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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

$\displaystyle \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

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

For these special $ \omega$ we can also compute

$\displaystyle \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

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

where $ s_x^2$ 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

$\displaystyle \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 $ a^2+b^2=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 $ (M^TM)^{-1} M^T X$ where $ 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

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


$\displaystyle 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

$\displaystyle \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

$\displaystyle 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

$\displaystyle \vert{\hat X}(\omega)\vert^2

Here are some periodogram plots for some data sets:

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Richard Lockhart