## Abstract

In numbers of industrial fields, many filtering algorithms of industrial signals, mechanism-based modeling methods and control strategies are based on the hypothesis of white noise. However, some researchers propose that the colored noise is closer to the real noise than the white noise. Then, whether the noise is the white noise, the colored noise or other else? And what is the intrinsic dynamic characteristics of the noise? In this paper, noise signals of thermal variables from rotary kiln are extracted and their chaotic, statistical and multifractal characteristics are analyzed to answer the two questions. Based on the experimental results, it is the first time to discover that they are not the white noise or the monofractal colored noise but have the high-dimensional chaotic characteristic, that is, they are determinate and predictable for short term theoretically. However, some models are failed to predict them. Then, further experimental results imply those noise signals have both persistent and anti-persistent multifractal characteristics. In particular, the latter is a reason to failed predictions of noise signals. Moreover, it is firstly discovered that the multifractality of each noise signal is generated mainly by the long-term temporal correlation. Finally, two ideas about modeling multifractality of noise from rotary kiln are proposed as the future work.

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## Acknowledgements

This work was supported in part by the National Natural Sciences Foundation of China (Nos. 61672216, 61673162), in part by the Natural Science Foundation of Hunan Province (No. 2018JJ2056), in part by the Research Committee at University of Macau (No. MYRG2018-00136-FST) and in part by the Macau Science and Technology Development Fund (No. FDCT/189/2017/A3). We thank the editor and anonymous reviewers for valuable comments that improved this article. We also thank Professor Charles L. Webber, Professor Herwig Wendt and Post-Doctor Tim Ziemer for, respectively, providing profound insights about chaotic theory, multifractal theory and noise signal. In particular, we thank to Professor Michael Small for discussing with us about the surrogate analysis.

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## Appendix

### Appendix

### 1.1 The wavelet packet decomposition method

The steps of noise extraction based the wavelet packet decomposition (WPD) method are introduced as follows. Moreover, please see Ref. [46, 47] for the more details.

- Step 1.:
The wavelet packet decomposition (WPD) and the confirmation of the best basis. In the point of view of compression, the standard wavelet transform may not produce the best result, since it is limited to wavelet bases that increase by a power of two toward the low frequencies, that is, the best basis of the WPD should be found out after the WPD. Moreover, the best basis corresponds to the minimum entropy or maximum information for the distribution of coefficients [46]. In this paper, the first step can be realized using the command “wpdec” in MATLAB.

- Step 2.:
The confirmation of the thresholding value. Generally, small coefficients are mostly noises and large coefficients contain the actual signal. Then, a thresholding value should be proposed to distinguish small coefficients and large coefficients. The penalization method is proposed to calculate the universal thresholding value and can be realized using the command “wpbmpen” in MATLAB.

- Step 3.:
The reconstruction of the signal based on wavelet transform. Donoho [47] pointed out that only coefficients whose absolute value is higher than the predefined threshold value is retained. Then, in this paper, the soft thresholding method [47] is used to replace wavelet transform coefficients as follows:

$$\begin{aligned} Rw_i^p =\left\{ \begin{array}{ll} 0, &{}\quad \mathrm{if} \quad |w_i^p|<\mathrm{thr} \\ \mathrm{sign}(w_i^p)(|w_i^p|-thr), &{}\quad \mathrm{if}\quad |w_i^p|>\mathrm{thr} \end{array} \right. \end{aligned}$$(22)where the \(w_i^p\) is the wavelet transform coefficient of the

*i*th sub-signal in the*p*th level, the \(Rw_i^p\) is the replace wavelet transform coefficient of the*i*th sub-signal in the*p*th level and the thr is the universal threshold which is calculated by the penalization method [47]. After replacing wavelet transform coefficients, the retained coefficients are used to reconstruct the useful signal. And the denoise procedure is realized using the command “wpdencmp” in Matlab.- Step 4.:
The noise signal is extracted through subtracting the useful signal from the original signal.

### 1.2 The Gao method

The Gao method [65] is a moving average method in nature, and the illustration of noise extraction using the Gao method is shown in Fig. 13. Moreover, the procedure is introduced as follows.

- Step 1.:
From the head to the end,

*l*signal points are selected as a segment in turn.- Step 2.:
The average value of each segment is calculated.

- Step 3.:
The average value of each segment is subtracted by the

*l*signal values of relative segment. Then,*l*noise values in each segment are extracted.- Step 4.:
The whole noise time series are extracted.

The higher frequency of noise, the smaller the value of the *l* is selected [65]. In our paper, we set \(l=2\), that is, two points of signals are selected as a segment in turn and the average value \(\overline{s}_i=(s_i+s_{i+1})/2\) is calculated. Then, each value of noise is calculated through \(x_{i+1}=s_{i+1}- \overline{s}_i\).

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Lv, M., Zhang, X., Chen, H. *et al.* Chaotic and multifractal characteristic analysis of noise of thermal variables from rotary kiln.
*Nonlinear Dyn* **99**, 3089–3111 (2020). https://doi.org/10.1007/s11071-020-05466-0

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DOI: https://doi.org/10.1007/s11071-020-05466-0