/*******************************************************************************
 * Copyright (c) 2010 Haifeng Li
 *   
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *  
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 *******************************************************************************/

package smile.interpolation.variogram;

import smile.math.Function;

/**
 * In spatial statistics the theoretical variogram 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
 */
public interface Variogram extends Function {

}