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

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

load iddata9 z9

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

For more information on viewing the estimation report, see Estimation Report.

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.

load iddata9 z9

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.

load iddata9 z9

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