Transit Detection steve

Our single transit detection method, which we namedsteve, can be summed up in three

simple steps - find all the dips, characterise them, then sift out those that aren’t exoplanet transits. We did this by first iterating a box-function over all the lightcurves, by model- fitting, and finally by random forest classification.

Box method

Firstly, the depth of a "box" with given transit centre and duration was compared to out- of-transit data. In- and egress were masked to remove effects from an off-centre transit. Where the depth was greater than some threshold (set in this search as 4σ), the eclipse time was stored in an array, along with depth, duration, signal-to-noise, the fraction of the ’box’ covered by the photometry and the ratio of errors in the dip compared to the out-of- transit. This was iterated along the lightcurve in steps 0.13 times the transit duration, and the duration was scaled up from 1.95 hours to 1.2 days, increasing by 25% each time. The

red noise was calculated by binning the lightcurve to each transit duration timescale and calculating the RMS, enabling the depth to be compared to the correlated noise in the data. Once box detections had been run on the entire lightcurve, the positions of concur- rent detections within 2 hours of each others (but at different durations) were combined into a single detection, taking that with the largest SNR.

Model Fit

Two models were then run on each detection higher than some threshold of SNRr. After

performing machine learning on the box-search detections (without fitting), we determined SNRr = 1.5 to be the best threshold, as below this level false positives dominated real injections by a ratio>1000 : 1.

The first model was a transit with depth, centre, duration and transit shape found by chi-square minimisation. In real transits, the lightcurve shape is a function of both limb darkening (which varies with stellar type) and impact parameter. To encapsulate this as a single one dimensional "shape" parameter, an array of transit models were created using the

transitpython package1for a range of stellar types (limb darkening found using the V band and the models of Claret and Bloemen (2011), see Figure 4.1). The transits were then normalised to have a depth (found as the median of the central 40% of transit compared to the out-of-transit normalised flux) and duration of unity. After stacking according to the flux at at arbitrary phase of 15% of Td (which aligns the transits mostly by impact parameter), the models were interpolated in 2D both for each of 400 phases across transit and against normalised phase. This resulted in a continuous 2D surface of normalised flux vs. normalised phase and transit shape. Hence transit shape gives a slice in flux and phase space which can be compared directly with the transit once it has been shifted for depth, duration and transit centre. This is performed with:

modeli = M ti−tcen 2TD ,t_shape ! −1 ! ∆F ! +1.0 (4.1)

WhereM is the 2D model transit array, ti is the individual time, tcen the transit centre,

TD the duration, tshape the shape parameter and ∆F the depth. Testing has shown that

the resulting 4-parameter transit model produces adequate fits for any and all transits in photometry with reasonable levels of noise (SNR<10).

In the case of the long-duration eclipse search, models for transiting planets do not adequately represent the likely eclipse shapes caused by large dusty bodies. To correct for

this, we adapted the model to include Gaussian dips, adding a Gaussian ofµ = 0.0 and

σ = 0.4 in a smooth transition (using interpolation) from the most V-shaped high-impact

Figure 4.1: 1D shape parameter used in injections and fitting for steve Long-duration planet search. Light green models represent the added Gaussian-dip models for the long- duration eclipse search.

parameter fit used in the transit model (models in green in Figure 4.1). Modifications of depth and duration therefore produce any possible symmetrical Gaussian dip.

To model non-transiting phenomena, we used a two-line model with a "step", sim- ilar to the non-transiting models used by Foreman-Mackey et al. (2016) to distinguish be- tween real and unreal signals. This was physically motivated by looking at false-positive signals, which often show only a trend at the end-of-night (and therefore best fit by a straight line), a V-shape from variability or grazing eclipsing binaries, a flux drop with no in- or egress, etc. Fitting this model involved minimising five parameters - the gradient and inter- cept of each line and the position of the "step".

In both the transit and the step function case, we performed minimisation with

scipy’s "optimize" tool, starting at with three different central starting positions across the initially detected ’duration’, to allow any off-centre step or transit to find the best fit. The lowest reduced chi-square of each minimised model was then used to determine the best fit. The ratio of these minimum reduced chi-squares (which approximates to the "f- distribution" between the models) was then calculated and used as a parameter in the Ran- dom Forest Classifier (RFC).

Machine Learning

Machine learning is an excellent method of recovering known signals from a large dataset with minimal human involvement, (see section 2.11). We applied this method to detect single transit events from the large number of candidate signals detected by thestevetransit search (Section 4.2.4).

The Random Forest Classifier package of scikit learn (Pedregosa et al., 2011) was used, with 500 trees and 16 jobs. We initialised the detection array with the known clas- sifications by comparing the position of the injections with the detections ofsteve’s box search and model fit sections. We determined an injection recovery to be any detection within±0.3 transit durations of the transit centre, and with more than 3 in-transit lightcurve points. The detection array was then split into two with one set used to initially train the random forest, followed by a test set to determine how well the RFC would perform on unseen data. To utilise the most of the low numbers of detected injections, we performed training and testing on both halfs and predicted the opposing half of the data in an isolated manner.

17 different statistics were used, generated from the transit-fitting procedure. These were:

• Recovered eclipse depth (both from the initial box-model, "dep", and from the sec- ondary transit-fit, "fit_tr_dep")

• The SNR with respect to both the white noise of the out-of-transit data ("snr_w") and the red noise of the whole lightcurve at that duration ("snr_r").

• "dur" - The recovered eclipse duration from the initial box-model,.

• "Nconc_all" - The number of concurrent transits detected globally within 2 hours of the detectedt_cen

• "SIGMA_XS" - the weights from the sysrem run, which uses the variance of stellar photometry as a function of image, star and field to use the most stable photometry and down-weight the most variable (e.g. clouds and stellar variability).

• "MAG_MEAN" - Stellar magnitude

• "tran_cover" - The proportion of the initial box model dip that was covered by data.

• "err_ratio" - The ratio of the median in-transit errorbar with the median out-of-transit errors, .

• "n_conc" - The number of detections at that transit centre (but different transit dura- tions) from the box model

• "S_max" - The maximum detection of all transits SNR multiplied by the square root of the number of detections

• "fit_f_ratio" - The ratio of box-model fit to the systematic step-model.

• "fit_tr_conv" - whether the transit model converged during transit fitting.

• "fit_tr_dep_offset", "fit_tr_dur_offset" & "fit_tr_tcen_offset" are the offsets in depth, duration and centre between the initial box model and the latter transit model.

Once trained on the data, we predicted the classifications of the test set in order to determine the recovery rate. The RFC is such that the probability of class membership can be determined from the number of different tree iterations that produced each classification. Hence, it is possible to tweak this probability to include more detections, but also more false-positives. For example, setting Ptransit=0.5 as a threshold may substantially increase the number of detections compared to Ptransit=1.0, while only marginally increasing the ratio of real signals to false positives.

The long-duration eclipse search was performed almost identically to that described above for the Single Transits. Both injections and the fitting procedure were modified to also including non-transit shaped gaussian dips (see Figure 4.1).