From 4614294614e6f6b38e0ca86e77871e75b4c26071 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 16 Mar 2021 15:48:10 +0100 Subject: add new slides --- vorlesungen/slides/8/markov/google.tex | 123 ------------------------- vorlesungen/slides/8/markov/irreduzibel.tex | 136 ---------------------------- vorlesungen/slides/8/markov/markov.tex | 111 ----------------------- vorlesungen/slides/8/markov/pf.tex | 53 ----------- vorlesungen/slides/8/markov/stationaer.tex | 57 ------------ 5 files changed, 480 deletions(-) delete mode 100644 vorlesungen/slides/8/markov/google.tex delete mode 100644 vorlesungen/slides/8/markov/irreduzibel.tex delete mode 100644 vorlesungen/slides/8/markov/markov.tex delete mode 100644 vorlesungen/slides/8/markov/pf.tex delete mode 100644 vorlesungen/slides/8/markov/stationaer.tex (limited to 'vorlesungen/slides/8/markov') diff --git a/vorlesungen/slides/8/markov/google.tex b/vorlesungen/slides/8/markov/google.tex deleted file mode 100644 index d1ec31d..0000000 --- a/vorlesungen/slides/8/markov/google.tex +++ /dev/null @@ -1,123 +0,0 @@ -% -% google.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Google-Matrix} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick] - -\def\r{2.4} -\coordinate (A) at (0,0); -\coordinate (B) at (0:\r); -\coordinate (C) at (60:\r); -\coordinate (D) at (120:\r); -\coordinate (E) at (180:\r); - -\foreach \a in {2,...,5}{ - \fill[color=white] ({60*(\a-2)}:\r) circle[radius=0.2]; - \draw ({60*(\a-2)}:\r) circle[radius=0.2]; - \node at ({60*(\a-2)}:\r) {$\a$}; -} -\fill[color=white] (A) circle[radius=0.2]; -\draw (A) circle[radius=0.2]; -\node at (A) {$1$}; - -{\color<6>{red} - \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B); - \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); -} - -{\color<7>{red} - \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C); - \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) to[out=-150,in=-30] (E); -} - -{\color<8>{red} - \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) to[out=-90,in=30] (A); - \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) to[out=-30,in=90] (B); -} - -{\color<9>{red} - \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (C); - \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (A); - \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E); -} - -{\color<10>{red} - \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A); - \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) to[out=90,in=-150] (D); -} - -\end{tikzpicture} -\end{center} -\vspace{-10pt} -\renewcommand{\arraystretch}{1.1} -\uncover<5->{ -\begin{align*} -H&=\begin{pmatrix} -\uncover<6->{0 } - &\uncover<7->{0 } - &\uncover<8->{{\color<8>{red}\frac{1}{2}}} - &\uncover<9->{{\color<9>{red}\frac{1}{3}}} - &\uncover<10->{{\color<10>{red}\frac{1}{2}}}\\ -\uncover<6->{{\color<6>{red}\frac{1}{2}}} - &\uncover<7->{0 } - &\uncover<8->{{\color<8>{red}\frac{1}{2}}} - &\uncover<9->{0 } - &\uncover<10->{0 }\\ -\uncover<6->{{\color<6>{red}\frac{1}{2}}} - &\uncover<7->{{\color<7>{red}\frac{1}{2}}} - &\uncover<8->{0 } - &\uncover<9->{{\color<9>{red}\frac{1}{3}}} - &\uncover<10->{0 }\\ -\uncover<6->{0 } - &\uncover<7->{0 } - &\uncover<8->{0 } - &\uncover<9->{0 } - &\uncover<10->{{\color<10>{red}\frac{1}{2}}}\\ -\uncover<6->{0 } - &\uncover<7->{{\color<7>{red}\frac{1}{2}}} - &\uncover<8->{0 } - &\uncover<9->{{\color<9>{red}\frac{1}{3}}} - &\uncover<10->{0 } -\end{pmatrix} -\\ -\uncover<11->{ -h_{ij} -&= -\frac{1}{\text{Anzahl Links ausgehend von $j$}} -} -\end{align*}} -\end{column} -\begin{column}{0.48\textwidth} -\begin{block}{Aufgabe} -Bestimme die Wahrscheinlichkeit $p(i)$, mit der sich ein Surfer -auf der Website $i$ befindet -\end{block} -\uncover<2->{ -\begin{block}{Navigation} -$p(i) = P(i,\text{vor Navigation})$, -\uncover<3->{$p'(i)=P(i,\text{nach Navigation})$} -\uncover<4->{ -\[ -p'(i) = \sum_{j=1}^n h_{ij} p(j) -\]} -\end{block}} -\vspace{-15pt} -\begin{block}{Freier Wille} -\vspace{-12pt} -\[ -G = \alpha H + (1-\alpha)\frac{UU^t}{n} -\] -Google-Matrix -\end{block} -\end{column} -\end{columns} -\end{frame} diff --git a/vorlesungen/slides/8/markov/irreduzibel.tex b/vorlesungen/slides/8/markov/irreduzibel.tex deleted file mode 100644 index 87e90e4..0000000 --- a/vorlesungen/slides/8/markov/irreduzibel.tex +++ /dev/null @@ -1,136 +0,0 @@ -% -% irreduzibel.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\frametitle{Irreduzible Markovkette} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\r{2} -\coordinate (A) at ({\r*cos(0*60)},{\r*sin(0*60)}); -\coordinate (B) at ({\r*cos(1*60)},{\r*sin(1*60)}); -\coordinate (C) at ({\r*cos(2*60)},{\r*sin(2*60)}); -\coordinate (D) at ({\r*cos(3*60)},{\r*sin(3*60)}); -\coordinate (E) at ({\r*cos(4*60)},{\r*sin(4*60)}); -\coordinate (F) at ({\r*cos(5*60)},{\r*sin(5*60)}); - -\uncover<-2>{ -\draw (A) -- (B); -\draw (A) -- (C); -\draw (A) -- (D); -\draw (A) -- (E); -\draw (A) -- (F); - -\draw (B) -- (A); -\draw (B) -- (C); -\draw (B) -- (D); -\draw (B) -- (E); -\draw (B) -- (F); - -\draw (C) -- (A); -\draw (C) -- (B); -\draw (C) -- (D); -\draw (C) -- (E); -\draw (C) -- (F); - -\draw (D) -- (A); -\draw (D) -- (B); -\draw (D) -- (C); -\draw (D) -- (E); -\draw (D) -- (F); - -\draw (E) -- (A); -\draw (E) -- (B); -\draw (E) -- (C); -\draw (E) -- (D); -\draw (E) -- (F); - -\draw (F) -- (A); -\draw (F) -- (B); -\draw (F) -- (C); -\draw (F) -- (D); -\draw (F) -- (E); -} - -\uncover<3->{ - -\draw[->,color=black!30,shorten >= 0.15cm,line width=3pt] (A) to[out=90,in=-30] (B); -\draw[->,color=black!70,shorten >= 0.15cm,line width=3pt] (A) -- (C); -\draw[->,color=black!20,shorten >= 0.15cm,line width=3pt] (B) -- (A); -\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt] (B) to[out=150,in=30] (C); -\draw[->,color=black!20,shorten >= 0.15cm,line width=3pt] (B) to[out=-90,in=-150,distance=1cm] (B); -\draw[->,color=black!50,shorten >= 0.15cm,line width=3pt] (C) to[out=-60,in=180] (A); -\draw[->,color=black!50,shorten >= 0.15cm,line width=3pt] (C) -- (B); - -\draw[->,color=black!40,shorten >= 0.15cm,line width=3pt] - (D) to[out=-90,in=150] (E); -\draw[->,color=black!30,shorten >= 0.15cm,line width=3pt] - (E) -- (D); -\draw[->,color=black!70,shorten >= 0.15cm,line width=3pt] - (E) to[out=-30,in=-150] (F); -\draw[->,color=black!40,shorten >= 0.15cm,line width=3pt] - (F) -- (E); -\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt] - (F) to[out=120,in=0] (D); -\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt] - (D) -- (F); -} - -\fill[color=white] (A) circle[radius=0.2]; -\fill[color=white] (B) circle[radius=0.2]; -\fill[color=white] (C) circle[radius=0.2]; -\fill[color=white] (D) circle[radius=0.2]; -\fill[color=white] (E) circle[radius=0.2]; -\fill[color=white] (F) circle[radius=0.2]; - -\draw (A) circle[radius=0.2]; -\draw (B) circle[radius=0.2]; -\draw (C) circle[radius=0.2]; -\draw (D) circle[radius=0.2]; -\draw (E) circle[radius=0.2]; -\draw (F) circle[radius=0.2]; - -\node at (A) {$1$}; -\node at (B) {$2$}; -\node at (C) {$3$}; -\node at (D) {$4$}; -\node at (E) {$5$}; -\node at (F) {$6$}; - -\end{tikzpicture} -\end{center} -\uncover<2->{% -\begin{block}{Irreduzibel} -Graph zusammenhängend $\Rightarrow$ -Keine Zerlegung in Teilgraphen möglich -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<3->{% -\begin{block}{Reduzibel} -Die Zustandsmenge zerfällt in zwei disjunkte Teilmengen $V=V_1\cup V_2$ -und es gibt keine Übergängen zwischen den Mengen: -\uncover<4->{% -\begin{align*} -P -&= -\begin{pmatrix*}[l] -0 &0.2&0.5& & & \\ -0.3&0.2&0.5& & & \\ -0.7&0.6&0 & & & \\ - & & &0 &0.3&0.4\\ - & & &0.4&0 &0.6\\ - & & &0.6&0.7&0 -\end{pmatrix*} -\end{align*}}% -\uncover<5->{% -$P$ zerfällt in zwei Blöcke die unabhängig voneinander analysiert werden können -} -\end{block}} -\end{column} -\end{columns} -\end{frame} diff --git a/vorlesungen/slides/8/markov/markov.tex b/vorlesungen/slides/8/markov/markov.tex deleted file mode 100644 index e92ff0f..0000000 --- a/vorlesungen/slides/8/markov/markov.tex +++ /dev/null @@ -1,111 +0,0 @@ -% -% markov.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\begin{frame}[t] -\frametitle{Markovketten} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick] - -\def\r{2.2} - -\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)}); -\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)}); -\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)}); -\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)}); -\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)}); - -\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!40] - (A) -- (C); -\draw[color=white,line width=8pt] (B) -- (D); -\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!80] - (B) -- (D); - -\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!60] - (A) -- (B); -\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!20] - (B) -- (C); -\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black] - (C) -- (D); -\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black] - (D) -- (E); -\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black] - (E) -- (A); - -\fill[color=white] (A) circle[radius=0.2]; -\fill[color=white] (B) circle[radius=0.2]; -\fill[color=white] (C) circle[radius=0.2]; -\fill[color=white] (D) circle[radius=0.2]; -\fill[color=white] (E) circle[radius=0.2]; - -\draw (A) circle[radius=0.2]; -\draw (B) circle[radius=0.2]; -\draw (C) circle[radius=0.2]; -\draw (D) circle[radius=0.2]; -\draw (E) circle[radius=0.2]; - -\node at (A) {$1$}; -\node at (B) {$2$}; -\node at (C) {$3$}; -\node at (D) {$4$}; -\node at (E) {$5$}; - -\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 0.6$}; -\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 0.2$}; -\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 1$}; -\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 1$}; -\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 1$}; -\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 0.4$}; -\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 0.8$}; - -\end{tikzpicture} -\end{center} -\vspace{-10pt} -\uncover<7->{% -\begin{block}{Verteilung} -\begin{itemize} -\item<8-> -Welche stationäre Verteilung auf den Knoten stellt sich ein? -\item<9-> -$P(i)=?$ -\end{itemize} -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<2->{% -\begin{block}{\strut\mbox{Übergang\only<3->{s-/Wahrscheinlichkeit}smatrix}} -$P_{ij} = P(i | j)$, Wahrscheinlichkeit, in den Zustand $i$ überzugehen, -\begin{align*} -P -&= -\begin{pmatrix} - & & & &1\phantom{.0}\\ -0.6& & & & \\ -0.4&0.2& & & \\ - &0.8&1\phantom{.0}& & \\ - & & &1\phantom{.0}& -\end{pmatrix} -\end{align*} -\end{block}} -\vspace{-10pt} -\uncover<4->{% -\begin{block}{Eigenschaften} -\begin{itemize} -\item<5-> $P_{ij}\ge 0\;\forall i,j$ -\item<6-> Spaltensumme: -\( -\displaystyle -\sum_{i=1}^n P_{ij} = 1\;\forall j -\) -\end{itemize} -\end{block}} -\end{column} -\end{columns} -\end{frame} diff --git a/vorlesungen/slides/8/markov/pf.tex b/vorlesungen/slides/8/markov/pf.tex deleted file mode 100644 index da2ef2b..0000000 --- a/vorlesungen/slides/8/markov/pf.tex +++ /dev/null @@ -1,53 +0,0 @@ -% -% pf.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\frametitle{Perron-Frobenius-Theorie} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Positive Matrizen und Vektoren} -$P\in M_{m\times n}(\mathbb{R})$ -\begin{itemize} -\item<2-> -$P$ heisst positiv, $P>0$, wenn $p_{ij}>0\;\forall i,j$ -\item<3-> -$P\ge 0$, wenn $p_{ij}\ge 0\;\forall i,j$ -\end{itemize} -\end{block} -\uncover<4->{% -\begin{block}{Beispiele} -\begin{itemize} -\item<5-> -Adjazenzmatrix $A(G)$ -\item<6-> -Gradmatrix $D(G)$ -\item<7-> -Wahrscheinlichkeitsmatrizen -\end{itemize} -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<8->{% -\begin{block}{Satz} -Es gibt einen positiven Eigenvektor $p$ von $P$ zum Eigenwert $1$ -\end{block}} -\uncover<9->{% -\begin{block}{Satz} -$P$ irreduzible Matrix, $P\ge 0$, hat einen Eigenvektor $p$, $p\ge 0$, -zum Eigenwert $1$ -\end{block}} -\uncover<10->{% -\begin{block}{Potenzmethode} -Falls $P\ge 0$ einen eindeutigen Eigenvektor $p$ hat\uncover<11->{, -dann konveriert die rekursiv definierte Folge -\[ -p_{n+1}=\frac{Pp_n}{\|Pp_n\|}, p_0 \ge 0, p_0\ne 0 -\]}% -\uncover<12->{$\displaystyle\lim_{n\to\infty} p_n = p$} -\end{block}} -\end{column} -\end{columns} -\end{frame} diff --git a/vorlesungen/slides/8/markov/stationaer.tex b/vorlesungen/slides/8/markov/stationaer.tex deleted file mode 100644 index 92fab16..0000000 --- a/vorlesungen/slides/8/markov/stationaer.tex +++ /dev/null @@ -1,57 +0,0 @@ -% -% stationaer.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\frametitle{Stationäre Verteilung} -%\vspace{-15pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Zeitentwicklung} -\begin{itemize} -\item<2-> -$P$ eine Wahrscheinlichkeitsmatrix -\item<3-> -$p_0\in\mathbb{R}^n$ Verteilung zur Zeit $t=0$ bekannt -\item<4-> -$p_k\in\mathbb{R}^n$ Verteilung zur Zeit $t=k$ -\end{itemize} -\uncover<5->{% -Entwicklungsgesetz -\begin{align*} -P(i,t=k) -&= -\sum_{j=1}^n P_{ij} P(j,t=k-1) -\\ -\uncover<6->{ -p_k &= Pp_{k-1} -} -\end{align*}} -\end{block} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<7->{% -\begin{block}{Stationär} -Bedingung: $p_{k\mathstrut} = p_{k-1}$ -\uncover<8->{ -\begin{align*} -\Rightarrow -Pp &= p -\end{align*}}\uncover<9->{% -Eigenvektor zum Eigenwert $1$} -\end{block}} -\uncover<10->{% -\begin{block}{Fragen} -\begin{enumerate} -\item<11-> -Gibt es eine stationäre Verteilung? -\item<12-> -Gibt es einen Eigenvektor mit Einträgen $\ge 0$? -\item<13-> -Gibt es mehr als eine Verteilung? -\end{enumerate} -\end{block}} -\end{column} -\end{columns} -\end{frame} -- cgit v1.2.1