Final prediction error 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... 

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. 

An important point is the evaluation of the models. While most methods select the best model at the basis of a criterion like adjusted R2, AIC, BIC, or Mallow s Cp (see Section 4.2.4), the resulting optimal model must not necessarily be optimal for prediction. These criteria take into consideration the residual sum of squared errors (RSS), and they penalize for a larger number of variables in the model. However, selection of the final best model has to be based on an appropriate evaluation scheme and on an appropriate performance measure for the prediction of new cases. A final model selection based on fit-criteria (as mostly used in variable selection) is not acceptable. 

In this paper I have attempted to demonstrate a method for the development of hyperbolic rate models that are adequate for the design of chemical reactors. The method is rapid and overcomes most of the problems that historically have hampered the development of such models for complex reactions. I have shown that the quality of fit of a model to error-containing data is a poor criterion for model discrimination, and that several models may predict almost equally well. This, of course, has been known for a long time, but it has not been widely recognized that the model that fits the data least well may be the best model, and that the converse also may be true. In the final analysis... 

Structural identification, i.e. selection of the model type and structure, is always an arbitrary research decision. What is helpful is autocorrelation and spectrum analysis (detection of the intervals). Generally, the simplest possible model is chosen. A series of information criteria (algorithms) exist that may help in this process, usually defined as a combination of the model error and the number of model parameters, such as the AIC criterion (Akaike s information criterion), the criterion of the final error of the prediction, Ravelli Vulpiani criterion or Schwarz s BIC criterion (Bayesian information criterion comparison of log likelihood of specific models corrected by the number of estimated parameters and the number of observations). 

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