# Root Cause Explanation of Outliers under Noisy Mechanisms

Phuoc Nguyen1, Truyen Tran1, Sunil Gupta1, Thin Nguyen1, Svetha Venkatesh1

###### Abstract

Identifying root causes of anomalies in causal processes is vital across
disciplines. Once identified, one can isolate the root causes and implement
necessary measures to restore the normal operation. Causal processes are often
modelled as graphs with entities being nodes and their paths/interconnections
as edge. Existing work only consider the contribution of nodes in the
generative process, thus can not attribute the outlier score to the edges of
the mechanism if the anomaly occurs in the connections. In this paper, we
consider both individual edge and node of each mechanism when identifying the
root causes. We introduce a noisy functional causal model to account for this
purpose. Then, we employ Bayesian learning and inference methods to infer the
noises of the nodes and edges. We then represent the functional form of a
target outlier leaf as a function of the node and edge noises. Finally, we
propose an efficient gradient-based attribution method to compute the anomaly
attribution scores which scales linearly with the number of nodes and edges.
Experiments on simulated datasets and two real-world scenario datasets show
better anomaly attribution performance of the proposed method compared to the
baselines. Our method scales to larger graphs with more nodes and edges.

## Introduction

Understanding the root causes behind anomalies in complex network systems
holds significant importance across various disciplines, ranging from science
to industry (Dhaou et al. 2021; Pool et al. 2020; Yi and Park 2021). Once the
causes are identified, one can isolate and implement necessary measures to
restore the normal operation of the process. Early fault recognition would
prevent costly damage to operations, services, and products. For example, in
complex network systems within modern manufacturing industries and information
services, the cost of system failure is notably high (Ni et al. 2017),
reaching as much as $\$20,000$ per minute of downtime in an automotive
manufacturing plant (Djurdjanovic, Lee, and Ni 2003). The monitoring data and
logs obtained from the processing nodes of these systems often contain noise
attributed to random fluctuations in nodes and the links between them. In
communication systems, for instance, random delays between nodes can arise
from bandwidth limitations and network overhead (Zhang, Branicky, and Phillips
2001). Given the intricate dependencies between monitoring nodes and the
substantial volume of data involved, manual analysis of root causes becomes
impractical.

Recent methods for root cause analysis (RCA) (Budhathoki et al. 2021, 2022)
are based on a given causal structure of the system and a learned functional
causal model (Peters, Janzing, and Schölkopf 2017) to explain the anomalous
observations at a leaf node. These methods work by first detecting the
anomalous leaf, then utilising the causal structure and counterfactual
reasoning attribute the outlier scores to ancestor nodes (Budhathoki et al.
2022). The causal structure of a system is a powerful tool enabling the formal
analysis of the root cause of unexpected event (Budhathoki et al. 2022). It is
a directed acyclic graph (DAG) composed of nodes representing system
components and directed edges representing causal links or dependency
connections from parent nodes to child nodes. The observational data of nodes
are assumed to have additive noise, the edge connections are assumed to be
noise-free (Budhathoki et al. 2022). However, in computer networking systems,
e.g., the connection between components can be noisy or faulty due to varying
workloads, faulty hardware, wear and tear, or signal interference. Therefore,
the root cause of an outlier can also arise from an edge in addition to a
node. This raises the question of whether an anomalous edge could be detected
by existing algorithms. In a recent work, Ni et al. (2017) attempted to detect
faulty edges in those networks. Nevertheless, to the best of our knowledge,
the study of noisy causal links as root causes has not been considered.

In this paper, we aim to fill this gap by generalising existing frameworks to
allow the detection of anomalous edges as well as anomalous nodes. We consider
changes at both the individual edges and model them via a Bayesian linear
regression, where the noise in the causal edges is represented as the
distribution of the regression weights.

In addition, we propose a causal contribution score called Bayesian Integrated
Gradient of Edge and Node noise (BIGEN) to attribute a leaf anomaly score to
the ancestor nodes and edges. First, given observations of outliers, we infer
the noise values of the causal edges represented by Bayesian linear regression
coefficients using the mode of the posterior distribution (MAP estimate)
(Bishop and Nasrabadi 2006). Second, we infer the noise values of the nodes
and edges given the outlier observations. Third, we use an attribution method
to compute the contribution of each noise term to the outlier score. Existing
works for explaining outliers use Shapley-based attribution methods that
require summing over all possible subsets of the ancestor nodes and edges
(Sundararajan and Najmi 2020; Covert and Lee 2021; Budhathoki et al. 2022).
This is computationally expensive, even for approximate methods, when applied
to large graphs. In this work, we apply the integrated gradient (IG) method
(Sundararajan and Najmi 2020) along the path from some references to the
target noise for calculating the attribution scores. We show that this
attribution method is efficient for scaling to graphs with thousands of nodes
in attributing the root causes, compared to the baseline methods based on
Shapley values (Sundararajan and Najmi 2020).

Our contributions are:

1\. A framework for identifying the sources of changes in both nodes and edges
to reduce operational costs.

2\. Modelling the causal connections using Bayesian linear regression to
enable the inference of edge noises and applying the Shapley-based attribution
framework.

3\. Introducing a new causal contribution score called BIGEN to efficiently
attribute a leaf anomaly to the ancestor nodes and edges.

4\. Demonstrating the effectiveness of the proposed methods on random graph
datasets and two real-world scenario datasets.

## Preliminaries

### Outlier Scores

Based on information theory, Budhathoki et al. (2022) introduced an outlier
score that calibrates all probabilistic outlier scores. The authors defined
the outlier score by characterising the tail probability of an event $X=x$
based on the distribution of some score space, such as negative log likelihood
or $z$-score, as follow:

$\displaystyle S_{X}(x)$ $\displaystyle=-\log P\left\\{-\log p(X)\geq-\log
p(x)\right\\}$ (1) $\displaystyle\text{or,~{}}S_{X}(x)$ $\displaystyle=-\log
P\left\\{|X-\mu_{x}|\geq|x-\mu_{X}|\right\\}$ (2)

### Functional Causal Mechanisms (FCMs)

Given a causal graph represented as a DAG, its functional causal mechanisms
(Pearl 2009; Peters, Janzing, and Schölkopf 2017) can be described by the
following set of equations. For each node $j$:

$\displaystyle X_{j}$ $\displaystyle=f_{j}(\text{Pa}_{j},\epsilon_{j})$ or,
$X_{j}$ $\displaystyle=\sum_{i\in
PA_{j}}W_{ij}X_{i}+\epsilon_{j},\qquad\text{when $f_{j}$ is linear}$

where $PA_{j}$ is the node indices of the parents of node $j$. These equations
represent the dependence of a node $X_{j}$ on its parent nodes
$X_{Pa_{j}}=\\{X_{i}:i\in\text{Pa}{}_{j}\\}$ and $\epsilon_{j}$ which is an
additive noise random variable independent of $X_{i}$. If we assume $f_{j}$ is
linear, given an observation data matrix $X$, we can fit a linear regression
model for this causal model to learn the weight parameters $W_{ij}$. A leaf
outlier $X_{n}=f_{n}(\text{Pa}_{n},\epsilon_{n})$ can be recursively expressed
as a function depending on all the noise variables as follows:

$X_{n}=f(\epsilon_{1},\dots,\epsilon_{n})=f(\boldsymbol{\epsilon})$

where $P_{\boldsymbol{\epsilon}}=P_{\epsilon_{1}}\times\dots\times
P_{\epsilon_{n}}$.

### Shapley-based RCA of outliers

Budhathoki et al. (2022) employed the concept of Shapley values (Shapley et
al. 1953) from cooperative game theory and randomised experiment to measure
the contribution of a noise term to the target outlier score for an observed
outlier at the leaf. The Shapley value, in the context of a coalition game, is
an axiom-based method which uniquely divides (attributes) a pay-off among the
players (the noise variables $\\{\epsilon_{j}\\}$ in our study) (Sundararajan
and Najmi 2020).

#### Shapley Classic

The originally proposed Shapley value (Shapley et al. 1953) for each player
$j$ is computed as the weighted average of its marginal contributions for all
possible coalitions (subsets of players), both with and without the player
$j$. Let $P={1,\dots,d}$ denote the set of players, $Q\subseteq P$ be a
subset, $|Q|$ denote its cardinality, and $v$ be the value function. The
Shapley value for the player $j$ is defined as:

$\phi_{j}(v)=\sum_{Q\subseteq P\backslash
j}\frac{|Q|!(|P|-|Q|-1)}{P!}\left(v(Q\cup i)-v(Q)\right)$ (3)

#### Sampling

This method approximates the Shapley values as a solution to the weighted
least squares problem (Janzing, Minorics, and Blöbaum 2020; Covert and Lee
2021):

$\displaystyle\min_{\phi_{1},\dots,\phi_{n}}\sum_{Q\subseteq
P}w(Q)\left(u(Q)-v(Q)\right)$ (4) $\displaystyle
w(Q)=\frac{d-1}{\left(\begin{array}[]{c}d\\\ |Q|\end{array}\right)|Q|(d-|Q|)}$
(7)

where the weighting function $w(Q)$ depends only on the cardinality of the
subset $Q$, and $u(S)=\sum_{i\in S}\phi_{i}$ represents the approximation
function.

#### Permutation

This method approximates the Shapley values using Monte Carlo samples of data
and random permutations (Štrumbelj and Kononenko 2014). It first draws a
random instance $\epsilon^{m}$ from the data, then chooses a random
permutation of the independent variables, and finally computes the marginal
contribution as:

$\phi_{j}(v)=\frac{1}{M}\sum_{m=1}^{M}\left(v(\epsilon_{+j}^{m})-v(\epsilon_{-j}^{m})\right)$
(8)

where $v(\epsilon_{+j}^{m})$ is the prediction for $\epsilon$, but with a
random number of variables replaced by the values from $\epsilon^{m}$, except
for $\epsilon_{j}$, and $v(\epsilon_{-j}^{m})$ similarly, but $\epsilon_{j}$
is also replaced by $\epsilon_{j}^{m}$.

#### Value function

The value function $v$ in this study is the outlier score $S(x_{n})$ at a leaf
node (Budhathoki et al. 2022).

## Methods

We first present our framework for RCA of an observed anomaly at a leaf node.
We formulate a noisy mechanism to model the noise in the edge connection
between component nodes. This allows the leaf node to be expressed as a
function of both node and edge noise dependent model. Finally, we present the
integrated gradient (IG) based contribution score of ancestor edge and node
noises.

### Framework

For concreteness, let us consider a computer network system111Our method is
applicable to all structural causal processes, e.g., in a manufacturing
process with sensor readings at each nodes as the variables. as a running
example. We have continuous observations from the network components
represented as the variables $X_{1},\dots,X_{n}$. The causal relationships
between these component variables are known and given in the form of a
directed acyclic graph (DAG). There is edge noise in each connection from
$X_{j}$ to parent to describe potential noise between each parent-child pair.
As example, this can be a random delay in the connection between two servers
due to a router malfunction in between. Based on the causal mechanism (Peters,
Janzing, and Schölkopf 2017), we introduce noise to each causal link from each
parent node $X_{i}\in\text{Pa}{}_{j}$ to $X_{j}$ in the causal graph to create
a noisy mechanism, where $\text{Pa}{}_{j}$ is the set of parent nodes of node
$j$.

######  Definition 1.

Noisy Causal Mechanisms.

For node $j$ , a noisy mechanism is a generative process,

$\displaystyle\xi_{\cdot j}$ $\displaystyle\sim\mathcal{N}(0,\alpha^{-1}I)$
(9) $\displaystyle W_{\cdot j}$ $\displaystyle=\mu_{\cdot j}+\xi_{\cdot j}$
(10) $\displaystyle\epsilon_{j}$ $\displaystyle\sim\mathcal{N}(0,\beta^{-1})$
(11) $\displaystyle X_{j}$ $\displaystyle=f_{j}(W_{\cdot
j}^{T}X_{\text{Pa}{}_{j}})+\epsilon_{j}$ (12)

where $f_{j}$ can be a nonlinear function, $\xi_{\cdot j}$ and $\epsilon_{j}$
are Gaussian edge and node noises with fixed variances $\alpha^{-1}$ and
$\beta^{-1}I$ ($\alpha$ and $\beta$ are the precision parameters), $\mu_{\cdot
j}$ is the mean of $p(W_{\cdot j})$ with $W_{ij}$ being the causal link from
node $i$ to node $j$ for $i\in\text{Pa}{}_{j}$, and we use $\cdot$ (dot) to
denote the index varying in the set of parent node $\text{Pa}{}_{j}$.

We use this noisy mechanism to model the generative process of the system
during normal operations. During testing, suppose some node or edge of the
system behave strangely, either the node noise $\epsilon_{j}$ or some edge
noise $\xi_{ij}$ is interfered with, causing anomalous observations at
downstream nodes and leaf nodes.

######  Definition 2.

Abnormal Causal Mechanism

An abnormal causal mechanism of a node $j$ is a similar generative process as
the noisy causal mechanism of the node $j$ in Definition 1 but with the noise
distribution $p(\epsilon_{j})$ or $p(\xi_{ij})$ interfered.

During abnormal operation, as defined in Definition 2, the underlying
anomalous noises $\xi_{ij}$ and $\epsilon_{j}$ render the generative processes
defined in Eq. 12 to generate anomalous observations of the system. The RCA
module then collects these anomalies along with their ancestors’ observations,
denoted as $X^{\prime}$, for analysing and identifying the root causes using
an attribution algorithm. Note that this generalises the approach presented in
(Budhathoki et al. 2022) which only considers node anomalies.

#### Fitting the Noisy Mechanism

In this paper, we model a noisy mechanism using Bayesian linear regression,
allowing us to compute its posterior analytically. We use training data
collected during the normal operation of the system to fit the posterior of
the noisy mechanism, as described in the following theorem.

######  Theorem 3.

Posterior distribution of noisy mechanism

For each node $j$, let $(X_{\text{Pa}{}_{j}},X_{j})$ be the set of input-
output observations of the noisy mechanism as defined in Definition 1.
Assuming fixed $\alpha$, and $\beta$ precision parameters. If the prior for
the weights of the causal links and the regression output are respectively:

$\displaystyle W_{\cdot j}$ $\displaystyle\sim\mathcal{N}\left(\mu_{\cdot
j}^{0},\alpha^{-1}I\right)$ $\displaystyle X_{j}$
$\displaystyle\sim\mathcal{N}\left(W_{\cdot
j}^{T}X_{\text{Pa}{}_{j}},\beta^{-1}I\right),$

then the posterior distribution of the weight vector $W_{\cdot j}$ of the
mechanism $j$ is given by:

$\displaystyle W_{\cdot j}|X_{j}$ $\displaystyle\sim\mathcal{N}(\mu_{\cdot
j},H)\,\,\text{for}$ $\displaystyle\mu_{\cdot j}$
$\displaystyle=H^{-1}\left(\alpha\mu_{\cdot j}^{0}+\beta
X_{\text{Pa}{}_{j}}^{T}X_{j}\right)$ $\displaystyle H$ $\displaystyle=\alpha
I+\beta X_{\text{Pa}{}_{j}}^{T}X_{\text{Pa}{}_{j}}.$

###### Proof.

Refer to Section 3.3 in (Bishop and Nasrabadi 2006). ∎

######  Proposition 4.

MAP estimate of the noisy mechanism

Given the data and the prior in Theorem 3, the maximum a posteriori (MAP)
estimate of the weights $W_{\cdot j}$ is

$W_{\cdot j}^{MAP}=\mu_{\cdot j}=H^{-1}(\alpha\mu_{\cdot j}^{0}+\beta
X_{\text{Pa}{}_{j}}^{T}X_{j})$

###### Proof.

By definition, the MAP estimate is the mode of the posterior distribution
(Bassett and Deride 2019). Since the posterior is a Gaussian, its mode
coincides with its mean. ∎

### Noise dependent reparametrisation

When considering RCA of a leaf outlier
$X_{n}=f_{n}(\text{Pa}_{n},\epsilon_{n},\xi_{\cdot n})$, it is convenient to
re-parameterise $X_{n}$ as a function of the node and edge noises
$(\epsilon,\xi)$ to facilitate the attribution algorithms later on. We can
recursively rewrite each ancestor of $X_{n}$ in Eq. 12 up to the root node and
arrive at a function of these noises

$X_{n}=g(\epsilon,\xi)$ (13)

where $P_{\boldsymbol{\epsilon}}=\prod_{i}P_{\epsilon_{i}}$, and
$P_{\xi}=\prod P_{\xi_{ij}}$ are independent noises. Using this re-
parameterisation, we can attribute the root cause of these anomalies directly
to these noise variables. Given a batch of abnormal values $X^{\prime}$ with
$x^{\prime}_{n}$ being the observed anomaly at the leaf node $X_{n}$, we first
infer the noise $\xi^{\prime}$, and $\epsilon^{\prime}$ as follows.

#### Edge noise estimation

We employ the MAP estimate of the posterior from Proposition 4 to compute the
new edge weights $W_{\cdot j}^{\prime}$ for each mechanism $j$. We then use
the posterior weights estimated from the training data as the new prior.
Finally, the estimated edge noise is defined as:

$\xi_{\cdot j}^{\prime}=W_{\cdot j}^{\prime}-W_{\cdot j}$ (14)

Note that if $X^{\prime}$ comes from the same distribution as $X$, i.e., the
generative process is normal, the new edge weights $W_{\cdot j}^{\prime}$ will
coincide with the mode $W_{\cdot j}$ of the fitted edge weight posterior,
rendering the edge noise being close to a zero vector.

#### Node noise estimation

We use Eq. 12 to estimate the node noise as

$\displaystyle\epsilon_{j}$ $\displaystyle=X_{j}-f_{j}\left(W_{\cdot
j}^{T}X_{\text{Pa}{}_{j}}\right).$ (15)

### Integrated Gradient of Edge and Node noise

We introduce a Bayesian Integrated Gradient of Edge and Node noise (BIGEN in
short) to attribute a leaf anomaly score to ancestor nodes and edges. Instead
of using subsets as in Shapley values methods, we use noise reference to
explain the root cause. This approach offers a performance gain compared to
previous methods. Specifically, we select a reference node noise
$\epsilon^{\prime}$ from the normal dataset and the mean edge noise
$\xi^{\prime}=W$. Additional reference points can also be chosen to calculate
multiple contribution scores, followed by averaging, to enhance the accuracy
of the score.

Let $f$ be the score function in Eq. 1 which is a continuous and
differentiable function. Then $\phi_{i}(x,x^{\prime},f)$ is defined as the
integral of the gradient of $f$ along the straight-line path between $x$ and
$x^{\prime}$. Formally, we define the path between $x$ and $x^{\prime}$ as
$\gamma(t)=tx+(1-t)x^{\prime}$ for $t\in[0,1].$ Then, the Integrated Gradient
(IG) for the $i$th feature $x_{i}$ is defined as:

$\displaystyle\text{IG}{}_{i}(x,x^{\prime},f)$
$\displaystyle=(x-x^{\prime})\int_{t=0}^{1}\frac{\partial
f(\gamma(t))}{\partial\gamma_{i}(t)}\frac{\partial\gamma_{i}(t)}{\partial
t}dt$ $\displaystyle=(x_{i}-x_{i}^{\prime})\int_{t=0}^{1}\frac{\partial
f(x^{\prime}+t(x-x^{\prime}))}{\partial x_{i}}dt$ (16)

This score integrates the gradient along the path from the reference to the
inferred noise to calculate the attribution of each node and edge towards the
observed anomaly score. In our case, with two noise variables $\epsilon$ and
$\xi$, computing the IG for one requires marginalising over the other noise
variable. Therefore, we adjust this IG to fit into our attribution to node and
edge noises as follows:

$\displaystyle\text{IG}{}_{i}(\epsilon,\epsilon^{\prime},f)$
$\displaystyle=\mathbb{E}_{\xi}\left[\epsilon_{i}-\epsilon_{i}^{\prime}\right]\int_{t=0}^{1}\frac{\partial
f_{t}}{\partial\epsilon_{i}}dt$ (17)
$\displaystyle\text{IG}{}_{ij}(\xi,\xi^{\prime},f)$
$\displaystyle=\mathbb{E}_{\epsilon}\left[\xi_{ij}-\xi_{ij}^{\prime}\right]\int_{t=0}^{1}\frac{\partial
f_{t}}{\partial\xi_{ij}}dt$ (18)

This method is more advantageous than subset sampling, as it is linear and
only dependent on the number of discretised steps in the path. In contrast,
Shapley-based method (Sundararajan and Najmi 2020) requires summing over all
possible subsets of the ancestor nodes and edges, which grows exponentially
with the number of nodes and edges.

#### Score function

Since the joint distribution of nodes and edges is Gaussian, the conditional
distribution of the normal observations at the leaf node is also Gaussian. In
this case, the outlier score using the negative log-likelihood feature in Eq.
1 is equivalent to the outlier score with the $z$-score feature in Eq. 2. An
efficient evaluation function can be derived using the error function as
follows:

$\displaystyle S_{X}(x)$ $\displaystyle=-\log P\left\\{-\log p(X)\geq-\log
p(x)\right\\}$ $\displaystyle=-\log
P\left\\{\frac{|X-\mu_{X}|^{2}}{2\sigma_{X}^{2}}\geq\frac{|x-\mu_{X}|^{2}}{2\sigma_{X}^{2}}\right\\}$
$\displaystyle=-\log
P\left\\{|\frac{X-\mu_{X}}{\sigma_{X}}|\geq|\frac{x-\mu_{X}}{\sigma_{X}}|\right\\}$
$\displaystyle=-\log\\{1-\text{$\Phi$}(z)\\}=-\log\text{$\Phi$}(-z)$ (19)

where $z=|\frac{x-\mu_{X}}{\sigma_{X}}|$,
$\text{\text{$\Phi$}}(z)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z}e^{-\frac{t^{2}}{2}}dt$
is the standard normal cumulative distribution function, and
$(\mu_{X},\sigma_{X})$ represents the maximum likelihood estimate of the
marginal mean and standard deviation of the marginal distribution of the
target node.

#### Causal graph and noisy FCM

While our focus is not on solving causal discovery, we assume a causal graph
is given. For each fixed weight noise vector, we have a set of FCMs with
additive node noise and their FCM parameters can be learned from normal
observation data (Peters, Janzing, and Schölkopf 2017). Budhathoki et al.
(2022) observed that when representing the FCM of a target leaf node as a
functional of all node noise, each noise vector takes on the role of selecting
a deterministic mechanism. We generalise this idea and represent the FCMs as
functionals of the edge and node noises, which similarly play the role of
choosing the deterministic mechanisms. Inferred noises of outlier edges or
nodes, therefore, will select outlier mechanisms in either cases. We build
upon the success of Budhathoki et al. (2022)’s approach, to infer functions
and noise from data, and show that counterfactual contribution score by
changing the noise term w.r.t. a reference is effective for causal attribution
at both the node and edge levels. This approach utilises Pearl’s third ladder
of causation (Pearl 2009) to enhance our understanding of the system’s causes
and effects.

#### Shapley values and IGs

Budhathoki et al. (2022) computed Shapley value contributions numerically by
averaging over all orderings sets. However for larger number of variables,
approximation is needed to be practical. In the experiments, we sample
orderings instead of using all orderings and compare Shapley with early
stopping (Shapley), subset sampling (Sampling), and methods based on a fixed
number of randomly generated permutations (Permutation). For the IGs, however,
no subsets of intervention are needed but rather a gradient path is taken
along the path from some reference noises to the target noises. For each noise
vector, BIGEN requires one forward pass and one backward pass. Since we use a
small fixed number of references, this computation scales linearly with the
number of nodes and edges in the subgraph. One advantage of BIGEN over Shapley
methods is that it can be applied to nonlinear (noisy) FCMs since the
contribution score is still linear w.r.t. the gradients, thus satisfying the
efficiency axiom (Shapley et al. 1953).

## Related Work

Causal structure discovery-based techniques have been recently used to find
the root cause(s) of faults in cloud applications (Arnold, Liu, and Abe 2007;
Wang et al. 2018). More recently, Budhathoki et al. (2022) introduced a causal
structure based root cause explanation for outlier using Shapley values. The
authors represented the dependence of an outlier leaf directly in terms of the
noise of ancestor nodes. This approach facilitates intervention on the noise
distributions. Their work, however, assumes that the outlier does not
originate from the connections in the functional causal models. In real
computer network systems, e.g., the connections may be broken or malfunction,
causing anomalies in downstream nodes. In our present work, we relax this
constraint and consider also changes in the weights as one possible source of
the root cause, in addition to the node anomalies.

Budhathoki et al. (2021) model a distribution change of a mechanism for
certain nodes and use a new dataset under this new mechanism to fit a new
model. In contrast, our work models the uncertainty in the edges of the
mechanisms and allows for temporary anomaly edge noise causing a faulty batch
of observations. Under Bayesian view, we employ the MAP estimate of the faulty
edge noise deviation from the normal edge prior. By estimating noise in both
edges and nodes, we can quantify the contribution of each noise term to the
outlier score observed at the leaf. This can be viewed as intervention and
counterfactual analysis on the nodes and edges. To the best of our knowledge,
we are the first to explain anomalies in both nodes and edges based on the
given causal structure. A different approach from our work, which combines RCA
and causal discovery in one framework, is proposed by Ikram et al. (2022).

In the recent domain of explainable AI, methods based on Shapley value
(Shapley et al. 1953; Sundararajan and Najmi 2020) have gained increasing
popularity. These methods use an axiomatic approach to design attribute
functions with desirable properties, e.g., being fair, unique, and efficient.
These methods explain prediction outcomes by attributing the prediction score
back through the deep neural networks to the input features (Erion et al.
2021; Yang et al. 2022). Among them, integral-based attribution methods are
the most efficient, as they use only a reference to represent the absence of
the input signal, rather than a random feature from the training data
(Lundberg and Lee 2017; Sundararajan and Najmi 2020; Samek et al. 2021).
However, these methods are designed for explaining neural network predictions,
which differs from our goal of RCA.

## Experiments

We run experiments on random graph datasets and two real-world settings,
namely a micro cloud service and a supply chain scenarios. We compare our
methods against three baselines described in the Preliminaries section, as
well as a naive approach:

  1. 1.

Shapley (classic): This method employs Shapley values as defined in Eq. 3 for
the contribution of each node.

  2. 2.

Sampling. This method calculates the Shapley values by selecting random
subsets and weighted least squares regression (Janzing, Minorics, and Blöbaum
2020; Covert and Lee 2021), as indicated in Eq. 4.

  3. 3.

Permutation: This method computes the Shapley values through permutation
sampling (Štrumbelj and Kononenko 2014), as shown in Eq. 8.

  4. 4.

Naive: This method uses the marginal distribution, i.e., the observational
distribution, of each node $X_{j}$ to compute the Shapley values.

  5. 5.

BIGEN (ours): We use the contribution scores outlined in Eq. 17, 18 to assess
the node and edge contributions.

For all methods, we use the outlier score in Eq. 19 and assess the
contribution to this score by each ancestor node and edge. For the edge score
of the baselines, we use the outer product of the node scores to estimate the
edge score $s_{edge}=s_{node}s_{node}^{T}$, where
$s_{node}=(s_{1},\dots,s_{n})$ represents the contributions of all the
ancestor nodes (including the target node). This score quantifies the level of
anomaly associated with an edge $e_{ij}$ by combining the scores observed at
nodes $i$ and $j$.

We adhere to the approach of Budhathoki et al. (2022) and employ NDCG@$k$
(Järvelin and Kekäläinen 2017) to gauge rankings based on graded relevance of
outcomes. NDCG@$k$ yields values within the $[0,1]$ range, with higher scores
indicating that highly “relevant” root causes are assigned higher ranks. To
establish the ground truth relevance of all nodes, we assign zero relevance
scores to non-root causes, and invert the ranking of injected root causes,
akin to the methodology in (Budhathoki et al. 2022).

### Random graph datasets

We randomly generate 1,000 causal graphs with varying number of nodes in the
range from 10 to 10000 nodes. The noisy causal mechanisms follow Definition 1
with $\alpha_{j}^{-1}=1$, $\beta_{ij}^{-1}=0.01$, and
$\mu_{ij}\sim|\mathcal{N}(0,1)|$. We draw normal data from these noisy FCMs,
following their generative process. For the abnormal data, we randomly select
a target node $X_{n}$ from this causal graph. We then choose among its
ancestor node either $k\in[1,\dots,m]$ root-cause nodes, or $l\in[1,\dots,m]$
root-cause edges, or $k+l$ root-cause nodes and edges. Here, $m$ is chosen to
be $10\%$ of the number of nodes in the subgraph. We inject outlier noises
into the nodes and edges to create the ground truths as follows: The outlier
node noise $\epsilon_{j}$ of node $j$ is randomly drawn from
$\mathcal{N}(a,b)$, where $a$ is drawn from $\pm\text{Uniform}(3,5)$ and $b$
is drawn from $\text{Uniform}(3,5)$. The outlier edge noise $\xi_{ij}$ of each
node $j$ is randomly drawn from $\mathcal{N}(am_{j},bs_{j})$, where $a$ is
drawn from $\pm\text{Uniform}(3,5)$, $b$ is drawn from $\text{Uniform}(3,5)$,
and $m_{j}$ represents the maximum magnitude of the current weights $w_{ij}$
(i.e., $m_{j}=\max_{i}(|w_{ij}|)$), and $s_{j}$ represents the maximum
standard deviation $\sigma_{ij}$ (i.e., $s_{j}=\max_{i}(\sigma_{ij})$).

Fig. 1 presents a comparison of the results obtained from all methods. On
average, BIGEN outperforms all the baselines in detecting the actual root
causes of outliers in the top $k$ candidates, at over NDCG 0.9. The Shapley,
Sampling, and Permutation methods can overall detect root causes of outliers,
at around NDCG 0.83, with similar performance across them. The Shapley method
seems to perform better than Sampling and Permutation in node and edge anomaly
attributions on average. The Naive method, on the other hand, could not
attribute the root causes correctly. Table 1 shows the details of node and
edge attribution results for the case when both type of anomaly are present.
Overall, the edge score rankings are on average lower than that of the node
scores. This suggests that it is greater difficulty in attributing root causes
when edge anomalies are present.

Figure 1: NDCG@$k$ ranking of root cause detection on random graphs. | Nodes | Edges | Nodes + Edges
---|---|---|---
Shapley* | $0.815\pm 0.045$ | $0.820\pm 0.110$ | $0.818\pm 0.077$
Sampling | $0.847\pm 0.095$ | $0.772\pm 0.137$ | $0.809\pm 0.116$
Permut. | $0.845\pm 0.115$ | $0.820\pm 0.118$ | $0.833\pm 0.116$
Naive | $0.691\pm 0.134$ | $0.536\pm 0.303$ | $0.633\pm 0.222$
BIGEN | $\mathbf{0.898\pm 0.085}$ | $\mathbf{0.938\pm 0.105}$ | $\mathbf{0.918\pm 0.095}$
Table 1: NDCG@$k$ for root cause detection in random graphs. (*) Due to
exponential runtime, for >20 nodes, we compute Shapley values using early
stopping when the contribution score does not change much (Blöbaum et al.
2022).

#### Runtime

Next, we compare the runtime of BIGEN to Shapley-based methods. Fig. 2 shows
the runtime of all methods on a Ubuntu 20.04 workstation with an Intel Xeon
E5-1650 v4 CPU and 46Gb RAM. Notably, both the BIGEN and Naive RCA methods
exhibit linear time complexities relative to the number of upstream nodes,
completing computations in less than a minute for target nodes with roughly
200 ancestors. In contrast, the Shapley method demonstrates exponential
complexity, requiring over 2 hours for cases involving more than 20 nodes.
This disparity arises due to the Shapley method’s necessity to iterate through
all conceivable subsets for the computation of the weighted average of each
subset’s contribution score. The Sampling and Permutation methods tend to
exhibit polynomial time complexities, taking just slightly over 1 hour to
compute for a target outlier node featuring around 200 ancestor nodes. The
Naive method independently computes the contribution of each node, while BIGEN
capitalises on gradient information and baseline noises to compute the
contribution of each noise term within the given context.

Figure 2: Comparing wall-clock runtime (in minutes) between BIGEN and Causal-
RCA methods with increasing number of nodes. The attribution complexity of
BIGEN and Naive is $O(d+e)$, while Shapley is $O(2^{(d+e)})$. Sampling and
Permutation methods tend to have polynomial time.

The key takeaway from this experiment is that BIGEN provides relevant top-$k$
ranking across values of $k$ while reducing the computation time
significantly. This is because BIGEN can model the uncertainty in the causal
edges to handle situations with noisy mechanisms.

### Root causes of observed latencies in cloud services

In this experiment, we study the root causes of unexpected observed latencies
in a microservice of an online shop. Microservices are building blocks for
complex mobile and Internet-of-Things (IoT) applications. Therefore, it is
essential to keep service access delay minimal (Guo, Tang, and Tang 2022). Any
unusual delay should be quickly studied to identify the root causes and
resolve the problem (Ikram et al. 2022). Often, the delay at a node is
composed of the communication delay between itself and each parent node (due
to network bandwidth, load, and other turbulences) and execution delay of each
node (due to the request complexity, node computational power and load).

We use the microservice architecture described in (Blöbaum et al. 2022). We
assume the delay noise in each node $X_{j}$ is a Gaussian noise with unknown
variance $\alpha_{j}^{-1}$ and in each edge $W_{ij}$ is a Gaussian noise with
unknown variance $\beta_{ij}^{-1}$. We thus make a realistic assumption and
allow for noisy connections to account for the above scenarios. The task is to
explain the root cause of an unwanted observed latency at the customer end in
processing an online order. This service involves multiple other web-services
described by a causal dependency graph involving ten other services (Blöbaum
et al. 2022), part of the causal graph is shown in Fig. 4 in the Appendix.
Assuming that we observe latencies in the order confirmation of the Website
leaf node, we also assume all services are synchronised.

For the abnormal data of the target node Website we select among its ancestor
node either $k\in[1,\dots,3]$ root-cause nodes or $l\in[1,\dots,3]$ root-cause
edges or $k+l$ root-cause nodes and edges, then we inject anomalous noises by
random sampling the noise outside $3\sigma$ ranges of their normal operation
distributions to create the ground truths.

We use similar models as in the previous section for this experiment, i.e.,
assuming the causal graph is given then fit the noisy FCMs models to the
training data collected during normal operations.

#### Results

Table 2 displays the outcomes achieved by all methods, revealing the NDCG@k
values for root cause detection within a micro cloud service environment. It
shows that the Shapley-based methods can detect the root cause in both the
nodes and edges. However, the edge score ranking is lower than that of the
node score. This suggests that the deterministic weights methods fall short in
high weight noise applications. However, the Shapley method shows better
attribution results than Sampling and Permutation. The Naive method can detect
node root causes but not edges root causes. The BIGEN, in contrast, can model
the uncertainty in the causal links therefore can account better the
contribution of the root causes to the observed outliers at leaves. Overall,
BIGEN shows better root cause attributions across nodes, edges, and both
compared to the baselines.

| Nodes | Edges | Nodes + Edges
---|---|---|---
Shapley | $0.904\pm 0.047$ | $0.778\pm 0.091$ | $0.841\pm 0.069$
Sampling | $0.918\pm 0.050$ | $0.715\pm 0.140$ | $0.816\pm 0.095$
Permut. | $0.914\pm 0.048$ | $0.726\pm 0.112$ | $0.820\pm 0.080$
Naive | $0.797\pm 0.102$ | $0.465\pm 0.199$ | $0.631\pm 0.150$
BIGEN | $0.910\pm 0.060$ | $\mathbf{0.924\pm 0.076}$ | $\mathbf{0.917\pm 0.068}$
Table 2: NDCG@$k$ for RCA in a micro cloud service.

Fig. 3 shows the NDCG raking at different $k$ between the methods. It shows
that BIGEN can rank the root cause better on average than all methods for
across top-$k\geq 2$ values. The increasing NDCG@$k$ scores of all methods
with larger $k$ values show that the relevant root causes have more chance to
appear in the top-$k$ results.

Figure 3: NDCG@$k$ when varying the number of $k$, error bars not included to
keep the figure uncluttered.

### Root Causes of Outliers in a Supply Chain

In this experiment, we apply our methods to model the noisy interactions
between businesses. The behaviour in supply chains usually includes complex
interaction of organisation structure, and time delays between decision and
implementation (Power 2005). To gain competitive advantage, it is critical
that businesses need to optimise their supply chain operations to ensure the
flow of physical goods between trading partners. Thus, it is important that
delays to this flow can be identified and rectified in a reliable and timely
manner. Often, business decisions are made based on predictions of real
demands and constraints and are not a deterministic process (Power 2005).

We study RCA of outliers in a supply chain process shown in Fig. 5 in the
Appendix. In this process, a retailer need to submit orders to a vendor based
on its inventory constraint of current stock and the demand forecasting of
near future sale to fulfil future customers’ purchases faster. The vendors
will then confirm the retailer’s purchase orders with a random delay. When
these orders are confirmed, the goods will be sent to the retailer again with
some random delay and may arrive at different times. The random delays can be
due to various overhead costs such as decisions of the vendors’ manager,
variable packaging and shipping operations (Croson et al. 2014). We assume
noisy linear FCMs for the supply chain process where the noise term of each
node is a Gamma distribution to mimic real-world settings, i.e., heavily-
tailed behaviour. We assume Gaussian noises for the edges to account for the
fluctuations in the causal links between nodes.

For the abnormal data of the target node Received we randomly select among its
ancestor nodes either $k\in\\{1,2\\}$ nodes or $l\in\\{1,2\\}$ edges or
$k=1,l=1$ node and edge, then inject anomalous noises by random sampling the
node noise $\epsilon\sim\text{Uniform}(3,5)$ and edge connection noise
$\xi\sim\text{Uniform}(3,5)$ then collect the outliers.

#### Results

Table 3 presents the outcomes of all methods employed for root cause detection
within a supply chain context. The result shows good performance among all
methods with our proposed method BIGEN has highest detection ranking followed
by Shapley methods, then Naive-RCA. Notably, our method can accurately
identify the anomalous link in the process, at NDCG@$k$ of $0.98$ on average.
This confirms the effectiveness of the noisy FCM models and the BIGEN
attribution method.

| Nodes | Edges | Nodes + Edges
---|---|---|---
Shapley | $0.891\pm 0.049$ | $0.927\pm 0.072$ | $0.909\pm 0.061$
Sampling | $0.890\pm 0.050$ | $0.925\pm 0.074$ | $0.908\pm 0.062$
Permut. | $0.892\pm 0.049$ | $0.926\pm 0.072$ | $0.909\pm 0.061$
Naive | $0.856\pm 0.055$ | $0.915\pm 0.091$ | $0.886\pm 0.073$
BIGEN | $0.890\pm 0.052$ | $\mathbf{0.980\pm 0.045}$ | $\mathbf{0.935\pm 0.048}$
Table 3: NDCG@$k$ for RCA in a supply chain.

## Conclusions

We have introduced a framework for identifying the root causes of unexpected
events observed at leaf nodes in causal generative processes. Through
modelling noises in both nodes and edges, we proposed noisy functional causal
models that enable the inference of both types of noises, making them suitable
for the application of attribution algorithms. Furthermore, we have developed
an efficient attribution score based on integrated gradient, which can be
readily applied to graphs with thousands of nodes. Our experimental results
demonstrated the effectiveness of our proposed methods.

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# Top Quark Mass and Cross Sections in ATLAS and CMS

Matteo M. Defranchis - for the ATLAS and CMS Collaborations  Experimental
Physics Department, CERN, Geneva (Switzerland)

###### Abstract

With the large data set delivered during the second run of the CERN LHC,
inclusive and differential measurements of the top quark-antiquark production
cross section at the ATLAS and CMS experiments often reach a precision that is
comparable or superior to that of the corresponding theoretical calculations.
The results of these measurements can be used to test the validity of
perturbative quantum chromodynamics (QCD), and to precisely extract the values
of parameters of the QCD Lagrangian such as the top quark mass
(${m_{\mathrm{t}}}$). The value of ${m_{\mathrm{t}}}$ can also be measured,
with higher precision, by fully or partially reconstructing the invariant mass
of the top quark decay products. However, the results of these measurements
lack a clear theoretical interpretation. In these proceedings, the most recent
measurements of the top quark mass and cross sections by ATLAS and CMS are
presented.

##  1 Introduction

At the CERN LHC, top quarks are mainly produced in quark-antiquark
($\mathrm{t\bar{t}}$) pairs via the mechanism of gluon fusion. The
$\mathrm{t\bar{t}}$ production cross section ($\sigma_{\mathrm{t\bar{t}}}$)
can be calculated in quantum chromodynamics (QCD), either inclusively or
differentially, using a perturbative approach. The inclusive
$\sigma_{\mathrm{t\bar{t}}}$ is currently known at the next-to-next-to-leading
order (NNLO) in perturbation theory, including next-to-next-to-leading
logarithmic (NNLL) corrections, with a precision of about 5% [3]. The ATLAS
[1] and CMS [2] experiments have performed measurements of the inclusive
$\sigma_{\mathrm{t\bar{t}}}$ at different centre-of-mass energies, often
outperforming the precision of the theoretical calculations. In this
contribution, two new measurements [4, 5] of $\sigma_{\mathrm{t\bar{t}}}$ at
$\sqrt{\mathrm{s}}=5.02\;\mathrm{TeV}$ by the ATLAS and CMS Collaborations,
respectively, are presented for the first time.

With the large data set delivered by the LHC during Run 2, differential and
multi-differential measurements have become increasingly more precise in the
less populated phase space, including the one with highly boosted top quarks.
Differential measurements are particularly sensitive to the details of the
parton distribution functions (PDFs) of the proton, and can be used to test
the validity of theoretical calculations and Monte Carlo (MC) predictions, or
to extract the values of QCD parameters such as ${m_{\mathrm{t}}}$ or the
strong coupling constant [6]. In this contribution, a new measurement [7] of
single and double-differential $\mathrm{t\bar{t}}$ cross sections in the
$\ell$+jets final state by the CMS Collaboration is presented for the first
time. This is also the first analysis in which the resolved and boosted
distributions are unfolded simultaneously. Significant progress has also been
made on the theoretical side, as single- and double-differential calculations
are becoming available at NNLO for an increasing number of variables [8].

The value of ${m_{\mathrm{t}}}$ can also be measured directly by comparing MC
predictions to variables related to the invariant mass of the top quark decay
products. Unlike indirect measurements, the direct ones lack a clear
theoretical interpretation due to the modelling of non-perturbative effects in
the MC simulation. In this contribution, a direct measurement of
${m_{\mathrm{t}}}$ obtained by the CMS Collaboration using single top events
[9] is presented for the first time. The relation between the results of
direct and indirect approaches can be investigated using measurements of the
invariant mass distribution of large-area jets containing the decay products
of a boosted top quark, such as the one in Ref. [10]. In fact, the value of
${m_{\mathrm{t}}}$ can be extracted from such measurements both with a direct
approach, i.e. by comparing to distributions obtained with MC generator, and
by comparing to calculations obtained in the framework of soft collinear
effective field theories [11].

##  2 Cross section measurements and extractions of the top quark mass

In this contribution, two new measurements of the inclusive
$\sigma_{\mathrm{t\bar{t}}}$ at $\sqrt{\mathrm{s}}=5.02\;\mathrm{TeV}$ by the
ATLAS and CMS Collaborations, respectively, are presented [4, 5]. Both
measurements are based on a small data set delivered by the LHC in 2017,
corresponding to 257 and 304 $\mathrm{pb}^{-1}$ of proton-proton (pp)
collisions for ATLAS and CMS, respectively. The ATLAS analysis consists of a
maximum-likelihood fit in the dileptonic final states, while the CMS result is
obtained using an event count method in the $\mathrm{e\mu}$ final state. The
CMS result is then combined with a previous measurement obtained in the
$\ell$+jets final state using a smaller data set collected in 2015 [12]. The
total uncertainty in the measured $\sigma_{\mathrm{t\bar{t}}}$ is 7.5 (7.9)%
for ATLAS (CMS). The precision of both measurements is limited by the
statistical uncertainty, and the results are in good agreement with the state-
of-the-art theoretical predictions, as shown in Figure 1.

Figure 1: ATLAS [4] (left) and CMS [5] (right) measurement of
$\sigma_{\mathrm{t\bar{t}}}$ at $\sqrt{\mathrm{s}}=5.02\;\mathrm{TeV}$
compared to NNLO+NNLL theoretical predictions [3] with different sets of
parton distribution functions.

To date, the most precise result of $\sigma_{\mathrm{t\bar{t}}}$ at
$\sqrt{\mathrm{s}}=13\;\mathrm{TeV}$ was obtained by the ATLAS Collaboration
in the $\mathrm{e\mu}$ final state using 36.1$\;\mathrm{fb}^{-1}$ of pp
collision data [13] and has been used to determine the value of the top quark
pole mass with a precision of about 2$\;\mathrm{GeV}$, limited by the QCD
scale uncertainties. The total uncertainty in the measured
$\sigma_{\mathrm{t\bar{t}}}$ is 2.4%, significantly smaller than the one of
the NNLO+NNLL calculation [3]. Several other measurements at
$\sqrt{\mathrm{s}}=13\;\mathrm{TeV}$ have been performed by the ATLAS and CMS
Collaborations in different decay channels [14, 15], including the one with
with hadronically decaying $\tau$ leptons which is used by the CMS
Collaboration to perform a test of the lepton universality in W boson decays
[16].

Figure 2: Measured normalised double-differential $\mathrm{t\bar{t}}$ cross
section as a function of the ${p_{\mathrm{t}}}$ of the hadronically-decaying
top quark and of the ${p_{\mathrm{t}}}$ of the $\mathrm{t\bar{t}}$ system,
unfolded to the parton level in the full phase space [7]. The measurement is
compared to state-of-the-art MC generators and NNLO theoretical predictions
(MATRIX [8]).

Single- and double-differential measurements of $\sigma_{\mathrm{t\bar{t}}}$
at $\sqrt{\mathrm{s}}=13\;\mathrm{TeV}$ are also performed by the two
Collaborations in the various decay channels, both in the resolved and boosted
regimes. The measured cross sections are presented both absolute and
normalized to the total cross section, and are unfolded to the particle and
the parton level. Particle-level results are obtained in the fiducial phase
space and can be used to test the predictions of different MC generators,
while parton-level results are usually extrapolated to the full phase space
and can be compared to fixed-order theoretical predictions. Furthermore,
measurements of multi-differential cross sections can provide information
about the correlation between different variables. A significant over-
prediction of the spectrum of the transverse momentum (${p_{\mathrm{t}}}$) of
the top quark in the boosted regime has been observed by both experiments in
several measurements [10, 17, 18, 19], while double-differential measurements
by the ATLAS Collaboration also highlight significant mismodelling of the
${p_{\mathrm{t}}}$ spectrum of the $\mathrm{t\bar{t}}$ system [18, 20].

In this contribution, a new measurement of single- and double-differential
$\mathrm{t\bar{t}}$ cross sections at $\sqrt{\mathrm{s}}=13\;\mathrm{TeV}$ by
the CMS Collaboration [7] is presented for the first time. The analysis is
performed in the $\ell$+jets final state using the full CMS Run 2 data set,
corresponding to a total integrated luminosity of 137$\;\mathrm{fb}^{-1}$.
This is the first analysis in which the distributions in the resolved and
boosted topologies are unfolded simultaneously. This is achieved by means of a
minimum $\chi^{2}$ method with profiled systematic uncertainties, which also
allows the total uncertainty in the measured cross sections to be
significantly reduced. The inclusive $\sigma_{\mathrm{t\bar{t}}}$ is also
measured with a precision of 3.1%, to date the most precise result in the
$\ell$+jets channel. The measured $\sigma_{\mathrm{t\bar{t}}}$ is found to be
in good agreement with the state-of-the-art theoretical prediction, and the
compatibility between the measured distributions and MC or fixed-order
theoretical predictions is also quantitatively assessed. Similarly to previous
ATLAS and CMS measurements, a softer-than-predicted spectrum of the top quark
${p_{\mathrm{t}}}$ is observed, as shown, e.g. in Figure 2.

Finally, a differential measurement of the $\mathrm{t\bar{t}}$ production
cross section as a function of the invariant mass of the $\mathrm{t\bar{t}}$
system ($m_{\mathrm{t\bar{t}}}$) is used by the CMS Collaboration to perform
the first investigation of the running of the top quark mass [21], an effect
analogous to the better-known running of the strong coupling constant. The
cross section is measured using 35.9$\;\mathrm{fb}^{-1}$ of pp collision data
at $\sqrt{\mathrm{s}}=13\;\mathrm{TeV}$ and is unfolded to the parton level by
means of a maximum-likelihood method. The running is then extracted by
comparing the measured cross section to NLO theoretical predictions in the
modified minimal subtraction ($\overline{\mathrm{MS}}$) renormalisation
scheme. The result is found to be in good agreement with the one-loop solution
of the corresponding renormalisation group equation, as shown in Figure 3.
Furthermore, a hypothetical no-running scenario is excluded at above 95%
confidence level.

Figure 3: Measured scale dependence (running) of the value of top quark mass
in the $\overline{\mathrm{MS}}$ scheme as a function of
$\mu_{m}=m_{\mathrm{t\bar{t}}}/2$, obtained using NLO theoretical predictions
with dynamic (full dots) or fixed (hollow squares) QCD scales [21]. The result
is in good agreement with the one-loop solution of the corresponding
renormalisation group equation (line).

##  3 Direct measurements of the top quark mass

Direct measurements of ${m_{\mathrm{t}}}$ are performed by fully or partially
reconstructing the invariant mass of the top quark decay products. In this
contribution, the two most recent results by ATLAS and CMS are presented. Both
analyses are performed in a different phase space compared to most direct
measurements of ${m_{\mathrm{t}}}$, and therefore provide additional
information for future combinations of ATLAS and CMS results.

The CMS measurement, which is presented here for the first time, consists of a
functional fit to the reconstructed invariant mass of the top quark decay
products in single top events in the $t$-channel (left-hand side of Figure 4),
and is performed using 35.9$\;\mathrm{fb}^{-1}$ of pp collision data [9]. The
top quark is required to decay leptonically, and the charge of the
reconstructed lepton is used to distinguish between event containing top
quarks and those containing top antiquarks. The value of ${m_{\mathrm{t}}}$ is
measured both inclusively, resulting in ${m_{\mathrm{t}}}=172.13\pm
0.77\;\mathrm{GeV}$, and separately for top quarks and antiquarks. The
inclusive result is in good agreement with previous ATLAS and CMS
measurements, as shown in Figure 4 (right), and is limited by the uncertainty
in the jet energy scale calibration. The ratio and difference between the
values of the top quark and antiquark masses are measured to be
$R_{{m_{\mathrm{t}}}}=m_{\mathrm{\bar{t}}}/{m_{\mathrm{t}}}=0.995\pm 0.006$
and
$\Delta{m_{\mathrm{t}}}={m_{\mathrm{t}}}-m_{\mathrm{\bar{t}}}=0.83\,^{+0.77}_{-1.01}\;\mathrm{GeV}$,
respectively, in good agreement with the hypothesis of conservation of the CPT
symmetry.

Figure 4: Reconstructed invariant mass of the top quark decay products in
single top $t$-channel events (left), and measured value of ${m_{\mathrm{t}}}$
compared to previous ATLAS and CMS results (right) [9].

The ATLAS measurement, performed with $\mathrm{t\bar{t}}$ candidate events in
the $\ell$+jets final state, is based on the reconstruction of the
$m_{\ell\mu}$ variable, defined as the invariant mass between the lepton from
the top quark decay and a soft, non-isolated muon originating from the
leptonic decay of a B hadron inside reconstructed a jet [22]. When the
$\mathrm{t\bar{t}}$ system is correctly reconstructed, the jet containing the
B hadron originates from the b quark from the leptonically-decaying top quark.
This measurement, performed with 36.1$\;\mathrm{fb}^{-1}$ of pp collision
data, is less sensitive to the calibration of the jet energy scale compared to
conventional direct measurements, and therefore carries additional information
when performing a combination of measurements. A dedicated algorithm was
developed in the scope of this measurement to identify the soft lepton
originated from the B hadron decay. Since this measurement is very sensitive
to the details of the b quark fragmentation, the parameters of the Bowler–Lund
fragmentation function have been re-optimized by means of a fit to LEP and SLC
data. In a dedicated study [23], the parameters derived in the context of this
analysis are found to well describe other distributions sensitive to the b
quark fragmentation. The top quark mass is measured to be
${m_{\mathrm{t}}}=178.48\pm 0.40\,(\mathrm{stat})\pm
0.67\,(\mathrm{syst})\;\mathrm{GeV}$, in agreement with the combination of
previous ATLAS measurements [24] within 2.2 standard deviations. The precision
of the measurement is limited by the uncertainty in the branching ratio of B
hadrons.

##  4 Summary

In this contribution, the most recent measurements of top quark mass and cross
section at the ATLAS and CMS experiments at the CERN LHC are reviewed,
including several new results. These include two measurements of the
$\mathrm{t\bar{t}}$ production cross section in proton-proton collisions at
$\sqrt{\mathrm{s}}=5.02\;\mathrm{TeV}$ by the ATLAS and CMS Collaborations
based on a small dataset delivered by the CERN LHC in 2017. The results are in
good agreement with state-of-the-art theoretical calculations, and are limited
by the statistical uncertainty of the data. The CMS Collaboration has also
published a measurement of single- and double-differential $\mathrm{t\bar{t}}$
cross sections in the $\ell$+jets final state based on the full LHC Run 2
dataset, where the distributions in the resolved and boosted topologies are
unfolded simultaneously for the first time. This is achieved by means of a
profiled $\chi^{2}$ fit, which also allows the impact of systematic
uncertainties to be significantly reduced. The $\mathrm{t\bar{t}}$ cross
section is also measured inclusively with a total uncertainty of 3.1%, the
most precise result in this channel, to date. In this analysis, a
significantly softer-than-predicted spectrum of the top quark transverse
momentum is observed, consistent with several previous measurements. Finally,
a direct measurement of the top quark mass using single top quark events in
the $t$-channel is presented for the first time. The value of the top quark
mass is measured both inclusively and separately for top quarks and
antiquarks, which allows for a test of the conservation of the CPT symmetry.

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