Documentation

# ar

Estimate parameters of AR model or ARI model for scalar time series

## Description

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sys = ar(y,n) estimates the parameters of an AR idpoly model sys of order n using a least-squares method. The model properties include covariances (parameter uncertainties) and estimation goodness of fit.

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sys = ar(y,n,approach,window) uses the algorithm specified by approach and the prewindowing and postwindowing specification in window. To specify window while accepting the default value for approach, use [] in the third position of the syntax.

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sys = ar(y,n,___,Name,Value) specifies additional options using one or more name-value pair arguments. For instance, using the name-value pair argument 'IntegrateNoise',1 estimates an ARI model, which is useful for systems with nonstationary disturbances. Specify Name,Value after any of the input argument combinations in the previous syntaxes.

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sys = ar(y,n,___,opt) specifies estimation options using the option set opt.

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[sys,refl] = ar(y,n,approach,___) returns an AR model along with the reflection coefficients refl when approach is the lattice-based method 'burg' or 'gl'.

## Examples

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Estimate an AR model and compare its response with the measured output.

Load the data, which contains the time series z9 with noise.

Estimate a fourth-order AR model.

sys = ar(z9,4)
sys =
Discrete-time AR model: A(z)y(t) = e(t)
A(z) = 1 - 0.8369 z^-1 - 0.4744 z^-2 - 0.06621 z^-3 + 0.4857 z^-4

Sample time: 0.0039062 seconds

Parameterization:
Polynomial orders:   na=4
Number of free coefficients: 4
Use "polydata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Estimated using AR ('fb/now') on time domain data "z9".
Fit to estimation data: 79.38%
FPE: 0.5189, MSE: 0.5108

The output displays the polynomial containing the estimated parameters alongside other estimation details. Under Status, Fit to estimation data shows that the estimated model has 1-step-ahead prediction accuracy above 75%.

You can find additional information about the estimation results by exploring the estimation report, sys.Report. For instance, you can retrieve the parameter covariance.

covar = sys.Report.Parameters.FreeParCovariance
covar = 4×4

0.0015   -0.0015   -0.0005    0.0007
-0.0015    0.0027   -0.0008   -0.0004
-0.0005   -0.0008    0.0028   -0.0015
0.0007   -0.0004   -0.0015    0.0014

Given a sinusoidal signal with noise, compare the spectral estimates of Burg's method with those found using the forward-backward approach.

Generate an output signal and convert it into an iddata object.

y = sin([1:300]') + 0.5*randn(300,1);
y = iddata(y);

Estimate fourth-order AR models using Burg's method and using the default forward-backward approach. Plot the model spectra together.

sys_b = ar(y,4,'burg');
sys_fb = ar(y,4);
spectrum(sys_b,sys_fb)
legend('Burg','Forward-Backward')

The two responses match closely throughout most of the frequency range.

Estimate an ARI model, which includes an integrator in the noise source.

Load the data, which contains the time series z9 with noise.

Integrate the output signal.

y = cumsum(z9.y);

Estimate an AR model with 'IntegrateNoise' set to true. Use the least-squares method 'ls'. Because y is a vector and not an iddata object, specify Ts.

Ts = z9.Ts;
sys = ar(y,4,'ls','Ts',Ts,'IntegrateNoise',true);

Predict the model output using 5-step prediction and compare the result with the integrated output signal y.

compare(y,sys,5)

Modify the default options for the AR function.

Load the data, which contains a time series z9 with noise.

Modify the default options so that the function uses the 'ls' approach and does not estimate covariance.

opt = arOptions('Approach','ls','EstimateCovariance',false)
opt =
Option set for the ar command:

Approach: 'ls'
Window: 'now'
DataOffset: 0
EstimateCovariance: 0
MaxSize: 250000

Description of options

Estimate a fourth-order AR model using the updated options.

sys = ar(z9,4,opt);

Retrieve reflection coefficients and loss functions when using Burg's method.

Lattice-based approaches such, as Burg's method 'burg' and geometric lattice 'gl', compute reflection coefficients and corresponding loss function values as part of the estimation process. Use a second output argument to retrieve these values.

Generate an output signal and convert it into an iddata object.

y = sin([1:300]') + 0.5*randn(300,1);
y = iddata(y);

Estimate a fourth-order AR model using Burg's method and include an output argument for the reflection coefficients.

[sys,refl] = ar(y,4,'burg');
refl
refl = 2×5

0   -0.3562    0.4430    0.5528    0.2385
0.8494    0.7416    0.5960    0.4139    0.3904

## Input Arguments

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Time-series data, specified as one of the following:

• An iddata object that contains a single output channel and an empty input channel.

• A double column vector containing output-channel data. When you specify y as a vector, you must also specify the sample time Ts.

Model order, specified as a positive integer. The value of n determines the number of A parameters in the AR model.

Example: ar(idy,2) computes a second-order AR model from the single-channel iddata object idy

Algorithm for computing the 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.

All of these algorithms are variants of the least-squares method. For more information, see Algorithms .

Example: ar(idy,2,'ls') computes an AR model using the least-squares approach

Prewindowing and postwindowing outside the measured time interval (past and future values), specified as one of the following values:

• 'now': No windowing. This value is the default except when you set approach to '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': Prewindowing and postwindowing. The software uses this value whenever you select the Yule-Walker approach 'yw', regardless of your window specification.

• 'prw': Prewindowing. Missing past values are replaced with zeros so that the summation in the criteria can start at time equal to zero.

Example: ar(idy,2,'yw','ppw') computes an AR model using the Yule-Walker approach with prewindowing and postwindowing.

Estimation options for AR model identification, specified as an arOptions option set. opt specifies the following options:

• Estimation approach

• Data windowing technique

• Data offset

• Maximum number of elements in a segment of data

For more information, see arOptions. For an example, see Modify Default Options.

### 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.

Example: 'IntegrateNoise',true adds an integrator in the noise source

Sample time, specified as the comma-separated pair consisting of 'Ts' and the sample time in seconds. If y is a double vector, then you must specify 'Ts'.

Example: ar(y_signal,2,'Ts',0.08) computes a second-order AR model with sample time of 0.08 seconds

Noise-channel integration option for estimating ARI models, specified as the comma-separated pair consisting of 'IntegrateNoise' and a logical. Noise integration is useful in cases where the disturbance is nonstationary.

When using 'IntegrateNoise', you must also integrate the output-channel data. For an example, see ARI Model.

## Output Arguments

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AR or ARI model that fits the given estimation data, returned as a discrete-time idpoly model object. This model is created using the specified model orders, delays, and estimation options.

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.

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 the following fields:

FieldDescription
FitPercent

Normalized root mean squared error (NRMSE) measure of how well the response of the model fits the estimation data, expressed as a percentage.

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.

OptionsUsed

Option set used for estimation. If no custom options were configured, this is a set of default options. See arOptions for more information.

RandState

State of the random number stream at the start of estimation. Empty, [], if randomization was not used during estimation. For more information, see rng in the MATLAB® documentation.

DataUsed

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

FieldDescription
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' — Zero-order hold maintains a piecewise-constant input signal between samples.

• 'foh' — 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 [].

Reflection coefficients and loss functions, returned as a 2-by-2 array. 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. For an example, see Retrieve Reflection Coefficients for Burg's Method.

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### AR (Autoregressive) Model

The AR model structure has no input, and is given by the following equation:

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

This model structure accommodates estimation for scalar time-series data, which have no input channel. The structure is a special case of the ARX structure.

### ARI (Autoregressive Integrated) Model

The ARI model is an AR model with an integrator in the noise channel. The ARI model structure is given by the following equation:

$A\left(q\right)y\left(t\right)=\frac{1}{1-{q}^{-1}}e\left(t\right)$

## Algorithms

AR and ARI 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 with no windowing
Correlation methodYule-Walker approach with prewindowing and postwindowing
Covariance methodLeast squares approach with no windowing. arx uses this routine

## References

[1] Marple, S. L., Jr. Chapter 8. Digital Spectral Analysis with Applications. Englewood Cliffs, NJ: Prentice Hall, 1987.