Understanding transformations and trends

Kriging as a predictor does not require that your data have a normal distribution. However, as you saw in Understanding different kriging models, normality is necessary to obtain quantile and probability maps for ordinary, simple, and universal kriging. When considering only predictors that are formed from weighted averages, kriging is the best unbiased predictor whether or not your data is normally distributed. However, if the data is normally distributed, kriging is the best predictor among all unbiased predictors, not only those that are weighted averages.

Kriging also relies on the assumption that all of the random errors are second-order stationarity, which is an assumption that the random errors have zero mean and the covariance between any two random errors depends only on the distance and direction that separates them, not their exact locations.

Transformations and trend removal can help justify assumptions of normality and stationarity. Prediction, using ordinary, simple, and universal kriging for general Box-Cox, arcsine, and log transformations, is called trans-Gaussian kriging. The log transformation is a special case of a Box-Cox transformation, but it has special prediction properties and is known as lognormal kriging.

Transformation and trend for primary variable

In the table below, the transformations and trend options available for each kriging method are shown for the primary variable. The table also shows whether the transformation or the trend removal is performed first when both are selected.

Kriging type

BAL

NST

Trend

Ordinary Kriging

Yes (1st if TR)

No

TR (2nd if BAL)

Simple Kriging

Yes

Yes

No

Universal Kriging

Yes (1st if T)

No

T (2nd if BAL)

Indicator Kriging

No

No

No

Probability Kriging*

No

No

No

Disjunctive Kriging

Yes (1st if TR)

Yes (2nd if TR)

TR (1st if NST, 2nd if BAL)

Primary variable transformations and trend options

*Note: For probability kriging, the primary variable is composed of indicators of the original variable; the original variable is then considered a secondary variable for cokriging.

Transformation and trend for secondary variable (cokriging)

In the table below, the transformation and trend options available for each kriging method are shown for the secondary variable. The table also shows whether the transformation or the trend removal is performed first when both are selected.

Kriging type

BAL

NST

Trend

Ordinary Kriging

Yes (1st if TR)

No

TR (2nd if BAL)

Simple Kriging

Yes

Yes

No

Universal Kriging

Yes (1st if T)

No

T (2nd if BAL)

Indicator Kriging

No

No

No

Probability Kriging

Yes (1st if TR)

No

TR (2nd if BAL)

Disjunctive Kriging

Yes (1st if TR)

Yes (2nd if TR)

TR (1st if NST, 2nd if BAL)

Secondary variable transformation and trend options

Definitions

• Primary variable: Variable to be predicted when using kriging or cokriging
• Secondary variables: Covariables (not predicted) when using cokriging
• Trend: Fixed effects composed of spatial coordinates used in a linear model

Abbreviations

• BAL—Box-Cox, arcsine, and log transformations
• NST—Normal score transformation
• SV—Variable (covariables for cokriging)
• T (internal trend)
• TR—Removal (external trend)

2/11/2011