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- Box Least Squares (BLS) Periodogram

# Box Least Squares (BLS) Periodogram

The “box least squares” (BLS) periodogram [1] is a statistical tool used for detecting transiting exoplanets and eclipsing binaries in time series photometric data. The main interface to this implementation is the `BoxLeastSquares` class.

## Mathematical Background

The BLS method finds transit candidates by modeling a transit as a periodic upside down top hat with four parameters: period, duration, depth, and a reference time. In this implementation, the reference time is chosen to be the mid-transit time of the first transit in the observational baseline. These parameters are shown in the following sketch:

(png, svg, pdf)

Assuming that the uncertainties on the measured flux are known, independent, and Gaussian, the maximum likelihood in-transit flux can be computed as

\[y_\mathrm{in} = \frac{\sum_\mathrm{in} y_n/{\sigma_n}^2}{\sum_\mathrm{in} 1/{\sigma_n}^2}\]

where \(y_n\) are the brightness measurements, \(\sigma_n\) are the associated uncertainties, and both sums are computed over the in-transit data points.

Similarly, the maximum likelihood out-of-transit flux is

\[y_\mathrm{out} = \frac{\sum_\mathrm{out} y_n/{\sigma_n}^2}{\sum_\mathrm{out} 1/{\sigma_n}^2}\]

where these sums are over the out-of-transit observations. Using these results, the log likelihood of a transit model (maximized over depth) at a given period \(P\), duration \(\tau\), and reference time \(t_0\) is

\[\log \mathcal{L}(P,\,\tau,\,t_0) = -\frac{1}{2}\,\sum_\mathrm{in}\frac{(y_n-y_\mathrm{in})^2}{{\sigma_n}^2} -\frac{1}{2}\,\sum_\mathrm{out}\frac{(y_n-y_\mathrm{out})^2}{{\sigma_n}^2} + \mathrm{constant}\]

This equation might be familiar because it is proportional to the “chi squared” \(\chi^2\) for this model and this is a direct consequence of our assumption of Gaussian uncertainties.

This \(\chi^2\) is called the “signal residue” by [1], so maximizing the log likelihood over duration and reference time is equivalent to computing the box least squares spectrum from [1].

In practice, this is achieved by finding the maximum likelihood model over a grid in duration and reference time as specified by the `durations` and `oversample` parameters for the `power` method.

Behind the scenes, this implementation minimizes the number of required calculations by pre-binning the observations onto a fine grid following [1] and [2].

## Basic Usage

The transit periodogram takes as input time series observations where the timestamps `t` and the observations `y` (usually brightness) are stored as `numpy` arrays or `Quantity` objects. If known, error bars `dy` can also optionally be provided.

### Example

To evaluate the periodogram for a simulated data set:

    >>> import numpy as np
    >>> import astropy.units as u
    >>> from astropy.timeseries import BoxLeastSquares
    >>> rng = np.random.default_rng(42)
    >>> t = rng.uniform(0, 20, 2000)
    >>> y = np.ones_like(t) - 0.1*((t%3)<0.2) + 0.01*rng.standard_normal(len(t))
    >>> model = BoxLeastSquares(t * u.day, y, dy=0.01)
    >>> periodogram = model.autopower(0.2)

The output of the `astropy.timeseries.BoxLeastSquares.autopower` method is a `BoxLeastSquaresResults` object with several useful attributes, the most useful of which are generally the `period` and `power` attributes.

This result can be plotted using matplotlib:

    >>> import matplotlib.pyplot as plt
    >>> plt.plot(periodogram.period, periodogram.power)

(png, svg, pdf)

In this figure, you can see the peak at the correct period of three days.

## Objectives

By default, the `power` method computes the log likelihood of the model fit and maximizes over reference time and duration. It is also possible to use the signal-to-noise ratio with which the transit depth is measured as an objective function.

### Example

To compute the log likelihood of the model fit, call `power` or `autopower` with `objective='snr'` as follows:

    >>> model = BoxLeastSquares(t * u.day, y, dy=0.01)
    >>> periodogram = model.autopower(0.2, objective="snr")

(png, svg, pdf)

This objective will generally produce a periodogram that is qualitatively similar to the log likelihood spectrum, but it has been used to improve the reliability of transit search in the presence of correlated noise.

## Period Grid

The transit periodogram is always computed on a grid of periods and the results can be sensitive to the sampling. As discussed in [1], the performance of the transit periodogram method is more sensitive to the period grid than the `LombScargle` periodogram.

This implementation of the transit periodogram includes a conservative heuristic for estimating the required period grid that is used by the `autoperiod` and `autopower` methods and the details of this method are given in the API documentation for `autoperiod`.

### Example

It is possible to provide a specific period grid as follows:

    >>> model = BoxLeastSquares(t * u.day, y, dy=0.01)
    >>> periods = np.linspace(2.5, 3.5, 1000) * u.day
    >>> periodogram = model.power(periods, 0.2)

(png, svg, pdf)

However, if the period grid is too coarse, the correct period might be missed.

    >>> model = BoxLeastSquares(t * u.day, y, dy=0.01)
    >>> periods = np.linspace(0.5, 10.5, 15) * u.day
    >>> periodogram = model.power(periods, 0.2)

(png, svg, pdf)

## Peak Statistics

To help in the transit vetting process and to debug problems with candidate peaks, the `compute_stats` method can be used to calculate several statistics of a candidate transit.

Many of these statistics are based on the VARTOOLS package described in [2]. This will often be used as follows to compute stats for the maximum point in the periodogram:

    >>> model = BoxLeastSquares(t * u.day, y, dy=0.01)
    >>> periodogram = model.autopower(0.2)
    >>> max_power = np.argmax(periodogram.power)
    >>> stats = model.compute_stats(periodogram.period[max_power],
    ...                             periodogram.duration[max_power],
    ...                             periodogram.transit_time[max_power])

This calculates a dictionary with statistics about this candidate. Each entry in this dictionary is described in the documentation for `compute_stats`.

## Literature References

[1] (1,2,3,4,5)

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)