nearly done

arbeit/bibma.bib

@@ -236,3 +236,21 @@
   year={2016},
   url={https://arxiv.org/abs/1604.00772}
 }
+@article{eiben1999parameter,
+  title={Parameter control in evolutionary algorithms},
+  author={Eiben, {\'A}goston E and Hinterding, Robert and Michalewicz, Zbigniew},
+  journal={IEEE Transactions on evolutionary computation},
+  volume={3},
+  number={2},
+  pages={124--141},
+  year={1999},
+  publisher={IEEE},
+  url={https://www.researchgate.net/profile/Marc_Schoenauer/publication/223460374_Parameter_Control_in_Evolutionary_Algorithms/links/545766440cf26d5090a9b951.pdf},
+}
+@article{rechenberg1973evolutionsstrategie,
+  title={Evolutionsstrategie Optimierung technischer Systeme nach Prinzipien der biologischen Evolution},
+  author={Rechenberg, Ingo},
+  year={1973},
+  publisher={Frommann-Holzboog}
+}
+

arbeit/ma.md

@@ -1,5 +1,5 @@
 ---
-fontsize: 12pt
+fontsize: 11pt
 ---
 
 \chapter*{How to read this Thesis}
@@ -23,8 +23,6 @@ Unless otherwise noted the following holds:
 
 # Introduction
 
-\improvement[inline]{Mehr Bilder}
-
 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
@@ -35,6 +33,13 @@ layouting of circuit boards or stacking of 3D--objects). Moreover these are
 typically not static environments but requirements shift over time or from case
 to case.
 
+\begin{figure}[hbt]
+\centering
+\includegraphics[width=\textwidth]{img/Evo_overview.png}
+\caption{Example of the use of evolutionary algorithms in automotive design
+(from \cite{anrichterEvol}).}
+\end{figure}
+
 Evolutionary algorithms cope especially well with these problem domains while
 addressing all the issues at hand\cite{minai2006complex}. One of the main
 concerns in these algorithms is the formulation of the problems in terms of a
@@ -65,6 +70,12 @@ varies from context to context\cite{richter2015evolvability}. As a consequence
 there is need for some criteria we can measure, so that we are able to compare different
 representations to learn and improve upon these.
 
+\begin{figure}[hbt]
+\centering
+\includegraphics[width=\textwidth]{img/deformations.png}
+\caption{Example of RBF--based deformation and FFD targeting the same mesh.}
+\end{figure}
+
 One example of such a general representation of an object is to generate random
 points and represent vertices of an object as distances to these points --- for
 example via \acf{RBF}. If one (or the algorithm) would move such a point the
@@ -96,7 +107,7 @@ take an abstract look at the definition of \ac{FFD} for a one--dimensional line
 Then we establish some background--knowledge of evolutionary algorithms (in
 \ref{sec:back:evo}) and why this is useful in our domain (in
 \ref{sec:back:evogood}) followed by the definition of the different evolvability
-criteria established in \cite{anrichterEvol} (in \ref {sec:back:rvi}).
+criteria established in \cite{anrichterEvol} (in \ref {sec:intro:rvi}).
 
 In Chapter \ref{sec:impl} we take a look at our implementation of \ac{FFD} and
 the adaptation for 3D--meshes that were used. Next, in Chapter \ref{sec:eval},
@@ -319,13 +330,20 @@ The main algorithm just repeats the following steps:
   of $\mu$ individuals.
 
 All these functions can (and mostly do) have a lot of hidden parameters that
-can be changed over time.
+can be changed over time. A good overview of this is given in
+\cite{eiben1999parameter}, so we only give a small excerpt here.
 
-\improvement[inline]{Genauer: Welche? Wo? Wieso? ...}
+For example the mutation can consist of merely a single $\sigma$ determining the
+strength of the gaussian defects in every parameter --- or giving a different
+$\sigma$ to every part. An even more sophisticated example would be the \glqq 1/5
+success rule\grqq \ from \cite{rechenberg1973evolutionsstrategie}.
 
-<!--One can for example start off with a high
-mutation rate that cools off over time (i.e. by lowering the variance of a
-gaussian noise).-->
+Also in selection it may not be wise to only take the best--performing
+individuals, because it may be that the optimization has to overcome a barrier
+of bad fitness to achieve a better local optimum.
+
+Recombination also does not have to be mere random choosing of parents, but can
+also take ancestry, distance of genes or grouping into account.
 
 ## Advantages of evolutionary algorithms
 \label{sec:back:evogood}
@@ -346,8 +364,8 @@ are shown in figure \ref{fig:probhard}.
 Most of the advantages stem from the fact that a gradient--based procedure has
 only one point of observation from where it evaluates the next steps, whereas an
 evolutionary strategy starts with a population of guessed solutions. Because an
-evolutionary strategy modifies the solution randomly, keeping the best solutions
-and purging the worst, it can also target multiple different hypothesis at the
+evolutionary strategy modifies the solution randomly, keeping some solutions
+and purging others, it can also target multiple different hypothesis at the
 same time where the local optima die out in the face of other, better
 candidates.
 
@@ -371,16 +389,18 @@ converge to the same solution.
 As we have established in chapter \ref{sec:back:ffd}, we can describe a
 deformation by the formula
 $$
-\vec{V} = \vec{U}\vec{P}
+\vec{S} = \vec{U}\vec{P}
 $$
-where $\vec{V}$ is a $n \times d$ matrix of vertices, $\vec{U}$ are the (during
+where $\vec{S}$ is a $n \times d$ matrix of vertices^[We use $\vec{S}$ in this
+notation, as we will use this parametrization of a source--mesh to manipulate
+$\vec{S}$ into a target--mesh $\vec{T}$ via $\vec{P}$], $\vec{U}$ are the (during
 parametrization) calculated deformation--coefficients and $P$ is a $m \times d$ matrix
 of control--points that we interact with during deformation.
 
 We can also think of the deformation in terms of differences from the original
 coordinates
 $$
-\Delta \vec{V} = \vec{U} \cdot \Delta \vec{P}
+\Delta \vec{S} = \vec{U} \cdot \Delta \vec{P}
 $$
 which is isomorphic to the former due to the linear correlation in the
 deformation. One can see in this way, that the way the deformation behaves lies
@@ -443,7 +463,6 @@ The definition for an *improvement potential* $P$ is\cite{anrichterEvol}:
 $$
 \mathrm{potential}(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F
 $$
-\unsure[inline]{ist das $^2$ richtig?}
 given some approximate $n \times d$ fitness--gradient $\vec{G}$, normalized to
 $\|\vec{G}\|_F = 1$, whereby $\|\cdot\|_F$ denotes the Frobenius--Norm.
 
@@ -499,10 +518,9 @@ $$
 $$
 and do a gradient--descend to approximate the value of $u$ up to an $\epsilon$ of $0.0001$.
 
-For this we use the Gauss--Newton algorithm\cite{gaussNewton}
-\todo[inline]{rewrite. falsch und wischi-waschi. Least squares?}
-as the solution to
-this problem may not be deterministic, because we usually have way more vertices
+For this we employ the Gauss--Newton algorithm\cite{gaussNewton}, which
+converges into the least--squares solution. An exact solution of this problem is
+impossible most of the times, because we usually have way more vertices
 than control points ($\#v~\gg~\#c$).
 
 ## Adaption of \ac{FFD} for a 3D--Mesh
@@ -754,10 +772,9 @@ 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}
-\mathrm{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)
+\mathrm{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{s}_i\|^2 \right)
 \label{eq:reg3d}
 \end{equation}
-\unsure[inline]{was ist $\vec{\overline{s}_j}$? Zentrum? eigentlich $s_i$?}
 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
@@ -765,8 +782,10 @@ 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?}
+material that will get more flexible per iteration. As a side--effect this also
+limits the effects of overagressive movement of the control--points in the
+beginning of the fitting process and thus should limit the generation of
+ill--defined grids mentioned in section \ref{sec:impl:grid}.
 
 # Evaluation of Scenarios
 \label{sec:res}
@@ -812,14 +831,15 @@ For our setup we first compute the coefficients of the deformation--matrix and
 use then the formulas for *variability* and *regularity* to get our predictions.
 Afterwards we solve the problem analytically to get the (normalized) correct
 gradient that we use as guess for the *improvement potential*. To check we also
-consider a distorted gradient $\vec{g}_{\textrm{d}}$
+consider a distorted gradient $\vec{g}_{\mathrm{d}}$
 $$
-\vec{g}_{\textrm{d}} = \frac{\vec{g}_{\textrm{c}} + \mathbb{1}}{\|\vec{g}_{\textrm{c}} + \mathbb{1}\|}
+\vec{g}_{\mathrm{d}} = \frac{\mu \vec{g}_{\mathrm{c}} + (1-\mu)\mathbb{1}}{\|\mu \vec{g}_{\mathrm{c}} + (1-\mu) \mathbb{1}\|}
 $$
-where $\mathbb{1}$ is the vector consisting of $1$ in every dimension and
-$\vec{g}_\textrm{c} = \vec{p^{*}} - \vec{p}$ the calculated correct gradient. As
-we always start with a gradient of $\mathbb{0}$ this shortens to
-$\vec{g}_\textrm{c} = \vec{p^{*}}$.
+where $\mathbb{1}$ is the vector consisting of $1$ in every dimension,
+$\vec{g}_\mathrm{c} = \vec{p^{*}} - \vec{p}$ is the calculated correct gradient,
+and $\mu$ is used to blend between $\vec{g}_\mathrm{c}$ and $\mathbb{1}$. As
+we always start with a gradient of $p = \mathbb{0}$ this means shortens
+$\vec{g}_\mathrm{c} = \vec{p^{*}}$.
 
 \begin{figure}[ht]
 \begin{center}
@@ -836,12 +856,21 @@ randomly inside the x--y--plane. As self-intersecting grids get tricky to solve
 with our implemented newtons--method we avoid the generation of such
 self--intersecting grids for our testcases (see section \ref{3dffd}).
 
-To achieve that we select a uniform distributed number $r \in [-0.25,0.25]$ per
-dimension and shrink the distance to the neighbours (the smaller neighbour for
-$r < 0$, the larger for $r > 0$) by the factor $r$^[Note: On the Edges this
-displacement is only applied outwards by flipping the sign of $r$, if
-appropriate.].
-\improvement[inline]{update!! gaussian, not uniform!!}
+To achieve that we generated a gaussian distributed number with $\mu = 0, \sigma=0.25$
+and clamped it to the range $[-0.25,0.25]$. We chose such an $r \in [-0.25,0.25]$
+per dimension and moved the control--points by that factor towards their
+respective neighbours^[Note: On the Edges this displacement is only applied
+outwards by flipping the sign of $r$, if appropriate.]. 
+
+In other words we set
+\begin{equation*}
+p_i =
+\begin{cases}
+  p_i + (p_i - p_{i-1}) \cdot r, & \textrm{if } r \textrm{ negative} \\
+  p_i + (p_{i+1} - p_i) \cdot r, & \textrm{if } r \textrm{ positive}
+\end{cases}
+\end{equation*}
+in each dimension separately.
 
 An Example of such a testcase can be seen for a $7 \times 4$--grid in figure
 \ref{fig:example1d_grid}.
@@ -957,12 +986,22 @@ grid--resolutions}
 \label{fig:1dimp}
 \end{figure}
 
-\improvement[inline]{write something about it..}
+The improvement potential should correlate to the quality of the
+fitting--result. We plotted the results for the tested grid-sizes $5 \times 5$,
+$7 \times 7$ and $10 \times 10$ in figure \ref{fig:1dimp}. We tested the
+$4 \times 7$ and $7 \times 4$ grids as well, but omitted them from the plot.
 
-- spearman 1 (p=0)
-- gradient macht keinen unterschied
-- $UU^+$ scheint sehr kleine EW zu haben, s. regularität
-- trotzdem sehr gutes kriterium - auch ohne Richtung.
+Additionally we tested the results for a distorted gradient described in 
+\ref{sec:proc:1d} with a $\mu$--value of $0.25$, $0.5$, $0,75$, and $1.0$ for
+the $5 \times 5$ grid and with a $\mu$--value of $0.5$ for all other cases.
+
+All results show the identical *very strong* and *significant* correlation with
+a Spearman--coefficient of $- r_S = 1.0$ and p--value of $0$.
+
+These results indicate, that $\|\mathbb{1} - \vec{U}\vec{U}^{+}\|_F$ is close to $0$,
+reducing the impacts of any kind of gradient. Nevertheless, the improvement
+potential seems to be suited to make estimated guesses about the quality of a
+fit, even lacking an exact gradient.
 
 ## Procedure: 3D Function Approximation
 \label{sec:proc:3dfa}
@@ -1010,9 +1049,10 @@ 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 \improvement{Beschreiben 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.].
+regular grid and move the control-points in the exact same random manner between
+their neighbours as described in section \ref{sec:proc:1d}, 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}
@@ -1244,7 +1284,14 @@ in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,
 \label{tab:3dimp}
 \end{table}
 
-\begin{figure}[!htb]
+Comparing to the 1D--scenario, we do not know the optimal solution to the given
+problem and for the calculation we only use the initial gradient produced by the
+initial correlation between both objects. This gradient changes with every
+iteration and will be off our first guess very quickly. This is the reason we
+are not trying to create artificially bad gradients, as we have a broad range in
+quality of such gradients anyway.
+
+\begin{figure}[htb]
 \centering
 \includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
 \caption[Improvement potential for different 3D--grids]{
@@ -1254,11 +1301,76 @@ indicate trends.}
 \label{fig:resimp3d}
 \end{figure}
 
+We plotted our findings on the improvement potential in a similar way as we did
+before with the regularity. In figure \ref{fig:resimp3d} one can clearly see the
+correlation and the spread within each setup and the behaviour when we increase
+the number of control--points.
 
-# Schluss
+Along with this we also give the spearman--coefficients along with their
+p--values in table \ref{tab:3dimp}. Within one scenario we only find a *weak* to
+*moderate* correlation between the improvement potential and the fitting error,
+but all findings (except for $7 \times 4 \times 4$ and $6 \times 6 \times 6$)
+are significant.
+
+If we take multiple datasets into account the correlation is *very strong* and 
+*significant*, which is good, as this functions as a litmus--test, because the
+quality is naturally tied to the number of control--points.
+
+All in all the improvement potential seems to be a good and sensible measure of
+quality, even given gradients of varying quality.
+
+\improvement[inline]{improvement--potential vs. steps ist anders als in 1d! Plot
+und zeigen!}
+
+
+# Discussion and outlook
 \label{sec:dis}
 
-- Regularity ist kacke für unser setup. Bessere Vorschläge? EW/EV?
+In this thesis we took a look at the different criteria for evolvability as
+introduced by Richter et al.\cite{anrichterEvol}, namely *variability*,
+*regularity* and *improvement potential* under different setup--conditions.
+Where Richter et al. used \acf{RBF}, we employed \acf{FFD} to set up a
+low--complexity parametrization of a more complex vertex--mesh.
+
+In our findings we could show in the 1D--scenario, that there were statistically
+significant very strong correlations between *variability and fitting error*
+($0.94$) and *improvement--potential and fitting error* ($1.0$) with
+comparable results than Richter et al. (with $0.31$ to $0.88$
+for the former and $0.75$ to $0.99$ for the latter), whereas we found
+only weak correlations for *regularity and convergence--speed* ($0.28$)
+opposed to Richter et al. with $0.39$ to $0.91$.^[We only took statistically
+*significant* results into consideration when compiling these numbers. Details
+are given in the respective chapters.]
+
+For the 3D--scenario our results show a very strong, significant  correlation
+between *variability and fitting error* with $0.89$ to $0.94$, which are pretty
+much in line with the findings of Richter et al. ($0.65$ to $0.95$). The
+correlation between *improvement potential and fitting error* behave similar,
+with our findings having a significant coefficient of $0.3$ to $0.95$ depending
+on the grid--resolution compared to the $0.61$ to $0.93$ from Richter et al.
+In the case of the correlation of *regularity and convergence speed* we found
+very different (and often not significant) correlations and anti--correlations
+ranging from $-0.25$ to $0.46$, whereas Richter et al. reported correlations
+between $0.34$ to $0.87$.
+
+Taking these results into consideration, one can say, that *variability* and
+*improvement potential* are very good estimates for the quality of a fit using
+\acf{FFD} as a deformation function.
+
+One reason for the bad or erratic behaviour of the *regularity*--criterion could
+be that in an \ac{FFD}--setting we have a likelihood of having control--points
+that are only contributing to the whole parametrization in negligible amounts.
+This results in very small right singular values of the deformation--matrix
+$\vec{U}$ that influence the condition--number and thus the *regularity* in a
+significant way. Further research is needed to refine *regularity* so that these
+problems get addressed.
+
+Richter et al. also compared the behaviour of direct and indirect manipulation
+in \cite{anrichterEvol}, whereas we merely used an indirect \ac{FFD}--approach.
+As direct manipulations tend to perform better than indirect manipulations, the
+usage of \acf{DM--FFD} could also work better with the criteria we examined.
+
+\improvement[inline]{write more outlook/further research}
 
 \improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
 Direktlinks des Autors.\newline

arbeit/ma.tex

@@ -2,8 +2,9 @@
 % abstracton : Abstract mit Ueberschrift
 \documentclass[
 a4paper,				% default
-12pt, 					% default = 11pt
-BCOR10mm,				% Bindungskorrektur bei Klebebindung 6mm, bei Lochen BCOR8.25mm
+11pt, 					% default = 11pt
+DIV=calc,
+BCOR6mm,				% Bindungskorrektur bei Klebebindung 6mm, bei Lochen BCOR8.25mm
 twoside,				% default, 2seitig
 titlepage,
 % pagesize=auto
@@ -168,8 +169,6 @@ Unless otherwise noted the following holds:
 
 \chapter{Introduction}\label{introduction}
 
-\improvement[inline]{Mehr Bilder}
-
 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.
@@ -181,6 +180,13 @@ circuit boards or stacking of 3D--objects). Moreover these are typically
 not static environments but requirements shift over time or from case to
 case.
 
+\begin{figure}[hbt]
+\centering
+\includegraphics[width=\textwidth]{img/Evo_overview.png}
+\caption{Example of the use of evolutionary algorithms in automotive design
+(from \cite{anrichterEvol}).}
+\end{figure}
+
 Evolutionary algorithms cope especially well with these problem domains
 while addressing all the issues at hand\cite{minai2006complex}. One of
 the main concerns in these algorithms is the formulation of the problems
@@ -214,6 +220,12 @@ from context to context\cite{richter2015evolvability}. As a consequence
 there is need for some criteria we can measure, so that we are able to
 compare different representations to learn and improve upon these.
 
+\begin{figure}[hbt]
+\centering
+\includegraphics[width=\textwidth]{img/deformations.png}
+\caption{Example of RBF--based deformation and FFD targeting the same mesh.}
+\end{figure}
+
 One example of such a general representation of an object is to generate
 random points and represent vertices of an object as distances to these
 points --- for example via \acf{RBF}. If one (or the algorithm) would
@@ -247,7 +259,7 @@ establish some background--knowledge of evolutionary algorithms (in
 \ref{sec:back:evo}) and why this is useful in our domain (in
 \ref{sec:back:evogood}) followed by the definition of the different
 evolvability criteria established in \cite{anrichterEvol} (in
-\ref {sec:back:rvi}).
+\ref {sec:intro:rvi}).
 
 In Chapter \ref{sec:impl} we take a look at our implementation of
 \ac{FFD} and the adaptation for 3D--meshes that were used. Next, in
@@ -486,9 +498,21 @@ The main algorithm just repeats the following steps:
 \end{itemize}
 
 All these functions can (and mostly do) have a lot of hidden parameters
-that can be changed over time.
+that can be changed over time. A good overview of this is given in
+\cite{eiben1999parameter}, so we only give a small excerpt here.
 
-\improvement[inline]{Genauer: Welche? Wo? Wieso? ...}
+For example the mutation can consist of merely a single \(\sigma\)
+determining the strength of the gaussian defects in every parameter ---
+or giving a different \(\sigma\) to every part. An even more
+sophisticated example would be the \glqq 1/5 success rule\grqq ~from
+\cite{rechenberg1973evolutionsstrategie}.
+
+Also in selection it may not be wise to only take the best--performing
+individuals, because it may be that the optimization has to overcome a
+barrier of bad fitness to achieve a better local optimum.
+
+Recombination also does not have to be mere random choosing of parents,
+but can also take ancestry, distance of genes or grouping into account.
 
 \section{Advantages of evolutionary
 algorithms}\label{advantages-of-evolutionary-algorithms}
@@ -512,8 +536,8 @@ Most of the advantages stem from the fact that a gradient--based
 procedure has only one point of observation from where it evaluates the
 next steps, whereas an evolutionary strategy starts with a population of
 guessed solutions. Because an evolutionary strategy modifies the
-solution randomly, keeping the best solutions and purging the worst, it
-can also target multiple different hypothesis at the same time where the
+solution randomly, keeping some solutions and purging others, it can
+also target multiple different hypothesis at the same time where the
 local optima die out in the face of other, better candidates.
 
 \improvement[inline]{Verweis auf MO-CMA etc. Vielleicht auch etwas
@@ -539,15 +563,18 @@ deformations}\label{criteria-for-the-evolvability-of-linear-deformations}
 
 As we have established in chapter \ref{sec:back:ffd}, we can describe a
 deformation by the formula \[
-\vec{V} = \vec{U}\vec{P}
-\] where \(\vec{V}\) is a \(n \times d\) matrix of vertices, \(\vec{U}\)
-are the (during parametrization) calculated deformation--coefficients
-and \(P\) is a \(m \times d\) matrix of control--points that we interact
-with during deformation.
+\vec{S} = \vec{U}\vec{P}
+\] where \(\vec{S}\) is a \(n \times d\) matrix of vertices\footnote{We
+  use \(\vec{S}\) in this notation, as we will use this parametrization
+  of a source--mesh to manipulate \(\vec{S}\) into a target--mesh
+  \(\vec{T}\) via \(\vec{P}\)}, \(\vec{U}\) are the (during
+parametrization) calculated deformation--coefficients and \(P\) is a
+\(m \times d\) matrix of control--points that we interact with during
+deformation.
 
 We can also think of the deformation in terms of differences from the
 original coordinates \[
-\Delta \vec{V} = \vec{U} \cdot \Delta \vec{P}
+\Delta \vec{S} = \vec{U} \cdot \Delta \vec{P}
 \] which is isomorphic to the former due to the linear correlation in
 the deformation. One can see in this way, that the way the deformation
 behaves lies solely in the entries of \(\vec{U}\), which is why the
@@ -614,9 +641,8 @@ in the given direction.
 The definition for an \emph{improvement potential} \(P\)
 is\cite{anrichterEvol}: \[
 \mathrm{potential}(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F
-\] \unsure[inline]{ist das $^2$ richtig?} given some approximate
-\(n \times d\) fitness--gradient \(\vec{G}\), normalized to
-\(\|\vec{G}\|_F = 1\), whereby \(\|\cdot\|_F\) denotes the
+\] given some approximate \(n \times d\) fitness--gradient \(\vec{G}\),
+normalized to \(\|\vec{G}\|_F = 1\), whereby \(\|\cdot\|_F\) denotes the
 Frobenius--Norm.
 
 \chapter{\texorpdfstring{Implementation of
@@ -670,10 +696,10 @@ v_x \overset{!}{=} \sum_i N_{i,d,\tau_i}(u) c_i
 \] and do a gradient--descend to approximate the value of \(u\) up to an
 \(\epsilon\) of \(0.0001\).
 
-For this we use the Gauss--Newton algorithm\cite{gaussNewton}
-\todo[inline]{rewrite. falsch und wischi-waschi. Least squares?} as the
-solution to this problem may not be deterministic, because we usually
-have way more vertices than control points (\(\#v~\gg~\#c\)).
+For this we employ the Gauss--Newton algorithm\cite{gaussNewton}, which
+converges into the least--squares solution. An exact solution of this
+problem is impossible most of the times, because we usually have way
+more vertices than control points (\(\#v~\gg~\#c\)).
 
 \section{\texorpdfstring{Adaption of \ac{FFD} for a
 3D--Mesh}{Adaption of  for a 3D--Mesh}}\label{adaption-of-for-a-3dmesh}
@@ -958,11 +984,10 @@ al.\cite[Section 3.2]{aschenbach2015} on similar models and was shown to
 lead to a more precise fit. The Laplacian
 
 \begin{equation}
-\mathrm{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)
+\mathrm{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{s}_i\|^2 \right)
 \label{eq:reg3d}
 \end{equation}
 
-\unsure[inline]{was ist $\vec{\overline{s}_j}$? Zentrum? eigentlich $s_i$?}
 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
@@ -972,8 +997,10 @@ 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?}
+iteration. As a side--effect this also limits the effects of
+overagressive movement of the control--points in the beginning of the
+fitting process and thus should limit the generation of ill--defined
+grids mentioned in section \ref{sec:impl:grid}.
 
 \chapter{Evaluation of Scenarios}\label{evaluation-of-scenarios}
 
@@ -1025,12 +1052,14 @@ deformation--matrix and use then the formulas for \emph{variability} and
 \emph{regularity} to get our predictions. Afterwards we solve the
 problem analytically to get the (normalized) correct gradient that we
 use as guess for the \emph{improvement potential}. To check we also
-consider a distorted gradient \(\vec{g}_{\textrm{d}}\) \[
-\vec{g}_{\textrm{d}} = \frac{\vec{g}_{\textrm{c}} + \mathbb{1}}{\|\vec{g}_{\textrm{c}} + \mathbb{1}\|}
+consider a distorted gradient \(\vec{g}_{\mathrm{d}}\) \[
+\vec{g}_{\mathrm{d}} = \frac{\mu \vec{g}_{\mathrm{c}} + (1-\mu)\mathbb{1}}{\|\mu \vec{g}_{\mathrm{c}} + (1-\mu) \mathbb{1}\|}
 \] where \(\mathbb{1}\) is the vector consisting of \(1\) in every
-dimension and \(\vec{g}_\textrm{c} = \vec{p^{*}} - \vec{p}\) the
-calculated correct gradient. As we always start with a gradient of
-\(\mathbb{0}\) this shortens to \(\vec{g}_\textrm{c} = \vec{p^{*}}\).
+dimension, \(\vec{g}_\mathrm{c} = \vec{p^{*}} - \vec{p}\) is the
+calculated correct gradient, and \(\mu\) is used to blend between
+\(\vec{g}_\mathrm{c}\) and \(\mathbb{1}\). As we always start with a
+gradient of \(p = \mathbb{0}\) this means shortens
+\(\vec{g}_\mathrm{c} = \vec{p^{*}}\).
 
 \begin{figure}[ht]
 \begin{center}
@@ -1048,13 +1077,24 @@ grids get tricky to solve with our implemented newtons--method we avoid
 the generation of such self--intersecting grids for our testcases (see
 section \ref{3dffd}).
 
-To achieve that we select a uniform distributed number
-\(r \in [-0.25,0.25]\) per dimension and shrink the distance to the
-neighbours (the smaller neighbour for \(r < 0\), the larger for
-\(r > 0\)) by the factor \(r\)\footnote{Note: On the Edges this
-  displacement is only applied outwards by flipping the sign of \(r\),
-  if appropriate.}.
-\improvement[inline]{update!! gaussian, not uniform!!}
+To achieve that we generated a gaussian distributed number with
+\(\mu = 0, \sigma=0.25\) and clamped it to the range \([-0.25,0.25]\).
+We chose such an \(r \in [-0.25,0.25]\) per dimension and moved the
+control--points by that factor towards their respective
+neighbours\footnote{Note: On the Edges this displacement is only applied
+  outwards by flipping the sign of \(r\), if appropriate.}.
+
+In other words we set
+
+\begin{equation*}
+p_i =
+\begin{cases}
+  p_i + (p_i - p_{i-1}) \cdot r, & \textrm{if } r \textrm{ negative} \\
+  p_i + (p_{i+1} - p_i) \cdot r, & \textrm{if } r \textrm{ positive}
+\end{cases}
+\end{equation*}
+
+in each dimension separately.
 
 An Example of such a testcase can be seen for a \(7 \times 4\)--grid in
 figure \ref{fig:example1d_grid}.
@@ -1177,19 +1217,26 @@ grid--resolutions}
 \label{fig:1dimp}
 \end{figure}
 
-\improvement[inline]{write something about it..}
+The improvement potential should correlate to the quality of the
+fitting--result. We plotted the results for the tested grid-sizes
+\(5 \times 5\), \(7 \times 7\) and \(10 \times 10\) in figure
+\ref{fig:1dimp}. We tested the \(4 \times 7\) and \(7 \times 4\) grids
+as well, but omitted them from the plot.
 
-\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}
+Additionally we tested the results for a distorted gradient described in
+\ref{sec:proc:1d} with a \(\mu\)--value of \(0.25\), \(0.5\), \(0,75\),
+and \(1.0\) for the \(5 \times 5\) grid and with a \(\mu\)--value of
+\(0.5\) for all other cases.
+
+All results show the identical \emph{very strong} and \emph{significant}
+correlation with a Spearman--coefficient of \(- r_S = 1.0\) and p--value
+of \(0\).
+
+These results indicate, that \(\|\mathbb{1} - \vec{U}\vec{U}^{+}\|_F\)
+is close to \(0\), reducing the impacts of any kind of gradient.
+Nevertheless, the improvement potential seems to be suited to make
+estimated guesses about the quality of a fit, even lacking an exact
+gradient.
 
 \section{Procedure: 3D Function
 Approximation}\label{procedure-3d-function-approximation}
@@ -1244,10 +1291,11 @@ 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
-\improvement{Beschreiben 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.}.
+before, we create a regular grid and move the control-points in the
+exact same random manner between their neighbours as described in
+section \ref{sec:proc:1d}, 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}
@@ -1491,7 +1539,15 @@ in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,
 \label{tab:3dimp}
 \end{table}
 
-\begin{figure}[!htb]
+Comparing to the 1D--scenario, we do not know the optimal solution to
+the given problem and for the calculation we only use the initial
+gradient produced by the initial correlation between both objects. This
+gradient changes with every iteration and will be off our first guess
+very quickly. This is the reason we are not trying to create
+artificially bad gradients, as we have a broad range in quality of such
+gradients anyway.
+
+\begin{figure}[htb]
 \centering
 \includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
 \caption[Improvement potential for different 3D--grids]{
@@ -1501,15 +1557,84 @@ indicate trends.}
 \label{fig:resimp3d}
 \end{figure}
 
-\chapter{Schluss}\label{schluss}
+We plotted our findings on the improvement potential in a similar way as
+we did before with the regularity. In figure \ref{fig:resimp3d} one can
+clearly see the correlation and the spread within each setup and the
+behaviour when we increase the number of control--points.
+
+Along with this we also give the spearman--coefficients along with their
+p--values in table \ref{tab:3dimp}. Within one scenario we only find a
+\emph{weak} to \emph{moderate} correlation between the improvement
+potential and the fitting error, but all findings (except for
+\(7 \times 4 \times 4\) and \(6 \times 6 \times 6\)) are significant.
+
+If we take multiple datasets into account the correlation is \emph{very
+strong} and \emph{significant}, which is good, as this functions as a
+litmus--test, because the quality is naturally tied to the number of
+control--points.
+
+All in all the improvement potential seems to be a good and sensible
+measure of quality, even given gradients of varying quality.
+
+\improvement[inline]{improvement--potential vs. steps ist anders als in 1d! Plot
+und zeigen!}
+
+\chapter{Discussion and outlook}\label{discussion-and-outlook}
 
 \label{sec:dis}
 
-\begin{itemize}
-\tightlist
-\item
-  Regularity ist kacke für unser setup. Bessere Vorschläge? EW/EV?
-\end{itemize}
+In this thesis we took a look at the different criteria for evolvability
+as introduced by Richter et al.\cite{anrichterEvol}, namely
+\emph{variability}, \emph{regularity} and \emph{improvement potential}
+under different setup--conditions. Where Richter et al. used \acf{RBF},
+we employed \acf{FFD} to set up a low--complexity parametrization of a
+more complex vertex--mesh.
+
+In our findings we could show in the 1D--scenario, that there were
+statistically significant very strong correlations between
+\emph{variability and fitting error} (\(0.94\)) and
+\emph{improvement--potential and fitting error} (\(1.0\)) with
+comparable results than Richter et al. (with \(0.31\) to \(0.88\) for
+the former and \(0.75\) to \(0.99\) for the latter), whereas we found
+only weak correlations for \emph{regularity and convergence--speed}
+(\(0.28\)) opposed to Richter et al. with \(0.39\) to
+\(0.91\).\footnote{We only took statistically \emph{significant} results
+  into consideration when compiling these numbers. Details are given in
+  the respective chapters.}
+
+For the 3D--scenario our results show a very strong, significant
+correlation between \emph{variability and fitting error} with \(0.89\)
+to \(0.94\), which are pretty much in line with the findings of Richter
+et al. (\(0.65\) to \(0.95\)). The correlation between \emph{improvement
+potential and fitting error} behave similar, with our findings having a
+significant coefficient of \(0.3\) to \(0.95\) depending on the
+grid--resolution compared to the \(0.61\) to \(0.93\) from Richter et
+al. In the case of the correlation of \emph{regularity and convergence
+speed} we found very different (and often not significant) correlations
+and anti--correlations ranging from \(-0.25\) to \(0.46\), whereas
+Richter et al. reported correlations between \(0.34\) to \(0.87\).
+
+Taking these results into consideration, one can say, that
+\emph{variability} and \emph{improvement potential} are very good
+estimates for the quality of a fit using \acf{FFD} as a deformation
+function.
+
+One reason for the bad or erratic behaviour of the
+\emph{regularity}--criterion could be that in an \ac{FFD}--setting we
+have a likelihood of having control--points that are only contributing
+to the whole parametrization in negligible amounts. This results in very
+small right singular values of the deformation--matrix \(\vec{U}\) that
+influence the condition--number and thus the \emph{regularity} in a
+significant way. Further research is needed to refine \emph{regularity}
+so that these problems get addressed.
+
+Richter et al. also compared the behaviour of direct and indirect
+manipulation in \cite{anrichterEvol}, whereas we merely used an indirect
+\ac{FFD}--approach. As direct manipulations tend to perform better than
+indirect manipulations, the usage of \acf{DM--FFD} could also work
+better with the criteria we examined.
+
+\improvement[inline]{write more outlook/further research}
 
 \improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
 Direktlinks des Autors.\newline

arbeit/template.tex

@@ -3,7 +3,8 @@
 \documentclass[
 a4paper,				% default
 $if(fontsize)$$fontsize$,$endif$ 					% default = 11pt
-BCOR10mm,				% Bindungskorrektur bei Klebebindung 6mm, bei Lochen BCOR8.25mm
+DIV=calc,
+BCOR6mm,				% Bindungskorrektur bei Klebebindung 6mm, bei Lochen BCOR8.25mm
 twoside,				% default, 2seitig
 titlepage,
 % pagesize=auto

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+200.375,0.0256024,0.00111111,0.933085,0.977449,0.953322,0.994814,163,209.762,0.0341017
+176.392,0.02493,0.00111111,0.941094,0.980148,0.958909,0.995434,246,185.182,0.024624
+215.099,0.0251406,0.00111111,0.928167,0.975792,0.949892,0.994432,182,225.686,0.0172172
+196.048,0.0244025,0.00111111,0.934529,0.977936,0.95433,0.994926,171,205.79,0.0216345
+192.129,0.0250595,0.00111111,0.935842,0.978378,0.955245,0.995027,163,201.653,0.0246749
+189.835,0.025235,0.00111111,0.936682,0.978661,0.955831,0.995092,113,198.984,0.0430281
+205.107,0.0256549,0.00111111,0.931504,0.976916,0.952219,0.994691,165,214.844,0.0304616
+193.362,0.0281933,0.00111111,0.935426,0.978238,0.954955,0.994995,133,202.594,0.0230947
+176.783,0.0253994,0.00111111,0.940963,0.980104,0.958818,0.995424,165,185.619,0.0298644
+201.911,0.0267396,0.00111111,0.932571,0.977276,0.952964,0.994774,247,211.904,0.0184367
+187.159,0.0264083,0.00111111,0.937498,0.978936,0.9564,0.995156,185,195.98,0.0301479
+184.049,0.0259232,0.00111111,0.938551,0.979291,0.957135,0.995237,232,192.979,0.0173422
+204.792,0.0254979,0.00111111,0.93161,0.976952,0.952293,0.994699,167,214.577,0.0239699
+199.555,0.0258068,0.00111111,0.933358,0.977541,0.953513,0.994835,157,209.134,0.0261165
+190.76,0.0261192,0.00111111,0.936414,0.978571,0.955644,0.995072,227,200.294,0.023586
+186.16,0.0270901,0.00111111,0.937832,0.979049,0.956633,0.995181,154,194.941,0.0553742
+191.062,0.0244287,0.00111111,0.936195,0.978497,0.955491,0.995055,206,199.917,0.0297287
+195.72,0.0264288,0.00111111,0.934639,0.977973,0.954406,0.994934,181,205.385,0.0241264
+194.606,0.0260995,0.00111111,0.935048,0.978111,0.954691,0.994966,183,204.299,0.0164021
+200.328,0.0273905,0.00111111,0.9331,0.977454,0.953333,0.994815,195,210.307,0.0270821
+194.583,0.0264801,0.00111111,0.935019,0.978101,0.954671,0.994963,167,203.889,0.0308113
+183.311,0.025923,0.00111111,0.938783,0.97937,0.957297,0.995255,167,192.409,0.0247795

dokumentation/evolution1d/adv-lamb.gnuplot.fit.log

@@ -0,0 +1,184 @@
+
+
+*******************************************************************************
+Fri Oct 27 21:50:01 2017
+
+
+FIT:    data read from "adv-lamb.csv" every ::1 using 2:5
+        format = x:z
+        #datapoints = 100
+        residuals are weighted equally (unit weight)
+
+function used for fitting: f(x)
+fitted parameters initialized with current variable values
+
+
+
+ Iteration 0
+ WSSR        : 0.227572          delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.707341
+
+initial set of free parameter values
+
+a               = 1
+b               = 1
+
+After 5 iterations the fit converged.
+final sum of squares of residuals : 0.000107016
+rel. change during last iteration : -2.47553e-06
+
+degrees of freedom    (FIT_NDF)                        : 98
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.00104499
+variance of residuals (reduced chisquare) = WSSR/ndf   : 1.092e-06
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+a               = -0.00321702      +/- 0.1044       (3244%)
+b               = 0.978108         +/- 0.002685     (0.2745%)
+
+
+correlation matrix of the fit parameters:
+
+               a      b      
+a               1.000 
+b              -0.999  1.000 
+
+
+*******************************************************************************
+Fri Oct 27 21:50:01 2017
+
+
+FIT:    data read from "adv-lamb.csv" every ::1 using 4:5
+        format = x:z
+        #datapoints = 100
+        residuals are weighted equally (unit weight)
+
+function used for fitting: g(x)
+fitted parameters initialized with current variable values
+
+
+
+ Iteration 0
+ WSSR        : 91.541            delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.967948
+
+initial set of free parameter values
+
+aa              = 1
+bb              = 1
+
+After 6 iterations the fit converged.
+final sum of squares of residuals : 1.03526e-11
+rel. change during last iteration : -9.82363e-11
+
+degrees of freedom    (FIT_NDF)                        : 98
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 3.25022e-07
+variance of residuals (reduced chisquare) = WSSR/ndf   : 1.05639e-13
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aa              = 0.337001         +/- 1.059e-05    (0.003142%)
+bb              = 0.662998         +/- 9.898e-06    (0.001493%)
+
+
+correlation matrix of the fit parameters:
+
+               aa     bb     
+aa              1.000 
+bb             -1.000  1.000 
+
+
+*******************************************************************************
+Fri Oct 27 21:50:01 2017
+
+
+FIT:    data read from "adv-lamb.csv" every ::1 using 4:6
+        format = x:z
+        #datapoints = 100
+        residuals are weighted equally (unit weight)
+
+function used for fitting: h(x)
+fitted parameters initialized with current variable values
+
+
+
+ Iteration 0
+ WSSR        : 96.0949           delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.967948
+
+initial set of free parameter values
+
+aaa             = 1
+bbb             = 1
+
+After 6 iterations the fit converged.
+final sum of squares of residuals : 1.22269e-11
+rel. change during last iteration : -1.20095e-10
+
+degrees of freedom    (FIT_NDF)                        : 98
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 3.5322e-07
+variance of residuals (reduced chisquare) = WSSR/ndf   : 1.24764e-13
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aaa             = 0.69757          +/- 1.151e-05    (0.00165%)
+bbb             = 0.30243          +/- 1.076e-05    (0.003557%)
+
+
+correlation matrix of the fit parameters:
+
+               aaa    bbb    
+aaa             1.000 
+bbb            -1.000  1.000 
+
+
+*******************************************************************************
+Fri Oct 27 21:50:01 2017
+
+
+FIT:    data read from "adv-lamb.csv" every ::1 using 3:6
+        format = x:z
+        #datapoints = 100
+        residuals are weighted equally (unit weight)
+
+function used for fitting: i(x)
+fitted parameters initialized with current variable values
+
+
+
+ Iteration 0
+ WSSR        : 0.21759           delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.707107
+
+initial set of free parameter values
+
+aaaa            = 1
+bbbb            = 1
+
+After 3 iterations the fit converged.
+final sum of squares of residuals : 0.000458526
+rel. change during last iteration : -2.92992e-11
+
+degrees of freedom    (FIT_NDF)                        : 98
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.00216306
+variance of residuals (reduced chisquare) = WSSR/ndf   : 4.67884e-06
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aaaa            = 0.999948         +/- 1.728e+14    (1.728e+16%)
+bbbb            = 0.953403         +/- 1.92e+11     (2.014e+13%)
+
+
+correlation matrix of the fit parameters:
+
+               aaaa   bbbb   
+aaaa            1.000 
+bbbb           -1.000  1.000 

dokumentation/evolution1d/adv-lamb.gnuplot.log

@@ -0,0 +1,349 @@
+
+
+ Iteration 0
+ WSSR        : 0.227572          delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.707341
+
+initial set of free parameter values
+
+a               = 1
+b               = 1
+/
+
+ Iteration 1
+ WSSR        : 0.00021325        delta(WSSR)/WSSR   : -1066.16
+ delta(WSSR) : -0.227359         limit for stopping : 1e-05
+ lambda	  : 0.0707341
+
+resultant parameter values
+
+a               = 0.998581
+b               = 0.952591
+/
+
+ Iteration 2
+ WSSR        : 0.000203712       delta(WSSR)/WSSR   : -0.0468247
+ delta(WSSR) : -9.53874e-06      limit for stopping : 1e-05
+ lambda	  : 0.00707341
+
+resultant parameter values
+
+a               = 0.978909
+b               = 0.952859
+/
+
+ Iteration 3
+ WSSR        : 0.000117743       delta(WSSR)/WSSR   : -0.730133
+ delta(WSSR) : -8.59683e-05      limit for stopping : 1e-05
+ lambda	  : 0.000707341
+
+resultant parameter values
+
+a               = 0.323908
+b               = 0.969698
+/
+
+ Iteration 4
+ WSSR        : 0.000107016       delta(WSSR)/WSSR   : -0.10024
+ delta(WSSR) : -1.07273e-05      limit for stopping : 1e-05
+ lambda	  : 7.07341e-05
+
+resultant parameter values
+
+a               = -0.00159147
+b               = 0.978066
+/
+
+ Iteration 5
+ WSSR        : 0.000107016       delta(WSSR)/WSSR   : -2.47553e-06
+ delta(WSSR) : -2.6492e-10       limit for stopping : 1e-05
+ lambda	  : 7.07341e-06
+
+resultant parameter values
+
+a               = -0.00321702
+b               = 0.978108
+
+After 5 iterations the fit converged.
+final sum of squares of residuals : 0.000107016
+rel. change during last iteration : -2.47553e-06
+
+degrees of freedom    (FIT_NDF)                        : 98
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.00104499
+variance of residuals (reduced chisquare) = WSSR/ndf   : 1.092e-06
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+a               = -0.00321702      +/- 0.1044       (3244%)
+b               = 0.978108         +/- 0.002685     (0.2745%)
+
+
+correlation matrix of the fit parameters:
+
+               a      b      
+a               1.000 
+b              -0.999  1.000 
+
+
+ Iteration 0
+ WSSR        : 91.541            delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.967948
+
+initial set of free parameter values
+
+aa              = 1
+bb              = 1
+/
+
+ Iteration 1
+ WSSR        : 0.00229828        delta(WSSR)/WSSR   : -39829.2
+ delta(WSSR) : -91.5387          limit for stopping : 1e-05
+ lambda	  : 0.0967948
+
+resultant parameter values
+
+aa              = 0.524975
+bb              = 0.492041
+/
+
+ Iteration 2
+ WSSR        : 2.92366e-05       delta(WSSR)/WSSR   : -77.6096
+ delta(WSSR) : -0.00226904       limit for stopping : 1e-05
+ lambda	  : 0.00967948
+
+resultant parameter values
+
+aa              = 0.513146
+bb              = 0.498339
+/
+
+ Iteration 3
+ WSSR        : 7.21159e-07       delta(WSSR)/WSSR   : -39.5412
+ delta(WSSR) : -2.85155e-05      limit for stopping : 1e-05
+ lambda	  : 0.000967948
+
+resultant parameter values
+
+aa              = 0.364666
+bb              = 0.637138
+/
+
+ Iteration 4
+ WSSR        : 1.28467e-11       delta(WSSR)/WSSR   : -56134.8
+ delta(WSSR) : -7.21146e-07      limit for stopping : 1e-05
+ lambda	  : 9.67948e-05
+
+resultant parameter values
+
+aa              = 0.337053
+bb              = 0.66295
+/
+
+ Iteration 5
+ WSSR        : 1.03526e-11       delta(WSSR)/WSSR   : -0.24091
+ delta(WSSR) : -2.49406e-12      limit for stopping : 1e-05
+ lambda	  : 9.67948e-06
+
+resultant parameter values
+
+aa              = 0.337001
+bb              = 0.662998
+/
+
+ Iteration 6
+ WSSR        : 1.03526e-11       delta(WSSR)/WSSR   : -9.82363e-11
+ delta(WSSR) : -1.017e-21        limit for stopping : 1e-05
+ lambda	  : 9.67948e-07
+
+resultant parameter values
+
+aa              = 0.337001
+bb              = 0.662998
+
+After 6 iterations the fit converged.
+final sum of squares of residuals : 1.03526e-11
+rel. change during last iteration : -9.82363e-11
+
+degrees of freedom    (FIT_NDF)                        : 98
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 3.25022e-07
+variance of residuals (reduced chisquare) = WSSR/ndf   : 1.05639e-13
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aa              = 0.337001         +/- 1.059e-05    (0.003142%)
+bb              = 0.662998         +/- 9.898e-06    (0.001493%)
+
+
+correlation matrix of the fit parameters:
+
+               aa     bb     
+aa              1.000 
+bb             -1.000  1.000 
+
+
+ Iteration 0
+ WSSR        : 96.0949           delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.967948
+
+initial set of free parameter values
+
+aaa             = 1
+bbb             = 1
+/
+
+ Iteration 1
+ WSSR        : 0.00241131        delta(WSSR)/WSSR   : -39850.7
+ delta(WSSR) : -96.0925          limit for stopping : 1e-05
+ lambda	  : 0.0967948
+
+resultant parameter values
+
+aaa             = 0.513505
+bbb             = 0.479371
+/
+
+ Iteration 2
+ WSSR        : 2.95208e-05       delta(WSSR)/WSSR   : -80.6818
+ delta(WSSR) : -0.00238179       limit for stopping : 1e-05
+ lambda	  : 0.00967948
+
+resultant parameter values
+
+aaa             = 0.520572
+bbb             = 0.467888
+/
+
+ Iteration 3
+ WSSR        : 7.2817e-07        delta(WSSR)/WSSR   : -39.5411
+ delta(WSSR) : -2.87926e-05      limit for stopping : 1e-05
+ lambda	  : 0.000967948
+
+resultant parameter values
+
+aaa             = 0.669772
+bbb             = 0.328416
+/
+
+ Iteration 4
+ WSSR        : 1.47452e-11       delta(WSSR)/WSSR   : -49382.6
+ delta(WSSR) : -7.28156e-07      limit for stopping : 1e-05
+ lambda	  : 9.67948e-05
+
+resultant parameter values
+
+aaa             = 0.697518
+bbb             = 0.302478
+/
+
+ Iteration 5
+ WSSR        : 1.22269e-11       delta(WSSR)/WSSR   : -0.205964
+ delta(WSSR) : -2.5183e-12       limit for stopping : 1e-05
+ lambda	  : 9.67948e-06
+
+resultant parameter values
+
+aaa             = 0.69757
+bbb             = 0.30243
+/
+
+ Iteration 6
+ WSSR        : 1.22269e-11       delta(WSSR)/WSSR   : -1.20095e-10
+ delta(WSSR) : -1.46839e-21      limit for stopping : 1e-05
+ lambda	  : 9.67948e-07
+
+resultant parameter values
+
+aaa             = 0.69757
+bbb             = 0.30243
+
+After 6 iterations the fit converged.
+final sum of squares of residuals : 1.22269e-11
+rel. change during last iteration : -1.20095e-10
+
+degrees of freedom    (FIT_NDF)                        : 98
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 3.5322e-07
+variance of residuals (reduced chisquare) = WSSR/ndf   : 1.24764e-13
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aaa             = 0.69757          +/- 1.151e-05    (0.00165%)
+bbb             = 0.30243          +/- 1.076e-05    (0.003557%)
+
+
+correlation matrix of the fit parameters:
+
+               aaa    bbb    
+aaa             1.000 
+bbb            -1.000  1.000 
+
+
+ Iteration 0
+ WSSR        : 0.21759           delta(WSSR)/WSSR   : 0
+ delta(WSSR) : 0                 limit for stopping : 1e-05
+ lambda	  : 0.707107
+
+initial set of free parameter values
+
+aaaa            = 1
+bbbb            = 1
+/
+
+ Iteration 1
+ WSSR        : 0.0004639         delta(WSSR)/WSSR   : -468.045
+ delta(WSSR) : -0.217126         limit for stopping : 1e-05
+ lambda	  : 0.0707107
+
+resultant parameter values
+
+aaaa            = 0.999948
+bbbb            = 0.953634
+/
+
+ Iteration 2
+ WSSR        : 0.000458526       delta(WSSR)/WSSR   : -0.0117211
+ delta(WSSR) : -5.37441e-06      limit for stopping : 1e-05
+ lambda	  : 0.00707107
+
+resultant parameter values
+
+aaaa            = 0.999948
+bbbb            = 0.953403
+/
+
+ Iteration 3
+ WSSR        : 0.000458526       delta(WSSR)/WSSR   : -2.92992e-11
+ delta(WSSR) : -1.34345e-14      limit for stopping : 1e-05
+ lambda	  : 0.000707107
+
+resultant parameter values
+
+aaaa            = 0.999948
+bbbb            = 0.953403
+
+After 3 iterations the fit converged.
+final sum of squares of residuals : 0.000458526
+rel. change during last iteration : -2.92992e-11
+
+degrees of freedom    (FIT_NDF)                        : 98
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 0.00216306
+variance of residuals (reduced chisquare) = WSSR/ndf   : 4.67884e-06
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+
+aaaa            = 0.999948         +/- 1.728e+14    (1.728e+16%)
+bbbb            = 0.953403         +/- 1.92e+11     (2.014e+13%)
+
+
+correlation matrix of the fit parameters:
+
+               aaaa   bbbb   
+aaaa            1.000 
+bbbb           -1.000  1.000 
+Warning: empty x range [0.00111111:0.00111111], adjusting to [0.0011:0.00112222]

dokumentation/evolution1d/adv-lamb.gnuplot.script

@@ -0,0 +1,26 @@
+set datafile separator ","
+f(x)=a*x+b
+fit f(x) "adv-lamb.csv" every ::1 using 2:5 via a,b
+set terminal png
+set xlabel 'Regularity'
+set ylabel 'Iterations'
+set output "adv-lamb_regularity-vs-steps.png"
+plot     "adv-lamb_25.csv" every ::1 using 2:5 title "\lambda = 0.25",     "adv-lamb_05.csv" every ::1 using 2:5 title "\lambda = 0.5",     "adv-lamb_75.csv" every ::1 using 2:5 title "\lambda = 0.75",     "adv-lamb_1.csv" every ::1 using 2:5 title "\lambda = 1",     f(x) title "lin. fit" lc rgb "black"
+g(x)=aa*x+bb
+fit g(x) "adv-lamb.csv" every ::1 using 4:5 via aa,bb
+set xlabel 'Improvement potential'
+set ylabel 'Iteration'
+set output "adv-lamb_improvement-vs-steps.png"
+plot     "adv-lamb_25.csv" every ::1 using 4:5 title "\lambda = 0.25",     "adv-lamb_05.csv" every ::1 using 4:5 title "\lambda = 0.5",     "adv-lamb_75.csv" every ::1 using 4:5 title "\lambda = 0.75",     "adv-lamb_1.csv" every ::1 using 4:5 title "\lambda = 1",     g(x) title "lin. fit" lc rgb "black"
+h(x)=aaa*x+bbb
+fit h(x) "adv-lamb.csv" every ::1 using 4:6 via aaa,bbb
+set xlabel 'Improvement potential'
+set ylabel 'Fitting error'
+set output "adv-lamb_improvement-vs-evo-error.png"
+plot     "adv-lamb_25.csv" every ::1 using 4:6 title "\lambda = 0.25",     "adv-lamb_05.csv" every ::1 using 4:6 title "\lambda = 0.5",     "adv-lamb_75.csv" every ::1 using 4:6 title "\lambda = 0.75",     "adv-lamb_1.csv" every ::1 using 4:6 title "\lambda = 1",     h(x) title "lin. fit" lc rgb "black"
+i(x)=aaaa*x+bbbb
+fit i(x) "adv-lamb.csv" every ::1 using 3:6 via aaaa,bbbb
+set xlabel 'Variability'
+set ylabel 'Fitting error'
+set output "adv-lamb_variability-vs-evo-error.png"
+plot     "adv-lamb_25.csv" every ::1 using 3:6 title "\lambda = 0.25",     "adv-lamb_05.csv" every ::1 using 3:6 title "\lambda = 0.5",     "adv-lamb_75.csv" every ::1 using 3:6 title "\lambda = 0.75",     "adv-lamb_1.csv" every ::1 using 3:6 title "\lambda = 1",     i(x) title "lin. fit" lc rgb "black"

dokumentation/evolution1d/adv-lamb.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing adv-lamb.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+  x y
+x 1 1
+y 1 1
+
+n= 100 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+  x y
+x 1 1
+y 1 1
+
+n= 100 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for regularity vs. steps"
+     x    y
+x 1.00 0.02
+y 0.02 1.00
+
+n= 100 
+
+
+P
+  x     y    
+x       0.872
+y 0.872      
+[1] "spearman for variability vs. evolution-error"
+    x   y
+x   1 NaN
+y NaN   1
+
+n= 100 
+
+
+P
+  x y
+x    
+y    

dokumentation/evolution1d/adv-lamb_05.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing adv-lamb_05.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+   x  y
+x  1 -1
+y -1  1
+
+n= 100 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+     x    y
+x 1.00 0.14
+y 0.14 1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.1641
+y 0.1641       
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.05
+y -0.05  1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.6033
+y 0.6033       
+[1] "spearman for variability vs. evolution-error"
+    x   y
+x   1 NaN
+y NaN   1
+
+n= 100 
+
+
+P
+  x y
+x    
+y    

dokumentation/evolution1d/adv-lamb_1.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing adv-lamb_1.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+   x  y
+x  1 -1
+y -1  1
+
+n= 100 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+     x    y
+x 1.00 0.14
+y 0.14 1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.1646
+y 0.1646       
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.05
+y -0.05  1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.6033
+y 0.6033       
+[1] "spearman for variability vs. evolution-error"
+    x   y
+x   1 NaN
+y NaN   1
+
+n= 100 
+
+
+P
+  x y
+x    
+y    

dokumentation/evolution1d/adv-lamb_25.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing adv-lamb_25.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+   x  y
+x  1 -1
+y -1  1
+
+n= 100 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+     x    y
+x 1.00 0.14
+y 0.14 1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.1664
+y 0.1664       
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.05
+y -0.05  1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.6033
+y 0.6033       
+[1] "spearman for variability vs. evolution-error"
+    x   y
+x   1 NaN
+y NaN   1
+
+n= 100 
+
+
+P
+  x y
+x    
+y    

dokumentation/evolution1d/adv-lamb_75.spearman

@@ -0,0 +1,49 @@
+[1] "================ Analyzing adv-lamb_75.csv"
+[1] "spearman for improvement-potential vs. evolution-error"
+   x  y
+x  1 -1
+y -1  1
+
+n= 100 
+
+
+P
+  x  y 
+x     0
+y  0   
+[1] "spearman for improvement-potential vs. steps"
+     x    y
+x 1.00 0.14
+y 0.14 1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.1646
+y 0.1646       
+[1] "spearman for regularity vs. steps"
+      x     y
+x  1.00 -0.05
+y -0.05  1.00
+
+n= 100 
+
+
+P
+  x      y     
+x        0.6033
+y 0.6033       
+[1] "spearman for variability vs. evolution-error"
+    x   y
+x   1 NaN
+y NaN   1
+
+n= 100 
+
+
+P
+  x y
+x    
+y