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Creating a surface with geostatistical techniques
Cokriging uses information on several variable types. The main variable of interest is Z_{1}, and both autocorrelation for Z_{1} and cross-correlations between Z_{1} and all other variable types are used to make better predictions. It is appealing to use information from other variables to help make predictions, but it comes at a price. Cokriging requires much more estimation, including estimating the autocorrelation for each variable as well as all cross-correlations. Theoretically, you can do no worse than kriging because if there is no cross-correlation, you can fall back on autocorrelation for Z_{1}. However, each time you estimate unknown autocorrelation parameters, you introduce more variability, so the gains in precision of the predictions may not be worth the extra effort.
Ordinary Cokriging assumes the models:
Z_{1}(s) = µ_{1} + ε_{1}(s)
Z_{2}(s) = µ_{2} + ε_{2}(s),
where µ
_{1} and µ
_{2} are unknown constants. Notice that now you have two types of random errors, ε
_{1}(
s) and ε
_{2}(
s), so there is autocorrelation for each of them and cross-correlation between them. Ordinary Cokriging attempts to predict Z
_{1}(
s
_{0}), just like Ordinary Kriging, but it uses information in the covariate {
_{2}(
s)} in an attempt to do a better job. For example, the following figure has the same data that was used for Ordinary Kriging, only here a second variable is added.
Notice that the data Z
_{1} and Z
_{2} appears autocorrelated. Also notice that when Z
_{1} is below its mean µ
_{1}, then Z
_{2} is often above its mean µ
_{2}, and vice versa. Thus, Z
_{1} and Z
_{2} appear to have negative cross-correlation. In this example, each location
s had both Z
_{1}(
s) and Z
_{2}(
s); however, this is not necessary, and each variable type can have its own unique set of locations. The main variable of interest is Z
_{1}, and both autocorrelation and cross-correlation are used to make better predictions.
The other cokriging methods, including Universal, Simple, Indicator, Probability, and Disjunctive, are all generalizations of the foregoing methods to the case where you have multiple datasets. For example, Indicator Cokriging can be implemented by using several
thresholds for your data, then using the binary data on each threshold to predict the threshold of primary interest. In this way, it is similar to
Probability Kriging but can be more robust to outliers and other erratic data.
Cokriging can use either
semivariograms or covariances (the mathematical forms used to express autocorrelation) and
crosscovariance (the mathematical form used to express cross-correlation), use
transformations and
remove trends, and allow for
measurement error in the same situations as those of the various kriging methods: Ordinary Kriging, Simple Kriging, and Universal Kriging.