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zeAs&-89ZCkJ%x6iBZE~w+zp`G6cYD`iUkyUND4EAvtmvGL%QBg&w0(&1Ml{JBlh4~ zHmPLkF%+m}hCLMRUc;S7%&{9uv%Wi#UUXa8$|axh_t;?(nRetOG4 O{`u1er;1Npz4bpj=|eOC literal 0 HcmV?d00001 diff --git a/sections/simulation.tex b/sections/simulation.tex index df22942..546542e 100644 --- a/sections/simulation.tex +++ b/sections/simulation.tex @@ -175,7 +175,13 @@ \subsection{Wahrscheinlichkeitsverteilungen} \subsubsection{Normalverteilung} $X \sim N(100, 10'000)$ \\ \includegraphics[width=0.25\textwidth]{pictures/normalverteilung} - \subsubsection{Empirische Verteilung} + \subsubsection{Poissonverteilung} + $X \sim P(4)$ \\ + \includegraphics[width=0.25\textwidth]{pictures/poissonverteilung} +\end{multicols} + +\subsubsection{Empirische Verteilung} +\begin{multicols}{2} \begin{tabular}{|l|l|} \hline \textbf{Zufallszahl} & \textbf{Wahrscheinlichkeit} \\ \hline @@ -184,20 +190,19 @@ \subsection{Wahrscheinlichkeitsverteilungen} 4 & 0.1 \\ \hline 8 & 0.15 \\ \hline 16 & 0.05 \\ \hline - \end{tabular} + \end{tabular} \\ + \textbf{Vorteile:} + \begin{compactitem} + \item Basiert auf realen Daten + \item Alle möglichen Formen sind möglich + \end{compactitem} + \textbf{Nachteile:} + \begin{compactitem} + \item Es können keine Zufallszahlen erzeugt werden, die nicht schon in der Vergangenheit aufgetreten sind. Daher braucht man eine lange Historie. + \item Empirische Verteilungen kann man nicht kompakt auf Papier darstellen. + \end{compactitem} \end{multicols} -\subsubsection{Empirische Verteilung - Vor- / Nachteile} -\begin{compactitem} - \item Basiert auf realen Daten - \item Alle möglichen Formen sind möglich -\end{compactitem} -\textbf{Nachteile:} -\begin{compactitem} - \item Es können keine Zufallszahlen erzeugt werden, die nicht schon in der Vergangenheit aufgetreten sind. Daher braucht man eine lange Historie. - \item Empirische Verteilungen kann man nicht kompakt auf Papier darstellen. -\end{compactitem} - \subsubsection{Parameterschätzungen Gefahren} \begin{multicols}{2} \begin{compactitem} @@ -208,5 +213,99 @@ \subsubsection{Parameterschätzungen Gefahren} \item Falsche Interpretation des vorhandenen Datenmaterials \end{compactitem} \end{multicols} + +\subsection{Zufallszahlen} +\textbf{Grobe Definition:} Eine Bit-Reihenfolge ist willkürlich, wenn sie auf keine Weise ein Muster enthält. \\ +\textbf{Zufallszahlengenerator:} Erzeugt Zufallszahlen zwischen 0 und 1 (und wandelt sie bei Bedarf in Zufallszahlen aus der gewünschten Wahrscheinlichkeitsverteilung um). \\ \\ +\textbf{Anforderungen:} +\begin{multicols}{2} + \begin{compactitem} + \item Unabhängigkeit (auch die Elemente jeder Teilfolge) + \item Gleichverteilung (empirische Verteilung weitgehend konstant) + \item Besetzungsdichte (keine Lücken) + \item Keine Periodizität + \item Schneller und speichereffizienter Generator + \item allenfalls Reproduzierbarkeit + \end{compactitem} +\end{multicols} +\textbf{Bemerkung:} Wirkliche Zufallszahlen sind nicht reproduzierbar. In den meisten Fällen will man die Zufallszahlen aber reproduzieren können. Wir sprechen in dem Fall von Pseudo – Zufallszahlen. - \ No newline at end of file +\subsubsection{Deterministische Reihenfolge} +\begin{compactitem} + \item Jede Zufallszahl wird direkt aus ihrem unmittelbaren Vorgänger erzeugt. + \item Die Erzeugung von Zufallszahlen startet mit einem initialen Wert. Dieser erste Wert des Zahlenflusses wird als Seed bezeichnet. + \item Der Seedwert legt den gesamten Zahlenfluss eindeutig fest. D.h., wenn die Seedwerte identisch sind, sind die Zahlenflüsse, die sie auslösen, es auch. +\end{compactitem} + +\subsubsection{Lineare Kongruenzmethode} +\textbf{Formel:} $X_{i+1}=(aX_i+c)\mod m$ mit $a$, $c$, $m$ und $X_0$ als Integer und $m > 0$, $a < m$, $c < m$ und $X_0 < m$ \\ +somit gilt: $u_i = \frac{X_i}{m-1}\sim U(0, 1)$ \\ +Die Qualität des Zufallszahlengenerators hängt stark von a, c und m ab. Die wichtigsten Parameter sind a und m; Meistens wählt man sehr hohe (Prim-)Zahlen. \\ +\textbf{Problem:} In den allermeisten Fällen brauchen wir keine uniformverteile Zufallszahlen zwischen 0 und 1 sondern Zufallszahlen die aus einer anderen Verteilung stammen. + +\subsubsection{Intervallmethode (Diskrete Zielverteilung)} +\begin{minipage}[h]{0.6\textwidth} + \begin{tabular}{|l|l|l|l|} + \hline + \textbf{i} & \textbf{Zufallszahl}($r_i$) & \textbf{Wahrscheinlichkeit} & \textbf{Intervall} ($c_{i-1}$, $c_i$) \\ \hline + 1 & 1 & 0.4 & [0.00, 0.4) \\ \hline + 2 & 2 & 0.3 & [0.40, 0.7) \\ \hline + 3 & 4 & 0.15 & [0.70, 0.85) \\ \hline + 4 & 5 & 0.1 & [0.85, 0.95) \\ \hline + 5 & 18 & 0.05 & [0.95, 1.0) \\ \hline + \end{tabular} +\end{minipage} +\begin{minipage}[h]{0.3\textwidth} + \begin{lstlisting}[mathescape=true, tabsize=2] +x= U(0,1); // Standard Zufallszahl + // zwischen 0 und 1 +for i = 1 : n + if $c_{i-1}$ =< x < $c_i$ + returnValue = $r_i$; + break; + end +end + \end{lstlisting} +\end{minipage} + +\subsubsection{Inversionsmethode (Kontinuierliche Zielverteilung)} +\begin{compactenum} + \item Generiere $x\sim U(0, 1)$ + \item Berechne $y = F_y^{-1}(x)$ +\end{compactenum} +\begin{example} + Uniformverteilung im Bereich [a, b] + \begin{multicols}{2} + \begin{compactitem} + \item Gegeben: $x\sim U(0, 1)$ + \item Gesucht: $y\sim U(a, b)$ + \item Transformation: $y = a + (b ‐ a)x$ + \item Beachte: $F_y(y) = \left\{\begin{array}{ll} + 0 & y < 0 \\ + \frac{y-a}{b-a} & a \leq y < b \\ + 1 & y \geq b \\ + \end{array}\right.$ + \end{compactitem} + \includegraphics[width=0.33\textwidth]{pictures/inversionsmethode} + \end{multicols} +\end{example} + +\begin{multicols}{2} + \begin{example} + Exponentionalverteilung mit Parameter $\lambda$ \\ + \begin{compactitem} + \item Gegeben: $x\sim U(0, 1)$ + \item Gesucht: $y\sim \exp (\lambda)$ + \item Beachte: $\begin{array}{l} + F_y(y) = 1-e^{-\lambda y} \\ + x = 1-e^{-\lambda y} \\ + e^{-\lambda y} = 1-x\\ + y = -\frac{1}{\lambda}\ln (1-x) + \end{array}$ + \end{compactitem} + \end{example} + \begin{example} + Empirische Verteilung \\ + \includegraphics[width=1\textwidth]{pictures/inversionsmethode2} + \end{example} +\end{multicols}