Cross-validation

Cross-validation, sometimes called rotation estimation , is the statistical practice of partitioning a sample of data into subsets such that the analysis is initially performed on a single subset, while the other subset(s) are retained for subsequent use in confirming and validating the initial analysis.

The initial subset of data is called the training set; the other subset(s) are called validation or testing sets.

The theory of cross-validation was inaugurated by Seymour Geisser. It is important in guarding against testing hypotheses suggested by the data ("Type III error"), especially where further samples are hazardous, costly or impossible (uncomfortable science) to collect.

Holdout validation
Holdout validation is not cross-validation in the common sense, because the data never are crossed over. Observations are chosen randomly from the initial sample to form the validation data, and the remaining observations are retained as the training data. Normally, less than a third of the initial sample is used for validation data.

K-fold cross-validation
In K-fold cross-validation, the original sample is partitioned into K subsamples. Of the K subsamples, a single subsample is retained as the validation data for testing the model, and the remaining K − 1 subsamples are used as training data. The cross-validation process is then repeated K times (the folds), with each of the K subsamples used exactly once as the validation data. The K results from the folds then can be averaged (or otherwise combined) to produce a single estimation.

Leave-one-out cross-validation
As the name suggests, leave-one-out cross-validation (LOOCV) involves using a single observation from the original sample as the validation data, and the remaining observations as the training data. This is repeated such that each observation in the sample is used once as the validation data. This is the same as a K-fold cross-validation with K being equal to the number of observations in the original sample. In the case of Tikhonov regularization, an efficient algorithm has been found.

Error estimation
The parameter estimation error can be computed. Common error metrics are the mean average error (MAE) and the root mean square error (RMSE), respectively the estimated variance and standard deviation of the cross validation.