## 2 Background Knowledge
### Gaussian Processes Regression  
GPR [25; 26; 27] is one of the representative nonparametric Bayesian algorithms. GPR is a technique of estimating and updating the frequency of occurrence, using a Gaussian nonlinear distribution function with a mean function and a covariance function. GPR algorithm has been used in many research fields, as provided in Table 3.  
GPR has been used in many fields, as it has high prediction measurement for changing environment based on a Bayesian framework, and enables probabilistic analyses using Gaussian distribution. Figure 2 illustrates the process for estimating the final output value \(Y\), using an \(f_{i}\) at the input value \(X_{*}\) after producing the intermediate value \(f_{i}\) by Bayesian renewal to convert the input value \(X_{i}\) to the observed value \(Y_{i}\). Equations (8)-(11) summarize the general GPR model. \[ \begin{array}{l}\text{y}=f+\epsilon,\\ \text{where, }\ \epsilon\sim\text{N}\big{(}0,\sigma_{y}^{2}\big{)}\end{array}  \tag{8}\]  
\[ \begin{array}{l}\text{D}=\{(X_{i},Y_{i}),\ \text{i}=1,\ldots,\text{a}\} \end{array}  \tag{9}\]  
\[ \begin{array}{l}f\sim\text{GP}(\text{m}(X),\text{k}(X,X_{*}))\\ \text{where}\ \text{m}(X)=\text{E}[f(X)]\\ \text{and}\ \text{k}(X,X_{*})=\text{E}[(f(X)-\text{m}(X))(f(X_{*})-\text{m}(X_{*}))] \end{array}  \tag{10}\]  
In Equation (8), \(\epsilon\) is a noise parameter with a mean of 0 and a variance of \(\sigma_{y}^{2}\) subsequent to the Gaussian distribution, and \(I\) is an Identity matrix considering the data dimension. Equation (10) is the \"distribution over functions\" [33] for converting to the target value \(Y\), if a training data is assigned with input vector \(X\) and output vector \(Y\) as shown in Equation (9), which is a Gaussian process distribution modeled with mean \(\text{m}(X)\) and variance covariance \(\text{k}(X,X_{*})\). \(\text{k}(X,X_{*})\) is a covariance function (\(\text{cov}(f(X),f(X_{*}))=\)), and indicates the similarity measure between both data. In general, a special kernel function is modeled as a radial basis kernel of Equation (11). \[\text{k}(X,X_{*})=\exp(-\frac{1}{2}|X-X_{*}|^{2}) \tag{11}\]  
Equation (10) can be mapped to Equation (12) with the condition of the absence of noise, and then the pre-distribution is expressed as Equation (13). \[f_{*}\sim\text{N}(0,\text{k}(X_{*},X_{*})) \tag{12}\]  
Figure 2: Schematic graph of a Gaussian processes regression. \begin{table}
\begin{tabular}{l l} \hline \hline
**Research Studies Using GPR** & **Application Fields** \\ \hline [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON] [28] & The time and space prediction of an infectious diseases \\ [PERSON] and [PERSON] [29] & Sea level temperature reconstruction \\ [PERSON] and [PERSON] [30] & Kinetics model estimation \\ [PERSON], [PERSON] and [PERSON] [31] & Efficient building field formation by estimating indoor environment fields \\ [PERSON], [PERSON], [PERSON] and [PERSON] [32] & Wind prediction for energy efficiency \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Research applications using GPR. \[\left[ \begin{array}{c}f\\ f_{*}\end{array} \right]\sim\mathrm{N}\!\left(0,\left[ \begin{array}{cc}k(X,X)&k(X,X _{*})\\ k(X_{*},X)&k(X_{*},X_{*})\end{array} \right]\right) \tag{13}\]  
However, the value obtained by the measurement of the actual data contains noise in general. In this manner, the \(Y\) value is expressed as Equation (14). \[Y\sim\mathrm{N}\!\left(0,\mathrm{k}+\sigma_{y}^{2}\mathrm{I}\right) \tag{14}\]  
This indicates that the estimated \(Y\) using a general GPR model follows the attributes of multivariate normal distribution with mean 0 and covariance \(\mathrm{k}+\sigma_{y}^{2}\mathrm{I}\). This derived GPR model is used as a prediction model using Bayesian framework. When the function value \(f_{*}(X_{*})\) is predicted for the test data \(X_{*}\), the pre-distribution for \(f_{*}(X_{*})\) considering the occurring noises is modeled using Equation (15). \[\left[ \begin{array}{c}Y\\ f_{*}\end{array} \right]\sim\mathrm{N}\!\left(0,\left[