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dokumentation/evolution3d/errors.gnuplot.log create mode 100644 dokumentation/evolution3d/errors.gnuplot.script create mode 100644 dokumentation/evolution3d/errors_improvement-vs-evo-error.png create mode 100644 dokumentation/evolution3d/errors_improvement-vs-steps.png create mode 100644 dokumentation/evolution3d/errors_regularity-vs-steps.png create mode 100644 dokumentation/evolution3d/errors_variability-vs-evo-error.png diff --git a/arbeit/Makefile b/arbeit/Makefile index a388c4b..c511d30 100644 --- a/arbeit/Makefile +++ b/arbeit/Makefile @@ -4,7 +4,7 @@ all: ma.md bibma.bib template.tex settings/abkuerzungen.tex settings/commands.te xelatex -interaction batchmode ma.tex || true bibtexu ma xelatex -interaction batchmode ma.tex || true - while test `cat ma.log | grep -e "\(Rerun to get citations correct\)" | wc -l` -gt 0 ; do \ + while test `cat ma.log | grep -e "Rerun to get \(citations correct\|cross-references right\)" | wc -l` -gt 0 ; do \ rm ma.log && (xelatex -interaction batchmode ma.tex || true) \ done rm -f ma.aux ma.idx ma.lof ma.lot ma.out ma.tdo ma.toc ma.bbl ma.blg ma.loa diff --git a/arbeit/ma.md b/arbeit/ma.md index b6f5495..597be84 100644 --- a/arbeit/ma.md +++ b/arbeit/ma.md @@ -61,17 +61,18 @@ strongly tied to the notion of *evolvability*\cite{wagner1996complex}, as the parametrization of the problem has serious implications on the convergence speed and the quality of the solution\cite{Rothlauf2006}. However, there is no consensus on how *evolvability* is defined and the meaning -varies from context to context\cite{richter2015evolvability}, so there is need -for some criteria we can measure, so that we are able to compare different +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. 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 -object will get deformed locally (due to the \ac{RBF}). As this results in a -simple mapping from the parameter-space onto the object one can try out -different representations of the same object and evaluate the *evolvability*. -This is exactly what Richter et al.\cite{anrichterEvol} have done. +object will get deformed only locally (due to the \ac{RBF}). As this results in +a simple mapping from the parameter-space onto the object one can try out +different representations of the same object and evaluate which criteria may be +suited to describe this notion of *evolvability*. This is exactly what Richter +et al.\cite{anrichterEvol} have done. As we transfer the results of Richter et al.\cite{anrichterEvol} from using \acf{RBF} as a representation to manipulate geometric objects to the use of @@ -94,17 +95,16 @@ take an abstract look at the definition of \ac{FFD} for a one--dimensional line (in \ref{sec:back:ffdgood}). 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}). -In a third step we take a look at the definition of the different evolvability -criteria established in \cite{anrichterEvol}. +\ref{sec:back:evogood}) followed by the definition of the different evolvability +criteria established in \cite{anrichterEvol} (in \ref {sec:back: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}, we describe the different scenarios we use to -evaluate the different evolvability--criteria incorporating all aspects -introduced in Chapter \ref{sec:back}. Following that, we evaluate the results in -Chapter \ref{sec:res} with further on discussion in Chapter \ref{sec:dis}. +the adaptation for 3D--meshes that were used. Next, in Chapter \ref{sec:eval}, +we describe the different scenarios we use to evaluate the different +evolvability--criteria incorporating all aspects introduced in Chapter +\ref{sec:back}. Following that, we evaluate the results in +Chapter \ref{sec:res} with further on discussion, summary and outlook in +Chapter \ref{sec:dis}. # Background @@ -124,10 +124,10 @@ The main idea of \ac{FFD} is to create a function $s : [0,1[^d \mapsto parametrized by some special control points $p_i$ and an constant attribution--function $a_i(u)$, so $$ -s(u) = \sum_i a_i(u) p_i +s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i} $$ can be thought of a representation of the inside of the convex hull generated by -the control points where each point can be accessed by the right $u \in [0,1[$. +the control points where each point can be accessed by the right $u \in [0,1[^d$. \begin{figure}[!ht] \begin{center} @@ -138,9 +138,9 @@ corresponding deformation to generate a deformed objet} \label{fig:bspline} \end{figure} -In the example in figure \ref{fig:bspline}, the control--points are indicated as -red dots and the color-gradient should hint at the $u$--values ranging from -$0$ to $1$. +In the 1--dimensional example in figure \ref{fig:bspline}, the control--points +are indicated as red dots and the color-gradient should hint at the $u$--values +ranging from $0$ to $1$. We now define a \acf{FFD} by the following: Given an arbitrary number of points $p_i$ alongside a line, we map a scalar @@ -299,6 +299,7 @@ that terminates the optimization. Biologically speaking the set $I$ corresponds to the set of possible *Genotypes* while $M$ represents the possible observable *Phenotypes*. +\improvement[inline]{Erklären, was das ist. Quellen!} The main algorithm just repeats the following steps: @@ -318,20 +319,20 @@ 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. 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). +can be changed over time. + +\improvement[inline]{Genauer: Welche? Wo? Wieso? ...} + + ## Advantages of evolutionary algorithms \label{sec:back:evogood} The main advantage of evolutionary algorithms is the ability to find optima of -general functions just with the help of a given fitness--function. With this -most problems of simple gradient--based procedures, which often target the same -error--function which measures the fitness, as an evolutionary algorithm, but can -easily get stuck in local optima. - -Components and techniques for evolutionary algorithms are specifically known to +general functions just with the help of a given fitness--function. Components +and techniques for evolutionary algorithms are specifically known to help with different problems arising in the domain of optimization\cite{weise2012evolutionary}. An overview of the typical problems are shown in figure \ref{fig:probhard}. @@ -345,11 +346,14 @@ 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, keeps the best solutions -and purges the worst, it can also target multiple different hypothesis at the +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 local optima die out in the face of other, better candidates. +\improvement[inline]{Verweis auf MO-CMA etc. Vielleicht auch etwas +ausführlicher.} + If an analytic best solution exists and is easily computable (i.e. because the error--function is convex) an evolutionary algorithm is not the right choice. Although both converge to the same solution, the analytic one is usually faster. @@ -357,8 +361,9 @@ Although both converge to the same solution, the analytic one is usually faster. But in reality many problems have no analytic solution, because the problem is either not convex or there are so many parameters that an analytic solution (mostly meaning the equivalence to an exhaustive search) is computationally not -feasible. Here evolutionary optimization has one more advantage as you can at -least get suboptimal solutions fast, which then refine over time. +feasible. Here evolutionary optimization has one more advantage as one can at +least get suboptimal solutions fast, which then refine over time and still +converge to the same solution. ## Criteria for the evolvability of linear deformations \label{sec:intro:rvi} @@ -366,26 +371,26 @@ least get suboptimal solutions fast, which then refine over time. As we have established in chapter \ref{sec:back:ffd}, we can describe a deformation by the formula $$ -V = UP +\vec{V} = \vec{U}\vec{P} $$ -where $V$ is a $n \times d$ matrix of vertices, $U$ are the (during +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. We can also think of the deformation in terms of differences from the original coordinates $$ -\Delta V = U \cdot \Delta P +\Delta \vec{V} = \vec{U} \cdot \Delta \vec{P} $$ 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 $U$, which is why the three criteria focus on this. +solely in the entries of $\vec{U}$, which is why the three criteria focus on this. ### Variability In \cite{anrichterEvol} *variability* is defined as -$$V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n},$$ -whereby $\vec{U}$ is the $n \times m$ deformation--Matrix \unsure{Nicht $(n\cdot d) \times m$? Wegen $u,v,w$?} used to map the $m$ +$$\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n},$$ +whereby $\vec{U}$ is the $n \times m$ deformation--Matrix used to map the $m$ control points onto the $n$ vertices. Given $n = m$, an identical number of control--points and vertices, this @@ -395,10 +400,20 @@ the solution is to trivially move every control--point onto a target--point. In praxis the value of $V(\vec{U})$ is typically $\ll 1$, because as there are only few control--points for many vertices, so $m \ll n$. +This criterion should correlate to the degrees of freedom the given +parametrization has. This can be seen from the fact, that +$\mathrm{rank}(\vec{U})$ is limited by $\min(m,n)$ and --- as $n$ is constant +--- can never exceed $n$. + +The rank itself is also interesting, as control--points could theoretically be +placed on top of each other or be linear dependent in another way --- but will +in both cases lower the rank below the number of control--points $m$ and are +thus measurable by the *variability*. + ### Regularity *Regularity* is defined\cite{anrichterEvol} as -$$R(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}}$$ +$$\mathrm{regularity}(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}}$$ where $\sigma_{min}$ and $\sigma_{max}$ are the smallest and greatest right singular value of the deformation--matrix $\vec{U}$. @@ -416,17 +431,19 @@ the notion of locality\cite{weise2012evolutionary,thorhauer2014locality}. ### Improvement Potential In contrast to the general nature of *variability* and *regularity*, which are -agnostic of the fitness--function at hand the third criterion should reflect a -notion of potential. +agnostic of the fitness--function at hand, the third criterion should reflect a +notion of the potential for optimization, taking a guess into account. -As during optimization some kind of gradient $g$ is available to suggest a -direction worth pursuing we use this to guess how much change can be achieved in +Most of the times some kind of gradient $g$ is available to suggest a +direction worth pursuing; either from a previous iteration or by educated +guessing. We use this to guess how much change can be achieved in the given direction. The definition for an *improvement potential* $P$ is\cite{anrichterEvol}: $$ -P(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F +\mathrm{potential}(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F $$ +\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. @@ -482,7 +499,9 @@ $$ $$ 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} as the solution to +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 ($\#v~\gg~\#c$). @@ -548,12 +567,27 @@ $$J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \ri and use Cramers rule for inverting the small Jacobian and solving this system of linear equations. +As there is no strict upper bound of the number of iterations for this +algorithm, we just iterate it long enough to be within the given +$\epsilon$--error above. This takes --- depending on the shape of the object and +the grid --- about $3$ to $5$ iterations that we observed in practice. + +Another issue that we observed in our implementation is, that multiple local +optima may exist on self--intersecting grids. We solve this problem by defining +self--intersecting grids to be *invalid* and do not test any of them. + +This is not such a big problem as it sounds at first, as self--intersections +mean, that control--points being further away from a given vertex have more +influence over the deformation than control--points closer to this vertex. Also +this contradicts the notion of locality that we want to achieve and deemed +beneficial for a good behaviour of the evolutionary algorithm. + ## Deformation Grid \label{sec:impl:grid} As mentioned in chapter \ref{sec:back:evo}, the way of choosing the representation to map the general problem (mesh--fitting/optimization in our -case) into a parameter-space it very important for the quality and runtime of +case) into a parameter-space is very important for the quality and runtime of evolutionary algorithms\cite{Rothlauf2006}. Because our control--points are arranged in a grid, we can accurately represent @@ -561,10 +595,10 @@ each vertex--point inside the grids volume with proper B--Spline--coefficients between $[0,1[$ and --- as a consequence --- we have to embed our object into it (or create constant "dummy"-points outside). -The great advantage of B--Splines is the locality, direct impact of each +The great advantage of B--Splines is the local, direct impact of each control point without having a $1:1$--correlation, and a smooth deformation. While the advantages are great, the issues arise from the problem to decide -where to place the control--points and how many. +where to place the control--points and how many to place at all. \begin{figure}[!tbh] \centering @@ -578,8 +612,8 @@ control--points.} \end{figure} One would normally think, that the more control--points you add, the better the -result will be, but this is not the case for our B--Splines. Given any point $p$ -only the $2 \cdot (d-1)$ control--points contribute to the parametrization of +result will be, but this is not the case for our B--Splines. Given any point +$\vec{p}$ only the $2 \cdot (d-1)$ control--points contribute to the parametrization of that point^[Normally these are $d-1$ to each side, but at the boundaries the number gets increased to the inside to meet the required smoothness]. This means, that a high resolution can have many control-points that are not @@ -587,29 +621,37 @@ contributing to any point on the surface and are thus completely irrelevant to the solution. We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the four red -central points are not relevant for the parametrization of the circle. +central points are not relevant for the parametrization of the circle. This +leads to artefacts in the deformation--matrix $\vec{U}$, as the columns +corresponding to those control--points are $0$. -\unsure[inline]{erwähnen, dass man aus $\vec{D}$ einfach die Null--Spalten -entfernen kann?} +This leads to useless increased complexity, as the parameters corresponding to +those points will never have any effect, but a naive algorithm will still try to +optimize them yielding numeric artefacts in the best and non--terminating or +ill--defined solutions^[One example would be, when parts of an algorithm depend +on the inverse of the minimal right singular value leading to a division by $0$.] +at worst. + +One can of course neglect those columns and their corresponding control--points, +but this raises the question why they were introduced in the first place. We +will address this in a special scenario in \ref{sec:res:3d:var}. For our tests we chose different uniformly sized grids and added noise onto each control-point^[For the special case of the outer layer we only applied noise away from the object, so the object is still confined in the convex hull of the control--points.] to simulate different starting-conditions. -\unsure[inline]{verweis auf DM--FFD?} - -# Scenarios for testing evolvability criteria using \acf{FFD} +# Scenarios for testing evolvability criteria using \ac{FFD} \label{sec:eval} In our experiments we use the same two testing--scenarios, that were also used by \cite{anrichterEvol}. The first scenario deforms a plane into a shape originally defined in \cite{giannelli2012thb}, where we setup control-points in -a 2--dimensional manner merely deform in the height--coordinate to get the +a 2--dimensional manner and merely deform in the height--coordinate to get the resulting shape. In the second scenario we increase the degrees of freedom significantly by using -a 3--dimensional control--grid to deform a sphere into a face. So each control +a 3--dimensional control--grid to deform a sphere into a face, so each control point has three degrees of freedom in contrast to first scenario. ## Test Scenario: 1D Function Approximation @@ -642,10 +684,10 @@ As the starting-plane we used the same shape, but set all $z$--coordinates to $0$, yielding a flat plane, which is partially already correct. -Regarding the *fitness--function* $f(\vec{p})$, we use the very simple approach +Regarding the *fitness--function* $\mathrm{f}(\vec{p})$, we use the very simple approach of calculating the squared distances for each corresponding vertex \begin{equation} -\textrm{f(\vec{p})} = \sum_{i=1}^{n} \|(\vec{Up})_i - t_i\|_2^2 = \|\vec{Up} - \vec{t}\|^2 \rightarrow \min +\mathrm{f}(\vec{p}) = \sum_{i=1}^{n} \|(\vec{Up})_i - t_i\|_2^2 = \|\vec{Up} - \vec{t}\|^2 \rightarrow \min \end{equation} where $t_i$ are the respective target--vertices to the parametrized source--vertices^[The parametrization is encoded in $\vec{U}$ and the initial @@ -662,7 +704,7 @@ the correct gradient in which the evolutionary optimizer should move. \label{sec:test:3dfa} Opposed to the 1--dimensional scenario before, the 3--dimensional scenario is much more complex --- not only because we have more degrees of freedom on each -control point, but also because the *fitness--function* we will use has no known +control point, but also, because the *fitness--function* we will use has no known analytic solution and multiple local minima. \begin{figure}[ht] @@ -683,12 +725,13 @@ these Models can be seen in figure \ref{fig:3dtarget}. Opposed to the 1D--case we cannot map the source and target--vertices in a one--to--one--correspondence, which we especially need for the approximation of the fitting--error. Hence we state that the error of one vertex is the distance -to the closest vertex of the other model. +to the closest vertex of the other model and sum up the error from the +respective source and target. We therefore define the *fitness--function* to be: \begin{equation} -f(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} - +\mathrm{f}(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} - \vec{s_i}\|_2^2}_{\textrm{source-to-target--distance}} + \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} - \vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}} @@ -711,9 +754,10 @@ As regularization-term we add a weighted Laplacian of the deformation that has been used before by Aschenbach et al.\cite[Section 3.2]{aschenbach2015} on similar models and was shown to lead to a more precise fit. The Laplacian \begin{equation} -\textrm{regularization}(\vec{P}) = \frac{1}{\sum_i A_i} \sum_{i=1}^n A_i \cdot \left( \sum_{\vec{s_j} \in \mathcal{N}(\vec{s_i})} w_j \cdot \|\Delta \vec{s_j} - \Delta \vec{\overline{s}_j}\|^2 \right) +\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) \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 @@ -731,19 +775,20 @@ To compare our results to the ones given by Richter et al.\cite{anrichterEvol}, we also use Spearman's rank correlation coefficient. Opposed to other popular coefficients, like the Pearson correlation coefficient, which measures a linear relationship between variables, the Spearmans's coefficient assesses \glqq how -well an arbitrary monotonic function can descripbe the relationship between two +well an arbitrary monotonic function can describe the relationship between two variables, without making any assumptions about the frequency distribution of the variables\grqq\cite{hauke2011comparison}. As we don't have any prior knowledge if any of the criteria is linear and we are just interested in a monotonic relation between the criteria and their -predictive power, the Spearman's coefficient seems to fit out scenario best. +predictive power, the Spearman's coefficient seems to fit out scenario best and +was also used before by Richter et al.\cite{anrichterEvol} For interpretation of these values we follow the same interpretation used in \cite{anrichterEvol}, based on \cite{weir2015spearman}: The coefficient intervals $r_S \in [0,0.2[$, $[0.2,0.4[$, $[0.4,0.6[$, $[0.6,0.8[$, and $[0.8,1]$ are classified as *very weak*, *weak*, *moderate*, *strong* and *very strong*. We -interpret p--values smaller than $0.1$ as *significant* and cut off the +interpret p--values smaller than $0.01$ as *significant* and cut off the precision of p--values after four decimal digits (thus often having a p--value of $0$ given for p--values $< 10^{-4}$). @@ -772,7 +817,9 @@ $$ \vec{g}_{\textrm{d}} = \frac{\vec{g}_{\textrm{c}} + \mathbb{1}}{\|\vec{g}_{\textrm{c}} + \mathbb{1}\|} $$ where $\mathbb{1}$ is the vector consisting of $1$ in every dimension and -$\vec{g}_\textrm{c} = \vec{p^{*}}$ the calculated correct gradient. +$\vec{g}_\textrm{c} = \vec{p^{*}} - \vec{p}$ the calculated correct gradient. As +we always start with a gradient of $\mathbb{0}$ this shortens to +$\vec{g}_\textrm{c} = \vec{p^{*}}$. \begin{figure}[ht] \begin{center} @@ -787,11 +834,7 @@ We then set up a regular 2--dimensional grid around the object with the desired grid resolutions. To generate a testcase we then move the grid--vertices 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. - -This is a reasonable thing to do, as self-intersecting grids violate our desired -property of locality, as the then farther away control--point has more influence -over some vertices as the next-closer. +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 @@ -810,20 +853,22 @@ analytical solution to the given problem--set. We use this to experimentally evaluate the quality criteria we introduced before. As an evolutional optimization is partially a random process, we use the analytical solution as a stopping-criteria. We measure the convergence speed as number of iterations the -evolutional algorithm needed to get within $1.05\%$ of the optimal solution. +evolutional algorithm needed to get within $1.05 \times$ of the optimal solution. We used different regular grids that we manipulated as explained in Section \ref{sec:proc:1d} with a different number of control points. As our grids have to be the product of two integers, we compared a $5 \times 5$--grid with $25$ control--points to a $4 \times 7$ and $7 \times 4$--grid with $28$ -control--points. This was done to measure the impact an \glqq improper\grqq +control--points. This was done to measure the impact an \glqq improper\grqq \ setup could have and how well this is displayed in the criteria we are examining. Additionally we also measured the effect of increasing the total resolution of the grid by taking a closer look at $5 \times 5$, $7 \times 7$ and $10 \times 10$ grids. -\begin{figure}[ht] +### Variability + +\begin{figure}[tbh] \centering \includegraphics[width=0.7\textwidth]{img/evolution1d/variability_boxplot.png} \caption[1D Fitting Errors for various grids]{The squared error for the various @@ -832,8 +877,6 @@ Note that $7 \times 4$ and $4 \times 7$ have the same number of control--points. \label{fig:1dvar} \end{figure} -### Variability - Variability should characterize the potential for design space exploration and is defined in terms of the normalized rank of the deformation matrix $\vec{U}$: $V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n}$, whereby $n$ is the number of @@ -844,11 +887,12 @@ grid), we have merely plotted the errors in the boxplot in figure It is also noticeable, that although the $7 \times 4$ and $4 \times 7$ grids have a higher variability, they perform not better than the $5 \times 5$ grid. -Also the $7 \times 4$ and $4 \times 7$ grids differ distinctly from each other, -although they have the same number of control--points. This is an indication the -impact a proper or improper grid--setup can have. We do not draw scientific -conclusions from these findings, as more research on non-squared grids seem -necessary.\todo{machen wir die noch? :D} +Also the $7 \times 4$ and $4 \times 7$ grids differ distinctly from each other +with a mean$\pm$sigma of $233.09 \pm 12.32$ for the former and $286.32 \pm 22.36$ for the +latter, although they have the same number of control--points. This is an +indication of an impact a proper or improper grid--setup can have. We do not +draw scientific conclusions from these findings, as more research on non-squared +grids seem necessary. Leaving the issue of the grid--layout aside we focused on grids having the same number of prototypes in every dimension. For the $5 \times 5$, $7 \times 7$ and @@ -857,7 +901,7 @@ between the variability and the evolutionary error. ### Regularity -\begin{figure}[ht] +\begin{figure}[tbh] \centering \includegraphics[width=\textwidth]{img/evolution1d/55_to_1010_steps.png} \caption[Improvement potential and regularity vs. steps]{\newline @@ -952,12 +996,13 @@ control--points.} For the next step we then halve the regularization--impact $\lambda$ (starting at $1$) of our *fitness--function* (\ref{eq:fit3d}) and calculate the next incremental solution $\vec{P^{*}} = \vec{U^+}\vec{T}$ with the updated -correspondences to get our next target--error. We repeat this process as long as -the target--error keeps decreasing and use the number of these iterations as -measure of the convergence speed. As the resulting evolutional error without -regularization is in the numeric range of $\approx 100$, whereas the -regularization is numerically $\approx 7000$ we need at least $10$ to $15$ iterations -until the regularization--effect wears off. +correspondences (again, mapping each vertex to its closest neighbour in the +respective other model) to get our next target--error. We repeat this process as +long as the target--error keeps decreasing and use the number of these +iterations as measure of the convergence speed. As the resulting evolutional +error without regularization is in the numeric range of $\approx 100$, whereas +the regularization is numerically $\approx 7000$ we need at least $10$ to $15$ +iterations until the regularization--effect wears off. The grid we use for our experiments is just very coarse due to computational limitations. We are not interested in a good reconstruction, but an estimate if @@ -965,7 +1010,7 @@ the mentioned evolvability criteria are good. In figure \ref{fig:setup3d} we show an example setup of the scene with a $4\times 4\times 4$--grid. Identical to the 1--dimensional scenario before, we create a -regular grid and move the control-points \todo{wie?} random between their +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.]. @@ -1009,6 +1054,7 @@ control--points.} \end{figure} ### Variability +\label{sec:res:3d:var} \begin{table}[tbh] \centering @@ -1046,13 +1092,14 @@ deformation--matrix. \centering \includegraphics[width=0.8\textwidth]{img/evolution3d/variability2_boxplot.png} \caption[Histogram of ranks of high--resolution deformation--matrices]{ -Histogram of ranks of various $10 \times 10 \times 10$ grids. +Histogram of ranks of various $10 \times 10 \times 10$ grids with $1000$ +control--points each. } \label{fig:histrank3d} \end{figure} Overall the correlation between variability and fitness--error were -*significantly* and showed a *very strong* correlation in all our tests. +*significant* and showed a *very strong* correlation in all our tests. The detailed correlation--coefficients are given in table \ref{tab:3dvar} alongside their p--values. @@ -1111,21 +1158,20 @@ between regularity and number of iterations for the 3D fitting scenario. Displayed are the negated Spearman coefficients with the corresponding p--values in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$). \newline Note: Not significant results are marked in \textcolor{red}{red}.} -\label{tab:3dvar} +\label{tab:3dreg} \end{table} - - Opposed to the predictions of variability our test on regularity gave a mixed result --- similar to the 1D--case. -In half scenarios we have a *significant*, but *weak* to *moderate* correlation -between regularity and number of iterations. On the other hand in the scenarios -where we increased the number of control--points, namely $125$ for the -$5 \times 5 \times 5$ grid and $216$ for the $6 \times 6 \times 6$ grid we found -a *significant*, but *weak* anti--correlation, which seem to contradict the -findings/trends for the sets with $64$, $80$, and $112$ control--points (first -two rows of table \ref{tab:3dvar}). +In roughly half of the scenarios we have a *significant*, but *weak* to *moderate* +correlation between regularity and number of iterations. On the other hand in +the scenarios where we increased the number of control--points, namely $125$ for +the $5 \times 5 \times 5$ grid and $216$ for the $6 \times 6 \times 6$ grid we found +a *significant*, but *weak* **anti**--correlation when taking all three tests into +account^[Displayed as $Y \times Y \times Y$], which seem to contradict the +findings/trends for the sets with $64$, $80$, and $112$ control--points +(first two rows of table \ref{tab:3dreg}). Taking all results together we only find a *very weak*, but *significant* link between regularity and the number of iterations needed for the algorithm to @@ -1135,21 +1181,76 @@ converge. \centering \includegraphics[width=\textwidth]{img/evolution3d/regularity_montage.png} \caption[Regularity for different 3D--grids]{ -**BLINDTEXT** -} +Plots of regularity against number of iterations for various scenarios together +with a linear fit to indicate trends.} \label{fig:resreg3d} \end{figure} -As can be seen from figure \ref{fig:resreg3d}, we can observe\todo{things}. +As can be seen from figure \ref{fig:resreg3d}, we can observe that increasing +the number of control--points helps the convergence--speeds. The +regularity--criterion first behaves as we would like to, but then switches to +behave exactly opposite to our expectations, as can be seen in the first three +plots. While the number of control--points increases from red to green to blue +and the number of iterations decreases, the regularity seems to increase at +first, but then decreases again on higher grid--resolutions. + +This can be an artefact of the definition of regularity, as it is defined by the +inverse condition--number of the deformation--matrix $\vec{U}$, being the +fraction $\frac{\sigma_{\mathrm{min}}}{\sigma_{\mathrm{max}}}$ between the +least and greatest right singular value. + +As we observed in the previous section, we cannot +guarantee that each control--point has an effect (see figure \ref{fig:histrank3d}) +and so a small minimal right singular value occurring on higher +grid--resolutions seems likely the problem. + +Adding to this we also noted, that in the case of the $10 \times 10 \times +10$--grid the regularity was always $0$, as a non--contributing control-point +yields a $0$--column in the deformation--matrix, thus letting +$\sigma_\mathrm{min} = 0$. A better definition for regularity (i.e. using the +smallest non--zero right singular value) could solve this particular issue, but +not fix the trend we noticed above. ### Improvement Potential +\begin{table}[tbh] +\centering +\begin{tabular}{c|c|c|c} + & $5 \times 4 \times 4$ & $7 \times 4 \times 4$ & $\mathrm{X} \times 4 \times 4$ \\ +\cline{2-4} + & 0.3 (0.0023) & \textcolor{red}{0.23} (0.0233) & 0.89 (0) \B \\ +\cline{2-4} +\multicolumn{4}{c}{} \\[-1.4em] +\hline +$4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \times 4 \times \mathrm{X}$ \T \\ +\hline +0.5 (0) & 0.38 (0) & 0.32 (0.0012) & 0.9 (0) \B \\ +\hline +\multicolumn{4}{c}{} \\[-1.4em] +\cline{2-4} + & $5 \times 5 \times 5$ & $6 \times 6 \times 6$ & $\mathrm{Y} \times \mathrm{Y} \times \mathrm{Y}$ \T \\ +\cline{2-4} + & 0.47 (0) & \textcolor{red}{-0.01} (0.8803) & 0.89 (0) \B \\ +\cline{2-4} +\multicolumn{4}{c}{} \\[-1.4em] +\cline{2-4} +\multicolumn{3}{c}{} & all: 0.95 (0) \T +\end{tabular} +\caption[Correlation between improvement--potential and fitting--error for 3D]{Correlation +between improvement--potential and fitting--error for the 3D fitting scenario. +Displayed are the negated Spearman coefficients with the corresponding p--values +in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$). +\newline Note: Not significant results are marked in \textcolor{red}{red}.} +\label{tab:3dimp} +\end{table} + \begin{figure}[!htb] \centering \includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png} \caption[Improvement potential for different 3D--grids]{ -**BLINDTEXT** -} +Plots of improvement potential against error given by our fitness--function +after convergence together with a linear fit of each of the plotted data to +indicate trends.} \label{fig:resimp3d} \end{figure} @@ -1160,4 +1261,5 @@ As can be seen from figure \ref{fig:resreg3d}, we can observe\todo{things}. - Regularity ist kacke für unser setup. Bessere Vorschläge? EW/EV? \improvement[inline]{Bibliotheksverzeichnis links anpassen. 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CKxUf& diff --git a/arbeit/ma.tex b/arbeit/ma.tex index 8680b27..c0a08e0 100644 --- a/arbeit/ma.tex +++ b/arbeit/ma.tex @@ -210,18 +210,19 @@ evolution is strongly tied to the notion of the problem has serious implications on the convergence speed and the quality of the solution\cite{Rothlauf2006}. However, there is no consensus on how \emph{evolvability} is defined and the meaning varies -from context to context\cite{richter2015evolvability}, so there is need -for some criteria we can measure, so that we are able to compare -different representations to learn and improve upon these. +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. 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 object will get deformed locally (due to the +move such a point the object will get deformed only locally (due to the \ac{RBF}). As this results in a simple mapping from the parameter-space onto the object one can try out different representations of the same -object and evaluate the \emph{evolvability}. This is exactly what -Richter et al.\cite{anrichterEvol} have done. +object and evaluate which criteria may be suited to describe this notion +of \emph{evolvability}. This is exactly what Richter et +al.\cite{anrichterEvol} have done. As we transfer the results of Richter et al.\cite{anrichterEvol} from using \acf{RBF} as a representation to manipulate geometric objects to @@ -244,18 +245,17 @@ for a one--dimensional line (in \ref{sec:back:ffd}) and discuss why this is a sensible deformation function (in \ref{sec:back:ffdgood}). 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}). In a third step we take a look at the -definition of the different evolvability criteria established in -\cite{anrichterEvol}. +\ref{sec:back:evogood}) followed by the definition of the different +evolvability criteria established in \cite{anrichterEvol} (in +\ref {sec:back: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}, we describe the different scenarios we -use to evaluate the different evolvability--criteria incorporating all -aspects introduced in Chapter \ref{sec:back}. Following that, we -evaluate the results in Chapter \ref{sec:res} with further on discussion -in Chapter \ref{sec:dis}. +\ac{FFD} and the adaptation for 3D--meshes that were used. Next, in +Chapter \ref{sec:eval}, we describe the different scenarios we use to +evaluate the different evolvability--criteria incorporating all aspects +introduced in Chapter \ref{sec:back}. Following that, we evaluate the +results in Chapter \ref{sec:res} with further on discussion, summary and +outlook in Chapter \ref{sec:dis}. \chapter{Background}\label{background} @@ -275,10 +275,10 @@ The main idea of \ac{FFD} is to create a function \(s : [0,1[^d \mapsto \mathbb{R}^d\) that spans a certain part of a vector--space and is only linearly parametrized by some special control points \(p_i\) and an constant attribution--function \(a_i(u)\), so \[ -s(u) = \sum_i a_i(u) p_i +s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i} \] can be thought of a representation of the inside of the convex hull generated by the control points where each point can be accessed by the -right \(u \in [0,1[\). +right \(u \in [0,1[^d\). \begin{figure}[!ht] \begin{center} @@ -289,9 +289,9 @@ corresponding deformation to generate a deformed objet} \label{fig:bspline} \end{figure} -In the example in figure~\ref{fig:bspline}, the control--points are -indicated as red dots and the color-gradient should hint at the -\(u\)--values ranging from \(0\) to \(1\). +In the 1--dimensional example in figure~\ref{fig:bspline}, the +control--points are indicated as red dots and the color-gradient should +hint at the \(u\)--values ranging from \(0\) to \(1\). We now define a \acf{FFD} by the following:\\ Given an arbitrary number of points \(p_i\) alongside a line, we map a @@ -458,7 +458,7 @@ space \(M\) (usually \(M = \mathbb{R}\)) along a convergence--function Biologically speaking the set \(I\) corresponds to the set of possible \emph{Genotypes} while \(M\) represents the possible observable -\emph{Phenotypes}. +\emph{Phenotypes}. \improvement[inline]{Erklären, was das ist. Quellen!} The main algorithm just repeats the following steps: @@ -486,9 +486,9 @@ 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. 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). +that can be changed over time. + +\improvement[inline]{Genauer: Welche? Wo? Wieso? ...} \section{Advantages of evolutionary algorithms}\label{advantages-of-evolutionary-algorithms} @@ -497,15 +497,10 @@ algorithms}\label{advantages-of-evolutionary-algorithms} The main advantage of evolutionary algorithms is the ability to find optima of general functions just with the help of a given -fitness--function. With this most problems of simple gradient--based -procedures, which often target the same error--function which measures -the fitness, as an evolutionary algorithm, but can easily get stuck in -local optima. - -Components and techniques for evolutionary algorithms are specifically -known to help with different problems arising in the domain of -optimization\cite{weise2012evolutionary}. An overview of the typical -problems are shown in figure \ref{fig:probhard}. +fitness--function. Components and techniques for evolutionary algorithms +are specifically known to help with different problems arising in the +domain of optimization\cite{weise2012evolutionary}. An overview of the +typical problems are shown in figure \ref{fig:probhard}. \begin{figure}[!ht] \includegraphics[width=\textwidth]{img/weise_fig3.png} @@ -517,10 +512,13 @@ 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, keeps the best solutions and purges the worst, it can -also target multiple different hypothesis at the same time where the +solution randomly, keeping the best solutions and purging the worst, 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 +ausführlicher.} + If an analytic best solution exists and is easily computable (i.e.~because the error--function is convex) an evolutionary algorithm is not the right choice. Although both converge to the same solution, @@ -530,8 +528,9 @@ But in reality many problems have no analytic solution, because the problem is either not convex or there are so many parameters that an analytic solution (mostly meaning the equivalence to an exhaustive search) is computationally not feasible. Here evolutionary optimization -has one more advantage as you can at least get suboptimal solutions -fast, which then refine over time. +has one more advantage as one can at least get suboptimal solutions +fast, which then refine over time and still converge to the same +solution. \section{Criteria for the evolvability of linear deformations}\label{criteria-for-the-evolvability-of-linear-deformations} @@ -540,27 +539,26 @@ 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 \[ -V = UP -\] where \(V\) is a \(n \times d\) matrix of vertices, \(U\) 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{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. We can also think of the deformation in terms of differences from the original coordinates \[ -\Delta V = U \cdot \Delta P +\Delta \vec{V} = \vec{U} \cdot \Delta \vec{P} \] 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 \(U\), which is why the three -criteria focus on this. +behaves lies solely in the entries of \(\vec{U}\), which is why the +three criteria focus on this. \subsection{Variability}\label{variability} In \cite{anrichterEvol} \emph{variability} is defined as -\[V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n},\] whereby \(\vec{U}\) -is the \(n \times m\) deformation--Matrix -\unsure{Nicht $(n\cdot d) \times m$? Wegen $u,v,w$?} used to map the -\(m\) control points onto the \(n\) vertices. +\[\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n},\] +whereby \(\vec{U}\) is the \(n \times m\) deformation--Matrix used to +map the \(m\) control points onto the \(n\) vertices. Given \(n = m\), an identical number of control--points and vertices, this quotient will be \(=1\) if all control points are independent of @@ -570,10 +568,21 @@ onto a target--point. In praxis the value of \(V(\vec{U})\) is typically \(\ll 1\), because as there are only few control--points for many vertices, so \(m \ll n\). +This criterion should correlate to the degrees of freedom the given +parametrization has. This can be seen from the fact, that +\(\mathrm{rank}(\vec{U})\) is limited by \(\min(m,n)\) and --- as \(n\) +is constant --- can never exceed \(n\). + +The rank itself is also interesting, as control--points could +theoretically be placed on top of each other or be linear dependent in +another way --- but will in both cases lower the rank below the number +of control--points \(m\) and are thus measurable by the +\emph{variability}. + \subsection{Regularity}\label{regularity} \emph{Regularity} is defined\cite{anrichterEvol} as -\[R(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}}\] +\[\mathrm{regularity}(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}}\] where \(\sigma_{min}\) and \(\sigma_{max}\) are the smallest and greatest right singular value of the deformation--matrix \(\vec{U}\). @@ -593,18 +602,21 @@ locality\cite{weise2012evolutionary,thorhauer2014locality}. \subsection{Improvement Potential}\label{improvement-potential} In contrast to the general nature of \emph{variability} and -\emph{regularity}, which are agnostic of the fitness--function at hand -the third criterion should reflect a notion of potential. +\emph{regularity}, which are agnostic of the fitness--function at hand, +the third criterion should reflect a notion of the potential for +optimization, taking a guess into account. -As during optimization some kind of gradient \(g\) is available to -suggest a direction worth pursuing we use this to guess how much change -can be achieved in the given direction. +Most of the times some kind of gradient \(g\) is available to suggest a +direction worth pursuing; either from a previous iteration or by +educated guessing. We use this to guess how much change can be achieved +in the given direction. The definition for an \emph{improvement potential} \(P\) is\cite{anrichterEvol}: \[ -P(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F -\] given some approximate \(n \times d\) fitness--gradient \(\vec{G}\), -normalized to \(\|\vec{G}\|_F = 1\), whereby \(\|\cdot\|_F\) denotes the +\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. \chapter{\texorpdfstring{Implementation of @@ -658,7 +670,8 @@ 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} as the +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 (\(\#v~\gg~\#c\)). @@ -727,13 +740,31 @@ With the Gauss--Newton algorithm we iterate via the formula and use Cramers rule for inverting the small Jacobian and solving this system of linear equations. +As there is no strict upper bound of the number of iterations for this +algorithm, we just iterate it long enough to be within the given +\(\epsilon\)--error above. This takes --- depending on the shape of the +object and the grid --- about \(3\) to \(5\) iterations that we observed +in practice. + +Another issue that we observed in our implementation is, that multiple +local optima may exist on self--intersecting grids. We solve this +problem by defining self--intersecting grids to be \emph{invalid} and do +not test any of them. + +This is not such a big problem as it sounds at first, as +self--intersections mean, that control--points being further away from a +given vertex have more influence over the deformation than +control--points closer to this vertex. Also this contradicts the notion +of locality that we want to achieve and deemed beneficial for a good +behaviour of the evolutionary algorithm. + \section{Deformation Grid}\label{deformation-grid} \label{sec:impl:grid} As mentioned in chapter \ref{sec:back:evo}, the way of choosing the representation to map the general problem (mesh--fitting/optimization in -our case) into a parameter-space it very important for the quality and +our case) into a parameter-space is very important for the quality and runtime of evolutionary algorithms\cite{Rothlauf2006}. Because our control--points are arranged in a grid, we can accurately @@ -742,10 +773,11 @@ B--Spline--coefficients between \([0,1[\) and --- as a consequence --- we have to embed our object into it (or create constant ``dummy''-points outside). -The great advantage of B--Splines is the locality, direct impact of each +The great advantage of B--Splines is the local, direct impact of each control point without having a \(1:1\)--correlation, and a smooth deformation. While the advantages are great, the issues arise from the -problem to decide where to place the control--points and how many. +problem to decide where to place the control--points and how many to +place at all. \begin{figure}[!tbh] \centering @@ -760,7 +792,7 @@ control--points.} One would normally think, that the more control--points you add, the better the result will be, but this is not the case for our B--Splines. -Given any point \(p\) only the \(2 \cdot (d-1)\) control--points +Given any point \(\vec{p}\) only the \(2 \cdot (d-1)\) control--points contribute to the parametrization of that point\footnote{Normally these are \(d-1\) to each side, but at the boundaries the number gets increased to the inside to meet the required smoothness}. This means, @@ -770,10 +802,21 @@ irrelevant to the solution. We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the four red central points are not relevant for the parametrization of the -circle. +circle. This leads to artefacts in the deformation--matrix \(\vec{U}\), +as the columns corresponding to those control--points are \(0\). -\unsure[inline]{erwähnen, dass man aus $\vec{D}$ einfach die Null--Spalten -entfernen kann?} +This leads to useless increased complexity, as the parameters +corresponding to those points will never have any effect, but a naive +algorithm will still try to optimize them yielding numeric artefacts in +the best and non--terminating or ill--defined solutions\footnote{One + example would be, when parts of an algorithm depend on the inverse of + the minimal right singular value leading to a division by \(0\).} at +worst. + +One can of course neglect those columns and their corresponding +control--points, but this raises the question why they were introduced +in the first place. We will address this in a special scenario in +\ref{sec:res:3d:var}. For our tests we chose different uniformly sized grids and added noise onto each control-point\footnote{For the special case of the outer layer @@ -781,23 +824,21 @@ onto each control-point\footnote{For the special case of the outer layer confined in the convex hull of the control--points.} to simulate different starting-conditions. -\unsure[inline]{verweis auf DM--FFD?} - \chapter{\texorpdfstring{Scenarios for testing evolvability criteria using -\acf{FFD}}{Scenarios for testing evolvability criteria using }}\label{scenarios-for-testing-evolvability-criteria-using} +\ac{FFD}}{Scenarios for testing evolvability criteria using }}\label{scenarios-for-testing-evolvability-criteria-using} \label{sec:eval} In our experiments we use the same two testing--scenarios, that were also used by \cite{anrichterEvol}. The first scenario deforms a plane into a shape originally defined in \cite{giannelli2012thb}, where we -setup control-points in a 2--dimensional manner merely deform in the +setup control-points in a 2--dimensional manner and merely deform in the height--coordinate to get the resulting shape. In the second scenario we increase the degrees of freedom significantly -by using a 3--dimensional control--grid to deform a sphere into a face. -So each control point has three degrees of freedom in contrast to first +by using a 3--dimensional control--grid to deform a sphere into a face, +so each control point has three degrees of freedom in contrast to first scenario. \section{Test Scenario: 1D Function @@ -835,12 +876,12 @@ As the starting-plane we used the same shape, but set all \(z\)--coordinates to \(0\), yielding a flat plane, which is partially already correct. -Regarding the \emph{fitness--function} \(f(\vec{p})\), we use the very -simple approach of calculating the squared distances for each +Regarding the \emph{fitness--function} \(\mathrm{f}(\vec{p})\), we use +the very simple approach of calculating the squared distances for each corresponding vertex \begin{equation} -\textrm{f(\vec{p})} = \sum_{i=1}^{n} \|(\vec{Up})_i - t_i\|_2^2 = \|\vec{Up} - \vec{t}\|^2 \rightarrow \min +\mathrm{f}(\vec{p}) = \sum_{i=1}^{n} \|(\vec{Up})_i - t_i\|_2^2 = \|\vec{Up} - \vec{t}\|^2 \rightarrow \min \end{equation} where \(t_i\) are the respective target--vertices to the parametrized @@ -862,9 +903,9 @@ Approximation}\label{test-scenario-3d-function-approximation} \label{sec:test:3dfa} Opposed to the 1--dimensional scenario before, the 3--dimensional scenario is much more complex --- not only because we -have more degrees of freedom on each control point, but also because the -\emph{fitness--function} we will use has no known analytic solution and -multiple local minima. +have more degrees of freedom on each control point, but also, because +the \emph{fitness--function} we will use has no known analytic solution +and multiple local minima. \begin{figure}[ht] \begin{center} @@ -884,12 +925,13 @@ Both of these Models can be seen in figure \ref{fig:3dtarget}. Opposed to the 1D--case we cannot map the source and target--vertices in a one--to--one--correspondence, which we especially need for the approximation of the fitting--error. Hence we state that the error of -one vertex is the distance to the closest vertex of the other model. +one vertex is the distance to the closest vertex of the other model and +sum up the error from the respective source and target. We therefore define the \emph{fitness--function} to be: \begin{equation} -f(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} - +\mathrm{f}(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} - \vec{s_i}\|_2^2}_{\textrm{source-to-target--distance}} + \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} - \vec{t_i}\|_2^2}_{\textrm{target-to-source--distance}} @@ -916,10 +958,11 @@ al.\cite[Section 3.2]{aschenbach2015} on similar models and was shown to lead to a more precise fit. The Laplacian \begin{equation} -\textrm{regularization}(\vec{P}) = \frac{1}{\sum_i A_i} \sum_{i=1}^n A_i \cdot \left( \sum_{\vec{s_j} \in \mathcal{N}(\vec{s_i})} w_j \cdot \|\Delta \vec{s_j} - \Delta \vec{\overline{s}_j}\|^2 \right) +\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) \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 @@ -941,21 +984,22 @@ al.\cite{anrichterEvol}, we also use Spearman's rank correlation coefficient. Opposed to other popular coefficients, like the Pearson correlation coefficient, which measures a linear relationship between variables, the Spearmans's coefficient assesses \glqq how well an -arbitrary monotonic function can descripbe the relationship between two +arbitrary monotonic function can describe the relationship between two variables, without making any assumptions about the frequency distribution of the variables\grqq\cite{hauke2011comparison}. As we don't have any prior knowledge if any of the criteria is linear and we are just interested in a monotonic relation between the criteria and their predictive power, the Spearman's coefficient seems to fit out -scenario best. +scenario best and was also used before by Richter et +al.\cite{anrichterEvol} For interpretation of these values we follow the same interpretation used in \cite{anrichterEvol}, based on \cite{weir2015spearman}: The coefficient intervals \(r_S \in [0,0.2[\), \([0.2,0.4[\), \([0.4,0.6[\), \([0.6,0.8[\), and \([0.8,1]\) are classified as \emph{very weak}, \emph{weak}, \emph{moderate}, \emph{strong} and \emph{very strong}. We -interpret p--values smaller than \(0.1\) as \emph{significant} and cut +interpret p--values smaller than \(0.01\) as \emph{significant} and cut off the precision of p--values after four decimal digits (thus often having a p--value of \(0\) given for p--values \(< 10^{-4}\)). @@ -984,8 +1028,9 @@ use as guess for the \emph{improvement potential}. To check we also consider a distorted gradient \(\vec{g}_{\textrm{d}}\) \[ \vec{g}_{\textrm{d}} = \frac{\vec{g}_{\textrm{c}} + \mathbb{1}}{\|\vec{g}_{\textrm{c}} + \mathbb{1}\|} \] where \(\mathbb{1}\) is the vector consisting of \(1\) in every -dimension and \(\vec{g}_\textrm{c} = \vec{p^{*}}\) the calculated -correct gradient. +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 \(\vec{g}_\textrm{c} = \vec{p^{*}}\). \begin{figure}[ht] \begin{center} @@ -1000,11 +1045,8 @@ We then set up a regular 2--dimensional grid around the object with the desired grid resolutions. To generate a testcase we then move the grid--vertices 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. - -This is a reasonable thing to do, as self-intersecting grids violate our -desired property of locality, as the then farther away control--point -has more influence over some vertices as the next-closer. +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 @@ -1026,22 +1068,23 @@ to experimentally evaluate the quality criteria we introduced before. As an evolutional optimization is partially a random process, we use the analytical solution as a stopping-criteria. We measure the convergence speed as number of iterations the evolutional algorithm needed to get -within \(1.05\%\) of the optimal solution. +within \(1.05 \times\) of the optimal solution. We used different regular grids that we manipulated as explained in Section \ref{sec:proc:1d} with a different number of control points. As our grids have to be the product of two integers, we compared a \(5 \times 5\)--grid with \(25\) control--points to a \(4 \times 7\) and \(7 \times 4\)--grid with \(28\) control--points. This was done to -measure the impact an \glqq improper\grqq -setup could have and how well this is displayed in the criteria we are -examining. +measure the impact an \glqq improper\grqq ~ setup could have and how +well this is displayed in the criteria we are examining. Additionally we also measured the effect of increasing the total resolution of the grid by taking a closer look at \(5 \times 5\), \(7 \times 7\) and \(10 \times 10\) grids. -\begin{figure}[ht] +\subsection{Variability}\label{variability-1} + +\begin{figure}[tbh] \centering \includegraphics[width=0.7\textwidth]{img/evolution1d/variability_boxplot.png} \caption[1D Fitting Errors for various grids]{The squared error for the various @@ -1050,8 +1093,6 @@ Note that $7 \times 4$ and $4 \times 7$ have the same number of control--points. \label{fig:1dvar} \end{figure} -\subsection{Variability}\label{variability-1} - Variability should characterize the potential for design space exploration and is defined in terms of the normalized rank of the deformation matrix \(\vec{U}\): @@ -1063,11 +1104,12 @@ plotted the errors in the boxplot in figure \ref{fig:1dvar} It is also noticeable, that although the \(7 \times 4\) and \(4 \times 7\) grids have a higher variability, they perform not better than the \(5 \times 5\) grid. Also the \(7 \times 4\) and \(4 \times 7\) -grids differ distinctly from each other, although they have the same -number of control--points. This is an indication the impact a proper or -improper grid--setup can have. We do not draw scientific conclusions -from these findings, as more research on non-squared grids seem -necessary.\todo{machen wir die noch? :D} +grids differ distinctly from each other with a mean\(\pm\)sigma of +\(233.09 \pm 12.32\) for the former and \(286.32 \pm 22.36\) for the +latter, although they have the same number of control--points. This is +an indication of an impact a proper or improper grid--setup can have. We +do not draw scientific conclusions from these findings, as more research +on non-squared grids seem necessary. Leaving the issue of the grid--layout aside we focused on grids having the same number of prototypes in every dimension. For the @@ -1077,7 +1119,7 @@ variability and the evolutionary error. \subsection{Regularity}\label{regularity-1} -\begin{figure}[ht] +\begin{figure}[tbh] \centering \includegraphics[width=\textwidth]{img/evolution1d/55_to_1010_steps.png} \caption[Improvement potential and regularity vs. steps]{\newline @@ -1185,14 +1227,15 @@ control--points.} For the next step we then halve the regularization--impact \(\lambda\) (starting at \(1\)) of our \emph{fitness--function} (\ref{eq:fit3d}) and calculate the next incremental solution -\(\vec{P^{*}} = \vec{U^+}\vec{T}\) with the updated correspondences to -get our next target--error. We repeat this process as long as the -target--error keeps decreasing and use the number of these iterations as -measure of the convergence speed. As the resulting evolutional error -without regularization is in the numeric range of \(\approx 100\), -whereas the regularization is numerically \(\approx 7000\) we need at -least \(10\) to \(15\) iterations until the regularization--effect wears -off. +\(\vec{P^{*}} = \vec{U^+}\vec{T}\) with the updated correspondences +(again, mapping each vertex to its closest neighbour in the respective +other model) to get our next target--error. We repeat this process as +long as the target--error keeps decreasing and use the number of these +iterations as measure of the convergence speed. As the resulting +evolutional error without regularization is in the numeric range of +\(\approx 100\), whereas the regularization is numerically +\(\approx 7000\) we need at least \(10\) to \(15\) iterations until the +regularization--effect wears off. The grid we use for our experiments is just very coarse due to computational limitations. We are not interested in a good @@ -1201,10 +1244,10 @@ are good. In figure \ref{fig:setup3d} we show an example setup of the scene with a \(4\times 4\times 4\)--grid. Identical to the 1--dimensional scenario -before, we create a regular grid and move the control-points \todo{wie?} -random between their neighbours, but in three instead of two -dimensions\footnote{Again, we flip the signs for the edges, if necessary - to have the object still in the convex hull.}. +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.}. \begin{figure}[!htb] \includegraphics[width=\textwidth]{img/3d_grid_resolution.png} @@ -1250,6 +1293,8 @@ control--points.} \subsection{Variability}\label{variability-2} +\label{sec:res:3d:var} + \begin{table}[tbh] \centering \begin{tabular}{c|c|c|c} @@ -1288,13 +1333,14 @@ the variability via the rank of the deformation--matrix. \centering \includegraphics[width=0.8\textwidth]{img/evolution3d/variability2_boxplot.png} \caption[Histogram of ranks of high--resolution deformation--matrices]{ -Histogram of ranks of various $10 \times 10 \times 10$ grids. +Histogram of ranks of various $10 \times 10 \times 10$ grids with $1000$ +control--points each. } \label{fig:histrank3d} \end{figure} Overall the correlation between variability and fitness--error were -\emph{significantly} and showed a \emph{very strong} correlation in all +\emph{significant} and showed a \emph{very strong} correlation in all our tests. The detailed correlation--coefficients are given in table \ref{tab:3dvar} alongside their p--values. @@ -1354,20 +1400,22 @@ between regularity and number of iterations for the 3D fitting scenario. Displayed are the negated Spearman coefficients with the corresponding p--values in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$). \newline Note: Not significant results are marked in \textcolor{red}{red}.} -\label{tab:3dvar} +\label{tab:3dreg} \end{table} Opposed to the predictions of variability our test on regularity gave a mixed result --- similar to the 1D--case. -In half scenarios we have a \emph{significant}, but \emph{weak} to -\emph{moderate} correlation between regularity and number of iterations. -On the other hand in the scenarios where we increased the number of -control--points, namely \(125\) for the \(5 \times 5 \times 5\) grid and -\(216\) for the \(6 \times 6 \times 6\) grid we found a -\emph{significant}, but \emph{weak} anti--correlation, which seem to +In roughly half of the scenarios we have a \emph{significant}, but +\emph{weak} to \emph{moderate} correlation between regularity and number +of iterations. On the other hand in the scenarios where we increased the +number of control--points, namely \(125\) for the +\(5 \times 5 \times 5\) grid and \(216\) for the \(6 \times 6 \times 6\) +grid we found a \emph{significant}, but \emph{weak} +\textbf{anti}--correlation when taking all three tests into +account\footnote{Displayed as \(Y \times Y \times Y\)}, which seem to contradict the findings/trends for the sets with \(64\), \(80\), and -\(112\) control--points (first two rows of table \ref{tab:3dvar}). +\(112\) control--points (first two rows of table \ref{tab:3dreg}). Taking all results together we only find a \emph{very weak}, but \emph{significant} link between regularity and the number of iterations @@ -1377,22 +1425,79 @@ needed for the algorithm to converge. \centering \includegraphics[width=\textwidth]{img/evolution3d/regularity_montage.png} \caption[Regularity for different 3D--grids]{ -**BLINDTEXT** -} +Plots of regularity against number of iterations for various scenarios together +with a linear fit to indicate trends.} \label{fig:resreg3d} \end{figure} -As can be seen from figure \ref{fig:resreg3d}, we can -observe\todo{things}. +As can be seen from figure \ref{fig:resreg3d}, we can observe that +increasing the number of control--points helps the convergence--speeds. +The regularity--criterion first behaves as we would like to, but then +switches to behave exactly opposite to our expectations, as can be seen +in the first three plots. While the number of control--points increases +from red to green to blue and the number of iterations decreases, the +regularity seems to increase at first, but then decreases again on +higher grid--resolutions. + +This can be an artefact of the definition of regularity, as it is +defined by the inverse condition--number of the deformation--matrix +\(\vec{U}\), being the fraction +\(\frac{\sigma_{\mathrm{min}}}{\sigma_{\mathrm{max}}}\) between the +least and greatest right singular value. + +As we observed in the previous section, we cannot guarantee that each +control--point has an effect (see figure \ref{fig:histrank3d}) and so a +small minimal right singular value occurring on higher grid--resolutions +seems likely the problem. + +Adding to this we also noted, that in the case of the +\(10 \times 10 \times 10\)--grid the regularity was always \(0\), as a +non--contributing control-point yields a \(0\)--column in the +deformation--matrix, thus letting \(\sigma_\mathrm{min} = 0\). A better +definition for regularity (i.e.~using the smallest non--zero right +singular value) could solve this particular issue, but not fix the trend +we noticed above. \subsection{Improvement Potential}\label{improvement-potential-2} +\begin{table}[tbh] +\centering +\begin{tabular}{c|c|c|c} + & $5 \times 4 \times 4$ & $7 \times 4 \times 4$ & $\mathrm{X} \times 4 \times 4$ \\ +\cline{2-4} + & 0.3 (0.0023) & \textcolor{red}{0.23} (0.0233) & 0.89 (0) \B \\ +\cline{2-4} +\multicolumn{4}{c}{} \\[-1.4em] +\hline +$4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \times 4 \times \mathrm{X}$ \T \\ +\hline +0.5 (0) & 0.38 (0) & 0.32 (0.0012) & 0.9 (0) \B \\ +\hline +\multicolumn{4}{c}{} \\[-1.4em] +\cline{2-4} + & $5 \times 5 \times 5$ & $6 \times 6 \times 6$ & $\mathrm{Y} \times \mathrm{Y} \times \mathrm{Y}$ \T \\ +\cline{2-4} + & 0.47 (0) & \textcolor{red}{-0.01} (0.8803) & 0.89 (0) \B \\ +\cline{2-4} +\multicolumn{4}{c}{} \\[-1.4em] +\cline{2-4} +\multicolumn{3}{c}{} & all: 0.95 (0) \T +\end{tabular} +\caption[Correlation between improvement--potential and fitting--error for 3D]{Correlation +between improvement--potential and fitting--error for the 3D fitting scenario. +Displayed are the negated Spearman coefficients with the corresponding p--values +in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$). +\newline Note: Not significant results are marked in \textcolor{red}{red}.} +\label{tab:3dimp} +\end{table} + \begin{figure}[!htb] \centering \includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png} \caption[Improvement potential for different 3D--grids]{ -**BLINDTEXT** -} +Plots of improvement potential against error given by our fitness--function +after convergence together with a linear fit of each of the plotted data to +indicate trends.} \label{fig:resimp3d} \end{figure} @@ -1407,7 +1512,8 @@ observe\todo{things}. \end{itemize} \improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt -Direktlinks des Autors.} +Direktlinks des Autors.\newline +Außerdem bricht url über Seitengrenzen den Seitenspiegel.} % \backmatter \cleardoublepage diff --git a/dokumentation/evolution1d/R_mean_med_sigma.sh b/dokumentation/evolution1d/R_mean_med_sigma.sh new file mode 100755 index 0000000..a09e4d1 --- /dev/null +++ b/dokumentation/evolution1d/R_mean_med_sigma.sh @@ -0,0 +1,26 @@ +#!/bin/bash + +# regularity,variability,improvement,"Evolution error",steps +# 6.57581e-05,0.00592209,0.622392,113.016,2368 + +if [[ -f "$2" ]]; then + +R -q --slave --vanilla < " +fi + diff --git a/dokumentation/evolution1d/errors.csv b/dokumentation/evolution1d/errors.csv index 41400f0..001cfd6 100644 --- a/dokumentation/evolution1d/errors.csv +++ b/dokumentation/evolution1d/errors.csv @@ -1,4 +1,4 @@ -"5x5","7x4","4x7","7x7","10x10" +"5x5","4x7","7x4","7x7","10x10" 218.554,280.917,211.096,126.241,15.0742 215.888,315.729,233.828,110.962,19.0281 274.375,264.639,205.276,125.853,11.8948 diff 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a/dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.gnuplot.fit.log b/dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.gnuplot.fit.log index a612cb1..99e8453 100644 --- a/dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.gnuplot.fit.log +++ b/dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.gnuplot.fit.log @@ -1,7 +1,7 @@ ******************************************************************************* -Sun Oct 1 20:12:40 2017 +Fri Oct 27 14:09:07 2017 FIT: data read from "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 @@ -47,7 +47,7 @@ b -0.992 1.000 ******************************************************************************* -Sun Oct 1 20:12:40 2017 +Fri Oct 27 14:09:07 2017 FIT: data read from "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 @@ -93,7 +93,7 @@ bb -1.000 1.000 ******************************************************************************* -Sun Oct 1 20:12:40 2017 +Fri Oct 27 14:09:07 2017 FIT: data read from "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 @@ -136,3 +136,49 @@ correlation matrix of the fit parameters: aaa bbb aaa 1.000 bbb -1.000 1.000 + + +******************************************************************************* +Fri Oct 27 14:09:07 2017 + + +FIT: data read from "20170926_3dFit_4x4x4_100times.csv" every ::1 using 2:4 + 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 : 1.39893e+06 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707119 + +initial set of free parameter values + +aaaa = 1 +bbbb = 1 + +After 3 iterations the fit converged. +final sum of squares of residuals : 2447.69 +rel. change during last iteration : -3.53005e-11 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 4.99764 +variance of residuals (reduced chisquare) = WSSR/ndf : 24.9765 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaaa = 1.69981 +/- 9.656e+16 (5.681e+18%) +bbbb = 119.169 +/- 5.718e+14 (4.798e+14%) + + +correlation matrix of the fit parameters: + + aaaa bbbb +aaaa 1.000 +bbbb -1.000 1.000 diff --git a/dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.gnuplot.log b/dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.gnuplot.log index 801e404..260ef1d 100644 --- a/dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.gnuplot.log +++ b/dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.gnuplot.log @@ -270,3 +270,69 @@ correlation matrix of the fit parameters: aaa bbb aaa 1.000 bbb -1.000 1.000 + + + Iteration 0 + WSSR : 1.39893e+06 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707119 + +initial set of free parameter values + +aaaa = 1 +bbbb = 1 +/ + + Iteration 1 + WSSR : 2482.26 delta(WSSR)/WSSR : -562.573 + delta(WSSR) : -1.39645e+06 limit for stopping : 1e-05 + lambda : 0.0707119 + +resultant parameter values + +aaaa = 1.69632 +bbbb = 118.581 +/ + + Iteration 2 + WSSR : 2447.69 delta(WSSR)/WSSR : -0.0141217 + delta(WSSR) : -34.5656 limit for stopping : 1e-05 + lambda : 0.00707119 + +resultant parameter values + +aaaa = 1.69981 +bbbb = 119.169 +/ + + Iteration 3 + WSSR : 2447.69 delta(WSSR)/WSSR : -3.53005e-11 + delta(WSSR) : -8.64047e-08 limit for stopping : 1e-05 + lambda : 0.000707119 + +resultant parameter values + +aaaa = 1.69981 +bbbb = 119.169 + +After 3 iterations the fit converged. +final sum of squares of residuals : 2447.69 +rel. change during last iteration : -3.53005e-11 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 4.99764 +variance of residuals (reduced chisquare) = WSSR/ndf : 24.9765 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaaa = 1.69981 +/- 9.656e+16 (5.681e+18%) +bbbb = 119.169 +/- 5.718e+14 (4.798e+14%) + + +correlation matrix of the fit parameters: + + aaaa bbbb +aaaa 1.000 +bbbb -1.000 1.000 +Warning: empty x range [0.00592209:0.00592209], adjusting to [0.00586287:0.00598131] diff --git a/dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.gnuplot.script b/dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.gnuplot.script index ec815f2..1e39ddf 100644 --- a/dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.gnuplot.script +++ b/dokumentation/evolution3d/20170926_3dFit_4x4x4_100times.gnuplot.script @@ -2,19 +2,25 @@ set datafile separator "," f(x)=a*x+b fit f(x) "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 via a,b set terminal png -set xlabel 'regularity' -set ylabel 'steps' +set xlabel 'Regularity' +set ylabel 'Number of iterations' set output "20170926_3dFit_4x4x4_100times_regularity-vs-steps.png" plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 title "data", f(x) title "lin. fit" lc rgb "black" g(x)=aa*x+bb fit g(x) "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 via aa,bb -set xlabel 'improvement potential' -set ylabel 'steps' +set xlabel 'Improvement potential' +set ylabel 'Number of iterations' set output "20170926_3dFit_4x4x4_100times_improvement-vs-steps.png" plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 title "data", g(x) title "lin. fit" lc rgb "black" h(x)=aaa*x+bbb fit h(x) "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 via aaa,bbb -set xlabel 'improvement potential' -set ylabel 'evolution error' +set xlabel 'Improvement potential' +set ylabel 'error given by fitness-function' set output "20170926_3dFit_4x4x4_100times_improvement-vs-evo-error.png" plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 title "data", h(x) title "lin. fit" lc rgb "black" +i(x)=aaaa*x+bbbb +fit i(x) "20170926_3dFit_4x4x4_100times.csv" every ::1 using 2:4 via aaaa,bbbb +set xlabel 'Variability' +set ylabel 'error given by fitness-function' +set output 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lzu*l^RCOvd9kro*+YHaNPBQflM=I0jhI(eYCE6|#{|7on5G?=z literal 0 HcmV?d00001 diff --git a/dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.gnuplot.fit.log b/dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.gnuplot.fit.log index 2c351b1..066d286 100644 --- a/dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.gnuplot.fit.log +++ b/dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.gnuplot.fit.log @@ -1,7 +1,7 @@ ******************************************************************************* -Sun Oct 1 20:12:42 2017 +Fri Oct 27 14:09:07 2017 FIT: data read from "20170926_3dFit_5x5x5_100times.csv" every ::1 using 1:5 @@ -47,7 +47,7 @@ b -0.970 1.000 ******************************************************************************* -Sun Oct 1 20:12:42 2017 +Fri Oct 27 14:09:07 2017 FIT: data read from "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:5 @@ -93,7 +93,7 @@ bb -1.000 1.000 ******************************************************************************* -Sun Oct 1 20:12:42 2017 +Fri Oct 27 14:09:07 2017 FIT: data read from "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:4 @@ -136,3 +136,49 @@ correlation matrix of the fit parameters: aaa bbb aaa 1.000 bbb -1.000 1.000 + + +******************************************************************************* +Fri Oct 27 14:09:07 2017 + + +FIT: data read from "20170926_3dFit_5x5x5_100times.csv" every ::1 using 2:4 + 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 : 582860 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707154 + +initial set of free parameter values + +aaaa = 1 +bbbb = 1 + +After 3 iterations the fit converged. +final sum of squares of residuals : 4883.49 +rel. change during last iteration : -7.32216e-12 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 7.05915 +variance of residuals (reduced chisquare) = WSSR/ndf : 49.8315 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaaa = 1.87923 +/- 6.796e+16 (3.616e+18%) +bbbb = 77.0146 +/- 7.861e+14 (1.021e+15%) + + +correlation matrix of the fit parameters: + + aaaa bbbb +aaaa 1.000 +bbbb -1.000 1.000 diff --git a/dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.gnuplot.log b/dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.gnuplot.log index 3ffcc49..1d3b7ce 100644 --- a/dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.gnuplot.log +++ b/dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.gnuplot.log @@ -226,3 +226,69 @@ correlation matrix of the fit parameters: aaa bbb aaa 1.000 bbb -1.000 1.000 + + + Iteration 0 + WSSR : 582860 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707154 + +initial set of free parameter values + +aaaa = 1 +bbbb = 1 +/ + + Iteration 1 + WSSR : 4897.8 delta(WSSR)/WSSR : -118.005 + delta(WSSR) : -577962 limit for stopping : 1e-05 + lambda : 0.0707154 + +resultant parameter values + +aaaa = 1.87486 +bbbb = 76.6364 +/ + + Iteration 2 + WSSR : 4883.49 delta(WSSR)/WSSR : -0.00292946 + delta(WSSR) : -14.306 limit for stopping : 1e-05 + lambda : 0.00707154 + +resultant parameter values + +aaaa = 1.87923 +bbbb = 77.0146 +/ + + Iteration 3 + WSSR : 4883.49 delta(WSSR)/WSSR : -7.32216e-12 + delta(WSSR) : -3.57577e-08 limit for stopping : 1e-05 + lambda : 0.000707154 + +resultant parameter values + +aaaa = 1.87923 +bbbb = 77.0146 + +After 3 iterations the fit converged. +final sum of squares of residuals : 4883.49 +rel. change during last iteration : -7.32216e-12 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 7.05915 +variance of residuals (reduced chisquare) = WSSR/ndf : 49.8315 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaaa = 1.87923 +/- 6.796e+16 (3.616e+18%) +bbbb = 77.0146 +/- 7.861e+14 (1.021e+15%) + + +correlation matrix of the fit parameters: + + aaaa bbbb +aaaa 1.000 +bbbb -1.000 1.000 +Warning: empty x range [0.0115666:0.0115666], adjusting to [0.0114509:0.0116823] diff --git a/dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.gnuplot.script b/dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.gnuplot.script index 595e1dc..c4d5911 100644 --- a/dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.gnuplot.script +++ b/dokumentation/evolution3d/20170926_3dFit_5x5x5_100times.gnuplot.script @@ -2,19 +2,25 @@ set datafile separator "," f(x)=a*x+b fit f(x) "20170926_3dFit_5x5x5_100times.csv" every ::1 using 1:5 via a,b set terminal png -set xlabel 'regularity' -set ylabel 'steps' +set xlabel 'Regularity' +set ylabel 'Number of iterations' set output "20170926_3dFit_5x5x5_100times_regularity-vs-steps.png" plot "20170926_3dFit_5x5x5_100times.csv" every ::1 using 1:5 title "data", f(x) title "lin. fit" lc rgb "black" g(x)=aa*x+bb fit g(x) "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:5 via aa,bb -set xlabel 'improvement potential' -set ylabel 'steps' +set xlabel 'Improvement potential' +set ylabel 'Number of iterations' set output "20170926_3dFit_5x5x5_100times_improvement-vs-steps.png" plot "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:5 title "data", g(x) title "lin. fit" lc rgb "black" h(x)=aaa*x+bbb fit h(x) "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:4 via aaa,bbb -set xlabel 'improvement potential' -set ylabel 'evolution error' +set xlabel 'Improvement potential' +set ylabel 'error given by fitness-function' set output "20170926_3dFit_5x5x5_100times_improvement-vs-evo-error.png" plot "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:4 title "data", h(x) title "lin. fit" lc rgb "black" +i(x)=aaaa*x+bbbb +fit i(x) "20170926_3dFit_5x5x5_100times.csv" every ::1 using 2:4 via aaaa,bbbb +set xlabel 'Variability' +set ylabel 'error given by fitness-function' 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a/dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.gnuplot.fit.log +++ b/dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.gnuplot.fit.log @@ -1,7 +1,7 @@ ******************************************************************************* -Sat Oct 7 11:48:58 2017 +Fri Oct 27 14:09:07 2017 FIT: data read from "20171005_3dFit_7x4x4_100times.csv" every ::1 using 1:5 @@ -47,7 +47,7 @@ b -0.972 1.000 ******************************************************************************* -Sat Oct 7 11:48:58 2017 +Fri Oct 27 14:09:07 2017 FIT: data read from "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:5 @@ -93,7 +93,7 @@ bb -1.000 1.000 ******************************************************************************* -Sat Oct 7 11:48:58 2017 +Fri Oct 27 14:09:07 2017 FIT: data read from "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:4 @@ -136,3 +136,49 @@ correlation matrix of the fit parameters: aaa bbb aaa 1.000 bbb -1.000 1.000 + + +******************************************************************************* +Fri Oct 27 14:09:07 2017 + + +FIT: data read from "20171005_3dFit_7x4x4_100times.csv" every ::1 using 2:4 + 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 : 716707 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707145 + +initial set of free parameter values + +aaaa = 1 +bbbb = 1 + +After 3 iterations the fit converged. +final sum of squares of residuals : 5014.73 +rel. change during last iteration : -8.78131e-12 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 7.15337 +variance of residuals (reduced chisquare) = WSSR/ndf : 51.1707 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaaa = 1.87421 +/- 6.575e+16 (3.508e+18%) +bbbb = 85.3528 +/- 6.814e+14 (7.983e+14%) + + +correlation matrix of the fit parameters: + + aaaa bbbb +aaaa 1.000 +bbbb -1.000 1.000 diff --git a/dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.gnuplot.log b/dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.gnuplot.log index 6a913c4..884c173 100644 --- a/dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.gnuplot.log +++ b/dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.gnuplot.log @@ -259,3 +259,69 @@ correlation matrix of the fit parameters: aaa bbb aaa 1.000 bbb -1.000 1.000 + + + Iteration 0 + WSSR : 716707 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707145 + +initial set of free parameter values + +aaaa = 1 +bbbb = 1 +/ + + Iteration 1 + WSSR : 5032.35 delta(WSSR)/WSSR : -141.42 + delta(WSSR) : -711675 limit for stopping : 1e-05 + lambda : 0.0707145 + +resultant parameter values + +aaaa = 1.86986 +bbbb = 84.9331 +/ + + Iteration 2 + WSSR : 5014.73 delta(WSSR)/WSSR : -0.00351279 + delta(WSSR) : -17.6157 limit for stopping : 1e-05 + lambda : 0.00707145 + +resultant parameter values + +aaaa = 1.87421 +bbbb = 85.3528 +/ + + Iteration 3 + WSSR : 5014.73 delta(WSSR)/WSSR : -8.78131e-12 + delta(WSSR) : -4.40359e-08 limit for stopping : 1e-05 + lambda : 0.000707145 + +resultant parameter values + +aaaa = 1.87421 +bbbb = 85.3528 + +After 3 iterations the fit converged. +final sum of squares of residuals : 5014.73 +rel. change during last iteration : -8.78131e-12 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 7.15337 +variance of residuals (reduced chisquare) = WSSR/ndf : 51.1707 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaaa = 1.87421 +/- 6.575e+16 (3.508e+18%) +bbbb = 85.3528 +/- 6.814e+14 (7.983e+14%) + + +correlation matrix of the fit parameters: + + aaaa bbbb +aaaa 1.000 +bbbb -1.000 1.000 +Warning: empty x range [0.0103637:0.0103637], adjusting to [0.0102601:0.0104673] diff --git a/dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.gnuplot.script b/dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.gnuplot.script index fd5042d..77851c0 100644 --- a/dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.gnuplot.script +++ b/dokumentation/evolution3d/20171005_3dFit_7x4x4_100times.gnuplot.script @@ -2,19 +2,25 @@ set datafile separator "," f(x)=a*x+b fit f(x) "20171005_3dFit_7x4x4_100times.csv" every ::1 using 1:5 via a,b set terminal png -set xlabel 'regularity' -set ylabel 'steps' +set xlabel 'Regularity' +set ylabel 'Number of iterations' set output "20171005_3dFit_7x4x4_100times_regularity-vs-steps.png" plot "20171005_3dFit_7x4x4_100times.csv" every ::1 using 1:5 title "data", f(x) title "lin. fit" lc rgb "black" g(x)=aa*x+bb fit g(x) "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:5 via aa,bb -set xlabel 'improvement potential' -set ylabel 'steps' +set xlabel 'Improvement potential' +set ylabel 'Number of iterations' set output "20171005_3dFit_7x4x4_100times_improvement-vs-steps.png" plot "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:5 title "data", g(x) title "lin. fit" lc rgb "black" h(x)=aaa*x+bbb fit h(x) "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:4 via aaa,bbb -set xlabel 'improvement potential' -set ylabel 'evolution error' +set xlabel 'Improvement potential' +set ylabel 'error given by fitness-function' set output "20171005_3dFit_7x4x4_100times_improvement-vs-evo-error.png" plot "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:4 title "data", h(x) title "lin. fit" lc rgb "black" +i(x)=aaaa*x+bbbb +fit i(x) "20171005_3dFit_7x4x4_100times.csv" every ::1 using 2:4 via aaaa,bbbb +set xlabel 'Variability' +set ylabel 'error given by fitness-function' +set output "20171005_3dFit_7x4x4_100times_variability-vs-evo-error.png" +plot "20171005_3dFit_7x4x4_100times.csv" every ::1 using 2:4 title "data", i(x) title "lin. fit" lc rgb "black" diff --git 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from "20171007_3dFit_all.csv" every ::1 using 3:4 @@ -139,7 +139,7 @@ bbb -0.997 1.000 ******************************************************************************* -Sat Oct 7 12:11:35 2017 +Fri Oct 27 14:09:07 2017 FIT: data read from "20171007_3dFit_all.csv" every ::1 using 2:4 diff --git a/dokumentation/evolution3d/20171007_3dFit_all.gnuplot.script b/dokumentation/evolution3d/20171007_3dFit_all.gnuplot.script index 0997e79..eacb317 100644 --- a/dokumentation/evolution3d/20171007_3dFit_all.gnuplot.script +++ b/dokumentation/evolution3d/20171007_3dFit_all.gnuplot.script @@ -2,25 +2,25 @@ set datafile separator "," f(x)=a*x+b fit f(x) "20171007_3dFit_all.csv" every ::1 using 1:5 via a,b set terminal png -set xlabel 'regularity' -set ylabel 'steps' +set xlabel 'Regularity' +set ylabel 'Number of iterations' set output "20171007_3dFit_all_regularity-vs-steps.png" -plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 title "20170926_3dFit_4x4x4_100times.csv", "20170926_3dFit_5x5x5_100times.csv" every ::1 using 1:5 title "20170926_3dFit_5x5x5_100times.csv", "20171005_3dFit_4x4x5_100times.csv" every ::1 using 1:5 title "20171005_3dFit_4x4x5_100times.csv", "20171005_3dFit_7x4x4_100times.csv" every ::1 using 1:5 title "20171005_3dFit_7x4x4_100times.csv", f(x) title "lin. fit" lc rgb "black" +plot "20171007_3dFit_all.csv" every ::1 using 1:5 title "data", f(x) title "lin. fit" lc rgb "black" g(x)=aa*x+bb fit g(x) "20171007_3dFit_all.csv" every ::1 using 3:5 via aa,bb -set xlabel 'improvement potential' -set ylabel 'steps' +set xlabel 'Improvement potential' +set ylabel 'Number of iterations' set output "20171007_3dFit_all_improvement-vs-steps.png" -plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 title "20170926_3dFit_4x4x4_100times.csv", "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:5 title "20170926_3dFit_5x5x5_100times.csv", "20171005_3dFit_4x4x5_100times.csv" every ::1 using 3:5 title "20171005_3dFit_4x4x5_100times.csv", "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:5 title "20171005_3dFit_7x4x4_100times.csv", g(x) title "lin. fit" lc rgb "black" +plot "20171007_3dFit_all.csv" every ::1 using 3:5 title "data", g(x) title "lin. fit" lc rgb "black" h(x)=aaa*x+bbb fit h(x) "20171007_3dFit_all.csv" every ::1 using 3:4 via aaa,bbb -set xlabel 'improvement potential' -set ylabel 'evolution error' +set xlabel 'Improvement potential' +set ylabel 'error given by fitness-function' set output "20171007_3dFit_all_improvement-vs-evo-error.png" -plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 title "20170926_3dFit_4x4x4_100times.csv", "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:4 title "20170926_3dFit_5x5x5_100times.csv", "20171005_3dFit_4x4x5_100times.csv" every ::1 using 3:4 title "20171005_3dFit_4x4x5_100times.csv", "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:4 title "20171005_3dFit_7x4x4_100times.csv", h(x) title "lin. fit" lc rgb "black" +plot "20171007_3dFit_all.csv" every ::1 using 3:4 title "data", h(x) title "lin. fit" lc rgb "black" i(x)=aaaa*x+bbbb fit i(x) "20171007_3dFit_all.csv" every ::1 using 2:4 via aaaa,bbbb -set xlabel 'variability' -set ylabel 'evolution error' +set xlabel 'Variability' +set ylabel 'error given by fitness-function' set output "20171007_3dFit_all_variability-vs-evo-error.png" -plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 2:4 title "20170926_3dFit_4x4x4_100times.csv", "20170926_3dFit_5x5x5_100times.csv" every ::1 using 2:4 title "20170926_3dFit_5x5x5_100times.csv", "20171005_3dFit_4x4x5_100times.csv" every ::1 using 2:4 title "20171005_3dFit_4x4x5_100times.csv", "20171005_3dFit_7x4x4_100times.csv" every ::1 using 2:4 title "20171005_3dFit_7x4x4_100times.csv", i(x) title "lin. fit" lc rgb "black" +plot "20171007_3dFit_all.csv" every ::1 using 2:4 title "data", i(x) title "lin. fit" lc rgb "black" diff --git a/dokumentation/evolution3d/20171007_3dFit_all_improvement-vs-evo-error.png 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+******************************************************************************* +Fri Oct 27 14:09:07 2017 + + +FIT: data read from "20171013_3dFit_4x4x7_100times.csv" every ::1 using 1:5 + 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 : 4.17059e+08 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707107 + +initial set of free parameter values + +a = 1 +b = 1 + +After 7 iterations the fit converged. +final sum of squares of residuals : 1.87465e+07 +rel. change during last iteration : -3.45833e-09 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 437.368 +variance of residuals (reduced chisquare) = WSSR/ndf : 191291 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +a = -1.04278e+07 +/- 2.15e+06 (20.62%) +b = 2804.5 +/- 174.4 (6.22%) + + +correlation matrix of the fit parameters: + + a b +a 1.000 +b -0.968 1.000 + + +******************************************************************************* +Fri Oct 27 14:09:07 2017 + + +FIT: data read from "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:5 + 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 : 4.16784e+08 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.860178 + +initial set of free parameter values + +aa = 1 +bb = 1 + +After 4 iterations the fit converged. +final sum of squares of residuals : 2.30326e+07 +rel. change during last iteration : -2.46431e-06 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 484.795 +variance of residuals (reduced chisquare) = WSSR/ndf : 235026 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aa = 7203.94 +/- 7544 (104.7%) +bb = -3004.38 +/- 5226 (173.9%) + + +correlation matrix of the fit parameters: + + aa bb +aa 1.000 +bb -1.000 1.000 + + +******************************************************************************* +Fri Oct 27 14:09:07 2017 + + +FIT: data read from "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:4 + 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 : 770224 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.860178 + +initial set of free parameter values + +aaa = 1 +bbb = 1 + +After 5 iterations the fit converged. +final sum of squares of residuals : 3831.77 +rel. change during last iteration : -2.81552e-12 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 6.25298 +variance of residuals (reduced chisquare) = WSSR/ndf : 39.0997 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaa = -284.393 +/- 97.31 (34.22%) +bbb = 286.203 +/- 67.4 (23.55%) + + +correlation matrix of the fit parameters: + + aaa bbb +aaa 1.000 +bbb -1.000 1.000 + + +******************************************************************************* +Fri Oct 27 14:09:07 2017 + + +FIT: data read from "20171013_3dFit_4x4x7_100times.csv" every ::1 using 2:4 + 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 : 782212 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707145 + +initial set of free parameter values + +aaaa = 1 +bbbb = 1 + +After 3 iterations the fit converged. +final sum of squares of residuals : 4165.76 +rel. change during last iteration : -1.15562e-11 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 6.51979 +variance of residuals (reduced chisquare) = WSSR/ndf : 42.5077 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaaa = 1.91405 +/- 1.245e+17 (6.505e+18%) +bbbb = 89.1974 +/- 1.29e+15 (1.447e+15%) + + +correlation matrix of the fit parameters: + + aaaa bbbb +aaaa 1.000 +bbbb -1.000 1.000 diff --git a/dokumentation/evolution3d/20171013_3dFit_4x4x7_100times.gnuplot.log b/dokumentation/evolution3d/20171013_3dFit_4x4x7_100times.gnuplot.log new file mode 100644 index 0000000..0f30cae --- /dev/null +++ b/dokumentation/evolution3d/20171013_3dFit_4x4x7_100times.gnuplot.log @@ -0,0 +1,338 @@ + + + Iteration 0 + WSSR : 4.17059e+08 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707107 + +initial set of free parameter values + +a = 1 +b = 1 +/ + + Iteration 1 + WSSR : 2.32566e+07 delta(WSSR)/WSSR : -16.9329 + delta(WSSR) : -3.93802e+08 limit for stopping : 1e-05 + lambda : 0.0707107 + +resultant parameter values + +a = 0.291954 +b = 1975.6 +/ + + Iteration 2 + WSSR : 2.32468e+07 delta(WSSR)/WSSR : -0.000422514 + delta(WSSR) : -9822.08 limit for stopping : 1e-05 + lambda : 0.00707107 + +resultant parameter values + +a = -86.0202 +b = 1985.48 +/ + + Iteration 3 + WSSR : 2.32393e+07 delta(WSSR)/WSSR : -0.000320176 + delta(WSSR) : -7440.69 limit for stopping : 1e-05 + lambda : 0.000707107 + +resultant parameter values + +a = -8710.18 +b = 1986.15 +/ + + Iteration 4 + WSSR : 2.25787e+07 delta(WSSR)/WSSR : -0.0292598 + delta(WSSR) : -660649 limit for stopping : 1e-05 + lambda : 7.07107e-05 + +resultant parameter values + +a = -805199 +b = 2048.71 +/ + + Iteration 5 + WSSR : 1.87911e+07 delta(WSSR)/WSSR : -0.201566 + delta(WSSR) : -3.78764e+06 limit for stopping : 1e-05 + lambda : 7.07107e-06 + +resultant parameter values + +a = -9.39061e+06 +b = 2723.04 +/ + + Iteration 6 + WSSR : 1.87465e+07 delta(WSSR)/WSSR : -0.00237512 + delta(WSSR) : -44525.2 limit for stopping : 1e-05 + lambda : 7.07107e-07 + +resultant parameter values + +a = -1.04266e+07 +b = 2804.4 +/ + + Iteration 7 + WSSR : 1.87465e+07 delta(WSSR)/WSSR : -3.45833e-09 + delta(WSSR) : -0.0648317 limit for stopping : 1e-05 + lambda : 7.07107e-08 + +resultant parameter values + +a = -1.04278e+07 +b = 2804.5 + +After 7 iterations the fit converged. +final sum of squares of residuals : 1.87465e+07 +rel. change during last iteration : -3.45833e-09 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 437.368 +variance of residuals (reduced chisquare) = WSSR/ndf : 191291 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +a = -1.04278e+07 +/- 2.15e+06 (20.62%) +b = 2804.5 +/- 174.4 (6.22%) + + +correlation matrix of the fit parameters: + + a b +a 1.000 +b -0.968 1.000 + + + Iteration 0 + WSSR : 4.16784e+08 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.860178 + +initial set of free parameter values + +aa = 1 +bb = 1 +/ + + Iteration 1 + WSSR : 2.32036e+07 delta(WSSR)/WSSR : -16.962 + delta(WSSR) : -3.9358e+08 limit for stopping : 1e-05 + lambda : 0.0860178 + +resultant parameter values + +aa = 948.6 +bb = 1318.67 +/ + + Iteration 2 + WSSR : 2.31176e+07 delta(WSSR)/WSSR : -0.00372074 + delta(WSSR) : -86014.6 limit for stopping : 1e-05 + lambda : 0.00860178 + +resultant parameter values + +aa = 2665.07 +bb = 139.584 +/ + + Iteration 3 + WSSR : 2.30326e+07 delta(WSSR)/WSSR : -0.00369116 + delta(WSSR) : -85017 limit for stopping : 1e-05 + lambda : 0.000860178 + +resultant parameter values + +aa = 7086.74 +bb = -2923.19 +/ + + Iteration 4 + WSSR : 2.30326e+07 delta(WSSR)/WSSR : -2.46431e-06 + delta(WSSR) : -56.7593 limit for stopping : 1e-05 + lambda : 8.60178e-05 + +resultant parameter values + +aa = 7203.94 +bb = -3004.38 + +After 4 iterations the fit converged. +final sum of squares of residuals : 2.30326e+07 +rel. change during last iteration : -2.46431e-06 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 484.795 +variance of residuals (reduced chisquare) = WSSR/ndf : 235026 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aa = 7203.94 +/- 7544 (104.7%) +bb = -3004.38 +/- 5226 (173.9%) + + +correlation matrix of the fit parameters: + + aa bb +aa 1.000 +bb -1.000 1.000 + + + Iteration 0 + WSSR : 770224 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.860178 + +initial set of free parameter values + +aaa = 1 +bbb = 1 +/ + + Iteration 1 + WSSR : 4287.26 delta(WSSR)/WSSR : -178.654 + delta(WSSR) : -765937 limit for stopping : 1e-05 + lambda : 0.0860178 + +resultant parameter values + +aaa = 40.5365 +bbb = 60.6977 +/ + + Iteration 2 + WSSR : 4061.95 delta(WSSR)/WSSR : -0.0554706 + delta(WSSR) : -225.318 limit for stopping : 1e-05 + lambda : 0.00860178 + +resultant parameter values + +aaa = -48.3014 +bbb = 122.669 +/ + + Iteration 3 + WSSR : 3831.93 delta(WSSR)/WSSR : -0.0600269 + delta(WSSR) : -230.019 limit for stopping : 1e-05 + lambda : 0.000860178 + +resultant parameter values + +aaa = -278.294 +bbb = 281.979 +/ + + Iteration 4 + WSSR : 3831.77 delta(WSSR)/WSSR : -4.00769e-05 + delta(WSSR) : -0.153566 limit for stopping : 1e-05 + lambda : 8.60178e-05 + +resultant parameter values + +aaa = -284.391 +bbb = 286.202 +/ + + Iteration 5 + WSSR : 3831.77 delta(WSSR)/WSSR : -2.81552e-12 + delta(WSSR) : -1.07884e-08 limit for stopping : 1e-05 + lambda : 8.60178e-06 + +resultant parameter values + +aaa = -284.393 +bbb = 286.203 + +After 5 iterations the fit converged. +final sum of squares of residuals : 3831.77 +rel. change during last iteration : -2.81552e-12 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 6.25298 +variance of residuals (reduced chisquare) = WSSR/ndf : 39.0997 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaa = -284.393 +/- 97.31 (34.22%) +bbb = 286.203 +/- 67.4 (23.55%) + + +correlation matrix of the fit parameters: + + aaa bbb +aaa 1.000 +bbb -1.000 1.000 + + + Iteration 0 + WSSR : 782212 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707145 + +initial set of free parameter values + +aaaa = 1 +bbbb = 1 +/ + + Iteration 1 + WSSR : 4185.02 delta(WSSR)/WSSR : -185.908 + delta(WSSR) : -778027 limit for stopping : 1e-05 + lambda : 0.0707145 + +resultant parameter values + +aaaa = 1.9095 +bbbb = 88.7586 +/ + + Iteration 2 + WSSR : 4165.76 delta(WSSR)/WSSR : -0.00462295 + delta(WSSR) : -19.2581 limit for stopping : 1e-05 + lambda : 0.00707145 + +resultant parameter values + +aaaa = 1.91405 +bbbb = 89.1974 +/ + + Iteration 3 + WSSR : 4165.76 delta(WSSR)/WSSR : -1.15562e-11 + delta(WSSR) : -4.81405e-08 limit for stopping : 1e-05 + lambda : 0.000707145 + +resultant parameter values + +aaaa = 1.91405 +bbbb = 89.1974 + +After 3 iterations the fit converged. +final sum of squares of residuals : 4165.76 +rel. change during last iteration : -1.15562e-11 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 6.51979 +variance of residuals (reduced chisquare) = WSSR/ndf : 42.5077 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaaa = 1.91405 +/- 1.245e+17 (6.505e+18%) +bbbb = 89.1974 +/- 1.29e+15 (1.447e+15%) + + +correlation matrix of the fit parameters: + + aaaa bbbb +aaaa 1.000 +bbbb -1.000 1.000 +Warning: empty x range [0.0103637:0.0103637], adjusting to [0.0102601:0.0104673] diff --git a/dokumentation/evolution3d/20171013_3dFit_4x4x7_100times.gnuplot.script b/dokumentation/evolution3d/20171013_3dFit_4x4x7_100times.gnuplot.script new file mode 100644 index 0000000..c572cee --- /dev/null +++ b/dokumentation/evolution3d/20171013_3dFit_4x4x7_100times.gnuplot.script @@ -0,0 +1,26 @@ +set datafile separator "," +f(x)=a*x+b +fit f(x) "20171013_3dFit_4x4x7_100times.csv" every ::1 using 1:5 via a,b +set terminal png +set xlabel 'Regularity' +set ylabel 'Number of iterations' +set output "20171013_3dFit_4x4x7_100times_regularity-vs-steps.png" +plot "20171013_3dFit_4x4x7_100times.csv" every ::1 using 1:5 title "data", f(x) title "lin. fit" lc rgb "black" +g(x)=aa*x+bb +fit g(x) "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:5 via aa,bb +set xlabel 'Improvement potential' +set ylabel 'Number of iterations' +set output "20171013_3dFit_4x4x7_100times_improvement-vs-steps.png" +plot "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:5 title "data", g(x) title "lin. fit" lc rgb "black" +h(x)=aaa*x+bbb +fit h(x) "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:4 via aaa,bbb +set xlabel 'Improvement potential' +set ylabel 'error given by fitness-function' +set output "20171013_3dFit_4x4x7_100times_improvement-vs-evo-error.png" +plot "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:4 title "data", h(x) title "lin. fit" lc rgb "black" +i(x)=aaaa*x+bbbb +fit i(x) "20171013_3dFit_4x4x7_100times.csv" every ::1 using 2:4 via aaaa,bbbb +set xlabel 'Variability' +set ylabel 'error given by fitness-function' +set output "20171013_3dFit_4x4x7_100times_variability-vs-evo-error.png" +plot "20171013_3dFit_4x4x7_100times.csv" every ::1 using 2:4 title "data", i(x) title "lin. fit" lc rgb "black" diff --git a/dokumentation/evolution3d/20171013_3dFit_4x4x7_100times_improvement-vs-evo-error.png b/dokumentation/evolution3d/20171013_3dFit_4x4x7_100times_improvement-vs-evo-error.png new file mode 100644 index 0000000000000000000000000000000000000000..f02a4a9f8f224d325333d869a3c3b17cab41f9c4 GIT binary patch literal 6251 zcmb_=c|4R~`~S=|GL{Tu7a3b+7ea`!Q`Rtp82b`Y zh-`x**$dgtcj|e*ujluBUa#MOzx#DB=Q`K9?(;s^d7tfyLmTRxqvfInfk5ZwF|B)HGwUkI`@Mw&o+ufbU14!}<&<1uR 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equally (unit weight) + +function used for fitting: f(x) +fitted parameters initialized with current variable values + + + + Iteration 0 + WSSR : 8.41899e+07 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707107 + +initial set of free parameter values + +a = 1 +b = 1 + +After 7 iterations the fit converged. +final sum of squares of residuals : 8.72636e+06 +rel. change during last iteration : -9.16821e-11 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 298.403 +variance of residuals (reduced chisquare) = WSSR/ndf : 89044.5 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +a = -1.15579e+06 +/- 1.468e+06 (127%) +b = 1020.47 +/- 194.2 (19.03%) + + +correlation matrix of the fit parameters: + + a b +a 1.000 +b -0.988 1.000 + + +******************************************************************************* +Fri Oct 27 14:09:07 2017 + + +FIT: data read from "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:5 + 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 : 8.40737e+07 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.850561 + +initial set of free parameter values + +aa = 1 +bb = 1 + +After 4 iterations the fit converged. +final sum of squares of residuals : 7.83163e+06 +rel. change during last iteration : -4.61976e-06 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 282.692 +variance of residuals (reduced chisquare) = WSSR/ndf : 79914.6 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aa = 10438.1 +/- 3028 (29%) +bb = -6107.9 +/- 2024 (33.14%) + + +correlation matrix of the fit parameters: + + aa bb +aa 1.000 +bb -1.000 1.000 + + +******************************************************************************* +Fri Oct 27 14:09:08 2017 + + +FIT: data read from "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:4 + 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 : 997151 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.850561 + +initial set of free parameter values + +aaa = 1 +bbb = 1 + +After 4 iterations the fit converged. +final sum of squares of residuals : 4984.54 +rel. change during last iteration : -5.99309e-06 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 7.1318 +variance of residuals (reduced chisquare) = WSSR/ndf : 50.8626 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaa = -241.373 +/- 76.38 (31.64%) +bbb = 262.595 +/- 51.06 (19.44%) + + +correlation matrix of the fit parameters: + + aaa bbb +aaa 1.000 +bbb -1.000 1.000 + + +******************************************************************************* +Fri Oct 27 14:09:08 2017 + + +FIT: data read from "20171013_3dFit_5x4x4_100times.csv" every ::1 using 2:4 + 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 : 1.01036e+06 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707126 + +initial set of free parameter values + +aaaa = 1 +bbbb = 1 + +After 3 iterations the fit converged. +final sum of squares of residuals : 5492.5 +rel. change during last iteration : -1.13205e-11 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 7.48638 +variance of residuals (reduced chisquare) = WSSR/ndf : 56.0459 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaaa = 1.74202 +/- 9.406e+16 (5.4e+18%) +bbbb = 101.237 +/- 6.963e+14 (6.878e+14%) + + +correlation matrix of the fit parameters: + + aaaa bbbb +aaaa 1.000 +bbbb -1.000 1.000 diff --git a/dokumentation/evolution3d/20171013_3dFit_5x4x4_100times.gnuplot.log b/dokumentation/evolution3d/20171013_3dFit_5x4x4_100times.gnuplot.log new file mode 100644 index 0000000..c14dfe4 --- /dev/null +++ b/dokumentation/evolution3d/20171013_3dFit_5x4x4_100times.gnuplot.log @@ -0,0 +1,327 @@ + + + Iteration 0 + WSSR : 8.41899e+07 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707107 + +initial set of free parameter values + +a = 1 +b = 1 +/ + + Iteration 1 + WSSR : 8.78343e+06 delta(WSSR)/WSSR : -8.58509 + delta(WSSR) : -7.54065e+07 limit for stopping : 1e-05 + lambda : 0.0707107 + +resultant parameter values + +a = 1.01743 +b = 865.06 +/ + + Iteration 2 + WSSR : 8.78156e+06 delta(WSSR)/WSSR : -0.000212651 + delta(WSSR) : -1867.41 limit for stopping : 1e-05 + lambda : 0.00707107 + +resultant parameter values + +a = -8.53399 +b = 869.381 +/ + + Iteration 3 + WSSR : 8.78147e+06 delta(WSSR)/WSSR : -1.03771e-05 + delta(WSSR) : -91.1264 limit for stopping : 1e-05 + lambda : 0.000707107 + +resultant parameter values + +a = -962.938 +b = 869.506 +/ + + Iteration 4 + WSSR : 8.77338e+06 delta(WSSR)/WSSR : -0.000922382 + delta(WSSR) : -8092.41 limit for stopping : 1e-05 + lambda : 7.07107e-05 + +resultant parameter values + +a = -89117.8 +b = 881.03 +/ + + Iteration 5 + WSSR : 8.72691e+06 delta(WSSR)/WSSR : -0.00532471 + delta(WSSR) : -46468.3 limit for stopping : 1e-05 + lambda : 7.07107e-06 + +resultant parameter values + +a = -1.04065e+06 +b = 1005.42 +/ + + Iteration 6 + WSSR : 8.72636e+06 delta(WSSR)/WSSR : -6.27725e-05 + delta(WSSR) : -547.775 limit for stopping : 1e-05 + lambda : 7.07107e-07 + +resultant parameter values + +a = -1.15565e+06 +b = 1020.45 +/ + + Iteration 7 + WSSR : 8.72636e+06 delta(WSSR)/WSSR : -9.16821e-11 + delta(WSSR) : -0.000800051 limit for stopping : 1e-05 + lambda : 7.07107e-08 + +resultant parameter values + +a = -1.15579e+06 +b = 1020.47 + +After 7 iterations the fit converged. +final sum of squares of residuals : 8.72636e+06 +rel. change during last iteration : -9.16821e-11 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 298.403 +variance of residuals (reduced chisquare) = WSSR/ndf : 89044.5 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +a = -1.15579e+06 +/- 1.468e+06 (127%) +b = 1020.47 +/- 194.2 (19.03%) + + +correlation matrix of the fit parameters: + + a b +a 1.000 +b -0.988 1.000 + + + Iteration 0 + WSSR : 8.40737e+07 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.850561 + +initial set of free parameter values + +aa = 1 +bb = 1 +/ + + Iteration 1 + WSSR : 8.69722e+06 delta(WSSR)/WSSR : -8.66674 + delta(WSSR) : -7.53765e+07 limit for stopping : 1e-05 + lambda : 0.0850561 + +resultant parameter values + +aa = 483.004 +bb = 542.6 +/ + + Iteration 2 + WSSR : 8.08871e+06 delta(WSSR)/WSSR : -0.0752288 + delta(WSSR) : -608504 limit for stopping : 1e-05 + lambda : 0.00850561 + +resultant parameter values + +aa = 5007.98 +bb = -2477.97 +/ + + Iteration 3 + WSSR : 7.83167e+06 delta(WSSR)/WSSR : -0.0328215 + delta(WSSR) : -257047 limit for stopping : 1e-05 + lambda : 0.000850561 + +resultant parameter values + +aa = 10373.7 +bb = -6064.85 +/ + + Iteration 4 + WSSR : 7.83163e+06 delta(WSSR)/WSSR : -4.61976e-06 + delta(WSSR) : -36.1803 limit for stopping : 1e-05 + lambda : 8.50561e-05 + +resultant parameter values + +aa = 10438.1 +bb = -6107.9 + +After 4 iterations the fit converged. +final sum of squares of residuals : 7.83163e+06 +rel. change during last iteration : -4.61976e-06 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 282.692 +variance of residuals (reduced chisquare) = WSSR/ndf : 79914.6 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aa = 10438.1 +/- 3028 (29%) +bb = -6107.9 +/- 2024 (33.14%) + + +correlation matrix of the fit parameters: + + aa bb +aa 1.000 +bb -1.000 1.000 + + + Iteration 0 + WSSR : 997151 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.850561 + +initial set of free parameter values + +aaa = 1 +bbb = 1 +/ + + Iteration 1 + WSSR : 5722.22 delta(WSSR)/WSSR : -173.259 + delta(WSSR) : -991429 limit for stopping : 1e-05 + lambda : 0.0850561 + +resultant parameter values + +aaa = 44.3933 +bbb = 71.0688 +/ + + Iteration 2 + WSSR : 5196.8 delta(WSSR)/WSSR : -0.101105 + delta(WSSR) : -525.422 limit for stopping : 1e-05 + lambda : 0.00850561 + +resultant parameter values + +aaa = -85.3429 +bbb = 158.291 +/ + + Iteration 3 + WSSR : 4984.57 delta(WSSR)/WSSR : -0.0425783 + delta(WSSR) : -212.235 limit for stopping : 1e-05 + lambda : 0.000850561 + +resultant parameter values + +aaa = -239.522 +bbb = 261.358 +/ + + Iteration 4 + WSSR : 4984.54 delta(WSSR)/WSSR : -5.99309e-06 + delta(WSSR) : -0.0298728 limit for stopping : 1e-05 + lambda : 8.50561e-05 + +resultant parameter values + +aaa = -241.373 +bbb = 262.595 + +After 4 iterations the fit converged. +final sum of squares of residuals : 4984.54 +rel. change during last iteration : -5.99309e-06 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 7.1318 +variance of residuals (reduced chisquare) = WSSR/ndf : 50.8626 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaa = -241.373 +/- 76.38 (31.64%) +bbb = 262.595 +/- 51.06 (19.44%) + + +correlation matrix of the fit parameters: + + aaa bbb +aaa 1.000 +bbb -1.000 1.000 + + + Iteration 0 + WSSR : 1.01036e+06 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 0.707126 + +initial set of free parameter values + +aaaa = 1 +bbbb = 1 +/ + + Iteration 1 + WSSR : 5517.37 delta(WSSR)/WSSR : -182.123 + delta(WSSR) : -1.00484e+06 limit for stopping : 1e-05 + lambda : 0.0707126 + +resultant parameter values + +aaaa = 1.73833 +bbbb = 100.739 +/ + + Iteration 2 + WSSR : 5492.5 delta(WSSR)/WSSR : -0.00452841 + delta(WSSR) : -24.8723 limit for stopping : 1e-05 + lambda : 0.00707126 + +resultant parameter values + +aaaa = 1.74202 +bbbb = 101.237 +/ + + Iteration 3 + WSSR : 5492.5 delta(WSSR)/WSSR : -1.13205e-11 + delta(WSSR) : -6.21776e-08 limit for stopping : 1e-05 + lambda : 0.000707126 + +resultant parameter values + +aaaa = 1.74202 +bbbb = 101.237 + +After 3 iterations the fit converged. +final sum of squares of residuals : 5492.5 +rel. change during last iteration : -1.13205e-11 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 7.48638 +variance of residuals (reduced chisquare) = WSSR/ndf : 56.0459 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaaa = 1.74202 +/- 9.406e+16 (5.4e+18%) +bbbb = 101.237 +/- 6.963e+14 (6.878e+14%) + + +correlation matrix of the fit parameters: + + aaaa bbbb +aaaa 1.000 +bbbb -1.000 1.000 +Warning: empty x range [0.00740261:0.00740261], adjusting to [0.00732858:0.00747664] diff --git a/dokumentation/evolution3d/20171013_3dFit_5x4x4_100times.gnuplot.script b/dokumentation/evolution3d/20171013_3dFit_5x4x4_100times.gnuplot.script new file mode 100644 index 0000000..3af7ad3 --- /dev/null +++ b/dokumentation/evolution3d/20171013_3dFit_5x4x4_100times.gnuplot.script @@ -0,0 +1,26 @@ +set datafile separator "," +f(x)=a*x+b +fit f(x) "20171013_3dFit_5x4x4_100times.csv" every ::1 using 1:5 via a,b +set terminal png +set xlabel 'Regularity' +set ylabel 'Number of iterations' +set output "20171013_3dFit_5x4x4_100times_regularity-vs-steps.png" +plot "20171013_3dFit_5x4x4_100times.csv" every ::1 using 1:5 title "data", f(x) title "lin. fit" lc rgb "black" +g(x)=aa*x+bb +fit g(x) "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:5 via aa,bb +set xlabel 'Improvement potential' +set ylabel 'Number of iterations' +set output "20171013_3dFit_5x4x4_100times_improvement-vs-steps.png" +plot "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:5 title "data", g(x) title "lin. fit" lc rgb "black" +h(x)=aaa*x+bbb +fit h(x) "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:4 via aaa,bbb +set xlabel 'Improvement potential' +set ylabel 'error given by fitness-function' +set output "20171013_3dFit_5x4x4_100times_improvement-vs-evo-error.png" +plot "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:4 title "data", h(x) title "lin. fit" lc rgb "black" +i(x)=aaaa*x+bbbb +fit i(x) "20171013_3dFit_5x4x4_100times.csv" every ::1 using 2:4 via aaaa,bbbb +set xlabel 'Variability' +set ylabel 'error given by fitness-function' +set output "20171013_3dFit_5x4x4_100times_variability-vs-evo-error.png" +plot "20171013_3dFit_5x4x4_100times.csv" 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z_DK3jxv`@-j0Ew4x@EFy8|7}ejuyJ!Nhp0jWYiuNEi(5GPR&ISfa@thI`fq7K#@;p zvPW*Bd83eb`1AGr$Y_y8-c-Va?bob{Q9aiDr7XBB3lsS3S=ICM_^oHv}iLJ-@^xb0qU_s8#2(S=yi72v4Hy++>OwseuQDR!ET}Rqpa_sfW@79O9FSz zVV$8-CqGBxae;E)T28(SC?VExR$R>Ng#Y|q)T$&8F^GBjwMf>?7(b^){S{m1PxYJ| wZRvVy04`#p(t+98upm;2BG13EqMqMo^9c*Tq{)`JJ%2c)qhX+4q=qB^3&OQ*)c^nh diff --git a/dokumentation/evolution3d/Xx4x4.gnuplot.fit.log b/dokumentation/evolution3d/Xx4x4.gnuplot.fit.log index bb17725..89568ef 100644 --- a/dokumentation/evolution3d/Xx4x4.gnuplot.fit.log +++ b/dokumentation/evolution3d/Xx4x4.gnuplot.fit.log @@ -1,7 +1,7 @@ ******************************************************************************* -Wed Oct 25 19:14:30 2017 +Fri Oct 27 14:12:05 2017 FIT: data read from "Xx4x4.csv" every ::1 using 1:5 @@ -47,7 +47,7 @@ b -0.934 1.000 ******************************************************************************* -Wed Oct 25 19:14:30 2017 +Fri Oct 27 14:12:05 2017 FIT: data read from "Xx4x4.csv" every ::1 using 3:5 @@ -93,7 +93,7 @@ bb -0.999 1.000 ******************************************************************************* -Wed Oct 25 19:14:30 2017 +Fri Oct 27 14:12:05 2017 FIT: data read from "Xx4x4.csv" every ::1 using 3:4 @@ -139,7 +139,7 @@ bbb -0.999 1.000 ******************************************************************************* -Wed Oct 25 19:14:30 2017 +Fri Oct 27 14:12:05 2017 FIT: data read from "Xx4x4.csv" every ::1 using 2:4 diff --git a/dokumentation/evolution3d/Xx4x4.gnuplot.script b/dokumentation/evolution3d/Xx4x4.gnuplot.script index 3d5f05d..5aef7fe 100644 --- a/dokumentation/evolution3d/Xx4x4.gnuplot.script +++ b/dokumentation/evolution3d/Xx4x4.gnuplot.script @@ -2,25 +2,25 @@ set datafile separator "," f(x)=a*x+b fit f(x) "Xx4x4.csv" every ::1 using 1:5 via a,b set terminal png -set xlabel 'regularity' -set ylabel 'steps' +set xlabel 'Regularity' +set ylabel 'Number of iterations' set output "Xx4x4_regularity-vs-steps.png" plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 title "4x4x4" pt 2, "20171013_3dFit_5x4x4_100times.csv" every ::1 using 1:5 title "5x4x4" pt 2, "20171005_3dFit_7x4x4_100times.csv" every ::1 using 1:5 title "7x4x4" pt 2, f(x) title "lin. fit" lc rgb "black" g(x)=aa*x+bb fit g(x) "Xx4x4.csv" every ::1 using 3:5 via aa,bb -set xlabel 'improvement potential' -set ylabel 'steps' +set xlabel 'Improvement potential' +set ylabel 'Number of iterations' set output "Xx4x4_improvement-vs-steps.png" plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 title "4x4x4" pt 2, "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:5 title "5x4x4" pt 2, "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:5 title "7x4x4" pt 2, g(x) title "lin. fit" lc rgb "black" h(x)=aaa*x+bbb fit h(x) "Xx4x4.csv" every ::1 using 3:4 via aaa,bbb -set xlabel 'improvement potential' -set ylabel 'evolution error' +set xlabel 'Improvement potential' +set ylabel 'Error given by fitness-function' set output "Xx4x4_improvement-vs-evo-error.png" plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 title "4x4x4" pt 2, "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:4 title "5x4x4" pt 2, "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:4 title "7x4x4" pt 2, h(x) title "lin. fit" lc rgb "black" i(x)=aaaa*x+bbbb fit i(x) "Xx4x4.csv" every ::1 using 2:4 via aaaa,bbbb -set xlabel 'variability' -set ylabel 'evolution error' +set xlabel 'Variability' +set ylabel 'Error given by fitness-function' set output "Xx4x4_variability-vs-evo-error.png" plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 2:4 title "4x4x4" pt 2, "20171013_3dFit_5x4x4_100times.csv" every ::1 using 2:4 title "5x4x4" pt 2, "20171005_3dFit_7x4x4_100times.csv" every ::1 using 2:4 title "7x4x4" pt 2, i(x) title "lin. fit" lc rgb "black" diff --git a/dokumentation/evolution3d/Xx4x4_improvement-vs-evo-error.png b/dokumentation/evolution3d/Xx4x4_improvement-vs-evo-error.png index 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******************************************************************************* -Wed Oct 25 19:09:05 2017 +Fri Oct 27 14:12:27 2017 FIT: data read from "all.csv" every ::1 using 3:5 @@ -93,7 +93,7 @@ bb -0.995 1.000 ******************************************************************************* -Wed Oct 25 19:09:05 2017 +Fri Oct 27 14:12:27 2017 FIT: data read from "all.csv" every ::1 using 3:4 @@ -139,7 +139,7 @@ bbb -0.995 1.000 ******************************************************************************* -Wed Oct 25 19:09:05 2017 +Fri Oct 27 14:12:27 2017 FIT: data read from "all.csv" every ::1 using 2:4 diff --git a/dokumentation/evolution3d/all.gnuplot.script b/dokumentation/evolution3d/all.gnuplot.script index 0175d8f..811c34c 100644 --- a/dokumentation/evolution3d/all.gnuplot.script +++ b/dokumentation/evolution3d/all.gnuplot.script @@ -2,25 +2,25 @@ set datafile separator "," f(x)=a*x+b fit f(x) "all.csv" every ::1 using 1:5 via a,b set terminal png -set xlabel 'regularity' -set ylabel 'steps' +set xlabel 'Regularity' +set ylabel 'Number of iterations' set output "all_regularity-vs-steps.png" plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 1:5 title "4x4x4" pt 2, "20171013_3dFit_5x4x4_100times.csv" every ::1 using 1:5 title "5x4x4" pt 2, "20171005_3dFit_7x4x4_100times.csv" every ::1 using 1:5 title "7x4x4" pt 2, "20171005_3dFit_4x4x5_100times.csv" every ::1 using 1:5 title "4x4x5" pt 2, "20171013_3dFit_4x4x7_100times.csv" every ::1 using 1:5 title "4x4x7" pt 2, "20170926_3dFit_5x5x5_100times.csv" every ::1 using 1:5 title "5x5x5" pt 2, "20171021-evolution3D_6x6_100Times.csv" every ::1 using 1:5 title "6x6x6" pt 2, f(x) title "lin. fit" lc rgb "black" g(x)=aa*x+bb fit g(x) "all.csv" every ::1 using 3:5 via aa,bb -set xlabel 'improvement potential' -set ylabel 'steps' +set xlabel 'Improvement potential' +set ylabel 'Number of iterations' set output "all_improvement-vs-steps.png" plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:5 title "4x4x4" pt 2, "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:5 title "5x4x4" pt 2, "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:5 title "7x4x4" pt 2, "20171005_3dFit_4x4x5_100times.csv" every ::1 using 3:5 title "4x4x5" pt 2, "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:5 title "4x4x7" pt 2, "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:5 title "5x5x5" pt 2, "20171021-evolution3D_6x6_100Times.csv" every ::1 using 3:5 title "6x6x6" pt 2, g(x) title "lin. fit" lc rgb "black" h(x)=aaa*x+bbb fit h(x) "all.csv" every ::1 using 3:4 via aaa,bbb -set xlabel 'improvement potential' -set ylabel 'evolution error' +set xlabel 'Improvement potential' +set ylabel 'Error given by fitness-function' set output "all_improvement-vs-evo-error.png" plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 3:4 title "4x4x4" pt 2, "20171013_3dFit_5x4x4_100times.csv" every ::1 using 3:4 title "5x4x4" pt 2, "20171005_3dFit_7x4x4_100times.csv" every ::1 using 3:4 title "7x4x4" pt 2, "20171005_3dFit_4x4x5_100times.csv" every ::1 using 3:4 title "4x4x5" pt 2, "20171013_3dFit_4x4x7_100times.csv" every ::1 using 3:4 title "4x4x7" pt 2, "20170926_3dFit_5x5x5_100times.csv" every ::1 using 3:4 title "5x5x5" pt 2, "20171021-evolution3D_6x6_100Times.csv" every ::1 using 3:4 title "6x6x6" pt 2, h(x) title "lin. fit" lc rgb "black" i(x)=aaaa*x+bbbb fit i(x) "all.csv" every ::1 using 2:4 via aaaa,bbbb -set xlabel 'variability' -set ylabel 'evolution error' +set xlabel 'Variability' +set ylabel 'Error given by fitness-function' set output "all_variability-vs-evo-error.png" plot "20170926_3dFit_4x4x4_100times.csv" every ::1 using 2:4 title "4x4x4" pt 2, "20171013_3dFit_5x4x4_100times.csv" every ::1 using 2:4 title "5x4x4" pt 2, "20171005_3dFit_7x4x4_100times.csv" every ::1 using 2:4 title "7x4x4" pt 2, "20171005_3dFit_4x4x5_100times.csv" every ::1 using 2:4 title "4x4x5" pt 2, "20171013_3dFit_4x4x7_100times.csv" every ::1 using 2:4 title "4x4x7" pt 2, 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z_s?n$cjzJR1W~V*`Rby+VVf_9{>FysEg!BGso^+>mYh;%xW0O3ljB zU+!}|4~|CWqVo=M+13{MoO-BxR+(<3SqqteP92)RxDJtlyr>wPc)hl@~$GJa1z{psx2_$6?f_r#Si>X1NE56vi(vIy)2yf ga}xSLKBMw3N!AMUm^&Q7ytkO diff --git a/dokumentation/evolution3d/combine.sh b/dokumentation/evolution3d/combine.sh index ee08351..6c003e6 100755 --- a/dokumentation/evolution3d/combine.sh +++ b/dokumentation/evolution3d/combine.sh @@ -10,8 +10,8 @@ set datafile separator "," f(x)=a*x+b fit f(x) "$data" every ::1 using 1:5 via a,b set terminal png -set xlabel 'regularity' -set ylabel 'steps' +set xlabel 'Regularity' +set ylabel 'Number of iterations' set output "${png}_regularity-vs-steps.png" plot \ "$2" every ::1 using 1:5 title "$3" pt 2, \ @@ -20,8 +20,8 @@ plot \ f(x) title "lin. fit" lc rgb "black" g(x)=aa*x+bb fit g(x) "$data" every ::1 using 3:5 via aa,bb -set xlabel 'improvement potential' -set ylabel 'steps' +set xlabel 'Improvement potential' +set ylabel 'Number of iterations' set output "${png}_improvement-vs-steps.png" plot \ "$2" every ::1 using 3:5 title "$3" pt 2, \ @@ -30,8 +30,8 @@ plot \ g(x) title "lin. fit" lc rgb "black" h(x)=aaa*x+bbb fit h(x) "$data" every ::1 using 3:4 via aaa,bbb -set xlabel 'improvement potential' -set ylabel 'evolution error' +set xlabel 'Improvement potential' +set ylabel 'Error given by fitness-function' set output "${png}_improvement-vs-evo-error.png" plot \ "$2" every ::1 using 3:4 title "$3" pt 2, \ @@ -40,8 +40,8 @@ plot \ h(x) title "lin. fit" lc rgb "black" i(x)=aaaa*x+bbbb fit i(x) "$data" every ::1 using 2:4 via aaaa,bbbb -set xlabel 'variability' -set ylabel 'evolution error' +set xlabel 'Variability' +set ylabel 'Error given by fitness-function' set output "${png}_variability-vs-evo-error.png" plot \ "$2" every ::1 using 2:4 title "$3" pt 2, \ diff --git a/dokumentation/evolution3d/combine7.sh b/dokumentation/evolution3d/combine7.sh index 5d76548..276c550 100755 --- a/dokumentation/evolution3d/combine7.sh +++ b/dokumentation/evolution3d/combine7.sh @@ -10,8 +10,8 @@ set datafile separator "," f(x)=a*x+b fit f(x) "$data" every ::1 using 1:5 via a,b set terminal png -set xlabel 'regularity' -set ylabel 'steps' +set xlabel 'Regularity' +set ylabel 'Number of iterations' set output "${png}_regularity-vs-steps.png" plot \ "$2" every ::1 using 1:5 title "$3" pt 2, \ @@ -24,8 +24,8 @@ plot \ f(x) title "lin. fit" lc rgb "black" g(x)=aa*x+bb fit g(x) "$data" every ::1 using 3:5 via aa,bb -set xlabel 'improvement potential' -set ylabel 'steps' +set xlabel 'Improvement potential' +set ylabel 'Number of iterations' set output "${png}_improvement-vs-steps.png" plot \ "$2" every ::1 using 3:5 title "$3" pt 2, \ @@ -38,8 +38,8 @@ plot \ g(x) title "lin. fit" lc rgb "black" h(x)=aaa*x+bbb fit h(x) "$data" every ::1 using 3:4 via aaa,bbb -set xlabel 'improvement potential' -set ylabel 'evolution error' +set xlabel 'Improvement potential' +set ylabel 'Error given by fitness-function' set output "${png}_improvement-vs-evo-error.png" plot \ "$2" every ::1 using 3:4 title "$3" pt 2, \ @@ -52,8 +52,8 @@ plot \ h(x) title "lin. fit" lc rgb "black" i(x)=aaaa*x+bbbb fit i(x) "$data" every ::1 using 2:4 via aaaa,bbbb -set xlabel 'variability' -set ylabel 'evolution error' +set xlabel 'Variability' +set ylabel 'Error given by fitness-function' set output "${png}_variability-vs-evo-error.png" plot \ "$2" every ::1 using 2:4 title "$3" pt 2, \ diff --git a/dokumentation/evolution3d/errors.gnuplot.fit.log b/dokumentation/evolution3d/errors.gnuplot.fit.log new file mode 100644 index 0000000..2eae556 --- /dev/null +++ b/dokumentation/evolution3d/errors.gnuplot.fit.log @@ -0,0 +1,184 @@ + + +******************************************************************************* +Fri Oct 27 14:09:08 2017 + + +FIT: data read from "errors.csv" every ::1 using 1:5 + 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 : 129069 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 84.3477 + +initial set of free parameter values + +a = 1 +b = 1 + +After 6 iterations the fit converged. +final sum of squares of residuals : 4993.5 +rel. change during last iteration : -5.46363e-11 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 7.13821 +variance of residuals (reduced chisquare) = WSSR/ndf : 50.9541 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +a = -0.0931363 +/- 0.1443 (154.9%) +b = 96.4721 +/- 17.21 (17.84%) + + +correlation matrix of the fit parameters: + + a b +a 1.000 +b -0.999 1.000 + + +******************************************************************************* +Fri Oct 27 14:09:08 2017 + + +FIT: data read from "errors.csv" every ::1 using 3:5 + 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 : 38697.6 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 71.7898 + +initial set of free parameter values + +aa = 1 +bb = 1 + +After 6 iterations the fit converged. +final sum of squares of residuals : 5010.73 +rel. change during last iteration : -1.443e-13 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 7.15052 +variance of residuals (reduced chisquare) = WSSR/ndf : 51.1299 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aa = 0.0270058 +/- 0.09648 (357.3%) +bb = 82.6379 +/- 9.795 (11.85%) + + +correlation matrix of the fit parameters: + + aa bb +aa 1.000 +bb -0.997 1.000 + + +******************************************************************************* +Fri Oct 27 14:09:08 2017 + + +FIT: data read from "errors.csv" every ::1 using 3:4 + 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 : 27023.7 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 71.7898 + +initial set of free parameter values + +aaa = 1 +bbb = 1 + +After 6 iterations the fit converged. +final sum of squares of residuals : 4159.2 +rel. change during last iteration : -2.19108e-13 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 6.51466 +variance of residuals (reduced chisquare) = WSSR/ndf : 42.4408 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaa = -0.0345469 +/- 0.0879 (254.4%) +bbb = 92.7152 +/- 8.924 (9.625%) + + +correlation matrix of the fit parameters: + + aaa bbb +aaa 1.000 +bbb -0.997 1.000 + + +******************************************************************************* +Fri Oct 27 14:09:08 2017 + + +FIT: data read from "errors.csv" every ::1 using 2:4 + 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 : 30294.4 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 72.9129 + +initial set of free parameter values + +aaaa = 1 +bbbb = 1 + +After 6 iterations the fit converged. +final sum of squares of residuals : 4165.22 +rel. change during last iteration : -6.01785e-13 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 6.51938 +variance of residuals (reduced chisquare) = WSSR/ndf : 42.5023 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaaa = -0.0109066 +/- 0.09721 (891.3%) +bbbb = 90.3395 +/- 10.02 (11.09%) + + +correlation matrix of the fit parameters: + + aaaa bbbb +aaaa 1.000 +bbbb -0.998 1.000 diff --git a/dokumentation/evolution3d/errors.gnuplot.log b/dokumentation/evolution3d/errors.gnuplot.log new file mode 100644 index 0000000..e520487 --- /dev/null +++ b/dokumentation/evolution3d/errors.gnuplot.log @@ -0,0 +1,392 @@ + + + Iteration 0 + WSSR : 129069 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 84.3477 + +initial set of free parameter values + +a = 1 +b = 1 +/ + + Iteration 1 + WSSR : 6564.6 delta(WSSR)/WSSR : -18.6613 + delta(WSSR) : -122504 limit for stopping : 1e-05 + lambda : 8.43477 + +resultant parameter values + +a = 0.708029 +b = 0.999863 +/ + + Iteration 2 + WSSR : 6554.01 delta(WSSR)/WSSR : -0.00161544 + delta(WSSR) : -10.5876 limit for stopping : 1e-05 + lambda : 0.843477 + +resultant parameter values + +a = 0.70464 +b = 1.23013 +/ + + Iteration 3 + WSSR : 6005.48 delta(WSSR)/WSSR : -0.0913382 + delta(WSSR) : -548.53 limit for stopping : 1e-05 + lambda : 0.0843477 + +resultant parameter values + +a = 0.549306 +b = 19.7746 +/ + + Iteration 4 + WSSR : 4995.1 delta(WSSR)/WSSR : -0.202276 + delta(WSSR) : -1010.39 limit for stopping : 1e-05 + lambda : 0.00843477 + +resultant parameter values + +a = -0.067621 +b = 93.426 +/ + + Iteration 5 + WSSR : 4993.5 delta(WSSR)/WSSR : -0.000319669 + delta(WSSR) : -1.59627 limit for stopping : 1e-05 + lambda : 0.000843477 + +resultant parameter values + +a = -0.0931258 +b = 96.4708 +/ + + Iteration 6 + WSSR : 4993.5 delta(WSSR)/WSSR : -5.46363e-11 + delta(WSSR) : -2.72827e-07 limit for stopping : 1e-05 + lambda : 8.43477e-05 + +resultant parameter values + +a = -0.0931363 +b = 96.4721 + +After 6 iterations the fit converged. +final sum of squares of residuals : 4993.5 +rel. change during last iteration : -5.46363e-11 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 7.13821 +variance of residuals (reduced chisquare) = WSSR/ndf : 50.9541 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +a = -0.0931363 +/- 0.1443 (154.9%) +b = 96.4721 +/- 17.21 (17.84%) + + +correlation matrix of the fit parameters: + + a b +a 1.000 +b -0.999 1.000 + + + Iteration 0 + WSSR : 38697.6 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 71.7898 + +initial set of free parameter values + +aa = 1 +bb = 1 +/ + + Iteration 1 + WSSR : 8562.61 delta(WSSR)/WSSR : -3.51936 + delta(WSSR) : -30134.9 limit for stopping : 1e-05 + lambda : 7.17898 + +resultant parameter values + +aa = 0.829791 +bb = 1.00677 +/ + + Iteration 2 + WSSR : 8489.56 delta(WSSR)/WSSR : -0.00860526 + delta(WSSR) : -73.0548 limit for stopping : 1e-05 + lambda : 0.717898 + +resultant parameter values + +aa = 0.820734 +bb = 1.84213 +/ + + Iteration 3 + WSSR : 5851.66 delta(WSSR)/WSSR : -0.450794 + delta(WSSR) : -2637.9 limit for stopping : 1e-05 + lambda : 0.0717898 + +resultant parameter values + +aa = 0.41725 +bb = 42.9138 +/ + + Iteration 4 + WSSR : 5010.8 delta(WSSR)/WSSR : -0.167809 + delta(WSSR) : -840.86 limit for stopping : 1e-05 + lambda : 0.00717898 + +resultant parameter values + +aa = 0.030744 +bb = 82.2573 +/ + + Iteration 5 + WSSR : 5010.73 delta(WSSR)/WSSR : -1.54001e-05 + delta(WSSR) : -0.0771658 limit for stopping : 1e-05 + lambda : 0.000717898 + +resultant parameter values + +aa = 0.0270062 +bb = 82.6378 +/ + + Iteration 6 + WSSR : 5010.73 delta(WSSR)/WSSR : -1.443e-13 + delta(WSSR) : -7.23048e-10 limit for stopping : 1e-05 + lambda : 7.17898e-05 + +resultant parameter values + +aa = 0.0270058 +bb = 82.6379 + +After 6 iterations the fit converged. +final sum of squares of residuals : 5010.73 +rel. change during last iteration : -1.443e-13 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 7.15052 +variance of residuals (reduced chisquare) = WSSR/ndf : 51.1299 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aa = 0.0270058 +/- 0.09648 (357.3%) +bb = 82.6379 +/- 9.795 (11.85%) + + +correlation matrix of the fit parameters: + + aa bb +aa 1.000 +bb -0.997 1.000 + + + Iteration 0 + WSSR : 27023.7 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 71.7898 + +initial set of free parameter values + +aaa = 1 +bbb = 1 +/ + + Iteration 1 + WSSR : 8641.55 delta(WSSR)/WSSR : -2.12719 + delta(WSSR) : -18382.2 limit for stopping : 1e-05 + lambda : 7.17898 + +resultant parameter values + +aaa = 0.867036 +bbb = 1.00818 +/ + + Iteration 2 + WSSR : 8549.83 delta(WSSR)/WSSR : -0.0107272 + delta(WSSR) : -91.716 limit for stopping : 1e-05 + lambda : 0.717898 + +resultant parameter values + +aaa = 0.857153 +bbb = 1.94665 +/ + + Iteration 3 + WSSR : 5220.55 delta(WSSR)/WSSR : -0.637727 + delta(WSSR) : -3329.29 limit for stopping : 1e-05 + lambda : 0.0717898 + +resultant parameter values + +aaa = 0.403866 +bbb = 48.0879 +/ + + Iteration 4 + WSSR : 4159.3 delta(WSSR)/WSSR : -0.255151 + delta(WSSR) : -1061.25 limit for stopping : 1e-05 + lambda : 0.00717898 + +resultant parameter values + +aaa = -0.0303472 +bbb = 92.2877 +/ + + Iteration 5 + WSSR : 4159.2 delta(WSSR)/WSSR : -2.34157e-05 + delta(WSSR) : -0.0973908 limit for stopping : 1e-05 + lambda : 0.000717898 + +resultant parameter values + +aaa = -0.0345465 +bbb = 92.7151 +/ + + Iteration 6 + WSSR : 4159.2 delta(WSSR)/WSSR : -2.19108e-13 + delta(WSSR) : -9.11314e-10 limit for stopping : 1e-05 + lambda : 7.17898e-05 + +resultant parameter values + +aaa = -0.0345469 +bbb = 92.7152 + +After 6 iterations the fit converged. +final sum of squares of residuals : 4159.2 +rel. change during last iteration : -2.19108e-13 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 6.51466 +variance of residuals (reduced chisquare) = WSSR/ndf : 42.4408 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaa = -0.0345469 +/- 0.0879 (254.4%) +bbb = 92.7152 +/- 8.924 (9.625%) + + +correlation matrix of the fit parameters: + + aaa bbb +aaa 1.000 +bbb -0.997 1.000 + + + Iteration 0 + WSSR : 30294.4 delta(WSSR)/WSSR : 0 + delta(WSSR) : 0 limit for stopping : 1e-05 + lambda : 72.9129 + +initial set of free parameter values + +aaaa = 1 +bbbb = 1 +/ + + Iteration 1 + WSSR : 7542.11 delta(WSSR)/WSSR : -3.0167 + delta(WSSR) : -22752.3 limit for stopping : 1e-05 + lambda : 7.29129 + +resultant parameter values + +aaaa = 0.854383 +bbbb = 1.0057 +/ + + Iteration 2 + WSSR : 7488.45 delta(WSSR)/WSSR : -0.00716584 + delta(WSSR) : -53.661 limit for stopping : 1e-05 + lambda : 0.729129 + +resultant parameter values + +aaaa = 0.84683 +bbbb = 1.71093 +/ + + Iteration 3 + WSSR : 5195.8 delta(WSSR)/WSSR : -0.44125 + delta(WSSR) : -2292.65 limit for stopping : 1e-05 + lambda : 0.0729129 + +resultant parameter values + +aaaa = 0.466748 +bbbb = 40.9842 +/ + + Iteration 4 + WSSR : 4165.38 delta(WSSR)/WSSR : -0.247377 + delta(WSSR) : -1030.42 limit for stopping : 1e-05 + lambda : 0.00729129 + +resultant parameter values + +aaaa = -0.00497837 +bbbb = 89.7269 +/ + + Iteration 5 + WSSR : 4165.22 delta(WSSR)/WSSR : -3.81126e-05 + delta(WSSR) : -0.158748 limit for stopping : 1e-05 + lambda : 0.000729129 + +resultant parameter values + +aaaa = -0.0109059 +bbbb = 90.3394 +/ + + Iteration 6 + WSSR : 4165.22 delta(WSSR)/WSSR : -6.01785e-13 + delta(WSSR) : -2.50657e-09 limit for stopping : 1e-05 + lambda : 7.29129e-05 + +resultant parameter values + +aaaa = -0.0109066 +bbbb = 90.3395 + +After 6 iterations the fit converged. +final sum of squares of residuals : 4165.22 +rel. change during last iteration : -6.01785e-13 + +degrees of freedom (FIT_NDF) : 98 +rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 6.51938 +variance of residuals (reduced chisquare) = WSSR/ndf : 42.5023 + +Final set of parameters Asymptotic Standard Error +======================= ========================== + +aaaa = -0.0109066 +/- 0.09721 (891.3%) +bbbb = 90.3395 +/- 10.02 (11.09%) + + +correlation matrix of the fit parameters: + + aaaa bbbb +aaaa 1.000 +bbbb -0.998 1.000 diff --git a/dokumentation/evolution3d/errors.gnuplot.script b/dokumentation/evolution3d/errors.gnuplot.script new file mode 100644 index 0000000..f60ca49 --- /dev/null +++ b/dokumentation/evolution3d/errors.gnuplot.script @@ -0,0 +1,26 @@ +set datafile separator "," +f(x)=a*x+b +fit f(x) "errors.csv" every ::1 using 1:5 via a,b +set terminal png +set xlabel 'Regularity' +set ylabel 'Number of iterations' +set output "errors_regularity-vs-steps.png" +plot "errors.csv" every ::1 using 1:5 title "data", f(x) title "lin. fit" lc rgb "black" +g(x)=aa*x+bb +fit g(x) "errors.csv" every ::1 using 3:5 via aa,bb +set xlabel 'Improvement potential' +set ylabel 'Number of iterations' +set output "errors_improvement-vs-steps.png" +plot "errors.csv" every ::1 using 3:5 title "data", g(x) title "lin. fit" lc rgb "black" +h(x)=aaa*x+bbb +fit h(x) "errors.csv" every ::1 using 3:4 via aaa,bbb +set xlabel 'Improvement potential' +set ylabel 'error 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+10,26 @@ set datafile separator "," f(x)=a*x+b fit f(x) "$data" every ::1 using 1:5 via a,b set terminal png -set xlabel 'regularity' -set ylabel 'steps' +set xlabel 'Regularity' +set ylabel 'Number of iterations' set output "${png}_regularity-vs-steps.png" plot "$data" every ::1 using 1:5 title "data", f(x) title "lin. fit" lc rgb "black" g(x)=aa*x+bb fit g(x) "$data" every ::1 using 3:5 via aa,bb -set xlabel 'improvement potential' -set ylabel 'steps' +set xlabel 'Improvement potential' +set ylabel 'Number of iterations' set output "${png}_improvement-vs-steps.png" plot "$data" every ::1 using 3:5 title "data", g(x) title "lin. fit" lc rgb "black" h(x)=aaa*x+bbb fit h(x) "$data" every ::1 using 3:4 via aaa,bbb -set xlabel 'improvement potential' -set ylabel 'evolution error' +set xlabel 'Improvement potential' +set ylabel 'error given by fitness-function' set output "${png}_improvement-vs-evo-error.png" plot "$data" every ::1 using 3:4 title "data", h(x) title "lin. fit" lc rgb 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