Splines and Kernel Smoothing
Spline and kernel smoothing methods represent a wide class of smoothing interpolation methods. One method of interpolation is to fit a polynomial surface the measured data points, where a global polynomial encompasses the entire area of interest, but may not contain sufficient detail to capture small scale variations. Local polynomials can be used to capture small scale variation, but they may not apply universally. A piecewise polynomial fitting combines adjacent local polynomial models into a patchwork model; the models can be generated using different polynomial functions but their boundaries must line up for an even joining.
Aligning boundaries is accomplished through the use of splines, which describe how the joined model behaves at the boundary between the different polynomial functions. Splines allow a smooth transition and a flexible interpolation surface. A spline is a special type of piecewise polynomial, in which the smoothness of the interpolation is controlled by a smoothing parameter. There are many different types of spline that behave similarly, including smoothing splines, regression splines, and penalized splines. The smoothing factor controls the tradeoff between fitting the data exactly and the degree of smoothness in the interpolation.
In contrast to spline smoothing, kernel smoothing is a type of moving average interpolation, in which the kernel function provides the weight that each data point receives in the average. The degree of smoothing depends on the choice of kernel function and kernel radius. Increasing the kernel radius results in a smoother interpolation as the moving average is performed over a larger area. Kernel smoothing is also called local regression.
Splines and kernel smoothers are simple to use and are readily available in many software packages. Unlike advanced spatial interpolation methods, such as kriging, splines and kernel smoothers do not require estimation of a statistical model of spatial correlation. For mapping as part of exploratory spatial data analysis, all that is needed is a list of locations and values, as well as a smoothing factor or kernel function and radius. A smoothing factor of zero will produce an interpolation spline that exactly interpolates the data. Larger smoothing factor values will smooth the spline to a greater degree. Similarly, the kernel radius controls the smoothness of the kernel smoother. These smoothing parameters can be chosen manually based on a visual appraisal of the resulting maps or through cross-validation.
