Hundreds of new periodic signals detected in the first year of TESS with the weirddetector

As the weirddetector searches for coherent, low duty-cycle periodic signals, the algorithm folds on many trial periods and searches for the period values which exhibit the smallest dispersion of fluxes. We note that phase dispersion minimization is not a novel concept–it has been used in other algorithms such as the Plavchan periodogram (Plavchan et al., 2008; Parks et al., 2014). In the case of TESS, which monitors each field for ${\simeq}27$ days, we opt for a period grid ranging from 0.25 days up to 15 days uniform in $\log P$ (24,567 total period values). Just as was found with Kepler though (Wheeler & Kipping, 2019), the algorithm is sensitive to periods outside the period grid if there is a harmonic of the true period within the grid (which is the case for some of our flagged signals later).

To quantify the relative success at phase dispersion minimization of a given trial period, the weirddetector examines the chi-squared ($\chi^{2}$, which reflects how scattered the values are) and kurtosis ($\kappa$, which reflects the weight of a distribution’s tail relative to its peak). The algorithm then calculates excess kurtosis, $\kappa^{\prime}=\kappa-\kappa_{\mathrm{Gaussian}}$ and chi-squared less than the median, $\Delta\chi^{2}=\mathrm{median}(\chi^{2})-\chi^{2}_{\mathrm{period}}$. These two metrics are multiplied to form a score strength, such that we require both perform well to yield a putative signal. To normalize the scores, we re-calculate $\kappa^{\prime}(P)\Delta\chi^{2}(P)$ along a grid of periods but for a scrambled version of the original data (which should have no periodicity), and evaluate a sliding standard deviation of the result, $\sigma(P)$. Using these quantities, the algorithm defines a final merit function, $\zeta$, for each period value, given by $\zeta(P)=\kappa^{\prime}(P)\Delta\chi^{2}(P)/\sigma(P)$.

The above value of the merit function gives us intuition about the ideal signal the weirddetector is searching for. To maximize the merit function of a period $P_{\mathrm{ideal}}$, the phase curve should be much more coherent at $P_{\mathrm{ideal}}$ than at other “random" values of $P$ (i.e. large decrease in $\chi^{2}$); the phase curve should hover around a constant value with only one (or a few) short excursion(s) from the baseline (i.e. high excess kurtosis $\kappa^{\prime}$); and these two conditions should otherwise have low variation at nearby $P$-values not equal to $P_{\mathrm{ideal}}$, maintaining small values for $\Delta\chi^{2}$ and $\kappa^{\prime}$ to demonstrate there is, in fact, a distinct signal with a specific period more likely than the rest (i.e. low $\sigma(P)$).

The weirddetector is highly versatile and generalizable, and was able to run on TESS data after appropriate changes were made from the implementation for Kepler. Following are the details that are sensitive to the particular telescope gathering the data.

Kepler observed a 115 square-degree patch of sky, at a cadence of 30 minutes for 17 quarters of up to 90 days each. TESS, on the other hand, generates light curves with a cadence of 2 minutes and will observe 26 different sectors for ${\simeq}$27 days each. TESS, while it spends only a month on the targets of each sector (barring targets that receive extra coverage as a result of overlap between sectors), will survey 85% of the sky, covering roughly 400 times as many unique stars as Kepler. As a result, there will be far more data to search through; however, by virtue of the shorter baseline, we are more limited in the range of periods for which we can detect signals.

In addition, having fewer data points means the weirddetector will be more sensitive to outlier events when calculating both $\chi^{2}$ and $\kappa$, such as stellar flares or instrument-caused deviations. As a result, we were more stringent in our outlier rejection when considering the phase-folding part of the algorithm to limit one-time events from inducing spurious signal detections or leading to false positives by unfavorably skewing $\kappa$. Additionally, while we do not expect $\kappa$ or $\sigma$ to change much, $\chi^{2}$ (and therefore $\Delta\chi^{2}$) is orders of magnitude lower in TESS because of the fewer points in the light curves. This means that $\zeta$ drops accordingly; however, because the absolute values of $\zeta$ do not matter as much as the relative values for determining the most promising period values, and we take the step of dividing by $\sigma(P)$, our analysis is little-affected.

We also had to adjust the bandwidth used for our median-filter detrending algorithm to better reflect the duration of the signals we were searching for while considering the shorter range of trial periods we were folding on. In addition, the presence of TESS’s regular data gaps posed an issue, as Kepler data suffered far fewer gaps per unit time. However, as these changes did not require fundamentally altering the algorithm, their treatment is discussed in the beginning of Section 3.

One major limitation of the weirddetector is its tendency to flag rational fractions or multiples (“aliases") of the true period of a signal. This is unavoidable when using a folding technique with such relaxed requirements for signal shape, and the issue is not specific to TESS light curves.

Here, we partially corrected for this error by attempting to automatically flag the correct period when an integer multiple was identified in the case of single-dip signals. It is important to note that a) our technique only accounts for flagged periods that are an integer multiple of the true period (which we chose to target because it is the most common case of aliasing); and b) we still need to manually examine the data to determine if the period detected is truly the correct one. Still, our technique helps us identify signals by suggesting the correct period more frequently.

For a single-dip event with true period $P_{\mathrm{true}}$ and flagged period $P_{\mathrm{flagged}}$, there will be $\;n\;=(P_{\mathrm{flagged}}/P_{\mathrm{true}})\;:\;n\;\in\;\mathbb{Z}^{+}$ dips in the phase curve folded with $P_{\mathrm{flagged}}$. By automatically detecting the number of dips $n$, we can therefore retrieve the correct period by multiplying. We identify a dip by the following: examine the points in folded time-order in 200-point sliding windows. If greater than one-third of the points in this window have a flux value that falls under $1.5\sigma$ below the median, note the window as the beginning of a dip. Then, wait for the flux values to return to normal before looking for another dip by flagging the end of a dip when greater than three-quarters of the window has a value within $\sigma$ of the median (see Figure 1). Note that the specific time picked out by the sliding window is not important, as this approach does not need to consider the duration or shape or the dip–only the number of significant dips $n$.

The specific numerical values were chosen heuristically to handle the aforementioned case of aliasing– by far the most common case within our data. Though a similar approach may work in principle for regular increases in flux, there are too many confounding factors (particularly flares) to have a reliable solution to that problem.

We ran the weirddetector on all light curve files in sectors 1 through 13 available at MAST (248,301 not including overlap of targets between sectors; 223,087 unique targets after considering overlap). We remove probable outliers from each PDC light curve by removing points more than $6*\mathrm{MAD}$ away from the median in an 11-point rolling window, where $\mathrm{MAD}$ is the median absolute deviation: $\mathrm{median}(|F_{i}-\mathrm{median({F})}|)$. (As one of the key assumptions of weirddetector is the presence of Gaussian noise, we can use $\mathrm{MAD}$ as an estimator of the standard deviation with $\sigma=1.4826*\mathrm{MAD}$, meaning our threshold is also roughly equal to $4\sigma$). We also removed all points with non-zero quality flags in the FITS files.

In each sector, we split each PDC time series into two segments (“semi-sectors”) before and after the ${\sim}1$ day data gap. Within each segment, we used linear interpolation to fill in missing data and detrended with a moving median filter, using a 1 day bandwidth, to remove long-term trends, following (Wheeler & Kipping, 2019). We decided on the 1-day window as it is both sufficiently shorter than the duration of observation (27 days per target) and longer than the duration of our signals of interest (most are of order $10^{-1}$ days). This also acts to suppress long duty-cycle signals associated with stellar rotation. After detrending, we removed the interpolated points and recombined the segments. The weirddetector was then run on the $24,567$ trial periods for 248,301 unique light curves.

In order to preserve only the most likely signal candidates and remove artifacts of the data analysis, some cuts were made to the resultant data. The first one we note is that upon examining a scatter plot of a light curve’s $\log\zeta_{\mathrm{peak}}$ versus $\log P$, one immediately sees piling up a certain frequencies. A similar effect was reported by Wheeler & Kipping (2019) on the Kepler data. These are immediately suspicious as spurious common modes, rather than astrophysical signals. We therefore proceeded to design a means to filter out these suspicious periods.

Since the distribution of periods guessed by the weirddetector was at first not log-uniform (in particular with large saturation towards the upper end), we cut the 1,000 most commonly appearing periods to achieve a more uniform distribution (Figures 2b-c). Though the precise number chosen as a cutoff was somewhat ad-hoc, it accomplished the desired goal of picking out a set of signals with a more uniform distribution of $\log(P)$.

Even after cuts, there are a few peculiarities in the distribution worth addressing. There is a low density of points at $\log(P)\approx 0.8$; we posit these are caused by phasing of periods which are almost exactly half ($\approx 6.5$) of the 13-day semi-sector baseline, causing dips to frequently fall within the one-day gap. We also note a concentration of high-$\zeta_{\mathrm{peak}}$ points close to $\log(P)=-0.3$; these could be caused by a common mode, as they are unlikely to be astrophysical considering their distance from the rest of the distribution. The high-$\zeta$ signals are denser at the higher end of $P$-values even after cuts, which is likely due to lower $\chi^{2}$ values caused by the lesser dispersions associated with folding on longer periods; this trend contributes significantly to the aliasing effect of rational multiples $P_{\mathrm{true}}$ as noted in Section 2.3

Since our goal is to find new signals, known variables from the Villanova eclipsing binary catalogue (Prsa, 2020), the ASAS-SN catalog of variable stars (Shappee, 2014; Jayasinghe et al., 2018, 2019), the International Variable Stars Index (VSX, Watson et al. (2006)), and the Tess Objects of Interest (TOIs) (NASA Exoplanet Archive list of current TOIs) were also removed. Altogether, these cuts comprised roughly 13% of the data, leaving 215,871 of the 248,301 targets intact.

We consider a signal to be “interesting” if its periodogram has a most-significant peak such that $\zeta\geq\zeta_{\mathrm{threshold}}$, again following Wheeler & Kipping (2019). We choose $\zeta_{\mathrm{threshold}}$ not independently at some fixed value, but set it to pick out the top decile of candidates, after cutting both common periods and already-discovered signals, for manual inspection. We look only at the most-significant peak (and rational fractions) as found by the previously described method, to generate phase curves with the highest likelihood of containing the correct period.

After filtering, we flag 377 previously-unidentified targets (dubbed Weird Objects of Interest, or WOIs) that appear to contain some periodic signal. The signals were selected manually based on whether their phase curves showed a clear signal (or multiple clear signals, in the case of aliases). $\zeta_{\mathrm{peak}}$ values of the WOIs fall within the range of $10^{3}$ to $10^{5}$, providing evidence that the selected periods are several orders of magnitude better than naive guessing. Additionally, we examined distributions of $\Delta\chi^{2}$ and $\kappa$ for the WOIs, which showed no dependence on each other, assuring us that the high $\zeta_{\mathrm{peak}}$ values are not artifacts. In Table LABEL:tab:WOIs, we present the $\zeta_{\mathrm{peak}}$ and periods, along with accompanying information, for the WOI signals reported in this work.

To ensure each signal was not an artifact of the median-filter detrending, we applied different detrending algorithms to the raw light curves and re-folded on the flagged period. We applied the implementation of Cosine Filtering with Autocorrelation Minimization (CoFiAM; Kipping et al. 2013) found in the MoonPy Python library, as well as phasma, a non-parametric phase-folding algorithm (Jansen & Kipping, 2018). Example phase curves presented in Figures 3 & 4 have been detrended with phasma. We also queried the TIC-8 catalog (Stassun et al., 2018) for these objects to report their basic stellar properties, which is provided in the figure panels where available. We provide moving median, CoFiAM and phasma detrended phase curves for the 337 signals, as well as a CSV form of Table LABEL:tab:WOIs at this URL.

To quantify the weirddetector’s performance, we analyzed its effectiveness at recovering signals from the Villanova eclipsing binary and TOI catalogues. At the time of writing, 62% of currently-documented TOIs (1267/2044) and 97% (1801/1855) of Villanova eclipsing binaries fell within the top decile of $\zeta_{\mathrm{peak}}$ values. The missing targets in each catalog are expected, as ideally a model-based approach would be best for finding transiting planets and eclipsing binaries, respectively, whereas the weirddetector is sensitive to all manner of periodic signals. We note also that all of the missing targets from the Villanova EB catalogue have periods falling outside our period grid ($>15$ days), explaining the algorithm’s failure to recover them. We maintained a high recall (the fraction of catalogued true signals recovered) percentage at the cost of precision (the fraction of flagged signals which we know or believe are real), as we wanted to minimize false negatives over false positives to find as many interesting signals as possible. Of course, one could vary the fraction of signals we consider from 10% to modify the precision and recall percentages (Figure 5).