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# ivar

AR model estimation using instrumental variable method

## Syntax

sys = ivar(data,na)
sys = ivar(data,na,nc)
sys = ivar(data,na,nc,max_size)

## Description

sys = ivar(data,na) estimates an AR polynomial model, sys, using the instrumental variable method and the time series data data. na specifies the order of the A polynomial.

An AR model is represented by the equation:

$A\left(q\right)y\left(t\right)=e\left(t\right)$

In the above model, e(t) is an arbitrary process, assumed to be a moving average process of order nc, possibly time varying. nc is assumed to be equal to na. Instruments are chosen as appropriately filtered outputs, delayed nc steps.

sys = ivar(data,na,nc) specifies the value of the moving average process order, nc, separately.

sys = ivar(data,na,nc,max_size) specifies the maximum size of matrices formed during estimation.

## Input Arguments

 data Estimation time series data. data must be an iddata object with scalar output data only. na Order of the A polynomial nc Order of the moving average process representing e(t). max_size Maximum matrix size. max_size specifies the maximum size of any matrix formed by the algorithm for estimation. Specify max_size as a reasonably large positive integer. Default: 250000

## Output Arguments

 sys Identified polynomial model. sys is an AR idpoly model which encapsulates the identified polynomial model.

## Examples

Compare spectra for sinusoids in noise, estimated by the IV method and by the forward-backward least squares method.

```y = iddata(sin([1:500]'*1.2) + sin([1:500]'*1.5) + ...
0.2*randn(500,1),[]);
miv = ivar(y,4);
mls = ar(y,4);
spectrum(miv,mls)
```

## References

[1] Stoica, P., et al. Optimal Instrumental Variable Estimates of the AR-parameters of an ARMA Process, IEEE Trans. Autom. Control, Volume AC-30, 1985, pp. 1066–1074.