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ar - Estimate parameters of AR model for scalar time series

Syntax

m = ar(y,n) [m,refl] = ar(y,n,approach,window) [m,refl] = ar(y,n,approach,window,'P1',V1,...,'PN',VN)

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

Lets you choose the algorithm for computing the least squares AR model from the following options:

'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

Lets you specify how to use information about the data outside the measured time interval (past and future values). The following windowing options are available:

'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.

'P1',V1,...,'PN',VN

Pairs of property names and property values can include any of the following.

Property Name Property Value Description

Property Name	Property Value	Description
`'Covariance'`	`'None'` Suppresses the calculation of the covariance matrix. [ ] Empty. Square matrix containing covariance values of size equal to the length of the parameter vector	Specifies calculation of uncertainties in parameter estimates.
`'MaxSize'`	Integer	See `Algorithm Properties` for the description.
`'Ts'`	Real positive number	Sets the sampling time and overrides the sampling time of `y`.

'Covariance'

'None' Suppresses the calculation of the covariance matrix.
[ ]
Empty.
Square matrix containing covariance values of size equal to the length of the parameter vector

Specifies calculation of uncertainties in parameter estimates.

'MaxSize' Integer See Algorithm Properties for the description.

'Ts' Real positive number Sets the sampling time and overrides the sampling time of y.

Description

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

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

[m,refl] = 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,refl] = ar(y,n,approach,window,'P1',V1,...,'PN',VN) returns an idpoly model m and the variable refl using additional windowing criteria.

Remarks

The AR model structure is given by the following equation:

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.

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

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)

References

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

`Algorithm Properties`
`arx`
`EstimationInfo`
`etfe`
`idpoly`
`ivar`
`pem`
`spa`
`step`