# 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`, and possibly time varying. The function assumes that `nc` is equal to `na`. Instruments are chosen as appropriately filtered outputs, delayed `nc` steps.```

example

````sys = ivar(data,na,nc)` specifies the value of the moving average process order `nc` independently.```
````sys = ivar(data,na,nc,max_size)` specifies the maximum size of matrices formed during estimation.```

## Examples

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Construct output data by combining sinusoidal signals and noise.

```y = iddata(sin([1:500]'*1.2) + sin([1:500]'*1.5) + 0.2*randn(500,1),[]); plot(y)```

Estimate fourth-order models using the IV method (`ivar`) and the forward-backward least-squares method (`ar`).

```sysiv = ivar(y,4); sysls = ar(y,4);```

Compare the spectra of the two models.

```spectrum(sysiv,sysls) legend('ivar','ar')```

The model estimated with `ivar` exhibits the two distinct spectral peaks. The model estimated with `ar` exhibits only one peak.

## Input Arguments

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Time-series data, specified as an `iddata` object that contains a single output channel and an empty input channel.

Order of the A polynomial, specified as a positive integer.

Order of the moving average process that represents e(t), specified as a positive integer.

Maximum matrix size for any matrix formed by the algorithm for estimation, specified as a positive integer. Specify `max_size` as a reasonably large value.

## Output Arguments

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Identified polynomial model, returned as a discrete-time `idpoly` model object.

Information about the estimation results and options used is stored in the `Report` property of the model. `Report` has the following fields.

Report FieldDescription
`Status`

Summary of the model status, which indicates whether the model was created by construction or obtained by estimation

`Method`

Estimation command used

`InitialCondition`

Handling of initial conditions during model estimation, returned as one of the following values:

• `'zero'` — The initial conditions were set to zero.

• `'estimate'` — The initial conditions were treated as independent estimation parameters.

This field is especially useful to view how the initial conditions were handled when the `InitialCondition` option in the estimation option set is `'auto'`.

`Fit`

Quantitative assessment of the estimation, returned as a structure. See Loss Function and Model Quality Metrics for more information on these quality metrics. The structure has these fields.

• `FitPercent` — Normalized root mean squared error (NRMSE) measure of how well the response of the model fits the estimation data, expressed as the percentage fitpercent = 100(1-NRMSE)

• `LossFcn` — Value of the loss function when the estimation completes

• `MSE` — Mean squared error (MSE) measure of how well the response of the model fits the estimation data

• `FPE` — Final prediction error for the model

• `AIC` — Raw Akaike Information Criteria (AIC) measure of model quality

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

• `nAIC` — Normalized AIC

• `BIC` — Bayesian Information Criteria (BIC)

`Parameters`

Estimated values of model parameters

`DataUsed`

Attributes of the data used for estimation, returned as a structure with the following fields.

• `Name` — Name of the data set

• `Type` — Data type

• `Length` — Number of data samples

• `Ts` — Sample time

• `InterSample` — Input intersample behavior, returned as one of the following values:

• `'zoh'` — A zero-order hold maintains a piecewise-constant input signal between samples.

• `'foh'` — A first-order hold maintains a piecewise-linear input signal between samples.

• `'bl'` — Band-limited behavior specifies that the continuous-time input signal has zero power above the Nyquist frequency.

• `InputOffset` — Offset removed from time-domain input data during estimation. For nonlinear models, it is `[]`.

• `OutputOffset` — Offset removed from time-domain output data during estimation. For nonlinear models, it is `[]`.

For more information on using `Report`, see Estimation Report.

## References

[1] Stoica, P., T. Soderstrom, and B. Friedlander. Optimal Instrumental Variable Estimates of the AR Parameters of an ARMA Process. IEEE Transactions on Automatic Control 30, no. 11 (November 1985): 1066–74, https://doi.org/10.1109/TAC.1985.1103839.

## Version History

Introduced before R2006a