savepoint

arbeit/ma.md

@@ -28,7 +28,7 @@ Unless otherwise noted the following holds:
 Many modern industrial design processes require advanced optimization methods
 due to the increased complexity resulting from more and more degrees of freedom
 as methods refine and/or other methods are used. Examples for this are physical
-domains like aerodynamic (i.e. drag), fluid dynamics (i.e. throughput of liquid)
+domains like aerodynamics (i.e. drag), fluid dynamics (i.e. throughput of liquid)
 --- where the complexity increases with the temporal and spatial resolution of
 the simulation --- or known hard algorithmic problems in informatics (i.e.
 layouting of circuit boards or stacking of 3D--objects). Moreover these are
@@ -45,7 +45,7 @@ representation (the *genome*) can be challenging.
 
 This translation is often necessary as the target of the optimization may have
 too many degrees of freedom. In the example of an aerodynamic simulation of drag
-onto an object, those objects--designs tend to have a high number of vertices to
+onto an object, those object--designs tend to have a high number of vertices to
 adhere to various requirements (visual, practical, physical, etc.). A simpler
 representation of the same object in only a few parameters that manipulate the
 whole in a sensible matter are desirable, as this often decreases the
@@ -425,7 +425,7 @@ the given direction.
 
 The definition for an *improvement potential* $P$ is\cite{anrichterEvol}:
 $$
-P(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec(G)\|^2_F
+P(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F
 $$
 given some approximate $n \times d$ fitness--gradient $\vec{G}$, normalized to
 $\|\vec{G}\|_F = 1$, whereby $\|\cdot\|_F$ denotes the Frobenius--Norm.
@@ -549,6 +549,7 @@ and use Cramers rule for inverting the small Jacobian and solving this system of
 linear equations.
 
 ## Deformation Grid
+\label{sec:impl:grid}
 
 As mentioned in chapter \ref{sec:back:evo}, the way of choosing the
 representation to map the general problem (mesh--fitting/optimization in our
@@ -565,19 +566,9 @@ control point without having a $1:1$--correlation, and a smooth deformation.
 While the advantages are great, the issues arise from the problem to decide
 where to place the control--points and how many.
 
-One would normally think, that the more control--points you add, the better the
+\begin{figure}[!tbh]
-result will be, but this is not the case for our B--Splines. Given any point $p$
+\centering
-only the $2 \cdot (d-1)$ control--points contribute to the parametrization of
-that point^[Normally these are $d-1$ to each side, but at the boundaries the
-number gets increased to the inside to meet the required smoothness].
-This means, that a high resolution can have many control-points that are not
-contributing to any point on the surface and are thus completely irrelevant to
-the solution.
-
-\begin{figure}[!ht]
-\begin{center}
 \includegraphics{img/enoughCP.png}
-\end{center}
 \caption[Example of a high resolution control--grid]{A high resolution
 ($10 \times 10$) of control--points over a circle. Yellow/green points
 contribute to the parametrization, red points don't.\newline
@@ -586,6 +577,14 @@ control--points.}
 \label{fig:enoughCP}
 \end{figure}
 
+One would normally think, that the more control--points you add, the better the
+result will be, but this is not the case for our B--Splines. Given any point $p$
+only the $2 \cdot (d-1)$ control--points contribute to the parametrization of
+that point^[Normally these are $d-1$ to each side, but at the boundaries the
+number gets increased to the inside to meet the required smoothness].
+This means, that a high resolution can have many control-points that are not
+contributing to any point on the surface and are thus completely irrelevant to
+the solution.
 
 We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the four red 
 central points are not relevant for the parametrization of the circle.
@@ -619,14 +618,14 @@ In this scenario we used the shape defined by Giannelli et al.\cite{giannelli201
 which is also used by Richter et al.\cite{anrichterEvol} using the same
 discretization to $150 \times 150$ points for a total of $n = 22\,500$ vertices. The
 shape is given by the following definition
-$$
+\begin{equation}
 t(x,y) =
 \begin{cases}
 0.5 \cos(4\pi \cdot q^{0.5}) + 0.5 & q(x,y) < \frac{1}{16},\\
 2(y-x) & 0 < y-x < 0.5,\\
 1 & 0.5 < y - x
 \end{cases}
-$$
+\end{equation}<!-- </> -->
 with $(x,y) \in [0,2] \times [0,1]$ and $q(x,y)=(x-1.5)^2 + (y-0.5)^2$, which we have
 visualized in figure \ref{fig:1dtarget}.
 
@@ -645,9 +644,9 @@ correct.
 
 Regarding the *fitness--function* $f(\vec{p})$, we use the very simple approach
 of calculating the squared distances for each corresponding vertex
-$$
+\begin{equation}
 \textrm{f(\vec{p})} = \sum_{i=1}^{n} \|(\vec{Up})_i - t_i\|_2^2 = \|\vec{Up} - \vec{t}\|^2 \rightarrow \min
-$$
+\end{equation}
 where $t_i$ are the respective target--vertices to the parametrized
 source--vertices^[The parametrization is encoded in $\vec{U}$ and the initial
 position of the control points. See \ref{sec:ffd:adapt}] with the current
@@ -659,6 +658,108 @@ This formula is also the least--squares approximation error for which we
 can compute the analytic solution $\vec{p^{*}} = \vec{U^+}\vec{t}$, yielding us
 the correct gradient in which the evolutionary optimizer should move.
 
+## Test Scenario: 3D Function Approximation
+\label{sec:test:3dfa}
+Opposed to the 1--dimensional scenario before, the 3--dimensional scenario is
+much more complex --- not only because we have more degrees of freedom on each
+control point, but also because the *fitness--function* we will use has no known
+analytic solution and multiple local minima.
+
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=0.9\textwidth]{img/3dtarget.png}
+\end{center}
+\caption[3D source and target meshes]{\newline
+Left: The sphere we start from with 10\,807 vertices\newline
+Right: The face we want to deform the sphere into with 12\,024 vertices.}
+\label{fig:3dtarget}
+\end{figure}
+
+First of all we introduce the set up: We have given a triangulated model of a
+sphere consisting of $10\,807$ vertices, that we want to deform into a
+the target--model of a face with a total of $12\,024$ vertices. Both of
+these Models can be seen in figure \ref{fig:3dtarget}.
+
+Opposed to the 1D--case we cannot map the source and target--vertices in a
+one--to--one--correspondence, which we especially need for the approximation of
+the fitting--error. Hence we state that the error of one vertex is the distance
+to the closest vertex of the other model.
+
+We therefore define the *fitness--function* to be:
+
+\begin{equation}
+f(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} -
+\vec{s_i}\|_2^2}_{\textrm{source-to-target--distance}}
++ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
+\vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}}
++ \lambda \cdot \textrm{regularization}(\vec{P})
+\label{eq:fit3d}
+\end{equation}
+
+where $\vec{c_T(s_i)}$ denotes the target--vertex that is corresponding to the
+source--vertex $\vec{s_i}$ and $\vec{c_S(t_i)}$ denotes the source--vertex that
+corresponds to the target--vertex $\vec{t_i}$. Note that the target--vertices
+are given and fixed by the target--model of the face we want to deform into,
+whereas the source--vertices vary depending on the chosen parameters $\vec{P}$,
+as those get calculated by the previously introduces formula $\vec{S} = \vec{UP}$
+with $\vec{S}$ being the $n \times 3$--matrix of source--vertices, $\vec{U}$ the
+$n \times m$--matrix of calculated coefficients for the \ac{FFD} --- analog to
+the 1D case --- and finally $\vec{P}$ being the $m \times 3$--matrix of the
+control--grid defining the whole deformation.
+
+As regularization-term we add a weighted Laplacian of the deformation that has
+been used before by Aschenbach et al.\cite[Section 3.2]{aschenbach2015} on
+similar models and was shown to lead to a more precise fit. The Laplacian
+\begin{equation}
+\textrm{regularization}(\vec{P}) = \frac{1}{\sum_i A_i} \sum_{i=1}^n A_i \cdot \left( \sum_{\vec{s_j} \in \mathcal{N}(\vec{s_i})} w_j \cdot \|\Delta \vec{s_j} - \Delta \vec{\overline{s}_j}\|^2 \right)
+\label{eq:reg3d}
+\end{equation}
+is determined by the cotangent weighted displacement $w_j$ of the to $s_i$
+connected vertices $\mathcal{N}(s_i)$ and $A_i$ is the Voronoi--area of the corresponding vertex
+$\vec{s_i}$. We leave out the $\vec{R}_i$--term from the original paper as our
+deformation is merely linear.
+
+This regularization--weight gives us a measure of stiffness for the material
+that we will influence via the $\lambda$--coefficient to start out with a stiff
+material that will get more flexible per iteration.
+\unsure[inline]{Andreas: hast du nen cite, wo gezeigt ist, dass das so sinnvoll ist?}
+
+# Evaluation of Scenarios
+\label{sec:res}
+
+To compare our results to the ones given by Richter et al.\cite{anrichterEvol},
+we also use Spearman's rank correlation coefficient. Opposed to other popular
+coefficients, like the Pearson correlation coefficient, which measures a linear
+relationship between variables, the Spearmans's coefficient assesses \glqq how
+well an arbitrary monotonic function can descripbe the relationship between two
+variables, without making any assumptions about the frequency distribution of
+the variables\grqq\cite{hauke2011comparison}.
+
+As we don't have any prior knowledge if any of the criteria is linear and we are
+just interested in a monotonic relation between the criteria and their
+predictive power, the Spearman's coefficient seems to fit out scenario best.
+
+For interpretation of these values we follow the same interpretation used in
+\cite{anrichterEvol}, based on \cite{weir2015spearman}: The coefficient
+intervals $r_S \in [0,0.2[$, $[0.2,0.4[$, $[0.4,0.6[$, $[0.6,0.8[$, and $[0.8,1]$ are
+classified as *very weak*, *weak*, *moderate*, *strong* and *very strong*. We
+interpret p--values smaller than $0.1$ as *significant* and cut off the
+precision of p--values after four decimal digits (thus often having a p--value
+of $0$ given for p--values $< 10^{-4}$).
+<!-- </> -->
+
+As we are looking for anti--correlation (i.e. our criterion should be maximized
+indicating a minimal result in --- for example --- the reconstruction--error)
+instead of correlation we flip the sign of the correlation--coefficient for
+readability and to have the correlation--coefficients be in the
+classification--range given above.
+
+For the evolutionary optimization we employ the CMA--ES (covariance matrix
+adaptation evolution strategy) of the shark3.1 library \cite{shark08}, as this
+algorithm was used by \cite{anrichterEvol} as well. We leave the parameters at
+their sensible defaults as further explained in
+\cite[Appendix~A: Table~1]{hansen2016cma}.
+
 ## Procedure: 1D Function Approximation
 \label{sec:proc:1d}
 
@@ -702,144 +803,6 @@ appropriate.].
 An Example of such a testcase can be seen for a $7 \times 4$--grid in figure
 \ref{fig:example1d_grid}.
 
-## Test Scenario: 3D Function Approximation
-
-Opposed to the 1--dimensional scenario before, the 3--dimensional scenario is
-much more complex --- not only because we have more degrees of freedom on each
-control point, but also because the *fitness--function* we will use has no known
-analytic solution and multiple local minima.
-
-\begin{figure}[ht]
-\begin{center}
-\includegraphics[width=0.7\textwidth]{img/3dtarget.png}
-\end{center}
-\caption[3D source and target meshes]{\newline
-Left: The sphere we start from with 10\,807 vertices\newline
-Right: The face we want to deform the sphere into with 12\,024 vertices.}
-\label{fig:3dtarget}
-\end{figure}
-
-First of all we introduce the set up: We have given a triangulated model of a
-sphere consisting of 10\,807 vertices, that we want to deform into a
-the target--model of a face with a total of 12\,024 vertices. Both of
-these Models can be seen in figure \ref{fig:3dtarget}.
-
-Opposed to the 1D--case we cannot map the source and target--vertices in a
-one--to--one--correspondence, which we especially need for the approximation of
-the fitting--error. Hence we state that the error of one vertex is the distance
-to the closest vertex of the other model.
-
-We therefore define the *fitness--function* to be:
-$$
-f(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} -
-\vec{s_i}\|_2^2}_{\textrm{source-to-target--distance}}
-+ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
-\vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}}
-+ \lambda \cdot \textrm{regularization}(\vec{P})
-$$
-where $\vec{c_T(s_i)}$ denotes the target--vertex that is corresponding to the
-source--vertex $\vec{s_i}$ and $\vec{c_S(t_i)}$ denotes the source--vertex that
-corresponds to the target--vertex $\vec{t_i}$. Note that the target--vertices
-are given and fixed by the target--model of the face we want to deform into,
-whereas the source--vertices vary depending on the chosen parameters $\vec{P}$,
-as those get calculated by the previously introduces formula $\vec{S} = \vec{UP}$
-with $\vec{S}$ being the $n \times 3$--matrix of source--vertices, $\vec{U}$ the
-$n \times m$--matrix of calculated coefficients for the \ac{FFD} --- analog to
-the 1D case --- and finally $\vec{P}$ being the $m \times 3$--matrix of the
-control--grid defining the whole deformation.
-
-As regularization-term we add a weighted Laplacian of the deformation that has
-been used before by Aschenbach et al.\cite[Section 3.2]{aschenbach2015} on
-similar models and was shown to lead to a more precise fit. The Laplacian
-$$
-\textrm{regularization}(\vec{P}) = \frac{1}{\sum_i A_i} \sum_{i=1}^n A_i \cdot \left( \sum_{\vec{s_j} \in \mathcal{N}(\vec{s_i})} w_j \cdot \|\Delta \vec{s_j} - \Delta \vec{\overline{s}_j}\|^2 \right)
-$$
-is determined by the cotangent weighted displacement $w_j$ of the to $s_i$
-connected vertices $\mathcal{N}(s_i)$ and $A_i$ is the Voronoi--area of the corresponding vertex
-$\vec{s_i}$. We leave out the $\vec{R}_i$--term from the original paper as our
-deformation is merely linear.
-
-This regularization--weight gives us a measure of stiffness for the material
-that we will influence via the $\lambda$--coefficient to start out with a stiff
-material that will get more flexible per iteration.
-\unsure[inline]{Andreas: hast du nen cite, wo gezeigt ist, dass das so sinnvoll ist?}
-
-## Procedure: 3D Function Approximation
-
-Initially we set up the correspondences $\vec{c_T(\dots)}$ and $\vec{c_S(\dots)}$ to be
-the respectively closest vertices of the other model. We then calculate the
-analytical solution given these correspondences via $\vec{P^{*}} = \vec{U^+}\vec{T}$,
-and also use the first solution as guessed gradient for the calculation of the
-*improvement--potential*, as the optimal solution is not known.
-We then let the evolutionary algorithm run up within $1.05$ times the error of 
-this solution and afterwards recalculate the correspondences $\vec{c_T(\dots)}$
-and $\vec{c_S(\dots)}$.
-For the next step we then halve the regularization--impact $\lambda$ and
-calculate the next incremental solution $\vec{P^{*}} = \vec{U^+}\vec{T}$ with
-the updated correspondences to get our next target--error.
-We repeat this process as long as the target--error keeps decreasing.
-
-\begin{figure}[ht]
-\begin{center}
-\includegraphics[width=\textwidth]{img/example3d_grid.png}
-\end{center}
-\caption[Example of a 3D--grid]{\newline Left: The 3D--setup with a $4\times
-4\times 4$--grid.\newline Right: The same grid after added noise to the
-control--points.}
-\label{fig:setup3d}
-\end{figure}
-
-The grid we use for our experiments is just very coarse due to computational
-limitations. We are not interested in a good reconstruction, but an estimate if
-the mentioned evolvability criteria are good.
-
-In figure \ref{fig:setup3d} we show an example setup of the scene with a
-$4\times 4\times 4$--grid. Identical to the 1--dimensional scenario before, we create a
-regular grid and move the control-points uniformly random between their
-neighbours, but in three instead of two dimensions^[Again, we flip the signs for
-the edges, if necessary to have the object still in the convex hull.].
-
-As is clearly visible from figure \ref{fig:3dtarget}, the target--model has many
-vertices in the facial area, at the ears and in the neck--region. Therefore we
-chose to increase the grid-resolutions for our tests in two different dimensions
-and see how well the criteria predict a suboptimal placement of these
-control-points.
-
-# Evaluation of Scenarios
-\label{sec:res}
-
-To compare our results to the ones given by Richter et al.\cite{anrichterEvol},
-we also use Spearman's rank correlation coefficient. Opposed to other popular
-coefficients, like the Pearson correlation coefficient, which measures a linear
-relationship between variables, the Spearmans's coefficient assesses \glqq how
-well an arbitrary monotonic function can descripbe the relationship between two
-variables, without making any assumptions about the frequency distribution of
-the variables\grqq\cite{hauke2011comparison}.
-
-As we don't have any prior knowledge if any of the criteria is linear and we are
-just interested in a monotonic relation between the criteria and their
-predictive power, the Spearman's coefficient seems to fit out scenario best.
-
-For interpretation of these values we follow the same interpretation used in
-\cite{anrichterEvol}, based on \cite{weir2015spearman}: The coefficient
-intervals $r_S \in [0,0.2[$, $[0.2,0.4[$, $[0.4,0.6[$, $[0.6,0.8[$, and $[0.8,1]$ are
-classified as *very weak*, *weak*, *moderate*, *strong* and *very strong*. We
-interpret p--values smaller than $0.1$ as *significant* and cut off the
-precision of p--values after four decimal digits (thus often having a p--value
-of $0$ given for p--values $< 10^{-4}$).
-
-As we are looking for anti--correlation (i.e. our criterion should be maximized
-indicating a minimal result in --- for example --- the reconstruction--error)
-instead of correlation we flip the sign of the correlation--coefficient for
-readability and to have the correlation--coefficients be in the
-classification--range given above.
-
-For the evolutionary optimization we employ the CMA--ES (covariance matrix
-adaptation evolution strategy) of the shark3.1 library \cite{shark08}, as this
-algorithm was used by \cite{anrichterEvol} as well. We leave the parameters at
-their sensible defaults as further explained in
-\cite[Appendix~A: Table~1]{hansen2016cma}.
-
 ## Results of 1D Function Approximation
 
 In the case of our 1D--Optimization--problem, we have the luxury of knowing the
@@ -894,21 +857,6 @@ between the variability and the evolutionary error.
 
 ### Regularity
 
-\begin{table}[bht]
-\centering
-\begin{tabular}{c|c|c|c|c}
-$5 \times 5$ & $7 \times 4$ & $4 \times 7$ & $7 \times 7$ & $10 \times 10$\\
-\hline
-$0.28$ ($0.0045$) & \textcolor{red}{$0.21$} ($0.0396$) & \textcolor{red}{$0.1$} ($0.3019$) & \textcolor{red}{$0.01$} ($0.9216$) & \textcolor{red}{$0.01$} ($0.9185$)
-\end{tabular}
-\caption[Correlation 1D Regularity/Steps]{Spearman's correlation (and p-values)
-between regularity and convergence speed for the 1D function approximation
-problem.\newline
-Not significant entries are marked in red.
-}
-\label{tab:1dreg}
-\end{table}
-
 \begin{figure}[ht]
 \centering
 \includegraphics[width=\textwidth]{img/evolution1d/55_to_1010_steps.png}
@@ -920,6 +868,21 @@ dataset.}
 \label{fig:1dreg}
 \end{figure}
 
+\begin{table}[b]
+\centering
+\begin{tabular}{c|c|c|c|c}
+$5 \times 5$ & $7 \times 4$ & $4 \times 7$ & $7 \times 7$ & $10 \times 10$\\
+\hline
+$0.28$ ($0.0045$) & \textcolor{red}{$0.21$} ($0.0396$) & \textcolor{red}{$0.1$} ($0.3019$) & \textcolor{red}{$0.01$} ($0.9216$) & \textcolor{red}{$0.01$} ($0.9185$)
+\end{tabular}
+\caption[Correlation 1D Regularity/Steps]{Spearman's correlation (and p-values)
+between regularity and convergence speed for the 1D function approximation
+problem.
+\newline Note: Not significant results are marked in \textcolor{red}{red}.
+}
+\label{tab:1dreg}
+\end{table}
+
 Regularity should correspond to the convergence speed (measured in
 iteration--steps of the evolutionary algorithm), and is computed as inverse
 condition number $\kappa(\vec{U})$ of the deformation--matrix.
@@ -936,13 +899,11 @@ improvement-potential against the steps next to the regularity--plot. Our theory
 is that the *very strong* correlation ($-r_S = -0.82, p=0$) between
 improvement--potential and number of iterations hints that the employed
 algorithm simply takes longer to converge on a better solution (as seen in
-figure \ref{fig:1dvar} and \ref{fig:1dimp}) offsetting any gain the regularity--measurement could
+figure \ref{fig:1dvar} and \ref{fig:1dimp}) offsetting any gain the
-achieve.
+regularity--measurement could achieve.
 
 ### Improvement Potential
 
-- Alle Spearman 1 und p-value 0.
-
 \begin{figure}[ht]
 \centering
 \includegraphics[width=0.8\textwidth]{img/evolution1d/55_to_1010_improvement-vs-evo-error.png}
@@ -952,36 +913,246 @@ grid--resolutions}
 \label{fig:1dimp}
 \end{figure}
 
-<!-- ![Improvement potential vs steps](img/evolution1d/20170830-evolution1D_5x5_100Times-all_improvement-vs-steps.png) -->
+\improvement[inline]{write something about it..}
-<!--  -->
+
-<!-- ![Improvement potential vs evolutional error](img/evolution1d/20170830-evolution1D_5x5_100Times-all_improvement-vs-evo-error.png) -->
+- spearman 1 (p=0)
-<!--  -->
+- gradient macht keinen unterschied
-<!-- ![Regularity vs steps](img/evolution1d/20170830-evolution1D_5x5_100Times-all_regularity-vs-steps.png) -->
+- $UU^+$ scheint sehr kleine EW zu haben, s. regularität
+- trotzdem sehr gutes kriterium - auch ohne Richtung.
+
+## Procedure: 3D Function Approximation
+\label{sec:proc:3dfa}
+
+As explained in section \ref{sec:test:3dfa} in detail, we do not know the
+analytical solution to the global optimum. Additionally we have the problem of
+finding the right correspondences between the original sphere--model and the
+target--model, as they consist of $10\,807$ and $12\,024$ vertices respectively,
+so we cannot make a one--to--one--correspondence between them as we did in the
+one--dimensional case.
+
+Initially we set up the correspondences $\vec{c_T(\dots)}$ and $\vec{c_S(\dots)}$ to be
+the respectively closest vertices of the other model. We then calculate the
+analytical solution given these correspondences via $\vec{P^{*}} = \vec{U^+}\vec{T}$,
+and also use the first solution as guessed gradient for the calculation of the
+*improvement--potential*, as the optimal solution is not known.
+We then let the evolutionary algorithm run up within $1.05$ times the error of 
+this solution and afterwards recalculate the correspondences $\vec{c_T(\dots)}$
+and $\vec{c_S(\dots)}$.
+
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=\textwidth]{img/example3d_grid.png}
+\end{center}
+\caption[Example of a 3D--grid]{\newline Left: The 3D--setup with a $4\times
+4\times 4$--grid.\newline Right: The same grid after added noise to the
+control--points.}
+\label{fig:setup3d}
+\end{figure}
+
+For the next step we then halve the regularization--impact $\lambda$ (starting
+at $1$) of our *fitness--function* (\ref{eq:fit3d}) and calculate the next
+incremental solution $\vec{P^{*}} = \vec{U^+}\vec{T}$ with the updated
+correspondences to get our next target--error. We repeat this process as long as
+the target--error keeps decreasing and use the number of these iterations as
+measure of the convergence speed. As the resulting evolutional error without
+regularization is in the numeric range of $\approx 100$, whereas the
+regularization is numerically $\approx 7000$ we need at least $10$ to $15$ iterations
+until the regularization--effect wears off.
+
+The grid we use for our experiments is just very coarse due to computational
+limitations. We are not interested in a good reconstruction, but an estimate if
+the mentioned evolvability criteria are good.
+
+In figure \ref{fig:setup3d} we show an example setup of the scene with a
+$4\times 4\times 4$--grid. Identical to the 1--dimensional scenario before, we create a
+regular grid and move the control-points \todo{wie?} random between their
+neighbours, but in three instead of two dimensions^[Again, we flip the signs for
+the edges, if necessary to have the object still in the convex hull.].
+
+\begin{figure}[!htb]
+\includegraphics[width=\textwidth]{img/3d_grid_resolution.png}
+\caption[Different resolution of 3D grids]{\newline
+Left: A $7 \times 4 \times 4$ grid suited to better deform into facial
+features.\newline
+Right: A $4 \times 4 \times 7$ grid that we expect to perform worse.}
+\label{fig:3dgridres}
+\end{figure}
+
+As is clearly visible from figure \ref{fig:3dgridres}, the target--model has many
+vertices in the facial area, at the ears and in the neck--region. Therefore we
+chose to increase the grid-resolutions for our tests in two different dimensions
+and see how well the criteria predict a suboptimal placement of these
+control-points.
 
 ## Results of 3D Function Approximation
 
+In the 3D--Approximation we tried to evaluate further on the impact of the
+grid--layout to the overall criteria. As the target--model has many vertices in
+concentrated in the facial area we start from a $4 \times 4 \times 4$ grid and
+only increase the number of control points in one dimension, yielding a
+resolution of $7 \times 4 \times 4$ and $4 \times 4 \times 7$ respectively. We
+visualized those two grids in figure \ref{fig:3dgridres}.
 
-\begin{figure}[!ht]
+To evaluate the performance of the evolvability--criteria we also tested a more
-\includegraphics[width=\textwidth]{img/evolution3d/4x4xX_montage.png}
+neutral resolution of $4 \times 4 \times 4$, $5 \times 5 \times 5$, and $6 \times 6 \times 6$ ---
-\caption{Results 3D for 4x4xX}
+similar to the 1D--setup.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=0.7\textwidth]{img/evolution3d/variability_boxplot.png}
+\caption[3D Fitting Errors for various grids]{The fitting error for the various
+grids we examined.\newline
+Note that the number of control--points is a product of the resolution, so $X
+\times 4 \times 4$ and $4 \times 4 \times X$ have the same number of
+control--points.}
+\label{fig:3dvar}
 \end{figure}
 
-\begin{figure}[!ht]
+### Variability
-\includegraphics[width=\textwidth]{img/evolution3d/Xx4x4_montage.png}
+
-\caption{Results 3D for Xx4x4}
+\begin{table}[tbh]
+\centering
+\begin{tabular}{c|c|c|c}
+$4 \times 4 \times \mathrm{X}$ & $\mathrm{X} \times 4 \times 4$ & $\mathrm{Y} \times \mathrm{Y} \times \mathrm{Y}$ & all \\
+\hline
+0.89 (0) & 0.9 (0) & 0.91 (0) & 0.94 (0)
+\end{tabular}
+\caption[Correlation between variability and fitting error for 3D]{Correlation
+between variability and fitting error for the 3D fitting scenario.\newline
+Displayed are the negated Spearman coefficients with the corresponding p-values
+in brackets for three cases of increasing variability ($\mathrm{X} \in [4,5,7],
+\mathrm{Y} \in [4,5,6]$).
+\newline Note: Not significant results are marked in \textcolor{red}{red}.}
+\label{tab:3dvar}
+\end{table}
+
+Similar to the 1D case all our tested matrices had a constant rank (being
+$m = x \cdot y \cdot z$ for a $x \times y \times z$ grid), so we again have merely plotted
+the errors in the boxplot in figure \ref{fig:3dvar}.
+
+As expected the $\mathrm{X} \times 4 \times 4$ grids performed
+slightly better than their $4 \times 4 \times \mathrm{X}$ counterparts with a
+mean$\pm$sigma of $101.25 \pm 7.45$ to $102.89 \pm 6.74$ for $\mathrm{X} = 5$ and
+$85.37 \pm 7.12$ to $89.22 \pm 6.49$ for $\mathrm{X} = 7$.
+
+Interestingly both variants end up closer in terms of fitting error than we
+anticipated, which shows that the evolutionary algorithm we employed is capable
+of correcting a purposefully created \glqq bad\grqq \ grid. Also this confirms,
+that in our cases the number of control--points is more important for quality
+than their placement, which is captured by the variability via the rank of the
+deformation--matrix.
+
+\begin{figure}[hbt]
+\centering
+\includegraphics[width=0.8\textwidth]{img/evolution3d/variability2_boxplot.png}
+\caption[Histogram of ranks of high--resolution deformation--matrices]{
+Histogram of ranks of various $10 \times 10 \times 10$ grids.
+}
+\label{fig:histrank3d}
 \end{figure}
 
-\begin{figure}[!ht]
+Overall the correlation between variability and fitness--error were
-\includegraphics[width=\textwidth]{img/evolution3d/YxYxY_montage.png}
+*significantly* and showed a *very strong* correlation in all our tests.
-\caption{Results 3D for YxYxY for Y $\in [4,5,6]$}
+The detailed correlation--coefficients are given in table \ref{tab:3dvar}
+alongside their p--values.
+
+As introduces in section \ref{sec:impl:grid} and visualized in figure
+\ref{fig:enoughCP}, we know, that not all control points have to necessarily
+contribute to the parametrization of our 3D--model. Because we are starting from
+a sphere, some control-points are too far away from the surface to contribute
+to the deformation at all.
+
+One can already see in 2D in figure \ref{fig:enoughCP}, that this effect
+starts with a regular $9 \times 9$ grid on a perfect circle. To make sure we
+observe this, we evaluated the variability for 100 randomly moved $10 \times 10 \times 10$
+grids on the sphere we start out with.
+
+As the variability is defined by $\frac{\mathrm{rank}(\vec{U})}{n}$ we can
+easily recover the rank of the deformation--matrix $\vec{U}$. The results are
+shown in the histogram in figure \ref{fig:histrank3d}. Especially in the centre
+of the sphere and in the corners of our grid we effectively loose
+control--points for our parametrization.
+
+This of course yields a worse error as when those control--points would be put
+to use and one should expect a loss in quality evident by a higher
+reconstruction--error opposed to a grid where they are used. Sadly we could not
+run a in--depth test on this due to computational limitations.
+
+Nevertheless this hints at the notion, that variability is a good measure for
+the overall quality of a fit.
+
+### Regularity
+
+\begin{table}[tbh]
+\centering
+\begin{tabular}{c|c|c|c}
+ & $5 \times 4 \times 4$ & $7 \times 4 \times 4$ & $\mathrm{X} \times 4 \times 4$ \\
+\cline{2-4}
+ & \textcolor{red}{0.15} (0.147) & \textcolor{red}{0.09} (0.37) & 0.46 (0) \B \\
+\cline{2-4}
+\multicolumn{4}{c}{} \\[-1.4em]
+\hline
+$4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \times 4 \times \mathrm{X}$ \T \\
+\hline
+0.38 (0) & \textcolor{red}{0.17} (0.09) & 0.40 (0) & 0.46 (0) \B \\
+\hline
+\multicolumn{4}{c}{} \\[-1.4em]
+\cline{2-4}
+ & $5 \times 5 \times 5$ & $6 \times 6 \times 6$ & $\mathrm{Y} \times \mathrm{Y} \times \mathrm{Y}$ \T \\
+\cline{2-4}
+ & \textcolor{red}{-0.18} (0.0775) & \textcolor{red}{-0.13} (0.1715) & -0.25 (0) \B \\
+\cline{2-4}
+\multicolumn{4}{c}{} \\[-1.4em]
+\cline{2-4}
+\multicolumn{3}{c}{} & all: 0.15 (0) \T
+\end{tabular}
+\caption[Correlation between regularity and iterations for 3D]{Correlation
+between regularity and number of iterations for the 3D fitting scenario.
+Displayed are the negated Spearman coefficients with the corresponding p--values
+in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
+\newline Note: Not significant results are marked in \textcolor{red}{red}.}
+\label{tab:3dvar}
+\end{table}
+
+
+
+Opposed to the predictions of variability our test on regularity gave a mixed
+result --- similar to the 1D--case.
+
+In half scenarios we have a *significant*, but *weak* to *moderate* correlation
+between regularity and number of iterations. On the other hand in the scenarios
+where we increased the number of control--points, namely $125$ for the
+$5 \times 5 \times 5$ grid and $216$ for the $6 \times 6 \times 6$ grid we found
+a *significant*, but *weak* anti--correlation, which seem to contradict the
+findings/trends for the sets with $64$, $80$, and $112$ control--points (first
+two rows of table \ref{tab:3dvar}).
+
+Taking all results together we only find a *very weak*, but *significant* link
+between regularity and the number of iterations needed for the algorithm to
+converge.
+
+\begin{figure}[!htb]
+\centering
+\includegraphics[width=\textwidth]{img/evolution3d/regularity_montage.png}
+\caption[Regularity for different 3D--grids]{
+**BLINDTEXT**
+}
+\label{fig:resreg3d}
+\end{figure}
+
+As can be seen from figure \ref{fig:resreg3d}, we can observe\todo{things}.
+
+### Improvement Potential
+
+\begin{figure}[!htb]
+\centering
+\includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
+\caption[Improvement potential for different 3D--grids]{
+**BLINDTEXT**
+}
+\label{fig:resimp3d}
 \end{figure}
 
-<!-- ![Improvement potential vs steps](img/evolution3d/20170926_3dFit_both_improvement-vs-steps.png) -->
-<!--  -->
-<!-- ![Improvement potential vs evolutional -->
-<!-- error](img/evolution3d/20170926_3dFit_both_improvement-vs-evo-error.png) -->
-<!--  -->
-<!-- ![Regularity vs steps](img/evolution3d/20170926_3dFit_both_regularity-vs-steps.png) -->
 
 # Schluss
 \label{sec:dis}

arbeit/ma.tex

@@ -35,6 +35,7 @@ xcolor=dvipsnames,
 \setlength{\parskip}{12pt plus6pt minus2pt} % dafür abstand zwischen absäzen
 % \renewcommand{\familydefault}{\sfdefault}
 \setstretch{1.5} % 1.5-facher zeilenabstand
+\renewcommand{\arraystretch}{1.5} % größere Abstände in Tabellen etc.
 
 %%%%%%%%%%%%%%% Header - Footer %%%%%%%%%%%%%%%
 % ### Fr 2 Seitig (option twopage):
@@ -172,7 +173,7 @@ Unless otherwise noted the following holds:
 Many modern industrial design processes require advanced optimization
 methods due to the increased complexity resulting from more and more
 degrees of freedom as methods refine and/or other methods are used.
-Examples for this are physical domains like aerodynamic (i.e.~drag),
+Examples for this are physical domains like aerodynamics (i.e.~drag),
 fluid dynamics (i.e.~throughput of liquid) --- where the complexity
 increases with the temporal and spatial resolution of the simulation ---
 or known hard algorithmic problems in informatics (i.e.~layouting of
@@ -191,7 +192,7 @@ representation (the \emph{genome}) can be challenging.
 
 This translation is often necessary as the target of the optimization
 may have too many degrees of freedom. In the example of an aerodynamic
-simulation of drag onto an object, those objects--designs tend to have a
+simulation of drag onto an object, those object--designs tend to have a
 high number of vertices to adhere to various requirements (visual,
 practical, physical, etc.). A simpler representation of the same object
 in only a few parameters that manipulate the whole in a sensible matter
@@ -601,7 +602,7 @@ can be achieved in the given direction.
 
 The definition for an \emph{improvement potential} \(P\)
 is\cite{anrichterEvol}: \[
-P(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec(G)\|^2_F
+P(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F
 \] given some approximate \(n \times d\) fitness--gradient \(\vec{G}\),
 normalized to \(\|\vec{G}\|_F = 1\), whereby \(\|\cdot\|_F\) denotes the
 Frobenius--Norm.
@@ -728,6 +729,8 @@ system of linear equations.
 
 \section{Deformation Grid}\label{deformation-grid}
 
+\label{sec:impl:grid}
+
 As mentioned in chapter \ref{sec:back:evo}, the way of choosing the
 representation to map the general problem (mesh--fitting/optimization in
 our case) into a parameter-space it very important for the quality and
@@ -744,6 +747,17 @@ control point without having a \(1:1\)--correlation, and a smooth
 deformation. While the advantages are great, the issues arise from the
 problem to decide where to place the control--points and how many.
 
+\begin{figure}[!tbh]
+\centering
+\includegraphics{img/enoughCP.png}
+\caption[Example of a high resolution control--grid]{A high resolution
+($10 \times 10$) of control--points over a circle. Yellow/green points
+contribute to the parametrization, red points don't.\newline
+An Example--point (blue) is solely determined by the position of the green
+control--points.}
+\label{fig:enoughCP}
+\end{figure}
+
 One would normally think, that the more control--points you add, the
 better the result will be, but this is not the case for our B--Splines.
 Given any point \(p\) only the \(2 \cdot (d-1)\) control--points
@@ -754,18 +768,6 @@ that a high resolution can have many control-points that are not
 contributing to any point on the surface and are thus completely
 irrelevant to the solution.
 
-\begin{figure}[!ht]
-\begin{center}
-\includegraphics{img/enoughCP.png}
-\end{center}
-\caption[Example of a high resolution control--grid]{A high resolution
-($10 \times 10$) of control--points over a circle. Yellow/green points
-contribute to the parametrization, red points don't.\newline
-An Example--point (blue) is solely determined by the position of the green
-control--points.}
-\label{fig:enoughCP}
-\end{figure}
-
 We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the
 four red central points are not relevant for the parametrization of the
 circle.
@@ -805,14 +807,18 @@ In this scenario we used the shape defined by Giannelli et
 al.\cite{giannelli2012thb}, which is also used by Richter et
 al.\cite{anrichterEvol} using the same discretization to
 \(150 \times 150\) points for a total of \(n = 22\,500\) vertices. The
-shape is given by the following definition \[
+shape is given by the following definition
+
+\begin{equation}
 t(x,y) =
 \begin{cases}
 0.5 \cos(4\pi \cdot q^{0.5}) + 0.5 & q(x,y) < \frac{1}{16},\\
 2(y-x) & 0 < y-x < 0.5,\\
 1 & 0.5 < y - x
 \end{cases}
-\] with \((x,y) \in [0,2] \times [0,1]\) and
+\end{equation}
+
+with \((x,y) \in [0,2] \times [0,1]\) and
 \(q(x,y)=(x-1.5)^2 + (y-0.5)^2\), which we have visualized in figure
 \ref{fig:1dtarget}.
 
@@ -831,9 +837,13 @@ already correct.
 
 Regarding the \emph{fitness--function} \(f(\vec{p})\), we use the very
 simple approach of calculating the squared distances for each
-corresponding vertex \[
+corresponding vertex
+
+\begin{equation}
 \textrm{f(\vec{p})} = \sum_{i=1}^{n} \|(\vec{Up})_i - t_i\|_2^2 = \|\vec{Up} - \vec{t}\|^2 \rightarrow \min
-\] where \(t_i\) are the respective target--vertices to the parametrized
+\end{equation}
+
+where \(t_i\) are the respective target--vertices to the parametrized
 source--vertices\footnote{The parametrization is encoded in \(\vec{U}\)
   and the initial position of the control points. See
   \ref{sec:ffd:adapt}} with the current deformation--parameters
@@ -847,6 +857,120 @@ can compute the analytic solution \(\vec{p^{*}} = \vec{U^+}\vec{t}\),
 yielding us the correct gradient in which the evolutionary optimizer
 should move.
 
+\section{Test Scenario: 3D Function
+Approximation}\label{test-scenario-3d-function-approximation}
+
+\label{sec:test:3dfa} Opposed to the 1--dimensional scenario before, the
+3--dimensional scenario is much more complex --- not only because we
+have more degrees of freedom on each control point, but also because the
+\emph{fitness--function} we will use has no known analytic solution and
+multiple local minima.
+
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=0.9\textwidth]{img/3dtarget.png}
+\end{center}
+\caption[3D source and target meshes]{\newline
+Left: The sphere we start from with 10\,807 vertices\newline
+Right: The face we want to deform the sphere into with 12\,024 vertices.}
+\label{fig:3dtarget}
+\end{figure}
+
+First of all we introduce the set up: We have given a triangulated model
+of a sphere consisting of \(10\,807\) vertices, that we want to deform
+into a the target--model of a face with a total of \(12\,024\) vertices.
+Both of these Models can be seen in figure \ref{fig:3dtarget}.
+
+Opposed to the 1D--case we cannot map the source and target--vertices in
+a one--to--one--correspondence, which we especially need for the
+approximation of the fitting--error. Hence we state that the error of
+one vertex is the distance to the closest vertex of the other model.
+
+We therefore define the \emph{fitness--function} to be:
+
+\begin{equation}
+f(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} -
+\vec{s_i}\|_2^2}_{\textrm{source-to-target--distance}}
++ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
+\vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}}
++ \lambda \cdot \textrm{regularization}(\vec{P})
+\label{eq:fit3d}
+\end{equation}
+
+where \(\vec{c_T(s_i)}\) denotes the target--vertex that is
+corresponding to the source--vertex \(\vec{s_i}\) and \(\vec{c_S(t_i)}\)
+denotes the source--vertex that corresponds to the target--vertex
+\(\vec{t_i}\). Note that the target--vertices are given and fixed by the
+target--model of the face we want to deform into, whereas the
+source--vertices vary depending on the chosen parameters \(\vec{P}\), as
+those get calculated by the previously introduces formula
+\(\vec{S} = \vec{UP}\) with \(\vec{S}\) being the \(n \times 3\)--matrix
+of source--vertices, \(\vec{U}\) the \(n \times m\)--matrix of
+calculated coefficients for the \ac{FFD} --- analog to the 1D case ---
+and finally \(\vec{P}\) being the \(m \times 3\)--matrix of the
+control--grid defining the whole deformation.
+
+As regularization-term we add a weighted Laplacian of the deformation
+that has been used before by Aschenbach et
+al.\cite[Section 3.2]{aschenbach2015} on similar models and was shown to
+lead to a more precise fit. The Laplacian
+
+\begin{equation}
+\textrm{regularization}(\vec{P}) = \frac{1}{\sum_i A_i} \sum_{i=1}^n A_i \cdot \left( \sum_{\vec{s_j} \in \mathcal{N}(\vec{s_i})} w_j \cdot \|\Delta \vec{s_j} - \Delta \vec{\overline{s}_j}\|^2 \right)
+\label{eq:reg3d}
+\end{equation}
+
+is determined by the cotangent weighted displacement \(w_j\) of the to
+\(s_i\) connected vertices \(\mathcal{N}(s_i)\) and \(A_i\) is the
+Voronoi--area of the corresponding vertex \(\vec{s_i}\). We leave out
+the \(\vec{R}_i\)--term from the original paper as our deformation is
+merely linear.
+
+This regularization--weight gives us a measure of stiffness for the
+material that we will influence via the \(\lambda\)--coefficient to
+start out with a stiff material that will get more flexible per
+iteration.
+\unsure[inline]{Andreas: hast du nen cite, wo gezeigt ist, dass das so sinnvoll ist?}
+
+\chapter{Evaluation of Scenarios}\label{evaluation-of-scenarios}
+
+\label{sec:res}
+
+To compare our results to the ones given by Richter et
+al.\cite{anrichterEvol}, we also use Spearman's rank correlation
+coefficient. Opposed to other popular coefficients, like the Pearson
+correlation coefficient, which measures a linear relationship between
+variables, the Spearmans's coefficient assesses \glqq how well an
+arbitrary monotonic function can descripbe the relationship between two
+variables, without making any assumptions about the frequency
+distribution of the variables\grqq\cite{hauke2011comparison}.
+
+As we don't have any prior knowledge if any of the criteria is linear
+and we are just interested in a monotonic relation between the criteria
+and their predictive power, the Spearman's coefficient seems to fit out
+scenario best.
+
+For interpretation of these values we follow the same interpretation
+used in \cite{anrichterEvol}, based on \cite{weir2015spearman}: The
+coefficient intervals \(r_S \in [0,0.2[\), \([0.2,0.4[\), \([0.4,0.6[\),
+\([0.6,0.8[\), and \([0.8,1]\) are classified as \emph{very weak},
+\emph{weak}, \emph{moderate}, \emph{strong} and \emph{very strong}. We
+interpret p--values smaller than \(0.1\) as \emph{significant} and cut
+off the precision of p--values after four decimal digits (thus often
+having a p--value of \(0\) given for p--values \(< 10^{-4}\)).
+
+As we are looking for anti--correlation (i.e.~our criterion should be
+maximized indicating a minimal result in --- for example --- the
+reconstruction--error) instead of correlation we flip the sign of the
+correlation--coefficient for readability and to have the
+correlation--coefficients be in the classification--range given above.
+
+For the evolutionary optimization we employ the CMA--ES (covariance
+matrix adaptation evolution strategy) of the shark3.1 library
+\cite{shark08}, as this algorithm was used by \cite{anrichterEvol} as
+well. We leave the parameters at their sensible defaults as further
+explained in \cite[Appendix~A: Table~1]{hansen2016cma}.
+
 \section{Procedure: 1D Function
 Approximation}\label{procedure-1d-function-approximation}
 
@@ -893,155 +1017,6 @@ neighbours (the smaller neighbour for \(r < 0\), the larger for
 An Example of such a testcase can be seen for a \(7 \times 4\)--grid in
 figure \ref{fig:example1d_grid}.
 
-\section{Test Scenario: 3D Function
-Approximation}\label{test-scenario-3d-function-approximation}
-
-Opposed to the 1--dimensional scenario before, the 3--dimensional
-scenario is much more complex --- not only because we have more degrees
-of freedom on each control point, but also because the
-\emph{fitness--function} we will use has no known analytic solution and
-multiple local minima.
-
-\begin{figure}[ht]
-\begin{center}
-\includegraphics[width=0.7\textwidth]{img/3dtarget.png}
-\end{center}
-\caption[3D source and target meshes]{\newline
-Left: The sphere we start from with 10\,807 vertices\newline
-Right: The face we want to deform the sphere into with 12\,024 vertices.}
-\label{fig:3dtarget}
-\end{figure}
-
-First of all we introduce the set up: We have given a triangulated model
-of a sphere consisting of 10,807 vertices, that we want to deform into a
-the target--model of a face with a total of 12,024 vertices. Both of
-these Models can be seen in figure \ref{fig:3dtarget}.
-
-Opposed to the 1D--case we cannot map the source and target--vertices in
-a one--to--one--correspondence, which we especially need for the
-approximation of the fitting--error. Hence we state that the error of
-one vertex is the distance to the closest vertex of the other model.
-
-We therefore define the \emph{fitness--function} to be: \[
-f(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} -
-\vec{s_i}\|_2^2}_{\textrm{source-to-target--distance}}
-+ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
-\vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}}
-+ \lambda \cdot \textrm{regularization}(\vec{P})
-\] where \(\vec{c_T(s_i)}\) denotes the target--vertex that is
-corresponding to the source--vertex \(\vec{s_i}\) and \(\vec{c_S(t_i)}\)
-denotes the source--vertex that corresponds to the target--vertex
-\(\vec{t_i}\). Note that the target--vertices are given and fixed by the
-target--model of the face we want to deform into, whereas the
-source--vertices vary depending on the chosen parameters \(\vec{P}\), as
-those get calculated by the previously introduces formula
-\(\vec{S} = \vec{UP}\) with \(\vec{S}\) being the \(n \times 3\)--matrix
-of source--vertices, \(\vec{U}\) the \(n \times m\)--matrix of
-calculated coefficients for the \ac{FFD} --- analog to the 1D case ---
-and finally \(\vec{P}\) being the \(m \times 3\)--matrix of the
-control--grid defining the whole deformation.
-
-As regularization-term we add a weighted Laplacian of the deformation
-that has been used before by Aschenbach et
-al.\cite[Section 3.2]{aschenbach2015} on similar models and was shown to
-lead to a more precise fit. The Laplacian \[
-\textrm{regularization}(\vec{P}) = \frac{1}{\sum_i A_i} \sum_{i=1}^n A_i \cdot \left( \sum_{\vec{s_j} \in \mathcal{N}(\vec{s_i})} w_j \cdot \|\Delta \vec{s_j} - \Delta \vec{\overline{s}_j}\|^2 \right)
-\] is determined by the cotangent weighted displacement \(w_j\) of the
-to \(s_i\) connected vertices \(\mathcal{N}(s_i)\) and \(A_i\) is the
-Voronoi--area of the corresponding vertex \(\vec{s_i}\). We leave out
-the \(\vec{R}_i\)--term from the original paper as our deformation is
-merely linear.
-
-This regularization--weight gives us a measure of stiffness for the
-material that we will influence via the \(\lambda\)--coefficient to
-start out with a stiff material that will get more flexible per
-iteration.
-\unsure[inline]{Andreas: hast du nen cite, wo gezeigt ist, dass das so sinnvoll ist?}
-
-\section{Procedure: 3D Function
-Approximation}\label{procedure-3d-function-approximation}
-
-Initially we set up the correspondences \(\vec{c_T(\dots)}\) and
-\(\vec{c_S(\dots)}\) to be the respectively closest vertices of the
-other model. We then calculate the analytical solution given these
-correspondences via \(\vec{P^{*}} = \vec{U^+}\vec{T}\), and also use the
-first solution as guessed gradient for the calculation of the
-\emph{improvement--potential}, as the optimal solution is not known. We
-then let the evolutionary algorithm run up within \(1.05\) times the
-error of this solution and afterwards recalculate the correspondences
-\(\vec{c_T(\dots)}\) and \(\vec{c_S(\dots)}\). For the next step we then
-halve the regularization--impact \(\lambda\) and calculate the next
-incremental solution \(\vec{P^{*}} = \vec{U^+}\vec{T}\) with the updated
-correspondences to get our next target--error. We repeat this process as
-long as the target--error keeps decreasing.
-
-\begin{figure}[ht]
-\begin{center}
-\includegraphics[width=\textwidth]{img/example3d_grid.png}
-\end{center}
-\caption[Example of a 3D--grid]{\newline Left: The 3D--setup with a $4\times
-4\times 4$--grid.\newline Right: The same grid after added noise to the
-control--points.}
-\label{fig:setup3d}
-\end{figure}
-
-The grid we use for our experiments is just very coarse due to
-computational limitations. We are not interested in a good
-reconstruction, but an estimate if the mentioned evolvability criteria
-are good.
-
-In figure \ref{fig:setup3d} we show an example setup of the scene with a
-\(4\times 4\times 4\)--grid. Identical to the 1--dimensional scenario
-before, we create a regular grid and move the control-points uniformly
-random between their neighbours, but in three instead of two
-dimensions\footnote{Again, we flip the signs for the edges, if necessary
-  to have the object still in the convex hull.}.
-
-As is clearly visible from figure \ref{fig:3dtarget}, the target--model
-has many vertices in the facial area, at the ears and in the
-neck--region. Therefore we chose to increase the grid-resolutions for
-our tests in two different dimensions and see how well the criteria
-predict a suboptimal placement of these control-points.
-
-\chapter{Evaluation of Scenarios}\label{evaluation-of-scenarios}
-
-\label{sec:res}
-
-To compare our results to the ones given by Richter et
-al.\cite{anrichterEvol}, we also use Spearman's rank correlation
-coefficient. Opposed to other popular coefficients, like the Pearson
-correlation coefficient, which measures a linear relationship between
-variables, the Spearmans's coefficient assesses \glqq how well an
-arbitrary monotonic function can descripbe the relationship between two
-variables, without making any assumptions about the frequency
-distribution of the variables\grqq\cite{hauke2011comparison}.
-
-As we don't have any prior knowledge if any of the criteria is linear
-and we are just interested in a monotonic relation between the criteria
-and their predictive power, the Spearman's coefficient seems to fit out
-scenario best.
-
-For interpretation of these values we follow the same interpretation
-used in \cite{anrichterEvol}, based on \cite{weir2015spearman}: The
-coefficient intervals \(r_S \in [0,0.2[\), \([0.2,0.4[\), \([0.4,0.6[\),
-\([0.6,0.8[\), and \([0.8,1]\) are classified as \emph{very weak},
-\emph{weak}, \emph{moderate}, \emph{strong} and \emph{very strong}. We
-interpret p--values smaller than \(0.1\) as \emph{significant} and cut
-off the precision of p--values after four decimal digits (thus often
-having a p--value of \(0\) given for p--values \(< 10^{-4}\)).
-
-As we are looking for anti--correlation (i.e.~our criterion should be
-maximized indicating a minimal result in --- for example --- the
-reconstruction--error) instead of correlation we flip the sign of the
-correlation--coefficient for readability and to have the
-correlation--coefficients be in the classification--range given above.
-
-For the evolutionary optimization we employ the CMA--ES (covariance
-matrix adaptation evolution strategy) of the shark3.1 library
-\cite{shark08}, as this algorithm was used by \cite{anrichterEvol} as
-well. We leave the parameters at their sensible defaults as further
-explained in \cite[Appendix~A: Table~1]{hansen2016cma}.
-
 \section{Results of 1D Function
 Approximation}\label{results-of-1d-function-approximation}
 
@@ -1072,7 +1047,7 @@ resolution of the grid by taking a closer look at \(5 \times 5\),
 \caption[1D Fitting Errors for various grids]{The squared error for the various
 grids we examined.\newline
 Note that $7 \times 4$ and $4 \times 7$ have the same number of control--points.}
-\label{fig:1dfiterr}
+\label{fig:1dvar}
 \end{figure}
 
 \subsection{Variability}\label{variability-1}
@@ -1083,7 +1058,7 @@ deformation matrix \(\vec{U}\):
 \(V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n}\), whereby \(n\) is the
 number of vertices. As all our tested matrices had a constant rank
 (being \(m = x \cdot y\) for a \(x \times y\) grid), we have merely
-plotted the errors in the boxplot in figure \ref{fig:1dfiterr}
+plotted the errors in the boxplot in figure \ref{fig:1dvar}
 
 It is also noticeable, that although the \(7 \times 4\) and
 \(4 \times 7\) grids have a higher variability, they perform not better
@@ -1102,21 +1077,6 @@ variability and the evolutionary error.
 
 \subsection{Regularity}\label{regularity-1}
 
-\begin{table}[bht]
-\centering
-\begin{tabular}{c|c|c|c|c}
-$5 \times 5$ & $7 \times 4$ & $4 \times 7$ & $7 \times 7$ & $10 \times 10$\\
-\hline
-$0.28$ ($0.0045$) & \textcolor{red}{$0.21$} ($0.0396$) & \textcolor{red}{$0.1$} ($0.3019$) & \textcolor{red}{$0.01$} ($0.9216$) & \textcolor{red}{$0.01$} ($0.9185$)
-\end{tabular}
-\caption[Correlation 1D Regularity/Steps]{Spearman's correlation (and p-values)
-between regularity and convergence speed for the 1D function approximation
-problem.\newline
-Not significant entries are marked in red.
-}
-\label{tab:1dreg}
-\end{table}
-
 \begin{figure}[ht]
 \centering
 \includegraphics[width=\textwidth]{img/evolution1d/55_to_1010_steps.png}
@@ -1128,6 +1088,21 @@ dataset.}
 \label{fig:1dreg}
 \end{figure}
 
+\begin{table}[b]
+\centering
+\begin{tabular}{c|c|c|c|c}
+$5 \times 5$ & $7 \times 4$ & $4 \times 7$ & $7 \times 7$ & $10 \times 10$\\
+\hline
+$0.28$ ($0.0045$) & \textcolor{red}{$0.21$} ($0.0396$) & \textcolor{red}{$0.1$} ($0.3019$) & \textcolor{red}{$0.01$} ($0.9216$) & \textcolor{red}{$0.01$} ($0.9185$)
+\end{tabular}
+\caption[Correlation 1D Regularity/Steps]{Spearman's correlation (and p-values)
+between regularity and convergence speed for the 1D function approximation
+problem.
+\newline Note: Not significant results are marked in \textcolor{red}{red}.
+}
+\label{tab:1dreg}
+\end{table}
+
 Regularity should correspond to the convergence speed (measured in
 iteration--steps of the evolutionary algorithm), and is computed as
 inverse condition number \(\kappa(\vec{U})\) of the deformation--matrix.
@@ -1145,17 +1120,12 @@ improvement-potential against the steps next to the regularity--plot.
 Our theory is that the \emph{very strong} correlation
 (\(-r_S = -0.82, p=0\)) between improvement--potential and number of
 iterations hints that the employed algorithm simply takes longer to
-converge on a better solution (as seen in figure \ref{fig:1dimp})
+converge on a better solution (as seen in figure \ref{fig:1dvar} and
-offsetting any gain the regularity--measurement could achieve.
+\ref{fig:1dimp}) offsetting any gain the regularity--measurement could
+achieve.
 
 \subsection{Improvement Potential}\label{improvement-potential-1}
 
-\begin{itemize}
-\tightlist
-\item
-  Alle Spearman 1 und p-value 0.
-\end{itemize}
-
 \begin{figure}[ht]
 \centering
 \includegraphics[width=0.8\textwidth]{img/evolution1d/55_to_1010_improvement-vs-evo-error.png}
@@ -1165,22 +1135,265 @@ grid--resolutions}
 \label{fig:1dimp}
 \end{figure}
 
+\improvement[inline]{write something about it..}
+
+\begin{itemize}
+\tightlist
+\item
+  spearman 1 (p=0)
+\item
+  gradient macht keinen unterschied
+\item
+  \(UU^+\) scheint sehr kleine EW zu haben, s. regularität
+\item
+  trotzdem sehr gutes kriterium - auch ohne Richtung.
+\end{itemize}
+
+\section{Procedure: 3D Function
+Approximation}\label{procedure-3d-function-approximation}
+
+\label{sec:proc:3dfa}
+
+As explained in section \ref{sec:test:3dfa} in detail, we do not know
+the analytical solution to the global optimum. Additionally we have the
+problem of finding the right correspondences between the original
+sphere--model and the target--model, as they consist of \(10\,807\) and
+\(12\,024\) vertices respectively, so we cannot make a
+one--to--one--correspondence between them as we did in the
+one--dimensional case.
+
+Initially we set up the correspondences \(\vec{c_T(\dots)}\) and
+\(\vec{c_S(\dots)}\) to be the respectively closest vertices of the
+other model. We then calculate the analytical solution given these
+correspondences via \(\vec{P^{*}} = \vec{U^+}\vec{T}\), and also use the
+first solution as guessed gradient for the calculation of the
+\emph{improvement--potential}, as the optimal solution is not known. We
+then let the evolutionary algorithm run up within \(1.05\) times the
+error of this solution and afterwards recalculate the correspondences
+\(\vec{c_T(\dots)}\) and \(\vec{c_S(\dots)}\).
+
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=\textwidth]{img/example3d_grid.png}
+\end{center}
+\caption[Example of a 3D--grid]{\newline Left: The 3D--setup with a $4\times
+4\times 4$--grid.\newline Right: The same grid after added noise to the
+control--points.}
+\label{fig:setup3d}
+\end{figure}
+
+For the next step we then halve the regularization--impact \(\lambda\)
+(starting at \(1\)) of our \emph{fitness--function} (\ref{eq:fit3d}) and
+calculate the next incremental solution
+\(\vec{P^{*}} = \vec{U^+}\vec{T}\) with the updated correspondences to
+get our next target--error. We repeat this process as long as the
+target--error keeps decreasing and use the number of these iterations as
+measure of the convergence speed. As the resulting evolutional error
+without regularization is in the numeric range of \(\approx 100\),
+whereas the regularization is numerically \(\approx 7000\) we need at
+least \(10\) to \(15\) iterations until the regularization--effect wears
+off.
+
+The grid we use for our experiments is just very coarse due to
+computational limitations. We are not interested in a good
+reconstruction, but an estimate if the mentioned evolvability criteria
+are good.
+
+In figure \ref{fig:setup3d} we show an example setup of the scene with a
+\(4\times 4\times 4\)--grid. Identical to the 1--dimensional scenario
+before, we create a regular grid and move the control-points \todo{wie?}
+random between their neighbours, but in three instead of two
+dimensions\footnote{Again, we flip the signs for the edges, if necessary
+  to have the object still in the convex hull.}.
+
+\begin{figure}[!htb]
+\includegraphics[width=\textwidth]{img/3d_grid_resolution.png}
+\caption[Different resolution of 3D grids]{\newline
+Left: A $7 \times 4 \times 4$ grid suited to better deform into facial
+features.\newline
+Right: A $4 \times 4 \times 7$ grid that we expect to perform worse.}
+\label{fig:3dgridres}
+\end{figure}
+
+As is clearly visible from figure \ref{fig:3dgridres}, the target--model
+has many vertices in the facial area, at the ears and in the
+neck--region. Therefore we chose to increase the grid-resolutions for
+our tests in two different dimensions and see how well the criteria
+predict a suboptimal placement of these control-points.
+
 \section{Results of 3D Function
 Approximation}\label{results-of-3d-function-approximation}
 
-\begin{figure}[!ht]
+In the 3D--Approximation we tried to evaluate further on the impact of
-\includegraphics[width=\textwidth]{img/evolution3d/4x4xX_montage.png}
+the grid--layout to the overall criteria. As the target--model has many
-\caption{Results 3D for 4x4xX}
+vertices in concentrated in the facial area we start from a
+\(4 \times 4 \times 4\) grid and only increase the number of control
+points in one dimension, yielding a resolution of
+\(7 \times 4 \times 4\) and \(4 \times 4 \times 7\) respectively. We
+visualized those two grids in figure \ref{fig:3dgridres}.
+
+To evaluate the performance of the evolvability--criteria we also tested
+a more neutral resolution of \(4 \times 4 \times 4\),
+\(5 \times 5 \times 5\), and \(6 \times 6 \times 6\) --- similar to the
+1D--setup.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=0.7\textwidth]{img/evolution3d/variability_boxplot.png}
+\caption[3D Fitting Errors for various grids]{The fitting error for the various
+grids we examined.\newline
+Note that the number of control--points is a product of the resolution, so $X
+\times 4 \times 4$ and $4 \times 4 \times X$ have the same number of
+control--points.}
+\label{fig:3dvar}
 \end{figure}
 
-\begin{figure}[!ht]
+\subsection{Variability}\label{variability-2}
-\includegraphics[width=\textwidth]{img/evolution3d/Xx4x4_montage.png}
+
-\caption{Results 3D for Xx4x4}
+\begin{table}[tbh]
+\centering
+\begin{tabular}{c|c|c|c}
+$4 \times 4 \times \mathrm{X}$ & $\mathrm{X} \times 4 \times 4$ & $\mathrm{Y} \times \mathrm{Y} \times \mathrm{Y}$ & all \\
+\hline
+0.89 (0) & 0.9 (0) & 0.91 (0) & 0.94 (0)
+\end{tabular}
+\caption[Correlation between variability and fitting error for 3D]{Correlation
+between variability and fitting error for the 3D fitting scenario.\newline
+Displayed are the negated Spearman coefficients with the corresponding p-values
+in brackets for three cases of increasing variability ($\mathrm{X} \in [4,5,7],
+\mathrm{Y} \in [4,5,6]$).
+\newline Note: Not significant results are marked in \textcolor{red}{red}.}
+\label{tab:3dvar}
+\end{table}
+
+Similar to the 1D case all our tested matrices had a constant rank
+(being \(m = x \cdot y \cdot z\) for a \(x \times y \times z\) grid), so
+we again have merely plotted the errors in the boxplot in figure
+\ref{fig:3dvar}.
+
+As expected the \(\mathrm{X} \times 4 \times 4\) grids performed
+slightly better than their \(4 \times 4 \times \mathrm{X}\) counterparts
+with a mean\(\pm\)sigma of \(101.25 \pm 7.45\) to \(102.89 \pm 6.74\)
+for \(\mathrm{X} = 5\) and \(85.37 \pm 7.12\) to \(89.22 \pm 6.49\) for
+\(\mathrm{X} = 7\).
+
+Interestingly both variants end up closer in terms of fitting error than
+we anticipated, which shows that the evolutionary algorithm we employed
+is capable of correcting a purposefully created \glqq bad\grqq ~grid.
+Also this confirms, that in our cases the number of control--points is
+more important for quality than their placement, which is captured by
+the variability via the rank of the deformation--matrix.
+
+\begin{figure}[hbt]
+\centering
+\includegraphics[width=0.8\textwidth]{img/evolution3d/variability2_boxplot.png}
+\caption[Histogram of ranks of high--resolution deformation--matrices]{
+Histogram of ranks of various $10 \times 10 \times 10$ grids.
+}
+\label{fig:histrank3d}
 \end{figure}
 
-\begin{figure}[!ht]
+Overall the correlation between variability and fitness--error were
-\includegraphics[width=\textwidth]{img/evolution3d/YxYxY_montage.png}
+\emph{significantly} and showed a \emph{very strong} correlation in all
-\caption{Results 3D for YxYxY for Y $\in [4,5,6]$}
+our tests. The detailed correlation--coefficients are given in table
+\ref{tab:3dvar} alongside their p--values.
+
+As introduces in section \ref{sec:impl:grid} and visualized in figure
+\ref{fig:enoughCP}, we know, that not all control points have to
+necessarily contribute to the parametrization of our 3D--model. Because
+we are starting from a sphere, some control-points are too far away from
+the surface to contribute to the deformation at all.
+
+One can already see in 2D in figure \ref{fig:enoughCP}, that this effect
+starts with a regular \(9 \times 9\) grid on a perfect circle. To make
+sure we observe this, we evaluated the variability for 100 randomly
+moved \(10 \times 10 \times 10\) grids on the sphere we start out with.
+
+As the variability is defined by \(\frac{\mathrm{rank}(\vec{U})}{n}\) we
+can easily recover the rank of the deformation--matrix \(\vec{U}\). The
+results are shown in the histogram in figure \ref{fig:histrank3d}.
+Especially in the centre of the sphere and in the corners of our grid we
+effectively loose control--points for our parametrization.
+
+This of course yields a worse error as when those control--points would
+be put to use and one should expect a loss in quality evident by a
+higher reconstruction--error opposed to a grid where they are used.
+Sadly we could not run a in--depth test on this due to computational
+limitations.
+
+Nevertheless this hints at the notion, that variability is a good
+measure for the overall quality of a fit.
+
+\subsection{Regularity}\label{regularity-2}
+
+\begin{table}[tbh]
+\centering
+\begin{tabular}{c|c|c|c}
+ & $5 \times 4 \times 4$ & $7 \times 4 \times 4$ & $\mathrm{X} \times 4 \times 4$ \\
+\cline{2-4}
+ & \textcolor{red}{0.15} (0.147) & \textcolor{red}{0.09} (0.37) & 0.46 (0) \B \\
+\cline{2-4}
+\multicolumn{4}{c}{} \\[-1.4em]
+\hline
+$4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \times 4 \times \mathrm{X}$ \T \\
+\hline
+0.38 (0) & \textcolor{red}{0.17} (0.09) & 0.40 (0) & 0.46 (0) \B \\
+\hline
+\multicolumn{4}{c}{} \\[-1.4em]
+\cline{2-4}
+ & $5 \times 5 \times 5$ & $6 \times 6 \times 6$ & $\mathrm{Y} \times \mathrm{Y} \times \mathrm{Y}$ \T \\
+\cline{2-4}
+ & \textcolor{red}{-0.18} (0.0775) & \textcolor{red}{-0.13} (0.1715) & -0.25 (0) \B \\
+\cline{2-4}
+\multicolumn{4}{c}{} \\[-1.4em]
+\cline{2-4}
+\multicolumn{3}{c}{} & all: 0.15 (0) \T
+\end{tabular}
+\caption[Correlation between regularity and iterations for 3D]{Correlation
+between regularity and number of iterations for the 3D fitting scenario.
+Displayed are the negated Spearman coefficients with the corresponding p--values
+in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
+\newline Note: Not significant results are marked in \textcolor{red}{red}.}
+\label{tab:3dvar}
+\end{table}
+
+Opposed to the predictions of variability our test on regularity gave a
+mixed result --- similar to the 1D--case.
+
+In half scenarios we have a \emph{significant}, but \emph{weak} to
+\emph{moderate} correlation between regularity and number of iterations.
+On the other hand in the scenarios where we increased the number of
+control--points, namely \(125\) for the \(5 \times 5 \times 5\) grid and
+\(216\) for the \(6 \times 6 \times 6\) grid we found a
+\emph{significant}, but \emph{weak} anti--correlation, which seem to
+contradict the findings/trends for the sets with \(64\), \(80\), and
+\(112\) control--points (first two rows of table \ref{tab:3dvar}).
+
+Taking all results together we only find a \emph{very weak}, but
+\emph{significant} link between regularity and the number of iterations
+needed for the algorithm to converge.
+
+\begin{figure}[!htb]
+\centering
+\includegraphics[width=\textwidth]{img/evolution3d/regularity_montage.png}
+\caption[Regularity for different 3D--grids]{
+**BLINDTEXT**
+}
+\label{fig:resreg3d}
+\end{figure}
+
+As can be seen from figure \ref{fig:resreg3d}, we can
+observe\todo{things}.
+
+\subsection{Improvement Potential}\label{improvement-potential-2}
+
+\begin{figure}[!htb]
+\centering
+\includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
+\caption[Improvement potential for different 3D--grids]{
+**BLINDTEXT**
+}
+\label{fig:resimp3d}
 \end{figure}
 
 \chapter{Schluss}\label{schluss}

arbeit/settings/commands.tex

@@ -95,6 +95,9 @@
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+\newcommand\T{\rule{0pt}{2.6ex}}       % Top strut
+\newcommand\B{\rule[-1.2ex]{0pt}{0pt}} % Bottom strut
+
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 % \newcommand\subdot[1]{\lower0.5em\hbox{$\stackrel{\displaystyle #1}{.}$}}
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arbeit/template.tex

@@ -35,6 +35,7 @@ xcolor=dvipsnames,
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@@ -1,101 +1,101 @@
 regularity,variability,improvement,"Evolution error",steps
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dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.error

@@ -0,0 +1,101 @@
+"Evolution error
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dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing 20170926_3dFit_4x4x4_100times.csv"
+[1] "Mean:"
+[1] 119.1789
+[1] "Median:"
+[1] 119.5255
+[1] "Sigma:"
+[1] 4.97234
+[1] "Range:"
+[1] 108.467 129.916

dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing 20170926_3dFit_4x4x4_100times.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+     x    y
+x  1.0 -0.5
+y -0.5  1.0
+
+n= 100 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+     x    y
+x 1.00 0.37
+y 0.37 1.00
+
+n= 100 
+
+
+P
+  x     y    
+x       2e-04
+y 2e-04      
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.38
+y -0.38  1.00
+
+n= 100 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for variability vs. evolution-error"
+    x   y
+x   1 NaN
+y NaN   1
+
+n= 100 
+
+
+P
+  x y
+x    
+y    

dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.csv

@@ -1,101 +1,101 @@
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dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.error

@@ -0,0 +1,101 @@
+"Evolution error
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dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing 20170926_3dFit_5x5x5_100times.csv"
+[1] "Mean:"
+[1] 77.03635
+[1] "Median:"
+[1] 76.541
+[1] "Sigma:"
+[1] 7.023404
+[1] "Range:"
+[1] 60.0022 97.4154

dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing 20170926_3dFit_5x5x5_100times.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+      x     y
+x  1.00 -0.47
+y -0.47  1.00
+
+n= 100 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+     x    y
+x 1.00 0.36
+y 0.36 1.00
+
+n= 100 
+
+
+P
+  x     y    
+x       3e-04
+y 3e-04      
+[1] "spearman for regularity vs. steps"
+     x    y
+x 1.00 0.18
+y 0.18 1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.0775
+y 0.0775       
+[1] "spearman for variability vs. evolution-error"
+    x   y
+x   1 NaN
+y NaN   1
+
+n= 100 
+
+
+P
+  x y
+x    
+y    

dokumentation/evolution3d/20171005_3dFit_4x4x5_100times.csv

@@ -1,101 +1,101 @@
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dokumentation/evolution3d/20171005_3dFit_4x4x5_100times.error

@@ -0,0 +1,101 @@
+"Evolution error
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dokumentation/evolution3d/20171005_3dFit_4x4x5_100times.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing 20171005_3dFit_4x4x5_100times.csv"
+[1] "Mean:"
+[1] 102.8913
+[1] "Median:"
+[1] 102.9855
+[1] "Sigma:"
+[1] 6.740435
+[1] "Range:"
+[1]  85.3402 119.6750

dokumentation/evolution3d/20171005_3dFit_4x4x5_100times.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing 20171005_3dFit_4x4x5_100times.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+      x     y
+x  1.00 -0.38
+y -0.38  1.00
+
+n= 100 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+     x    y
+x 1.00 0.27
+y 0.27 1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.0066
+y 0.0066       
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.17
+y -0.17  1.00
+
+n= 100 
+
+
+P
+  x    y   
+x      0.09
+y 0.09     
+[1] "spearman for variability vs. evolution-error"
+    x   y
+x   1 NaN
+y NaN   1
+
+n= 100 
+
+
+P
+  x y
+x    
+y    

dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.csv

@@ -1,101 +1,101 @@
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dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.error

@@ -0,0 +1,101 @@
+"Evolution error
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dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing 20171005_3dFit_7x4x4_100times.csv"
+[1] "Mean:"
+[1] 85.37222
+[1] "Median:"
+[1] 84.7745
+[1] "Sigma:"
+[1] 7.117153
+[1] "Range:"
+[1]  72.2744 105.0320

dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing 20171005_3dFit_7x4x4_100times.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+      x     y
+x  1.00 -0.23
+y -0.23  1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.0233
+y 0.0233       
+[1] "spearman for improvement-potential vs. steps"
+     x    y
+x 1.00 0.06
+y 0.06 1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.5405
+y 0.5405       
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.09
+y -0.09  1.00
+
+n= 100 
+
+
+P
+  x    y   
+x      0.37
+y 0.37     
+[1] "spearman for variability vs. evolution-error"
+    x   y
+x   1 NaN
+y NaN   1
+
+n= 100 
+
+
+P
+  x y
+x    
+y    

dokumentation/evolution3d/20171007_3dFit_all.csv

@@ -1,401 +1,401 @@
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dokumentation/evolution3d/20171007_3dFit_all.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing 20171007_3dFit_all.csv"
+[1] "Mean:"
+[1] 96.11968
+[1] "Median:"
+[1] 94.61025
+[1] "Sigma:"
+[1] 17.52693
+[1] "Range:"
+[1]  60.0022 129.9160

dokumentation/evolution3d/20171007_3dFit_all.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing 20171007_3dFit_all.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+      x     y
+x  1.00 -0.92
+y -0.92  1.00
+
+n= 400 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+      x     y
+x  1.00 -0.67
+y -0.67  1.00
+
+n= 400 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.09
+y -0.09  1.00
+
+n= 400 
+
+
+P
+  x      y     
+x        0.0863
+y 0.0863       
+[1] "spearman for variability vs. evolution-error"
+      x     y
+x  1.00 -0.91
+y -0.91  1.00
+
+n= 400 
+
+
+P
+  x  y 
+x     0
+y  0   

dokumentation/evolution3d/20171013_3dFit_4x4x7_100times.csv

@@ -1,101 +1,101 @@
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dokumentation/evolution3d/20171013_3dFit_4x4x7_100times.error

@@ -0,0 +1,101 @@
+"Evolution error
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dokumentation/evolution3d/20171013_3dFit_4x4x7_100times.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing 20171013_3dFit_4x4x7_100times.csv"
+[1] "Mean:"
+[1] 89.21728
+[1] "Median:"
+[1] 88.6566
+[1] "Sigma:"
+[1] 6.486783
+[1] "Range:"
+[1]  77.2599 116.6090

dokumentation/evolution3d/20171013_3dFit_4x4x7_100times.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing 20171013_3dFit_4x4x7_100times.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+      x     y
+x  1.00 -0.32
+y -0.32  1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.0012
+y 0.0012       
+[1] "spearman for improvement-potential vs. steps"
+     x    y
+x 1.00 0.13
+y 0.13 1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.2147
+y 0.2147       
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.41
+y -0.41  1.00
+
+n= 100 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for variability vs. evolution-error"
+    x   y
+x   1 NaN
+y NaN   1
+
+n= 100 
+
+
+P
+  x y
+x    
+y    

dokumentation/evolution3d/20171013_3dFit_5x4x4_100times.csv

@@ -1,101 +1,101 @@
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dokumentation/evolution3d/20171013_3dFit_5x4x4_100times.error

@@ -0,0 +1,101 @@
+"Evolution error
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dokumentation/evolution3d/20171013_3dFit_5x4x4_100times.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing 20171013_3dFit_5x4x4_100times.csv"
+[1] "Mean:"
+[1] 101.2503
+[1] "Median:"
+[1] 100.7065
+[1] "Sigma:"
+[1] 7.448474
+[1] "Range:"
+[1]  87.7666 130.8540

dokumentation/evolution3d/20171013_3dFit_5x4x4_100times.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing 20171013_3dFit_5x4x4_100times.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+     x    y
+x  1.0 -0.3
+y -0.3  1.0
+
+n= 100 
+
+
+P
+  x      y     
+x        0.0023
+y 0.0023       
+[1] "spearman for improvement-potential vs. steps"
+     x    y
+x 1.00 0.25
+y 0.25 1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.0124
+y 0.0124       
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.15
+y -0.15  1.00
+
+n= 100 
+
+
+P
+  x     y    
+x       0.147
+y 0.147      
+[1] "spearman for variability vs. evolution-error"
+    x   y
+x   1 NaN
+y NaN   1
+
+n= 100 
+
+
+P
+  x y
+x    
+y    

dokumentation/evolution3d/20171021-evolution3D_6x6_100Times.csv

@@ -1,111 +1,111 @@
 regularity,variability,improvement,"Evolution error",steps
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dokumentation/evolution3d/20171021-evolution3D_6x6_100Times.error

@@ -0,0 +1,111 @@
+"Evolution error
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dokumentation/evolution3d/20171021-evolution3D_6x6_100Times.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing 20171021-evolution3D_6x6_100Times.csv"
+[1] "Mean:"
+[1] 62.82964
+[1] "Median:"
+[1] 61.9227
+[1] "Sigma:"
+[1] 5.727806
+[1] "Range:"
+[1] 52.0186 82.5252

dokumentation/evolution3d/20171021-evolution3D_6x6_100Times.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing 20171021-evolution3D_6x6_100Times.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+     x    y
+x 1.00 0.01
+y 0.01 1.00
+
+n= 110 
+
+
+P
+  x      y     
+x        0.8803
+y 0.8803       
+[1] "spearman for improvement-potential vs. steps"
+    x   y
+x 1.0 0.1
+y 0.1 1.0
+
+n= 110 
+
+
+P
+  x      y     
+x        0.3009
+y 0.3009       
+[1] "spearman for regularity vs. steps"
+     x    y
+x 1.00 0.13
+y 0.13 1.00
+
+n= 110 
+
+
+P
+  x      y     
+x        0.1715
+y 0.1715       
+[1] "spearman for variability vs. evolution-error"
+    x   y
+x   1 NaN
+y NaN   1
+
+n= 110 
+
+
+P
+  x y
+x    
+y    

dokumentation/evolution3d/20171025-evolution3D_10x10x10_noFit.csv

@@ -0,0 +1,11 @@
+regularity,variability,improvement
+0,0.090682,0.913618
+0,0.0907745,0.913428
+0,0.0905894,0.916839
+0,0.090682,0.915586
+0,0.0907745,0.915006
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+0,0.0907745,0.912454
+0,0.0903118,0.914794
+0,0.0904969,0.912754
+0,0.0909596,0.912461

dokumentation/evolution3d/20171025-evolution3D_10x10x10_noFit.gnuplot.fit.log

@@ -0,0 +1,184 @@
+
+
+*******************************************************************************
+Wed Oct 25 16:01:21 2017
+
+
+FIT:    data read from "20171025-evolution3D_10x10x10_noFit.csv" every ::1 using 1:5
+        format = x:z
+        #datapoints = 6
+        residuals are weighted equally (unit weight)
+
+function used for fitting: f(x)
+fitted parameters initialized with current variable values
+
+
+
+ Iteration 0
+ WSSR        : 9.03463           delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.800174
+
+initial set of free parameter values
+
+a               = 1
+b               = 1
+
+After 4 iterations the fit converged.
+final sum of squares of residuals : 0.760112
+rel. change during last iteration : -7.81424e-14
+
+degrees of freedom    (FIT_NDF)                        : 4
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.435922
+variance of residuals (reduced chisquare) = WSSR/ndf   : 0.190028
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+a               = -0.500504        +/- 0.4333       (86.58%)
+b               = 0.50226          +/- 0.2295       (45.7%)
+
+
+correlation matrix of the fit parameters:
+
+               a      b      
+a               1.000 
+b              -0.632  1.000 
+
+
+*******************************************************************************
+Wed Oct 25 16:01:21 2017
+
+
+FIT:    data read from "20171025-evolution3D_10x10x10_noFit.csv" every ::1 using 3:5
+        format = x:z
+        #datapoints = 6
+        residuals are weighted equally (unit weight)
+
+function used for fitting: g(x)
+fitted parameters initialized with current variable values
+
+
+
+ Iteration 0
+ WSSR        : 9.042             delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.80039
+
+initial set of free parameter values
+
+aa              = 1
+bb              = 1
+
+After 4 iterations the fit converged.
+final sum of squares of residuals : 0.760537
+rel. change during last iteration : -7.73688e-14
+
+degrees of freedom    (FIT_NDF)                        : 4
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.436044
+variance of residuals (reduced chisquare) = WSSR/ndf   : 0.190134
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aa              = -0.499395        +/- 0.4329       (86.68%)
+bb              = 0.502057         +/- 0.2296       (45.72%)
+
+
+correlation matrix of the fit parameters:
+
+               aa     bb     
+aa              1.000 
+bb             -0.631  1.000 
+
+
+*******************************************************************************
+Wed Oct 25 16:01:21 2017
+
+
+FIT:    data read from "20171025-evolution3D_10x10x10_noFit.csv" every ::1 using 3:4
+        format = x:z
+        #datapoints = 6
+        residuals are weighted equally (unit weight)
+
+function used for fitting: h(x)
+fitted parameters initialized with current variable values
+
+
+
+ Iteration 0
+ WSSR        : 9.04152           delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.80039
+
+initial set of free parameter values
+
+aaa             = 1
+bbb             = 1
+
+After 4 iterations the fit converged.
+final sum of squares of residuals : 0.763537
+rel. change during last iteration : -7.73556e-14
+
+degrees of freedom    (FIT_NDF)                        : 4
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.436903
+variance of residuals (reduced chisquare) = WSSR/ndf   : 0.190884
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aaa             = -0.501106        +/- 0.4337       (86.55%)
+bbb             = 0.503355         +/- 0.23         (45.7%)
+
+
+correlation matrix of the fit parameters:
+
+               aaa    bbb    
+aaa             1.000 
+bbb            -0.631  1.000 
+
+
+*******************************************************************************
+Wed Oct 25 16:01:21 2017
+
+
+FIT:    data read from "20171025-evolution3D_10x10x10_noFit.csv" every ::1 using 2:4
+        format = x:z
+        #datapoints = 6
+        residuals are weighted equally (unit weight)
+
+function used for fitting: i(x)
+fitted parameters initialized with current variable values
+
+
+
+ Iteration 0
+ WSSR        : 9.04263           delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.800411
+
+initial set of free parameter values
+
+aaaa            = 1
+bbbb            = 1
+
+After 4 iterations the fit converged.
+final sum of squares of residuals : 0.763697
+rel. change during last iteration : -7.7194e-14
+
+degrees of freedom    (FIT_NDF)                        : 4
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.436949
+variance of residuals (reduced chisquare) = WSSR/ndf   : 0.190924
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aaaa            = -0.50098         +/- 0.4338       (86.59%)
+bbbb            = 0.50338          +/- 0.2301       (45.71%)
+
+
+correlation matrix of the fit parameters:
+
+               aaaa   bbbb   
+aaaa            1.000 
+bbbb           -0.632  1.000 

dokumentation/evolution3d/20171025-evolution3D_10x10x10_noFit.gnuplot.log

@@ -0,0 +1,304 @@
+
+
+ Iteration 0
+ WSSR        : 9.03463           delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.800174
+
+initial set of free parameter values
+
+a               = 1
+b               = 1
+/
+
+ Iteration 1
+ WSSR        : 1.04136           delta(WSSR)/WSSR   : -7.67579
+ delta(WSSR) : -7.99327          limit for stopping : 1e-05
+ lambda	  : 0.0800174
+
+resultant parameter values
+
+a               = 0.00294917
+b               = 0.398082
+/
+
+ Iteration 2
+ WSSR        : 0.760123          delta(WSSR)/WSSR   : -0.36999
+ delta(WSSR) : -0.281238         limit for stopping : 1e-05
+ lambda	  : 0.00800174
+
+resultant parameter values
+
+a               = -0.497122
+b               = 0.501019
+/
+
+ Iteration 3
+ WSSR        : 0.760112          delta(WSSR)/WSSR   : -1.53218e-05
+ delta(WSSR) : -1.16463e-05      limit for stopping : 1e-05
+ lambda	  : 0.000800174
+
+resultant parameter values
+
+a               = -0.500504
+b               = 0.50226
+/
+
+ Iteration 4
+ WSSR        : 0.760112          delta(WSSR)/WSSR   : -7.81424e-14
+ delta(WSSR) : -5.93969e-14      limit for stopping : 1e-05
+ lambda	  : 8.00174e-05
+
+resultant parameter values
+
+a               = -0.500504
+b               = 0.50226
+
+After 4 iterations the fit converged.
+final sum of squares of residuals : 0.760112
+rel. change during last iteration : -7.81424e-14
+
+degrees of freedom    (FIT_NDF)                        : 4
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.435922
+variance of residuals (reduced chisquare) = WSSR/ndf   : 0.190028
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+a               = -0.500504        +/- 0.4333       (86.58%)
+b               = 0.50226          +/- 0.2295       (45.7%)
+
+
+correlation matrix of the fit parameters:
+
+               a      b      
+a               1.000 
+b              -0.632  1.000 
+
+
+ Iteration 0
+ WSSR        : 9.042             delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.80039
+
+initial set of free parameter values
+
+aa              = 1
+bb              = 1
+/
+
+ Iteration 1
+ WSSR        : 1.04131           delta(WSSR)/WSSR   : -7.68331
+ delta(WSSR) : -8.0007           limit for stopping : 1e-05
+ lambda	  : 0.080039
+
+resultant parameter values
+
+aa              = 0.00287365
+bb              = 0.398135
+/
+
+ Iteration 2
+ WSSR        : 0.760548          delta(WSSR)/WSSR   : -0.369155
+ delta(WSSR) : -0.28076          limit for stopping : 1e-05
+ lambda	  : 0.0080039
+
+resultant parameter values
+
+aa              = -0.496029
+bb              = 0.50082
+/
+
+ Iteration 3
+ WSSR        : 0.760537          delta(WSSR)/WSSR   : -1.52182e-05
+ delta(WSSR) : -1.1574e-05       limit for stopping : 1e-05
+ lambda	  : 0.00080039
+
+resultant parameter values
+
+aa              = -0.499395
+bb              = 0.502057
+/
+
+ Iteration 4
+ WSSR        : 0.760537          delta(WSSR)/WSSR   : -7.73688e-14
+ delta(WSSR) : -5.88418e-14      limit for stopping : 1e-05
+ lambda	  : 8.0039e-05
+
+resultant parameter values
+
+aa              = -0.499395
+bb              = 0.502057
+
+After 4 iterations the fit converged.
+final sum of squares of residuals : 0.760537
+rel. change during last iteration : -7.73688e-14
+
+degrees of freedom    (FIT_NDF)                        : 4
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.436044
+variance of residuals (reduced chisquare) = WSSR/ndf   : 0.190134
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aa              = -0.499395        +/- 0.4329       (86.68%)
+bb              = 0.502057         +/- 0.2296       (45.72%)
+
+
+correlation matrix of the fit parameters:
+
+               aa     bb     
+aa              1.000 
+bb             -0.631  1.000 
+
+
+ Iteration 0
+ WSSR        : 9.04152           delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.80039
+
+initial set of free parameter values
+
+aaa             = 1
+bbb             = 1
+/
+
+ Iteration 1
+ WSSR        : 1.04503           delta(WSSR)/WSSR   : -7.65191
+ delta(WSSR) : -7.99648          limit for stopping : 1e-05
+ lambda	  : 0.080039
+
+resultant parameter values
+
+aaa             = 0.00194603
+bbb             = 0.399071
+/
+
+ Iteration 2
+ WSSR        : 0.763548          delta(WSSR)/WSSR   : -0.36865
+ delta(WSSR) : -0.281482         limit for stopping : 1e-05
+ lambda	  : 0.0080039
+
+resultant parameter values
+
+aaa             = -0.497734
+bbb             = 0.502116
+/
+
+ Iteration 3
+ WSSR        : 0.763537          delta(WSSR)/WSSR   : -1.52098e-05
+ delta(WSSR) : -1.16133e-05      limit for stopping : 1e-05
+ lambda	  : 0.00080039
+
+resultant parameter values
+
+aaa             = -0.501106
+bbb             = 0.503355
+/
+
+ Iteration 4
+ WSSR        : 0.763537          delta(WSSR)/WSSR   : -7.73556e-14
+ delta(WSSR) : -5.90639e-14      limit for stopping : 1e-05
+ lambda	  : 8.0039e-05
+
+resultant parameter values
+
+aaa             = -0.501106
+bbb             = 0.503355
+
+After 4 iterations the fit converged.
+final sum of squares of residuals : 0.763537
+rel. change during last iteration : -7.73556e-14
+
+degrees of freedom    (FIT_NDF)                        : 4
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.436903
+variance of residuals (reduced chisquare) = WSSR/ndf   : 0.190884
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aaa             = -0.501106        +/- 0.4337       (86.55%)
+bbb             = 0.503355         +/- 0.23         (45.7%)
+
+
+correlation matrix of the fit parameters:
+
+               aaa    bbb    
+aaa             1.000 
+bbb            -0.631  1.000 
+
+
+ Iteration 0
+ WSSR        : 9.04263           delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.800411
+
+initial set of free parameter values
+
+aaaa            = 1
+bbbb            = 1
+/
+
+ Iteration 1
+ WSSR        : 1.04513           delta(WSSR)/WSSR   : -7.65212
+ delta(WSSR) : -7.99749          limit for stopping : 1e-05
+ lambda	  : 0.0800411
+
+resultant parameter values
+
+aaaa            = 0.00204362
+bbbb            = 0.399044
+/
+
+ Iteration 2
+ WSSR        : 0.763709          delta(WSSR)/WSSR   : -0.368499
+ delta(WSSR) : -0.281426         limit for stopping : 1e-05
+ lambda	  : 0.00800411
+
+resultant parameter values
+
+aaaa            = -0.497607
+bbbb            = 0.50214
+/
+
+ Iteration 3
+ WSSR        : 0.763697          delta(WSSR)/WSSR   : -1.52103e-05
+ delta(WSSR) : -1.16161e-05      limit for stopping : 1e-05
+ lambda	  : 0.000800411
+
+resultant parameter values
+
+aaaa            = -0.50098
+bbbb            = 0.50338
+/
+
+ Iteration 4
+ WSSR        : 0.763697          delta(WSSR)/WSSR   : -7.7194e-14
+ delta(WSSR) : -5.89528e-14      limit for stopping : 1e-05
+ lambda	  : 8.00411e-05
+
+resultant parameter values
+
+aaaa            = -0.50098
+bbbb            = 0.50338
+
+After 4 iterations the fit converged.
+final sum of squares of residuals : 0.763697
+rel. change during last iteration : -7.7194e-14
+
+degrees of freedom    (FIT_NDF)                        : 4
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.436949
+variance of residuals (reduced chisquare) = WSSR/ndf   : 0.190924
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aaaa            = -0.50098         +/- 0.4338       (86.59%)
+bbbb            = 0.50338          +/- 0.2301       (45.71%)
+
+
+correlation matrix of the fit parameters:
+
+               aaaa   bbbb   
+aaaa            1.000 
+bbbb           -0.632  1.000 

dokumentation/evolution3d/20171025-evolution3D_10x10x10_noFit.gnuplot.script

@@ -0,0 +1,26 @@
+set datafile separator ","
+f(x)=a*x+b
+fit f(x) "20171025-evolution3D_10x10x10_noFit.csv" every ::1 using 1:5 via a,b
+set terminal png
+set xlabel 'regularity'
+set ylabel 'steps'
+set output "20171025-evolution3D_10x10x10_noFit_regularity-vs-steps.png"
+plot "20171025-evolution3D_10x10x10_noFit.csv" every ::1 using 1:5 title "data", f(x) title "lin. fit" lc rgb "black"
+g(x)=aa*x+bb
+fit g(x) "20171025-evolution3D_10x10x10_noFit.csv" every ::1 using 3:5 via aa,bb
+set xlabel 'improvement potential'
+set ylabel 'steps'
+set output "20171025-evolution3D_10x10x10_noFit_improvement-vs-steps.png"
+plot "20171025-evolution3D_10x10x10_noFit.csv" every ::1 using 3:5 title "data", g(x) title "lin. fit" lc rgb "black"
+h(x)=aaa*x+bbb
+fit h(x) "20171025-evolution3D_10x10x10_noFit.csv" every ::1 using 3:4 via aaa,bbb
+set xlabel 'improvement potential'
+set ylabel 'evolution error'
+set output "20171025-evolution3D_10x10x10_noFit_improvement-vs-evo-error.png"
+plot "20171025-evolution3D_10x10x10_noFit.csv" every ::1 using 3:4 title "data", h(x) title "lin. fit" lc rgb "black"
+i(x)=aaaa*x+bbbb
+fit i(x) "20171025-evolution3D_10x10x10_noFit.csv" every ::1 using 2:4 via aaaa,bbbb
+set xlabel 'variability'
+set ylabel 'evolution error'
+set output "20171025-evolution3D_10x10x10_noFit_variability-vs-evo-error.png"
+plot     "20171025-evolution3D_10x10x10_noFit.csv" every ::1 using 2:4 title "data",     i(x) title "lin. fit" lc rgb "black"

dokumentation/evolution3d/20171025-evolution3D_10x10x10_noFit.log

@@ -0,0 +1,180 @@
+info: using info log level
+info: Free_form_deformation_plugin loaded.
+info: Modelling_plugin loaded.
+info: Point_set_io_plugin loaded.
+info: Scene_graph_plugin loaded.
+info: Selection_plugin loaded.
+info: Surface_mesh_io_plugin loaded.
+GL error at "after Initialize": invalid enum
+
+info: GLEW errorcode: 0
+info: GLEW   1.13.0
+info: OpenGL 3.3.0 NVIDIA 384.90 (Core Profile)
+info: GLSL   3.30 NVIDIA via Cg compiler
+info: GeForce GTX 1080/PCIe/SSE2
+info: OpenCL 1.2 CUDA 9.0.194 
+info: 10807 Vertices, 21610 Faces.
+info: Loaded /home/sdressel/git/graphene/offs/source_ball_10807v_good_normed.off.
+info: 12024 Vertices, 23997 Faces.
+info: Loaded /home/sdressel/git/graphene/offs/target_mario_12024v_rem_normed.off.
+info: setting source
+info: setting target
+info: initialising Grid
+info: bbmin: -1.01856 -1.01771 -1.01575
+info: bbmax: 1.02125 1.02206 1.02408
+info: bbsize: 2.03981 2.03977 2.03983
+info: setting up10807Points
+info: worst iteration-count:3
+info: generating Cache
+info: cache size: 10807*1000=10807000
+info: done.
+info: perturbating...
+info: initialising Grid
+info: bbmin: -1.07522 -1.07437 -1.07241
+info: bbmax: 1.09379 1.09753 1.09917
+info: bbsize: 2.16901 2.17189 2.17158
+info: setting up10807Points
+info: worst iteration-count:4
+info: generating Cache
+info: cache size: 10807*1000=10807000
+info: done.
+info: entering fit..
+info: EVOL: Evolvability-criteria:
+info: EVOL: regularity:  0
+info: EVOL: variability: 0.090682
+info: EVOL: improvement: 0.913618
+info: perturbating...
+info: initialising Grid
+info: bbmin: -1.07522 -1.07437 -1.07241
+info: bbmax: 1.09449 1.09677 1.09756
+info: bbsize: 2.16971 2.17114 2.16997
+info: setting up10807Points
+info: worst iteration-count:4
+info: generating Cache
+info: cache size: 10807*1000=10807000
+info: done.
+info: entering fit..
+info: EVOL: Evolvability-criteria:
+info: EVOL: regularity:  0
+info: EVOL: variability: 0.0907745
+info: EVOL: improvement: 0.913428
+info: perturbating...
+info: initialising Grid
+info: bbmin: -1.07522 -1.07437 -1.07241
+info: bbmax: 1.09456 1.09481 1.09742
+info: bbsize: 2.16978 2.16918 2.16983
+info: setting up10807Points
+info: worst iteration-count:4
+info: generating Cache
+info: cache size: 10807*1000=10807000
+info: done.
+info: entering fit..
+info: EVOL: Evolvability-criteria:
+info: EVOL: regularity:  0
+info: EVOL: variability: 0.0905894
+info: EVOL: improvement: 0.916839
+info: perturbating...
+info: initialising Grid
+info: bbmin: -1.07522 -1.07437 -1.07241
+info: bbmax: 1.0952 1.0937 1.09694
+info: bbsize: 2.17042 2.16806 2.16935
+info: setting up10807Points
+info: worst iteration-count:4
+info: generating Cache
+info: cache size: 10807*1000=10807000
+info: done.
+info: entering fit..
+info: EVOL: Evolvability-criteria:
+info: EVOL: regularity:  0
+info: EVOL: variability: 0.090682
+info: EVOL: improvement: 0.915586
+info: perturbating...
+info: initialising Grid
+info: bbmin: -1.07522 -1.07437 -1.07241
+info: bbmax: 1.09532 1.0969 1.09633
+info: bbsize: 2.17054 2.17127 2.16874
+info: setting up10807Points
+info: worst iteration-count:4
+info: generating Cache
+info: cache size: 10807*1000=10807000
+info: done.
+info: entering fit..
+info: EVOL: Evolvability-criteria:
+info: EVOL: regularity:  0
+info: EVOL: variability: 0.0907745
+info: EVOL: improvement: 0.915006
+info: perturbating...
+info: initialising Grid
+info: bbmin: -1.07522 -1.07437 -1.07241
+info: bbmax: 1.09164 1.09026 1.09787
+info: bbsize: 2.16686 2.16463 2.17029
+info: setting up10807Points
+info: worst iteration-count:4
+info: generating Cache
+info: cache size: 10807*1000=10807000
+info: done.
+info: entering fit..
+info: EVOL: Evolvability-criteria:
+info: EVOL: regularity:  0
+info: EVOL: variability: 0.090867
+info: EVOL: improvement: 0.91456
+info: perturbating...
+info: initialising Grid
+info: bbmin: -1.07522 -1.07437 -1.07241
+info: bbmax: 1.09473 1.09554 1.09647
+info: bbsize: 2.16995 2.16991 2.16888
+info: setting up10807Points
+info: worst iteration-count:4
+info: generating Cache
+info: cache size: 10807*1000=10807000
+info: done.
+info: entering fit..
+info: EVOL: Evolvability-criteria:
+info: EVOL: regularity:  0
+info: EVOL: variability: 0.0907745
+info: EVOL: improvement: 0.912454
+info: perturbating...
+info: initialising Grid
+info: bbmin: -1.07522 -1.07437 -1.07241
+info: bbmax: 1.09276 1.09735 1.09676
+info: bbsize: 2.16799 2.17172 2.16917
+info: setting up10807Points
+info: worst iteration-count:4
+info: generating Cache
+info: cache size: 10807*1000=10807000
+info: done.
+info: entering fit..
+info: EVOL: Evolvability-criteria:
+info: EVOL: regularity:  0
+info: EVOL: variability: 0.0903118
+info: EVOL: improvement: 0.914794
+info: perturbating...
+info: initialising Grid
+info: bbmin: -1.07522 -1.07437 -1.07241
+info: bbmax: 1.09644 1.09619 1.09812
+info: bbsize: 2.17166 2.17056 2.17053
+info: setting up10807Points
+info: worst iteration-count:4
+info: generating Cache
+info: cache size: 10807*1000=10807000
+info: done.
+info: entering fit..
+info: EVOL: Evolvability-criteria:
+info: EVOL: regularity:  0
+info: EVOL: variability: 0.0904969
+info: EVOL: improvement: 0.912754
+info: perturbating...
+info: initialising Grid
+info: bbmin: -1.07522 -1.07437 -1.07241
+info: bbmax: 1.09505 1.09665 1.09904
+info: bbsize: 2.17027 2.17102 2.17146
+info: setting up10807Points
+info: worst iteration-count:4
+info: generating Cache
+info: cache size: 10807*1000=10807000
+info: done.
+info: entering fit..
+info: EVOL: Evolvability-criteria:
+info: EVOL: regularity:  0
+info: EVOL: variability: 0.0909596
+info: EVOL: improvement: 0.912461

dokumentation/evolution3d/20171025-evolution3D_10x10x10_noFit.var

@@ -0,0 +1,11 @@
+variability
+0.090682
+0.0907745
+0.0905894
+0.090682
+0.0907745
+0.090867
+0.0907745
+0.0903118
+0.0904969
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dokumentation/evolution3d/20171025-evolution3D_10x10x10_noFit_100Times.csv

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dokumentation/evolution3d/20171025-evolution3D_10x10x10_noFit_100Times.mms

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+[1] "================ Analyzing 20171025-evolution3D_10x10x10_noFit_100Times.csv"
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dokumentation/evolution3d/20171025-evolution3D_10x10x10_noFit_100Times.var

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+7.43768e-05,0.0103637,0.686236,89.5657,2487
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+0.000116654,0.0103637,0.703829,78.0427,2381
-4.85051e-05,0.0103637,0.685997,91.3327.,2003
+4.85051e-05,0.0103637,0.685997,91.3327,2003
-5.55332e-05,0.0103637,0.691656,116.609.,1170
+5.55332e-05,0.0103637,0.691656,116.609,1170
-6.60943e-05,0.0103637,0.693062,83.6798.,1744
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-5.83277e-05,0.0103637,0.692641,89.0655.,2169
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+9.19515e-05,0.0103637,0.696555,82.225,2525
-8.81229e-05,0.0103637,0.68317,90.5327.,2012
+8.81229e-05,0.0103637,0.68317,90.5327,2012
-5.85726e-05,0.0103637,0.692007,78.9694.,2646
+5.85726e-05,0.0103637,0.692007,78.9694,2646
-9.00751e-05,0.0103637,0.696617,83.061.,2168
+9.00751e-05,0.0103637,0.696617,83.061,2168
-9.74536e-05,0.0103637,0.701995,97.7498.,1565
+9.74536e-05,0.0103637,0.701995,97.7498,1565
-8.15851e-05,0.0103637,0.693622,87.9928.,1602
+8.15851e-05,0.0103637,0.693622,87.9928,1602
-0.000105786,0.0103637,0.702003,83.4737.,1711
+0.000105786,0.0103637,0.702003,83.4737,1711

dokumentation/evolution3d/4x4xX.error

@@ -0,0 +1,301 @@
+"Evolution error
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dokumentation/evolution3d/4x4xX.gnuplot.fit.log

@@ -1,7 +1,7 @@
 
 
 *******************************************************************************
-Mon Oct 16 12:22:40 2017
+Wed Oct 25 19:14:24 2017
 
 
 FIT:    data read from "4x4xX.csv" every ::1 using 1:5
@@ -47,7 +47,7 @@ b              -0.938  1.000
 
 
 *******************************************************************************
-Mon Oct 16 12:22:40 2017
+Wed Oct 25 19:14:24 2017
 
 
 FIT:    data read from "4x4xX.csv" every ::1 using 3:5
@@ -93,7 +93,7 @@ bb             -0.999  1.000
 
 
 *******************************************************************************
-Mon Oct 16 12:22:40 2017
+Wed Oct 25 19:14:24 2017
 
 
 FIT:    data read from "4x4xX.csv" every ::1 using 3:4
@@ -139,7 +139,7 @@ bbb            -0.999  1.000
 
 
 *******************************************************************************
-Mon Oct 16 12:22:40 2017
+Wed Oct 25 19:14:24 2017
 
 
 FIT:    data read from "4x4xX.csv" every ::1 using 2:4

dokumentation/evolution3d/4x4xX.gnuplot.script

@@ -5,22 +5,22 @@ set terminal png
 set xlabel 'regularity'
 set ylabel 'steps'
 set output "4x4xX_regularity-vs-steps.png"
-plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 title "4x4x4",     "20171005_3dFit_4x4x5_100times.csv" every ::1 using 1:5 title "4x4x5",     "20171013_3dFit_4x4x7_100times.csv" every ::1 using 1:5 title "4x4x7",     f(x) title "lin. fit" lc rgb "black"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 title "4x4x4" pt 2,     "20171005_3dFit_4x4x5_100times.csv" every ::1 using 1:5 title "4x4x5" pt 2,     "20171013_3dFit_4x4x7_100times.csv" every ::1 using 1:5 title "4x4x7" pt 2,     f(x) title "lin. fit" lc rgb "black"
 g(x)=aa*x+bb
 fit g(x) "4x4xX.csv" every ::1 using 3:5 via aa,bb
 set xlabel 'improvement potential'
 set ylabel 'steps'
 set output "4x4xX_improvement-vs-steps.png"
-plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 title "4x4x4",     "20171005_3dFit_4x4x5_100times.csv" every ::1 using 3:5 title "4x4x5",     "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:5 title "4x4x7",     g(x) title "lin. fit" lc rgb "black"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 title "4x4x4" pt 2,     "20171005_3dFit_4x4x5_100times.csv" every ::1 using 3:5 title "4x4x5" pt 2,     "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:5 title "4x4x7" pt 2,     g(x) title "lin. fit" lc rgb "black"
 h(x)=aaa*x+bbb
 fit h(x) "4x4xX.csv" every ::1 using 3:4 via aaa,bbb
 set xlabel 'improvement potential'
 set ylabel 'evolution error'
 set output "4x4xX_improvement-vs-evo-error.png"
-plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 title "4x4x4",     "20171005_3dFit_4x4x5_100times.csv" every ::1 using 3:4 title "4x4x5",     "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:4 title "4x4x7",     h(x) title "lin. fit" lc rgb "black"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 title "4x4x4" pt 2,     "20171005_3dFit_4x4x5_100times.csv" every ::1 using 3:4 title "4x4x5" pt 2,     "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:4 title "4x4x7" pt 2,     h(x) title "lin. fit" lc rgb "black"
 i(x)=aaaa*x+bbbb
 fit i(x) "4x4xX.csv" every ::1 using 2:4 via aaaa,bbbb
 set xlabel 'variability'
 set ylabel 'evolution error'
 set output "4x4xX_variability-vs-evo-error.png"
-plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 2:4 title "4x4x4",     "20171005_3dFit_4x4x5_100times.csv" every ::1 using 2:4 title "4x4x5",     "20171013_3dFit_4x4x7_100times.csv" every ::1 using 2:4 title "4x4x7",     i(x) title "lin. fit" lc rgb "black"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 2:4 title "4x4x4" pt 2,     "20171005_3dFit_4x4x5_100times.csv" every ::1 using 2:4 title "4x4x5" pt 2,     "20171013_3dFit_4x4x7_100times.csv" every ::1 using 2:4 title "4x4x7" pt 2,     i(x) title "lin. fit" lc rgb "black"

dokumentation/evolution3d/4x4xX.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing 4x4xX.csv"
+[1] "Mean:"
+[1] 103.7625
+[1] "Median:"
+[1] 103.3795
+[1] "Sigma:"
+[1] 13.69887
+[1] "Range:"
+[1]  77.2599 129.9160

dokumentation/evolution3d/4x4xX.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing 4x4xX.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+     x    y
+x  1.0 -0.9
+y -0.9  1.0
+
+n= 300 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+      x     y
+x  1.00 -0.13
+y -0.13  1.00
+
+n= 300 
+
+
+P
+  x      y     
+x        0.0232
+y 0.0232       
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.46
+y -0.46  1.00
+
+n= 300 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for variability vs. evolution-error"
+      x     y
+x  1.00 -0.89
+y -0.89  1.00
+
+n= 300 
+
+
+P
+  x  y 
+x     0
+y  0   

dokumentation/evolution3d/R_analysis.sh

@@ -0,0 +1,32 @@
+#!/bin/bash
+
+# regularity,variability,improvement,"Evolution error",steps
+# 6.57581e-05,0.00592209,0.622392,113.016,2368
+
+if [[ -f "$1" ]]; then
+
+R -q --slave --vanilla <<EOF
+print("================ Analyzing $1")
+library(Hmisc)
+print("spearman for improvement-potential vs. evolution-error")
+DF=as.matrix(read.csv("$1",header=TRUE))
+rcorr(DF[,3],DF[,4],type="spearman")
+
+print("spearman for improvement-potential vs. steps")
+DF=as.matrix(read.csv("$1",header=TRUE))
+rcorr(DF[,3],DF[,5],type="spearman")
+
+print("spearman for regularity vs. steps")
+DF=as.matrix(read.csv("$1",header=TRUE))
+rcorr(DF[,1],DF[,5],type="spearman")
+
+print("spearman for variability vs. evolution-error")
+DF=as.matrix(read.csv("$1",header=TRUE))
+rcorr(DF[,2],DF[,4],type="spearman")
+EOF
+
+else
+
+echo "Usage: $0 <Filename.csv>"
+fi
+

dokumentation/evolution3d/R_mean_med_sigma.sh

@@ -0,0 +1,26 @@
+#!/bin/bash
+
+# regularity,variability,improvement,"Evolution error",steps
+# 6.57581e-05,0.00592209,0.622392,113.016,2368
+
+if [[ -f "$2" ]]; then
+
+R -q --slave --vanilla <<EOF
+print("================ Analyzing $2")
+#library(Hmisc)
+DF=as.matrix(read.csv("$2",header=TRUE))
+print("Mean:")
+mean(DF[,$1])
+print("Median:")
+median(DF[,$1])
+print("Sigma:")
+sd(DF[,$1])
+print("Range:")
+range(DF[,$1])
+EOF
+
+else
+
+echo "Usage: $0 <column> <Filename.csv>"
+fi
+

dokumentation/evolution3d/Xx4x4.csv

@@ -1,301 +1,301 @@
 regularity,variability,improvement,"Evolution error",steps
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+7.8252e-05,0.0103637,0.684139,82.1324,773
-8.1221e-05,0.0103637,0.691527,90.3791.,611
+8.1221e-05,0.0103637,0.691527,90.3791,611
-0.000110163,0.0103637,0.702362,99.9413.,506
+0.000110163,0.0103637,0.702362,99.9413,506
-5.54961e-05,0.0103637,0.709284,72.5502.,882
+5.54961e-05,0.0103637,0.709284,72.5502,882
-7.37375e-05,0.0103637,0.696269,83.4268.,761
+7.37375e-05,0.0103637,0.696269,83.4268,761
-8.96068e-05,0.0103637,0.707139,87.4954.,393
+8.96068e-05,0.0103637,0.707139,87.4954,393
-5.39211e-05,0.0103637,0.696067,83.3203.,762
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-7.70122e-05,0.0103637,0.702879,91.7128.,613
+7.70122e-05,0.0103637,0.702879,91.7128,613

dokumentation/evolution3d/Xx4x4.error

@@ -0,0 +1,301 @@
+"Evolution error
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dokumentation/evolution3d/Xx4x4.gnuplot.fit.log

@@ -1,7 +1,7 @@
 
 
 *******************************************************************************
-Mon Oct 16 12:15:22 2017
+Wed Oct 25 19:14:30 2017
 
 
 FIT:    data read from "Xx4x4.csv" every ::1 using 1:5
@@ -47,7 +47,7 @@ b              -0.934  1.000
 
 
 *******************************************************************************
-Mon Oct 16 12:15:22 2017
+Wed Oct 25 19:14:30 2017
 
 
 FIT:    data read from "Xx4x4.csv" every ::1 using 3:5
@@ -93,7 +93,7 @@ bb             -0.999  1.000
 
 
 *******************************************************************************
-Mon Oct 16 12:15:22 2017
+Wed Oct 25 19:14:30 2017
 
 
 FIT:    data read from "Xx4x4.csv" every ::1 using 3:4
@@ -139,7 +139,7 @@ bbb            -0.999  1.000
 
 
 *******************************************************************************
-Mon Oct 16 12:15:22 2017
+Wed Oct 25 19:14:30 2017
 
 
 FIT:    data read from "Xx4x4.csv" every ::1 using 2:4

dokumentation/evolution3d/Xx4x4.gnuplot.script

@@ -5,22 +5,22 @@ set terminal png
 set xlabel 'regularity'
 set ylabel 'steps'
 set output "Xx4x4_regularity-vs-steps.png"
-plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 title "4x4x4",     "20171013_3dFit_5x4x4_100times.csv" every ::1 using 1:5 title "5x4x4",     "20171005_3dFit_7x4x4_100times.csv" every ::1 using 1:5 title "7x4x4",     f(x) title "lin. fit" lc rgb "black"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 title "4x4x4" pt 2,     "20171013_3dFit_5x4x4_100times.csv" every ::1 using 1:5 title "5x4x4" pt 2,     "20171005_3dFit_7x4x4_100times.csv" every ::1 using 1:5 title "7x4x4" pt 2,     f(x) title "lin. fit" lc rgb "black"
 g(x)=aa*x+bb
 fit g(x) "Xx4x4.csv" every ::1 using 3:5 via aa,bb
 set xlabel 'improvement potential'
 set ylabel 'steps'
 set output "Xx4x4_improvement-vs-steps.png"
-plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 title "4x4x4",     "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:5 title "5x4x4",     "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:5 title "7x4x4",     g(x) title "lin. fit" lc rgb "black"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 title "4x4x4" pt 2,     "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:5 title "5x4x4" pt 2,     "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:5 title "7x4x4" pt 2,     g(x) title "lin. fit" lc rgb "black"
 h(x)=aaa*x+bbb
 fit h(x) "Xx4x4.csv" every ::1 using 3:4 via aaa,bbb
 set xlabel 'improvement potential'
 set ylabel 'evolution error'
 set output "Xx4x4_improvement-vs-evo-error.png"
-plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 title "4x4x4",     "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:4 title "5x4x4",     "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:4 title "7x4x4",     h(x) title "lin. fit" lc rgb "black"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 title "4x4x4" pt 2,     "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:4 title "5x4x4" pt 2,     "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:4 title "7x4x4" pt 2,     h(x) title "lin. fit" lc rgb "black"
 i(x)=aaaa*x+bbbb
 fit i(x) "Xx4x4.csv" every ::1 using 2:4 via aaaa,bbbb
 set xlabel 'variability'
 set ylabel 'evolution error'
 set output "Xx4x4_variability-vs-evo-error.png"
-plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 2:4 title "4x4x4",     "20171013_3dFit_5x4x4_100times.csv" every ::1 using 2:4 title "5x4x4",     "20171005_3dFit_7x4x4_100times.csv" every ::1 using 2:4 title "7x4x4",     i(x) title "lin. fit" lc rgb "black"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 2:4 title "4x4x4" pt 2,     "20171013_3dFit_5x4x4_100times.csv" every ::1 using 2:4 title "5x4x4" pt 2,     "20171005_3dFit_7x4x4_100times.csv" every ::1 using 2:4 title "7x4x4" pt 2,     i(x) title "lin. fit" lc rgb "black"

dokumentation/evolution3d/Xx4x4.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing Xx4x4.csv"
+[1] "Mean:"
+[1] 101.9338
+[1] "Median:"
+[1] 101.4365
+[1] "Sigma:"
+[1] 15.31929
+[1] "Range:"
+[1]  72.2744 130.8540

dokumentation/evolution3d/Xx4x4.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing Xx4x4.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+      x     y
+x  1.00 -0.89
+y -0.89  1.00
+
+n= 300 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+      x     y
+x  1.00 -0.69
+y -0.69  1.00
+
+n= 300 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.46
+y -0.46  1.00
+
+n= 300 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for variability vs. evolution-error"
+     x    y
+x  1.0 -0.9
+y -0.9  1.0
+
+n= 300 
+
+
+P
+  x  y 
+x     0
+y  0   

dokumentation/evolution3d/YxYxY.error

@@ -0,0 +1,311 @@
+"Evolution error
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dokumentation/evolution3d/YxYxY.gnuplot.fit.log

@@ -1,7 +1,7 @@
 
 
 *******************************************************************************
-Mon Oct 23 12:09:55 2017
+Wed Oct 25 19:14:34 2017
 
 
 FIT:    data read from "YxYxY.csv" every ::1 using 1:5
@@ -47,7 +47,7 @@ b              -0.937  1.000
 
 
 *******************************************************************************
-Mon Oct 23 12:09:55 2017
+Wed Oct 25 19:14:34 2017
 
 
 FIT:    data read from "YxYxY.csv" every ::1 using 3:5
@@ -93,7 +93,7 @@ bb             -0.994  1.000
 
 
 *******************************************************************************
-Mon Oct 23 12:09:55 2017
+Wed Oct 25 19:14:34 2017
 
 
 FIT:    data read from "YxYxY.csv" every ::1 using 3:4
@@ -139,7 +139,7 @@ bbb            -0.994  1.000
 
 
 *******************************************************************************
-Mon Oct 23 12:09:55 2017
+Wed Oct 25 19:14:34 2017
 
 
 FIT:    data read from "YxYxY.csv" every ::1 using 2:4

dokumentation/evolution3d/YxYxY.gnuplot.script

@@ -5,22 +5,22 @@ set terminal png
 set xlabel 'regularity'
 set ylabel 'steps'
 set output "YxYxY_regularity-vs-steps.png"
-plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 title "4x4x4",     "20170926_3dFit_5x5x5_100times.csv" every ::1 using 1:5 title "5x5x5",     "20171021-evolution3D_6x6_100Times.csv" every ::1 using 1:5 title "6x6x6",     f(x) title "lin. fit" lc rgb "black"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 title "4x4x4" pt 2,     "20170926_3dFit_5x5x5_100times.csv" every ::1 using 1:5 title "5x5x5" pt 2,     "20171021-evolution3D_6x6_100Times.csv" every ::1 using 1:5 title "6x6x6" pt 2,     f(x) title "lin. fit" lc rgb "black"
 g(x)=aa*x+bb
 fit g(x) "YxYxY.csv" every ::1 using 3:5 via aa,bb
 set xlabel 'improvement potential'
 set ylabel 'steps'
 set output "YxYxY_improvement-vs-steps.png"
-plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 title "4x4x4",     "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:5 title "5x5x5",     "20171021-evolution3D_6x6_100Times.csv" every ::1 using 3:5 title "6x6x6",     g(x) title "lin. fit" lc rgb "black"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 title "4x4x4" pt 2,     "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:5 title "5x5x5" pt 2,     "20171021-evolution3D_6x6_100Times.csv" every ::1 using 3:5 title "6x6x6" pt 2,     g(x) title "lin. fit" lc rgb "black"
 h(x)=aaa*x+bbb
 fit h(x) "YxYxY.csv" every ::1 using 3:4 via aaa,bbb
 set xlabel 'improvement potential'
 set ylabel 'evolution error'
 set output "YxYxY_improvement-vs-evo-error.png"
-plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 title "4x4x4",     "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:4 title "5x5x5",     "20171021-evolution3D_6x6_100Times.csv" every ::1 using 3:4 title "6x6x6",     h(x) title "lin. fit" lc rgb "black"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 title "4x4x4" pt 2,     "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:4 title "5x5x5" pt 2,     "20171021-evolution3D_6x6_100Times.csv" every ::1 using 3:4 title "6x6x6" pt 2,     h(x) title "lin. fit" lc rgb "black"
 i(x)=aaaa*x+bbbb
 fit i(x) "YxYxY.csv" every ::1 using 2:4 via aaaa,bbbb
 set xlabel 'variability'
 set ylabel 'evolution error'
 set output "YxYxY_variability-vs-evo-error.png"
-plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 2:4 title "4x4x4",     "20170926_3dFit_5x5x5_100times.csv" every ::1 using 2:4 title "5x5x5",     "20171021-evolution3D_6x6_100Times.csv" every ::1 using 2:4 title "6x6x6",     i(x) title "lin. fit" lc rgb "black"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 2:4 title "4x4x4" pt 2,     "20170926_3dFit_5x5x5_100times.csv" every ::1 using 2:4 title "5x5x5" pt 2,     "20171021-evolution3D_6x6_100Times.csv" every ::1 using 2:4 title "6x6x6" pt 2,     i(x) title "lin. fit" lc rgb "black"

dokumentation/evolution3d/YxYxY.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing YxYxY.csv"
+[1] "Mean:"
+[1] 85.58962
+[1] "Median:"
+[1] 76.04925
+[1] "Sigma:"
+[1] 24.66794
+[1] "Range:"
+[1]  52.0186 129.9160

dokumentation/evolution3d/YxYxY.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing YxYxY.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+      x     y
+x  1.00 -0.89
+y -0.89  1.00
+
+n= 310 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+      x     y
+x  1.00 -0.81
+y -0.81  1.00
+
+n= 310 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for regularity vs. steps"
+     x    y
+x 1.00 0.25
+y 0.25 1.00
+
+n= 310 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for variability vs. evolution-error"
+      x     y
+x  1.00 -0.91
+y -0.91  1.00
+
+n= 310 
+
+
+P
+  x  y 
+x     0
+y  0   

dokumentation/evolution3d/all.csv

@@ -0,0 +1,911 @@
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+5.729e-05,0.019987,0.813128,64.2813,423
+5.05547e-05,0.019987,0.828796,61.7037,452
+6.5795e-05,0.019987,0.819006,59.3731,682
+4.81851e-05,0.019987,0.810773,65.7621,519
+4.71572e-05,0.019987,0.818411,63.3544,597
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+6.10619e-05,0.019987,0.817943,55.0987,1004
+4.4631e-05,0.019987,0.805461,64.6963,474

dokumentation/evolution3d/all.gnuplot.fit.log

@@ -0,0 +1,184 @@
+
+
+*******************************************************************************
+Wed Oct 25 19:09:05 2017
+
+
+FIT:    data read from "all.csv" every ::1 using 1:5
+        format = x:z
+        #datapoints = 910
+        residuals are weighted equally (unit weight)
+
+function used for fitting: f(x)
+fitted parameters initialized with current variable values
+
+
+
+ Iteration 0
+ WSSR        : 2.69952e+09       delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.707107
+
+initial set of free parameter values
+
+a               = 1
+b               = 1
+
+After 6 iterations the fit converged.
+final sum of squares of residuals : 5.77426e+08
+rel. change during last iteration : -1.48436e-07
+
+degrees of freedom    (FIT_NDF)                        : 908
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 797.454
+variance of residuals (reduced chisquare) = WSSR/ndf   : 635932
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+a               = -5.38736e+06     +/- 8.165e+05    (15.16%)
+b               = 1966.44          +/- 72.92        (3.708%)
+
+
+correlation matrix of the fit parameters:
+
+               a      b      
+a               1.000 
+b              -0.932  1.000 
+
+
+*******************************************************************************
+Wed Oct 25 19:09:05 2017
+
+
+FIT:    data read from "all.csv" every ::1 using 3:5
+        format = x:z
+        #datapoints = 910
+        residuals are weighted equally (unit weight)
+
+function used for fitting: g(x)
+fitted parameters initialized with current variable values
+
+
+
+ Iteration 0
+ WSSR        : 2.6977e+09        delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.857335
+
+initial set of free parameter values
+
+aa              = 1
+bb              = 1
+
+After 3 iterations the fit converged.
+final sum of squares of residuals : 3.3878e+08
+rel. change during last iteration : -3.41053e-07
+
+degrees of freedom    (FIT_NDF)                        : 908
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 610.824
+variance of residuals (reduced chisquare) = WSSR/ndf   : 373106
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aa              = -8301.78         +/- 310.7        (3.743%)
+bb              = 7184             +/- 213          (2.965%)
+
+
+correlation matrix of the fit parameters:
+
+               aa     bb     
+aa              1.000 
+bb             -0.995  1.000 
+
+
+*******************************************************************************
+Wed Oct 25 19:09:05 2017
+
+
+FIT:    data read from "all.csv" every ::1 using 3:4
+        format = x:z
+        #datapoints = 910
+        residuals are weighted equally (unit weight)
+
+function used for fitting: h(x)
+fitted parameters initialized with current variable values
+
+
+
+ Iteration 0
+ WSSR        : 8.64029e+06       delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.857335
+
+initial set of free parameter values
+
+aaa             = 1
+bbb             = 1
+
+After 3 iterations the fit converged.
+final sum of squares of residuals : 44319.2
+rel. change during last iteration : -3.6749e-06
+
+degrees of freedom    (FIT_NDF)                        : 908
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 6.98639
+variance of residuals (reduced chisquare) = WSSR/ndf   : 48.8097
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aaa             = -292.832         +/- 3.554        (1.214%)
+bbb             = 296.825          +/- 2.437        (0.8209%)
+
+
+correlation matrix of the fit parameters:
+
+               aaa    bbb    
+aaa             1.000 
+bbb            -0.995  1.000 
+
+
+*******************************************************************************
+Wed Oct 25 19:09:05 2017
+
+
+FIT:    data read from "all.csv" every ::1 using 2:4
+        format = x:z
+        #datapoints = 910
+        residuals are weighted equally (unit weight)
+
+function used for fitting: i(x)
+fitted parameters initialized with current variable values
+
+
+
+ Iteration 0
+ WSSR        : 8.75529e+06       delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.707146
+
+initial set of free parameter values
+
+aaaa            = 1
+bbbb            = 1
+
+After 4 iterations the fit converged.
+final sum of squares of residuals : 84605.2
+rel. change during last iteration : -1.31864e-06
+
+degrees of freedom    (FIT_NDF)                        : 908
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 9.65285
+variance of residuals (reduced chisquare) = WSSR/ndf   : 93.1775
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aaaa            = -4091.03         +/- 73.19        (1.789%)
+bbbb            = 136.014          +/- 0.7684       (0.5649%)
+
+
+correlation matrix of the fit parameters:
+
+               aaaa   bbbb   
+aaaa            1.000 
+bbbb           -0.909  1.000 

dokumentation/evolution3d/all.gnuplot.log

@@ -0,0 +1,304 @@
+
+
+ Iteration 0
+ WSSR        : 2.69952e+09       delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.707107
+
+initial set of free parameter values
+
+a               = 1
+b               = 1
+/
+
+ Iteration 1
+ WSSR        : 6.0511e+08        delta(WSSR)/WSSR   : -3.4612
+ delta(WSSR) : -2.09441e+09      limit for stopping : 1e-05
+ lambda	  : 0.0707107
+
+resultant parameter values
+
+a               = -9.15081
+b               = 1517.25
+/
+
+ Iteration 2
+ WSSR        : 6.05099e+08       delta(WSSR)/WSSR   : -1.84933e-05
+ delta(WSSR) : -11190.3          limit for stopping : 1e-05
+ lambda	  : 0.00707107
+
+resultant parameter values
+
+a               = -1036.65
+b               = 1518.17
+/
+
+ Iteration 3
+ WSSR        : 6.04072e+08       delta(WSSR)/WSSR   : -0.00169898
+ delta(WSSR) : -1.02631e+06      limit for stopping : 1e-05
+ lambda	  : 0.000707107
+
+resultant parameter values
+
+a               = -101864
+b               = 1526.56
+/
+
+ Iteration 4
+ WSSR        : 5.80578e+08       delta(WSSR)/WSSR   : -0.040467
+ delta(WSSR) : -2.34943e+07      limit for stopping : 1e-05
+ lambda	  : 7.07107e-05
+
+resultant parameter values
+
+a               = -3.56955e+06
+b               = 1815.15
+/
+
+ Iteration 5
+ WSSR        : 5.77427e+08       delta(WSSR)/WSSR   : -0.00545821
+ delta(WSSR) : -3.15172e+06      limit for stopping : 1e-05
+ lambda	  : 7.07107e-06
+
+resultant parameter values
+
+a               = -5.37788e+06
+b               = 1965.65
+/
+
+ Iteration 6
+ WSSR        : 5.77426e+08       delta(WSSR)/WSSR   : -1.48436e-07
+ delta(WSSR) : -85.7109          limit for stopping : 1e-05
+ lambda	  : 7.07107e-07
+
+resultant parameter values
+
+a               = -5.38736e+06
+b               = 1966.44
+
+After 6 iterations the fit converged.
+final sum of squares of residuals : 5.77426e+08
+rel. change during last iteration : -1.48436e-07
+
+degrees of freedom    (FIT_NDF)                        : 908
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 797.454
+variance of residuals (reduced chisquare) = WSSR/ndf   : 635932
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+a               = -5.38736e+06     +/- 8.165e+05    (15.16%)
+b               = 1966.44          +/- 72.92        (3.708%)
+
+
+correlation matrix of the fit parameters:
+
+               a      b      
+a               1.000 
+b              -0.932  1.000 
+
+
+ Iteration 0
+ WSSR        : 2.6977e+09        delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.857335
+
+initial set of free parameter values
+
+aa              = 1
+bb              = 1
+/
+
+ Iteration 1
+ WSSR        : 3.537e+08         delta(WSSR)/WSSR   : -6.62708
+ delta(WSSR) : -2.344e+09        limit for stopping : 1e-05
+ lambda	  : 0.0857335
+
+resultant parameter values
+
+aa              = -6338.16
+bb              = 5839.13
+/
+
+ Iteration 2
+ WSSR        : 3.3878e+08        delta(WSSR)/WSSR   : -0.0440412
+ delta(WSSR) : -1.49203e+07      limit for stopping : 1e-05
+ lambda	  : 0.00857335
+
+resultant parameter values
+
+aa              = -8296.31
+bb              = 7180.26
+/
+
+ Iteration 3
+ WSSR        : 3.3878e+08        delta(WSSR)/WSSR   : -3.41053e-07
+ delta(WSSR) : -115.542          limit for stopping : 1e-05
+ lambda	  : 0.000857335
+
+resultant parameter values
+
+aa              = -8301.78
+bb              = 7184
+
+After 3 iterations the fit converged.
+final sum of squares of residuals : 3.3878e+08
+rel. change during last iteration : -3.41053e-07
+
+degrees of freedom    (FIT_NDF)                        : 908
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 610.824
+variance of residuals (reduced chisquare) = WSSR/ndf   : 373106
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aa              = -8301.78         +/- 310.7        (3.743%)
+bb              = 7184             +/- 213          (2.965%)
+
+
+correlation matrix of the fit parameters:
+
+               aa     bb     
+aa              1.000 
+bb             -0.995  1.000 
+
+
+ Iteration 0
+ WSSR        : 8.64029e+06       delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.857335
+
+initial set of free parameter values
+
+aaa             = 1
+bbb             = 1
+/
+
+ Iteration 1
+ WSSR        : 65352.7           delta(WSSR)/WSSR   : -131.21
+ delta(WSSR) : -8.57494e+06      limit for stopping : 1e-05
+ lambda	  : 0.0857335
+
+resultant parameter values
+
+aaa             = -219.119
+bbb             = 246.318
+/
+
+ Iteration 2
+ WSSR        : 44319.4           delta(WSSR)/WSSR   : -0.474585
+ delta(WSSR) : -21033.3          limit for stopping : 1e-05
+ lambda	  : 0.00857335
+
+resultant parameter values
+
+aaa             = -292.627
+bbb             = 296.685
+/
+
+ Iteration 3
+ WSSR        : 44319.2           delta(WSSR)/WSSR   : -3.6749e-06
+ delta(WSSR) : -0.162868         limit for stopping : 1e-05
+ lambda	  : 0.000857335
+
+resultant parameter values
+
+aaa             = -292.832
+bbb             = 296.825
+
+After 3 iterations the fit converged.
+final sum of squares of residuals : 44319.2
+rel. change during last iteration : -3.6749e-06
+
+degrees of freedom    (FIT_NDF)                        : 908
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 6.98639
+variance of residuals (reduced chisquare) = WSSR/ndf   : 48.8097
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aaa             = -292.832         +/- 3.554        (1.214%)
+bbb             = 296.825          +/- 2.437        (0.8209%)
+
+
+correlation matrix of the fit parameters:
+
+               aaa    bbb    
+aaa             1.000 
+bbb            -0.995  1.000 
+
+
+ Iteration 0
+ WSSR        : 8.75529e+06       delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.707146
+
+initial set of free parameter values
+
+aaaa            = 1
+bbbb            = 1
+/
+
+ Iteration 1
+ WSSR        : 356710            delta(WSSR)/WSSR   : -23.5445
+ delta(WSSR) : -8.39858e+06      limit for stopping : 1e-05
+ lambda	  : 0.0707146
+
+resultant parameter values
+
+aaaa            = -135.643
+bbbb            = 98.21
+/
+
+ Iteration 2
+ WSSR        : 98176.3           delta(WSSR)/WSSR   : -2.63337
+ delta(WSSR) : -258534           limit for stopping : 1e-05
+ lambda	  : 0.00707146
+
+resultant parameter values
+
+aaaa            = -3207.68
+bbbb            = 127.583
+/
+
+ Iteration 3
+ WSSR        : 84605.3           delta(WSSR)/WSSR   : -0.160403
+ delta(WSSR) : -13570.9          limit for stopping : 1e-05
+ lambda	  : 0.000707146
+
+resultant parameter values
+
+aaaa            = -4088.49
+bbbb            = 135.99
+/
+
+ Iteration 4
+ WSSR        : 84605.2           delta(WSSR)/WSSR   : -1.31864e-06
+ delta(WSSR) : -0.111564         limit for stopping : 1e-05
+ lambda	  : 7.07146e-05
+
+resultant parameter values
+
+aaaa            = -4091.03
+bbbb            = 136.014
+
+After 4 iterations the fit converged.
+final sum of squares of residuals : 84605.2
+rel. change during last iteration : -1.31864e-06
+
+degrees of freedom    (FIT_NDF)                        : 908
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 9.65285
+variance of residuals (reduced chisquare) = WSSR/ndf   : 93.1775
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aaaa            = -4091.03         +/- 73.19        (1.789%)
+bbbb            = 136.014          +/- 0.7684       (0.5649%)
+
+
+correlation matrix of the fit parameters:
+
+               aaaa   bbbb   
+aaaa            1.000 
+bbbb           -0.909  1.000 

dokumentation/evolution3d/all.gnuplot.script

@@ -0,0 +1,26 @@
+set datafile separator ","
+f(x)=a*x+b
+fit f(x) "all.csv" every ::1 using 1:5 via a,b
+set terminal png
+set xlabel 'regularity'
+set ylabel 'steps'
+set output "all_regularity-vs-steps.png"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 title "4x4x4" pt 2,     "20171013_3dFit_5x4x4_100times.csv" every ::1 using 1:5 title "5x4x4" pt 2,     "20171005_3dFit_7x4x4_100times.csv" every ::1 using 1:5 title "7x4x4" pt 2,     "20171005_3dFit_4x4x5_100times.csv" every ::1 using 1:5 title "4x4x5" pt 2,     "20171013_3dFit_4x4x7_100times.csv" every ::1 using 1:5 title "4x4x7" pt 2,     "20170926_3dFit_5x5x5_100times.csv" every ::1 using 1:5 title "5x5x5" pt 2,     "20171021-evolution3D_6x6_100Times.csv" every ::1 using 1:5 title "6x6x6" pt 2,     f(x) title "lin. fit" lc rgb "black"
+g(x)=aa*x+bb
+fit g(x) "all.csv" every ::1 using 3:5 via aa,bb
+set xlabel 'improvement potential'
+set ylabel 'steps'
+set output "all_improvement-vs-steps.png"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 title "4x4x4" pt 2,     "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:5 title "5x4x4" pt 2,     "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:5 title "7x4x4" pt 2,     "20171005_3dFit_4x4x5_100times.csv" every ::1 using 3:5 title "4x4x5" pt 2,     "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:5 title "4x4x7" pt 2,     "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:5 title "5x5x5" pt 2,     "20171021-evolution3D_6x6_100Times.csv" every ::1 using 3:5 title "6x6x6" pt 2,     g(x) title "lin. fit" lc rgb "black"
+h(x)=aaa*x+bbb
+fit h(x) "all.csv" every ::1 using 3:4 via aaa,bbb
+set xlabel 'improvement potential'
+set ylabel 'evolution error'
+set output "all_improvement-vs-evo-error.png"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 title "4x4x4" pt 2,     "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:4 title "5x4x4" pt 2,     "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:4 title "7x4x4" pt 2,     "20171005_3dFit_4x4x5_100times.csv" every ::1 using 3:4 title "4x4x5" pt 2,     "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:4 title "4x4x7" pt 2,     "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:4 title "5x5x5" pt 2,     "20171021-evolution3D_6x6_100Times.csv" every ::1 using 3:4 title "6x6x6" pt 2,     h(x) title "lin. fit" lc rgb "black"
+i(x)=aaaa*x+bbbb
+fit i(x) "all.csv" every ::1 using 2:4 via aaaa,bbbb
+set xlabel 'variability'
+set ylabel 'evolution error'
+set output "all_variability-vs-evo-error.png"
+plot     "20170926_3dFit_4x4x4_100times.csv" every ::1 using 2:4 title "4x4x4" pt 2,     "20171013_3dFit_5x4x4_100times.csv" every ::1 using 2:4 title "5x4x4" pt 2,     "20171005_3dFit_7x4x4_100times.csv" every ::1 using 2:4 title "7x4x4" pt 2,     "20171005_3dFit_4x4x5_100times.csv" every ::1 using 2:4 title "4x4x5" pt 2,     "20171013_3dFit_4x4x7_100times.csv" every ::1 using 2:4 title "4x4x7" pt 2,     "20170926_3dFit_5x5x5_100times.csv" every ::1 using 2:4 title "5x5x5" pt 2,     "20171021-evolution3D_6x6_100Times.csv" every ::1 using 2:4 title "6x6x6" pt 2,     i(x) title "lin. fit" lc rgb "black"

dokumentation/evolution3d/all.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing all.csv"
+[1] "Mean:"
+[1] 96.96886
+[1] "Median:"
+[1] 97.8825
+[1] "Sigma:"
+[1] 20.3298
+[1] "Range:"
+[1]  52.0186 130.8540

dokumentation/evolution3d/all.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing all.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+      x     y
+x  1.00 -0.95
+y -0.95  1.00
+
+n= 910 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+      x     y
+x  1.00 -0.69
+y -0.69  1.00
+
+n= 910 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.15
+y -0.15  1.00
+
+n= 910 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for variability vs. evolution-error"
+      x     y
+x  1.00 -0.94
+y -0.94  1.00
+
+n= 910 
+
+
+P
+  x  y 
+x     0
+y  0   

dokumentation/evolution3d/combine.sh

@@ -14,9 +14,9 @@ set xlabel 'regularity'
 set ylabel 'steps'
 set output "${png}_regularity-vs-steps.png"
 plot \
-    "$2" every ::1 using 1:5 title "$3", \
+    "$2" every ::1 using 1:5 title "$3" pt 2, \
-    "$4" every ::1 using 1:5 title "$5", \
+    "$4" every ::1 using 1:5 title "$5" pt 2, \
-    "$6" every ::1 using 1:5 title "$7", \
+    "$6" every ::1 using 1:5 title "$7" pt 2, \
     f(x) title "lin. fit" lc rgb "black"
 g(x)=aa*x+bb
 fit g(x) "$data" every ::1 using 3:5 via aa,bb
@@ -24,9 +24,9 @@ set xlabel 'improvement potential'
 set ylabel 'steps'
 set output "${png}_improvement-vs-steps.png"
 plot \
-    "$2" every ::1 using 3:5 title "$3", \
+    "$2" every ::1 using 3:5 title "$3" pt 2, \
-    "$4" every ::1 using 3:5 title "$5", \
+    "$4" every ::1 using 3:5 title "$5" pt 2, \
-    "$6" every ::1 using 3:5 title "$7", \
+    "$6" every ::1 using 3:5 title "$7" pt 2, \
     g(x) title "lin. fit" lc rgb "black"
 h(x)=aaa*x+bbb
 fit h(x) "$data" every ::1 using 3:4 via aaa,bbb
@@ -34,9 +34,9 @@ set xlabel 'improvement potential'
 set ylabel 'evolution error'
 set output "${png}_improvement-vs-evo-error.png"
 plot \
-    "$2" every ::1 using 3:4 title "$3", \
+    "$2" every ::1 using 3:4 title "$3" pt 2, \
-    "$4" every ::1 using 3:4 title "$5", \
+    "$4" every ::1 using 3:4 title "$5" pt 2, \
-    "$6" every ::1 using 3:4 title "$7", \
+    "$6" every ::1 using 3:4 title "$7" pt 2, \
     h(x) title "lin. fit" lc rgb "black"
 i(x)=aaaa*x+bbbb
 fit i(x) "$data" every ::1 using 2:4 via aaaa,bbbb
@@ -44,9 +44,9 @@ set xlabel 'variability'
 set ylabel 'evolution error'
 set output "${png}_variability-vs-evo-error.png"
 plot \
-    "$2" every ::1 using 2:4 title "$3", \
+    "$2" every ::1 using 2:4 title "$3" pt 2, \
-    "$4" every ::1 using 2:4 title "$5", \
+    "$4" every ::1 using 2:4 title "$5" pt 2, \
-    "$6" every ::1 using 2:4 title "$7", \
+    "$6" every ::1 using 2:4 title "$7" pt 2, \
     i(x) title "lin. fit" lc rgb "black"
 EOD
 ) > "${png}.gnuplot.script" 

dokumentation/evolution3d/combine7.sh

@@ -0,0 +1,71 @@
+#!/bin/bash
+
+if [[ $# -eq 0 ]]; then
+    echo "usage: $0 <DATA.csv>"
+else
+    data="$1";
+    png="`echo $1 | sed -s "s/\.csv$//"`" # strip ending
+    (cat <<EOD
+set datafile separator ","
+f(x)=a*x+b
+fit f(x) "$data" every ::1 using 1:5 via a,b
+set terminal png
+set xlabel 'regularity'
+set ylabel 'steps'
+set output "${png}_regularity-vs-steps.png"
+plot \
+    "$2" every ::1 using 1:5 title "$3" pt 2, \
+    "$4" every ::1 using 1:5 title "$5" pt 2, \
+    "$6" every ::1 using 1:5 title "$7" pt 2, \
+    "$8" every ::1 using 1:5 title "$9" pt 2, \
+    "${10}" every ::1 using 1:5 title "${11}" pt 2, \
+    "${12}" every ::1 using 1:5 title "${13}" pt 2, \
+    "${14}" every ::1 using 1:5 title "${15}" pt 2, \
+    f(x) title "lin. fit" lc rgb "black"
+g(x)=aa*x+bb
+fit g(x) "$data" every ::1 using 3:5 via aa,bb
+set xlabel 'improvement potential'
+set ylabel 'steps'
+set output "${png}_improvement-vs-steps.png"
+plot \
+    "$2" every ::1 using 3:5 title "$3" pt 2, \
+    "$4" every ::1 using 3:5 title "$5" pt 2, \
+    "$6" every ::1 using 3:5 title "$7" pt 2, \
+    "$8" every ::1 using 3:5 title "$9" pt 2, \
+    "${10}" every ::1 using 3:5 title "${11}" pt 2, \
+    "${12}" every ::1 using 3:5 title "${13}" pt 2, \
+    "${14}" every ::1 using 3:5 title "${15}" pt 2, \
+    g(x) title "lin. fit" lc rgb "black"
+h(x)=aaa*x+bbb
+fit h(x) "$data" every ::1 using 3:4 via aaa,bbb
+set xlabel 'improvement potential'
+set ylabel 'evolution error'
+set output "${png}_improvement-vs-evo-error.png"
+plot \
+    "$2" every ::1 using 3:4 title "$3" pt 2, \
+    "$4" every ::1 using 3:4 title "$5" pt 2, \
+    "$6" every ::1 using 3:4 title "$7" pt 2, \
+    "$8" every ::1 using 3:4 title "$9" pt 2, \
+    "${10}" every ::1 using 3:4 title "${11}" pt 2, \
+    "${12}" every ::1 using 3:4 title "${13}" pt 2, \
+    "${14}" every ::1 using 3:4 title "${15}" pt 2, \
+    h(x) title "lin. fit" lc rgb "black"
+i(x)=aaaa*x+bbbb
+fit i(x) "$data" every ::1 using 2:4 via aaaa,bbbb
+set xlabel 'variability'
+set ylabel 'evolution error'
+set output "${png}_variability-vs-evo-error.png"
+plot \
+    "$2" every ::1 using 2:4 title "$3" pt 2, \
+    "$4" every ::1 using 2:4 title "$5" pt 2, \
+    "$6" every ::1 using 2:4 title "$7" pt 2, \
+    "$8" every ::1 using 2:4 title "$9" pt 2, \
+    "${10}" every ::1 using 2:4 title "${11}" pt 2, \
+    "${12}" every ::1 using 2:4 title "${13}" pt 2, \
+    "${14}" every ::1 using 2:4 title "${15}" pt 2, \
+    i(x) title "lin. fit" lc rgb "black"
+EOD
+) > "${png}.gnuplot.script" 
+    gnuplot "${png}.gnuplot.script" 2> "${png}.gnuplot.log"
+    mv fit.log "${png}.gnuplot.fit.log"
+fi

dokumentation/evolution3d/errors.csv

@@ -0,0 +1,111 @@
+"4x4x4","4x4x5","5x4x4","4x4x7","7x4x4","5x5x5","6x6x6"
+113.016,104.911,91.4882,85.1149,105.032,73.9627,56.9424
+118.796,102.122,95.001,82.2095,91.0336,79.3303,65.8672
+127.157,100.463,97.72,93.8283,74.0894,72.8373,61.6979
+121.613,111.329,99.0429,99.4121,77.0829,60.0032,64.1851
+119.539,104.712,102.273,84.7928,84.9413,80.1321,55.1204
+119.512,100.048,98.0028,89.4784,103.909,66.1526,63.4494
+118.285,102.05,101.929,98.0445,88.0771,74.6032,56.8508
+126.579,98.3549,103.969,82.4413,93.4708,71.3161,68.7883
+122.512,93.4038,92.5069,95.9898,83.1573,71.9377,63.1337
+118.473,104.028,118.88,82.9514,80.9548,70.127,71.6163
+121.998,102.965,96.2138,95.6417,78.1902,61.7195,71.1037
+115.313,102.822,100.164,97.597,84.7296,86.6101,55.0916
+118.034,97.9662,95.9282,92.8455,94.4156,77.235,63.109
+120.003,100.341,96.3737,87.554,84.0696,70.3058,56.7978
+123.494,109.73,107.927,88.0962,83.4435,73.3268,64.0039
+122.989,99.6362,98.0377,92.764,88.5741,72.8603,63.2723
+123.757,113.29,94.9011,92.6177,77.5017,70.5867,57.7703
+110.67,112.572,102.223,91.9667,82.8934,79.8851,61.9955
+121.232,86.3175,126.177,82.2732,89.5173,89.169,55.9687
+120.979,91.637,101.879,94.741,92.6297,86.3505,78.0966
+115.394,107.27,90.0009,87.4631,88.5844,77.8282,65.5659
+116.928,110.797,101.026,99.0457,82.1173,70.418,65.5294
+125.294,99.0484,94.5618,83.0585,73.2,82.3889,52.8169
+111.635,94.2222,99.2481,87.7612,84.76,80.3257,59.6491
+114.681,104.417,94.1741,92.4952,73.4001,85.6612,54.1868
+111.935,96.4144,102.894,87.4672,92.3227,76.2377,64.4377
+111.392,108.23,103.993,85.5196,84.2123,76.8062,64.634
+115.346,98.5045,96.549,96.1244,84.2583,75.0253,61.125
+119.055,93.4554,90.1281,80.3879,78.5571,73.6296,59.6531
+117.452,110.564,104.444,88.557,89.2675,71.8145,61.0567
+123.379,104.082,104.542,80.9192,81.8836,82.1172,66.6297
+115.053,95.1024,107.051,88.0159,83.885,68.4228,70.4683
+123.994,100.132,102.128,92.5849,91.469,74.5628,63.4839
+118.576,102.451,105.315,95.6485,81.0534,82.4297,58.2558
+116.894,116.302,104.097,82.6378,90.3739,74.6568,66.0124
+120.129,98.5173,96.8133,86.2912,85.6114,76.5874,57.8232
+126.708,104.428,99.8496,87.6295,91.687,82.2821,63.7635
+114.595,104.909,103.385,93.7818,90.1008,71.321,58.0721
+120.056,106.289,108.874,81.2872,89.7677,80.1887,60.9659
+122.057,106.708,112.975,92.884,79.4512,60.0022,62.2392
+116.846,108.022,113.068,85.1577,86.5543,73.5674,75.0921
+124.35,97.8253,99.4226,82.4601,88.7329,68.7547,55.2029
+118.43,114.829,115.742,88.7562,91.8764,97.4154,82.5252
+116.574,115.054,94.4964,85.7044,85.9747,81.1639,60.4597
+128.474,106.6,103.45,90.0389,82.843,82.2747,58.192
+111.111,105.157,116.123,90.6256,83.5281,70.415,65.2124
+120.004,93.2856,98.1676,86.5225,78.0961,72.564,60.0963
+119.866,98.8994,105.069,93.0686,104.49,70.1961,75.3887
+119.676,117.524,108.953,86.9607,88.2734,83.8507,80.3804
+128.957,97.0705,91.997,91.6042,102.032,82.3622,55.2663
+108.467,95.9897,95.3832,100.349,85.7869,74.3285,60.0725
+123.353,104.384,87.7666,78.4994,90.7034,84.2108,62.3384
+123.301,104.01,97.0353,89.8504,89.582,69.7744,64.2813
+114.838,104.585,99.1089,83.6403,83.1789,82.1376,61.7037
+118.611,98.8101,95.5062,101.654,75.4405,74.2818,59.3731
+109.616,104.262,130.854,99.3329,79.2881,78.989,65.7621
+113.565,95.1758,101.333,92.788,86.13,82.2812,63.3544
+120.01,109.61,96.735,83.6513,85.3256,74.0614,56.9314
+125.441,96.3427,98.5471,85.7803,83.3977,83.257,59.2639
+122.541,106.331,107.22,83.9198,84.789,72.454,73.3526
+123.169,103.753,97.6237,97.1194,88.6137,71.9554,66.3558
+117.344,106.698,105.503,86.66,82.1479,93.8159,60.2468
+112.674,97.9397,110.648,95.519,75.152,64.5602,59.9975
+118.787,106.656,95.6589,94.3315,82.7487,81.597,66.071
+112.061,102.299,98.2938,89.9138,84.791,74.8947,66.9982
+110.262,104.71,103.401,89.7127,93.4055,71.937,56.1226
+116.504,99.6516,110.527,88.7631,74.1192,82.8423,66.7656
+109.607,85.3402,96.7363,86.5084,83.2161,66.0827,65.502
+122.198,103.875,95.9747,93.2268,94.8049,79.8222,59.5304
+124.527,99.7871,93.9381,86.4354,72.2744,80.2208,59.7764
+120.886,99.823,101.823,91.7037,93.935,76.7288,62.1932
+119.176,110.554,115.05,89.9394,96.7835,75.2866,61.2462
+121.081,114.952,101.306,82.762,86.4099,75.6383,75.8608
+121.038,99.7046,102.493,81.379,87.3714,79.3073,69.8036
+112.392,96.1227,94.8579,80.2224,77.0284,69.5742,64.1075
+121.148,94.9822,108.05,79.8926,74.9314,76.4946,66.444
+117.053,109.207,101.873,95.5281,87.7317,92.7633,55.9495
+123.651,106.855,113.606,87.6941,91.005,83.3937,74.2188
+128.446,110.589,97.6295,101.649,76.2978,83.8943,61.8076
+115.527,96.4926,94.5646,85.3415,84.0268,73.5603,65.2168
+121.456,104.201,89.2764,86.2938,82.958,76.8801,57.2958
+127.418,95.364,97.445,101.265,80.859,70.703,52.0186
+113.541,95.9782,101.923,84.5136,78.3619,69.1493,66.3049
+116.805,109.152,103.308,90.5252,87.6379,79.6539,68.9979
+123.244,99.7984,92.9774,90.4494,82.0693,83.4016,64.4292
+123.095,106.249,95.1595,94.5348,75.4678,88.5801,57.3039
+115.9,91.3925,101.54,77.2599,85.7208,85.8292,60.8438
+124.024,102.095,106.698,89.5657,78.6163,72.5448,61.5872
+114.685,108.615,108.192,78.0427,97.2452,82.2857,58.9557
+113.428,100.235,101.598,91.3327,80.0625,83.8247,58.0811
+117.456,102.616,102.393,116.609,85.7072,78.3849,61.8499
+122.15,114.163,105.018,83.6798,75.0646,84.6062,58.8907
+124.576,107.106,94.0851,89.0655,82.1324,75.7662,60.5339
+113.63,94.8621,97.1255,82.225,90.3791,78.4256,66.9486
+126.608,91.0518,100.425,90.5327,99.9413,68.3999,57.7944
+129.916,112.361,100.988,78.9694,72.5502,72.1529,62.5237
+119.673,119.675,100.096,83.061,83.4268,78.0544,60.6411
+116.233,102.147,95.9289,97.7498,87.4954,89.1144,72.3431
+125.086,103.006,89.3173,87.9928,83.3203,86.8059,61.1932
+112.897,104.759,103.315,83.4737,91.7128,75.2054,59.7782
+,,,,,,58.5321
+,,,,,,60.3221
+,,,,,,62.104
+,,,,,,65.3453
+,,,,,,61.6116
+,,,,,,62.1421
+,,,,,,64.1642
+,,,,,,59.2356
+,,,,,,55.0987
+,,,,,,64.6963

dokumentation/evolution3d/errors.mms

@@ -0,0 +1,9 @@
+[1] "================ Analyzing errors.csv"
+[1] "Mean:"
+[1] NA
+[1] "Median:"
+[1] NA
+[1] "Sigma:"
+[1] NA
+[1] "Range:"
+[1] NA NA

dokumentation/evolution3d/errors.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing errors.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+      x     y
+x  1.00 -0.09
+y -0.09  1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.3979
+y 0.3979       
+[1] "spearman for improvement-potential vs. steps"
+     x    y
+x 1.00 0.11
+y 0.11 1.00
+
+n= 100 
+
+
+P
+  x     y    
+x       0.258
+y 0.258      
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.09
+y -0.09  1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.3491
+y 0.3491       
+[1] "spearman for variability vs. evolution-error"
+      x     y
+x  1.00 -0.01
+y -0.01  1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.9057
+y 0.9057       

dokumentation/evolution3d/images.sh

@@ -26,6 +26,14 @@ set xlabel 'improvement potential'
 set ylabel 'evolution error'
 set output "${png}_improvement-vs-evo-error.png"
 plot "$data" every ::1 using 3:4 title "data", h(x) title "lin. fit" lc rgb "black"
+i(x)=aaaa*x+bbbb
+fit i(x) "$data" every ::1 using 2:4 via aaaa,bbbb
+set xlabel 'variability'
+set ylabel 'evolution error'
+set output "${png}_variability-vs-evo-error.png"
+plot \
+    "$data" every ::1 using 2:4 title "data", \
+    i(x) title "lin. fit" lc rgb "black"
 EOD
 ) > "${png}.gnuplot.script" 
     gnuplot "${png}.gnuplot.script" 2> "${png}.gnuplot.log"