## 2 Materials and Methods
### Physicochemical and Metal Concentrations Analysis  
Temperature, pH, electric conductivity (EC), and total dissolved solids (TDS) of the samples were determined onsite using a multi-parameter water analyzer (HANNA HI 9811-5) with HI1285-5 probe (electrode) and HI7007, HI70031, HI70032, buffer solutions for calibration [31]. The HI7007, HI70031, HI70032 solution were used for pH, EC and TDS calibration, respectively. While HI700661 solution was used for cleaning the electrode. The physicochemical values detected in groundwater were compared to the permissible limits specified in the 2017 Philippine National Standards for Drinking Water (PNSDW) [32] and the WHO Drinking Water Guidelines [33; 34]. These water parameters were used in the hybrid machine learning technique.  
The heavy metal concentration analysis employed the use of portable handheld Olympus Vanta X-Ray Fluorescence Spectrometry. This pXRF is a rapid onsite accurate elemental analyzer that could be used for various environmental media including water [35; 36; 37; 38; 39]. The pXRF was set on geochem mode and recorded metals concentration in ppm (mg/L) detected from GW samples. Target metals were Ba, Cu, Fe, Mn and Zn.  
Figure 2: Map of the Study Area and Sampling Sites.  
Spatial Concentration Mapping Using Machine Learning Informed Empirical Bayesian Kriging (EBK) Method  
Hybrid machine learning--geostatistical interpolation (MLGI) method was employed. The EBK technique was used to produce spatial concentration maps of the GW's physicochemical properties and heavy metal concentrations. By sub-setting and replicating observed data, the EBK automates the most time-consuming aspects of constructing a viable kriging model. EBK provides a distribution of semi-variogram models and compensates for semi-variogram estimate uncertainty. The EBK is more realistic and superior to other current geostatistical modeling methods owing to its dependence on limited maximum likelihood estimation. This is in contrast to other existing kriging models that rely on weighted least squares estimation. EBK has many significant benefits, including a low need for interactive modeling, more accurate prediction of standard error and projection for small datasets as compared to other traditional kriging techniques, and exact prediction of substantially non-static data [40].  
The Artificial Neural Network (ANN) approach is a subset of methods for artificial intelligence inspired by biological neurons. It is capable of quickly acquiring patterns and forecasting the result of a problem in a multi-dimensional environment. ANN models are trained using datasets [41] to show the efficacy. The training algorithm and the transfer function that was utilized in the model are two critical components of the ANN model. The Levenberg--Marquardt (LM) algorithm was chosen as the training algorithm since it is the quickest function for training a network, and the hyperbolic tangent sigmoid function was used as the transfer function because it is the recommended transfer function for rapid processes [42; 43].  
Particle Swarm Optimization (PSO) is a population-based stochastic optimization technique inspired by biological communities' collaborative nature. The PSO is initiated using a community of randomly generated particles as solution options. It looks for global optima via iterations in which particles with their own velocity fly around the search space following the current optimum particles, which is the best approach for finding the best solution. The PSO was integrated to the ANN to determine the weights and biases which gives the minimum error [44].  
This hybrid technique integrated to the EBK method generated the spatial concentration maps of the target study area. The Neural Network--Particle Swarm Optimization (NN-PSO) approach was applied to generate the spatial concentration maps of physicochemical parameters and HM concentrations.