ArcGIS Tutorial: Semivariogram vs. covariance functions

The Semivariogram and covariance functions quantify the assumption that neighboring things are more similar than things in the distance. Both Semivariogram and covariance measure the strength of statistical correlations as distance functions.

The process of modeling semi-variance functions and covariance functions fits the semivariogram or covariance curves with empirical data. The goal is to achieve the best fit and to incorporate the cognition of the phenomenon into the model. The model can then be used for predictions.

The orientation autocorrelation in the browsing data when the model is fitted. abutment, range and nugget are important characteristics of the model. If there is a measurement error in your data, use the measurement error model. Follow this link to learn how to fit a model to an empirical semivariogram.

　　Semi-variant function

The semivariogram is defined as

Gamma (SI,SJ) =? var (z (SI)-Z (SJ)),

where Var is the variance.

If the two position Si and SJ, in D (SI, SJ) distance measurement close to each other, then you would like the two locations similar, so as to reduce the difference of two position z (SI)-Z (SJ) size. When Si and SJ are gradually increasing in distance, they become more and more dissimilar, and their value Z (SI)-Z (SJ) will also increase in difference. You can see this in this case, which shows the analytic diagram of a typical semivariogram.

　　

Note that the variance of the difference increases with the distance, so the semivariogram can be considered a dissimilarity function. The terms that are often associated with this function are also available in geostatistical Analyst. The height the Semivariogram achieves when it is stationary is called the abutment. It is usually made up of two parts: The origin is discontinuous (called the Nugget effect) and the bias abutment, and both form the abutment. The nugget effect can be subdivided into measurement errors and micro-scale changes. The nugget effect is the sum of the measurement error and the micro-scale variation, since either component can be zero, so the nugget effect can be formed entirely by one component or another component. The range is the distance at which the semivariogram reaches the stationary abutment.

　　covariance function

The covariance function is defined as

C (SI, SJ) = CoV (Z (SI), z (SJ)),

Where CoV is the covariance.

Covariance is a scaled version of relevance. So when two locations, Si and SJ are close to each other, you want the two locations to be similar, and their covariance (correlations) will become larger. When Si and SJ are gradually increasing in distance, they become more and more dissimilar, and their covariance becomes zero. In this case, you can see the analytic diagram of the typical covariance function.

　　

Note that the covariance function decreases with distance, so it can be treated as a similarity function.

　　The relationship between the semivariogram and the covariance function

The following relationship exists between the semivariogram and the covariance function:

Γ (SI, SJ) = Sill-c (Si, SJ),

can see the relationship. Because of this equality relationship, you can use either of the two functions in geostatistical Analyst to perform predictions. (All Semivariogram in Geostatistical Analyst has a abutment.) )

Semivariogram and covariance are not arbitrary functions. To make predictions have nonnegative kriging standard errors, only some of the functions can be used as semivariogram and covariance. Geostatistical Analyst offers a variety of acceptable options that you can try for different options for your data. You can also get the model by adding multiple models at the same time-this construct provides an effective model that you can add up to four models in Geostatistical Analyst. There are instances where the covariance function does not exist when the semivariogram is present. For example, there is a linear semivariogram, but it has no abutment and does not have a corresponding covariance function. Only models with a base station are used in Geostatistical Analyst. There are no rules that must be followed when choosing the best Semivariogram model. You can view the empirical semivariogram or covariance function and choose the model that looks right. You can also use validation and cross-validation as a guide.

ArcGIS Tutorial: Semivariogram vs. covariance functions