Scalar Weak Gravity Conjecture in Super Yang-Mills Inflationary Model
Jafar Sadeghi⋆111Email<EMAIL_ADDRESS>Mohammad Reza Alipour ⋆222Email:
<EMAIL_ADDRESS>Saeed Noori Gashti⋆333Email:
<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran

###### Abstract

In this article, we want to check four inflation models, such as composite NJL
inflation (NJLI), Glueball inflation(GI), super Yang-Mills inflation (SYMI),
and Orientifold inflation (OI), with two conjectures of the swampland program:
scalar weak gravity conjecture (SWGC) and strong scalar weak gravity
conjecture (SSWGC) since all these models violate the dS swampland
conjecture(DSC) but are compatible with further refining de Sitter swampland
conjecture (FRDSSC) through manual adjustment of free parameters of the
mentioned conjecture. We want to study the simultaneous compatibility of each
model with these two new conjectures. Despite being consistent with (FRDSSC),
we find that all models are not compatible with the other conjectures of the
Swampland program in all regions, and these conjectures are only satisfied in
a specific area. Also, due to the presence of constant parameter $(\phi_{0})$
in the higher orders derivatives, the (SYMI) and (OI) among all the models are
more compatible with all conjectures of the swampland program. These models
can provide a more significant amount of satisfaction with all of them. They
can be suitable and accurate inflation models for a more profound examination
of universe developments. We determined a particular region for these models
is compatible with (FRDSSC), (SWGC), and (SSWGC) simultaneously.

Keywords: further refining the de sitter swampland conjecture, inflation
models, scalar weak gravity conjecture, strong scalar weak gravity conjecture.

###### Contents

  1. 1 Introduction
  2. 2 Overview of inflationary models
    1. 2.1 Model I: NJLI
    2. 2.2 Model II: GI
    3. 2.3 Model III: SYMI
    4. 2.4 Model IV: OI
  3. 3 SWGC and SSWGC on inflation Models
    1. 3.1 Model I
    2. 3.2 Model II
    3. 3.3 Model III
  4. 4 Concluding remarks

##  1 Introduction

Swampland program has recently been presented to evaluate and prove string
theory to investigate effective low energy theories related to quantum
gravity. In recent years, many efforts have been made to develop the theory of
everything, and perhaps string theory is one of the most well-known theories
in this route. Since, the string theory is explained very clearly, we can
expect various consequences from it in cosmology. As a result, a wide range of
ideas and structures in the literature takes an in-depth look at the
cosmological implications of string theory[2, 5, 8, 19]. In string theory,
many possible vacuums are created, which are one of the exciting aspects of
the theory’s cosmology; the set of these also forms the string theory
landscape. The important question in this regard is which of the effective low
energy theories can be compatible with the string theory[2, 5, 8, 19].
Therefore, the swampland program was introduced. This program includes many
conjectures, including weak gravity conjecture (WGC), dS and AdS conjectures,
SWGC, SSWGC, TCC, etc[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. The
collection of effective low-energy theories compatible with quantum gravity
lives in the landscape. A broader area surrounds the landscape, and the set of
theories incompatible with quantum gravity is placed in this area, referred to
as the swampland. Many researchers have accepted string theory as a theory
that determines quantum gravity. Therefore, effective low-energy theories
compatible with these swampland conjectures are aligned with quantum gravity,
which can be of great help in finding a solution to advance this great
question that has occupied researchers for years, namely, quantum gravity. So
far, much work has been examined about the swampland program in literature;
you can see Ref.s[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39] for further study. The swampland program has played an important
role in finding phenomena consistent with quantum gravity. It has been used in
various parts of physics, including black holes, inflation, and dark energy.
Recently, many improvements have been made in the swampland program, which can
be used to solve many cosmological problems. Among them, we can mention the de
Sitter and refined de Sitter swampland conjecture that it is possible to find
models compatible with quantum gravity by using the derivative of scalar field
potentials [45, 46]. One investigate the four-dimensional theory of a real
field $\varphi^{i}$ coupled to gravity, whose dynamics can be controlled using
a scalar potential $V(\varphi^{j})$ and whose action is as follows [46, 47],

$S=\int_{4D}d^{4}x\sqrt{-g}\left[-\frac{1}{2}M_{p}^{2}R+\frac{1}{2}g^{\mu\nu}h_{ij}\partial_{\mu}\varphi^{i}\partial_{\nu}\varphi^{j}-V\right]$
(1)

where $g$ is the matrix metric in the four-dimensional space, $M_{p}$ is the
Planck mass, $R$ is the Riemann curvature in the four-dimensional space, and
$h_{ij}$ is the metric of the field space. Therefore, we examine different
phenomenological models, such as inflation, described by action (1). Among the
conjectures of the swampland program used to investigate cosmology is the de
Sitter and refined de Sitter swampland conjecture, which state that the
effective theories of quantum gravity placed in the landscape must satisfy at
least one of the following constraints [48, 49],

$|\nabla V|\geq\frac{c_{1}}{M_{p}}V,\hskip
12.0ptmin(\nabla_{i}\nabla_{j}V)\leq-\frac{c_{2}}{M_{pl}^{2}}V$ (2)

The above equations for the $V>0$ can be rewritten in terms of the slow-roll
parameters as follows,

$\sqrt{2\epsilon_{V}}\geq c_{1},\hskip 12.0ptor\hskip 12.0pt\eta_{V}\leq-
c_{2}$ (3)

where $c_{1}$ and $c_{2}$ are both positive and order of one, i.e.,
$c_{1}=c_{2}=\mathcal{O}(1)$. Also, the left side of equation (2) is related
to the main swampland conjecture. Recently, David Andriot and Christoph Roupec
combined the two of the de Sitter swampland conjecture and refined and
formulated them, which is called further refining de Sitter swampland
conjecture. It states that an effective low-energy theory of quantum gravity
that consider the action equation(1) must satisfy the following relation [46,
47],

$\bigg{(}M_{p}\frac{|\nabla
V|}{V}\bigg{)}^{q}-aM_{p}^{2}\frac{min(\nabla_{i}\nabla_{j}V)}{V}\geq b,$ (4)

where $a+b=1$, $a,b>0$, $q>2$. $a$, $b$, and $q$ are its free constant
parameters that create a restriction for this conjecture. Many inflationary
models have been investigated using this conjecture. The advantage of this
conjecture over the old conjecture is that, unlike the refined dS conjecture,
it doesn’t sound inconsistent with the slow-roll single-field inflationary
models [47]. One of the other important conjectures of the swampland program
is the weak gravity conjecture $WGC$, which states that gravity is the weakest
force. Also, Palti generalized the WGC and showed the scalar field forces are
stronger than gravity [50, 51]. Considering a particle $h$ with mass $m$
coupled to a light scalar $\varphi$ whose mass is a function of the scalar, in
that case, the scalar weak gravity conjecture ($SWGC$) states that the
intermediate force is stronger than gravity, and assuming
$m^{2}=V^{\prime\prime}=\frac{\partial^{2}V}{\partial\varphi^{2}}$, we have
the following condition for SWGC,

$(V^{(3)})^{2}\geq\frac{(V^{(2)})^{2}}{M_{p}^{2}},$ (5)

where the power number in the parentheses means the order of the derivative
relative to $\varphi$. Also, Eduardo Gonzalo and Luis E.Ibáñez suggested a
strong version of SWGC, i.e., SSWGC [52] which expresses that the potential of
any canonically normalized real scalar $V(\varphi)$ must satisfy any value of
the field restriction:

$2(V^{(3)})^{2}-V^{(2)}V^{(4)}\geq\frac{(V^{(2)})^{2}}{M_{p}^{2}}.$ (6)

In this article, we are trying to test the considered inflation models with
$SWGC$ and $SSWGC$ conjectures to find the model compatible with quantum
gravity. Therefore, according to all the above explanations, we organize the
article.
In section 2, we overview the inflation models such as (NJLI), (GI), (SYMI)
and (OI) in 4 subsections. In section 3, we challenge these inflationary
models with two conjectures of the swampland program, i.e., the SWGC and
SSWGC. We will discuss the compatibility or incompatibility of each model with
mentioned conjecture and determine the consistent regions. We compare the
results with each other. Finally, we describe the outcomes in Section 4.

##  2 Overview of inflationary models

In this section, according to [1], we briefly introduce four inflation models
composite NJL Inflation(NJLI), Glueball inflation(GI), super Yang-Mills
inflation (SYMI), and Orientifold inflation (OI). We review the results of
each model’s compatibility with the two swampland conjectures described in
[1]. Then, in the next section, using the potential of each model, we will
check other important conjectures of the swampland program, namely (SWG) and
(SSWG). Finally, we will introduce the results thoroughly, and the best model
that has the most compatibility with all conjectures presented as well as the
best inflation model to examine the universe development.

###  2.1 Model I: NJLI

The action expressing the inflationary model that the inflation has a non-
minimally coupling with gravity is defined as follows in Jordan’s framework
[1, 40],

$\begin{split}&S_{J}=\int
d^{4}x\sqrt{-g}\bigg{(}-\frac{1}{2}M_{p}^{2}R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{\xi
R}{2}\Big{[}\varphi^{2}-\frac{\upsilon^{2}}{2}\Big{]}-V_{J}(\varphi)\bigg{)}\\\
&V_{J}(\varphi)=-\frac{1}{2}m^{2}_{\varphi}\varphi^{2}+\frac{1}{2}\lambda\varphi^{4},\end{split}$
(7)

where ($\upsilon$) and ($\varphi$) specify vacuum expectation value and
inflation field. Also, the index (J) shows the Jordan frame. The action, as
mentioned earlier, can be transformed into an Einstein framework by applying
conformal transformation with a new canonical normalized field as a minimally
coupled form. Hence, this conformal transformation is expressed in the
following form [40, 41],

$\begin{split}\widetilde{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu}=\bigg{(}1+\frac{\xi(\varphi^{2}-\upsilon^{2}/2)}{M_{p}^{2}}\bigg{)}g_{\mu\nu}.\end{split}$
(8)

The action in equation (7) is rewritten in Einstein’s framework, where the
index (E) is the characteristic of Einstein’s frame,

$\begin{split}S_{E}=\int
d^{4}x\sqrt{-g}\bigg{(}-\frac{1}{2}M_{p}^{2}R+\frac{1}{2}\Omega^{-4}\Big{(}\Omega^{2}+\frac{6\xi\varphi^{2}}{M_{p}^{2}}\Big{)}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-U(\varphi)\bigg{)}\\\
\end{split}$ (9)

where

$\begin{split}&\Omega^{2}=\bigg{(}1+\frac{\xi(\varphi^{2}-\upsilon^{2}/2)}{M_{p}^{2}}\bigg{)}\\\
&U(\varphi)\equiv\Omega^{-4}V_{J}(\varphi)\end{split}$ (10)

According to [1] by introducing a new canonically normalized scalar field, we
will have,

$\begin{split}\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi(\varphi)\partial_{\nu}\chi(\varphi)=\frac{1}{2}\big{(}\frac{d\chi}{d\varphi}\big{)}^{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi,\end{split}$
(11)

where

$\begin{split}\big{(}\frac{d\chi}{d\varphi}\big{)}=\sqrt{\Omega^{-4}\Big{(}\Omega^{2}+\frac{6\xi\varphi^{2}}{M_{p}^{2}}\Big{)}}.\end{split}$
(12)

In the limit $\xi\varphi^{2}\ll M_{p}^{2}$, that is, the small field values,
the potential for the new field becomes the original field, which is not valid
for the $\xi\varphi^{2}\gg M_{p}^{2}$. Therefore, the field solution
($\varphi$) is rewritten according to the new field $\chi$ in the following
form [1, 40],

$\begin{split}\varphi\simeq\frac{M_{p}}{\sqrt{\xi}}\exp\Big{(}\frac{\chi}{\sqrt{6}M_{p}}\Big{)}.\end{split}$
(13)

Thus, the effective potential is also expressed as follows,

$\begin{split}U(\chi)\simeq\frac{\lambda
M_{p}^{4}}{2\xi^{2}}\bigg{(}1+\exp\Big{[}-\frac{2\chi}{\sqrt{6}M_{p}}\Big{]}\bigg{)}^{-2}.\end{split}$
(14)

The authors in [1] challenged this inflationary model according to one of the
conjectures of the swampland program. It was found that the model is in strong
tension with dS swampland conjectures because
$C_{1}=C_{2}\neq\mathcal{O}(1)$[1]. Therefore, they checked the model with
another conjecture: further refining dS swampland conjecture. By manually
adjusting the parameters of the mentioned conjecture, namely a, b and q, they
showed that the mentioned model is compatible with it. In the next section, we
will examine this model with other conjectures of the swampland program,i.e.,
(SWGC) and (SSWGC), and explain the results in detail.

###  2.2 Model II: GI

We provide a brief description of this model. According to [1], the action of
the corresponding model is expressed in the following form, where the model
has a general non-minimal coupling to gravity [42, 43],

$\begin{split}S=\int
d^{4}x\sqrt{-g}\bigg{(}-\frac{M_{p}^{2}+\xi\Lambda^{2}(\phi/\phi_{0})^{2}}{2}R+L_{GI}\bigg{)},\end{split}$
(15)

where,

$\begin{split}&L_{GI}=\varphi^{-3/2}\partial_{\mu}\varphi\partial^{\mu}\varphi-\frac{\varphi}{2}\ln\big{(}\frac{\varphi}{\Lambda^{4}}\big{)}\\\
&\frac{\varphi}{\Lambda^{4}}=\big{(}\frac{\phi}{\phi_{0}}\big{)}^{4}\\\
&\phi_{0}=4\sqrt{2}\Lambda,\end{split}$ (16)

which $\Lambda$ is called mass scale, and parameter $\xi$ characterized the
coupling to gravity, respectively. According to the above explanations, the
action in Einstein’s frame is rewritten as follows [1, 42, 43],

$\begin{split}S=\int
d^{4}x\sqrt{-g}\bigg{(}-\frac{1}{2}M_{p}^{2}R+\Omega^{-2}\bigg{[}1+\frac{3\xi^{2}\Lambda^{2}(\phi/\phi_{0})^{2}}{16M_{p}^{2}}\Omega^{-2}\bigg{]}\big{(}\frac{\Lambda}{\phi_{0}}\big{)}^{2}\partial_{\mu}\phi\partial^{\mu}\phi-\Omega^{-4}V_{GI}\bigg{)}\end{split}$
(17)

where
$\Omega^{2}=\big{(}M_{P}^{2}+\xi\Lambda^{2}(\phi/\phi_{0})^{2}\big{)}/M_{p}^{2}$.
According to $\xi\neq 0$ and the large field limit, the mentioned equation is
reduced to $\Omega^{2}\simeq\xi\Lambda^{2}(\phi/\phi_{0})^{2}/M_{p}^{2}$, and
the potential is calculated [1, 42, 43],

$\begin{split}V_{GI}=2\Lambda^{4}(\phi/\phi_{0})^{4}\ln(\phi/\phi_{0})\end{split}$
(18)

where $\phi_{0}\equiv 4\sqrt{2}\Lambda$. According to the definition, we will
consider a canonically normalized field $\chi$ associated with ($\phi$),

$\begin{split}\frac{1}{2}\widetilde{g}^{\mu\nu}\partial_{\mu}\chi(\phi)\partial_{\nu}\chi(\phi)=\frac{1}{2}\big{(}\frac{d\chi}{d\phi}\big{)}^{2}\widetilde{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\end{split}$
(19)

where

$\begin{split}\frac{1}{2}\big{(}\frac{d\chi}{d\phi}\big{)}^{2}=\Omega^{-2}\bigg{[}1+\frac{3\xi^{2}\Lambda^{2}(\phi/\phi_{0})^{2}}{16M_{p}^{2}}\Omega^{-2}\bigg{]}\big{(}\frac{\Lambda}{\phi_{0}}\big{)}^{2}\end{split}$
(20)

By considering a condition as $\chi\propto\ln\phi$, the potential reduces to
$\Omega^{-4}V_{GI}\propto\ln\phi$ [1, 42, 43]. Hence, we will have the
following equation with respect to the canonically normalized field,

$\begin{split}S_{E}=\int
d^{4}x\sqrt{-g}\bigg{(}-\frac{1}{2}M_{p}^{2}R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-
U_{GI}(\chi)\bigg{)},\end{split}$ (21)

where

$\begin{split}U_{GI}(\chi)=\Omega^{-4}V_{GI}(\varphi).\end{split}$ (22)

The potential of this model in Einstein’s frame is rewritten in terms of field
($\phi$) according to [1], slow-roll analysis and the large field regime
$N_{c}^{2}\xi\Lambda^{2}(\phi/\phi_{0})^{2}\gg M_{p}^{2}$,

$\begin{split}U_{GI}(\phi)=\frac{2M_{p}^{4}}{\xi^{2}}\ln\big{(}\frac{\phi}{\phi_{0}}\big{)}\end{split}$
(23)

Similar to the previous situation, the authors in[1] concluded that this model
is incompatible with the dS conjecture and does not satisfy it. Then they
applied FRDSSC to this model and proved complete compatibility between the
model and this conjecture by manually adjusting the free parameters. We will
also challenge this model with two other conjectures of the swampland program
and describe the results in detail.

###  2.3 Model III: SYMI

We also briefly explain this model, which you can see [1] for further study.
The action in Jordan’s frame for this model is generally expressed in the
following form according to the non-minimally coupled gravity structure [42],

$\begin{split}S_{J}=\int
d^{4}x\sqrt{-g}\bigg{(}-\frac{M^{2}+N_{c}^{2}\xi\Lambda^{2}(\phi/\phi_{0})^{2}}{2}R+L_{SYM}\bigg{)}\end{split}$
(24)

where

$\begin{split}&L_{SYM}=-\frac{N_{c}^{2}}{\alpha}(\varphi\varphi^{\dagger})^{-2/3}\partial_{\mu}\varphi\partial^{\mu}\varphi^{\dagger}-\frac{4\alpha
N_{c}^{2}}{9}(\varphi\varphi^{\dagger})^{2/3}\ln\big{(}\frac{\varphi}{\Lambda^{3}}\big{)}\ln\big{(}\frac{\varphi^{\dagger}}{\Lambda^{3}}\big{)}\\\
&\frac{\varphi}{\Lambda^{3}}=\big{(}\frac{\phi}{\phi_{0}}\big{)}^{3}\\\
&\phi_{0}=3N_{c}\big{(}\frac{2}{\alpha}\big{)}^{1/2}\Lambda.\end{split}$ (25)

In the above equations, $\alpha$ is a constant parameter, and $\Lambda$ is the
mass scale. We also focus on the real part of the inflation field ie
($\varphi=\varphi^{\dagger}$) [1, 42]. The above action is rewritten in
Einstein’s framework [42],

$\begin{split}&S_{E}=\int
d^{4}x\sqrt{-g}\bigg{(}-\frac{1}{2}M_{p}^{2}R+\frac{9N_{c}^{2}}{\alpha}\Omega^{-2}\bigg{[}1+\frac{\alpha
N_{c}^{2}\xi^{2}}{3M_{p}^{2}}\Omega^{-2}\Lambda^{2}\big{(}\frac{\phi}{\phi_{0}}\big{)}^{2}\bigg{]}\\\
&\times\big{(}\frac{\Lambda}{\phi_{0}}\big{)}^{2}\partial_{\mu}\phi\partial^{\mu}\phi-\Omega^{-4}V_{SYM}\bigg{)}\end{split}$
(26)

Assuming $\xi\neq 0$, $\Omega^{2}\simeq
N_{c}^{2}\xi\Lambda^{2}(\phi/\phi_{0})^{2}/M_{p}^{2}$ and according to the
explanations in[1], we will have

$\begin{split}V_{SYM}(\phi)=4\alpha
N_{c}^{2}\Lambda^{4}\big{(}\frac{\phi}{\phi_{0}}\big{)}^{4}\ln^{2}\big{(}\frac{\phi}{\phi_{0}}\big{)}\end{split}$
(27)

By introducing a canonically normalized field related to the $\phi$ through
the following relations [1],

$\begin{split}\frac{1}{2}\widetilde{g}^{\mu\nu}\partial_{\mu}\chi(\phi)\partial_{\nu}\chi(\phi)=\frac{1}{2}\big{(}\frac{d\chi}{d\phi}\big{)}^{2}\widetilde{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\end{split}$
(28)

where

$\begin{split}\frac{1}{2}\big{(}\frac{d\chi}{d\phi}\big{)}^{2}=\frac{9N_{c}^{2}}{\alpha}\Omega^{-2}\bigg{[}1+\frac{\alpha
N_{c}^{2}\xi^{2}}{3M_{p}^{2}}\Omega^{-2}\Lambda^{2}\big{(}\frac{\phi}{\phi_{0}}\big{)}^{2}\bigg{]}\big{(}\frac{\Lambda}{\phi_{0}}\big{)}^{2}\end{split}$
(29)

Thus, we can rewrite the action in Einstein’s frame in terms of the
canonically normalized field as follows,

$\begin{split}S_{E}=\int
d^{4}x\sqrt{-g}\bigg{(}-\frac{1}{2}M_{p}^{2}R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-
U_{SYM}(\chi)\bigg{)}\end{split}$ (30)

where

$\begin{split}U_{SYM}(\chi)=\Omega^{-4}V_{SYM}(\chi)\end{split}$ (31)

According to [1] and slow-roll analysis of the potential, also taking into
account the large field regime
$N_{c}^{2}\xi\Lambda^{2}\big{(}\phi/\phi_{0}\big{)}^{2}\gg M_{p}$, the model’s
potential in Einstein’s frame is calculated in terms of the field $\phi$ [44],

$\begin{split}U_{SYM}(\phi)=\frac{4\alpha}{N_{c}^{2}}\frac{M_{p}^{4}}{\xi^{2}}\ln^{2}\big{(}\frac{\phi}{\phi_{0}}\big{)}.\end{split}$
(32)

The authors showed in [1] that this model, similar to the previous models,
violates the dS swampland conjecture and agrees and is compatible with FRDSSC.
Therefore, in this article, we challenge this model concerning other
conjectures of the swampland. So we can introduce the best model that is more
compatible with all conjectures of the swampland program.

###  2.4 Model IV: OI

To introduce the final model, we start by expressing the action in the Jordan
frame by considering non-minimal coupled to gravity; hence we have [42],

$\begin{split}S_{J}=\int
d^{4}x\sqrt{-g}\bigg{(}-\frac{M^{2}+N_{c}^{2}\xi\Lambda^{2}\big{(}\frac{\phi}{\phi_{0}}\big{)}^{2}}{2}R+L_{OI}\bigg{)}\end{split}$
(33)

where

$\begin{split}&L_{OI}=-\frac{N_{c}^{2}}{\alpha_{OI}}(\varphi\varphi^{\dagger})^{-2/3}\partial_{\mu}\varphi\partial^{\mu}\varphi^{\dagger}-\frac{4\alpha_{OI}N_{c}^{2}}{9}(\varphi\varphi^{\dagger})^{2/3}\bigg{[}\ln\big{(}\frac{\varphi}{\Lambda^{3}}\big{)}\ln\big{(}\frac{\varphi^{\dagger}}{\Lambda^{3}}\big{)}-\beta\bigg{]}\\\
&\frac{\varphi}{\Lambda^{3}}=\big{(}\frac{\phi}{\phi_{0}}\big{)}^{3}\\\
&\phi_{0}=3N_{c}\big{(}\frac{2}{\alpha}\big{)}^{1/2}\Lambda\end{split}$ (34)

where M is a mass scale, $\beta=\mathcal{O}(1/N_{c})$ and
$\varphi=\varphi^{\dagger}$, the above action in Einstein’s frame is in the
following form.

$\begin{split}&S_{E}=\int
d^{4}x\sqrt{-g}\bigg{[}-\frac{1}{2}M_{p}^{2}R+\frac{9N_{c}^{2}}{\alpha}\Omega^{-2}\bigg{(}1+\frac{\alpha
N_{c}^{2}\xi^{2}}{3M_{p}^{2}}\Omega^{-2}\Lambda^{2}\big{(}\frac{\phi}{\phi_{0}}\big{)}^{2}\bigg{)}\big{(}\frac{\Lambda}{\phi_{0}}\big{)}^{2}\\\
&\times\partial_{\mu}\phi\partial^{\mu}\phi-\Omega^{-4}V_{OI}\bigg{]}\end{split}$
(35)

Here, taking into account conditions such as $\xi\neq 0$ and with respect
to[1], we will have,

$\begin{split}V_{OI}(\phi)=4\alpha
N_{c}^{2}\Lambda^{4}\big{(}\frac{\phi}{\phi_{0}}\big{)}^{4}\bigg{[}\ln^{2}(\frac{\phi}{\phi_{0}}\big{)}-\frac{\beta}{9}\bigg{]}\end{split}$
(36)

It is possible to introduce a canonically normalized field related to the
$\phi$ with respect to the following equation,

$\begin{split}\frac{1}{2}\widetilde{g}^{\mu\nu}\partial_{\mu}\chi(\phi)\partial_{\nu}\chi(\phi)=\frac{1}{2}\big{(}\frac{d\chi}{d\phi}\big{)}^{2}\widetilde{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi,\end{split}$
(37)

where

$\begin{split}\frac{1}{2}\big{(}\frac{d\chi}{d\phi}\big{)}^{2}=\frac{9N_{c}^{2}}{\alpha}\Omega^{-2}\bigg{(}1+\frac{\alpha
N_{c}^{2}\xi^{2}}{3M_{p}^{2}}\Omega^{-2}\Lambda^{2}\big{(}\frac{\phi}{\phi_{0}}\big{)}^{2}\bigg{)}\big{(}\frac{\Lambda}{\phi_{0}}\big{)}^{2}.\end{split}$
(38)

According to the canonically normalized field, we have,

$\begin{split}S_{E}=\int
d^{4}x\sqrt{-g}\bigg{(}-\frac{1}{2}M_{p}^{2}R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-
U_{OI}(\chi)\bigg{)}\end{split}$ (39)

where

$\begin{split}U_{OI}(\chi)=\Omega^{-4}V_{OI}(\varphi)\end{split}$ (40)

Like the previous models, assuming conditions such as slow-roll analysis of
the potential, the large field regime
$N_{c}^{2}\xi\Lambda^{2}\big{(}\phi/\phi_{0}\big{)}^{2}\gg M^{2}$, and
according to [1], the final model potential is also calculated in the Einstein
frame as follows,

$\begin{split}U_{OI}(\phi)=\frac{4\alpha}{N_{c}^{2}}\frac{M_{P}^{4}}{\xi^{2}}\bigg{[}\ln^{2}\big{(}\frac{\phi}{\phi_{0}}\big{)}-\frac{\beta}{9}\bigg{]}\end{split}$
(41)

Like the previous models, this final model satisfies the FRDSSC while it is
inconsistent with the dS swampland conjecture. Hence, we face four models in
different structures, all of which violate the dS swampland conjecture and
satisfy FRDSSC. Therefore, we challenge all these models with other swampland
conjectures to introduce a model compatible with all conjectures of the
swampland, which can be considered a suitable model for further investigations
of universe evolutions.

##  3 SWGC and SSWGC on inflation Models

In this section, we apply the SWGC and SSWGC of the swampland program to the
mentioned inflation models. Considering that all mentioned models satisfy the
FRDSSC, we intend to choose the best model with the highest compatibility with
all of these swampland conjectures through further investigation. We will also
explain the results in detail.

###  3.1 Model I

According to the SWGC and SSWGC in equations (5), (6) and using the potential
of Model I (NJLI) in equation (14), we will have, we put $M_{p}=1$,

$U^{(1)}(\chi)\simeq\frac{2\lambda}{\sqrt{6}\xi^{2}}\frac{e^{\frac{2\chi}{\sqrt{6}}}}{(1+e^{\frac{2\chi}{\sqrt{6}}})^{3}},$
(42)
$U^{(2)}(\chi)\simeq\frac{-2\lambda}{3\xi^{2}}\frac{e^{\frac{4}{\sqrt{6}}}\chi}{(1+e^{\frac{2\chi}{\sqrt{6}}})^{4}}{(-2+e^{\frac{2\chi}{\sqrt{6}}})},$
(43)
$U^{(3)}(\chi)\simeq\frac{4\lambda}{3\sqrt{6}\xi^{2}}\frac{e^{\frac{4}{\sqrt{6}}\chi}(4-7e^{\frac{2\chi}{\sqrt{6}}}+e^{\frac{4\chi}{\sqrt{6}}})}{(1+e^{\frac{2\chi}{\sqrt{6}}})^{5}},$
(44)
$U^{(4)}(\chi)\simeq\frac{-8\lambda}{9\xi^{2}}\frac{e^{\frac{4}{\sqrt{6}}\chi}(-8+33e^{\frac{2\chi}{\sqrt{6}}}-18e^{\frac{4\chi}{\sqrt{6}}}+e^{\frac{6\chi}{\sqrt{6}}})}{(1+e^{\frac{2\chi}{\sqrt{6}}})^{6}}.$
(45)

Now, we put the above equation in the $SWGC$ equation(5). So, we get the
following relationship,

$\frac{4\lambda^{2}e^{\frac{8\chi}{\sqrt{6}}}}{9\xi^{4}(1+e^{\frac{2\chi}{\sqrt{6}}})^{10}}\left[-(2+e^{\frac{2\chi}{\sqrt{6}}}-e^{\frac{4\chi}{\sqrt{6}}})^{2}+\frac{2}{3}(4-7e^{\frac{2\chi}{\sqrt{6}}}+e^{\frac{4\chi}{\sqrt{6}}})^{2}\right]\geq
0$ (46)

The above relation is greater than zero if the following condition is met,

$(\frac{2}{\sqrt{6}}+1)e^{\frac{4\chi}{\sqrt{6}}}-(\frac{14}{\sqrt{6}}+1)e^{\frac{2\chi}{\sqrt{6}}}+(\frac{8}{\sqrt{6}}+2)\geq
0$ (47)

Now, we use the change of variable $y=e^{\dfrac{2\chi}{\sqrt{6}}}$. A 2nd-
degree equation is obtained in the following form,

$f(y)=(\frac{2}{\sqrt{6}}+1)y^{2}-(\frac{14}{\sqrt{6}}+1)y+(\frac{8}{\sqrt{6}}+2)\geq
0.$ (48)

First, we get the points where the function $f(y)=0$. We also calculate its
minimum point. So, we will have,

$f(y)=0\longrightarrow(y=1.129,y=2.568)\qquad\frac{\partial f(y)}{\partial
y}=0\longrightarrow y_{min}=1.849,f(y_{min})=-0.941.$ (49)

Since, the minimum point is negative, $f(y)$ has a negative value in the
interval $1.129<y<2.568$, so the SWGC is not satisfied in this range. We see
the SWGC is met in the $\chi<0.148$ and $\chi>1.154$. Next, we check the SSWGC
in equation (6). So, we will have,

$\frac{4\lambda^{2}\exp\big{(}\frac{8\chi}{\sqrt{6}}\big{)}}{27\xi^{4}(1+\exp\big{(}\frac{2\chi}{\sqrt{6}}\big{)})^{10}}\bigg{[}20-88\exp\big{(}\frac{2\chi}{\sqrt{6}}\big{)}+99\exp\big{(}\frac{4\chi}{\sqrt{6}}\big{)}-\exp\big{(}\frac{8\chi}{\sqrt{6}}\big{)}-10\exp\big{(}\sqrt{6}\chi\big{)}\bigg{]}\geq
0.$ (50)

The above relationship is established if that,

$F(\chi)=\bigg{[}20-88\exp\big{(}\frac{2\chi}{\sqrt{6}}\big{)}+99\exp\big{(}\frac{4\chi}{\sqrt{6}}\big{)}-\exp\big{(}\frac{8\chi}{\sqrt{6}}\big{)}-10\exp\big{(}\sqrt{6}\chi\big{)}\bigg{]}\geq
0.$

As a result, we will plot a figure to check it.

Figure 1: $F(\chi)$

From the Figure 1, we find that when $\chi\leq 2.068$, $F(\chi)\geq 0$, SSWGC
is established. Therefore, comparing the two conjectures, we can see that
these two conjectures can be satisfied when $1.154\leq\chi\leq 2.068$ and
$\chi\leq 0.148$.

###  3.2 Model II

We go through a similar process for all models to determine their
compatibility with the mentioned conjectures. Therefore, with respect to the
equations (5) and (6), also the potential of model II (GI) in equation(23),
one can calculate,

$U_{GI}^{1}(\phi)=\frac{2}{\xi^{2}\phi}$ (51)
$U_{GI}^{2}(\phi)=-\frac{2}{\xi^{2}\phi^{2}}$ (52)
$U_{GI}^{3}(\phi)=\frac{4}{\xi^{2}\phi^{3}}$ (53)
$U_{GI}^{4}(\phi)=-\frac{12}{\xi^{2}\phi^{4}}$ (54)

Now, we first consider the SWGC. By putting the above relation in the
equation(5), one can obtain

$\frac{4}{\xi^{2}\phi^{6}}(4-\phi^{2})\geq 0$ (55)

According to the above relation, the SWGC is satisfied when $\phi<2$. Next, we
examine the SSWGC,

$\frac{4}{\xi^{2}\phi^{6}}(2-\phi^{2})\geq 0$ (56)

According to the above relation, when $\phi<\sqrt{2}$, the SSWGC will be
satisfied. Therefore, both them (SWGC) and (SSWGC) are satisfied by sharing
two conjectures, when $\phi<\sqrt{2}$.

###  3.3 Model III

We apply the conjectures to model III (SYMI), so with respect to equation (5),
(6) and (32), we have

$U^{1}_{SYM}(\phi)=\frac{8\alpha}{N_{c}^{2}\xi^{2}\phi}(\ln(\frac{\phi}{\phi_{0}}))$
(57)
$U^{2}_{SYM}(\phi)=\frac{8\alpha}{N_{c}^{2}\xi^{2}\phi^{2}}\big{(}1-\ln\big{(}\frac{\phi}{\phi_{0}}\big{)}\big{)}$
(58)
$U^{3}_{SYM}(\phi)=\frac{8\alpha}{N_{c}^{2}\xi^{2}\phi^{3}}\big{(}-3+2\ln\big{(}\frac{\phi}{\phi_{0}}\big{)}\big{)}$
(59)
$U^{4}_{SYM}(\phi)=\frac{8\alpha}{N_{c}^{2}\xi^{2}\phi^{4}}\big{(}11-6\ln\big{(}\frac{\phi}{\phi_{0}}\big{)}\big{)}$
(60)

First, we consider SWGC. So, one can obtain,

$\frac{64\alpha^{2}}{N_{c}^{4}\xi^{4}\phi^{6}}F(\phi)\geq 0,\hskip
12.0ptF(\phi)=\bigg{[}\big{(}3-2\ln\big{(}\frac{\phi}{\phi_{0}}\big{)}\big{)}^{2}-\phi^{2}\big{(}1-\ln\big{(}\frac{\phi}{\phi_{0}}\big{)}\big{)}^{2}\bigg{]}.$
(61)

Next, we discuss the SSWGC and reach the following equation,

$\frac{64\alpha^{2}}{N_{c}^{4}\xi^{4}\phi^{6}}G(\phi)\geq 0,\hskip
12.0ptG(\phi)=\bigg{[}7-\phi^{2}+(2\phi^{2}-7)\ln\big{(}\frac{\phi}{\phi_{0}}\big{)}+(2-\phi^{2})\big{(}\ln\big{(}\frac{\phi}{\phi_{0}}\big{)}\big{)}^{2}\bigg{]}.$
(62)

In order for the two conjectures met, they must be $F(\phi)\geq 0$ and
$G(\phi)\geq 0$. We need to find their common points to see in which interval
of $\phi$ these conjectures are valid. Since, $F(\phi)$ and $G(\phi)$ are
complex relations, we are trying to examine using the plot to find the points
compatible with these conjectures.

(a)  (b)  (c)

Figure 2: $(a)\phi_{0}=2\qquad,\qquad(b)\phi_{0}=2.37\qquad
and\qquad(c)\phi_{0}=2.6$

According to the above plots, both functions are positive for different
$\phi_{0}$ in different ranges, and both conjectures are valid in these
ranges. As seen in the above figures, for $\phi_{0}\leq 2.37$, two curves meet
the $\phi$ axis at only one point, and the two conjectures are compatible only
in one interval. For example, in figure $2(a)$ in the range of $\phi\leq
6.141$ and for figure $2(b)$ in the range of $\phi\leq 7.184$, the two
conjectures are compatible. If for $\phi_{0}>2.37$, two curves intersect the
$\phi$ axis at three points, in this case, these two conjectures are
compatible in the two regions. For example, in figure $2(c)$, the two
conjectures are not compatible in the intervals of $\phi\leq 2.701$ and
$5.142\leq\phi\leq 7.822$. We can also find that when $\phi_{0}$ takes
different values, we have various allowed ranges and regions.

(a)  (b)

Figure 3: $(a)\phi_{0}=3\qquad,\qquad(b)\phi_{0}=4$

For $\phi_{0}=3$, which is shown in figure $3(a)$, two conjectures are
satisfied between $\phi\leq 2.394$ and $6.556\leq\phi\leq 8.948$. Also, two
conjectures are met for $\phi_{0}=4$ in the range of $\phi\leq 2.141$ and
$9.559\leq\phi\leq 11.675$. The potential structure for the fourth model
equation(41) in Einstein’s framework is similar to model III in equation (32).
Since, the SWGC and SSWGC of the swampland program are proportional to the
derivatives of higher orders, as is apparent in the equations (5) and (6).
Therefore, for the fourth model, the same results as model III are obtained,
and thus the last two models satisfy the FRDSSC, SWGC, and SSWGC,
respectively. Thus, these two models are more compatible with the swampland
program, which is somehow related to quantum gravity. They can be considered
desirable models to investigate the universe’s evolution. They are considered
favorable inflation models in terms of string theory structure. This way,
other cosmological applications of these models can be investigated more
profoundly and compared with the latest observable data.

##  4 Concluding remarks

In this article, we want to check four inflation models, such as composite NJL
Inflation(NJLI), Glueball Inflation(GI), super Yang-Mills Inflation (SYMI),
and Orientifold Inflation (OI), with two conjectures of the swampland program:
scalar weak gravity conjecture (SWGC) and strong scalar weak gravity
conjecture (SSWGC). Since all these models violate the dS swampland
conjecture(DSC) but are compatible with (FRDSSC) through manual adjustment of
free parameters of the mentioned conjecture. We studied the simultaneous
compatibility of each model with these two new conjectures. Despite being
consistent with (FRDSSC), we found that all models are not compatible with the
other conjectures of the Swampland program in all regions, and these
conjectures are only satisfied in a specific area. Also, Due to the presence
of constant parameter $(\phi_{0})$ in the higher orders derivatives, the
(SYMI) and (OI) among all the models are more compatible with all conjectures
of the swampland program. They can provide a more significant amount of
satisfaction with all of them. They can be suitable and accurate inflation
models for a more profound examination of universe developments. We determined
a particular region for these models are compatible with (FRDSSC), (SWGC), and
(SSWGC) simultaneously.
We can ask some questions, such as What are the consequences of these
conjectures for other models and theories? are there inflationary models that
satisfy all conjectures in all regions? Is it possible to extend these
conjectures to be consistent with any of the inflationary theories be
compatible? We have left the examination of these questions to future work.

Conflict of Interest
The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.
Data Availability Statement
Data sharing is not applicable to this article as no datasets were generated
or analyzed during the current study.

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11institutetext:  Intel Technology Poland, Gdansk, Poland
11email<EMAIL_ADDRESS>
# Quantization Aware Training, ERNIE and Kurtosis Regularizer:
an empirical study

Andrea Zanetti

###### Abstract

Pre-trained language models like ERNIE or BERT [2] are currently used in many
applications. These models come with a set of pre-trained weights typically
obtained in unsupervised/self-supervised modality on a huge amount of data.
After that, they are fine-tuned on a specific task. Applications then use
these models for inference, and often some additional constraints apply, like
low power-budget or low latency between input and output. The main avenue to
meet these additional requirements for the inference settings, is to use low
precision computation (e.g. INT8 rather than FP32), but this comes with the
cost of deteriorating the functional performance (e.g. accuracy) of the model.
Some approaches have been developed to tackle the problem and go beyond the
limitations of the PTO (Post-Training Quantization), more specifically the QAT
(Quantization Aware Training, see [4]) is a procedure that “interferes” with
the training process in order to make it affected (or simply ‘disturbed’) by
the quantization phase during the training itself. Besides QAT, recently
Intel-Habana Labs have proposed an additional and more direct way to make the
training results more robust to subsequent quantization which uses a
regularizer, therefore changing the loss function that drives the training
procedure. But their proposal does not work “out of the box” for pre-trained
models like ERNIE, for example. In this short paper we show why this is not
happening (for the ERNIE case) and we propose a very basic way to deal with
it, sharing as well some initial results (increase in final INT8 accuracy)
that might be of interest to practitioners willing to use ERNIE in their
applications, in low precision regime.

##  1 Introduction: Quantization Aware Training - what it is and why it is
useful

Quantization Aware Training (QAT) is a framework for training neural networks
that modifies the training procedure inserting (in the forward pass) fake-
quantize operations that simulate the actual quantization of the computations
(and of data used for it), typically leaving the backward pass untouched. The
overall goal of this technique is to make the final numeric format -that will
be used at inference time- affect the training procedure, in such a way to
make the final solution found by the learning algorithm more compatible and
thus better performant with the target low precision numeric format.

Among the entities that are quantized, most often there are some -if not all-
the parameters of the network. As mentioned above, in doing so the training
procedure is pushed to converge towards solutions (aka set of parameters) that
are likely to be more robust -in terms of degradation of performance- with
respect the quantization of the model itself. However, there are very many
possible ways to quantize a model. Typically, the initial numerical format is
FP32, which we assume here to be the format used for training, whereas the
destination numeric format, that we want to use at inference time, could be
FP16 or more commonly, INT8. For simplicity here we focus on a quantization
strategy that is simple but at the same time widely used, and therefore of
practical importance; we will steer our attention on the symmetric uniform
FP32 - to - INT8 quantization case, in which a tensor of FP32 numbers is
represented by a single FP32 numbers (scale) and an INT8 tensor of the same
size of the original FP32 tensor.

But how this mapping works? There are many techniques for choosing the most
appropriate representation of a FP32 tensor using a FP32 scale and a
corresponding tensor of INT8 numbers. For example, we can choose the scale
using the ABS_MAX approach as depicted in fig.1 (but the reader must be warned
that other approaches are possible):

Figure 1:  Example of the MAX_ABS approach to choose the scale for the
conversion of a tensor from FP32 into INT8. Figure 2:  Symmetric quantization:
visualization of the overall process.

In a nutshell, it can be said the QAT procedure makes the training procedure
aware of the quantization step and it does that in an implicit way due to the
fact that the quantization step is experienced by the network during training.

However, we may want to add an explicit push for the training towards specific
characteristics of the tensors that are useful for the purpose of using at
interference time our model with a reduce precision. But what characteristics
for our tensors should we enforce in order to make the quantization of a model
the least harmful possible? And how practically implement this enforcement?

##  2 A very smart idea from our colleagues in Intel-Habana Labs

Intel-Habana Labs have recently published a paper [1] which considers the
problem of steering the training procedure in such a way that the training
itself is guided to finding solutions (that is, a set of parameters in
parameter space) such that the subsequent quantization of the model is
minimally harmful. Their paper considers the case of uniform symmetrical
quantization and shows that in such settings, among all possible
distributions, the uniform distribution is the one that yields the minimal
quantization sensitivity (theorem 4 in the paper). The technique they devised
to guide the training is to add a regularizer to the loss function which is
computed using the distribution of all the weight tensors in the model at
hand. More precisely, they enforce the preference of the training procedure
for solutions which have uniformly distributed weight tensors adding a
regularizer that is obtained from the sum of the Kurtosis of each weight
tensors. As a reminder, given a set of n numbers (for example, a tensor T in
the Deep Learning parlance), its Kurtosis is calculated as follows, with the
operator ’$E$’ being the probabilistic expectation:

$\displaystyle K_{x}$
$\displaystyle=\frac{E\left[\left(x-\mu\right)^{4}\right]}{\left(E\left[\left(x-\mu\right)^{2}\right]\right)^{2}}$
$\displaystyle=E\left[\left(x-\mu\right)^{4}\right]\left(E\left[\left(x-\mu\right)^{2}\right]\right)^{-2}$
(1)

Even though the relation between distributions and Kurtosis values is not a
bijection, it turns out to be practically useful to take the value of Kurtosis
of the uniform distribution as a proxy of uniformity; that is, forcing the
training procedure to choose weight tensors which have the same Kurtosis value
as the uniform distribution is likely to push those tensors towards a more
uniform distribution. This fact, that in theory is not guaranteed, in practice
is well established especially for the models trained from scratch, where the
initial parameters distributions are typically Normal-like and surely well
verified for the models considered in the Intel-Habana Labs paper, which are
mostly based on convolutional architectures. Their paper [1] concludes with
the statement:

“This work focuses on weights but can also be used for activations. KURE (the
regularization based on Kurtosis) can be extended to other domains such as
recommendation systems and NLP models”

So, along with a QAT procedure, it is now also possible to deal with Kurtosis
regularizer to explicitly “tell” the training procedure to prefer more
uniformly distributed tensors. But will the Kurtosis regularized work out of
the box also for pre-trained models that need fine-tuning before being
eventually quantized and deployed on the target device? In addition, when
considering both strategies (QAT and Kurtosis regularizer) is better to use
them at the same time or pipeline them? The ambition of this short paper is
not to provide precise answer to these questions, but to share the results of
some experiments we have done in the context of QAT with an important pre-
trained model such as ERNIE, and the insertion of KURE, that is name given to
the Kurtosis regularizer, in the process

##  3 A special case of practical importance: applying kurtosis to pretrained
models – the ERNIE case

Here we present a case of application of Kurtosis as regularizer that is not
considered in the reference paper [1], but which is of practical interest.
The problem we consider is the following: pretrained models such as BERT or
ERNIE come with a lot of knowledge encoded in their parameters, thanks to the
huge amount of data that has been used for the pretraining itself, and the
smart pretraining techniques used. Typically, in order to fully leverage their
transfer learning potential, intermediate users of such models add one or few
layers on top of these models and do some fine tuning of the whole Neural
Network so obtained in order to get a final model for their task. The final
model then might be deployed on devices where the power budget is constrained
or where the memory footprint is critical, or even where the latency should be
minimized. And here is where QAT comes into play in order to get a low-
precision version of the ERNIE fine-tuned model that can be deployed on the
target device. Clearly the hope is to retain as much as possible the
performance of the FP32-original model into the final INT8 model. For all
these cases, the typical operational procedure is to:

  1. 1.

Obtain, for example, the ERNIE pretrained model

  2. 2.

Fine-tuning it on specific application-relevant data

  3. 3.

More fine-tuning but now in QAT modality, acquiring all the parameters
necessary for the subsequent quantization

  4. 4.

Quantize the model

Note that step 2) and 3) can be collapsed. Once reached point 4) it is
possible to deploy the obtained model in low-precision inference mode, for
example using INT8 as numerical format. But with these settings, where could
we add the Kurtosis regularizer?

We tested the options to add Kurtosis Regularizer to step 2) or adding it to
step 2) and and 3) when collapsed, gathering better results in the latter
case. However, it must be noticed that the naïve application of the Kurtosis
regularizer to pre-trained models such ERNIE may not always work out of the
box. The reason why this is like so (for example in the ERNIE case), it is
connected to the distribution that ERNIE parameters have after the
pretraining. Indeed, if we calculate the overall cost component due to the
Kurtosis regularizer using the Ernie pretrained parameters we obtain a number
that is in the order of 10E10. See fig. 10 for a visualization of the Kurtosis
values of the pretrained weight tensors in the ERNIE case. This is a huge
number and whatever cost function is, it is very unlikely that components of
the gradients in such an unbalanced scenario will be appropriately steering
the training, and in deeds the training does not proceed in this case. Of
course, taking this as black box issue, one could try to find an appropriate
lambda (regularizer coefficient) but this is not very practical (we tried!)
and does not offer any clarification or understanding of what is going on. And
in practice, it does not work at all! Therefore, a bit more of investigation
is necessary in order to understand why the Kurtosis behaves like so in the
ERNIE case.

In fig. 3 we show an idea about how Kurtosis could be computed using the
PaddlePaddle Deep Learning framework, showing also a possible way to select
what tensors to include in the computation, based on the tensors’ name.

Figure 3:  Concept of how Kurtosis could be added to a loss function in
Paddle, showing also how to select a subset of tensors for its computation.

The overall Kurtosis applied to ERNIE parameters is the sum the Kurtosis of
each parameter tensor used at each layer of the network. In the context of a
model which encompasses many, say M, tensors of parameters, we can then
consider the total (or the average) Kurtosis, which is computed as the sum of
the Kurtosis for each tensor of parameters.

$\displaystyle
TotalModelKurtosis=\sum\limits_{j=1}^{M}Kurtosis\left(T_{k}\right)$ (2)

Looking at each tensor distribution independently, it can be noticed that most
of the tensors after the ERNIE pre-training have Kurtosis which is manageable
and within reasonable orders of magnitude (say, values are within 1-100)
whereas some tensors happen to have Kurtosis in the order of 10E7 or higher.

We can visualize the distribution of those tensors to better understand what
it is going on. As we will see, looking at the list of tensors and their
kurtosis, it turns out that there is a regularity in the distribution of the
Kurtosis values. But let’s not run ahead of ourselves: before delving into the
details of the distribution of Kurtosis values, for the sake of clarify it is
worth reviewing some concepts about ERNIE and BERT models. They basically
consist of a pile (6, 12, 24 or even more) of transformer-encoder blocks, and
they are pretrained for different tasks and on a huge amount of data. Please
check the ERNIE paper [2] for more information. Each encoder block encompasses
a multihead-attention part and a feedforward part, as depicted in fig. 5.

Figure 4:  Transformer architecture (encoder on the left, decoder on the
right) as proposed in [3]. Figure 5:  Transformer Encoder block and Decoder
block. ERNIE and BERT are composed of only encoder blocks. The picture is
taken from https://jalammar.github.io/illustrated-transformer/.

Therefore, analyzing the distribution of the tensors of parameters corresponds
to analyzing the distribution of the parameters used in the self-attention
sub-layer and in the feed-forward sub-layer of each block. More precisely,
each self-attention sub-layer is composed of 4 parametrized steps, represented
in grey in figure 6, whereas each feed-forward Linear sub-layer is composed of
two parametrized steps, as described in the transformer paper already
mentioned in figure 7.

Figure 6:  Multihead attention scheme from [3]. Figure 7:  Extract from [3]

Therefore, analyzing the distribution of the tensors of parameters corresponds
to analyzing the distribution of the parameters used in the self-attention
sub-layer and in the feed-forward sub-layer of each block, in other words, of
each of the 6 parametrized “steps” in every ERNIE block. A very intuitive way
to do it, it is just to visualize it! So let’s have a look at the parameters
distributions for all the 12 blocks which we call also layers (as opposed to
sub-layers used above) of the Ernie model we used for our experiments (see
figure 8

Figure 8:  Distribution of weighs after ERNIE pretraining

In figure 8, from left to right we show the distribution from each encoder
block (or layer) of the following:

  1. 1.

feed-forward sub-layer: parameter tensor for 1st Fully Connected

  2. 2.

feed-forward sub-layer: parameter tensor for 2nd Fully Connected

  3. 3.

(Multihead) self-attention sub-layer: parameter tensor for Attention Keys

  4. 4.

(Multihead) self-attention sub-layer: parameter tensor for Attention outputs

  5. 5.

(Multihead) self-attention sub-layer: parameter tensor for Attention Queries

  6. 6.

(Multihead) self-attention sub-layer: parameter tensor for Attention Values

As it can be even visually noticed, the Ernie pretraining creates a regular
pattern of distributions across blocks and (looking closely) it can be noticed
that the parameter tensor for the 2nd Fully Connected in each feed-forward
sub-layer has a peculiar distribution, quite peaked and with big very long
tails of “outliers”. Outliers are not clearly visible in the histogram chart
above, but their presence is the reason why Tensorboard displays a very big
interval around the peak in the histogram. The existence of wide tails can be
better seen using the distribution tab in Tensorboard, to get (for the first
block) figure 9.

Figure 9:  Distribution of weighs after ERNIE pretraining (alternative view)

Again, we see that the second Fully Connected in the feed-forward sublayer of
the block is extremely peaked with wide tails. For such distribution that is
very peaked with wide tails, Kurtosis assumes extremely high values due the
division by the standard deviation, and therefore it becomes less
straightforward to enjoy the effect of Kurtosis regularizer on the fine-tuning
step of Ernie.

This can be clearly visualized with a chart that shows how the scale of
Kurtosis change as we include critical tensors of parameters in the analysis.

Figure 10:  Kurtosis in log10 scale: it can be seen that some of the tensors
(it is FFN_FC_1 weight tensor in each block) have Kurtosis of orders of
magnitude higher than the others. Note: from left to right we move from the
top layer to the bottom layer in ERNIE. There are 12 tensors with Kurtosis
above 100 (above 2 in log10 scale): these are the FFN_FC_1 weight tensor in
each ERNIE block, with ERNIE being composed of 12 blocks

##  4 Results and Conclusions

We now summarize what we have presented so far: we have found that the Ernie
pretraining generates a pattern of distributions in the model parameters such
that the Fully Connected operations, done as last operation of each ERNIE
block, have a peculiar distribution with a peaked core and wide tails. This
condition is the reason why the overall Kurtosis of the model becomes huge.
Other parameter tensors have Kurtosis that is in the same range of magnitude,
making the Kurtosis regularizer easily applicable for them.

The most basic approach to tackle the situation and to make it practical to
use Kurtosis regularizer is then to exclude those “critical” tensors from the
overall Kurtosis regularizer computation. This will have the effect to make
all the other tensors more “uniformly” distributed at the end of the fine-
tuning process, leaving those critical tensors, namely the 2nd Fully Connected
tensors in each ERNIE block, adapt during (QAT) fine tuning but without being
forced to move towards a more distributed profile. Please note that we do not
investigate here the reason why the pretraining originates this regularity in
the tensor distributions: this is a very interesting topic that we leave to
future investigations.

One additional point is worth noting: many scenarios are possible, depending
on the deployment target we have. Beyond the combination of QAT and Kurtosis,
the kind of quantization used, the set of quantizable operations is a “degree
of freedom” too. In this work we take the position that we want to quantize
matmul and mul operations, being ERNIE not based on convolutions. We report
here some numbers relative to what we get in our tests using the Chinese part
of the xnli dataset for a task of sentiment classification.

Figure 11:  examples of data entries in xnli dataset

Here we compare 2 cases:

  1. 1.

Pre-trained ERNIE followed by fine tuning in QAT with selected Kurtosis, and
lambda 0.5, on 60000 batches, with batch-size = 20

  2. 2.

Pre-trained ERNIE followed by fine tuning in QAT with NO Kurtosis on 60000
batches, with batch-size = 20

Figure 12:  effect of Kurtosis regularizer we obtained in our tests

So the addition of Kurtosis regularizer, applied selectively in the case of
ERNIE, has led to an increase to the final INT8 accuracy of 1.2We are aware of
the lack of generality of these results, and their sub-optimality due to the
reduced length of the fine-tuning process and the non-optimal batch size used
(both due to the limitations of the computational resources we used to for
this study) but, in spite of all that, the accuracy gap we obtained in tests
is an interesting indicator of the benefits brought by the Kurtosis
regularizer also for pre-trained models.

To effectively visualize the effect of a QAT fine tuning with or without
Kurtosis, fig. 13 shows the distribution of parameter values changes during
the training in the 2 cases.

Figure 13:  LEFT: QAT fine tuning of ERNIE with Kurtosis and lambda=0.5 RIGHT:
QAT fine tuning of Ernie without kurtosis. It can be noticed how the presence
of the Kurtosis regularizer progressively shapes the tensor distribution (in
this case we show the “key” part of the Key-Query-Value multi-attention
component in layer/block 0 of Ernie. The same effect is present in all tensors
that are included in the regularizer

## References

  * [1] Robust Quantization: One Model to Rule Them All, Shkolnik et al., arXiv:2002.07686
  * [2] ERNIE: Enhanced Language Representation with Informative Entities, Zhang et al. arXiv:1905.07129
  * [3] Attention is all you need, Aswani et al., arXiv:1706.03762
  * [4] Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference, Benoit et al., arXiv:1712.05877


# Assessing course difficulty and the effect of weather in amateur cross
country running races

Kevin J. Wilson1 & Nina Wilson2
1School of Mathematics, Statistics & Physics, Newcastle University, UK
2Biostatistics Research Group, Population Health Sciences Institute, Newcastle
University, UK

###### Abstract

Cross country running races are different to track and road races in that the
courses are not typically accurately measured and the condition of the course
can have a strong effect on the finish times of the participants. In this
paper we investigate these effects by modelling the finish times of all
participants in 28 cross country running races over 5 seasons in the North
East of England. We model the natural logarithm of the finish times using
linear mixed effects models for both the senior men’s and senior women’s
races. We investigate the effects of weather and underfoot conditions using
windspeed and rainfall as covariates, fit distance as a covariate, and
investigate the effect of time via the season of the race, in particular
investigating any evidence of a pre- to post-Covid effect. We use random
athlete effects to model the participant to participant variability and
identify the most difficult courses using random course effects. The
statistical inference is Bayesian. We assess model adequacy by comparing
samples from the posterior predictive distribution of finish times to the
observed distribution of finish times in each race. We find strong differences
between the difficulty of the courses, effects of rainfall in the month of the
race and the previous month to increase finish times and an effect of
increasing distance increasing finish times. We find no evidence that
windspeed affects finish times.

Keywords: linear mixed effects models, Bayesian inference, endurance running,
race times

##  1 Introduction

Cross country racing is a form of running which takes place in the winter
months over grass, fields, and other soft surfaces. It differs from road and
track racing in that distances tend not to be exactly measured, and finish
times are not directly comparable from year to year on the same course due to
differing weather and underfoot conditions. Cross country racing is an elite
discipline which attracts many of the top endurance runners in the world, with
National, Continental and World Championships taking place each year. In
addition, it is a popular amateur sport, with junior and local leagues
operating within individual countries. For example, in England there are 39
counties which organise cross country championships each year.

In this paper, we consider one amateur cross country league, the North East
Harrier League (NEHL) in England, which organises a series of 6 cross country
races each season between September and March. Using the results of all races
staged by the league between the 2017/18 and 2022/23 seasons, we investigate
the effects of different courses, different seasons (to investigate a possible
time trend), the weather and the underfoot conditions. We also take into
account the fact that the same individuals take part in multiple races within
and between seasons.

There has been a body of work establishing the effects of weather and
environmental conditions on running performances, particularly in marathons
(Suping et al., 1992; Ely et al., 2007; Vihma, 2010; Knechtle et al., 2019;
Berke, 2019; Weiss et al., 2022). While most focus on effects of high
temperatures and humidity, not relevant to cross country running in the North
East of England, Knechtle et al. (2019) investigated performances in the
Boston marathon and found significant negative effects of windspeed and
rainfall. Similarly, Vihma (2010) found a negative effect of rainfall on the
marathon performance of male runners, although they attributed this effect to
the negative correlation between rainfall and air temperature. Berke (2019)
found effects of distance and elevation on the pace of runners over different
distances in road races.

Researchers have developed prediction models for endurance running events,
typically either for track or road races (Keogh et al., 2019; Alvero-Cruz et
al., 2020). In a review article, Alvero-Cruz et al. (2020) found that
prediction models considered to date were mainly focused on laboratory testing
variables such as VO2max, training variables such as training pace and
training load and anthropometric variables such as fat mass and skinfolds.
Keogh et al. (2019) performed a review of 114 prediction models for marathon
times. They found that while there were some models with high $R^{2}$ values
based on training, laboratory and anthropometric variables, important
variables such as course gradient, sex, and weather conditions were typically
not included, limiting the usefulness of such models.

More statistical investigations into endurance running and racing have
examined the impact the development and widespread adoption of super-shoes has
had on endurance racing times for elite athletes via extreme value models
(Arderiu and de Fondeville, 2022) and developed a predictive framework to
estimate the probability a runner will reach the next checkpoint and expected
passage time at the checkpoint in ultra-distance trail races (Fogliato et al.,
2021). Stival et al. (2023) considered the prediction of race times for elite
runners in track events in the presence of missing data and proposed a latent
class matrix-variate state space model, demonstrating that taking into account
missing data patterns improves predictive performance.

In this paper we examine the difficulty of each course in the NEHL, based on
all of the senior (individuals at least 18 years old) finish times over a 5
season period. We also look to see if there is evidence of the effect of
weather and underfoot conditions on finish times, and investigate whether
there is a time-effect, in particular looking to see if there is any evidence
of a pre- to post-Covid difference. We consider men’s and women’s results
separately, both to identify any sex-based differences and because men and
women run different distances in the NEHL - with the longer men’s races
requiring more endurance than the shorter women’s races. To investigate these
questions we

  1. 1.

Perform exploratory data analysis on all of the data for men and women in the
five seasons of the NEHL considered.

  2. 2.

Fit linear mixed effects models via Bayesian inference to the men’s and
women’s data separately, investigating both log finish times and log pace as
response variables, and including a distance covariate to allow for linear
effects of distance on (log) finish times.

  3. 3.

Investigate summaries of the posterior predictive distributions of finish
times to assess model adequacy, and validate the inference from the linear
mixed effects models.

The rest of the paper is structured as follows. In Section 2 we provide a
description of the data used in the analyses in the paper and then undertake
an initial exploratory analysis. In Section 3 we describe the linear mixed
effects models, their implementation via Bayesian inference and the results
from the modelling. In Section 3.3 we assess model adequacy via an assessment
of the posterior predictive distributions of finish times. In Section 4 we
discuss the results and conclude the paper.

##  2 Data and exploratory data analysis

###  2.1 Data

We consider all senior men’s and women’s results from the NEHL between the
2017/18 season and the 2022/23 season. The data are available publicly via the
league’s website www.harrierleague.com. In all, there are 14,067 men’s finish
times and 10,515 women’s times across 28 races on 8 different courses, made up
of 2668 unique male and 2116 unique female athletes. A breakdown of the number
of finishers by age group across all races is provided in Figure 6 in the
Appendix. The courses visited during the 5 season period are named Alnwick,
Aykley Heads, Druridge Bay, Gosforth, Herrington, Lambton, Thornley and
Wrekenton. Note that there are two races missing from the 2019/20 season,
which were cancelled due to the Covid-19 pandemic, and there were no races in
the 2020/21 season.

Alongside the finish time in the dataset we also have an identification number
for each athlete, the course used, the NEHL season of the race, the distance
of the race, the windspeed for the race and the month and year the race took
place. The distances for the men’s races were taken via GPS using a Garmin
Forerunner 235 watch. Each race is a lapped course, where men run three laps
(approximately 10km) and women two laps (approximately 6.5km). The women’s
distances were taken to be $2/3$ of the men’s distances. The windspeeds were
taken to be the windspeed recorded at the Durham weather station, within the
NEHL area, at 2pm on the day of the race, taken from www.visualcrossing.com.
In addition, we have data on the recorded monthly rainfall at the Durham
weather station in each month of the period, taken from
https://www.metoffice.gov.uk/pub/data/weather/uk/climate/stationdata/durhamdata.txt.
This will be our proxy for the underfoot conditions in the races.

There are no missing data in the dataset. The finish times are accurately
recorded using chip timing. There will be some measurement error in the race
distances due to the nature of wearable GPS devices, and the windspeeds and
rainfall as a result of them being measured at a central weather station
rather than on the individual courses themselves. We will not model these
measurement errors specifically, although future work could consider this.

###  2.2 Exploratory data analysis

Initially we consider the overall distributions of finish times for men and
women. These are given in the top row of Figure 1. We see that in both cases
the finish times have a unimodal distribution with a positive skew. In the
bottom row of Figure 1 we provide histograms of the natural logarithm of the
finish times for mean and women. We see that these are much more symmetrical
around the centre of the distribution. Note that in both sets of plots the
women’s times are typically faster than the men’s times due the shorter
distance of the women’s event in the NEHL.

Figure 1: Top: the distribution of finish times across all 28 races for men
(left, red) and women (right, blue). Bottom: the distribution of the natural
logarithm of finish times for men (left, red) and women (right, blue).

In Figure 2 we provide boxplots of the distribution of the natural logarithm
of the finish times for each course for men (top) and women (bottom). We see
the same patterns of finish times across men and women - Druridge Bay,
Gosforth and Wrekenton have the lowest median finish times and Lambton the
highest. We see, however, substantial overlap between the distributions of
finish times between all of the 8 different courses. There are outliers in the
upper tail of each distribution indicating that there is still some slight
positive skew present, representing runners who are substantially slower than
a typical runner.

Figure 2: Boxplots of the distributions of the natural logarithm of finish
times on each course for men (top) and women (bottom). The abbreviations of
the course names stand for: Aln -Alnwick, Ayk - Aykley Heads, Dru - Druridge
Bay, Gos - Gosforth, Herr - Herrington, Lam - Lambton, Tho - Thornley, Wre -
Wrekenton.

In Figure 3 we plot the windspeed and distance against the log finish times
for both men and women. In each case the plots are made up of the median log
finish time (points) and interquartile range (vertical bars) for each race in
each season on the y-axis, and the windspeed or distance on the x-axis. We see
that for both men (red, top) and women (blue, bottom) there is no clear
relationship between the windspeed and the distribution of finish times based
on these data. However, for both men and women there appears to be a positive
linear relationship between the distance of the race and the finish time on
the log scale - the further the race, the slower the time on average. The
distributions of finish times for each race show very similar patterns for men
and women, although there is a shift in the y-axis values.

Figure 3: The median and inter-quartile range of the log finish times for each
race plotted against the windspeed (left) and distance (right) for men (top,
red) and women (bottom, blue).

In the Appendix we also include a set of plots (Figure 7) of the rainfall in
the month of each race and the rainfall in the previous month, against the
median and inter-quartile ranges of log finish times. While we see similar
patterns between men and women, we see no clear relationship between rainfall
and finish times.

##  3 Statistical modelling

###  3.1 The linear mixed effects model

Our aim is to understand the relative difficulty of each of the eight courses
used by the NEHL. We would also like to understand the effect of the
covariates windspeed, distance and rainfall on the finish times of runners in
NEHL races, and to see if we can detect any season to season variation.
Finally, the same runners take part in multiple races across the five seasons,
and so we need to make provision for the lack of independence of observations
on the same individuals.

Thus we propose the following linear mixed effects model. The response
$Y_{ijk}$ for individual $i$ on course $j$ in season $k$ is defined to be the
natural logarithm of the finish time $T_{ijk}$, that is
$Y_{ijk}=\log(T_{ijk})$. We assume
$Y_{ijk}\mid\mu_{ijk},\tau\sim\textrm{N}(\mu_{ijk},1/\tau)$, where $\tau$ is
the observation precision and the mean $\mu_{ijk}$ takes the form of a linear
predictor

$\displaystyle\mu_{ijk}$ $\displaystyle=$
$\displaystyle\lambda+\alpha_{i}+\beta_{j}+\delta_{k}+\gamma(D_{jk}-\bar{d})+\hskip
3.0pt\lambda(W_{jk}-\bar{w})+\rho_{m}R_{m(jk)}+\rho_{m-1}R_{m(jk)-1},$

where $\lambda$ is an overall mean,
$\alpha_{i}\sim\textrm{N}(0,1/\tau_{\alpha})$ is a random athlete effect,
$\beta_{j}\sim\textrm{N}(0,1/\tau_{\beta})$ is a random course effect,
$\delta_{k}\sim\textrm{N}(0,1/\tau_{\delta})$ is a random season effect,
$\gamma$ is a fixed effect of distance, $\lambda$ is a fixed windspeed effect,
$\rho_{m}$ is a fixed effect of rainfall in the current month and $\rho_{m-1}$
is a fixed effect of rainfall in the previous month. The covariates $D_{jk}$
and $W_{jk}$ are the distance and windspeed of course $j$ in season $k$.
$R_{m(jk)}$ represents the rainfall in month the race at course $j$ in season
$k$. The terms $\bar{d}$ and $\bar{w}$ are chosen values representing average
distance and average windspeed, to centre the covariates.

That is, we treat the athlete, course and season as random effects centred on
zero a priori, consider the linear effects of distance and windspeed on the
log scale and include fixed effects of rainfall in both the current and
previous month, to represent that underfoot conditions are a result of the
weather conditions over a number of weeks leading up to the race. By including
individual terms for each month, rather than summing them, we hope to
understand the relative contribution of each.

Inference is Bayesian and so a prior distribution is required. In practice, we
assign independent marginal prior distributions to each model parameter except
the rainfall effects, which are correlated in the prior. We give each fixed
effect independent Normal prior distributions

$\displaystyle\gamma\sim\textrm{N}(m_{\gamma},v_{\gamma}),~{}\lambda\sim\textrm{N}(m_{\lambda},v_{\lambda}),$
$\displaystyle\rho_{m}\sim\textrm{N}(m_{\rho_{m}},v_{\rho_{m}}),~{}\rho_{m-1}\sim\textrm{N}(\phi
m_{\rho_{m}},v_{\rho_{m-1}}),$

where we choose relatively large prior variances in each case. The rainfall
effects are correlation through their means, which both depend on
$m_{\rho_{m}}$, the mean effect of rainfall in the current month, which is
given a Normal prior distribution with a mean of zero. The effect of rainfall
in the previous month has mean $\phi m_{\rho_{m}}$, where $\phi<1$ would
represent a stronger effect of rainfall in the current month than the previous
month. We give $\phi$ a gamma prior distribution,
$\phi\sim\textrm{Gamma}(a_{\phi},b_{\phi})$, where $a_{\phi}\leq b_{\phi}$.

The precisions from the random effects are given independent gamma priors, to
provide some conjugacy and speed up inference. Specifically,

$\displaystyle\tau_{\alpha}\sim\textrm{Gamma}(a_{\alpha},b_{\alpha}),~{}\tau_{\beta}\sim\textrm{Gamma}(a_{\beta},b_{\beta})~{}\tau_{\delta}\sim\textrm{Gamma}(a_{\delta},b_{\delta})$

where the hyper-parameters are to be chosen. In order for each of our
parameters to be identifiable in the inference, we require corner constraints.
That is, we constrain $\alpha_{1}=\beta_{1}=\delta_{1}=0.$ Thus all other
random effects are interpreted relative to these values.

Inference is conducted using Markov Chain Monte Carlo (MCMC) methods using the
`rjags` package in R (Plummer, 2022), which implements JAGS software.
Specifically, JAGS uses a combination of Gibbs sampling, Metropolis-Hastings
sampling and slice sampling steps to sample from the posterior distribution.
In the analyses that follow we discard an initial 10,000 samples as burn-in,
before running a further 1,000,000 iterations and thinning by 100, to obtain
10,000 approximately independent samples from the posterior distribution.
Convergence and mixing is checked using traceplots and effective sample sizes.
All analyses run in minutes on a standard desktop or laptop computer. Hyper-
parameters are chosen to give relatively diffuse prior distributions.

In the results that follow in the next section we report a slightly simpler
version of the model, with linear predictor

$\displaystyle\mu_{ijk}$ $\displaystyle=$
$\displaystyle\lambda+\alpha_{i}+\beta_{j}+\delta_{k}+\gamma(D_{jk}-\bar{d})+\rho_{m}R_{m(jk)}+\rho_{m-1}R_{m(jk)-1},$

that is, with no windspeed effect. This is because all models that we have
fitted show no discernible effect of windspeed, with posterior distributions
for windspeed centred around zero. See Table 2 in the Appendix for posterior
summaries from this model.

We also fitted the model with a slightly different response variable, using
log pace $Y_{ijk}=\log(T_{ijk}/D_{jk})$ instead of log finish time, to
understand if our course effects were being driven by distance in spite of the
distance covariate in the model. The results were qualitatively the same, and
so we are confident that this is not the case. A summary of these results are
provided in Table 3 in the Appendix.

###  3.2 Results

First we consider the course effects. The left hand side of Figure 4 provides
the posterior mean and 95% posterior symmetric intervals for each of the
courses for both men and women. We note that all effects are compared to
Alnwick. We see that for men the most difficult courses are Herrington and
Thornley, with the easiest courses being Druridge Bay and Gosforth. The
ordering is almost identical for women, with relatively similar effects
observed to the men, typically slightly higher. In general, we see much
stronger differences between courses than in the exploratory analysis, now
that the individual athlete effects and covariate effects have been removed.
We provide the individual athlete effects in Figure 8 in the Appendix. We see
a large spread in athlete abilities in both the men’s and women’s races, with
more evidence of a small number of very slow and very fast runners, compared
to the rest, in the women’s race.

Figure 4: Left: the posterior mean and 95% posterior symmetric intervals for
the course effects for men (red) and women (blue). Right: the posterior mean
and 95% posterior symmetric intervals for the season effects for men (red) and
women (blue).

We provide a plot of the posterior season effects for men (red) and women
(blue) in the form of the posterior means and 95% posterior symmetric
intervals in right hand side of Figure 4, for each of the five seasons in the
data. We see no evidence of a systematic difference pre- and post-Covid, with
the effects for both women and men being similar in Seasons 17/18, 18/19 and
22/23 and being positive in Seasons 19/20 and 21/22, representing slightly
slower finish times on average. While we may have surmised that this may be
due to either a lack of fitness in runners, or a different cohort made up of a
higher percentage of new runners in 21/22, this would not explain the positive
season effect in 19/20 prior to the pandemic. Overall, we conclude that there
are no obvious temporal trends present.

We can investigate the effects of the covariates in the model. We plot their
posterior densities in Figure 5 for men (red) and women (blue). We see that
for rainfall in the current and previous month the effect is stronger for men
than for women.The opposite is the case for distance. In each case the
qualitative interpretation of the effect is the same for men and women.

Figure 5: The posterior densities for the covariates distance (top left),
rainfall in the current month (top right) and rainfall in the previous month
(bottom left) for men (red) and women (blue).

  * •

The distance effect is positive, so as distance increases, so does the log
finish time. Based on the posterior means of the distance effects, a 0.1 mile
increase in distance would lead to an increase in finish time of 1 minute 5
seconds for a man running a 47 minute race and 1 minute 26 seconds for a woman
running a 38 minute race.

  * •

The rainfall effect in the current month is positive, so as rainfall increases
so do log finish times. Based on the posterior means of the current rainfall
effects, a 10mm increase in the current month’s rainfall would lead to an
increase in finish time of 25 seconds for a man running a 47 minute race and
18 seconds for a woman running a 38 minute race.

  * •

The rainfall effect in the previous month is positive, so as rainfall
increases so do log finish times. Based on the posterior means of the previous
rainfall effects, a 10mm increase in the previous month’s rainfall would lead
to an increase in finish time of 22 seconds for a man running a 47 minute race
and 14 seconds for a woman running a 38 minute race.

###  3.3 Model checking

We assess the fit of our models to the data in this section using posterior
predictive distributions. We compare the distribution of (log) finish times
based on a sample from the posterior predictive distributions of the (log)
linear mixed effects models to the observed distributions of finish times for
all of the races in the 17/18 to 22/23 seasons of the NEHL. We do so via the
histograms of the full distributions of finish times and summary statistics
from these distributions. This will allow us to see if our models are
adequately capturing the variation in finish times in the data, validating the
inferences made in the previous section.

We sample results from each of the 28 races in the five NEHL seasons
considered, from the posterior predictive distribution, utilising the
information on the course, athletes, season and covariates. We overlay the
resulting distribution of log finish times for each race (blue) and the
observed distribution of log finish times (red) in Figures 9 and 10 for men
and women respectively in the Appendix.

We see that for both the men’s and women’s races there are no obvious
discrepancies between the sample of log finish times from the posterior
predictive distributions and the observed log finish times. They are
remarkably similar. We investigate this further by considering summaries from
these two distributions, reported in Table 1, for three randomly selected
races: Aykley Heads in 2017, Druridge Bay in 2022 and Wrekenton in 2021.

Men
---
Course | Year | Obs/Pred | Min | LQ | Med | UQ | Max
Aykley Heads | 2017 | Obs | 34.47 | 42.06 | 46.02 | 51.32 | 77.20
|  | Pred | 30.64 | 42.67 | 46.40 | 52.17 | 83.83
Druridge Bay | 2022 | Obs | 31.58 | 39.72 | 43.42 | 46.98 | 74.68
|  | Pred | 30.32 | 39.66 | 43.42 | 48.04 | 72.22
Wrekenton | 2021 | Obs | 31.95 | 39.99 | 43.98 | 48.18 | 85.55
|  | Pred | 30.59 | 39.73 | 43.30 | 47.61 | 64.77
Women
Course | Year | Obs/Pred | Min | LQ | Med | UQ | Max
Aykley Heads | 2017 | Obs | 27.02 | 34.12 | 37.05 | 40.91 | 60.00
|  | Pred | 27.32 | 34.38 | 37.78 | 41.73 | 61.86
Druridge Bay | 2022 | Obs | 31.58 | 39.72 | 43.42 | 46.98 | 74.68
|  | Pred | 21.99 | 31.01 | 34.50 | 37.86 | 57.44
Wrekenton | 2021 | Obs | 23.82 | 31.38 | 34.72 | 38.85 | 62.10
|  | Pred | 23.21 | 31.42 | 35.11 | 39.05 | 58.23
Table 1: Summary statistics from the posterior predictive distribution of log
finish times and the distribution of observed log finish times for three
randomly chosen races for both men (top) and women (bottom).

Again, we observe no systematic discrepancies between the summary statistics
resulting from the sample from the model and the observations. This is
particularly true for the quartiles, although even the minimum and maximum
finish times for the races are sometimes more extreme in the predictions and
sometimes more extreme in the observations.

##  4 Discussion

In this paper we have considered the modelling of cross country finish times
for amateur athletes in the NEHL in the North East of England. We conducted an
exploratory data analysis which indicated potential differences in the
distributions of finish times between different courses and a potential
positive effect of distance on finish times, although the inter-individual
variability in finish times made inferring relationships difficult. It was
even more difficult to identify any potential relationship between either
windspeed or rainfall with finish times.

We fitted a linear mixed effects model using Bayesian inference via MCMC, with
random athlete, course and season effects and fixed effects of distance,
windspeed and rainfall, both in the month of the race and the previous month.
Windspeed was found to have no effect and was removed. While this contrasts
with the results in Knechtle et al. (2019) who found a negative association
between windspeed and performance in the Boston marathon, this could
potentially be accounted for by the windspeed measurements used being from a
central weather station which could be a number of miles from the race venue.
Boston is also a point to point course, and so a headwind would be a headwind
for the whole race, whereas in the NEHL athletes run a looped course, and so
would experience both a headwind and a tailwind on each lap.

We found highly variable athlete effects indicating the wide range of
individuals taking part in the NEHL. There was more variation in the tails for
the female runners, indicting that there were a small number of very slow and
very fast women compared to the rest of the runners. The course effects were
varied, with a similar ordering for the men and women. Herrington and Thornley
were the hardest courses, with Druridge Bay and Gosforth being the easiest. We
note that this contrasts with the distributions of raw finish times, which
essentially ordered the courses by length. These findings are consistent with
the informally gathered qualitative opinions expressed by runners in the NEHL.

Distance, rainfall in the month of the race and rainfall in the previous month
were found to have positive effects on finish times, in decreasing order of
strength. This is consistent with findings in Knechtle et al. (2019); Vihma
(2010); Berke (2019) for road races. In cross country rainfall leading up to
the event is particularly important, as it can cause the course to become
muddy and slippery, making running much more difficult and physically
demanding.

We found quite different effects in the five seasons of the NEHL considered.
There was no obvious pre- to post-Covid effect. Again the effects were
consistent between male and female runners. Possible reasons for the season to
season variability could be that the condition of the courses and the
influence of weather are not fully captured by our rainfall variables or that
that the overall populations of runners were systematically changing in terms
of their fitness in different seasons (e.g., if the placement of races were
close to other important races, meaning runners were tired).

Model adequacy was assessed using in-sample validation - comparing the overall
distribution and summary statistics from the model’s posterior predictive
distribution to the observed distribution of finish times. We found good
agreement between samples from the posterior predictive distribution and the
data for both men and women, indicating good model fit. While this is
sufficient for our purposes of inference, if we were interested in using the
model for predictive purposes for individual runners we would need to also
undertake out-of-sample validation, for example using the results from the
23/24 NEHL season.

This paper is the first to our knowledge that uses statistical models to make
inferences about some of the most important factors influencing finish times
of cross country runners. There are some extensions to this work that are
worth considering. One to use the inferences from this paper to propose a
predictive model for amateur cross country finish times. Another is to extend
the model to include more weather and condition covariates such as temperature
and snowfall (not uncommon at the NEHL). For example there may be an
interaction between rainfall and temperature, as frozen ground may change the
effect of rainfall on finish times. Additional covariates such as elevation
could improve predictive performance, but were not considered for this work
since they affect the difficulty of the course. There was an implicit
assumption that all runners were remaining of a similar ability in the data.
This is likely to be reasonable over a 5-year period, but if further years of
data were used it would be sensible to have some discounting of older results
in the athlete effects, as ageing runners are likely to get slower over time
and younger runners will typically get faster. This would also help to allow
for other factors affecting the ability of individual runners, such as female
runners returning after giving birth.

## References

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## Appendix

Figure 6: The number of finishers by age group across all races for men (top, red) and women (bottom, blue). MU20 represents men 18-20 years old, Msen represents men 20-35, MV35 represents men 36-40, etc. The convention is the same for women, with F replacing M. Women aged 18-20 run in a separate race. Figure 7: The median and inter-quartile range of the log finish times for each race plotted against the rainfall in the current month (left) and rainfall in the previous month (right) for men (top, red) and women (bottom, blue). | Men | Women
---|---|---
Effect | LQ | Med | UQ | LQ | Med | UQ
Alnwick | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
Aykley Heads | 0.016 | 0.017 | 0.019 | 0.029 | 0.031 | 0.033
Druridge Bay | -0.070 | -0.068 | -0.066 | -0.059 | -0.057 | -0.055
Gosforth | -0.045 | -0.043 | -0.041 | -0.032 | -0.029 | -0.026
Herrington | 0.147 | 0.150 | 0.154 | 0.187 | 0.191 | 0.195
Lambton | -0.005 | -0.003 | -0.002 | 0.000 | 0.002 | 0.003
Thornley Hall | 0.149 | 0.153 | 0.156 | 0.176 | 0.181 | 0.185
Wrekenton | -0.010 | -0.006 | -0.003 | 0.014 | 0.018 | 0.021
Season 17/18 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
Season 18/19 | 0.002 | 0.004 | 0.005 | -0.002 | 0.000 | 0.002
Season 19/20 | 0.031 | 0.032 | 0.034 | 0.007 | 0.009 | 0.011
Season 21/22 | 0.044 | 0.045 | 0.046 | 0.026 | 0.028 | 0.029
Season 22/23 | -0.001 | 0.000 | 0.002 | -0.010 | -0.008 | -0.007
Distance | 0.219 | 0.224 | 0.228 | 0.359 | 0.368 | 0.375
Windspeed | -0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
Current rainfall | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001
Previous rainfall | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001
Table 2: The lower quartile (LQ), median (Med) and upper quartile (UQ) of the marginal posterior distribution for each fixed and random effect for the linear mixed effects model including windspeed, for both men and women. | Men | Women
---|---|---
Effect | LQ | Med | UQ | LQ | Med | UQ
Alnwick | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
Aykley Heads | 0.018 | 0.020 | 0.021 | 0.018 | 0.020 | 0.021
Druridge Bay | -0.065 | -0.064 | -0.062 | -0.065 | -0.064 | -0.062
Gosforth | -0.043 | -0.040 | -0.037 | -0.043 | -0.040 | -0.037
Herrington | 0.153 | 0.158 | 0.162 | 0.153 | 0.158 | 0.162
Lambton | -0.001 | 0.000 | 0.002 | -0.001 | 0.000 | 0.002
Thornley Hall | 0.155 | 0.160 | 0.165 | 0.155 | 0.160 | 0.165
Wrekenton | -0.001 | 0.004 | 0.008 | -0.001 | 0.004 | 0.008
Season 17/18 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
Season 18/19 | -0.001 | 0.000 | 0.002 | -0.001 | 0.000 | 0.002
Season 19/20 | 0.030 | 0.031 | 0.033 | 0.030 | 0.031 | 0.033
Season 21/22 | 0.042 | 0.043 | 0.045 | 0.042 | 0.043 | 0.045
Season 22/23 | 0.000 | 0.001 | 0.002 | 0.000 | 0.001 | 0.002
Distance | 0.063 | 0.069 | 0.075 | 0.063 | 0.069 | 0.075
Current rainfall | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001
Previous rainfall | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001
Table 3: The lower quartile (LQ), median (Med) and upper quartile (UQ) of the
marginal posterior distribution for each fixed and random effect for the
linear mixed effects model with response variable log pace, for both men and
women. Figure 8: The posterior mean and 95% posterior symmetric intervals for
the athlete effects for men (red, left) and women (blue, right). Figure 9:
Histograms of the distribution of men’s log finish times from a sample from
the posterior predictive distribution (blue) and the observed log finish times
(red) for each of the 28 races used to fit the model. Figure 10: Histograms of
the distribution of women’s log finish times from a sample from the posterior
predictive distribution (blue) and the observed log finish times (red) for
each of the 28 races used to fit the model.