Akaike information criterion

The Akaike information criterion is a measure of the relative goodness of fit of a statistical model. It was developed by Hirotsugu Akaike, under the name of "an information criterion" (AIC), and was first published by Akaike in 1974. It is grounded in the concept of information entropy, in effect offering a relative measure of the information lost when a given model is used to describe reality. It can be said to describe the tradeoff between bias and variance in model construction, or loosely speaking between accuracy and complexity of the model.

AIC values provide a means for model selection. AIC does not provide a test of a model in the sense of testing a null hypothesis; i.e. AIC can tell nothing about how well a model fits the data in an absolute sense. If all the candidate models fit poorly, AIC will not give any warning of that.

Definition

In the general case, the AIC is

${\mathit {AIC}}=2k-2\ln(L)\,$

where k is the number of parameters in the statistical model, and L is the maximized value of the likelihood function for the estimated model.

Given a set of candidate models for the data, the preferred model is the one with the minimum AIC value. Hence AIC not only rewards goodness of fit, but also includes a penalty that is an increasing function of the number of estimated parameters. This penalty discourages overfitting (increasing the number of free parameters in the model improves the goodness of the fit, regardless of the number of free parameters in the data-generating process).

AIC is founded in information theory. Suppose that the data is generated by some unknown process f. We consider two candidate models to represent f: g₁ and g₂. If we knew f, then we could find the information lost from using g₁ to represent f by calculating the Kullback–Leibler divergence, D_KL(f,g₁); similarly, the information lost from using g₂ to represent f would be found by calculating D_KL(f,g₂). We would then choose the candidate model that minimized the information loss.

We cannot choose with certainty, because we do not know f. Akaike (1974) showed, however, that we can estimate, via AIC, how much more (or less) information is lost by g₁ than by g₂. It is remarkable that such a simple formula for AIC results. The estimate, though, is only valid asymptotically; if the number of data points is small, then some correction is often necessary (see AICc, below).

How to apply AIC in practice

AIC estimates relative support for a model. To apply this in practice, we start with a set of candidate models, and then find the models' corresponding AIC values. There will almost always be information lost due to using one of the candidate models to represent the "true" model. We wish to select, from among R candidate models, the model that minimizes the information loss. We cannot do this exactly, but we can minimize the estimated information loss.

Denote the AIC values of the candidate models by AIC₁, AIC₂, AIC₃, …, AIC_R. Let AIC_min be the minimum of those values. Then exp((AIC_min−AIC_i)/2) can be interpreted as the relative probability that the ith model minimizes the (estimated) information loss.

As an example, suppose that there were three models in the candidate set, with AIC values 100, 102, and 110. Then the second model is exp((100−102)/2) = 0.368 times as probable as the first model to minimize the information loss, and the third model is exp((100−110)/2) = 0.007 times as probable as the first model to minimize the information loss. In this case, we might omit the third model from further consideration and take a weighted average of the first two models, with weights 1 and 0.368, respectively. Statistical inference would then be based on the weighted multimodel.

If all the models in the candidate set have the same number of parameters, then using AIC might at first appear to be very similar to using the likelihood-ratio test. There are, however, important distinctions. In particular, the likelihood-ratio test is valid only for nested models whereas AIC (and AICc) has no such restriction.

The quantity exp((AIC_min−AIC_i)/2) is the relative likelihood of model i.

AICc

AICc is AIC with a correction for finite sample sizes:

$AICc=AIC+{\frac {2k(k+1)}{n-k-1}}$

where n denotes the sample size. Thus, AICc is AIC with a greater penalty for extra parameters.

Burnham & Anderson (2002) strongly recommend using AICc, rather than AIC, if n is small or k is large. Since AICc converges to AIC as n gets large, AICc generally should be employed regardless. Using AIC, instead of AICc, when n is not many times larger than k², increases the probability of selecting models that have too many parameters, i.e. of overfitting. The probability of AIC overfitting can be substantial, in some cases.

Brockwell & Davis (p. 273) advise using AICc as the primary criterion in selecting the orders of an ARMA model for time series. McQuarrie & Tsai ground their high opinion of AICc on extensive simulation work with regression and time series.

AICc was first proposed by Hurvich & Tsai (1989). Different derivations of it are given by Brockwell & Davis, Burnham & Anderson, and Cavanaugh. All the derivations assume a univariate linear model with normally-distributed errors (conditional upon regressors); if that assumption does not hold, then the formula for AICc will usually change. Further discussion of this, with examples of other assumptions, is given by Burnham & Anderson (2002, ch.7). In particular, bootstrap estimation is usually feasible.

Note that when all the models in the candidate set have the same k, then AICc and AIC will give identical (relative) valuations. In that situation, then, AIC can always be used.

Relevance to chi-squared fitting

Often, one wishes to select amongst competing models where the likelihood functions assume that the underlying errors are normally distributed and independent. This assumption leads to $\chi ^{2}$ model fitting.

For $\chi ^{2}$ fitting, the likelihood is given by

$L=\prod _{i=1}^{n}\left({\frac {1}{2\pi \sigma _{i}^{2}}}\right)^{1/2}\exp \left(-\sum _{i=1}^{n}{\frac {(y_{i}-f(\mathbf {x} ))^{2}}{2\sigma _{i}^{2}}}\right)$

$\therefore \ln(L)=\ln \left(\prod _{i=1}^{n}\left({\frac {1}{2\pi \sigma _{i}^{2}}}\right)^{1/2}\right)-{\frac {1}{2}}\sum _{i=1}^{n}{\frac {(y_{i}-f(\mathbf {x} ))^{2}}{\sigma _{i}^{2}}}$

$\therefore \ln(L)=C-\chi ^{2}/2\,$,

where C is a constant independent of the model used, and dependent only on the use of particular data points. i.e. it does not change if the data do not change.

The AIC is therefore given by $AIC=2k-2\ln(L)=2k-2(C-\chi ^{2}/2)=2k-2C+\chi ^{2}\,$. As only differences in AIC are meaningful, the constant C can be ignored, allowing us to take $AIC=\chi ^{2}+2k$ for model comparisons. This form is often convenient, because most model-fitting programs produce $\chi ^{2}$ as a statistic for the fit.

Another convenient form arises if the σ_i are assumed to be identical and the residual sum of squares (RSS) is available. Then we get AIC = n ln(RSS/n) + 2k + C, where again C can be ignored in model comparisons.

Bayesian information criterion

The AIC penalizes the number of parameters less strongly than does the Bayesian information criterion (BIC), which was independently developed by Akaike and by Schwarz in 1978, using Bayesian formalism. Akaike's version of BIC was originally denoted ABIC (for "a Bayesian Information Criterion") or referred to as Akaike's Bayesian Information Criterion.

A comparison of AIC/AICc and BIC is given by Burnham & Anderson (2002, sect. 6.4). The authors argue that AIC/AICc has theoretical advantages over BIC. Firstly, because AIC/AICc is derived from principles of information. Secondly, because the (Bayesian) derivation of BIC has a prior of 1/R (where R is the number of candidate models), which is "not sensible", since the prior should be a decreasing function of k. The authors also show that AIC and AICc can be derived in the same Bayesian framework as BIC, just by using a different prior. Additionally, they present a few simulation studies that suggest AICc tends to have practical/performance advantages over BIC. See additionally Burnham & Anderson (2004).

Further comparison of AIC and BIC, in the context of regression, is given by Yang (2005). In particular, AIC is asymptotically optimal in selecting the model with the least mean squared error, under the assumption that the exact "true" model is not in the candidate set (as is virtually always the case in practice); BIC is not asymptotically optimal. Yang further shows that the rate at which AIC converges to the optimum is, in a certain sense, the best possible.

See also

• Deviance information criterion
• Focused information criterion
• Occam's Razor

References

• Akaike, Hirotugu (1974). "A new look at the statistical model identification". IEEE Transactions on Automatic Control. 19 (6): 716–723. doi:10.1109/TAC.1974.1100705. MR 0423716.
• Akaike, Hirotsugu (1980). "Likelihood and the Bayes procedure", Bayesian Statistics, Ed. J.M. Bernardo et al., Valencia: University Press. p.143-166.
• Anderson, D.R. (2008), Model Based Inference in the Life Sciences, Springer.
• Brockwell, P.J., and Davis, R.A. (2009). Time Series: Theory and Methods, 2nd ed. Springer.
• Burnham, K. P., and Anderson, D.R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. Springer-Verlag. ISBN 0-387-95364-7. [This has over 10000 citations on Google Scholar.]
• Burnham, K. P., and Anderson, D.R. (2004), "Multimodel inference: understanding AIC and BIC in Model Selection", Sociological Methods and Research, 33: 261-304.
• Cavanaugh, J.E. (1997). "Unifying the derivations of the Akaike and corrected Akaike information criteria", Statistics and Probability Letters, 31: 201-208.
• Claeskens, G, and N.L. Hjort (2008). Model Selection and Model Averaging, Cambridge.
• Fang, Yixin (2011). "Asymptotic equivalence between cross-validations and Akaike Information Criteria in mixed-effects models", Journal of Data Science, 9:15-21.
• Hurvich, C. M., and Tsai, C.-L. (1989). "Regression and time series model selection in small samples", Biometrika, 76: 297–307.
• Lukacs, P.M., et al. (2007). "Concerns regarding a call for pluralism of information theory and hypothesis testing", Journal of Applied Ecology, 44:456–460. doi:10.1111/j.1365-2664.2006.01267.x
• McQuarrie, A. D. R., and Tsai, C.-L. (1998). Regression and Time Series Model Selection. World Scientific. ISBN 981023242X
• Takeuchi, K. (1976). "Distribution of informational statistics and a criterion of model fitting", Suri-Kagaku (Mathematical Sciences), 153: 12–18. (In Japanese).
• Yang, Y. (2005). "Can the strengths of AIC and BIC be shared?", Biometrika, 92: 937-950.

External links

• Hirotogu Akaike comments on how he arrived at the AIC, in This Week's Citation Classic (21 December 1981)
• AIC (Aalto University)
• Example Calculation (University of Georgia)
• Model Selection (University of Iowa)