Bootstrap prediction error estimates from

Cross validation and bootstrap techniques can be applied for a statistically based estimation of the optimum number of PCA components. The idea is to randomly split the data into training and test data. PCA is then applied to the training data and the observations from the test data are reconstmcted using 1 to m PCs. The prediction error to the real test data can be computed. Repeating this procedure many times indicates the distribution of the prediction errors when using 1 to m components, which then allows deciding on the optimal number of components. For more details see Section 3.7.1. 

Determination of the optimum complexity of a model is an important but not always an easy task, because the minimum of measures for the prediction error for test sets is often not well marked. In chemometrics, the complexity is typically controlled by the number of PLS or PCA components, and the optimum complexity is estimated by CV (Section 4.2.5). Several strategies are applied to determine a reasonable optimum complexity from the prediction errors which may have been obtained by CV (Figure 4.4). CV or bootstrap allows an estimation of the prediction error for each object of the calibration set at each considered model complexity. 

It is intuitive that the predictabihty of the dependent variables into the training data set from which a model was estimated will be optimistic, when compared to predicting into an external data set. In such a case, the prediction errors will have a downward bias. Therefore, a method that estimates predictability for external data is needed and this can be executed via the bootstrap. 

This optimism represented the underestimation of the squared prediction error that was expected to occur when the model was applied to the data from which it was derived. In a final step, the average optimism across all bootstrap iterations was estimated and added to the SPE estimated when the Mo was applied to Do. This resulted in an improved estimate of the absolute prediction error (SPEimp). 

Standard deviation, MSE, and bias of all methods. The small k chosen for two-fold CV and split sample with p = is due to the reduced training set size. For r-fold CV, a significant decrease in prediction error, bias, and MSE is seen as v increases from 2 to 10. Tenfold CV has a slightly decreased error estimate compared to LOOCV as well as a smaller standard deviation, bias, and MSE however, the LOOCV k is smaller than that of 10-fold CV. Repeated 5-fold CV decreases the standard deviation and MSE over 5-fold CV however, values for the bias and k are slightly larger. In comparison to 10-fold CV, the 0.632-1- bootstrap has a smaller standard deviation and MSE with a larger prediction error, bias, and k. 

Figure 3. Reconstructions of (A) diatom-based and (B) chrysophyte-based monomeric Al for Big Moose Lake, and diatom-based monomeric Al for (C) Deep Lake, (D) Upper Wallface Pond, and (E) Windfall Pond in the Adirondack Mountains, New York. Reconstructions are bounded by bootstrapping estimates of the root mean-squared error of prediction for each sample. Bars to the right of each reconstruction indicate historical (H) and Chaoborus-based (C) reconstructions of fishery resources. The historical fish records are not continuous, unlike the paleolimnological records. Intervals older than 1884 are dated by extrapolation. (Reproduced with permission from reference 10.

RF [29,30] is an ensemble of unpruned classification trees separately grown from bootstrap samples of the training data set. A subset of nitry input variables is randomly selected as candidates to determine the best possible split at each node during the tree induction. The final prediction is generally made by aggregating the outputs of all the ntree trees generated in the forest. The unbiased out-of-bag (OOB) estimate of the generalization error is used to internally evaluate the prediction performance ofRF. 

Prior to Harwood s work, the existence of a Bootstrap effect in copolymerization was considered but rejected after the failure of efforts to correlate polymer-solvent interaction parameters with observed solvent effects. Kamachi, for instance, estimated the interaction between polymer and solvent by calculating the difference between their solubility parameters. He found that while there was some correlation between polymer-solvent interaction parameters and observed solvent effects for methyl methacrylate, for vinyl acetate there was none. However, it should be noted that evidence for radical-solvent complexes in vinyl acetate systems is fairly strong (see Section 3), so a rejection of a generalized Bootstrap model on the basis of evidence from vinyl acetate polymerization is perhaps unwise. Kratochvil et al." investigated the possible influence of preferential solvation in copolymerizations and concluded that, for systems with weak non-specific interactions, such as STY-MMA, the effect of preferential solvation on kinetics was probably comparable to the experimental error in determining the rate of polymerization ( 5%). Later, Maxwell et al." also concluded that the origin of the Bootstrap effect was not likely to be bulk monomer-polymer thermodynamics since, for a variety of monomers, Flory-Huggins theory predicts that the monomer ratios in the monomer-polymer phase would be equal to that in the bulk phase. 

ABSTRACT Soil hydraulic properties parameters are the crucial input parameters in water and solute transport modeling in the vadose zone. Pedotransfer functions are an alternative to direct measurement for obtaining soil hydraulic properties. In this study, Pedotransfer functions were established from particle-size distribution, bulk density, and organic matter using linear regression method with bootstrap analysis, then its prediction performance were further compared to artificial neural network and Vereecken with the Mean Error (ME) and the Mean Absolute Errors (MAE). The developed models were ranked the best models for estimation of hydraulic parameters with ME and MAE were less than O.lOcm /cm whereas Vereecken performed the worse results for hydraulic parameters, especially ME and MAE for parameters of a reached 0.74 1/cm and 0.63 1/cm, respectively. Function uncertainty evaluation was performed in Hydrus-ID model to simulate soil water, the developed pedotransfer functions and artificial neural network provide similar higher level of accuracy and precision with ME. 

Fig. 8.5 PLS validation plots showing for each predicted variable (i.e., sensory descriptor) the root mean squared error of prediction (RMSEP) over the first five model dimensions. RMSEP values were obtained from a leave-one-out bootstrapping algorithm, and both the cross-validated estimate black solid line) and the bias-adjusted eross-validation estimate ned doited line) are shown [38]...

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