Documentation

# 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 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.

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', 'EstimateCovariance', false);
model = ar(y, N, opt);

## 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.