Robust Multivariate Analysis by David J. Olive

Author:David J. Olive
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

where is the estimate of when is deleted from the n training cases . Note that is the proportion of training cases that are misclassified by the n leave one out rules. If is the number of cases correctly classified by leave one out classification, then .

For KNN, find the K cases in the training data closest to not including . Then compute the leave one out cross validation error rate as in Definition 8.15.

Assume that the training data is a random sample from the G populations so that as for . Hence is a consistent estimator of . Following Devroye and Wagner (1982), when the test error rate of KNN method converges in probability to L where and is the test error rate of the Bayes classifier. If and as , then the KNN method converges to the Bayes classifier in that the KNN test error rate . Then the leave one out cross validation error rate is a good estimator of in that was usually an upper bound on for small .

For the method below, and the validation set or hold-out set is the small part of the data. Typically, 10% or 20% of the data is randomly selected to be in the validation set. Note that the DA method is only computed once to compute the error rate.

Definition 8.16.

The validation set approach has . Let the validation set contain cases , say. Then the validation set error rate is

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