# BoxLeastSquares¶

class astropy.stats.BoxLeastSquares(t, y, dy=None)[source] [edit on github]

Bases: object

Compute the box least squares periodogram

This method is a commonly used tool for discovering transiting exoplanets or eclipsing binaries in photometric time series datasets. This implementation is based on the “box least squares (BLS)” method described in [1] and [2].

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.

References

 [1] (1, 2) Kovacs, Zucker, & Mazeh (2002), A&A, 391, 369 (arXiv:astro-ph/0206099)
 [2] (1, 2) Hartman & Bakos (2016), Astronomy & Computing, 17, 1 (arXiv:1605.06811)

Examples

Generate noisy data with a transit:

>>> rand = np.random.RandomState(42)
>>> t = rand.uniform(0, 10, 500)
>>> y = np.ones_like(t)
>>> y[np.abs((t + 1.0)%2.0-1)<0.08] = 1.0 - 0.1
>>> y += 0.01 * rand.randn(len(t))


Compute the transit periodogram on a heuristically determined period grid and find the period with maximum power:

>>> model = BoxLeastSquares(t, y)
>>> results = model.autopower(0.16)
>>> results.period[np.argmax(results.power)]
1.9923406038842544


Compute the periodogram on a user-specified period grid:

>>> periods = np.linspace(1.9, 2.1, 5)
>>> results = model.power(periods, 0.16)
>>> results.power
array([0.01421067, 0.02842475, 0.10867671, 0.05117755, 0.01783253])


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.day
>>> y = y * u.dimensionless_unscaled
>>> model = BoxLeastSquares(t, y)
>>> results = model.autopower(0.16 * u.day)
>>> results.period.unit
Unit("d")
>>> results.power.unit
Unit(dimensionless)


Methods Summary

 autoperiod(duration[, minimum_period, …]) Determine a suitable grid of periods autopower(duration[, objective, method, …]) Compute the periodogram at set of heuristically determined periods compute_stats(period, duration, transit_time) Compute descriptive statistics for a given transit model model(t_model, period, duration, transit_time) Compute the transit model at the given period, duration, and phase power(period, duration[, objective, method, …]) Compute the periodogram for a set of periods transit_mask(t, period, duration, transit_time) Compute which data points are in transit for a given parameter set

Methods Documentation

autoperiod(duration, minimum_period=None, maximum_period=None, minimum_n_transit=3, frequency_factor=1.0)[source] [edit on github]

Determine a suitable grid of periods

This method uses a set of heuristics to select a conservative period grid that is uniform in frequency. This grid might be too fine for some user’s needs depending on the precision requirements or the sampling of the data. The grid can be made coarser by increasing frequency_factor.

Parameters: duration : float, array-like or Quantity The set of durations that will be considered. minimum_period, maximum_period : float or Quantity, optional The minimum/maximum periods to search. If not provided, these will be computed as described in the notes below. minimum_n_transits : int, optional If maximum_period is not provided, this is used to compute the maximum period to search by asserting that any systems with at least minimum_n_transits will be within the range of searched periods. Note that this is not the same as requiring that minimum_n_transits be required for detection. The default value is 3. frequency_factor : float, optional A factor to control the frequency spacing as described in the notes below. The default value is 1.0. period : array-like or Quantity The set of periods computed using these heuristics with the same units as t.

Notes

The default minimum period is chosen to be twice the maximum duration because there won’t be much sensitivity to periods shorter than that.

The default maximum period is computed as

maximum_period = (max(t) - min(t)) / minimum_n_transits


ensuring that any systems with at least minimum_n_transits are within the range of searched periods.

The frequency spacing is given by

df = frequency_factor * min(duration) / (max(t) - min(t))**2


so the grid can be made finer by decreasing frequency_factor or coarser by increasing frequency_factor.

autopower(duration, objective=None, method=None, oversample=10, minimum_n_transit=3, minimum_period=None, maximum_period=None, frequency_factor=1.0)[source] [edit on github]

Compute the periodogram at set of heuristically determined periods

This method calls BoxLeastSquares.autoperiod() to determine the period grid and then BoxLeastSquares.power() to compute the periodogram. See those methods for documentation of the arguments.

compute_stats(period, duration, transit_time)[source] [edit on github]

Compute descriptive statistics for a given transit model

These statistics are commonly used for vetting of transit candidates.

Parameters: period : float or Quantity The period of the transits. duration : float or Quantity The duration of the transit. transit_time : float or Quantity The mid-transit time of a reference transit. stats : dict A dictionary containing several descriptive statistics: depth: The depth and uncertainty (as a tuple with two values) on the depth for the fiducial model. depth_odd: The depth and uncertainty on the depth for a model where the period is twice the fiducial period. depth_even: The depth and uncertainty on the depth for a model where the period is twice the fiducial period and the phase is offset by one orbital period. depth_half: The depth and uncertainty for a model with a period of half the fiducial period. depth_phased: The depth and uncertainty for a model with the fiducial period and the phase offset by half a period. harmonic_amplitude: The amplitude of the best fit sinusoidal model. harmonic_delta_log_likelihood: The difference in log likelihood between a sinusoidal model and the transit model. If harmonic_delta_log_likelihood is greater than zero, the sinusoidal model is preferred. transit_times: The mid-transit time for each transit in the baseline. per_transit_count: An array with a count of the number of data points in each unique transit included in the baseline. per_transit_log_likelihood: An array with the value of the log likelihood for each unique transit included in the baseline.
model(t_model, period, duration, transit_time)[source] [edit on github]

Compute the transit model at the given period, duration, and phase

Parameters: t_model : array-like or Quantity Times at which to compute the model. period : float or Quantity The period of the transits. duration : float or Quantity The duration of the transit. transit_time : float or Quantity The mid-transit time of a reference transit. y_model : array-like or Quantity The model evaluated at the times t_model with units of y.
power(period, duration, objective=None, method=None, oversample=10)[source] [edit on github]

Compute the periodogram for a set of periods

Parameters: period : array-like or Quantity The periods where the power should be computed duration : float, array-like or Quantity The set of durations to test objective : {‘likelihood’, ‘snr’}, optional The scalar that should be optimized to find the best fit phase, duration, and depth. This can be either 'likelihood' (default) to optimize the log-likelihood of the model, or 'snr' to optimize the signal-to-noise with which the transit depth is measured. method : {‘fast’, ‘slow’}, optional The computational method used to compute the periodogram. This is mainly included for the purposes of testing and most users will want to use the optimized 'fast' method (default) that is implemented in Cython. 'slow' is a brute-force method that is used to test the results of the 'fast' method. oversample : int, optional The number of bins per duration that should be used. This sets the time resolution of the phase fit with larger values of oversample yielding a finer grid and higher computational cost. results : BoxLeastSquaresResults The periodogram results as a BoxLeastSquaresResults object. ValueError If oversample is not an integer greater than 0 or if objective or method are not valid.
transit_mask(t, period, duration, transit_time)[source] [edit on github]

Compute which data points are in transit for a given parameter set

Parameters: t_model : array-like or Quantity Times where the mask should be evaluated. period : float or Quantity The period of the transits. duration : float or Quantity The duration of the transit. transit_time : float or Quantity The mid-transit time of a reference transit. transit_mask : array-like A boolean array where True indicates and in transit point and False indicates and out-of-transit point.