From 92ff396b3684781d291fe7171338ed5a73549ef6 Mon Sep 17 00:00:00 2001 From: paul-loedige Date: Sun, 13 Feb 2022 13:11:20 +0100 Subject: [PATCH] Kernel Regression abgeschlossen --- Glossary.tex | 41 ++++++++------- Preface.tex | 12 +++-- .../Trees_and_Forests.tex | 4 +- chapters/Kernel_Methods/Kernel-Regression.tex | 49 +++++++++++++++++- images/gaussian_kernel_model_selection.png | Bin 0 -> 13468 bytes images/kernel_regression_comparison.pdf | Bin 0 -> 117292 bytes 6 files changed, 79 insertions(+), 27 deletions(-) create mode 100644 images/gaussian_kernel_model_selection.png create mode 100644 images/kernel_regression_comparison.pdf diff --git a/Glossary.tex b/Glossary.tex index 5cf285b..9122fa8 100644 --- a/Glossary.tex +++ b/Glossary.tex @@ -84,15 +84,16 @@ %-------------------- %add new key -\glsaddstoragekey{unit}{}{\glsentryunit} +%\glsaddstoragekey{unit}{}{\glsentryunit} \glsnoexpandfields -\newcommand{\newnom}[5]{ +%\newcommand{\newnom}[5]{ +\newcommand{\newnom}[4]{ \newglossaryentry{#1}{ name={#2}, symbol={#3}, description={#4}, - unit={#5}, + %unit={#5}, type=nomenclature, sort={#1} } @@ -108,23 +109,23 @@ %use nomenclature entry (use in equation) \newcommand{\nomeq}[1]{\glslink{#1}{\glsentrysymbol{#1}}} -\newnom{summed_squared_error}{\gls{SSE}}{\text{\glsxtrshort{SSE}}}{\glsxtrfull{SSE}}{} -\newnom{mean_squared_error}{\gls{MSE}}{\text{\glsxtrshort{MSE}}}{\glsxtrfull{MSE}}{} -\newnom{residual_sum_squares}{\gls{RSS}}{\text{\glsxtrshort{RSS}}}{\glsxtrfull{RSS}}{} -\newnom{gaussian_noise}{Gausches Rauschen}{\epsilon}{zufällige (normalverteilte) Abweichung}{} -\newnom{vector_valued_function}{vektorwertige Funktion}{\phi(\bm{x})}{vektorwertige Funktion der des Eingangsvektor $\bm{x}$}{} -\newnom{regularization_factor}{Regularisierungsfaktor}{\lambda}{}{} -\newnom{identity_matrix}{Identitätsmatrix}{\bm{I}}{$\begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 1 \end{bmatrix}$}{} -\newnom{probability_mass_function}{Probability Mass Function}{p(x)}{Wahrscheinlichkeitsdichte-\slash\,Wahrscheinlichkeitsmassefunktion}{} -\newnom{mean}{arithmetisches Mittel}{\mu}{}{} -\newnom{mean-vector}{Mittelwerts-Vektor}{\bm{\mu}}{}{} -\newnom{covariance}{Kovarianz-Matrix}{\bm{\Sigma}}{}{} -\newnom{variance}{Varianz}{\sigma^2}{$\mathbb{E}_p[(X-\nomeq{mean})$]}{} -\newnom{sigmoid}{Sigmoid Function}{\sigma}{}{} -\newnom{learning_rate}{Learning Rate}{\eta}{}{} -\newnom{kernel_matrix}{Kernel Matrix}{\bm{K}}{}{} -\newnom{kernel_function}{Kernel Function}{k}{}{} -\newnom{kernel_vector}{Kernel Vector}{\bm{k}}{}{} +\newnom{summed_squared_error}{\gls{SSE}}{\text{\glsxtrshort{SSE}}}{\glsxtrfull{SSE}} +\newnom{mean_squared_error}{\gls{MSE}}{\text{\glsxtrshort{MSE}}}{\glsxtrfull{MSE}} +\newnom{residual_sum_squares}{\gls{RSS}}{\text{\glsxtrshort{RSS}}}{\glsxtrfull{RSS}} +\newnom{gaussian_noise}{Gausches Rauschen}{\epsilon}{zufällige (normalverteilte) Abweichung} +\newnom{vector_valued_function}{vektorwertige Funktion}{\bm\phi(\bm{x})}{vektorwertige Funktion der des Eingangsvektor $\bm{x}$} +\newnom{regularization_factor}{Regularisierungsfaktor}{\lambda}{} +\newnom{identity_matrix}{Identitätsmatrix}{\bm{I}}{$\begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 1 \end{bmatrix}$} +\newnom{probability_mass_function}{Probability Mass Function}{p(x)}{Wahrscheinlichkeitsdichte-\slash\,Wahrscheinlichkeitsmassefunktion} +\newnom{mean}{arithmetisches Mittel}{\mu}{} +\newnom{mean-vector}{Mittelwerts-Vektor}{\bm{\mu}}{} +\newnom{covariance}{Kovarianz-Matrix}{\bm{\Sigma}}{} +\newnom{variance}{Varianz}{\sigma^2}{$\mathbb{E}_p[(X-\nomeq{mean})$]} +\newnom{sigmoid}{Sigmoid Function}{\sigma}{} +\newnom{learning_rate}{Learning Rate}{\eta}{} +\newnom{kernel_matrix}{Kernel Matrix}{\bm{K}}{} +\newnom{kernel_function}{Kernel Function}{k}{} +\newnom{kernel_vector}{Kernel Vector}{\bm{k}}{} \shorthandoff{"} \makeglossaries diff --git a/Preface.tex b/Preface.tex index 1bdc9f4..c596ffd 100644 --- a/Preface.tex +++ b/Preface.tex @@ -25,17 +25,21 @@ \renewcommand{\glsgroupskip}{}%avoids grouping the elements by alphabetical order \renewenvironment{theglossary}{% Change the table type --> 4 columns \renewcommand*{\arraystretch}{1.5} - \begin{longtable}{>{\centering}p{.1\textwidth} >{\arraybackslash}p{.225\textwidth} p{.475\textwidth}>{\centering\arraybackslash}p{.1\textwidth}}}% + %\begin{longtable}{>{\centering}p{.1\textwidth} >{\arraybackslash}p{.225\textwidth} p{.475\textwidth}>{\centering\arraybackslash}p{.1\textwidth}}}% + \begin{longtable}{>{\centering}p{.1\textwidth} >{\arraybackslash}p{.225\textwidth} p{.575\textwidth}}}% {\end{longtable}}% % \renewcommand*{\glossaryheader}{% Change the table header - \bfseries\large Symbol & \bfseries\large Bezeichnung & \large\bfseries Beschreibung & \large\bfseries Einheit\\ + %\bfseries\large Symbol & \bfseries\large Bezeichnung & \large\bfseries Beschreibung & \large\bfseries Einheit\\ + \bfseries\large Symbol & \bfseries\large Bezeichnung & \large\bfseries Beschreibung\\ \hline\endhead}% \renewcommand*{\glossentry}[2]{% Change the displayed items - \boldmath\ensuremath{\glossentrysymbol{##1}} + %\boldmath\ensuremath{\glossentrysymbol{##1}} + \ensuremath{\glossentrysymbol{##1}} & \glstarget{##1}{\hspace*{0pt}\glossentryname{##1}} % & \glossentrydesc{##1} - & \glsentryunit{##1}\tabularnewline + %& \glsentryunit{##1} + \tabularnewline }% } \printglossary[type=nomenclature, nonumberlist, style=symbunitlong] diff --git a/chapters/Classical_Supervised_Learning/Trees_and_Forests.tex b/chapters/Classical_Supervised_Learning/Trees_and_Forests.tex index ef0bfc3..26607c4 100644 --- a/chapters/Classical_Supervised_Learning/Trees_and_Forests.tex +++ b/chapters/Classical_Supervised_Learning/Trees_and_Forests.tex @@ -81,7 +81,7 @@ welchen Anteil die Klasse $k$ auf der linken Seite des Splits hat. \subsubsection{Classification Tree}% \label{ssub:Classification Tree} \includegraphics[width=.6\textwidth]{classification_tree.png} -{\color{red} Herleitung Vorlesung 03 Seite 24-31} +{\color{red} Herleitung Vorlesung 04 Seite 24-31} \subsubsection{Regression Tree}% \label{ssub:Regression Tree} @@ -96,7 +96,7 @@ Predict (log) prostate specific antigen from \end{itemize} } \vspace*{30mm} -{\color{red} Herleitung Vorlesung 03 Seite 32-36} +{\color{red} Herleitung Vorlesung 04 Seite 32-36} \section{Random Forests}% \label{sec:Random Forests} diff --git a/chapters/Kernel_Methods/Kernel-Regression.tex b/chapters/Kernel_Methods/Kernel-Regression.tex index 720e875..c3481ca 100644 --- a/chapters/Kernel_Methods/Kernel-Regression.tex +++ b/chapters/Kernel_Methods/Kernel-Regression.tex @@ -1,5 +1,52 @@ \chapter{Kernel-Regression}% \label{cha:Kernel-Regression} +Die Kernel Regression ist das Äquivalent der Linear \nameref{sub:Ridge Regression} (\cref{sub:Ridge Regression}), +weshalb es auch oft als Kernel Ridge Regression bezeichnet wird. +Die Linear Ridge Regression ist allerdings für den linearen Feature Space gedacht +und lässt sich nicht direkt in einem Feature Space mit unendlicher Dimension anwenden. -WEITER AUF FOLIE 294 +Mithilfe eines mathematischen Tricks (aus dem Matrix Cookook) lässt sich die Lösung der Ridge Regression so umstellen, +dass statt einer $d\times d$ Matrix lediglich eine $N\times N$ Matrix invertiert werden muss: +\begin{equation} + \bm w^* = \underbrace{(\bm\Phi^T\bm\Phi + \nomeq{regularization_factor}\nomeq{identity_matrix})^{-1}}_{\text{$d\times d$ matrix inversion}}\bm\Phi^T\bm y + = \bm\Phi^T\underbrace{(\bm\Phi\bm\Phi^T + \nomeq{regularization_factor}\nomeq{identity_matrix})^{-1}}_{\text{$N\times N$ matrix inversion}}\bm y +\end{equation} +Nun erlaubt es die Verwendung einer \nomf{kernel_matrix} (\cref{cha:Kernel Basics}), +die Gleichung weiter zu vereinfachen: +\begin{equation} + \bm w^* = \bm\Phi^T\underbrace{(\bm\Phi\bm\Phi^T + \nomeq{regularization_factor}\nomeq{identity_matrix})^{-1}}_{\text{$N\times N$ matrix inversion}}\bm y + = \bm\Phi^T \underbrace{(\nomeq{kernel_matrix} + \nomeq{regularization_factor}\nomeq{identity_matrix})^{-1}\bm y}_{\bm \alpha} + = \bm\Phi^T \bm\alpha +\end{equation} +Allerdings besteht weiterhin das Problem, +dass $\bm w^* \mathbb{R}^d$ eine potentiell unendlich große Dimension hat +und daher nicht dargestellt oder abgespeichert werden kann. +Allerdings ermöglicht es und die Beschreibung mithilfe des Kernels, +eine Funktion $f(\bm x)$, +die $\bm w^*$ verwendet auszuwerten: +\begin{equation} + f(\bm x) = \nomeq{vector_valued_function}^T\bm w^* + = \nomeq{vector_valued_function}^T\bm\Phi^T\bm\alpha + = \nomeq{kernel_vector}(\bm x)^T\bm\alpha + = \sum_i \alpha_i \nomeq{kernel_function}(\bm x_i,\bm x) +\end{equation} +Die Lösung der Kernel Ridge Regression wird daher gegeben durch: +\begin{equation} \label{eq:kernel_ridge_regression_solution} + f^*(\bm x) = \nomeq{kernel_vector}(\bm x)^T (\nomeq{kernel_matrix} + \nomeq{regularization_factor}\nomeq{identity_matrix})^{-1}\bm y +\end{equation} +\section{Selecting the hyper-parameters}% +\label{sub:Selecting the hyper-parameters} +Die Auswahl der passenden Hyperparameter (z.B. \nomsym{variance} für den \nameref{sub:Gaussian Kernel}) ist ein Model Selection Problem (\cref{cha:Model Selection}). +\begin{figure}[H] + \centering + \includegraphics[width=0.5\textwidth]{gaussian_kernel_model_selection.png} + \caption{\nameref{cha:Model Selection} Problem für einen \nameref{sub:Gaussian Kernel}} + \label{fig:gaussian_kernel_model_selection} +\end{figure} + +\section{Examples and comparison to \glsxtrshort{RBF} regression}% +\label{sec:Examples and comparison to RBF regression} +\begin{center} + \includegraphics[width=.9\textwidth]{kernel_regression_comparison.pdf} +\end{center} diff --git a/images/gaussian_kernel_model_selection.png b/images/gaussian_kernel_model_selection.png new file mode 100644 index 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