next up previous contents
Next: Akaike Information Criterion (AIC) Up: Autoregressive (AR) Method Previous: Yule-Walker equation

AR Spectrum

 
Wiener-Khintchine theorem
For any stationary stochastic process with an autocovariance function $\gamma(k)$, there exists a function p(f) such that
\begin{displaymath}
\gamma(k) = \int e^{2 \pi ikf} p(f) df.
 \end{displaymath} (11)
The function p(f) is the spectral density function, or simply the spectrum. The quantity p(f)df represents the contribution to variance of components within frequencies in the range (f, f + df).

The inverse relation is
\begin{displaymath}
p(f) = \sum_{k = -\infty}^{\infty} \gamma(k) e^{-2 \pi ikf}
 = \gamma(0) + 2 \sum_{k = 1}^{\infty} \gamma(k) \cos(2 \pi kf)\end{displaymath} (12)

For AR(1), the spectrum is
\begin{displaymath}
p(f) = \frac{\sigma_Z^2}{\vert 1 - \alpha e^{-2 \pi if} \vert^2}
 = \frac{\sigma_Z^2}{1 - 2 \alpha \cos 2 \pi f + \alpha^2}\end{displaymath} (13)

For the general case AR(p) the spectrum is
\begin{displaymath}
p(f) = \frac{\sigma_Z^2}{\vert 1 - \alpha e^{-2 \pi if}
 - \cdots - \alpha e^{-2 \pi ipf} \vert^2}\end{displaymath} (14)

Estimating spectrum by using the AR model is sometimes called the parametric approach, opposed to the Fourier transform as the non-parametric approach.

The number of peaks in an AR(p) spectrum is approximately p/2. Estimating the AR order is an important issue (see the example below).

Except for seismological studies, there are very few cases that one would need an AR order larger than 20.

Example: AR spectrum of geomagnetic pulsations. Figure 4 shows the spectra of different AR oders for a pulsation event. As the AR order becomes large, many peaks can be seen in the spectrum, but most of them may not be meaningful. Note that this work was done at the time when the AIC had not been well publicized.

  
Figure 4. Burg maximum entropy spectrum of geomagnetic micropulsations (From Radoski et al. [1975]).


next up previous contents
Next: Akaike Information Criterion (AIC) Up: Autoregressive (AR) Method Previous: Yule-Walker equation