smile.interpolation.variogram.package-info Maven / Gradle / Ivy
/*
* Copyright (c) 2010-2021 Haifeng Li. All rights reserved.
*
* Smile is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* Smile is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with Smile. If not, see 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.
*
* @author Haifeng Li
*/
package smile.interpolation.variogram;