Leave-one-out-cross-validation LOOCV

The cross-validation approach can also be used to estimate the predictive ability of a calibration model. One method of cross-validation is leave-one-out cross-validation (LOOCV). Leave-one-out cross-validation is performed by estimating n calibration models, where each of the n calibration samples is left out one at a time in turn. The resulting calibration models are then used to estimate the sample left out, which acts as an independent validation sample and provides an independent prediction of each y, value, y(i), where the notation i indicates that the /th sample was left out during model estimation. This process of leaving a sample out is repeated until all of the calibration samples have been left out. The predictions y(i) can be used in Equation 5.10 to estimate the RMSECV. However, LOOCV has been shown to determine models that are overfitting (too many parameters are included) [7, 8], The same is true for v-fold cross-validation, where the calibrations set is split into... 

With V = N (the total number of samples), commonly referred to as leave-one-out cross-validation (LOOCV), each CV run uses N I samples for training and the remaining one sample for testing. Since the N training samples are so similar in each run, CV is approximately unbiased for the tme prediction error. However, the variance of the true prediction error would be large and it is also computationally expensive. In practice, the preference of the number of folds depends on the size of the dataset. Tenfold CV has become generally accepted. 

Most current studies report an internal validation accuracy without an independent validation set. When there are a sufficiently large number of samples, the whole dataset can be split into two, one for training and one for testing (validation) this method is called hold-out validation. When the number of samples is limited, leave-one-out cross validation (LOOCV) is a popular technique. Here, a procedure is repeated N times, and each time a different sample is left out and used for testing the model learned from the remaining (N - 1) samples. The accuracy of... 

Fork random partition of m. If Jt = m, the method is called leave-one-out cross-validation (LOOCV). A predicting function / is calculated for each i m, using the observations from m i = 0,..., - 1, i + 1,..., m - 1 exclusively for learning. The formula for the sum of squared residuals is simplified to... 

With their natural logarithm, the data in Table 2 and Table 3 were input to the Eq. (1) to build early-warning models in R environment (Wold, et al., 2001, Mevik Wehrens, 2007), respectively. The model parameters in Chongqing City and Ningbo City were shown in Table 5. One was called Chongqing Model, the other was named Ningbo model. All the models were assessed by Leave-One-Out Cross Validation method (LOOCV), and the maximum model error was less than 15%. 

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