# LombScargle¶

class astropy.timeseries.LombScargle(t, y, dy=None, fit_mean=True, center_data=True, nterms=1, normalization='standard')[source]

Compute the Lomb-Scargle Periodogram.

This implementations here are based on code presented in [R14388b5a5a57-1] and [R14388b5a5a57-2]; if you use this functionality in an academic application, citation of those works would be appreciated.

Parameters: t : array_like or Quantity sequence of observation times y : array_like or Quantity sequence of observations associated with times t dy : float, array_like or Quantity (optional) error or sequence of observational errors associated with times t fit_mean : bool (optional, default=True) if True, include a constant offset as part of the model at each frequency. This can lead to more accurate results, especially in the case of incomplete phase coverage. center_data : bool (optional, default=True) if True, pre-center the data by subtracting the weighted mean of the input data. This is especially important if fit_mean = False nterms : int (optional, default=1) number of terms to use in the Fourier fit normalization : {‘standard’, ‘model’, ‘log’, ‘psd’}, optional Normalization to use for the periodogram.

References

 [R14388b5a5a57-1] Vanderplas, J., Connolly, A. Ivezic, Z. & Gray, A. Introduction to astroML: Machine learning for astrophysics. Proceedings of the Conference on Intelligent Data Understanding (2012)
 [R14388b5a5a57-2] VanderPlas, J. & Ivezic, Z. Periodograms for Multiband Astronomical Time Series. ApJ 812.1:18 (2015)

Examples

Generate noisy periodic data:

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


Compute the Lomb-Scargle periodogram on an automatically-determined frequency grid & find the frequency of max power:

>>> frequency, power = LombScargle(t, y).autopower()
>>> frequency[np.argmax(power)]  # doctest: +FLOAT_CMP
1.0016662310392956


Compute the Lomb-Scargle periodogram at a user-specified frequency grid:

>>> freq = np.arange(0.8, 1.3, 0.1)
>>> LombScargle(t, y).power(freq)  # doctest: +FLOAT_CMP
array([0.0204304 , 0.01393845, 0.35552682, 0.01358029, 0.03083737])


If the inputs are astropy Quantities with units, the units will be validated and the outputs will also be Quantities with appropriate units:

>>> from astropy import units as u
>>> t = t * u.s
>>> y = y * u.mag
>>> frequency, power = LombScargle(t, y).autopower()
>>> frequency.unit
Unit("1 / s")
>>> power.unit
Unit(dimensionless)


Note here that the Lomb-Scargle power is always a unitless quantity, because it is related to the $$\chi^2$$ of the best-fit periodic model at each frequency.

Attributes Summary

Methods Summary

 autofrequency(self[, samples_per_peak, …]) Determine a suitable frequency grid for data. autopower(self[, method, method_kwds, …]) Compute Lomb-Scargle power at automatically-determined frequencies. design_matrix(self, frequency[, t]) Compute the design matrix for a given frequency distribution(self, power[, cumulative]) Expected periodogram distribution under the null hypothesis. false_alarm_level(self, false_alarm_probability) Level of maximum at a given false alarm probability. false_alarm_probability(self, power[, …]) False alarm probability of periodogram maxima under the null hypothesis. from_timeseries(timeseries[, …]) Initialize a periodogram from a time series object. model(self, t, frequency) Compute the Lomb-Scargle model at the given frequency. model_parameters(self, frequency[, units]) Compute the best-fit model parameters at the given frequency. offset(self) Return the offset of the model power(self, frequency[, normalization, …]) Compute the Lomb-Scargle power at the given frequencies.

Attributes Documentation

available_methods = ['auto', 'slow', 'chi2', 'cython', 'fast', 'fastchi2', 'scipy']

Methods Documentation

autofrequency(self, samples_per_peak=5, nyquist_factor=5, minimum_frequency=None, maximum_frequency=None, return_freq_limits=False)[source]

Determine a suitable frequency grid for data.

Note that this assumes the peak width is driven by the observational baseline, which is generally a good assumption when the baseline is much larger than the oscillation period. If you are searching for periods longer than the baseline of your observations, this may not perform well.

Even with a large baseline, be aware that the maximum frequency returned is based on the concept of “average Nyquist frequency”, which may not be useful for irregularly-sampled data. The maximum frequency can be adjusted via the nyquist_factor argument, or through the maximum_frequency argument.

Parameters: samples_per_peak : float (optional, default=5) The approximate number of desired samples across the typical peak nyquist_factor : float (optional, default=5) The multiple of the average nyquist frequency used to choose the maximum frequency if maximum_frequency is not provided. minimum_frequency : float (optional) If specified, then use this minimum frequency rather than one chosen based on the size of the baseline. maximum_frequency : float (optional) If specified, then use this maximum frequency rather than one chosen based on the average nyquist frequency. return_freq_limits : bool (optional) if True, return only the frequency limits rather than the full frequency grid. frequency : ndarray or Quantity The heuristically-determined optimal frequency bin
autopower(self, method='auto', method_kwds=None, normalization=None, samples_per_peak=5, nyquist_factor=5, minimum_frequency=None, maximum_frequency=None)[source]

Compute Lomb-Scargle power at automatically-determined frequencies.

Parameters: method : string (optional) specify the lomb scargle implementation to use. Options are: ‘auto’: choose the best method based on the input ‘fast’: use the O[N log N] fast method. Note that this requires evenly-spaced frequencies: by default this will be checked unless assume_regular_frequency is set to True. ‘slow’: use the O[N^2] pure-python implementation ‘cython’: use the O[N^2] cython implementation. This is slightly faster than method=’slow’, but much more memory efficient. ‘chi2’: use the O[N^2] chi2/linear-fitting implementation ‘fastchi2’: use the O[N log N] chi2 implementation. Note that this requires evenly-spaced frequencies: by default this will be checked unless assume_regular_frequency is set to True. ‘scipy’: use scipy.signal.lombscargle, which is an O[N^2] implementation written in C. Note that this does not support heteroskedastic errors. method_kwds : dict (optional) additional keywords to pass to the lomb-scargle method normalization : {‘standard’, ‘model’, ‘log’, ‘psd’}, optional If specified, override the normalization specified at instantiation. samples_per_peak : float (optional, default=5) The approximate number of desired samples across the typical peak nyquist_factor : float (optional, default=5) The multiple of the average nyquist frequency used to choose the maximum frequency if maximum_frequency is not provided. minimum_frequency : float (optional) If specified, then use this minimum frequency rather than one chosen based on the size of the baseline. maximum_frequency : float (optional) If specified, then use this maximum frequency rather than one chosen based on the average nyquist frequency. frequency, power : ndarrays The frequency and Lomb-Scargle power
design_matrix(self, frequency, t=None)[source]

Compute the design matrix for a given frequency

Parameters: frequency : float the frequency for the model t : array_like or Quantity or Time, length n_samples times at which to compute the model (optional). If not specified, then the times and uncertainties of the input data are used X : np.ndarray (len(t), n_parameters) The design matrix for the model at the given frequency.
distribution(self, power, cumulative=False)[source]

Expected periodogram distribution under the null hypothesis.

This computes the expected probability distribution or cumulative probability distribution of periodogram power, under the null hypothesis of a non-varying signal with Gaussian noise. Note that this is not the same as the expected distribution of peak values; for that see the false_alarm_probability() method.

Parameters: power : array_like The periodogram power at which to compute the distribution. cumulative : bool (optional) If True, then return the cumulative distribution. dist : np.ndarray The probability density or cumulative probability associated with the provided powers.
false_alarm_level(self, false_alarm_probability, method='baluev', samples_per_peak=5, nyquist_factor=5, minimum_frequency=None, maximum_frequency=None, method_kwds=None)[source]

Level of maximum at a given false alarm probability.

This gives an estimate of the periodogram level corresponding to a specified false alarm probability for the largest peak, assuming a null hypothesis of non-varying data with Gaussian noise.

Parameters: false_alarm_probability : array-like The false alarm probability (0 < fap < 1). maximum_frequency : float The maximum frequency of the periodogram. method : {‘baluev’, ‘davies’, ‘naive’, ‘bootstrap’}, optional The approximation method to use; default=’baluev’. method_kwds : dict, optional Additional method-specific keywords. power : np.ndarray The periodogram peak height corresponding to the specified false alarm probability.

Notes

The true probability distribution for the largest peak cannot be determined analytically, so each method here provides an approximation to the value. The available methods are:

• “baluev” (default): the upper-limit to the alias-free probability, using the approach of Baluev (2008) [1].
• “davies” : the Davies upper bound from Baluev (2008) [1].
• “naive” : the approximate probability based on an estimated effective number of independent frequencies.
• “bootstrap” : the approximate probability based on bootstrap resamplings of the input data.

Note also that for normalization=’psd’, the distribution can only be computed for periodograms constructed with errors specified.

References

 [1] (1, 2) Baluev, R.V. MNRAS 385, 1279 (2008)
false_alarm_probability(self, power, method='baluev', samples_per_peak=5, nyquist_factor=5, minimum_frequency=None, maximum_frequency=None, method_kwds=None)[source]

False alarm probability of periodogram maxima under the null hypothesis.

This gives an estimate of the false alarm probability given the height of the largest peak in the periodogram, based on the null hypothesis of non-varying data with Gaussian noise.

Parameters: power : array-like The periodogram value. method : {‘baluev’, ‘davies’, ‘naive’, ‘bootstrap’}, optional The approximation method to use. maximum_frequency : float The maximum frequency of the periodogram. method_kwds : dict (optional) Additional method-specific keywords. false_alarm_probability : np.ndarray The false alarm probability

Notes

The true probability distribution for the largest peak cannot be determined analytically, so each method here provides an approximation to the value. The available methods are:

• “baluev” (default): the upper-limit to the alias-free probability, using the approach of Baluev (2008) [1].
• “davies” : the Davies upper bound from Baluev (2008) [1].
• “naive” : the approximate probability based on an estimated effective number of independent frequencies.
• “bootstrap” : the approximate probability based on bootstrap resamplings of the input data.

Note also that for normalization=’psd’, the distribution can only be computed for periodograms constructed with errors specified.

References

 [1] (1, 2) Baluev, R.V. MNRAS 385, 1279 (2008)
classmethod from_timeseries(timeseries, signal_column_name=None, uncertainty=None, **kwargs)

Initialize a periodogram from a time series object.

If a binned time series is passed, the time at the center of the bins is used. Also note that this method automatically gets rid of NaN/undefined values when initalizing the periodogram.

Parameters: signal_column_name : str The name of the column containing the signal values to use. uncertainty : str or float or Quantity, optional The name of the column containing the errors on the signal, or the value to use for the error, if a scalar. **kwargs Additional keyword arguments are passed to the initializer for this periodogram class.
model(self, t, frequency)[source]

Compute the Lomb-Scargle model at the given frequency.

The model at a particular frequency is a linear model: model = offset + dot(design_matrix, model_parameters)

Parameters: t : array_like or Quantity, length n_samples times at which to compute the model frequency : float the frequency for the model y : np.ndarray, length n_samples The model fit corresponding to the input times
model_parameters(self, frequency, units=True)[source]

Compute the best-fit model parameters at the given frequency.

The model described by these parameters is:

$y(t; f, \vec{\theta}) = \theta_0 + \sum_{n=1}^{\tt nterms} [\theta_{2n-1}\sin(2\pi n f t) + \theta_{2n}\cos(2\pi n f t)]$

where $$\vec{\theta}$$ is the array of parameters returned by this function.

Parameters: frequency : float the frequency for the model units : bool If True (default), return design matrix with data units. theta : np.ndarray (n_parameters,) The best-fit model parameters at the given frequency.
offset(self)[source]

Return the offset of the model

The offset of the model is the (weighted) mean of the y values. Note that if self.center_data is False, the offset is 0 by definition.

Returns: offset : scalar
power(self, frequency, normalization=None, method='auto', assume_regular_frequency=False, method_kwds=None)[source]

Compute the Lomb-Scargle power at the given frequencies.

Parameters: frequency : array_like or Quantity frequencies (not angular frequencies) at which to evaluate the periodogram. Note that in order to use method=’fast’, frequencies must be regularly-spaced. method : string (optional) specify the lomb scargle implementation to use. Options are: ‘auto’: choose the best method based on the input ‘fast’: use the O[N log N] fast method. Note that this requires evenly-spaced frequencies: by default this will be checked unless assume_regular_frequency is set to True. ‘slow’: use the O[N^2] pure-python implementation ‘cython’: use the O[N^2] cython implementation. This is slightly faster than method=’slow’, but much more memory efficient. ‘chi2’: use the O[N^2] chi2/linear-fitting implementation ‘fastchi2’: use the O[N log N] chi2 implementation. Note that this requires evenly-spaced frequencies: by default this will be checked unless assume_regular_frequency is set to True. ‘scipy’: use scipy.signal.lombscargle, which is an O[N^2] implementation written in C. Note that this does not support heteroskedastic errors. assume_regular_frequency : bool (optional) if True, assume that the input frequency is of the form freq = f0 + df * np.arange(N). Only referenced if method is ‘auto’ or ‘fast’. normalization : {‘standard’, ‘model’, ‘log’, ‘psd’}, optional If specified, override the normalization specified at instantiation. fit_mean : bool (optional, default=True) If True, include a constant offset as part of the model at each frequency. This can lead to more accurate results, especially in the case of incomplete phase coverage. center_data : bool (optional, default=True) If True, pre-center the data by subtracting the weighted mean of the input data. This is especially important if fit_mean = False. method_kwds : dict (optional) additional keywords to pass to the lomb-scargle method power : ndarray The Lomb-Scargle power at the specified frequency