fix proof image

DSD/qrs/main.tex

@@ -689,19 +689,24 @@ This leads to believe that \gls{mad} balances the tradeoff between simple, expla
 \section{Discussion}\label{sec:discussion}
 In this section we highlight specific aspects of the proposed solution.
 
-\textbf{Dynamic Window Placement vs Fixed Windows: }
+\textbf{Dynamic Window vs Fixed Windows: }
 One of the core mechanism of \gls{mad} is the ability to choose the best fitting window to calssify each sample.
 This mechanism is crucial to overcome some of the shortcomings of a traditional \gls{1nn}.
 It is important to understand the advantages of this dynamic window placement to fully appreciate the performances of \gls{mad}
 Figure~\ref{fig:proof} illustrates test case that focuses on the comarison between the two methodes.
 In this figure, the top graph represent the near-perfect classification of the trace into different classes by \gls{mad}.
 To make the results more comparables, the $\alpha$ parameter of \gls{mad} was set to $\infty$ to avoid the distance threshold mechanism and focuse on the dynamic window placement.
-The bottom graph represent the classification by a \gls{1nn} and it illustrates the three types of errors that \gls{mad} aims to overcome.
+The middle graph represent the classification by a \gls{1nn} and it illustrates the three types of errors that \gls{mad} aims to overcome.
+The bottom graph represent the predicted state fro each sample by each method with -~1 the UNKNOWN state and $[0-4]$ the possible states of the trace.
 
 \begin{itemize}
-    \item Transition Bleeding Error: Around transitions, tends to miss the exact transition timing and miss-classify samples. This si explained by the rigidity of the window around the sample. At the transition time, the two halfs of the window are competing to match different states. Depending on the shape of the states involved, it may require much more than half of the window to prefer the new state, leading to miss-classified samples around the transition. In contrast, \gls{mad} will always choose a window that fully matches either of the states and that is not across the transition, avoiding the transition error.
-    \item Out-of-phase Error: When a states is describe by multiple iterations of a periodic pattern, the match between a window and the trace varies dramatically every half-period. When the window is in phase with the pattern, the match is maximal and \gls{1nn} perfectly fills its role. However, when the window and the pattern are out of phase, the match is minimal and the nearest neighbor may be a flat pattern at the average level of the pattern. \gls{mad} avoids this error by moving the window by, at most, half a period to ensure a perfect match with the periodic pattern.
-    \item Unknown-Edges Error: Because of the fixed nature of the window of a \gls{1nn}, every sample that is less than a half window away from either end cannot be classified. This error is not so important in most cases where edge sample are less important and many solutions are available to solve this issue. However, \gls{mad} naturally solve this issue by shifting the window in the valid range until the edge. \agd{check figure to make DSD go to the end}
+    \item Transition Bleeding Error: Around transitions, tends to miss the exact transition timing and miss-classify samples.
+        This si explained by the rigidity of the window around the sample.
+        At the transition time, the two halfs of the window are competing to match different states.
+        Depending on the shape of the states involved, it may require much more than half of the window to prefer the new state, leading to miss-classified samples around the transition.
+        In contrast, \gls{mad} will always choose a window that fully matches either of the states and that is not across the transition, avoiding the transition error.
+    \item Out-of-phase Error: When a states is describe by multiple iterations of a periodic pattern, the match between a window and the trace varies dramatically every half-period. When the window is in phase with the pattern, the match is maximal and \gls{1nn} perfectly fills its role. However, when the window and the pattern are out of phase, the match is minimal and the nearest neighbor may be a flat pattern at the average level of the pattern. This error manifests itself through predictions switching between two values at half the period of the pattern. \gls{mad} avoids this error by moving the window by, at most, half a period to ensure a perfect match with the periodic pattern.
+    \item Unknown-Edges Error: Because of the fixed nature of the window of a \gls{1nn}, every sample that is less than a half window away from either end cannot be classified. This error is not so important in most cases where edge sample are less important and many solutions are available to solve this issue. However, \gls{mad} naturally solve this issue by shifting the window only in the valid range until the edge.
 \end{itemize}
 
 There surely are other methods than \gls{mad} to sovle these issues.
@@ -713,10 +718,9 @@ The dynamic window placement also avoid increased complexity by requiring the sa
 \begin{figure*}
 \centering
 \includegraphics[width=0.9\textwidth]{images/proof.pdf}
-    \caption{Classification comparison between MAD and 1-NN.}
+    \caption{Classification comparison between MAD and 1-NN with examples of prediction error from 1-NN highlighted. Top graph is \gls{mad}, middle graph is 1-NN, bottom graph is the prediction vector of both methods.}
 \label{fig:proof}
 \end{figure*}
-\agd{make figure better. Add rects around types of errors}