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STAT 804: Lecture 21 Notes

The Periodogram

The sample covariance between a series X and is

Using the identity and formulas for geometric sums the mean of the sines can be evaluated. When for an integer k, not 0, we find that so that the sample covariance is simply

For these special we can also compute

so that the sample correlation between X and is just

where is the sample variance .

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

so that we are simply choosing coefficients a and b to maximize the correlation between X and subject to the condition . Since correlations are scale invariant we can drop the condition on a and b and miximize the correlation between X and the linear combination of sine and cosine. This problem is solved by linear regression; the coefficients are given by where M is the T by 2 design matrix filled in with the sines and cosines. In fact and we see that the desired regression coefficients are


The covariance between X and this best linear combination is

But in fact

which is just the modulus of the discrete Fourier transform divided by T.

Definition: The periodogram is the function

Here are some periodogram plots for some data sets:

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Richard Lockhart
Fri Nov 14 10:12:00 PST 1997