begin results for SIR

.vscode/settings.json

@@ -1,5 +1,6 @@
 {
     "cSpell.words": [
-        "PINN"
+        "PINN",
+        "Thuringia"
     ]
 }

chapters/chap04/chap04.tex

@@ -98,7 +98,7 @@ in which the COVID-19 disease was the most active and severe.
         and~\Cref{eq:modSIR}. The Germany data is taken from the death case
         data set. Exemplatory we show illustrations of the datasets of Schleswig
         Holstein, Berlin, and Thuringia. For the other states see~\Cref{chap:appendix} }
-    \label{fig:datasets}
+    \label{fig:datasets_sir}
 \end{figure}
 
 The PINN that we employ consists of seven hidden layers with twenty neurons
@@ -117,6 +117,52 @@ setups that we describe in this section.
 
 \subsection{Results    4}
 \label{sec:sir:results}
+
+\begin{figure}[t]
+    \centering
+    \includegraphics[width=0.7\textwidth]{reproducability.pdf}
+    \caption{Visualization of all 5 predictions for the synthetic dataset,
+        compared to the true values of $\alpha = \nicefrac{1}{3}$ and $\beta = \nicefrac{1}{2}$}
+    \label{fig:reprod}
+\end{figure}
+
+In this section we describe the results, that we obtain from the conducted
+experiments, that we describe in the preceding section. First we show the
+results for the synthetic dataset and look at the accuracy and reproducibility.
+Then we present and discuss the results for the German states and Germany.\\
+
+The results of the experiment regarding the synthetic data can be seen
+in~\Cref{table:alpha_beta_synth} and in~\Cref{fig:reprod}.~\Cref{fig:reprod}
+shows the values of $\beta$ and $\alpha$ of each iteration compared to the true
+values of $\beta=\nicefrac{1}{2}$ and $\alpha=\nicefrac{1}{3}$. In~\Cref{table:alpha_beta_synth}
+we present the mean $\mu$ and standard variation $\sigma$ of both values across
+all 5 iterations.\\
+
+\begin{table}[h]
+    \begin{center}
+        \begin{tabular}{ccc|ccc}
+            true $\alpha$ & $\mu(\alpha)$ & $\sigma(\alpha)$ & true $\beta$ & $\mu(\beta)$ & $\sigma(\beta)$ \\
+            \hline
+            0.3333        & 0.3334        & 0.0011           & 0.5000       & 0.5000       & 0.0017          \\
+        \end{tabular}
+        \caption{The mean $\mu$ and standard variation $\sigma$ across the 5
+            independent iterations of training our PINNs with the synthetic dataset.}
+        \label{table:alpha_beta_synth}
+    \end{center}
+\end{table}
+From the results we can see that the model is able to approximate the correct
+parameters for the small, synthetic dataset in each of the 5 iterations. Even
+though the predicted value is never exactly correct, the standard deviation is
+negligible small and taking the mean of multiple iterations yields an almost
+perfect result.\\
+
+In~\Cref{table:alpha_beta} we present the results of the training for the
+real-world data. These are presented from top to bottom, in the order of the
+community identification number, with the last entry being Germany. $\mu$ and
+$\sigma$ are both calculated across all 5 iterations of our experiment. We can
+see that the values of \emph{Hamburg} have the highest standard deviation, while
+\emph{Mecklenburg Vorpommern} has the smallest $\sigma$.\\
+
 \begin{table}[h]
     \begin{center}
         \begin{tabular}{c|cc|cc}
@@ -145,6 +191,28 @@ setups that we describe in this section.
         \label{table:alpha_beta}
     \end{center}
 \end{table}
+
+In~\Cref{fig:alpha_beta_mean_std} we visualize the means and standard variations
+in contrast to the national values. The states with the highest transmission rate
+values are Thuringia, Saxony Anhalt and Mecklenburg West-Pomerania. It is also,
+visible that all six of the eastern states have a higher transmission rate than
+Germany. These results may be explainable with the ratio of vaccinated individuals\footnote{\url{https://impfdashboard.de/}}.
+The eastern state have a comparably low complete vaccination ratio, accept for
+Berlin. While Berlin has a moderate vaccination ratio, it is also a hub of
+mobility, which means that contact between individuals happens much more often. This is also a reason for Hamburg being a state with an above national standard rate of transmission.
+\\
+
+
+
+We visualize these numbers in~\Cref{fig:alpha_beta_mean_std},
+where all means and standard variations are plotted as points, while the values
+for Germany are also plotted as lines to make a classification easier. It is
+visible that Hamburg, Baden-Württemberg, Bayern and all six of the states that
+lie in the eastern part of Germany have a higher transmission rate $\beta$ than
+overall Germany. Furthermore, it can be observed, that all values for the
+recovery $\alpha$ seem to be correlating to the value of $\beta$, which can be
+explained with the assumption that we make when we preprocess the data using the
+recovery queue by setting the recovery time to 14 days.
 \begin{figure}[h]
     \centering
     \includegraphics[width=\textwidth]{mean_std_alpha_beta_res.pdf}
@@ -192,50 +260,87 @@ we restrict the data points to an interval of 1200 days starting from March 09.
 \begin{figure}[h]
     %\centering
     \setlength{\unitlength}{1cm} % Set the unit length for coordinates
-    \begin{picture}(12, 9.5) % Specify the size of the picture environment (width, height)
-        \put(1.5, 4.5){
+    \begin{picture}(12, 14.5) % Specify the size of the picture environment (width, height)
+        \put(0, 10){
             \begin{subfigure}{0.3\textwidth}
                 \centering
-                \includegraphics[width=\textwidth]{SIR_synth.pdf}
+                \includegraphics[width=\textwidth]{I_synth.pdf}
+                \caption{Synthetic data}
                 \label{fig:synthetic_I}
             \end{subfigure}
         }
-        \put(8, 4.5){
+        \put(4.75, 10){
             \begin{subfigure}{0.3\textwidth}
                 \centering
-                \includegraphics[width=\textwidth]{datasets_states/Germany_SIR_14.pdf}
-                \label{fig:germany_I}
+                \includegraphics[width=\textwidth]{datasets_states/Germany_I_14.pdf}
+                \caption{Germany with $\alpha=\nicefrac{1}{14}$}
+                \label{fig:germany_I_14}
+            \end{subfigure}
+        }
+        \put(9.5, 10){
+            \begin{subfigure}{0.3\textwidth}
+                \centering
+                \includegraphics[width=\textwidth]{datasets_states/Germany_I_5.pdf}
+                \caption{Germany with $\alpha=\nicefrac{1}{5}$}
+                \label{fig:germany_I_5}
+            \end{subfigure}
+        }
+        \put(0, 5){
+            \begin{subfigure}{0.3\textwidth}
+                \centering
+                \includegraphics[width=\textwidth]{datasets_states/Nordrhein_Westfalen_I_14.pdf}
+                \caption{NRW with $\alpha=\nicefrac{1}{14}$}
+                \label{fig:schleswig_holstein_I_14}
+            \end{subfigure}
+        }
+        \put(4.75, 5){
+            \begin{subfigure}{0.3\textwidth}
+                \centering
+                \includegraphics[width=\textwidth]{datasets_states/Hessen_I_14.pdf}
+                \caption{Hessen with $\alpha=\nicefrac{1}{14}$}
+                \label{fig:berlin_I_14}
+            \end{subfigure}
+        }
+        \put(9.5, 5){
+            \begin{subfigure}{0.3\textwidth}
+                \centering
+                \includegraphics[width=\textwidth]{datasets_states/Thueringen_I_14.pdf}
+                \caption{Thüringen with $\alpha=\nicefrac{1}{14}$}
+                \label{fig:thüringen_I_14}
             \end{subfigure}
         }
         \put(0, 0){
             \begin{subfigure}{0.3\textwidth}
                 \centering
-                \includegraphics[width=\textwidth]{datasets_states/Schleswig_Holstein_SIR_14.pdf}
-                \label{fig:schleswig_holstein_I}
+                \includegraphics[width=\textwidth]{datasets_states/Nordrhein_Westfalen_I_5.pdf}
+                \caption{NRW with $\alpha=\nicefrac{1}{5}$}
+                \label{fig:schleswig_holstein_I_5}
             \end{subfigure}
         }
         \put(4.75, 0){
             \begin{subfigure}{0.3\textwidth}
                 \centering
-                \includegraphics[width=\textwidth]{datasets_states/Berlin_SIR_14.pdf}
-                \label{fig:berlin_I}
+                \includegraphics[width=\textwidth]{datasets_states/Hessen_I_5.pdf}
+                \caption{Hessen with $\alpha=\nicefrac{1}{5}$}
+                \label{fig:berlin_I_5}
             \end{subfigure}
         }
         \put(9.5, 0){
             \begin{subfigure}{0.3\textwidth}
                 \centering
-                \includegraphics[width=\textwidth]{datasets_states/Thueringen_SIR_14.pdf}
-                \label{fig:thüringen_I}
+                \includegraphics[width=\textwidth]{datasets_states/Thueringen_I_5.pdf}
+                \caption{Thüringen with $\alpha=\nicefrac{1}{5}$}
+                \label{fig:thüringen_I_5}
             \end{subfigure}
         }
 
     \end{picture}
-    \caption{Synthetic and real-world training data. The synthetic data is
-        generated with $\alpha=\nicefrac{1}{3}$ and $\beta=\nicefrac{1}{2}$
-        and~\Cref{eq:modSIR}. The Germany data is taken from the death case
-        data set. Exemplatory we show illustrations of the datasets of Schleswig
-        Holstein, Berlin, and Thuringia. For the other states see~\Cref{chap:appendix} }
-    \label{fig:datasets}
+    \caption{Visualization of the datasets for the training process.
+        Illustration (a) is the synthetic data. For the real-world data we use a
+        dataset with $\alpha=\nicefrac{1}{14}$ and $\alpha=\nicefrac{1}{5}$ each.
+        (b) and (c) for Germany, (d) and (g) for Nordrhein-Westfalen (NRW), (e) and (h)
+        for Hessen, and (f) and (i) for Thüringen.}
+    \label{fig:i_datasets}
 \end{figure}
 
 For this task the chosen architecture of the neural network consists of 4 hidden