Kernel density estimation is an important nonparametric technique to estimate density from point-based or line-based data. It has been widely used for various purposes, such as point or line data smoothing, risk mapping, and hot spot detection. It applies a kernel function on each observation (point or line) and spreads the observation over the kernel window. The kernel density estimate at a location will be the sum of the fractions of all observations at that location. In a GIS environment, kernel density estimation usually results in a density surface where each cell is rendered based on the kernel density estimated at the cell center. The result of kernel density estimation could vary substantially depending on the choice of kernel function or kernel bandwidth, with the latter having a greater impact. When applying a fixed kernel bandwidth over all of the observations, undersmoothing of density may occur in areas with only sparse observation while oversmoothing may be found in other areas. To solve this issue, adaptive or variable bandwidth approaches have been suggested.
Spatial interpolation methods use the measured values at given locations to estimate the values at unsampled locations, for example, in computing raster digital elevation models from scattered measured elevations. Since this problem does not have a unique solution, many approaches have been developed to accomplish this task. Methods based on linear superposition of radial basis functions (RBF) centered at the data points include multivariate splines that simultaneously minimize the sum of the deviations from the measured points and the smoothness seminorm referred to also as a roughness penalty. The thin plate spline minimizes the 2D surface curvature and mimics a thin steel plate forced to pass through the data points: its equilibrium shape minimizes the bending energy which is closely related to the surface curvature. There are many generalizations such as spline with tension that controls the plate stiffness, while regularized spline enables direct calculations of surface gradients and curvatures making it effective for terrain modeling with simultaneous topographic analysis. Trivariate splines are used to interpolate meteorological variables with influence of topography. The RBF splines are related to universal kriging with the choice of the covariance function determined by the smoothness seminorm. Multiquadric RBF methods are similar in formulation and performance to splines.