# Noise Cancellation Method Based on TVF-EMD with Bayesian Parameter Optimization

^{*}

## Abstract

**:**

_{2}) concentration signal, we proposed a noise cancellation method, based on time-varying, filtering-based empirical mode decomposition (TVF-EMD) with Bayesian optimization (BO). The adaptive parameters of TVF-EMD, that is, bandwidth threshold $\xi $ and B-spline order $n$, were determined by the BO algorithm, and the correlation coefficient for the kurtosis index (CCKur) constituted the objective function. Initially, the objective function CCKur was introduced to systematically identify anomalous signals while preserving signal feature extraction between the modes and the input signal. Subsequently, the proposed signal noise cancellation model based on TVF-EMD and the BO algorithm were employed, along with the Hurst exponent, to extract the sensitive mode. An examination of the optimization indices of the decomposed intrinsic mode functions (IMFs), namely CC, Kur, MI, EE, EEMI, and CCKur, revealed that the synthetic measurement index CCKur and objective function fitness were reasonable and effective. The proposed method exhibited better signal cancellation performance, compared to that of TVF-EMD with the default values, EMD, the moving average method, and the exponential smoothing method.

## 1. Introduction

_{2}or equipment sampling failures, the CO

_{2}-monitoring signal becomes noisy, making it difficult to distinguish important signal features from noise. Therefore, the noise cancellation of the CO

_{2}-monitoring signal is important.

_{2}concentration signals. However, the EMD methods applied for noise reduction are subject to modal aliasing and endpoint effects, known as the mode-mixing issue [12]. In other words, EMD is prone to noise. Based on the framework of EMD, the improved methods, such as ensemble EMD [13] and noise-assisted MEMD [14], are proposed to solve the intermittence problem of EMD. However, there are still many problems with these methods, such as unselectable parameters of noise and failure to separate modes.

_{2}concentrations in buildings is a typical non-stationary signal. These signal-denoising methods, such as Fourier transform, EMD, ensemble EMD, and noise-assisted MEMD, are not suitable for processing non-stationary signals. The wavelet transform is suitable for dealing with non-stationary, time-varying signals, but there is the problem of difficult wavelet basis selection. The TVF-EMD method can solve the shortcomings of the above methods. Combined with the hyperparameter algorithm, the TVF-EMD method can match the appropriate parameters more quickly to achieve accurate noise reduction of the indoor CO

_{2}concentration.

_{2}concentration signals from two different sets of functional building spaces demonstrated that the method was successful in canceling noise and could be effective in CO

_{2}-monitoring engineering applications.

## 2. Preliminaries

#### 2.1. Time-Varying Filter Empirical Mode Decomposition

- Local cutoff frequency rearrangement

- 2.
- Shifting process for TVF-EMD

#### 2.2. Bayesian Optimization

## 3. Adopted Methodology

_{2}signal decomposition must be at least two. After several attempts, it was discovered that the bandwidth threshold $\xi $ met $0\le \xi \le 0.4$, and $1\le n\le 16$ satisfied the requirements. The same number of modes was obtained with the typically used standard EMD, and the maximum number of modes for the TVF-EMD was set to $K={\mathrm{log}}_{2}N$ [4].

_{2}concentration signal $x\left(t\right)$ and set the parameter population $X$ to the TVF-EMD model. Concurrently, initialize the parameters of BO algorithms and population $X$, including the bandwidth threshold $\xi $ and B-spline order $n$.

_{2}signal.

## 4. Results and Discussion

#### 4.1. Data Acquisition

_{2}concentration was analyzed after denoising the CO

_{2}signals, which were obtained using pump-activated CO

_{2}concentration detectors installed in offices and university classrooms. The CO

_{2}signals were sampled at a frequency of 5 s and monitored continuously for 24 h. Figure 3 illustrates the CO

_{2}signal from an $8\text{}\mathrm{m}\times 8\text{}\mathrm{m}\times 3\text{}\mathrm{m}$ regular-sized office accommodating approximately six to ten people. The detector measured the CO

_{2}concentration at a height of 2.8 m in the room. The specifications used are listed in Table 1.

#### 4.2. Analysis of the Simulated Signal

#### 4.2.1. Simulation and Comparison

_{2}signal was then processed by the proposed TVF-EMD model, based on the BO algorithm. The combination of the bandwidth threshold $\xi $, B-spline order $n$, and the maximum $CCKur$ of the BO algorithm were $\xi =0.31$, $n=7$, and $CCKu{r}_{max}=0.49$, respectively. Figure 4b shows the signal decomposition results obtained using the optimized TVF-EMD model.

#### 4.2.2. The Effects of TVF-EMD Parameters

_{2}concentration signal $x\left(t\right)$ and the effect of TVF-EMD parameters on the decomposition results are shown in Figure 8. The bandwidth threshold $\xi $ reached 0.95 in the decomposition results, indicating that the selection of the optimization index should be sensitive to the change in $\xi $. Therefore, we analyzed the results of CC, Kur, MI, EE, EEMI, and $CCKur$ under varying $\xi $. As shown in Figure 9, when $\xi $ assumed values of 0.1, 0.2, and 0.31 (the optimal parameter), eight, nine, and seven IMFs were obtained by decomposition, respectively. When $\xi $ was at least four, two IMFs were obtained, indicating that mode aliasing and under-decomposition occurred. When $\xi $ increased from 0.1 to 0.31, the MI, EE, and EEMI indices of the IMFs did not change significantly, suggesting that these indices were insensitive to the variation in $\xi $. The CC, Kur, and $CCKur$ indices were sensitive to $\xi $; hence, they can be selected as indices for the best parameters. When $\xi =0.31$, the sum of the $CCKur$ indices of IMF1 to IMF9 was the largest, which indicates that $CCKur$ can be regarded as the best optimization model. The results showed that (1) $\xi $ had to be at least 4 for mode aliasing and under-decomposition to occur; (2) $CC$, $Kur$, and $CCKur$ indices were sensitive to $\xi $, whereas MI, EE, and EEMI indices were not; (3) the proposed $CCKur$ index and objective function $\underset{\gamma \in \left\{\xi ,n\right\}}{\mathrm{max}}\left\{CCKur\right\}$ were effective; and (4) the value of 0.31 for $\xi $ was reasonable and correct.

#### 4.3. Validation of the Proposed Method

_{2}signals studied from the university classroom had a capacity of 100 students. As shown in Table 2, the proposed noise cancellation model yielded the largest absolute SNR value. The MSE, RES, and NRMSE of the proposed method were all smaller than those obtained by the TVF-EMD, EMD, moving average, and exponential smoothing methods, indicating that the proposed noise cancellation model had superior noise reduction quality for the CO

_{2}concentration signal.

## 5. Conclusions

_{2}concentration signal of a building. The Bayesian algorithm was used to optimally estimate the TVF-EMD parameters, namely the bandwidth threshold $\xi $ and B-spline order $n$, and the adaptive matching of the given CO

_{2}concentration signal. The main conclusions can be summarized as follows:

_{2}concentration signals.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Longo, A.; Bianchi, S.; Plastino, W. tvf-EMD based time series analysis of 7Be sampled at the CTBTO-IMS network. Phys. A Stat. Mech. Its Appl.
**2019**, 523, 908–914. [Google Scholar] [CrossRef] [Green Version] - Apte, M.G.; Fisk, W.J.; Daisey, J.M. Associations between indoor CO
_{2}concentrations and sick building syndrome symptoms in US office buildings: An analysis of the 1994-1996 BASE study data. Indoor Air-Int. J. Indoor Air Qual. Clim.**2000**, 10, 246–257. [Google Scholar] [CrossRef] [Green Version] - Muscatiello, N.; McCarthy, A.; Kielb, C.; Hsu, W.H.; Hwang, S.A.; Lin, S. Classroom conditions and CO
_{2}concentrations and teacher health symptom reporting in 10 New York State Schools. Indoor Air**2015**, 25, 157–167. [Google Scholar] [CrossRef] [PubMed] - Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.L.C.; Shih, H.H.; Zheng, Q.N.; Yen, N.C.; Tung, C.C.; Liu, H.H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.
**1998**, 454, 903–995. [Google Scholar] [CrossRef] - Wang, W.J.; McFadden, P.D. Application of wavelets to gearbox vibration signals for fault detection. J. Sound Vib.
**1996**, 192, 927–939. [Google Scholar] [CrossRef] - Huang, N.E.; Shen, Z.; Long, S.R. A new view of nonlinear water waves: The Hilbert Spectrum. Annu. Rev. Fluid Mech.
**1999**, 31, 417–457. [Google Scholar] [CrossRef] [Green Version] - Montesinos, M.E.; Muñoz-Cobo, J.L.; Pérez, C. Hilbert–Huang analysis of BWR neutron detector signals: Application to DR calculation and to corrupted signal analysis. Ann. Nucl. Energy
**2003**, 30, 715–727. [Google Scholar] [CrossRef] - Tian, P.; Cao, X.; Liang, J.; Zhang, L.; Yi, N.; Wang, L.; Cheng, X. Improved empirical mode decomposition based denoising method for lidar signals. Opt. Commun.
**2014**, 325, 54–59. [Google Scholar] [CrossRef] - Wu, S.; Liu, Z.; Liu, B. Enhancement of lidar backscatters signal-to-noise ratio using empirical mode decomposition method. Opt. Commun.
**2006**, 267, 137–144. [Google Scholar] [CrossRef] - Torrence, C.; Compo, G.P. A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc.
**1998**, 79, 61–78. [Google Scholar] [CrossRef] - Huang, N.E.; Wu, Z.; Long, S.R. Hilbert-Huang transform. Scholarpedia
**2008**, 3, 2544. [Google Scholar] [CrossRef] - Li, H.; Li, Z.; Mo, W. A time varying filter approach for empirical mode decomposition. Signal Process.
**2017**, 138, 146–158. [Google Scholar] [CrossRef] - Wu, Z.; Huang, N.E. Ensemble empirical mode decomposition: A noise-assisted data analysis method. Adv. Adapt. Data Anal.
**2009**, 1, 1–41. [Google Scholar] [CrossRef] - Ur Rehman, N.; Mandic, D.P. Filter bank property of multivariate empirical mode decomposition. IEEE Trans. Signal Process.
**2011**, 59, 2421–2426. [Google Scholar] [CrossRef] - Rehman, N.; Mandic, D.P. Multivariate empirical mode decomposition. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2010**, 466, 1291–1302. [Google Scholar] [CrossRef] - Stoica, P.; Gershman, A.B. Maximum-likelihood DOA estimation by data-supported grid search. IEEE Signal Process. Lett.
**1999**, 6, 273–275. [Google Scholar] [CrossRef] - Bellman, R.E.; Dreyfus, S.E. Applied Dynamic Programming; Princeton University Press: Princeton, NJ, USA, 2015; Volume 2050. [Google Scholar] [CrossRef]
- Bergstra, J.; Bengio, Y. Random search for hyper-parameter optimization. J. Mach. Learn. Res.
**2012**, 13, 281–305. [Google Scholar] - Yang, X.-S. (Ed.) Chapter 6—Genetic Algorithms. In Nature-Inspired Optimization Algorithms, 2nd ed.; Academic Press: Cambridge, MA, USA, 2021; pp. 91–100. [Google Scholar] [CrossRef]
- Frazier, P.I. A tutorial on Bayesian optimization. arXiv
**2018**, arXiv:1807.02811. [Google Scholar] - Shahriari, B.; Swersky, K.; Wang, Z.; Adams, R.P.; De Freitas, N. Taking the human out of the loop: A review of Bayesian optimization. Proc. IEEE
**2015**, 104, 148–175. [Google Scholar] [CrossRef] [Green Version] - Jones, D.R.; Schonlau, M.; Welch, W.J. Efficient global optimization of expensive black-box functions. J. Glob. Optim.
**1998**, 13, 455–492. [Google Scholar] [CrossRef] - Mirjalili, S. The ant lion optimizer. Adv. Eng. Softw.
**2015**, 83, 80–98. [Google Scholar] [CrossRef] - Marini, F.; Walczak, B. Particle swarm optimization (PSO). A tutorial. Chemom. Intell. Lab. Syst.
**2015**, 149, 153–165. [Google Scholar] [CrossRef] - Mirjalili, S.; Lewis, A. The whale optimization algorithm. Adv. Eng. Softw.
**2016**, 95, 51–67. [Google Scholar] [CrossRef] - Krishnanand, K.N.; Ghose, D. Glowworm swarm optimization for simultaneous capture of multiple local optima of multimodal functions. Swarm Intell.
**2009**, 3, 87–124. [Google Scholar] [CrossRef] - Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolf optimizer. Adv. Eng. Softw.
**2014**, 69, 46–61. [Google Scholar] [CrossRef] [Green Version] - Zhou, C.; Xiong, Z.; Bai, H.; Xing, L.; Jia, Y.; Yuan, X. Parameter-Adaptive TVF-EMD Feature Extraction Method Based on Improved GOA. Sensors
**2022**, 22, 7195. [Google Scholar] [CrossRef] - Bergstra, J.; Bardenet, R.; Bengio, Y.; Kégl, B. Algorithms for hyper-parameter optimization. Adv. Neural Inf. Process. Syst.
**2011**, 24, 2546–2554. [Google Scholar] - Bergstra, J.; Yamins, D.; Cox, D. Making a science of model search: Hyperparameter optimization in hundreds of dimensions for vision architectures. In Proceedings of the 30th International Conference on Machine Learning, Atlanta, GA, USA, 17–19 June 2013; pp. 115–123. [Google Scholar]
- Ozaki, Y.; Tanigaki, Y.; Watanabe, S.; Nomura, M.; Onishi, M. Multiobjective Tree-Structured Parzen Estimator. J. Artif. Intell. Res.
**2022**, 73, 1209–1250. [Google Scholar] [CrossRef] - Zhang, S.; Xu, F.; Hu, M.; Zhang, L.; Liu, H.; Li, M. A novel denoising algorithm based on TVF-EMD and its application in fault classification of rotating machinery. Measurement
**2021**, 179, 109337. [Google Scholar] [CrossRef] - Wang, J.; Liu, P.; Lu, S.; Zhou, M.; Chen, X. Decentralized plant-wide monitoring based on mutual information-Louvain decomposition and support vector data description diagnosis. ISA Trans.
**2023**, 133, 42–52. [Google Scholar] [CrossRef] - Gao, S.; Ren, Y.; Zhang, Y.; Li, T. Fault diagnosis of rolling bearings based on improved energy entropy and fault location of triangulation of amplitude attenuation outer raceway. Measurement
**2021**, 185, 109974. [Google Scholar] [CrossRef]

Parameters | Value |
---|---|

Sampling frequency | 5 s |

Measurement accuracy | 1 ppm |

Error range | 5% |

Time sample length | 24 h |

Index | Proposed TVF-EMD $(\mathit{\xi}=0.015,\mathit{n}=4)$ | TVF-EMD with Default Values $(\mathit{\xi}=0.1,\mathit{n}=26)$ | EMD | Moving Average (Window = 40) | Exponential Smoothing $(\mathit{\alpha}=0.5)$ |
---|---|---|---|---|---|

SNRabs | 71.376 | 71.296 | 71.141 | 71.251 | 71.239 |

MSE | 7.877 | 95.976 | 3217.423 | 154.281 | 21.473 |

RSE | 0.000276 | 0.00336 | 0.112 | 0.00540 | 0.000752 |

NRMSE | 0.0166 | 0.0581 | 0.3614 | 0.0736 | 0.0274 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yu, M.; Yuan, H.; Li, K.; Deng, L.
Noise Cancellation Method Based on TVF-EMD with Bayesian Parameter Optimization. *Algorithms* **2023**, *16*, 296.
https://doi.org/10.3390/a16060296

**AMA Style**

Yu M, Yuan H, Li K, Deng L.
Noise Cancellation Method Based on TVF-EMD with Bayesian Parameter Optimization. *Algorithms*. 2023; 16(6):296.
https://doi.org/10.3390/a16060296

**Chicago/Turabian Style**

Yu, Miaomiao, Hongyong Yuan, Kaiyuan Li, and Lizheng Deng.
2023. "Noise Cancellation Method Based on TVF-EMD with Bayesian Parameter Optimization" *Algorithms* 16, no. 6: 296.
https://doi.org/10.3390/a16060296