This example shows how to estimate Autoregressive Integrated Moving Average or ARIMA models.

Models of time series containing non-stationary trends (seasonality) are sometimes required. One category of such models are the ARIMA models. These models contain a fixed integrator in the noise source. Thus, if the governing equation of an ARMA model is expressed as *A(q)y(t)=Ce(t)*, where *A(q)* represents the auto-regressive term and *C(q)* the moving average term, the corresponding model of an ARIMA model is expressed as

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

where the term $$\frac{1}{1-{q}^{-1}}$$ represents the discrete-time integrator. Similarly, you can formulate the equations for ARI and ARIX models.

Using time-series model estimation commands `ar`

, `arx`

and `armax`

you can introduce integrators into the noise source `e(t)`

. You do this by using the `IntegrateNoise`

parameter in the estimation command.

The estimation approach does not account any constant offsets in the time-series data. The ability to introduce noise integrator is not limited to time-series data alone. You can do so also for input-output models where the disturbances might be subject to seasonality. One example is the polynomial models of ARIMAX structure:

$$A(q)y(t)=B(q)u(t)+\frac{C(q)}{(1-{q}^{-1})}e(t)$$

See the `armax`

reference page for examples.

Estimate an ARI model for a scalar time-series with linear trend.

load iddata9 z9 Ts = z9.Ts; y = cumsum(z9.y); model = ar(y,4,'ls','Ts',Ts,'IntegrateNoise', true); % 5 step ahead prediction compare(y,model,5)

Estimate a multivariate time-series model such that the noise integration is present in only one of the two time series.

load iddata9 z9 Ts = z9.Ts; y = z9.y; y2 = cumsum(y); % artificially construct a bivariate time series data = iddata([y, y2],[],Ts); na = [4 0; 0 4]; nc = [2;1]; model1 = armax(data, [na nc], 'IntegrateNoise',[false; true]); % Forecast the time series 100 steps into future yf = forecast(model1,data(1:100), 100); plot(data(1:100),yf)

If the outputs were coupled ( `na`

was not a diagonal matrix), the situation will be more complex and simply adding an integrator to the second noise channel will not work.