#
Wavelet Fisher’s Information Measure of 1=f ^{α} Signals

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## Abstract

**:**

## 1. Introduction

## 2. $1/{f}^{\alpha}$ Signals: Definition and Wavelet Analysis

#### 2.1. $1/{f}^{\alpha}$ Processes

#### 2.2. Wavelet Analysis of $1/{f}^{\alpha}$ Signals

#### 2.3. Wavelet-Based Probability Densities

## 3. Wavelet-Based Fisher’s Information Measure

#### 3.1. Time-Domain Fisher’s Information Measure

#### 3.2. Wavelet-Based Fisher’s Information Measure

**Figure 1.**Theoretical plane of wavelet Fisher’s information for $1/{f}^{\alpha}$ signals. When $\alpha >1$ increases exponentially and when $\alpha <-1$ it converges to 1.

#### 3.3. Applications of Wavelet Fisher’s Information Measure

## 4. Detection of Structural Changes in the Mean

#### 4.1. The Problem of Level-Shift Detection

#### 4.2. Level-Shift Detection in $1/f$ Signals with Wavelet FIM

## 5. Results and Discussion

**Figure 2.**Wavelet Fisher’s information of a fractional Gaussian noise with $H=0.1$ and embedded jumps. Top left plot displays the fGn signal with a single level-shift and top right plot with a more elaborate combination of jumps. Bottom plots represent their corresponding wavelet Fisher’s information.

**Figure 3.**Wavelet FIM of anticorrelated fractional Gaussian noises with a single level-shift. (

**a**) fGn with $H=0.2$; (

**b**) fGn with $H=0.3$; (

**c**) fGn with $H=0.4$; and (

**d**) fGn with $H=0.5$.

**Figure 4.**Wavelet FIM of correlated stationary fractional Gaussian noises with a single level-shift at ${t}_{b}=32768$. (

**a**) fGn with $H=0.6$; (

**b**) fGn with $H=0.7$; (

**c**) fGn with $H=0.8$ and the wavelet FIM of a fGn with $H=0.9$ is displayed in (

**d**).

**Figure 5.**Wavelet FIM of anticorrelated stationary fractional Gaussian noises with multiple mean level-shifts. (

**a**) fGn with $H=0.2$; (

**b**) fGn with $H=0.3$; (

**c**) fGn with $H=0.4$ and the wavelet FIM of a fGn with $H=0.5$ is displayed in (

**d**).

**Figure 6.**Wavelet FIM of correlated stationary fractional Gaussian noises with multiple mean level-shifts. (

**a**) fGn with $H=0.6$; (

**b**) fGn with $H=0.7$; (

**c**) fGn with $H=0.8$ and the wavelet FIM of a fGn with $H=0.9$ is displayed in (

**d**).

## 6. Conclusions

## Acknowledgements

## References

- Beran, J. Statistics for Long-Memory Processes; Chapman & Hall/CRC Press: Boca Raton, FL, USA, 1994. [Google Scholar]
- Caccia, D.C.; Percival, D.B.; Cannon, M.; Raymond, G.; Bassingthwaighte, J.B. Analyzing exact fractal time series: Evaluating dispersional analysis and rescaled range methods. Phys. A
**1997**, 246, 609–632. [Google Scholar] [CrossRef] - Thurner, S.; Lowen, S.B.; Feurstein, M.C.; Heneghan, C.; Feichtinger, H.G.; Teich, M.C. Analysis, synthesis and estimation of fractal-rate stochastic point processes. Fractals
**1997**, 5, 565–595. [Google Scholar] [CrossRef] - Samorodnitsky, G.; Taqqu, M. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance; Chapman & Hall/CRC Press: Boca Raton, FL, USA, 1994. [Google Scholar]
- Frieden, B.R.; Hughes, R.J. Spectral 1/f noise derived from extremized physical information. Phys. Rev. E
**1994**, 49, 2644–2649. [Google Scholar] [CrossRef] - Leland, W.E.; Taqqu, M.S.; Willinger, W.; Wilson, D. On the self-similar nature of ethernet traffic (extended version). IEEE/ACM Trans. Netw.
**1994**, 2, 1–15. [Google Scholar] [CrossRef] - Perez, D.G.; Zunino, L.; Garavaglia, M.; Rosso, O.A. Wavelet entropy and fractional brownian motion time series. Phys. A
**2006**, 365, 282–288. [Google Scholar] [CrossRef] - Zunino, L.; Perez, D.G.; Garavaglia, M.; Rosso, O.A. Wavelet entropy of stochastic processes. Phys. A
**2007**, 379, 503–512. [Google Scholar] [CrossRef] - Serinaldi, F. Use and misuse of some Hurst parameter estimators applied to stationary and non-stationary financial time series. Phys. A
**2010**, 389, 2770–2781. [Google Scholar] [CrossRef] - Shen, H.; Zhu, Z.; Lee, T. Robust estimation of the self-similarity parameter in network traffic using the wavelet transform. Signal Process.
**2007**, 87, 2111–2124. [Google Scholar] [CrossRef] - Stoev, S; Taqqu, M.S.; Park, C.; Marron, J.S. On the wavelet spectrum diagnostic for hurst parameter estimation in the analysis of internet traffic. Comput. Netw.
**2005**, 48, 423–445. [Google Scholar] [CrossRef] - Kowalski, A.M.; Plastino, A.; Casas, M. Generalized complexity and classical quantum transition. Entropy
**2009**, 11, 111–123. [Google Scholar] [CrossRef] [Green Version] - Ramirez-Pacheco, J.; Torres-Roman, D. Cosh window behaviour of wavelet Tsallis q-entropies in 1/f
^{α}signals. Electron. Lett.**2011**, 47, 186–187. [Google Scholar] [CrossRef] - Percival, D.B. Stochastic models and statistical analysis for clock noise. Metrologia
**2003**, 40, S289–S304. [Google Scholar] [CrossRef] - Lee, I.W.C.; Fapojuwo, A.O. Stochastic processes for computer network traffic modelling. Comput. Commun.
**2005**, 29, 1–23. [Google Scholar] [CrossRef] - Flandrin, P. Wavelet analysis and synthesis of fractional brownian motion. IEEE Trans. Inf. Theory
**1992**, 38, 910–917. [Google Scholar] [CrossRef] - Malamud, B.D.; Turcotte, D.L. Self-affine time series: Measures of weak and strong persistence. J. Stat. Plann. Inference
**1999**, 80, 173–196. [Google Scholar] [CrossRef] - Eke, A.; Hermán, P.; Bassingthwaighte, J.B.; Raymond, G.; Percival, D.B.; Cannon, M.; Balla, I.; Ikrényi, C. Physiological time series: Distinguishing fractal noises and motions. Pflugers Arch.
**2000**, 439, 403–415. [Google Scholar] [CrossRef] [PubMed] - Lowen, S.B.; Teich, M.C. Estimation and simulation of fractal stochastic point processes. Fractals
**1995**, 3, 183–210. [Google Scholar] [CrossRef] - Hudgins, L.; Friehe, C.A.; Mayer, M.E. Wavelet transforms and atmospheric turbulence. Phys. Rev. Lett.
**1993**, 71, 3279–3283. [Google Scholar] [CrossRef] [PubMed] - Cohen, A.; Kovacevic, J. Wavelets: The mathematical background. Proc. IEEE
**1996**, 84, 514–522. [Google Scholar] [CrossRef] - Pesquet-Popescu, B. Statistical properties of the wavelet decomposition of certain non-gaussian self-similar processes. Signal Process.
**1999**, 75, 303–322. [Google Scholar] [CrossRef] - Abry, P.; Veitch, D. Wavelet analysis of long-range dependent traffic. IEEE Trans. Inf. Theory
**1998**, 44, 2–15. [Google Scholar] [CrossRef] - Veitch, D; Abry, P. A wavelet based joint estimator of the parameters of long-range dependence. IEEE Trans. Inf. Theory
**1999**, 45, 878–897. [Google Scholar] [CrossRef] - Eke, A.; Hermán, P.; Kocsis, L; Kozak, L.R. Fractal characterization of complexity in temporal physiological signals. Physiol. Meas.
**2002**, 23, R1–R38. [Google Scholar] [CrossRef] [PubMed] - Quiroga, R.Q.; Rosso, O.A.; Basar, E.; Schurmann, M. Wavelet entropy in event-related potentials: A new method shows ordering of EEG oscillations. Biol. Cybern.
**2001**, 84, 291–299. [Google Scholar] [CrossRef] [PubMed] - Martin, M.T.; Penini, F.; Plastino, A. Fisher’s information and the analysis of complex signals. Phys. A
**1999**, 256, 173–180. [Google Scholar] [CrossRef] - Martin, M.T.; Perez, J.; Plastino, A. Fisher information and non-linear dynamics. Phys. A
**2001**, 291, 523–532. [Google Scholar] [CrossRef] - Telesca, L.; Lapenna, V.; Lovallo, M. Fisher information measure of geoelectrical signals. Phys. A
**2005**, 351, 637–644. [Google Scholar] [CrossRef] - Romera, E.; Sánchez-Moreno, P.; Dehesa, J.S. The Fisher information of single-particle systems with a central potential. Chem. Phys. Lett.
**2005**, 414, 468–472. [Google Scholar] [CrossRef] - Luo, S. Quantum fisher information and uncertainty relation. Lett. Math. Phys.
**2000**, 53, 243–251. [Google Scholar] [CrossRef] - Vignat, C.; Bercher, J.-F. Analysis of signals in the fisher-shannon information plane. Phys. Lett. A
**2003**, 312, 27–33. [Google Scholar] [CrossRef] - Deligneres, D.; Ramdani, S.; Lemoine, L.; Torre, K.; Fortes, M.; Ninot, G. Fractal analyses of short time series: A re-assessment of classical methods. J. Math. Psychol.
**2006**, 50, 525–544. [Google Scholar] [CrossRef] - Castiglioni, P.; Parato, G.; Civijian, A; Quintin, L.; di Rienzo, M. Local scale exponents of blood pressure and heart rate variability by detrended fluctuation analysis: Effects of posture, exercise and aging. IEEE Trans. Biomed. Eng.
**2009**, 56, 675–684. [Google Scholar] [CrossRef] [PubMed] - Esposti, F.; Ferrario, M.; Signorini, M.G. A blind method for the estimation of the hurst exponent in time series: Theory and methods. Chaos
**2008**, 18, 033126. [Google Scholar] [CrossRef] [PubMed] - Rea, W.; Reale, M.; Brown, J.; Oxley, L. Long-memory or shifting means in geophysical time series? Math. Comput. Simul.
**2011**, 81, 1441–1453. [Google Scholar] [CrossRef] - Capelli, C.; Penny, R.N.; Rea, W. Detecting multiple mean breaks at unknown points with atheoretical regression trees. Math. Comput. Simul.
**2008**, 78, 351–356. [Google Scholar] [CrossRef] - Davies, R.B.; Harte, D.S. Tests for hurst effect. Biometrika
**1987**, 74, 95–101. [Google Scholar] [CrossRef] - Cannon, M.J.; Percival, D.B.; Caccia, D.C.; Raymond, G.M.; Bassingthwaighte, J.B. Evaluating scaled windowed variance for estimating the hurst coefficient of time series. Phys. A
**1996**, 241, 606–626. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Ramírez-Pacheco, J.; Torres-Román, D.; Rizo-Dominguez, L.; Trejo-Sanchez, J.; Manzano-Pinzón, F.
Wavelet Fisher’s Information Measure of 1=*f ^{α}* Signals.

*Entropy*

**2011**,

*13*, 1648-1663. https://doi.org/10.3390/e13091648

**AMA Style**

Ramírez-Pacheco J, Torres-Román D, Rizo-Dominguez L, Trejo-Sanchez J, Manzano-Pinzón F.
Wavelet Fisher’s Information Measure of 1=*f ^{α}* Signals.

*Entropy*. 2011; 13(9):1648-1663. https://doi.org/10.3390/e13091648

**Chicago/Turabian Style**

Ramírez-Pacheco, Julio, Deni Torres-Román, Luis Rizo-Dominguez, Joel Trejo-Sanchez, and Francisco Manzano-Pinzón.
2011. "Wavelet Fisher’s Information Measure of 1=*f ^{α}* Signals"

*Entropy*13, no. 9: 1648-1663. https://doi.org/10.3390/e13091648