From ac7e7cfef5e1ea07e2e735af2afcd995022afc6d Mon Sep 17 00:00:00 2001
From: sutymate <sutymate@fit.cvut.cz>
Date: Fri, 7 May 2021 19:14:37 +0200
Subject: [PATCH 1/2] expand SA for n, ks, ko

---
 report/MatejSutyBP.pdf | Bin 4427305 -> 6465659 bytes
 report/MatejSutyBP.tex | 133 ++++++++++++++++++++++++-----------------
 2 files changed, 79 insertions(+), 54 deletions(-)

diff --git a/report/MatejSutyBP.pdf b/report/MatejSutyBP.pdf
index 4f03364abd1b81df678e15ba55b1514843369b40..d8b1ec3278d0a1c16761d620a356e1aff4c49fd2 100644
GIT binary patch
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diff --git a/report/MatejSutyBP.tex b/report/MatejSutyBP.tex
index d26f09a..a25d8ba 100644
--- a/report/MatejSutyBP.tex
+++ b/report/MatejSutyBP.tex
@@ -529,14 +529,24 @@ indicators
 							\chapter{Sensitivity analysis}\label{chapter_analysis_design}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-This chapter presents research performed on input parameters using sensitivity analysis in OptiSLang software.
-
+This chapter presents research based on quantitative and qualitative analysis of the input parameters. On one hand, the quantitative analysis measures the influence of the individual input parameters on the simulation using sensitivity analysis in OptiSLang software. On the other hand, qualitative analysis of the model explains the observable structures that form during the simulation and show how the parameters are linked together. The qualities of the model were researched by observing running simulations and by statistical methods of analysis. 
 
 
 \section{Input parameters}
-The course of the simulation is determined by input parameters. Some parameters, such as number of agents $n$, size of the room $W \times H$ or location of the \textit{exit}, are not subject to calibration as they correspond to the real world setting and can't be manipulated. The number of agents $n$, however, can show how the model responds to the congestion phenomenon --- more agents are approaching the exit than are being evacuated/egressed through the exit.
+The course of the simulation is affected by input parameters. Some input parameters, such as number of agents $n$, size of the room $W \times H$ or location of the \textit{exit}, are not subject to calibration of the model as they correspond to the real world setting and can't be manipulated. The number of agents $n$, however, can show how the model responds to the congestion phenomenon --- more agents are approaching the exit than are being evacuated/egressed through the exit.
+
+
+Individual agents are assigned different aggressivity parameter $\gamma$. With higher value, the agent is more likely to win the conflict. The influence of $\gamma$ was researched in Article~\ref{aggressivity}.
+
+Simulations in this thesis always assign $\gamma$ to agents uniformly from the interval $[0,1]$ with granularity $m=10$, thus $\forall{}\gamma\in{}\{0.0,0.1,\dots{},0.9,1.0\}$. These values are generated by PRNG $G_1$ initialized by seed $I$, which also generates the positions of the agents on the grid. Each iteration has seed $I=1245$.
+
+The sensitivity parameters ($k_S,~k_D,~k_O$) are also input parameters. These parameters are uniformly assigned to all agents (if not stated other), and they affect their behavior, for example their solution of conflicts.
+Both the aggressivity $\gamma$ and sensitivity parameters can be assigned uniformly to agents in groups as can be found in simulation in Section~\ref{other_discovery}.
+
+
+The friction parameter $\mu{}$ is a global parameter of the model that affects the stochastic occurrence of blocking occasions from conflicts.
+All simulations in this thesis use the new $strategy B$ for $choosing destination cell$ and unchanged $strategy A$ for $solving conflicts$, because $strategy B$ was ruled out as described in Section~\ref{other_discovery}.
 
-The influence of the sensitivity parameters ($k_S,~k_D,~k_O,~\mu$) on the simulation needs to be analyzed.
 
 \subsection{Numeric intervals}
 In Equation~\ref{dest_strat_mix} the sensitivity to occupancy $k_O$, the sensitivity to diagonal movement $k_D$, and in conflict solution, the friction parameter $\mu$, can be set to values from interval $[0,1]$. This interval can be studied using discrete values that cover the range. Contrary to this, the sensitivity to the \textit{static field} $k_S$ can be any positive real number and requires different approach as will be described later.
@@ -588,33 +598,30 @@ J_{ij} &= \frac{m}{\Delta{t}}
 
 
 \section{Methods of analysis}
-To measure the influence of the parameters, OptiSLang software uses Metamodel of Optimal Prognosis (MOP). At first it calculates the quality of each available model (Polynomial, MLS, Kriging) using Coefficient of Prognosis(COP) and then chooses the one with the highest quality. This model then evaluates the simulation data --- input parameters and output --- using cross validation and approximates the influence of input variables.
-\textit{The variance contribution of a single input variable is quantified by the product of the CoP and the total effect sensitivity index estimated from the approximation model.}~\cite{MOP_COP2011}
+Sensitivity analysis was performed on OptiSLang software developed by Dynardo GmbH. The graphs exported from OptiSLang have this information in the caption. Other graphs are made using Matplotlib.
 
-\subsubsection*{Ansys OptiSLang}
-The sensitivity analysis of input parameters was performed on OptiSLang. This software was developed by Dynardo GmbH. \textit{TBA: More text will be added when I use the software. }
+
+To measure the influence of the parameters, OptiSLang uses Metamodel of Optimal Prognosis (MOP). The metamodel can be Polynomial, MLS or Krigins model. At first, OptiSLang calculates the quality of each available model (Polynomial, MLS, Kriging) using Coefficient of Prognosis(COP) and then chooses the one with the highest quality. COP measures the ability of the metamodel to predict the data. Evacuation in this model is stochastic and the total evacuation time can be different for the same input parameters. Metamodel uses cross validation and other methods to make better prognoses. The MOP with highest COP then evaluates the simulation data --- input parameters $k_S, k_O, k_D, \mu{}$ and the simulation response $T_\text{TET}$ --- and creates a 3D projection that approximates the influence of individual input parameters on $T_\text{TET}$.
+\textit{The variance contribution of a single input variable is quantified by the product of the COP and the total effect sensitivity index estimated from the approximation model.}~\cite{MOP_COP2011}
 
 \section{Simulations}
 At first, the simulations were observed on the Mesa framework web interface, that is depicted in~Figure~\ref{MESA-map}. They were used mostly during implementation of the model and in testing the new strategies.
-Later, batch simulations on wide range of intervals were performed. The input parameters of simulations $S_1,\ S_2,\  S_3,\ S_4$ are in Table~\ref{table_simulations} below. All simulations had the same model configuration of grid $15 \times 15$, exit placed at border $(0, 8)$, same seed $I = 1245$ for PRNG $G_1$. Each iteration had different seed $J$ for PRNG $G_2$, that solves conflicts and selects destination cells.
-
-%...
-
+Later, batch simulations on wide range of intervals were performed. All simulations had the same model configuration of grid $15 \times 15$, exit placed at border $(0, 8)$, same seed $I = 1245$ for PRNG $G_1$. Each iteration had different seed $J$ for PRNG $G_2$, that solves conflicts and selects destination cells.
+In the Table~\ref{table_simulations} are input parameters of simulations for SA in OptiSLang. Other analyses have configurations in the text.
+The parameter $k_S$ can be any positive real number. The first simulations showed that $T_\text{TET}$ does not change very much for values higher than $5$ so this value was set as the upper range of $k_S$.
 \begin{table}[h!]
   \begin{center}
     \label{tab:table1}
     \begin{tabular}{|l|c|l|l|l|l|}
     \hline
      \multicolumn{2} {|c|}{} & \multicolumn{4} {c|}{Simulation name}\\
-      \hline            \textbf{Parameters} & \textbf{Step} & $S_1$ & $S_2$ & $S_3$  & $S_4$ \\
+      \hline            \textbf{Parameters} & \textbf{Step} & $S_2$ & $S_3$ & $S_4$  & $S_4$ \\
       \hline
 	  iterations & - & 3 & 1 & 1 & 5\\
-	  $n$ agents & 10 & 70 & [50, 90] & 70 & [10, 220]\\
-      $k_S$ & 0.1 & [0.3, 5.0] & 2.5 & [0.3, 5.0] & 1.5\\
-	  $k_O$ & 0.2 & [0.0, 1.0] & [0.0, 1.0] & [0.0, 1.0] & 0.3\\
-      $k_D$ & 0.3 & [0.0, 0.9] & [0.0, 0.9] & [0.0, 0.9] & 0.5\\
-	  $\mu$ & 0.3 & [0.0, 0.9] & [0.0, 0.9] & [0.0, 0.9] & 0.5\\
-	  $strategy$ & - & B & B & A & B\\
+      $k_S$ & 0.1 & [0.3, 5.0] & [1.5, 4.5] & \{1.5, 3.0, 4.5\} & 1.5\\
+	  $k_O$ & 0.1 & [0.0, 1.0] & [0.0, 1.0] & [0.0, 1.0] & 0.3\\
+      $k_D$ & 0.2 & [0.0, 0.1] & [0.0, 1.0] & [0.0, 0.9] & 0.5\\
+	  $\mu$ & 0.2 & [0.0, 0.1] & [0.0, 1.0] & [0.0, 0.9] & 0.5\\
 	  \hline
     \end{tabular}
   \end{center}
@@ -622,19 +629,26 @@ Later, batch simulations on wide range of intervals were performed. The input pa
   \label{table_simulations}
 \end{table}
 
+In the 3D and COP graphs exported from OptiSLang, the axes are renamed with capitals $S, O, D, F, T$ and mean $\protect{S=k_S, O=k_O, D=k_D, , F=\mu{}, T=T_\text{TET}}$. 3D graphs approximate the $T_\text{TET}$ with data from simulations and plot them on a plane. COP graphs shows how much individual parameters  $S, O, D, F, T$  contribute to the prediction of $T_\text{TET}$. The number in percent at the top of COP (for example on the right in Figure~\ref{ks_influence}) shows much many results from the simulations MOP was able to approximate. 
+%...
 %...
 
 \newpage
 \chapter{Results}
 
-This in general provides two types of result. The graph of influence based on COP and approximate MOP.
+The results of quantitative analysis in this chapter are explained by graphs from OptiSLang: the 3D plot of $T_\text{TET}$ and COP of individual parameters. The discoveries of qualitative analysis, such as linear dependency of $T_\text{TET}$ on number of agents $n$, are depicted by other graphs, for example histograms or boxplots.
 
 \section{Number of agents $\protect{n}$}
-As expected, the increase of total evacuation time $T_\text{TET}$ is linear depending on the number of agents \textit{n}. Figure~\ref{n_dep} demonstrates the linear dependency of $T_\text{TET}$ on $n$. 
-This test was performed on a grid  $15\times15$, with \textit{exit} placed in the middle of the border at position $(0, 8)$. The sensitivity parameters for blue boxplots are $k_O=0.9,~k_D=0.5,~\mu=0.9$ and for red boxplots $k_O=0.9,~k_D=0.5,~\mu=0.1$.
-Number of agents $n \in{\{10, 20, 30,\dots,210\}}$. Simulation was interated $30$ times.
+It was expected, that the increase of total evacuation time $T_\text{TET}$ is linear and depends on the number of agents \textit{n}. Figure~\ref{n_dep} demonstrates the linear dependency of $T_\text{TET}$ on $n$.
+In both figures, the simulations with higher friction $\mu{}=0.9$ show increasing variance of $T_\text{TET}$.\newline
+
 
+The sensitivity parameters for blue boxplots are $k_O=0.9,~k_D=0.5,~\mu=0.9$ and for red boxplots $k_O=0.9,~k_D=0.5,~\mu=0.1$. Each simulation was repeated $30$ times with differente seed $J$ for $G_2$.
+Horizontal axis $x$ is the number of agents $n$ in tens and vertical axis $y$ is total evacuation time $T_\text{TET}$. The boxplots show the variance in $T_\text{TET}$ for each $n$. There are two graphs with different $k_S\in{}\{1.5, 3.5\}$. This shows that the influence of $n$ is significant and linear, and cannot be compared to influences of friction or sensitivity parameters.\newline
 
+
+This happens because with more agents the conflicts occur more frequently and this results in more blocking occasions. Contrary to this, the boxplots in red are rather consistent. These figures demonstrate linear dependency of $T_\text{TET}$ on number of agents. The angle of the linearity is influence by other parameters.
+\pagebreak
 \begin{figure}[h!]
 	\begin{center}
 		\begin{subfigure}[t]{0.49\textwidth}
@@ -646,10 +660,11 @@ Number of agents $n \in{\{10, 20, 30,\dots,210\}}$. Simulation was interated $30
 	    \caption{$\protect{k_S=3.5}$}  		
   		\end{subfigure}
   	\end{center}
-	\caption{Linear dependency of TET on number of agents.}
+	\caption{Linear dependency of TET on number of agents. Higher friction, blue boxplots, increases variance of the results. Two graphs with different $\protect{k_S}$ show linear dependence regardless of this parameter.}
 	\label{n_dep}
 \end{figure}
 
+
 \newpage
 \section{Sensitivity to static field $\protect{k_S}$}
 Notice the $P_S$ member in Equation~\ref{dest_strat_ps} from \textit{strategy for choosing destination cell}.
@@ -658,31 +673,41 @@ P_S(y) = \frac{\exp(-k_SS(y))(1 - k_DD(y))}{\sum_{z\in{N}}  \exp(-k_SS(z))(1 - k
 \end{equation*}
 $P_S$ grows exponentially with $k_S$. The influence of $k_S$ can be very strong for high numbers, and marginal for values lower than $1$. It was noticed during the testing phases of the model implementation, when the simulations with very low $k_S$ values lasted very long. Visual examination of simulation exposed the erratic movement of agents that did not progress to the exit. 
 
-Figure~\ref{ks_influence} shows very high total evacuation time on axis $T$ for $k_S<1$ on axis $S$. The influence of $\mu{}$ on axis $F$ might seem marginal, but as will be presented later, the friction parameter $\mu{}$ plays an important role in the course of the evacuation.
+Figure~\ref{ks_influence} is the 3D plot of approximated $T_\text{TET}$ from simulation $S_2$ in Table~\ref{table_simulations}, exported from OptiSLang. The black dots are the values of $T_\text{TET}$ from the simulations. Vertical axis $T$ is the $T_\text{TET}$, on horizontal axis $F$ are values of friction $\mu{}$ and horizontal axis $S$ holds the values of sensitivity to static field $k_S$.  This figure shows very high $T_\text{TET}$ on axis $T$ for $k_S<1$ on axis $S$. For~$k_S~\gg{}~4.5$ the influence of $k_O, k_D$ is non-existent. Because $T_\text{TET}$ for these values is meaningless and steep slope lowers the resolution of the graph, the interval of $k_S$ was limited to $[1.5, 4.5]$.  The influence of $\mu{}$ on axis $F$ might seem marginal, but, as will be presented later, the friction parameter $\mu{}$ plays an important role in the course of the evacuation.
 
 \begin{figure}[h!]
 	\begin{center}
-		\begin{subfigure}[b]{\linewidth}
+		\begin{subfigure}[b]{0.6\linewidth}
 	    \includegraphics[width=\linewidth]{img/visuals/03-50/3D lq SF.png}
   		\end{subfigure}
+		\begin{subfigure}[b]{0.38\linewidth}
+	    \includegraphics[width=\linewidth]{img/visuals/03-50/COP 03-50.pdf}
+  		\end{subfigure}
   	\end{center}
-	\caption{Influence of $\protect{k_S}$ on total evacuation time.}
+	\caption{Left: 3D graph of total evacuation time, exported from OptiSLang.  Right: COP of input parameters, exported from OptiSLang. Data from simulation $\protect{S_2}$. 
+	For~$\protect{k_S<1.5}$ the total evacuation time is erratic and meaningless, which lead to limit of the range of $\protect{k_S~\text{to}~[1.5, 4.5]}$.}
 	\label{ks_influence}
 \end{figure}
+\pagebreak
+% Horizontal axes are parameters friction $\protect{\mu{}~\text{and}~k_S.}$
 
-
-The values $k_S\in{}[1.5, 4.5]$  bring meaningful results and allow other parameters to influence the simulation. For~$k_S~\gg{}~4.5$ the influence of $k_O, k_D$ is marginal. For further simulations $k_S$ was assigned the constant values of $k_S\in{}\{1.5,3.5,4.5\}$.  Parameters $k_O, k_D, \mu{}$ were set to cover the $[0,1]$ range in uniform discreet steps as follows: $k_0~\in{}\{0.1,0.2,\dots{},1.0\}$ and~$k_D,\mu{}~\in{}\{0.0, 0.2,\dots{},1.0\}$.
+The values $k_S\in{}[1.5, 4.5]$  bring meaningful results and allow other parameters to influence the simulation as can be seen in Figure~\ref{limited_ks}.
 
 \begin{figure}[h!]
 	\begin{center}
 		\begin{subfigure}[b]{0.6\linewidth}
-	    \includegraphics[width=\linewidth]{img/visuals/03-50/COP 03-50.pdf}
+	    \includegraphics[width=\linewidth]{img/visuals/15-45/3D 15-45 lq SF.png}
+  		\end{subfigure}
+		\begin{subfigure}[b]{0.38\linewidth}
+	    \includegraphics[width=\linewidth]{img/visuals/15-45/COP15-45.pdf}
   		\end{subfigure}
   	\end{center}
-	\caption{Irrelevant, maybe in appendix.}
-	\label{low_ks}
+	\caption{Left: 3D graph of $\protect{T_\text{TET}}$ from OptiSLang. Right: COP of input parameters, exported from OptiSLang. Data from simulation $\protect{S_3}$. With limited range of $\protect{k_S}$, the contribution of individual parameters in COP, on the right, is well distributed. \textit{Note: The angle of view is different compared to Figure~\ref{ks_influence}, axes $F$ and $S$ are switched.}}
+	\label{limited_ks}
 \end{figure}
 
+According to COP the most significant parameter is friction $\mu$. 
+
 
 \newpage
 \section{Sensitivity to occupancy $\protect{k_O}$}
@@ -694,7 +719,7 @@ Lower values of $k_O$ allow the agent to choose an occupied cell as his target c
     		\includegraphics[width=\linewidth]{img/vector/figX3.pdf}
 	\end{subfigure}
 \end{center}
-\caption{influence of ko and mu.}
+\caption{Different sets of parameters $\protect{k_O, \mu}$ and how they affect the $\protect{T_\text{TET}}$. Vertical axis is averaged $\protect{T_\text{TET}}$ from three simulations and horizontal axis is variable $\protect{k_S}$. Lower $\protect{k_O}$ increases evacuation time and the same applies to $\protect{\mu}$.}
 \label{ko_friction}  
 \end{figure}
 
@@ -705,18 +730,13 @@ The agents inside the cluster seldom move. Even if they choose an occupied cell,
 
 Counter to this, low $k_O$ can create \textit{queue} structures. Transition probabilities, influenced by the \textit{strategy for choosing destination cell} and low $k_O$, are much higher for cells closer to exit --- cells with lower \textit{static} field values. Agents are prone to ignore the empty cells on the sides of the queue and they prefer to stay in the queue. Even though the movement inside the queue is rather consistent, the total evacuation time increases. When a queue is formed, two clusters are formed as well, one at both ends of the queue. This can be noticed in Figure~\ref{exhausted_agents} in Appendix. One cluster forms at the exit, and because two or more agents attempt to enter the exit at the same time, blocking occasions might occur and total evacuation time will be increased. The cluster at the other end fills the queue and the agents inside it are mostly stationary. If more agents were approaching the exit simultaneously, the total evacuation time would be lower.
 
-Due to the these factors, total evacuation time increases with lower $k_O$, as can be seen in~Figure~\ref{ko_influence} and in Figure~\ref{ko_friction}.
-
-\begin{figure}[h!]
-	\begin{center}
-		\begin{subfigure}[b]{1\linewidth}
-	    \includegraphics[width=\linewidth]{img/visuals/15/3D 15 lq OD.png}
-  		\end{subfigure}
-  	\end{center}
-	\caption{kO depends on kD and can increase TET.}
-	\label{ko_influence}
-\end{figure}
+Figure~\ref{ko_friction} shows the characteristics of four constant parameters sets with variable $k_S\in{}[1.0, 5.0]$ on $x~axis$. Each set of parameters was simulated 3 times and the vertical $y~axis$ is the averaged $T_\text{TET}$ from  these simulations. Even though the lines are not very smooth, it needs to be said that the resolution of $T_\text{TET}$ is high --- from 80 to 125. 
+All sets have the same $k_D=0.5$. 
 
+The green line shows how parameters $k_O=0.9, \mu{}=0.9$ are affected by $k_S$. The relative position of the green line, compared to others, shows that by allowing agents to overtake queue (high $k_O$) and with low number of conflicts (low $\mu$), the evacuation times are the shortest (81 to 105 epochs).
+Contrary to this, with same $k_O$ and higher $\mu{}=0.9$, the evacuations take longer (105 to 117 epochs) and the results for various $k_S$ vary a lot due to more frequent conflicts.
+The remaining two sets have low $k_0=0.1$, which minimizes overtakes and forces the agents to stay in \textit{queues}.This results in longer evacuations. The higher $\mu=0.9$ in the orange line is rough. It should be noticed, that with increasing $k_S$, the evacuations take longer than in the beginning (115 epochs for low $k_S$ and 125 epochs for higher $k_S$). 
+In the end the total evacuation time is increased with lower $k_O$ and it also increases with higher $\mu$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -764,7 +784,10 @@ Even though the microscopic behavior of agents can be significantly affected by
 
 \begin{figure}[h!]
 \begin{center}
-\begin{subfigure}[b]{0.8\linewidth}
+\begin{subfigure}[b]{0.6\linewidth}
+      \includegraphics[width=\linewidth]{img/visuals/45/3D 45 lq OD.png}
+  \end{subfigure}
+\begin{subfigure}[b]{0.39\linewidth}
     \includegraphics[width=\linewidth]{img/visuals/45/COP45.pdf}
   \end{subfigure}
 \end{center}
@@ -773,18 +796,19 @@ Even though the microscopic behavior of agents can be significantly affected by
 \end{figure}
 
 
-
 \begin{figure}[h!]
-\begin{center}
-\begin{subfigure}[b]{0.8\linewidth}
-    \includegraphics[width=\linewidth]{img/visuals/45/3D 45 lq OD.png}
-  \end{subfigure}
-\end{center}
- \caption{High $\protect{k_S=4.5}$. Compared to other parameters, influence of $\protect{k_D}$ is marginal.}
- \label{kd_influence}
+	\begin{center}
+		\begin{subfigure}[b]{1\linewidth}
+	    \includegraphics[width=\linewidth]{img/visuals/15/3D 15 lq OD.png}
+  		\end{subfigure}
+  	\end{center}
+	\caption{kO is influenced by kD and can increase TET.}
+	\label{ko_influence}
 \end{figure}
 
 
+
+
 \newpage
 \section{Friction parameter $\protect{\mu}$}
 As was depicted in Figure~\ref{ko_friction}, $\mu{}$ and $k_O$ have major influence on $T_\text{TET}$.
@@ -806,7 +830,7 @@ Friction $\mu{}$ affects the evacuation regardless of the other parameters $k_O$
 
 
 \newpage
-\section{Other discovery: heterogeneity}
+\section{Other discovery: heterogeneity}\label{other_discovery}
 
 \begin{figure}[h!]
 \begin{center}
@@ -972,6 +996,7 @@ The model uses a stochastic selection for several processes in the simulation. I
 	\item[COP] Coefficient of Prognosis
 	\item[PRNG] Pseudo-random number generator
 	\item[TET] Total evacuation time
+	\item[MLS] Method of least squares
 \end{description}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-- 
GitLab


From 3d63cd3650c324ccac83abf122de6f20979feec9 Mon Sep 17 00:00:00 2001
From: sutymate <sutymate@fit.cvut.cz>
Date: Tue, 11 May 2021 11:03:13 +0200
Subject: [PATCH 2/2] aaa

---
 report/MatejSutyBP.pdf | Bin 6465659 -> 6220256 bytes
 report/MatejSutyBP.tex | 230 +++++++++++++++++++++++------------------
 2 files changed, 132 insertions(+), 98 deletions(-)

diff --git a/report/MatejSutyBP.pdf b/report/MatejSutyBP.pdf
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diff --git a/report/MatejSutyBP.tex b/report/MatejSutyBP.tex
index a25d8ba..41f9604 100644
--- a/report/MatejSutyBP.tex
+++ b/report/MatejSutyBP.tex
@@ -283,6 +283,7 @@ One step $s_i$ of model takes $\Delta{t_i}=[t_i, t_{i+1}]$ time. Time-frame $t_i
 \subsubsection*{CELLS}\label{cells}
 Cell $y$ is an element of grid $G$ in position $(y_1, y_2):~y_1\in{\{0,1,\dots{},W-1\}},$ $y_2~\in{\{0,1,\dots{},H-1\}}$.
 Each cell can be occupied by one agent at a given time-frame. Agents can move in 8 directions to cells in Moore neighborhood or decide to stay in the same cell.
+%CHANGE THIS CHANGE THIS CHANGE THIS NOT NORMALIZED
 Cell $y$ holds the normalized value of static potential $S_{norm}(y)\in{[0,1]}$. These values are normalized to fit the lowest value at exit $e: S(e) = 0$ and the highest value of cell $y: S(y) = q, q\in{\mathbb{R}}$ that is furthest away from the exit.
 For cell $y$ and exit $e$ in grid $G$ is $S(y) = L_1(y, e)$ where $L_1(y, e)$ is Manhattan distance~\cite{distance_metric} from cell $y$ to exit $e$.
 
@@ -313,7 +314,7 @@ Strategy $A$ was described by Pavel Hrab{\' a}k and Marek Buk{\'a}{\v c}ek in Ar
 
 
 \begin{equation}\label{dest_strat_old}
-P(y\mid x) = \frac{\exp(-k_SS(y))(1 - k_OO(y))(1 - k_DD(y))} {\sum_{z\in{N}}  \exp(-k_SS(z))(1 - k_OO(z)(1 - k_DD(z)}
+P(y\mid x) = \frac{\exp(-k_SS(y))(1 - k_OO(y))(1 - k_DD(y))} {\sum_{z\in{N}}  \exp(-k_SS(z))(1 - k_OO(z)(1 - k_DD(z))}
 \end{equation}
 
 
@@ -327,11 +328,11 @@ P(y\mid x) = k_OP_O(y) + (1 - k_O)P_S(y)
 \end{equation}
 The term $P_O$ contributes to attractiveness with focus on occupancy. Notice the missing parameter $k_O$ in $(1-O(y))$ which is different from strategy $A$. Term is normalized across neighbor cells of $x$.
 \begin{equation}\label{dest_strat_po}
-P_O(y) = \frac{\exp(-k_SS(y))(1 - O(y))(1 - k_DD(y))}{\sum_{z\in{N}}  \exp(-k_SS(z))(1 - O(z)(1 - k_DD(z)}
+P_O(y) = \frac{\exp(-k_SS(y))(1 - O(y))(1 - k_DD(y))}{\sum_{z\in{N}}  \exp(-k_SS(z))(1 - O(z)(1 - k_DD(z))}
 \end{equation}
 The term $P_S$ takes into account only the static potential $S(y)$ and makes the agent move in the correct direction to the exit. This term is also normalized across neighbor cells of $x$.
 \begin{equation}\label{dest_strat_ps}
-P_S(y) = \frac{\exp(-k_SS(y))(1 - k_DD(y))}{\sum_{z\in{N}}  \exp(-k_SS(z))(1 - k_DD(z)}
+P_S(y) = \frac{\exp(-k_SS(y))(1 - k_DD(y))}{\sum_{z\in{N}}  \exp(-k_SS(z))(1 - k_DD(z))}
 \end{equation}
 \end{itemize}
 \pagebreak
@@ -549,7 +550,7 @@ All simulations in this thesis use the new $strategy B$ for $choosing destinatio
 
 
 \subsection{Numeric intervals}
-In Equation~\ref{dest_strat_mix} the sensitivity to occupancy $k_O$, the sensitivity to diagonal movement $k_D$, and in conflict solution, the friction parameter $\mu$, can be set to values from interval $[0,1]$. This interval can be studied using discrete values that cover the range. Contrary to this, the sensitivity to the \textit{static field} $k_S$ can be any positive real number and requires different approach as will be described later.
+In Equation~(\ref{dest_strat_mix}) the sensitivity to occupancy $k_O$, the sensitivity to diagonal movement $k_D$, and in conflict solution, the friction parameter $\mu$, can be set to values from interval $[0,1]$. This interval can be studied using discrete values that cover the range. Contrary to this, the sensitivity to the \textit{static field} $k_S$ can be any positive real number and requires different approach as will be described later.
 
 
 
@@ -606,7 +607,7 @@ To measure the influence of the parameters, OptiSLang uses Metamodel of Optimal
 
 \section{Simulations}
 At first, the simulations were observed on the Mesa framework web interface, that is depicted in~Figure~\ref{MESA-map}. They were used mostly during implementation of the model and in testing the new strategies.
-Later, batch simulations on wide range of intervals were performed. All simulations had the same model configuration of grid $15 \times 15$, exit placed at border $(0, 8)$, same seed $I = 1245$ for PRNG $G_1$. Each iteration had different seed $J$ for PRNG $G_2$, that solves conflicts and selects destination cells.
+Later, batch simulations on wide range of intervals were performed. All simulations had the same model configuration of grid $15 \times 15$, exit placed at border $(0, 8)$ and same seed $I = 1245$ for PRNG $G_1$, which assigns individual values of aggressivity $\gamma$ to agents and places them in the grid. Each iteration had different seed $J$ for PRNG $G_2$, that solves conflicts and selects destination cells. With different $J$ seeds, the course of the simulation varies and can be generalized.
 In the Table~\ref{table_simulations} are input parameters of simulations for SA in OptiSLang. Other analyses have configurations in the text.
 The parameter $k_S$ can be any positive real number. The first simulations showed that $T_\text{TET}$ does not change very much for values higher than $5$ so this value was set as the upper range of $k_S$.
 \begin{table}[h!]
@@ -629,14 +630,14 @@ The parameter $k_S$ can be any positive real number. The first simulations showe
   \label{table_simulations}
 \end{table}
 
-In the 3D and COP graphs exported from OptiSLang, the axes are renamed with capitals $S, O, D, F, T$ and mean $\protect{S=k_S, O=k_O, D=k_D, , F=\mu{}, T=T_\text{TET}}$. 3D graphs approximate the $T_\text{TET}$ with data from simulations and plot them on a plane. COP graphs shows how much individual parameters  $S, O, D, F, T$  contribute to the prediction of $T_\text{TET}$. The number in percent at the top of COP (for example on the right in Figure~\ref{ks_influence}) shows much many results from the simulations MOP was able to approximate. 
+In the 3D and COP graphs exported from OptiSLang, the axes are renamed with capitals $S, O, D, F, T$ and mean $S~=~k_S,\ O~=~k_O,\ D~=~k_D,\ F~=~\mu{}$ and $T~=~T_\text{TET}$. 3D graphs approximate the $T_\text{TET}$ with data from simulations and plot them on a plane. COP graphs shows how much individual parameters  $S, O, D, F, T$  contribute to the prediction of $T_\text{TET}$. The number in percent at the top of COP (for example on the right in Figure~\ref{ks_influence}) shows much many results from the simulations MOP was able to approximate. 
 %...
 %...
 
 \newpage
 \chapter{Results}
 
-The results of quantitative analysis in this chapter are explained by graphs from OptiSLang: the 3D plot of $T_\text{TET}$ and COP of individual parameters. The discoveries of qualitative analysis, such as linear dependency of $T_\text{TET}$ on number of agents $n$, are depicted by other graphs, for example histograms or boxplots.
+The results of  SA in this chapter are explained by graphs from OptiSLang: the 3D plot of $T_\text{TET}$ and COP of individual parameters. The discoveries of qualitative analysis, such as linear dependency of $T_\text{TET}$ on number of agents $n$, are depicted by other graphs, for example histograms or boxplots.
 
 \section{Number of agents $\protect{n}$}
 It was expected, that the increase of total evacuation time $T_\text{TET}$ is linear and depends on the number of agents \textit{n}. Figure~\ref{n_dep} demonstrates the linear dependency of $T_\text{TET}$ on $n$.
@@ -651,11 +652,11 @@ This happens because with more agents the conflicts occur more frequently and th
 \pagebreak
 \begin{figure}[h!]
 	\begin{center}
-		\begin{subfigure}[t]{0.49\textwidth}
+		\begin{subfigure}[b]{0.49\textwidth}
 	    \includegraphics[width=\textwidth]{img/vector/s11-3-33.pdf}
  		\caption{$\protect{k_S=1.5}$}
   		\end{subfigure}
-  		\begin{subfigure}[t]{0.49\textwidth}
+  		\begin{subfigure}[b]{0.49\textwidth}
 	    \includegraphics[width=\textwidth]{img/vector/s12-6-55.pdf}
 	    \caption{$\protect{k_S=3.5}$}  		
   		\end{subfigure}
@@ -669,11 +670,11 @@ This happens because with more agents the conflicts occur more frequently and th
 \section{Sensitivity to static field $\protect{k_S}$}
 Notice the $P_S$ member in Equation~\ref{dest_strat_ps} from \textit{strategy for choosing destination cell}.
 \begin{equation*}
-P_S(y) = \frac{\exp(-k_SS(y))(1 - k_DD(y))}{\sum_{z\in{N}}  \exp(-k_SS(z))(1 - k_DD(z)}
+P_S(y) = \frac{\exp(-k_SS(y))(1 - k_DD(y))}{\sum_{z\in{N}}  \exp(-k_SS(z))(1 - k_DD(z))}
 \end{equation*}
 $P_S$ grows exponentially with $k_S$. The influence of $k_S$ can be very strong for high numbers, and marginal for values lower than $1$. It was noticed during the testing phases of the model implementation, when the simulations with very low $k_S$ values lasted very long. Visual examination of simulation exposed the erratic movement of agents that did not progress to the exit. 
 
-Figure~\ref{ks_influence} is the 3D plot of approximated $T_\text{TET}$ from simulation $S_2$ in Table~\ref{table_simulations}, exported from OptiSLang. The black dots are the values of $T_\text{TET}$ from the simulations. Vertical axis $T$ is the $T_\text{TET}$, on horizontal axis $F$ are values of friction $\mu{}$ and horizontal axis $S$ holds the values of sensitivity to static field $k_S$.  This figure shows very high $T_\text{TET}$ on axis $T$ for $k_S<1$ on axis $S$. For~$k_S~\gg{}~4.5$ the influence of $k_O, k_D$ is non-existent. Because $T_\text{TET}$ for these values is meaningless and steep slope lowers the resolution of the graph, the interval of $k_S$ was limited to $[1.5, 4.5]$.  The influence of $\mu{}$ on axis $F$ might seem marginal, but, as will be presented later, the friction parameter $\mu{}$ plays an important role in the course of the evacuation.
+Figure~\ref{ks_influence} is the 3D plot of approximated $T_\text{TET}$, from simulation $S_2$ in Table~\ref{table_simulations}, exported from OptiSLang. The black dots are the values of $T_\text{TET}$ from the simulations. \textit{MLS} model was used as MOP, superior to \textit{polynomial} model in terms of capturing COP. Due to too many simulation data \textit{Krigin} model was not allowed. Vertical axis $T$ is the $T_\text{TET}$, on horizontal axis $F$ are values of friction $\mu{}$ and horizontal axis $S$ holds the values of sensitivity to static field $k_S$.  This figure shows very high $T_\text{TET}$ on axis $T$ for $k_S<1$ on axis $S$. For~$k_S~\gg{}~4.5$ the influence of $k_O, k_D$ is non-existent. Because $T_\text{TET}$ for these values is pointlessly high and steep slope lowers the resolution of the 3D graph, the interval of $k_S$ was limited to $[1.5, 4.5]$. In Figure~\ref{ko_friction} can be spotted how $T_\text{TET}$ does not change much for values higher than 4.5. The influence of $\mu{}$ on axis $F$ might seem marginal, but, as will be presented later, the friction parameter $\mu{}$ plays an important role in the course of the evacuation.
 
 \begin{figure}[h!]
 	\begin{center}
@@ -691,12 +692,12 @@ Figure~\ref{ks_influence} is the 3D plot of approximated $T_\text{TET}$ from sim
 \pagebreak
 % Horizontal axes are parameters friction $\protect{\mu{}~\text{and}~k_S.}$
 
-The values $k_S\in{}[1.5, 4.5]$  bring meaningful results and allow other parameters to influence the simulation as can be seen in Figure~\ref{limited_ks}.
+The values $k_S\in{}[1.5, 4.5]$  bring meaningful results and allow other parameters to influence the simulation as can be seen in Figure~\ref{limited_ks}. According to COP the most significant parameter is friction $\mu$. SA of $k_S$ was performed using \textit{MLS} model for MOP. Krigin model was disabled because of high computational time. The following SA on simulations $S_4, S_5$ use \textit{Krigin} model.
 
 \begin{figure}[h!]
 	\begin{center}
 		\begin{subfigure}[b]{0.6\linewidth}
-	    \includegraphics[width=\linewidth]{img/visuals/15-45/3D 15-45 lq SF.png}
+	    \includegraphics[width=\linewidth]{img/visuals/15-45/3D 15-45 lq SF2.png}
   		\end{subfigure}
 		\begin{subfigure}[b]{0.38\linewidth}
 	    \includegraphics[width=\linewidth]{img/visuals/15-45/COP15-45.pdf}
@@ -706,7 +707,7 @@ The values $k_S\in{}[1.5, 4.5]$  bring meaningful results and allow other parame
 	\label{limited_ks}
 \end{figure}
 
-According to COP the most significant parameter is friction $\mu$. 
+
 
 
 \newpage
@@ -719,7 +720,7 @@ Lower values of $k_O$ allow the agent to choose an occupied cell as his target c
     		\includegraphics[width=\linewidth]{img/vector/figX3.pdf}
 	\end{subfigure}
 \end{center}
-\caption{Different sets of parameters $\protect{k_O, \mu}$ and how they affect the $\protect{T_\text{TET}}$. Vertical axis is averaged $\protect{T_\text{TET}}$ from three simulations and horizontal axis is variable $\protect{k_S}$. Lower $\protect{k_O}$ increases evacuation time and the same applies to $\protect{\mu}$.}
+\caption{Different sets of parameters $\protect{k_O, \mu}$ and how they affect $\protect{T_\text{TET}}$. Vertical axis is averaged $\protect{T_\text{TET}}$ from three simulations and horizontal axis is variable $\protect{k_S}$. Lower $\protect{k_O}$ and higher $\protect{\mu}$ increase evacuation time.}
 \label{ko_friction}  
 \end{figure}
 
@@ -748,7 +749,7 @@ In the end the total evacuation time is increased with lower $k_O$ and it also i
 \begin{figure}[h!]
 	\begin{center}
 		\begin{subfigure}[b]{\linewidth}
-    		\includegraphics[width=\linewidth]{img/raster/Response surface 3D plot kriggins 25.png}
+    		\includegraphics[width=\linewidth]{img/raster/Response surface 3D plot Krigin 25.png}
  	 	\end{subfigure}
 	\end{center}
 	\caption{Start of the simulation.}
@@ -757,11 +758,11 @@ In the end the total evacuation time is increased with lower $k_O$ and it also i
 \begin{figure}[h!]
 \begin{center}
 \begin{subfigure}[b]{0.8\linewidth}
-    \includegraphics[width=\linewidth]{img/raster/Coefficient of Prognosis 40 kriggins.pdf}
+    \includegraphics[width=\linewidth]{img/raster/Coefficient of Prognosis 40 Krigin.pdf}
   \end{subfigure}
   
 \begin{subfigure}[b]{0.8\linewidth}
-    \includegraphics[width=\linewidth]{img/raster/Response surface 3D plot 40 kriggins revers.png}
+    \includegraphics[width=\linewidth]{img/raster/Response surface 3D plot 40 Krigin revers.png}
   \end{subfigure}
 \end{center}
  \caption{Start of the simulation.}
@@ -779,9 +780,6 @@ In the end the total evacuation time is increased with lower $k_O$ and it also i
 \section{Sensitivity to diagonal movement $\protect{k_D}$}
 Agent that moves to adjacent diagonal cells increases his inner time period by $\frac{3}{2}$ of nominal movement duration. At the start of the simulation, agents closest to the exit have a free path, and move rapidly  towards exit. Movement in diagonal directions allows agents, that are not in a straight line from the \textit{exit}, to get there with less movements. However, they are penalized by increased movement duration --- after two diagonal movements the agents are 'exhausted' and do not move. Especially at the start of the simulation, with low $k_D$, agents can move in waves, as can be seen in Figure~\ref{exhausted_agents} in Appendix~\ref{exhausted_appendix}. Diagonal movement is also useful when a queue or other congestion structure forms. The agents can overtake the queue when they move diagonally. 
 
-Even though the microscopic behavior of agents can be significantly affected by $k_D$, the effect on the $T_\text{TET}$ is marginal. Sensitivity analysis using MOP in Figure~\ref{kd_influence} shows that $k_D$ has very low influence on the total evacuation time for high $k_S$.
-
-
 \begin{figure}[h!]
 \begin{center}
 \begin{subfigure}[b]{0.6\linewidth}
@@ -791,18 +789,30 @@ Even though the microscopic behavior of agents can be significantly affected by
     \includegraphics[width=\linewidth]{img/visuals/45/COP45.pdf}
   \end{subfigure}
 \end{center}
- \caption{High $\protect{k_S=4.5}$. Compared to other parameters, influence of $\protect{k_D}$ is marginal.}
+ \caption{Left: 3D graph of $\protect{T_\text{TET}}$ from OptiSLang. Right: COP of input parameters, exported from OptiSLang. Data from simulation $\protect{S_4}, k_S=4.5$. 
+ Compared to other parameters, influence of $\protect{k_D}$ is marginal. \textit{Note: MOP selected Krigin model.}}
+
  \label{kd_influence}
 \end{figure}
 
+Even though the microscopic behavior of agents can be significantly affected by $k_D$, the general effect on the $T_\text{TET}$ is marginal. SA using COP in Figure~\ref{limited_ks} shows that $k_D$ has very low influence (6\%) on the total evacuation time.
+Figure~\ref{kd_influence} is 3D graph and COP, exported from OptiSLang, SA of simulation $S_4$ with fixed $k_S=4.5$, which is on the upper end of $k_S$ limit described before. It can be seen, that $k_O$ and $k_D$ have minor influence and the prevalent input parameter is friction $\mu$.
+\pagebreak
+
 
+When $k_S=1.5$, the value on the lower limit of the range, on simulation $S_4$, parameters $k_O, k_D$ have noticeably increased COP. Looking at 3D graph in Figure~\ref{ko_influence}, compared to $k_S=4.5$ in Figure~\ref{kd_influence}, also the range of $T_\text{TET}$ has increased upper limit: from $T_\text{TET}\in{}[84, 100]$ ($k_S=4.5$) to $T_\text{TET}\in{}[88, 118]$ ($k_S=1.5$). When agents calculate transition probabilities of adjacent cells more evenly, they choose suboptimal cells more frequently and increate $T_\text{TET}$. The graphs, comparing new and old strategy for choosing destination cell, in Appendix~\ref{appendix_comparison_strategy_choosing} demonstrate how high $k_S$ can obliterate the chance to move to suboptimal cell.
 \begin{figure}[h!]
 	\begin{center}
-		\begin{subfigure}[b]{1\linewidth}
+		\begin{subfigure}[b]{0.6\linewidth}
 	    \includegraphics[width=\linewidth]{img/visuals/15/3D 15 lq OD.png}
   		\end{subfigure}
+  		\hfill
+		\begin{subfigure}[b]{0.38\linewidth}
+	    \includegraphics[width=\linewidth]{img/visuals/15/COP15.pdf}
+  		\end{subfigure}
   	\end{center}
-	\caption{kO is influenced by kD and can increase TET.}
+	\caption{Left: 3D graph of $\protect{T_\text{TET}}$ from OptiSLang. Right: COP of input parameters, exported from OptiSLang. Data from simulation $\protect{S_4},\ k_S=1.5$. Agents calculate transition probability to adjacent cells more evenly and are more sensitive to input parameters.
+ \textit{Note: MOP selected Krigin model.}}
 	\label{ko_influence}
 \end{figure}
 
@@ -815,61 +825,54 @@ As was depicted in Figure~\ref{ko_friction}, $\mu{}$ and $k_O$ have major influe
 Friction parameter $\mu{}$ affects the number of blocking occasions and directly increases the total evaction time. With higher number of agents $n$ the blocking occasions occur more frequently, because more agents create more conflicts. This results in increasing variance in $T_\text{TET}$ as demonstrated in~Figure~\ref{n_dep}. The blue box-plots have high $\mu{}=0.9$ and show increasing variance of $T_\text{TET}$. Contrary to this, the red box-plots with $\mu{}=0.1$ have rather constant variance of $T_\text{TET}$.
 
 
-Friction $\mu{}$ affects the evacuation regardless of the other parameters $k_O$ and $k_D$. For $k_S=4.5$ the influence of other parameters is minimalized due to strong $static$ sensitivity but $\mu{}$ keeps its influence as can be seen in~Figure~\ref{ks45}.
+Friction $\mu{}$ affects the evacuation regardless of the other parameters $k_O$ and $k_D$. For $k_S=4.5$ the influence of other parameters ($k_O, k_D$) is minimalized due to strong $static$ sensitivity $k_S$ but $\mu{}$ retains its influence as can be seen in~Figure~\ref{ks45}.
 
 \begin{figure}[h!]
 \begin{center}
-\begin{subfigure}[b]{\linewidth}
-    \includegraphics[width=\linewidth]{img/visuals/45/3D 45 lq OF.png}
-
-\end{subfigure}
+	\begin{subfigure}[b]{0.6\linewidth}
+    		\includegraphics[width=\linewidth]{img/visuals/45/3D 45 lq OF.png}
+	\end{subfigure}
+	\begin{subfigure}[b]{0.38\linewidth}
+    		\includegraphics[width=\linewidth]{img/visuals/45/COP45.pdf}
+	\end{subfigure}
 \end{center}
-\caption{Friction.}
+\caption{Left: 3D graph of $\protect{T_\text{TET}}$ from OptiSLang. Right: COP of input parameters, exported from OptiSLang. Data from simulation $\protect{S_4}, k_S=4.5$. Friction $\protect{\mu}$ has major influence while COP of $\protect{k_O, k_D}$ are minimized in response to high $\protect{k_S}$. \textit{Note: MOP selected Krigin model.}}
 \label{ks45}
 \end{figure}	
 
 
 \newpage
-\section{Other discovery: heterogeneity}\label{other_discovery}
+\section{Other discovery: heterogeneity in parameters}\label{other_discovery}
+The evacuation model in this thesis can assign different parameters to individual agents by PRNG $G_2$ or assign different parameters to groups of agents. To see how the model responds to homogeneous and heterogeneous parameters, simulations $S_{6.1}$ and $S_{6.2}$ were performed. In both simulations grid $15\times{}15$ was used, with \textit{exit} placed at $(0, 8)$ and populated with $70$ agents. Simulations were repeated 1000 times,
+%        'S': 2,        'O': 0.5,        'D': 0.5,        'F': 0.1,
+
 
+First simulation $S_{6.1}$ with set of input parameters $k_S=2.0,\ k_D=0.5$ and low $\mu{}=0.1$.
+This simulation was, at first, run with \textit{homogeneous} $k_O=0.5$ for all agents. The blue histogram in Figure~\ref{s61_hetero} shows distribution of $T_\text{TET}$ from 1000 iterations of this simulation. Most iterations, more than 300, resulted in $T_\text{TET}\approx 85$ and all iterations had $T_\text{TET}\in{[80, 96]}$.
+
+
+This simulation was then rerun, also repeated 1000 times, with \textit{heterogeneous} distribution of $k_O$. The $70$ agents were split to two groups of $35$. First groups was assigned $k_O=0.1$ and second group $k_0=0.9$, so the average $k_O$ of all agents was $0.5$, identical to the previous run. The orange histogram in Figure~\ref{s61_hetero} shows that this simulation resulted in shorter evacuations --- all iteration had $T_\text{TET}\in{[80, 91]}$.
+
+
+Another simulation $S_{6.2}$ was performed, this time with increased friction $\mu=0.9$, other parameters remained unchanged $k_S=2.0, k_D=0.5$. Identical to $S_{6.1}$, simulation was repeated 1000 times --- once for \textit{homogeneous} and then for \textit{heterogeneous} $k_O$ of agents. Figure~\ref{s62_hetero} shows that distribution of $T_\text{TET}$ for \textit{homogeneous} $k_O$, blue histogram, is similar to \textit{heterogenenous} $k_O$ for two groups of agents, orange histogram.
 \begin{figure}[h!]
 \begin{center}
-\begin{subfigure}[b]{0.8\linewidth}
+\begin{subfigure}[b]{0.49\linewidth}
     \includegraphics[width=\linewidth]{img/vector/heteroA2.pdf}
-	\caption{Low friction}
+	\caption{Simulation $\protect{S_{6.1}}$, low friction $\protect{\mu=0.1}$.}
+	\label{s61_hetero}
 \end{subfigure}
-  
-\begin{subfigure}[b]{0.8\linewidth}
+  \hfill
+\begin{subfigure}[b]{0.49\linewidth}
     \includegraphics[width=\linewidth]{img/vector/heteroB2.pdf}
-    	\caption{High friction}
+	\caption{Simulation $\protect{S_{6.2}}$, high friction $\protect{\mu=0.9}$}
+	\label{s62_hetero}
 \end{subfigure}
 \end{center}
+\caption{Simulations with heterogeneous and homogeneous distribution of parameter $\protect{k_O}$, repeated 1000 times. Blue histograms show $\protect{T_\text{TET}}$ for homogeneous $\protect{k_O=0.5}$. Simulations with two groups of agents with with different $\protect{k_O=0.1}$ and $\protect{k_O=0.9}$ are captured in orange histograms.}
 \end{figure}	
 
 
-\section{Contribution to variance in TET}
-Why are there different results????
-
-
-\begin{figure}[h!]
-  	\begin{subfigure}[t]{0.5\textwidth}
-	    \includegraphics[width=\textwidth]{img/visuals/15/COP15.pdf}
-	    \caption{$\protect{k_S=1.5}$}  		
-  	\end{subfigure}
-	\begin{subfigure}[t]{0.5\linewidth}
-	    \includegraphics[width=\linewidth]{img/visuals/45/COP45.pdf}
-	    \caption{$\protect{k_S=4.5}$}  		
-	\end{subfigure}
-  	\begin{subfigure}[t]{0.55\textwidth}
-	    	\includegraphics[width=\textwidth]{img/visuals/30/COP30.pdf}
-	    	\caption{$\protect{k_S=3.0}$}  		
-  	\end{subfigure}
-	\label{cop}
-\end{figure}
-
-\newpage
-\paragraph*{SOMETGHIN}
-\newpage
 
 
 
@@ -1000,38 +1003,21 @@ The model uses a stochastic selection for several processes in the simulation. I
 \end{description}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-							\chapter{Figures}\label{appendix_chapter_comparison_strategy}
+							\chapter{Further graphical output}\label{appendix_chapter_comparison_strategy}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{figure}[h!]
-\begin{center}
-\begin{subfigure}[b]{0.8\linewidth}
-    \includegraphics[width=\linewidth]{img/vector/hmap1.pdf}
-  \end{subfigure}
-  
-\begin{subfigure}[b]{0.8\linewidth}
-    \includegraphics[width=\linewidth]{img/vector/hmap2.pdf}
-  \end{subfigure}
-\end{center}
- \caption{Start of the simulation.}
-\end{figure}
-
 
+\section{Diagonal movement, waves}\label{exhausted_appendix}
+\begin{wrapfigure}[]{o}{0.6\textwidth}
+	\centering
+		\includegraphics[width=\linewidth]{img/raster/lowD.pdf}
+	\centering
+  \caption{Waves are forming at the start due to low $\protect{k_D=0}$ and high $\protect{k_O=1}$. Pink agents are penalized for repeated diagonal movement.}
+  \label{exhausted_agents}
+\end{wrapfigure}
+Mesa framework allows visualization of evacuation with individual agents, who have color based on their situation. The grid consists of rectangular cells, that have darker color when closer to the exit --- according to the $L_1$ distance. Figure~\ref{exhausted_agents} shows the formation of waves at the start of the evacuation. Agents are not allowed to select an occupied cell, $k_0=1$, and diagonal movement is allowed, $k_D=0$. The first wave of agents, closest to the exit on the right in dark color area, have pink color, as they already advanced by two diagonal steps. Their inner time period was increased by $2\cdot{}\frac{3}{2}=3$ in step 2 thus they were not allowed to move in step 3.
 
-
-\begin{figure}[h!]
-\begin{center}
-  \begin{subfigure}[b]{0.8\linewidth}
-    \includegraphics[width=\linewidth]{img/vector/hmap3.pdf}
-  \end{subfigure}
-  \vfill
-  \begin{subfigure}[b]{0.8\linewidth}
-    \includegraphics[width=\linewidth]{img/vector/hmap4.pdf}
-  \end{subfigure}
-\end{center}
- \caption{Later stages of the simulation.}
-\end{figure}
-
-\section{Comparison of new and old strategy for choosing destination cell}\label{appendix_section_comparison_strategy}
+\section{Strategies for choosing destination cell}\label{appendix_section_comparison_strategy}
+In new proposed \textit{strategy A for choosing destination cell}, parameter $k_O$ influences the selection of destination cell more evenly. Graphs in Figure~\ref{appendix_comparison_strategy_choosing} below are 100\% stacked graphs of individual transition probabilities of adjacent cells in situation with blocked exit, see Figure~\ref{fig_strat_choosing}. The horizontal axis is variable $k_0\in{\{0.0,0.1,\dots,1.0\}}$. The legend in the upper left corner assigns colors to cell numbers.
 \begin{figure}[h!]
   \begin{subfigure}[b]{0.5\linewidth}
     \includegraphics[width=\linewidth]{img/vector/2NEW-15-05.pdf}
@@ -1049,6 +1035,13 @@ The model uses a stochastic selection for several processes in the simulation. I
     \includegraphics[width=\linewidth]{img/vector/2OLD-25-05.pdf}
     \caption{Old strategy. $\protect{k_S=2.5,~k_D=0.5}$}
   \end{subfigure}
+    \caption{Comparison of strategies for choosing destination cell.}
+  \label{appendix_comparison_strategy_choosing}
+\end{figure}
+
+\begin{comment}
+
+\begin{figure}[h!]
   \begin{subfigure}[b]{0.5\linewidth}
     \includegraphics[width=\linewidth]{img/vector/2NEW-35-05.pdf}
     \caption{New strategy. $\protect{k_S=3.5,~k_D=0.5}$}
@@ -1057,12 +1050,20 @@ The model uses a stochastic selection for several processes in the simulation. I
     \includegraphics[width=\linewidth]{img/vector/2OLD-35-05.pdf}
     \caption{Old strategy. $\protect{k_S=3.5,~k_D=0.5}$}
   \end{subfigure}
-  \caption{Comparison of strategies for choosing destination cell.}
-  \label{appendix_comparison_strategy_choosing}
+      \caption{Comparison of strategies for choosing destination cell.}
 \end{figure}
+\end{comment}
 
+
+COMMENT OUT?
+\pagebreak
 \section{Congestion structures}\label{exit_structure}
-text
+During the implementation of the evacuation model, certain patterns were noticed, namely the formation of \textit{queues} and \textit{cones} at the exit.
+
+
+In Figures~\ref{queue_forms} and \ref{cone_forms} are two visuals. The heatmap on the left shows the results from 100 iterations of simulation with grid $30 \times 30$, populated with $150$ agents with exit placed at $(0, 15)$ and homogeneous input parameters $k_S=2.5, k_D=0.5, \mu{}=0.7$. From all iterations, the iteration $M$ with longest evacuation time $T_\text{TET} = m$ was selected. In epoch $\{1,2,3,\dots,m\}$ each cell of the heatmap grid was assigned color based on number of agents present in that cell. At the start of the evacuation, epoch 1, all iterations had agents in the same cells so they all had white color. Later individual iterations moved agents to different places and some cells were occupied in more iterations than others. These frequent cells, where many iterations placed their agents, had brighter color. Contrary to this, cells never visited had black color. The colorbar on the right next to the heatmap shows the resolution of agents occupancy in cells. The brighter the color of a cell, the more agents were present in that cell in that epoch.
+
+
 \begin{figure}[h!]
   \begin{center}
   \begin{subfigure}[b]{0.6\linewidth}
@@ -1073,8 +1074,16 @@ text
   \end{subfigure}
   \end{center}
   \caption{Queue forms at the exit due to low $\protect{k_O}$.}
+  \label{queue_forms}
 \end{figure}
 
+
+
+The simulations with low $k_O=0.1$ produced heatmaps where kind of a \textit{queue} can be seen in Figure~\ref{queue_forms}. One concrete occasion of this phenomena is on the right --- visualization of one evacuation simulation in Mesa framework with identical parameters to simulation mentioned before.
+
+
+In comparison, with high $k_O=0.9$, the heatmap in Figure~\ref{cone_forms} shows a formation of a cone structure at the exit, where agents are clustered and wait for evacuation. One the right is one instance of evacuation simulation in Mesa framework, parameters are identical to the simulation with heatmap.
+
 \begin{figure}[h!]
   \begin{center}
   \begin{subfigure}[b]{0.6\linewidth}
@@ -1085,21 +1094,46 @@ text
   \end{subfigure}
   \end{center}
   \caption{Cone forms at the exit due to high $\protect{k_O}$.}
+  \label{cone_forms}
 \end{figure}
 
-\section{Diagonal movement, waves}\label{exhausted_appendix}
+
+
+\begin{comment}
+referovat sa s KS na kvalitativnu, preoc som zvolil ks 1.5 az 4.5
+\end{comment}
 
 \begin{figure}[h!]
-  \begin{center}
-  \begin{subfigure}[b]{0.6\linewidth}
-    \includegraphics[width=\linewidth]{img/raster/lowD.pdf}
+\begin{center}
+\begin{subfigure}[b]{0.49\linewidth}
+    \includegraphics[width=\linewidth]{img/vector/hmap1.pdf}
   \end{subfigure}
-  \end{center}
-  \caption{Waves are forming at the start due to low $\protect{k_D=0}$ and high $\protect{k_O=1}$. Pink agents are penalized for repeated diagonal movement.}
-  \label{exhausted_agents}
+  \hfill
+\begin{subfigure}[b]{0.49\linewidth}
+    \includegraphics[width=\linewidth]{img/vector/hmap2.pdf}
+  \end{subfigure}
+\end{center}
+ \caption{Start of the simulation.}
 \end{figure}
 
 
+
+
+\begin{figure}[h!]
+\begin{center}
+  \begin{subfigure}[b]{0.8\linewidth}
+    \includegraphics[width=\linewidth]{img/vector/hmap3.pdf}
+  \end{subfigure}
+  \vfill
+  \begin{subfigure}[b]{0.8\linewidth}
+    \includegraphics[width=\linewidth]{img/vector/hmap4.pdf}
+  \end{subfigure}
+\end{center}
+ \caption{Later stages of the simulation.}
+\end{figure}
+
+
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 							\chapter{Contents of enclosed CD}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-- 
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