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), state-space (
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.
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,
zero. The governing equation of
sys is A(q)y(t) = e(t).
You can access the A polynomial using
the estimated variance of the noise e(t) using