From f679f8b3b6d06ee8cacde1e37f6880a06d67304a Mon Sep 17 00:00:00 2001 From: paul-loedige Date: Thu, 10 Feb 2022 17:44:56 +0100 Subject: [PATCH] Mathematische Grundlagen in eigenen Teil ausgelagert und bis Seite 114 zusammengefasst. --- Glossary.tex | 5 +- ML_Zusammenfassung.tex | 5 + .../Linear_Regression.tex | 96 +----------------- .../Lineare_Algebra.tex | 90 ++++++++++++++++ .../Probability_Theory.tex | 61 +++++++++++ images/chain_rule.png | Bin 0 -> 108089 bytes images/conditional_and_joint_distribution.png | Bin 0 -> 190961 bytes images/sum_rule.png | Bin 0 -> 177751 bytes 8 files changed, 160 insertions(+), 97 deletions(-) create mode 100644 chapters/Mathematische_Grundlagen/Lineare_Algebra.tex create mode 100644 chapters/Mathematische_Grundlagen/Probability_Theory.tex create mode 100644 images/chain_rule.png create mode 100644 images/conditional_and_joint_distribution.png create mode 100644 images/sum_rule.png diff --git a/Glossary.tex b/Glossary.tex index 10cea49..262dfea 100644 --- a/Glossary.tex +++ b/Glossary.tex @@ -66,7 +66,7 @@ \newcommand{\noms}[1]{\glsentryname{#1}\xspace} \newcommand{\nomS}[1]{\Glsentryname{#1}\xspace} %use nomenclature entry (symbol only) -\newcommand{\nomsym}[1]{\texorpdfstring{\glslink{#1}{\boldmath\ensuremath{\glsentrysymbol{#1}}}}{#1}\xspace} +\newcommand{\nomsym}[1]{\texorpdfstring{\glslink{#1}{\ensuremath{\glsentrysymbol{#1}}}}{#1}\xspace} %use nomenclature entry (use in equation) \newcommand{\nomeq}[1]{\glslink{#1}{\glsentrysymbol{#1}}} @@ -74,7 +74,8 @@ \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{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} \shorthandoff{"} diff --git a/ML_Zusammenfassung.tex b/ML_Zusammenfassung.tex index 7c0f30f..0b935d3 100644 --- a/ML_Zusammenfassung.tex +++ b/ML_Zusammenfassung.tex @@ -31,6 +31,11 @@ \part{Einleitung} \input{chapters/Einleitung.tex} + \part{Mathematische Grundlagen} + \label{part:Mathematische Grundlagen} + \input{chapters/Mathematische_Grundlagen/Lineare_Algebra.tex} + \input{chapters/Mathematische_Grundlagen/Probability_Theory.tex} + \part{Classical Supervised Learning} \label{part:Classical Supervised Learning} \input{chapters/Classical_Supervised_Learning/Linear_Regression.tex} diff --git a/chapters/Classical_Supervised_Learning/Linear_Regression.tex b/chapters/Classical_Supervised_Learning/Linear_Regression.tex index 888de0c..1979e3a 100644 --- a/chapters/Classical_Supervised_Learning/Linear_Regression.tex +++ b/chapters/Classical_Supervised_Learning/Linear_Regression.tex @@ -1,100 +1,6 @@ \chapter{Linear Regression}% \label{cha:Linear Regression} -Das Ziel von linearer Regression ist es eine Gerade zu finden, -die eine Menge von Eingabedatenpunkten am besten approximiert. - -\section{Lineare Algebra}% -\label{sec:Lineare Algebra} - -\subsection{Vektoren}% -\label{sec:Vektoren} -\begin{itemize} - \item Alle Vektoren werden \textbf{fett} geschrieben: - $\underbrace{x=1}_{\text{Skalar}},\qquad\underbrace{\bm{x}=\begin{bmatrix} 1\\2\\4 \end{bmatrix}}_{\text{Vektor}}$ - \item ein Vektor $\bm{x}$ ist immer ein Spaltenvektor: - $\bm{x}=\begin{bmatrix} 1\\2\\4 \end{bmatrix}$ - \item ein transponierter Vektor $\bm{x}^T$ ist immer ein Reihenvektor: - $\bm{x}^T=\begin{bmatrix} 1 & 2 & 4 \end{bmatrix}$ - \item Multiplikation eines Vektors mit einem Skalar: - $2\begin{bmatrix} 1\\2\\4 \end{bmatrix} = \begin{bmatrix} 2\\4\\8\end{bmatrix}$ - \item Vektoraddition: - $\begin{bmatrix} 1\\2\\4 \end{bmatrix} + \begin{bmatrix} 2\\1\\4 \end{bmatrix} = \begin{bmatrix} 3\\3\\8 \end{bmatrix}$ - \item Skalarprodukt $\langle \bm{v},\bm{w}\rangle$ der Vektoren $\bm{v}=\begin{bmatrix} 1\\2\\4 \end{bmatrix}$ und $\bm{w}=\begin{bmatrix} 2\\4\\8 \end{bmatrix}$: - $\langle \bm{v},\bm{w}\rangle = 1\cdot 2 + 2\cdot 4 + 4\cdot 8 = 42$ - \item Länge eines Vektors: - $||\bm{v}|| = \langle \bm{v},\bm{v}\rangle^{\frac{1}{2}} = (1^2 + 2^2 + 4^2)^{\frac{1}{2}} = \sqrt{21}$ -\end{itemize} - -\subsection{Matrizen}% -\label{sec:Matrizen} -\begin{itemize} - \item Matrizen werden \uppercase{groß} und \textbf{fett} geschrieben: - $\bm{X}=\begin{bmatrix} 1&3\\2&3\\4&7 \end{bmatrix}\qquad\bm{A}=\begin{bmatrix}1&3&5&4\\2&3&7&2 \end{bmatrix}$ - \item Mehrere Vektoren können zu einer Matrix zusammengefasst werden. - Hierbei werden die Vektoren meistens zu Zeilen und repräsentieren einen Einzelnen Datensatz: - $$\bm{x}_1 = \begin{bmatrix} 37\\72\\175 \end{bmatrix} \qquad \bm{x}_2 = \begin{bmatrix} 10\\30\\61 \end{bmatrix}\qquad \bm{x}_3 = \begin{bmatrix} 25\\65\\121 \end{bmatrix}\qquad \bm{x}_4 = \begin{bmatrix} 66\\67\\175 \end{bmatrix}$$ - $$\bm{X} = \begin{bmatrix} \bm{x}_1^T\\\bm{x}_2^T\\\bm{x}_3^T\\\bm{x}_4^T \end{bmatrix} = \begin{bmatrix} 37&72&175\\10&30&61\\25&65&121\\66&67&175 \end{bmatrix}$$ - \item Multiplikation mit einem Skalar: - $ 3\bm{M} = 3\begin{bmatrix} 3 & 4 & 5 \\ 1 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 9 & 12 & 15\\ 3 & 0 & 3 \end{bmatrix}$ - \item Addition von Matrizen: - $ \bm{M} + \bm{N} = \begin{bmatrix} 3 & 4 & 5 \\ 1 & 0 & 1 \end{bmatrix} + \begin{bmatrix} 1 & 2 & 1 \\ 3 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 4 & 6 & 6 \\ 4 & 1 & 2 \end{bmatrix}$ - \item Transponierte Matrizen: - $ \bm{M} = \begin{bmatrix} 3 & 4 & 5 \\ 1 & 0 & 1 \end{bmatrix}, \bm{M}^T = \begin{bmatrix} 3 & 1\\ 4 & 0\\ 5 & 1 \end{bmatrix}$ - \item Matrix-Vektor-Produkt: - $$ \bm{Wv} = \underbrace{\begin{bmatrix} \bm{w}_1 & \cdots & \bm{w}_n \end{bmatrix}}_{\bm{W}}\underbrace{\begin{bmatrix} v_1 \\ \vdots \\ v_n \end{bmatrix}}_{\bm{v}} = \begin{bmatrix} v_1\bm{w}_1 + \cdots + v_n\bm{w}_n \end{bmatrix} $$ - $$ \bm{Wv} = \begin{bmatrix} 3 & 4 & 5 \\ 1 & 0 & 1 \end{bmatrix}\begin{bmatrix} 3\\0\\2 \end{bmatrix} = \begin{bmatrix} 3\cdot1 + 4\cdot0 + 5\cdot2 \\ 1\cdot1 + 0\cdot0 + 1\cdot2 \end{bmatrix} = \begin{bmatrix} 13\\3 \end{bmatrix}$$ - \item Matrix-Matrix-Produkt: - $$ \bm{WV} = \begin{bmatrix} 3 & 4 & 5\\ 1 & 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0\\0 & 3\\2 & 4 \end{bmatrix} = - \begin{bmatrix} 3\cdot1+4\cdot0+5\cdot2 & 3\cdot0+4\cdot3+5\cdot4\\ 1\cdot1+0\cdot0+1\cdot2 & 1\cdot0+0\cdot3+1\cdot4 \end{bmatrix} = \begin{bmatrix} 13 & 32\\ 3 & 4 \end{bmatrix}$$ - \begin{itemize} - \item Dimensionen: $\underbrace{m\times n}_{\bm{W}}\cdot\underbrace{n\times j}_{\bm{V}} = \underbrace{m\times j}_{\bm{U}}$ - \item nicht kommutativ: $\bm{VW} \neq \bm{WV}$ - \item Assoziativ: $\bm{V}(\bm{WX}) = (\bm{VW})\bm{X}$ - \item Transponiertes Produkt: $(\bm{VW})^T = \bm{W}^T\bm{V}^T$ - \end{itemize} - \item Reihen"~ und Spaltendurchschnitt: - $$\bm{X} = \begin{bmatrix} \bm{X}_{1,1} & \cdots & \bm{X}_{1,m}\\ \vdots & \ddots & \vdots\\ \bm{X}_{n,1} & \cdots & \bm{X}_{n,m} \end{bmatrix}$$ - \begin{itemize} - \item Vektor von Reihen-Durchschnitten: - $$\begin{bmatrix} \frac{1}{m}\sum_{i=1}^{m} X_{1,i}\\ \vdots \\ \frac{1}{m}\sum_{i=1}^{m} X_{n,i} \end{bmatrix} = \bm{X}\begin{bmatrix} \frac{1}{m}\\\vdots\\\frac{1}{m} \end{bmatrix} = \bm{Xa}, - \text{ mit } \bm{a} = \begin{bmatrix} \frac{1}{m}\\\vdots\\\frac{1}{m} \end{bmatrix}$$ - \item Vektor von Spalten-Durchschnitten: - $$ \begin{bmatrix} \frac{1}{n}\sum_{i=1}^{n} X_{i,1} & \cdots & \frac{1}{n}\sum_{i=1}^{n}X_{i,m} \end{bmatrix} = \begin{bmatrix} \frac{1}{n} & \cdots & \frac{1}{n} \end{bmatrix}\bm{X} = - \bm{b}^T\bm{X}, \text{ mit }\bm{b} = \begin{bmatrix} \frac{1}{n}\\\vdots\\\frac{1}{n} \end{bmatrix}$$ - \end{itemize} - \item Matrix Inverse: - \begin{itemize} - \item Definition: $\bm{WW}^{-1} = \bm{I},\qquad\bm{W}^{-1}\bm{W} = \bm{I}$ - \item Identiätsmatrix $\bm{I} = \begin{bmatrix} 1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 1 \end{bmatrix}$ - \item \textbf{nur} quadratische Matrizen können invertiert werden - \end{itemize} -\end{itemize} - -\subsection{Matrix-Calculus}% -\label{sec:Matrix-Calculus} -\begin{itemize} - \item Die Ableitung einer skalare Funktion, die auf einem Vektor ausgewertet wird ergibt einen Gradientenvektor: - $$\nabla_{\bm{x}}f = \frac{\partial f(\bm{x})}{\partial \bm{x}} = \begin{bmatrix} \frac{\partial f(\bm{x})}{\partial x_1} \\\vdots\\ \frac{\partial f(\bm{x})}{\partial x_d}\end{bmatrix}$$ - \item Die Ableitung einer Vektor-Funktion, die auf einem Vektor ausgewertet wird ergibt eine Jacobische Matrix - $$\nabla_{\bm{x}}\bm{f} = \frac{\partial \bm{f}(\bm{x})}{\partial \bm{x}} - = \begin{bmatrix} \frac{\partial f_1(\bm{x})}{\partial x_1} & \cdots & \frac{\partial f_k(\bm{x})}{\partial x_1} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_1(\bm{x})}{\partial x_d} & \cdots & \frac{\partial f_k(\bm{x})}{\partial x_d}\end{bmatrix}$$ - \item die Ableitung einer skalaren Funktion, die auf einer Matrix ausgewertet wird ergibt eine Matrix: - $$\nabla_{\bm{W}}f = \frac{\partial f(\bm{W})}{\partial \bm{W}} - = \begin{bmatrix} \frac{\partial f(\bm{W})}{\partial W_{11}} & \cdots & \frac{\partial f(\bm{W})}{\partial W_{1d}} \\ \vdots & \ddots & \vdots \\ \frac{\partial f(\bm{W})}{\partial W_{k1}} & \cdots & \frac{\partial f(\bm{W})}{\partial W_{kd}}\end{bmatrix}$$ - \item die Ableitung einer Vektor-Funktion, die auf einer Matrix ausgewertet wird ergibt einen 3D-Tensor (sehr kompliziert, wird (fast) nie benötigt) - \item Grundlegende Formeln:\\ - \begin{tabular}{l c|c|c} - & \bfseries Skalar & \bfseries Vektor & \bfseries Matrix\\ - &&&\\ - \textbf{Linear} & $\frac{\partial a x}{\partial x} = a$ & $\nabla_{\bm{x}}\bm{Ax} = \bm{A}^T$ & \begin{tabular}{@{}c@{}} $\nabla_{\bm{X}}\bm{a}^T\bm{Xb} = \bm{ab}^T $ \\ $\nabla_{\bm{X}}tr(\bm{AXB}) = \bm{A}^T\bm{B}^T$ \end{tabular}\\ - &&&\\ - \textbf{Quadratisch} & $\frac{\partial x^2}{\partial x} = 2x$ & $\begin{aligned} \nabla_{\bm{x}}\bm{x}^T\bm{x} &= 2\bm{x} \\ \nabla_{\bm{x}}\bm{x}^T\bm{Ax} &= (\bm{A}^T + \bm{A})\bm{x}\\&=2\bm{Ax}\text{ wenn $A$ symmetrisch ist }\end{aligned}$ & - \end{tabular} -\end{itemize} - -\section{Regression}% -\label{sec:Regression} Das Ziel einer Regression ist es eine kontinuierliche Funktion $y=f(x)+\nomeq{gaussian_noise}$ zu lernen. Im Falle der linearen Regression bedeutet das, dass versucht wird eine Gerade zu finden, @@ -113,7 +19,7 @@ Die Regression verfolgt hierbei zumeist das Ziel die Summe oder den Durchschnitt \nomeq{summed_squared_error} = \sum_{i=1}^{N}(y_i-f(x_i))^2 \end{equation} -\subsection{Regression für d-dimensionale Eingabevektoren}% +\section{Regression für d-dimensionale Eingabevektoren}% \label{sub:Regression für d-dimensionale Eingabevektoren} Wenn die Eingangswerte durch einen d-dimensionalen Vektor $\bm{x}$ dargestellt werden, ergibt sich die folgende Funktion: diff --git a/chapters/Mathematische_Grundlagen/Lineare_Algebra.tex b/chapters/Mathematische_Grundlagen/Lineare_Algebra.tex new file mode 100644 index 0000000..8cd549f --- /dev/null +++ b/chapters/Mathematische_Grundlagen/Lineare_Algebra.tex @@ -0,0 +1,90 @@ +\chapter{Lineare Algebra}% +\label{cha:Lineare Algebra} + +\section{Vektoren}% +\label{sec:Vektoren} +\begin{itemize} + \item Alle Vektoren werden \textbf{fett} geschrieben: + $\underbrace{x=1}_{\text{Skalar}},\qquad\underbrace{\bm{x}=\begin{bmatrix} 1\\2\\4 \end{bmatrix}}_{\text{Vektor}}$ + \item ein Vektor $\bm{x}$ ist immer ein Spaltenvektor: + $\bm{x}=\begin{bmatrix} 1\\2\\4 \end{bmatrix}$ + \item ein transponierter Vektor $\bm{x}^T$ ist immer ein Reihenvektor: + $\bm{x}^T=\begin{bmatrix} 1 & 2 & 4 \end{bmatrix}$ + \item Multiplikation eines Vektors mit einem Skalar: + $2\begin{bmatrix} 1\\2\\4 \end{bmatrix} = \begin{bmatrix} 2\\4\\8\end{bmatrix}$ + \item Vektoraddition: + $\begin{bmatrix} 1\\2\\4 \end{bmatrix} + \begin{bmatrix} 2\\1\\4 \end{bmatrix} = \begin{bmatrix} 3\\3\\8 \end{bmatrix}$ + \item Skalarprodukt $\langle \bm{v},\bm{w}\rangle$ der Vektoren $\bm{v}=\begin{bmatrix} 1\\2\\4 \end{bmatrix}$ und $\bm{w}=\begin{bmatrix} 2\\4\\8 \end{bmatrix}$: + $\langle \bm{v},\bm{w}\rangle = 1\cdot 2 + 2\cdot 4 + 4\cdot 8 = 42$ + \item Länge eines Vektors: + $||\bm{v}|| = \langle \bm{v},\bm{v}\rangle^{\frac{1}{2}} = (1^2 + 2^2 + 4^2)^{\frac{1}{2}} = \sqrt{21}$ +\end{itemize} + +\section{Matrizen}% +\label{sec:Matrizen} +\begin{itemize} + \item Matrizen werden \uppercase{groß} und \textbf{fett} geschrieben: + $\bm{X}=\begin{bmatrix} 1&3\\2&3\\4&7 \end{bmatrix}\qquad\bm{A}=\begin{bmatrix}1&3&5&4\\2&3&7&2 \end{bmatrix}$ + \item Mehrere Vektoren können zu einer Matrix zusammengefasst werden. + Hierbei werden die Vektoren meistens zu Zeilen und repräsentieren einen Einzelnen Datensatz: + $$\bm{x}_1 = \begin{bmatrix} 37\\72\\175 \end{bmatrix} \qquad \bm{x}_2 = \begin{bmatrix} 10\\30\\61 \end{bmatrix}\qquad \bm{x}_3 = \begin{bmatrix} 25\\65\\121 \end{bmatrix}\qquad \bm{x}_4 = \begin{bmatrix} 66\\67\\175 \end{bmatrix}$$ + $$\bm{X} = \begin{bmatrix} \bm{x}_1^T\\\bm{x}_2^T\\\bm{x}_3^T\\\bm{x}_4^T \end{bmatrix} = \begin{bmatrix} 37&72&175\\10&30&61\\25&65&121\\66&67&175 \end{bmatrix}$$ + \item Multiplikation mit einem Skalar: + $ 3\bm{M} = 3\begin{bmatrix} 3 & 4 & 5 \\ 1 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 9 & 12 & 15\\ 3 & 0 & 3 \end{bmatrix}$ + \item Addition von Matrizen: + $ \bm{M} + \bm{N} = \begin{bmatrix} 3 & 4 & 5 \\ 1 & 0 & 1 \end{bmatrix} + \begin{bmatrix} 1 & 2 & 1 \\ 3 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 4 & 6 & 6 \\ 4 & 1 & 2 \end{bmatrix}$ + \item Transponierte Matrizen: + $ \bm{M} = \begin{bmatrix} 3 & 4 & 5 \\ 1 & 0 & 1 \end{bmatrix}, \bm{M}^T = \begin{bmatrix} 3 & 1\\ 4 & 0\\ 5 & 1 \end{bmatrix}$ + \item Matrix-Vektor-Produkt: + $$ \bm{Wv} = \underbrace{\begin{bmatrix} \bm{w}_1 & \cdots & \bm{w}_n \end{bmatrix}}_{\bm{W}}\underbrace{\begin{bmatrix} v_1 \\ \vdots \\ v_n \end{bmatrix}}_{\bm{v}} = \begin{bmatrix} v_1\bm{w}_1 + \cdots + v_n\bm{w}_n \end{bmatrix} $$ + $$ \bm{Wv} = \begin{bmatrix} 3 & 4 & 5 \\ 1 & 0 & 1 \end{bmatrix}\begin{bmatrix} 3\\0\\2 \end{bmatrix} = \begin{bmatrix} 3\cdot1 + 4\cdot0 + 5\cdot2 \\ 1\cdot1 + 0\cdot0 + 1\cdot2 \end{bmatrix} = \begin{bmatrix} 13\\3 \end{bmatrix}$$ + \item Matrix-Matrix-Produkt: + $$ \bm{WV} = \begin{bmatrix} 3 & 4 & 5\\ 1 & 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0\\0 & 3\\2 & 4 \end{bmatrix} = + \begin{bmatrix} 3\cdot1+4\cdot0+5\cdot2 & 3\cdot0+4\cdot3+5\cdot4\\ 1\cdot1+0\cdot0+1\cdot2 & 1\cdot0+0\cdot3+1\cdot4 \end{bmatrix} = \begin{bmatrix} 13 & 32\\ 3 & 4 \end{bmatrix}$$ + \begin{itemize} + \item Dimensionen: $\underbrace{m\times n}_{\bm{W}}\cdot\underbrace{n\times j}_{\bm{V}} = \underbrace{m\times j}_{\bm{U}}$ + \item nicht kommutativ: $\bm{VW} \neq \bm{WV}$ + \item Assoziativ: $\bm{V}(\bm{WX}) = (\bm{VW})\bm{X}$ + \item Transponiertes Produkt: $(\bm{VW})^T = \bm{W}^T\bm{V}^T$ + \end{itemize} + \item Reihen"~ und Spaltendurchschnitt: + $$\bm{X} = \begin{bmatrix} \bm{X}_{1,1} & \cdots & \bm{X}_{1,m}\\ \vdots & \ddots & \vdots\\ \bm{X}_{n,1} & \cdots & \bm{X}_{n,m} \end{bmatrix}$$ + \begin{itemize} + \item Vektor von Reihen-Durchschnitten: + $$\begin{bmatrix} \frac{1}{m}\sum_{i=1}^{m} X_{1,i}\\ \vdots \\ \frac{1}{m}\sum_{i=1}^{m} X_{n,i} \end{bmatrix} = \bm{X}\begin{bmatrix} \frac{1}{m}\\\vdots\\\frac{1}{m} \end{bmatrix} = \bm{Xa}, + \text{ mit } \bm{a} = \begin{bmatrix} \frac{1}{m}\\\vdots\\\frac{1}{m} \end{bmatrix}$$ + \item Vektor von Spalten-Durchschnitten: + $$ \begin{bmatrix} \frac{1}{n}\sum_{i=1}^{n} X_{i,1} & \cdots & \frac{1}{n}\sum_{i=1}^{n}X_{i,m} \end{bmatrix} = \begin{bmatrix} \frac{1}{n} & \cdots & \frac{1}{n} \end{bmatrix}\bm{X} = + \bm{b}^T\bm{X}, \text{ mit }\bm{b} = \begin{bmatrix} \frac{1}{n}\\\vdots\\\frac{1}{n} \end{bmatrix}$$ + \end{itemize} + \item Matrix Inverse: + \begin{itemize} + \item Definition: $\bm{WW}^{-1} = \bm{I},\qquad\bm{W}^{-1}\bm{W} = \bm{I}$ + \item Identiätsmatrix $\bm{I} = \begin{bmatrix} 1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 1 \end{bmatrix}$ + \item \textbf{nur} quadratische Matrizen können invertiert werden + \end{itemize} +\end{itemize} + +\section{Matrix-Calculus}% +\label{sec:Matrix-Calculus} +\begin{itemize} + \item Die Ableitung einer skalare Funktion, die auf einem Vektor ausgewertet wird ergibt einen Gradientenvektor: + $$\nabla_{\bm{x}}f = \frac{\partial f(\bm{x})}{\partial \bm{x}} = \begin{bmatrix} \frac{\partial f(\bm{x})}{\partial x_1} \\\vdots\\ \frac{\partial f(\bm{x})}{\partial x_d}\end{bmatrix}$$ + \item Die Ableitung einer Vektor-Funktion, die auf einem Vektor ausgewertet wird ergibt eine Jacobische Matrix + $$\nabla_{\bm{x}}\bm{f} = \frac{\partial \bm{f}(\bm{x})}{\partial \bm{x}} + = \begin{bmatrix} \frac{\partial f_1(\bm{x})}{\partial x_1} & \cdots & \frac{\partial f_k(\bm{x})}{\partial x_1} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_1(\bm{x})}{\partial x_d} & \cdots & \frac{\partial f_k(\bm{x})}{\partial x_d}\end{bmatrix}$$ + \item die Ableitung einer skalaren Funktion, die auf einer Matrix ausgewertet wird ergibt eine Matrix: + $$\nabla_{\bm{W}}f = \frac{\partial f(\bm{W})}{\partial \bm{W}} + = \begin{bmatrix} \frac{\partial f(\bm{W})}{\partial W_{11}} & \cdots & \frac{\partial f(\bm{W})}{\partial W_{1d}} \\ \vdots & \ddots & \vdots \\ \frac{\partial f(\bm{W})}{\partial W_{k1}} & \cdots & \frac{\partial f(\bm{W})}{\partial W_{kd}}\end{bmatrix}$$ + \item die Ableitung einer Vektor-Funktion, die auf einer Matrix ausgewertet wird ergibt einen 3D-Tensor (sehr kompliziert, wird (fast) nie benötigt) + \item Grundlegende Formeln:\\ + \begin{tabular}{l c|c|c} + & \bfseries Skalar & \bfseries Vektor & \bfseries Matrix\\ + &&&\\ + \textbf{Linear} & $\frac{\partial a x}{\partial x} = a$ & $\nabla_{\bm{x}}\bm{Ax} = \bm{A}^T$ & \begin{tabular}{@{}c@{}} $\nabla_{\bm{X}}\bm{a}^T\bm{Xb} = \bm{ab}^T $ \\ $\nabla_{\bm{X}}tr(\bm{AXB}) = \bm{A}^T\bm{B}^T$ \end{tabular}\\ + &&&\\ + \textbf{Quadratisch} & $\frac{\partial x^2}{\partial x} = 2x$ & $\begin{aligned} \nabla_{\bm{x}}\bm{x}^T\bm{x} &= 2\bm{x} \\ \nabla_{\bm{x}}\bm{x}^T\bm{Ax} &= (\bm{A}^T + \bm{A})\bm{x}\\&=2\bm{Ax}\text{ wenn $A$ symmetrisch ist }\end{aligned}$ & + \end{tabular} +\end{itemize} + + diff --git a/chapters/Mathematische_Grundlagen/Probability_Theory.tex b/chapters/Mathematische_Grundlagen/Probability_Theory.tex new file mode 100644 index 0000000..1c9046f --- /dev/null +++ b/chapters/Mathematische_Grundlagen/Probability_Theory.tex @@ -0,0 +1,61 @@ +\chapter{Probability Theory}% +\label{cha:Probability Theory} +Eine Funktion \nomsym{probability_mass_function}, +die die Wahrscheinlichkeit angibt, +dass $X$ den Wert $x$ annimmt, +wird als \noms{probability_mass_function} bezeichnet. +Eine gültige \noms{probability_mass_function} muss folgende Eigenschaften erfüllen: +\begin{itemize} + \item weist jedem $x\in X$ einen Wert zu + \item nicht-negativ + \item die Summe aller Wahrscheinlichkeiten ist 1 +\end{itemize} + +Zwei Wahrscheinlichkeitsdichtefunktionen können dabei auf verschiedene Arten miteinander zusammenhängen: +\begin{itemize} + \item \textbf{Joint Distribution} $p(x,y)$: Die Wahrscheinlichkeit das $X=x$ und $Y=y$ + \item \textbf{Conditional Distribution} $p(x|y)$: Die Wahrscheinlichkeit für $X=x$, wenn $Y=y$ gegeben ist +\end{itemize} +\begin{figure}[H] + \centering + \includegraphics[width=.6\linewidth]{images/conditional_and_joint_distribution.png} + \caption{Conditional and Joint Distribution} + \label{fig:conditional_and_joint_distribution} +\end{figure} + +\section{Rules of Probability}% +\label{sec:Rules of Probability} +\paragraph{Summenregel}% +\label{par:Summenregel} +\begin{align} \label{eq:sum_rule} + p(x) &= \sum_y p(x,y)\\ + p(x_1) &= \sum_{x_2}\sum_{x_3}\cdots\sum_{x_D} p(x_1,\dots,x_D) +\end{align} +\begin{figure}[H] + \centering + \includegraphics[width=0.6\textwidth]{images/sum_rule.png} + \caption{Summenregel} + \label{fig:sum_rule} +\end{figure} + +\paragraph{Ketten"~\slash\,Produktregel}% +\label{par:Ketten-/Produktregel} +\begin{align}\label{eq:chain_rule} + p(x,y) &= p(x|y)p(y) \\ + p(x_1,\dots,x_D) &= p(x_1)p(x_2|x_1)\dots p(x_D|x_1,\dots, x_{D-1}) +\end{align} +\begin{figure}[H] + \centering + \includegraphics[width=0.7\textwidth]{images/chain_rule.png} + \caption{Ketten"~\slash\,Produktregel} + \label{fig:chain_rule} +\end{figure} + +\subsection{Bayes Rule}% +\label{sub:Bayes Rule} +Die Regel von Bayes ist eine der wichtigsten Regeln der Wahrscheinlichkeitstheorie und essentiell im Bereich des Maschinellen Lernens. +\begin{equation} \label{eq:Bayes Rule} + p(x|y) = \dfrac{p(y|x)p(x)}{p(y)} = \dfrac{p(y|x)p(x)}{\sum_{x'}p(y|x')p(x')} +\end{equation} + +Weiter auf Seite 114 diff --git a/images/chain_rule.png b/images/chain_rule.png new file mode 100644 index 0000000000000000000000000000000000000000..0645c282048ef31446a1580a38e9b02e6ad50b54 GIT binary patch literal 108089 zcmc$`cR1Gn`!}v7g;GQj(va+A3ninJ5!tfImR)9r3Q3Y3ijeG0Rw&{^A$v9`d+*Wx zxcYoQfBydZ{*L>;j-#V@Z(Ls2^Ywh5=VP5Oe`Q5E%Dwb^Nk~X2uUwW^At52vBq7<> zOST>VhwT8#5Bzo0RP&0Nf&vLU{+oY zwvOg@PRDrpxX%i4^Pa_*pY8bfBQ*GqD}20se4@O3qGx%JofQxjI4jC0Kr*Ju7eIWI zE7BKl+<7_C?|O%(y>s`BSYG*?WxdO%^1r%Im%n>_h}`Ak4I1(Uwg-|jA0JSX2OQt7 zB7ITWZNKjYw`ot+qUq)`Z;I(+?kip2{C=(9_3$tgJm+ec%i_M==lCT_GyD|s3je)g z8EvNb{r4XvPCIv2|MwrSu3k7s@!x;E|9|_1*X3d)U$vRNR=&T!^k}8+Qy|@!7@^<< 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