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.
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 (
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.
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