public interface Variogram
extends smile.math.Function
2γ(x,y) is a function describing the degree
of spatial dependence of a spatial random field or stochastic process
Z(x). It is defined as the expected squared increment
of the values between locations x and y:
2γ(x,y) = E(|Z(x)-Z(y)|2)
where γ(x,y) itself is called the semivariogram.
In case of a stationary process the variogram and semivariogram can be
represented as a function
γs(h) = γ(0, 0 + h) of the difference
h = y - x between locations only, by the following relation:
γ(x,y) = γs(y - x).
In Kriging interpolation or Gaussian process regression, we employ this kind
of variogram as an estimation of the mean square variation of the
interpolation/fitting function. For interpolation, even very crude
variogram estimate works fine.
The variogram characterizes the spatial continuity or roughness of a data set. Ordinary one dimensional statistics for two data sets may be nearly identical, but the spatial continuity may be quite different. Variogram analysis consists of the experimental variogram calculated from the data and the variogram model fitted to the data. The experimental variogram is calculated by averaging one half the difference squared of the z-values over all pairs of observations with the specified separation distance and direction. It is plotted as a two-dimensional graph. The variogram model is chosen from a set of mathematical functions that describe spatial relationships. The appropriate model is chosen by matching the shape of the curve of the experimental variogram to the shape of the curve of the mathematical function.