# aic

Akaike’s Information Criterion for estimated model

## 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

collapse all

Estimate a transfer function model.

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.

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.

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

collapse all

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

collapse all

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 kth estimated model modelk.

collapse all

### 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-Hall PTR, 1999. See sections about the statistical framework for parameter estimation and maximum likelihood method and comparing model structures.