# A critical analysis on the sensitivity enhancement of surface plasmon
resonance sensors with graphene

Aline dos S. Almeida  D. A. Bahamon  School of Engineering, Mackenzie
Presbyterian University, São Paulo - 01302-907, Brazil  MackGraphe, Mackenzie
Presbyterian Institute, São Paulo - 01302-907, Brazil  Nuno M. R. Peres
Physics Department - Minho University, Gualtar’s Campus - 4710-057 Braga,
Portugal  Christiano J. S. de Matos<EMAIL_ADDRESS>School of
Engineering, Mackenzie Presbyterian University, São Paulo - 01302-907, Brazil
MackGraphe, Mackenzie Presbyterian Institute, São Paulo - 01302-907, Brazil

###### Abstract

The use of graphene in surface plasmon resonance sensors, covering a metallic
(plasmonic) film, has a number of demonstrated advantages, such protecting the
film against corrosion/oxidation and facilitating the introduction of
functional groups for selective sensing. Recently, a number of works have
claimed that few-layer graphene can also increase the sensitivity of the
sensor. However, graphene was treated as an isotropic thin film, with an out-
of-plane refractive index that is identical to the in-plane index. Here, we
critically examine the role of single and few layers of graphene in the
sensitivity enhancement of surface plasmon resonance sensors. Graphene is
introduced over the metallic film via three different descriptions: as an
atomic-thick two-dimensional sheet, as a thin effective isotropic material
(same conductivity in the three coordinate directions), and as an non-
isotropic layer (different conductivity in the perpendicular direction to the
two-dimensional plane). We find that only the isotropic layer model, which is
known to be incorrect for the optically modelling of graphene, provides
sizeable sensitivity increases, while the other, more accurate, models lead to
negligible contribution to the sensitivity.

Surface Plasmon Resonance, Sensitivity, Graphene,biosensors

††preprint: APS/123-QED

##  I Introduction

Lately, much attention has been paid to the development of new chemical and
biological sensors based on low cost and fast diagnosisBiosensorReview1:21 ;
BiosensorReview2:20 ; BiosensorSpeci1:21 ; BiosensorSpeci2:21 . Surface
plasmon resonance (SPR) is a high sensitivity, real time, and label free
technique with great potential for sensing applicationsPathak:19 . Surface
plasmon polaritons (SPPs) are surface waves propagating at a metal/dielectric
interface and arise from the electromagnetic field coupling to dipolar
excitation of free electrons in the metal. In the resonance condition, these
waves are highly sensitive to changes in dielectric function the surrounding
environment Pathak:19 ; Maier . In particular, for SPR biosensors, reversible
adsorption of biomolecules (analyte) onto the sensing surface
(metal/dielectric interface) changes the local refractive index (RI), leading
to changes in the SPR condition Tang:10 .

A conventional SPR biosensor consists of a thin metallic film deposited on one
side of a prism. The metallic film is typically made up from gold, which
supports the propagation of SPPs at the visible wavelength range and has good
resistance to oxidation and corrosion in wet environments. However, sensing
specificity usually requires preparing the surface (e.g., using alkanethiol in
the case of gold Choi:10 ) to receive a linker molecule which, in turn, allows
for the fixation of the required bio-receptor Choi:10 ; Mok:08 ; Taninaka:10 .
In this sense, adding a graphene layer to the metal surface may simplify
functionalization steps, as graphene can work as a bio-functional surface due
to ${\pi}-{\pi}$ interactions with carbon-based ring structures that are
widely present in bio-molecules Song:10 . Besides, it has already been
experimentally demonstrated that graphene can protect the surface against
oxidation and corrosion, which is particularly important when more reactive
metals, such as copper, a cheaper alternative to gold, is used as the
plasmonic material GRIGORENKO:14 ; GRIGORENKO:19 .

In addition to the mentioned advantages, recently there have been theoretical
reports that seem to indicate that graphene layers could improve gold’s
Chen:19  and silver’s PAL:21 ; Choi:11  SPR sensitivity. Nevertheless
calculated sensitivity increases diverge significantly from report to report,
ranging from 1 to 200${\%}$ for few-layer graphene Bhavsar:19 ; Wu:10 ;
SAIFUR:17 . We note that these discrepancies are rather intriguing, as the
same physical parameters and SPR system geometry seem to have been used, with
only one report presenting a slight difference: a thin binding layer between
graphene and the analyte, producing some absorption.Wu:10  We also highlight
that all three reports treat graphene as an ultrathin isotropic material,
which is notorious for yielding un-physical electromagnetic results Matthes:16
; Majerus:18 ; Valuev16 ; Oliveira:15 . According to Refs. Valuev16 ;
Majerus:18 , modeling graphene as an isotropic thin film results in an
artificial plasmonic resonance (of a metallic nature), since the same
refractive index is adopted in and out of the plane where graphene is placed.
In this case, the electric field of light in the _p_ -polarization, which has
a component that is normal to the graphene plane, generates a non-physical
current inside the graphene thin film along the perpendicular direction.

In this work, to access the actual impact of graphene on the SPR sensor
sensitivity. We model graphene in a number of ways and compare the obtained
results. First, graphene is modeled as a surface conductivity, in which case
it is treated as a strictly 2D material. Subsequently, graphene is treated as
an ultrathin isotropic film, and, finally, as an ultrathin anisotropic film.
Our results show that the selected modelling approach directly impacts on the
observed sensitivity enhancement, with significant enhancements obtained only
in the isotropic case. Furthermore, in the case of an isotropic ultrathin
layer, the results highlight the importance of the choice of the value for
graphene’s refractive index, with results diverging more than $\sim 50\%$
among them.

Figure 1: Schematic representation of the SPR sensor.

##  II Model Description

The proposed SPR sensor structure is identical to that examined in
Refs.Bhavsar:19 ; SAIFUR:17 , and is based on the Kretschmann configuration.
As sketched in Fig. 1, it consists of a multilayer structure: a SF11 glass
prism, a 50-nm-thick gold film, a graphene layer, and the analyte. In this
configuration, the SPR resonance appears as a dip in the reflectance of _p_
-polarized light impinging on the prism at a certain angle larger than the
critical angle of the prism-analyte interface. Thus, the SPR condition is
easily calculated through the reflectance of the multilayer structure
($R=|r_{1...N}|^{2}$). Note that we treat the structure as a three-layer
system when graphene is modelled as a surface conductivity, and as a four-
layer system when graphene is taken to be a thin film.

For the three-layer system (2D graphene model; see Fig.2(a)), the first layer
is the SF11 glass prism (semi-infinite) with refractive index ${n_{1}}$ =
1.7786 coated by a gold film with refractive index ${n_{2}}$ = 0.1834 + 3.4332
_i_ Bhavsar:19  and the third layer is the aqueous analyte that is considered
semi-infinite (water refractive index taken to be 1.332). In the four-layer
case (3D graphene model presented in Fig. 2(b)), graphene is considered as the
third layer with thickness $d_{g}$ and refractive index $n^{g}_{3}$, while the
semi-infinite region of the analyte is now considered as the fourth layer,
with refractive index $n_{4}$. The $n^{g}_{3}$ parameter will be discussed in
detail below, for the isotropic and anisotropic cases. The reflection
coefficients for the three- and four-layer systems are respectively given by:

$\displaystyle r_{123}=\ \frac{r_{{12}\ }\ +\ r_{{23}}\ e^{i2\phi_{2}}\ }{{1\
+\ r}_{{12}}\ r_{{23}}\ \ e^{i2\phi_{2}}}$ (1)

and

$\displaystyle r_{{1234}}=\frac{\left(r_{{12}}+r_{{23}}e^{i2\phi_{2}}\right)\
+\left(r_{12}r_{23}+e^{i2\phi_{2}}\right)r_{{34}}e^{i2\phi_{3}}}{\left({1+r}_{{12}}\
r_{{23}}\ e^{i2\phi_{2}}\right)+\left(r_{{23}}+r_{{12}\
}e^{i2\phi_{2}}\right)r_{{34}}e^{i2\phi_{3}}},$ (2)

where $r_{ij}$ are the Fresnel reflection coefficients for _p_ -polarized
waves between layers $i$ and $j$, $\phi_{i}=k_{zi}d_{i}$ and
$k_{zi}=\sqrt{n^{2}_{i}\frac{\omega^{2}}{c^{2}}-k^{2}_{xi}}$. Given that the
layered systems present translational symmetry along the _x_ -direction,
$k_{xi}=k_{x}=n_{1}\left(\frac{\omega}{c}\right)\sin\theta_{i}$. $\omega$ and
$c$ are respectively the angular frequency and the speed of light in vacuum.

The angle corresponding to the reflectance minimum is called SPR angle
$\left(\theta_{SPR}\right)$. The sensing mechanism consists of monitoring the
$\theta_{SPR}$ shift as the refractive index of the analyte (sensing medium)
changes WIJAYA:11 . We assume that the refractive index of the aqueous analyte
$n_{3(4)}$ varies from 1.332 to 1.35 in steps of ${\Delta n}={5\times
10^{-6}}$. The sensor’s sensitivity is defined as
$S=\frac{d\theta_{SPR}}{dn_{3(4)}}$, in units of ∘/RIU (where RIU stands for
Refractive Index Units), and is calculated for each studied case. To clearly
access the effect of the graphene modelling on the performance of the
sensitivity, the thickness of the gold layer ($d_{2}$) and the wavelength of
the incident light ($\lambda=633$ nm) are kept constant.

Figure 2: (a) Three-layer system: graphene as an atomic sheet with a surface
conductivity $\sigma_{g}$. (b) Four-layer system: graphene as a thin layer
film with thickness dg and refractive index $n^{g}_{3}$.

###  II.1 Graphene Modeling

In the three-layer case, graphene is modelled as an atomically thin film. Its
surface conductivity ($\sigma_{g}$) is introduced in the boundary conditions
of the electromagnetic fields at the metal-analyte interface to obtain the
Fresnel reflection coefficient

$\displaystyle r_{23}=\frac{k_{z_{2}}\epsilon_{3}\ -\ k_{z_{3}}\epsilon_{2}\
+\
\frac{k_{z_{2}}k_{z_{3}}\sigma_{g}}{\epsilon_{0}\omega}}{k_{z_{2}}\epsilon_{3}\
+\ k_{z_{3}}\epsilon_{2}\ +\
\frac{k_{z_{2}}k_{z_{3}}\sigma_{g}}{\epsilon_{0}\omega}}.$ (3)

In this expression, $\epsilon_{0}$ is the vacuum permittivity. In the visible
region of the electromagnetic spectrum ${\sigma_{g}\approx\sigma}_{0}\ =\
\frac{e^{2}}{4\hbar}$ Nair:08 ; Peres:15 , with $e$ being the electron charge
and $\hbar$ the reduced Planck constant.

In the isotropic 3D model, graphene is considered a thin film with an
effective thickness $d_{g}=0.34$ nm Bhavsar:19 ; Wu:10 ; SAIFUR:17 ;
Oliveira:15 ; Ni:07  and refractive index

$\displaystyle n^{g}_{3}=\sqrt{1+i\frac{\sigma_{g}}{\epsilon_{0}\omega
d_{g}}}=2+1.7119i.$ (4)

Finally, in the anisotropic case, graphene is taken to have different in-plane
and out-of-plane values for the diagonal dielectric tensor components.
Specifically, for the in-plane response, the refractive index was considered
to have the same value as in the isotropic case
$n^{g}_{3\text{in}}=n_{xx}=n_{yy}=\sqrt{\epsilon^{g}_{3\text{in}}}=\sqrt{1+i\frac{\sigma_{g}}{\epsilon_{0}\omega
d_{g}}}$. The out-of-plane refractive index was assumed to be
$n^{g}_{3z}=\sqrt{\epsilon^{g}_{3z}}=\sqrt{2.5}$. Oliveira:15 ; Gao:12 ;
Kwon:14  Considering that the reflecting surface of the anisotropic thin film
coincides with graphene’s basal plane, the coupling between the _s_ \- and _p_
-polarized waves is zero. This ensures that the reflectance of the four-layer
system with the anisotropic graphene continues to be calculated by Eq. 2, but
with $\phi_{3}=k_{z3e}d_{3}$,

$\displaystyle
k_{z3e}=\sqrt{\left(\epsilon^{g}_{3\text{in}}\frac{\omega^{2}}{c^{2}}-\frac{\epsilon^{g}_{3\text{in}}}{\epsilon^{g}_{3z}}k_{x}^{2}\right)},$
(5)

and

$\displaystyle
r_{i3}=\frac{\left(n^{g}_{3\text{in}}\right)^{2}k_{zi}-n^{2}_{i}k_{z3e}}{\left(n^{g}_{3\text{in}}\right)^{2}k_{zi}+n^{2}_{i}k_{z3e}}.$
(6)

##  III Results and Discussion

Figure 3:  (a) Calculated reflectance $(\text{R}_{123(1234)})$ as a function
of the incident angle $(\theta_{i})$ for analyte refractive indices of 1.332
and 1.342 (red and black curves, respectively). The vertical lines indicate
the positions of $(\theta_{SPR})$ for the system without graphene. Graphene
modelled as a surface conductivity (dashed lines); and as isotropic
$(n^{g}_{3})$ (dotted lines) and anisotropic $(n^{g}_{3\text{in}})$ (dashed-
dotted lines) films. (b) Zoom in the the minimum reflectance region of (a).

The effect of the different approaches used to model the graphene monolayer is
presented in Fig.3(a), which shows the reflectance for the three-layer
($R_{123}$) and the four-layer ($R_{1234}$) systems as functions of the angle
of incidence, $\theta_{i}$, for two different analyte refractive indices:
$n_{3(4)}=1.332~{}\text{and}~{}1.342$ (red and black curves, respectively).
The continuous lines correspond to the response of the sensor without
graphene; the dashed lines present the response with graphene as a sheet
conductivity (2D model); the dotted lines stand for the 3D isotropic case; and
the dashed-dotted lines represent the 3D anisotropic model.

At first sight, the four models produce similar reflectance curves, with
graphene only slightly modifying the response of the sensor. However, the zoom
in the reflectance curves provided in Fig. 3(b) shows observable differences
produced by the different graphene models, particularly in the shift of
$\theta_{SPR}$ with the analyte refractive index, as well as in the amplitude
of the reflectance at the plasmon resonance condition. For example, in the
system based only on gold’s SPR, the SPR corresponds to a minimum reflectance
of $\sim 0.71\%$ for the analyte refractive index of 1.332 (solid red curve in
Fig. 3). By adding graphene monolayer and modeling it either through the 2D
model or the anisotropic thin film model, the reflectance minimum increases to
$\sim 1.9\%$ due to the absorption by the sheet of graphene, while modelling
graphene as a thin isotropic layer produces not only the largest shift of
$\theta_{SPR}$, but also the largest change in the reflectance minimum, which
increases to $\sim 2.6\%$.

However, the most important result of the present work is to quantify the
sensitivity of the sensor (i.e., the SPR shift with analyte index change),
which is shown in table 1 for the different graphene models ($S_{gr}$), as
well as for the sensor without graphene ($S_{Au}$). The sensitivity increase
provided by graphene, given by $\Delta
S=\left((S_{gr}-S_{Au})/S_{Au}\right)\times 100$, is also presented for each
case. When modelling monolayer graphene as an atomic sheet (2D model) or as an
anisotropic thin film (3D anisotropic) there is minimal change in the system’s
sensitivity relative to case of gold only. In stark contrast, modelling
monolayer graphene as an isotropic film, which is known to lead to unrealistic
results, as discussed Oliveira:15 , yields a sensitivity increase that is 3
orders of magnitude higher than with the 2D model.

Table 1: Sensitivity and sensitivity increase for the SPR sensor without and with a monolayer of graphene modeled in the three different examined ways. Model |  $S_{Au(gr)}(^{\circ}/\text{RIU}$) |  $\Delta S$ (%)
---|---|---
Gold | 70.8951 | 0
2D | 70.8962 | 0.0016
3D Iso. | 71.4515 | 0.7848
3D Aniso. | 70.8339 | -0.086

Having established that the way a single sheet of graphene is modelled affects
the obtained (theoretical) sensor sensitivity, we now investigate the
sensitivity as a function of the number of graphene layers. To model more than
one layer, previous works treating graphene as a thin film PAL:21 ; Choi:11 ;
Bhavsar:19 ; Wu:10 ; SAIFUR:17  have used the transfer matrix method, with
each graphene layer represented by a individual matrix. This strategy is
equivalent to the one we proposed using eq. 2, given that the multiplication
of transfer matrices of the N-layers of graphene ultimately leads to an
effective layer of thickness $L=N\times d_{g}=N\times 0.34$ nm, with the same
refractive index $n^{g}_{3}$ of the monolayer. In the 2D model, we take the
optical conductivity of multilayer graphene to be
$\sigma_{t}=N\times\sigma_{g}$, where interlayer interactions are neglected
(this has been shown experimentally to be a rather accurate description of
multilayer graphene conductivity in the visible range Nair:08 ). Figure 4
shows the obtained $\Delta S$ as functions of the number of graphene layers.

When graphene is modeled through the 2D or the 3D anisotropic models (green
and cyan lines in Fig. 4, respectively), the addition of graphene layers
practically does not affect the sensitivity of the system based only on gold’s
SPR. However, when the 3D isotropic model is applied, the increase is rather
large (dark blue line, Fig. 4). Also, by changing the graphene refractive
index to the one used in Bhavsar:19 ; Wu:10 ; SAIFUR:17
($n^{g}_{3}=3+1.1491i$), we get an even larger sensitivity increase (red line,
Fig. 4), which agrees with that reported by Ref. Bhavsar:19 . We believe this
sensitivity enhancement, obtained solely with the isotropic model, is related
to the fact that in this case the graphene film would effectively reduce the
absolute value of the isotropic dielectric constant of gold at the surface,
which is known to increase the sensitivity to changes in the analyte’s
refractive index POCKRAND:78 ; Raether:88 . Indeed, if we artificially reduce
gold’s dielectric constant in our model without graphene, we also obtain an
increase in the system’s sensitivity. We, thus, highlight that modelling
graphene as two-dimensional surface or as an anisotropic thin film are more
accurate alternatives, and are found to lead to no significant SPR sensor
sensitivity increases.

Figure 4: Calculated sensitivity increase for the addition of graphene layers
compared with the system based only on gold. Graphene as an: atomic sheet
(green), isotropic film, $n^{g}_{3}=3+1.1491i$ (dark blue) and
$n^{g}_{3}=2+1.7119i$ (red); anisotropic film, $n^{g}_{3\text{in}}=2+1.7119i$
(cyan).

##  IV Conclusions

In this paper we theoretically studied the influence of graphene modelling to
the sensitivity obtained in a refractometer based on gold’s SPR. When graphene
is modeled as a surface (2D model) or an anisotropic thin film (anisotropic 3D
model) the system’s sensitivity is practically unaffected by the presence and
number of layers of graphene. However, when graphene is modelled as an
isotropic thin film (isotropic 3D model), a significant sensitivity increase
is observed, specially when the refractive index used is the same as that by
previous worksBhavsar:19 ; Wu:10 ; SAIFUR:17 . Treating graphene as an
isotropic film is known to lead to a number of previously reported un-physical
results. Therefore, we suggest that modelling graphene through the 2D or 3D
anisotropic models are more accurate alternatives, which must be adopted for
the study of graphene-assisted SPR sensors.

##  V Acknowledgements

This work was funded by FAPESP (grant nos. 2018/07276-5 and 2018/25339-4), the
Brazilian Nanocarbon Institute of Science and Technology (INCT/Nanocarbon),
and CAPES-PrInt (grant no. 88887.310281/2018-00). NMRP acknowledges PORTUGAL
2020, FEDER, and the FCT through projects: UIDB/04650/2020 strategic project,
QML-HEP - CERN/FIS-COM/0004/2021, PTDC/FIS-MAC/2045/2021, and POCI-01-
0145-FEDER-028114, and the European Commission through the project
GrapheneDriven Revolutions in ICT and Beyond (Ref. No. 881603, CORE 3).

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# Swampland conjectures and single field inflation in modified cosmological
scenarios

Oem Trivedi<EMAIL_ADDRESS>School of Arts and Sciences, Ahmedabad
University,Ahmedabad 380009,India

###### Abstract

Swampland Conjectures have attracted quite some interest in the Cosmological
Community.They have been shown to have wide ranging implications,like
Constraints on Inflationary Models, Primordial Black Holes, Energy to name a
few.Particularly,their implications on Single Field Inflationary Models in
General Relativity Based Cosmology has gathered huge attention.Swampland
Conjectures in their usual form have been shown to be incompatible with these
kind of Single Field Models, or have been shown to induce severe Fine Tuning
in these Inflationary Models for them to be consistent with the Conjectures.
In this work, we show that a Large Class of Single Field Inflationary Models
can in fact bypass the problems faced by Inflationary Paradigms in GR Based
Cosmology. We use the Exact Solution Approach to Inflation for the same
purpose and show how String Theoretic Motivations of the Swampland Conjectures
can be in perfect symphony with various Single Field Inflationary Models in
Modified Cosmological Scenarios.

††preprint: APS/123-QED

###### Contents

  1. I Introduction
  2. II The Exact Solution Approach to Inflation
  3. III Consistency of Single Field Models in Modified Cosmologies
  4. IV Concluding Remarks and Discussion
  5. V Acknowledgments

##  I Introduction

The idea of Cosmic Inflation has achieved a tremendous amount of success in
describing the Early Universe. Inflation initially captured attention with
it’s solutions to the fine tuning problems of Standard Big Bang Cosmology Guth
(1981); Sato (1981); Starobinskii (1979); Linde (1995); Baumann (2009). The
optimism regarding Inflationary Cosmology kept on increasing as it’s various
observational predictions about the Early Universe were shown to be very
precise in accordance with various satellite experiments, a trend which
continued even in the Planck 18’ results Aghanim et al. (2018); Akrami et al.
(2018).A variety of Inflationary Models are supported by the observational
data ,and amongst the most famous of these paradigms are Single Field
Inflationary Models.In these models, Inflation is driven by a single Scalar
Field, popularly referred to as the ”Inflaton” Baumann (2009). While the usual
paradigm of Single Field Models considers only a real scalar field which
dominates the energy density of the Universe in Early Times, there are some
other regimes where non standard considerations are taken about the field
nature or the way it interacts Berera (1995); Golovnev et al. (2008).But still
the conventional single field models (which some would like to call
”Supercooled Inflationary Models” Berera (1995)) are very well supported
observationally and are very prevalent in theoretical studies too.

String Theory has attracted a resounding amount of attention in the
Cosmological Community ever since it’s inception McAllister and Silverstein
(2008); Baumann and McAllister (2015); Gasperini (2007); Heckman et al.
(2019a, b).As one of the most well studied theories which promise to describe
” Quantum Gravity”,one may expect String Theory to appropriately describe the
workings of the Early Universe. Indeed it has been found that the ”String
Landscape”(the set of all possible vacua admitted by the theory), is quite
large going to the order of $10^{500}$. Hence one can expect that String
Theory can be a good candidate for a ” Theory of Everything” for our
Universe.But then the question of how one could distinguish between effective
field theories which are consistent and those which are non consistent with
String Theory. To this end, Cumrun Vafa in Vafa (2005) coined the term
”Swampland”, referring to the class of Effective Field Theories incompatible
with a self consistent Theory of Quantum Gravity. Further, to distinguish
between theories in the swampland and those compatible with self consistent
Quantum Gravity, Vafa and his co authors proposed a number of ”Swampland
Conjectures” Ooguri and Vafa (2007, 2016); McNamara and Vafa (2019); Bedroya
and Vafa (2019). The prominent Swampland Conejctures which gathered immediate
interest in the context of Inflationary Cosmology were :

$1$ : Swampland Distance Conjecture (SDC) : This conjecture limits the field
space of validity of any effective field theory Ooguri and Vafa (2007). This
sets a maximum range traversable by the scalar fields in an EFT as

$\Delta\phi\leq d\sim\mathcal{O}(1)$ (1)

where we are working in the Planck Units $m_{p}=1$ where $m_{p}$ is the
reduced Planck’s constant, d is some constant of $\mathcal{O}(1)$ , and $\phi$
is the Scalar Field of the EFT.

$2$ Swampland De Sitter Conjecture (SDSC) : This Conjecture states that it is
not possible to create dS Vacua in String Theory Ooguri and Vafa (2016). The
conjecture is a result of the observation that it has been very hard to
generate dS Vacua in String Theory Dasgupta et al. (2019); Danielsson and Riet
(2018)( While it has been shown that creating dS Vacua in String Theory is
possible in some schemes ,like the KKLT Construction Kachru et al. (2003)).
The Conjecture sets a lower bound on the gradient of Scalar Potentials in an
EFT ,

$\frac{|V^{\prime}|}{V}\geq c\sim\mathcal{O}(1)$ (2)

where c is some constant of $\mathcal{O}(1)$ , and V is the scalar Field
Potential. Another ” refined ” form of the Swampland De Sitter Conjecutre
(RSDSC) places constraints on the Hessian of the Scalar Potential (a finding
which first appeared in Garg and Krishnan (2018) and later in Ooguri et al.
(2019) ). Expressed in $m_{p}=1$ units, it reads

$\frac{V^{\prime\prime}}{V}\leq-c^{\prime}\sim\mathcal{O}(1)$ (3)

where $c^{\prime}$ is again some constant of Order 1.

These criterion have quite strong implications across different fields of
physics. It’s pretty revealing consequences on the Standard Model were
discussed in March-Russell and Petrossian-Byrne (2020).Here it was shown that
although the SM is compatible with the conjectures so far, changing the value
of various a-priori free parameters, such as masses of quarks and the Higgs
boson, would plunge the SM into the swamp. The most exciting consequence of
this might be to tie certain unexplained features of the SM, such as the
notoriously problematic lightness of the Higgs to quantum gravity.These
conjectures also have a very interesting say on the nature of Dark Energy
Agrawal et al. (2018); Odintsov and Oikonomou (2019); Chiang et al. (2019) and
Dark Matter Kawasaki and Takhistov (2018); Agrawal et al. (2019).Further,it
has been shown that Single Field Inflation in a Universe described by General
Relativity based Cosmology is incompatible with the Swampland Criterion, in a
particular the De Sitter condition, for a general class of potentialsKinney et
al. (2019). The disagreement can be seen in essence through the slow roll
parameters for Single Field GR Based Inflation. The primary slow roll
parameters of usual Single Field Inflation are the famous $\epsilon$ and
$\eta$ parameters, and it is well known that for Inflation to occur both of
them should be very less than unity Baumann (2009). Defining $\eta$ and
$\epsilon$, in Planck Units the condition is given as

$\epsilon=\frac{1}{2}\left(\frac{V^{\prime}}{V}\right)^{2}$ (4)
$\eta=\frac{V^{\prime\prime}}{V}$ (5)

It is immediately seen from the definitions of the $\epsilon$ parameter (4) ,
and the De Sitter Conjecture (2) , that these conditions are in conflict with
each other. Furthermore, a new Swampland Criteria by the name of ” Trans
Planckian Censorship Conjecture(TCC)” Bedroya and Vafa (2019) , implies that
Singe Field GR Based Inflationary Models would have to be severely fine tuned
for them to not be in the Swampland Bedroya et al. (2020). A lot of work has
been done to further understand the relations of this conjecture with
Inflation Dhuria and Goswami (2019); Brahma (2020); Mizuno et al. (2020); Cai
and Wang (2020); Kamali and Brandenberger (2020).While these works seem to
show that Inflationary Cosmology and Swampland Criterion are in direct logger
heads, hence a lot of research has gone into understanding whether there could
be ways to reduce, if not fully resolve, the conflict between the conjecture
and Inflationary Models. A lot of work has gone into resolving the conflict
between Single Field Inflation and the Conjectures. In Geng (2020) it was
found that in a certain regime on the parameter space the back reaction from
emergent modes due to distance conjecture can render slow roll consistent with
de-sitter conjecture and its refinement.In Scalisi (2020) the implications of
SDC on Inflation were discussed for the tower of states having spin larger
than 1 and it was shown that the only way to avoid conflict with Single Field
Inflation was to have a cut off spin value.Scalisi and Valenzuela (2019) also
discussed about the implications on Inflationary $\alpha$ attractors. Brahma
et al. (2020) discussed the general properties required for avoiding the
swampland in single-field inflation while relationships between different
Swampland Conjectures in an Inflating Space-Time were pointed out in Brahma
and Hossain (2019). While in Ashoorioon (2019), it was shown how one can
salvage Single Field Inflation given the Swampland Conjectures using excited
initial states for tensor perturbations. While this great amount of literature
has produced some really insightful and encouraging results, Fine Tuning of
key physical parameters still remains a big issue for Single Field Inflation
in a GR Based Cosmology given the Swampland Conjectures. However, Inflation
stands on more amicable standings with the conjectures in certain other
different regimes of Inflation besides Single Field GR Based Models. Multi
Field Inflation has been shown to be consistent with the Swampland Conjectures
(3) and (4). Achúcarro and Palma (2019); Bravo et al. (2020); Damian and
Loaiza-Brito (2019). The paradigm of Warm Inflation, even for Single Field
Models, is consistent with the Swampland Conjectures Berera and Calderón
(2019); Motaharfar et al. (2019); Das (2019); Das et al. (2020) .

Alongside the Usual GR Based Cosmology, an enticing possibility for
Inflationary Cosmology in recent decades has been the study of Inflation in
various Modified Cosmological Scenarios. These include the Randall-Sundrum and
other Braneworld Scenarios from String Theory itself Randall and Sundrum
(1999a, b); Gogberashvili (2002), Cosmologies due to Modified Gravitational
Scenarios Cai et al. (2008); Tsallis and Cirto (2013); Gomez et al. (2011) ,
Loop Quantum Cosmologies Ashtekar et al. (2006a, b, c) amongst others. The
paradigm of Single Scalar Field Inflation has been studied substantially in
all the above mentioned regimes as well. Braneworld Inflation has been studied
both in the Supercooled and Warm Regimes Setare et al. (2014); Hawkins and
Lidsey (2001); Choudhury et al. (2003); Kallosh and Linde (2003); Del Campo
and Herrera (2007); Cid et al. (2007); Matsui and Takahashi (2019), Loop
Quantum Inflation was explored in Bhardwaj et al. (2019) while del Campo
(2012) studied the modified Chern-Simons Cosmology. It has also been shown
that Single Field Inflation in Modified Cosmological Scenarios are quite
easily consistent with Swampland Conjectures Kamali et al. (2020); Lin et al.
(2019); Blumenhagen et al. (2017); Odintsov and Oikonomou (2020); Bernardo
(2020); Adhikari et al. (2020). Single Field Supercooled Inflation in non GR
Based Cosmological Scenarios being in extremely good agreements with Swampland
Conjectures might suggest that a Self Consistent Quantum Gravitational theory
would point towards the Early Universe having a different Cosmological Setup
than General Relativity (considering that the Swampland Conjectures indeed
describe conditions a self consistent theory of Gravity would uphold).

In this paper, we show that a wide variety of Single Scalar Field Inflationary
Models in Modified Cosmological Scenarios can satisfy the Swampland Criteria.
In Section 2 , we will briefly describe the Exact Solution Approach to
Inflation del Campo (2012) which we will use in the whole of our paper. In
Section 3 , use the Exact Solution Approach and show how Single Field
Inflation can satisfy Swampland Constraints for a general class of modified
cosmological scenarios. We conclude our work in Section 4 with comments on the
scope of applicability of our analysis.

##  II The Exact Solution Approach to Inflation

The Exact Solution approach for Single Field (Supercooled) Inflationary Models
was introduced by Del Campo in del Campo (2012). The crucial point of note is
that a Generalized Friedmann Equation of the form,

$F(H)=\frac{8\pi}{3m_{p}^{2}}\rho_{\phi}$ (6)

where we have written the Friedmann Equation in the units used in del Campo
(2012) , $c=\hbar=1$ and $\rho_{\phi}$ is the energy density of the Inflaton
Field and $F(H)$ is a General Function of the Hubble Parameter $H=H(\phi)$ (
$\phi$ dependent Hubble Parameters were famously first considered in the
Hamilton Jacobi Approach to Inflation of Kinney Kinney (1997) ) . This
Equation is quite general in the sense that one can arrive at the Friedmann
Equation of different cosmologies for different $F(H)$. For Instance, the
Braneworld Scenario Friedmann Equation is attained for Shiromizu et al. (2000)
for

$F(H)=\left(\frac{8\pi\lambda}{3m_{p}^{2}}\right)\bigg{[}\sqrt{1+\left(\frac{3m_{p}^{2}}{4\pi\lambda}\right)H^{2}}-1\bigg{]}$
(7)

The Chern-Simons Model Gomez et al. (2011) can be attained for

$F(H)=H^{2}-\alpha H^{4}$ (8)

Various Modified Gravitational Models can also be encapsulted in (5) . Like
L(R) gravity for $L(R)=R-\frac{\alpha^{2}}{6R}$ can be given by for $F(H)$ as

$F(H)=\frac{6H^{2}-\frac{\alpha}{2}}{\frac{11}{8}-\frac{9}{4\alpha}H^{2}}$ (9)

And similarly, one can even get Loop Quantum Cosmology Ashtekar et al. (2006c)
with (5).

As is well known during Slow Roll Inflation, the field velocity
$\dot{\phi}=\frac{d\phi}{dt}$ is very less than the Scalar Potential
$V(\phi)$,

$\dot{\phi}^{2}<<V(\phi)$ (10)

which allows us to write $\rho_{\phi}\approx V(\phi)$. The Equation of Motion
which the scalar Field Satisfies is the usual Klein Gordon Equation,

$\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)=0$ (11)

where $\ddot{\phi}=\frac{d^{2}\phi}{dt^{2}}$

Based on these basics ideas, the Field Velocity is given by

$\dot{\phi}=-\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}H^{\prime}}{H}$ (12)

one can further find the $\epsilon$ and $\eta$ parameters to be,

$\epsilon=\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}}{H}\left(\frac{H^{\prime}}{H}\right)^{2}$
(13)
$\eta=\frac{m_{p}^{2}}{8\pi}\frac{F_{{}_{\prime}H}}{H}\frac{H^{\prime\prime}}{H}$
(14)

where $F_{{}_{\prime}H}=\frac{\partial F}{\partial H}$ and
$H^{\prime}=\frac{dH}{d\phi}$ The Number of e-folds is given by,

$N=\frac{8\pi}{m_{p}^{2}}\int_{\phi_{e}}^{\phi}\frac{H^{2}}{H^{\prime}F_{{}_{\prime}H}}d\phi$
(15)

One can further define the horizon flow equations and comment on the attractor
nature of the Inflationary Model in question using this approach, as sh own in
del Campo (2012).

The spectra of Scalar and Tensor Perturbations are of special interest for any
Inflationary Model as they allow the model to be tested using observational
data Baumann (2009). Using the definition of the slow roll parameters and the
usual formulation of the amplitude squared scalar Perturbations at horizon
exit $k=aH$,

$P_{\mathcal{R}}(k)=\left(\frac{H}{|\dot{\phi}|}\right)^{2}\left(\frac{H}{2\pi}\right)_{aH=k}$
(16)

we get the scalar spectral index as ,

$n_{s}=1+\frac{dP_{\mathcal{R}}(k)}{dlnk}=1+2\eta-2\epsilon\left(3-H\frac{F_{{\prime}HH}}{F_{{}_{\prime}H}}\right)$
(17)

where $F_{{}_{\prime}HH}=\frac{\partial^{2}F}{\partial H^{2}}$.
Similarly the squared amplitude of Tensor Perturbations is given by ,

$P_{\mathcal{T}}(k)=\frac{16\pi}{m_{p}^{2}}\left(\frac{H}{2\pi}\right)_{aH=k}^{2}$
(18)

And hence the tensor spectral index is given as,

$n_{T}=-2\epsilon$ (19)

The tensor to scalar ratio is hence given by,

$r=2\frac{F_{{}_{\prime}H}}{H}\epsilon$ (20)

While one can also get the Running of both scalar and tensor spectral index,
we will not pursue it at this time. The reader can refer to del Campo (2012)
for any additional details about this approach.

Now, we have cleared the basics of the Exact Solution Appraoch which we will
be using in the next section.

##  III Consistency of Single Field Models in Modified Cosmologies

We will use the Planck Units $m_{p}=\sqrt{\frac{1}{8\pi G}}=1$. So our
Generalized Friedmann Equation (5) takes the form,

$F(H)=\frac{\rho_{\phi}}{3}$ (21)

Which during Inflation($\dot{\phi}^{2}<<V(\phi)$ ) becomes ,

$F(H)=\frac{V}{3}$ (22)

This allows us to write the Gradient of the derivative of the scalar potential
with respect to $\phi$ as ,

$F_{{}_{\prime}H}H^{\prime}=\frac{V^{\prime}}{3}$ (23)

This allows us to write the ratio

$\frac{|V^{\prime}|}{V}=\frac{F_{{}_{\prime}H}H^{\prime}}{F}$ (24)

For now, let’s call the above ratio k

$k=\frac{V^{\prime}}{V}=\frac{F_{{}_{\prime}H}H^{\prime}}{F}$

. Now, the $\epsilon$ parameter in $m_{p}=1$ units is given by,

$\epsilon=\frac{F_{{}_{\prime}H}}{H}\left(\frac{H^{\prime}}{H}\right)^{2}$
(25)

Now, $\epsilon$ can be written in terms of k as ,

$\epsilon=\frac{FH^{\prime}k}{H^{3}}$ (26)

Now, one of the central points of disagreement between Single Field
Inflationary Models in GR Based Cosmologies and the Swampland conjectures is
the logger heads the smallness of $\epsilon$ parameter condition,
$\epsilon<<1$ , is with the Swampland De Sitter Conjecture (2). This conflict
is , however, not evident in various investigations of Inflation in Modified
Cosmological Scenarios Kamali et al. (2020); Lin et al. (2019); Blumenhagen et
al. (2017). Keeping that in mind, if we now write the De Sitter Swampland
Conjecture in a Modified Cosmology given by a Friedmann Equation of the form
(21) in terms of the ratio k, we have the De Sitter Criterion as

$k\geq c\sim\mathcal{O}(1)$ (27)

This tells us that in order to satisfy the De Sitter Swampland Conjecture in a
Modified Cosmological Scenario we need ,

$\frac{F_{{}_{\prime}H}H^{\prime}}{F}\geq c$ (28)

So now, it is evident that

$\epsilon<<1$ (29)

even for $k\geq\mathcal{O}(1)$ .
Hence, a large class of cosmologies which are within the scope of the
generalized Friedmann Equation (21) can easily be consistent with the De
Sitter Swampland Conjecture. For GR Based Cosmology, $F(H)=H^{2}$, one
recovers the conflict between the concerned conjecture and single Field
Inflation as the Exact Solution approach reduces to Kinney’s Hamilton Jacobi
Approach in that limit Kinney (1997).

Another point of Conflict with the Conjectures and Single Field Inflationary
Models in GR Based Cosmologies is the issue of insufficient e-fold number
Kinney et al. (2019). It can be shown that the Number of e-folds for Single
Field GR Based Inflationary Models can be given by, Lin et al. (2019) ,

$N\approx\frac{V\Delta\phi}{V^{\prime}}$ (30)

If we move in accordance with the Swampland Conjectures, then
$N<\mathcal{O}(1)$ . This is a serious problem , as the most recent data on
Inflation Akrami et al. (2018) requires around 50 to 60 Number of e-folds. It
is again clearly evident that the Swampland Conjectures are in conflict with
Single Field Inflationary Models in GR Based Cosmology. But once again, it has
been shown that this problem does not arise in Modified Cosmological Scenarios
Kamali et al. (2020); Lin et al. (2019); Blumenhagen et al. (2017) . Now, we
will look at the same problem within the view of the Exact Solution Approach
and show that this issue does not arise in a large class of Modified
Cosmological Scenarios.

The e-fold Number for Inflation del Campo (2012) expressed in $m_{p}=1$ units
is ,

$N=\int_{\phi_{e}}^{\phi}\frac{H^{2}}{F_{{}_{\prime}H}H^{\prime}}d\phi$ (31)

The above expression for the e-fold number can be rewritten as

$N=\int_{\phi_{e}}^{\phi}\frac{H^{2}}{F}\frac{1}{\frac{F_{{}_{\prime}H}H^{\prime}}{F}}d\phi$
(32)

Thus, e-fold number can be approximately given as ,

$N\approx\frac{H^{2}}{F}\frac{\Delta\phi}{\frac{F_{{}_{\prime}H}H^{\prime}}{F}}=\frac{H^{2}}{F}\frac{\Delta\phi}{k}$
(33)

Now, in accordance to the Distance Conjecture and De Sitter Conjecture ,
$\Delta\phi\leq\mathcal{O}(1)$ , $k\geq 1$ respectively. We see that in
contrast to Single Field Inflation in GR Based Cosmology, the e-fold number
for Single Field Models in a Modified Cosmological Scenario can be high enough
for sufficient Inflation to occur. Again , in the GR Limit $F=H^{2}$ the
problem of insufficient e-fold number persists but in a generalized scenario
it can be appropriately high.

Another issue with Single Field Inflationary Models in GR Based Cosmology and
Swampland Conjectures is the order of the parameter c in the definition of the
De Sitter Conjecture. From String Theory motivated constructions, c in (2)
should be of the order of unity. While it was shown in Kinney et al. (2019)
that in order for the conjectures and Single field Models to be consistent
with the value of the scalar spectral index of Planck 18’ data Akrami et al.
(2018) , $c\sim\mathcal{O}(0.1)$. This shows a stark contrast between the
String Theory Motivations of the Conjecture and their consistency with the
observational data for Inflationary Cosmology.

We now show that the String Theoretic Motivations of the Swampland Conjectures
can indeed be consistent with the Observational Data for Inflationary Models
in Modified Cosmological Scenarios. From the equation (16) for $n_{s}$ and the
definitions of the $\epsilon$ and $\eta$ parameters (12-13) in $m_{p}=1$ units
, we can express the scalar spectral index as,

$n_{s}=1+2\left(\frac{F_{{}_{\prime}H}}{H}\frac{H^{\prime\prime}}{H}\right)-2\frac{F_{{}_{\prime}H}}{H}\left(\frac{H^{\prime}}{H}\right)^{2}\big{(}3-H\frac{F_{{}_{\prime}HH}}{F_{{}_{\prime}H}}\big{)}$
(34)

Mow, the above equation can be rewritten as,

$n_{s}=1+2\frac{F_{{}_{\prime}H}H^{\prime}}{F}\frac{FH^{\prime\prime}}{H^{\prime}H^{2}}-2\frac{F_{{}_{\prime}H}H^{\prime}}{F}\frac{FH^{\prime}}{H^{3}}\bigg{(}3-H\frac{F_{{}_{\prime}HH}}{F_{{}_{\prime}H}}\bigg{)}$
(35)

And finally after some rearrangement, one can arrive at the following relation

$\left(\frac{(1-n_{s})H^{2}}{2F}\right)\bigg{[}\frac{1}{\left(3-H\frac{F_{{}_{\prime}HH}}{F_{{}_{\prime}H}}\right)\frac{H^{\prime}}{H}-\frac{H^{\prime\prime}}{H^{\prime}}}\bigg{]}=\frac{F_{{}_{\prime}H}H^{\prime}}{F}\geq
c$ (36)

The above relation allows us to have an upper bound on c in terms of the
observed value of the scalar spectral index in some Cosmological Scenario
described by (21). Alongside this, we now make use of the Refined De Sitter
Conjecture (3) to comment on the relationship between $\eta$ and 4
$c^{\prime}$. Using (22) , we can write,

$F_{{}_{\prime}HH}{H^{\prime}}^{2}+F_{{}_{\prime}H}H^{\prime\prime}=\frac{V^{\prime\prime}}{3}$
(37)

In the light of the above relation, one can express (3) in terms of $\eta$
(14) as

$\frac{1}{F_{{}_{\prime}H}}\bigg{[}F_{{}_{\prime}HH}H^{\prime}+\eta\frac{F_{{}_{\prime}H}H^{2}}{H^{\prime}}\bigg{]}\leq-c^{\prime}$
(38)

Equations (36) and (38) alongside (26) are especially important in
understanding the distinction between implications of Swampland Conjectures
for Single Field Inflation in GR Based Cosmology and in more generalized
Scenarios. It was shown systematically in Kinney et al. (2019) how (2) and (3)
lead the $\epsilon$ and $\eta$ parameters in Single Field GR Based
Inflationary Models to be constrained as

$\epsilon\geq\frac{c^{2}}{2}$ $\eta\leq-c^{\prime}$

These relations lead in turn to

$1-n_{s}=[3c^{2};2c^{\prime}]$ (39)

Then, the authors in Kinney et al. (2019) used the data for the scalar
spectral index and the tensor to scalar ratio Akrami et al. (2018) to show
that $c\leq\mathcal{O}(0.1)$ and $c^{\prime}\leq\mathcal{O}(0.01)$ . This led
them to categorically demonstrate that there is a severe conflict with the
String Motivated Definitions of c and $c^{\prime}$ being order 1, and the
Observational Data for Inflation which constrain them to be smaller orders in
order for Single Scalar Field models in GR Based Cosmologies to be compatible
with them.

We now see from equations (38) , (36) and (26) that in a general modified
cosmology described by (22) the strict relations of $\epsilon$ and $\eta$ to c
and $c^{\prime}$ do not hold. (26) shows that c can indeed be some order 1
term and still $\epsilon<<1$. Similarly (36) shows that the scalar spectral
index is not directly related to an upper bound for c , as found for Single
Field Models in Kinney et al. (2019). Rather, in a General Cosmology the free
parameters of the models can be put in limits to fit up well with the
observational data without violating in the Order 1 nature of c. Observational
data also constrains $\eta\leq 0$ and this fact can also be easily satisfied
alongside the Swampland Definition of $c^{\prime}$ in a General Cosmological
Scenario as shown in (38). Equation (38) also shows that, similar to the
result of $\epsilon$ parameter in a general Cosmology, one cannot set a lower
bound on the $\eta$ parameter directly using $c^{\prime}$. This shows that the
disagreement found in Kinney et al. (2019) between the orders of c and
$c^{\prime}$ from the String Theory Motivated Cosmological and that allowed by
the Observational Data would not arise in a General Modified Cosmological
Scenario.

##  IV Concluding Remarks and Discussion

In conclusion, we have shown using quite a general treatment that Single Field
Inflationary Models can indeed be compatible with the Swampland De Sitter and
Distance Conjectures in a wide class of cosmologies where the Scalar Field
follows the usual Klein Gordon Form. We began by discussing in brief about the
Swampland Conjectures and their implications on Inflationary Cosmology, after
which we briefly discussed the Exact Solution Approach to Inflation. Then
using that approach , we showed systematically how Single Field Inflation in
Modified Cosmological Scenarios can bypass the problems it faces in GR Based
Cosmology. We showed how the $\epsilon$ parameter for Inflation can still
appropriately small for Inflation to occur and how the e-fold number can still
be to the scales required by the latest observational data Akrami et al.
(2018). We then showed that both the observational data for Inflation and
String Theoretic Definitions of the c and $c^{\prime}$ parameters of the De
Sitter Conjectures and it’s refinement , respectively, can agree for Single
Field Inflation in Modified Cosmological Scenarios. Hence, in essence, we have
shown that Single Field Inflation is still very much compatible with the
Swampland Conjectures in a wide Class of Modified Cosmological Scenarios. One
crucial point which we would like to elaborate here is that in obtaining the
equation of the inflaton field we have assumed that the matter, specified by
the inflaton scalar field, enters into the action Lagrangian in such a way
that its variation in a Friedmann-Robertson-Walker-Lemaitre background metric
leads to the Klein-Gordon equation, expressed by (11). Therefore our method is
only applicable to theories where the background metric alongside the
perturbations, are not modified.This means that Horava-Lifshitz theories of
gravity Mukohyama (2010) or theories of similar plight are beyond the scope of
our approach. Hence, we cannot comment on the compatibility of Inflationary
Regimes in such theories with the Swampland Conjectures.

##  V Acknowledgments

The author of this paper would like to tremendously thank Dr. Sunny Vagnozzi
(Newton-Kavli fellow at the Kavli Institute of Cosmology (KICC) and the
Institute of Astronomy, University of Cambridge) for insightful comments on
the paper. The author cannot express enough gratitude for this help.

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# Shapelet-based Model-agnostic Counterfactual Local Explanations for Time
Series Classification

Qi Huang , Wei Chen11footnotemark: 1, Thomas Bäck, Niki van Stein These
authors contribute equally to the paperCorresponding author

###### Abstract

In this work, we propose a model-agnostic instance-based post-hoc
explainability method for time series classification. The proposed algorithm,
namely Time-CF, leverages shapelets and TimeGAN to provide counterfactual
explanations for arbitrary time series classifiers. We validate the proposed
method on several real-world univariate time series classification tasks from
the UCR Time Series Archive. The results indicate that the counterfactual
instances generated by Time-CF when compared to state-of-the-art methods,
demonstrate better performance in terms of four explainability metrics:
closeness, sensibility, plausibility, and sparsity.

## Introduction

In recent years, the performance of machine learning models has become
increasingly impressive. Advances in the capability of data computation, the
development of sophisticated algorithms, and the proliferation of high-quality
datasets have all contributed to this upward trend (Sarker 2021a, b). Machine
learning models and methods are being leveraged in various aspects of daily
life and science, such as automated driving, banking, healthcare, physics, and
geoscience, where intelligible explanations for their decision-making
processes are essential for establishing public trust over utilized opaque
models (Rudin 2019). In addition, these methods can also generate new insights
for researchers in various domains. Especially in safety-critical applications
such as medical diagnosis and self-driving cars, mistakes in predictions could
be fatal; thus, there is an urgent need either to provide post-hoc
explanations based on the decisions made by the models or to build inherently
explainable models before predictions (ante-hoc).

In response to the opacity property of machine learning models, the field of
eXplainable AI (XAI) has emerged to address this issue. Although XAI methods
have been proposed and effectively applied to static data such as images,
text, and tabular data, there has been relatively less work on time series
data (Schlegel et al. 2019). The existing XAI approaches for time series
analysis can be categorized into three types regarding the level of
explanation: point-level, subsequence-level, and instance-level. Among the
existing methods, counterfactual explanations for instance-level have recently
gained wider popularity (Theissler et al. 2022).

As its name suggests, an instance-level counterfactual explanation for a to-
be-explained (target) time series is an artificially generated instance,
derived from the target, that does not exist in original observations but
contradicts the original facts e.g., predicted to be a different class in
classification. Perturbing the target time series, where minimal changes are
preferred, is one of the most common techniques for creating counterfactual
explanations for time series instances (Li et al. 2022; Li, Boubrahimi, and
Hamd 2022). Explainability is hence achieved by highlighting the difference
between the original instance and its generated counterpart. The focus of this
research is to use a Generative Adversarial Network (GAN) to fabricate
counterfactual explanations that can explain the predictive behavior of time
series classifiers on an instance level.

Figure 1: The flowchart of Time-CF in binary classification. The process
describes how the counterfactual explanation of an instance labeled as Class
A, is synthesized by TimeGAN and shapelet transform, based on instances from
Class B.

## Related Works

In this work, we focus on counterfactual-based XAI methods for Time Series
Classification (TSC). TSC stands for a type of machine learning task where a
machine learning model (classifier) shall be learned to maximally correctly
assign labels to a set of time series instances. We refer the readers to the
survey paper regarding XAI for time series classification (Theissler et al.
2022) for additional information.

Wachter et al. are pioneers in generating counterfactual instances to
elucidate anomalies (Wachter, Mittelstadt, and Russell 2017). This involves
simultaneously minimizing a prediction loss function that drives the
prediction on counterfactuals towards a different class and a distance loss
function that ensures the similarity between counterfactuals and to-be-
explained instances. Recent efforts to use counterfactual instances as
explanations are mostly derived from this work. Native-Guide (Delaney, Greene,
and Keane 2021a) is another counterfactual-based method, introduced by Delaney
et al, following four desirable properties: Sparsity, Proximity, Plausibility,
and Diversity. In essence, they find the nearest-unlike neighbor of the to-be-
explained instance and generate a counterfactual instance for explanation. MG-
CF (Li, Boubrahimi, and Hamd 2022), SG-CF (Li et al. 2022) are also extensions
of (Wachter, Mittelstadt, and Russell 2017), but incorporate shapelets into
consideration. Specifically, MG-CF generates counterfactual instances by using
shapelet Transform (Hills et al. 2014) to extract all potential shapelet
candidates and then replace the same part of the to-be-explained instance.
Moreover, SG-CF adds an additional term to the loss function used in (Wachter,
Mittelstadt, and Russell 2017), which ensures high Euclidean-based similarity
between the counterfactual instance and the shapelet, thus providing more
robust explanations. Besides directly perturbing the original sequence which
is essentially injecting random noise to the region-of-interest, (Lang et al.
2022) and (Dantas, Marcos, and Ienco 2023) propose to employ the generative
adversarial networks (Goodfellow et al. 2020) to fabricate entire
counterfactual instances, which also enlightens us in developing our method.

## Methodology

Having discussed related work on counterfactual-based instance-level XAI for
time series classification, we now introduce our proposed algorithm, Time-CF.
A flowchart of the algorithm under binary classification is shown in Figure 1.
An example outcome of Time-CF is displayed in Figure 2, where only a
subsequence (the blue curve) of the to-be-explained instance is altered (to
the orange curve).

The process starts with extracting a designated number of shapelet candidates
of various lengths from the time series instances. Shapelets are subsequences
of a time series instance that can be extracted by shapelet transform (Hills
et al. 2014; Bostrom and Bagnall 2017). When it comes to instance-level
explanations, shapelets can be directly projected onto the original time
series, thus providing more informative and intuitive explanations that
require minimal domain knowledge. A typical Shapelet Transform (ST) takes as
input the training set and outputs extracted shapelets. The transform is
composed of three steps: candidate extraction, computation of information gain
per candidate, and shapelet selection. As a consequence of the selection,
shapelets with less discriminative information (w.r.t. class labels) are
filtered out, and those with relatively high discriminative power are retained
for the perturbation steps. In this study, we utilize the Random Shapelet
Transform (RST), which accommodates time limitations and random candidate
extraction, thus providing time-saving and flexibility advantages in
comparison to the complete shapelet transform. After shapelet extraction, the
collected candidates are sorted by their respective information gain in
descending order and then filtered to the top N (the number of retained
shapelets). Inspired by the uprising interest in utilizing generative models
for XAI in time series analysis (Van Looveren et al. 2021; Lang et al. 2022;
Dantas, Marcos, and Ienco 2023), we likewise alternatively apply generative
adversarial networks dedicated for time series data namely, TimeGAN (Yoon,
Jarrett, and Van der Schaar 2019), as our counterfactual generation model.

Figure 2: An example of a time series counterfactual instance, where the blue
curve represents the to-be-explained shapelet of the electrocardiogram
instance from the UCR ECG200 (Chen et al. 2015). The orange curve stands for
its generated counterfactual subsequence, which drives the classifier to alter
its prediction.

Data: shapelet candidates: $S$, a to-be-explained instance with class label
$l$: $T$, the length of $T$: $N$, a to-be-explained classifier $f$, the
maximum number of generated sequences: $M$

1

Result: a collection of counterfactual instances: $C$

2

$C\leftarrow\emptyset$ ;

$\triangleright$ A set of counterfactual instances for $T$

$F\leftarrow\emptyset$ ;

$\triangleright$ A set of generated fake instances

3 Train a TimeGAN with instances of other labels;

4

5

6for  _$i\leftarrow 1$ to $M$_ do

$\mathbf{x}\leftarrow\left[x_{1},x_{2},\ldots,x_{N}\right]^{T}\sim\text{Uniform}[0,1]$
;

$\triangleright$ Sample the initial vector

$T_{fake}\leftarrow\text{TimeGAN}(x)$ ;

$\triangleright$ Transform $x$ to a fake sequence

7 $F\leftarrow F\cup\\{T_{fake}\\}$

8for  _$s$ in $S$_ do

$\text{start\\_pos},s_{\text{len}}\leftarrow\text{s.info}$ ;

$\triangleright$ Get the starting index and length of $s$

9 if  _s.class == l_ then

10 $\triangleright$ Retrieve shapelets that belong to the same interval as $s$
from each $T_{fake}$ in $F$

11 $S_{fake}\leftarrow\text{crop}(\text{start\\_pos},s_{\text{len}},$F$)$;

12

13 for  _$s_{j}$ in $S_{fake}$_ do

$T_{fake}\leftarrow\text{replace}(T,s_{j},\text{start\\_pos},s_{\text{len}})$
;

$\triangleright$ Replace the target subsequence in $T$ with $s_{j}$

14 if  _$f(T_{fake})\neq s.class$_ then

15 $C\leftarrow C\cup\\{T_{fake}\\}$

16

17

18

return $C$

Algorithm 1 Generating counterfactual explanations of a classifier $f$ for a
time series instance

Algorithm 1 depicts how the counterfactual instances are generated. With a
given to-be-explained time series instance $T$ with class label $l$, we first
train a TimeGAN using all instances from the training data except for those
with label $l$. The trained TimeGAN is then used to generate artificial
instances. The final output sequences of TimeGAN can closely resemble the
sequence-level distribution of the training data while preserving the within-
sequence stepwise transition (Yoon, Jarrett, and Van der Schaar 2019). As
depicted in lines 5 to 7 of the pseudo-code, a total of $M$ fake instances,
each of length $N$, are obtained based on $M$ initial vectors. These initial
vectors are uniformly sampled from $[0,1]^{N}$. Next, the algorithm iterates
over the shapelet candidate set $S$. For each candidate $s$, it is essential
to identify its start and end positions within the corresponding instance.
Further, the $crop()$ function is executed to extract the artificial shapelets
$S_{fake}$ by cropping each interval with the same time range (start and end
positions) in each generated fake instance $T_{fake}$ from $F$, that is, an
instance from the same class. The final process includes iterating through the
set of fake shapelets and using the $replace()$ method to replace the
contiguous time points (ranging from the start position and end position
stored previously) of the to-be-explained time series instance $T$, with one
of the (fake) shapelets generated by TimeGAN. A synthetic time series instance
$T_{fake}$ is considered as a counterfactual only if it flips the prediction
of the to-be-explained classifier, says, the predictive label of $T_{fake}$ is
no longer $l$. Moreover, this step reserves all the fake counterfactual time
series instances for diverse explanations. In case of finding multiple
counterfactual instances, the final recommended one is determined by:

$T_{c}\leftarrow\underset{\ \ \ \ T_{f}\in C}{\text{argmin}}\ d(T_{f},T)$ (1)

where $d$ is a (distance) metric defined over the given time series.
Specifically, we minimize the Hamming distance in this study to ensure minimum
perturbation actions.

## Experimental Setup

Time-Cf is evaluated on four types of classifiers (Convolutional Neural
Networks (CNNs), Diverse Representation Canonical Interval Forest (DrCIF),
Catch22 with Random Forest (RF), and K Nearest Neighbor (KNN) Classifier. Each
classifier is trained with four binary classification time series data from
the UCR time series classification archive (Chen et al. 2015). They are
ECG200, an electrocardiogram; Wafer, sensor signals during silicon wafer
processing for semiconductor fabrication; FordA, readings of engine noise for
outlier detection; MoteStrain, sensor measurements of humidity or temperature.
The results obtained by Time-CF on these four datasets are compared with those
of Native Guide (Delaney, Greene, and Keane 2021a) and Mlxtend (Raschka 2018),
where the former is a well-established counterfactual-based method (see
related works) and the latter is an open-source library tailored for
interpreting machine learning classifiers.

## Analysis of Results

Although there is no consensus on the evaluation of counterfactual-based
methods, closeness, sparsity, plausibility, and diversity have been widely
used in recent studies (Delaney, Greene, and Keane 2021a; Mothilal, Sharma,
and Tan 2020; Li, Boubrahimi, and Hamd 2022; Li et al. 2022). In a similar
vein, we utilize the first three measures and replace diversity with
sensibility in our evaluation since the existence of instance-wise
explanations is more crucial and questionable, as observed in our study (see
Fig. 4).

### Closeness

Figure 3: The results of L1-norm closeness. A low value is preferred, as it
indicates less scale change.

It measures the shape similarity between the counterfactual instance and the
original instance in Manhattan distances (Delaney, Greene, and Keane 2021b).
Figure 3 shows that the counterfactual instances made by our method are the
closest to the instances being explained in all datasets. This reveals Time-
CF’s proficiency in perturbing the minimal, contiguous segment of the original
instances.

### Sensibility

Sensibility serves as a metric to determine the extent to which
counterfactual-based methodologies are model-agnostic or can explain any
classifier. It records the proportion of instances per classifier that are
explained by the given XAI approach. The results in Fig. 4 demonstrate that
both mlxtend and Native-Guide fail to effectively handle a variety of
classifiers across these datasets. In contrast, with the exception of the
extremely imbalanced dataset Wafer, our method can aptly discern the
informative features in time series data, thereby positioning itself as a
model-agnostic tool that can, in theory, clarify any classifier.

Figure 4: Assessment of sensibility, where higher values (darker in plot) are
preferable. The figure displays the ratios of each post-hoc explainer in
successfully finding counterfactual instances across four datasets.

### Plausibility

Figure 5:  Assessment of plausibility, where a lower value is preferable. The
y-axis quantifies the outlier ratios of the generated counterfactual
instances.

In this experiment, we delve deep into the percentage of generated instances
identified as out of distribution. Given our emphasis on the importance of
plausibility when generating counterfactual instances, we exclusively retain
those counterfactuals with the lowest outlier rates to ensure heightened
plausibility. As depicted in Fig. 5, our method outperforms others in terms of
plausibility in dataset Wafer. However, it shows worse performance than the
others when using an imbalanced dataset. Note that this metric is dependent on
the outlier detection algorithm, which in this study is the Isolation Forest.

### Sparsity

Lastly, curve graph in Fig. 6 contrasts the sparsity level of each XAI method.
When comparing the sparsity levels among mlxtend (blue), Native-Guide (green)
and Time-CF (yellow), our approach consistently showcases superior
performance. It is evident that both mlxtend and Native-Guide require
modifications to nearly all the time steps to alter the class of the to-be-
explained instance from one to another, with values approaching a sparsity of
0%. In contrast, our approach only perturbs a minor segment, thereby offering
not only clearer visualization for analytical objectives but also a more
intuitive comprehension of which portions primarily influence the classifier.

Figure 6: Measurement for sparsity (the higher, the better). This metric
indicates the extent to which the number of time steps is altered. A value of
0 means that the counterfactual instances are derived from the original
instances that have alterations at every time point.

## Conclusion

In this work, a model-agnostic, counterfactual-based XAI method, Time-CF is
introduced to explain any prediction for any classifier concerning time series
classification tasks. Time-CF chiefly utilizes shapelets and TimeGAN to
deliver meaningful counterfactual explanations, exhibiting exceptional
performance in terms of closeness, sensibility, plausibility, and sparsity
measures. The incorporation of shapelets indicates that a contiguous segment
of the time series instance is perturbed, thus offering intuitive insights for
the analysis of anomalies for end-users. In addition, TimeGAN ensures that at
least one counterfactual explanation is generated, provided the training set
is both ample and not extremely imbalanced. TimeGAN also guarantees that the
explanations produced adhere to the data distribution.

In future work, we aim to refine our methodology to cater to multi-class and
multivariate time series classification tasks. Moreover, a more profound
exploration of the synergies between TimeGAN and Shapelet Transform (ST) will
be undertaken to address the suboptimal performance observed with imbalanced
datasets, and a wider set of experiments will be conducted to assess the
behavior and performance of the proposed method against other purely GAN-based
methods (Lang et al. 2023).

## Acknowledgement

This publication is part of the project XAIPre (with project number 19455) of
the research program Smart Industry 2020 which is (partly) financed by the
Dutch Research Council (NWO).

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