# A Note on Wavelet-Based Estimator of the Hurst Parameter

## Abstract

**:**

## 1. Introduction

- The changes of bias and variance with all different Hs, different data lengths, different ${j}_{1}$s and different wavelets;
- The relations of selected ${j}_{1}$ with data length and H;

- The initialization for initial approximation wavelet coefficients which introduces errors in used detailed wavelet coefficients.
- The inaccurate bias correction caused by correlations of wavelet coefficients.
- The method of simulation for FBM is not enough exact that the empirical bias is caused.

## 2. Wavelet-Based Estimator

#### 2.1. Definitions and Properties

**Lemma**

**1.**

**Remark**

**1.**

- For fixed j, the ${d}_{X}(j,\xb7)$ are independent and identically distributed;
- The processes ${d}_{X}(j,\xb7)$ and ${d}_{X}({j}^{\prime},\xb7)$, $j\ne {j}^{\prime}$, are independent.

#### 2.2. Two Wavelet-Based Estimators

#### **The First Estimator**

#### **The Second Estimator**

#### **Explicit Formula of theTwo Estimators**

#### **Variance Comparison**

#### 2.3. Calculation of Wavelet Coefficients

**Remark**

**2.**

#### 2.4. The Initialization Method

## 3. Simulation of FBM

#### **The Cholesky Method**

#### **The Circulant Embedding Method**

## 4. Simulation Results and Discussions

#### 4.1. Selection of Parameters

#### 4.2. Results and Discussions on Empirical Bias

- The initialization for ${a}_{X}(0,k)$ given in (27) introduces errors in ${d}_{X}(j,k)$, and the initialization errors are significant on small octaves but decrease with increasing j.
- The inaccurate bias correction for $\mathbb{E}{log}_{2}S\left(j\right)\ne {log}_{2}\mathbb{E}S\left(j\right)$ (under independent assumptions) caused by correlations of wavelet coefficients.
- The method of simulation for FBM is not enough exact that the empirical bias is caused.

- The increase of N and change of wavelet made no improvements to the empirical bias. The chosen of biorthogonal wavelet makes the empirical bias worse.
- The increase of ${j}_{1}$ leads to the decrease of empirical bias.
- The empirical bias increases with the decrease of H. when choosing ${j}_{1}=3$ and $N=3$, the empirical bias of estimator $\widehat{{H}_{1}}$ can be ignored for $H\ge 0.4$. So the estimator $\widehat{{H}_{1}}$ is suitable to detect the long-range dependence (can be described by $H>0.5$).

#### **Comparison of Estimators**

#### **Comparison of Simulation Methods**

#### 4.3. Analysis of the Initialization Method

#### 4.4. Analysis of Noise Effects

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Li, Q.; Liang, S.Y.; Yang, J.; Li, B. Long range dependence prognostics for bearing vibration intensity chaotic time series. Entropy
**2016**, 18, 23. [Google Scholar] [CrossRef] [Green Version] - Liu, C.; Yang, Z.; Shi, Z.; Ma, J.; Cao, J. A gyroscope signal denoising method based on empirical mode decomposition and signal reconstruction. Sensors
**2019**, 19, 5064. [Google Scholar] [CrossRef] [Green Version] - Li, X.; Chen, W.; Chan, C.; Li, B.; Song, X. Multi-sensor fusion methodology for enhanced land vehicle positioning. Inform. Fusion
**2019**, 46, 51–62. [Google Scholar] [CrossRef] - Dou, C.; Wei, X.; Lin, J. Fault diagnosis of gearboxes using nonlinearity and determinism by generalized Hurst exponents of shuffle and surrogate data. Entropy
**2018**, 20, 364. [Google Scholar] [CrossRef] [Green Version] - Wu, L.; Chen, L.; Ding, Y.; Zhao, T. Testing for the source of multifractality in water level records. Physica A
**2018**, 508, 824–839. [Google Scholar] [CrossRef] - Graves, T.; Gramacy, R.; Watkins, N.; Franzke, C. A brief history of long memory: Hurst, Mandelbrot and the road to ARFIMA, 1951–1980. Entropy
**2017**, 19, 437. [Google Scholar] [CrossRef] [Green Version] - Hurst, H.E. Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng.
**1951**, 116, 770–799. [Google Scholar] - Mandelbrot, B.; Van Ness, J. Fractional Brownian Motions, Fractional Noises and Applications. SIAM Rev.
**1968**, 10, 422–437. [Google Scholar] [CrossRef] - Deng, Z.; Wang, J.; Liang, X.; Liu, N. Function extension based real-time wavelet de-noising method for projectile attitude measurement. Sensors
**2020**, 20, 200. [Google Scholar] [CrossRef] [Green Version] - He, K.; Xia, Z.; Si, Y.; Lu, Q.; Peng, Y. Noise reduction of welding crack AE signal based on EMD and wavelet packet. Sensors
**2020**, 20, 761. [Google Scholar] [CrossRef] [Green Version] - Nicolis, O.; Mateu, J.; Contreras-Reyes, J.E. Wavelet-based entropy measures to characterize two-dimensional fractional Brownian fields. Entropy
**2020**, 22, 196. [Google Scholar] [CrossRef] [Green Version] - Li, J.; Ke, L.; Du, Q. Classification of heart sounds based on the wavelet fractal and twin support vector machine. Entropy
**2019**, 21, 472. [Google Scholar] [CrossRef] [Green Version] - 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. [Google Scholar] [CrossRef] [Green Version] - Flandrin, P. Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans. Inf. Theory
**1992**, 38, 910–917. [Google Scholar] [CrossRef] - Abry, P.; Gonçalvès, P.; Flandrin, P. Wavelets, spectrum analysis and 1/f processes. In Wavelets and Statistics; Antoniadis, A., Oppenheim, G., Eds.; Springer: New York, NY, USA, 1995; Section 2; Volume 103, pp. 15–29. [Google Scholar] [CrossRef]
- Delbeke, L.; Van Assche, W. A wavelet based estimator for the parameter of self-similarity of fractional Brownian motion. In Proceedings of the 3rd International Conference on Approximation and Optimization in the Caribbean (Puebla, 1995), Puebla, Mexico, 8–13 October 1995; Volume 24, pp. 65–76. [Google Scholar]
- 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] - Abry, P.; Flandrin, P.; Taqqu, M.; Veitch, D. Wavelets for the analysis, estimation and synthesis of scaling data. In Self-Similar Network Traffic and Performance Evaluation; Park, K., Willinger, W., Eds.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2000; pp. 39–88. [Google Scholar]
- Abry, P.; Flandrin, P.; Taqqu, M.S.; Veitch, D. Self-similarity and long-range dependence through the wavelet lens. In Theory and Applications of Long-Range Dependence; Doukhan, P., Oppenheim, G., Taqqu, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; pp. 527–556. [Google Scholar]
- Abry, P.; Helgason, H.; Pipiras, V. Wavelet-based analysis of non-Gaussian long-range dependent processes and estimation of the Hurst parameter. Lith. Math. J.
**2011**, 51, 287–302. [Google Scholar] [CrossRef] - Rea, W.; Oxley, L.; Reale, M.; Brown, J. Estimators for long range dependence: An empirical study. arXiv
**2009**, arXiv:0901.0762. [Google Scholar] - Soltani, S.; Simard, P.; Boichu, D. Estimation of the self-similarity parameter using the wavelet transform. Signal Process.
**2004**, 84, 117–123. [Google Scholar] [CrossRef] - Shen, H.; Zhu, Z.; Lee, T.C. Robust estimation of the self-similarity parameter in network traffic using wavelet transform. Signal Process.
**2007**, 87, 2111–2124. [Google Scholar] [CrossRef] - Park, J.; Park, C. Robust estimation of the Hurst parameter and selection of an onset scaling. Stat. Sin.
**2009**, 19, 1531–1555. [Google Scholar] - Feng, C.; Vidakovic, B. Estimation of the Hurst exponent using trimean estimators on nondecimated wavelet coefficients. arXiv
**2017**, arXiv:1709.08775. [Google Scholar] - Kang, M.; Vidakovic, B. MEDL and MEDLA: Methods for assessment of scaling by medians of log-squared nondecimated wavelet coefficients. arXiv
**2017**, arXiv:1703.04180. [Google Scholar] - Wu, L.; Ding, Y. Estimation of self-similar Gaussian fields using wavelet transform. Int. J. Wavelets Multiresolut. Inf. Process.
**2015**, 13, 1550044. [Google Scholar] [CrossRef] - Wu, L.; Ding, Y. Wavelet-based estimator for the Hurst parameters of fractional Brownian sheet. Acta Math. Sci.
**2017**, 37B, 205–222. [Google Scholar] [CrossRef] [Green Version] - Wu, L.; Ding, Y. Wavelet-based estimations of fractional Brownian sheet: Least squares versus maximum likelihood. J. Comput. Appl. Math.
**2020**, 371, 112609. [Google Scholar] [CrossRef] - Bardet, J.M.; Lang, G.; Oppenheim, G.; Philippe, A.; Stoev, S.; Taqqu, M.S. Semi-parametric estimation of the long-range dependence parameter: A survey. In Theory and Applications of Long-Range Dependence; Doukhan, P., Oppenheim, G., Taqqu, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; pp. 557–577. [Google Scholar]
- Tewfik, A.H.; Kim, M. Correlation structure of the discrete wavelet coefficients of fractional Brownian motion. IEEE Trans. Inf. Theory
**1992**, 38, 904–909. [Google Scholar] [CrossRef] - Dijkerman, R.W.; Mazumdar, R.R. On the correlation structure of the wavelet coefficients of fractional Brownian motion. IEEE Trans. Inf. Theory
**1994**, 40, 1609–1612. [Google Scholar] [CrossRef] - Abry, P.; Delbeke, L.; Flandrin, P. Wavelet based estimator for the self-similarity parameter of α-stable processes. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Phoenix, AZ, USA, 15–19 March 1999; Volume 3, pp. 1729–1732. [Google Scholar] [CrossRef]
- Veitch, D.; Taqqu, M.S.; Abry, P. Meaningful MRA initialization for discrete time series. Signal Process.
**2000**, 80, 1971–1983. [Google Scholar] [CrossRef] - Abry, P.; Flandrin, P. On the initialization of the discrete wavelet transform algorithm. IEEE Signal Process. Lett.
**1994**, 1, 32–34. [Google Scholar] [CrossRef] [Green Version] - Veitch, D.; Abry, P.; Taqqu, M.S. On the automatic selection of the onset of scaling. Fractals
**2003**, 11, 377–390. [Google Scholar] [CrossRef] - Papoulis, A.; Pillai, S.U. Probability, Random Variables and Stochastic Processes; Tata McGraw-Hill Education: New York, NY, USA, 2002. [Google Scholar]
- Davies, R.B.; Harte, D.S. Tests for Hurst effect. Biometrika
**1987**, 74, 95–101. [Google Scholar] [CrossRef] - Dieker, T. Simulation of Fractional Brownian Motion. Master’s Thesis, University of Twente, Amsterdam, The Netherlands, 2004. [Google Scholar]
- Kroese, D.P.; Botev, Z.I. Spatial process simulation. In Stochastic Geometry, Spatial Statistics and Random Fields; Spodarev, E., Ed.; Springer: Berlin/Heidelberg, Germany, 2015; pp. 369–404. [Google Scholar] [CrossRef]
- Dietrich, C.; Newsam, G.N. Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput.
**1997**, 18, 1088–1107. [Google Scholar] [CrossRef]

**Figure 1.**The Bias, Std, RMSE for estimators: ${j}_{1}$ is the lower bound of octaves js. Std is the standard deviation, Bias $=\mathbb{E}\widehat{H}-H$, RMSE is the square root of MSE. The values of Std, Bias, and RMSE are the estimated versions of those for 1000 independent copies of FBM with length $n={2}^{18}$. The used wavelet is the Daubechies wavelet with $N=3$ vanishing moments.

**Figure 2.**The Bias, Std, RMSE for estimators: n is the data length. Std is the standard deviation, Bias $=\mathbb{E}\widehat{H}-H$, RMSE is the square root of MSE. The values of Std, Bias, and RMSE are the estimated versions of those for 1000 independent copies of FBM with length n. The lower bound of octaves js is chosen ${j}_{1}=3$. The used wavelet is the Daubechies wavelet with $N=3$ vanishing moments.

**Figure 3.**The Bias, Std, RMSE for estimators: N is the number of vanishing moments of Daubechies wavelet. Std is the standard deviation, Bias $=\mathbb{E}\widehat{H}-H$, RMSE is the square root of MSE. The values of Std, Bias and RMSE are the estimated versions of those for 1000 independent copies of FBM with length $n={2}^{18}$. The lower bound of octaves js is chosen ${j}_{1}=3$.

**Figure 4.**The Bias, Std, RMSE for estimators: db3 stands for Daubechies wavelet with three vanishing moments, sym4 stands for Symlets wavelet with four vanishing moments, dmey stands for discrete Meyer wavelet, bior3.1 stands for biorthogonal spline wavelets with orders ${N}_{r}=3$ (vanishing moments) and ${N}_{d}=1$. The values of Std, Bias and RMSE are the estimated versions of those for 1000 independent copies of FBM with length $n={2}^{18}$. The lower bound of octaves js is chosen ${j}_{1}=3$.

**Figure 5.**The Bias and Std for estimators: M1 denotes the first estimator, M2 denotes the second estimator. Std is the standard deviation, Bias $=\mathbb{E}\widehat{H}-H$. The values of Std and Bias are the estimated versions of those for 1000 independent copies of FBM with length $n={2}^{18}$.

**Figure 6.**The Bias for estimators: circ denotes the circulant embedding method for simulation of FBM, chol denotes the Cholesky method for simulation of FBM. Bias $=\mathbb{E}\widehat{H}-H$. The values of Bias are the estimated versions of those for 1000 independent copies of FBM with length $n={2}^{12}$. The lower bound of octaves js is chosen ${j}_{1}=3$. The used wavelet is the Daubechies wavelet with $N=3$ vanishing moments.

**Figure 7.**The Bias and Std for estimators: Init1 denotes the initialization by itself (or by (27)), Init2 denotes the initialization denoted by (29). Std is the standard deviation, Bias $=\mathbb{E}\widehat{H}-H$. The values of Std and Bias are the estimated versions of those for 1000 independent copies of FBM with length $n={2}^{18}$.

**Figure 8.**The Bias, Std, RMSE for estimators: orig stands for original series without noise, gau stands for Gaussian noise, unm stands for uniform noise, cau stands for Cauchy noise. The values of Std, Bias and RMSE are the estimated versions of those for 1000 independent copies of FBM with noise. The data length $n={2}^{18}$. The lower bound of octaves js is chosen ${j}_{1}=3$. The used wavelet is the Daubechies wavelet with $N=3$ vanishing moments

**Table 1.**Estimation quality for FBM series. On the left, the ${j}_{1}$ for minimum MSE and its Bias, Std, RMSE is given. On the right, the same quantities with ${j}_{1}=3$ are also given for comparison. RMSE is the square root of MSE. All the results are the estimated versions of Bias, Std, RMSE for 1000 independent copies of FBM with length $n={2}^{18}$. The used wavelet is the Daubechies wavelet with $N=3$ vanishing moments.

H | ${\mathit{j}}_{1}^{\mathit{MSE}}$ | Bias | Std | RMSE | ${\mathit{j}}_{1}$ | Bias | Std | RMSE |
---|---|---|---|---|---|---|---|---|

0.05 | 7 | −0.0122 | 0.0133 | 0.0180 | 3 | −0.1450 | 0.0030 | 0.1451 |

0.10 | 6 | −0.0087 | 0.0091 | 0.0126 | 3 | −0.0801 | 0.0031 | 0.0801 |

0.15 | 6 | −0.0046 | 0.0090 | 0.0101 | 3 | −0.0499 | 0.0030 | 0.0500 |

0.20 | 5 | −0.0056 | 0.0064 | 0.0085 | 3 | −0.0330 | 0.0031 | 0.0331 |

0.25 | 5 | −0.0032 | 0.0063 | 0.0071 | 3 | −0.0226 | 0.0032 | 0.0228 |

0.30 | 5 | −0.0019 | 0.0063 | 0.0066 | 3 | −0.0160 | 0.0032 | 0.0163 |

0.35 | 4 | −0.0038 | 0.0045 | 0.0059 | 3 | −0.0115 | 0.0032 | 0.0119 |

0.40 | 4 | −0.0023 | 0.0047 | 0.0052 | 3 | −0.0081 | 0.0033 | 0.0087 |

0.45 | 4 | −0.0019 | 0.0048 | 0.0052 | 3 | −0.0060 | 0.0033 | 0.0068 |

0.50 | 4 | −0.0012 | 0.0048 | 0.0049 | 3 | −0.0044 | 0.0033 | 0.0056 |

0.55 | 3 | −0.0030 | 0.0034 | 0.0046 | 3 | −0.0030 | 0.0034 | 0.0046 |

0.60 | 3 | −0.0025 | 0.0036 | 0.0044 | 3 | −0.0025 | 0.0036 | 0.0044 |

0.65 | 3 | −0.0018 | 0.0034 | 0.0038 | 3 | −0.0018 | 0.0034 | 0.0038 |

0.70 | 3 | −0.0014 | 0.0035 | 0.0038 | 3 | −0.0014 | 0.0035 | 0.0038 |

0.75 | 3 | −0.0013 | 0.0037 | 0.0039 | 3 | −0.0013 | 0.0037 | 0.0039 |

0.80 | 3 | −0.0008 | 0.0036 | 0.0037 | 3 | −0.0008 | 0.0036 | 0.0037 |

0.85 | 3 | −0.0006 | 0.0037 | 0.0038 | 3 | −0.0006 | 0.0037 | 0.0038 |

0.90 | 3 | −0.0006 | 0.0037 | 0.0038 | 3 | −0.0006 | 0.0037 | 0.0038 |

0.95 | 3 | −0.0006 | 0.0038 | 0.0038 | 3 | −0.0006 | 0.0038 | 0.0038 |

**Table 2.**Estimation quality for FBM series. On the left, the ${j}_{1}$ for minimum MSE and its Bias, Std, RMSE is given. On the right, the same quantities with ${j}_{1}=3$ are also given for comparison. RMSE is the square root of MSE. All the results are the estimated versions of Bias, Std, RMSE for 1000 independent copies of FBM. The used wavelet is the Daubechies wavelet with $N=3$ vanishing moments.

H | n | ${\mathit{j}}_{1}^{\mathit{MSE}}$ | Bias | Std | RMSE | ${\mathit{j}}_{1}$ | Bias | Std | RMSE |
---|---|---|---|---|---|---|---|---|---|

${2}^{10}$ | 2 | −0.0632 | 0.0473 | 0.0789 | 3 | −0.0239 | 0.0776 | 0.0811 | |

${2}^{12}$ | 3 | −0.0220 | 0.0305 | 0.0376 | 3 | −0.0220 | 0.0305 | 0.0376 | |

0.3 | ${2}^{14}$ | 4 | −0.0080 | 0.0202 | 0.0217 | 3 | −0.0186 | 0.0136 | 0.0231 |

${2}^{16}$ | 4 | −0.0063 | 0.0096 | 0.0115 | 3 | −0.0167 | 0.0065 | 0.0179 | |

${2}^{18}$ | 5 | −0.0019 | 0.0063 | 0.0066 | 3 | −0.0160 | 0.0032 | 0.0163 | |

${2}^{10}$ | 2 | −0.0276 | 0.0479 | 0.0553 | 3 | −0.0078 | 0.0779 | 0.0783 | |

${2}^{12}$ | 2 | −0.0231 | 0.0202 | 0.0307 | 3 | −0.0073 | 0.0312 | 0.0320 | |

0.5 | ${2}^{14}$ | 3 | −0.0048 | 0.0142 | 0.0149 | 3 | −0.0048 | 0.0142 | 0.0149 |

${2}^{16}$ | 3 | −0.0050 | 0.0068 | 0.0085 | 3 | −0.0050 | 0.0068 | 0.0085 | |

${2}^{18}$ | 4 | −0.0012 | 0.0048 | 0.0049 | 3 | −0.0044 | 0.0033 | 0.0056 | |

${2}^{10}$ | 2 | −0.0109 | 0.0526 | 0.0537 | 3 | −0.0056 | 0.0878 | 0.0879 | |

${2}^{12}$ | 2 | −0.0061 | 0.0233 | 0.0241 | 3 | −0.0010 | 0.0359 | 0.0359 | |

0.8 | ${2}^{14}$ | 2 | −0.0062 | 0.0106 | 0.0123 | 3 | −0.0015 | 0.0159 | 0.0160 |

${2}^{16}$ | 3 | −0.0010 | 0.0074 | 0.0075 | 3 | −0.0010 | 0.0074 | 0.0075 | |

${2}^{18}$ | 3 | −0.0008 | 0.0036 | 0.0037 | 3 | −0.0008 | 0.0036 | 0.0037 |

H | 0.05 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.50 |
---|---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathrm{Var}}\widehat{{H}_{2}}/\widehat{\mathrm{Var}}\widehat{{H}_{1}}$ | 2.41 | 2.18 | 2.32 | 2.27 | 2.16 | 2.19 | 2.27 | 2.22 | 2.18 | 2.13 |

H | 0.55 | 0.60 | 0.65 | 0.70 | 0.75 | 0.80 | 0.85 | 0.90 | 0.95 | |

$\widehat{\mathrm{Var}}\widehat{{H}_{2}}/\widehat{\mathrm{Var}}\widehat{{H}_{1}}$ | 1.92 | 2.08 | 2.00 | 1.94 | 1.76 | 1.95 | 1.71 | 1.89 | 1.88 |

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**MDPI and ACS Style**

Wu, L.
A Note on Wavelet-Based Estimator of the Hurst Parameter. *Entropy* **2020**, *22*, 349.
https://doi.org/10.3390/e22030349

**AMA Style**

Wu L.
A Note on Wavelet-Based Estimator of the Hurst Parameter. *Entropy*. 2020; 22(3):349.
https://doi.org/10.3390/e22030349

**Chicago/Turabian Style**

Wu, Liang.
2020. "A Note on Wavelet-Based Estimator of the Hurst Parameter" *Entropy* 22, no. 3: 349.
https://doi.org/10.3390/e22030349