## 2 Mathematical Background
### Gaussian Process  
Mathematically, the Gaussian process (GP) can be defined as an infinite collection of random variables where any of the finite subsets can be treated as Gaussian (i.e., normal) distribution [17; 18]. A GP model is fully expressed or specified by a mean function \(m(x)\) in a vector and a covariance \(k(x,x^{\prime})\) in the form of a matrix. GP is a probabilistic and nonparametric method for fitting any functional forms based on observations of the domain. This method differs from most of the other black-box identification approaches. GP does not try to approximate the model by trying to fit the parameters of the chosen basis function, but it searches the relationship among the given knowledge or information. This stochastic approach can provide predictive mean and variance. The mean represents the most likely outputs, whereas the variance plays a role as the uncertainty's quantifier. The magnitude of variance is incorporated into the GP prediction and inference. It does reflect the amount and quality of the data distribution, and it is an essential numerical feature when it comes to differentiating the GP regression from any other computational intelligence methods.  
GP regression is a well-known technique and has been adopted in many scientific and engineering fields, such as machine learning, computer experiments, geostatistics, electronics, etc. Its convenient properties in modeling make GP successful in demonstrating outstanding performances in numerous applications. Researchers in [19] employed GP regression for predicting the concentration of ozone in an area of Bourgas, Bulgaria. The prediction was based on the measurements of the concentration of ozone, phenol, sulphur  
Figure 2: Machine learning (ML) process. ML utilizes sets of inputs, the corresponding outputs, and interested inputs to yield the predictive outputs.  
dioxide, and benzene in the air (hourly). Researchers in [20] studied the capability of GP in estimating load-bearing pile capacity. This research work compared the performance of GP prediction with other approaches, and GP worked very well in predicting the pile capacity. In a wireless networking application, researchers in [21] inferred that, besides resulting in low-cost outcomes, GP was also capable of obtaining low complex proximity reports for a received-signal-strength (RSS) threshold. Next, GP was also used by researchers in [22] to explain the uncertainty in estimating the state of health (SOH) of lithium-ion batteries. Additionally, according to researchers in [23], GP can successfully decrease the complexity of the computer vision with a better performance of reconstructions.  
In [24], GP was hybridized with particle swarm optimization (PSO) to do advanced prediction of roadway broken road zone based on data collected in China. The study inferred that the hybrid model showed the best performance with higher variance accounted for (VAF), higher coefficient of determination (\(R^{2}\)), and lower root mean squared error (RMSE) compared to the other methods such as multiple linear regression (MLR) and artificial neural network (ANN). For geophysics problems, researchers in [25] mentioned that GP was able to yield reliable information of permeability and porosity in well analysis. GP was also used by researchers in [26] to predict the presence of a thin hydrocarbon layer by utilizing its predictive variance for quantifying the uncertainties of EM responses. Furthermore, GP was exploited for seismic signal detection as well. According to [27], the rate of detection when using GP was higher compared to other methods and the rate of false alarm was lower especially for the weak seismic signals. Researchers in [28] proved that GP was successfully capable of being exploited in processing EM responses by calibrating the stochastic and the computer experiment models for magnitude versus offset (MVO) analysis.