From 628100b97268a2f54d71316ab320a6b3ce49efaa Mon Sep 17 00:00:00 2001
From: Arthur Grisel-Davy <agriseld@uwaterloo.ca>
Date: Wed, 14 Jun 2023 11:34:01 -0400
Subject: [PATCH] add notation illustration

---
 PhD/research_proposal/futurwork.tex           |   34 +-
 PhD/research_proposal/glossaries.tex          |    1 +
 .../images/ssmm_illustration.pdf              |  Bin 0 -> 27966 bytes
 .../images/ssmm_illustration.svg              | 1280 +++++++++++++++++
 PhD/research_proposal/proposal.tex            |    2 +
 5 files changed, 1306 insertions(+), 11 deletions(-)
 create mode 100644 PhD/research_proposal/images/ssmm_illustration.pdf
 create mode 100644 PhD/research_proposal/images/ssmm_illustration.svg

diff --git a/PhD/research_proposal/futurwork.tex b/PhD/research_proposal/futurwork.tex
index 18db596..faff6b1 100644
--- a/PhD/research_proposal/futurwork.tex
+++ b/PhD/research_proposal/futurwork.tex
@@ -37,28 +37,40 @@ The work on \gls{dsd} is the fundation for the planned development of more speci
 \end{figure}
 
 \section{Single-Source, Multi-Measure}
-The global power consumption of a machine does not always tell the full story about its activity.
+The global power consumption of a machine does not fully describe its activity.
 In an embedded system, the power consumption can be attributed to different components, each with its specific activity.
-For the simplest systems performing one specific task --- such as \gls{rtu} ---, the activity of each component is often correlated.
-If the system is in a mode \textit{A} then each component is in mode \textit{A}, and the global power consumption will display the \textit{mode A} pattern.
+For the simplest systems performing one specific task, the activity of each component is often correlate with each other.
+If the system is in a Mode \textit{A} then each component is in Mode \textit{A}, and the global power consumption will display the Mode \textit{A} pattern.
 For more complex systems, different components can be in different modes to accommodate the multi-tasking nature of the global activity.
-In this case, if the first component is in mode \textit{A} but the second is in mode \textit{B}, this indicates a different global activity than if both are in the same mode.
-For example, if the bootup sequence of a general-purpose computer shows a high \gls{cpu} activity but a low storage activity, it could indicate a failure to boot or an attacker booting the system from external storage.
+In this case, if the first component is in Mode \textit{A} but the second is in Mode \textit{B}, this indicates a different global activity than if both are in the same mode.
+For example, if the bootup sequence of a general-purpose computer shows a significant \gls{cpu} activity but no \gls{hdd} activity, it could indicate a failure to boot or an attacker booting the system from external storage.
 Access to each component's individual power consumption opens the way to a more granular understanding of the machine's activity.
-However, the new nature of the captured data requires an evolution of the techniques developed before.
+However, the multivariate aspect of the captured data requires an evolution of the detection techniques.
 
 \subsection{Problem Statement}
 Differentiating between the different components to better understand the activity of a machine is a valuable capability associated with a new problem.
 
 \begin{problem-statement}[Single-Source Multi-Measure]
-    Given a discretized time series $t$ and a set of $n$ components for each of $m$ patterns $P=\{\{\chi\},\{P_{11},\dots, P_{1n}\},\dots, \{P_{m1},\dots, P_{mn}\}\}$, identify an injective mapping $m_{SSMM}:\mathbb{N}\longrightarrow P$ such that every sample $t[i]$\agd{fix equation overflow}
+    Given a discretized, multivariate time series $ts$ and a set of $n$ components for each of $m$ patterns $P=\{\{\chi\},P_1=\{P_{1,1},\dots, P_{1,n}\},\dots,$
+    $P_m=\{P_{m,1},\dots, P_{m,n}\}\}$, identify an injective mapping $m_{SSMM}:\mathbb{N}\longrightarrow P$ such that every sample $ts[i]$
     maps to exactly one set of pattern components in $P$ with the condition that the sample matches an occurrence of the set of patterns in $t$.
 \end{problem-statement}
 
-The time series $t$ is a discretized, multivariate, real-valued time series.
-Each sample $t[i]$ is a vector or $n$ component representing the value of each dimension of $t$ at a point in time.
-Each pattern in $P$ contain multiple pattern components and represent a global pattern across all dimension of $t$.
-Thus, the number of components of each pattern must be equal to the dimensions of $t$.
+The time series $ts$ is a discretized, multivariate, real-valued time series.
+$ts$ is composed of $n$ dimensions with the $j^{th}$ dimension referred to as $ts_j$.
+Each sample $ts[i]$ is a vector or $n$ component representing the value of each dimension of $t$ at a point in time.
+The items of the set $P$ are sets of patterns $P_j$ with $j\in[1,m]$.
+Each set of patterns $P_j$ is associated with one component of a global pattern.
+In other words, each component $P_{j,k}$ represent a the pattern $j$ along the $k^{th}$ dimension of $ts$.
+Thus, the number of components of each pattern must be equal to the dimensions of $ts$.
+Figure \ref{fig:notation} illustrate the $ts$ and $P$ objects.
+
+\begin{figure}
+    \centering
+    \includegraphics[width=0.9\textwidth]{images/ssmm_illustration.pdf}
+    \caption{Notations for the multivariate time series and the patterns set.}
+    \label{fig:notation}
+\end{figure}
 
 \subsection{Applications}
 The goal of the multi-measure setup is dual.
diff --git a/PhD/research_proposal/glossaries.tex b/PhD/research_proposal/glossaries.tex
index 1a90094..391db2f 100644
--- a/PhD/research_proposal/glossaries.tex
+++ b/PhD/research_proposal/glossaries.tex
@@ -31,3 +31,4 @@
 \newacronym{cnn}{CNN}{Convolutional Neural Network}
 \newacronym{dpa}{DPA}{Differential Power Analysis}
 \newacronym{ics}{ICS}{Industrial Control System}
+\newacronym{hdd}{HDD}{Hard Drive Disk}
diff --git a/PhD/research_proposal/images/ssmm_illustration.pdf b/PhD/research_proposal/images/ssmm_illustration.pdf
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diff --git a/PhD/research_proposal/proposal.tex b/PhD/research_proposal/proposal.tex
index 9fee643..7a3b7fe 100644
--- a/PhD/research_proposal/proposal.tex
+++ b/PhD/research_proposal/proposal.tex
@@ -5,6 +5,7 @@
 % University of Waterloo, 200 University Ave. W., Waterloo, Ontario, Canada
 % FOR ASSISTANCE, please send mail to helpdesk@uwaterloo.ca
 
+
 % DISCLAIMER
 % To the best of our knowledge, this template satisfies the current uWaterloo thesis requirements.
 % However, it is your responsibility to assure that you have met all requirements of the University and your particular department.
@@ -202,6 +203,7 @@
 \input{futurwork}
 \input{timetable}
 \input{conclusion}
+
 %----------------------------------------------------------------------
 % END MATERIAL
 % Bibliography, Appendices, Index, etc.