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\fancyfoot[L]{\emph{History of Media Studies}, vol. 1, 2021}

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\title[Emerging Digital Transitions in the Arab World]{Emerging Digital Transitions in the Arab World: Implications for the Region’s Communication Studies} % longtitle shouldn't be necessary

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\noindent\LARGE\emph{Launch essay
} \hspace{85mm}\includegraphics[height=1cm]{logo3.png}\\
\noindent\hrulefill\\
\vspace*{1em}
\noindent{\Huge{Emerging Digital Transitions in the Arab\\\noindent World: Implications for the Region’s\\\noindent Communication Studies\par}}

\vspace*{1.5em}

\noindent\LARGE{Mohammad Ayish} \href{https://orcid.org/0000-0002-1262-078X}{\includegraphics[height=0.5cm]{orcid.png}}\par}\marginnote{\emph{Mohammad Ayish, ``Emerging Digital Transitions in the Arab World: Implications for the Region’s Communication Studies,'' \emph{History of Media Studies} 1 (2021), \href{https://doi.org/10.32376/d895a0ea.69b4fc2d}{https://doi.org/ 10.32376/d895a0ea.69b4fc2d}.} \vspace*{0.75em}}
\vspace*{0.5em}
\noindent{{\large\emph{American University of Sharjah}, \href{mailto:mayish@aus.edu}{mayish@aus.edu}\par}} \marginnote{\href{https://creativecommons.org/licenses/by-nc/4.0/}{\includegraphics[height=0.5cm]{by-nc.png}}}

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\vspace*{1em}

\newthought{Throughout its eighty-year} history, communication studies in the Arab
World has generally been shaped by a convergence of national political
agendas, cultural concerns, technological transitions, and
Western-centered perspectives on modernization, dependency,
globalization and empowerment.\footnote{Mohammad Ayish, ``Arab Television Goes Commercial: A Case Study of the
  Middle East Broadcasting Centre,'' \emph{Gazette} 59, no. 6 (1997):
  473.
}
During the first three decades of post-colonial independence, the
region's media education programs, scholarship, and professional
orientations were hugely informed by both modernization and dependency
perspectives, while the globalization perspective was widely popular in
the 1990s, coinciding with the end of the Cold War and the advent of
transnational satellite television. The empowerment perspective, on the
other hand, was gaining momentum only at the dawn of the
21\textsuperscript{st} Century, as the Arab World came to gradually
embrace digital and online communications at institutional and personal
levels. While the region's governments and business sectors saw a huge
potential in digital/online communications as key forces of political
stability and economic prosperity, the region's communities at large
seemed to view them as enablers of greater democratization and
grassroots political engagement. Across the region in general and in the
Arabian Gulf countries in particular, huge investments have been
channeled into building world-class digital communications
infrastructures, allowing greater citizen access to an emerging virtual
public sphere. At universities and research communities, this ``digital
phase'' has been marked by new pedagogical, conceptual, and
methodological shifts in media education programs; new research
activities; and new professional orientations towards

\enlargethispage{2\baselineskip}

\vspace*{2em}

\noindent{\emph{History of Media Studies}, vol. 1, 2021}

 \end{titlepage}

\noindent
cyber-communications and virtual engagements. I argue here that while
digital transitions will continue to define the Arab World's political,
cultural, and economic living experiences for many years to come, they
are bound to have significant implications for media studies in the
region.

\hypertarget{communication-studies-and-its-engagement-with-modernization-dependency-and-globalization-a-historical-overview}{%
\section{Communication Studies and its Engagement with Modernization,
Dependency and Globalization: A Historical
Overview}\label{communication-studies-and-its-engagement-with-modernization-dependency-and-globalization-a-historical-overview}}

Although the American University in Cairo was the first institution of
higher education in the Arab World to host a journalism program in 1935,
the distinctive features of media education and scholarship in the
region did not take shape until the onset of the post-colonial era in
which newly-independent Arab states saw mass media as key engines of
socio-economic transformation and political integration. For these new
governments, mass communication channels, especially radio, were
instrumental in communicating ``developmental'' messages to
mostly-illiterate national populations in line with Lerner's
modernization framework of national
development.\footnote{Daniel Lerner, \emph{The Passing of Traditional Societies: Modernizing
  the Middle East} (Glencoe, IL: Free Press, 1958).
} Around that concept,
the region's media education and scholarship were harnessed to prepare
new generations of professional communicators and researchers who
strongly believed in the power of media to bring about positive
socio-economic and cultural changes within their under-developed
communities. In the 1960s and 1970s, universities in countries such as
Egypt, Tunisia, Jordan, and Morocco became home to journalism and mass
communication programs and to what came to be known as ``developmental
communication'' research traditions.

In the meantime, communication studies in some Arab countries aligned
with the former Soviet bloc---such as Egypt, Iraq, Syria, Libya, and
Algeria---promoted a more critical media outlook typical of the
dependency perspective. Media education programs and scholarship in
those countries focused on topics such as propaganda, media imperialism,
and Western information hegemony. This communication studies tradition
was hugely sustained by UNESCO discussions of a New World Information
and Communication Order (NWICO) that addressed one-way information flows
from the North (Western nations) to the South (Third World nations). It
promoted a view of media as guardians of national identity and tools of
political mobilization and national integration. Egyptian scholar Awatef
Abdul Rahman of Cairo University has argued that international media
channels were used to undermine pan-Arabism as a political ideology in
the region.\footnote{Awatef Abdul Rahman, \emph{Communication and Cultural Dependency
  Issues in the Third World} {[}in Arabic{]} (Kuwait: National Council
  for Culture and Arts, 1984).
}

In the 1990s, communication studies in the Arab World were bound to be
impacted by emerging globalization, itself enabled by the rise of
transnational communication technologies such as satellite television.
Many Arab media scholars argued that globalization was posing a serious
threat to Arab-Islamic values and traditions and called on national
media to maintain indigenous cultures and languages. Amin's critical
view of globalization as a brutal and hegemonic process enabled by
modern communications technologies\footnote{Jalal Amin, \emph{Globalization} {[}in Arabic{]} (Cairo: Dar Al
  Ma'aref, 1996).
}
generated a good deal of interest among academics and policy makers
across the region. Media education curricula came to include new courses
on globalization and satellite television, while research works sought
to throw light on how transnational media were impacting the region's
communications\footnote{Mohammad Ayish, ``Arab Television Goes Commercial: A Case Study of the
  Middle East Broadcasting Centre,'' \emph{Gazette} 59, no. 6 (1997):
  473-494.
} and socio-political
and cultural norms.\footnote{Douglas Boyd, \emph{Broadcasting in the Middle East} (Des Moines,
  Iowa: Iowa State University Press, 1999).
}

\hypertarget{the-current-state-of-communication-studies-the-empowerment-perspective}{%
\section{The Current State of Communication Studies: the Empowerment
Perspective}\label{the-current-state-of-communication-studies-the-empowerment-perspective}}

As the new century dawned on the Arab region, it was clear that digital
and online communications were presenting a serious challenge to
traditional, state-operated mass media. The ability of social media
platforms to give a voice to the average person in a state-controlled
communication environment was hailed across the region as highly
empowering. And the implications for communications studies have been
quite immense. Universities across the region came to introduce new
curricula aiming at educating new generations of young communicators in
digital media techniques and concepts. Topics such as \emph{social
media, citizen journalism, media convergence, digital photography,
digital video, online journalism, visual storytelling, virtual public
relations, digital advertising, new media theories, social analytics,
big data, website design, webcasting, blogging,} and \emph{cybermedia
ethics and laws} have made headlines in the region's media education
curricular maps.\footnote{Mohammad Ayish, ``Arab Media Studies'' in \emph{The International
  History of Communication Study}, ed. Peter Simonson and David Park
  (New York: Routledge, 2016), 476.
}

During my thirty-five year career as a media educator and researcher, I
have seen communication studies in the Arab World adapt dynamically at
the regional, national, and global scale to significant political,
economic, and technological developments. But the rush to engage with
digital and online communications on the part of media educators,
scholars, and professionals seems remarkably unprecedented. If long-time
state domination of the region's media landscape was perceived as the
black tunnel in which the region's communications have historically
evolved, digital empowerment came to be seen as ``the light at the end
of that tunnel.'' Three political/military developments that fostered
the region's perceptions of the power of social media and digital
communications in the past two decades seem to be cases in point: the
Anglo-American invasion of Iraq (2003), the Arab Spring (2011-2013), and
the Palestinian-Israeli military conflict of May 2021. The US occupation
of Iraq in April 2003 came to receive wide coverage on emerging online
national and international news platforms that were widely perceived
across the region as more diverse and freer than state-operated media.
Ralph Berenger, who had worked for 13 years at both the American
University of Cairo and the American University of Sharjah initiated his
``Media Go to Cyber War'' scholarly project that culminated in a
valuable collection of research works dealing with cyber media's
handling of the Iraq conflict.\footnote{Ralph Berenger, ed., \emph{Cybermedia Go to War: Role of Converging
  Media During and After the 2003 Iraq War} (Spokane, Washington:
  Marquette Books, 2006).
} The
real spike in scholarly interest in online media in the Arab World,
however, came during the second decade of the century and was triggered
mainly by upheavals across the region referred to as the ``Arab
Spring.'' Social media platforms accessible to the region's populations
were widely viewed as catalysts for the popular uprisings. The region's
researchers drew on empowerment theory to address social media
contributions to the Arab Spring, but the majority of those studies were
descriptive and drew on historical and conventional quantitative
methods.\footnote{ Adam Smidi and Saif Shahin, ``Social Media and Social Mobilization in
  the Middle East: A Survey of Research on the Arab Spring,''
  \emph{India Quarterly} 73, no. 2 (June 2017): 196--209.} Finally, the third
development most likely to attract scholarly attention in the region has
been the May 8-14, 2021 outbreak of violence in Palestine. During this
conflict, social media was viewed as a key tool for mobilizing public
opinion, and the resulting conversation about the role of social media
in politics has raised high expectations about what cyberspace could
deliver in times of crisis, particularly when state-run communications
are failing to provide the full picture of the conflict.

\hypertarget{implications-for-communication-studies}{%
\section{Implications for Communication
Studies}\label{implications-for-communication-studies}}

It is clear that the history of communication studies in the Arab World
will continue to be defined by regional and global political,
technological, and socio-cultural developments. The region's growing
investments in digital communications have converged with regional
political conflicts, social upheavals, and cultural concerns to position
digital and online features at the center of the region's communication
education and scholarship. These developments offer both opportunities
and challenges for media educators, scholars, and professional
practitioners. Media education programs could benefit from the
empowering aspects of digital communications in enhancing knowledge
about political participation, democratization, multiculturalism, and
diversity in their curricula. In research, media studies could also make
use of the emerging field of digital humanities and especially big data
analytics for long-term historical investigations of media behavior
across time as well as for enhancing qualitative aspects of scholarship.

On the other hand, communication studies in the region will continue to
face the challenge of freedom deficiency in the region. We have to
remember that as much as digital communications have provided
individuals with access to a global public sphere, they have also
enabled states to tighten their grip on the media landscape. Cybermedia
laws enacted over the past decades across the Arab World could be hugely
detrimental to the development of sustainable communication studies in
areas of education, research, and professional practice. In addition,
media educators should be sensitized to the fact that teaching online
and digital media is not a technical concern but a pursuit that also
involves significant social and cultural knowledge that students should
engage in. In this context, the liberal arts education model that
enables students' exposure to knowledge from the humanities and social
sciences should be maintained in digitally-centered media education
programs. In digital communication-focused scholarship, the region's
researchers should be mindful of that fact that algorithm-driven
analysis would be no alternative to their interpretations of data to
understand the human living experience.

\section{Bibliography}\label{bibliography}

\begin{hangparas}{.25in}{1}

Abdul Rahman, Awatef. \emph{Communication and Cultural Dependency Issues
in the Third World}. {[}In Arabic.{]} Kuwait: National Council for
Culture and Arts, 1984.

Amin, Jalal. \emph{Globalization}. {[}In Arabic.{]} Cairo: Dar Al
Ma'aref, 1996.

Ayish, Mohammad. ``Arab Television Goes Commercial: A Case Study of the
Middle East Broadcasting Centre.'' \emph{Gazette} 59, no. 6 (1997):
473-494. \url{https://doi.org/10.1177/0016549297059006004}.

Ayish, Mohammad. ``Arab Media Studies.'' In \emph{The International
History of Communication Study}, edited by Peter Simonson and David
Park, 474-493. New York: Rutledge, 2016.

Berenger, Ralph, ed. \emph{Cybermedia Go to War: Role of Converging
Media During and After the 2003 Iraq War}. Spokane, Washington:
Marquette Books, 2006.

Boyd, Douglas. \emph{Broadcasting in the Middle East}. Des Moines: Iowa
State University Press, 1999.

Lerner, Daniel. \emph{The Passing of Traditional Societies: Modernizing
the Middle East}. New York: Free Press, 1958.

Smidi, Adam, and Saif Shahin. ``Social Media and Social Mobilization in
the Middle East: A Survey of Research on the Arab Spring.'' \emph{India
Quarterly} 73, no. 2 (June 2017): 196--209.
\url{https://doi.org/10.1177/0974928417700798}.

\end{hangparas}

\end{document}

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text:
\section{Results}
The results presented for each scenario include the energy produced, reactor
deployment schedule, enriched
uranium mass, \gls{HALEU} mass, and \gls{SWU} capacity required as a function of time.

\subsection{Scenario 1}
The amount of energy supplied by the \glspl{LWR} and the number of \glspl{LWR}
deployed as a function of time are shown in Figure \ref{fig:energy_rx_1}.
\glspl{LWR} are first deployed in August of 1967, and the last
\gls{LWR} is deployed in June of 2016 and decommissioned in July 2076. The
maximum number of
\glspl{LWR} deployed at one time is 109. The energy produced by the
\glspl{LWR} follows the number of reactors deployed. The maximum energy
produced by the \glspl{LWR} is 102.46 GWe-y, and they produce 91.82 GWe-y
in 2025.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{figures/energy_scenario1.png}
    \caption{Energy supplied by \glspl{LWR} compared to the number of
    \glspl{LWR} deployed in Scenario 1.}
    \label{fig:energy_rx_1}
\end{figure}

Next, Figure \ref{fig:fuel_1} shows the mass of enriched uranium sent to
the \glspl{LWR} at each time step. The \glspl{LWR} are
defined to have an 18 month cycle length, with a third of the uranium
mass replaced at each refueling outage. However, when a new reactor
is deployed it is deployed with an entire core of uranium which leads
to the large increases in the mass of uranium sent to the reactors, such
as in 1983 and 2016. An average of 96.2 MTU/month and a maximum of 513.7 MTU
are sent to the \glspl{LWR}. Prior to 2025, the average mass
of enriched uranium sent to the \glspl{LWR} is 157.6 MTU/month.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{figures/fuelsupply_scenarios_1.png}
    \caption{Mass of enriched uranium sent to reactors in Scenario 1.}
    \label{fig:fuel_1}
\end{figure}

Finally, the \gls{SWU} capacity to produce fuel for the scenario is shown in
Figure \ref{fig:swu_1}. The \gls{SWU} capacity required as a function of
time follows the mass of uranium sent to the reactors, since the \gls{SWU}
is calculated based on those transactions. An average of 0.74$\times 10^6$
kg-\gls{SWU}/month and a maximum of 3.95$\times 10^6$ kg-\gls{SWU} are
required for this scenario. Prior to 2025, an average of 1.21$\times 10^6$
kg-\gls{SWU}/month is required to enrich the uranium sent to the \glspl{LWR}.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{figures/totalswu_scenarios_1.png}
    \caption{Total \gls{SWU} capacity required to produce fuel sent to the
    reactors at each time step in Scenario 1.}
    \label{fig:swu_1}
\end{figure}

The deployment and decommissioning schedule of the \glspl{LWR} in this
scenario is
applied to each of the other scenarios, and therefore the material
requirements of each of the following scenarios are the same as those
of this scenario prior to 2025.

\subsection{Scenario 2}
The energy produced by each type of reactor, the transition energy demand,
and the deployment schedule of the \glspl{MMR} for Scenario 2 are shown in
Figure \ref{fig:energy_rx_2}. The first \glspl{MMR} are deployed in
October 2031, and the
maximum number deployed at one time is
9,182.

Once the transition begins in 2025, there are some time periods in which
the energy demand of the scenario is
not met. The first of these is between 2030-2050, with a maximum deficit
of 5.7855 GWe-y in 2032. Other periods in which the energy demand is not met
correspond to the decommissioning of \glspl{MMR} at the end of their lifetime
and the deployment of new reactors, such as from 2062-2069.
The deployment of \glspl{MMR} in 2031 contributes to the inability to
meet the energy demand of the scenario between 2030-2050, because the energy
produced by
\glspl{LWR} decreases to 89.35 GWe-y in 2030, despite a demand of 91.82
GWe-y, before the \glspl{MMR} are
deployed. Therefore, the reactors are deployed in a reactionary fashion to
past energy production, as opposed to in response to forecasted energy
production.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{figures/energy_scenario2.png}
    \caption{Energy supplied by each type of reactor compared to the number of
    \glspl{MMR} deployed in Scenario 2.}
    \label{fig:energy_rx_2}
\end{figure}

Next, Figure \ref{fig:fuel_2} shows the mass of enriched uranium sent to all
reactors in the scenario and just to the \glspl{MMR}. The \glspl{MMR}
do not require refueling during their lifetime, so uranium is only
sent to these reactors when they are deployed. This causes the
large increases in uranium sent to these reactors and the periods of
no uranium sent to them. An average of 104.94 MTU/month and a maximum
of 781.41 MTU are sent to all of the reactors in the scenario after 2025.
This average is lower than the average mass of enriched uranium
sent to the \glspl{LWR} in Scenario 1, but the maximum exceeds the maximum
mass of enriched uranium sent to \glspl{LWR} in Scenario 1 by 267.71 MT.
An average of 48.50 MTU/month and a maximum of 719.25
MTU are sent to the \glspl{MMR} once they are deployed in 2031. The average
mass of enriched uranium sent to the
\glspl{MMR} is lower than the average mass of uranium sent to the \glspl{LWR}
prior to 2025. This is despite a much higher maximum and multiple peaks higher
than
average mass sent to the \glspl{LWR}. The lower average is explained by the
multiple time periods in
which no \gls{HALEU} is sent to the \glspl{MMR} due to the lack of refueling.

\begin{figure}
    \centering
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/fuelsupply_scenarios_2.png}
        \caption{Mass of enriched uranium sent to all reactors.}
        \label{fig:totalfuel_2}
    \end{subfigure}
    \hspace{0.8cm}
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/advancedRX_fuelsupply_scenarios_2.png}
        \caption{Mass of enriched uranium sent to \glspl{MMR}.}
        \label{fig:haleu_2}
    \end{subfigure}
    \caption{Uranium mass sent to reactors in Scenario 2.}
    \label{fig:fuel_2}
\end{figure}

Figure \ref{fig:swu_2} shows the \gls{SWU} capacity needed to
enrich uranium for all reactors in the scenario, and to enrich uranium for
just the \glspl{MMR}. There is a large difference in the \gls{SWU}
capacity needed to enrich uranium for \glspl{LWR} and \glspl{MMR}. This
is because the \glspl{MMR} require a much larger mass and
a higher enrichment level. An average of 1.37$\times 10^6$ kg-\gls{SWU}/month
and a maximum of 20.3$\times 10^6$ kg-\gls{SWU}
are needed to enrich the uranium sent to \glspl{MMR}. These values are both
higher than the
average and maximum \gls{SWU} capacity needed to enrich fuel for the
\glspl{LWR} in Scenario 1.

\begin{figure}
    \centering
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/totalswu_scenarios_2.png}
        \caption{Total \gls{SWU} required to enrich uranium sent to all reactors at each time step.}
        \label{fig:totalswu_2}
    \end{subfigure}
    \hspace{0.8cm}
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/haleuSWU_scenarios_2.png}
        \caption{\gls{SWU} required to enrich uranium sent to \glspl{MMR} at each time step.}
        \label{fig:haleuswu_2}
    \end{subfigure}
    \caption{\gls{SWU} required to enrich natural uranium in Scenario 2.}
    \label{fig:swu_2}
\end{figure}

\subsection{Scenario 3}
For Scenario 3, Figure \ref{fig:energy_rx_3} shows the number of Xe-100 reactors deployed,
the energy produced by each reactor type, and the energy demand.
Xe-100 reactors are deployed in October 2031, which is the same as when
\glspl{MMR}
are deployed in Scenario 2. Scenario 3 deploys a maximum of 1,225 Xe-100
reactors.

The same deficit between the energy produced and demand between
2030-2050 that was observed in Scenario 2 is present in the energy produced
in this scenario. However, there are no significant
differences (more than 1 GWe-y for this work) between the energy
produced and demand of the scenario
after 2050 because the Xe-100 reactors have a longer lifetime, and there are
no decreases in energy due to decommissioning of reactors. After 2050, the
maximum difference between the energy produced and demand is 0.057 GWe-y.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{figures/energy_scenario3.png}
    \caption{Energy supplied by each type of reactor compared to the number of
    Xe-100s deployed in Scenario 3.}
    \label{fig:energy_rx_3}
\end{figure}

Comparing the mass of uranium sent to all of the reactors and just the Xe-100
reactors, shown in Figure \ref{fig:fuel_3}, the Xe-100 reactors
require significantly less fuel than what is sent to the \glspl{LWR},
despite there being more Xe-100 reactors than \glspl{LWR}. Uranium
is sent to the Xe-100 reactors every six months in the simulations,
while it is sent to the \glspl{LWR} every 18 months. An average of
74.98 MTU/month and a maximum of 342.58 MTU are sent to all of the reactors
in this scenario starting in 2025. Both values are much
lower than in Scenario 1. An average of 23.81
MTU/month and a maximum of 105.67 MTU are sent to the Xe-100 reactors in
this scenario. Each of these numbers are lower than the
average and maximum masses of enriched uranium
in Scenario 1, and the \gls{HALEU} mass sent to the \glspl{MMR} in
Scenario 2.

\begin{figure}
    \centering
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/fuelsupply_scenarios_3.png}
        \caption{Mass of enriched uranium sent to all reactors.}
        \label{fig:totalfuel_3}
    \end{subfigure}
    \hspace{0.8cm}
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/advancedRX_fuelsupply_scenarios_3.png}
        \caption{Mass of enriched uranium sent to Xe-100s.}
        \label{fig:haleu_3}
    \end{subfigure}
    \caption{Uranium mass sent to reactors in Scenario 3.}
    \label{fig:fuel_3}
\end{figure}

Figure \ref{fig:swu_3} shows the \gls{SWU} capacity needed to enrich
uranium for all of the reactors in the scenario, and for the uranium sent
to just the Xe-100 reactors. There is no substantial change in the
\gls{SWU} capacity needed to enrich uranium between Scenarios 1 and 3,
despite the higher enrichment level of uranium
sent to the Xe-100 reactors. This is because the Xe-100 reactors receive
a lower average mass of enriched uranium at each time step. Enriching uranium
for the Xe-100 reactors requires an average of
0.82$\times 10^6$ kg-\gls{SWU}/month and a maximum of 3.64$\times 10^6$
kg-\gls{SWU}.

\begin{figure}
    \centering
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/totalswu_scenarios_3.png}
        \caption{\gls{SWU} required to enrich uranium sent to all reactors.}
        \label{fig:totalswu_3}
    \end{subfigure}
    \hspace{0.8cm}
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/haleuSWU_scenarios_3.png}
        \caption{\gls{SWU} required to enrich uranium sent to Xe-100s.}
        \label{fig:haleuswu_3}
    \end{subfigure}
    \caption{\gls{SWU} required to enrich natural uranium in Scenario 3.}
    \label{fig:swu_3}
\end{figure}

\subsection{Scenario 4}
Figure \ref{fig:energy_rx_4} shows the number of \glspl{MMR} deployed, the
energy produced by each type of reactor, and the energy demand of this
scenario with 1\% growth. \glspl{MMR} are deployed in January 2030, and the maximum
number of \glspl{MMR} is 17,496.

There is a deficit between the energy produced and
demand from 2026-2046, with a maximum difference of 4.51 GWe-y in 2032.
There are no other times in which the energy demand is not met by a
significant amount (more than 1 GWe-y), including when the \glspl{MMR} are
decommissioned. After 2047 electricity is generally produced in surplus,
up to 1.64 GWe-y.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{figures/energy_scenario4.png}
    \caption{Energy supplied by each type of reactor compared to the number of
    \glspl{MMR} deployed in Scenario 4.}
    \label{fig:energy_rx_4}
\end{figure}

Figure \ref{fig:fuel_4} shows the mass of enriched uranium sent to all the
reactors in the scenario, and to just the \glspl{MMR}
in the scenario. An average of 144.36 MTU/month and a maximum of 796.71 MTU
are sent to all the reactors starting in 2025 in this scenario. An average of
75.62 MTU/month and a maximum of 755.60 MTU are sent to just the \glspl{MMR}
in the scenario once they are deployed. The mass of \gls{HALEU}
required by the \glspl{MMR} in this scenario is higher than
in Scenario 2 due to the higher energy
demand and the additional \glspl{MMR} deployed. The
average mass of \gls{HALEU} sent to the \glspl{MMR} in this scenario is
slightly lower than the average mass sent to the \glspl{LWR} in Scenario 1
due to the multiple time steps in which no uranium is sent to the \glspl{MMR}.

\begin{figure}
    \centering
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/fuelsupply_scenarios_4.png}
        \caption{Mass of enriched uranium sent to all reactors.}
        \label{fig:totalfuel_4}
    \end{subfigure}
    \hspace{0.8cm}
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/advancedRX_fuelsupply_scenarios_4.png}
        \caption{Mass of enriched uranium sent to \glspl{MMR}.}
        \label{fig:haleu_4}
    \end{subfigure}
    \caption{Uranium mass sent to reactors in Scenario 4.}
    \label{fig:fuel_4}
\end{figure}

Figure \ref{fig:swu_4} shows the \gls{SWU} capacity required to
enrich uranium for all the reactors in the scenario, and
only for the \glspl{MMR}. For the same reasons described for
Scenario 2, the \gls{SWU} capacity required to enrich uranium
for the \glspl{MMR} is much higher than the capacity needed to
enrich uranium for the \glspl{LWR}. An average of 2.19$\times 10^6$
kg-\gls{SWU}/month and a maximum of 21.5$\times 10^6$ kg-\gls{SWU}
are required to enrich the uranium that is sent to \glspl{MMR}. These values
are higher than the \gls{SWU}
capacity required to enrich uranium for the \glspl{MMR} in
Scenario 2. The average is slightly higher than the average \gls{SWU}
capacity required to enrich uranium for the \glspl{LWR} before 2025 in
Scenario 1.

\begin{figure}
    \centering
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/totalswu_scenarios_4.png}
        \caption{\gls{SWU} required to enrich uranium sent to all reactors.}
        \label{fig:totalswu_4}
    \end{subfigure}
    \hspace{0.8cm}
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/haleuSWU_scenarios_4.png}
        \caption{\gls{SWU} required to enrich uranium sent to \glspl{MMR}.}
        \label{fig:haleuswu_4}
    \end{subfigure}
    \caption{\gls{SWU} required to enrich natural uranium in Scenario 4.}
    \label{fig:swu_4}
\end{figure}

\subsection{Scenario 5}
Figure \ref{fig:energy_rx_5} shows the number of Xe-100 reactors, the
energy produced by each type of reactor, and the energy demand of the
scenario. The Xe-100 reactors are deployed in January 2030, the
same as when
\glspl{MMR} are deployed in Scenario 4. The maximum number of Xe-100
reactors deployed in the scenario is 2,339.

There is a small deficit in the energy produced and demand from
2026-2046, the same as what is observed in Scenario 4. After this
initial difference, there are no further significant (more than
1 GWe-y) differences between
the energy produced and demand. For most years after 2046 there
is a surplus of energy, up to 1.64 GWe-y.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{figures/energy_scenario5.png}
    \caption{Energy supplied by each type of reactor compared to the number of
    Xe-100s deployed in Scenario 5.}
    \label{fig:energy_rx_5}
\end{figure}

Figure \ref{fig:fuel_5} shows the mass of enriched uranium sent the all
the reactors in the scenario and the \gls{HALEU} mass sent just to the
Xe-100 reactors. There is a clear increase in the mass of \gls{HALEU} sent
to the Xe-100 reactors as time goes on, but this mass is still
low compared to the mass of enriched uranium sent to the \glspl{LWR} in
the scenario. An average of 95.46 MTU/month and a maximum of 347.28 MTU
are sent to all of the reactors in this scenario after 2025. These
values are both lower than the uranium mass sent to the
\glspl{LWR} in Scenario 1. An average of
38.11 MTU/month and a maximum of 117.98 MTU are sent to the Xe-100 reactors.
These metrics are all much lower than what is observed
for fueling the \glspl{MMR} in Scenario 4.

\begin{figure}
    \centering
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/fuelsupply_scenarios_5.png}
        \caption{Mass of enriche uranium sent to all reactors.}
        \label{fig:totalfuel_5}
    \end{subfigure}
    \hspace{0.8cm}
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/advancedRX_fuelsupply_scenarios_5.png}
        \caption{Mass of enriched uranium sent to Xe-100s.}
        \label{fig:haleu_5}
    \end{subfigure}
    \caption{Uranium mass sent to reactors in Scenario 5.}
    \label{fig:fuel_5}
\end{figure}

Finally, Figure \ref{fig:swu_5} shows the \gls{SWU} capacity required
to enrich the uranium sent to all of the reactors, and just the Xe-100
reactors. The \gls{SWU} capacity needed
to enrich uranium for the Xe-100 reactors starts out at a similar
amount as the \glspl{LWR}, but
increases as the energy demand and the number of reactors increases.
The \gls{SWU} capacity required to enrich uranium
for the Xe-100 reactors becomes higher than the capacity
required to enrich uranium for \glspl{LWR}, despite the Xe-100
reactors requiring a lower mass of uranium. This is due to the higher
enrichment level of the fuel for the Xe-100 reactors. An average of
1.36$\times 10^6$ kg-\gls{SWU}/month and a maximum of
4.11$\times 10^6$ kg-\gls{SWU} are required to enrich the uranium sent
to the Xe-100 reactors. The average \gls{SWU} capacity
required to enrich uranium for the Xe-100 reactors is a little lower
than the average capacity needed to enrich uranium for the \glspl{MMR}
in Scenario 4, but the maximum capacity required for this scenario is much
lower than in Scenario 4. These values are slightly higher
than what is observed to enrich uranium for the \glspl{LWR} in Scenario 1
prior to 2025.

\begin{figure}
    \centering
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/totalswu_scenarios_5.png}
        \caption{Total \gls{SWU} required to enrich uranium sent to all reactors at each time step.}
        \label{fig:totalswu_5}
    \end{subfigure}
    \hspace{0.8cm}
    \begin{subfigure}{0.5\textwidth}
        \centering
        \includegraphics[scale=0.5]{figures/haleuSWU_scenarios_5.png}
        \caption{\gls{SWU} required to enrich uranium sent to Xe-100s at each time step.}
        \label{fig:haleuswu_5}
    \end{subfigure}
    \caption{\gls{SWU} required to enrich natural uranium in Scenario 5.}
    \label{fig:swu_5}
\end{figure}

ext: tex


text:
\documentclass[a4paper, 10pt]{article}
\usepackage{graphicx}
\usepackage{caption}
\usepackage{subcaption}
\usepackage[hidelinks]{hyperref}

\graphicspath{ {./img/} }
\begin{document}

\title{\bf ISODS: An ISODATA-Based Initial Centroid Algorithm}
\author{
      Noah Gardner$^1$,
      Sahar Yarmohammadtoosky$^2$, \\
      Anthony Phan$^3$,
      and Chih-Cheng Hung$^4$\\[1ex]
      \sl \small Kennesaw State University, Marietta, GA, USA$^{1, 3, 4}$\\
      \sl \small Iran University of Science and Technology, Tehran,
      Iran$^2$\\[1ex]
      \small \href{mailto:ngardn10@students.kennesaw.edu}{$^1$ngardn10@students.kennesaw.edu},
      \small \href{mailto:yarmohamadishr@gmail.com}{$^2$yarmohamadishr@gmail.com}, \\
      \small \href{mailto:aphan5@students.kennesaw.edu}{$^3$aphan5@students.kennesaw.edu},
      \small \href{mailto:chung1@kennesaw.edu}{$^4$chung1@kennesaw.edu} \\
}
\date{}
\maketitle

\begin{abstract}
      \noindent
      Iterative Self-Organizing Data Analysis Technique (ISODATA) is a
      clustering algorithm based on the K-Means algorithm that intends to find
      the correct number of clusters through merging and splitting. It is
      difficult to determine the correct number of clusters in many real-world
      applications, in particular, remote sensing image interpretation. Compared
      to K-Means, the ISODATA algorithm should be resilient to initial cluster
      centers. However, ISODATA is also sensitive to initial cluster centers
      which impacts the performance of the algorithm. This work proposes using
      the ISODATA approach of splitting clusters to select quality initial
      cluster centers, called Iterative Self-Organizing Data Split (ISODS). The
      proposed approach is evaluated on several benchmark UCI datasets as well
      as the Salinas-A hyperspectral image dataset. The results are evaluated
      using standard clustering evaluation parameters. When ISODS is used
      together with ISODATA, the performance is improved and the generated
      clusters more closely resemble the expected number of clusters.\\
      {\bf Keywords :} Clustering, ISODATA, Initial Centroids, Remote Sensing
\end{abstract}

% =========================================================================== %

\section{Introduction}

Remote sensing technology is used when the data collection takes place at a
remote distance from the subject matter, such as the study of daily weather and
long-term climate change, land cover monitoring, ecosystem modeling of
vegetation, and so on \cite{Jensen1996,Mallapragada2018}. Multispectral imaging
technology is one of the instruments for remote sensing used since the late
1960s \cite{Khan2018,Landgrebe1999}. However, multispectral imaging systems use
a low number of spectral bands (three to six spectral bands in different
systems). Increasing spectral resolution in the hyperspectral images (HSI) with
several hundred spectral bands of the observational scene in a single
acquisition makes it possible to acquire detailed examination of observational
scenes \cite{Khan2018}. The ground-based version of hyperspectral images has
gained immense interest. It has been applied to numerous electronic imaging
applications, including food inspection, forensic science, medical surgery and
diagnosis, and military applications
\cite{Govender2009,Adam2009,Fischer2006,Carrasco2003}. Because of the difficulty
of obtaining prior knowledge in remote sensing studies, unsupervised
classification and cluster analysis have great importance compared to supervised
classification \cite{Abbas2016}.

Unsupervised learning is a machine learning technique used when the learning
algorithm has no pre-existing labels and looks for patterns in the data set with
minimum human supervision and without external guidance. In contrast to
supervised learning, where a training set of inputs and observations are given
to the method, unsupervised learning uses only the inputs and should find an
interesting pattern in the data set. Two main approaches are used in
unsupervised learning, named principal component \cite{Mallapragada2018} and
cluster analysis \cite{Xu2015}. In the clustering method, unlabeled and unsorted
data points are grouped and segmented according to similarities and differences
to find structure in the inputs, group it into meaningful clusters, and
understand the data set better. There are different algorithms for Clustering,
such as K-Means \cite{Lloyd1982,Macqueen1967}, fuzzy C-Means, ISODATA
\cite{Ball1965}, and so on. In the K-Means algorithm, the goal is to minimize
the sum of the squared distances points have in the data set from its center.
The problem with the K-Means algorithm is that it requires a priori knowledge
about the number of classes in the data set, and choosing the cluster centers
randomly may not result in optimal partitioning of the data set.

One of the most popular clustering algorithms is called ISODATA clustering,
based on the K-means algorithm with some variations. The ISODATA method's
preference in comparison with the K-means algorithm is that the ISODATA
algorithm uses splitting and merging forms for clusters. The ISODATA method
allows starting from arbitrary cluster centers and obtains better partitioning
results than the K-Means algorithm, but it is not entirely resilient to initial
cluster centers. In this paper, a novel method using the ISODATA approach of
splitting clusters to select quality initial cluster centers is proposed.

The paper is organized as follows: Section 2 introduces K-Means and ISODATA
clustering algorithms. The new proposed algorithm is explained in Section 3. We
present the experimental results in Section 4. Finally, Section 5 contains
concluding remarks.

% =========================================================================== %

\section{Background}

\subsection{K-Means Algorithm}

The K-Means algorithm is an unsupervised clustering algorithm and aims to
partition $N$ observations into $K$ clusters \cite{Lloyd1982,Macqueen1967}. Data
points in the same cluster are very similar to each other and have less
variation compared to data points assigned to other clusters. The user defines
$K$, the number of expected clusters. Following is a description of the K-Means
algorithm:

\begin{itemize}
      \item \textbf{Step 1:} Choose $K$ cluster centers from the data set.
      \item \textbf{Step 2:} Assign each element of the data set to the cluster
            based on the minimum distance from the cluster center.
      \item \textbf{Step 3:} Re-calculate new cluster centers for each cluster
            after the assignment in \textbf{Step 2} by computing the mean of all
            patterns assigned to each cluster.
      \item \textbf{Step 4:} Repeat \textbf{Step 2 and Step 3} until clustering
            convergence occurs, or the set number of iterations is exceeded.
\end{itemize}

For selecting the initial cluster centers in \textbf{Step 1}, there are two
standard methods: Random Centers, and K-Means++ \cite{Arthur2007}. In Random
Centers, $K$ samples are randomly chosen from the dataset to serve as the
initial cluster centers. In K-Means++, $1$ sample is randomly chosen from the
dataset to serve as the initial cluster center. Then, $K-1$ samples are
iteratively selected, where the probability of a sample being chosen as a
cluster center is the distance of the sample from previously chosen centers
\cite{Arthur2007}. The K-Means algorithm is susceptible to initial starting
cluster centers, where different initial values can result in a different
classification \cite{Franti2018}.

\subsection{ISODATA Algorithm}

ISODATA is another frequently used clustering algorithm and is based on the
K-Means algorithm \cite{Ball1965}. Like K-Means, cluster centers are iteratively updated and
reassigned to the samples in the dataset. However, ISODATA determines the number
of clusters dynamically and allows for a different number of cluster centers. In
contrast, the K-Means algorithm requires a fixed user-set parameter of the
number of expected clusters. The advantage found within ISODATA allows for
discovery of a different number of clusters, which is achieved through
discarding clusters with a small number of member samples, and merging and
splitting clusters that meet certain parameters. There are three main ISODATA
parameters:

\begin{itemize}
      \item $\theta n$ - minimum number of cluster members, which determines
            whether the cluster is discarded,
      \item $\theta e$ - maximum standard deviation allowed for a cluster, which
            determines whether the cluster is split,
      \item $\theta c$ - minimum distance between clusters, which determines
            whether two clusters are merged.
\end{itemize}

Additionally, the user must set $K$, the maximum number of clusters to generate.
The ISODATA algorithm is briefly explained below:

\begin{itemize}
      \item \textbf{Step 1:} Choose $K$ cluster centers from the data set.
      \item \textbf{Step 2:} Assign each element of the dataset to the cluster
            based on the minimum distance from the cluster center.
      \item \textbf{Step 3:} Re-calculate new cluster centers for each cluster
            by computing the mean of all patterns assigned to each cluster.
      \item \textbf{Step 4:} Discard clusters that have fewer than ($\theta n$)
            members. If any clusters were discarded, move to \textbf{Step 7}.
      \item \textbf{Step 5:} If this iteration is an odd numbered iteration or
            the number of clusters is less than or equal to  $1/2 * K$, then
            split clusters which have a standard deviation in a dimension
            greater than ($\theta e$). Move to \textbf{Step 7}.
      \item \textbf{Step 6:} If this iteration is an even numbered iteration or
            the number of clusters is greater than or equal to $2 * K$, then
            merge clusters which have an intercluster distance less than
            ($\theta c$).
      \item \textbf{Step 7:} If this is not the last iteration, start the
            process again at step 2. If this is the last iteration or clustering
            convergence occurs, then end the process.
\end{itemize}

There are some disadvantages to the ISODATA algorithm. First, it is sensitive to
the parameters $\theta n$, $\theta e$, and $\theta c$. These parameters should
be fine-tuned for optimal performance based on the attributes of the dataset
being tested. In addition, similar to K-Means, it is sensitive to initial
cluster centers, although less so than K-Means due to the extra processes of
discarding, splitting, and merging. The initial centroids can affect both the
quality of clusters produced and the number of iterations required to converge
\cite{Franti2018}.

% =========================================================================== %

\section{Proposed Method}

The proposed clustering algorithm's objective is to improve the result we obtain
from the clustering algorithms by targeting the initial cluster centers.
Inspired by the ISODATA algorithm, Iterative Self-Organizing Data Split (ISODS)
uses the splitting process from ISODATA to create the required initial cluster
centers to use for a clustering algorithm. The inputs for the algorithm are the
number of cluster centers $K$ as an integer, and the data to generate centers
from.

The algorithm starts by selecting $1$ sample as an initial cluster center. In
the next step, each data is assigned to the cluster and the mean of the cluster
is updated. Then, by the ISODATA splitting process, each cluster is split along
the dimension with the highest standard deviation. The assignments for the new
clusters are calculated based on distance and the centers are updated. This
process is continued until either the number of clusters is greater than or
equal to $K$. If the number of clusters is greater than $K$, the clusters with
the fewest members are discarded one at a time, and the assignments and centers
are updated after each rejection. The algorithm is outlined below:

\begin{itemize}
      \item \textbf{Step 1:} Choose one observation as a cluster center from the
            dataset.
      \item \textbf{Step 2:} Assign each element of the dataset to the cluster
            based on the minimum distance from the cluster center.
      \item \textbf{Step 3:} Re-calculate new cluster centers for each cluster
            by computing the mean of all patterns assigned to each cluster.
      \item \textbf{Step 4:} If the number of clusters is equal to $K$, then the
            process is terminated. If the number of clusters is greater than
            $K$, then the clusters with the fewest members is discarded, and the
            algorithm moves to \textbf{Step 2}.
      \item \textbf{Step 5:} Calculate the standard deviation of each cluster in
            each dimension. Then, split each cluster along the dimension with
            the greatest standard deviation and move to \textbf{Step 2}.
\end{itemize}

This method began as a modified ISODATA process. However, the modification was
unique in that there is a clear distinction from the proposed change and the
beginning of the original process. Although more complex than other methods of
generating initial cluster centers, this approach guarantees separation between
clusters. As a result, ISODS can be used to generate clusters for other
clustering algorithms, such as K-Means.

% =========================================================================== %

\section{Results}

Due to the response of the ISODATA algorithm to the discarding, merging, and
splitting parameters, these parameters should be carefully selected. Since the
goal of this research is to explore the impact of initial cluster centers on
ISODATA, we use a relatively simple approach for generating ISODATA parameters.
Based on the available training data with included labels, each class is
summarized with the mean and standard deviation of that class, as well as the
number of members in each class. Typically, since we are working with
unsupervised learning algorithms, class information may not be available.

To determine $\theta n$, the minimum number of cluster members used for
discarding, we find the class with the fewest members and use the number of
members in that class as $\theta n$. For $\theta e$, the maximum standard
deviation allowed for a cluster, we follow a similar approach and find the
maximum standard deviation within a class. Finally, for $\theta c$, we calculate
the distance between the means of each class, and use the minimum distance as
$\theta c$.

For the first experimental results shown in Table \ref{tab:ucitable}, we explore
applying ISODS to ISODATA and K-Means by using benchmark UCI \textit{Iris,}
\textit{Wine,} and \textit{Glass} datasets \cite{Dua2019}. We also show results
from applying Random Centers and K-Means++ Centers. For the \textit{Glass}
dataset, the data was Z-Score normalized (based on Z-Score of the training data)
because ISODATA can have problems converging on the raw dataset. We used $80\%$
of the data for training and $20\%$ of the data for testing to evaluate the
models. Following standard practice, the models were trained on the training
data, then fit to the test data. For the ISODATA parameters, we used the
approach of summarizing the class data based on the information in the training
data. The output labels (cluster assignments) were compared to the ground truth
labels contained in the test data using pair-counting measures.

To evaluate the cluster results, external cluster validation measures are used.
We utilize pair-counting measures which rely on counting matching pairs between
two sets of data to determine agreement and disagreement \cite{Rezaei2016}. The
pair-counting measures applied are \textit{Rand index} (RI - agreement
probability), \textit{adjusted Rand index} (ARI), \textit{Mirkin's index} (MI -
disagreement probability), and \textit{Hubert's index} (HI - $(P(agree)$ -
$P(disagree))$) \cite{Hubert1985, Meila2003}. When comparing ground truth labels
to cluster assignments, RI can serve as an estimation of the accuracy. ARI is
similar to the RI, though it is corrected for chance and thus does not count
randomly generated labels \cite{Rand1971}.

The ISODATA algorithm does not have a determined number of clusters to generate.
Instead, the user sets the max clusters desired, and the algorithm will generate
a number of clusters between and the max and half the max number desired. Since
the output depends on the data, the ISODATA parameters, and the initial centers,
we include a column to depict the average number of clusters.

\begin{table}[ht]
      \caption{UCI benchmark dataset results averaged over 1000 iterations of
            ISODATA comparing three different initial cluster center algorithms
            (\textit{Init.}). ARI - adjusted Rand index; RI - Rand index; MI -
            Mirkin's index; HI - Hubert's index.}
      \begin{tabular}{ |p{1.25cm}||p{1.25cm} p{1.25cm} p{1.25cm} p{1.25cm} p{1.25cm} p{1.25cm}|}
            \hline
            \multicolumn{7}{|c|}{\textbf{ISODATA}}                               \\
            \hline
            Init. & Iter.  & Clusters & ARI             & RI    & MI    & HI     \\
            \hline
            \multicolumn{7}{|c|}{\textit{Iris}}                                  \\
            \hline
            ISODS & 13.46  & 2.599    & \textbf{0.646}  & 0.826 & 0.173 & 0.653  \\
            KM++  & 15.43  & 2.544    & 0.628           & 0.816 & 0.184 & 0.631  \\
            Rand  & 11.90  & 2.41     & 0.607           & 0.802 & 0.198 & 0.604  \\
            \hline
            \multicolumn{7}{|c|}{\textit{Glass (Normalized)}}                    \\
            \hline
            ISODS & 3.19   & 1.066    & \textbf{0.0018} & 0.266 & 0.734 & -0.468 \\
            KM++  & 4.02   & 1.028    & 0.0011          & 0.265 & 0.735 & -0.470 \\
            Rand  & 3.69   & 1.055    & 0.0013          & 0.266 & 0.733 & -0.467 \\
            \hline
            \multicolumn{7}{|c|}{\textit{Wine}}                                  \\
            \hline
            ISODS & 9.16   & 1.836    & \textbf{0.313}  & 0.618 & 0.382 & 0.236  \\
            KM++  & 8.76   & 1.816    & 0.307           & 0.612 & 0.388 & 0.224  \\
            Rand  & 9.26   & 1.810    & 0.304           & 0.608 & 0.391 & 0.217  \\
            \hline
            \multicolumn{7}{|c|}{\textbf{K-Means}}                               \\
            \hline
            \multicolumn{7}{|c|}{\textit{Iris}}                                  \\
            \hline
            ISODS & 2.29   & 3        & \textbf{0.623}  & 0.833 & 0.167 & 0.667  \\
            KM++  & 4.149  & 3        & 0.584           & 0.811 & 0.189 & 0.621  \\
            Rand  & 29.402 & 3        & 0.569           & 0.797 & 0.203 & 0.593  \\
            \hline
            \multicolumn{7}{|c|}{\textit{Glass (Normalized)}}                    \\
            \hline
            ISODS & 5.651  & 6        & 0.164           & 0.665 & 0.335 & 0.329  \\
            KM++  & 4.866  & 6        & \textbf{0.173}  & 0.643 & 0.357 & 0.285  \\
            Rand  & 77.178 & 6        & 0.151           & 0.634 & 0.366 & 0.268  \\
            \hline
            \multicolumn{7}{|c|}{\textit{Wine}}                                  \\
            \hline
            ISODS & 3.145  & 3        & \textbf{0.882}  & 0.947 & 0.053 & 0.895  \\
            KM++  & 5.388  & 3        & 0.839           & 0.926 & 0.074 & 0.851  \\
            Rand  & 24.775 & 3        & 0.834           & 0.920 & 0.080 & 0.840  \\
            \hline
      \end{tabular}
      \label{tab:ucitable}
\end{table}

Following the results on the benchmark datasets, we applied ISODATA to the
Salinas-A hyperspectral image dataset \cite{Salinas}. Salinas-A is a small
subscene of the Salinas image. It has a size of $86$ pixels by $83$ pixels, each
pixel contains $224$ bands, and each pixel is assigned to $1$ of $6$ classes,
disregarding the background. Due to the large number of dimensions in this
dataset, prinicipal component analysis (PCA) is used to extract features
\cite{Rodarmel2002}. In this experimental setup, we use $3$ prinicipal
components as the input to the model. We utilize the same 80/20 test-train split
as the previous experiment and Z-Score Normalize the data. Additionally, the
same ISODATA parameter selection approach is employed. Then, we run ISODATA for
$1000$ iterations using ISODS, K-Means++ Centers, and Random Centers. We save
the best performing model (based on test data performance) for each initial
cluster center approach and fit each pixel to the original dataset to obtain the
pictures shown in Figure \ref{fig:salinas}. Finally, the external cluster
validation measures are shown in Table \ref{tab:salinastable}.

\begin{table}[ht]
      \caption{Salinas-A dataset average and best results over 1000 iterations
            of ISODATA comparing three different initial cluster center
            algorithms (\textit{Init.}). ARI - adjusted Rand index; RI - Rand
            index; MI - Mirkin's index; HI - Hubert's index.}
      \begin{tabular}{ |p{1.25cm}||p{1.25cm} p{1.25cm} p{1.25cm} p{1.25cm} p{1.25cm} p{1.25cm}|}
            \hline
            \multicolumn{7}{|c|}{\textbf{ISODATA} - \textit{Salinas-A}}        \\
            \hline
            Init. & Iter. & Clusters & ARI            & RI    & MI    & HI     \\
            \hline
            \multicolumn{7}{|c|}{\textit{Average}}                             \\
            \hline
            ISODS & 2.772 & 2.42     & \textbf{0.156} & 0.388 & 0.612 & -0.224 \\
            KM++  & 3.177 & 1.097    & 0.019          & 0.228 & 0.772 & -0.544 \\
            Rand  & 3.693 & 1.584    & 0.084          & 0.308 & 0.691 & -0.383 \\
            \hline
            \multicolumn{7}{|c|}{\textit{Best}}                                \\
            \hline
            ISODS & 2     & 6        & 0.758          & 0.921 & 0.078 & 0.842  \\
            KM++  & 8     & 6        & 0.781          & 0.930 & 0.070 & 0.860  \\
            Rand  & 14    & 6        & \textbf{0.798} & 0.933 & 0.066 & 0.867  \\
            \hline
      \end{tabular}
      \label{tab:salinastable}
\end{table}

\begin{figure}[ht]
      \centering
      \begin{subfigure}[b]{0.24\textwidth}
            \centering
            \includegraphics[width=\linewidth]{salinasgt.png}
            \caption{}
            \label{fig:salinasgt}
      \end{subfigure}
      \begin{subfigure}[b]{0.24\textwidth}
            \centering
            \includegraphics[width=\linewidth]{risodata.png}
            \caption{}
            \label{fig:salinasr}
      \end{subfigure}
      \centering
      \begin{subfigure}[b]{0.24\textwidth}
            \centering
            \includegraphics[width=\linewidth]{kisodata.png}
            \caption{}
            \label{fig:salinask}
      \end{subfigure}
      \begin{subfigure}[b]{0.24\textwidth}
            \centering
            \includegraphics[width=\linewidth]{iisodata.png}
            \caption{}
            \label{fig:salinasi}
      \end{subfigure}\hfill \caption{Heatmap showing cluster assignments of each
            pixel using ISODATA with different initial cluster center methods.
            (a) Salinas-A Ground Truth; (b) Random Centers (ARI=0.798); KM++
            Centers (ARI=0.781); ISODS Centers (ARI=0.758).}
      \label{fig:salinas}
\end{figure}

% =========================================================================== %

\section{Conclusion}
This study presents a method for generating initial cluster centers based on a
splitting approach, Iterative Self-Organizing Data Split (ISODS). We tested ISODS on
bechmark datasets as well as the Salinas-A hyperspectral image dataset. Both
tests on average shows the performance of ISODS using external cluster
validation measures, especially on the \textit{adjusted Rand index} measure.
Additonally, ISODS may decrease the number of iterations required for a
clustering algorithm such as ISODATA or K-Means to converge. However, due to the
complexity of the ISODS approach compared to other initial centroid algorithms,
specifically calculating the means of the clusters, the context should be
considered when applying this method. When used in conjunction with ISODATA,
ISODS shows an average number of clusters generated closer to the expected
output.

% =========================================================================== %

\subsection*{Acknowledgements}
This research was supported by the College of Computing and Software Engineering
at Kennesaw State University. We also want to thank our Laboratory for Machine
Vision and Security Research for providing us resources and mentoring.

\vspace{5mm}

\bibliographystyle{unsrt}
\bibliography{paper}

\end{document}


text:
\chapter{Background} \label{chap:back}
%\section*{} \label{sec:...}
%\begin{dummied}
%\end{dummied}

\section*{}
\begin{dummied}
\end{dummied}

%%\section{Problem Statement} \label{sec:back_}
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\section{Shamanic Interface} \label{sec:shamanic_interface}
    The proposal for the Shamanic Interfaces was originally contemplated in a journal article by Leonel Morgado \cite{MOR2013} focused on the limitations and potential of then current approaches to full-body interaction and methods of issuing commands to systems by the users. Some reasoning behind the need for and the approach of the new paradigm is backed by use cases, and some expected benefits outlined throughout.

    Morgado makes a distinction between categories of interactions based on the degree of mimickry involved with the commands, with the most frequent forms of interaction on those systems closely resembling common day to day human actions. These movements face two primary difficulties, the lack of precision and need for a degree of tolerance with how well the mimickry is performed due to the wide-amplitude analogue nature of body motion, and then further, a barrier of entry based on the user's physical capabilities, particularly unsurmountable to those with special needs or handicapped. On the other end of the scale, for commands that are based on non-kinesthetic concepts, names the learning aspect as its primary challenge, as given these commands must be taught in isolation by the system before effective usage, and many may have a valid yet entirely arbitrary metaphorical reasoning behind their choice.

    The proposed solution to these concerns, Shamanic Interface, attempts to leverage existing knowledge and meaning the users already have from their personal background in the process of interaction. The idea is, as opposed to delivering standard sets of operations and motions per system, performing a stronger separation between the gesture the user performs and the gestural commands, which are the intent perceived by the system, such that the experience of interaction may be more closely tied to the individual and their expectations in a manner that feels more natural. The approach is not entirely unique, since other existing proposals have shown similar concerns and demonstrated the need for gesture and command decoupling\cite{VATAVU}. However, the argument was made that simply allowing users to define their own sets of gestures and migrating these between systems wouldn't be enough to naturally overcome the escalating complexity inherent to the creation and addition of more and more commads. This focus on user cultural kinesis and meta-communication aspires to be the key behind reaching a more easily discoverable, learnable and remembered set of gestures.

    As mentioned above, some of the more impactful contributions the SI could beget would favor users who suffer from balance impairments, are handicapped or are in some other fashion impeded from natural interaction through standard sets of gestures. Another subset of users that could see an opportunity for ease of interaction are those who have a lesser understanding of technology or even level of education. The adoption of non-kinestesic concept should be simplified given these have a prior backing for their understanding. The operation of the systems in augmented reality spaces would also appear more inconspicous to other people in the vicinity of its user, as their actions would appear more natural.

    Further work was performed with the concept of the SI. With the paper \begin{dummied}Cite Here\end{dummied}, a team including Morgado\begin{dummied}...\end{dummied} Finally, this thesis follows up on a prior work by \begin{dummied}Cite Here\end{dummied} where a research tool was developed for the purpose of testing and developing the concept of Shamanic Interfaces. The developed application was in a working condition, capable of identifying cultural gestures, however it specifies performing the actual tests as a requirement among future work.

\section{\dummyText{User Interaction}} \label{sec:back_}
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\section{Cultural Emblems} \label{sec:back_}
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\section{} \label{sec:back_}
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\section{Summary} \label{sec:sota_summary}
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text:
\chapter{Nonparametric and Exact Hypothesis Tests}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Mann-Whitney Test}

All of these variants of the T-test make assumptions about the normality of the data. Sometimes you want to compare the means of two groups but you aren't sure whether the normal assumption holds. In this case, you might want to try a \textbf{nonparametric} alternative to the two-sample T-test called the \textbf{Mann-Whitney Test}, or Wilcoxon Rank Sum Test.

Some interesting points about nonparametric tests:
\begin{itemize}
\item There are no distributional assumptions.
\item Most nonparametric methods replace data by their ranks, which are invariant to any monotonic transformation of the data.
\item Compare this to the T-test: If you use the log-transform of the data instead of the original values and do a T-test on them, the $p$-value can change. But with ranks that doesn't happen.
\item Replacing the data by ranks also makes the test less sensitive to outliers.
\item But remember, there is no free lunch: You will always lose power relative to a parametric test if you use the nonparametric alternative in a case where the distributional assumptions of the parametric test are true.
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Fisher's Exact Test}

For small sample sizes, the assumptions underlying Pearson's chi-squared test are no longer true. In these cases, it is common to replace the chi-squared test with something called \textbf{Fisher's Exact Test}, which calculates an exact $p$-value directly by considering every possible outcome.

% tie to discussion of the log rank test in survival analysis

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Tests of Normality}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Binomial Test}

The \textbf{binomial test} is a simple test of the probability of success in a Bernoulli experiment (see Sections~\ref{sect:bernoulli} and \ref{sect:binomial}).

\paragraph{Example: Diabetes Prevalence with Different Sample Sizes} The prevalence of diabetes in the U.S. population in 2020 was 10.5\% (Source: CDC's National Diabetes Statistics Report, 2020). Imagine that you test a random sample of $10$ people from your small Appalachian town and none of them are hypertensive.

\noindent Let's conduct a hypothesis test to examine this result.

\begin{enumerate}
\item \textit{Create an initial \textbf{research hypothesis}}. ``The prevalence of hypertension in this community is different from that of the U.S. population at large.''
\item \textit{State the \textbf{null hypothesis}}. Let $\mu_c$ be the prevalence of hypertension in the Appalachian community and $\mu_0$ be its prevalence for the general U.S. population.
\begin{align*}
H_0: & \mu_c = \mu_0 \\
H_a: & \mu_c \neq \mu_0
\end{align*}

\begin{question}{}
These statements about the null and alternative hypotheses in this example look the same as in the previous example, but they mean something different here. What is the difference between $\mu_c$ and $\mu_0$ in the previous example vs. this one?
\end{question}

\item \textit{List statistical {assumptions}}. We assume that the individuals we tested were chosen independently from the population of the Appalachian town.
\item \textit{Decide on an appropriate test and test statistic}. Our test statistic in this case is going to be the \textbf{Z-statistic}, which measures the deviation of the sample mean from the population mean in units of the \textbf{standard deviation of the sample mean}, $\sigma/\sqrt{n}$:
$$ Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \qquad \text{where} \qquad \bar{x} = \frac{1}{n} \sum_{i=1}^n x^{(i)}$$
In our case, $n = 20$ because $\bar{x}$, our sample mean, is an average of 20 samples.
\item \textit{Derive the distribution of the test statistic under the null}. The Z-statistic follows a \textbf{standard normal} distribution under the null, which is a normal distribution with $\mu=0$ and $\sigma=1$. To see this, remember that the distribution of $\bar{x}$ (blue distribution in the figure) under the null is $\mathcal{N}(\mu_0, \sigma/\sqrt{n})$. When you calculate the Z-statistic, you shift that distribution by a distance $\mu_0$ so it is centered at zero, then adjust its width to 1.0 by dividing by $\sigma/\sqrt{n}$.
\item \textit{Select a {significance level} under which you'll reject the null}. For the purposes of this example, we will choose $\alpha = 0.05$ (5\% chance of a type I error). The null distribution of the Z-statistic is shown below. The vertical dotted black lines are situated at the \textbf{critical values} that produce $\alpha = 0.05$ (the area under the null distribution that is outside those lines is 0.05).
\item \textit{Compute the observed value of the test statistic from the data.} The observed value of the test statistic is:
$$ Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} = \frac{125.45 - 139.75}{21.40/\sqrt{20}} = -2.99. $$
\item \textit{Decide whether or not to reject the null hypothesis.} The value of our test statistic falls outside the region contained by the critical values (the \textbf{acceptance region}), so we reject the null at this value of $\alpha$.
\end{enumerate}

max_stars_repo_name: blpercha/mcds-notes

max_issues_repo_name: blpercha/mcds-notes

max_forks_repo_name: blpercha/mcds-notes


text:
%!TEX root=../mythesis.tex
% Appendix A

\chapter{Proofs for Part I or Chapter 3} % Main appendix title

\label{AppendixA} % For referencing this appendix elsewhere, use \ref{AppendixA}

%\lhead{Appendix A. \emph{Appendix Title Here}} % This is for the header on each page - perhaps a shortened title
\section{Proof of Lemma}
\label{pf:ApprGradSys}

$$\psi^{av}(\theta)=\frac{1}{T}\int^{T}_0[\psi(\theta+\mu(\tau))+C]\otimes\frac{\mu(\tau)}{a}d\tau$$

\section{Proof of another Lemma}
\begin{equation}
\begin{aligned}
\gamma_1(\left\|x\right\|)\leq W(t,x)\leq \gamma_2(\left\|x\right\|)\\
\frac{\partial{W}}{\partial{t}}+\frac{\partial{W}}{\partial{x}}\phi(t,x,0)\leq-\gamma_3(\left\|x\right\|)
\end{aligned}
\end{equation}


text:
%\chapter{Derivation of Transfer Matrices for Electric Field Propagation in Planar Quasi-1D Waveguide From Maxwell's Equations}
\chapter{Transfer Matrices for Electric Field Propagation}
\label{sec:appendix_derivation_transfer_matrices_quasi1d}

In the following derivation, the transfer matrix method\cite{1981_MacKinnon_scaling}\cite{1992_Pendry}\cite{2003_Kettemann} is developed from Maxwell's equations\cite{1999_Jackson} for metallic waveguides with scatterers. Before starting, assumptions necessary for the derivation are enumerated.
\begin{itemize}
\item Metallic boundaries for the surfaces of the waveguide result in no leakage of electric field at the edges. These boundaries occur in the direction perpendicular to propagation $y$, which extends from $y=0$ to $y=W\lambda$. In the direction of propagation along $z$, the waveguide has open boundaries at $z=0$ and $z=L/\lambda$. The metallic boundaries are infinite along direction $x$.
\item Scattering potentials are initially assumed to be $\delta$-functions, and are later reduced to a finite sum of Fourier components. This scattering potential description gives rise to transport behavior similar to that in physical disordered media.
%   \item non-physical
%   \item infinite number of closed channels
\item No inelastic scattering occurs; there is no energy loss due to scattering, and phase can remain coherent.% [scatterers only affect amplitude].
\item No noise (spontaneous emission) is included. The primary interest in the modeling the transition from phenomena described by diffusion to that of Anderson localization. %Also, experimentally noise can be suppressed. % CITE
%The transmission with gain will be slightly different compared to real output.
\item Only transverse-magnetic (TM) waves are assumed incident. The electric field oscillates perpendicular to the plane of the 2D waveguide~
%``For EM waves propagating in the $z,y$ plane, the $s$ ($E$ field parallel to the $x$ axis) and $p$ ($E$ field perpendicular to the $x$ axis) polarized waves can be described by two decoupled wave equations.''
\cite{1996_soukoulis_dis2d}.
\item The absorption and gain mechanism is purely mathematical: no atomic level modeling is included. This is the appeal of working in the mesoscopic regime, namely that atomic-based scattering descriptions are not necessary.
%\item %, other than the capability to be selectively incident on a specific channel.
%\item By choosing planar quasi-1D geometry (and thus scalar waves), we implicitly assume that polarization will not significantly alter transport phenomena. Although planar geometry may be experimentally realizable, it is not as popular as 3D quasi-1D
\end{itemize}
No input beam properties are initially assumed. The incident energy can be a plane wave, or any other distribution.

%\section{From Maxwell equations to wave equation}
%Starting with Maxwell's equations, the wave equation for electromagnetic waves is derived to demonstrate that only a single polarization is being studied. Ampere's Law (with current $\vec{J}=0$ and $D=\epsilon_0 E$)\cite{1999_Jackson}, %Eq. I.1b, page 2
%\begin{equation}
%\vec{\nabla} \times \vec{{\cal H}} = \epsilon_0 \frac{\partial\vec{E}}{\partial t}
%\label{eq:amperes_law}
%\end{equation}
%Faraday's Law \cite{1999_Jackson}, %Eq. I.1a, page 2]
%\begin{equation}
%\vec{\nabla} \times \vec{E} = -\mu_0 \frac{\partial\vec{{\cal H}}}{\partial t}
%\label{eq:faradays_law}
%\end{equation}
%Taking the temporal derivative of Eq.~\ref{eq:amperes_law}
%\begin{equation}
%\vec{\nabla} \times \frac{\partial\vec{{\cal H}}}{\partial t} = \epsilon_0 \frac{\partial^2\vec{E}}{\partial t^2}
%\label{eq:temporal_derivative_of_amperes_law}
%\end{equation}
%Taking the curl of Eq.~\ref{eq:faradays_law}
%\begin{equation}
% \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \left(\vec{\nabla} \times \frac{\partial\vec{{\cal H}}}{\partial t} \right)
%\label{eq:curl_of_faradays_law}
%\end{equation}
%Substitute Eq.~\ref{eq:temporal_derivative_of_amperes_law} into Eq.~\ref{eq:curl_of_faradays_law}
%\begin{equation}
%  \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2\vec{E}}{\partial t^2}
%\end{equation}
%Vector identity: $\vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla} (\vec{\nabla}\cdot \vec{E})-\nabla^2 \vec{E}$. Since we assume no source, we have the wave equation

The wave equation can be derived from Maxwell's equations (not shown), yielding
\begin{equation}
\nabla^2 E = \frac{1}{c^2} \frac{\partial^2\vec{E}}{\partial t^2}
\label{eq:general_wave_equ}
\end{equation}
where $\mu_0 \epsilon_0 = \frac{1}{c^2}$.
%[See 1996 Soukoulis et al PRB v53 n13 pg 8340, section II]
% Eq.~\ref{eq:general_wave_equ} ``is identical with the scalar wave equation.'' The $p$-polarized wave equation involves ${\cal H}$.

\section{Time Independent Wave Equation}
Assuming electric field variables are separable,
\begin{equation}
E(\vec{r},t) = E(\vec{r}) e^{i\omega t}
\label{eq:separation_of_variables_E_field}
\end{equation}
the field is simplified by also assuming monochromatic and continuous wave (CW). Substituting  Eq.~\ref{eq:separation_of_variables_E_field} into the right side of Eq.~\ref{eq:general_wave_equ}, time dependence can be canceled.
\begin{equation}
\nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
\label{eq:wave_equation_electric_field}
\end{equation}
where $\frac{\omega}{c}=k$. Although the following results will appear to be ``time independent,'' the time dependence can be reintroduced by multiplying both sides by $e^{i\omega t}$. Effectively the same as assuming $t=0$.

\section{Separation of Variables}
Convert from general $\vec{r}$ to two-dimensional Cartesian coordinates (since the transfer matrices for a planar quasi-1D waveguide are desired): $\vec{r} = z \hat{i}+y\hat{j}$. Let $W\equiv$~width and $L\equiv$~length of waveguide.
%\begin{equation}
%\vec{r} = z \hat{i}+y\hat{j}
%\end{equation}
%and the Laplacian is
%\begin{equation}
%\nabla^2 = \frac{\partial^2}{\partial z^2} + \frac{\partial^2}{\partial y^2}
%\label{eq:laplacian_cartesian}
%\end{equation}

%unjustified claim
The $z$ and $y$ components of the field are independent, the separation of variables applies spatially.
\begin{equation}
E(\vec{r}) = E(z,y) = \sum^\infty_{n=1} E_n(z) \chi_n(y)
\label{eq:spacialseparation}
\end{equation}
where the sum is over all channels. For $\delta$-function scatterers, there can be an infinite number of closed channels.

Now the wave equation (Eq.~\ref{eq:wave_equation_electric_field}) is
\begin{equation}
\nabla^2 E(z,y) = - \frac{\omega^2}{c^2} E(z,y)
\label{eq:wave_equation_electric_field_cartesian}
\end{equation}
Apply Laplacian %(Eq.~\ref{eq:laplacian_cartesian})
and separation (Eq.~\ref{eq:spacialseparation})
\begin{equation}
\sum^\infty_{n=1} \left[ \frac{\partial^2E_n(z)}{\partial z^2} \chi_n(y) + E_n(z) \frac{\partial^2 \chi_n(y)}{\partial y^2} \right] = \\
- \frac{\omega^2}{c^2} \sum^\infty_{n=1} E_n(z) \chi_n(y)
\label{eq:summation_variable_separation_cartesian}
\end{equation}

\section{Perpendicular Component Solution}
The solution to the differential equation perpendicular to the direction of propagation is found from the auxiliary equation for each channel
\begin{equation}
\left(\frac{\partial^2}{\partial y^2} + k_{\bot n}^2 \right) \chi_n(y) = 0
\end{equation}
Boundary conditions for metallic waveguide: Electric field $E$ is zero at the boundaries, $\chi_n(0) = \chi_n(W) = 0$.
%\begin{equation}
%\chi_n(0) = \chi_n(W) = 0
%\end{equation}
The normalized solution is the familiar
\begin{equation}
\chi_n(y) = \sqrt{\frac{2}{W}} \sin (k_{\bot n} y)
\end{equation}
where
%\begin{equation}
$k_{\bot n} \equiv \frac{n \pi}{W}$.
%\label{eq:k_perpendicular}
%\end{equation}
As a check of normalization, for $m=n$
\begin{equation}
\int^W_0 \chi_n^2(y) dy = \frac{2}{W} \int^W_0 \sin^2 (k_{\bot n} y) = \frac{2}{W} \frac{1}{2} W = 1
\end{equation}
and if $m \neq n$, solutions are orthogonal
\begin{equation}
\int^W_0 \chi_n(y)\chi_m(y) dy = 0
\end{equation}
Thus, for general $n$ and $m$,
\begin{equation}
\int^W_0 \chi_n(y)\chi_m(y) dy = \delta_{n,m}
\label{eq:convert_to_kronecker}
\end{equation}

\section{Parallel Component Solution}
For the solution parallel to the direction of propagation of Eq.~\ref{eq:wave_equation_electric_field_cartesian}, the $z$-component starts with
\begin{equation}
\frac{\partial^2 E_n(z)}{\partial z^2} - k_{\bot n}^2 E_n(z) = - \frac{\omega^2}{c^2} E_n(z)
\end{equation}
Re-arrange and introduce a new variable
\begin{equation}
\frac{\partial^2 E_n(z)}{\partial z^2} + k_{\parallel n}^2 E_n(z) = 0
\label{eq:zcomponentdiffequ}
\end{equation}
where
\begin{equation}
k_{\parallel n}^2 \equiv \frac{\omega^2}{c^2} - k_{\bot n}^2
\label{eq:k_parallel}
\end{equation}
Note: $k_{\parallel n}^2$ can be positive (corresponding to open channels) or negative (closed channels). If negative, then $k_{\parallel n}$ is imaginary, denoted $k_{\parallel n} = i \kappa_{\parallel n}$ for $n > N_{open}$. Open channels propagate forward, with velocity decreasing as channel index increases. Closed channels decrease in amplitude exponentially.

%The summation in Eq.~\ref{eq:summation_variable_separation_cartesian} can be split into open and closed channels
%\begin{equation}
%\sum_{n=1}^\infty = \sum_{n=1}^{N_o} + \sum_{n=N_{o}+1}^\infty
%\end{equation}

%\begin{tabular}{cc}
%$ n \leq N_o $ \quad \quad \quad & $ k_{\parallel n}^2 = \frac{\omega^2}{c^2} - \left(\frac{n \pi}{W}\right)^2 > 0 $ \\
%$ n > N_o $ \quad \quad \quad & $ k_{\parallel n}^2 < 0 $ \\
%\end{tabular}

Separate electric field components into left(-) and right (+) traveling plane waves (two solutions to the second order differential equation)
\begin{equation}
\begin{gathered}
\text{Open: \ }  E_n(z) = E_n^+ \exp(i k_{\parallel n} z) + E_n^- \exp(-i k_{\parallel n} z) \\
\text{Closed: \ }   E_n(z) = E_n^+ \exp(-\kappa_{\parallel n} z) + E_n^- \exp(\kappa_{\parallel n} z)
\end{gathered}
\label{eq:Eleftandrightpropagating}
\end{equation}
where $i \kappa \equiv k$

\section{Waveguide With Scatterers}
Up to this point, an empty waveguide has been considered. For scattering, replace $\frac{\omega^2}{c^2}$ of the wave equation \ref{eq:wave_equation_electric_field_cartesian} with a spacial Sellmeier equation
\begin{equation}
\frac{\omega^2}{c^2} (1 + \alpha \delta(z-z_0,y-y_0))
\label{eq:scatterer}
\end{equation}
where $\delta(z-z_0,y-y_0) \equiv \delta(z-z_0) \delta(y-y_0)$ is the scattering potential and $\alpha$ is the scattering strength. $\alpha$ can be complex; then the real part is the strength and the imaginary component is gain or absorption.

%Note: this new term changes Eqs.~\ref{eq:amperes_law},\ref{eq:general_wave_equ}, and \ref{eq:wave_equation_electric_field} by introducing a non-unity refractive index.

To determine transport of light past a scattering potential, apply continuity of electric field $E$ and its derivative. The following carries out matching component-wise derivative.

Assuming the scattering potential is located at cross-section $z$ (inside the waveguide $0<z<L$), and the electric field just before or after the scatterer (at $z \pm \Delta$) is a sum of independent channel components.
\begin{equation}
E(z \pm \Delta, y) = \sum_{n=1}^\infty E_n(z \pm \Delta) \chi_n(y)
\end{equation}
Applying Eq.~\ref{eq:scatterer} to Eq.~\ref{eq:zcomponentdiffequ}, the wave equation becomes
\begin{equation}
\sum_{n=1}^\infty \left( E_n^{\prime\prime} \chi_n + k_{\parallel n}^2 E_n \chi_n + \alpha \frac{\omega^2}{c^2} \delta(z-z_0,y-y_0) E_n \chi_n \right) = 0
\label{eq:doubleprimeEz}
\end{equation}

Multiply Eq.~\ref{eq:doubleprimeEz} by $ \chi_m $ and $ \int_0^W dy $. By applying Eq.~\ref{eq:convert_to_kronecker} and letting $A_{m,n}(y_0)=\chi_m(y_0) \chi_n(y_0)$,
\begin{equation}
\sum_{n=1}^\infty \left( E_n^{\prime\prime} \delta_{nm} + k_{\parallel n}^2 E_n \delta_{nm} + \alpha \frac{\omega^2}{c^2} E_n \delta(z_0) A_{nm}(y_0)  \right) = 0
\end{equation}
Apply the summation over $n$, which eliminates the Kronecker deltas.
\begin{equation}
 E_m^{\prime\prime} + k_{\parallel m}^2 E_m + \alpha \frac{\omega^2}{c^2} E_n \delta(z-z_0) \sum_{n=1}^\infty A_{nm}(y_0) = 0
\end{equation}
Integrate over $z$ from $(z-\Delta)$ to $(z+\Delta)$ and let $\Delta\rightarrow0$.
\begin{equation}
\begin{gathered}
\int_{z_0-\Delta}^{z_0+\Delta} E_m^{\prime\prime}(z) dz + k_{\parallel m} ^2 \int_{z_0-\Delta}^{z_0+\Delta}
E_m(z) dz +\\ \alpha \frac{\omega^2}{c^2} \sum_{n=1}^\infty A_{n,m}(y_0) \int_{z_0-\Delta}^{z_0+\Delta} \delta(z_0) E_n dz = 0
\end{gathered}
\end{equation}

To do the second term integration, assume that for small $\Delta$, $E(z) \approx E(z_0)$.
\begin{equation}
E_m^{\prime}(z_0 + \Delta) - E_m^{\prime}(z_0 - \Delta) + k_{\parallel m}^2 E_m(z_0) 2 \Delta + \alpha \frac{\omega^2}{c^2} \sum_{n=1}^\infty A_{n,m}(y_0) E_n(z_0) = 0
\end{equation}

Since $\Delta \rightarrow 0$, then $2 \Delta$ is really small, so that term is dropped.

To conclude, for a given channel $m$, electric field and the field derivative on both sides of the scatterer must match
\begin{equation}
\begin{gathered}
%\boxed{
E_m(z_0+\Delta) = E_m(z_0-\Delta) \\
%}
%\\
%\boxed{
E_m^{\prime}(z_0+\Delta) = E_m^{\prime}(z_0-\Delta) - \alpha \frac{\omega^2}{c^2} \sum_{n=1}^\infty A_{n,m}(y_0) E_n(z_0)
%}
\end{gathered}
\end{equation}

Note that the $\delta$ function scatterer has been eliminated, and $A_{n,m}$ can form an array (the ``scattering matrix'').

\begin{equation}
\begin{gathered}
\left( \begin{array}{cc}
\hat{I} & 0 \\
-\alpha \frac{\omega^2}{c^2}A_{mn}(y_0) & \hat{I} \\
\end{array} \right)
\left( \begin{array}{c}
E_{1..N_{max}}(z_0-\Delta) \\
\frac{1}{\kappa_{\parallel 1..N_{max}}} E_{1..N_{max}}^{\prime}(z_0-\Delta)
\end{array} \right)
=\\
\left( \begin{array}{c}
E_{1..N_{max}}(z_0+\Delta) \\
\frac{1}{\kappa_{\parallel 1..N_{max}}} E_{1..N_{max}}^{\prime}(z_0+\Delta)
\end{array} \right)
\end{gathered}
\end{equation}
Due to the form of the matrix, the determinant is always unity (only the diagonal contributes non-zero terms) regardless of the elements in the lower left quadrant. Elements of the lower left quadrant are
\begin{equation}
 -\alpha \frac{\omega^2}{c^2} \frac{2}{W} \sin(k_{\perp m} y_0) \sin(k_{\perp n} y_0)
\end{equation}
Note that the scattering matrix is real unless $\alpha$ or $\omega$ are complex.

\section{Free Space Propagation of Open Channels}
For open channels ($n \leq N_o$), field $E_n$ and derivative of field $ \frac{1}{k_{\parallel n}}E_n^{\prime}$ are more convenient basis than ``left traveling'' $E_n^-(z)$ and ``right traveling'' $E_n^+(z)$. First, the connection between the two basis is found. Starting from Eq.~\ref{eq:Eleftandrightpropagating}, electric field $E(z)$ is the solution to a second order differential equation, so it has two solutions.
\begin{equation}
\begin{gathered}
E_n(z) = E_n^+ \exp(i k_{\parallel n} z) + E_n^- \exp(-i k_{\parallel n} z) \\
E_n^{\prime}(z) = i k_{\parallel n} E_n^+ \exp(i k_{\parallel n} z) - i k_{\parallel n}E_n^- \exp(-i k_{\parallel n} z)
\label{eq:Eleftandrightpropagating_again}
\end{gathered}
\end{equation}
Solving for left- and right-traveling wave components,
\begin{equation}
\begin{gathered}
E_n^+(z) = \frac{1}{2} \left( E_n(z)+\frac{1}{i} \frac{1}{k_{\parallel n}} E_n^{\prime}(z) \right) \exp(-i k_{\parallel n} z) \\
E_n^-(z) = \frac{1}{2} \left( E_n(z)-\frac{1}{i} \frac{1}{k_{\parallel n}} E_n^{\prime}(z) \right) \exp(i k_{\parallel n} z)
\label{eq:Eleftright}
\end{gathered}
\end{equation}

To preemptively clear up notation confusion, in previous steps $\Delta$ was used to denote a small ($\Delta \rightarrow 0$) distance from the scatterer. Here $\Delta z$ will be used to signify a finite displacement in position along the $z$ axis.

The field and derivative of field is translated over distance $\Delta z$ from the original position $z$. First, substitute the shift into Eq.~\ref{eq:Eleftandrightpropagating_again}
\begin{equation}
E_n(z+\Delta z) = E_n^+ \exp(i k_{\parallel n} (z+\Delta z)) + E_n^- \exp(-i k_{\parallel n} (z+\Delta z))
\end{equation}
Then substitute Eq.~\ref{eq:Eleftright}
\begin{equation}
\begin{gathered}
E_n(z+\Delta z) = \frac{1}{2} \left( E_n(z) + \frac{1}{i} \frac{1}{k_{\parallel n}} E_n^{\prime}(z) \right) \exp(i k_{\parallel n} z) + \\
\frac{1}{2} \left( E_n(z) - \frac{1}{i} \frac{1}{k_{\parallel n}} E_n^{\prime}(z) \right) \exp(-i k_{\parallel n} z)
\end{gathered}
\end{equation}
Reducing leaves how to shift an electric field over distance $\Delta z$.
\begin{equation}
%\boxed{
E_n(z+\Delta z) =E_n(z) \cos(k_{\parallel n} \Delta z) + \frac{1}{k_{\parallel n}} E_n^{\prime} \sin(k_{\parallel n} \Delta z)
%}
\label{eq:open_channel_field_transfer}
\end{equation}
Similarly,
\begin{equation}
\frac{1}{k_{\parallel n}} E_n^{\prime}(z+\Delta z) = i E_n^+ \exp(i k_{\parallel n} (z+\Delta z)) -i E_n^- \exp(-i k_{\parallel n} (z+\Delta z))
\end{equation}
Then substitute Eq.~\ref{eq:Eleftright}
\begin{equation}
\begin{gathered}
\frac{1}{k_{\parallel n}} E_n^{\prime}(z+\Delta z) =
\frac{i}{2} \left( E_n(z) + \frac{1}{i} \frac{1}{k_{\parallel n}} E_n^{\prime}(z) \right) \exp(i k_{\parallel n} z) -\\
\frac{i}{2} \left( E_n(z) - \frac{1}{i} \frac{1}{k_{\parallel n}} E_n^{\prime}(z) \right) \exp(-i k_{\parallel n} z)
\end{gathered}
\end{equation}

\begin{equation}
%\boxed{
\frac{1}{k_{\parallel n}} E^{\prime}(z+\Delta z)=- E_n(z) \sin(k_{\parallel n} \Delta z) + \frac{1}{k_{\parallel n}} E_n^{\prime} \cos(k_{\parallel n} \Delta z)
%}
\label{eq:open_channel_deriv_transfer}
\end{equation}

\section{Free-space Propagation of Closed Channels}
For closed channels ($n > N_o$), change of $i$ results in hyperbolic trig functions.
\begin{equation}
\begin{gathered}
E_n(z) = E_n^+ \exp(-\kappa_{\parallel n} z) + E_n^- \exp(\kappa_{\parallel n} z) \\
E_n^{\prime}(z) = -\kappa_{\parallel n} E_n^+ \exp(-\kappa_{\parallel n} z) + \kappa_{\parallel n}E_n^- \exp(\kappa_{\parallel n} z)
\end{gathered}
\end{equation}
Recalling that $k_{\parallel n} = i \kappa_{\parallel n}$, then
\begin{equation}
\begin{gathered}
E_n^+(z) = \frac{1}{2} \left( E_n(z)- \frac{1}{\kappa_{\parallel n}} E_n^{\prime}(z) \right) \exp(\kappa_{\parallel n} z) \\
E_n^-(z) = \frac{1}{2} \left( E_n(z)+ \frac{1}{\kappa_{\parallel n}} E_n^{\prime}(z) \right) \exp(-\kappa_{\parallel n} z)
\end{gathered}
\end{equation}

Shifting the field by $\Delta z$
%\begin{equation}
%E_n(z+\Delta z) = E_n^+ \exp(-\kappa_{\parallel n}(z+\Delta z)) + E_n^- \exp(\kappa_{\parallel n}(z+\Delta z))
%\end{equation}
%\begin{equation}
%E_n(z+\Delta z) = \frac{1}{2} \left( E_n(z)- \frac{1}{\kappa_{\parallel n}} E_n^{\prime}(z) \right) \exp(-\kappa_{\parallel n} z) +
%\frac{1}{2} \left( E_n(z)+ \frac{1}{\kappa_{\parallel n}} E_n^{\prime}(z) \right) \exp(\kappa_{\parallel n} z)
%\end{equation}
\begin{equation}
%\boxed{
E_n(z+\Delta z) = E_n(z) \cosh(\kappa_{\parallel n}\Delta z) + \frac{1}{\kappa_{\parallel n}} E_n^{\prime}(z) \sinh(\kappa_{\parallel n}\Delta z)
%}
\end{equation}
and
%\begin{equation}
%\frac{1}{\kappa_{\parallel n}}E_n^{\prime}(z+\Delta z) = -E_n^+ \exp(-\kappa_{\parallel n}(z+\Delta z)) + E_n^- \exp(\kappa_{\parallel n}(z+\Delta z))
%\end{equation}
%\begin{equation}
%\begin{gathered}
%\frac{1}{\kappa_{\parallel n}}E_n^{\prime}(z+\Delta z) =-\frac{1}{2} \left( E_n(z)- \frac{1}{\kappa_{\parallel n}} E_n^{\prime}(z) \right) \exp(-\kappa_{\parallel n} z) + \\
%\frac{1}{2} \left( E_n(z)+ \frac{1}{\kappa_{\parallel n}} E_n^{\prime}(z) \right) \exp(\kappa_{\parallel n} z)
%\end{gathered}
%\end{equation}
\begin{equation}
%\boxed{
\frac{1}{\kappa_{\parallel n}}E_n^{\prime}(z+\Delta z) =E_n(z) \sinh(\kappa_{\parallel n}\Delta z) + \frac{1}{\kappa_{\parallel n}} E_n^{\prime}(z) \cosh(\kappa_{\parallel n}\Delta z)
%}
\end{equation}

To summarize,
\begin{equation}
\begin{gathered}
E_n(z+\Delta z) = E_n(z) \cosh(\kappa_{\parallel n}\Delta z) + \frac{1}{\kappa_{\parallel n}} E_n^{\prime}(z) \sinh(\kappa_{\parallel n}\Delta z) \\
\frac{1}{\kappa_{\parallel n}}E_n^{\prime}(z+\Delta z) =E_n(z) \sinh(\kappa_{\parallel n}\Delta z) + \frac{1}{\kappa_{\parallel n}} E_n^{\prime}(z) \cosh(\kappa_{\parallel n}\Delta z)
\label{eq:closed_channel_transfer}
\end{gathered}
\end{equation}

From Eq.~\ref{eq:open_channel_field_transfer}, \ref{eq:open_channel_deriv_transfer}, and \ref{eq:closed_channel_transfer} the ``free space propagation matrix'' can be constructed. The array would be of rank $2 n_{max}$ ($n_{max}=N_o+N_c$). The determinant of this matrix is alway unity (regardless of argument) because terms can be factored into $\sin^2x +\cos^2=1$ for each channel. Thus, for both free and scattering matrices, the determinant is unity regardless of free space separation $\Delta z$ or real (passive) and complex (active media) dielectric values.

With this description of matrices for scattering planes and for free space propagation, the transmission of electric field and its derivative can be computed for a metallic waveguide containing arbitrarily placed scatterers. The set of matrices is multiplied together to form a matrix $\hat{S}$ of size $2n_{max}$. Then the vector describing the initial incident field and derivative of field for each channel $n$ is multiplied by $\hat{S}$ to give the transmission and reflection matrices (one vector for each channel $n$).

This approach of multiply matrices together inherently results in numerical instability~\cite{1968_Osedelec} due to the limited precision of computational devices. Therefore, the self-embedding technique~\cite{1999_yamilov_selfembed,1976_Bellman_Wing_embedding} is used to periodically renormalize the product.

\begin{comment}
\begin{equation}
\left( \begin{array}{cccccccc}
\cos(k_{\parallel 1}\Delta z)   & 0 & 0 & 0 & \sin(k_{\parallel 1}\Delta z)   & 0 & 0 & 0 \\
0 & \cos(k_{\parallel N_0}\Delta z) & 0 & 0 & 0 & \sin(k_{\parallel N_0}\Delta z) & 0 & 0 \\

0 & 0 & \cosh(k_{\parallel N_0+1}\Delta z)   & 0 & 0 & 0 & \sinh(k_{\parallel n}\Delta z) & 0  \\
0 & 0 & 0 & \cosh(k_{\parallel n_{max}}\Delta z) & 0 & 0 & 0 & \sinh(k_{\parallel n}\Delta z) \\

-\sin(k_{\parallel n}\Delta z) & 0 & \cos(k_{\parallel n}\Delta z) & 0 \\
-\sin(k_{\parallel n}\Delta z) & 0 & \cos(k_{\parallel n}\Delta z) & 0 \\

0 & \sinh(k_{\parallel n}\Delta z) & 0 & \cosh(k_{\parallel n}\Delta z) \\
0 & \sinh(k_{\parallel n}\Delta z) & 0 & \cosh(k_{\parallel n}\Delta z) \\
\end{array} \right)
\end{equation}

\begin{equation}
\left( \begin{array}{cccc}
\cos(k_{\parallel 1..N_0}\Delta z)  & 0 & \sin(k_{\parallel 1}\Delta z) & 0 \\
0 & \cosh(k_{\parallel N_0+1..n_{max}}\Delta z) & 0 & \sinh(k_{\parallel N_0+1..n_{max}}\Delta z)  \\
-\sin(k_{\parallel 1..N_0}\Delta z) & 0 & \cos(k_{\parallel 1..N_0}\Delta z) & 0 \\
0 & \sinh(k_{\parallel N_0+1..n_{max}}\Delta z) & 0 & \cosh(k_{\parallel N_0+1..n_{max}}\Delta z) \\
\end{array} \right)
\left( \begin{array}{c}
E_1 \\
\vdots \\
E_N \\
E_{N+1} \\
\vdots \\
E_{N_{max}} \\
\frac{1}{k_{\parallel 1}} E_1^{\prime} \\
\vdots \\
\frac{1}{k_{\parallel N}} E_N^{\prime} \\
\frac{1}{\kappa_{\parallel N+1}} E_{N+1}^{\prime} \\
\vdots \\
\frac{1}{\kappa_{\parallel N_{max}}} E_{N_{max}}^{\prime}
\end{array} \right)
\end{equation}
Note that the derivative portion of the free space propagation is where ``energy loss'' occurs (in passive $N_{open}$ systems).

\section{Boundary Condition}

\begin{equation}
v(z) =
\left( \begin{array}{c}
E_1 \\
\vdots \\
E_N \\
E_{N+1} \\
\vdots \\
E_{N_{max}} \\
\frac{1}{k_{\parallel 1}} E_1^{\prime} \\
\vdots \\
\frac{1}{k_{\parallel N}} E_N^{\prime} \\
\frac{1}{\kappa_{\parallel N+1}} E_{N+1}^{\prime} \\
\vdots \\
\frac{1}{\kappa_{\parallel N_{max}}} E_{N_{max}}^{\prime}
\end{array} \right)
= \left( \begin{array}{c}
\vec{E}_{open}(z) \\
\vec{E}_{closed}(z) \\
\vec{D}_{open}(z) \\
\vec{D}_{open}(z) \\
\end{array} \right)
\end{equation}

\begin{equation}
\hat{M}(0,L) \vec{v}(0) = \vec{v}(L)
\end{equation}

\begin{equation}
 E(y,z,t) = \left\{
\begin{array}{l l}
E_{in} e^{i(\vec{r}\cdot \vec{k} - \omega t)} + E_{r} e^{-i(\vec{r}\cdot \vec{k} + \omega t)} & \quad \mbox{ if $x<0$} \\
E_t e^{i(\vec{r}\cdot \vec{k} - \omega t)}  & \quad \mbox{ if $x>L$} \\ \end{array} \right.
\end{equation}

The transmission coefficient is $T=\frac{|E_t|^2}{|E_{in}|^2}$.
\end{comment}