Estimate ARIMA Models

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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(问)y(t)=Ce(t), where(问)represents the auto-regressive term andC(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 commandsar,arxandarmaxyou can introduce integrators into the noise sourcee(t). You do this by using theIntegrateNoiseparameter in the estimation command.

的评估方法ch 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 thearmaxreference page for examples.

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

loadiddata9z9Ts = z9.Ts; y = cumsum(z9.y); model = ar(y,4,'ls','Ts',Ts,'IntegrateNoise', true);% 5 step ahead predictioncompare(y,model,5)

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

loadiddata9z9Ts = z9.Ts; y = z9.y; y2 = cumsum(y);% artificially construct a bivariate time seriesdata = 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 futureyf = forecast(model1,data(1:100), 100); plot(data(1:100),yf)

If the outputs were coupled (nawas not a diagonal matrix), the situation will be more complex and simply adding an integrator to the second noise channel will not work.

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