# Documentation

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# ar

Estimate parameters of AR model for scalar time series

## Syntax

```m = ar(y,n)[m,ref1] = ar(y,n,approach,window)m= ar(y,n,Name,Value)m= ar(y,n,___,opt)```

## Description

 Note:   Use for scalar time series only. For multivariate data, use `arx`.

`m = ar(y,n)` returns an `idpoly` model `m`.

```[m,ref1] = ar(y,n,approach,window)``` returns an `idpoly` model `m` and the variable `refl`. For the two lattice-based approaches, `'burg'` and `'gl'`, `refl` stores the reflection coefficients in the first row, and the corresponding loss function values in the second row. The first column of `refl` is the zeroth-order model, and the `(2,1)` element of `refl` is the norm of the time series itself.

`m= ar(y,n,Name,Value)` specifies model structure attributes using one or more `Name,Value` pair arguments.

`m= ar(y,n,___,opt)` specifies the estimations options using `opt`.

## Input Arguments

 `y` `iddata` object that contains the time-series data (one output channel). `n` Scalar that specifies the order of the model you want to estimate (the number of A parameters in the AR model). `approach` Algorithm for computing the least squares AR model, specified as one of the following values: `'burg'`: Burg's lattice-based method. Solves the lattice filter equations using the harmonic mean of forward and backward squared prediction errors.`'fb'`: (Default) Forward-backward approach. Minimizes the sum of a least- squares criterion for a forward model, and the analogous criterion for a time-reversed model.`'gl'`: Geometric lattice approach. Similar to Burg's method, but uses the geometric mean instead of the harmonic mean during minimization. `'ls'`: Least-squares approach. Minimizes the standard sum of squared forward-prediction errors.`'yw'`: Yule-Walker approach. Solves the Yule-Walker equations, formed from sample covariances. `window` Use of information about the data outside the measured time interval (past and future values), specified as one of the following values: `'now'`: (Default) No windowing. This value is the default except when the `approach` argument is `'yw'`. Only measured data is used to form regression vectors. The summation in the criteria starts at the sample index equal to `n+1`.`'pow'`: Postwindowing. Missing end values are replaced with zeros and the summation is extended to time `N+n` (`N` is the number of observations).`'ppw'`: Pre- and postwindowing. Used in the Yule-Walker approach.`'prw'`: Prewindowing. Missing past values are replaced with zeros so that the summation in the criteria can start at time equal to zero. `opt` Estimation options. `opt` is an options set that specifies the following: data offsetscovariance handlingestimation approachestimation window Use `arOptions` to create the options set.

### Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside single quotes (`' '`). You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

 `'Ts'` Positive scalar that specifies the sample time. Use when you specify `Y` as double vector rather than an `IDDATA` object. `'IntegrateNoise'` Boolean value that specifies whether the noise source contains an integrator or not. Use it to create "ARI" structure models: $Ay=\frac{e}{\left(1-{z}^{-1}\right)}$ Default: false

## Output Arguments

 `m` An `idpoly` model. `ref1` An 2–by-2 array. The first row stores the reflection coefficients, and the second row stores the corresponding loss function values. The first column of `refl` is the zeroth-order model, and the (2,1) element of `refl` is the norm of the time series itself.

## Examples

Given a sinusoidal signal with noise, compare the spectral estimates of Burg's method with those found from the forward-backward approach and no-windowing method on a Bode plot.

```y = sin([1:300]') + 0.5*randn(300,1); y = iddata(y); mb = ar(y,4,'burg'); mfb = ar(y,4); bode(mb,mfb) ```

Estimate an ARI model.

```load iddata9 z9 Ts = z9.Ts; y = cumsum(z9.y); model = ar(y, 4, 'ls', 'Ts', Ts, 'IntegrateNoise', true) compare(y,model,5) % 5 step ahead prediction```

Use option set to choose `'ls'` estimation approach and to specify that covariance matrix should not be estimated.

```y = rand(100,1); opt = arOptions('Approach', 'ls', 'EstCovar', false); model = ar(y, N, opt); ```

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### Algorithms

The AR model structure is given by the following equation:

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

AR model parameters are estimated using variants of the least-squares method. The following table summarizes the common names for methods with a specific combination of `approach` and `window` argument values.

MethodApproach and Windowing
Modified Covariance Method(Default) Forward-backward approach and no windowing.
Correlation MethodYule-Walker approach, which corresponds to least squares plus pre- and postwindowing.
Covariance MethodLeast squares approach with no windowing. `arx` uses this routine.

## References

Marple, Jr., S.L., Digital Spectral Analysis with Applications, Prentice Hall, Englewood Cliffs, 1987, Chapter 8.