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arbeit/bibma.bib
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@ -236,3 +236,21 @@
year={2016}, year={2016},
url={https://arxiv.org/abs/1604.00772} 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}
}

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@ -1,5 +1,5 @@
--- ---
fontsize: 12pt fontsize: 11pt
--- ---
\chapter*{How to read this Thesis} \chapter*{How to read this Thesis}
@ -23,8 +23,6 @@ Unless otherwise noted the following holds:
# Introduction # Introduction
\improvement[inline]{Mehr Bilder}
Many modern industrial design processes require advanced optimization methods Many modern industrial design processes require advanced optimization methods
due to the increased complexity resulting from more and more degrees of freedom 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 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 typically not static environments but requirements shift over time or from case
to 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 Evolutionary algorithms cope especially well with these problem domains while
addressing all the issues at hand\cite{minai2006complex}. One of the main 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 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 there is need for some criteria we can measure, so that we are able to compare different
representations to learn and improve upon these. 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 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 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 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 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:evo}) and why this is useful in our domain (in
\ref{sec:back:evogood}) followed by the definition of the different evolvability \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 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}, 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. of $\mu$ individuals.
All these functions can (and mostly do) have a lot of hidden parameters that 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 Also in selection it may not be wise to only take the best--performing
mutation rate that cools off over time (i.e. by lowering the variance of a individuals, because it may be that the optimization has to overcome a barrier
gaussian noise).--> 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 ## Advantages of evolutionary algorithms
\label{sec:back:evogood} \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 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 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 starts with a population of guessed solutions. Because an
evolutionary strategy modifies the solution randomly, keeping the best solutions evolutionary strategy modifies the solution randomly, keeping some solutions
and purging the worst, it can also target multiple different hypothesis at the 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 same time where the local optima die out in the face of other, better
candidates. candidates.
@ -371,16 +389,18 @@ converge to the same solution.
As we have established in chapter \ref{sec:back:ffd}, we can describe a As we have established in chapter \ref{sec:back:ffd}, we can describe a
deformation by the formula 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 parametrization) calculated deformation--coefficients and $P$ is a $m \times d$ matrix
of control--points that we interact with during deformation. of control--points that we interact with during deformation.
We can also think of the deformation in terms of differences from the original We can also think of the deformation in terms of differences from the original
coordinates 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 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 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 \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 given some approximate $n \times d$ fitness--gradient $\vec{G}$, normalized to
$\|\vec{G}\|_F = 1$, whereby $\|\cdot\|_F$ denotes the Frobenius--Norm. $\|\vec{G}\|_F = 1$, 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$. 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} For this we employ the Gauss--Newton algorithm\cite{gaussNewton}, which
\todo[inline]{rewrite. falsch und wischi-waschi. Least squares?} converges into the least--squares solution. An exact solution of this problem is
as the solution to impossible most of the times, because we usually have way more vertices
this problem may not be deterministic, because we usually have way more vertices
than control points ($\#v~\gg~\#c$). than control points ($\#v~\gg~\#c$).
## Adaption of \ac{FFD} for a 3D--Mesh ## 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 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 similar models and was shown to lead to a more precise fit. The Laplacian
\begin{equation} \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} \label{eq:reg3d}
\end{equation} \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$ 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 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 $\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 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 that we will influence via the $\lambda$--coefficient to start out with a stiff
material that will get more flexible per iteration. material that will get more flexible per iteration. As a side--effect this also
\unsure[inline]{Andreas: hast du nen cite, wo gezeigt ist, dass das so sinnvoll ist?} 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 # Evaluation of Scenarios
\label{sec:res} \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. use then the formulas for *variability* and *regularity* to get our predictions.
Afterwards we solve the problem analytically to get the (normalized) correct 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 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 where $\mathbb{1}$ is the vector consisting of $1$ in every dimension,
$\vec{g}_\textrm{c} = \vec{p^{*}} - \vec{p}$ the calculated correct gradient. As $\vec{g}_\mathrm{c} = \vec{p^{*}} - \vec{p}$ is the calculated correct gradient,
we always start with a gradient of $\mathbb{0}$ this shortens to and $\mu$ is used to blend between $\vec{g}_\mathrm{c}$ and $\mathbb{1}$. As
$\vec{g}_\textrm{c} = \vec{p^{*}}$. we always start with a gradient of $p = \mathbb{0}$ this means shortens
$\vec{g}_\mathrm{c} = \vec{p^{*}}$.
\begin{figure}[ht] \begin{figure}[ht]
\begin{center} \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 with our implemented newtons--method we avoid the generation of such
self--intersecting grids for our testcases (see section \ref{3dffd}). 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 To achieve that we generated a gaussian distributed number with $\mu = 0, \sigma=0.25$
dimension and shrink the distance to the neighbours (the smaller neighbour for and clamped it to the range $[-0.25,0.25]$. We chose such an $r \in [-0.25,0.25]$
$r < 0$, the larger for $r > 0$) by the factor $r$^[Note: On the Edges this per dimension and moved the control--points by that factor towards their
displacement is only applied outwards by flipping the sign of $r$, if respective neighbours^[Note: On the Edges this displacement is only applied
appropriate.]. outwards by flipping the sign of $r$, if appropriate.].
\improvement[inline]{update!! gaussian, not uniform!!}
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 An Example of such a testcase can be seen for a $7 \times 4$--grid in figure
\ref{fig:example1d_grid}. \ref{fig:example1d_grid}.
@ -957,12 +986,22 @@ grid--resolutions}
\label{fig:1dimp} \label{fig:1dimp}
\end{figure} \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) Additionally we tested the results for a distorted gradient described in
- gradient macht keinen unterschied \ref{sec:proc:1d} with a $\mu$--value of $0.25$, $0.5$, $0,75$, and $1.0$ for
- $UU^+$ scheint sehr kleine EW zu haben, s. regularität the $5 \times 5$ grid and with a $\mu$--value of $0.5$ for all other cases.
- trotzdem sehr gutes kriterium - auch ohne Richtung.
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 ## Procedure: 3D Function Approximation
\label{sec:proc:3dfa} \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 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 $4\times 4\times 4$--grid. Identical to the 1--dimensional scenario before, we create a
regular grid and move the control-points \improvement{Beschreiben wie} random between their regular grid and move the control-points in the exact same random manner between
neighbours, but in three instead of two dimensions^[Again, we flip the signs for their neighbours as described in section \ref{sec:proc:1d}, but in three instead
the edges, if necessary to have the object still in the convex hull.]. 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] \begin{figure}[!htb]
\includegraphics[width=\textwidth]{img/3d_grid_resolution.png} \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} \label{tab:3dimp}
\end{table} \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 \centering
\includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png} \includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
\caption[Improvement potential for different 3D--grids]{ \caption[Improvement potential for different 3D--grids]{
@ -1254,11 +1301,76 @@ indicate trends.}
\label{fig:resimp3d} \label{fig:resimp3d}
\end{figure} \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} \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 \improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
Direktlinks des Autors.\newline Direktlinks des Autors.\newline

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@ -2,8 +2,9 @@
% abstracton : Abstract mit Ueberschrift % abstracton : Abstract mit Ueberschrift
\documentclass[ \documentclass[
a4paper, % default a4paper, % default
12pt, % default = 11pt 11pt, % 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 twoside, % default, 2seitig
titlepage, titlepage,
% pagesize=auto % pagesize=auto
@ -168,8 +169,6 @@ Unless otherwise noted the following holds:
\chapter{Introduction}\label{introduction} \chapter{Introduction}\label{introduction}
\improvement[inline]{Mehr Bilder}
Many modern industrial design processes require advanced optimization Many modern industrial design processes require advanced optimization
methods due to the increased complexity resulting from more and more methods due to the increased complexity resulting from more and more
degrees of freedom as methods refine and/or other methods are used. 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 not static environments but requirements shift over time or from case to
case. 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 Evolutionary algorithms cope especially well with these problem domains
while addressing all the issues at hand\cite{minai2006complex}. One of while addressing all the issues at hand\cite{minai2006complex}. One of
the main concerns in these algorithms is the formulation of the problems 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 there is need for some criteria we can measure, so that we are able to
compare different representations to learn and improve upon these. 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 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 random points and represent vertices of an object as distances to these
points --- for example via \acf{RBF}. If one (or the algorithm) would 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:evo}) and why this is useful in our domain (in
\ref{sec:back:evogood}) followed by the definition of the different \ref{sec:back:evogood}) followed by the definition of the different
evolvability criteria established in \cite{anrichterEvol} (in 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 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 \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} \end{itemize}
All these functions can (and mostly do) have a lot of hidden parameters 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 \section{Advantages of evolutionary
algorithms}\label{advantages-of-evolutionary-algorithms} 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 procedure has only one point of observation from where it evaluates the
next steps, whereas an evolutionary strategy starts with a population of next steps, whereas an evolutionary strategy starts with a population of
guessed solutions. Because an evolutionary strategy modifies the guessed solutions. Because an evolutionary strategy modifies the
solution randomly, keeping the best solutions and purging the worst, it solution randomly, keeping some solutions and purging others, it can
can also target multiple different hypothesis at the same time where the also target multiple different hypothesis at the same time where the
local optima die out in the face of other, better candidates. local optima die out in the face of other, better candidates.
\improvement[inline]{Verweis auf MO-CMA etc. Vielleicht auch etwas \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 As we have established in chapter \ref{sec:back:ffd}, we can describe a
deformation by the formula \[ 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, \(\vec{U}\) \] where \(\vec{S}\) is a \(n \times d\) matrix of vertices\footnote{We
are the (during parametrization) calculated deformation--coefficients use \(\vec{S}\) in this notation, as we will use this parametrization
and \(P\) is a \(m \times d\) matrix of control--points that we interact of a source--mesh to manipulate \(\vec{S}\) into a target--mesh
with during deformation. \(\vec{T}\) 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 We can also think of the deformation in terms of differences from the
original coordinates \[ 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 \] 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 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 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} \(P\) The definition for an \emph{improvement potential} \(P\)
is\cite{anrichterEvol}: \[ is\cite{anrichterEvol}: \[
\mathrm{potential}(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F \mathrm{potential}(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F
\] \unsure[inline]{ist das $^2$ richtig?} given some approximate \] given some approximate \(n \times d\) fitness--gradient \(\vec{G}\),
\(n \times d\) fitness--gradient \(\vec{G}\), normalized to normalized to \(\|\vec{G}\|_F = 1\), whereby \(\|\cdot\|_F\) denotes the
\(\|\vec{G}\|_F = 1\), whereby \(\|\cdot\|_F\) denotes the
Frobenius--Norm. Frobenius--Norm.
\chapter{\texorpdfstring{Implementation of \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 \] and do a gradient--descend to approximate the value of \(u\) up to an
\(\epsilon\) of \(0.0001\). \(\epsilon\) of \(0.0001\).
For this we use the Gauss--Newton algorithm\cite{gaussNewton} For this we employ the Gauss--Newton algorithm\cite{gaussNewton}, which
\todo[inline]{rewrite. falsch und wischi-waschi. Least squares?} as the converges into the least--squares solution. An exact solution of this
solution to this problem may not be deterministic, because we usually problem is impossible most of the times, because we usually have way
have way more vertices than control points (\(\#v~\gg~\#c\)). more vertices than control points (\(\#v~\gg~\#c\)).
\section{\texorpdfstring{Adaption of \ac{FFD} for a \section{\texorpdfstring{Adaption of \ac{FFD} for a
3D--Mesh}{Adaption of for a 3D--Mesh}}\label{adaption-of-for-a-3dmesh} 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 lead to a more precise fit. The Laplacian
\begin{equation} \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} \label{eq:reg3d}
\end{equation} \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 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 \(s_i\) connected vertices \(\mathcal{N}(s_i)\) and \(A_i\) is the
Voronoi--area of the corresponding vertex \(\vec{s_i}\). We leave out 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 This regularization--weight gives us a measure of stiffness for the
material that we will influence via the \(\lambda\)--coefficient to material that we will influence via the \(\lambda\)--coefficient to
start out with a stiff material that will get more flexible per start out with a stiff material that will get more flexible per
iteration. iteration. As a side--effect this also limits the effects of
\unsure[inline]{Andreas: hast du nen cite, wo gezeigt ist, dass das so sinnvoll ist?} 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} \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 \emph{regularity} to get our predictions. Afterwards we solve the
problem analytically to get the (normalized) correct gradient that we problem analytically to get the (normalized) correct gradient that we
use as guess for the \emph{improvement potential}. To check we also use as guess for the \emph{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 \] where \(\mathbb{1}\) is the vector consisting of \(1\) in every
dimension and \(\vec{g}_\textrm{c} = \vec{p^{*}} - \vec{p}\) the dimension, \(\vec{g}_\mathrm{c} = \vec{p^{*}} - \vec{p}\) is the
calculated correct gradient. As we always start with a gradient of calculated correct gradient, and \(\mu\) is used to blend between
\(\mathbb{0}\) this shortens to \(\vec{g}_\textrm{c} = \vec{p^{*}}\). \(\vec{g}_\mathrm{c}\) 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{figure}[ht]
\begin{center} \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 the generation of such self--intersecting grids for our testcases (see
section \ref{3dffd}). section \ref{3dffd}).
To achieve that we select a uniform distributed number To achieve that we generated a gaussian distributed number with
\(r \in [-0.25,0.25]\) per dimension and shrink the distance to the \(\mu = 0, \sigma=0.25\) and clamped it to the range \([-0.25,0.25]\).
neighbours (the smaller neighbour for \(r < 0\), the larger for We chose such an \(r \in [-0.25,0.25]\) per dimension and moved the
\(r > 0\)) by the factor \(r\)\footnote{Note: On the Edges this control--points by that factor towards their respective
displacement is only applied outwards by flipping the sign of \(r\), neighbours\footnote{Note: On the Edges this displacement is only applied
if appropriate.}. outwards by flipping the sign of \(r\), if appropriate.}.
\improvement[inline]{update!! gaussian, not uniform!!}
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 An Example of such a testcase can be seen for a \(7 \times 4\)--grid in
figure \ref{fig:example1d_grid}. figure \ref{fig:example1d_grid}.
@ -1177,19 +1217,26 @@ grid--resolutions}
\label{fig:1dimp} \label{fig:1dimp}
\end{figure} \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} Additionally we tested the results for a distorted gradient described in
\tightlist \ref{sec:proc:1d} with a \(\mu\)--value of \(0.25\), \(0.5\), \(0,75\),
\item and \(1.0\) for the \(5 \times 5\) grid and with a \(\mu\)--value of
spearman 1 (p=0) \(0.5\) for all other cases.
\item
gradient macht keinen unterschied All results show the identical \emph{very strong} and \emph{significant}
\item correlation with a Spearman--coefficient of \(- r_S = 1.0\) and p--value
\(UU^+\) scheint sehr kleine EW zu haben, s. regularität of \(0\).
\item
trotzdem sehr gutes kriterium - auch ohne Richtung. These results indicate, that \(\|\mathbb{1} - \vec{U}\vec{U}^{+}\|_F\)
\end{itemize} 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 \section{Procedure: 3D Function
Approximation}\label{procedure-3d-function-approximation} 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 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 \(4\times 4\times 4\)--grid. Identical to the 1--dimensional scenario
before, we create a regular grid and move the control-points before, we create a regular grid and move the control-points in the
\improvement{Beschreiben wie} random between their neighbours, but in exact same random manner between their neighbours as described in
three instead of two dimensions\footnote{Again, we flip the signs for section \ref{sec:proc:1d}, but in three instead of two
the edges, if necessary to have the object still in the convex hull.}. dimensions\footnote{Again, we flip the signs for the edges, if necessary
to have the object still in the convex hull.}.
\begin{figure}[!htb] \begin{figure}[!htb]
\includegraphics[width=\textwidth]{img/3d_grid_resolution.png} \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} \label{tab:3dimp}
\end{table} \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 \centering
\includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png} \includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
\caption[Improvement potential for different 3D--grids]{ \caption[Improvement potential for different 3D--grids]{
@ -1501,15 +1557,84 @@ indicate trends.}
\label{fig:resimp3d} \label{fig:resimp3d}
\end{figure} \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} \label{sec:dis}
\begin{itemize} In this thesis we took a look at the different criteria for evolvability
\tightlist as introduced by Richter et al.\cite{anrichterEvol}, namely
\item \emph{variability}, \emph{regularity} and \emph{improvement potential}
Regularity ist kacke für unser setup. Bessere Vorschläge? EW/EV? under different setup--conditions. Where Richter et al. used \acf{RBF},
\end{itemize} 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 \improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
Direktlinks des Autors.\newline Direktlinks des Autors.\newline

3
arbeit/template.tex
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@ -3,7 +3,8 @@
\documentclass[ \documentclass[
a4paper, % default a4paper, % default
$if(fontsize)$$fontsize$,$endif$ % default = 11pt $if(fontsize)$$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 twoside, % default, 2seitig
titlepage, titlepage,
% pagesize=auto % pagesize=auto

101
dokumentation/evolution1d/20171027-evolution1D_5x5_100Times_adv-lamb.csv Normal file
View File

@ -0,0 +1,101 @@
"Least squares",regularity,variability,improvement,improvement.5,improvement.75,improvement.25,steps,"Evolution error",sigma
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196.451,0.0244511,0.00111111,0.93446,0.977913,0.954281,0.99492,157,205.896,0.0228021
189.09,0.0262443,0.00111111,0.936855,0.97872,0.955952,0.995106,159,198.496,0.0265
190.532,0.0261834,0.00111111,0.936372,0.978557,0.955615,0.995068,237,199.725,0.0171699
190.508,0.0256818,0.00111111,0.93638,0.97856,0.95562,0.995069,187,200.021,0.0279554
195.427,0.0276415,0.00111111,0.934737,0.978006,0.954474,0.994942,233,205.078,0.0239691
187.043,0.0248676,0.00111111,0.937537,0.978949,0.956427,0.995159,147,195.885,0.0307528
183.789,0.0266311,0.00111111,0.938623,0.979316,0.957185,0.995243,191,192.902,0.0200394
178.989,0.0248993,0.00111111,0.940228,0.979856,0.958305,0.995367,218,187.778,0.0188421
196.752,0.0272486,0.00111111,0.934294,0.977857,0.954166,0.994907,203,206.385,0.0247391
184.537,0.0266148,0.00111111,0.938374,0.979232,0.957011,0.995223,192,193.482,0.0207295
201.489,0.0259811,0.00111111,0.932757,0.977339,0.953094,0.994788,209,211.501,0.020084
199.803,0.0263512,0.00111111,0.933275,0.977513,0.953455,0.994828,261,209.303,0.0238203
196.761,0.0264791,0.00111111,0.934291,0.977856,0.954164,0.994907,142,206.22,0.0255644
198.277,0.0267124,0.00111111,0.933786,0.977685,0.953811,0.994868,182,207.938,0.0336422
208.415,0.0264034,0.00111111,0.9304,0.976544,0.951449,0.994605,137,218.736,0.0313946
207.674,0.0252068,0.00111111,0.930647,0.976628,0.951621,0.994625,176,217.926,0.0150835
188.854,0.0239334,0.00111111,0.936932,0.978746,0.956005,0.995112,223,198.057,0.0253543
219.268,0.0250067,0.00111111,0.926775,0.975323,0.94892,0.994324,159,230.187,0.0183427
204.945,0.0249211,0.00111111,0.931558,0.976935,0.952257,0.994695,176,215.127,0.0234684
203.53,0.0253226,0.00111111,0.932031,0.977094,0.952587,0.994732,198,213.673,0.016402
190.941,0.02429,0.00111111,0.936235,0.978511,0.955519,0.995058,121,200.174,0.0488998
188.434,0.026564,0.00111111,0.937085,0.978797,0.956112,0.995124,139,197.224,0.022272
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196.368,0.025791,0.00111111,0.934422,0.9779,0.954255,0.994917,173,206.172,0.0266269
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200.662,0.02717,0.00111111,0.932989,0.977417,0.953255,0.994806,166,210.591,0.0191685
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208.013,0.0253904,0.00111111,0.930534,0.976589,0.951542,0.994616,148,217.876,0.0354772
198.242,0.02544,0.00111111,0.933797,0.977689,0.953819,0.994869,220,207.826,0.0198802
190.364,0.0261942,0.00111111,0.936428,0.978576,0.955654,0.995073,229,199.813,0.017872
199.888,0.0260878,0.00111111,0.933248,0.977504,0.953436,0.994826,196,209.821,0.0433316
192.861,0.0268398,0.00111111,0.935594,0.978295,0.955072,0.995008,191,202.42,0.0178469
192.098,0.0262855,0.00111111,0.935856,0.978383,0.955255,0.995028,183,201.417,0.0248602
181.505,0.0252787,0.00111111,0.939556,0.97963,0.957836,0.995315,186,190.047,0.0284528
187.323,0.0267028,0.00111111,0.937443,0.978918,0.956362,0.995151,183,196.374,0.0358928
184.677,0.0255025,0.00111111,0.938327,0.979216,0.956979,0.99522,198,193.685,0.024289
205.213,0.0252094,0.00111111,0.931469,0.976905,0.952195,0.994688,194,215.462,0.0177816
202.104,0.0274611,0.00111111,0.932507,0.977254,0.952919,0.994769,145,212.139,0.0193782
188.727,0.0261381,0.00111111,0.936977,0.978761,0.956037,0.995115,146,197.984,0.022975
195.625,0.0255377,0.00111111,0.934671,0.977984,0.954428,0.994936,157,204.831,0.0274209
192.408,0.0243666,0.00111111,0.935745,0.978346,0.955178,0.99502,184,201.998,0.0295321
194.585,0.0257087,0.00111111,0.935029,0.978104,0.954678,0.994964,209,204.132,0.0176607
212.338,0.0246703,0.00111111,0.929237,0.976153,0.950638,0.994515,209,222.918,0.0241369
181.9,0.0245309,0.00111111,0.939254,0.979528,0.957625,0.995292,172,190.866,0.0173801
193.352,0.0258004,0.00111111,0.93543,0.978239,0.954958,0.994995,181,202.647,0.0281892
201.972,0.0257228,0.00111111,0.932551,0.977269,0.95295,0.994772,159,211.847,0.0197761
189.39,0.024665,0.00111111,0.936753,0.978686,0.955881,0.995098,213,198.75,0.0378911
213.861,0.024889,0.00111111,0.928915,0.976044,0.950413,0.99449,164,224.289,0.0270203
208.41,0.0265947,0.00111111,0.930403,0.976545,0.951451,0.994606,178,218.633,0.0445227
197.211,0.0248719,0.00111111,0.934141,0.977805,0.954059,0.994895,214,206.585,0.0256113
193.28,0.0264211,0.00111111,0.935595,0.978295,0.955073,0.995008,145,202.822,0.0209876
184.13,0.0247761,0.00111111,0.938843,0.97939,0.957339,0.99526,154,192.806,0.0281888
214.345,0.0254178,0.00111111,0.928419,0.975877,0.950067,0.994452,204,224.986,0.0233669
198.282,0.0243538,0.00111111,0.933783,0.977685,0.953809,0.994868,159,207.701,0.017893
190.332,0.026492,0.00111111,0.936439,0.978579,0.955661,0.995073,265,199.844,0.0225251
187.448,0.0255424,0.00111111,0.937614,0.978975,0.956481,0.995165,174,196.687,0.0194459
186.94,0.0237614,0.00111111,0.937571,0.978961,0.956451,0.995161,205,196.03,0.0327835
204.373,0.0291347,0.00111111,0.931749,0.976999,0.95239,0.99471,154,214.317,0.0221349
189.99,0.0245251,0.00111111,0.936553,0.978618,0.955741,0.995082,243,199.197,0.0280832
193.767,0.0250692,0.00111111,0.935291,0.978193,0.954861,0.994985,208,203.173,0.0213713
207.374,0.0247131,0.00111111,0.930747,0.976661,0.951691,0.994632,208,217.691,0.0213977
201.25,0.0258389,0.00111111,0.932792,0.977351,0.953118,0.994791,143,210.68,0.0255796
212.58,0.0246806,0.00111111,0.929009,0.976075,0.950479,0.994498,166,223.133,0.0236094
191.974,0.025784,0.00111111,0.93589,0.978395,0.955279,0.995031,157,201.275,0.0330152
185.387,0.0252672,0.00111111,0.938097,0.979138,0.956818,0.995202,189,194.38,0.0257366
212.023,0.0263675,0.00111111,0.929196,0.976139,0.950609,0.994512,145,222.547,0.0246763
186.682,0.026184,0.00111111,0.937657,0.97899,0.956512,0.995168,186,195.909,0.0179976
182.965,0.0268297,0.00111111,0.938978,0.979435,0.957433,0.99527,161,191.875,0.0360357
204.758,0.0256502,0.00111111,0.931784,0.977011,0.952414,0.994713,135,214.718,0.0244786
195.023,0.0240548,0.00111111,0.934872,0.978051,0.954569,0.994952,236,204.506,0.0168769
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
1 Least squares regularity variability improvement improvement.5 improvement.75 improvement.25 steps Evolution error sigma
2 179.603 0.026188 0.00111111 0.940021 0.979787 0.958161 0.995351 253 188.254 0.0232041
3 196.451 0.0244511 0.00111111 0.93446 0.977913 0.954281 0.99492 157 205.896 0.0228021
4 189.09 0.0262443 0.00111111 0.936855 0.97872 0.955952 0.995106 159 198.496 0.0265
5 190.532 0.0261834 0.00111111 0.936372 0.978557 0.955615 0.995068 237 199.725 0.0171699
6 190.508 0.0256818 0.00111111 0.93638 0.97856 0.95562 0.995069 187 200.021 0.0279554
7 195.427 0.0276415 0.00111111 0.934737 0.978006 0.954474 0.994942 233 205.078 0.0239691
8 187.043 0.0248676 0.00111111 0.937537 0.978949 0.956427 0.995159 147 195.885 0.0307528
9 183.789 0.0266311 0.00111111 0.938623 0.979316 0.957185 0.995243 191 192.902 0.0200394
10 178.989 0.0248993 0.00111111 0.940228 0.979856 0.958305 0.995367 218 187.778 0.0188421
11 196.752 0.0272486 0.00111111 0.934294 0.977857 0.954166 0.994907 203 206.385 0.0247391
12 184.537 0.0266148 0.00111111 0.938374 0.979232 0.957011 0.995223 192 193.482 0.0207295
13 201.489 0.0259811 0.00111111 0.932757 0.977339 0.953094 0.994788 209 211.501 0.020084
14 199.803 0.0263512 0.00111111 0.933275 0.977513 0.953455 0.994828 261 209.303 0.0238203
15 196.761 0.0264791 0.00111111 0.934291 0.977856 0.954164 0.994907 142 206.22 0.0255644
16 198.277 0.0267124 0.00111111 0.933786 0.977685 0.953811 0.994868 182 207.938 0.0336422
17 208.415 0.0264034 0.00111111 0.9304 0.976544 0.951449 0.994605 137 218.736 0.0313946
18 207.674 0.0252068 0.00111111 0.930647 0.976628 0.951621 0.994625 176 217.926 0.0150835
19 188.854 0.0239334 0.00111111 0.936932 0.978746 0.956005 0.995112 223 198.057 0.0253543
20 219.268 0.0250067 0.00111111 0.926775 0.975323 0.94892 0.994324 159 230.187 0.0183427
21 204.945 0.0249211 0.00111111 0.931558 0.976935 0.952257 0.994695 176 215.127 0.0234684
22 203.53 0.0253226 0.00111111 0.932031 0.977094 0.952587 0.994732 198 213.673 0.016402
23 190.941 0.02429 0.00111111 0.936235 0.978511 0.955519 0.995058 121 200.174 0.0488998
24 188.434 0.026564 0.00111111 0.937085 0.978797 0.956112 0.995124 139 197.224 0.022272
25 192.602 0.0257663 0.00111111 0.935682 0.978325 0.955134 0.995015 194 202.147 0.0170614
26 189.234 0.0234601 0.00111111 0.936806 0.978703 0.955918 0.995102 138 198.141 0.0424643
27 196.368 0.025791 0.00111111 0.934422 0.9779 0.954255 0.994917 173 206.172 0.0266269
28 206.929 0.0257442 0.00111111 0.930896 0.976711 0.951795 0.994644 177 217.236 0.0313754
29 208.073 0.0248913 0.00111111 0.930514 0.976583 0.951528 0.994614 171 218.32 0.0186665
30 196.852 0.02525 0.00111111 0.934261 0.977846 0.954142 0.994905 162 206.082 0.0317539
31 196.103 0.0269448 0.00111111 0.934512 0.97793 0.954318 0.994924 213 205.887 0.0319141
32 200.662 0.02717 0.00111111 0.932989 0.977417 0.953255 0.994806 166 210.591 0.0191685
33 192.795 0.0248511 0.00111111 0.935616 0.978302 0.955088 0.99501 188 202.024 0.0216186
34 201.416 0.0255063 0.00111111 0.932737 0.977332 0.953079 0.994787 248 211.435 0.0186956
35 192.352 0.0236198 0.00111111 0.935764 0.978352 0.955191 0.995021 184 201.584 0.027124
36 183.207 0.0272397 0.00111111 0.938825 0.979384 0.957326 0.995258 167 191.923 0.0400563
37 208.013 0.0253904 0.00111111 0.930534 0.976589 0.951542 0.994616 148 217.876 0.0354772
38 198.242 0.02544 0.00111111 0.933797 0.977689 0.953819 0.994869 220 207.826 0.0198802
39 190.364 0.0261942 0.00111111 0.936428 0.978576 0.955654 0.995073 229 199.813 0.017872
40 199.888 0.0260878 0.00111111 0.933248 0.977504 0.953436 0.994826 196 209.821 0.0433316
41 192.861 0.0268398 0.00111111 0.935594 0.978295 0.955072 0.995008 191 202.42 0.0178469
42 192.098 0.0262855 0.00111111 0.935856 0.978383 0.955255 0.995028 183 201.417 0.0248602
43 181.505 0.0252787 0.00111111 0.939556 0.97963 0.957836 0.995315 186 190.047 0.0284528
44 187.323 0.0267028 0.00111111 0.937443 0.978918 0.956362 0.995151 183 196.374 0.0358928
45 184.677 0.0255025 0.00111111 0.938327 0.979216 0.956979 0.99522 198 193.685 0.024289
46 205.213 0.0252094 0.00111111 0.931469 0.976905 0.952195 0.994688 194 215.462 0.0177816
47 202.104 0.0274611 0.00111111 0.932507 0.977254 0.952919 0.994769 145 212.139 0.0193782
48 188.727 0.0261381 0.00111111 0.936977 0.978761 0.956037 0.995115 146 197.984 0.022975
49 195.625 0.0255377 0.00111111 0.934671 0.977984 0.954428 0.994936 157 204.831 0.0274209
50 192.408 0.0243666 0.00111111 0.935745 0.978346 0.955178 0.99502 184 201.998 0.0295321
51 194.585 0.0257087 0.00111111 0.935029 0.978104 0.954678 0.994964 209 204.132 0.0176607
52 212.338 0.0246703 0.00111111 0.929237 0.976153 0.950638 0.994515 209 222.918 0.0241369
53 181.9 0.0245309 0.00111111 0.939254 0.979528 0.957625 0.995292 172 190.866 0.0173801
54 193.352 0.0258004 0.00111111 0.93543 0.978239 0.954958 0.994995 181 202.647 0.0281892
55 201.972 0.0257228 0.00111111 0.932551 0.977269 0.95295 0.994772 159 211.847 0.0197761
56 189.39 0.024665 0.00111111 0.936753 0.978686 0.955881 0.995098 213 198.75 0.0378911
57 213.861 0.024889 0.00111111 0.928915 0.976044 0.950413 0.99449 164 224.289 0.0270203
58 208.41 0.0265947 0.00111111 0.930403 0.976545 0.951451 0.994606 178 218.633 0.0445227
59 197.211 0.0248719 0.00111111 0.934141 0.977805 0.954059 0.994895 214 206.585 0.0256113
60 193.28 0.0264211 0.00111111 0.935595 0.978295 0.955073 0.995008 145 202.822 0.0209876
61 184.13 0.0247761 0.00111111 0.938843 0.97939 0.957339 0.99526 154 192.806 0.0281888
62 214.345 0.0254178 0.00111111 0.928419 0.975877 0.950067 0.994452 204 224.986 0.0233669
63 198.282 0.0243538 0.00111111 0.933783 0.977685 0.953809 0.994868 159 207.701 0.017893
64 190.332 0.026492 0.00111111 0.936439 0.978579 0.955661 0.995073 265 199.844 0.0225251
65 187.448 0.0255424 0.00111111 0.937614 0.978975 0.956481 0.995165 174 196.687 0.0194459
66 186.94 0.0237614 0.00111111 0.937571 0.978961 0.956451 0.995161 205 196.03 0.0327835
67 204.373 0.0291347 0.00111111 0.931749 0.976999 0.95239 0.99471 154 214.317 0.0221349
68 189.99 0.0245251 0.00111111 0.936553 0.978618 0.955741 0.995082 243 199.197 0.0280832
69 193.767 0.0250692 0.00111111 0.935291 0.978193 0.954861 0.994985 208 203.173 0.0213713
70 207.374 0.0247131 0.00111111 0.930747 0.976661 0.951691 0.994632 208 217.691 0.0213977
71 201.25 0.0258389 0.00111111 0.932792 0.977351 0.953118 0.994791 143 210.68 0.0255796
72 212.58 0.0246806 0.00111111 0.929009 0.976075 0.950479 0.994498 166 223.133 0.0236094
73 191.974 0.025784 0.00111111 0.93589 0.978395 0.955279 0.995031 157 201.275 0.0330152
74 185.387 0.0252672 0.00111111 0.938097 0.979138 0.956818 0.995202 189 194.38 0.0257366
75 212.023 0.0263675 0.00111111 0.929196 0.976139 0.950609 0.994512 145 222.547 0.0246763
76 186.682 0.026184 0.00111111 0.937657 0.97899 0.956512 0.995168 186 195.909 0.0179976
77 182.965 0.0268297 0.00111111 0.938978 0.979435 0.957433 0.99527 161 191.875 0.0360357
78 204.758 0.0256502 0.00111111 0.931784 0.977011 0.952414 0.994713 135 214.718 0.0244786
79 195.023 0.0240548 0.00111111 0.934872 0.978051 0.954569 0.994952 236 204.506 0.0168769
80 200.375 0.0256024 0.00111111 0.933085 0.977449 0.953322 0.994814 163 209.762 0.0341017
81 176.392 0.02493 0.00111111 0.941094 0.980148 0.958909 0.995434 246 185.182 0.024624
82 215.099 0.0251406 0.00111111 0.928167 0.975792 0.949892 0.994432 182 225.686 0.0172172
83 196.048 0.0244025 0.00111111 0.934529 0.977936 0.95433 0.994926 171 205.79 0.0216345
84 192.129 0.0250595 0.00111111 0.935842 0.978378 0.955245 0.995027 163 201.653 0.0246749
85 189.835 0.025235 0.00111111 0.936682 0.978661 0.955831 0.995092 113 198.984 0.0430281
86 205.107 0.0256549 0.00111111 0.931504 0.976916 0.952219 0.994691 165 214.844 0.0304616
87 193.362 0.0281933 0.00111111 0.935426 0.978238 0.954955 0.994995 133 202.594 0.0230947
88 176.783 0.0253994 0.00111111 0.940963 0.980104 0.958818 0.995424 165 185.619 0.0298644
89 201.911 0.0267396 0.00111111 0.932571 0.977276 0.952964 0.994774 247 211.904 0.0184367
90 187.159 0.0264083 0.00111111 0.937498 0.978936 0.9564 0.995156 185 195.98 0.0301479
91 184.049 0.0259232 0.00111111 0.938551 0.979291 0.957135 0.995237 232 192.979 0.0173422
92 204.792 0.0254979 0.00111111 0.93161 0.976952 0.952293 0.994699 167 214.577 0.0239699
93 199.555 0.0258068 0.00111111 0.933358 0.977541 0.953513 0.994835 157 209.134 0.0261165
94 190.76 0.0261192 0.00111111 0.936414 0.978571 0.955644 0.995072 227 200.294 0.023586
95 186.16 0.0270901 0.00111111 0.937832 0.979049 0.956633 0.995181 154 194.941 0.0553742
96 191.062 0.0244287 0.00111111 0.936195 0.978497 0.955491 0.995055 206 199.917 0.0297287
97 195.72 0.0264288 0.00111111 0.934639 0.977973 0.954406 0.994934 181 205.385 0.0241264
98 194.606 0.0260995 0.00111111 0.935048 0.978111 0.954691 0.994966 183 204.299 0.0164021
99 200.328 0.0273905 0.00111111 0.9331 0.977454 0.953333 0.994815 195 210.307 0.0270821
100 194.583 0.0264801 0.00111111 0.935019 0.978101 0.954671 0.994963 167 203.889 0.0308113
101 183.311 0.025923 0.00111111 0.938783 0.97937 0.957297 0.995255 167 192.409 0.0247795

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*******************************************************************************
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

349
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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]

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dokumentation/evolution1d/adv-lamb.gnuplot.script Normal file
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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"

49
dokumentation/evolution1d/adv-lamb.spearman Normal file
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[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

49
dokumentation/evolution1d/adv-lamb_05.spearman Normal file
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[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

49
dokumentation/evolution1d/adv-lamb_1.spearman Normal file
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@ -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

49
dokumentation/evolution1d/adv-lamb_25.spearman Normal file
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@ -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

49
dokumentation/evolution1d/adv-lamb_75.spearman Normal file
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[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

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