger, A.L. Mosaic organization of DNA nucleotides. Phys. Rev. E
**1994**, 49, 1685–1689. [Google Scholar] [CrossRef] - Ashkenazy, Y.; Ivanov, P.C.; Havlin, S.; Peng, C.K.; Goldberger, A.L.; Stanley, H.E. Magnitude and Sign Correlations in Heartbeat Fluctuations. Phys. Rev. Lett.
**2001**, 86, 1900–1903. [Google Scholar] [CrossRef] [PubMed] - Gomez-Extremera, M.; Carpena, P.; Ivanov, P.C.; Bernaola-Galván, P.A. Magnitude and sign of long-range correlated time series: Decomposition and surrogate signal generation. Phys. Rev. E
**2016**, 93, 042201. [Google Scholar] [CrossRef] [PubMed] - Ashkenazy, Y.; Havlin, S.; Ivanov, P.C.; Peng, C.K.; Schulte-Frohlinde, V.; Stanley, H.E. Magnitude and sign scaling in power-law correlated time series. Physica A
**2003**, 323, 19–41. [Google Scholar] [CrossRef] - Kalisky, T.; Ashkenazy, Y.; Havlin, S. Volatility of linear and nonlinear time series. Phys. Rev. E
**2005**, 72, 011913. [Google Scholar] [CrossRef] [PubMed] - Kantelhardt, J.W.; Ashkenazy, Y.; Ivanov, P.C.; Bunde, A.; Havlin, S.; Penzel, T.; Peter, J.H.; Stanley, H.E. Characterization of sleep stages by correlations in the magnitude and sign of heartbeat increments. Phys. Rev. E
**2002**, 65, 051908. [Google Scholar] [CrossRef] [PubMed] - Bernaola-Galván, P.A.; Gómez-Extremera, M.; Romance, A.R.; Carpena, P. Correlations in magnitude series to assess nonlinearities: Application to multifractal models and heartbeat fluctuations. arXiv
**2017**. [Google Scholar] - Zhu, L.; Jin, N.D.; Gao, Z.K.; Zong, Y.B.; Zhai, L.S.; Wang, Z.Y. Magnitude and sign correlations in conductance fluctuations of horizontal oil water two-phase flow. J. Phys. Conf. Ser.
**2012**, 364, 012067. [Google Scholar] [CrossRef] - Makse, H.A.; Davies, G.W.; Havlin, S.; Ivanov, P.C.; King, P.R.; Stanley, H.E. Long-range correlations in permeability fluctuations in porous rock. Phys. Rev. E
**1996**, 54, 3129–3134. [Google Scholar] [CrossRef] - Makse, H.A.; Havlin, S.; Ivanov, P.C.; King, P.R.; Prakash, S.; Stanley, H.E. Pattern formation in sedimentary rocks: Connectivity, permeability, and spatial correlations. Physica A
**1996**, 233, 587–605. [Google Scholar] [CrossRef] - Bartos, I.; Jánosi, I.M. Nonlinear correlations of daily temperature records over land. Nonlinear Process. Geophys.
**2006**, 13, 571–576. [Google Scholar] [CrossRef] - Li, Q.; Fu, Z.; Yuan, N.; Xie, F. Effects of non-stationarity on the magnitude and sign scaling in the multi-scale vertical velocity increment. Physica A
**2014**, 410, 9–16. [Google Scholar] [CrossRef] - Liu, Y.; Gopikrishnan, P.; Cizeau, P.; Meyer, M.; Peng, C.K.; Stanley, H.E. Statistical properties of the volatility of price fluctuations. Phys. Rev. E
**1999**, 60, 1390–1400. [Google Scholar] [CrossRef] - Beran, J. Statistics for Long-Memory Processes; Chapman and Hall/CRC: Boca Raton, FL, USA, 1994. [Google Scholar]
- Coronado, A.V.; Carpena, P. Size effects on correlation measures. J. Biol. Phys.
**2005**, 31, 121–133. [Google Scholar] [CrossRef] [PubMed] - Bryce, R.M.; Sprague, K.B. Revisiting detrended fluctuation analysis. Sci. Rep.
**2012**, 2, 315. [Google Scholar] [CrossRef] [PubMed] - Allegrini, P.; Barbi, M.; Grigolini, P.; West, B.J. Dynamical model for DNA sequences. Phys. Rev. E
**1995**, 52, 5281–5296. [Google Scholar] [CrossRef] - Molchan, G.M. Maximum of fractional Brownian motion: Probabilities of small values. Commun. Math. Phys.
**1999**, 205, 97–111. [Google Scholar] [CrossRef] - Rangarajan, G.; Ding, M.Z. Integrated approach to the assessment of long range correlation in time series data. Phys. Rev. E
**2000**, 61, 4991–5001. [Google Scholar] [CrossRef] - Hu, K.; Ivanov, P.C.; Chen, Z.; Carpena, P.; Stanley, H.E. Effect of trends on detrended fluctuation analysis. Phys. Rev. E
**2001**, 64, 011114. [Google Scholar] [CrossRef] [PubMed] - Chen, Z.; Hu, K.; Carpena, P.; Bernaola-Galván, P.; Stanley, H.E.; Ivanov, P.C. Effect of nonlinear filters on detrended fluctuation analysis. Phys. Rev. E
**2005**, 71, 011104. [Google Scholar] [CrossRef] [PubMed] - Blázquez, M.T.; Anguiano, M.; de Saavedra, F.A.; Lallena, A.M.; Carpena, P. Study of the human postural control system during quiet standing using detrended fluctuation analysis. Phys. A
**2009**, 388, 1857–1866. [Google Scholar] [CrossRef] - Carpena, P.; Bernaola-Galván, P.; Coronado, A.V.; Hackenberg, M.; Oliver, J.L. Identifying characteristic scales in the human genome. Phys. Rev. E
**2007**, 75, 032903. [Google Scholar] [CrossRef] [PubMed] - Karlin, S.; Brendel, V. Patchiness and correlations in DNA sequences. Science
**1993**, 259, 677–680. [Google Scholar] [CrossRef] [PubMed] - Höll, M.; Kantz, H. The relationship between the detrended fluctuation analysis and the autocorrelation function of a signal. Eur. Phys. J. B
**2015**, 88, 327. [Google Scholar] [CrossRef] - Talkner, P.; Weber, R.O. Power spectrum and detrended fluctuation analysis: Application to daily temperatures. Phys. Rev. E
**2000**, 62, 150–160. [Google Scholar] [CrossRef] - Carretero-Campos, C.; Bernaola-Galván, P.; Ivanov, P.C.; Carpena, P. Phase transitions in the first-passage time of scale-invariant correlated processes. Phys. Rev. E
**2012**, 85, 011139. [Google Scholar] [CrossRef] [PubMed] - Makse, H.A.; Havlin, S.; Schwartz, M.; Stanley, H.E. Method for generating long-range correlations for large systems. Phys. Rev. E
**1996**, 53, 5445–5449. [Google Scholar] [CrossRef] - Bernaola-Galván, P.; Oliver, J.L.; Hackenberg, M.; Coronado, A.V.; Ivanov, P.C.; Carpena, P. Segmentation of time series with long-range fractal correlations. Eur. Phys. J. B
**2012**, 85, 211. [Google Scholar] [CrossRef] [PubMed] - Apostolov, S.S.; Izrailev, F.M.; Makarov, N.M.; Mayzelis, Z.A.; Melnyk, S.S.; Usatenko, O.V. The Signum function method for the generation of correlated dichotomic chains. J. Phys. A Math. Theor.
**2008**, 41, 175101. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) a fractional Gaussian noise-type time series, $\left\{{x}_{i}\right\}$, generated using the Fourier Filtering Method (FFM) with ${\alpha}_{\mathrm{in}}=0.85$ and $N={2}^{8}$; (

**b**) the sign series $\left\{{s}_{i}\right\}$ obtained from $\left\{{x}_{i}\right\}$; and (

**c**) the magnitude (or volatility) series $\left\{{m}_{i}\right\}$ obtained from $\left\{{x}_{i}\right\}$.

**Figure 2.**The sign scaling exponent ${\alpha}_{s}$ and the magnitude scaling exponent ${\alpha}_{m}$ as a function of the scaling exponent ${\alpha}_{\mathrm{in}}$ of the original signal. Top panel: results of Fluctuation Analysis (FA); bottom panel: results of Detrended Fluctuation Analysis (DFA). We have used time series with $N={2}^{20}$, and we have generated 200 series for any ${\alpha}_{\mathrm{in}}$ value.

**Figure 3.**The sign (

**a**) and magnitude (

**b**) autocorrelation functions, ${C}_{m}\left(r\right)$ and ${C}_{s}\left(r\right)$, as a function of the values of the autocorrelation function of the original time series, $C\left(r\right)$. The solid lines correspond to the exact analytical results (Equations (27) and (28)), and the dashed lines to the first term in the corresponding Taylor expansions (Equations (29) and (30)).

**Figure 4.**Autocorrelation function of the original time series ($C\left(r\right)$), of the sign series (${C}_{s}\left(r\right)$) and of the magnitude series (${C}_{m}\left(r\right)$) obtained numerically for synthetic fractional Gaussian noises (fGns) generated with different ${\alpha}_{\mathrm{in}}$ values. (

**a**–

**d**) correspond to situations where the original fGn presents positive correlations (${\alpha}_{\mathrm{in}}>0.5$); (

**e**,

**f**) show examples in which the fGn is anticorrelated (${\alpha}_{\mathrm{in}}<0.5$). As, in this case, both $C\left(r\right)$ and ${C}_{s}\left(r\right)$ are negative, (

**e**,

**f**) show the absolute value of $C\left(r\right)$ and ${C}_{s}\left(r\right)$. In all cases, the lines correspond to the analytical values of ${C}_{s}\left(r\right)$ (solid lines) and ${C}_{m}\left(r\right)$ (dotted lines) obtained from $C\left(r\right)$ using Equations (29) and (30). For any ${\alpha}_{\mathrm{in}}$ value, we generate 100 fGns of length $N={2}^{24}$ and average the corresponding $C\left(r\right)$, ${C}_{s}\left(r\right)$ and ${C}_{m}\left(r\right)$.

**Figure 5.**Magnitude and sign scaling exponents, ${\alpha}_{m}$ and ${\alpha}_{s}$ provided by FA and DFA (symbols), and the correct results for ${\alpha}_{m}$ and ${\alpha}_{s}$ (lines) expected from the corresponding autocorrelation functions.

**Figure 6.**FA and DFA Fluctuation functions for the magnitude series obtained from two fGns with ${\alpha}_{\mathrm{in}}=0.65$ and $0.8$, and $N={2}^{23}$. The numeric results of applying FA and DFA to the corresponding time series are shown in symbols, and the lines correspond to the analytical results obtained from Equations (43) and (46). We also include dashed lines corresponding to the correct scaling ${\ell}^{2{\alpha}_{\mathrm{in}}-1}$, which gives ${\ell}^{0.6}$ and ${\ell}^{0.3}$ for ${\alpha}_{\mathrm{in}}=0.8$ and $0.65$, respectively, and to the spurious scaling ${\ell}^{1/2}$. Note how FA and DFA predict the correct scaling exponent ${\alpha}_{m}=0.6$ for ${\alpha}_{\mathrm{in}}=0.8$ but fail for ${\alpha}_{\mathrm{in}}=0.65$ since, instead of ${\alpha}_{m}=0.3$, both techniques give ${\alpha}_{m}=1/2$.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Carpena, P.; Gómez-Extremera, M.; Carretero-Campos, C.; Bernaola-Galván, P.; Coronado, A.V.
Spurious Results of Fluctuation Analysis Techniques in Magnitude and Sign Correlations. *Entropy* **2017**, *19*, 261.
https://doi.org/10.3390/e19060261

**AMA Style**

Carpena P, Gómez-Extremera M, Carretero-Campos C, Bernaola-Galván P, Coronado AV.
Spurious Results of Fluctuation Analysis Techniques in Magnitude and Sign Correlations. *Entropy*. 2017; 19(6):261.
https://doi.org/10.3390/e19060261

**Chicago/Turabian Style**

Carpena, Pedro, Manuel Gómez-Extremera, Concepción Carretero-Campos, Pedro Bernaola-Galván, and Ana V. Coronado.
2017. "Spurious Results of Fluctuation Analysis Techniques in Magnitude and Sign Correlations" *Entropy* 19, no. 6: 261.
https://doi.org/10.3390/e19060261