## 3 Feature Selection and Decision
### GP Based on Bayesian Inference  
This section describes the GP regression [31] used to train and test the proposed algorithm. Owing to the size of the paper, the description of the conventional algorithms is omitted. The GP algorithm is a robust, flexible, nonparametric Bayesian algorithm used in supervised ML [31]. To train the GP algorithm, the explanatory and response variables should be prepared as the input and output data, respectively, as \(\mathbf{D}=\{x_{i},y_{i}\}_{i=1}^{1}\), \(x\in\mathbf{R}^{1\times D}\), and \(y\in\mathbf{R}^{1\times 1}\). Here, we used a mapping function \(f_{m}=f(x)\) to estimate \(y\) given \(x\). Hence, we assumed that the response variable \(y\) was acquired using the corresponding \(x^{\mathrm{T}}w\) by including noise, as follows  
\[y=x^{\mathrm{T}}w+\varepsilon,\quad\varepsilon\backsim\mathbf{N}(0,\sigma^{2} \mathrm{I}) \tag{14}\]  
The weighted vectors \(w\) and variance \(\sigma^{2}\) were acquired from the resampled signal dataset. The GP algorithm estimates the response variable based on Gaussian processes (GPs) using the mapping function \(f_{m}(x)\) and explicit essential functions \(\beta\). \[f_{m}(x)\backsim\mathbf{GP}(0,\mathrm{k}(x,x^{{}^{\prime}})) \tag{15}\]  
where \(f_{m}(x)\) is acquired from a zero-mean GP algorithm using a covariance function \(\mathrm{k}(x,x^{{}^{\prime}})\)[32]. Hence, we can obtain the mapping function \(f_{m}(x)=\beta(x)^{\mathrm{T}}w\). The mean function of the input data can be defined as the expected value of the mapping function \(\theta(x)=\mathbf{E}[f_{m}(x)]\). A latent variable covariance function obtains the smoothness of the response variables, and the basic function projects the input data \(x\) into the dimensional feature space. \[\mathrm{k}(x,x^{{}^{\prime}})=\mathbf{E}[(f_{m}(x)-\theta(x))(f_{m}(x^{{}^{ \prime}})-\theta(x^{{}^{\prime}}))^{\mathrm{T}}] \tag{16}\]  
We defined the expected value of (16) as  
\[\mathrm{k}(x,x^{{}^{\prime}}|\eta)\approx\sigma^{2}\exp\biggl{(}-\frac{\|x-x^ {{}^{\prime}}\|^{2}}{2\eta^{2}}\biggr{)} \tag{17}\]  
where k is a kernel for the GP [31], \(\eta\) is a hyperparameter, and \(\sigma^{2}\) is a variance based on resampled signals. In the study, we used exponential squares as the kernel, as in (17). Thus, the kernel decided the properties of the mapping function \(f_{m}(x)\). We could define an instance of response variables \(\mathbf{y}\) using the Bayesian inference based GP as  
\[p(y_{i}|f_{m}(x_{i}),x_{i})\backsim\mathbf{N}\bigl{(}y_{i}|\beta(x_{i})^{ \mathrm{T}}w+f_{m}(x_{i}),\sigma^{2}\bigr{)} \tag{18}\]  
where \(\beta(x_{i})\) denotes a basic function transforming the original explanatory variable \(x\) into a new variable \(\beta(x)\). Thus, we determined \(\mathbf{\Theta}=\{w,\eta,\sigma^{2}\}\) from the dataset \(\mathbf{D}\), and the marginal likelihood was expressed as  
\[p(y|x)=p(y|x,\mathbf{\Theta})\approx\mathbf{N}(y|\mathbf{\Omega}w,\mathrm{k} (x,x^{{}^{\prime}}|\eta)+\sigma^{2}\mathrm{I}), \tag{19}\]  
Generally, the local maxima for the hyperparameter \(\mathbf{\Theta}\) can be determined and used to train the GP algorithm. In addition, choosing an appropriate kernel depends on hypotheses, such as the smoothness and expected patterns of the data. By maximizing the log marginal likelihood, we could estimate the hyperparameter \(\mathbf{\Theta}\), as follows  
\[\log p(y|x,\mathbf{\Theta})=-\frac{1}{2}\log\big{|}\mathrm{k}(x, x^{{}^{\prime}}|\eta)+\sigma^{2}\mathrm{I}|-\frac{1}{2}i\log 2\pi\\ -\frac{1}{2}\bigl{(}y-\mathbf{\Omega}w\bigr{)}^{\mathrm{T}}\bigl{[} \mathrm{k}(x,x^{{}^{\prime}}|\eta)+\sigma^{2}\mathrm{I}\bigr{]}^{-1}(y- \mathbf{\Omega}w) \tag{20}\]  
where \(\mathrm{k}(x,x^{{}^{\prime}}|\eta)\) is the kernel matrix and \(\mathbf{\Omega}\) denotes the matrix of the explicit basic function. Herein, we applied a penalty-fitting scale to represent the logarithmic likelihood and maximized it using a gradient approach using optimization techniques.