<P> In summary, cross-validation combines (averages) measures of fit (prediction error) to derive a more accurate estimate of model prediction performance . </P> <P> Suppose we have a model with one or more unknown parameters, and a data set to which the model can be fit (the training data set). The fitting process optimizes the model parameters to make the model fit the training data as well as possible . If we then take an independent sample of validation data from the same population as the training data, it will generally turn out that the model does not fit the validation data as well as it fits the training data . This is called overfitting, and is particularly likely to happen when the size of the training data set is small, or when the number of parameters in the model is large . Cross-validation is a way to predict the fit of a model to a hypothetical validation set when an explicit validation set is not available . </P> <P> Linear regression provides a simple illustration of overfitting . In linear regression we have real response values y,..., y, and n p - dimensional vector covariates x,..., x . The components of the vectors x are denoted x,..., x . If we use least squares to fit a function in the form of a hyperplane y = a + β x to the data (x, y), we could then assess the fit using the mean squared error (MSE). The MSE for given estimated parameter values a and β on the training set (x, y) is </P> <Dl> <Dd> 1 n ∑ i = 1 n (y i − a − β T x i) 2 = 1 n ∑ i = 1 n (y i − a − β 1 x i 1 − ⋯ − β p x i p) 2 (\ displaystyle (\ frac (1) (n)) \ sum _ (i = 1) ^ (n) (y_ (i) - a - (\ boldsymbol (\ beta)) ^ (T) \ mathbf (x) _ (i)) ^ (2) = (\ frac (1) (n)) \ sum _ (i = 1) ^ (n) (y_ (i) - a - \ beta _ (1) x_ (i1) - \ dots - \ beta _ (p) x_ (ip)) ^ (2)) </Dd> </Dl>

Which of the following value of k will have least leave one out cross validation accuracy