Akaike information criterion

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The Akaike information criterion (AIC) is a measure of the relative quality of a statistical model for a given set of data. As such, AIC provides a means for model selection.

AIC deals with the trade-off between the goodness of fit of the model and the complexity of the model. It is founded on information theory: it offers a relative estimate of the information lost when a given model is used to represent the process that generates the data.

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

Definition[edit]

For any statistical model, the AIC value is

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

where k is the number of parameters in the model, and L is the maximized value of the likelihood function for the 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. The penalty discourages overfitting (increasing the number of parameters in the model almost always improves the goodness of the fit).

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, DKL(fg1); similarly, the information lost from using g2 to represent f could be found by calculating DKL(fg2). 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. 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[edit]

To apply AIC 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 (i.e. the process that generates the data). We wish to select, from among R candidate models, the model that minimizes the information loss. We cannot choose with certainty, but we can minimize the estimated information loss.

Denote the AIC values of the candidate models by AIC1, AIC2, AIC3, …, AICR. Let AICmin be the minimum of those values. Then exp((AICmin−AICi)/2) can be interpreted as the relative probability that the ith model minimizes the (estimated) information loss.[1]

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; similarly, the third model is exp((100−110)/2) = 0.007 times as probable as the first model to minimize the information loss.

In this example, we would omit the third model from further consideration. We then have three options: (1) gather more data, in the hope that this will allow clearly distinguishing between the first two models; (2) simply conclude that the data is insufficient to support selecting one model from among the first two; (3) take a weighted average of the first two models, with weights 1 and 0.368, respectively, and then do statistical inference based on the weighted multimodel.[2]

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

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.[3]

AICc[edit]

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.[4] 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.[5]

Brockwell & Davis (1991, p. 273) advise using AICc as the primary criterion in selecting the orders of an ARMA model for time series. McQuarrie & Tsai (1998) 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 (1991), Burnham & Anderson, and Cavanaugh (1997). 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.

Relationship to chi-square fits and GLMs[edit]

Chi-square fits[edit]

Often, one wishes to select amongst competing models where the likelihood functions assume that the underlying errors are normally distributed (with mean zero) 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(x_i))^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(x_i))^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.

Another convenient form arises if the σi are assumed to be identical (i.e. \sigma_i = \sigma) and σ is unknown and has to be fitted. Then one can write AIC = n \ln( RSS/n) + 2 k + C, where RSS = \sum_{i=1}^n (y_i-f(x_i))^2 is the residual sum of squares. Again C can be ignored in model comparisons. Because σ is fitted, it has to be counted as a parameter when calculating k.[6]

GLMs[edit]

When all the candidate models are generalized linear models, a reasonable approach to selecting a model is to choose the model that maximizes the likelihood given the expected, not the actual, response \mu, given by \mathcal{L}(\beta;\mu). This avoids improvements in model performance coming from fitting parameters to error terms, as can occur if \mathcal{L}(\beta;\mathbf{y}) is instead maximized. The likelihood \mathcal{L}(\beta;\mu) can be easily estimated, and the model with the maximum estimate turns out to be the model with the minimum AIC.[7]

History[edit]

The Akaike information criterion was developed by Hirotugu Akaike, under the name of "an information criterion". It was first published by Akaike in 1974.[8] The paper was the 73rd most cited paper of all time in a Thomson Reuters list, having received more than 14,000 citations as of October 2014.[9]

The original derivation of AIC relied upon some strong assumptions. Takeuchi (1976) showed that the assumptions could be made much weaker. Takeuchi's work, however, was in Japanese, and was not widely known outside Japan for many years.

AICc was originally proposed for linear regression (only) by Sugiura (1978). That instigated the work of Hurvich & Tsai (1989), and several further papers by the same authors, which extended the situations in which AICc could be applied. The work of Hurvich & Tsai contributed to the decision to publish a second edition of the volume by Brockwell & Davis (1991), which is the standard reference for linear time series; the second edition states, "our prime criterion for model selection [among ARMA(p,q) models] will be the AICc".[10]

The volume by Burnham & Anderson (2002) was the first attempt to set out the information-theoretic approach in a general context. It includes an English exposition of the work of Takeuchi. The volume led to far greater use of the information-theoretic approach, and it now has over 20000 citations on Google Scholar.

Akaike originally called his approach an "entropy maximization principle", as the approach is founded on the concept entropy in information theory. Burnham & Anderson (2002, ch. 2) discuss and expand on this, and trace the approach back to the work of Ludwig Boltzmann on thermodynamics. Briefly, minimizing AIC in a statistical model is essentially equivalent to maximizing entropy in a thermodynamic system. In other words, the information-theoretic approach in statistics is essentially applying the Second Law of Thermodynamics.

Comparison with other model selection methods[edit]

Comparison with BIC[edit]

The AIC penalizes the number of parameters less strongly than does the Bayesian information criterion (BIC). A comparison of AIC/AICc and BIC is given by Burnham & Anderson (2002, §6.4). The authors show that AIC and AICc can be derived in the same Bayesian framework as BIC, just by using a different prior. The authors also argue that AIC/AICc has theoretical advantages over BIC. First, because AIC/AICc is derived from principles of information; BIC is not, despite its name. Second, because the (Bayesian-framework) 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. Additionally, they present a few simulation studies that suggest AICc tends to have practical/performance advantages over BIC. See too 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 under the assumption. Yang further shows that the rate at which AIC converges to the optimum is, in a certain sense, the best possible.

Comparison with cross-validation[edit]

For ordinary linear regression models, leave-one-out cross-validation is asymptotically equivalent to the AIC criterion.[11] Such asymptotic equivalence also holds for mixed-effect models.[12]

See also[edit]

Notes[edit]

  1. ^ Burnham & Anderson 2002, §6.4.5
  2. ^ Burnham & Anderson 2002
  3. ^ Burnham & Anderson 2002, §2.12.4
  4. ^ Burnham & Anderson 2004
  5. ^ Claeskens & Hjort 2008, §8.3
  6. ^ Burnham & Anderson 2002, pp. 12, 63
  7. ^ Wood 2006, §2.1.4
  8. ^ Akaike 1974
  9. ^ Van Noordon R, Maher B, Nuzzo R. The Top 100 papers. Nature 2014;514:550-553 Numbers from Thomson Reuters; extracted 7 October 2014 http://www.nature.com/polopoly_fs/1.16224!/menu/main/topColumns/topLeftColumn/pdf/514550a.pdf
  10. ^ Brockwell & Davis 1991, p. 273
  11. ^ Stone 1977
  12. ^ Fang 2011

References[edit]

Further reading[edit]

  • Anderson, D. R. (2008), Model Based Inference in the Life Sciences, Springer .
  • Liu, W.; Yang, Y. (2011), Parametric or nonparametric?, Annals of Statistics 39: 2074–2102, doi:10.1214/11-AOS899 
  • Lukacs, Paul M.; Thompson, William L.; Kendall, William L.; Gould, William R.; Doherty, Paul F. Jr.; Burnham, Kenneth P.; Anderson, David R. (5 March 2007), Concerns regarding a call for pluralism of information theory and hypothesis testing, Journal of Applied Ecology 44 (2): 456–460, doi:10.1111/j.1365-2664.2006.01267.x 

External links[edit]