next up previous
Postscript version of these notes

STAT 804

Lecture 21 Notes

The Periodogram

The sample covariance between a series X and $\sin(2\pi \omega t +
\phi)$ is

\begin{displaymath}\frac{1}{T}\sum_{t=0}^{T-1} X_t \sin(2\pi \omega t +\phi)
- {\bar X} \frac{1}{T}\sum_{t=0}^{T-1}\sin(2\pi \omega t +\phi)

Using the identity $\sin(\theta) = (e^{i\theta}-e^{-i\theta})/(2i)$ and formulas for geometric sums the mean of the sines can be evaluated. When $\omega=k/T$ for an integer k, not 0, we find that $\sum_{t=0}^{T-1}\sin(2\pi \omega t +\phi)=0$ so that the sample covariance is simply

\begin{displaymath}\frac{1}{T}\sum_{t=0}^{T-1} X_t \sin(2\pi \omega t +\phi) \, .

For these special $\omega$ we can also compute

\begin{displaymath}\sum_{t=0}^{T-1} \sin^2(2\pi \omega t +\phi) = T/2

so that the sample correlation between X and $\sin(2\pi \omega t +
\phi)$ is just

\begin{displaymath}\frac{\frac{1}{T}\sum_{t=0}^{T-1} X_t \sin(2\pi \omega t

where sx2 is the sample variance $\sum (X_t-{\bar X})^2/T$.

Consider now adjusting $\phi$ to maximize this correlation. The sine can be rewritten as

\begin{displaymath}\cos(\phi)\sin(2\pi \omega t) + \sin(\phi)\cos(2\pi \omega t)

so that we are simply choosing coefficients a and b to maximize the correlation between X and $ a\sin(2\pi \omega t) + b X_t \cos(2\pi \omega t) $subject to the condition a2+b2=1. Since correlations are scale invariant we can drop the condition on a and b and maximize the correlation between X and the linear combination of sine and cosine. This problem is solved by linear regression; the coefficients are given by (MTM)-1 MT Xwhere M is the T by 2 design matrix filled in with the sines and cosines. In fact $M^TM =\frac{T}{2}I_{T\times T}$ and we see that the desired regression coefficients are

\begin{displaymath}a= \frac{2}{T} \sum_{t=0}^{T-1} X_t \sin(2\pi \omega t)


\begin{displaymath}b= \frac{2}{T} \sum_{t=0}^{T-1} X_t \sin(2\pi \omega t) \, .

The covariance between X and this best linear combination is

\begin{displaymath}\frac{1}{T} \left\{ a\sum_{t=0}^{T-1} X_t \sin(2\pi \omega t) +
b\sum_{t=0}^{T-1} X_t \sin(2\pi \omega t)\right\} =(a^2+b^2)/2

But in fact

\begin{displaymath}a^2+b^2 = \left\vert\frac{1}{T} \sum_{t=0}^{T-1} X_t \exp(2\pi\omega t i)\right\vert^2

which is just the modulus of the discrete Fourier transform ${\hat
X}(\omega)$ divided by T.

Definition: The periodogram is the function

\begin{displaymath}\vert{\hat X}(\omega)\vert^2

Here are some periodogram plots for some data sets:

next up previous

Richard Lockhart