## Documentation Center |

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`.

Was this topic helpful?