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The question is regarding the Matlab implementation. As we can see here, the crossval function expects to receive a full trained model. For example, my data consist of 100 observations and we would like to build a model that classify each observation to "1" or "-1" using the SVM classifier. So if we like to cross validate we first should fit a classifier for all 100 observations (this results a ClassificationSVM model trained using fitcsvm) and then we input the crossval function with it.

What I don't understand is, why Matlab's implementation require to first train the model before cross validating? since the standard CV process trains on each k − 1 subsamples (and don't use the full fitted model).

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If you have a model that is already trained on the full data, it is a ClassificationSVM object. There is a crossval method that you can use to perform cross-validation on it to assess its generalization error.

If you haven't trained a model yet and you want to do cross validation up front, you can use the 'CrossVal' option to the fitcsvm function, and set its value to 'on'.

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