# fpe

Akaike’s Final Prediction Error for estimated model

## Description

example

value = fpe(model) returns the Final Prediction Error (FPE) value for the estimated model.

value = fpe(model1,...,modeln) returns the FPE value for multiple estimated models.

## Examples

collapse all

Estimate a transfer function model.

np = 2;
sys = tfest(z1,np);

Compute the Final Prediction Error (FPE) value.

value = fpe(sys)
value = 1.7252

Alternatively, use the Report property of the model to access the value.

sys.Report.Fit.FPE
ans = 1.7252

Estimate multiple Output-Error (OE) models and use Akaike's Final Prediction Error (FPE) 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 = fpe(models{:});
[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:

## Output Arguments

collapse all

Final Prediction Error (FPE) value, 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 Final Prediction Error (FPE)

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.

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:

$FPE=\mathrm{det}\left(\frac{1}{N}\sum _{1}^{N}e\left(t,{\stackrel{^}{\theta }}_{N}\right){\left(e\left(t,{\stackrel{^}{\theta }}_{N}\right)\right)}^{T}\right)\left(\frac{1+d}{N}}{1-d}{N}}\right)$

where:

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

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

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

• d is the number of estimated parameters.

If number of parameters exceeds the number of samples, FPE is not computed when model estimation is performed (model.Report.FPE is empty). The fpe command returns NaN.

## Tips

• The software computes and stores the FPE value during model estimation. If you want to access this value, see the Report.Fit.FPE property of the model.

## References

[1] Ljung, L. System Identification: Theory for the User, Upper Saddle River, NJ, Prentice-Hall PTR, 1999. See sections 7.4 and 16.4.