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AR with AIC

The AIC criterion has become a standard tool in time series model fitting, and its computation is available in many time series programs. To illustrate how AIC works for the AR model, we consider a noisy signal

\begin{displaymath}
y_t = \sin(2 \pi t / 10) + \epsilon_t\end{displaymath}

where $\epsilon_t \sim N(0, 0.64)$.In this example the noise is slightly stronger than the sine wave (Figure 6).
     
Figure 6. An example of a noisy signal that contains a sine wave and random noise.

In  Figure 7, the AR orders are determined by minimizing AIC. For the noise part, the best model, AR(0), agrees with the formulation. The AR(10) spectrum of the combined signal also clearly reveals the wave signal at f = 0.1.
  
Figure 7. (Top) Fourier power spectra for the sine wave, random noise, and the combined signal from Figure 6. (Bottom) AR spectra. AR orders are determined by minimizing AIC.