finish last results

chapters/chap04/chap04.tex

@@ -144,10 +144,14 @@ all five 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          \\
+        \begin{tabular}{ccc ccc}
+            \toprule
+            \multicolumn{3}{c}{$\alpha$} & \multicolumn{3}{c}{$\beta$}                                         \\
+            \cmidrule{1-3}\cmidrule{4-6}
+            true                         & $\mu$                       & $\sigma$ & true   & $\mu$  & $\sigma$ \\
+            \midrule
+            0.3333                       & 0.3334                      & 0.0011   & 0.5000 & 0.5000 & 0.0017   \\
+            \bottomrule
         \end{tabular}
         \caption{The mean $\mu$ and standard deviation $\sigma$ across the 5
             independent iterations of training our PINNs with the synthetic dataset.}
@@ -161,7 +165,7 @@ While the predicted value is not precisely accurate, the standard deviation is
 sufficiently small, and taking the mean of multiple iterations produces an
 almost perfect result.\\
 
-In~\Cref{table:alpha_beta} we present the results of the training for the
+In~\Cref{table:state_mean_std} we present the results of the training for the
 real-world data. The results are presented from top to bottom, in the order of
 the community identification number, with the last entry being Germany. Both
 the mean $\mu$ and the standard deviation $\sigma$ are calculated across all
@@ -171,30 +175,34 @@ has the lowest $\sigma$.\\
 
 \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          \\
-            Lower Saxony           & 0.0735        & 0.0014           & 0.0962       & 0.0018          \\
-            Bremen                 & 0.0588        & 0.0018           & 0.0795       & 0.0025          \\
-            North Rhine-Westphalia & 0.0780        & 0.0009           & 0.1001       & 0.0011          \\
-            Hesse                  & 0.0653        & 0.0016           & 0.0854       & 0.0020          \\
-            Rhineland-Palatinate   & 0.0808        & 0.0016           & 0.1036       & 0.0018          \\
-            Baden-Württemberg      & 0.0862        & 0.0014           & 0.1132       & 0.0016          \\
-            Bavaria                & 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          \\
-            Saxony                 & 0.0797        & 0.0017           & 0.1073       & 0.0022          \\
-            Saxony-Anhalt          & 0.0932        & 0.0019           & 0.1207       & 0.0027          \\
-            Thuringia              & 0.0952        & 0.0011           & 0.1248       & 0.0016          \\
-            Germany                & 0.0803        & 0.0012           & 0.1044       & 0.0014          \\
+        \begin{tabular}{lccccc}
+            \toprule
+                                   & \multicolumn{2}{c}{$\alpha$} & \multicolumn{2}{c}{$\beta$} &                                        \\
+            \cmidrule{2-3}\cmidrule{4-5}
+            state name             & $\mu$                        & $\sigma$                    & $\mu$  & $\sigma$ & $e_{\text{synth}}$ \\
+            \midrule
+            Schleswig Holstein     & 0.0771                       & 0.0010                      & 0.0966 & 0.0013   & 0.0849             \\
+            Hamburg                & 0.0847                       & 0.0035                      & 0.1077 & 0.0037   & 0.0948             \\
+            Lower Saxony           & 0.0735                       & 0.0014                      & 0.0962 & 0.0018   & 0.0774             \\
+            Bremen                 & 0.0588                       & 0.0018                      & 0.0795 & 0.0025   & 0.0933             \\
+            North Rhine-Westphalia & 0.0780                       & 0.0009                      & 0.1001 & 0.0011   & 0.0777             \\
+            Hesse                  & 0.0653                       & 0.0016                      & 0.0854 & 0.0020   & 0.1017             \\
+            Rhineland-Palatinate   & 0.0808                       & 0.0016                      & 0.1036 & 0.0018   & 0.0895             \\
+            Baden-Württemberg      & 0.0862                       & 0.0014                      & 0.1132 & 0.0016   & 0.0796             \\\addlinespace
+            Bavaria                & 0.0809                       & 0.0021                      & 0.1106 & 0.0027   & 0.0952             \\
+            Saarland               & 0.0746                       & 0.0021                      & 0.0996 & 0.0024   & 0.1080             \\
+            Berlin                 & 0.0901                       & 0.0008                      & 0.1125 & 0.0008   & 0.0667             \\
+            Brandenburg            & 0.0861                       & 0.0008                      & 0.1091 & 0.0010   & 0.0724             \\
+            Mecklenburg-Vorpommern & 0.0910                       & 0.0007                      & 0.1167 & 0.0008   & 0.0540             \\
+            Saxony                 & 0.0797                       & 0.0017                      & 0.1073 & 0.0022   & 0.1109             \\
+            Saxony-Anhalt          & 0.0932                       & 0.0019                      & 0.1207 & 0.0027   & 0.0785             \\
+            Thuringia              & 0.0952                       & 0.0011                      & 0.1248 & 0.0016   & 0.0837             \\\addlinespace
+            Germany                & 0.0803                       & 0.0012                      & 0.1044 & 0.0014   & 0.0804             \\
+            \bottomrule
         \end{tabular}
         \caption{Mean and standard deviation across the 5 iterations, that we
             conducted for each German state and Germany as the whole country.}
-        \label{table:alpha_beta}
+        \label{table:state_mean_std}
     \end{center}
 \end{table}
 
@@ -259,12 +267,12 @@ real-world data.\\
 For the purposes of validation, we create a synthetic dataset, by setting the parameter
 of $\alpha$ and the reproduction value each to a specific values, and solving~\Cref{eq:reduced_sir_ODE}
 for a given time interval. We set $\alpha=\nicefrac{1}{3}$ and $\Rt$ to the
-values as can be seen in~\Cref{fig:synthetic_I_r_t} as well as the population
+values as can be seen in~\Cref{fig:Rt_dataset} 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 use this
 dataset to demonstrate, that our method is working on a simple and minimal
 dataset.\\ To obtain a dataset of the infectious group, consisting of the
-real-world data, we we processed the data of the dataset
+real-world data, we processed the data of the dataset
 \emph{COVID-19-Todesfälle in Deutschland} to extract the number of infections
 in Germany as a whole. For the German states, we use the data of \emph{SARS-CoV-2
     Infektionen in Deutschland}. In the preprocessing stage, we employ a constant
@@ -298,6 +306,7 @@ we restrict the data points to an interval of 1200 days, beginning on March 09.
         infectious group (left) and the corresponding true reproduction value
         $\Rt$ (right) for the synthetic data. The lower graphic exemplary
         illustrates the different curves for Germany.}
+    \label{fig:Rt_dataset}
 \end{figure}
 
 In order to achieve the desired output, the selected neural network
@@ -319,13 +328,13 @@ reduced SIR model and the reproduction number $\Rt$. First, we present
 our findings for the synthetic dataset. Then, we provide and discuss the
 results for the real-world data.\\
 
-\Cref{fig:synth_results} illustrates that the model successfully learns the
-synthetic training data, with an error of $e_{\text{synth}} = 0.0016$. Meanwhile,
-the prediction for the reproduction number $\Rt$ for the synthetic data, is accuracy,
-while having by far the highest standard deviation in the first 30 days. The
-error concerning the reproduction number is $e_{\Rt} = 0.0521$.
+\Cref{fig:synth_results} illustrates the results of our experiments conducted on
+the synthetic dataset, which can be seen in~\Cref{fig:Rt_dataset}. It is evident
+that the model is capable of learning the infection data across all data points.
+The error for this is, $e_{\text{synth}} = 0.0016$, which is of a negligible
+magnitude.\\
 
-\begin{figure}[t]
+\begin{figure}[h]
     \centering
     \begin{subfigure}{0.45\textwidth}
         \includegraphics[width=\textwidth]{synthetic_I_prediction.pdf}
@@ -341,34 +350,120 @@ error concerning the reproduction number is $e_{\Rt} = 0.0521$.
         standard deviation.}
 \end{figure}
 
+An examination of the predictions for the representation value $\Rt$ reveals
+that here as well, the model is capable of accurately delineating the value at
+each time point. However, during the first 30 days, the standard deviation is
+exhibits an upward trend, while during the final 120 days, the predictions
+demonstrate remarkable precision. The overall prediction of $\Rt$ has an error
+of $e_{\Rt} = 0.0521$.\\
+
+In~\Cref{fig:state_results}, we present the graphs of $\Rt$ for the state with
+the highest value of $\beta$, namely Thuringia, and for the state with the lowest
+transmission rate $\beta$, namely Bremen. Further visualizations of the results
+can be found in~\Cref{chap:appendix}. In all datasets, the graphs with $\alpha =
+    \nicefrac{1}{5}$ are of a smaller size than those with
+$\alpha = \nicefrac{1}{14}$. This is due to the fact that the individuals are
+being moved to the removed compartment at a faster rate. Resulting, it can be
+observed that the value of $\Rt$ is constantly remaining closer to the threshold
+of $\Rt=1$, while the reproduction number for datasets with $\alpha = \nicefrac{1}{14}$
+reaches values of up to 1.6. In states with higher values of $\beta$, the period
+during which the value of $\Rt$ is above the threshold of one 1 is longer, but
+the peak is lower. In states with a lower transmission rate, the period above 1
+is shorter, but the peak value is higher.\\
+
 \begin{figure}[t]
     \centering
     \begin{subfigure}{0.45\textwidth}
-        \includegraphics[width=\textwidth]{Germany_R_t_statistics.pdf}
+        \includegraphics[width=\textwidth]{I_prediction/Thueringen_I_prediction.pdf}
     \end{subfigure}
     \quad
     \begin{subfigure}{0.45\textwidth}
-        \includegraphics[width=\textwidth]{Germany_I_prediction.pdf}
-    \end{subfigure}
-    \vskip\baselineskip
-    \begin{subfigure}{0.45\textwidth}
-        \includegraphics[width=\textwidth]{R_t/Sachsen_Anhalt_R_t_statistics.pdf}
+        \includegraphics[width=\textwidth]{I_prediction/Bremen_I_prediction.pdf}
     \end{subfigure}
-    \quad
     \begin{subfigure}{0.45\textwidth}
         \includegraphics[width=\textwidth]{R_t/Thueringen_R_t_statistics.pdf}
     \end{subfigure}
-    \vskip\baselineskip
-    \begin{subfigure}{0.45\textwidth}
-        \includegraphics[width=\textwidth]{R_t/Bremen_R_t_statistics.pdf}
-    \end{subfigure}
     \quad
     \begin{subfigure}{0.45\textwidth}
-        \includegraphics[width=\textwidth]{R_t/Hessen_R_t_statistics.pdf}
+        \includegraphics[width=\textwidth]{R_t/Bremen_R_t_statistics.pdf}
     \end{subfigure}
     \label{fig:state_results}
-    \caption{text}
+    \caption{Visualization of the prediction of the training and  the graphs of
+        $\Rt$ for Thuringia (left) and Bremen (right) with both
+        $\alpha = \nicefrac{1}{14}$ and $\alpha = \nicefrac{1}{5}$. Events like
+        the peak of an influential variant are marked horizontally.}
 \end{figure}
 
+\Cref{table:state_error} presents data regarding the discrepancy between the
+predicted and actual values from the dataset for compartment $I$. It is evident,
+that the error for all experiments falls within a range of values that is not
+negligible and will have an influence on the resulting reproduction values that
+are learned while fitting the data. A comparison of the results for the various
+values of $\alpha$ reveals that the errors associated with $\alpha = \nicefrac{1}{14}$
+are consistently smaller, with the exception of Saxony and Germany. This can be
+attributed to the differing sizes of infection counts, particularly in relation
+to the normalization factor $C$. The model is unable to learn effectively if the
+values of the data loss $\mathcal{L}_{\text{data}}$ are too large or too small
+at the beginning.\\
+
+As illustrated in~\Cref{fig:state_results}, the training data is overlaid with the
+corresponding prediction of the model. We can observe that the prediction, though
+an exact reconstruction, accurately captures the general trajectory of the
+pandemic. The model's prediction demonstrates an ability to capture larger
+peaks, exhibiting a tendency to ignore smaller changes. This suggests that the
+prediction of the model is capable show the rough outline of the progression of COVID-19. In the
+beginning, the majority of predictions below $\Rt=1$, indicating an outbreak.
+As we observed in the synthetic data, the model exhibits a higher standard
+deviation at the boundaries. In the graphs, we mark the
+peaks of the most severe COVID-19 variants in Germany. While the peaks of the
+Alpha and Delta variants are clearly visible in the data, the model does not
+learn these, and thus they are not reflected in the results. The peak of the
+Omicron variant  represents the culmination of the COVID-19 pandemic in Germany
+and can be identified as the most prominent peak in the dataset. Immediately preceding this peak, we observe the highest
+value of the reproduction number across all states. This phenomenon can be explained, by
+number  of individuals infected by one infectious person reaching its peak. In
+some states the peaks of other Omicron variants after the maximum peak are visible (see Thuringia).\\
+
+The experiments demonstrate, that our model encounteres difficulties in learning the data for the
+states and Germany and consequently in predicting the reproduction values for each dataset.
+Nonetheless, the predictions illustrate the general trends of the most impactful
+events of the COVID-19 pandemic.\\
+
+\begin{table}[t]
+    \begin{center}
+        \begin{tabular}{lcc}
+            \toprule
+                                   & \multicolumn{2}{c}{$e_I$}                            \\
+            \cmidrule{2-3}
+            state name             & $\alpha=\nicefrac{1}{14}$ & $\alpha=\nicefrac{1}{5}$ \\
+            \midrule
+            Schleswig Holstein     & 0.2005                    & 0.2514                   \\
+            Hamburg                & 0.3045                    & 0.3357                   \\
+            Lower Saxony           & 0.2140                    & 0.3082                   \\
+            Bremen                 & 0.2370                    & 0.3838                   \\
+            North Rhine-Westphalia & 0.1718                    & 0.2460                   \\
+            Hesse                  & 0.2736                    & 0.3172                   \\
+            Rhineland-Palatinate   & 0.2442                    & 0.2674                   \\
+            Baden-Württemberg      & 0.1984                    & 0.2958                   \\\addlinespace
+            Bavaria                & 0.1928                    & 0.2825                   \\
+            Saarland               & 0.2554                    & 0.4676                   \\
+            Berlin                 & 0.1885                    & 0.2948                   \\
+            Brandenburg            & 0.2023                    & 0.2571                   \\
+            Mecklenburg-Vorpommern & 0.1518                    & 0.3272                   \\
+            Saxony                 & 0.3382                    & 0.2807                   \\
+            Saxony-Anhalt          & 0.1959                    & 0.2564                   \\
+            Thuringia              & 0.1401                    & 0.2221                   \\\addlinespace
+            Germany                & 0.3371                    & 0.2533                   \\
+            \bottomrule
+        \end{tabular}
+        \caption{This table displays all average values of the error $e_{\text{synth}}$
+            for all German states and Germany. The average is formed across all
+            10 iteration.}
+        \label{table:state_error}
+    \end{center}
+\end{table}
+
+
+
 
 % -------------------------------------------------------------------

thesis.tex

@@ -72,10 +72,10 @@
 %--------------------------------------------------
 %--------------------------------------------------
 
-\appendix
+%\appendix
 
 % if you do not have appendix sections, comment this include command out
-%\include{./chapters/appendix/appendix}
+\include{./chapters/appendix/appendix}
 
 \singlespacing