.. _book_fig_chapter10_fig_LS_sg_comparison:


Generalized vs Standard Lomb-Scargle
------------------------------------
Figure 10.16

A comparison of the standard and generalized Lomb-Scargle periodograms for a
signal y(t) = 10 + sin(2pi t/P) with P = 0.3, corresponding to omega_0 ~ 21.
This example is, in some sense, a worst-case scenario for the standard
Lomb-Scargle algorithm because there are no sampled points during the times
when ytrue < 10, which leads to a gross overestimation of the mean. The bottom
panel shows the Lomb-Scargle and generalized Lomb-Scargle periodograms for
these data; the generalized method recovers the expected peak as the highest
peak, while the standard method incorrectly chooses the peak at omega ~ 17.6
(because it is higher than the true peak at omega_0 ~ 21). The dotted lines
show the 1% and 5% significance levels for the highest peak in the generalized
periodogram, determined by 1000 bootstrap resamplings (see Section 10.3.2).

Note: This Plot Contains an Error
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
After the book was in press, a reader pointed out that this plot contains a
typo.  Instead of passing the noisy data to the Lomb-Scargle routine, we
had passed the underlying, non-noisy data.  This caused an over-estimate
of the Lomb-Scargle power.

Because of this, we add two extra plots to this script: the first reproduces
the current plot without the typo.  In it, we see that for the noisy data,
the period is not detected for just ~30 points within ten periods.

In the second additional plot, we increase the baseline and the number of
points by a factor of ten.  With this configuration, the peak is detected,
and the qualitative aspects of the above discussion hold true.

We regret the error!



.. rst-class:: horizontal


.. image:: ../images/chapter10/fig_LS_sg_comparison_1.png
    :align: center
    :scale: 100



.. image:: ../images/chapter10/fig_LS_sg_comparison_2.png
    :align: center
    :scale: 100



.. image:: ../images/chapter10/fig_LS_sg_comparison_3.png
    :align: center
    :scale: 100



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