# A Cognitive Radio Enabled RF/FSO Communication Model for Aerial Relay
Networks: Possible Configurations and Opportunities

Eylem Erdogan, , Ibrahim Altunbas, ,
Nihat Kabaoglu, and Halim Yanikomeroglu E. Erdogan and N. Kabaoglu are with
the Department of Electrical and Electronics Engineering, Istanbul Medeniyet
University, Istanbul, Turkey, (e-mails: {eylem.erdogan,
nihat.kabaoglu}@medeniyet.edu.tr). I. Altunbas is with the Department of
Electronics and Communication Engineering, Istanbul Technical University,
Istanbul, Turkey, (e-mail: ibraltunbas@itu.edu.tr).H. Yanikomeroglu is with
the Department of Systems and Computer Engineering, Carleton University,
Ottawa, ON, Canada, (e-mail: halim@sce.carleton.ca).

###### Abstract

Two emerging technologies, cognitive radio (CR) and free-space optical (FSO)
communication, have created much interest both in academia and industry
recently as they can fully utilize the spectrum while providing cost-efficient
secure communication. In this article, motivated by the mounting interest in
CR and FSO systems and by their ability to be rapidly deployed for civil and
military applications, particularly in emergency situations, we propose a CR
enabled radio frequency (RF)/FSO communication model for an aerial relay
network. In the proposed model, CR enabled RF communication is employed for a
ground-to-air channel to exploit the advantages of CR, including spectrum
efficiency, multi-user connectivity, and spatial diversity. For an air-to-air
channel, FSO communication is used, since the air-to-air path can provide
perfect line-of-sight connectivity, which is vital for FSO systems. Finally,
for an air-to-ground channel, a hybrid RF/FSO communication system is
employed, where the RF communication functions as a backup for the FSO
communication in the presence of adverse weather conditions. The proposed
communication model is shown to be capable of fully utilizing the frequency
spectrum, while effectively dealing with RF network problems of spectrum
mobility and underutilization, especially for emergency conditions when
multiple unmanned aerial vehicles (UAVs) are deployed.

###### Index Terms:

Aerial relay networks (ARNs), cognitive radio (CR), free space optical (FSO)
communication, unmanned aerial vehicle (UAV).

##  I INTRODUCTION

Over the past few years, there has been a mounting interest in the efficient
use of the frequency spectrum as wireless data traffic has been increasing at
a tremendous rate due to the proliferation of user equipment. The ever
increasing demand for spectrum efficiency has trigged new spectrally efficient
technologies including cognitive radio (CR) and free-space optical (FSO)
communication. Among them, CR can utilize the spectrum access, and it offers
notable advantages including energy efficiency, lower delay, and higher
throughput. FSO communication can be used for high rate point-to-point
communications over long distances. Compared to radio frequency (RF)
communication, FSO systems operate over a much higher bandwidth (over $300$
GHz of unlicensed spectrum band); moreover, they have very narrow laser beams
that can provide inherent security and robustness to interference.

CR technology can be realized by using three spectrum sharing paradigms:
underlay, overlay and interweave. In the underlay approach, an unlicensed
secondary user (SU) can use the same spectrum as a licensed primary user (PU)
without creating any interference. Hence, energy efficient dynamic spectrum
sharing can be enabled. In the overlay approach, the SU helps the PU by
sharing its resources (e.g., time slot, power) to improve the communication
capabilities of the PU. Finally, in the interweave approach, the SU uses the
spectrum band of the PU when no active licensed users are in the system. In
this technique, advanced spectrum sensing technologies aided with artificial
intelligence are required to avoid unwanted interference. With the aid of
these paradigms, CR can utilize the frequency spectrum efficiently [1].

FSO communication, on the other hand, can be described as the transmission of
information by using light emitting diodes or laser beams with the aid of
optical carriers over two stationary points operating in the near infrared
(IR) band. In the current literature, FSO communication is usually enabled at
the rooftops of the buildings to provide line-of-sight (LOS) connectivity [2].
Compared to its RF counterpart, FSO communication can boost the overall
performance of wireless systems by enabling high data rates, lower delay,
inherent security, and easy deployment. On account of this potential, FSO
systems are expected to be used in many areas including campus connectivity,
video surveillance, and monitoring. One of the major challenges in FSO systems
is atmospheric turbulence induced fading, which is caused by adverse weather
conditions. Specifically, fog, snow and rain can cause random fluctuations in
the amplitude and phase of the optical transmission [3]. One of the solutions
presented in the literature is to use FSO and RF communication together to
exploit the advantages of optical communication, while mitigating the
turbulence induced fading. In the so-called “mixed” RF/FSO communication, FSO
systems can be employed with its RF counterpart in a dual-hop or multi-hop
configuration with the aid of well-known relaying approaches. By contrast, in
“hybrid” RF/FSO communication, RF transmission can be used as a backup for the
FSO communication, and the system can switch between RF and FSO depending on
the weather conditions [4].

Like CR and FSO systems, aerial relay (AR) aided communication is another
promising technology for 5G+ networks. ARs can operate as flying base stations
and provide perfect line-of-sight (LOS) connectivity to ground users and other
ARs while being agile enough to maneuver and adjust their altitudes depending
on weather conditions. ARs use amplify-and-forward (AF) and decode-and-forward
relaying (DF) strategies to improve communication reliability and coverage
[5]. Owing to this potential, ARs have many fields of application, including
traffic monitoring, search and rescue missions, remote sensing, and IoT
scenarios where they can provide internet access to anywhere and anytime.
Moreover, in future 5G+ networks, ARs are not only expected to be used for
emergency situations, such as floods and earthquakes, but also for major
political, sports, and music events. Indeed, “Project Loon” has already
started to deploy ARs at altitudes of about $20$ km to provide greater
connectivity to unserved and underserved communities [6].

One of the the most important challenges in AR aided communication is the use
of the frequency spectrum. The current spectrum is overcrowded, and the
unlicensed frequency bands can create additional bandwidth and data rate
problems for ARs. To solve these problems and to create a spectrally efficient
AR network, this article proposes combining two emerging technologies, CR and
FSO communication with an AR network, consisting of low-altitude unmanned
aerial vehicle (UAV) relay nodes (URNs) and high-altitude platform station
(HAPS) systems, as demonstrated in Fig. 1. More precisely, we propose to use a
CR enabled RF transmission for the ground-to-air (GtA) channel to create
spectrally efficient multi-user connectivity, whereas FSO communication is
used at the air-to-air (AtA) link, where a continuous LOS connection is
available with the aid of acousto-optic and electro-optic FSO transceivers
[7]. Finally, for the air-to-ground (AtG) channel, hybrid RF/FSO communication
is used, where the RF communication functions as a backup for the FSO
communication, creating continuous connectivity without any performance loss.

##  II Background

###  II-A Types of ARs

ARs can be classified into two groups on the basis of their altitudes: HAPS
systems and URNs as they act as intermediate relay nodes for integrated access
and backhaul by using AF and DF relaying strategies [8]. URNs can be deployed
at an altitude of up to $300$ m, whereas HAPS systems can be located in the
stratosphere at $17$-$22$ km above the surface [9]. HAPS systems have wider
coverage (about $200$-$300$ km in radius) and longer endurance (up to three
months with the aid of solar energy panels). In HAPS communication, balloons,
airships, airplanes and zeppelins may be used. Among them, balloons and
zeppelins can hover at a fixed point that allows LOS conditions needed for the
optical communication [10]. For example, in Project Loon, essential base
station components were redesigned to be light enough to be carried by a
balloon. Even though HAPS communications have attracted considerable interest
in academia and industry, HAPS systems can be vulnerable to the harsh weather
conditions of stratosphere, where temperatures drop to $-90$∘C and wind speeds
reach up to $100$ km per hour. In addition, the use of the RF spectrum can
create additional problems for HAPS nodes, especially in crowded metropolitan
areas, and deployment time can be critical in the event of emergency
situations.

URNs are flexible, highly maneuverable and can be deployed very quickly,
unlike HAPS systems, making them suitable for emergency conditions, including
earthquakes, floods, and search and rescue missions. URNs can be categorized
as rotary-wing and fixed-wing URNs. Rotary-wing URNs have rotor blades and
there are quadcopter (four rotors) and hexacopter (six rotors) designs; they
can hover in a given area in the sky, which makes them preferable for LOS
communication. Fixed-wing URNs are faster than rotary-wing URNs; however, they
need a runway for takeoffs and landings. In addition, fixed-wing URNs can be
exposed to significant Doppler spread as their relative velocities are very
high. Another important challenge associated with URN communication is service
time. URNs should be fully charged before being deployed, and they can only be
used for a short time.

###  II-B Channel Models

####  II-B1 Ground-to-Air (GtA) Channel

##### Path Loss

In RF communication, the path loss (PL) experienced by GtA or AtG paths can be
expressed as $V(r)=\zeta r^{\rho}$, where $\zeta$ is the PL at the reference
point, $\rho$ is the PL exponent, $r=\sqrt{d^{2}+h^{2}}$ is the distance
between ground station and URNs, $d$ is the horizontal hop distance and $h$
shows the altitude of the URN [5]. In [5], three different types of
deterministic PL models were proposed. In the Type A PL model, a quadcopter
URN was used in the experiments and empirical results showed that the signal
levels dropped quickly in the AtG or GtA channels compared to the AtA channel,
which can be modeled with free-space PL. In the Type B PL model, URNs
experienced shadowing and scattering effects due to man-made structures as
they moved through urban areas. Finally, in the Type C PL model, the URN
altitude was taken as $1\leq h\leq 120$ meters, and the PL for the AtG path
was obtained by considering only the altitude of the URN.

##### Fading Models

In AR aided RF communication, almost all existing fading models including
Loo’s model, Rayleigh, Rician, Nakagami-_m_ and Weibull have been used so far
to model AtG or GtA paths [11]. More precisely, Loo’s model is a composite
channel model, in which the LOS component is modeled by a Log-normal
distribution, whereas the multi-path components follow a Rician distribution.
The Rayleigh channel, which is a special case of Rician and Nakagami-_m_ , is
used for scattering environments. In the literature, Rayleigh distribution was
adopted to the AR networks. Rician can be used to model LOS fluctuations of
the channel, and is suitable for URNs and HAPS systems. In addition, Nakagami-
_m_ and Weibull distributions can characterize HAPS communications [11].

It is noteworthy that, even though these classical fading models can be used
for ARs including URNs and HAPS systems, they neglect the shadowing effect
that can block the AtG or GtA communication due to huge mountains or manmade
constructions as the control center or ground station may be located inside
urban areas including metropolitan centers. For this reason, Shadowed-Rician
and Shadowed-Rayleigh distributions can be more appropriate for modeling AtG
and GtA communications. Specifically, Shadowed-Rayleigh can be used in highly
shadowed places like shopping malls or skyscrapers where users do not have a
LOS communication with HAPS systems or satellites. Likewise, in regions where
LOS connectivity can be established, Shadowed-Rician model can be good fit for
HAPS or satellite communications.

Figure 1: Graphical Illustration of CR enabled RF/FSO communication model for
Aerial Relay Network

####  II-B2 Air-to-Air (AtA) Channel

##### Atmospheric Attenuation

Atmospheric attenuation, which can be caused by scattering and absorption
during the propagation of the beam, is one of the major drawbacks in AtA FSO
communication. In general, atmospheric attenuation ($I_{l}$) is deterministic,
and it can be modeled by using Beer-Lambert’s law as $I_{l}=\exp(-\sigma L)$
[12], where $L$ is the propagation distance between ARs and $\sigma$ is the
attenuation coefficient which can be calculated with the aid of Kim’s model
[13]. Table I presents the attenuation coefficients and visibility parameters
that can be used to determine the atmospheric attenuation for various fog
conditions, when the wavelength is set to $\lambda=1550$ nm.

TABLE I: Atmospheric attenuation and visibility parameters for different fog conditions at $\lambda=1550$ nm Fog | Visibility (km) | Attenuation coefficient (dB/km)
---|---|---
Dense | $0.05$ | $339.62$
Thick | $0.20$ | $84.90$
Moderate | $0.50$ | $33.96$
Light | $0.77$ | $16.67$
Thin | $1.90$ | $4.59$

##### Atmospheric Turbulence

In FSO communication, atmospheric turbulence ($I_{a}$) can be described as the
random fluctuations of the received signal due to adverse weather conditions,
including fog, snow, heavy rain, thunderstorms, etc. For AtA FSO
communication, classical turbulence induced fading channels, including Log-
normal and Gamma-Gamma have been introduced in the literature. Log-normal
fading is appropriate for weak turbulence conditions, whereas Gamma-Gamma can
be used in all turbulence regimes. Moreover, $\mathcal{K}$, Malaga
($\mathcal{M}$) and Exponentiated Weibull (EW) fading can be used in AtA
communication. Specifically, $\mathcal{K}$ distribution can precisely model
strong turbulence induced fading, whereas $\mathcal{M}$ is a general
distribution model that can cover classical turbulence channels, including
Log-normal, Gamma-Gamma, and $\mathcal{K}$ [14], [15]. Compared to its
counterparts, EW can be adopted for the AtA communication as it can model
weak-to-strong turbulence induced fading for various aperture sizes, including
point-like receive apertures ($3$ mm), when aperture averaging takes place
[16]. EW can be an important model for the AtA communication channel as URNs
can have limited aperture sizes, and a point-like aperture can be employed in
URNs [17].

##### Geometric Loss

Geometric loss, which can be caused by the divergence of the beam between
transmitter and receiver, is another important drawback in AtA communications.
In geometric loss, the light spreads out as it moves from the transmitter to
the photo detector, and the received power can be reduced . Another important
cause of geometric loss is the angle of arrival fluctuations, which is caused
by the position and orientation of ARs. In the literature, geometric loss has
been studied for UAV aided FSO front-haul channels, where deterministic and
statistical geometric loss models were developed for fixed and variable URN
positions [18].

By considering atmospheric attenuation, atmospheric turbulence and geometrical
loss ($I_{g}$), the aggregated channel model for FSO communication can be
expressed as $I=I_{a}I_{g}I_{l}$ [19].

####  II-B3 Air-to-Ground (AtG) Channel

In the AtG channel, a hybrid RF/FSO relaying procedure is employed. To this
end, atmospheric attenuation and turbulence induced fading can be modeled
similarly as given in AtA channel. In this model, geometric loss can be
compensated as the destination ground station is in a fixed location. In the
RF transmission, which is a backup for the FSO communication, GtA PL and
fading models can be used. Specifically, Shadowed-Rician can be more
appropriate in the RF communication as the ground station can be located in a
highly shadowed area.

##  III Proposed System and Possible Configurations

Consider a catastrophic emergency situation like a large earthquake where the
entire communication system collapses. In such a scenario, a massive number of
ARs, including URNs and HAPS systems, could be deployed in a short period of
time to provide continuous communication connectivity and to help in the
search rescue missions. Under such conditions, an important challenge, is the
use of the frequency spectrum. So far, UAVs have been operating in the
unlicensed band (IEEE L-Band, ISM band, etc.). However, these spectrum bands
have become overcrowded as many other wireless systems, including WiMAX and
Bluetooth, are using the same spectrum. To solve the spectrum scarcity
problem, and to provide continuous connectivity, we propose combining CR
enabled RF transmission and FSO communication for AR networks. The following
subsections describe the proposed system and its advantages.

###  III-A System Model

For the GtA channel, CR enabled RF communication is established to provide
multi-user connectivity while enhancing the utilization of the
spectrum111Please note that, it is almost impossible to use hybrid RF/FSO
communication model in the GtA channel as user equipments are limited with an
RF antenna and the user mobility can prohibit the LOS communication which is
indispensable for the FSO communication.. In this model, the spectrum is
utilized through spectrum sharing, where users/citizens behave like secondary
nodes that would like to communicate with the other users in the system as
depicted in Fig. 1. In the underlay mode, SUs can communicate with the URNs by
obeying the power constraints of the primary network, as long as the SU
related interference remains below the predetermined threshold. In return,
they may pay a roaming fee for being served by the primary network. In the
overlay mode, multi-user connectivity can be established with the aid of SUs.
More precisely, SUs can help the primary network to send a confidential
message (governmental information) by creating a virtual multi-input single-
output structure, and in return SUs can be served when the primary network is
idle. The interweave mode, by contrast, is feasible when there are a limited
number of PUs in the system. In CR enabled RF transmission, spectrum mobility
and spectrum sensing are two important limiting factors as SUs need continuous
and reliable connectivity with the AR networks. Spectrum sharing can be
realized by using several different approaches, including energy detection,
autocorrelation based detection, matched filter based detection, and Euclidean
distance based detection. Among these techniques, energy detection can be
preferable for the proposed setup as it depends only on the received signal
energy. On the other hand, in spectrum mobility, the SUs should move from one
licensed band to another when the aggregate interference exceeds the
predefined threshold. The spectrum handoff decision is therefore very
important in CR enabled RF transmission for AR networks to provide minimum
delay [20].

For the AtA channel, FSO communication can be employed between URNs and HAPS
systems as they can provide perfect LOS connectivity. In this communication
model, we assume that the ground signal is received by the hovering URNs,
especially the ones that are closer to the ground terminal [21]. The URNs
first filter out the DC components from the received signal, and then the
electrical signal is converted to the optical signal by using subcarrier
intensity modulation with proper biasing to ensure that the optical signal is
positive. It is important to note that this conversion operation is simple and
cheap as it only needs an optical transceiver, and almost all ARs can be
equipped with this material. After the conversion is completed, URNs form a
cooperation with other URNs or HAPS systems depending on the weather
conditions or the other necessities. To do so, all URNs are equipped with an
RF antenna and an optical transceiver, whereas HAPS systems can be employed
with one or more optical transceivers as they have more available space. To
exploit the benefits of the cooperation, different relaying protocols
including serial, parallel, and bidirectional relaying can be used for the AtA
channel [22]. For example, serial relaying can be used between URNs when they
are at similar heights, particularly in cases of search and rescue missions,
when urgent action is needed. Parallel relaying can be used in the presence of
harsh weather conditions when URNs can be aided by HAPS systems to increase
the diversity and reliability of communication. On the other hand, as optical
communication allows full duplex communication, bidirectional relaying can be
used between different URNs and HAPS systems especially when urgent and
important governmental messages are needed to be shared with survivors. In
serial relaying, DF protocol can be preferable as AF relaying increases the
total noise, whereas in parallel relaying, the AF mode can be beneficial for
improving the overall performance of the system. In bidirectional relaying, on
the other hand, AF and DF transmission can be used as both can reduce the
total transmission time into two time slots.

For the AtG channel, hybrid RF/FSO communication is employed where FSO is used
as a primary transmission model as long as an LOS path is formed with the
ground station. In the presence of adverse weather conditions or when the
ground station is located in a shadowed urban area, RF communication can be
used as a backup for the optical communication. As mentioned above, all URNs
are readily employed with RF antennas and FSO apertures to convert the
electrical signal into optical in the GtA channel and the optical signal into
electrical in the AtG channel. Likewise, in the presence of heavy weather
conditions, like dense fog, heavy snow or to improve reliability and
diversity, URNs can transmit both electrical and optical information to the
ground station. Hence, the overall spatial diversity can be doubled. In
addition, more than one URN can be used for the AtG channel to further enhance
the diversity gain, reliability, and the overall system performance. It is
important to note that the ground station has the ability to convert the
optical signal into an electrical signal by using a high speed photodiode that
can deliver RF electrical signal output. Thereafter, it can combine all
replicas of the RF received signals that are transmitted from different URNs
by using appropriate combining methods, like maximum ratio combining, equal
gain combining, or selection combining. Finally, the ground station delivers
the primary and the secondary messages to the primary and secondary core
networks.

###  III-B Advantages

Battery Lifetime: As mentioned before, to convert the RF signal into an
optical signal, all URNs should be outfitted with an RF antenna, a photodiode
or an optical modulator. This procedure is energy efficient. In addition, FSO
communication further improves battery life of ARs as laser beams (consisting
of LEDs) spend less energy in the communication process.

Enhanced Performance: The proposed system model can enhance overall
performance of the AR networks as it exploits all the benefits of RF/FSO
transmission. More precisely, a CR enabled GtA channel can provide diversity
gain by allowing multi-user connectivity. The AtA channel can enable fast and
ubiquitous communication by forming different cooperation strategies between
URNs and HAPS systems, including serial, parallel, and bidirectional relaying.
Finally, the AtG channel can provide diversity and enhanced performance as the
information can be transmitted through RF and FSO antennas to the ground
station.

Efficient Spectrum Usage: Both CR and FSO communication can utilize the
frequency spectrum. More precisely, CR allows dynamic spectrum access in which
both PUs and SUs can use the same spectrum, whereas FSO communication can
remedy the problem of spectrum scarcity by allowing license free spectrum with
Gigabit data rates. Therefore, FSO communication with its CR RF counterpart
can be a spectrally efficient transmission model for AR networks.

Cost-Effective: Dynamic spectrum access can bring a new perspective to the use
of the wireless spectrum efficiently. This inevitably results in cheaper
spectrum access. In addition, using simple FSO transceivers on URNs and HAPS
systems, is inexpensive and can be easily implemented.

Inherent Security: AR networks can be more prone to security threats,
especially since it is easy to tackle the information from a URN that is
hovering at a specific altitude. However, due to the specific nature of FSO
systems, the eavesdropper can not tackle the information from an optical beam.
Hence, a secure FSO communication can be established between AR nodes.

Figure 2: a) Overlay RF/FSO transmission with parallel relaying b) Overlay
RF/FSO transmission with serial relaying c) Underlay RF/FSO transmission with
parallel relaying d) Underlay RF/FSO transmission with serial relaying.

##  IV Performance Evaluation

In this section, we consider four practical CR RF/FSO relaying modes including
underlay/overlay serial and parallel DF relaying222In this section DF mode is
used to make a fair comparison of serial and parallel relaying schemes.
scenarios as depicted in Fig. 2. To quantify the performance of the proposed
schemes, we present the statistical properties of the signal-to-noise ratio
(SNR) for GtA, AtA and AtG channels, and provide outage probability analysis.
Thereafter, simulation parameters are introduced, numerical examples are
illustrated, and important design guidelines are outlined.

###  IV-A Statistical Properties of the SNR

####  IV-A1 GtA Communication

As depicted in Fig. 2, CR enabled RF transmission is adopted in the GtA
channel, where both overlay and underlay schemes are used depending on the
availability of the primary network. In the overlay mode333In the overlay
mode, we assume that the primary network is idle, so that the secondary users
can communicate with the AR network without any power limitations., the
signal-to-noise ratio (SNR) of the GtA channel can be expressed as

$\displaystyle\gamma_{\text{GtA}}^{\text{o}}=\frac{P_{s}|g|^{2}}{P_{n}V(r)},$
(1)

where $P_{s}$ and $P_{n}$ are the transmit and noise powers, $V(r)$ is the
path-loss variable defined in Section II, and $|g|$ is the channel magnitude,
which is distributed with Nakagami-_m_ fading. In the underlay mode, the SNR
of the GtA channel can be written as [23]

$\displaystyle\gamma_{\text{GtA}}^{\text{u}}=\frac{P_{s}}{P_{n}V(r)}\times\frac{|g|^{2}}{E[|h_{I}|^{2}]},$
(2)

where $E[\cdot]$ shows the expectation operator and $h_{I}$ is the secondary-
primary interference term which can be obtained through mean-value power
allocation model presented in [23, Sect. 4.4]. In the presence of primary-
secondary interference, the effective SNR of the GtA channel can be expressed
as

$\displaystyle\gamma_{\text{GtA}}^{\text{u,ef}}=\frac{\gamma_{\text{GtA}}^{\text{u}}}{1+\mathcal{I}_{P}},$
(3)

where $\mathcal{I}_{P}$ shows the fixed primary-secondary interference SNR.
For underlay and overlay modes, the cumulative distribution function (CDF) of
the SNR can be expressed as

$\displaystyle
F_{\gamma_{\text{GtA}}}(\gamma)=1-e^{-\frac{m\gamma}{\bar{\gamma}_{\text{GtA}}}}\sum_{v=0}^{m-1}\Big{(}\frac{m\gamma}{\bar{\gamma}_{\text{GtA}}}\Big{)}^{v}\frac{1}{v!},$
(4)

where
$\bar{\gamma}_{\text{GtA}}=\bar{\gamma}_{\text{GtA}}^{\text{o}}=\frac{P_{s}}{P_{n}V(r)}$
for the overlay mode and
$\bar{\gamma}_{\text{GtA}}=\bar{\gamma}_{\text{GtA}}^{\text{u},\text{ef}}=\frac{\frac{P_{s}}{P_{n}V(r)E[|h_{I}|^{2}]}}{1+\mathcal{I}_{P}}$
for the underlay scheme.

####  IV-A2 AtA Communication

In AtA communication, FSO transmission is adopted between ARs where
communication is affected both from atmospheric attenuation and turbulence
induced fading which is distributed with EW fading. In this case, the
aggregated channel model can be written as444In the AR aided communication,
geometric losses can be compensated by using tracking enabled electro-optic
and acousto-optic FSO transceivers [7]. Thereby, we neglect the geometric loss
in this work. $I=I_{a}I_{l}$, and the SNR of AtA channel can be expressed as

$\displaystyle\gamma_{\text{AtA}}=\frac{P_{s}}{P_{n}}I_{a}^{2}|I_{l}|^{2},$
(5)

and the CDF of SNR for AtA channel can be obtained as [16]

$\displaystyle
F_{\gamma_{\text{AtA}}}(\gamma)=\Bigg{(}1-\exp\Bigg{[}-\Bigg{(}\frac{\gamma}{\eta^{2}\bar{\gamma}_{\text{AtA}}}\Bigg{)}^{\beta/2}\Bigg{]}\Bigg{)}^{\alpha},$
(6)

where $\bar{\gamma}_{\text{AtA}}=\frac{P_{s}}{P_{n}}I_{a}^{2}E[|I_{l}|^{2}]$
is the average SNR of the AtA channel, $\alpha$ and $\beta$ are the shape
parameters and $\eta$ is the scale parameter of the EW fading channel.

####  IV-A3 AtG Communication

In AtG communication, hybrid RF/FSO communication model is used where FSO is
adopted as the primary transmission model as long as an LOS path is formed
with the ground station. Therefore, the CDF of AtG channel can be obtained
very similar to (6)

###  IV-B Outage Probability Analysis

Outage probability ($P_{\text{out}}$) is an important performance metric that
can assess the overall performance of a wireless system. It can be defined as
the probability of SNR that will fall below a predetermined threshold
($\gamma_{\text{th}}$) for acceptable communication quality. Mathematically,
it can be formulated as

$\displaystyle
P_{\text{out}}=\Pr\Big{(}\gamma_{e2e}<\gamma_{\text{th}}\Big{)},$ (7)

where $\gamma_{e2e}$ is the end-to-end (e2e) SNR. In this section, the outage
probability (OP) analysis is pursued for DF serial and parallel relaying
networks.

For a DF serial relaying system, the e2e SNR can be expressed as

$\displaystyle\gamma_{e2e}^{\text{S}}=\min\Big{(}\gamma_{1},\gamma_{2},\ldots,\gamma_{N}\Big{)},$
(8)

where $N$ denotes the number of hops and $\gamma_{j}$, $j\in\\{1,N\\}$ shows
the SNR of the $j$-th hop. Similarly, for a parallel dual-hop DF relaying, the
e2e SNR can be expressed as

$\displaystyle\gamma_{e2e}^{\text{P}}=\max_{1\leq k\leq
K}\Big{(}\min\Big{(}\gamma_{1,k},\gamma_{2,k}\Big{)}\Big{)},$ (9)

where $\gamma_{1,k}$ and $\gamma_{2,k}$ shows the SNRs of the first and second
hops, and $K$ denotes the number of relays that are available for
communication. By substituting (8) and (9) into (7), the outage probabilities
for serial and parallel DF relaying schemes can be obtained as [22]

$\displaystyle
P_{\text{out}}=\begin{cases}1-\prod_{j=1}^{N}(1-F_{\gamma_{j}}(\gamma_{\text{th}})),&\text{Serial
R.,}\\\
\prod_{k=1}^{K}\Big{(}1-(1-F_{\gamma_{1,k}}(\gamma_{\text{th}}))(1-F_{\gamma_{2,k}}(\gamma_{\text{th}}))\Big{)},&\text{Parallel
R.,}\end{cases}$ (10)

where $F_{\gamma_{z,k}}(\gamma_{\text{th}})$, $z\in\\{1,2\\}$ and
$F_{\gamma_{j}}(\gamma_{\text{th}})$ denotes CDF of the SNR per hop. By
substituting (4) and (6) into (10), and after several manipulations, the OPs
for underlay/overlay serial and parallel relaying can be obtained.

###  IV-C Simulation Setup

In this section, we provide the simulation parameters that are used to
illustrate the practical CR RF/FSO relaying scenarios, which are depicted in
Fig. 2. In the simulations, we assume that the FSO communication is affected
both from turbulence induced fading and atmospheric attenuation, whereas the
RF communication is impaired from path-loss and multi-path fading.
Furthermore, in parallel relaying, the HAPS aided communication link can be
affected from the atmospheric attenuation more than URNs, by contrast, URN to
URN communication can experience strong turbulence effects as they are
deployed at a lower altitude. Based on this assumption, the average SNRs are
assumed equal. In the calculation, transmit power $P_{t}$ and noise power
$P_{n}$ are set to $P_{t}=32$ dBm, and $P_{n}=-100$ dBm, respectively.
Moreover, the altitude of the HAPS and URNs are set to $h_{s}=19000$ m, and
$h_{s}=200$ m, respectively, and the SNR threshold is given as
$\gamma_{\text{th}}=3$ dB for acceptable communication quality. In the overlay
mode, primary-secondary interference can be neglected as it uses precise
spectrum sensing techniques, whereas in the underlay mode, fixed
$\mathcal{I}_{P}$ values varying from $0$ dB to $5$ dB, are used. Moreover, in
FSO communication, Hufnagel-Valley day model is used to obtain the refractive-
index structure parameter [16], and thin fog weather condition ($\sigma=4.59$)
is used to obtain atmospheric attenuation variable $I_{l}$. On the contrary,
in RF communication, Type A PL model [5] is used and the PL variable is
obtained by setting $\alpha=2.32$, $\beta=\Big{(}\frac{2\pi
f_{c}}{c}\Big{)}^{2}$, $f_{c}=2$ GHz carrier frequency, and Nakagami-_m_
fading parameter is set to $m=4$. Unless otherwise stated, Table II presents
all simulation parameters.

TABLE II: LIST OF PARAMETERS AND VALUES Parameter | Value
---|---
Optical wavelength ($\lambda$) |  $1550$ nm
SNR threshold ($\gamma_{\text{th}}$) |  $3$ dB
Nakagami-_m_ severity parameter ($m$) | 4
Optical to electrical conversion ratio | $1$
Primary-secondary interference SNR ($\mathcal{I}_{P}$) |  $0$-$5$ dB
Horizontal hop distance ($d$) |  $2500$ m
URN altitude ($h_{u}$) |  $200$ m
HAPS altitude ($h_{s}$) |  $19$ km
Carrier frequency ($f_{c}$) |  $2$ GHz
Attenuation coefficient ($\sigma$) | $4.5859$
Wind speed ($v_{g}$) |  $21$ m/s

###  IV-D Numerical Illustrations

Fig. 3 shows the OP as a function of horizontal hop distance ($d$) for the
proposed overlay/underlay serial and parallel relaying systems, when the
primary-secondary interference is set to $\mathcal{I}_{P}=5$ dB. As we can
see, parallel relaying outperforms serial relaying both in underlay and
overlay mode, as it forms a cooperation to enhance the system performance. We
also note that, underlay mode performs worse than its overlay counterpart due
to high $\mathcal{I}_{P}$ levels. Furthermore, OP monotonically increases when
$d$ goes from $1500$ m to $2600$ m. Then, due to huge atmospheric attenuation
and PL effects, the OP performance worsens and when $d$ reaches to $3300$ m,
the OP saturates and reaches to a outage ceiling where $P_{\text{out}}\approx
1$. Therefore, as can be observed from the figure, the optimum OP performance
can be obtained when $d<2000$ m, where $P_{\text{out}}<10^{-4}$ for the
overlay mode.

Figure 3: Outage probability versus horizontal hop distance of
overlay/underlay serial and parallel relay networks. Figure 4:  Outage
probability versus URN altitude of overlay and underlay serial relaying.

Fig. 4 illustrates the OP performance of underlay and overlay modes for serial
relaying as a function of the URN altitude ($h_{u}$) for different
$\mathcal{I}_{P}$ levels, when $d=2500$ m. As can be observed, the theoretical
outage probability results, which are shown with solid lines, are in good
agreement with the marker symbols, which are generated by the simulations.
Interestingly, the outage performance of the proposed scheme enhances with
respect to the $h_{u}$ as the refractive-index structure parameter is directly
related with URN altitude. More precisely, URN to URN communication can be
less affected from the atmospheric turbulence induced fading when they are
deployed at higher altitudes. On the contrary, when $h_{u}$ is higher than
$200$ m, the OP performance of the proposed scheme worsens slightly due to
huge PL and atmospheric attenuation effects. Thereby, placing the URNs at
about $200$ m can provide optimum OP performance for the proposed scheme.
Finally, it can be observed from the figure that, as $\mathcal{I}_{P}$
increases, the OP performance worsens in the underlay scheme. Therefore, the
overlay scheme can be preferable for the GtA channel especially when the
primary network is idle.

###  IV-E Design Guidelines

In this subsection, we shed light on the design of practical CR enabled RF/FSO
AR networks by providing some important design guidelines.

  * •

As can be seen in Fig. 3 and 4, the overlay method can be preferable in GtA
communication as it allows multi-user access to the URNs through the primary
spectrum without causing any interference.

  * •

In the AtA channel, DF relaying can be preferable when serial relaying is used
as it does not enhance the noise level like AF relaying. On the other hand,
when parallel or bidirectional relaying is used, the AF protocol can be
preferable for enhancing the system performance.

  * •

To provide a perfect LOS connection, URNs can be positioned close enough to
each other in the presence of adverse weather conditions. In addition, HAPS
systems can aid URNs in parallel relaying mode.

  * •

In the AtG channel, multiple URNs can be used to serve as a backhaul.
Moreover, to provide reliable communication, both RF and FSO communications
can be used simultaneously, depending on the urgency of the status or weather
conditions.

  * •

Using FSO communication for the AtA channel can further improve the battery
life of the URNs and HAPS systems. Moreover, FSO transceivers are cheap, easy
to deploy, and consume less battery power as they use simple laser beams.

  * •

As all URNs are fitted with both RF and FSO antennas, hybrid RF/FSO
communication model can be used for all URNs as needed. It is worth noting
that hybrid RF/FSO communication uses more battery power and thus decreases
the battery life of ARs quicker.

##  V Conclusion

This article has proposed a new transmission model for aerial relay networks
by combining spectrally efficient cognitive radio with FSO communication. More
precisely, we showed how employing cognitive radio enabled RF communication
for the ground-to-air channel, FSO communication for the air-to-air path, and
a hybrid RF/FSO communication system for the air-to-ground channel can
leverage the advantages of RF/FSO transmission, while effectively addressing
the problem of spectrum scarcity. As we observed, the proposed setup utilizes
the frequency spectrum, enhances data rates, and provides inherent security.

## References

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# Invariant idempotent $\ast$-measures

Nataliya Mazurenko<EMAIL_ADDRESS>Department of Mathematics and Computer
Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str.,
Ivano-Frankivsk, 76025, Ukraine  ,  Khrystyna Sukhorukova
<EMAIL_ADDRESS>Ivan Franko Lviv National University, 1
Universytetska str., 79000, Lviv, Ukraine  and  Mykhailo Zarichnyi
<EMAIL_ADDRESS>Institute of Mathematics, College of Natural Sciences,
University of Rzeszów, Pigonia 1, 35-310 Rzeszów, Poland

###### Abstract.

The notion of $\ast$-idempotent measure is a modification of the notion of
idempotent measure defined for every triangular norm $\ast$. We prove
existence and uniqueness of invariant $\ast$-idempotent measures for iterated
function systems on compact metric spaces.

###### Key words and phrases:

triangular norms, non-additive measure, invariant measure

###### 2020 Mathematics Subject Classification:

28A80, 54B20

##  1\. Introduction

Idempotent mathematics is a part of mathematics in which one of the ordinary
arithmetic operations in $\mathbb{R}$ is replaced by an idempotent operation
(e.g., maximum).

The notion of probability measure has its counterparts in idempotent
mathematics. The idempotent measures (also called Maslov measures) are
introduced in [7]. The topological and categorical aspects of the theory of
idempotent measures are considered in [12]. Also, in [2] theory of the max-min
measures, which can also be regarded as idempotent counterparts of probability
measures, is developed.

Some analogous classes of measures, namely the so called *-measures, for any
triangular norm $\ast$, are considered in [9].

In [8] the invariant idempotent measures for iterated function systems are
defined and the existence and uniqueness theorem for such measures is proved.
The proof is based on functional representation of measures. In [3] the
authors proved the existence and uniqueness of invariant idempotent measures
by using Zaitov’s metric [11] as well as modified Bazylevych-Repovš-Zarichnyi
metric on the space of idempotent measures [1] and applying the Banach
contraction principle.

In the present note we provide a simple proof of existence and uniqueness of
invariant idempotent measures that works also for the $\ast$-idempotent
measures in the sense of [9].

##  2\. Preliminaries

###  2.1. $\ast$-measures

A triangular norm (t-norm) is a continuous, associative, commutative and
monotonic operation on the unit segment $\mathbb{I}=[0,1]$ for which 1 is a
unit. (see, e.g., [6]).

Let $X$ be a compact Hausdorff space. By $C(X,\mathbb{I})$ we denote the set
of all continuous functions from $X$ to $\mathbb{I}$. Given $c\in\mathbb{I}$,
we denote by $c_{X}\in C(X,\mathbb{I})$ the constant function with the value
$c$. A functional $\mu\colon C(X,\mathbb{I})\to\mathbb{I}$ is called an
idempotent $\ast$-measure [9] if

  1. (1)

$\mu(c_{X})=c$;

  2. (2)

$\mu(\lambda\ast\varphi)=\lambda\ast\mu(\varphi)$;

  3. (3)

$\mu(\varphi\vee\psi)=\mu(\varphi)\vee\mu(\psi)$,

for all $c\in\mathbb{I}$ and $\varphi,\psi\in C(X,\mathbb{I})$. (Hereafter
$\vee$ denotes the maximum.)

Let $M^{\ast}(X)$ denote the set of all $\ast$-measures on $X$.

Given $\mu\in M^{\ast}(X)$, $\varphi_{1},\dots,\varphi_{n}\in C(X,\mathbb{I})$
and $\varepsilon>0$, we define

$O\langle\mu;\varphi_{1},\dots,\varphi_{n};\varepsilon\rangle=\\{\nu\in
M^{\ast}(X)\mid|\mu(\varphi_{i})-\nu(\varphi_{i})|<\varepsilon,\
i=1,\dots,n\\}.$

Then the sets of the form
$O\langle\mu;\varphi_{1},\dots,\varphi_{n};\varepsilon\rangle$, where $\mu\in
M^{\ast}(X)$, $\varphi_{1},\dots,\varphi_{n}\in C(X,\mathbb{I})$, $n\in
C(X,\mathbb{N}$ and $\varepsilon>0$, comprise a base of the weak* topology on
$M^{\ast}(X)$.

Given a continuous map $f\colon X\to Y$ of compact Hausdorff space, and
$\mu\in M^{\ast}(X)$, define $M^{\ast}(f)(\mu):C(X,\mathbb{I})\to\mathbb{I}$
as follows:

$M^{\ast}(f)(\mu)(\varphi)=\mu(\varphi f).$

Actually, $M^{\ast}$ is a functor from the category $\mathbf{Comp}$ of compact
Hausdorff spaces and continuous maps to itself.

###  2.2. Hyperspaces

Let $X$ be a compact Hausdorff space. By $\exp X$ we denote the set of all
nonempty compact subsets in $X$. Given $U_{1},\dots,U_{n}\subset X$, let

$\langle U_{1},\dots,U_{n}\rangle=\\{A\in\exp X\mid
A\subset\cup_{i=1}^{n}U_{i},\ A\cap U_{i}\neq\emptyset,\ i=1,\dots,n\\}.$

The family

$\\{\langle U_{1},\dots,U_{n}\rangle\mid U_{1},\dots,U_{n}\text{ are open},\
n\in\mathbb{N}\\}$

is known to be a base of the Vietoris topology on the set $\exp X$. The
obtained topological space is called the hyperspace of $X$.

If $(X,d)$ is a compact metric space, then the set $\exp X$ is endowed with
the Hausdorff metric $d_{H}$,

$d_{H}(A,B)=\max\left\\{\sup_{x\in A}\inf_{y\in B}d(x,y),\;\sup_{y\in
B}\inf_{x\in A}d(x,y)\right\\}.$

The Hausdorff metric is known to generate the Vietoris topology.

In the following lemma, the sup metric is considered on the product.

######  Lemma 2.1.

Let $X,Y$ be compact metric spaces, $\mathrm{pr}_{Y}\colon X\times Y\to Y$ be
the projections. Let $A,B\in\exp(X\times Y)$. If
$\mathrm{pr}_{Y}(A)=\mathrm{pr}_{Y}(B)$, then

$d_{H}(A,B)\leq\mathrm{diam}(X).$

If $f\colon X\to Y$ is a map, then the map $\exp f\colon\exp X\to\exp Y$ is
defined as follows: $\exp f(A)=f(A)$, $A\in\exp X$.

Denote by $\bar{M}(X)$ the set of all $A\in\exp(X\times\mathbb{I})$ satisfying
the following conditions:

  1. (1)

$A\cap(X\times\\{1\\})\neq\emptyset$;

  2. (2)

$X\times\\{0\\}\subset A$;

  3. (3)

$A$ is saturated, i.e., if $(x,t)\in A$, then $(x,s)\in A$ for every
$s\in[0,t]$.

Let $f\colon X\to Y$ be a map. Define the map
$\bar{M}(f)\colon\bar{M}(X)\to\bar{M}(Y)$ by the formula:

$\bar{M}(f)(A)=\exp(f\times 1_{\mathbb{I}})(A)\cup(Y\times\\{0\\}).$

Actually, $\bar{M}$ is a functor in the category $\mathbf{Comp}$.

The functors $M^{\ast}$ and $\bar{M}$ are known to be isomorphic for all
triangular norms $\ast$ (see [9]). Note that this is an isomorphism of
functors, but not of more complicated structures, e.g., monads (see [10]). For
the sake of reader’s convenience, in the sequel we deal with $\bar{M}$ having
a triangular norm $\ast$ in mind.

Given $A\in\bar{M}(X)$ and $r\in\mathbb{I}$, define

$r\bar{\ast}A=\\{(x,r\ast t)\mid(x,t)\in A\\}.$

##  3\. Iterated Function Systems

Let $(X,d)$ be a compact metric space. A map $f\colon X\to X$ is called a
contraction if there exists $c\in[0,1)$ such that $d(f(x),f(y))\leq cd(x,y)$,
for all $x,y\in X$. Here, $c$ is a contraction constant of $f$.

An Iterated Function System is a collection $f_{i}\colon X\to X$ of
contractions.

Let $\lambda_{1},\dots,\lambda_{n}\in[0,1]$ be such that
$\vee_{i=1}^{n}\lambda_{i}=1$.

Define $\Psi\colon\bar{M}(X)\to\bar{M}(X)$ as follows:

$\Psi(A)=\cup_{i=1}^{n}\lambda_{i}\bar{\ast}\bar{M}(f_{i})(A).$

An element $A\in\bar{M}(X)$ is called invariant if $\Psi(A)=A$.

######  Lemma 3.1.

There exists $A\in\bar{M}(X)$ such that $\Psi(A)\subset A$.

###### Proof.

There exists $B\in\exp X$ such that $f_{i}(B)\subset B$ for every
$i=1,2,\dots,n$ (see [5], Section 3.1, Theorem (3), Proof of Existence). Then
$A=(B\times\mathbb{I})\cup(X\times\\{0\\})$ is as required.

∎

######  Theorem 3.2.

There is a unique invariant element in $\bar{M}(X)$.

###### Proof.

Let $A$ be as in Lemma 3.1. Then

$A\supset\Psi(A)\supset\Psi^{2}(A)\supset\Psi^{3}(A)\supset\dots$

and clearly $A_{\infty}=\cap_{i=1}^{\infty}\Psi^{i}(A)$ is an invariant set.

We are going to prove the uniqueness. Suppose that $A,A^{\prime}$ are
invariant elements. Let $c>0$ be a contraction constant for all $f_{i}$,
$i=1,\dots,k$. Then, for any natural $n$,

$\displaystyle d_{H}$
$\displaystyle(A,A^{\prime})=d_{H}(\Psi^{n}(A),\Psi^{n}(A^{\prime}))$
$\displaystyle\leq
d_{H}(\bigcup\\{(\lambda_{i_{1}}\ast\dots\ast\lambda_{i_{n}})\bar{\ast}\bar{M}(f_{i_{1}}\dots
f_{i_{n}})(A)\mid(i_{1},\dots,i_{n})\in\\{1,\dots,k\\}^{n}\\},$
$\displaystyle\bigcup\\{(\lambda_{i_{1}}\ast\dots\ast\lambda_{i_{n}})\bar{\ast}\bar{M}(f_{i_{1}}\dots
f_{i_{n}})(A^{\prime})\mid(i_{1},\dots,i_{n})\in\\{1,\dots,k\\}^{n}\\})$
$\displaystyle\leq\max\\{d_{H}((\lambda_{i_{1}}\ast\dots\ast\lambda_{i_{n}})\bar{\ast}\bar{M}(f_{i_{1}}\dots
f_{i_{n}})(A),(\lambda_{i_{1}}\ast\dots\ast\lambda_{i_{n}})\bar{\ast}\bar{M}(f_{i_{1}}\dots
f_{i_{n}})(A^{\prime}))$
$\displaystyle\mid(i_{1},\dots,i_{n})\in\\{1,\dots,k\\}^{n}\\}$
$\displaystyle\leq$
$\displaystyle\max\\{d_{H}((\lambda_{i_{1}}\ast\dots\ast\lambda_{i_{n}})\bar{\ast}((f_{i_{1}}\dots
f_{i_{n}})\times
1_{\mathbb{I}})(A),(\lambda_{i_{1}}\ast\dots\ast\lambda_{i_{n}})\bar{\ast}((f_{i_{1}}\dots
f_{i_{n}})\times 1_{\mathbb{I}})(A^{\prime}))$
$\displaystyle\mid(i_{1},\dots,i_{n})\in\\{1,\dots,k\\}^{n}\\}$
$\displaystyle\leq c^{n}\mathrm{diam}(X)$

by Lemma 2.1. Therefore $A=A^{\prime}$.

∎

##  4\. Remarks

Non-finite IFSs are considered in [4]. We believe that our result can also be
extended over a non-finite (e.g., compact) case.

Some interesting examples of invariant idempotent measures are presented in
[3]. This corresponds to the case of the multiplication t-norm. It is an open
question to find meaningful examples of invariant $\ast$-idempotent measures,
for another triangular norms $\ast$.

## References

  * [1] L. Bazylevych, D. Repovš, M. Zarichnyi, Spaces of idempotent measures of compact metric spaces. Topology Appl. 157, 1 (2010), 136–144.
  * [2] V. Brydun, M. Zarichnyi, Spaces of max-min measures on compact Hausdorff spaces, Fuzzy Sets and Systems, 396 (2020), 138–151.
  * [3] R.D. da Cunha, E.R. Oliveira, F. Strobin, Existence of invariant idempotent measures by contractivity of idempotent Markov operators. J. Fixed Point Theory Appl. 25,1 (2023), Paper No.8, 11.
  * [4] R.D. da Cunha, E.R. Oliveira, F. Strobin, Fuzzy-set approach to invariant idempotent measures, Fuzzy Sets and Systems, Volume 457, Issue C, 2023, 46–65, https://doi.org/10.1016/j.fss.2022.06.008
  * [5] J. E. Hutchinson, Fractals and self-similarity Indiana Univ. Math. J. 30 (1981), 713–747.
  * [6] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms. Dordrecht: Kluwer. (2000), ISBN 0-7923-6416-3.
  * [7] V.N. Kolokoltsov, V.P. Maslov, Idempotent Analysis and Its Applications. - Springer-Science+Business Media, B.V., 1997, 318 pp., ISBN 0-7923-4509-6
  * [8] N. Mazurenko, M. Zarichnyi, Invariant idempotent measures, Carpathian Math. Publ. 10, 1(2018), 172–178.
  * [9] Kh. Sukhorukova, Spaces of non-additive measures generated by tri- angular norms. Matematychni Studii, 59(2), 215–224.
  * [10] Kh.O. Sukhorukova, M.M. Zarichnyi, On $\ast$-measure monads on the category of ultrametric spaces, Carpathian Math. Publ. 2022, 14 (2), 429–436. doi:10.15330/cmp.14.2.429-436
  * [11] A.A. Zaitov, On a metric of the space of idempotent probability measures. Appl. Gen. Topol. 21(2020), no.1, 35–51.
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[table]capposition=top

Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

CLEF 2021 – Conference and Labs of the Evaluation Forum, September 21–24,
2021, Bucharest, Romania

<EMAIL_ADDRESS>]

<EMAIL_ADDRESS>]

<EMAIL_ADDRESS>]

# O2D2: Out-Of-Distribution Detector to Capture Undecidable Trials in
Authorship Verification

Benedikt Boenninghoff  Ruhr University Bochum, Germay  Robert M. Nickel
Bucknell University, USA  Dorothea Kolossa

(2021)

# Notebook for PAN at CLEF 2021

Benedikt Boenninghoff  Ruhr University Bochum, Germay  Robert M. Nickel
Bucknell University, USA  Dorothea Kolossa

(2021)

###### Abstract

The PAN 2021 authorship verification (AV) challenge is part of a three-year
strategy, moving from a cross-topic/closed-set AV task to a cross-topic/open-
set AV task over a collection of fanfiction texts. In this work, we present a
novel hybrid neural-probabilistic framework that is designed to tackle the
challenges of the 2021 task. Our system is based on our 2020 winning
submission, with updates to significantly reduce sensitivities to topical
variations and to further improve the system’s calibration by means of an
uncertainty adaptation layer. Our framework additionally includes an out-of-
distribution detector (O2D2) for defining non-responses. Our proposed system
outperformed all other systems that participated in the PAN 2021 AV task.

###### keywords:

Authorship Verification Out-Of-Distribution Detection Open-Set

##  1 Introduction

In this paper we are proposing a significant extension to the authorship
verification (AV) system presented in [1]. The work is part of the PAN 2021 AV
shared task[2], for which the PAN organizers provided the challenge
participants with a publicly available dataset of fanfiction.

Fanfiction texts are fan-written extensions of well-known story lines, in
which the so-called fandom topic describes the principal subject of the
literary document (e.g. _Harry Potter_). The use of fanfiction as a genre has
three major advantages. Firstly, the abundance of texts written in this genre
makes it feasible to collect a large training dataset and, therefore, to build
more complex authorship verification (AV) systems based on modern deep
learning techniques, which will hopefully boost progress in this research
area. Additionally, fanfictional documents also come with meaningful meta-data
like topical information, which can be used to investigate the topical
interference in authorship analysis. Lastly, although the documents are
usually produced by non-professional writers, contrary to social media
messages, they usually follow standard grammatical and spelling conventions.
This allows participants to incorporate pretrained models for, e.g., part-of-
speech tagging, and to reliably extract traditional stylometric features [3].

The previous edition of the PAN AV task dealt with cross-fandom/closed-set AV
[4]. The objective of the cross-fandom AV task is to automatically decide
whether two fanfictional documents covering different fandoms belong to the
same author. The term closed-set refers to the fact that the test dataset,
which is not publicly available, only contains trials from a subset of the
authors and fandoms provided in the training data.

To increase the level of difficulty, the current PAN AV challenge moved from a
closed-set task to an open-set task in 2021, while the training dataset is
identical to that of the previous year [5]. In this scenario, the new test
data contains only authors and fandoms that were not included in the training
data. We thus expect a covariate shift between training and testing data, i.e.
the distribution of our neural stylometric representations extracted from the
training data is expected to be different from the distribution of the test
data representations. It was implicitly shown in [4], and our experiments
confirm this analysis, that such a covariate shift, due to topic variability,
is a major cause of errors.

##  2 System Overview

The overall structure of our revised system111The source code is accessible
online: https://github.com/boenninghoff/pan_2020_2021_authorship_verification
is shown in Fig. 1. It expands our winning system from 2020 as follows:
Suppose we have a pair of documents $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$
with an associated ground-truth hypothesis $\mathcal{H}_{a}$ for
$a\in\\{0,1\\}$. The value of $a$ indicates, whether the two documents were
written by the same author ($a=1$) or by different authors ($a=0$). Our task
can formally be expressed as a mapping
$f\\!\\!:\\!\\!\\{\mathcal{D}_{1},\mathcal{D}_{2}\\}\longrightarrow
p\in[0,1]$. The estimated label $\widehat{a}$ is obtained from a threshold
test applied to the output prediction $p$. In our case, we choose
$\widehat{a}=1$ if $p>0.5$ and $\widehat{a}=0$ if $p<0.5$. The PAN 2020/21
shared tasks also permit the return of a non-response (in addition to
$\widehat{a}=1$ and $\widehat{a}=0$) in cases of high uncertainty [4], e.g.
when $p$ is close to 0.5. In this work, we therefore define three hypotheses:

$\displaystyle\mathcal{H}_{0}:~{}~{}$ The two documents were written by two
different persons, $\displaystyle\mathcal{H}_{1}:~{}~{}$ The two documents
were written by the same person, $\displaystyle\mathcal{H}_{2}:~{}~{}$
Undecidable, trial does not suffice to establish authorship.

Figure 1: Our proposed hybrid neural-probabilistic framework for the PAN 2021
cross-fandom open-set authorship verification task.

In [1], we introduced the concept of linguistic embedding vectors (LEVs). To
obtain these, we perform neural feature extraction followed by deep metric
learning (DML) to encode the stylistic characteristics of a pair of documents
into a pair of fixed-length and topic-invariant stylometric representations.
Given the LEVs, a Bayes factor scoring (BFS) layer computes the posterior
probability for a trial. This discriminative two-covariance model was
introduced in [6]. As a new component, we propose an uncertainty adaptation
layer (UAL). This idea is adopted from [7], aiming to find and correct wrongly
classified trials of the BFS layer, to model its noise behavior, and to return
re-calibrated posteriors.

For the decision whether to accept $\mathcal{H}_{0}$/$\mathcal{H}_{1}$, or to
return a non-response, i.e. $\mathcal{H}_{2}$, it is desirable that the value
of the posterior $p$ reliably reflects the uncertainty of the decision-making
process. We may roughly distinguish two different types of uncertainty [8]: In
AV, aleatoric or data uncertainty is associated with properties of the
document pairs. Examples are topical variations or the intra- and inter-author
variabilities. Aleatoric uncertainty generally can not be reduced, but it can
be addressed (to a certain extent) by returning a non-response (i.e.
hypothesis $\mathcal{H}_{2}$) if it is too large to allow for a reliable
decision. To accomplish this, and inspired by [9], we incorporate a feed-
forward network for out-of-distribution detection (O2D2), which is trained on
a dataset that is different, i.e. disjoint w.r.t. authors and fandoms, from
the training set used to optimize the DML, BFS and UAL components.

Additionally, epistemic or model uncertainty characterizes uncertainty in the
model parameters. Examples are unseen authors or topics. Epistemic uncertainty
can be reduced through a substantial increase in the amount of training data,
i.e. an increase in the number of training pairs. We capture epistemic
uncertainty in our work through the proposed O2D2 approach and also by
extending our model to an ensemble. We expect all models to behave similarly
for known authors or topics, but the output predictions may be widely
dispersed for pairs under covariate shift [10].

The training procedure consists of two stages: In the first stage, we
simultaneously train the DML, BFS and UAL components. In the second stage, we
learn the parameters of the O2D2 model.

##  3 Dataset Splits for the PAN 2021 AV Task

The text preprocessing strategies, including tokenization and pair re-
sampling, are comprehensively described in [11]. The fanfictional dataset for
the PAN 2020/21 AV tasks are described in [4, 5]. In the following, we report
on the various dataset splits that we employed for our PAN 2021 submission.

Each document pair is characterized by a tuple $(a,f)$, where $a\in\\{0,1\\}$
denotes the authorship similarity label and $f\in\\{0,1\\}$ describes the
equivalent for the fandom. We assign each document pair to one of the
following author-fandom subsets222SA=same author, DA=different authors,
SF=same fandom, DF=different fandoms SA_SF, SA_DF, DA_SF, and DA_DF given its
label tuple $(a,f)$.

As shown in [11], one of the difficulties working with the provided
small/large PAN datasets is that each author generally contributes only with a
small number of documents. As a result, we observe a high degree of overlap in
the re-sampled subsets of same-author trials. We decided to work only with the
large dataset this year and split the documents into three disjoint (w.r.t.
authorship and fandom) sets. Overlapping documents, where author and fandom
belong to different sets, are removed. The splits are summarized in Fig. 2 and
Table 1. Altogether, the following datasets have been involved in the PAN 2021
shared task, to train the model components, tune the hyper-parameter and for
testing:

  * •

The training set is identical to the one used in [11] and was employed for the
first stage, i.e., to train the DML, BFS and UAL components simultaneously.
During training we re-sampled the pairs epoch-wise such that all documents
contribute equally to the neural network training in each epoch. The numbers
of training pairs provided in Table 1 therefore vary in each epoch.

  * •

The calibration set has been used for the second stage, i.e., to train
(calibrate) the O2D2 model. During training, we again re-sampled the pairs in
each epoch and limited the total number of pairs in the different-authors
subsets to partly balance the dataset.

  * •

The purpose of the validation set is to tune the hyper-parameters of the O2D2
stage and to report the final evaluation metrics for all stages in Section 5.

  * •

The development set is identical to the evaluation set in[11] and was used to
tune the hyper-parameters during the training of the first stage. This dataset
contains pairs from the calibration and validation sets. However, due to the
pair re-sampling strategy in [11], documents may appear in different subsets
and varied document pairs may be sampled. It thus does not represent a union
of the calibration and validation sets.

  * •

Finally, the PAN 2021 evaluation set, which is not publicly available, has
been used to test our submission and to compare it with the proposed
frameworks of all other participants.

Note that both, the validation and development set in Table 1 only contain
SA_DF and DA_SF pairs, for reasons discussed in Section 5. The pairs of these
sets are sampled once and then kept fixed.

Figure 2: Disjoint splits of the large PAN 2020/21 training set. Table 1:
Numbers of (re-)sampled pairs for all datasets.

Dataset | SA_SF | SA_DF | DA_SF | DA_DF
---|---|---|---|---
Training set | 16,045 | 28,500 | 64,300 | 42,730
Calibration set | 2,100 | 2,715 | 4,075 | 4,075
Validation set | 0 | 2,280 | 3075 | 0
Development set | 0 | 5,215 | 7,040 | 0

##  4 Methodologies

In this section, we briefly describe all components of our neural-
probabilistic model. Sections 4.1 through 4.4 repeat information that is
already provided in [11] to provide proper context.

###  4.1 Neural Feature Extraction and Deep Metric Learning

Feature extraction and deep metric learning are realized in the form of a
Siamese network, feeding both input documents through exactly the same
function.

####  4.1.1 Neural Feature Extraction:

The system passes token and character embeddings into a two-tiered
bidirectional LSTM network with attentions,

$\displaystyle\boldsymbol{x}_{i}=\text{NeuralFeatureExtraction}_{\boldsymbol{\theta}}\big{(}\boldsymbol{E}_{i}^{w},\boldsymbol{E}_{i}^{c}\big{)},$
(1)

where $\boldsymbol{\theta}$ contains all trainable parameters,
$\boldsymbol{E}^{w}_{i}$ represents word embeddings and
$\boldsymbol{E}^{c}_{i}$ represents character embeddings. A comprehensive
description is given in [12].

####  4.1.2 Deep Metric Learning:

We feed the document embeddings $\boldsymbol{x}_{i}$ in Eq. (1) into a metric
learning layer,
$\boldsymbol{y}_{i}=\tanh\big{(}\boldsymbol{W}^{\textsc{DML}}\boldsymbol{x}_{i}+\boldsymbol{b}^{\textsc{DML}}\big{)},$
which yields the two LEVs $\boldsymbol{y}_{1}$ and $\boldsymbol{y}_{2}$ via
the trainable parameters $\boldsymbol{\psi}=\\{\boldsymbol{W}^{\textsc{DML}},$
$\boldsymbol{b}^{\textsc{DML}}\\}$. We then compute the Euclidean distance
between both LEVs,
$d(\boldsymbol{y}_{1},\boldsymbol{y}_{2})=\left\lVert\boldsymbol{y}_{1}-\boldsymbol{y}_{2}\right\rVert_{2}^{2}.$
In [11], we introduced a new probabilistic version of the contrastive loss:
Given the Euclidean distance of the LEVs, we apply a kernel function

$\displaystyle
p_{\text{DML}}(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})=\exp\big{(}-\gamma~{}d(\boldsymbol{y}_{1},\boldsymbol{y}_{2})^{\alpha}\big{)},$
(2)

where $\gamma$ and $\alpha$ can be seen as both, hyper-parameters or trainable
variables. The loss then is given by

$\displaystyle\begin{split}\mathcal{L}^{\text{DML}}_{\boldsymbol{\theta},\boldsymbol{\psi}}=a\cdot\max\big{\\{}\tau_{s}-p_{\text{DML}}(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2}),0\big{\\}}^{2}+(1-a)\cdot\max\left\\{p_{\text{DML}}(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})-\tau_{d},0\right\\}^{2},\end{split}\vspace{-0.4in}$
(3)

where we set $\tau_{s}=0.91$ and $\tau_{d}=0.09$.

###  4.2 Deep Bayes Factor Scoring

We assume that the LEVs stem from a Gaussian generative model that can be
decomposed as $\boldsymbol{y}=\boldsymbol{s}+\boldsymbol{n}$, where
$\boldsymbol{n}$ characterizes a noise term. We assume that the writing
characteristics of the author lie in a latent stylistic variable
$\boldsymbol{s}$. The probability density functions for $\boldsymbol{s}$ and
$\boldsymbol{n}$ are modeled as Gaussian distributions. We outlined in [1] how
to compute the likelihoods for both hypotheses. The verification score for a
trial is then given by the log-likelihood ratio:
$\text{score}(\boldsymbol{y}_{1},\boldsymbol{y}_{2})=\log
p(\boldsymbol{y}_{1},\boldsymbol{y}_{2}|\mathcal{H}_{1})-\log
p(\boldsymbol{y}_{1},\boldsymbol{y}_{2}|\mathcal{H}_{0})$. Assuming
$p(\mathcal{H}_{1})=p(\mathcal{H}_{0})=\frac{1}{2}$, the probability for a
same-author trial is calculated as [1]:

$\displaystyle
p_{\text{BFS}}(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})=\frac{p(\boldsymbol{y}_{1},\boldsymbol{y}_{2}|\mathcal{H}_{1})}{p(\boldsymbol{y}_{1},\boldsymbol{y}_{2}|\mathcal{H}_{1})+p(\boldsymbol{y}_{1},\boldsymbol{y}_{2}|\mathcal{H}_{0})}=\text{Sigmoid}\big{(}\text{score}(\boldsymbol{y}_{1},\boldsymbol{y}_{2})\big{)}$
(4)

We reduce the dimension of the LEVs via
$\boldsymbol{y}_{i}^{\textsc{BFS}}=\text{tanh}\big{(}\boldsymbol{W}^{\textsc{BFS}}\boldsymbol{y}_{i}+\boldsymbol{b}^{\textsc{BFS}}\big{)}$
to ensure numerically stable inversions of the matrices [1]. We rewrite Eq.
(4) as

$\displaystyle
p_{\text{BFS}}(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})=\text{Sigmoid}\big{(}\text{score}(\boldsymbol{y}_{1}^{\textsc{BFS}},\boldsymbol{y}_{2}^{\textsc{BFS}})\big{)}$
(5)

and incorporate Eq. (5) into the binary cross entropy,

$\displaystyle\mathcal{L}^{\text{BFS}}_{\boldsymbol{\phi}}=a\cdot\log\left\\{p_{\text{BFS}}(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})\right\\}+(1-a)\cdot\log\left\\{1-p_{\text{BFS}}(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})\right\\},$
(6)

where all trainable parameters are denoted with
$\boldsymbol{\phi}=\big{\\{}\boldsymbol{W}^{\textsc{BFS}},\boldsymbol{b}^{\textsc{BFS}},\boldsymbol{W},\boldsymbol{B},\boldsymbol{\mu}\big{\\}}$.

###  4.3 Uncertainty Modeling and Adaptation

Now, we treat the posteriors of the BFS component as noisy outcomes and
rewrite Eq. (5) as
$p_{\text{BFS}}(\widehat{\mathcal{H}}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})$
to emphasize that this represents an estimated posterior. We firstly have to
find a single representation for both LEVs, which is done by
$\boldsymbol{y}^{\textsc{UAL}}=\tanh\big{(}\boldsymbol{W}^{\textsc{UAL}}\big{(}\boldsymbol{y}_{1}-\boldsymbol{y}_{2}\big{)}^{\circ
2}+\boldsymbol{b}^{\textsc{UAL}}\big{)},$ where $(\cdot)^{\circ 2}$ denotes
the element-wise square. Next, we compute a $2\times 2$ confusion matrix as
follows

$\displaystyle
p(\mathcal{H}_{j}|\widehat{\mathcal{H}}_{i},\boldsymbol{y}_{1},\boldsymbol{y}_{2})=\frac{\exp\big{(}\boldsymbol{w}_{ji}^{T}~{}\boldsymbol{y}^{\textsc{BFS}}+b_{ji}\big{)}}{\sum\limits_{i^{\prime}\in\\{0,1\\}}\exp\big{(}\boldsymbol{w}_{ji^{\prime}}^{T}~{}\boldsymbol{y}^{\textsc{BFS}}+b_{ji^{\prime}}\big{)}}\quad\text{
for }i,j\in\\{0,1\\}.$ (7)

The term
$p(\mathcal{H}_{j}|\widehat{\mathcal{H}}_{i},\boldsymbol{y}_{1},\boldsymbol{y}_{2})$
defines the conditional probability of the true hypothesis $\mathcal{H}_{j}$
given the hypothesis $\widehat{\mathcal{H}}_{i}$ assigned by the BFS. We can
then define the final output predictions as:

$\displaystyle
p_{\text{UAL}}(\mathcal{H}_{j}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})=\sum\limits_{i\in\\{0,1\\}}p(\mathcal{H}_{j}|\widehat{\mathcal{H}}_{i},\boldsymbol{y}_{1},\boldsymbol{y}_{2})\cdot
p_{\text{BFS}}(\widehat{\mathcal{H}}_{i}|\boldsymbol{y}_{1},\boldsymbol{y}_{2}).$
(8)

The loss consists of two terms, the negative log-likelihood of the ground-
truth hypothesis and a regularization term,

$\displaystyle\begin{split}\mathcal{L}^{\text{UAL}}_{\boldsymbol{\lambda}}=-\log
p_{\text{UAL}}(\mathcal{H}_{j}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})+\beta~{}\sum_{i\in\\{0,1\\}}\sum_{j\in\\{0,1\\}}p(\mathcal{H}_{j}|\widehat{\mathcal{H}}_{i},\boldsymbol{y}_{1},\boldsymbol{y}_{2})\cdot\log
p(\mathcal{H}_{j}|\widehat{\mathcal{H}}_{i},\boldsymbol{y}_{1},\boldsymbol{y}_{2}),\end{split}$
(9)

with trainable parameters denoted by
$\boldsymbol{\lambda}=\big{\\{}\boldsymbol{W}^{\textsc{UAL}},\boldsymbol{b}f^{\textsc{UAL}},\boldsymbol{w}_{ji},\boldsymbol{b}_{ji}|j,i\in\\{0,1\\}\big{\\}}$.
The regularization term, controlled by $\beta$, follows the maximum entropy
principle to penalize the confusion matrix for returning over-confident
posteriors [13].

###  4.4 Combined Loss Function:

All components are optimized independently w.r.t. the following combined loss:

$\displaystyle\begin{split}\mathcal{L}_{\boldsymbol{\theta},\boldsymbol{\psi},\boldsymbol{\phi},\boldsymbol{\lambda}}=\mathcal{L}^{\text{DML}}_{\boldsymbol{\theta},\boldsymbol{\psi}}+\mathcal{L}^{\text{BFS}}_{\boldsymbol{\phi}}+\mathcal{L}^{\text{UAL}}_{\boldsymbol{\lambda}}.\end{split}$
(10)

###  4.5 Out-of-Distribution Detector (O2D2)

Following [9], we incorporate a second neural network to detect undecidable
trials. We treat the training procedure as a binary verification task. Given
the learned DML, BFS and UAL components, the estimated authorship labels are
obtained via

$\displaystyle\widehat{a}=\arg\max\big{[}p_{\text{UAL}}(\mathcal{H}_{0}|\boldsymbol{y}_{1},\boldsymbol{y}_{2}),~{}p_{\text{UAL}}(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})\big{]}.$
(11)

Now, we can define the binary O2D2 labels as follows:

$l^{\text{O2D2}}=\begin{cases}1,&\text{if
}a\neq\widehat{a}\text{~{}~{}~{}or~{}~{}~{}}0.5-\epsilon\leq
p_{\text{UAL}}(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})\leq
0.5+\epsilon,\\\ 0,&\text{otherwise}.\end{cases}$ (12)

The model-dependent hyper-parameter $\epsilon\in[0.05,0.15]$ is optimized on
the validation set w.r.t the PAN 2021 metrics. The input of O2D2, noted as
$\boldsymbol{y}^{\text{O2D2}}$, is a concatenated vector of the LEVs, i.e.
$\big{(}\boldsymbol{y}_{1}-\boldsymbol{y}_{2}\big{)}^{\circ 2}$ and
$\big{(}\boldsymbol{y}_{1}+\boldsymbol{y}_{2}\big{)}^{\circ 2}$, and the
confusion matrix. This vector is fed into a three-layer architecture,

$\displaystyle\begin{split}\boldsymbol{h}_{1}&=\tanh\big{(}\boldsymbol{W}^{\textsc{O2D2}}_{1}\boldsymbol{y}^{\textsc{O2D2}}+\boldsymbol{b}^{\textsc{O2D2}}_{1}\big{)},\\\
\boldsymbol{h}_{2}&=\tanh\big{(}\boldsymbol{W}^{\textsc{O2D2}}_{2}\boldsymbol{h}_{1}+\boldsymbol{b}^{\textsc{O2D2}}_{2}\big{)},\\\
p_{\text{O2D2}}(\mathcal{H}_{2}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})&=\text{Sigmoid}\big{(}\boldsymbol{W}^{\textsc{O2D2}}_{3}\boldsymbol{h}_{2}+\boldsymbol{b}^{\textsc{O2D2}}_{3}\big{)}.\end{split}$
(13)

All trainable parameters are summarized in
$\boldsymbol{\Gamma}=\big{\\{}\boldsymbol{W}^{\textsc{O2D2}}_{i},\boldsymbol{b}^{\textsc{O2D2}}_{i}|i\in\\{1,2,3\\}\big{\\}}$.
The obtained prediction for hypothesis $\mathcal{H}_{2}$ is inserted into the
cross-entropy loss,

$\displaystyle\begin{split}\mathcal{L}^{\text{O2D2}}_{\boldsymbol{\Gamma}}=l^{\text{O2D2}}\cdot\log\left\\{p_{\text{O2D2}}(\mathcal{H}_{2}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})\right\\}+(1-l^{\text{O2D2}})\cdot\log\left\\{1-p_{\text{O2D2}}(\mathcal{H}_{2}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})\right\\}.\end{split}$
(14)

Figure 3: Averaged accuracy curves (including mean and standard deviation) for
the authorship and fandom verification outputs during training.

###  4.6 Ensemble Inference

As a last step, an ensemble is constructed from $M$ trained models,
$\mathcal{M}_{1},\ldots,\mathcal{M}_{M}$, with $M$ being an odd number. Since
all models are randomly initialized and trained on different re-sampled pairs
in each epoch, we expect to obtain a slightly different set of weights/biases,
which in turn produces different posteriors, especially for pairs under
covariate shift. We propose a majority voting for the non-responses. More
precisely, the ensemble returns a non-response, if

$\displaystyle\sum_{m=1}^{M}\mathbbm{1}\big{[}p_{\text{O2D2}}(\mathcal{H}_{2}|\boldsymbol{y}_{1},\boldsymbol{y}_{2},\mathcal{M}_{m})\geq
0.5\big{]}>\left\lfloor\frac{M}{2}\right\rfloor,$ (15)

where $\mathbbm{1}[\cdot]$ denotes the indicator function. Otherwise, we
define a subset of confident models,
$\mathcal{M}_{c}=\\{\mathcal{M}|~{}p_{\text{O2D2}}(\mathcal{H}_{2}|\boldsymbol{y}_{1},\boldsymbol{y}_{2},\mathcal{M})<0.5\\}$,
and return the averaged posteriors of its elements,

$\displaystyle\mathbb{E}\big{[}p_{\text{UAL}}(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})\big{]}=\frac{1}{|\mathcal{M}_{c}|}\sum_{\mathcal{M}\in\mathcal{M}_{c}}p_{\text{UAL}}(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2},\mathcal{M}).$
(16)

Our submitted system consisted of an ensemble with $M=21$ trained models.

##  5 Experiments

Table 2: Averaged results (including mean and standard deviation) of the UAL
framework for different subset combinations on the calibration dataset.

Subsets | PAN 2021 Evaluation Metrics
---|---
AUC |  c@1 | f_05_u |  F1 |  Brier | overall
SA_SF \+ DA_DF | $99.8\pm 0.0$ | $97.5\pm 0.2$ | $97.2\pm 0.3$ | $97.5\pm 0.2$ | $98.1\pm 0.1$ | $98.0\pm 0.2$
SA_SF \+ DA_SF | $99.6\pm 0.1$ | $95.9\pm 0.4$ | $94.8\pm 0.6$ | $96.0\pm 0.4$ | $97.1\pm 0.2$ | $96.7\pm 0.4$
SA_DF \+ DA_DF | $98.1\pm 0.1$ | $92.4\pm 0.3$ | $94.8\pm 0.3$ | $92.1\pm 0.3$ | $94.2\pm 0.2$ | $94.3\pm 0.2$
SA_DF \+ DA_SF | $97.1\pm 0.1$ | $90.9\pm 0.3$ | $92.3\pm 0.6$ | $90.6\pm 0.3$ | $93.1\pm 0.2$ | $92.8\pm 0.3$

Table 3: Averaged calibration results (including mean and standard deviation)
of the UAL framework for different subsets on the calibration dataset.

Subsets | Calibration Metrics
---|---
acc |  conf |  ECE |  MCE
SA_SF \+ DA_DF | $98.4\pm 0.3$ | $97.4\pm 1.0$ | $1.1\pm 0.9$ | $5.7\pm 3.6$
SA_SF \+ DA_SF | $96.1\pm 0.6$ | $95.6\pm 1.0$ | $1.3\pm 0.8$ | $9.9\pm 4.0$
SA_DF \+ DA_DF | $92.4\pm 0.3$ | $93.4\pm 1.2$ | $1.6\pm 0.6$ | $6.2\pm 2.5$
SA_DF \+ DA_SF | $90.9\pm 0.3$ | $92.4\pm 1.2$ | $2.0\pm 0.7$ | $7.4\pm 2.9$

The PAN evaluation metrics and procedure are described in [4, 5, 14]. To
capture the calibration capacity, we also provide the accuracy (acc),
confidence score (conf), expected calibration error (ECE) and maximum
calibration error (MCE) [15]. All confidence values lie within the interval
$[0.5,1]$, since we are solving a binary classification task. Hence, to obtain
confidence scores, the posterior values are transformed w.r.t. the estimated
authorship label, showing
$p(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})$ if $\widehat{a}=1$
and $1-p(\mathcal{H}_{1}|\boldsymbol{y}_{1},\boldsymbol{y}_{2})$ if
$\widehat{a}=0$. For both metrics, the confidence interval is discretized into
a fixed number of bins. The ECE then reflects the average absolute error
between confidence and accuracy of all bins, while the MCE returns the maximum
absolute error. For acc and conf, we perform weighted macro-averaging w.r.t.
the number of trials in each bin.

Inspired by the promising results in domain-adversarial training of neural
networks in [16, 17], we also experimented with an adversarial fandom
verifier: Starting with the document embeddings in Eq. (1), we fed this vector
into the author verification system (including DML, BFS and UAL) and into an
additional fandom verifier, which is placed parallel to the author
verification system. It has the same architecture but includes a gradient
reversal layer and different trainable parameters. However, in these
experiments, we did not achieve any significant improvements by domain-
adversarial training. Therefore, we independently optimized the fandom
verifier by stopping the flow of the gradients from the fandom verifier to the
authorship verification components, so that the training of the fandom
verifier does not affect the target system at all. Fig. 3 shows the obtained
epoch-wise accuracies during training. It can be seen that the fandom accuracy
stays around $55$%, which indicates that the training strategy yields nearly
topic-invariant stylometric representations, even without domain-adversarial
training.

###  5.1 Results on the Calibration Dataset

We first evaluated the UAL component on the calibration set (without non-
responses) and calculated the respective PAN metrics for different
combinations of the author-fandom subsets. Results are shown in Table 2. To
guarantee that the calculated metrics are not biased by an imbalanced dataset,
we reduced the number of pairs to the smallest number of pairs of all subsets.
Thus, all results in Table 2 were computed from $2\times 2,100$ pairs.
Unsurprisingly, best performance was obtained for the least challenging SA_SF
\+ DA_DF pairs and the worst performance was seen for the most challenging
SA_DF \+ DA_SF pairs. We continued to optimize our system w.r.t this most
challenging subset combination in particular, even though we specifically
expect to see SA_DF \+ DA_DF pairs in the PAN 2021 evaluation set.

Table 3 additionally provides the corresponding calibration metrics.
Analogously to the PAN metrics, the ECE consistently increases from the least
to the most challenging data scenarios. Interestingly, our system is under-
confident for SA_SF pairs, i.e. conf $<$ acc. The predictions then change to
be over-confident (conf $>$ acc) for SA_DF pairs.

Table 4: Results for PAN 2021 evaluation metrics on the validation datset.

Model | PAN 2021 Evaluation Metrics
---|---
AUC |  c@1 | f_05_u |  F1 |  Brier | overall
single | DML | $97.2\pm 0.1$ | $91.3\pm 0.3$ | $90.5\pm 0.6$ | $89.6\pm 0.4$ | $93.2\pm 0.4$ | $92.4\pm 0.2$
BFS | $97.1\pm 0.1$ | $91.0\pm 0.3$ | $90.7\pm 0.8$ | $89.2\pm 0.5$ | $93.2\pm 0.1$ | $92.3\pm 0.2$
UAL | $97.2\pm 0.1$ | $91.3\pm 0.3$ | $90.7\pm 0.5$ | $89.6\pm 0.4$ | $93.5\pm 0.2$ | $92.5\pm 0.2$
O2D2 | $97.1\pm 0.1$ | $93.8\pm 0.2$ | $88.1\pm 0.6$ | $93.5\pm 0.3$ | $93.4\pm 0.1$ | $93.2\pm 0.2$
ensemble | $97.8$ | $92.5$ | $92.1$ | $90.9$ | $94.3$ | $93.5$
ensemble + O2D2 | $97.7$ | $94.8$ | $90.0$ | $94.5$ | $94.2$ | $94.2$

Table 5: Results for the calibration metrics on the validation dataset.

Model | Calibration Metrics
---|---
acc |  conf |  ECE |  MCE
single | DML | $91.3\pm 0.3$ | $87.9\pm 2.7$ | $3.4\pm 2.7$ | $~{}9.0\pm 3.6$
BFS | $91.0\pm 0.3$ | $90.0\pm 2.3$ | $2.3\pm 1.5$ | $~{}6.2\pm 3.0$
UAL | $91.3\pm 0.3$ | $92.3\pm 1.2$ | $1.6\pm 0.6$ | $~{}5.8\pm 2.2$
O2D2 | $91.4\pm 0.3$ | $90.9\pm 1.1$ | $2.3\pm 0.5$ | $10.7\pm 2.7$
ensemble | $92.5$ | $91.2$ | $1.2$ | $2.9$
ensemble + O2D2 | $92.6$ | $91.8$ | $1.5$ | $10.2$

(a) SA_DF pairs without O2D2.

(b) DA_SF pairs without O2D2.

(c) SA_DF pairs including O2D2.

(d) DA_SF pairs including O2D2.

Figure 4: Posterior histograms on the validation dataset.

###  5.2 Results on the Validation Dataset

Next, we separately provide experimental results for all system components on
the validation dataset, since O2D2 has been trained on the calibration
dataset. The first four rows in Tables 4 and 5 summarize the PAN metrics and
the corresponding calibration measures averaged over all ensembles models.

The overall score of the UAL component in the third row of Table 4 is on par
with the DML and BFS components and slightly lower compared to the
corresponding UAL score measured on the calibration dataset in Table 2.
Nevertheless, we do not observe significant differences in the metrics for
both datasets, which shows the robustness and generalization of our system.

Going from the third to the fourth row in Table 4, it can be observed that the
overall score, boosted by c@1 and F1, significantly increases from $92.5$ to
$93.2$. Hence, the model performs better if we take undecidable trials into
account. However, the f_05_u score decreases, since it treats non-responses as
false negatives. The percentage of undecidable trials generally ranges from 8%
to 11%.

In Table 5, we see that both, the BFS and UAL components notably improve the
ECE and MCE metrics. However, an insertion of non-responses via O2D2
significantly increases the MCE. This can be explained by the posterior
histograms in Fig. 4. The plots (a) and (b) show the histograms for SA_DF and
DA_SF pairs without applying O2D2 to define non-responses. In contrast, plots
(c) and (d) present the corresponding histograms including the $0.5$-values of
non-responses. The effect of O2D2 is that most of the trials, whose posteriors
fall within the interval $[0.3,0.7]$, are eventually declared as undecidable.
Hence, the system correctly predicts nearly all of the remaining as
confidently assigned trials around $0.7$/$0.8$ for same-author pairs or
$0.2$/$0.3$ for different-author pairs. As a result, we see a large gap (i.e.
conf $<<$ acc) between the confidence score and the averaged accuracy in these
bins.

The last two rows in Tables 4 and 5 show the performance of the ensemble,
first without and then with non-responses, to show the effect of O2D2. On the
validation set, our ensemble with O2D2 returns non-responses in $9$% of the
test cases. Comparing the last two rows, we obtain the highest overall score
with our proposed framework, which ultimately presents our final submission.

###  5.3 Results on the PAN 2021 Evaluation Dataset

To conclude this section, we present our results on the official PAN 2021
evaluation set. The performance for both, the early-bird and the final
submission, can be found in Table 6. We also provide the reported result on
the PAN 2020 evaluation set for the predecessor model.

Unsurprisingly, the early-bird overall score (single model) on the PAN 2021
evaluation set is slightly higher, since it contains DA_DF pairs instead of
DA_SF pairs. The main difference is, unexpectedly, given by the f_05_u score,
which increases from $89.3$% to $94.6$%. In our opinion, this is caused by
returning a lower number of non-responses, which would also explain the lower
values for c@1 and F1.

Comparing the early-bird ($2^{nd}$ row) with the final submission ($4^{th}$
row), we can further significantly increase the overall score by $1.5\%$. We
assume that the ensemble now returns a higher number of non-responses, which
results in a slightly lower f_05_u score. Conversely, we can observe improved
values for the c@1, F1 and brier scores.

The last row displays the achieved PAN 2020 results. As can be seen, our final
submission ends up with a higher overall score (plus $2\%$) by significantly
improving all single metrics, although the PAN competition moved from a
closed-set to open-set shared task, illustrating the efficiency of the
proposed extensions.

Table 6: Results of the early-bird (first two rows) and the final submission
runs.

Dataset | Model type |  AUC |  c@1 |  f_05_u |  F1 |  brier | overall
---|---|---|---|---|---|---|---
Validation dataset | single-21 | $97.2$ | $93.6$ | $89.3$ | $92.9$ | $93.3$ | $93.3$
PAN 21 evaluation dataset | single-21 | $98.3$ | $92.6$ | $94.6$ | $92.1$ | $92.7$ | $94.0$
Validation dataset | ensemble-21 | $97.7$ | $94.8$ | $90.1$ | $94.4$ | $94.2$ | $94.2$
PAN 21 evaluation dataset | ensemble-21 | $98.7$ | $95.0$ | $93.8$ | $95.2$ | $94.5$ | $95.5$
PAN 20 evaluation dataset | ensemble-20 | $96.9$ | $92.8$ | $90.7$ | $93.6$ | - | $93.5$

##  6 Conclusion

In this work, we presented O2D2, which captures undecidable trials and
supports our hybrid neural-probabilistic end-to-end framework for authorship
verification. We made use of the early-bird submission to receive a
preliminary assessment of how the framework behaves on the novel open-set
evaluation. Finally, based on the presented results, we submitted an
O2D2-supported ensemble to the shared task, which clearly outperformed our own
system from 2020 as well as the new submissions to the PAN 2021 AV task.

These results support our hypothesis that modeling aleatoric and epistemic
uncertainty and using them for decision support is a beneficial strategy—not
just for responsible ML, which needs to be aware of the reliability of its
proposed decisions, but also, importantly, for achieving optimal performance
in real-life settings, where distributional shift is almost always hard to
avoid.

###### Acknowledgements.

This work was in significant parts performed on an HPC cluster at Bucknell
University through the support of the National Science Foundation, Grant
Number 1659397. Project funding was provided by the state of North Rhine-
Westphalia within the Research Training Group "SecHuman - Security for Humans
in Cyberspace" and by the Deutsche Forschungsgemeinschaft (DFG) under
Germany’s Excellence Strategy - EXC2092CaSa- 390781972.

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