Least-squares Spectral Analysis - The Lomb–Scargle Periodogram

The Lomb–Scargle Periodogram

Rather than just taking dot products of the data with sine and cosine waveforms directly, Scargle modified the standard periodogram formula to first find a time delay τ such that this pair of sinusoids would be mutually orthogonal at sample times tj, and also adjusted for the potentially unequal powers of these two basis functions, to obtain a better estimate of the power at a frequency, which made his modified periodogram method exactly equivalent to Lomb's least-squares method. The time delay τ is defined by the formula

\tan{2 \omega \tau} = \frac{\sum_j \sin 2 \omega t_j}{\sum_j \cos 2 \omega t_j}.

The periodogram at frequency ω is then estimated as:

P_x(\omega) = \frac{1}{2} \left( \frac { \left ^ 2} { \sum_j \cos^2 \omega ( t_j - \tau ) } + \frac {\left ^ 2} { \sum_j \sin^2 \omega ( t_j - \tau ) } \right)

which Scargle reports then has the same statistical distribution as the periodogram in the evenly-sampled case.

At any individual frequency ω, this method gives the same power as does a least-squares fit to sinusoids of that frequency, of the form

.

Read more about this topic:  Least-squares Spectral Analysis