# Documentation

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# aic

Akaike's Information Criterion for estimated model

## Syntax

• ``value = aic(model)``
example
• ``value = aic(model1,...,modeln)``
• ``value = aic(___,measure)``
example

## Description

example

````value = aic(model)` returns the normalized Akaike's Information Criterion (AIC) value for the estimated model.```
````value = aic(model1,...,modeln)` returns the normalized AIC values for multiple estimated models.```

example

````value = aic(___,measure)` specifies the type of AIC.```

## Examples

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Estimate a transfer function model.

```load iddata1 z1; np = 2; sys = tfest(z1,np); ```

Compute the normalized Akaike's Information Criterion value.

```value = aic(sys) ```
```value = 0.5453 ```

The value is also computed during model estimation. Alternatively, use the `Report` property of the model to access this value.

```sys.Report.Fit.nAIC ```
```ans = 0.5453 ```

Estimate a transfer function model.

```load iddata1 z1; np = 2; sys = tfest(z1,np); ```

Compute the normalized Akaike's Information Criterion (AIC) value. This syntax is equivalent to `aic_raw = aic(sys)`.

```aic_raw = aic(sys,'nAIC') ```
```aic_raw = 0.5453 ```

Compute the raw AIC value.

```aic_raw = aic(sys,'aic') ```
```aic_raw = 1.0150e+03 ```

Compute the sample-size corrected AIC value.

```aic_c = aic(sys,'AICc') ```
```aic_c = 1.0153e+03 ```

Compute the Bayesian Information Criteria (BIC) value.

```bic = aic(sys,'BIC') ```
```bic = 1.0372e+03 ```

These values are also computed during model estimation. Alternatively, use the `Report.Fit` property of the model to access these values.

```sys.Report.Fit ```
```ans = struct with fields: FitPercent: 70.7720 LossFcn: 1.6575 MSE: 1.6575 FPE: 1.7252 AIC: 1.0150e+03 AICc: 1.0153e+03 nAIC: 0.5453 BIC: 1.0372e+03 ```

Estimate multiple Output-Error (OE) models and use the small sample-size corrected Akaike's Information Criterion (AICc) value to pick the one with optimal tradeoff between accuracy and complexity.

```load iddata2 ```

Specify model orders varying in 1:4 range.

```nf = 1:4; nb = 1:4; nk = 0:4; ```

Estimate OE models with all possible combinations of chosen order ranges.

```NN = struc(nf,nb,nk); models = cell(size(NN,1),1); for ct = 1:size(NN,1) models{ct} = oe(z2, NN(ct,:)); end ```

Compute the small sample-size corrected AIC values for the models, and return the smallest value.

```V = aic(models{:},'AICc'); [Vmin,I] = min(V); ```

Return the optimal model that has the smallest AICc value.

```models{I} ```
```ans = Discrete-time OE model: y(t) = [B(z)/F(z)]u(t) + e(t) B(z) = 1.067 z^-2 F(z) = 1 - 1.824 z^-1 + 1.195 z^-2 - 0.2307 z^-3 Sample time: 0.1 seconds Parameterization: Polynomial orders: nb=1 nf=3 nk=2 Number of free coefficients: 4 Use "polydata", "getpvec", "getcov" for parameters and their uncertainties. Status: Estimated using OE on time domain data "z2". Fit to estimation data: 86.53% FPE: 0.9809, MSE: 0.9615 ```

## Input Arguments

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Identified model, specified as one of the following model objects:

Type of AIC, specified as one of the following values:

• `'nAIC'` — Normalized AIC

• `'aic'` — Raw AIC

• `'AICc'` — Small sample-size corrected AIC

• `'BIC'` — Bayesian Information Criteria

## Output Arguments

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Value of the quality measure, returned as a scalar or vector. For multiple models, `value` is a row vector where `value(k)` corresponds to the `k`th estimated model `modelk`.

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### Akaike's Information Criterion (AIC)

Akaike's Information Criterion (AIC) provides a measure of model quality obtained by simulating the situation where the model is tested on a different data set. After computing several different models, you can compare them using this criterion. According to Akaike's theory, the most accurate model has the smallest AIC. If you use the same data set for both model estimation and validation, the fit always improves as you increase the model order and, therefore, the flexibility of the model structure.

Akaike's Information Criterion (AIC) includes the following quality metrics:

• Raw AIC, defined as:

`$AIC=N*\mathrm{log}\left(\mathrm{det}\left(\frac{1}{N}\sum _{1}^{N}\epsilon \left(t,{\stackrel{^}{\theta }}_{N}\right){\left(\epsilon \left(t,{\stackrel{^}{\theta }}_{N}\right)\right)}^{T}\right)\right)+2{n}_{p}+N*\left({n}_{y}*\left(\mathrm{log}\left(2\pi \right)+1\right)\right)$`

where:

• N is the number of values in the estimation data set

• ε(t) is a ny-by-1 vector of prediction errors

• ${\theta }_{N}$ represents the estimated parameters

• np is the number of estimated parameters

• ny is the number of model outputs

• Small sample-size corrected AIC, defined as:

`$AICc=AIC+2{n}_{p}*\frac{{n}_{p}+1}{N-{n}_{p}-1}$`
• Normalized AIC, defined as:

`$nAIC=\mathrm{log}\left(\mathrm{det}\left(\frac{1}{N}\sum _{1}^{N}\epsilon \left(t,{\stackrel{^}{\theta }}_{N}\right){\left(\epsilon \left(t,{\stackrel{^}{\theta }}_{N}\right)\right)}^{T}\right)\right)+\frac{2{n}_{p}}{N}$`
• Bayesian Information Criteria, defined as:

`$BIC=N*\mathrm{log}\left(\mathrm{det}\left(\frac{1}{N}\sum _{1}^{N}\epsilon \left(t,{\stackrel{^}{\theta }}_{N}\right){\left(\epsilon \left(t,{\stackrel{^}{\theta }}_{N}\right)\right)}^{T}\right)\right)+N*\left({n}_{y}*\mathrm{log}\left(2\pi \right)+1\right)+{n}_{p}*\mathrm{log}\left(N\right)$`

### Tips

• The software computes and stores all types of Akaike's Information Criterion metrics during model estimation. If you want to access these values, see the `Report.Fit` property of the model.

## References

[1] Ljung, L. System Identification: Theory for the User, Upper Saddle River, NJ, Prentice-Hal PTR, 1999. See sections about the statistical framework for parameter estimation and maximum likelihood method and comparing model structures.