# Source code for astropy.timeseries.periodograms.lombscargle.core

"""Main Lomb-Scargle Implementation"""

import numpy as np

from .implementations import lombscargle, available_methods
from .implementations.mle import periodic_fit, design_matrix
from . import _statistics
from astropy import units
from astropy.time import Time, TimeDelta
from astropy import units as u
from astropy.timeseries.periodograms.base import BasePeriodogram

def has_units(obj):
return hasattr(obj, 'unit')

def get_unit(obj):
return getattr(obj, 'unit', 1)

def strip_units(*arrs):
strip = lambda a: None if a is None else np.asarray(a)
if len(arrs) == 1:
return strip(arrs[0])
else:
return map(strip, arrs)

[docs]class LombScargle(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 : 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. 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 :math:\\chi^2 of the best-fit periodic model at each frequency. 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) """ available_methods = available_methods() def __init__(self, t, y, dy=None, fit_mean=True, center_data=True, nterms=1, normalization='standard'): # If t is a TimeDelta, convert it to a quantity. The units we convert # to don't really matter since the user gets a Quantity back at the end # so can convert to any units they like. if isinstance(t, TimeDelta): t = t.to('day') # We want to expose self.t as being the times the user passed in, but # if the times are absolute, we need to convert them to relative times # internally, so we use self._trel and self._tstart for this. self.t = t if isinstance(self.t, Time): self._tstart = self.t[0] trel = (self.t - self._tstart).to(u.day) else: self._tstart = None trel = self.t self._trel, self.y, self.dy = self._validate_inputs(trel, y, dy) self.fit_mean = fit_mean self.center_data = center_data self.nterms = nterms self.normalization = normalization def _validate_inputs(self, t, y, dy): # Validate shapes of inputs if dy is None: t, y = np.broadcast_arrays(t, y, subok=True) else: t, y, dy = np.broadcast_arrays(t, y, dy, subok=True) if t.ndim != 1: raise ValueError("Inputs (t, y, dy) must be 1-dimensional") # validate units of inputs if any is a Quantity if any(has_units(arr) for arr in (t, y, dy)): t, y = map(units.Quantity, (t, y)) if dy is not None: dy = units.Quantity(dy) try: dy = units.Quantity(dy, unit=y.unit) except units.UnitConversionError: raise ValueError("Units of dy not equivalent " "to units of y") return t, y, dy def _validate_frequency(self, frequency): frequency = np.asanyarray(frequency) if has_units(self._trel): frequency = units.Quantity(frequency) try: frequency = units.Quantity(frequency, unit=1./self._trel.unit) except units.UnitConversionError: raise ValueError("Units of frequency not equivalent to " "units of 1/t") else: if has_units(frequency): raise ValueError("frequency have units while 1/t doesn't.") return frequency def _validate_t(self, t): t = np.asanyarray(t) if has_units(self._trel): t = units.Quantity(t) try: t = units.Quantity(t, unit=self._trel.unit) except units.UnitConversionError: raise ValueError("Units of t not equivalent to " "units of input self.t") return t def _power_unit(self, norm): if has_units(self.y): if self.dy is None and norm == 'psd': return self.y.unit ** 2 else: return units.dimensionless_unscaled else: return 1
[docs] def autofrequency(self, 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 : 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. Returns ------- frequency : ndarray or Quantity The heuristically-determined optimal frequency bin """ baseline = self._trel.max() - self._trel.min() n_samples = self._trel.size df = 1.0 / baseline / samples_per_peak if minimum_frequency is None: minimum_frequency = 0.5 * df if maximum_frequency is None: avg_nyquist = 0.5 * n_samples / baseline maximum_frequency = nyquist_factor * avg_nyquist Nf = 1 + int(np.round((maximum_frequency - minimum_frequency) / df)) if return_freq_limits: return minimum_frequency, minimum_frequency + df * (Nf - 1) else: return minimum_frequency + df * np.arange(Nf)
[docs] def autopower(self, 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 : 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. Returns ------- frequency, power : ndarrays The frequency and Lomb-Scargle power """ frequency = self.autofrequency(samples_per_peak=samples_per_peak, nyquist_factor=nyquist_factor, minimum_frequency=minimum_frequency, maximum_frequency=maximum_frequency) power = self.power(frequency, normalization=normalization, method=method, method_kwds=method_kwds, assume_regular_frequency=True) return frequency, power
[docs] def power(self, frequency, normalization=None, method='auto', assume_regular_frequency=False, method_kwds=None): """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 Returns ------- power : ndarray The Lomb-Scargle power at the specified frequency """ if normalization is None: normalization = self.normalization frequency = self._validate_frequency(frequency) power = lombscargle(*strip_units(self._trel, self.y, self.dy), frequency=strip_units(frequency), center_data=self.center_data, fit_mean=self.fit_mean, nterms=self.nterms, normalization=normalization, method=method, method_kwds=method_kwds, assume_regular_frequency=assume_regular_frequency) return power * self._power_unit(normalization)
def _as_relative_time(self, name, times): """ Convert the provided times (if absolute) to relative times using the current _tstart value. If the times provided are relative, they are returned without conversion (though we still do some checks). """ if isinstance(times, TimeDelta): times = times.to('day') if self._tstart is None: if isinstance(times, Time): raise TypeError('{0} was provided as an absolute time but ' 'the LombScargle class was initialized ' 'with relative times.'.format(name)) else: if isinstance(times, Time): times = (times - self._tstart).to(u.day) else: raise TypeError('{0} was provided as a relative time but ' 'the LombScargle class was initialized ' 'with absolute times.'.format(name)) return times
[docs] def model(self, 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 : array_like or Quantity, length n_samples times at which to compute the model frequency : float the frequency for the model Returns ------- y : np.ndarray, length n_samples The model fit corresponding to the input times See Also -------- design_matrix offset model_parameters """ frequency = self._validate_frequency(frequency) t = self._validate_t(self._as_relative_time('t', t)) y_fit = periodic_fit(*strip_units(self._trel, self.y, self.dy), frequency=strip_units(frequency), t_fit=strip_units(t), center_data=self.center_data, fit_mean=self.fit_mean, nterms=self.nterms) return y_fit * get_unit(self.y)
[docs] def offset(self): """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 """ y, dy = strip_units(self.y, self.dy) if dy is None: dy = 1 dy = np.broadcast_to(dy, y.shape) if self.center_data: w = dy ** -2.0 y_mean = np.dot(y, w) / w.sum() else: y_mean = 0 return y_mean * get_unit(self.y)
[docs] def model_parameters(self, frequency, units=True): r"""Compute the best-fit model parameters at the given frequency. The model described by these parameters is: .. math:: 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 :math:\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. Returns ------- theta : np.ndarray (n_parameters,) The best-fit model parameters at the given frequency. See Also -------- design_matrix model offset """ frequency = self._validate_frequency(frequency) t, y, dy = strip_units(self._trel, self.y, self.dy) if self.center_data: y = y - strip_units(self.offset()) dy = np.ones_like(y) if dy is None else np.asarray(dy) X = self.design_matrix(frequency) parameters = np.linalg.solve(np.dot(X.T, X), np.dot(X.T, y / dy)) if units: parameters = get_unit(self.y) * parameters return parameters
[docs] def design_matrix(self, frequency, t=None): """Compute the design matrix for a given frequency Parameters ---------- frequency : float the frequency for the model t : array_like or ~astropy.units.Quantity or ~astropy.time.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 Returns ------- X : np.ndarray (len(t), n_parameters) The design matrix for the model at the given frequency. See Also -------- model model_parameters offset """ if t is None: t, dy = strip_units(self._trel, self.dy) else: t, dy = strip_units(self._validate_t(self._as_relative_time('t', t)), None) return design_matrix(t, frequency, dy, nterms=self.nterms, bias=self.fit_mean)
[docs] def distribution(self, 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 : array_like The periodogram power at which to compute the distribution. cumulative : bool (optional) If True, then return the cumulative distribution. See Also -------- false_alarm_probability false_alarm_level Returns ------- dist : np.ndarray The probability density or cumulative probability associated with the provided powers. """ dH = 1 if self.fit_mean or self.center_data else 0 dK = dH + 2 * self.nterms dist = _statistics.cdf_single if cumulative else _statistics.pdf_single return dist(power, len(self._trel), self.normalization, dH=dH, dK=dK)
[docs] def false_alarm_probability(self, 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 : 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. Returns ------- 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. See Also -------- distribution false_alarm_level References ---------- .. [1] Baluev, R.V. MNRAS 385, 1279 (2008) """ if self.nterms != 1: raise NotImplementedError("false alarm probability is not " "implemented for multiterm periodograms.") if not (self.fit_mean or self.center_data): raise NotImplementedError("false alarm probability is implemented " "only for periodograms of centered data.") fmin, fmax = self.autofrequency(samples_per_peak=samples_per_peak, nyquist_factor=nyquist_factor, minimum_frequency=minimum_frequency, maximum_frequency=maximum_frequency, return_freq_limits=True) return _statistics.false_alarm_probability(power, fmax=fmax, t=self._trel, y=self.y, dy=self.dy, normalization=self.normalization, method=method, method_kwds=method_kwds)
[docs] def 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): """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. Returns ------- 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. See Also -------- distribution false_alarm_probability References ---------- .. [1] Baluev, R.V. MNRAS 385, 1279 (2008) """ if self.nterms != 1: raise NotImplementedError("false alarm probability is not " "implemented for multiterm periodograms.") if not (self.fit_mean or self.center_data): raise NotImplementedError("false alarm probability is implemented " "only for periodograms of centered data.") fmin, fmax = self.autofrequency(samples_per_peak=samples_per_peak, nyquist_factor=nyquist_factor, minimum_frequency=minimum_frequency, maximum_frequency=maximum_frequency, return_freq_limits=True) return _statistics.false_alarm_level(false_alarm_probability, fmax=fmax, t=self._trel, y=self.y, dy=self.dy, normalization=self.normalization, method=method, method_kwds=method_kwds)