## 3 Geospatial Informatics of the Geotechnical Database
### Hot Spot Analysis of the Geotechnical Database  
In contrast to the predicted number of occurrences within a random distribution, a hot spot may be defined as a region with a greater event concentration. The discovery of hot spots has been developed by the study of spatial points or spatial configurations [33]. The density of points within a certain region is compared to a completely random model in geospatial pattern analysis. In addition to density measurements and the grouping of correlated datasets, hot spot techniques often calculate the extent of the association between point events [46]. Spatial clustering methods such as kernel density estimation (KDE), local Moran's I, or Getis-Ord Gi* statistics are therefore required to avoid the fragmentation of hot spots. In this study, these hot spot analyses were applied to the geo-layer thickness and average \(N_{60}^{exp}\) (Figure 5). Figure 4: Outlier detection of bedrock depth data in the borehole dataset based on (**a**) the cross-validation-based method, (**b**) Anselin Local Moran’s I statistic-based method, and (**c**) outliers detected by both methods, applied to bedrock depth. Outlier and clusters classified by Anselin Local Moran’s I statistic: NS (Not significant); H-H (High–High cluster); H-L (High–Low outlier); L-H (Low–High outlier); L-L (Low–Low cluster). Figure 5: Comparison of three hot spot analysis algorithms (Kernel Density, Local Moran’s I, and Getis-Ord Gi* statistics) applied to the engineering geo-layer thickness and average \(N_{60}^{xy}\) for fill layer; alluvial soil; weathered soil; weathered rock. The KDE approach is a non-parametric procedure that uses a technique to estimate spatial density using a kernel function to fit a smoothly tapered surface to each point [47; 48]. The kernel density represents a smooth and continuous surface map because interpolation generates a continuous discrete pattern [49]. The kernel calculator can be defined for a particular set of observations from an unknown probability density function:  
\[\hat{f_{h}}(x)=\frac{1}{nh}\sum_{i=1}^{n}K\Big{(}\frac{x-x_{i}}{h}\Big{)} \tag{1}\]  
where \(x_{i}\) is the value of the variable \(x\) at location \(i\), \(n\) signifies the total number of locations, \(h\) is the smoothing parameter or bandwidth, \(K\) is the kernel, and \(\hat{f}\) is the estimator of the probability density function \(f\)[50]. After KDE analyses of the Daejeon dataset, each geo-layer shows localized densities that are focused on central and northern parts of the urban region of Daejeon city (Figure 5). In the case of average \(N_{60}^{exp}\) for each geo-layer, the density trends were more remarkable in the order AS \(>\) WR \(>\) FL \(>\) WS. The resemblance between the local density of a geo-layer's thickness and average \(N_{60}^{exp}\) suggests that the higher density areas of the geo-layer have the higher soil stiffness. KDE is only useful for aggregating spatially consistent grid data and is not suited to fragmented, location-based data [51]. Local Moran's I can also be used to classify hot spots using Getis-Ord Gi* statistics, but in practice, Getis-Ord Gi* statistics (or \(Gi^{*}\) statistics) have been shown to be superior. High concentrations and spatial outliers of geo-layer thickness and average \(N_{60}^{exp}\) (described by Anselin Local Moran's I statistic-based outlier analysis) are samples surrounded by normal or low-value samples [37; 44]. Local Moran's I is more convenient for locating high or low values and outlier statistically relevant clusters. It is defined as follows:  
\[I_{i}=\frac{x_{i}-\overline{X}}{S_{i}^{2}}\sum_{i=1,j\
eq i}^{n}w_{i,j}\Big{(} x_{i}-\overline{X}\Big{)} \tag{2}\]  
where \(\overline{X}\) is the average value of \(x\) with sample number \(n\), \(S_{i}^{2}\) is the variance of variable \(x\), and \(w_{i,j}\) is a weight that can be defined as the inverse of the distance between locations \(i\) and \(j\). The weight \(w_{i,j}\) can be calculated using the distance bands.