Akaike’s Information Criterion

I. Bondarenko, H. Van Malderen, B. Treiger, P. Van Espen and R. Van Grieken, Hierarchical cluster analysis with stopping rules built on Akaike s information criterion for aerosol particle classification based on electron probe X-ray microanalysis. Chemom. Intell. Lab. Syst., 22 (1994) 87-95. 

A number of performance criteria are not primarily dedicated to the users of a model but are applied in model generation and optimization. For instance, the mean squared error (MSE) or similar measures are considered for optimization of the number of components in PLS or PC A. For variable selection, the models to be compared have different numbers of variables in this case—and especially if a fit criterion is used—the performance measure must consider the number of variables appropriate measures are the adjusted squared correlation coefficient, adjR, or the Akaike S information criterion (AIC) see Section 4.2.3. 

Yamaoka K, Nakagawa T, Uno T (1978) Application of Akaike s Information criterion (AIC) in the evaluation of linear pharmacokinetic equations. J Pharmacokinet Biopharm 6 165-175. 

For comparison, we also use stepwise regression to select variables, specifically the R function step (R Development Core Team, 2005), which uses Akaike s information criterion (see Akaike, 1973). The selection from all 41 input variables... 

If the calculated F value is higher than the tabled F at the chosen level of significance (often 0.05), the use of the more detailed model is supported. Another commonly used parameter is the Akaike s Information Criterion (AIC) value. It is calculated for each model, and the model producing the lowest value (most negative value) is considered the better model. The AIC value is calculated using the number of data (n), WSS, and the number of parameters (m). 

In forward selection, the first variable selected for an entry into the constructed model is the one with the largest correlation with the dependent variable. Once the variable has been selected, it is evaluated on the basis of certain criteria. The most common ones are Mallows Cp or Akaike s information criterion. If the first selected variable meets the criterion for inclusion, then the forward selection continues, i.e. the statistics for the variables not in the equation are used to select the next one. The procedure stops, when no other variables are left that meet the entry criterion. 

Model comparison plays a central role in statistical learning and chemometrics. Performances of models need to be assessed using a given criterion based on which models can be compared. To our knowledge, there exist a variety of criteria that can be applied for model assessment, such as Akaike s information criterion (AIC) [1], Bayesian information criterion (BIC) [2], deviance information criterion (DlC),Mallow s Cp statistic, cross validation [3-6] and so on. There is a large body of literature that is devoted to these criteria. With the aid of a chosen criterion, different models can be comp>ared. For example, a model with a smaller AIC or BIC is preferred if AIC or BIC are chosen for model assessment. 

Bozdogan, H. (1987) Model selection and Akaike s information criterion (AiC) the generai theory and its analytical extensions. Psychometrika 52, 345-370. 

MSE is mean square error and AlC is Akaike s Information Criterion. QSARs with the smallest AlC are likely to have the most information regardless of the numb of independent variables. 

Mean absolute percent error between observed and predicted values in the prediction Akaike s Information Criterion Mean square error Adjusted r square... 

Another approach to model validation is to consider various information criteria that assess the trade-off between the number of parameters selected and the variance of the model. These criteria can be useful for automating the estimation of initial process parameters. However, any model obtained using such an approach still needs to be validated for normality and purpose. The most common information criteriMi is Akaike s information criterion, which for any time series model can be written as... 

Akaike s Irtformation Criterion (AIC) Akaike s information criterion seeks to find the global minimum between the variance and the number of parameters. It is defined as... 

The Akaike s Information Criterion (AIC) can be used for model selection and is applicable to compare models. The AIC value is calculated according to equation [6.2] ... 

Statistical analyses were carried out using the MIXED procedure for repeated measures in SAS (SAS release 9.1) with treatment group (1-4), and period (1-5) as the main effects, and interactions between the main effects. The covariance structure was UN (unstructured) and was chosen according Akaike s information criterion. 

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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