diff --git a/DSD/qrs/main.tex b/DSD/qrs/main.tex
index a41cb2c..8ea86f1 100644
--- a/DSD/qrs/main.tex
+++ b/DSD/qrs/main.tex
@@ -3,12 +3,22 @@
 %\input epsf
 \usepackage{graphicx}
 \usepackage{multirow}
+\usepackage{xcolor}
+\usepackage{booktabs}
+\usepackage{tabularx}
+\usepackage{algpseudocodex}
+\usepackage{algorithm}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{amsthm}
 \usepackage[toc,acronym,abbreviations,nonumberlist,nogroupskip]{glossaries-extra}
 
 \renewcommand\thesection{\arabic{section}} % arabic numerals for the sections
 \renewcommand\thesubsectiondis{\thesection.\arabic{subsection}.}% arabic numerals for the subsections
 \renewcommand\thesubsubsectiondis{\thesubsectiondis.\arabic{subsubsection}.}% arabic numerals for the subsubsections
 
+\newtheorem{problem-statement}{Problem Statement}
+
 \newcommand\agd[1]{{\color{red}$\bigstar$}\footnote{agd: #1}}
 \newcommand\SF[1]{{\color{blue}$\bigstar$}\footnote{sf: #1}}
 \newcommand{\cn}{{\color{purple}[citation needed]}}
@@ -157,4 +167,461 @@ The similarities are evaluated with Nearest-Neighbor rules or \gls{svm}.
 Their approach bears some interesting similarities with the one presented in this paper.
 However, they evaluate their work on the recognition of handwritten numerals, which is far from the use case we are interested in.
 
+\section{Problem Statement}\label{sec:statement}
+%\gls{mad} focuses on detecting the state of a time series at any point in time.
+We consider the problem from the point of view of multi-class, mono-label classification problem \cite{aly2005survey} for every sample in a time series.
+The problem is multi-class because multiple states can occur in one-time series, and therefore any sample is assigned one of multiple states.
+The problem is mono-label because only one state is assigned to each sample.
+The classification is a mapping from the samples space to the states space.
+
+\begin{problem-statement}[\gls{mad}]
+Given a discretized time series $t$ and a set of patterns $P=\{P_1,\dots, P_n\}$, identify a mapping $m:\mathbb{N}\longrightarrow P\cup \lambda$ such that every sample $t[i]$
+maps to a pattern in $P\cup \lambda$ with the condition that the sample matches an occurrence of the pattern in $t$.
+\end{problem-statement}
+
+The time series $t: \mathbb{N} \longrightarrow \mathbb{R}$ is a finite, discretized, mono-variate, real-valued time series.
+The patterns (also called training samples) $P_j \in P$ are of the same type as $t$.
+Each pattern $P_j$ can take any length denoted $N_j$.
+A sample $t[i]$ \textit{matches} a pattern $P_j \in P$ if there exists a substring of $t$, the length of $P_j$, that includes the sample, such that a similarity measure between this substring and $P_j$ is below a pre-defined threshold.
+The pattern $\lambda$ is the \textit{unknown} pattern assigned to the samples in $t$ that do not match any of the patterns in $P$.
+
+\begin{figure}
+\centering
+\includegraphics[width=0.45\textwidth]{images/overview.pdf}
+\caption{Illustration of the sample distance from one sample to each training example in a 2D space.}
+\label{fig:overview}
+\end{figure}
+
+\section{Proposed Solution: MAD}\label{sec:solution}
+\gls{mad}'s core idea separates it from other traditional sliding window algorithm.
+In \gls{mad}, the sample window around the sample to classify dynamically adapts for optimal context selection.
+This principle influences the design of the detector and requires the definition of new distance metrics.
+Because the patterns lengths may differ, our approach requires distance metrics that are robust to length variations.
+%For the following explanation, the pattern set $P$ refers to the provided patterns only $\{P\setminus \lambda\}$ --- unless specified otherwise.
+We first define the fundamental distance metric as the normalized Euclidean distance between two-time series $a$ and $b$ of the same length $N_a=N_b$
+\begin{equation}
+    nd(a,b) = \dfrac{EuclideanDist(a,b)}{N_a}
+\end{equation}
+
+Using this normalized distance $nd$, we define the distance from a sample $t[i]$ to a pattern $P_j \in P$.
+This is the sample distance $sd$ defined as
+\begin{equation}\label{eq:sd}
+    sd(i,P_j) = \min_{k\in [i-N_j,i+N_j])}(nd(t[i-k:i+k],P_j))
+\end{equation}
+
+%with $P_j$ the training sample corresponding to the state $j$, and $t$ the complete time series.
+Computing the distance $sd(i,P_j)$ requires to: (1) select every substring of $t$ of length $N_j$ that contains the sample $t[i]$, (2) evaluate their normalized distance to the pattern $P_j$, and (3) consider $sd(i,P_j)$ as the smallest of these distances.
+For simplicity, Equation~\ref{eq:sd} omits the border conditions for the range of $k$.
+When the sample position $i$ is less than $N_j$ or greater than $N_t-N_j$, the range adapts to only consider valid substrings.
+
+Our approach uses a threshold-based method to decide what label to assign to a sample.
+For each sample in $t$, the algorithm compares the distance $sd(i,P_j)$ to the threshold $T_j$.
+The sample receives the label $j$ associated with the pattern $P_j$ that results in the smallest distance $sd(i,P_j)$ with $sd(i,P_j)<T_j$.
+
+The minimum distance from the pattern $P_j$ to all other patterns $P_l$ with $l\neq i$ --- denoted $ID_j$ --- forms the basis of the threshold $T_j$.
+Intuitively, the patterns in $P$ represent most of the patterns expected in the trace.
+Thus, to decide that a substring matches a pattern $P_j$, it must match $P_j$ better than any other pattern $P_l$ with $l\neq i$ does.
+Otherwise, the distance metric justifies to assign the label of $P_j$ to a pattern of another label instead of the substring, which is counter-intuitive.
+The inter-distance between $P_j$ to $P_l$, defined as
+\begin{equation}
+ID(P_j,P_l) = \min_{i\in[0,N_l-N_j]} nd(P_j,P_l[i:i+N_j])
+\end{equation}
+represents the smallest distance between $P_j$ and any substring of length $N_j$ from $P_l$ --- with $N_l>N_j$.
+If $N_l<N_j$, then $ID(P_j,P_l) = ID(P_l,P_j)$.
+In other words, when computing the inter-distance between two patterns, we slide the short pattern along the length of the long one and compute the normalized distance at every position to finally consider only the smallest of these distances as the inter-distance.
+
+To fully define the threshold $T_j$, we introduce the shrinkage coefficient $\alpha$.
+This coefficient, multiplied with the smallest inter-distance $ID_j$, forms the threshold $T_j$.
+\begin{equation}
+T_j = \alpha\times ID_j = \alpha \min_{l\in[1;k];l\neq j} \{ID(e_j,e_l)\}
+\end{equation}
+The shrinkage coefficient $\alpha$ provides some control over the confidence of the detector.
+A small value shrinks the range of capture of each label more and will leave more samples classified as \textit{unknown}.
+A large value leaves less area for the \textit{unknown} state and forces the detector to choose a label, even for samples unlike any pattern.
+The \textit{unknown} label enables the detector to carry over the information of novelty to the output.
+In cases where a substring does not resemble any pattern --- for example, in cases of anomalies, or unforeseen activities ---, the ability to inform of novel patterns enables a more granular definition of security policies.
+
+Finally, we assign to each sample the label of the closest pattern with a distance lower than its threshold.
+\begin{equation}
+s_i = \underset{j\in[1,k]}{\arg\min}(sd(i,e_j) \textrm{ with } sd(i,e_j)<T_j)
+\end{equation}
+In the case where no distance is below the threshold, the sample defaults to the \textit{unknown} state.
+
+\subsection{Algorithm}
+The algorithm for \gls{mad} follows three steps:
+
+\begin{enumerate}
+\item Compute the inter-distances and threshold values for the pattern set. The algorithm can reuse the result from this step for all following detection with the same pattern set.
+\item For each sample $t[i]$, compute the sample distance to each pattern $\{sd(i,p) \forall p\in P\}$.
+\item Select the label by comparing the sample distances to the threshold.
+\end{enumerate}
+
+However, directly implementing this suite of operations is not optimal as it requires computing the distance from any substring to any pattern multiple times --- exactly once per sample in the substring.
+A more efficient solution considers each substring only once.
+In other words, iterating over the patterns rather than the samples is more efficient as it replaces distance computations with comparison operations.
+The efficient implementation follows the operations:
+
+\begin{enumerate}
+\item Compute the inter-distances and threshold values for the pattern set --- no optimization at this step.
+\item For every pattern $P_j$ of length $N_j$ in $P$, consider every substring of length $N_j$ in $t$ and compute the normalized distance $nd(t[i:i+N_j],P_j)$.
+\item For every sample in the substring, store the minimum of the previously stored and newly computed normalized distance as the sample distance.
+\item Select the label by comparing the sample distances to the thresholds.
+\end{enumerate}
+This results in the same final value for the sample distance $sd(i,P_j)$ with less computations of the normalized distance --- at the expense of cheaper comparison operations.
+Algorithm~\ref{alg:code} presents the implementation's pseudo-code.
+
+\begin{algorithm}
+\caption{Pseudo code for state detection.}
+\label{alg:code}
+\begin{algorithmic}[1]
+\Require $t$ the time series of length $N_t$, $P$ the set $n$ of patterns, $\alpha$ the shrinkage coefficient.
+\BeginBox
+\LComment{First part: computation of the thresholds.}
+\State $interDistances \gets nilMatrix(n,n)$
+\State $thresholds \gets nilList(n)$
+\For{$i \in [0,n-1]$}
+    \For{$j \in [0,n-1]$}
+        \If{$i\neq j$ and $interDistance[i,j] \neq Nil$}
+            \State $dist \gets ID(P[i],P[j]$
+            \State $interDistances[i,j] \gets dist$
+            \State $interDistances[j,i] \gets dist$
+        \EndIf
+    \EndFor
+    \State $thresholds[i] \gets min(interDistances[i,:])$
+\EndFor
+\EndBox
+
+\BeginBox
+\LComment{Second part: computation of the distances.}
+\State $distances \gets nilMatrix(n,N_t)$
+\State $labels \gets nilList(N_t)$
+
+\For{$i \in [0,n-1]$}
+    \For{$k \in [0,N_t-1]$}
+        \State $dist \gets nd(t[k:k+N_{P_i}], P_i)$
+        \For{$l\in [0,N_t-1]$}
+            \State $distances[i,k] \gets min(distances[i,k], dist)$
+        \EndFor
+    \EndFor
+\EndFor
+\EndBox
+
+\BeginBox
+\LComment{Third part: selection of the label based on the distances.}
+\For{$k \in [0,N_t-1]$}
+    \State $rowMin \gets Nil$
+    \State $distanceMin \gets \infty$
+    \For{$i \in [0,n-1]$}
+        \If{$distances[i,k] \leq thresholds[i]$}
+            \State $rowMin \gets i$
+            \State $distanceMin \gets distances[i,k]$
+        \EndIf
+    \EndFor
+    \State $labels[k] \gets rowMin$
+\EndFor
+\EndBox
+
+\State \Return $labels$
+
+\end{algorithmic}
+\end{algorithm}
+
+
+\subsection{Analysis}
+
+\textbf{Time-Efficiency:}
+\agd{Better time efficiency analysis and comparison with the efficiency of \gls{1nn}}
+The time efficiency of the algorithm is expressed as a function of the number of normalized distance computations and the number of comparison operations.
+Each part of the algorithm has its own time-efficiency expression with Algorithm~\ref{alg:code} showing each of the three parts.
+The first part, dedicated to the threshold computation, is polynomial in the number of patterns and linear in the length of each pattern.
+The second part, in charge of the distances computation, is linear in the number of patterns, the length of the time series, and the length of each pattern.
+Finally, the third part, focusing on the final label selection, is linear in both the length of the time series and the number of patterns.
+Overall, the actual detection computation --- second and third parts --- is linear in all input sizes.
+Adding an additional value to the time series triggers the computation of one more distance value per pattern, hence the linear relationship.
+Similarly, lengthening a pattern by one triggers one more comparison operation for each substring of the time series, hence the linear relationship.
+Concluding from the analysis, the additional operations introduced by \gls{mad} over the traditional \gls{1nn} do not significantly impact the time efficiency of the detection that remains linear.
+
+\textbf{Termination:}
+Every part of the algorithm terminates.
+The first part iterates on the patterns with two nested loops over the samples of two patterns.
+No instruction modifies the patterns that are all of finite lengths.
+Thus the loops always terminate.
+The second part iterates over the patterns and the time series with two nested loops.
+Similarly to the first part, the time series is finite and never altered.
+Thus the second part also terminates.
+Finally, the third part uses the sames loops as the second and also terminates.
+Overall, \gls{mad} always terminates for any finite time series and finite set of finite patterns.
+
+\textbf{Monotony of number of unknown sample}\agd{find better title}
+Explain that the number of unknown sample is monotonic as a function of alpha.
+Also, a sample that is classified as unknown will always remain unknown if alpha decreases.
+
+\section{Evaluation}
+The evaluation of \gls{mad} consists in the detection of the states for time series from various machines.
+We evaluate the performance of the proposed solution against other traditional methods to illustrate the capabilities and advantages of \gls{mad}.
+
+\subsection{Performance Metrics}
+We considered two metrics to illustrate the performance of \gls{mad}.
+Performance evaluations of labeling systems traditionally use the accuracy \cite{grandini2020metrics}.
+Accuracy is defined as the number of correctly classified samples divided by the total number of samples.
+However, accuracy only illustrates a part of the performances.
+In the context of state detection, we are interested in taking actions depending on the state of a system.
+Detecting the start and stop times of each state is not as important as detecting the correct list of occurrences of states.
+We are interested in making sure that the state is detected, even at the cost of some time inaccuracy.
+The Levenshtein distance~\cite{4160958} illustrates the classifier's performance at detecting the correct list of states from a time series.
+The Levenshtein distance is defined as the number of single-character edits --- insertions, deletions or substitutions --- between two strings.
+The Levenshtein distance could use the raw detected labels list as input.
+However, the raw label list embeds state detection time information, which the Levenshtein distance is very sensitive to.
+We first reduce the ground truth and the detected labels by removing immediate duplicate of labels.
+This reduction removes timing information yet conserves the global order of state occurrences.
+The Levenshtein distance between the ground truth and the detected labels is low if every state occurrence is correctly detected.
+Similarly the metric is high if states occurrences are missed, added, or miss-detected.
+To remove length bias and make the metric comparable across datasets, we normalize the raw Levenshtein distance and define it as
+\begin{equation}
+    levacc = \dfrac{Levenshtein(rgtruth,rlabels)}{max(rN_t,rN_l)}
+\end{equation}
+with $rgtruth$ and $rlabels$ respectively the reduced ground truth and reduced labels and $rN_t$ and $rN_l$ their length.
+The Levenshtein distance provides complementary insights in the performance of the detection in this specific use case.
+Figure~\ref{fig:metrics} illustrates the impact of an error on both metrics.
+It is important to notice that zero represents the best Levenshtein distance and one the worst --- contrary to the accuracy.
+
+\begin{figure}
+\centering
+\includegraphics[width=0.49\textwidth]{images/metric.pdf}
+\caption{Accuracy or Levenshtein distance alone are unable to illustrate all types of error. We consider both to provide a better evaluation of the performances.}
+\label{fig:metrics}
+\end{figure}
+
+
+\subsection{Dataset}\label{sec:dataset}
+\agd{include more datasets from REFIT. One per house would be perfect but simply more is already good. Add in annexe why other are rejected.}
+We evaluate the performance of \gls{mad} against eight time series.
+One is a simulated signal composed of sine waves of varying frequency and average.
+Four were captured in a lab environment on consumer-available machines (two NUC PCs and two wireless routers).
+Finally, two were extracted from the REFIT dataset \cite{278e1df91d22494f9be2adfca2559f92} and correspond to home appliances during real-life use.
+Table~\ref{tab:dataset} presents the times series and their characteristics.
+
+\begin{table}
+    \centering
+    \begin{tabular}{lcc}
+        Name & length & Number of states\\
+        \toprule
+        NUCPC-0 & 22700 & 11\\
+        NUCPC-1 & 7307  & 8\\
+        GENERATED & 15540 & 18\\
+        WAP-ASUS & 26880 & 18\\
+        WAP-LINKSYS & 22604 & 18\\
+        REFIT-H4A4 & 5366 & 17\\
+        REFIT-H4A1 & 100000 & 142\\
+        \bottomrule
+    \end{tabular}
+    \caption{Characteristics of the machines in the evaluation dataset.}
+    \label{tab:dataset}
+\end{table}
+
+The dataset aims to provide a diverse machine and state patterns to evaluate the performances.
+For each time series, we generated the ground truth by manually labeling all sections of the time series using a custom-made range selection tool based on a Matplotlib \cite{Hunter:2007} application.
+The dataset is publicly available \cite{zenodo}.
+
+\textbf{Lab Captures:}
+NUCPC-0, NUCPC-1, WAP-ASUS and WAP-LINKSYS correspond to lab-captured machine activity power consumption.
+A commercial solution \cite{palitronica}, placed in series with the main power cable, measures the global power consumption of the machine.
+We considered two types of machines.
+The NUCPC-* are small form factor general-purpose computers.
+The WAP-* are wireless access points from two different brands.
+The states to detect on these computing machines are \textit{powered off}, \textit{boot sequence}, and \textit{on}.
+With these states, it is possible to set up many security rules such as: \textit{"machine on after office hours"}, \textit{"X reboots in a row"} or \textit{"Coincident shutdown of Y machines within Z minutes"}.
+
+\textbf{GENERATED:}
+An algorithm generated the GENERATED time series following 3 steps.
+First, the algorithm randomly selects multiple frequency/average pairs.
+Second, the algorithm generates 18 segments, by selecting a pair and a random length.
+Finally, the algorithm concatenates the segments to form the complete time series.
+The patterns correspond to a minimal length example of each pair.
+This time series illustrates the capabilities of the proposed solution in a case where a simple threshold would fail.
+
+\textbf{REFIT:}
+In 215, D. Murray et. al \cite{278e1df91d22494f9be2adfca2559f92} created the REFIT dataset for \gls{nilm} research.
+This dataset is now widely used in this research area.
+REFIT is composed of the global consumption of 20 houses along with the specific consumption of nine appliances per house.
+The global house consumption does not fit the problem statement of this paper as multiple patterns overlap.
+However the individual consumption of some appliances fit the problem statement and two were selected.
+The REFIT-H4A1 is the first appliance of the fourth house and corresponds to a fridge.
+The REFIT-H4A4 is the fourth appliance of the fourth house and corresponds to a washing machine.
+The activity in this second time series was very sparse with long periods without consumption.
+The no consumption sections: are not a challenging --- i.e., all detectors perform well on this type of pattern ---, make the manual labeling more difficult, and level all results up.
+For this reason we removed large sections of inactivity between active segments to make the time series more challenging without tempering with the order of detector performances.
+
+\input{refit_table}
+
+\subsection{Alternative Methods}
+\agd{Explain better why the alternative methods are chosen.}
+\agd{explain how the svm and mlp are trained.}
+We implemented three alternative methods to compare with the performance of \gls{mad}.
+These methods are commonly deployed to detect patterns in a time series.
+The methods are: a \gls{1nn} detector, an \gls{svm} classifier, and an \gls{mlp} classifier.
+More complex solutions like \gls{rnn} or \gls{cnn} show good performances on time series analysis but require too much data to be applicable to one-shot classification.
+All alternative methods rely on a sliding window to extract substring to classify.
+The window is centred around the sample.
+This choice --- or any other placement of the window --- implies that some samples corresponding to the length of the longest pattern remain unclassified toward the ends.
+The stride of the window is a single sample to consider every possible window.
+Each extracted window is sent to the classifier, and the result is applied to the sample at the center of the window.
+The alternative detectors are not meant to handle variable-size time series as input.
+For the \gls{svm} and \gls{mlp} detectors, the window size is shorter than the shortest pattern.
+The training sample extraction algorithm slides the window along all patterns to extract all possible substrings.
+These substrings constitute the training dataset with multiple samples per pattern.
+The \gls{1nn} considers one window per pattern length around each sample.
+Every window is compared to its pattern, and the normalized Euclidean distance is considered for the decision.
+Overall, it is possible to adapt the methods to work with variable length patterns, but \gls{mad} is the only pattern-length-agnostic method by design.
+
+\subsection{Results}\label{sec:results}
+The benchmark consists in detecting the label of every sample for each time series with each method and compute the performance metrics.
+The detectors that require training (\gls{svm} and \gls{mlp}) were re-trained for every evaluation.
+Figure~\ref{fig:res} presents the results.
+\gls{mad} is consistently as or more accurate than the alternative method.
+The Levenshtein distance illustrates how \gls{mad} provides a smoother and less noisy labeling.
+This stability introduces less state detection errors that could falsly trigger security rules.
+With both performances metrics combined, \gls{mad} outperforms the other methods.
+
+\begin{figure*}
+\centering
+\includegraphics[width=\textwidth]{images/dsd_acc.pdf}
+\caption{Performances of the different methods on all the datasets.}
+\label{fig:res}
+\end{figure*}
+
+\subsection{Influence of $\alpha$}
+The shrink coefficient $\alpha$ is the only hyperparameter of the detector.
+Its default value is one.
+$\alpha$ controls the threshold of similarity that a substring should cross to get qualified as a match to a pattern.
+$\alpha$ takes its value in $\mathbb{R}_*^+$.
+The default value for $\alpha$ is one.
+This value follows the intuitive reasoning presented in Section~\ref{sec:solution}.
+
+To better understand the influence of the shrink coefficient, the algorithm can be perceived as a 2D area segmentation problem.
+Let us consider the 2D plane where each pattern has a position based on its shape.
+A substring to classify also has a position in the plane and a distance to each pattern (see bottom part of Figure~\ref{fig:overview}).
+During classification, the substring takes the label of the closest pattern.
+For any pattern $P_j$, the set of positions in the plane that are assigned to $P_j$ --- i.e., the set of positions for which $P_j$ is the closest pattern --- is called the area of attraction of $P_j$.
+In a classic \gls{1nn} context, every point in the plane is in the area of attraction of one pattern.
+
+This infinite area of attraction is not a desirable feature in this context.
+Let us consider now a time series exhibiting anomalous or unforeseen behavior.
+Some substrings in this time series do not resemble any of the provided pattern.
+In an infinite area of attraction context, the anomalous points are assigned to a pattern, even if they poorly match it.
+As a result, the behavior of the security rule can become unpredictable as anomalous points can receive a seemingly random label.
+
+A more desirable behavior of the state detection system is to inform of the presence of unpredicted behavior.
+This behavior naturally emerges when the areas of attraction of the patterns are limited to a finite size.
+The shrink coefficient $\alpha$ --- through the modification of the threshold $T_j$ --- provides control over the shrink of the areas of attraction.
+The lower the value of $\alpha$, the smaller the areas of attraction around each sample.
+Applying a coefficient to the thresholds produces a reduction of the radius of the area of attraction, not an homothety of the initial areas.
+In other words, the shrink does not preserve the shape of the area.
+For a value $\alpha < 0.5$, all areas become disks --- in the 2D representation --- and all shape information are lost.
+
+The impact of the $\alpha$ coefficient on the classification is monotonic and predictable.
+Because $\alpha$ influences the thresholds, changing $\alpha$ results in moving the transitions in the detected labels.
+In other words, a lower value of $\alpha$ expands the unknown segments while a higher value shrinks them until they disappear.
+Figure~\ref{fig:alpha_impact} illustrates the impact $\alpha$ on the width of unknown segments.
+The impact of $\alpha$ on the number of unknown sample is also monotonic.
+
+\begin{proof}
+We prove the monotony of the number of unknown samples as a function of $\alpha$ by induction.
+The base case is $\alpha=0$.
+In this case, the threshold for every pattern $P_j\in P$ is $T_j = \alpha\times ID_j = 0$.
+With every $T_j=0$, no sample can have a distance below the threshold and every sample is labeled as \textit{unknown}.
+
+For the induction case, let us consider $\alpha$ increasing from the value $\alpha_0$ to $\alpha_1 = \alpha_0 + \delta$ with $\delta \in \mathbb{R}_*^+$.
+The increasing of $\alpha$ induces the increase of every threshold $T$ from the value $T_0$ to $T_1$
+\begin{equation}
+\alpha_0 <\alpha_1 \rightarrow T_0 < T_1
+\end{equation}
+
+For every value of every threshold $T$ we can define a set of all samples below the threshold as $S_T$.
+When a threshold increases from $T_0$ to $T_1$, all the samples in $S_{T_0}$ also belong in $S_{T_1}$ by the transitivity of order in $\mathbb{R}_*^+$.
+It is also possible for samples to belong to $S_{T_1}$ but not to $S_{T_0}$ if their distance falls between $T_0$ and $T_1$.
+Hence, $S_{T_0}$ is a subset of $S_{T_1}$ and the cardinality of $S_T$ as a function of $T$ is monotonically non-decreasing.
+
+We conclude that the number of unknown samples --- i.e.,samples above every thresholds --- as a function of $\alpha$ is monotonically non-increasing.
+\end{proof}
+
+
+Figure~\ref{fig:alpha} presents the number of unknown samples in the classification of the NUCPC-1 time series based on the value of $\alpha$.
+
+
+\begin{figure}
+\centering
+\includegraphics[width=0.49\textwidth]{images/alpha.pdf}
+\caption{Evolution of the number of unknown samples based on the value of the shrink coefficient $\alpha$.}
+\label{fig:alpha}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics[width=0.49\textwidth]{images/alpha_impact.pdf}
+\caption{Behavior of the classifier with different values of $\alpha$. A lower value of $\alpha$ expands the unknown sections (orange sections)}
+\label{fig:alpha_impact}
+\end{figure}
+
+\begin{figure*}
+\centering
+\includegraphics[width=\textwidth]{images/areas.pdf}
+\caption{2D visualization of the areas of capture around each pattern as $\alpha$ changes. When $\alpha \ggg 2$, the areas of capture tends to equal these of a classic \gls{1nn}.}
+\label{fig:areas}
+\end{figure*}
+
+\section{Discussion}\label{sec:discussion}
+In this section we highlight specific aspects of the proposed solution.
+
+Side-channel based state detection enables a more robust security policy enforcement.
+Let us consider the classic case of some security policies in a company.
+The office hours are set between 8 am and 8 pm.
+Outside of office hours, a security policy specifies that no computer should be on --- or should not be awake.
+The traditional way of enforcing such policies would be to have a server evaluates the state of each computer remotely (via a PING command, for example) or to have an agent on each computer sending the state to a server.
+Both cases are highly susceptible to bypass.
+A local attacker could boot a system on a secondary OS and immediately disable all agents on the machine.
+A remote attacker could infect the machine and forge the reported data.
+Any attacker that can disable the network connection would make the activities invisible to the policy enforcement system.
+All of these methods have no impact on a side-channel intrusion detection system.
+Whatever the motivations of the attacker, there are no malicious operations that do not require the machine to consume power.
+The capability to detect the state of the system independently of the willingness of the system itself is a major step forward in enabling robust security policies enforcement on computing devices.
+
+The proposed method have some limitations that are important to acknowledge.
+The current version of \gls{mad} is tailored for a specific use case.
+The goal is to enable high-level security policies with a secure and reliable state detection of a machine from a time series.
+The purpose of the state detection is not anomaly or novelty detection at the time series level.
+For this reason, the patterns to be detected by \gls{mad} bear some limitations.
+First, the patterns must be distinct.
+If two patterns share a significant portion of time series, \gls{mad} will have an issue leading to unstable results.
+Second, the states must be hand selected.
+The data requirement is extremely low --- only one sample per pattern --- so the selected samples must be reliable.
+For now, a human expert decided on the best patterns to select.
+While there is nothing particularly difficult in the selection, it is still a highly manual process that we hope to automatize with future iterations.
+Finally, the states must be consistent.
+If a state has an unpredictable signature --- i.e., each occurence display a significantly different pattern ---, \gls{mad} will not be able to detect the occurences reliably.
+
+\gls{mad} is not limited to one-shot cases and can leverage more labeled data.
+\gls{mad} is based on a \gls{1nn}, so the evolution to \gls{knn} is natural.
+If more than one pattern is available for one state, \gls{mad} will apply the same detection method only with multiple patterns leading to the same label.
+The number of training samples per class can be unbalanced, and the training samples within a class can have different lengths.
+\gls{mad} preserves the versatility of a \gls{knn} solution in this regard.
+
+\gls{mad} remains time-efficient compared to a classic \gls{1nn}.
+Although there are more operations to perform to evaluate all possible windows around a sample, the impact on detection time is small.
+Over all the datasets considered, the time for \gls{mad} was, on average, 14\% higher than the time for the \gls{1nn}.
+\gls{mad} is also slower than \gls{svm} and faster than \gls{mlp}, but comparison to other methods is less relevant as computation time is highly sensitive to implementation, and no optimization was attempted.
+Finally, because \gls{mad} is distance-based and window-based, parallelization is naturally applicable and can significantly reduce the processing time.
+
+\section{Conclusion}
+We present \gls{mad}, a novel solution to enable high-level security policy enforcement from side channel information.
+Leveraging side channel information requires labeling samples to discover the state of the monitored system.
+Additionally, in the use cases where side-channels are leveraged, collecting large labeled datasets can be challenging.
+\gls{mad} is designed around three core features: low data requirement, flexibility of the detection capabilities, and stability of the results.
+Built as a variation of a traditional \gls{1nn}, \gls{mad} uses a dynamic window placement that always provides the most relevant context for sample classification.
+One hyper-parameter, $\alpha$, controls the confidence of the detector and the trade-off between un-classified and miss-classified samples.
+The comparison to traditional state detection methods highlights the potential of \gls{mad} for the pre-processing of raw data for security applications.
+
+\bibliographystyle{splncs04}
+\bibliography{biblio}
+
 \end{document}