From 4086da2f7abf7054acd5f44c49cb5ca92eef0b92 Mon Sep 17 00:00:00 2001 From: Jan Grewe Date: Sun, 8 Nov 2015 18:10:06 +0100 Subject: [PATCH] finished regression chapter --- regression/code/errorSurface.m | 31 +++ regression/code/{lsq_error.m => lsqError.m} | 2 +- regression/code/lsqGradient.m | 7 + regression/code/lsq_gradient.m | 7 - ...{mean_square_error.m => meanSquareError.m} | 0 regression/lecture/figures/error_gradient.pdf | Bin 0 -> 30541 bytes regression/lecture/figures/error_surface.pdf | Bin 0 -> 1734558 bytes regression/lecture/regression.tex | 197 +++++++++++++----- 8 files changed, 183 insertions(+), 61 deletions(-) create mode 100644 regression/code/errorSurface.m rename regression/code/{lsq_error.m => lsqError.m} (89%) create mode 100644 regression/code/lsqGradient.m delete mode 100644 regression/code/lsq_gradient.m rename regression/code/{mean_square_error.m => meanSquareError.m} (100%) create mode 100644 regression/lecture/figures/error_gradient.pdf create mode 100644 regression/lecture/figures/error_surface.pdf diff --git a/regression/code/errorSurface.m b/regression/code/errorSurface.m new file mode 100644 index 0000000..421c11c --- /dev/null +++ b/regression/code/errorSurface.m @@ -0,0 +1,31 @@ +clear +close all + +load('lin_regression.mat'); + +slopes = -5:0.25:5; +intercepts = -30:1:30; + +error_surf = zeros(length(slopes), length(intercepts)); + +for i = 1:length(slopes) + for j = 1:length(intercepts) + error_surf(i,j) = lsqError([slopes(i), intercepts(j)], x, y); + end +end + + +% plot the error surface +figure() +[N,M] = meshgrid(intercepts, slopes); +s = surface(M,N,error_surf); +xlabel('slope', 'rotation', 7.5) +ylabel('intercept', 'rotation', -22.5) +zlabel('error') +set(gca,'xtick', (-5:2.5:5)) +grid on +view(3) +set(gcf, 'paperunits', 'centimeters', 'papersize', [15, 15], ... + 'paperposition', [0., 0., 15, 15]) + +saveas(gcf, 'error_surface', 'pdf') \ No newline at end of file diff --git a/regression/code/lsq_error.m b/regression/code/lsqError.m similarity index 89% rename from regression/code/lsq_error.m rename to regression/code/lsqError.m index a648e48..b701b35 100644 --- a/regression/code/lsq_error.m +++ b/regression/code/lsqError.m @@ -1,4 +1,4 @@ -function error = lsq_error(parameter, x, y) +function error = lsqError(parameter, x, y) % Objective function for fitting a linear equation to data. % % Arguments: parameter, vector containing slope and intercept (1st and 2nd element) diff --git a/regression/code/lsqGradient.m b/regression/code/lsqGradient.m new file mode 100644 index 0000000..c848b81 --- /dev/null +++ b/regression/code/lsqGradient.m @@ -0,0 +1,7 @@ +function gradient = lsqGradient(parameter, x, y) +h = 1e-6; + +partial_m = (lsqError([parameter(1)+h, parameter(2)],x,y) - lsqError(parameter,x,y))/ h; +partial_n = (lsqError([parameter(1), parameter(2)+h],x,y) - lsqError(parameter,x,y))/ h; + +gradient = [partial_m, partial_n]; \ No newline at end of file diff --git a/regression/code/lsq_gradient.m b/regression/code/lsq_gradient.m deleted file mode 100644 index 6773c4e..0000000 --- a/regression/code/lsq_gradient.m +++ /dev/null @@ -1,7 +0,0 @@ -function gradient = lsq_gradient(parameter, x, y) -h = 1e-6; - -partial_m = (lsq_error([parameter(1)+h, parameter(2)],x,y) - lsq_error(parameter,x,y))/ h; -partial_n = (lsq_error([parameter(1), parameter(2)+h],x,y) - lsq_error(parameter,x,y))/ h; - -gradient = [partial_m, partial_n]; \ No newline at end of file diff --git a/regression/code/mean_square_error.m b/regression/code/meanSquareError.m similarity index 100% rename from regression/code/mean_square_error.m rename to regression/code/meanSquareError.m diff --git a/regression/lecture/figures/error_gradient.pdf b/regression/lecture/figures/error_gradient.pdf new file mode 100644 index 0000000000000000000000000000000000000000..159c684a7f3e70d56d99c0269a3a02efff401e0c GIT binary patch literal 30541 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zdW3w$=suUDnDt=H2%X92tOxm=y-GgkTlS6g;i4DXJ1*NUxzct~;<*mX8rf|bGc0Sw ztrpz}7Cqu2A5p)_N2GBjcjU0>5kf1XpTe?6au0Jx!Lmkf;(JFVT^?v9=5A?!B76D% zpa(4HxfNggfqs+g<}gZa$is4;8(q!{Sk6YJq;jspqWcIy_F79;^wbg4g-~+FySC>F|>Bpac60!E>=gOex!8$^y8 0.1} + benutzt werden): \begin{itemize} - \item Wie findet man die Extrempunkte in einer Kurve?\pause - \item Ableitung der Funktion auf Null setzen und nach x aufl\"osen. - \item Definition der Ableitung:\\ \vspace{0.25cm} - \begin{center} - $ f'(x) = \lim\limits_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h} $ - \vspace{0.25cm}\pause - \end{center} - \item Bei zwei Parametern $g(m,n)$ k\"onnen wie die partielle - Ableitung bez\"uglich eines Parameters benutzen um die - Ver\"anderung des Fehlers bei Ver\"anderung eines Parameters - auszuwerten. - \item Partielle Ableitung nach \code{m}?\\\pause - \vspace{0.25cm} - \begin{center} - $\frac{\partial g(m,n)}{\partial m} = \lim\limits_{h \rightarrow 0} \frac{g(m + h, n) - g(m,n)}{h}$ - \vspace{0.25cm} - \end{center} + \item Berechne den Gradienten an der akutellen Position $p_t$. + \item Gehe einen kleinen Schritt ($\epsilon = 0.01$) in die entgegensetzte Richtung des + Gradienten: + \[p_{t+1} = p_t - \epsilon \cdot \bigtriangledown g(m_t, n_t)\] \end{itemize} +\end{enumerate} -\section{Der Gradientenabstieg} +Abbildung \ref{gradientdescentfig} zeigt den Verlauf des +Gradientenabstiegs. Von einer Startposition aus wird die Position +solange ver\"andert, wie der Gradint eine bestimmte Gr\"o{\ss}e +\"uberschreitet. An den Stellen, an denen der Gradient sehr stark ist +ist auch die Ver\"anderung der Position gro{\ss} und der Abstand der +Punkte in Abbildung \ref{gradientdescentfig} gro{\ss}. +\begin{figure}[hb] + \includegraphics[width=0.6\columnwidth]{figures/gradient_descent} + \caption{\textbf{Gradientenabstieg.} Es wird von einer beliebigen + Position aus gestartet und der Gradient berechnet und die Position + ver\"andert. Jeder Punkt zeigt die Position nach jedem + Optimierungsschritt an.} \label{gradientdescentfig} +\end{figure} + +\clearpage +\begin{exercise}{gradientDescent.m}{} + Implementiere den Gradientenabstieg f\"ur das Problem der + Parameteranpassung der linearen Geradengleichung an die Messdaten in + der Datei \code{lin\_regression.mat}. + \begin{enumerate} + \item Merke Dir f\"ur jeden Schritt den Fehler zwischen + Modellvorhersage und Daten. + \item Erstelle eine Plot, der die Entwicklung des Fehlers als + Funktion der Optimierungsschritte zeigt. + \item Erstelle einen Plot, der den besten Fit in die Daten plottet. + \end{enumerate} +\end{exercise}