Generalized cross validation

Generalized cross-validation. To overcome some problems with ordinary cross-validation, Golub et al. (1979) have proposed the generalized cross-validation (GCV) which is a weighted version of CV ... 

To compute y several methods have been proposed. In this work, the Generalized Cross Validation (GCV) technique will be preferred. GCV is explained in detail elsewhere [13], The value of computed by means of GCV is the one that minimizes the following function ... 

Generalized cross-validation, gcv. The ratio of the residual sum of squares to the squared residual degrees of freedom ... 

The simplest and most general cross-validation procedure is the leave-one-out technique (LOO technique), where each object is taken away, one at a time. In this case, given n objects, n reduced models have to be calculated. This technique is particularly important as this deletion scheme is unique, and the predictive ability of the different models can be compared accurately. However, in several cases, the predictive ability obtained is too optimistic, particularly when the number of objects is quite large. This is because of a too small perturbation of the data when only one object is left out. 

In general, a regularization parameter should be chosen for each voxel. Since there may be thousands of voxels, the use of graphical or other methods requiring intervention is prohibitive. In the present work, an automatic, data-driven method is utilized to obtain a reliable estimate of the regularization parameter for each voxel. It is based on nonparametric statistical theory, which can incorporate a number of performance criteria, including unbiased prediction risk (UBPR),9 cross-validation (CV),15 and generalized cross-validation (GCV).16... 

During the selection of the number of hidden layer neurons, the desired tolerance should also be considered. In general, a tight tolerance requires that the selected network be trained with fewer hidden neurons. As mentioned earlier, cross-validation during training can be used to monitor the error progression, which subsequently serves as a guideline in the selection of the hidden layer neurons. 

Many people use the term PRESS to refer to the result of leave-one-out cross-validation. This usage is especially common among the community of statisticians. For this reason, the terms PRESS and cross-validation are sometimes used interchangeably. However, there is nothing inate in the definition of PRESS that need restrict it to a particular set of predictions. As a result, many in the chemometrics community use the term PRESS more generally, applying it to predictions other than just those produced during cross-validation. 

Traditional electrophoresis and capillary electrophoresis are competitive techniques as both can be used for the analysis of similar types of samples. On the other hand, whereas HPLC and GC are complementary techniques since they are generally applicable to different sample types, HPLC and CE are more competitive with each other since they are applicable to many of the same types of samples. Yet, they exhibit different selec-tivities and thus are very suitable for cross-validation studies. CE is well suited for analysis of both polar and nonpolar compounds, i.e. water-soluble and water-insoluble compounds. CE may separate compounds that have been traditionally difficult to handle by HPLC (e.g. polar substances, large molecules, limited size samples). 

The cross validation estimate of x for equally spaced data is defined to the minimizer of V (x) (IS. IB). Results were also given for the "generalized"°cross vaTTdation to compensate for unequally spaced data points (15-17) but will not be... 

The second possibility is called cross-validation. The test samples are measured by a reference method. Howevep the reference method cannot provide true values because measurement error occurs here as well. Nevertheless, well-characterized methods can provide generally accepted values, which are then compared to the ones obtained using the test calibration. Note that this comparison must be done using particularly suited regression methods, because the error for both methods will be in the same order of magnitude. Especially, cross-validation of CE and HPLC has been frequently reported. 

Calculated descriptors have generally fallen into two broad categories those that seek to model an experimentally determined or physical descriptor (such as ClogP or CpKJ and those that are purely mathematical [such as the Kier and Hall connectivity indices (4)]. Not surprisingly, the latter category has been heavily populated over the years, so much so that QSAR/QSPR practitioners have had to rely on model validation procedures (such as leave-k-out cross-validation) to avoid models built upon chance correlation. Of course, such procedures are far less critical when very few descriptors are used (such as with the Hansch, Leo, and Abraham descriptors) it can even be argued that they are unnecessary. 

Since most chemicals caused different phenotypic outcomes between the rats and rabbits, species-specific models were analyzed, with 251 chemicals evaluated in the rat model and 234 in the rabbit (Fig. 2). Cross-validation balanced accuracies in the resulting classification models were 71% for the rat model (12 features), and 74% for the rabbit model (7 features). Each model contained positive predictors or assay features generally affected by the developmental toxicants (as defined above) and negative predictors or assay features that were generally affected by the nondevelopmen-tal toxicants (as defined above). 

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