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# fpe

Akaike Final Prediction Error for estimated model

## Syntax

```fp = fpe(Model1,Model2,Model3,...)
```

## Description

Model is the name of an idtf, idgrey, idpoly, idproc, idss, idnlarx, idnlhw, or idnlgrey model object.

fp is returned as a row vector containing the values of the Akaike Final Prediction Error (FPE) for the different models.

## Definitions

Akaike's Final Prediction Error (FPE) criterion provides a measure of model quality 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 FPE.

 Note:   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 Final Prediction Error (FPE) is defined by the following equation:

$FPE=V\left(\frac{1+d}{N}}{1-d}{N}}\right)$

where V is the loss function, d is the number of estimated parameters, and N is the number of values in the estimation data set.

The toolbox assumes that the final prediction error is asymptotic for d<<N and uses the following approximation to compute FPE:

$FPE=V\left(1+2d}{N}\right)$

The loss function V is defined by the following equation:

$V=\mathrm{det}\left(\frac{1}{N}\sum _{1}^{N}\epsilon \left(t,{\stackrel{^}{\theta }}_{N}\right){\left(\epsilon \left(t,{\stackrel{^}{\theta }}_{N}\right)\right)}^{T}\right)$

where ${\theta }_{N}$ represents the estimated parameters.

## References

Sections 7.4 and 16.4 in Ljung (1999).