.. _stats-lombscargle:

*************************
Lomb-Scargle Periodograms
*************************

The Lomb-Scargle Periodogram (after Lomb [1]_, and Scargle [2]_)
is a commonly-used statistical tool designed to detect periodic signals
in unevenly-spaced observations.
The :class:`~astropy.stats.LombScargle` class is a unified interface to several
implementations of the Lomb-Scargle periodogram, including a fast *O[NlogN]*
implementation following the algorithm presented by Press & Rybicki [3]_.

The code here is adapted from the `astroml`_ package ([4]_, [5]_) and the `gatspy`_ package ([6]_, [7]_).
For a detailed practical discussion of the lomb-scargle periodogram, with code
examples based on astropy, see *Understanding the Lomb-Scargle Periodogram* [11]_, with associated code at https://github.com/jakevdp/PracticalLombScargle/.

.. _gatspy: http://www.astroml.org/gatspy/
.. _astroml: http://astroml.org/


Basic Usage
===========

.. Note::
   All frequencies in :class:`~astropy.stats.LombScargle` are **not** angular
   frequencies, but rather frequencies of oscillation; i.e. number of
   cycles per unit time.

The Lomb-Scargle periodogram is designed to detect periodic signals in
unevenly-spaced observations. For example, consider the following data:

>>> import numpy as np
>>> rand = np.random.RandomState(42)
>>> t = 100 * rand.rand(100)
>>> y = np.sin(2 * np.pi * t) + 0.1 * rand.randn(100)

These are 100 noisy measurements taken at irregular times, with a frequency
of 1 cycle per unit time.
The Lomb-Scargle periodogram, evaluated at frequencies chosen
automatically based on the input data, can be computed as follows
using the :class:`~astropy.stats.LombScargle` class:

>>> from astropy.stats import LombScargle
>>> frequency, power = LombScargle(t, y).autopower()

Plotting the result with matplotlib gives:

>>> import matplotlib.pyplot as plt  # doctest: +SKIP
>>> plt.plot(frequency, power)       # doctest: +SKIP

.. plot::

    from astropy.stats import LombScargle

    import numpy as np
    import matplotlib.pyplot as plt
    plt.style.use('ggplot')

    rand = np.random.RandomState(42)
    t = 100 * rand.rand(100)
    y = np.sin(2 * np.pi * t) + 0.1 * rand.randn(100)

    frequency, power = LombScargle(t, y).autopower()
    fig = plt.figure(figsize=(6, 4.5))
    plt.plot(frequency, power)

The periodogram shows a clear spike at a frequency of 1 cycle per unit time,
as we would expect from the data we constructed.

Measurement Uncertainties
-------------------------

The :class:`~astropy.stats.LombScargle` interface can also handle data with
measurement uncertainties.
For example, if all uncertainties are the same, you can pass a scalar:

>>> dy = 0.1
>>> frequency, power = LombScargle(t, y, dy).autopower()

If uncertainties vary from observation to observation, you can pass them as
an array:

>>> dy = 0.1 * (1 + rand.rand(100))
>>> y = np.sin(2 * np.pi * t) + dy * rand.randn(100)
>>> frequency, power = LombScargle(t, y, dy).autopower()

Gaussian uncertainties are assumed, and ``dy`` here specifies the standard
deviation (not the variance).

Periodograms and Units
----------------------
The :class:`~astropy.stats.LombScargle` interface properly handles
:class:`~astropy.units.Quantity` objects with units attached,
and will validate the inputs to make sure units are appropriate. For example:

>>> import astropy.units as u
>>> t_days = t * u.day
>>> y_mags = y * u.mag
>>> dy_mags = y * u.mag
>>> frequency, power = LombScargle(t_days, y_mags, dy_mags).autopower()
>>> frequency.unit
Unit("1 / d")
>>> power.unit
Unit(dimensionless)

We see that the output is dimensionless, which is always the case for the
standard normalized periodogram (for more on normalizations,
see :ref:`lomb-scargle-normalization` below).


Specifying the Frequency
------------------------
With the :func:`~astropy.stats.LombScargle.autopower` method used above,
a heuristic is applied to select
a suitable frequency grid. By default, the heuristic assumes that the width of
peaks is inversely proportional to the observation baseline, and that the
maximum frequency is a factor of 5 larger than the so-called "average Nyquist
frequency", computed based on the average observation spacing.

This heuristic is not universally useful, as the frequencies probed by
irregularly-sampled data can be much higher than the average Nyquist frequency.
For this reason, the heuristic can be tuned through keywords passed to the
:func:`~astropy.stats.LombScargle.autopower` method. For example:

>>> frequency, power = LombScargle(t, y, dy).autopower(nyquist_factor=2)
>>> len(frequency), frequency.min(), frequency.max()  # doctest: +FLOAT_CMP
(500, 0.0010189890448009111, 1.0179700557561102)

Here the highest frequency is two times the average Nyquist frequency.
If we increase the ``nyquist_factor``, we can probe higher frequencies:

>>> frequency, power = LombScargle(t, y, dy).autopower(nyquist_factor=10)
>>> len(frequency), frequency.min(), frequency.max()  # doctest: +FLOAT_CMP
(2500, 0.0010189890448009111, 5.0939262349597545)

Alternatively, we can use the :func:`~astropy.stats.LombScargle.power`
method to evaluate the periodogram at a user-specified set of frequencies:

>>> frequency = np.linspace(0.5, 1.5, 1000)
>>> power = LombScargle(t, y, dy).power(frequency)

Note that the fastest Lomb-Scargle implementation requires regularly-spaced
frequencies; if frequencies are irregularly-spaced, a slower method will be
used instead.

Frequency Grid Spacing
^^^^^^^^^^^^^^^^^^^^^^

One common issue with user-specified frequencies is inadvertently choosing
too coarse a grid, such that significant peaks lie between grid points and
are missed entirely.

For example, imagine you chose to evaluate your periodogram at 100 points:

>>> frequency = np.linspace(0.1, 1.9, 100)
>>> power = LombScargle(t, y, dy).power(frequency)
>>> plt.plot(frequency, power)   # doctest: +SKIP

.. plot::

    import numpy as np
    import matplotlib.pyplot as plt
    from astropy.stats import LombScargle

    rand = np.random.RandomState(42)
    t = 100 * rand.rand(100)
    dy = 0.1
    y = np.sin(2 * np.pi * t) + dy * rand.randn(100)

    frequency = np.linspace(0.1, 1.9, 100)
    power = LombScargle(t, y, dy).power(frequency)

    plt.style.use('ggplot')
    plt.figure(figsize=(6, 4.5))
    plt.plot(frequency, power)
    plt.xlabel('frequency')
    plt.ylabel('Lomb-Scargle Power')
    plt.ylim(0, 1)

From this plot alone, one might conclude that no clear periodic signal exists
in the data.
But this conclusion is in error: there is in fact a strong periodic signal,
but the periodogram peak falls in the gap between the chosen grid points!

A safer approach is to use the frequency heuristic to decide on the appropriate
grid spacing to use, optionally passing a minimum and maximum frequency to
the :func:`~astropy.stats.LombScargle.autopower` method:

>>> frequency, power = LombScargle(t, y, dy).autopower(minimum_frequency=0.1,
...                                                    maximum_frequency=1.9)
>>> len(frequency)
884
>>> plt.plot(frequency, power)   # doctest: +SKIP

.. plot::

    import numpy as np
    import matplotlib.pyplot as plt
    from astropy.stats import LombScargle

    rand = np.random.RandomState(42)
    t = 100 * rand.rand(100)
    dy = 0.1
    y = np.sin(2 * np.pi * t) + dy * rand.randn(100)

    frequency, power = LombScargle(t, y, dy).autopower(minimum_frequency=0.1,
                                                       maximum_frequency=1.9)

    plt.style.use('ggplot')
    plt.figure(figsize=(6, 4.5))
    plt.plot(frequency, power)
    plt.xlabel('frequency')
    plt.ylabel('Lomb-Scargle Power')
    plt.ylim(0, 1)

With a finer grid (here 884 points between 0.1 and 1.9),
it is clear that there is a very strong periodic signal in the data.

By default, the heuristic aims to have roughly five grid points across each
significant periodogram peak; this can be increased by changing the
``samples_per_peak`` argument:

>>> frequency, power = LombScargle(t, y, dy).autopower(minimum_frequency=0.1,
...                                                    maximum_frequency=1.9,
...                                                    samples_per_peak=10)
>>> len(frequency)
1767

Keep in mind that the width of the peak scales inversely with the baseline of
the observations (i.e. the difference between the maximum and minimum time),
and the required number of grid points will scale linearly with the size of
the baseline.

The Lomb-Scargle Model
----------------------
Under the hood, the Lomb-Scargle periodogram essentially fits a sinusoidal
model to the data at each frequency, with a larger power reflecting a better
fit. With this in mind, it is often helpful to plot the best-fit sinusoid
over the phased data.

This best-fit sinusoid can be computed using the :func:`~astropy.stats.LombScargle.model` method of the :class:`~astropy.stats.LombScargle` object:

>>> best_frequency = frequency[np.argmax(power)]
>>> t_fit = np.linspace(0, 1)
>>> y_fit = LombScargle(t, y, dy).model(t_fit, best_frequency)

We can then phase the data and plot the Lomb-Scargle model fit:

.. plot::

    import numpy as np
    import matplotlib.pyplot as plt
    plt.style.use('ggplot')

    from astropy.stats import LombScargle

    rand = np.random.RandomState(42)
    t = 100 * rand.rand(100)
    dy = 0.1
    y = np.sin(2 * np.pi * t) + dy * rand.randn(100)

    frequency, power = LombScargle(t, y, dy).autopower(minimum_frequency=0.1,
                                                       maximum_frequency=1.9)
    best_frequency = frequency[np.argmax(power)]
    phase_fit = np.linspace(0, 1)
    y_fit = LombScargle(t, y, dy).model(t=phase_fit / best_frequency,
                                        frequency=best_frequency)
    phase = (t * best_frequency) % 1

    fig, ax = plt.subplots(figsize=(6, 4.5))
    ax.errorbar(phase, y, dy, fmt='o', mew=0, capsize=0, elinewidth=1.5)
    ax.plot(phase_fit, y_fit, color='black')
    ax.invert_yaxis()
    ax.set(xlabel='phase',
           ylabel='magnitude',
           title='phased data at frequency={0:.2f}'.format(best_frequency))

Additional Arguments
--------------------
On initialization, :class:`~astropy.stats.LombScargle` takes a few additional
arguments which control the model for the data:

- ``center_data`` (``True`` by default) controls whether the ``y`` values are
  pre-centered before the algorithm fits the data.
  The only time it is really warranted to change the default
  is if you are computing the periodogram of a sequence
  of constant values to, e.g. estimate the window power spectrum for a series of
  observations.
- ``fit_mean`` (``True`` by default) controls whether the model fits for the
  mean of the data, rather than assuming the mean is zero. When ``fit_mean=True``,
  the periodogram is more robust than the original Lomb-Scargle formalism,
  particularly in the case of smaller sample sizes and/or
  data with nontrivial selection bias. In the literature, this model has
  variously been called the *date-compensated discrete Fourier transform*,
  the *floating-mean periodogram*, the  *generalized Lomb-Scargle method*,
  and likely other names as well.
- ``nterms`` (``1`` by default) controls how many Fourier terms are used in the
  model. As seen above, the standard Lomb-Scargle periodogram is equivalent to
  a single-term sinusoidal fit to the data at each frequency; the
  generalization is to expand this to a truncated Fourier series with multiple
  frequencies. While this can be very useful in some cases, in others the
  additional model complexity can lead to spurious periodogram peaks that
  outweigh the benefit of the more flexible model.

.. _lomb-scargle-normalization:

Periodogram Normalizations
==========================
There are several normalizations of the Lomb-Scargle periodogram found in the
literature. :class:`~astropy.stats.LombScargle` makes four options available,
via the ``normalization`` argument: ``normalization='standard'``
(the default), ``normalization='model'``, ``normalization='log'``,
and ``normalization='psd'``.
These normalizations are easiest to discuss in terms of least-squares fits around
a constant reference model :math:`M_{ref}` and a periodic model :math:`M(f)` at
each frequency, with best-fit sum-of-residuals that we'll denote by
:math:`\chi^2_{ref}` and :math:`\chi^2(f)` respectively.

Standard Normalization
----------------------
The default, the standard normalized periodogram, is normalized by the residuals
of the data around the constant reference model:

.. math::

   P_{standard}(f) = \frac{\chi^2_{ref} - \chi^2(f)}{\chi^2_{ref}}

This form of the normalization (``normalization='standard'``) is the default
choice used in :class:`~astropy.stats.LombScargle`.
The resulting power *P* is a dimensionless quantity that lies in the
range *0 ≤ P ≤ 1*.

Model Normalization
-------------------
Alternatively, the periodogram is sometimes normalized instead by the residuals
around the periodic model:

.. math::

   P_{model}(f) = \frac{\chi^2_{ref} - \chi^2(f)}{\chi^2(f)}

This form of the normalization can be specified with ``normalization='model'``.
As above, the resulting power is a dimensionless
quantity that lies in the range *0 ≤ P ≤ ∞*.

Logarithmic Normalization
-------------------------
Another form of normalization is to scale the periodogram logarithmically:

.. math::

   P_{log}(f) = \log \frac{\chi^2_{ref}}{\chi^2(f)}

This normalization can be specified with ``normalization='log'``, and the
resulting power is a dimensionless quantity in the range *0 ≤ P ≤ ∞*.

PSD Normalization (Unnormalized)
--------------------------------
Finally, it is sometimes useful to compute an unnormalized periodogram
(``normalization='psd'``):

.. math::

   P_{psd}(f) = \frac{1}{2}\left(\chi^2_{ref} - \chi^2(f)\right)

which in the no-uncertainty case will have units ``y.unit ** 2``.
This normalization is constructed to be
comparable to the standard Fourier power spectral density (PSD):

>>> ls = LombScargle(t_days, y_mags, normalization='psd')
>>> frequency, power = ls.autopower()
>>> power.unit
Unit("mag2")

Note, however, that the ``normalization='psd'`` result only has these units
*if uncertainties are not specified*. In the presence of uncertainties,
even the unnormalized PSD periodogram will be dimensionless; this is due to
the scaling of data by uncertainty within the Lomb-Scargle computation:

>>> # with uncertainties, PSD power is unitless
>>> ls = LombScargle(t_days, y_mags, dy_mags, normalization='psd')
>>> frequency, power = ls.autopower()
>>> power.unit
Unit(dimensionless)

The equivalence of the PSD-normalized periodogram and the Fourier PSD
in the unnormalized, no-uncertainty case can be confirmed by comparing
results directly for uniformly-sampled inputs.
We will first define a convenience function to compute the basic
Fourier periodogram for uniformly-sampled quantities:

>>> def fourier_periodogram(t, y):
...     N = len(t)
...     frequency = np.fft.fftfreq(N, t[1] - t[0])
...     y_fft = np.fft.fft(y.value) * y.unit
...     positive = (frequency > 0)
...     return frequency[positive], (1. / N) * abs(y_fft[positive]) ** 2

Next we compute the two versions of the PSD from uniformly-sampled data:

>>> t_days = np.arange(100) * u.day
>>> y_mags = rand.randn(100) * u.mag
>>> frequency, PSD_fourier = fourier_periodogram(t_days, y_mags)
>>> ls = LombScargle(t_days, y_mags, normalization='psd')
>>> PSD_LS = ls.power(frequency)

Examining the results, we see that the two outputs match:

>>> u.allclose(PSD_fourier, PSD_LS)
True

This equivalence is one reason the Lomb-Scargle periodogram is considered
to be an extension of the Fourier PSD.

For more information on the statistical properties of these normalizations,
see e.g. Baluev 2008 [8]_.


Peak Significance and False Alarm Probabilities
===============================================

.. Note::
   Interpretation of Lomb-Scargle peak significance via false alarm
   probabilities is a subtle subject, and the quantities computed below are
   easy to misinterpret or misuse if one is not careful. For a detailed
   discussion of periodogram peak significance, see see [11]_.

When using the Lomb-Scargle Periodogram to decide whether a signal contains a
periodic component, an important consideration is the significance of the
periodogram peak. This significance is usually expressed in terms of a
false alarm probability, which encodes the probability of measuring a
peak of a given height (or higher) conditioned on the assumption that
the data consists of Gaussian noise with no periodic component.

For example, let's simulate 60 observations of a sine wave with noise:

>>> t = 100 * rand.rand(60)
>>> dy = 1.0
>>> y = np.sin(2 * np.pi * t) + dy * rand.randn(60)
>>> ls = LombScargle(t, y, dy)
>>> freq, power = ls.autopower()
>>> print(power.max())  # doctest: +FLOAT_CMP
0.33814001958188855

The peak of the periodogram has a value of 0.33, but how significant is
this peak? We can address this question using the
:func:`~astropy.stats.LombScargle.false_alarm_probability` method:

.. doctest-requires:: scipy

  >>> ls.false_alarm_probability(power.max())  # doctest: +FLOAT_CMP
  0.0043217866919174324

What this tells us is that, under the assumption that there is no periodic
signal in the data, we will observe a peak this high or higher approximately
0.4% of the time, which gives a strong indication that a periodic signal is
present in the data.

.. Note::
  One must be careful in interpreting this probability: it is a measurement
  conditioned on the assumption of the null hypothesis of no signal; in symbols,
  one might write :math:`P({\rm data} \mid {\rm noise-only})`.

  It is tempting to interpret this quantity incorrectly; in particular, you
  might wish to say a statement like "there is an 0.4% chance that this data
  is noise only", but this is *not* a correct statement; in symbols, this
  statement describes the quantity :math:`P({\rm noise-only} \mid {\rm data})`,
  and in general :math:`P(A\mid B) \ne P(B\mid A)`.

  See [11]_ for a more detailed discussion of such caveats.

We might also wish to compute the required peak height to attain any given
false alarm probability, which can be done with the
:func:`~astropy.stats.LombScargle.false_alarm_level` method:

.. doctest-requires:: scipy

  >>> probabilities = [0.1, 0.05, 0.01]
  >>> ls.false_alarm_level(probabilities)  # doctest: +FLOAT_CMP
  array([0.25446627, 0.27436154, 0.31716182])

This tells us that to attain a 10% false alarm probability requires the highest
periodogram peak to be approximately 0.25; 5% requires 0.27, and 1% requires
0.32.

False Alarm Approximations
--------------------------

Although the false alarm probability at any particular frequency is analytically
computable, there is no closed-form analytic expression for the more relevant
quantity of the false alarm level of the *highest* peak in a particular
periodogram.
This must be either determined through bootstrap simulations, or approximated
by various means.

AstroPy provides four options for approximating the false alarm probability,
which can be chosen using the ``method`` keyword:

- ``method="baluev"`` (the default)
  implements the approximation proposed by Baluev 2008 [8]_,
  which employs extreme value statistics to compute an upper-bound of the false
  alarm probability for the alias-free case. Experiments show that the bound is
  also useful even for highly-aliased observing patterns.

.. doctest-requires:: scipy

    >>> ls.false_alarm_probability(power.max(), method='baluev')  # doctest: +FLOAT_CMP
    0.0043217866919174324

- ``method="bootstrap"`` implements a bootstrap simulation: effectively it
  computes many Lomb-Scargle periodograms on simulated data at the same
  observation times. The bootstrap approach can very accurately determine
  the false alarm probability, but is very computationally expensive:
  to estimate the level corresponding to a false alarm probability
  :math:`P_{false}`, it requires on order :math:`n_{boot} \approx 10/P_{false}`
  individual periodograms to be computed for the dataset.

.. doctest-requires:: scipy

    >>> ls.false_alarm_probability(power.max(), method='bootstrap')  # doctest: +SKIP
    0.0030000000000000027

- ``method="davies"`` is related to the Baluev method, but loses accuracy
  at large false alarm probabilities

.. doctest-requires:: scipy

    >>> ls.false_alarm_probability(power.max(), method='davies')  # doctest: +FLOAT_CMP
    0.0043311525763707216

- ``method="naive"`` is a simplistic method based on the assumption that
  well-separated areas in the periodogram are independent. In general, it
  provides a very poor estimate of the false alarm probability and should
  not be used in practice, but is included for completeness

.. doctest-requires:: scipy

    >>> ls.false_alarm_probability(power.max(), method='naive')  # doctest: +FLOAT_CMP
    0.0011693992470136049

The following figure compares these false alarm estimates at a range of
peak heights for 100 observations with a heavily-aliased observing pattern:

.. plot::

    import numpy as np
    import matplotlib.pyplot as plt
    plt.style.use('ggplot')

    from astropy.stats import LombScargle

    rng = np.random.RandomState(42)

    N = 100
    t = 5 * rng.rand(N)
    t -= 0.5 * (t % 1)  # create alias-inducing structure in the window function
    dy = 0.5 * (1 + rng.rand(N))
    y = dy * rng.randn(N)

    ls = LombScargle(t, y, dy, normalization='standard')
    z = np.linspace(1E-3, 0.15, 1000)

    def false_alarm(method):
        return ls.false_alarm_probability(z, method=method, maximum_frequency=5)

    fa_boot = ls.false_alarm_probability(z, method='bootstrap',
                                         maximum_frequency=5,
                                         method_kwds=dict(random_seed=42))

    fig, ax = plt.subplots(figsize=(6, 4.5))

    ax.plot(z, false_alarm('naive'), label='naive estimate')
    ax.plot(z, false_alarm('baluev'), label='Baluev estimate')
    ax.plot(z, false_alarm('davies'), ':k', label='Davies bound')
    ax.plot(z, fa_boot, '-k', label='bootstrap estimate')

    ax.legend(loc='lower left')
    ax.set(yscale='log',
           title='False Alarm Estimates (N=100)',
           xlim=(0, 0.15), ylim=(0.01, 1.5),
           xlabel='Value of Highest Periodogram Peak',
           ylabel='False Alarm Probability');

In general, one should use the bootstrap approach when computationally feasible,
and the Baluev approach otherwise.

In all of this, it is important to keep in mind a few caveats:

- False alarm probabilities are computed relative to a particular set of
  observing times, and a particular choice of frequency grid.
- False alarm probabilities are conditioned upon the null hypothesis of
  data with no periodic component, and in particular say nothing
  quantitative about whether the data are actually consistent with a
  periodic model.
- False alarm probabilities are not related to the question of whether the
  highest peak in a periodogram is the *correct* peak, and in particular
  are not especially useful in the case of observations with a strong
  aliasing pattern.

For a detailed discussion of these caveats and others when computing and
interpreting false alarm probabilities, please refer to [11]_.


Periodogram Algorithms
======================
The :class:`~astropy.stats.LombScargle` class makes available
several complementary implementations of the Lomb-Scargle Periodogram,
which can be selected using the ``method`` keyword of the Lomb-Scargle power.
By design all methods will return the same results (some approximate),
and each has its advantages and disadvantages.

For example, to compute a periodogram using the fast chi-square method
of Palmer (2009) [9]_, you can specify ``method='fastchi2'``:

    >>> frequency, power = LombScargle(t, y).autopower(method='fastchi2')

There are currently six methods available in the package:

``method='auto'``
-----------------
The ``auto`` method is the default, and will attempt to select the best option
from the following methods using heuristics driven by the input data.

``method='slow'``
-----------------
The ``slow`` method is a pure-Python implementation of the original Lomb-Scargle
periodogram ([1]_, [2]_), enhanced to account for observational noise,
and to allow a floating mean (sometimes called the *generalized periodogram*;
see e.g. [10]_). The method is not particularly fast, scaling approximately
as :math:`O[NM]` for :math:`N` data points and :math:`M` frequencies.

``method='cython'``
-------------------
The ``cython`` method is a cython implementation of the same algorithm used for
``method='slow'``. It is slightly faster than the pure-python implementation,
but much more memory-efficient as the size of the inputs grow. The computational
scaling is approximately :math:`O[NM]` for :math:`N` data points and
:math:`M` frequencies.

``method='scipy'``
------------------
The ``scipy`` method wraps the C implementation of the original Lomb-Scargle
periodogram which is available in :func:`scipy.signal.lombscargle`. This is
slightly faster than the ``slow`` method, but does not allow for errors in
data or extensions such as the floating mean. The scaling is approximately
:math:`O[NM]` for :math:`N` data points and :math:`M` frequencies.

``method='fast'``
-----------------
The ``fast`` method is a pure-Python implementation of the fast periodogram of
Press & Rybicki [3]_. It uses an *extirpolation* approach to approximate
the periodogram frequencies using a fast Fourier transform. As with the
``slow`` method, it can handle data errors and floating mean.
The scaling is approximately :math:`O[N\log M]` for :math:`N`
data points and :math:`M` frequencies.

``method='chi2'``
-----------------
The ``chi2`` method is a pure-Python implementation based on matrix algebra
(see, e.g. [7]_). It utilizes the fact that the Lomb-Scargle periodogram at
each frequency is equivalent to the least-squares fit of a sinusoid to the
data. The advantage of the ``chi2`` method is that it allows extensions of
the periodogram to multiple Fourier terms, specified by the ``nterms``
parameter. For the standard problem, it is slightly slower than
``method='slow'`` and scales as :math:`O[n_fNM]` for :math:`N` data points,
:math:`M` frequencies, and :math:`n_f` Fourier terms.

``method='fastchi2'``
---------------------
The fast chi-squared method of Palmer (2009) [9]_ is equivalent to the ``chi2`` method,
but the matrices are constructed using an FFT-based approach similar to that
of the ``fast`` method. The result is a relatively efficient periodogram
(though not nearly as efficient as the ``fast`` method) which can be extended
to multiple terms. The scaling is approximately :math:`O[n_f(M + N\log M)]`
for :math:`N` data points, :math:`M` frequencies, and :math:`n_f` Fourier terms.

Summary
-------
The following table summarizes the features of the above algorithms:

==============  ============================  =============  ===============  ========
method          Computational                 Observational  Bias Term        Multiple
                Scaling                       Uncertainties  (Floating Mean)  Terms
==============  ============================  =============  ===============  ========
``"slow"``      :math:`O[NM]`                 Yes            Yes              No
``"cython"``    :math:`O[NM]`                 Yes            Yes              No
``"scipy"``     :math:`O[NM]`                 No             No               No
``"fast"``      :math:`O[N\log M]`            Yes            Yes              No
``"chi2"``      :math:`O[n_fNM]`              Yes            Yes              Yes
``"fastchi2"``  :math:`O[n_f(M + N\log M)]`   Yes            Yes              Yes
==============  ============================  =============  ===============  ========

In the Computational Scaling column, :math:`N` is the number of data points,
:math:`M` is the number of frequencies, and :math:`n_f` is the number of
Fourier terms for a multi-term fit.

.. _lomb-scargle-example:

RR-Lyrae Example
================

An example of computing the periodogram for a more realistic dataset is
shown in the following figure. The data here consist of
50 nightly observations of a simulated RR Lyrae-like variable star,
with lightcurve shape that is more complicated than a simple sine wave:

.. plot::

    import numpy as np
    import matplotlib.pyplot as plt
    plt.style.use('ggplot')

    from astropy.stats import LombScargle


    def simulated_data(N, rseed=2, period=0.41, phase=0.0):
        """Simulate data based from a pre-computed empirical fit"""

        # coefficients from a 5-term Fourier fit to SDSS object 1019544
        coeffs = [-0.0191, 0.1375, -0.1968, 0.0959, 0.075,
                  -0.0686, 0.0307, -0.0045, -0.0421, 0.0216, 0.0041]

        rand = np.random.RandomState(rseed)
        t = phase + np.arange(N, dtype=float)
        t += 0.1 * rand.randn(N)
        dmag = 0.01 + 0.03 * rand.rand(N)

        omega = 2 * np.pi / period
        n = np.arange(1 + len(coeffs) // 2)[:, None]

        mag = (15 + dmag * rand.randn(N)
               + np.dot(coeffs[::2], np.cos(n * omega * t)) +
               + np.dot(coeffs[1::2], np.sin(n[1:] * omega * t)))

        return t, mag, dmag


    # generate data and compute the periodogram
    t, mag, dmag = simulated_data(50)
    ls = LombScargle(t, mag, dmag, normalization='standard')
    freq, PLS = ls.autopower(minimum_frequency=1 / 1.2,
                             maximum_frequency=1 / 0.2)
    best_freq = freq[np.argmax(PLS)]
    phase = (t * best_freq) % 1

    # compute the best-fit model
    phase_fit = np.linspace(0, 1)
    mag_fit = ls.model(t=phase_fit / best_freq,
                       frequency=best_freq)

    # set up the figure & axes for plotting
    fig, ax = plt.subplots(1, 2, figsize=(12, 5))
    fig.suptitle('Lomb-Scargle Periodogram (period=0.41 days)')
    fig.subplots_adjust(bottom=0.12, left=0.07, right=0.95)
    inset = fig.add_axes([0.78, 0.56, 0.15, 0.3])

    # plot the raw data
    ax[0].errorbar(t, mag, dmag, fmt='ok', elinewidth=1.5, capsize=0)
    ax[0].invert_yaxis()
    ax[0].set(xlim=(0, 50),
              xlabel='Observation time (days)',
              ylabel='Observed Magnitude')

    # plot the periodogram
    ax[1].plot(1. / freq, PLS)
    ax[1].set(xlabel='period (days)',
              ylabel='Lomb-Scargle Power',
              xlim=(0.2, 1.2),
              ylim=(0, 1));

    # plot the false-alarm levels
    z_false = ls.false_alarm_level(0.01, maximum_frequency=1 / 0.2,
                                   method='baluev')
    ax[1].axhline(z_false, linestyle='dotted', color='black')

    # plot the phased data & model in the inset
    inset.errorbar(phase, mag, dmag, fmt='.k', capsize=0)
    inset.plot(phase_fit, mag_fit)
    inset.invert_yaxis()
    inset.set_xlabel('phase')
    inset.set_ylabel('mag')


The dotted line shows the periodogram level corresponding to a maximum peak
false alarm probability of 1%.
This example demonstrates that for irregularly-sampled
data, the Lomb-Scargle periodogram can be sensitive to frequencies higher
than the average Nyquist frequency: the above data are sampled at
an average rate of roughly one observation per night, and the periodogram
relatively cleanly reveals the true period of 0.41 days.

Still, the periodogram has many spurious peaks, which are due to several factors:

1. Errors in observations lead to leakage of power from the true peaks.
2. The signal is not a perfect sinusoid, so additional peaks can indicate
   higher-frequency components in the signal.
3. The observations take place only at night, meaning that the survey window
   has non-negligible power at a frequency of 1 cycle per day.
   Thus we expect aliases to
   appear at :math:`f_{\rm alias} = f_{\rm true} + n f_{\rm window}` for integer
   values of :math:`n`. With a true period of 0.41 days and a 1-day signal
   in the observing window, the :math:`n=+1` and :math:`n=-1`
   aliases to lie at periods of 0.29 and 0.69 days, respectively:
   these aliases are prominent in the above plot.

The interaction of these effects means that in practice there is
no absolute guarantee that the highest peak corresponds to the best frequency,
and results must be interpreted carefully.
For a detailed discussion of these effects, see [11]_.


Literature References
=====================
.. [1] Lomb, N.R. *Least-squares frequency analysis of unequally spaced data*.
       Ap&SS 39 pp. 447-462 (1976)
.. [2] Scargle, J. D. *Studies in astronomical time series analysis. II -
       Statistical aspects of spectral analysis of unevenly spaced data*.
       ApJ 1:263 pp. 835-853 (1982)
.. [3] Press W.H. and Rybicki, G.B, *Fast algorithm for spectral analysis
       of unevenly sampled data*. ApJ 1:338, p. 277 (1989)
.. [4] Vanderplas, J., Connolly, A. Ivezic, Z. & Gray, A. *Introduction to
       astroML: Machine learning for astrophysics*. Proceedings of the
       Conference on Intelligent Data Understanding (2012)
.. [5]  Vanderplas, J., Connolly, A. Ivezic, Z. & Gray, A. *Statistics,
	Data Mining and Machine Learning in Astronomy*. Princeton Press (2014)}
.. [6] VanderPlas, J. *Gatspy: General Tools for Astronomical Time Series
       in Python* (2015) https://zenodo.org/record/14833
.. [7] VanderPlas, J. & Ivezic, Z. *Periodograms for Multiband Astronomical
       Time Series*. ApJ 812.1:18 (2015)
.. [8] Baluev, R.V. *Assessing Statistical Significance of Periodogram Peaks*
       MNRAS 385, 1279 (2008)
.. [9] Palmer, D. *A Fast Chi-squared Technique for Period Search of
       Irregularly Sampled Data*. ApJ 695.1:496 (2009)
.. [10] Zechmeister, M. and Kurster, M. *The generalised Lomb-Scargle
       periodogram. A new formalism for the floating-mean and Keplerian
       periodograms*, A&A 496, 577-584 (2009)
.. [11] VanderPlas, J. *Understanding the Lomb-Scargle Periodogram*
	arXiv:1703.09824 (2017)
	http://adsabs.harvard.edu/abs/2017arXiv170309824V
