# Noise Cancellation Method Based on TVF-EMD with Bayesian Parameter Optimization
## 1 Introduction  
Signals contain rich characteristic information; thus, signal processing analysis is crucial in the development of natural sciences, particularly in environmental monitoring [1]. There is a relationship between the indoor carbon dioxide concentration and respiratory mucosal symptoms [2; 3]; hence, it is essential to monitor and study carbon dioxide concentrations in various functional building spaces in real time. However, when there are sudden changes in background concentrations of CO\({}_{2}\) 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. There are various methods of signal denoising, and the most commonly used include smoothing, Fourier transform [4], wavelet theory [5], and Hilbert-Huang transform [6; 7]. Smoothing is a simple and convenient method that denoises the signal at the expense of reduced temporal resolution [8; 9]. In Fourier and wavelet transforms, the selection of the a priori basis function directly affects the results of signal noise reduction [9; 10], which is a major limitation in the application of these methods. The core of the Hilbert-Huang transform is empirical modal decomposition (EMD). The EMD model identifies the intrinsic oscillatory modes in the signal, based on the local characteristic timescale of the signal, and, accordingly, decomposes the raw signal into several intrinsic mode functions(IMFs) without requiring previous knowledge of original signal values [11]. Therefore, EMD is commonly adopted for analyzing adaptive, nonlinear, and nonstationary signal processing, such as indoor CO\({}_{2}\) 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. The time-varying, filtering-based empirical mode decomposition (TVF-EMD) was developed to address the shortcomings of the EMD model, with the shifting process completed by the B-spline approximation filter [12]. The three main features of TVF-EMD, compared with most existing methods, can be summarized as follows: (1) The TVF-EMD method can simultaneously address the issues of separation and intermittence [15]. (2) With a B-spline approximation filter integrated into the shifting process, TVF-EMD solves the issue of mode mixing and maintains time-varying features. (3) The enhanced stopping criterion improves the performance of the TVF-EMD model for low sampling rates. However, two significant effects, bandwidth threshold \(\xi\) and B-spline order \(n\), have direct impacts on the separation and filter performance of the time-varying filter, respectively [12]. A reasonable selection of a combination of the two parameters in advance enables the TVF-EMD model to resolve the mode-mixing problem, thereby achieving optimal noise reduction. Hence, the choice of parameters for the TVF-EMD is clearly important. Many optimization methods are used in hyperparameter estimation. Among the most widely used are the grid search [16; 17], random search [18], Genetic algorithm [19], and Bayesian optimization (BO) algorithms [20; 21]. Unlike grid and random searches, the framework of BO leverages information from existing data, and the current search for optimal values is based on previous search results [22]. In addition, there are swarm intelligence optimization technologies that imitate the behaviors of various organisms, such as ants [23], particle swarm optimization [24], fish schools [25], glow worm swarm optimization [26], and grey wolves [27], to achieve optimal parameter estimation. [PERSON] et al.