Final prediction error

Several regression parameters are measures suitable for choosing models to predict a dependent variable y from a potentially large set of independent variables. The generalized final prediction error criteria (FPE criteria) include many known parameters and are defined as [Krieger and Zhang, 2006]... 

Table Rl Final prediction errors derived from the generalized final prediction error criteria.

Among the final prediction error criteria, the most popular are detailed below together with some other criteria. 

Krieger, A.M. and Zhang, P. (2006) Generalized final prediction error criteria, in Encyclopedia of Statistical Sciences (eds S. Kotz, C.B. Read, N. Balakrishnan and B. Vidakovic), John Wiley Sons, Inc., New York, pp. 1—4. 

The loss function is also an estimate of the noise covariance 8, what explains the notation. Other criteria include penalties for model complexity like Akaike s final prediction error (FPE) criterion or Rissanen s minimum description length criterion. 

Final Prediction Error Criterion (FPE) The final prediction error criterion seeks to minimise the variance of the prediction errors with future data. It is defined as... 

It should also be acknowledged that in recent years computational quantum chemistry has achieved a number of predictions that have since been experimentelly confirmed (45-47). On the other hand, since numerous anomalies remain even within attempts to explain the properties of atoms in terms of quantum mechanics, the field of molecular quantum mechanics can hardly be regarded as resting on a firm foundation (48). Also, as many authors have pointed out, the vast majority of ab initio research judges its methods merely by comparison with experimental date and does not seek to establish internal criteria to predict error bounds theoretically (49-51). The message to chemical education must, therefore, be not to emphasize the power of quantum mechanics in chemistry and not to imply that it necessarily holds the final answers to difficult chemical questions (52). 

Note that the two samples with the largest concentration residuals have extreme temperature values 6ow), Because the model is extrapolating when predicting the caustic concentration of these samples, the slightly inflated prediction errors associated with these samples do not necessarily indicate a poor model or that the samples are outliers. Therefore, these samples are included when the final model is constructed and the final model temperature range is 50-70 C. 

Root Mean Square EtTor of Prediction (RMSEP) Plot (Model Diagnostic) The number of variables to include is finalized using a validation procedure that accounts for predictive ability. There are two approaches for calculating the prediction error internal cross-validation (e.g., Icave-one-out cross-validation with the calibration data) or external validation (i.e.. perform prediction... 

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. 

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). 

For each reduced data set, the model is calculated and responses for the deleted objects are predicted from the model. The squared differences between the true response and the predicted response for each object left out are added to PRESS ( prediction error sum of squares). From the final PRESS, the (or R ) and RMSEP ( root mean square error in prediction) values are usually calculated [Cruciani, Baroni et al, 1992]. 

The second way that an independent data set is used in validation is to fix the parameter estimates under the final model and then obtain summary measures of the goodness of fit under the independent data set (Kimko, 2001). For example, Mandema et al. (1996) generated a PopPK model for immediate release (IR) and con-trolled-release (CR) oxycodone after single dose administration. The plasma concentrations after four days administration were then compared to model predictions. The log-prediction error (LPE) between observed concentrations (Cp) and model predicted concentrations (Cp j was calculated as... 

Time GM Predictive value The error predictive value The final predictive value T statistic... 

Finally, the evidence of model class Cj is readily obtained by integrating the prediction-error variance ... 

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