# Classification of Fractal Signals Using Two-Parameter Non-Extensive Wavelet Entropy

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## Abstract

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## 1. Introduction

## 2. Materials and methods

#### 2.1. The Representation of Fractal Signals by Wavelets

#### 2.2. A Nonextensive Wavelet $(q,{q}^{\prime})$-Entropy of Fractal Signals

#### 2.3. The Behaviour of Wavelet $(q,{q}^{\prime})$-Entropy for Various $(q,{q}^{\prime})$ Pairs

#### 2.4. The Classification of Fractal Signals with Wavelet $(q,{q}^{\prime})$-Entropy

## 3. Results

#### 3.1. Experimental Results

#### 3.2. The Threshold for Long and Short Fractal Time Series

#### 3.3. Comparison with the Standard SSC Technique

#### 3.4. Computational Complexity

#### 3.5. Application to Financial Time Series

#### 3.6. Application to Physiological Time Series

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Wavelet $(q,{q}^{\prime})$-entropy for fractal signals of parameter $\alpha $. Parameters q and ${q}^{\prime}$ are set to $q=7$ and ${q}^{\prime}=4$ respectively. Length N is given in powers of 2.

**Figure 2.**Wavelet $(q,{q}^{\prime})$-entropy for fractal signals of parameter $\alpha $. Parameters q and ${q}^{\prime}$ are set to $q=-9$ and ${q}^{\prime}=-2$ respectively. Length N is given in powers of 2.

**Figure 3.**Wavelet $(q,{q}^{\prime})$-entropy for $1/{f}^{\alpha}$ signals. Parameters q and ${q}^{\prime}$ are set to $q=7$ and ${q}^{\prime}=4$. Scaling index range is $\alpha \in (-1,1)$ and the length of the signal ranges in the interval $N\in ({2}^{4},{2}^{13})$.

**Figure 4.**Wavelet entropy plane when $q<0$ (or ${q}^{\prime}<0$). Constant regions are observed in $\alpha \in (-{\alpha}_{\mathrm{coff}},{\alpha}_{\mathrm{coff}})$ and variable regions outside this interval.

**Figure 5.**Fractal signal classification scheme using wavelet $(q,{q}^{\prime})$-entropy and the biweight midvariance technique.

**Figure 6.**Classification of signals as stationary or nonstationary. Top plot: concatenated nonstationary and stationary signal, first half part is a nonstationary signal (fractional Brownian motion (fBm) with $\alpha =1.05$) and second half corresponds to a stationary one (fractional Gaussian noise (fGn) with $\alpha =0.95$). Bottom plot: wavelet $(q,{q}^{\prime})$-entropy of concatenated signal computed in sliding windows with $W=2048$, $\mathrm{\Delta}=256$ and $(q,{q}^{\prime})=(-0.99,-0.1)$.

**Figure 7.**Determination of the optimal threshold $thrsh$ for long and short time series. For convenience only the range $\alpha \in (0,1.2)$ is shown and every plot was obtained using $(q,{q}^{\prime})=(-0.99,-0.1)$ and $W=256$ and $\mathrm{\Delta}=64$.

**Figure 8.**Classification of synthesized fGn and fBm signals using wavelet $(q,{q}^{\prime})$-entropy and signal summation conversion (SSC). Left plot corresponds to signals with length $N={2}^{14}$ and right plot to signals with $N={2}^{11}$.

**Figure 9.**Classification of synthesized pure-power-law (PPL) signals as stationary or nonstationary using wavelet $(q,{q}^{\prime})$-entropy and SSC. Left plot corresponds to long signals with length $N={2}^{14}$ and right plot to short-length signals with $N={2}^{11}$.

**Figure 10.**Stock market closing indices. Top left plot displays the S&P index from January 1956 to 21 April 2017. Top right plot displays the Dow Jones Industrial Average time series from February 1985 to 21 April 2017. Finally bottom left and right plots display the indices for the NASDAQ and Nikkei indices.

**Figure 11.**Stride interval time series from four healthy, young adults walking during a period of an hour in a 400-m oval.

**Figure 12.**Average times for obtaining a classification using the proposed technique based on wavelet $(q,{q}^{\prime})$-entropies and the SSC. Left plot displays a comparison between these techniques while the plot on the right shows the classification times of SSC.

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**MDPI and ACS Style**

Ramírez-Pacheco, J.C.; Trejo-Sánchez, J.A.; Cortez-González, J.; Palacio, R.R.
Classification of Fractal Signals Using Two-Parameter Non-Extensive Wavelet Entropy. *Entropy* **2017**, *19*, 224.
https://doi.org/10.3390/e19050224

**AMA Style**

Ramírez-Pacheco JC, Trejo-Sánchez JA, Cortez-González J, Palacio RR.
Classification of Fractal Signals Using Two-Parameter Non-Extensive Wavelet Entropy. *Entropy*. 2017; 19(5):224.
https://doi.org/10.3390/e19050224

**Chicago/Turabian Style**

Ramírez-Pacheco, Julio César, Joel Antonio Trejo-Sánchez, Joaquin Cortez-González, and Ramón R. Palacio.
2017. "Classification of Fractal Signals Using Two-Parameter Non-Extensive Wavelet Entropy" *Entropy* 19, no. 5: 224.
https://doi.org/10.3390/e19050224