add second setup section

chapters/chap04/chap04.tex

@@ -44,7 +44,7 @@ The parameters are set to $\alpha = \nicefrac{1}{3}$ and $\beta = \nicefrac{1}{2
 The size of the population is $N = \expnumber{7.6}{6}$ and the initial amount of
 infectious individuals of is $I_0 = 10$. We simulate over 150 days and get a
 dataset of the form of~\Cref{fig:synthetic_SIR}.\\For the real-world RKI data we
-preprocess the row data data of each state and Germany separately using a
+preprocess the raw data of each state and Germany separately using a
 recovery queue with a recovery period of 14 days. As for the population size of
 each state we set it to the respective value counted at the end of 2019\footnote{\url{https://de.statista.com/statistik/kategorien/kategorie/8/themen/63/branche/demographie/\#overview}}.
 The initial number of infectious individuals is set to the number of infected
@@ -117,30 +117,34 @@ setups that we describe in this section.
 
 \subsection{Results    4}
 \label{sec:sir:results}
-\begin{center}
-    \begin{tabular}{c|cc|cc}
-                               & $\alpha$ & $\sigma(\alpha)$ & $\beta$ & $\sigma(\beta)$ \\
-        \hline
-        Schleswig Holstein     & 0.0771   & 0.0010           & 0.0966  & 0.0013          \\
-        Hamburg                & 0.0847   & 0.0035           & 0.1077  & 0.0037          \\
-        Niedersachsen          & 0.0735   & 0.0014           & 0.0962  & 0.0018          \\
-        Bremen                 & 0.0588   & 0.0018           & 0.0795  & 0.0025          \\
-        Nordrhein-Westfalen    & 0.0780   & 0.0009           & 0.1001  & 0.0011          \\
-        Hessen                 & 0.0653   & 0.0016           & 0.0854  & 0.0020          \\
-        Rheinland-Pfalz        & 0.0808   & 0.0016           & 0.1036  & 0.0018          \\
-        Baden-Württemberg      & 0.0862   & 0.0014           & 0.1132  & 0.0016          \\
-        Bayern                 & 0.0809   & 0.0021           & 0.1106  & 0.0027          \\
-        Saarland               & 0.0746   & 0.0021           & 0.0996  & 0.0024          \\
-        Berlin                 & 0.0901   & 0.0008           & 0.1125  & 0.0008          \\
-        Brandenburg            & 0.0861   & 0.0008           & 0.1091  & 0.0010          \\
-        Mecklenburg Vorpommern & 0.0910   & 0.0007           & 0.1167  & 0.0008          \\
-        Sachsen                & 0.0797   & 0.0017           & 0.1073  & 0.0022          \\
-        Sachsen-Anhalt         & 0.0932   & 0.0019           & 0.1207  & 0.0027          \\
-        Thüringen              & 0.0952   & 0.0011           & 0.1248  & 0.0016          \\
-        Germany                & 0.0803   & 0.0012           & 0.1044  & 0.0014          \\
-    \end{tabular}
-\end{center}
-
+\begin{table}[h]
+    \begin{center}
+        \begin{tabular}{c|cc|cc}
+                                   & $\mu(\alpha)$ & $\sigma(\alpha)$ & $\mu(\beta)$ & $\sigma(\beta)$ \\
+            \hline
+            Schleswig Holstein     & 0.0771        & 0.0010           & 0.0966       & 0.0013          \\
+            Hamburg                & 0.0847        & 0.0035           & 0.1077       & 0.0037          \\
+            Niedersachsen          & 0.0735        & 0.0014           & 0.0962       & 0.0018          \\
+            Bremen                 & 0.0588        & 0.0018           & 0.0795       & 0.0025          \\
+            Nordrhein-Westfalen    & 0.0780        & 0.0009           & 0.1001       & 0.0011          \\
+            Hessen                 & 0.0653        & 0.0016           & 0.0854       & 0.0020          \\
+            Rheinland-Pfalz        & 0.0808        & 0.0016           & 0.1036       & 0.0018          \\
+            Baden-Württemberg      & 0.0862        & 0.0014           & 0.1132       & 0.0016          \\
+            Bayern                 & 0.0809        & 0.0021           & 0.1106       & 0.0027          \\
+            Saarland               & 0.0746        & 0.0021           & 0.0996       & 0.0024          \\
+            Berlin                 & 0.0901        & 0.0008           & 0.1125       & 0.0008          \\
+            Brandenburg            & 0.0861        & 0.0008           & 0.1091       & 0.0010          \\
+            Mecklenburg Vorpommern & 0.0910        & 0.0007           & 0.1167       & 0.0008          \\
+            Sachsen                & 0.0797        & 0.0017           & 0.1073       & 0.0022          \\
+            Sachsen-Anhalt         & 0.0932        & 0.0019           & 0.1207       & 0.0027          \\
+            Thüringen              & 0.0952        & 0.0011           & 0.1248       & 0.0016          \\
+            Germany                & 0.0803        & 0.0012           & 0.1044       & 0.0014          \\
+        \end{tabular}
+        \caption{Mean and standard variation across the 5 iterations, that we
+            conducted for each German state and Germany as the whole country.}
+        \label{table:alpha_beta}
+    \end{center}
+\end{table}
 \begin{figure}[h]
     \centering
     \includegraphics[width=\textwidth]{mean_std_alpha_beta_res.pdf}
@@ -164,7 +168,84 @@ are described in~\Cref{sec:pinn:rsir}.
 \label{sec:rsir:setup}
 In this section we describe the choice of parameters and configuration for data
 generation, preprocessing and the neural networks. We use these setups to train
-the PINNs to find the reproduction number on both synthetic and real-world data.
+the PINNs to find the reproduction number on both synthetic and real-world data.\\
+
+For validation reasons we create a synthetic dataset, by setting the parameters
+of $\alpha$ and $\beta$ each to a specific value, and solving~\Cref{eq:modSIR}
+for a given time interval. We set $\alpha=\nicefrac{1}{3}$ and
+$\beta=\nicefrac{1}{2}$ as well as the population size $N=\expnumber{7.6}{6}$
+and the initial amount of infected people to $I_0=10$. Furthermore, we set our
+simulated time span to 150 days.We will use this dataset to show, that our
+method is working on a simple and minimal dataset.\\ For the real-world data we
+we processed the data of the dataset \emph{COVID-19-Todesfälle in Deutschland}
+to extract the number of infections in the whole of Germany, while we used the
+data of \emph{SARS-CoV-2 Infektionen in Deutschland} for the German states. For
+the preprocessing we use a constant rate for $\alpha$ to move individual into
+the removed compartment. First we choose $\alpha = \nicefrac{1}{14}$ as this is
+covers the time of recovery\footnote{\url{https://github.com/robert-koch-institut/SARS-CoV-2-Infektionen_in_Deutschland.git}}.
+Second we use $\alpha=\nicefrac{1}{5}$ since the peak of infectiousness is
+reached right in front or at 5 days into the infection\footnote{\url{https://www.infektionsschutz.de/coronavirus/fragen-und-antworten/ansteckung-uebertragung-und-krankheitsverlauf/}}.
+Just as in~\Cref{sec:sir} we set the population size $N$ of each state and
+Germany to the corresponding size at the end of 2019. Also, for the same reason
+we restrict the data points to an interval of 1200 days starting from March 09.
+2020.
+\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{subfigure}{0.3\textwidth}
+                \centering
+                \includegraphics[width=\textwidth]{SIR_synth.pdf}
+                \label{fig:synthetic_I}
+            \end{subfigure}
+        }
+        \put(8, 4.5){
+            \begin{subfigure}{0.3\textwidth}
+                \centering
+                \includegraphics[width=\textwidth]{datasets_states/Germany_SIR_14.pdf}
+                \label{fig:germany_I}
+            \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}
+            \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}
+            \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}
+            \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}
+\end{figure}
+
+For this task the chosen architecture of the neural network consists of 4 hidden
+layers with each 100 neurons. The activation function is the tangens
+hyperbolicus function tanh. We weight the data loss with a weight of
+$\expnumber{1}{6}$ into the total loss. The model is trained using a base
+learning rate of $\expnumber{1}{-3}$ with the same scheduler and optimizer as
+we use in~\Cref{sec:sir:setup}. We train the model for 20000 epochs. Also, we
+conduct each experiment 15 times to reduce the standard deviation.
+
 
 
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