Akaike’s

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. 

K Yamaoka, T Nakagawa, T Uno. Application of Akaike s information criteria (AIC) in the evaluation of linear pharmacokinetic model. J Pharmacokin Biopharm 6 165-175, 1978. 

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. 

Akaike, S. et al., A new ent-clerodane diterpene from the aerial parts of Baccharis gaudichaudiana, Chem. Pharm. Bull, 51, 197, 2003. 

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). 

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

M. Stone, An asymptotic equivalence of choice of model by cross validation and Akaike s criterion. J R Stat Soc 39 44 7 (1977). 

Akaike s criterion and its derivations has been called by some [see Verbeke and Molenberghs (2000) for example] as a minimization function plus a penalty term for the number of parameters being estimated. As more model parameters are added to a model, 2LL tends to decrease but 2 p increases. Hence, AIC may decrease to a point as more parameters are added to a model but eventually the penalty term dominates the equation and AIC begins to increase. Conceptually this fits into the concept of bias-variance trade-off or the trade-off between overfitting and underfitting. 

Akaike, S. Studies on the chemical constituents of tobacco plant. IV. Isolation of mesoinositol from fresh and flue-cured tobacco leaves J. Agr. Chem. Soc. Japan 33 (1959) 670-671, see Chem. Abstr. 62 (1976) 6816c, see Tobacco Abstr. 3 (1959) 418. 

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. 

Akaike, S., Sumino, M., Sekine, T., Seo, S., Kimura, N. Ikegami, F. (2003). A New ent-Clerodane Diterpene from the Aerial Parts of Baccharis gaudidtaudiana. Chemical Pharmaceutical Bulletin, Vol 51, No. 2 (February 2003), pp.197-199, ISSN 0009-2363. 

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

Each model was applied to sugar attenuation data taken from multiple brewing fermentations (assessed using high-pressure liquid chromatography). Three techniques Akaike s (corrected) Information Criterion, comparison of the coefficients of determination (r ) and absolute residual sum of squares (RSS) were used to compare the fit of each model. Ideally, the data would adhere to a simplistic, theoretically derived formula such as a low parameter symmetric model. Unfortunately, the variability in both shape and lag time for each individual sugar necessitated a more flexible... 

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. 

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... 

In most cases, techniques are used to vary both the structure and the model parameters to obtain the best or simplest model that gives the best fit. Various information criteria like Akaike s or Bayesian measures can be used that essentially get the simplest model for the best fit. It is a form of parsimony in model building. Many packages such as the MATLAB Identification Toolbox help in such modeling. 

This section illustrates application of the PRESS statistic for process model structure selection. Ljung (1987) used data collected from a laboratory-scale Process Trainer to illustrate various identification techniques and examined the sum of squared conventional residuals and Akaike s information theoretic criterion (AIC) for model structure selection. Two sets of input-output data collected from this process are available within MATLAB. We use the entire first set of data M = 1000), called DRYER2 in MATLAB, for this study. Two different model structures are examined here, namely the ARX and FIR model structures, with the objective to find the model within a pcurtic-ular structure that produces the smallest PRESS. 

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