Performing cross-validation and validation

Before you produce the final surface, you should have some idea of how well the model predicts the values at unknown locations. Cross-validation and validation help you make an informed decision as to which model provides the best predictions. The calculated statistics serve as diagnostics that indicate whether the model and/or its associated parameter values are reasonable.

Cross-validation and validation use the following idea—remove one or more data locations and predict their associated data using the data at the rest of the locations. In this way, you can compare the predicted value to the observed value and obtain useful information about the quality of your kriging model (for example, the semivariogram parameters and the searching neighborhood).


Cross-validation uses all the data to estimate the trend and autocorrelation models. It removes each data location one at a time and predicts the associated data value. For example, the diagram below shows 10 data points. Cross-validation omits a point (red point) and calculates the value of this location using the remaining 9 points (blue points). The predicted and actual values at the location of the omitted point are compared. This procedure is repeated for a second point, and so on. For all points, cross-validation compares the measured and predicted values. In a sense, cross-validation "cheats" a little by using all the data to estimate the trend and autocorrelation models. After completing cross-validation, some data locations may be set aside as unusual, requiring the trend and autocorrelation models to be refit.

Cross validation

Cross-validation is performed automatically and results are shown in the last step of the Geostatistical Wizard. Cross-validation can also be performed manually using the Cross Validation geoprocessing tool.


Validation first removes part of the data (call it the test dataset) then uses the rest of the data (call it the training dataset) to develop the trend and autocorrelation models to be used for prediction. In Geostatistical Analyst, you create the test and training datasets using the Create Subset tools. Other than that, the types of graphs and summary statistics used to compare predictions to true values are similar for both validation and cross-validation. Validation creates a model for only a subset of the data, so it does not directly check your final model, which should include all available data. Rather, validation checks whether a protocol of decisions is valid, for example, choice of semivariogram model, lag size, and search neighborhood. If the decision protocol works for validation, you can feel comfortable that it also works for the whole dataset.

Model validation can be performed using the GA Layer to Points geoprocessing tool. For more information on how to validate a model, refer to Using validation to assess models.


Geostatistical Analyst gives several graphs and summaries of the measured values versus the predicted values. A scatterplot of predicted values versus true values is given. You might expect that these should scatter around the 1:1 line (the black dashed line in the plot given below). However, the slope is usually less than 1. It is a property of kriging that tends to underpredict large values and overpredict small values, as shown in the following figure.

Scatter plot illustration

The fitted line through the scatter of points is given in blue with the equation given just below the plot. The error plot is the same as the prediction plot, except the measured values are subtracted from the predicted values. For the standardized error plot, the measured values are subtracted from the predicted values and divided by the estimated kriging standard errors. All three of these plots show how well kriging is predicting. If all the data was independent (no autocorrelation), all predictions would be the same (every prediction would be the mean of the measured data), so the blue line would be horizontal. With autocorrelation and a good kriging model, the blue line should be closer to the 1:1 (black dashed) line.

The final plot is a QQ plot. This shows the quantiles of the difference between the predicted and measured values and the corresponding quantiles from a standard normal distribution. If the errors of the predictions from their true values are normally distributed, the points should lie roughly along the gray line. If the errors are normally distributed, you can be confident of using methods that rely on normality (for example, quantile maps in simple kriging).

QQ Plot

Prediction error statistics

Finally, some summary statistics on the kriging prediction errors are given below. Use these as diagnostics.

Kriging summary statistics for prediction errors