% !TEX root = ./busty_transcription.tex
\section{Beyond Means in Gene Expression}
\label{sec:beyond_means}

In this section, our objective is to explore the same models considered in the
previous section, but now with reference to the question of how well they
describe the distribution of gene expression levels, with special reference to
the variance in these distributions. To that end, we repeat the same pattern as
in the previous section by examining the models one by one. In particular we
will focus on the Fano factor, defined as the variance/mean. This metric serves
as a powerful discriminatory tool to compare our different models to the null
model that the steady-state mRNA distribution must be Poisson, resulting a Fano
factor of one.

\subsection{Kinetic models for unregulated promoter noise}

Before we can tackle simple repression, we need an adequate phenomenological
model of constitutive expression. The literature abounds with options from which
we can choose, and we show several potential kinetic models for constitutive
promoters in Figure~\ref{fig2:constit_cartoons}(A). Let us consider
the suitability of each model for our purposes in turn.

\begin{figure}%[h!]
\centering
\includegraphics[width=\textwidth]{../../figures/main/fig02.pdf}
\caption{\textbf{Comparison of different models for noise in the constitutive promoter.}
(A) The left column depicts various plausible models for the dynamics of
constitutive promoters. In model (1), transcripts are produced in a Poisson
process~\cite{Sanchez2013, Jones2014}. Model (2) features explicit treatment of
RNAP binding/unbinding kinetics~\cite{Phillips2015a}. Model (3) is a more
detailed generalization of model (2), treating transcription initiation as a
multi-step process proceeding through closed and open
complexes~\cite{Mitarai2015}. Model (4) is somewhat analogous to (2) except with
the precise nature of active and inactive states left
ambiguous~\cite{Peccoud1995, Shahrezaei2008, Razo-Mejia2020}. Finally, model (5)
can be viewed as a certain limit of model (4) in which transcripts are produced
in bursts, and initiation of bursts is a Poisson process. The right column shows
the Fano factor $\nu$ (variance/mean) for each model. Note especially the
crucial diagnostic: (2) and (3) have $\nu$ strictly below 1, while only for (4)
and (5) can $\nu$ exceed 1. Models with Fano factors $\le 1$ cannot produce the
single-cell data observed in part (B) without introducing additional assumptions
and model complexity. (B) Data from \cite{Jones2014}. Mean mRNA count vs. Fano
factor (variance/mean) for different promoters as determined with
single-molecule mRNA Fluorescence \textit{in situ} Hybridization. The colorbar
indicates the predicted binding affinity of RNAP to the promoter sequence as
determined in \cite{Brewster2012}. Numbers serve for cross comparison with data
presented in Figure 3.}
\label{fig2:constit_cartoons}
\end{figure}

\subsubsection{Poisson Noise promoter}

The simplest model of constitutive expression that we can imagine is shown as
model 1 in Figure~\ref{fig2:constit_cartoons}(A) and assumes that transcripts
are produced as a Poisson process from a single promoter state. This is the
picture from Jones et.\ al.~\cite{Jones2014} that was used to interpret a
systematic study of gene expression noise over a series of promoters designed to
have different strengths, but no regulation. This model insists that the
``true'' steady-state mRNA distribution is Poisson, implying the Fano factor
$\nu$ must be 1. In~\cite{Jones2014}, the authors carefully attribute measured
deviations from Fano = 1 to intensity variability in fluorescence measurements,
gene copy number variation, and copy number fluctuations of the transcription
machinery, e.g., RNAP itself. In this picture, all the corrections to Poisson
behavior are derived as additive corrections to the Fano factor. This picture is
appealing in its simplicity, with only two parameters, the initiation rate $r$
and degradation rate $\gamma$. In other words, the model is not excessively
complex for the data at hand. But for many interesting questions, for instance
in the recent work~\cite{Razo-Mejia2020}, attributing all deviations from the
model to extrinsic noise sources, limits the kinds of predictions that can be
done. To make progress then we need a (slightly) more complex model than model 1
that would allow us to incorporate the non-Poissonian features of constitutive
promoters directly into a master equation formulation.

\subsubsection{Sub-Poissoninan noise promoters with RNAP escape}
A natural extension of the one-state promoter studied in the previous section is
to explicitly include an empty promoter state. This state allows for single RNAP
to bind and unbind from the promoter with rates $k_P^+$ and $k_P^-$,
respectively, before engaging in a transcriptional event. Once the RNAP
irreversibly escapes from the promoter to transcribe the mRNA, the promoter goes
back to this empty state. Models 2 and 3 from
Figure~\ref{fig2:constit_cartoons}(A) show two different versions of this
process. In model 2 the polymerase can directly engage in a transcription event
with rate $r$ after binding to the promoter, while model 3 includes an
irreversible step with rate $k_O$ that coarse-grains the known multi-step
process involved in transcriptional initiation~\cite{DeHaseth1998}.

Although physically intuitive given our understanding of how a single transcript
is produced, we are forced to discard both of these models on the basis that the
Fano factor for both of them is strictly $< 1$ as shown in
Figure~\ref{fig2:constit_cartoons}(A) (See Appendix~\ref{sec:non_bursty} for
detailed derivations). This is in disagreement with the experimental data
from~\cite{Jones2014}, reproduced in Figure~\ref{fig2:constit_cartoons}(B). All
of these unregulated promoters --which only differ in their promoter sequence--
show strictly super-Poissonian noise (Fano factor $> 1$). In fact, we suspect
\textit{any} model in which transcription proceeds through a multistep cycle
must necessarily have $\nu<1$. The intuitive argument compares the waiting time
distribution to traverse the cycle with the waiting time for a Poisson promoter
(model 1) with the same mean time. The latter is simply an exponential
distribution. The former is a convolution of multiple exponentials, and
intuitively the waiting time distribution for a multistep process should be more
peaked with a smaller fractional width than a single exponential with the same
mean. A less disperse waiting time distribution means transcription initiations
are more uniformly distributed in time relative to a Poisson process. Hence the
distribution of mRNA over a population of cells should be less variable compared
to Poisson, giving $\nu<1$. (In Appendix~\ref{sec:non_bursty} we present a more
precise version of the intuitive arguments in this paragraph).

\subsubsection{Noise in a Two-State Promoter with ``Active'' and ``Inactive''
States}

Inspired by~\cite{Razo-Mejia2020}, we next revisit an analog of model 2 in
Figure~\ref{fig2:constit_cartoons}(A), but as with
the analogous models considered in Section~\ref{section_02_means},
the interpretation of the two
states is changed. Rather than explicitly viewing them as RNAP bound and
unbound, we view them as ``active'' and ``inactive,'' which are able and unable
to initiate transcripts, respectively. We are noncommittal as to the microscopic
details of these states.

One interpretation~\cite{Chong2014, Sevier2016, Sevier2018} for the active and
inactive states is that they represent the promoter's supercoiling state:
transitions to the inactive state are caused by accumulation of positive
supercoiling, which inhibits transcription, and transitions back to ``active''
are caused by gyrase or other topoisomerases relieving the supercoiling. This is
an interesting possibility because it would mean the timescale for promoter
state transitions is driven by topoisomerase kinetics, not by RNAP kinetics.
From in vitro measurements, the former are suggested to be of order
minutes~\cite{Chong2014}. Contrast this with model 2, where the state
transitions are assumed to be governed by RNAP, which, assuming a copy number
per cell of order $10^3$, has a diffusion-limited association rate $k_{on} \sim
10^2~\text{s}^{-1}$ to a target promoter. Combined with known $K_d$'s of order
$\mu$M, this gives an RNAP dissociation rate $k_{off}$ of order $10^2$~s$^{-1}$.
As we will show below, however, there are some lingering puzzles with
interpreting this supercoiling hypothesis, so we leave it as a speculative
hypothesis and refrain from assigning definite physical meaning to the two
states in this model.

Intuitively one might expect that, since transcripts are produced as a Poisson
process only when the promoter is in one of the two states in this model,
transcription initiations should now be ``bunched'' in time, in contrast to the
``anti-bunching'' of models 2 and 3 above. One might further guess that this
bunching would lead to super-Poissonian noise in the mRNA distribution over a
population of cells.  Indeed, as shown in Appendix~\ref{sec:non_bursty}, a
calculation of the Fano factor produces
\begin{equation}
\nu = 1 + \frac{r k^-}{(k^+ + k^- + \gamma)(k^+ + k^-)},
\label{eq:model4_fano}
\end{equation}
which is strictly greater than 1, verifying the above intuition. Note we have
dropped the $P$ label on the promoter switching rates to emphasize that these
very likely do not represent kinetics of RNAP itself. This calculation can also
be sidestepped by noting that the model is mathematically equivalent to the
simple repression model from~\cite{Jones2014}, with states and rates relabeled
and reinterpreted.

How does this model compare to model 1 above? In model 1, all non-Poisson
features of the mRNA distribution were handled as extrinsic corrections. By
contrast, here the 3 parameter model is used to fit the full mRNA distribution
as measured in mRNA FISH experiments. In essence, all variability in the mRNA
distribution is regarded as ``intrinsic,'' arising either from stochastic
initiation or from switching between the two coarse-grained promoter states. The
advantage of this approach is that it fits neatly into the master equation
picture, and the parameters thus inferred can be used as input for more
complicated models with regulation by transcription factors.

While this seems promising, there is a major drawback for our purposes which was
already uncovered by the authors of~\cite{Razo-Mejia2020}: the statistical
inference problem is nonidentifiable, in the sense described in Section 4.3
of~\cite{Gelman2013}. What this means is that it is impossible to infer the
parameters $r$ and $k^-$ from the single-cell mRNA counts data
of~\cite{Jones2014} (as shown in Fig.~S2 of~\cite{Razo-Mejia2020}). Rather, only
the ratio $r/k^-$ could be inferred. In that work, the problem was worked around
with an informative prior on the ratio $k^-/k^+$. That approach is unlikely to
work here, as, recall, our entire goal in modeling constitutive expression is to
use it as the basis for a yet more complicated model, when we add on repression.
But adding more complexity to a model that is already poorly identified is a
fool's errand, so we will explore one more potential model.

\subsubsection{Noise Model for One-State Promoter with Explicit Bursts}

The final model we consider is inspired by the failure mode of model 4. The key
observation above was that, as found in~\cite{Razo-Mejia2020}, only two
parameters, $k^+$ and the ratio $r/k^-$, could be directly inferred from the
single-cell mRNA data from~\cite{Jones2014}. So let us take this seriously and
imagine a model where these are the only two model parameters. What would this
model look like?

To develop some intuition, consider model 4 in the limit $k^+ \ll k^- \lesssim
r$, which is roughly satisfied by the parameters inferred
in~\cite{Razo-Mejia2020}. In this limit, the system spends the majority of its
time in the inactive state, occasionally becoming active and making a burst of
transcripts. This should call to mind the well-known phenomenon of
transcriptional bursting, as observed in,
e.g.,~\cite{Golding2005,Chong2014,Senecal2014,Sevier2016,Sepulveda2016}. Let us
make this correspondence more precise. The mean dwell time in the active state
is $1/k^-$. While in this state, transcripts are produced at a rate $r$ per unit
time. So on average, $r/k^-$ transcripts are produced before the system switches
to the inactive state. Once in the inactive state, the system dwells there for
an average time $1/k^+$ before returning to the active state and repeating the
process. $r/k^-$ resembles an average burst size, and $1/k^+$ resembles the time
interval between burst events. More precisely, $1/k^+$ is the mean time between
the end of one burst and the start of the next, whereas $1/k^+ + 1/k^-$ would be
the mean interval between the start of two successive burst events, but in the
limit $k^+ \ll k^-$, $1/k^+ + 1/k^- \approx 1/k^+$. Note that this limit ensures
that the waiting time between bursts is approximately exponentially distributed,
with mean set by the only timescale left in the problem, $1/k^+$. If instead it
were the case that $k^+ \sim k^-$, then the waiting time $1/k^+ + 1/k^-$ would
have a peaked distribution, but this does not appear to be the case for any of
datasets from~\cite{Jones2014}.

Let us now verify this intuition with a precise derivation to check that $r/k^-$
is in fact the mean burst size and to obtain the full burst size distribution.
Consider first a constant, known dwell time $T$ in the active state. Transcripts
are produced at a rate $r$ per unit time, so the number of transcripts $n$
produced during $T$ is Poisson distributed with mean $rT$, i.e.,
\begin{equation}
    P(n\mid T) = \frac{(rT)^n}{n!} \exp(-rT).
\end{equation}
Since the dwell time $T$ is unobservable from single molecule mRNA counts, we
actually want $P(n)$, the dwell time distribution with no conditioning on $T$.
Basic rules of probability theory tell us we can write $P(n)$ in terms of
$P(n\mid T)$ as
\begin{equation}
    P(n) =\int_0^\infty P(n\mid T) P(T) dT.
\end{equation}
But we know the dwell time distribution $P(T)$, which is exponentially
distributed according to
\begin{equation}
    P(T) = k^- \exp(-T k^-),
\end{equation}
so $P(n)$ can be written as
\begin{equation}
    P(n) = k^- \frac{r^n}{n!}
            \int_0^\infty T^n\exp[-(r + k^-)T]\,dT.
\end{equation}
A standard integral table shows $\int_0^\infty x^n e^{-ax}\,dx = n!/a^{n+1}$, so
\begin{equation}
    P(n) = \frac{k^- r^n}{(k^- + r)^{n+1}}
        = \frac{k^-}{k^- + r}
            \left(\frac{r}{k^- + r}\right)^n
        = \frac{k^-}{k^- + r}
            \left(1 - \frac{k^-}{k^- + r}\right)^n,
\end{equation}
which is exactly the geometric distribution with standard parameter
$\theta\equiv k^-/(k^- + r)$ and domain $n \in \{0, 1, 2, \dots\}$.
The mean of the geometric distribution, with this convention, is
\begin{equation}
\langle n\rangle = \frac{1 - \theta}{\theta}
        = \left(1 - \frac{k^-}{k^- + r}\right)
                    \frac{k^- + r}{k^-}
        = \frac{r}{k^-},
\end{equation}
exactly as we guessed intuitively above.

So in taking the limit $r,k^-\rightarrow\infty$, $r/k^-\equiv b$, we obtain a
model which effectively has only a single promoter state, which initiates bursts
at rate $k^+$ (transitions to the active state, in the model 4 picture). The
master equation for mRNA copy number $m$ as derived in
Appendix~\ref{sec:non_bursty} takes the form
\begin{equation}
\begin{split}
{d \over dt}p(m,t) = & (m+1)\gamma p(m+1,t) - m\gamma p(m,t) \\
        &+ \sum_{m'=0}^{m-1} k_i p(m',t) \text{Geom}(m-m';b)
         - \sum_{m'=m+1}^\infty k_i p(m,t) \text{Geom}(m'-m;b),
\end{split}
\end{equation}
where we use $k_i$ to denote the burst initiation rate, $\text{Geom}(n;b)$ is
the geometric distribution with mean~$b$, i.e., $\text{Geom}(n;b) =
\frac{1}{1+b}\left(\frac{b}{1+b}\right)^n$ (with domain over nonnegative
integers as above). The first two terms are the usual mRNA degradation terms.
The third term enumerates all ways the system can produce a burst of transcripts
and arrive at copy number $m$, given that it had copy number $m'$ before the
burst. The fourth term allows the system to start with copy number $m$, then
produce a burst and end with copy number $m'$. In fact this last sum has trivial
$m'$ dependence and simply enforces normalization of the geometric distribution.
Carrying it out we have
\begin{equation}
\begin{split}
{d \over dt} p(m,t) = & (m+1)\gamma p(m+1,t) - m\gamma p(m,t) \\
        &+ \sum_{m'=0}^{m-1} k_i p(m',t) \text{Geom}(m-m';b)
            - k_i p(m,t),
\end{split}
\end{equation}
We direct readers again to Appendix~\ref{sec:gen_fcn_appdx} for further details.
This improves on model 4 in that now the parameters are easily inferred, as we
will see later, and have clean interpretations. The non-Poissonian features are
attributed to the empirically well-established phenomenological picture of
bursty transcription.

The big approximation in going from model 4 to 5 is that a burst is produced
instantaneously rather than over a finite time. If the true burst duration is
not short compared to transcription factor kinetic timescales, this could be a
problem in that mean burst size in the presence and absence of repressors could
change, rendering parameter inferences from the constitutive case inappropriate.
Let us make some simple estimates of this.

Consider the time delay between the first and final RNAPs in a burst initiating
transcription (\textit{not} the time to complete transcripts, which potentially
could be much longer.) If this timescale is short compared to the typical search
timescale of transcription factors, then all is well. The estimates from
deHaseth et.\ al.~\cite{DeHaseth1998} put RNAP's diffusion-limited on rate
around $\sim\text{few}\times10^{-2}~\text{nM}^{-1}~\text{s}^{-1}$ and polymerase
loading as high as $1~\text{s}^{-1}$. Then for reasonable burst sizes of $<10$,
it is reasonable to guess that bursts might finish initiating on a timescale of
tens of seconds or less (with another 30-60 sec to finish elongation, but that
does not matter here). A transcription factor with typical copy number of order
10 (or less) would have a diffusion-limited association rate of order
$(10~\text{sec})^{-1}$ \cite{Hammar2014}. Higher copy number TFs tend to have
many binding sites over the genome, which should serve to pull them out of
circulation and keep their effective association rates from rising too
large~\cite{Schmidt2016}. Therefore, there is \textit{perhaps} a timescale
separation possible between transcription factor association rates and burst
durations, but this assumption could very well break down, so we will have to
keep it in mind when we infer repressor rates from the Jones et.\ al.\
single-cell mRNA counts data later~\cite{Jones2014}.

In reflecting on these 5 models, the reader may feel that exploring a multitude
of potential models just to return to a very minimal phenomenological model of
bursty transcription may seem highly pedantic. But the purpose of the exercise
was to examine a host of models from the literature and understand why they are
insufficient, one way or another, for our purposes. Along the way we have
learned that the detailed kinetics of RNAP binding and initiating transcription
are probably irrelevant for setting the population distribution of mRNA. The
timescales are simply too fast, and as we will see later in
Figures~\ref{fig:constit_post_full} and~\ref{fig4:repressed_post_full},
the noise seems to be governed by slower timescales.
Perhaps in hindsight this is not surprising: intuitively, the degradation rate
$\gamma$ sets the fundamental timescale for mRNA dynamics, and any other
processes that substantially modulate the mRNA distribution should not differ
from $\gamma$ by orders of magnitude.


\chapter{Event Reconstruction}\label{chapter:event_reconstruction}

After physics interactions are detected inside of \gls{ATLAS}, are accepted by the trigger system, and are read out to disk they exist as raw detector level information and need to be reconstructed into detector-based representations of physical particles (physics objects) for analysis.
Low latency versions of reconstruction are done at the trigger level to make acceptance decisions.
However, the full event reconstruction is done ``offline'' later with more current detector models and with computationally expensive algorithms with the luxury of time to trade for great accuracy.
This also allows for reprocessing of data in the future with improved algorithms and models.
This chapter gives a high level overview of the physics objects that ATLAS defines and the methods used to make them.

\section{Tracks}\label{section:tracks}

One of the most common physics objects is charged tracks, as seen in \Cref{fig:track_cartoon}, representing charged particles passing through the subsystems of the \gls{inner detector}, as discussed in \Cref{sec:ATLAS_ID}, and \gls{muon spectrometer}.
These tracks are reconstructed from hits in these subsystems by using the hit coordinates as inputs to fitting algorithms that generally apply $\chi^{2}$ fitting techniques and Kalman filters to find the track that has the highest likelihood for the observed hits~\cite{Fruhwirth:1987fm,Fruhwirth:2003702,Strandlie:1999,Bugge:1981,Salzburger:2008cea}.

\begin{figure}[htbp]
 \centering
 \includegraphics[width=0.8\linewidth]{event_reconstruction/track_cartoon.png}
 \caption[Cartoon of comparison of the ideal charged track to a possible reconstructed charged track.]{%
  Cartoon of comparison of the ideal charged track (\textcolor{black}{black}) in ATLAS from four hits in the pixel layers and four hits in the strip detectors to a possible reconstructed charged track (\textcolor{red}{red}) that accounts for all of the hits except for one hit in the pixel detector~\cite{Salzburger:HammersAndNails}.}
 \label{fig:track_cartoon}
\end{figure}

\section{Electrons and Photons}\label{section:e_gamma}

Reconstruction of electrons begins with the signals arriving from the ECAL cells after they have been reshaped and sampled~\cite{PERF-2013-05}.
These signals are then converted to energy deposit clusters using hardware calibrations, where the ECAL is divided into a grid of towers of size $\Delta \eta \times \Delta \phi = 0.075 \times 0.175$ in the barrel and $\Delta \eta \times \Delta \phi = 0.125 \times 0.125$ in the end-caps to perform sliding-window scans~\cite{Lampl:1099735} to find local maxima to use as seeds for these clusters~\cite{PERF-2013-05}.
Well-reconstructed tracks from the \gls{inner detector} are then matched to the calorimeter clusters.
The matching of tracks is used to infer if the cluster deposits are from prompt%
\footnote{Prompt decays are where a particle originates from the primary interaction.
Prompt decays are distinguished from non-prompt by their decay time.
An example of a prompt electron would be those originating from $Z\to e^{+}e^{-}$, while a non-prompt electron would be an electron that originated from photon conversion from a radiated photon.}
electrons, electrons from photon conversion, or an unconverted photon~\cite{PERF-2013-05,PERF-2016-01}.
If there is no ID track matched to the cluster, the cluster is selected as an unconverted photon.
The fine granularity in $\eta$ of the silicon strip detectors is sufficient to provide event-by-event discrimination between single-photon showers and two overlapping showers originating from the decays of neutral hadrons (mostly $\pi^{0}$ and $\eta$ mesons) in the fiducial region $\abs{\eta} < 1.37$ and $1.52 < \abs{\eta} < 2.37$~\cite{PERF-2017-02}.
If the cluster is matched to a pair of oppositely charged ID tracks that are collinear from the primary vertex and additionally have signatures in the TRT that are consistent with electrons then the clusters are identified as a converted photon.
In all other cases where there are matched ID tracks to calorimeter clusters, the clusters are identified as electrons.

In addition, electron candidates have additional quality and isolation identification performed.
There are three levels of electron quality ``working points'': \texttt{Loose}, \texttt{Medium}, and \texttt{Tight}.
These working points are effectively minimum values of a multivariate likelihood ratio based discriminate which an analyst can then ``cut'' on to make acceptance decisions.
Isolation criteria are determined using a track-based isolation variable, $p_{T}^{\mathrm{varcone30}}$, and a calorimeter-based isolation variable, $E_{T}^{\mathrm{topcone20}}$.
The isolation criteria compare the scalar sum of the transverse momentum in a cone around the electron candidate track of size $\Delta R < 0.2$ for $E_{T}^{\mathrm{topcone20}}$ and $\Delta R < 0.3$ for $p_{T}^{\mathrm{varcone30}}$ and the transverse momentum of the electron candidate.
This provides an effective discriminant between prompt electron candidates and other electron candidates.

Electrons are also calibrated to data using the well known and characterized resonant decays of $J/\Psi \to e^{+}e^{-}$, $Z \to e^{+}e^{-}$, and $Z \to \ell^{+}\ell^{-}\gamma$.
This calibration is done to derive data-driven \glspl{scale factor} that can be applied to ensure uniformity in the detector response across different subsystems and regions and to establish systematic uncertainties that can cover discrepancies between simulation and data.

\section{Muons}\label{section:muons}

Reconstruction of muons uses tracks from both the \gls{inner detector} and the \gls{muon spectrometer}~\cite{PERF-2015-10}.
The tracks from the ID are reconstructed in the same manner as other charged particles, and the reconstruction algorithms in the MS look for hit patterns in each muon chamber to create unique chamber track segments.
Four muon candidate types are defined based on the combination of subdetectors used in the reconstruction.
Combined (CB) muons are constructed from combined, global refits of the ID and MS tracks by adding or removing tracks from the MS to improve the quality of the fit.
Segment-tagged (ST) muons are constructed by identifying tracks in the ID as muon tracks if there is a track in the \gls{MDT} or \gls{CSC}.
ST muon candidates are primarily low $p_{T}$ muons that do not traverse the entire MS.
Calorimeter-tagged (CT) muons are constructed from ID tracks and an ECAL deposit consistent with a minimum ionizing particle.
Extrapolated (ME) muons are constructed from MS tracks in at least two layers that point back to the primary vertex.
ME muon candidates are used to extend the acceptance of muon reconstruction outside the ID in the region of $2.5 < \abs{\eta} < 2.7$ where there is no ID coverage.
With regards to priority when there is overlap of muons of various categories, more information is preferred and so the (decreasing) priority ordering is CB, ST, CT.
If there is overlap with ME muons additional track quality information is required to resolve priority.

Like electrons, muon quality working points are established, and four are used in ATLAS.
For the \texttt{loose} working point all muon types are used.
This is useful for analyses where it is beneficial to maximize reconstruction efficiency above other concerns, such as $H \to 4\ell$.
The \texttt{medium} working point CB and ME tracks are used where there are at least three CB track hits and three ME layers.
The \texttt{tight} working point is designed to maximize muon purity and only used CB muons which have hits in at least two layers of the MS and also satisfy the \texttt{medium} working point requirements.
Finally, the high-$p_{T}$ working point requires CB muons passing the \texttt{medium} selection with at least three hits in the MS.
This working point is designed to maximize the momentum resolution for muon tracks with $p_{T} > 100~\GeV$.
Additionally, there are muon isolation requirements.
These isolation requirements use the track-based isolation variable, $p_{T}^{\mathrm{varcone30}}$, and a calorimeter-based isolation variable, $E_{T}^{\mathrm{topcone20}}$, described in \Cref{section:e_gamma}.
Muons are also further calibrated to data using $J/\Psi \to \mu^{+}\mu^{-}$ to cover the low $p_{T}$ spectrum and $Z \to \mu^{+}\mu^{-}$ for the high $p_{T}$ spectrum, as shown in \Cref{fig:dimuon_efficiency_scale_factor_uncert}.

\begin{figure}[htbp]
 \centering
 \includegraphics[width=\linewidth]{event_reconstruction/dimuon_efficiency_scale_factor_uncert.eps}
 \caption[Total uncertainty in the efficiency scale factor for Medium muons as a function of $p_{T}$ as obtained from $Z\to\mu\mu$ and $J/\Psi \to \mu\mu$ decays.]{%
  Total uncertainty in the efficiency scale factor for Medium muons as a function of $p_{T}$ as obtained from $Z\to\mu\mu$ (solid lines) and $J/\Psi \to \mu\mu$ (dashed lines) decays.
  The combined uncertainty is the sum in quadrature of the individual contributions~\cite{PERF-2015-10}.}
 \label{fig:dimuon_efficiency_scale_factor_uncert}
\end{figure}

\section{Jets}\label{section:jets}

As particles that carry \gls{QCD} color charge do not exist by themselves in isolation, but combine with other colored particles to form colorless composite hadrons, it is not possible to directly observe quarks or gluons in ATLAS.
Instead, this process of hadronization, and subsequent decays and rehadronization, creates a shower of both charged and neutral particles that hit the detector, creating charged tracks and energy deposits.
These hadronic showers are stochastic, and there is no way to give a full descriptive definition of them, though there is very impressive recent work on operational definitions~\cite{Metodiev:2018ftz,Komiske:2018vkc,Larkoski:2019nwj}.
Instead, clustering algorithms are used to create collections of tracks and energy deposits, known as ``jets,'' based off of the criteria of interest in the algorithm.
In a similar manner to how QCD's depth and richness as a theory requires proper treatment outside the scope of this thesis, jet physics is complex and deep enough to rightly be its own intense physics program at the LHC.
This section will only attempt to give an executive summary of the jet physics and techniques that are pertinent to my thesis analysis --- however for an exceedingly thorough overview see~\cite{Nachman:2016qyc}.

Of the jet clustering algorithms that are common in high energy physics~\cite{Salam:2009jx,Salam:2007xv} the most widely used are the $k_{t}$, Cambridge/Aachen, and anti-$k_{t}$ algorithms~\cite{Cacciari:2008gp,Soyez:2008pq}.
These algorithms are all similar in their approaches with variations on the features they emphasize.
The approach is to iteratively apply the following for all objects in the considered RoI:
\begin{enumerate}
 \item Compute the pairwise distance $d_{ij}$ between objects $i$ and $j$ where
       \[
        d_{ij} = \textrm{min}\left(k_{ti}^{2p}, k_{tj}^{2p}\right) \frac{\Delta_{ij}^{2}}{R^{2}}
       \]
       and $\Delta_{ij}^{2} = \left(y_{i} - y_{j}\right)^{2} + \left(\phi_{i} - \phi_{j}\right)^{2}$ for transverse momentum $k_{t}$, rapidity $y$, and azimuthal angle $\phi$, and parameters of choice $R$, which governs the size of the jet (though it is not a hard cutoff limit), and $p$, which governs the relative power of the energy versus geometrical $\left(\Delta_{ij}\right)$ scales~\cite{Cacciari:2008gp}.
 \item Compute the distance $d_{iB}$ between the object $i$ and the beamline $B$ where $d_{iB} = k_{ti}^{2p}$.
 \item Determine the smallest distance out of the set of distances $d_{ij}$ and $d_{iB}$, $d_{\mathrm{min}} \in \left\{\left\{d_{ij}\right\},\left\{d_{iB}\right\}\right\}$.
 \item If the minimum distance is between the object $i$ and the beamline, $d_{\mathrm{min}} \in \left\{d_{iB}\right\}$, label this object a jet and remove it.
 \item If the minimum distance is between object $i$ and $j$, $d_{\mathrm{min}} \in \left\{d_{ij}\right\}$, group these two objects into a new object $k$ which is added to the object collection, and remove objects $i$ and $j$.
\end{enumerate}
The above is repeated until there are no more remaining objects, at which point all objects have been assigned to a jet.
It is seen that the choice of free parameter $p$ then defines which algorithm is used, and what information the jet clustering algorithm prioritizes.
The choice of $p=1$ corresponds to the $k_{t}$ algorithm~\cite{Catani:1993hr}, which is seen to cluster softer --- less energetic --- objects together into progressively harder --- more energetic --- objects, as seen in \Cref{fig:jet_algorithms_kt}.
Working under the assumption that generally the hardest objects will be towards the center of the shower of particles, this can be seen as an ``outside-in'' clustering which will result in irregularly shaped (probably not very cone-like) jets.
Choosing $p=0$ corresponds to the Cambridge/Aachen jet algorithm~\cite{Dokshitzer:1997in}, which reduces the distance $d_{ij}$ to only include the angular information, $\Delta_{ij}$.
This means that softer splittings of the shower will be clustered into harder branches, which will again produce irregular shaped jets, as seen in \Cref{fig:jet_algorithms_Cambridge_Aachen}.
Choosing $p=-1$ corresponds to the anti-$k_{t}$ algorithm~\cite{Cacciari:2008gp}, where the hardest objects are clustered together first and then subsequently softer objects are added.
The anti-$k_{t}$ algorithm produces relatively regular cone shaped jets that are focused around a hard core, as seen in \Cref{fig:jet_algorithms_anti_kt}.
A typical choice of jet algorithm in ATLAS is the anti-$k_{t}$ algorithm with size parameter of $R=0.4$.

\begin{figure}[htbp]
 \centering
 \begin{subfigure}[t]{0.315\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/kt.eps}
  \caption[The sample parton-level event clustered with the $k_{t}$ jet algorithm.]{%
   The sample parton-level event clustered with the $k_{t}$ jet algorithm.}
  \label{fig:jet_algorithms_kt}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.315\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/Cambridge_Aachen.eps}
  \caption[The sample parton-level event clustered with the Cambridge/Aachen jet algorithm.]{%
   The sample parton-level event clustered with the Cambridge/Aachen jet algorithm.}
  \label{fig:jet_algorithms_Cambridge_Aachen}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.315\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/anti_kt.eps}
  \caption[The sample parton-level event clustered with the anti-$k_{t}$ jet algorithm.]{%
   The sample parton-level event clustered with the anti-$k_{t}$ jet algorithm.}
  \label{fig:jet_algorithms_anti_kt}
 \end{subfigure}%
 \caption[A sample parton-level event, together with many random soft ``ghosts,'' clustered with three different jet algorithms, illustrating the ``active'' catchment areas of the resulting hard jets.]{%
  A sample parton-level event (generated with Herwig), together with many random soft ``ghosts,'' clustered with the $k_{t}$, Cambridge/Aachen, and anti-$k_{t}$ jet algorithms, illustrating the ``active'' catchment areas of the resulting hard jets.
  For $k_{t}$ and Cambridge/Aachen the detailed shapes are in part determined by the specific set of ghosts used, and change when the ghosts are modified~\cite{Cacciari:2008gp}.}
 \label{fig:jet_algorithms}
\end{figure}

\subsection{Large Radius Jets}\label{subsection:largeR_jets}

For analyses that are dealing with very high momentum resonances, the resulting decay products can become highly collimated and be reconstructed as a single jet with a large radius parameter, $R$, (a ``\largeR{}'' jet) typically set to $R=1.0$~\cite{JETM-2018-02,SUSY-2016-13}, as seen in \Cref{fig:largeR_jet}.
For my thesis analysis \largeR{} jets were used, where the \largeR{} jets were reconstructed from topological clusters in the calorimeters using the anti-$k_{t}$ algorithm with radius parameter of $R=1.0$ and were trimmed~\cite{Krohn:2009th} to improve mass resolution and reduce dependence on pile-up.
The trimming is done by reclustering the \largeR{} jet constituents using the $k_{t}$ algorithm with a radius parameter of $R=0.2$ into subjets, and then removing any subjet that has $p_{T}$ less than $5\%$ of the \largeR{} parent jet's energy.

\begin{figure}
 \centering
 \includegraphics[width=0.6\textwidth]{event_reconstruction/largeR_jet.eps}
 \caption[The calorimeter clusters corresponding to a \largeR{} jet and its subjets in a simulated $\Zprime \to t\bar{t}$ decay with $m_{\Zprime}=1.75~\TeV$.]{%
  The calorimeter clusters corresponding to a \largeR{} jet and its subjets in a simulated $\Zprime \to t\bar{t}$ decay with $m_{\Zprime}=1.75~\TeV$.
  One of the top quarks decays hadronically, and the resulting clusters are matched to the top-quark decay products as well as the additional radiation.
  Jet constituents are shown as black dots~\cite{ATLAS-CONF-2014-003,Schwienhorst:2016efd}.}
 \label{fig:largeR_jet}
\end{figure}

\subsection{Variable Radius Track Jets}\label{subsection:VR_jets}

In recent years there have been dedicated efforts to improve jet algorithm techniques, especially in the high momentum regime.
As part of these efforts, variable radius (VR) jets~\cite{Krohn:2009zg,ATL-PHYS-PUB-2017-010} were introduced where the radius parameter, $R$, decreases as a function of the jet $p_{T}$,
\[
 R_{\mathrm{eff}} \left(p_{T}\right)= \frac{\rho}{p_{T}},
\]
where the dimensionful constant $\rho$ determines how fast the effective jet size decreases with the transverse momentum of the jet.
The choice of $\rho$ should be proportional to the mass of the resonance, and so should correctly reproduce the size of jets as long as $\rho \lesssim 2 p_{T}$.
Additional parameters $R_{\mathrm{min}}$ and $R_{\mathrm{max}}$ are used to impose
lower and upper cut-offs on the jet size,
\[
 R_{\mathrm{eff}} \left(p_{T}\right)= \mathrm{max}\left[\mathrm{min}\left(\frac{\rho}{p_{T}},R_{\mathrm{max}}\right),R_{\mathrm{min}}\right]\,.
\]
For my thesis analysis variable radius track jets~\cite{Zenz:2010hfa} were used with $\rho=30~\GeV$, $R_{\mathrm{min}}=0.02$, and $R_{\mathrm{max}}=0.4$, which in simulation gives the highest truth subjet double $b$-labelling efficiency for high $p_{T}$ Higgs bosons~\cite{ATL-PHYS-PUB-2017-010}, as seen in \Cref{fig:VR_jets_parameters}.
It is seen from \Cref{fig:VR_jets_lead_subjet}, \Cref{fig:VR_jets_sublead_subjet}, and \Cref{fig:Higgs_VR_DeltaR} that VR track jets are able to describe the topology of $\Hbb$ events across high $p_{T}$ spectrum much more accurately than fixed radius $R=0.2$ track jets, making their use an excellent choice for the analysis.

\begin{figure}[htbp]
 \centering
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/VR_jets_rho_values.eps}
  \caption[The efficiency for VR track jets with ${R_{\mathrm{min}}=0.02}$ and ${R_{\mathrm{max}} = 0.4}$ for several $\rho$ values.]{%
   The efficiency for VR track jets with ${R_{\mathrm{min}}=0.02}$ and ${R_{\mathrm{max}} = 0.4}$ for several $\rho$ values.}
  \label{fig:VR_jets_rho_values}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/VR_jets_Rmin.eps}
  \caption[The efficiency for VR track jets with ${\rho=30~\GeV}$ and ${R_{\mathrm{max}} = 0.4}$ for different values of $R_{\mathrm{min}}$.]{%
   The efficiency for VR track jets with ${\rho=30~\GeV}$ and ${R_{\mathrm{max}} = 0.4}$ for different values of $R_{\mathrm{min}}$.}
  \label{fig:VR_jets_Rmin}
 \end{subfigure}%

 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/VR_jets_Rmax.eps}
  \caption[The efficiency for VR track jets with ${\rho=30~\GeV}$ and ${R_{\mathrm{min}} = 0.02}$ for varying values of $R_{\mathrm{max}}$.]{%
   The efficiency for VR track jets with ${\rho=30~\GeV}$ and ${R_{\mathrm{min}} = 0.02}$ for varying values of $R_{\mathrm{max}}$.}
  \label{fig:VR_jets_Rmax}
 \end{subfigure}%
 \caption[Efficiency of subjet double $b$-labelling at the truth level of a Higgs jet as a function of the Higgs jet $p_{T}$.]{%
  Efficiency of subjet double $b$-labelling at the truth level of a Higgs jet as a function of the Higgs jet $p_{T}$.
  The efficiency for $R=0.2$ track jets is also included in all of the plots.
  The uncertainty bars include statistical uncertainties only~\cite{ATL-PHYS-PUB-2017-010}.}
 \label{fig:VR_jets_parameters}
\end{figure}

\begin{figure}[htbp]
 \centering
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/VR_jets_lead_subjet_low.eps}
  \caption[Distributions of the $\Delta R$ between leading subjets and matched truth $b$-hadrons for Higgs with low jet $p_{T}$ of ${250~\GeV < p_{T} < 400~\GeV}$.]{%
   Distributions of the $\Delta R$ between leading subjets and matched truth $b$-hadrons for Higgs with low jet $p_{T}$ of ${250~\GeV < p_{T} < 400~\GeV}$.}
  \label{fig:VR_jets_lead_subjet_low}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/VR_jets_lead_subjet_high.eps}
  \caption[Distributions of the $\Delta R$ between leading subjets and matched truth $b$-hadrons for Higgs with high jet $p_{T}$ of ${800~\GeV < p_{T} < 1000~\GeV}$.]{%
   Distributions of the $\Delta R$ between leading subjets and matched truth $b$-hadrons for Higgs with high jet $p_{T}$ of ${800~\GeV < p_{T} < 1000~\GeV}$.}
  \label{fig:VR_jets_lead_subjet_high}
 \end{subfigure}%
 \caption[Distributions of the $\Delta R$ between leading subjets and matched truth $b$-hadrons for two different Higgs jet $p_{T}$ bins.]{%
  Distributions of the $\Delta R$ between leading subjets and matched truth $b$-hadrons for low Higgs jet $p_{T}$ of ${250~\GeV < p_{T} < 400~\GeV}$ and high Higgs jet $p_{T}$ of ${800~\GeV < p_{T} < 1000~\GeV}$.
  The uncertainty bars include statistical uncertainties only.
  All algorithms have been normalized to an area corresponding to the fraction of signal jets which contain a leading subjet~\cite{ATL-PHYS-PUB-2017-010}.}
 \label{fig:VR_jets_lead_subjet}
\end{figure}

\begin{figure}[htbp]
 \centering
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/VR_jets_sublead_subjet_low.eps}
  \caption[Distributions of the $\Delta R$ between subleading subjets and matched truth $b$-hadrons for Higgs with low jet $p_{T}$ of ${250~\GeV < p_{T} < 400~\GeV}$.]{%
   Distributions of the $\Delta R$ between subleading subjets and matched truth $b$-hadrons for Higgs with low jet $p_{T}$ of ${250~\GeV < p_{T} < 400~\GeV}$.}
  \label{fig:VR_jets_sublead_subjet_low}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/VR_jets_sublead_subjet_high.eps}
  \caption[Distributions of the $\Delta R$ between subleading subjets and matched truth $b$-hadrons for Higgs with high jet $p_{T}$ of ${800~\GeV < p_{T} < 1000~\GeV}$.]{%
   Distributions of the $\Delta R$ between subleading subjets and matched truth $b$-hadrons for Higgs with high jet $p_{T}$ of ${800~\GeV < p_{T} < 1000~\GeV}$.}
  \label{fig:VR_jets_sublead_subjet_high}
 \end{subfigure}%
 \caption[Distributions of the $\Delta R$ between subleading subjets and matched truth $b$-hadrons for two different Higgs jet $p_{T}$ bins.]{%
  Distributions of the $\Delta R$ between subleading subjets and matched truth $b$-hadrons for low Higgs jet $p_{T}$ of ${250~\GeV < p_{T} < 400~\GeV}$ and high Higgs jet $p_{T}$ of ${800~\GeV < p_{T} < 1000~\GeV}$.
  The uncertainty bars include statistical uncertainties only.
  All algorithms have been normalized to an area corresponding to the fraction of signal jets which contain a leading subjet~\cite{ATL-PHYS-PUB-2017-010}.}
 \label{fig:VR_jets_sublead_subjet}
\end{figure}

\begin{figure}[htbp]
 \centering
 \includegraphics[width=0.8\linewidth]{event_reconstruction/Higgs_VR_DeltaR.eps}
 \caption[The $\Delta R$ between the two leading truth $b$-hadrons or subjets associated to Higgs jets as a function of Higgs jet $p_{T}$.]{%
  The $\Delta R$ between the two leading truth $b$-hadrons or subjets associated to Higgs jets as a function of Higgs jet $p_{T}$~\cite{Salzburger:HammersAndNails}.}
 \label{fig:Higgs_VR_DeltaR}
\end{figure}

\clearpage
\section{Flavor Tagging}\label{section:flavor_tagging}

Flavor tagging of jets is the process of determining a ``flavor'' label --- light, charm $(c)$, or bottom $(b)$ --- to characterize the type of hadrons in the hadronic shower that resulted in the jet.
Flavor tagging is vital in precision measurements in the top quark sector%
\footnote{Noting that the top quark has a branching fraction of $\mathcal{B}\left(t \to W b\right) = 0.96_{-0.066}^{+0.068}\left(\mathrm{stat.}\right)_{-0.052}^{+0.064}\left(\mathrm{syst.}\right)$~\cite{Abazov:2010tm,PhysRevD.98.030001}.}%
, in the search for the Higgs boson as well as new phenomena decaying to $b\bar{b}$ states, and, of particular importance to this thesis analysis, the suppression of background processes that contain predominantly light-flavor jets~\cite{PERF-2012-04}.
Of particular great interest in flavor tagging is $b$-tagging (labeling of jets containing $b$-hadrons --- $b$-jets), as $b$-hadrons are often produced in decays of heavy resonances that could be indicative of interesting new physics.
$b$-hadrons have a number of unique properties that distinguish them, as seen in \Cref{fig:b_jet}.
Notable among them is their long lifetime, discussed in \Cref{appendix:B_hadron_lifetimes}, of approximately $1.5~\mathrm{ps}$ which gives a characteristic length scale of $c\tau \sim 0.45~\mathrm{mm}$.
This is a significant enough flight distance of the $b$-hadron before it decays, that this subsequent hadron shower and jet is viewed as having a secondary vertex (SV) displaced from the original jet vertex.
The beam pipe that was installed with the \gls{IBL} in 2014 is mostly made of $0.8$-mm-thick beryllium with an inner radius of $23.5~\mathrm{mm}$ and outer radius of $24.3~\mathrm{mm}$~\cite{PERF-2015-07}.
Given the characteristic length scale of $b$-hadrons most%
\footnote{A typical $B$ meson $\left(m = 5~\GeV, \tau = 1.5~\mathrm{ps}\right)$ would require $p_{T} \gtrsim 280~\GeV$ to decay outside the beam pipe.}
decay inside of the beam pipe.
This secondary vertex is still detectable as the vertex resolution in ATLAS for $5$ to $10$ associated tracks --- typical of a SV in a $b$-jet~\cite{ATL-PHYS-PUB-2017-013} --- is approximately $65~\mu\mathrm{m}$ in $r$-$\phi$ by $110~\mu\mathrm{m}$ in $z$~\cite{Choi:2271033,ATL-PHYS-PUB-2015-026}, as seen in \Cref{fig:primary_vertex_resolution}.
$B$ mesons also have a mass of approximately $5~\GeV$.
Collectively, these properties can be exploited by $b$-tagging algorithms to discriminate $b$-jets from light or charm jets~\cite{ATL-PHYS-PUB-2015-022,ATL-PHYS-PUB-2016-012,ATL-PHYS-PUB-2017-013,PERF-2016-05}.

\begin{figure}[htbp]
 \centering
 \begin{subfigure}[t]{0.315\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/primary_vertex_resolution_x.pdf}
  \caption[The primary vertex resolution in $x$.]{%
   The primary vertex resolution in $x$.}
  \label{fig:primary_vertex_resolution_x}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.315\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/primary_vertex_resolution_y.pdf}
  \caption[The primary vertex resolution in $y$.]{%
   The primary vertex resolution in $y$.}
  \label{fig:primary_vertex_resolution_y}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.315\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/primary_vertex_resolution_z.pdf}
  \caption[The primary vertex resolution in $z$.]{%
   The primary vertex resolution in $z$.}
  \label{fig:primary_vertex_resolution_z}
 \end{subfigure}%
 \caption[The primary vertex resolution as a function of the average number of tracks in the two vertices used in the Split-Vertex method for low-$\mu$ data compared to Monte Carlo simulation.]{%
  The primary vertex resolution (with the reconstruction resolution corrected by the appropriate scale-factor) as a function of the average number of tracks in the two vertices used in the Split-Vertex method for low-$\mu$ data compared to Monte Carlo simulation (\textsc{Pythia8}).
  This measurement uses a subset of a low-$\mu$ dataset corresponding to an integrated luminosity of $216.9~\mu\textrm{b}^{-1}$~\cite{ATL-PHYS-PUB-2015-026}.}
 \label{fig:primary_vertex_resolution}
\end{figure}

\begin{figure}[htbp]
 \centering
 \centering
 \includegraphics[width=0.6\textwidth]{event_reconstruction/b_jet.pdf}
 \caption[Cartoon of a typical $b$-jet.]{%
  Cartoon of a typical $b$-jet containing a $b$-hadron decay vertex (\textcolor{track_blue}{blue}~\tikzdot{track_blue}) displaced from the primary $pp$ vertex (\textcolor{red}{red}~\tikzdot{red}), and a $c$-hadron decay vertex (\textcolor{orange}{orange}~\tikzdot{orange}) further displaced and often close to the $b$-hadron flight axis.
  Tracks from secondary (\textcolor{track_blue}{blue}) and tertiary (\textcolor{orange}{orange}) vertices have large impact parameters (\textcolor{IPgreen}{green}) with respect to the primary $pp$ vertex~\cite{Chisholm:bjet}.}
 \label{fig:b_jet}
\end{figure}

It is seen from \Cref{fig:signed_impact_parameters} that the transverse and longitudinal impact parameters --- respectively, $d_{0}$ and $z_{0}$ --- of $b$-jets tend to be positive, while $c$-jets and light-jets tend to have more impact parameters distributed more symmetrically around $0$.
As a result, these impact parameters can be used as inputs to discriminating algorithms.
For the data taking periods of my thesis analysis the main $b$-tagging algorithm used in ATLAS was the \texttt{MV2c10} \gls{BDT} based algorithm.%
\footnote{\texttt{MV2c10} is named to reflect that it is a multivariate algorithm with the fraction of $c$-jets in the training sample at roughly $10\%$.
 In reality the $c$-jet fraction is $7\%$ and the light-jet fraction is $93\%$ to give a good balance between light-jet and $c$-jet rejection.}
A \gls{BDT} is a machine learning algorithm that uses the weighted score of an ensemble of decision trees (flows of orthogonal selection criteria) to give a discriminant score.
The ensemble is iteratively constructed to emphasize areas of poor performance in previous iterations.
In addition to the quantities of the jet itself, \texttt{MV2c10} uses the output of other lower level $b$-tagging algorithms as inputs, as seen in \Cref{fig:MV2_inputs}.
These include the likelihood ratio based two-dimensional and three-dimensional impact parameter algorithms, IP2D and IP3D.
The IP2D and IP3D algorithms assume that the track IPs are uncorrelated.
The output of IP3D is shown in \Cref{fig:IP3D_LLR} for the 2017 and 2018 data taking optimization of using a training sample of $50\%$ $t\bar{t}$ and $50\%$ $\Zprime \to q\bar{q}$ for $q\in\left\{\mathrm{light}, c, b\right\}$ to cover a large $p_{T}$ range of jets.
The 2016 optimization used a training sample of $50\%$ $t\bar{t}$ and $50\%$ $\Zprime \to t\bar{t}$.
In the 2017 data taking a Recurrent Neural Network (RNN) impact parameter tagger (RNNIP)~\cite{ATL-PHYS-PUB-2017-003} was added as well, that exploits the correlations between the impact parameters between the tracks, as $b$-jets tend to have multiple highly significant IP tracks, while this is not the case for light flavor jets, as seen in \Cref{fig:RNNIP_track_significance}.
There are additional displaced secondary vertex finding algorithms (SV1), and Kalman filter algorithms (JetFitter) that exploit that roughly $90\%$ of $b$-jets contain a $c$-jet and so follow this decay chain.
Additionally a Soft Muon Tagger is also added in the 2017 data taking based on
the reconstruction of muons from the semileptonic decay of $b$-hadrons and $c$-hadrons.
The \texttt{MV2c10} BDT combines all these inputs and then gives a discriminant score indicative of how $b$-jet-like or how un-$b$-jet-like the inputs are given its training, as seen in \Cref{fig:MV2c10_BDT}.
The \texttt{MV2c10} BDT is trained using $t\bar{t}$ for the 2016 optimization, and a hybrid sample of $t\bar{t}$ and a dedicated high mass sample of $\Zprime$ decaying to jets, with nominal mass of $m = 4~\TeV$, to cover a wide $p_{T}$ spectrum for the 2017 optimization, as seen in \Cref{fig:hybrid_sample_jets}~\cite{ATL-PHYS-PUB-2017-013}.
The performance of the BDT is calibrated in data using jets that contain a muon, indicative of the semileptonic decay of a $b$-hadron, and a correction scale factor is derived~\cite{PERF-2016-05}.

\begin{figure}[htbp]
 \centering
 \centering
 \includegraphics[width=\textwidth]{event_reconstruction/MV2_inputs.pdf}
 \caption[Inputs to the high level $b$-tagging algorithm \texttt{MV2c10} for data taking in 2017 and 2018.]{%
  Inputs to the high level $b$-tagging algorithm \texttt{MV2c10} for data taking in 2017 and 2018~\cite{Feickert:ML4Jets2018}.
  The RNNIP was added in 2017; for the 2015 and 2016 data taking JetFitter was used instead.}
 \label{fig:MV2_inputs}
\end{figure}

\begin{figure}[htbp]
 \centering
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/IP3D_d0_2016.eps}
  \caption[Transverse impact parameter significance values for the 2016 configuration of the IP3D algorithm.]{%
   Data-MC comparisons of the transverse impact parameter significance values for the 2016 configuration of the IP3D algorithm.}
  \label{fig:IP3D_d0_2016}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/IP3D_d0_2017.eps}
  \caption[Transverse impact parameter significance values for the 2017 configuration of the IP3D algorithm.]{%
   Data-MC comparisons of the transverse impact parameter significance values for the 2017 configuration of the IP3D algorithm.}
  \label{fig:IP3D_d0_2017}
 \end{subfigure}%

 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/IP3D_z0_2016.eps}
  \caption[Longitudinal impact parameter significance values for the 2016 configuration of the IP3D algorithm.]{%
   Data-MC comparisons of the longitudinal impact parameter significance values for the 2016 configuration of the IP3D algorithm.}
  \label{fig:IP3D_z0_2016}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/IP3D_z0_2017.eps}
  \caption[Longitudinal impact parameter significance values for the 2017 configuration of the IP3D algorithm.]{%
   Data-MC comparisons of the longitudinal impact parameter significance values for the 2017 configuration of the IP3D algorithm.}
  \label{fig:IP3D_z0_2017}
 \end{subfigure}%
 \caption[Data-Monte Carlo comparisons of the transverse and longitudinal impact parameter significance values for IP3D selected tracks in the leading jet of the $Z\to\mu\mu + \textrm{jets}$ dominated sample.]{%
  Data-Monte Carlo comparisons of the transverse and longitudinal impact parameter significance values for IP3D selected tracks in the leading jet of a $Z\to\mu\mu + \textrm{jets}$ dominated sample~\cite{ATL-PHYS-PUB-2017-013}.}
 \label{fig:signed_impact_parameters}
\end{figure}

\begin{figure}[htbp]
 \centering
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/IP3D_LLR_ttbar.eps}
  \caption[Data-Monte Carlo comparison of the IP3D log-likelihood ratio using a $t\bar{t}$-dominated $e\mu$ sample.]{%
   Data-Monte Carlo comparison of the IP3D log-likelihood ratio using a $t\bar{t}$-dominated $e\mu$-dominated sample.}
  \label{fig:IP3D_LLR_ttbar}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/IP3D_LLR_Zjets.eps}
  \caption[Data-Monte Carlo comparison of the IP3D log-likelihood ratio using a $Z\to \mu^{+}\mu^{-}+\textrm{jets}$-dominated sample.]{%
   Data-Monte Carlo comparison of the IP3D log-likelihood ratio using a $Z\to \mu^{+}\mu^{-}+\textrm{jets}$-dominated sample.}
  \label{fig:IP3D_LLR_Zjets}
 \end{subfigure}%
 \caption[Data-Monte Carlo comparison of the log-likelihood ratio used to discriminate the $b$-jet from the light-flavor jet hypotheses in the IP3D $b$-tagging algorithm using different samples.]{%
  Data-Monte Carlo comparison of the log-likelihood ratio used to discriminate the $b$-jet from the light-flavor jet hypotheses in the IP3D $b$-tagging algorithm using a $t\bar{t}$-dominated $e\mu$ sample and a ${Z\to \mu^{+}\mu^{-}+\textrm{jets}}$-dominated sample~\cite{ATL-PHYS-PUB-2017-013}.}
 \label{fig:IP3D_LLR}
\end{figure}

\begin{figure}[htbp]
 \centering
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/d0_sig_correlations_bjet.pdf}
  \caption[The distribution of the $d_{0}$ significance for the leading $d_{0}$ significance track and subleading $d_{0}$ significance track in $b$-jet jets.]{%
   The distribution of the $d_{0}$ significance for the leading $d_{0}$ significance track and subleading $d_{0}$ significance track in $b$-jet jets.}
  \label{fig:d0_sig_correlations_bjet}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/d0_sig_correlations_lightjet.pdf}
  \caption[The distribution of the $d_{0}$ significance for the leading $d_{0}$ significance track and subleading $d_{0}$ significance track in light jets.]{%
   The distribution of the $d_{0}$ significance for the leading $d_{0}$ significance track and subleading $d_{0}$ significance track in light jets.}
  \label{fig:d0_sig_correlations_lightjet}
 \end{subfigure}%
 \caption[The distribution of the $d_{0}$ significance for the leading $d_{0}$ significance track and subleading $d_{0}$ significance track in $b$-jets and light jets.]{%
  The distribution of the $d_{0}$ significance for the leading $d_{0}$ significance track and subleading $d_{0}$ significance track in $b$-jets and light jets.
  The distributions were produced with 700 thousand $b$-jets and 1 million light jets, and each distribution is normalized to unity~\cite{ATL-PHYS-PUB-2017-003}.}
 \label{fig:RNNIP_track_significance}
\end{figure}

\begin{figure}[htbp]
 \centering
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/MV2c10_BDT_output.eps}
  \caption[The \texttt{MV2c10} output for $b$-jets, $c$-jets, and light-flavor jets in simulated $t\bar{t}$ events.]{%
   The \texttt{MV2c10} output for $b$-jets (solid line), $c$-jets (dashed line), and light-flavor jets (dotted line) in simulated $t\bar{t}$ events.}
  \label{fig:MV2c10_BDT_output}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/MV2c10_background_rejection.eps}
  \caption[The light-flavor jet and $c$-jet rejection factors as a function of the $b$-jet tagging efficiency of the \texttt{MV2c10} $b$-tagging algorithm.]{%
   The light-flavor jet (dashed line) and $c$-jet rejection factors (solid line) as a function of the $b$-jet tagging efficiency of the \texttt{MV2c10} $b$-tagging algorithm.}
  \label{fig:MV2c10_background_rejection}
 \end{subfigure}%
 \caption[The performance of the \texttt{MV2c10} BDT $b$-tagging algorithm for the 2016 optimization in simulated $t\bar{t}$ events.]{%
  The performance of the \texttt{MV2c10} BDT $b$-tagging algorithm for the 2016 optimization in simulated $t\bar{t}$ events.
  The performance was evaluated on $t\bar{t}$ events simulated using \textsc{Powheg} interfaced to \textsc{Pythia6}~\cite{PERF-2016-05}.}
 \label{fig:MV2c10_BDT}
\end{figure}

\begin{figure}[htbp]
 \centering
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/hybrid_sample_light_jets.eps}
  \caption[Distribution of the jet transverse momentum for light-flavor jets in the $\Zprime$ and $t\bar{t}$ sample.]{%
   Distribution of the jet transverse momentum for light-flavor jets in the $\Zprime$ and $t\bar{t}$ sample.}
  \label{fig:hybrid_sample_light_jets}
 \end{subfigure}%
 \quad
 \begin{subfigure}[t]{0.48\textwidth}
  \centering
  \includegraphics[width=\textwidth]{event_reconstruction/hybrid_sample_b_jets.eps}
  \caption[Distribution of the jet transverse momentum for $b$-jets in the $\Zprime$ and $t\bar{t}$ sample.]{%
   Distribution of the jet transverse momentum for $b$-jets in the $\Zprime$ and $t\bar{t}$ sample.}
  \label{fig:hybrid_sample_b_jets}
 \end{subfigure}%
 \caption[Distribution of the jet transverse momentum of the 2017 $\Zprime$ sample compared to that of $t\bar{t}$ events for light-flavor and $b$-jets.]{%
  Distribution of the jet transverse momentum of the 2017 $\Zprime$ sample compared to that of $t\bar{t}$ events for light-flavor and $b$-jets~\cite{ATL-PHYS-PUB-2017-013}.}
 \label{fig:hybrid_sample_jets}
\end{figure}

\clearpage
\section{Taus}\label{section:taus}

Tau leptons also produced in collisions in ATLAS, however, given their short lifetime they will decay into other SM particles before entering the detector subsystems and are reconstructed as other physics objects.
When taus decay they decay to hadrons approximately $64\%$ of the time~\cite{Gallinaro:2014eha}, and to other leptons $36\%$ of the time.
The leptonic decays are reconstructed as muons or electrons, and the hadronic decay modes, usually to pions, they are reconstructed as multi-pronged jets matched with tracks in the ID.
As hadronic decays of the tau also have a displaced secondary vertex they can be a source of fakes for $b$-jets.

\section{Missing Transverse Momentum}\label{section:MET}

Missing transverse momentum, $\MET$ --- or ``MET'' --- is the imbalance of momentum in the transverse plane of the event.
Any event that has neutrinos produced in it, such as events with $W \to \ell \nu_{\ell}$ processes, will have $\MET$ as neutrinos pass through ATLAS without interaction, escaping detection.
However, in events without neutrinos if other physics objects are not properly reconstructed there will still be some $\MET$ in the event due to acceptance and efficiency effects.

In closing, given the fully hadronic signature of the analysis signature in the high momentum regime, that will be described in \Cref{chapter:analysis}, and the use of $b$-tagging it is seen that the proper reconstruction of \largeR{} jets with $b$-tagged VR subjets is going to be of great importance.
Additionally, well reconstructed muons will play an important role in the establishment of a $t\bar{t}$ control region.


\chapter{Abstract}

There has been significant progress in Machine Learning (ML) and Natural
Language Processing (NLP) in recent years due to advancements in deep neural
networks. However, one of the main criticisms of these black-box models has been
their lack of transparency and weak explainability techniques. In this
thesis, we build upon the explainability of the SoPa model proposed by
\citet{schwartz2018sopa}. The strength of the SoPa model is that it incorporates
weighted finite-state automata into its neural architecture. However, its
existing explainability techniques are both localized and indirect. To address
this limitation, we developed a modified SoPa++ model that offers globalized and
direct explanations by simplification. We found that the SoPa++ model offers a
competitive accuracy range of 97.6-98.3$\%$, effective explanations by
simplification and interesting classification explanations on the Facebook
Multilingual Task Oriented Dialog (FMTOD) data set. This improved SoPa++ model
shows that black-box explainability can be effectively achieved without an
impractical compromise in performance.

% LocalWords:  NLP interpretability explainability automata FMTOD

%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
%%% End:


\section{Parameters}\label{sec:parameters}

\begin{itemize}
	\item \standout{Floorplan}:
		\begin{description}
			\item[userCount] \textit{(int)} The number \(N\) of
				users inside the network (default: \code{100}).
			\item[sizeX] \textit{(meters)} The horizontal size of
				the floorplan (default: \code{150m}).
			\item[sizeY] \textit{(meters)} The vertical size of the
				floorplan (default: \code{150m}).
			\item[indexStartingNode] \textit{(int distribution)} A
				random number generated from a distribution to
				select a random user to broadcast the message at
				the start of the simulation (default:
				\code{intuniform(0, userCount-1)}).
		\end{description}
	\item \standout{User}:
		\begin{description}
			\item[posX] \textit{(meters)} The X-axis position of the
				user in the floorplan (default: set by Floorplan
				as \code{uniform(0m, sizeX)}).
			\item[posY] \textit{(meters)} The Y-axis position of the
				user in the floorplan (default: set by Floorplan
				as \code{uniform(0m, sizeY)}).
			\item[sendOnStart] \textit{(bool)} Specifies if the user
				should send out the message at the start of
				simulation (default: \code{false}).
			\item[slotDuration] \textit{(seconds)} The duration of a
				time slot (default: \code{1s}).
			\item[broadcastRadius] \textit{(meters)} The broadcast
				radius \(R\) (default: \code{20m}).
			\item[hearWindow] \textit{(int)} The time window of
				\(T\) slots that the user should wait before
				relaying the message (default: \code{5}).
			\item[maxCopies] \textit{(int)} The maximum number of
				copies \(m\) that the user can receive to
				decide to relay the message (default: \code{3}).
				Note that our model counts also the first
				message as a ``copy'', but this is not an issue
				since the problem defined in \chref{ch:specs}
				says ``less than \(m\) copies'' (\emph{strict}
				constraint) while here we talk about the
				``maximum number of copies (including first
				message)'' (\emph{loose} constraint).
			\item[relayDelay] \textit{(int distribution)} The number
				of time slots \(\delta\) to add to the time
				window \(T\) before relaying the message
				(default: \code{intuniform(0, 3)}).
		\end{description}
	\item \standout{Oracle}:
		\begin{description}
			\item[slotDuration] \textit{(seconds)} The duration of a
				time slot. It should be equal to
				\code{User.slotDuration}.
			\item[timeout] \textit{(int)} The number of time slots
				with no network activity the oracle must wait
				before stopping the simulation. It should be
				\(\geq T + \max(\delta) + 1\).
		\end{description}
\end{itemize}

\begin{tcolorbox}[title=Note]
	Since users only operates at time slot intervals, the parameter
	\code{slotDuration} does not affect the behavior and the performance of
	the network in any way. The only effect is that times \exgratia{total
	broadcast time} are scaled, so we will fix this parameter to \(1s\). In
	this way, each second of simulation time represents a time slot and the
	analysis become easier since, by default, \omnetpp{} records emitted
	signals with the timestamp at with emission occur.

	Of course this is not the case in a real system (probably each slot is
	much more shorter than 1 second) but this decision will not break any
	consideration made by our analysis, except for the time scaling effect.

	In the following we may use the terms ``1 slot'' and ``1 second''
	interchangeably.
\end{tcolorbox}


\section{Model Functions}
\begin{itemize}
	\item \textbf{Compute Output Data}: This method computes the output structure for messaging. The miss angle is absolute distance between the desired body point and the specified structural vector. The azimuth angle is the angle between the y pointing axis and the desired pointing vector projected into the y/z plane.
\end{itemize}

\section{Model Assumptions and Limitations}

When the miss angle goes to 0$^{\circ}$ the azimuth angle is ill defined. The model will default to setting the azimuth angle to 0$^{\circ}$


\subsection{Introduction}

In practice we don't know distribution.

\subsection{Minimax}

\subsection{inimax regret}

\subsection{Robust decision making}

\subsection{Info-gap decision theory}

\subsection{Wald's maximin model}


\section{Related Work}

Related work tends to fall into distinct categories. Kalman filters allow for sensor integration with historical data, but rely on all inputs to be expressible as gaussian probabilities, or as absolute certainties.
2-dimensional probability grids have been used in robotics to assist with self-location, but existing techniques use the existing model as history only, rather than as a tool to enable the use of additional rules.
Finally, map-matching techniques attempt to attach all sensor positions to a known network, typically to enable routing applications.

\subsection{Kalman Filters}
  Kalman Filters~\cite{kalmanJBE60} allow for the efficient combination of sensor data and historical state by modelling all inputs as gaussian uncertainties. In addition, it can accommodate for unknown or random inputs by reducing certainty as forward propagation continues. However, because many rules cannot be expressed with gaussian uncertainty, Kalman filters are unable to be used for our proposed application.

\subsection{Probability Grids}
  Probability grids~\cite{burgardNCAI96} have been used in robotics to enable memory of environments, both for position estimation and for environment recognition. However, such grids are historically used to learn an environment, and tend to track probabilities other than the current location. This work uses probability meshes to track the object's current position in real-time.

\subsection{Map Matching}
  Map-Matching~\cite{brakatsoulasVLDB05} is the process of mapping arbitrary locations onto a network. As a typical example, GPS positions are mapped onto the road network, to allow for routing and turn-by-turn directions.
Most map-matching algorithms attempt to map current positions to networks, though some are designed to match an entire path. When an entire path must be matched to the network, greedy strategies are typically employed, occasionally with lookahead to reduce error.


\chapter{Results}\label{chapter:results}

To search for and quantify the significance of the production of signals $\left(V, H, \Zprime\right)$, and to set limits in the absence of observation, a Bayesian method is used to calculate a posterior likelihood as a function of number of signal events for the signal hypotheses under consideration~\cite{EXOT-2010-07}.
In this method, a final conditional probability of the parameters of interest given the observed data (the ``posterior'') is built by integrating over the nuisance parameters (``marginalization'') with a \gls{MCMC} procedure.
The posterior distribution is used to gauge the fitted signal statistical significance and to set $95\%$ credibility level (CL)%
\footnote{As discussed in \Cref{section:intervals_and_limits}, \emph{credibility levels} for Bayesian credible intervals are different from \emph{confidence levels} for frequentist confidence intervals.}
upper limits on the cross section times acceptance times efficiency.

\section{Measurement of Standard Model Signals}

To measure the Standard Model signals a model comprised of the Standard Model $\Vjets$, $\Hbb$, and $\ttbar$ signal templates along with the QCD multijet model is fit to the data.
The normalization of the $\ttbar$ component of the model is constrained with the scale factor obtained in the dedicated $\ttbar$ Control Region.
This fit simultaneously extracts the signal strengths of the $\Vjets$ and $\Hbb$ process, $\mu_{V}$ and $\mu_{H}$ respectively, which are unconstrained.
The comparison of the model post marginalization of the nuisance parameters to the data is seen in \Cref{fig:post_fit}.

\begin{figure}[htbp]
 \centering
 \includegraphics[width=\linewidth]{results/SRPostFit_mass}
 \caption[Plot of post-fit model with observed data and with model components subtracted.]{%
  The top panel shows the post-fit plot of the SM Higgs boson, $\Vjets$, $\ttbar$ and QCD model with the observed data.
  The middle panel shows the post-fit model and the data with the QCD and $\ttbar$ components of the model subtracted, highlighting the large resonance from $\Vjets$.
  The bottom panel shows the post-fit model and the data with the QCD, $\Vjets$, and $\ttbar$ components of the model subtracted, highlighting a small excess of events near $125~\GeV$~\cite{ATLAS-CONF-2018-052}.
 }
 \label{fig:post_fit}
\end{figure}

\clearpage
\subsection{Observation of boosted $V\to b\bar{b}$}

The observed signal strength for the $\Vjets$ process is
\[
 \mu_{V} = 1.5 \pm 0.22~\mathrm{(stat.)}^{+0.29}_{-0.25}~\mathrm{(syst.)} \pm 0.18~\mathrm{(th.)}\,,
\]
corresponding to an observed significance of $5\,\sigma$ with an expected significance of $4.8\,\sigma$.
This constitutes the first direct observation of boosted vector bosons decaying to bottom quark pairs in ATLAS for a center-of-mass energy of $\sqrt{s}=13~\TeV$ following the lower momentum Run-I measurement of high transverse momentum $\Zbb$~\cite{STDM-2013-04}.
From the results of simulation, seen in \Cref{table:cutflow}, the expected composition of the $\Vjets$ excess is approximately $24\%$ $W+\mathrm{jets}$ and approximately $76\%$ $Z+\mathrm{jets}$.

\subsection{Measurement of boosted $H\to b\bar{b}$}

For the $\Hbb$ process, the observed signal strength is
\[
 \mu_{H} = 5.8 \pm 3.1~\mathrm{(stat.)} \pm 1.9~\mathrm{(syst.)} \pm 1.7~\mathrm{(th.)}\,,
\]
which given the uncertainties is consistent with the background-only hypothesis at $1.6\,\sigma$ with an expected sensitivity of $0.28\,\sigma$.
This constitutes a measurement of boosted Higgs decaying to bottom quark pairs, though not a direct observation.

The combined posterior distributions of $\mu_{V}$ and $\mu_{H}$ is seen in \Cref{fig:signal_strength_contour}, showing the agreement between the best-fit values of the model and the Standard Model prediction of $\mu_{V} = \mu_{H} = 1$.
Given their respective uncertainties, the $\mu_{V}$ best-fit value is compatible with the SM prediction at $1.32\,\sigma$, and the $\mu_{H}$ best-fit value is compatible at $1.2\,\sigma$.

\begin{figure}[htbp]
 \centering
 \includegraphics[width=\linewidth]{results/contourFinal}
 \caption[Compatibility of best-fit values for $\mu_{H}$ and $\mu_{V}$ with Standard Model predicted values.]{%
  Combined posterior distributions of $\mu_{H}$ and $\mu_{V}$ from the signal region fit.
  It is seen that the best-fit values for the signal processes lie within the $2\,\sigma$ interval of the Standard Model prediction~\cite{ATLAS-CONF-2018-052}.}
 \label{fig:signal_strength_contour}
\end{figure}
\clearpage

\section{Limits on $\Zprime$ production}

Following the measurement of the Standard Model signals, a search for exotic signals in the \largeR jet mass distribution is performed.
The first step in the search is to apply the \BumpHunter{} search procedure~\cite{Aaltonen:2008vt,Choudalakis:2011qn}.
Given that the effect of the Higgs boson with a SM strength is smaller than the expected uncertainty on the $\Zprime$ limit the SM Higgs template is excluded from the model, and only the $\Vjets$ and $\ttbar$ templates are considered along with the QCD parametric model.
With this model, a fit to the data is performed with the full set of systematic uncertainties resulting in the best-fit values for the model nuisance parameters used in the analysis.
With these post-fit shapes, the \BumpHunter{} goodness-of-fit algorithm scans the mass range looking for significant deviations from the background-only model.
For the given model, the largest deviation from the background is found in the \largeR jet mass interval of $\left[115,130\right]~\GeV$, as seen in \Cref{fig:BumpHunter_scan}.
This deviation has a \BumpHunter{} global \pvalue{} of $0.54$, indicating that background-only model is quite consistent with the data.

The \BumpHunter{} test statistic, described in detail in~\cite{Choudalakis:2011qn}, is calculated for the given model and for pseudo-data sampled from the background only hypothesis (the null hypothesis, $H_{0}$) for various widths of fit windows.
The \BumpHunter{} \pvalue{} is then calculated from the most discrepant test statistic from all the window width choices in a treatment that creates a hypertest --- a union of multiple hypothesis tests --- allowing for accounting of the ``trial factor''~\cite{Gross:2010qma} in the calculation resulting in a global \pvalue{}, making the result quite robust.
For each bin the significance of the deviation between the model and the data is also calculated.
The significance for each bin is defined as the $z$-score for the observed Poisson \pvalue{}~\cite{Choudalakis:2012}, and given the integral form of the relationship between a Normal \pvalue{}, $p$, and the $z$-score, $z$,
\[
 p = \int_{z}^{\infty} \frac{1}{\sqrt{2\pi}}\,e^{-t^2/2}\,dt = 1 - \mathrm{CDF}\left(z\right) = \frac{1}{2}\left(1 - \mathrm{erf}\left(\frac{z}{\sqrt{2}}\right)\right)\,,
\]
shown in \Cref{fig:pvalues_from_zscores}, it is seen the $z$-score can be described numerically%
\footnote{The error function does not have a closed form solution.}
as
\[
 z\textrm{-score} = \sqrt{2}\, \mathrm{erf}^{-1}\left(1-2p\right).
\]

\begin{figure}[htbp]
 \centering
 \includegraphics[width=0.75\linewidth]{results/pvalues_from_zscores.eps}
 \caption[Plot of \pvalue{} as a function of $z$-score.]{%
  Plot of $\frac{1}{2}\left(1 - \mathrm{erf}\left(z/\sqrt{2}\,\right)\right)$, which describes the corresponding \pvalue{} for a given $z$-score.}\label{fig:pvalues_from_zscores}
\end{figure}

In the absence of excesses in the data that could correspond to new physics, $95\%$ credibility level upper limits, competitive with other published ATLAS limits~\cite{EXOT-2017-32} as seen in \Cref{fig:darkmatter_coupling_summary}, are set on signals from dark matter mediators with democratic decays to quarks for masses between $100~\GeV$ and $200~\GeV$.
These limits are shown in terms of cross section times branching ratio, acceptance and efficiency in \Cref{fig:cross_section_limits}, and in terms of the $g_{q}$ parameter%
\footnote{The limits on $g_{q}$ are determined from the limits on $\sigma \times \epsilon \times A \times BR$ which used signal events simulated with $g_{q} = 0.25$, meaning that as $\sigma\left(\Zprime \to q\bar{q}\right) \propto g_{q}^{2}$ that $g_{q} = 0.25 \sqrt{\sigma/\sigma_{g_{q} = 0.25}}$\,.}
that controls the coupling of the \gls{dark matter mediator} to quarks that determines the cross section in \Cref{fig:gq_limits}.
From these limits, an exotic dark matter mediator $\Zprime$ with $g_{q}=0.25$ is excluded for $m_{\Zprime} < 200~\GeV$ at the $95\%~\mathrm{CL}$.
As in seen in \Cref{fig:gq_limits}, the smallest coupling to quarks excluded at the $95\%~\mathrm{CL}$ is $g_{q} = 0.124$ for a mass hypothesis of $m_{\Zprime} = 105~\GeV$.
The limits become less restrictive at higher masses as the systematic uncertainty on the $t\bar{t}$ scale factor becomes dominant and greatly affects the total uncertainty, as seen in \Cref{table:systematic_uncertainties}.

\begin{figure}[htbp]
 \centering
 \includegraphics[width=\textwidth]{results/Final_Limits}
 \caption[The $95\%$ credibility level upper limit on the cross section times acceptance times branching ratio times efficiency for the $\Zprime$ dark matter mediator model.]{%
  The $95\%$ credibility level upper limit on the cross section times acceptance times branching ratio times efficiency obtained from the invariant mass distribution for the $\Zprime$ dark matter mediator model~\cite{ATLAS-CONF-2018-052}.}
 \label{fig:cross_section_limits}
\end{figure}

\begin{figure}[htbp]
 \centering
 \includegraphics[width=\textwidth]{results/gqlimit}
 \caption[The $95\%$ credibility level upper limit on the $g_{q}$ parameter for the $\Zprime$ dark matter mediator model.]{%
  The $95\%$ credibility level upper limit on the $g_{q}$ parameter that controls the decay width of the $\Zprime$ into SM quarks obtained from the invariant mass distribution for the $\Zprime$ dark matter mediator model~\cite{ATLAS-CONF-2018-052}.}
 \label{fig:gq_limits}
\end{figure}

\begin{figure}[htbp]
 \centering
 \includegraphics[width=\linewidth]{results/BH_final}
 \caption[\BumpHunter{} fit of post-fit background-only model to data.]{%
  The reconstructed mass distribution $m_{J}$ with the event reconstruction and selection as described in the text.
  The solid red line depicts the background prediction, consisting of the non-resonant dijet, $\Vjets$, and $\ttbar$ processes.
  The vertical blue lines indicate the most discrepant interval identified by the \BumpHunter{} algorithm between $115~\GeV$ and $130~\GeV$.
  Without including systematic uncertainties, the probability that fluctuations of the background model would produce an excess at least as significant as the one observed in the data anywhere in the distribution, the \BumpHunter{} probability, is $0.54$.
  The bottom panel shows the bin-by-bin significances~\cite{Choudalakis:2012} of the differences between the data and the fit, considering only statistical fluctuations~\cite{ATLAS-CONF-2018-052}.
 }
 \label{fig:BumpHunter_scan}
\end{figure}


%\documentstyle[11pt,../tools/coq-tex/coq]{article}
%\input{title}

%\include{macros}
%\begin{document}

%\coverpage{Co-inductive types in Coq}{Eduardo Gim\'enez}
\chapter[Co-inductive types in Coq]{Co-inductive types in Coq\label{Co-inductives}}

%\begin{abstract}
{\it Co-inductive} types are types whose elements may not be well-founded.
A formal study of the Calculus of Constructions extended by
co-inductive types has been presented
in \cite{Gim94}. It is based on the notion of
{\it guarded definitions} introduced by Th. Coquand
in \cite{Coquand93}. The implementation is by E. Gim\'enez.
%\end{abstract}

\section{A short introduction to co-inductive types}

We assume that the reader is rather familiar with inductive types.
These types are characterized by their {\it constructors}, which can be
regarded as the basic methods from which the elements
of the type can be built up. It is implicit in the definition
of an inductive type that
its elements are the result of a {\it finite} number of
applications of its constructors. Co-inductive types arise from
relaxing this implicit condition and admitting that an element of
the type can also be introduced by a non-ending (but effective) process
of construction defined in terms of the basic methods which characterize the
type. So we could think in the wider notion of types defined by
constructors (let us call them {\it recursive types}) and classify
them into inductive and co-inductive ones, depending on whether or not
we consider non-ending methods as admissible for constructing elements
of the type. Note that in both cases we obtain a ``closed type'', all whose
elements are pre-determined in advance (by the constructors). When we
know that $a$ is an element of a recursive type (no matter if it is
inductive or co-inductive) what we know is that it is the result of applying
one of the basic forms of construction allowed for the type.
So the more primitive way of eliminating an element of a recursive type is
by case analysis, i.e. by considering through which constructor it could have
been introduced. In the case of inductive sets, the additional knowledge that
constructors can be applied only a finite number of times provide
us with a more powerful way of eliminating their elements, say,
the principle of
induction. This principle is obviously not valid for co-inductive types,
since it is just the expression of this extra knowledge attached to inductive
types.

An example of a co-inductive type is the type of infinite sequences formed with
elements of type $A$, or streams for shorter. In Coq,
it can be introduced using the \verb!CoInductive! command~:
\begin{coq_example}
CoInductive Stream (A:Set) : Set :=
    cons : A -> Stream A -> Stream A.
\end{coq_example}

The syntax of this command is the same as the
command \verb!Inductive! (cf. section
\ref{gal_Inductive_Definitions}).
Definition of mutually co-inductive types are possible.

As was already said, there are not principles of
induction for co-inductive sets, the only way of eliminating these
elements is by case analysis.
In the example of streams, this elimination principle can be
used for instance to define the well known
destructors on streams $\hd : (\Str\;A)\rightarrow A$
and $\tl: (\Str\;A)\rightarrow (\Str\;A)$  :
\begin{coq_example}
Section Destructors.
Variable A : Set.
Definition hd (x:Stream A) := match x with
                              | cons a s => a
                              end.
Definition tl (x:Stream A) := match x with
                              | cons a s => s
                              end.
\end{coq_example}
\begin{coq_example*}
End Destructors.
\end{coq_example*}

\subsection{Non-ending methods of construction}

At this point the reader should have realized that we have left unexplained
what is a ``non-ending but effective process of
construction'' of a stream. In the widest sense, a
method is a non-ending process of construction if we can eliminate the
stream that it introduces, in other words, if we can reduce
any case analysis on it. In this sense, the following ways of
introducing a stream are not acceptable.
\begin{center}
$\zeros = (\cons\;\nat\;\nO\;(\tl\;\zeros))\;\;:\;\;(\Str\;\nat)$\\[12pt]
$\filter\;(\cons\;A\;a\;s) = \si\;\;(P\;a)\;\;\alors\;\;(\cons\;A\;a\;(\filter\;s))\;\;\sinon\;\;(\filter\;s) )\;\;:\;\;(\Str\;A)$
\end{center}
\noindent The former it is not valid since the stream can not be eliminated
to obtain its tail. In the latter, a stream is naively defined as
the result of erasing from another (arbitrary) stream
all the elements which does not verify a certain property $P$. This
does not always makes sense, for example it does not when all the elements
of the stream verify $P$, in which case we can not eliminate it to
obtain its head\footnote{Note that there is no notion of ``the empty
stream'', a stream is always infinite and build by a \texttt{cons}.}.
On the contrary, the following definitions are acceptable methods for
constructing a stream~:
\begin{center}
$\zeros = (\cons\;\nat\;\nO\;\zeros)\;\;:\;\;(\Str\;\nat)\;\;\;(*)$\\[12pt]
$(\from\;n) = (\cons\;\nat\;n\;(\from\;(\nS\;n)))\;:\;(\Str\;\nat)$\\[12pt]
$\alter  = (\cons\;\bool\;\true\;(\cons\;\bool\;\false\;\alter))\;:\;(\Str\;\bool)$.
\end{center}
\noindent The first one introduces a stream containing all the natural numbers
greater than a given one, and the second the stream which infinitely
alternates the booleans true and false.

In general it is not evident to realise when a definition can
be accepted or not. However, there is a class of definitions that
can be easily recognised as being valid :  those
where (1) all the recursive calls of the method  are done
after having explicitly mentioned  which is (at least) the first constructor
to start building the element, and (2) no other
functions apart from constructors are applied to recursive calls.
This class of definitions is usually
referred as {\it guarded-by-constructors}
definitions \cite{Coquand93,Gim94}.
The methods $\from$
and $\alter$ are examples of definitions which are guarded by constructors.
The definition of function $\filter$ is not, because there is no
constructor to guard
the recursive call in the {\it else} branch. Neither is the one of
$\zeros$, since there is function applied to the recursive call
which is not a constructor. However, there is a difference between
the definition of $\zeros$ and $\filter$. The former may be seen as a
wrong way of characterising an object which makes sense, and it can
be reformulated in an admissible way using the equation (*). On the contrary,
the definition of
$\filter$ can not be patched, since is the idea itself
of traversing an infinite
construction searching for an element whose existence is not ensured
which does not make sense.

Guarded definitions are exactly the kind of non-ending process of
construction which are allowed in Coq. The way of introducing
a guarded definition in Coq is using the special command
{\tt CoFixpoint}.  This command verifies that the definition introduces an
element of a co-inductive type, and checks if it is guarded by constructors.
If we try to
introduce the definitions above, $\from$ and $\alter$ will be accepted,
while $\zeros$ and $\filter$ will be rejected giving some explanation
about why.
\begin{coq_example}
CoFixpoint zeros  : Stream nat := cons nat 0%N (tl nat zeros).
CoFixpoint zeros  : Stream nat := cons nat 0%N zeros.
CoFixpoint from (n:nat) : Stream nat := cons nat n (from (S n)).
\end{coq_example}

As in the \verb!Fixpoint! command (see Section~\ref{Fixpoint}), it is possible
to introduce a block of mutually dependent methods. The general syntax
for this case is :

{\tt CoFixpoint {\ident$_1$}  :{\term$_1$} := {\term$_1'$}\\
     with\\
        \mbox{}\hspace{0.1cm} $\ldots$  \\
        with {\ident$_m$}   : {\term$_m$} := {\term$_m'$}}

\subsection{Non-ending methods and reduction}

The elimination of a stream introduced by a \verb!CoFixpoint! definition
is done lazily, i.e. its definition can be expanded only when it occurs
at the head of an application which is the argument of a case expression.
Isolately it is considered as a canonical expression which
is completely evaluated. We can test this using the command \verb!compute!
to calculate the normal forms of some terms~:
\begin{coq_example}
Eval compute in (from 0).
Eval compute in (hd nat (from 0)).
Eval compute in (tl nat (from 0)).
\end{coq_example}
\noindent Thus, the equality
$(\from\;n)\equiv(\cons\;\nat\;n\;(\from \; (\S\;n)))$
does not hold as definitional one. Nevertheless, it can be proved
as a propositional  equality, in the sense of Leibniz's equality.
The version {\it à la Leibniz} of the equality above follows from
a general lemma stating that eliminating and then re-introducing a stream
yields the same stream.
\begin{coq_example}
Lemma unfold_Stream :
   forall x:Stream nat, x = match x with
                            | cons a s => cons nat a s
                            end.
\end{coq_example}

\noindent The proof is immediate from the analysis of
the possible cases for $x$, which transforms
the equality in a trivial one.

\begin{coq_example}
olddestruct x.
trivial.
\end{coq_example}
\begin{coq_eval}
Qed.
\end{coq_eval}
The application of this lemma to  $(\from\;n)$ puts this
constant at the head of an application which is an  argument
of a case analysis, forcing its expansion.
We can test the type of this application using Coq's command \verb!Check!,
which infers the type of a given term.
\begin{coq_example}
Check (fun n:nat => unfold_Stream (from n)).
\end{coq_example}
 \noindent  Actually, The elimination of $(\from\;n)$ has actually
no effect, because it is followed by a re-introduction,
so the type of this application is in fact
definitionally equal to the
desired proposition. We can test this computing
the normal form of the application above to see its type.
\begin{coq_example}
Transparent unfold_Stream.
Eval compute in (fun n:nat => unfold_Stream (from n)).
\end{coq_example}

\section{Reasoning about infinite objects}

At a first sight, it might seem that
case analysis does not provide a very powerful way
of reasoning about infinite objects. In fact, what we can prove about
an infinite object using
only case analysis is just what we can prove unfolding its method
of construction a finite number of times, which is not always
enough. Consider for example the following method for appending
two streams~:
\begin{coq_example}
Variable A : Set.
CoFixpoint conc (s1 s2:Stream A) : Stream A :=
  cons A (hd A s1) (conc (tl A s1) s2).
\end{coq_example}

Informally speaking, we expect that for all pair of streams $s_1$ and $s_2$,
$(\conc\;s_1\;s_2)$
defines the ``the same'' stream as $s_1$,
in the sense that if we would be able to unfold the definition
``up to the infinite'', we would obtain definitionally equal normal forms.
However, no finite unfolding of the definitions gives definitionally
equal terms. Their equality can not be proved just using case analysis.

The weakness of the elimination principle proposed for infinite objects
contrast with the power provided by the inductive
elimination principles, but it is not actually surprising. It just means
that we can not expect to prove very interesting things about infinite
objects doing finite proofs. To take advantage of infinite objects we
have to consider infinite proofs as well. For example,
if we want to catch up the equality between $(\conc\;s_1\;s_2)$ and
$s_1$ we have to introduce first the type of the infinite proofs
of equality between streams. This is a
co-inductive type, whose elements are build up from a
unique constructor, requiring a proof of the equality of the
heads of the streams, and an (infinite) proof of the equality
of their tails.

\begin{coq_example}
CoInductive EqSt : Stream A -> Stream A -> Prop :=
    eqst :
      forall s1 s2:Stream A,
        hd A s1 = hd A s2 -> EqSt (tl A s1) (tl A s2) -> EqSt s1 s2.
\end{coq_example}
\noindent Now the equality of both streams can be proved introducing
an infinite object of type

\noindent $(\EqSt\;s_1\;(\conc\;s_1\;s_2))$ by a \verb!CoFixpoint!
definition.
\begin{coq_example}
CoFixpoint eqproof (s1 s2:Stream A) : EqSt s1 (conc s1 s2) :=
  eqst s1 (conc s1 s2) (eq_refl (hd A (conc s1 s2)))
    (eqproof (tl A s1) s2).
\end{coq_example}
\begin{coq_eval}
Reset eqproof.
\end{coq_eval}
\noindent Instead of giving an explicit definition,
we can use the proof editor of Coq to help us in
the construction of the proof.
A tactic \verb!Cofix! allows placing a \verb!CoFixpoint! definition
inside a proof.
This tactic introduces a variable in the context which has
the same type as the current goal, and its application stands
for a recursive call in the construction of the proof. If no name is
specified for this variable, the name of the lemma  is chosen by
default.
%\pagebreak

\begin{coq_example}
Lemma eqproof : forall s1 s2:Stream A, EqSt s1 (conc s1 s2).
cofix.
\end{coq_example}

\noindent An easy (and wrong!) way of finishing the proof is just to apply the
variable \verb!eqproof!, which has the same type as the goal.

\begin{coq_example}
intros.
apply eqproof.
\end{coq_example}

\noindent The ``proof'' constructed in this way
would correspond to the \verb!CoFixpoint! definition
\begin{coq_example*}
CoFixpoint eqproof  : forall s1 s2:Stream A, EqSt s1 (conc s1 s2) :=
  eqproof.
\end{coq_example*}

\noindent which is obviously non-guarded. This means that
we can use the proof editor to
define a method of construction which does not make sense. However,
the system will never accept to include it as part of the theory,
because the guard condition is always verified before saving the proof.

\begin{coq_example}
Qed.
\end{coq_example}

\noindent Thus, the user must be careful in the
construction of infinite proofs
with the tactic \verb!Cofix!. Remark that once it has been used
the application of tactics performing automatic proof search in
the environment (like for example \verb!Auto!)
could introduce unguarded recursive calls in the proof.
The command \verb!Guarded! verifies
that the guarded condition has been not violated
during the construction of the proof. This command can be
applied even if the proof term is not complete.

\begin{coq_example}
Restart.
cofix.
auto.
Guarded.
Undo.
Guarded.
\end{coq_example}

\noindent To finish with this example, let us restart from the
beginning and show how to construct an admissible proof~:

\begin{coq_example}
Restart.
 cofix.
\end{coq_example}

%\pagebreak

\begin{coq_example}
intros.
apply eqst.
trivial.
simpl.
apply eqproof.
Qed.
\end{coq_example}

\section{Experiments with co-inductive types}

Some examples involving co-inductive types are available with
the distributed system, in the theories library and in the contributions
of the Lyon site. Here we present a short description of their contents~:
\begin{itemize}
\item Directory \verb!theories/LISTS! :
        \begin{itemize}
                \item File \verb!Streams.v! : The type of streams and the
extensional equality between streams.
        \end{itemize}

\item Directory \verb!contrib/Lyon/COINDUCTIVES! :
      \begin{itemize}
                \item Directory \verb!ARITH! : An arithmetic where $\infty$
is an explicit constant of the language instead of a metatheoretical notion.
                \item Directory \verb!STREAM! :
      \begin{itemize}
                \item File \verb!Examples! :
Several examples of guarded definitions, as well as
of frequent errors in the introduction of a stream. A different
way of defining the extensional equality of two streams,
and the proofs showing that it is equivalent to the one in \verb!theories!.
                \item File \verb!Alter.v! : An example showing how
an infinite proof introduced by a guarded definition can be also described
using an operator of co-recursion \cite{Gimenez95b}.
                 \end{itemize}
\item Directory \verb!PROCESSES! : A proof of the alternating
bit protocol based on Pra\-sad's Calculus of Broadcasting Systems \cite{Prasad93},
and the verification of an interpreter for this calculus.
See \cite{Gimenez95b} for a complete description about this development.
        \end{itemize}
\end{itemize}

%\end{document}