Estimate an ARMAX model and view the fit of the model output to the estimation data.

Load the measurement data in iddata object z2.

load iddata2 z2

Estimate an ARMAX model with second-order $\mathit{A}$,$\mathit{B}$, and $\mathit{C}$ polynomials and a transport delay of one sample period.

na = 2;
nb = 2;
nc = 2;
nk = 1;
sys = armax(z2,[na nb nc nk])

sys =
Discrete-time ARMAX model: A(z)y(t) = B(z)u(t) + C(z)e(t)
  A(z) = 1 - 1.512 z^-1 + 0.7006 z^-2                    
                                                         
  B(z) = -0.2606 z^-1 + 1.664 z^-2                       
                                                         
  C(z) = 1 - 1.604 z^-1 + 0.7504 z^-2                    
                                                         
Sample time: 0.1 seconds
  
Parameterization:
   Polynomial orders:   na=2   nb=2   nc=2   nk=1
   Number of free coefficients: 6
   Use "polydata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                          
Estimated using ARMAX on time domain data "z2".  
Fit to estimation data: 85.89% (prediction focus)
FPE: 1.086, MSE: 1.054                           

The output displays the polynomial containing the estimated parameters alongside the estimation details. Under Status, Fit to estimation data shows that the estimated model has 1-step-ahead prediction accuracy above 80%.

Compare the model simulated output to the measured data.

compare(z2,sys)

The fit of the simulated model to the measured data is nearly the same as the estimation fit.

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

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

load iddata9 z9

Estimate a fourth-order ARMA model with a first-order $\mathit{C}$polynomial.

na = 4;
nc = 1;
sys = armax(z9,[na nc]);

Estimate a fourth-order AR model.

sys_ar = ar(z9,na);

Compare the model outputs with the measured data.

compare(z9,sys,sys_ar)

The ARMA model has the better fit to the data.

Estimate an ARMAX model from measured data and specify estimation options.

Load the data and create an iddata object. Initialize option set opt, and set options for Focus, SearchMethod, MaxIterations, and Display. Then estimate the ARMAX model using the updated option set.

load twotankdata;
z = iddata(y,u,0.2);
opt = armaxOptions;
opt.Focus = 'simulation';
opt.SearchMethod = 'lm';
opt.SearchOptions.MaxIterations = 10;
opt.Display = 'on';
sys = armax(z,[2 2 2 1],opt);

The termination conditions for measured component of the model shown in the progress viewer is that the maximum number of iterations were reached.

To improve results, re-estimate the model using a greater value for MaxIterations, or continue iterations on the previously estimated model as follows:

sys2 = armax(z,sys);
compare(z,sys,sys2)

where sys2 refines the parameters of sys to improve the fit to data.

Estimate a regularized ARMAX model by converting a regularized ARX model.

Load data.

load regularizationExampleData.mat m0simdata;

Estimate an unregularized ARMAX model of order 30.

m1 = armax(m0simdata(1:150),[30 30 30 1]);

Estimate a regularized ARMAX model by determining the Lambda value by trial and error.

opt = armaxOptions;
opt.Regularization.Lambda = 1;
m2 = armax(m0simdata(1:150),[30 30 30 1],opt);

Obtain a lower order ARMAX model by converting a regularized ARX model and then performing order reduction.

opt1 = arxOptions;
[L,R] = arxRegul(m0simdata(1:150),[30 30 1]);
opt1.Regularization.Lambda = L;
opt1.Regularization.R = R;
m0 = arx(m0simdata(1:150),[30 30 1],opt1);
mr = idpoly(balred(idss(m0),7));

Compare the model outputs against the data.

opt2 = compareOptions('InitialCondition','z');
compare(m0simdata(150:end),m1,m2,mr,opt2);

Estimate a fourth-order ARIMA model for univariate time-series data.

Load data that contains a time series with noise.

load iddata9 z9;

Integrate the output signal and use the result to replace the original output signal in z9.

z9.y = cumsum(z9.y);

Estimate a fourth-order ARIMA model with a first-order$\mathit{C}$polynomial by setting 'IntegrateNoise' to true.

model = armax(z9,[4 1],'IntegrateNoise',true); 

Predict the model output using 10-step ahead prediction, and compare the predicted output with the estimation data.

compare(z9,model,10)

Estimate ARMAX models of varying orders iteratively from measured data.

Load dryer2 data and perform estimation for combinations of polynomial orders na, nb, nc, and input delay nk.

load dryer2;
z = iddata(y2,u2,0.08,'Tstart',0);
na = 2:4;
nc = 1:2;
nk = 0:2;
models = cell(1,18);
ct = 1;
for i = 1:3
    na_ = na(i);
    nb_ = na_;
    for j = 1:2
        nc_ = nc(j);
        for k = 1:3
            nk_ = nk(k); 
            models{ct} = armax(z,[na_ nb_ nc_ nk_]);
            ct = ct+1;
        end
    end
end

Stack the estimated models and compare their simulated responses to the estimation data z.

models = stack(1,models{:});
compare(z,models)

Load the estimation data.

load iddata2 z2

Estimate a state-space model of order 3 from the estimation data.

sys0 = n4sid(z2,3);

Estimate an ARMAX model using the previously estimated state-space model to initialize the parameters.

sys = armax(z2,sys0);

Load the data.

load iddata1ic z1i

Estimate a second-order ARMAX model sys and return the initial conditions in ic.

na = 2;
nb = 2;
nc = 2;
nk = 1;
[sys,ic] = armax(z1i,[na nb nc nk]);
ic

ic = 
  initialCondition with properties:

     A: [2x2 double]
    X0: [2x1 double]
     C: [0 1]
    Ts: 0.1000

ic is an initialCondition object that encapsulates the free response of sys, in state-space form, to the initial state vector in X0. You can incorporate ic when you simulate sys with the z1i input signal and compare the response with the z1i output signal.