## 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
*λ* as its virtual input. The true measured input size of such
models is zero, and their governing equation takes the form:

*y*(*t*) =
H*e*(*t*)

Here, *y*(*t*) is the signal being
modeled and *H* is the transfer function that represents the
relationship between *y*(*t*) and
*e*(*t*).

The multivariate power spectrum *Φ* of the time series
*y*(*t*) is given by:

*Φ* =
*H*(*Λ**Ts*)*H'*

Here *Λ* is the noise variance
matrix 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`

), state-space (`idss`

, or `idgrey`

) model. Some particular types of
models are parametric autoregressive (AR), autoregressive and moving average (ARMA), and
autoregressive models with integrated moving average (ARIMA). For nonlinear time series
models, the toolbox supports nonlinear ARX models.

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.

To represent a time series vector or a matrix `s`

as an
`iddata`

object, use the following syntax:

y = iddata(s,[],Ts);

The following example illustrates a 4th order autoregressive model estimation for the
time series data `z9`

that is stored in file
`iddata9`

.

load iddata9 z9 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`

.