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## What Are Time-Series Models?

A time series is one or more measured output channels with no measured input. A time-series model, also called a signal model, is a dynamic system that is identified to fit a given signal or time series data. The time series can be multivariate, which leads to multivariate models.

A time series is modeled by assuming it to be the output of a system that takes a white noise signal e(t) of variance NV as its virtual input. The true measured input size of such models is zero, and their governing equation takes the form y(t) = He(t), where y(t) is the signal being modeled and H is the transfer function that represents the relationship between y(t) and e(t). The power spectrum of the time series is given by H*(NV*Ts)*H', where NV is the noise variance and Ts is the model sample time.

System Identification Toolbox™ software provides tools for modeling and forecasting time-series data. You can estimate both linear and nonlinear black-box and grey-box models for time-series data. A linear time-series model can be a polynomial (idpoly) or state-space (idss, idgrey) model. Some particular types of models are parametric autoregressive (AR), autoregressive and moving average (ARMA), and autoregressive models with integrated moving average (ARIMA).

You can estimate time-series spectra using both time- and frequency-domain data. Time-series spectra describe time-series variations using cyclic components at different frequencies.

The following example illustrates a 4th order autoregressive model estimation for time series data:

```load iddata9
sys = ar(z9,4);
```

Because the model has no measured inputs, size(sys,2) returns zero. The governing equation of sys is A(q)y(t) = e(t). You can access the A polynomial using sys.a and the estimated variance of the noise e(t) using sys.NoiseVariance.