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# LombScargle

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

  Bases: `BasePeriodogram`

  Compute the Lomb-Scargle Periodogram.

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

  Parameters:
    t numpy:array_like or `Quantity` [:ref: ‘time’]

      sequence of observation times

    y numpy:array_like or `Quantity`

      sequence of observations associated with times t

    dy `python:float`, numpy:array_like, or `Quantity`, optional

      error or sequence of observational errors associated with times t

    fit_mean bool, optional

      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

      if True, pre-center the data by subtracting the weighted mean of the input data. This is especially important if fit_mean = False

    nterms `python:int`, optional

      number of terms to use in the Fourier fit

    normalization{‘standard’, ‘model’, ‘log’, ‘psd’}, optional

      Normalization to use for the periodogram.

  References

  [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)

  [2]

  VanderPlas, J. & Ivezic, Z. Periodograms for Multiband Astronomical Time Series. ApJ 812.1:18 (2015)

  Examples

  Generate noisy periodic data:

      >>> rand = np.random.default_rng(42)
      >>> t = 100 * rand.random(100)
      >>> y = np.sin(2 * np.pi * t) + rand.standard_normal(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)]
      1.0007641728995051

  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)
      array([0.0792948 , 0.01778874, 0.25328167, 0.01064157, 0.01471387])

  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

  `available_methods` |

  Methods Summary

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

  Attributes Documentation

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

  Methods Documentation

  autofrequency(samples_per_peak=5, nyquist_factor=5, minimum_frequency=None, maximum_frequency=None, return_freq_limits=False)

    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 `python:float`, optional

        The approximate number of desired samples across the typical peak

      nyquist_factor `python:float`, optional

        The multiple of the average nyquist frequency used to choose the maximum frequency if maximum_frequency is not provided.

      minimum_frequency `python:float`, optional

        If specified, then use this minimum frequency rather than one chosen based on the size of the baseline.

      maximum_frequency `python: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.

    Returns:
      frequency `ndarray` or `Quantity` [:ref: ‘frequency’]

        The heuristically-determined optimal frequency bin

  autopower(method='auto', method_kwds=None, normalization=None, samples_per_peak=5, nyquist_factor=5, minimum_frequency=None, maximum_frequency=None)

    Compute Lomb-Scargle power at automatically-determined frequencies.

    Parameters:
      method `python:str`, 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 `python: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 `python:float`, optional

        The approximate number of desired samples across the typical peak

      nyquist_factor `python:float`, optional

        The multiple of the average nyquist frequency used to choose the maximum frequency if maximum_frequency is not provided.

      minimum_frequency `python:float` or `Quantity` [:ref: ‘frequency’], optional

        If specified, then use this minimum frequency rather than one chosen based on the size of the baseline. Should be `Quantity` if inputs to LombScargle are `Quantity`.

      maximum_frequency `python:float` or `Quantity` [:ref: ‘frequency’], optional

        If specified, then use this maximum frequency rather than one chosen based on the average nyquist frequency. Should be `Quantity` if inputs to LombScargle are `Quantity`.

    Returns:
      frequency, power `ndarray`

        The frequency and Lomb-Scargle power

  design_matrix(frequency, t=None)

    Compute the design matrix for a given frequency

    Parameters:
      frequency `python:float`

        the frequency for the model

      t numpy:array_like, `Quantity`, or `Time` (optional)

        Times (length `n_samples`) at which to compute the model. If not specified, then the times and uncertainties of the input data are used.

    Returns:
      X `array`

        The design matrix for the model at the given frequency. This should have a shape of (`len(t)`, `n_parameters`).

    See also

    `model`
    `model_parameters`
    `offset`
  distribution(power, cumulative=False)

    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 numpy:array_like

        The periodogram power at which to compute the distribution.

      cumulative bool, optional

        If True, then return the cumulative distribution.

    Returns:
      dist `np.ndarray`

        The probability density or cumulative probability associated with the provided powers.

    See also

    `false_alarm_probability`
    `false_alarm_level`
  false_alarm_level(false_alarm_probability, method='baluev', samples_per_peak=5, nyquist_factor=5, minimum_frequency=None, maximum_frequency=None, method_kwds=None)

    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 numpy:array_like

        The false alarm probability (0 < fap < 1).

      maximum_frequency `python:float`

        The maximum frequency of the periodogram.

      method{‘baluev’, ‘davies’, ‘naive’, ‘bootstrap’}, optional

        The approximation method to use; default=’baluev’.

      method_kwds `python:dict`, optional

        Additional method-specific keywords.

    Returns:
      power `np.ndarray`

        The periodogram peak height corresponding to the specified false alarm probability.

    See also

    `distribution`
    `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. The number of samples can be set with the method-specific keyword “n_bootstraps” (default=1000).

    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(power, method='baluev', samples_per_peak=5, nyquist_factor=5, minimum_frequency=None, maximum_frequency=None, method_kwds=None)

    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 numpy:array_like

        The periodogram value.

      method{‘baluev’, ‘davies’, ‘naive’, ‘bootstrap’}, optional

        The approximation method to use.

      maximum_frequency `python:float`

        The maximum frequency of the periodogram.

      method_kwds `python:dict`, optional

        Additional method-specific keywords.

    Returns:
      false_alarm_probability `np.ndarray`

        The false alarm probability

    See also

    `distribution`
    `false_alarm_level`

    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 initializing the periodogram.

    Parameters:
      signal_column_name `python:str`

        The name of the column containing the signal values to use.

      uncertainty `python:str` or `python: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(t, frequency)

    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 numpy:array_like or `Quantity` [:ref: ‘time’]

        Times (length `n_samples`) at which to compute the model.

      frequency `python:float`

        the frequency for the model

    Returns:
      y `np.ndarray`

        The model fit corresponding to the input times (will have length `n_samples`).

    See also

    `design_matrix`
    `offset`
    `model_parameters`
  model_parameters(frequency, units=True)

    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 `python:float`

        the frequency for the model

      units bool

        If True (default), return design matrix with data units.

    Returns:
      theta `np.ndarray` (n_parameters,)

        The best-fit model parameters at the given frequency.

    See also

    `design_matrix`
    `model`
    `offset`
  offset()

    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

    See also

    `design_matrix`
    `model`
    `model_parameters`
  power(frequency, normalization=None, method='auto', assume_regular_frequency=False, method_kwds=None)

    Compute the Lomb-Scargle power at the given frequencies.

    Parameters:
      frequency numpy:array_like or `Quantity` [:ref: ‘frequency’]

        frequencies (not angular frequencies) at which to evaluate the periodogram. Note that in order to use method=’fast’, frequencies must be regularly-spaced.

      method `python:str`, 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.

      method_kwds `python:dict`, optional

        additional keywords to pass to the lomb-scargle method

    Returns:
      power `ndarray`

        The Lomb-Scargle power at the specified frequency