# Noise Cancellation Method Based on TVF-EMD with Bayesian Parameter Optimization
## 3 Adopted Methodology  
The proposed optimized TVF-EMD method is based on the BO-TPE algorithm. It aims to search for optical combinations of parameters for the bandwidth threshold \(\xi\) and B-spline order \(n\) using the objective function, which determines the merits of the decomposition results. The kurtosis index \(Kur\) depends on the distribution density of the signal, which is highly sensitive to large amplitudes with dispersed distributions. A kurtosis index value between 0 and 3 indicates that the center peak of the signal is lower and broader, compared to the normal distribution represented by \(Kur=3(Kur\approx 3)\). In contrast, \(Kur>3\) indicates that the central peak of the signal is higher and sharper. Thus, a smaller kurtosis index is required for a more sensitive identification of outliers. However, to avoid excessive noise cancellation, the correlation coefficient (\(CC\)) is used to characterize the similarities between original and decomposing signals. Therefore, the synthetic measurement index, consisting of the kurtosis index and CC, was developed as an objective function for TVF-EMD parameter optimization. The synthetic measurement index, correlation coefficient for kurtosis index (\(CCKur\)), was calculated as follows:  
\[CCKur=\frac{|CC|}{Kur} \tag{6}\]  
\[CC:\rho_{x,imf}=\frac{\sum\limits_{i=1}^{N}{(x_{i}-\overline{x})\left(imf_{i}- \overline{imf}\right)}}{\sqrt{\sum\limits_{i=1}^{N}{(x_{i}-\overline{x})^{2} \left(imf_{i}-\overline{imf}\right)^{2}}}} \tag{7}\]  
\[Kur=\frac{1}{M}\sum\limits_{m=0}^{m-1}{x^{4}(m)/\left(\frac{1}{M}\sum\limits_{ m=0}^{m-1}{x^{4}(m)}\right)^{2}} \tag{8}\]  
Figure 1: Flowchart of Bayesian optimization. As the original BO algorithm was developed to determine the maximum value, the method used the maximum _CCKur_ between the original signal and the modes obtained by TVF-EMD as the fitness. Therefore, the maximization of _CCKur_ was the optimization problem expressed below:  
\[\left\{ \begin{matrix} fitness=\max_{\gamma\in\{\xi,n\}}\{CCKur\}\\ \xi\in(0,0.4]\\ n\in[1,16]\end{matrix} \right. \tag{9}\]  
where \(fitness\) is the objective function, and \(\gamma=\{\xi,n\}\) represents the parameter combination of the TVF-EMD method to be optimized. The CC of the original signal \(x(t)\), which is a function of mode \(imf(t)\) with the same length as \(N\), is described in Equation (7). Ensuring the reliability of the parameter optimization was essential, and the number of modes after CO\({}_{2}\) signal decomposition must be at least two. After several attempts, it was discovered that the bandwidth threshold \(\xi\) met \(0\leq\xi\leq 0.4\), and \(1\leq n\leq 16\) 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 \(K=\log_{2}N\)[4]. The autocorrelation properties of the signal and Hurst exponent \(H\) were evaluated to distinguish noise from the most relevant modes. When \(0\leq H<0.5\), two signals were anticorrelated. In contrast, \(H=0.5\) indicated white noise, and \(H>0.5\) represented a positive correlation. In this study, the threshold for the Hurst index was defined as 0.5 (\(H_{thr}=0.5\)). The detailed steps of the noise cancellation methodology were as follows:  
Step 1: Input the CO\({}_{2}\) concentration signal \(x(t)\) 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 \(n\). Step 2: Decompose the signal \(x(t)\) using the TVF-EMD model for the parameter combination of \(\xi\) and \(n\), and then calculate the IMFs to obtain the objective function \(fitness\), where the best fitness for each iteration of the BO algorithm is stored. Step 3: If the stored value of fitness satisfies the threshold, then save the optical parameters \(\xi\) and \(n\). Otherwise, \(l=l+1\), and continue Step 2 to update parameters \(\xi\) and \(n\) until the maximization of \(CCKur\) is up to requirements.