# QoQ: a Q-transform based test for Gravitational Wave transient events

Siddharth Soni1, Ethan Marx1, Erik Katsavounidis1, Reed Essick2,3,4 G. S.
Cabourn Davies5, Patrick Brockill6, Michael W. Coughlin7, Shaon Ghosh8,
Patrick Godwin9 1LIGO Laboratory, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  2Canadian Institute for Theoretical Astrophysics,
Toronto, ON M5S 3H8, Canada  3Department of Physics, University of Toronto,
Toronto, ON M5S 1A7, Canada  4David A. Dunlap Department of Astronomy,
University of Toronto, ON M5S 3H4, Canada  5University of Portsmouth,
Portsmouth, PO1 3FX, United Kingdom  6University of Wisconcin–Milwaukee,
Milwaukee, WI 53201, USA  7School of Physics and Astronomy, University of
Minnesota, Minneapolis, Minnesota 55455, USA  8Montclair State University, 1
Normal Ave, Montclair, NJ 07043  9LIGO Laboratory, California Institute of
Technology, Pasadena, CA 91125, USA<EMAIL_ADDRESS>
###### Abstract

The observation of transient gravitational waves is hindered by the presence
of transient noise, colloquially referred to as glitches. These glitches can
often be misidentified as gravitational waves by searches for unmodeled
transients using the excess-power type of methods and sometimes even excite
template waveforms for compact binary coalescences while using matched filter
techniques. They thus create a significant background in the searches. This
background is more critical in getting identified promptly and efficiently
within the context of real-time searches for gravitational-wave transients.
Such searches are the ones that have enabled multi-messenger astrophysics with
the start of the Advanced LIGO and Advanced Virgo data taking in 2015 and they
will continue to enable the field for further discoveries. With this work we
propose and demonstrate the use of a signal-based test that quantifies the
fidelity of the time-frequency decomposition of the putative signal based on
first principles on how astrophysical transients are expected to be registered
in the detectors and empirically measuring the instrumental noise. It is based
on the Q-transform and a measure of the occupancy of the corresponding time-
frequency pixels over select time-frequency volumes; we call it “QoQ”. Our
method shows a 40% reduction in the number of retraction of public alerts that
were issued by the LIGO-Virgo-KAGRA collaborations during the third observing
run with negligible loss in sensitivity. Receiver Operator Characteristic
measurements suggest the method can be used in online and offline searches for
transients, reducing their background significantly.

  *

##  1 Introduction and motivation

The direct detection of gravitational waves by ground-based interferometric
detectors, Advanced LIGO (aLIGO) [1] and Advanced Virgo (AdV) [2] has
burgeoned the field of gravitational-wave physics [3]. Together with that, the
first detection of a binary neutron star coalescence by LIGO-Virgo in
association with a Gamma-Ray Burst [4] has also inaugurated a new era in
multi-messenger observations enabled by gravitational-wave observations. Three
observing runs of the international network of gravitational-wave detectors
have already taken place starting in 2015 and until 2020. They are referred to
as “O1”, “O2” and “O3”; they resulted in about 100 gravitational-wave
candidates following thorough analyses of data conducted over these years [5,
6, 7, 8, 9, 10, 11, 12, 13]. These gravitational-wave candidates are the
result of collisions between compact binary objects such as black holes and
neutron stars. Additionally, during these observing runs, real-time searches
were also conducted by the collaborations to disseminate transient event
detection candidates to the broader astronomy community in order to enable
their multi-messenger observations [14]. During the O3 run, 80 public alerts
for gravitational-wave candidates have been distributed via the Gamma-Ray
Burst Coordination Network (GCN). Out of these, 24 were retracted on
timescales of hours to days following their initial release, primarily
reflecting manual examination of the events by the LIGO-Virgo Collaboration
and deeming them to be of non-astrophysical origin. Quantifying and automating
the retraction process for this kind of event in upcoming observing runs of
the LIGO-Virgo-KAGRA network of detectors has been the primary motivation and
goal of this work.

The output of ground-based gravitational-wave detectors is strain amplitude,
also referred to as $h(t)$. This captures with the highest sensitivity the
differential change in length of the kilometer-long arms of the
interferometer. The $h(t)$ is impacted by multiple sources of noise which can
be broadly classified into two categories, short duration non-Gaussian
transients, also known as glitches, and noise that behaves in a Gaussian and
stationary fashion over a given time and frequency range [15, 16, 17, 18].
These transients can adversely impact gravitational-wave (GW) searches both at
the detection confidence level as well as in their source parameter
estimation, including localization. [19, 20, 21, 22, 23, 24, 25, 26]. Aside
from their impact to searches, identification of transient noise may lead to
studies on their coupling mechanism in the detector and subsequent hardware
changes that can help remove or at least reduce such noise sources [27].

The task of identifying transient gravitational-wave candidate events utilizes
the $h(t)$ time-series. It is performed by a variety of transient-finding
search pipelines developed by the LIGO-Virgo-KAGRA collaborations, as well as
the broader community. In this paper, we focus on low-latency searches [28,
29, 30, 31]. These searches are primarily responsible for producing public GCN
alerts in close to real time in order to allow for their multi-messenger
follow-up. The LIGO-Virgo-KAGRA (LVK) collaboration also performs searches
offline [32, 33, 30, 34] with refined versions of their online pipelines. Such
refinement allows for, among other things, improved sensitivity, better noise
rejection, although it often comes with higher computational complexity and
slower turn-around times. The broader GW community outside the LVK also
analyses the interferometric data once they become publicly available, with
redundant and complementary results obtained. In the typical searches for
compact binary coalescences, the waveform of such signals can be modelled
using post-Newtonian and numerical relativity methods, thus resulting in a
family of signals spanning a space parameterized by masses and spins of the
binaries; this is referred to as template banks. The various search pipelines
employ what is known as match filtering to look forGW signals in the data, by
matching the incoming strain with waveforms in such template banks [35, 33,
30]. However, the presence of noise transients in the data complicates this
process as they may mask and sometimes mimic a true GW signal [4]. To remedy
this, the search pipelines use a chi-squared test to differentiate the time-
frequency distribution of power in real GW signals and noise transients [36,
37]. Even with all these precautions, transient noise can trigger a pipeline
alert for a GW candidate event, which consequently has to be retracted. The
decision to retract an event is carried out via human intervention with
minimal quantitative analysis.

In this article, we present a tool to distinguish binary black hole (BBH)
signals from transient noise based on the energy distribution in the signal
across the time-frequency plane, as obtained via the Q-transform [38]. In
Section 2, we discuss events retracted during O3. In Section 3, we look at the
energy distribution of Gaussian and transient noise and describe in detail the
Q-occupancy (QoQ) test. In Section 4, we discuss its application on O3 low
latency astrophysical candidates and retracted events. In Section 5, we
discuss our QoQ test analysis on the Mock Data Challenge (MDC) Injection Data
Set and PyCBC background events. In Section 6, we extend the analysis of
Section 4 to events found in offline analysis in O3. Finally we conclude and
summarize in Sec 7.

##  2 O3 retracted events

During the third Observing run (O3), a total of 80 events were identified in
low latency, of which 23 were later retracted as their origin was found to be
environmental or instrumental rather than astrophysical. An additional event
was retracted as the alert was sent out due to an error
111https://gracedb.ligo.org/superevents/S190405ar/view/ [8, 7, 39, 40, 41].
Offline re-analysis of the O3 data refined the admission criteria of
astrophysical events, including the requirement of the probability of an event
being astrophysical to be greater than $0.5$ and having a False Alarm Rate
(FAR) of less than 2 per day; this led to the GWTC-3 catalog [7]. Of the non-
retracted 56 events identified in low latency, $44$ events were also
identified by the offline re-analysis and thus included in the GWTC-3 catalog
[7]. A long-term solution to reduce the number of retractions is to identify
the transient noise coupling in the detector and remedy it by making hardware
changes [27]. During O3b, after the reaction chain tracking was employed at
LIGO Livingston Observatory (LLO) and LIGO Hanford Observatory (LHO) to
diminish the slow scattering transient noise, the average fraction of
retracted O3b public alerts before and after dropping from $0.55$ to $0.21$
[8]. However, reduction of transient noise is only sometimes possible as the
noise originates in the complex instrumentation and environment of the
detector, and new categories of transient noise may always show up as the
sensitivity of the detector improves and new subsystems get added [42, 43, 44,
45]. So, a need for prompt, quantifiable, and automated ability to address
noise transients and their impact on public alerts from the LIGO-Virgo-KAGRA
network remains.

(a) Retracted event S191117j [46]

(b) Retracted event S190808ae [47]

Figure 1: Q-transform visualizations of example retracted events from O3.
_Left_ : L1 data has a loud glitch at the time of event S191117j [46]. _Right_
: The retracted event S190808ae do not contain any visible presence of
transient noise.

Retracted events during O3 can be broadly classified into two categories:
events with clear evidence of loud transient noise in one or more detectors
and low SNR events with almost no visible presence of transient noise. Fig 1
shows an example from each category. About half of the O3 retractions belong
to the first category. We aim to separate these false positives from
astrophysical transient candidate events. The QoQ test explained in Section 4
later on in the paper is a method for distinguishing transient noise from
astrophysical events using the Q-transform and the time-frequency pixel-energy
distribution measure that it provides. There has been a growing interest among
researchers in solving such problems [48, 49, 50].

##  3 The Q-transform in gravitational wave data analysis

A commonly used analysis tool for identifying and characterizing transients
both in the gravitational-wave strain channel and in the wealth of auxiliary
channels invoked in the interferometry is the Q-transform [51, 52]. It is a
modification of the short Fourier transform that covers the time-frequency
plane with pixels of constant $Q$ (quality factor) by invoking analysis
windows with durations that vary inversely proportional to the frequency. It
can be shown that it is equivalent to a matched filter search for minimum
uncertainty waveforms [38, 53]. A search for gravitational-wave bursts using
the Q-transform was first implemented in the early science runs of initial
LIGO [38, 53]. The implementation of Q-transform as an unknown-morphology
transient-finding method generally tiles the time-frequency plane using a
range of quality factors $Q$. We will refer to these tiles as pixels. The
squared magnitude of their (discrete) Q-transform coefficient is a measure of
the signal energy, and we will refer to it as pixel energy. For white noise,
such pixel energy at a given frequency is exponentially distributed.
Transients are generally identifiable as outliers in the pixel energy
distribution via invoking a threshold and some clustering algorithm in order
to achieve a desired false alarm rate on the assumption of Gaussian (or
otherwise directly measured from the data) noise.

This can be seen in Fig 2 where we histogram the energy of all the pixels
generated from a 4s long Q-transform of 16 “noisy” and 16 “quiet” times of the
LIGO detectors. The noisy times refer to ones identified by the Omicron [54]
method as having a glitch of signal-to-noise (SNR) ratio of at or above $6$
within such time-window, while “quiet” times have been randomly sampled from
the LIGO detectors data taking corresponding to livetime over which no glitch
has been identified by Omicron.

Figure 2: Q-transform pixel-energy distribution comparison of detector data
around noisy, O3 catalog, and quiet times in the LIGO detectors. “Noisy” times
are identified by Omicron as being subject to a glitch while “quiet” times are
selected via randomly sampling times of the LIGO detectors data taking over
which no glitch by Omicron was identified. As compared to quiet times, noise
transients or short duration glitches appear in the tail of the pixel-energy
distribution.

###  3.1 Pixel occupancy

Our proposed test for gravitational-wave transient events uses as a
discriminant the fraction (in percentage) of pixels within a few time-
frequency volumes that exceed pre-set values of pixel energy. We refer to this
quantity as “occupancy” or “density” of pixels, thus the name of our method as
“Q-occupancy”, or QoQ. Astrophysical transients of known morphology, including
compact binary coalescences (CBC), have analytically calculable occupancy
values (given the astrophysical source parameters). For those of unknown
morphology, it is a priori impossible to discriminate them from non-Gaussian
noise artifacts via a single QoQ measurement. In this case, in order to
utilize QoQ, some basic (or non) assumptions need to be made on the physics
and the detection of the (unknown morphology) astrophysical signal one is
after. Astrophysical signals are generally louder than Gaussian noise but
usually not as loud as the glitches responsible for the retracted events. As
shown in Fig 2, the pixel-energy distribution of the true events extends
beyond the Gaussian noise but not as far as the transient noise. And the kind
of noise artifacts in gravitational-wave interferometers we are primarily
gearing our method to identify has an extended structure in the time-frequency
plane and often exceeds pixel energies expected from astrophysical populations
we currently know [15]. This is learned by the noise data the instruments
record and its implications to searches is assessed via standard confusion
matrix and Receiver Operating Characteristic (ROC) analysis under some
assumption on the astrophysical signal in consideration.

In this work we utilized the GWpy [55] and Omicron [56, 54] implementation of
the Q-transform. We have used a range of quality factors $Q$ from $4$ to $64$
and frequencies from $10$ Hz to $1024$ Hz to logarithmically tile the time-
frequency plane. Analysis of the h(t) time series via the Q-transform results
to “triggers” corresponding to transients present in the data. Such triggers
are described with a few parameters such as the central event time, central
frequency, quality factor Q, and signal-to-noise ratio; we will use these for
the main feature description of the transients we will be analyzing.
Additionally, the full time-frequency decomposition with all pixel energy
values across resolutions is made available and is what we use for deriving
the QoQ quantities.

Several options exist on how to define the time-frequency volume over which to
measure QoQ. We first need to pick a threshold on the pixel-energy at or above
which we will consider the pixels as “loud” and thus counted toward the QoQ.
We generally need to pick values above the Gaussian noise level, although the
exact threshold has to be informed by the character of the noise and the
corresponding impact on the misclassification of signals from astrophysical
populations. The overall time-frequency volume over which $h(t)$ is analyzed
can vary significantly depending on the astrophysical search within which
noise rejection via this method is being implemented. In this analysis, we
will focus on transient gravitational waves from binary black hole (BBH)
mergers typically lasting from tens of milliseconds to a few seconds.
Extension of the method to other astrophysical populations is straightforward.

We use a duration of 4 seconds set by the scale of BBH events for the
Q-transform and restrict the analysis to the frequency band of
$10$–$1024~{}\mathrm{Hz}$. The frequency span of the search is motivated by
the overall frequency content of noise artifacts in aLIGO and AdV as well as
that of the BBH and CBC – in general – signals. This Q-transform is then sub-
divided into $n$ (disconnected, i.e., mutually exclusive) time-frequency
regions spanning the original time-frequency volume. For each of these $n$
regions, the fraction (in percentage) of pixels above a given pixel-energy
threshold is computed. We will refer to this fraction as the QoQ value. If the
QoQ exceeds a certain threshold in a given region, we use this to infer that
the corresponding region is contaminated with noise. This procedure is shown
as a flow chart in Fig 4.

The method allows for a different QoQ threshold to be used in each of the $n$
sub-divisions of the time-frequency volume one starts with. In our approach,
currently, any one of such $n$ sub-divisions exceeding the QoQ threshold flags
the event for further inspection. Clearly, more complicated logic can be
implemented, making certain assumptions for the hypothetical astrophysical
signal as well as for the noise behavior that may be targeted via such an
approach. This procedure is shown as a flow chart in Fig 4. There is an
additional conceptually different approach in the sub-division of the original
time-frequency volume, namely in $m$ overlapping rather than disconnected
regions. We invoke this approach as well since, as we will show below, it
offers a unique handle in identifying astrophysical events that may be
overlapping with noise artifacts.

Figure 3:  Q occupancy distribution of 1 second window surrounding the event
time of O3 online public alerts and O3 retracted events, for different values
of pixel-energy thresholds. The retracted events with clear presence of
transient noise have much higher Q occupancy values compared to the
astrophysical candidates. For the O3 online candidates, the max Q occupancy is
below 4, for the retractions the distribution extends to Q occupancy of much
higher than 5 shown here.

To decide the values of pixel-energy and QoQ thresholds to be used in the QoQ
test, we plot the cumulative distribution of QoQ values in the 1-second window
surrounding the event time of O3 low latency candidates and O3 retracted
events. Fig 3 shows this QoQ distribution of O3 low latency candidates for
different pixel-energy thresholds and of O3 retracted events for a pixel-
energy threshold of 60. As we can see from this figure, in the case of O3
online candidates, for a pixel-energy value of 60, the maximum QoQ value for
the central 1-sec window is 2.0, we use these values as our thresholds for the
pixel-energy and QoQ, respectively for the QoQ analysis on O3 online
candidates next.

Figure 4: Flowchart of the QoQ test. Given an event time, the tool makes a 4
second long Q-transform. We then split the pixel-energy of this Q-transform
into multiple time-frequency regions and apply a pixel-energy threshold. This
gives us, for each time-freq region, fraction of pixels above the pixel-energy
threshold i.e. its Q-occupancy. The next step is to apply a QoQ threshold on
each of these regions. If any of these time-freq regions are above its QoQ
threshold, the event is flagged for bad data quality, otherwise it passes the
QoQ test.

##  4 O3 online candidates analysis

We apply the QoQ test on the O3 online candidates. This set includes 56 non
retracted and 23 retracted events. We first perform the overlapping and non
overlapping analysis individually in 4.1 and 4.2 respectively and then combine
the two analyses to obtain the results shown in Tab 1.

Figure 5: _Left_ : A Q-transform divided into 9 time frequency blocks. These
blocks which are 1, 2 and 4 seconds long are centered at $t=0$ and overlap in
time. They cover three frequency bands from $\mathrm{10-100}$ Hz,
$\mathrm{100-512}$ Hz and $\mathrm{512-1024}$ Hz. _Right_ : This table shows
the fraction of pixels with energy above the pixel-energy threshold of 60 in
each of the 9 time-frequency regions.

###  4.1 Overlapping time windows and O3 online candidates analysis

The frequency axis in the Q-transform shown on the left in Fig 5 is divided
into three bands, ($\mathrm{10-100}$ Hz), ($\mathrm{100-512}$ Hz), and
($\mathrm{512-1024}$ Hz) and the time axis into three windows (-0.5 s, 0.5 s),
(-1.0 s, 1.0 s) and (-2.0 s, 2.0 s). The intersection of these time and
frequency windows gives us a total of 9 regions. For each region, we calculate
the fraction of pixels in the given region with a pixel-energy value above the
pixel-energy threshold. The fractions for each region are shown on the right
in Fig 5 for a pixel-energy threshold of 60. For example, the value $2.3$ in
the first row and first column indicates that for the 1-second long window in
the frequency band, $10$–$100~{}\mathrm{Hz}$, $2.3$ percent of the pixels have
pixel-energy above 60. The QoQ values for GW signals are smaller in
comparison, and a suitable threshold can be used to separate the transient
noise-generated false alerts from true GW candidate events.

We use the QoQ thresholds of 2, 1, and 1 for the three time windows $t_{1}=1$
sec, $t_{2}=2$ sec, and $t_{3}=4$ sec respectively. To reduce the chances of
flagging a true GW signal, the threshold on the central 1-sec window is higher
as the majority of BBH signals fall within this. With these QoQ thresholds and
a pixel-energy threshold of 60, 9 out of the 23 retractions and 1 of the 56 O3
online transient astrophysical candidates are flagged. A contains more details
for each of the flagged events by the QoQ analysis at this stage Fig 6 shows
two O3 online events that our method classifies as noise. More than half of
the retracted events do not contain visible glitch activity, and the QoQ test
does not flag these events. Out of 23 retractions, 14 were not flagged as
noise by the QoQ test as they are low SNR events at one or both detectors.

(a) S191212q [57]

(b) S191213g [8, 58]

Figure 6: _Left_ : The H1 data quality around the time of retracted event
S191212q [57] is impaired by the presence of Scattering glitch. The transient
noise is clearly visible in the Q-transform of H1. _Right_ : The O3 online
event S191213g was found with FAR of 1.1 per year in low latency and a public
alert was issued. The surrounding data quality is heavily contaminated with
scattered light glitches. The S3 offline analysis did not identify this event
as a significant candidate [8].

###  4.2 Non overlapping time windows and O3 online candidates analysis

Another way to create the time-frequency regions in the Q-transform is with
windows that do not overlap in time. Compared to the overlapping analysis,
there is no change in frequency delineation, but the time axis is now split
into five windows: (-2.0 s, -1.5 s), (-1.5 s, -0.5 s), (-0.5 s, 0.5 s), (0.5
s, 1.5 s) and (1.5 s, 2.0 s). In this case, we get a total of 15 time-
frequency regions. And again, we calculate the fraction of pixels with a
pixel-energy value above a certain pixel-energy threshold for each region. For
example, fig 7 shows the 15 time-frequency regions with no overlap in time and
the QoQ of pixels for each region using a pixel-energy threshold of 60.

Figure 7: _Left_ : A Q-transform divided into 15 time frequency regions. There
is no overlap in time among these blocks. The first time window is half a
second long and extends from -2.0 to -1.5 sec, the next three windows are each
1 sec long and the final region is again 0.5 secs long and extends from 1.5
sec to 2.0 sec. The division along the frequency axes is similar to the
overlapping case. _Right_ : The table shows the fraction of pixels with energy
above the pixel-energy threshold of 60, for each of these regions. There is
some transient activity in the first one and a half second as seen in the
Q-transform on left, this is reflected in the table on right.

In many cases, environmental or instrumental noise artifacts adversely impact
the data quality a few seconds before or after a GW chirp signal. These events
may not be retracted, but they require closer inspection for any noise
coupling with the auxiliary channels. The gravitational waveforms resulting
from the merger of binary black holes are of the order of a few tens of
milliseconds to a few seconds in the sensitive frequency band of ground-based
detectors. This means any signal energy outside this time window can be
attributed to a non-astrophysical phenomenon. The non-overlapping time windows
method can be conveniently used to catch such events. Any events for which the
QoQ in the central time window (-0.5, 0.5) is above 2 (same as the overlapping
case) or in any of the non-central windows is above 0 can be flagged for
stricter event validation. In this qualitative way, the non-overlapping time
windows method differs from the overlapping time windows.

The QoQ thresholds used in this case are $[0,0,2,0,0]$, i.e. events with QoQ
above $2\%$ for the central window ($t3$) or any non-zero value in any of the
non-central windows will be flagged as transient noise.

The O3 retracted events flagged by this method are the same as those flagged
by the overlapping time windows method shown in 4.1. However, unlike the
overlapping case, this method finds the presence of transient noise in three
GW candidates found in low latency in O3. Unsurprisingly, none of the central
time window (t3) values are flagged since this window is equivalent to the
central window (t1) used in the overlapping method. However, the non-central
windows flag a total of three O3 online astrophysical candidates. All of these
three events are real GW signals polluted by the presence of noise artifacts
in their vicinity at one or both detectors. A contains more details on these
events. During O3, these events were examined for any potential correlation
with the auxiliary channels and underwent a careful process of offline event
validation [59, 60]. To summarize, the non-overlapping time windows method
effectively catches both false pipeline alerts due to transient noise as well
as real astrophysical signals polluted by nearby transient noise.

Based on this analysis, we can use binary decision-making as shown in Fig 8 to
predict the nature of a GW candidate alert. Suppose an event is flagged by
both the overlapping and the non-overlapping time windows analyses. In that
case, such an event is likely a noise transient, and caution should be
exercised before disseminating the information to the public. If the event is
only flagged by the non-overlapping time windows method, then it is likely an
astrophysical signal along with some transient noise nearby. Conversely, if an
event is not flagged by the non-overlapping method, then it is an
astrophysical event. Table 1 shows the results of QoQ test on O3 events found
in low latency and O3 retractions at different stages of the flowchart in Fig
8.

Figure 8: To evaluate the data quality of a GW candidate event, we first feed the event to Non Overlap QoQ test, if the event is not flagged at this stage, it is considered a “Signal”. However if the event is flagged, it is then analyzed by the Overlap QoQ test. If the event is flagged at this stage, it is considered a “Noise”, otherwise it is classified as a “Signal with presence of transient noise”.  | O3 online retractions | O3 online nonretracted alerts
---|---|---
Signal | $13/23$ | $52/56$
Signal + Noise | $1/23$ | $3/56$
Noise | $9/23$ | $1/56$
Table 1: Of the 23 O3 retracted public alerts, the QoQ test classified 9 as
“Noise” and 1 as “Signal $+$ Noise”. Out of the 56 O3 low latency
astrophysical events, 3 were classified as “Signal $+$ Noise”, and 1 event was
flagged as “Noise”. A visual inspection supports this classification. A
contains more details on these flagged retracted and O3 low latency events.

##  5 MDC Data and PyCBC background analysis

###  5.1 MDC Data

To more robustly assess the QoQ test response to GW events, we analyze a set
of BBH injections. We use the injection set from the O3 Mock Data Challenge
(MDC). This MDC set contains signals injected in the O3 LIGO Livingston (LLO)
and LIGO Hanford (LHO) data between Jan 1, 2020, and Feb 14, 2020. The dataset
contains injections with source frame masses between 1 $M_{\odot}$ and 100
$M_{\odot}$, injected isotropically in the sky. The purpose of these
injections is to evaluate the detection efficiency of the search pipelines,
perform latency and alert generation checks as well as test source parameter
estimation. The MDC injections will thus allow us to perform end-to-end
testing of the low latency search pipelines in preparation for O4 [61, 62].
Since the bigger goal of these MDC injections is to check pipeline
consistency, the distribution of their source properties is not astrophysical.
Compared to the catalog events, there is an excess of low luminosity distance,
and high mass events, as shown in Fig 9. We will correct for this when we draw
conclusions for our QoQ method, as ultimately, we need to assess the
performance on the astrophysical population.

Figure 9: Joint plot distribution of the source chirp mass and source
luminosity distance of the Catalog events and the MDC injections. The MDC set
contains multiple injections at much lower luminosity distance and much higher
chirp mass compared to expected values from an astrophysical population.

To isolate the injection and background data for binary black hole
coalescences, we use the thresholds: $m_{1}>5$ $M_{\odot}$ and $m_{2}>5$
$M_{\odot}$ where $m_{1}$ and $m_{2}$ refers to primary and secondary mass
with $m_{1}$ $>$ $m_{2}$. To ensure the high significance of events, we also
use injection events with $\rho>5$ where $\rho$ represents the signal-to-noise
ratio. With these cuts, we get a total of 1354 MDC Injections. Since the MDC
waveforms were injected into the O3 data, some of them are polluted by the
presence of very high SNR glitches near them. To enhance the purity of this
dataset, we removed those injections that contained omicron transients with
SNR above 50 within a 3-second window of the injection time. This reduced the
size of the injection set marginally from 1354 to 1331 injections. Finally, we
apply a threshold on network SNR of $\rho_{net}>10$ to characterize found
injections [63]. This brings the total number of mock injections to 1053.
Next, as shown in Fig 9 the MDC injection set consists of high mass and low
luminosity distance events and does not match the expected astrophysical
distribution. To obtain a set that more closely resembles an astrophysical
population, we use Importance Sampling described next.

###  5.2 Importance Sampling

Say we would like to calculate the expectation value of a random variable
f(x), where x is a random vector. Assuming continuous distribution and p(x)
the probability density, we can write the expectation value as:

$\mu_{f}=\int f(x)p(x)dx$ (1)

Now, let’s say we have another distribution defined by the probability density
$q(x)$, and we would like to calculate $\mu_{f}$ by drawing samples from this
new distribution $q(x)$. Given that $q(x)\neq 0$ whenever $f(x)p(x)\neq 0$, we
can take the above equation and modify it as:

$\mu_{f}=\int\frac{f(x)p(x)}{q(x)}q(x)dx$ (2)

The ratio $\frac{p(x)}{q(x)}$ weights the samples from this new distribution.
Certain elements from q(x) have a higher impact or they are more “important”
and the weights emphasize these elements while sampling. Essentially, this
allows to calculate the expectation value of samples drawn from q(x) by
adjusting their importance with respect to the original distribution p(x)
[64]. So for samples $X_{1}....X{n}$ the importance sampling algorithm can be
written as:

$\hat{\mu}=\frac{1}{n}\sum_{i=1}^{n}\frac{f(X_{i})p(X_{i})}{q(X_{i})}$ (3)

In our case, the $p(x)$ and $q(x)$ refers to the Catalog and MDC Injections
respectively and $f(x)$ is the QoQ test function as shown in Fig 8. We use
luminosity distance and source chirp mass to generate the kernel density
estimate, and so we can rewrite the above equation as:

$\hat{\mu}=\frac{1}{n}\sum_{i=1}^{n}\frac{f_{QoQ}(X_{i},Y_{i})p_{cat}(X_{i},Y_{i})}{q_{mdc}(X_{i},Y_{i})}$
(4)

An MDC injection sample whose luminosity distance and source chirp mass
closely resembles the respective distributions of Catalog events is assigned a
higher weight as opposed to injections that do not. This way, we can sample
all the MDC injections and calculate the weight-adjusted expectation value.

To calculate the density estimate, we use a Gaussian kernel:

$K(x,y)=\frac{1}{2\pi
w_{x}w_{y}}\frac{1}{n}\sum_{i=1}^{n}exp\left[-\frac{(x-x_{i})^{2}}{2w_{x}^{2}}-\frac{(y-y_{i})^{2}}{2w_{y}^{2}}\right]$
(5)

where $w_{x}$ and $w_{y}$ are the bandwidth values.

###  5.3 Q-occupancy analysis of BBH data

Each event follows the steps outlined in the flowchart shown in Fig 8. For
each MDC injection, we first calculate its weight
$\frac{p_{cat}(X_{i},Y_{i})}{q_{mdc}(X_{i},Y_{i})}$. We then apply the QoQ
test on this event as outlined in Fig 8. Any event that is not flagged by the
nonoverlap method is classified as a “Signal”. Any event that fails to pass
the nonoverlap method then goes through the overlap method. If the event is
flagged at this stage, it is then classified as “Noise”, otherwise, it is
classified as “Signal $+$ Noise”. We use a total of three sets of QoQ
thresholds, Thresh A, Thresh B and Thresh C. The non overlap and overlap Q
occupancy threshold values for Thresh A, Thresh B and Thresh C are
($[0,0,2,0,0]$,$[2,1,1]$), ($[0,0,3,0,0]$,$[3,1,1]$) and
($[0,0,4,0,0]$,$[4,1,1]$) respectively. The fraction of events flagged at each
stage of Fig 8, for each set of QoQ thresholds is shown in Table 2.

Ideally, we would not misclassify a true gravitational-wave signal as noise.
There are multiple reasons why a small percentage of MDC injections are
classified as noise. These chirp signals were injected into the actual O3 data
from LLO and LHO. And so, a number of these signals are polluted by the
presence of noise transients near them. The Q-transforms of the injection
events that are flagged by the non-overlap method but not by the overlap
method reveal the presence of nearby transient noise along with the chirp
morphology.

| Injections | Background
---|---|---
| Thresh A | Thresh B | Thresh C | Thresh A | Thresh B | Thresh C
Signal |  $95^{100.0}_{88.9}$% |  $95.6^{100.0}_{89.5}$% |  $96.2^{100.0}_{90.2}$% |  $52.7^{55.1}_{50.2}$% |  $55.1^{57.5}_{52.6}$% |  $57.2^{59.6}_{54.7}$%
Signal + Noise |  $2.9^{3.7}_{2.0}$% |  $2.3^{3.2}_{1.4}$% |  $1.7^{2.6}_{0.8}$% |  $4.2^{5.2}_{3.2}$% |  $4.1^{5.1}_{3.1}$% |  $4.1^{5.1}_{3.1}$%
Noise |  $2.1^{2.7}_{1.5}$% |  $2.1^{2.7}_{1.5}$% |  $2.1^{2.7}_{1.5}$% |  $43.1^{45.5}_{40.6}$% |  $40.8^{43.2}_{38.4}$% |  $38.7^{41.0}_{36.3}$%
Table 2:  This table shows the results on MDC Injections and PyCBC background
data for pixel-energy threshold of 60 and QoQ threshold values of 2, 3 and 4
(Thresh A, Thresh B and Thresh C respectively). For QoQ threhsold of 2, the
test classifies 95$\%$ of the Injection data as “Signal”, 2.9 $\%$ as “Signal
$+$ Noise” and 2.1 $\%$ as just “Signal”. The lower and upper limits cover the
95 $\%$ confidence interval for these values. Figure 13 shows these results
for other values of pixel-energy thresholds.

For the Thresh A case, about $2.0\%$ of the injection events are classified as
noise. Since these events come from the injection set, this fraction
represents the false positive in our analysis. Fig 10 compares the Network SNR
of these flagged injections with the GWTC Catalog. As we can see from this
comparison, these flagged MDC injections are very loud, and their SNR
distribution does not resemble the distribution expected from a set of
astrophysical events [65]. The importance sampling weight of all these events
is below 1. Furthermore, Fig 11 shows the Q-transforms of the three categories
of injections flagged at different stages of the QoQ test (“Signal”, “Signal
$+$ Noise”, and “Noise”).

Figure 10: About $2\%$ of the MDC injections were classified as “Noise” by the
QoQ test. This figure shows the network SNR distribution comparison between
these injections flagged as “Noise” and Catalog events. The histogram is
normalized so that the area under each distribution sums to 1. These flagged
injections have a much high network SNR, not expected from astrophysical
signals.

Figure 11: Q-transforms of BBH injections analyzed at different stages in the
Fig 8._Top left_ : An MDC injection classified as “Signal” by the QoQ test.
This event is not flagged by the non overlapping analysis. About $95\%$ of the
MDC injections are classified as “Signal” by the QoQ test. _Top right_ and
_Bottom left_ : These two injections are flagged by non overlapping analysis
but not by overlapping, and are classified as “Signal $+$ Noise”. For both of
these injections, we can observe the presence of transients in the L1 data.
About $3\%$ of the MDC injections are classified in this category by the QoQ
test. _Bottom right_ : This injection is classified as noise due to its very
high energy content. About $2\%$ of the MDC injections were classified as
“Noise” by the QoQ test.

###  5.4 PyCBC Background

In order to assess the QoQ’s test ability to reject false pipeline alerts, we
analyze a set of background events generated by PyCBC [32, 35]. We use the
PyCBC-broad analysis from GWTC PyCBC is a matched filtering pipeline that
identifies triggers by finding peaks in the matched filter SNR between the
data and a template bank of waveforms. Events are then constructed by matching
coincident sets of these triggers in the detector network. Each event is
compared to a background to assess the false alarm rate.

Background events are constructed through time shifts, where the triggers from
one detector are shifted in time by more than the gravitational-wave travel
time compared to other detectors, breaking the astrophysical coincidence. The
background events we use come from the PyCBC-broad analysis of the first half
of the third LVK observing run, O3a. These analyses were used to produce
results in GWTC-2.1 [7]. We use the PyCBC ‘exclusive’ background only, meaning
any triggers which appear in the un-shifted coincident sets of triggers are
not used, ensuring the background is not contaminated by potential
astrophysical signals. These background events mimic possible false
coincidences that could appear during an online observing run, providing a
useful data set to evaluate the QoQ test.

To obtain the background data for binary black hole coalescences, we use the
thresholds: $m_{1}>5$ $M_{\odot}$ and $m_{2}>5$ $M_{\odot}$ where $m_{1}$ and
$m_{2}$ refers to primary and secondary mass with $m_{1}$ $>$ $m_{2}$. And
similar to MDC injections, to ensure the high significance of events, we use
background events with $ifar>50$ years where ifar is the inverse false alarm
rate. With these cuts, we get a total of 1591 PyCBC background coincidences.
The importance sampling is not required for the background data since they are
not astrophysical events and are not expected to match the astrophysical
distribution of source properties.

####  5.4.1 Analysis of BBH Background flagged as noise

Depending on the Q-occupancy threshold, Table 2 shows that the test flags
between $39\%$ and $43\%$ of the background events as noise, while between
$53\%$ and $57\%$ are classified as “Signals”. As mentioned earlier in Section
2, O3 retractions events fall into two main categories: events with a clear
presence of transient noise and low SNR events with no visible transient noise
present. The majority of the background coincidences not flagged by the
Q-occupancy test belong to this latter category of events. Fig 12 shows pixel-
energy comparison of flagged and unflagged background coincidence. The Q scans
of these unflagged background coincidences do not contain considerable noise
above the pixel-energy threshold of 60 and thus were not picked up by the
Q-occupancy test. However, since the background events have $ifar>50$ years,
these are considered highly significant events by the PyCBC search pipeline.
With a Q-occupancy threshold of 2 and pixel-energy threshold of 60, the QoQ
test flagged $43\%$ of BBH background as “Noise”. This is very consistent with
the amount of O3 retractions classified as “Noise” by the QoQ test, as shown
in Table 1. This suggests that we can expect QoQ test to flag similar
percentage of false positives in O4.

Figure 12: A comparison of distribution of mean pixel energies between
unflagged background events and background events flagged as noise by the QoQ
analysis.

Figure 13: Here we show the variation in MDC Injections classified as “CBC” and PyCBC background classified as “Noise” as we change the pixel-energy threshold and Q-occupancy threshold values. The 95 $\%$ confidence interval error bars are shown for Q-occupancy threshold of 2. _Left_ : As we increase the pixel-energy threshold, a larger fraction of injections are classified as “CBC”. However, after pixel-energy threshold of 60, this change is rather small. _Right_ : As expected, increasing pixel-energy threshold results in smaller percentage of background events classified as “Noise”. | O3 online retractions | O3 online non retracted | O3 Catalog events
---|---|---|---
Signal | $13/23$ | $52/56$ | $67/74$
Signal + Noise | $1/23$ | $3/56$ | $7/74$
Noise | $9/23$ | $1/56$ | $0/74$
Table 3:  In this Table, we show the results of QoQ classification on O3
events that include offline and online events and on O3 Catalog events, along
with the retracted events. A contains more details on the O3 events classified
as Signal $+$ Noise, and as Noise by the QoQ test.

##  6 O3 offline analysis

In Table 1 we compare the results of the QoQ test on O3 retracted events with
non-retracted events found in low latency. Now we extend this comparison to
include the events in O3 Catalog. As opposed to online analyses, offline
analyses use cleaner and better-calibrated data over which the search is
performed. This leads to offline events not found in online analyses since
more computationally extensive methods can be used on an improved dataset.
Also, some of the events found in low latency may not pass the thresholds of
FAR and/or p-astro and thus do not appear in the Catalog list. The catalog
events list contains a total of 74 candidates, 44 of which are found in low
latency and 30 in offline analyses [7, 8]. Tab 3 shows the results on this
full set of events from O3. Out of the 74 events in O3 catalog list, the QoQ
test classifies 7 of these events as Signal with presence of transient noise,
while none of the events are classified as just noise. In the Q scans plotted
in 15 we can see the presence of transient noise near the candidate time for
each of these 7 events. Astrophysical candidates polluted with nearby
transient noise require a more careful event validation to rule out their
instrumental origin [15].

##  7 Discussion

The presence of short-duration transient noise adversely impacts the quality
of gravitational-wave strain data. The origin of these transients is in the
complicated web of detector hardware, and it is often challenging to find
their exact source. Apart from masking the GW signal morphology, the non-
Gaussian transient noise may sometimes mimic the astrophysical signals and end
up as search pipeline detection candidates. It is thus essential to build the
ability to identify and characterize potential noise artifacts in real time so
that false public alerts can be reduced. Our QoQ method approaches the
classification problem from first principles on how excess power in a time-
frequency decomposition is expected to appear for astrophysical signals vs
noise and derives straightforward criteria to separate events into three
(mutually exclusive) categories: signal, noise, and signal plus noise. Few
analysis parameters allow the method to tune to specific astrophysical sources
and/or noise artifacts. The application of the method we described here is
within the context of binary black hole candidate events as they are
identified by matched filtering (or otherwise) detection algorithms. Our
analysis starts with the Q-transformation of the strain data and thresholds on
the signal energy of the time-frequency pixels. Then it uses the pixel
occupancy over several time-frequency windows in order to perform the
classification problem. We tested QoQ extensively with both simulated GW
waveforms the LIGO-Virgo-KAGRA Collaborations introduced as part of testing
the low-latency alert infrastructure [61] as well as background data from the
search for binary black holes with PyCBC in O3 [32]. This allowed us to
explore various options for pixel energy thresholds and actual pixel occupancy
ones, leading to ROC curves.

The application of our method in the above dataset was mostly geared in
minimizing the false dismissal of GW-like events while maximizing the
rejection of noise. While there is room for further optimization, our present
choice of analysis thresholds leads to over 40% reduction of the background
(noise) while misidentifying as noise about 2% of GW-like signals. These
numbers are consistent with what we obtain when applying QoQ on actual public
alerts that were issued by the LIGO-Virgo-KAGRA collaboration in real-time
during O3 as well as with events in the O3 catalog as published by the LIGO-
Virgo-KAGRA collaborations [8]. The ability to reject additional noise events
with QoQ (and beyond the efficacy we currently have) faces challenges as about
half of the noise and/or background events show little signal energy (Signal-
to-Noise-Ratios below 10) in their time-frequency decomposition.

It is an efficient algorithm to run in real-time and integrate into workflows
for astronomical alert generation, mostly in the form of preventing false
alerts from dissemination. Additionally, its ability to identify the
simultaneous presence of signal and noise within an analysis window can
facilitate the workflow of further denoising/deglitching as well as follow-up
GW parameter estimation analyses. The occurrence of transient noise (glitches)
may introduce biases in the parameters estimated for a given GW candidate [21,
24, 25, 66, 8]. The ability to quantify and identify automatically such types
of events will further assist their analysis with parameter estimation
techniques. The enhanced sensitivity of the GW detectors in O4 will result in
an increased rate of both the GW candidates and likely new sources of noise
transients. Tools such as QoQ will be helpful in improving the purity of
resulting real-time alerts.

The authors would like to thank the low latency and detector characterization
working groups of the LIGO Scientific Collaboration for feedback while
carrying out this work. We would especially like to thank Geoffrey Mo and Deep
Chatterjee for their input. SS, EM and EK acknowledge support from the United
States National Science Foundation (NSF) under award PHY-1764464 to the LIGO
Laboratory and OAC-2117997 to the A3D3 Institute. GCD acknowledges the Science
and Technology Funding Council (STFC) for funding through grant ST/T000333/1.
M. W. Coughlin acknowledges NSF support under awards PHY-2010970 and
OAC-2117997. S. Ghosh acknowledges NSF support under award PHY-2110576. R.
Essick is supported by the Natural Sciences & Engineering Research Council of
Canada (NSERC). This material is based upon work supported by NSF’s LIGO
Laboratory which is a major facility fully funded by the National Science
Foundation. This research has made use of data and software obtained from
GWOSC (gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific
Collaboration, the Virgo Collaboration, and KAGRA Collaboration. The authors
gratefully acknowledge the support of the US NSF for the construction and
operation of the LIGO Laboratory and Advanced LIGO as well as STFC of the
United Kingdom, and the Max-Planck-Society for support of the construction of
Advanced LIGO. Additional support for Advanced LIGO was provided by the
Australian Research Council. Advanced LIGO was built under award PHY-0823459.
LIGO was constructed by the California Institute of Technology and
Massachusetts Institute of Technology with funding from the National Science
Foundation and operates under Cooperative Agreement PHY-1764464. The authors
are grateful for computational resources provided by the LIGO Laboratory and
supported by National Science Foundation Grants PHY-0757058 and PHY-0823459.

##  Appendix A QoQ analysis of O3 Catalog and retracted events

IFO | Freq band [Hz] | t1 | t2 | t3 | t4 | t5 | QoQ Flag | Superevent
---|---|---|---|---|---|---|---|---
L1 | $10-100$ | $42.0$ | $78.7$ | $19.8$ | $10.8$ | $0.0$ | Noise | S200308e [67]
L1 | $10-100$ | $0.0$ | $0.0$ | $17.2$ | $5.1$ | $0.0$ | Noise | S200108v [68]
L1 | $10-100$ | $0.0$ | $0.0$ | $0.0$ | $1.4$ | $5.1$ | Noise | S190405ar [69]
H1 | $10-100$ | $0.0$ | $0.6$ | $2.3$ | $0.0$ | $0.0$ | Noise | S191212q [57]
H1 | $10-100$ | 24.0 | 34.8 | 24.6 | 0.0 | 0.0 | Noise | S200106av [70]
L1 | $10-100$ | 7.1 | 6.0 | 1.7 | 0.0 | 0.0 | Noise | S200106au [70]
L1 | $10-100$ | 0.0 | 23.3 | 22.0 | 0.0 | 0.0 | Noise | S191117j [46]
L1 | $10-100$ | 0.0 | 24.6 | 0.2 | 0.0 | 0.0 | Noise | S191120at [71]
L1 | $10-100$ | 0.0 | 0.0 | 2.5 | 0.0 | 0.0 | Noise | S200116ah [72]
L1 | $10-100$ | 0.0 | 0.3 | 1.1 | 0.0 | 0.0 | Signal $+$ Noise | S191220af [73]
Table 4: QoQ Non Overlapping method analysis of the retracted events. The
table shows the pixel occupancy values of the flagged retracted events. The
first nine events in this table were also flagged by the Overlapping method,
as shown in Fig 8 and thus are classified as “Noise” by the QoQ test. The last
event S191220af was only flagged by the Non-Overlapping method and is thus
classifed as “Signal + Noise”.

(a) S190405ar

(b) S191117j

(c) S191120at

(d) S191212q

(e) S19200106av

(f) S200106au

(g) S200108v

(h) S200116ah

(i) S200308ae

(j) S191220af

Figure 14: O3 retracted events flagged as “Noise” by QoQ test. The last event S191220af is classfied as “Signal + Noise”. IFO | Freq band [Hz] | t1 | t2 | t3 | t4 | t5 | QoQ Flag | Superevent
---|---|---|---|---|---|---|---|---
L1 | $10-100$ | $2.0$ | $0.7$ | $0.0$ | $0.0$ | $0.0$ | Signal $+$ Noise |  $S190513bm$ [8, 74]
H1 | $10-100$ | $0.1$ | $0.1$ |  $0.$0 | $0.0$ | $0.0$ | Signal $+$ Noise |  $S191109d$ [8, 75]
L1 | $10-100$ | $0.0$ | $0.0$ | $0.0$ | $0.3$ | $0.0$ | Signal $+$ Noise |  $S200224ca$ [8, 76]
H1 | $10-100$ | $0.0$ | $0.0$ | $0.0$ | $0.0$ | $0.4$ | Signal $+$ Noise |  S191127p [8]
L1 | $10-100$ | $0.0$ | $0.0$ | $0.0$ | $0.0$ | $0.5$ | Signal $+$ Noise |  S200216br [8]
H1 | $10-100$ | $0.0$ | $1.7$ | $0.0$ | $0.0$ | $0.0$ | Signal $+$ Noise |  S190803e [8]
L1 | $10-100$ | $1.9$ | $0.4$ | $0.0$ | $0.0$ | $0.0$ | Signal $+$ Noise |  S190514n[8]
L1 | $10-100$ | $8.6$ | $3.7$ | $0.0$ | $1.3$ | $2.5$ | Noise |  $S191213g$[8, 58]
Table 5: QoQ Non Overlapping method analysis of the O3 Catalog events. The
table shows the pixel occupancy values of the 7 out of the 74 Catalog events
classified as “Signal + Noise” and 1 out of the 56 O3 online events classified
as “Noise” by the QoQ test. The candidate names in bold were not reported in
low latency and were found in offline analyses. None of these 7 events were
flagged by the Overlapping method, as shown in Fig 8 and thus are classified
as signal with transient noise. 4 of these events (S190513bm, S190514n,
S191109d and S191127p) required noise mitigation [8, 59]. The last event in
the table, S191213g was classifed as “Noise” as the L1 data is contaminated by
scattering noise just before and after the event time.

(a) S190513bm

(b) S190514n

(c) S190803e

(d) S191109d

(e) S191127p

(f) S200216br

(g) S200224ca

(h) S191213g

Figure 15: O3 Catalog events flagged as “Signal + Noise” and “Noise” by QoQ
test. The first 7 events shown here are classified as “Signal + Noise” while
the event S191213g is classifed as “Noise” due the presence of high SNR
scattering arches in the data.

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