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What is the best way to approach multiple-fold cross validation for a 1 nearest-neighbor model used for prediction?

A common approach to cross validation is to, for example, split the dataset into 10 folds, train on on 9 of them, and test on 1 (repeating for each of the 10 folds).

In a nearest-neighbors model the concepts of "training set" and "test set" do not apply in the same way that they do for a regression. An approximation of the above procedure would be to split the dataset into 10 folds, choose 1 fold as the "test set", and search for nearest neighbors in the remaining 9 (repeating for each fold). I could then calculate the MSE across the 10 folds by comparing predicted and actual responses. Is this a good approach to cross validation? Are there good alternative approaches in this case?

A specific alternative I am considering is doing splits that would leave more data in the test set than the training set. So, for example, I would choose 9 of the 10 folds as the "test set" and search for nearest neighbors in the remaining 1 (repeating for each fold). Is there an advantage/disadvantage to doing it this way?

Some specifics on my data: I have roughly 20K observations, roughly 30 features and am predicting multiple responses (roughly 100). I am interested in using cross validation for model selection / evaluation. Because I am using 1 nearest neighbors, I expect the variance of the model to be high. Multiple-fold cross validation is therefore desirable to get a better estimate of the prediction MSE.

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Usually, the larger the $k$ (the number of folds of the cross validation), the more accurate is the estimation of the RMSE. See Choice of K in K-fold cross-validation, per example.

In your case, performing a leave-one-out cross-validation (LOOCV) is not much more expansive than a ten fold! Indeed, in a ten fold, you will be doing predictions using 90% of $n$ lines. The overall number of operations will be $0.9n^2$. If you use all the training set (except the element you are trying to predict), you will be doing $n^2$ operations.

So, as the performance penalty for running a LOOCV is low (and this is true because you are using a KNN), I would probably use this.

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    $\begingroup$ I'm not sure this is good advice. While LOOCV is almost unbiased, it tends to have a high variance and has some pathological behaviours for some problems (especially likely to be a problem for a high-variance model). The reason why LOOCV is so often used for model selection is that in some cases you can compute it very efficiently. In the case of kNN if you are going to work out the training set error, you can work out the LOOCV error as a by-product almost for free. $\endgroup$ – Dikran Marsupial Jan 15 '16 at 9:52
  • $\begingroup$ RUser4512, good point. In general this might be a better approach. In my case I am also doing some pre-processing before searching for nearest neighbors, in order to determine the features to use and the weights for the distance metric. So LOOCV actually would be more expensive. $\endgroup$ – Iggy25 Jan 15 '16 at 17:06
  • $\begingroup$ I have updated the original post with a specific alternative I am considering. $\endgroup$ – Iggy25 Jan 15 '16 at 17:38

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