From b5084b78f44632f51eacd0546f43da969918e62f Mon Sep 17 00:00:00 2001 From: Jan Benda <jan.benda@uni-tuebingen.de> Date: Mon, 19 Oct 2015 17:35:27 +0200 Subject: [PATCH 1/4] Added exercises --- statistics-fabian/assignments/day1_001.pdf | Bin 53507 -> 0 bytes statistics-fabian/assignments/day1_002.pdf | Bin 52598 -> 0 bytes statistics-fabian/assignments/example001.csv | 43 ---- statistics-fabian/assignments/example002.csv | 186 ------------------ statistics/exercises/UT_WBMW_Black_RGB.pdf | Bin 0 -> 31288 bytes .../exercises/descriptivestatistics-01.tex | 162 +++++++++++++++ .../exercises/descriptivestatistics-02.tex | 164 +++++++++++++++ statistics/lecture/descriptivestatistics.tex | 4 +- 8 files changed, 328 insertions(+), 231 deletions(-) delete mode 100644 statistics-fabian/assignments/day1_001.pdf delete mode 100644 statistics-fabian/assignments/day1_002.pdf delete mode 100755 statistics-fabian/assignments/example001.csv delete mode 100755 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+\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry} +\pagestyle{headandfoot} +\header{{\bfseries\large \"Ubung 1}}{{\bfseries\large Deskriptive Statistik}}{{\bfseries\large 19. Oktober, 2015}} +\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email: +jan.grewe@uni-tuebingen.de} +\runningfooter{}{\thepage}{} + +\setlength{\baselineskip}{15pt} +\setlength{\parindent}{0.0cm} +\setlength{\parskip}{0.3cm} +\renewcommand{\baselinestretch}{1.15} + +\newcommand{\qt}[1]{\textbf{#1}\\} +\newcommand{\pref}[1]{(\ref{#1})} +\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}} +\newcommand{\code}[1]{\texttt{#1}} + +\newcommand{\continue}{\ifprintanswers% +\else +\vfill\hspace*{\fill}$\rightarrow$\newpage% +\fi} +\newcommand{\continuepage}{\ifprintanswers% +\newpage +\else +\vfill\hspace*{\fill}$\rightarrow$\newpage% +\fi} +\newcommand{\newsolutionpage}{\ifprintanswers% +\newpage% +\else +\fi} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} + +\vspace*{-6.5ex} +\begin{center} +\textbf{\Large Einf\"uhrung in die wissenschaftliche Datenverarbeitung}\\[1ex] +{\large Jan Grewe, Jan Benda}\\[-3ex] +Abteilung Neuroethologie \hfill --- \hfill Institut f\"ur Neurobiologie \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\ +\end{center} + +% Die folgenden Aufgaben dienen der Wiederholung, \"Ubung und +% Selbstkontrolle und sollten eigenst\"andig bearbeitet und gel\"ost +% werden. Die L\"osung soll in Form eines einzelnen Skriptes (m-files) +% im ILIAS hochgeladen werden. Jede Aufgabe sollte in einer eigenen +% ``Zelle'' gel\"ost sein. Die Zellen \textbf{m\"ussen} unabh\"angig +% voneinander ausf\"uhrbar sein. Das Skript sollte nach dem Muster: +% ``variablen\_datentypen\_\{nachname\}.m'' benannt werden +% (z.B. variablen\_datentypen\_mueller.m). + +\begin{itemize} +\item \"Uberzeuge dich von jeder einzelnen Zeile deines Codes, dass sie +auch wirklich das macht, was sie machen soll! Teste dies mit kleinen +Beispielen direkt in der Kommandozeile. +\item Versuche die L\"osungen der folgenden Aufgaben m\"oglichst in +sinnvolle kleine Funktionen herunterzubrechen. +\item Sobald etwas \"ahnliches mehr als einmal berechnet werden soll, +lohnt es sich eine Funktion daraus zu schreiben! +\item Teste rechenintensive \code{for} Schleifen zuerst mit einer kleinen +Anzahl von Wiederholungen, und benutze erst am Ende, wenn alles +stimmt, eine gro{\ss}e Anzahl von Wiederholungen, um eine gute +Statistik zu bekommen. +\item Benutze die Hilfsfunktion von matlab und das Internet, um +herauszufinden wie bestimmte \code{matlab} Funktionen zu verwenden +sind und was f\"ur M\"oglichkeiten sie bieten. +\item Auch zu inhaltlichen Konzepten bietet das Internet oft viele Antworten! +\end{itemize} + + +\begin{questions} + +\question \qt{Wahrscheinlichkeiten eines W\"urfels I} +Der Computer kann auch als W\"urfel verwendet werden! +\begin{parts} + \part Simuliere 10000 W\"urfe mit dem W\"urfel durch Erzeugung von + ganzzahligen Zufallszahlen mit den Augenzahlen $x_i = 1, 2, \ldots 6$ . + \part Berechne die Wahrscheinlichkeit $P(3)$ + des Auftretens der Augenzahl drei durch Bestimmung der Anzahl der Dreien im Datensatz.\\ + Entspricht das Ergebnis deiner Erwartung?\\ + \"Uberpr\"ufe auch die Wahrscheinlichkeit $P(x_i)$ der anderen Zahlen.\\ + Ist das ein fairer W\"urfel? + \part Speicher die berechneten Wahrscheinlichkeiten $P(x_i)$ f\"ur das Auftreten der + gew\"urfelten Zahlen in einem Vektor und benutze die \code{bar} Funktion, + um diese Wahrscheinlichkeiten als Funktion der Augenzahl zu plotten. + \part Erstelle in einem weiterem Plot ein entsprechendes normiertes Histogramm + mit der \code{hist} Funktion. + \part \extra Wie k\"onnte man einen gezinkten W\"urfel simulieren, bei dem die sechs + dreimal so h\"aufig wie die anderen Zahlen gew\"urfelt wird?\\ + Fertige von diesem W\"urfel ein Histogram aus 10000 W\"urfen an. +\end{parts} + + +\continue +\question \qt{Wahrscheinlichkeiten eines W\"urfels II} +Wir werten nun das Verhalten mehrerer W\"urfel aus. +\begin{parts} + \part Simuliere 20 W\"urfel, von denen jeder 100 mal geworfen wird + (jeder W\"urfel wird mit dem gleichen Zufallsgenerator simuliert). + \part Berechne aus diesem Datensatz f\"ur jeden W\"urfel ein normiertes Histogramm. + \part Bestimme den Mittelwert und die Standardabweichung f\"ur jede + Augenzahl gemittelt \"uber die W\"urfel. + \part Stelle das Ergebnis mit einem S\"aulenplot mit Fehlerbalken dar + (\code{bar} mit \code{errorbar} Funktionen). +\end{parts} + + +\question \qt{Wahrscheinlichkeiten der Normalverteilung} +Mit den folgenden Aufgaben wollen wir bestimmen, welcher Anteil eines +normalverteilten Datensatzes in bestimmten Grenzen symmetrisch um den +Mittelwert enthalten ist. +\begin{parts} + \part Erzeuge einen Datensatz $X = (x_1, x_2, ... x_n)$ aus + $n=10000$ normalverteilten Zufallszahlen mit Mittelwert $\mu=0$ und + Standardabweichung $\sigma=1$. + \part \label{onesigma} Wieviele dieser Daten sind maximal eine Standardabweichung vom Mittelwert entfernt?\\ + D.h. wieviele Datenwerte $x_i$ haben den Wert $-\sigma < x_i < +\sigma$?\\ + Wie gro{\ss} ist also die Wahrscheinlichkeit $P_{\pm\sigma}$ einen + Wert in diesem Interval zu erhalten? + \part \label{probintegral} Berechne numerisch diese + Wahrscheinlichkeit aus dem entsprechenden Integral + \[ P_{\pm\sigma}=\int_{x=\mu-\sigma}^{x=\mu+\sigma} p_g(x) \, dx \] + \"uber die Normalverteilung + \[ p_g(x) = + \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \; . \] + \"Uberpr\"ufe zuerst, ob tats\"achlich + \[ \int_{-\infty}^{+\infty} p_g(x) \, dx = 1 \; . \] + Warum muss das so sein? + \part Welcher Anteil der Daten ist in den Intervallen $\pm2\sigma$ sowie $\pm3\sigma$ + enthalten? + \part \label{givenfraction} Finde heraus in welchem Interval symmetrisch um den Mittelwert + 50\,\%, 90\,\%, 95\,\% bzw. 99\,\% der Daten enhalten sind. + \part Was passiert mit der Wahrscheinlichkeit eine Zahl in einem bestimmten Interval + zu ziehen, wenn dieses Intervall immer kleiner wird?\\ + Schreibe ein Programm, das dies illustriert.\\ + Wie gro{\ss} ist die Wahrscheinlichkeit $P(x_i=0.1234)$? + \part \extra Modifiziere den Code der Teilaufgaben \pref{onesigma} + -- \pref{givenfraction} so, dass er f\"ur Datens\"atze mit + beliebigen Mittelwerten und Standardabweichungen funktioniert.\\ + Teste den Code mit entsprechenden Zufallszahlen.\\ + Wie bekommt man mit \code{randn} Zufallszahlen mit beliebiger + Standardabweichung und Mittelwerten? +\end{parts} + + +\end{questions} + +\end{document} \ No newline at end of file diff --git a/statistics/exercises/descriptivestatistics-02.tex b/statistics/exercises/descriptivestatistics-02.tex new file mode 100644 index 0000000..3f92aa5 --- /dev/null +++ b/statistics/exercises/descriptivestatistics-02.tex @@ -0,0 +1,164 @@ +\documentclass[12pt,a4paper,pdftex]{exam} + +\usepackage[german]{babel} +\usepackage{natbib} +\usepackage{graphicx} +\usepackage[small]{caption} +\usepackage{sidecap} +\usepackage{pslatex} +\usepackage{amsmath} +\usepackage{amssymb} +\setlength{\marginparwidth}{2cm} +\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref} + +%%%%% text size %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry} +\pagestyle{headandfoot} +\header{{\bfseries\large \"Ubung 2}}{{\bfseries\large Deskriptive Statistik}}{{\bfseries\large 19. Oktober, 2015}} +\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email: +jan.grewe@uni-tuebingen.de} +\runningfooter{}{\thepage}{} + +\setlength{\baselineskip}{15pt} +\setlength{\parindent}{0.0cm} +\setlength{\parskip}{0.3cm} +\renewcommand{\baselinestretch}{1.15} + +\newcommand{\qt}[1]{\textbf{#1}\\} +\newcommand{\pref}[1]{(\ref{#1})} +\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}} +\newcommand{\code}[1]{\texttt{#1}} + +\newcommand{\continue}{\ifprintanswers% +\else +\vfill\hspace*{\fill}$\rightarrow$\newpage% +\fi} +\newcommand{\continuepage}{\ifprintanswers% +\newpage +\else +\vfill\hspace*{\fill}$\rightarrow$\newpage% +\fi} +\newcommand{\newsolutionpage}{\ifprintanswers% +\newpage% +\else +\fi} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} + +\vspace*{-6.5ex} +\begin{center} +\textbf{\Large Einf\"uhrung in die wissenschaftliche Datenverarbeitung}\\[1ex] +{\large Jan Grewe, Jan Benda}\\[-3ex] +Abteilung Neuroethologie \hfill --- \hfill Institut f\"ur Neurobiologie \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\ +\end{center} + +% Die folgenden Aufgaben dienen der Wiederholung, \"Ubung und +% Selbstkontrolle und sollten eigenst\"andig bearbeitet und gel\"ost +% werden. Die L\"osung soll in Form eines einzelnen Skriptes (m-files) +% im ILIAS hochgeladen werden. Jede Aufgabe sollte in einer eigenen +% ``Zelle'' gel\"ost sein. Die Zellen \textbf{m\"ussen} unabh\"angig +% voneinander ausf\"uhrbar sein. Das Skript sollte nach dem Muster: +% ``variablen\_datentypen\_\{nachname\}.m'' benannt werden +% (z.B. variablen\_datentypen\_mueller.m). + + +\begin{itemize} +\item \"Uberzeuge dich von jeder einzelnen Zeile deines Codes, dass sie +auch wirklich das macht, was sie machen soll! Teste dies mit kleinen +Beispielen direkt in der Kommandozeile. +\item Versuche die L\"osungen der folgenden Aufgaben m\"oglichst in +sinnvolle kleine Funktionen herunterzubrechen. +\item Sobald etwas \"ahnliches mehr als einmal berechnet werden soll, +lohnt es sich eine Funktion daraus zu schreiben! +\item Teste rechenintensive \code{for} Schleifen zuerst mit einer kleinen +Anzahl von Wiederholungen, und benutze erst am Ende, wenn alles +stimmt, eine gro{\ss}e Anzahl von Wiederholungen, um eine gute +Statistik zu bekommen. +\item Benutze die Hilfsfunktion von matlab und das Internet, um +herauszufinden wie bestimmte \code{matlab} Funktionen zu verwenden +sind und was f\"ur M\"oglichkeiten sie bieten. +\item Auch zu inhaltlichen Konzepten bietet das Internet oft viele Antworten! +\end{itemize} + +\begin{questions} + +\question \qt{Zentraler Grenzwertsatz} +Der Zentrale Grenzwertsatz besagt, dass die Summe von unabh\"angigen +und identisch verteilten (i.i.d. = independent and identically +distributed) Zufallsvariablen gegen die Normalverteilung konvergiert. + +Den Zentralen Grenzwertsatz wollen wir uns im Folgenden veranschaulichen. +\begin{parts} + \part Versuche dir klar zu machen, was der Zentrale Grenzwertsatz + bedeutet, und wie du vorgehen k\"onntest ein Programm zu + schreiben, das den Grenzwertsatz illustriert. + \part Erzeuge 10000 zwischen 0 und 1 gleichverteilte Zufallszahlen + (Funktion \code{rand}). + \part Plotte deren Wahrscheinlichkeitsdichte (normiertes Histogram). + \part Erzeuge weitere 10000 gleichverteilte Zufallszahlen und + addiere diese zu den bereits vorhandenen auf. + \part Plotte die Wahrscheinlichkeitsdichte der aufsummierten + Zufallszahlen. + \part Wiederhole Schritt (d) und (e) viele Male. + \part Vergleiche in einer Grafik die Wahrscheinlichkeitsdichte der + aufsummierten Zufallszahlen mit der Gaussfunktion + \[ p_g(x) = + \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\] + mit dem Mittelwert $\mu$ und der Standardabweichung $\sigma$ der + aufsummierten Zufallszahlen. + \part Wie \"andert sich der Mittelwert und die + Standardabweichung/Varianz + der aufsummierten Zufallszahlen?\\ + Wie h\"angen diese mit den Werten der urspr\"unglichen Verteilung + zusammen? + \part \extra \"Uberpr\"ufe den Grenzwertsatz in gleicher Weise mit exponentiell + verteilten Zufallszahlen (Funktion \code{rande}). +\end{parts} + + +\question \qt{Random Walk} +Im folgenden wollen wir einige Eigenschaften des Random Walks bestimmen. +\begin{parts} + \part Schreibe eine Funktion, die einen einzelnen Random Walk mit + Startwert 0 f\"ur $n$ Schritte und Wahrscheinlichkeit $p$ f\"ur + einen positiven Schritt als Vektor zur\"uckgibt. + \part Visualisiere jeweils 10 Random Walks mit $p=0.5$ zusammen in einem Plot + f\"ur $n=100$, $n=1000$ und $n=10000$ (drei Plots).\\ + Sch\"atze aus den Abbildungen ab, wie sich der Mittelwert und die Standardabweichung + des Random Walks mit der Zeit (Schritte) sich entwickelt. + \part \"Uberpr\"uefe deine Hypothese zum Mittelwert und zur + Standardabweichung, indem du von $m$ Random Walks ($m \ge 10$) f\"ur + jeden z.B. zehnten Schritt den Mittelwert und die Standardabweichung + \"uber die Positionen der $m$ Random Walks berechnest.\\ + Wie h\"angt also die Standardabweichung von der Anzahl der Schritte + ab? Wie entwickelt sich die Standardabweichung f\"ur eine sehr + gro{\ss}e Anzahl von Schritten? + \part \extra Erstelle eine Grafik, die die Verteilung der Position eines Random Walkers + zu drei verschiedenen Zeitpunkten zeigt. +\end{parts} + + +\question \qt{\extra 2D Random Walk} +Bisher hat sich unser Random Walker nur in einer Dimension bewegt +(nur vorw\"arts oder r\"uckw\"arts). Er kann aber auch in mehreren Dimensionen laufen!\\ +In zwei Dimensionen wird dazu in jedem Schritt eine weitere +Zufallszahl gezogen, die bestimmt ob er einen Schritt nach links oder +rechts gemacht hat. Die Bewegung nach vorne/hinten bzw. links/rechts +sind unabh\"angig voneinander. +\begin{parts} + \part Wie kann unter Verwendung unserer Funktion f\"ur den + eindimensionalen Random Walk ein zweidimensionaler Random Walk + simuliert werden? + \part Erstelle h\"ubsche Bilder, die zweidimensionalen Random + Walks verschiedener L\"ange (bis zu mindestens $n=1000000$) illustrieren. + \part Animationen sind auch sch\"on! z.B. mit dem \code{pause} Befehl. + \part Anstatt einfach den Weg des Random Walks zu zeichnen, kann man + sich auch merken, wie oft er an jeder Stelle vorbeigekommen ist und + mit einem Farbcode plotten. +\end{parts} + +\end{questions} + +\end{document} \ No newline at end of file diff --git a/statistics/lecture/descriptivestatistics.tex b/statistics/lecture/descriptivestatistics.tex index 3ae32ac..989dc0f 100644 --- a/statistics/lecture/descriptivestatistics.tex +++ b/statistics/lecture/descriptivestatistics.tex @@ -74,8 +74,8 @@ \newenvironment{definition}[1][]{\medskip\noindent\textbf{Definition}\ifthenelse{\equal{#1}{}}{}{ #1}:\newline}% {\medskip} -%\newcommand{\showlisting}{yes} -\newcommand{\showlisting}{no} +\newcommand{\showlisting}{yes} +%\newcommand{\showlisting}{no} \newcounter{theexercise} \setcounter{theexercise}{1} \newenvironment{exercise}[1][]{\medskip\noindent\textbf{\tr{Exercise}{\"Ubung} From 1e3b02b9a1bd2d17fede25c756dcbfca5134ff2e Mon Sep 17 00:00:00 2001 From: Jan Benda <jan.benda@uni-tuebingen.de> Date: Mon, 19 Oct 2015 18:53:52 +0200 Subject: [PATCH 2/4] Few updates on todays lecture --- statistics/code/gaussianbins.m | 7 +++--- statistics/code/gaussianpdf.m | 26 +++++++++++++------- statistics/code/mymedian.m | 10 ++++---- statistics/code/quartiles.m | 9 ++++--- statistics/lecture/Makefile | 7 +++++- statistics/lecture/descriptivestatistics.tex | 15 ++++++----- 6 files changed, 44 insertions(+), 30 deletions(-) diff --git a/statistics/code/gaussianbins.m b/statistics/code/gaussianbins.m index 5f3cc6d..effe1fd 100644 --- a/statistics/code/gaussianbins.m +++ b/statistics/code/gaussianbins.m @@ -1,6 +1,6 @@ -x = randn( 100, 1 ); -bins1 = -4:2:4; -bins2 = -4:0.5:4; +x = randn( 100, 1 ); % generate some data +bins1 = -4:2:4; % large bins +bins2 = -4:0.5:4; % small bins subplot( 1, 2, 1 ); hold on; hist( x, bins1 ); @@ -10,6 +10,7 @@ ylabel('Frequeny') hold off; subplot( 1, 2, 2 ); hold on; +% normalize to the rigtht bin size: hist( x, bins1, 1.0/(bins1(2)-bins1(1)) ); hist( x, bins2, 1.0/(bins2(2)-bins2(1)) ); xlabel('x') diff --git a/statistics/code/gaussianpdf.m b/statistics/code/gaussianpdf.m index 6c33dc6..259d6c9 100644 --- a/statistics/code/gaussianpdf.m +++ b/statistics/code/gaussianpdf.m @@ -1,22 +1,30 @@ % plot Gaussian pdf: -dx=0.1 +dx=0.1; x = [-4.0:dx:4.0]; p = exp(-0.5*x.^2)/sqrt(2.0*pi); hold on -plot(x,p, 'linewidth', 10 ) +plot(x, p, 'linewidth', 10) +% show area of integral: +area(x((x>=x1)&(x<=x2)), p((x>=x1)&(x<=x2)), 'FaceColor', 'r' ) +hold off % compute integral between x1 and x2: -x1=1.0 -x2=2.0 -P = sum(p((x>=x1)&(x<x2)))*dx +x1=1.0; +x2=2.0; +P = sum(p((x>=x1)&(x<x2)))*dx; +disp( [ 'The integral between ', num2str(x1, 1), ' and ', num2str(x2, 1), ' is ', num2str(P, 3) ] ); % draw random numbers: -r = randn( 10000, 1 ); -hist(r,x,1.0/dx) +%r = randn( 10000, 1 ); +%hist(r,x,1.0/dx) % check P: -Pr = sum((r>=x1)&(r<x2))/length(r) +Pr = sum((r>=x1)&(r<x2))/length(r); +disp( [ 'The probability of getting a number between ', num2str(x1, 1), ' and ', num2str(x2, 1), ' is ', num2str(Pr, 3) ] ); -hold off +% infinite integral: +P = sum(p)*dx; +disp( [ 'The integral between -infinity and +infinity is ', num2str(P, 3) ] ); +disp( [ 'I.e. the probability to get any number is ', num2str(P, 3) ] ); diff --git a/statistics/code/mymedian.m b/statistics/code/mymedian.m index d2af4d0..1f88193 100644 --- a/statistics/code/mymedian.m +++ b/statistics/code/mymedian.m @@ -1,13 +1,13 @@ function m = mymedian( x ) % returns the median of the vector x xs = sort( x ); - if ( length( xs ) == 0 ) + if ( length( xs ) == 0 ) % empty input vector m = NaN; - elseif ( rem( length( xs ), 2 ) == 0 ) + elseif ( rem( length( xs ), 2 ) == 0 ) % even number of data values index = length( xs )/2; - m = (xs( index ) + xs( index+1 ))/2; - else - index = (length( xs ) + 1)/2; + m = (xs( index ) + xs( index+1 ))/2; % average the two central elements + else % odd number of data values + index = (length( xs ) + 1)/2; % take the middle element m = xs( index ); end end diff --git a/statistics/code/quartiles.m b/statistics/code/quartiles.m index 3f9ebe5..8123733 100644 --- a/statistics/code/quartiles.m +++ b/statistics/code/quartiles.m @@ -1,13 +1,14 @@ function q = quartiles( x ) - % returns a vector with the first, second, and third quartile of the vector x + % returns a vector with the first, second, and third quartile + % of the vector x xs = sort( x ); - if ( length( xs ) == 0 ) + if ( length( xs ) == 0 ) % no data q = []; - elseif ( rem( length( xs ), 2 ) == 0 ) + elseif ( rem( length( xs ), 2 ) == 0 ) % even number of data index = length( xs )/2; m = (xs( index ) + xs( index+1 ))/2; q = [ round( xs(length(xs)/4) ), m, xs(round(3*length(xs)/4)) ]; - else + else % odd number of data index = (length( xs ) + 1)/2; m = xs( index ); q = [ round( xs(length(xs)/4) ), m, xs(round(3*length(xs)/4)) ]; diff --git a/statistics/lecture/Makefile b/statistics/lecture/Makefile index b68ae15..59963db 100644 --- a/statistics/lecture/Makefile +++ b/statistics/lecture/Makefile @@ -1,11 +1,16 @@ TEXFILES=$(wildcard *.tex) PDFFILES=$(TEXFILES:.tex=.pdf) +PYFILES=$(wildcard *.py) +PYPDFFILES=$(PYFILES:.py=.pdf) -pdf : $(PDFFILES) +pdf : $(PDFFILES) $(PYPDFFILES) $(PDFFILES) : %.pdf : %.tex pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true +$(PYPDFFILES) : %.pdf : %.py + python $< + clean : rm -f *~ $(TEXFILES:.tex=.aux) $(TEXFILES:.tex=.log) $(TEXFILES:.tex=.out) $(TEXFILES:.tex=.nav) $(TEXFILES:.tex=.snm) $(TEXFILES:.tex=.toc) $(TEXFILES:.tex=.vrb) diff --git a/statistics/lecture/descriptivestatistics.tex b/statistics/lecture/descriptivestatistics.tex index 989dc0f..bd20f5a 100644 --- a/statistics/lecture/descriptivestatistics.tex +++ b/statistics/lecture/descriptivestatistics.tex @@ -154,7 +154,7 @@ below. In particular the script should test data vectors of different length.} {Schreibe ein Skript, das testet ob die \code{mymedian} Funktion wirklich die Zahl zur\"uckgibt, \"uber - der genausoviele Datenwerte liegen wie darunter. Das Skript sollte + der genauso viele Datenwerte liegen wie darunter. Das Skript sollte insbesondere verschieden lange Datenvektoren testen.} \end{exercise} @@ -246,7 +246,7 @@ $A$ des Histogramms ist also \[ A = \sum_{i=1}^N ( n_i \cdot \Delta x ) = \Delta x \sum_{i=1}^N n_i \] und das normierte Histogramm hat die H\"ohe \[ p(x_i) = \frac{n_i}{\Delta x \sum_{i=1}^N n_i} \] -Es muss also nicht nur durch die Summe, sondern auch durch die Breite der Klassen $\Delta x$ +Es muss also nicht nur durch die Summe, sondern auch durch die Breite $\Delta x$ der Klassen geteilt werden. $p(x_i)$ kann keine Wahrscheinlichkeit sein, da $p(x_i)$ nun eine @@ -258,14 +258,14 @@ spricht von einer Wahrscheinlichkeitsdichte. \caption{\label{pdfprobabilitiesfig} Wahrscheinlichkeiten bei einer Wahrscheinlichkeitsdichtefunktion.} \end{figure} - -\begin{exercise} + +\begin{exercise}[gaussianpdf.m] \tr{Plot the Gaussian probability density}{Plotte die Gauss'sche Wahrscheinlichkeitsdichte } - \[ p_g(x) = 1/\sqrt{2\pi\sigma^2}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\] + \[ p_g(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\] \tr{What does it mean?}{Was bedeutet die folgende Wahrscheinlichkeit?} - \[ P(x_1 < x < x2) = \int_{x_1}^{x_2} p(x) \, dx \] + \[ P(x_1 < x < x2) = \int\limits_{x_1}^{x_2} p(x) \, dx \] \tr{How large is}{Wie gro{\ss} ist} - \[ \int_{-\infty}^{+\infty} p(x) \, dx \; ?\] + \[ \int\limits_{-\infty}^{+\infty} p(x) \, dx \; ?\] \tr{Why?}{Warum?} \end{exercise} @@ -392,7 +392,6 @@ spricht von einer Wahrscheinlichkeitsdichte. \end{document} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Statistics} What is "a statistic"? % dt. Sch\"atzfunktion From 2787e55094949b099fec2cf7a967ef27d5317cac Mon Sep 17 00:00:00 2001 From: Jan Benda <jan.benda@uni-tuebingen.de> Date: Tue, 20 Oct 2015 01:43:18 +0200 Subject: [PATCH 3/4] New bootstrap exercises --- statistics/code/bootstrapsem.m | 23 + statistics/code/bootstraptymus.m | 24 + statistics/code/boxwhisker.m | 3 + statistics/code/correlations.m | 14 + statistics/code/tdistribution.m | 32 + statistics/exercises/bootstrap-01.tex | 132 + statistics/exercises/thymusglandweights.dat | 10000 ++++++++++++++++ statistics/lecture/boxwhisker.py | 47 + statistics/lecture/boxwhiskerdata.npy | Bin 0 -> 3280 bytes statistics/lecture/descriptivestatistics.tex | 85 + .../figures/2012-10-29_16-26-05_771.jpg | Bin 0 -> 740827 bytes .../figures/2012-10-29_16-29-35_312.jpg | Bin 0 -> 395085 bytes .../figures/2012-10-29_16-41-39_523.jpg | Bin 0 -> 472511 bytes statistics/lecture/figures/srs1.png | Bin 0 -> 60051 bytes statistics/lecture/figures/srs2.png | Bin 0 -> 56148 bytes statistics/lecture/figures/srs3.png | Bin 0 -> 74342 bytes statistics/material/Hesterberg2003.pdf | Bin 0 -> 589657 bytes 17 files changed, 10360 insertions(+) create mode 100644 statistics/code/bootstrapsem.m create mode 100644 statistics/code/bootstraptymus.m create mode 100644 statistics/code/boxwhisker.m create mode 100644 statistics/code/correlations.m create mode 100644 statistics/code/tdistribution.m create mode 100644 statistics/exercises/bootstrap-01.tex create mode 100644 statistics/exercises/thymusglandweights.dat create mode 100644 statistics/lecture/boxwhisker.py create mode 100644 statistics/lecture/boxwhiskerdata.npy create mode 100755 statistics/lecture/figures/2012-10-29_16-26-05_771.jpg create mode 100755 statistics/lecture/figures/2012-10-29_16-29-35_312.jpg create mode 100755 statistics/lecture/figures/2012-10-29_16-41-39_523.jpg create mode 100644 statistics/lecture/figures/srs1.png create mode 100644 statistics/lecture/figures/srs2.png create mode 100644 statistics/lecture/figures/srs3.png create mode 100644 statistics/material/Hesterberg2003.pdf diff --git a/statistics/code/bootstrapsem.m b/statistics/code/bootstrapsem.m new file mode 100644 index 0000000..b5a7cbe --- /dev/null +++ b/statistics/code/bootstrapsem.m @@ -0,0 +1,23 @@ +nsamples = 1000 +resample = 500 + + x = randn( nsamples, 1 ); +sem = std(x)/sqrt(nsamples); + +mu = zeros( resample, 1 ); +for i = 1:resample + % resample: + xr = x(randi(nsamples, nsamples, 1)); + % compute statistics on sample: + mu(i) = mean(xr); +end +bootsem = std( mu ); + +hold on +hist( x, 20 ); +hist( mu, 20 ); +hold off + +disp(['bootstrap standard error: ', num2str(bootsem)]); +disp(['standard error: ', num2str(sem)]); + diff --git a/statistics/code/bootstraptymus.m b/statistics/code/bootstraptymus.m new file mode 100644 index 0000000..e92a5b7 --- /dev/null +++ b/statistics/code/bootstraptymus.m @@ -0,0 +1,24 @@ +resample = 500 + +load( 'thymusglandweights.dat' ); +x = thymusglandweights; +nsamples = length( x ); +sem = std(x)/sqrt(nsamples); + +mu = zeros( resample, 1 ); +for i = 1:resample + % resample: + xr = x(randi(nsamples, nsamples, 1)); + % compute statistics on sample: + mu(i) = mean(xr); +end +bootsem = std( mu ); + +hold on +hist( x, 20 ); +hist( mu, 20 ); +hold off + +disp(['bootstrap standard error: ', num2str(bootsem)]); +disp(['standard error: ', num2str(sem)]); + diff --git a/statistics/code/boxwhisker.m b/statistics/code/boxwhisker.m new file mode 100644 index 0000000..65e8f24 --- /dev/null +++ b/statistics/code/boxwhisker.m @@ -0,0 +1,3 @@ +x = randn(40, 10); +boxplot(x, 'whisker', 100.0 ); + diff --git a/statistics/code/correlations.m b/statistics/code/correlations.m new file mode 100644 index 0000000..7816457 --- /dev/null +++ b/statistics/code/correlations.m @@ -0,0 +1,14 @@ +n = 1000 +x = randn( n, 1 ); +y = randn( n, 1 ) + 0.2*x; +r = corr(x,y) + + nsamples = 500; + rs = zeros( nsamples, 1 ); +for i = 1:nsamples + xs = x(randi(n,n,1)); +ys = x(randi(n,n,1)); + rs(i) = corr(xs,ys); +end + +hist( rs, 20 ) diff --git a/statistics/code/tdistribution.m b/statistics/code/tdistribution.m new file mode 100644 index 0000000..9d8920b --- /dev/null +++ b/statistics/code/tdistribution.m @@ -0,0 +1,32 @@ +n = 100000 +x=randn(n, 1); + +nsamples = 3; +nmeans = 10000; +means = zeros( nmeans, 1 ); +sdevs = zeros( nmeans, 1 ); +students = zeros( nmeans, 1 ); +for i=1:nmeans + sample = x(randi(n, nsamples, 1)); + means(i) = mean(sample); + sdevs(i) = std(sample); + students(i) = mean(sample)/std(sample)*sqrt(nsamples); +end +sdev = 1.0 +msdev = std(means) + +% scatter( means, sdevs ) + +hold on; +dxg=0.01; +xmax = 10.0 +xmin = -xmax +xg = [xmin:dxg:xmax]; +pg = exp(-0.5*(xg/sdev).^2)/sqrt(2.0*pi)/sdev; +hold on +plot(xg, pg, 'r', 'linewidth', 4) + +bins = xmin:0.1:xmax; +hist(means, bins, 1.0/(bins(2)-bins(1)) ); +hold off; + diff --git a/statistics/exercises/bootstrap-01.tex b/statistics/exercises/bootstrap-01.tex new file mode 100644 index 0000000..a5860e1 --- /dev/null +++ b/statistics/exercises/bootstrap-01.tex @@ -0,0 +1,132 @@ +\documentclass[12pt,a4paper,pdftex]{exam} + +\usepackage[german]{babel} +\usepackage{natbib} +\usepackage{graphicx} +\usepackage[small]{caption} +\usepackage{sidecap} +\usepackage{pslatex} +\usepackage{amsmath} +\usepackage{amssymb} +\setlength{\marginparwidth}{2cm} +\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref} + +%%%%% text size %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry} +\pagestyle{headandfoot} +\header{{\bfseries\large \"Ubung 1}}{{\bfseries\large Bootstrap}}{{\bfseries\large 21. Oktober, 2015}} +\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email: +jan.benda@uni-tuebingen.de} +\runningfooter{}{\thepage}{} + +\setlength{\baselineskip}{15pt} +\setlength{\parindent}{0.0cm} +\setlength{\parskip}{0.3cm} +\renewcommand{\baselinestretch}{1.15} + +\newcommand{\qt}[1]{\textbf{#1}\\} +\newcommand{\pref}[1]{(\ref{#1})} +\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}} +\newcommand{\code}[1]{\texttt{#1}} + +\newcommand{\continue}{\ifprintanswers% +\else +\vfill\hspace*{\fill}$\rightarrow$\newpage% +\fi} +\newcommand{\continuepage}{\ifprintanswers% +\newpage +\else +\vfill\hspace*{\fill}$\rightarrow$\newpage% +\fi} +\newcommand{\newsolutionpage}{\ifprintanswers% +\newpage% +\else +\fi} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} + +\vspace*{-6.5ex} +\begin{center} +\textbf{\Large Einf\"uhrung in die wissenschaftliche Datenverarbeitung}\\[1ex] +{\large Jan Grewe, Jan Benda}\\[-3ex] +Abteilung Neuroethologie \hfill --- \hfill Institut f\"ur Neurobiologie \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\ +\end{center} + +\begin{itemize} +\item \"Uberzeuge dich von jeder einzelnen Zeile deines Codes, dass sie +auch wirklich das macht, was sie machen soll! Teste dies mit kleinen +Beispielen direkt in der Kommandozeile. +\item Versuche die L\"osungen der folgenden Aufgaben m\"oglichst in +sinnvolle kleine Funktionen herunterzubrechen. +\item Sobald etwas \"ahnliches mehr als einmal berechnet werden soll, +lohnt es sich eine Funktion daraus zu schreiben! +\item Teste rechenintensive \code{for} Schleifen zuerst mit einer kleinen +Anzahl von Wiederholungen, und benutze erst am Ende, wenn alles +stimmt, eine gro{\ss}e Anzahl von Wiederholungen, um eine gute +Statistik zu bekommen. +\item Benutze die Hilfsfunktion von matlab und das Internet, um +herauszufinden wie bestimmte \code{matlab} Funktionen zu verwenden +sind und was f\"ur M\"oglichkeiten sie bieten. +\item Auch zu inhaltlichen Konzepten bietet das Internet oft viele Antworten! +\end{itemize} + + +\begin{questions} + +\question \qt{Bootstrap des Standardfehlers} +\begin{parts} + \part Lade von Ilias die Datei \code{thymusglandweights.dat} herunter. + Darin befindet sich ein Datensatz vom Gewicht der Thymus Dr\"use in 14-Tage alten + H\"uhnerembryos in mg. + \part Lade diese Daten in Matlab (\code{load} Funktion). + \part Bestimme Histogramm, Mittelwert und Standardfehler aus den ersten 80 Datenpunkten. + \part Bestimme den Standardfehler aus den ersten 80 Datenpunkten durch 500 Mal Bootstrappen. + \part Bestimme das 95\,\% Konfidenzintervall f\"ur den Mittelwert + aus der Bootstrap Verteilung (\code{quantile()} Funktion) --- also + das Interval innerhalb dessen mit 95\,\% Wahrscheinlichkeit der + wahre Mittelwert liegen wird. + \part Benutze den ganzen Datensatz und die Bootstrapping Technik, um die Abh\"angigkeit + des Standardfehlers von der Stichprobengr\"o{\ss}e zu bestimmen. + \part Vergleiche mit der bekannten Formel f\"ur den Standardfehler $\sigma/\sqrt{n}$. +\end{parts} + + +\continue +\question \qt{Student t-Verteilung} +\begin{parts} +\part Erzeuge 100000 normalverteilte Zufallszahlen. +\part Ziehe daraus 1000 Stichproben vom Umfang $m$ (3, 5, 10, 50). +\part Berechne den Mittelwert $\bar x$ der Stichproben und plotte die Wahrscheinlichkeitsdichte +dieser Mittelwerte. +\part Vergleiche diese Wahrscheinlichkeitsdichte mit der Gausskurve. +\part Berechne ausserdem die Gr\"o{\ss}e $t=\bar x/(\sigma_x/\sqrt{m}$ +(Standardabweichung $\sigma_x$) und vergleiche diese mit der Normalverteilung mit Standardabweichung Eins. Ist $t$ normalverteilt, bzw. unter welchen Bedingungen ist $t$ Normalverteilt? +\end{parts} + + +\question \qt{Korrelationen} +\begin{parts} +\part Erzeuge 1000 korrelierte Zufallszahlen $x$, $y$ durch +\begin{verbatim} +n = 1000 +a = 0.2; +x = randn(n, 1); +y = randn(n, 1) + a*x; +\end{verbatim} +\part Erstelle einen Scatterplot der beiden Variablen. +\part Warum ist $y$ mit $x$ korreliert? +\part Berechne den Korrelationskoeffizienten zwischen $x$ und $y$. +\part Was m\"usste man tun, um die Korrelationen zwischen den $x$-$y$ +Paaren zu zerst\"oren? +\part Mach genau dies 1000 mal und berechne jedes Mal den Korrelationskoeffizienten. +\part Bestimme die Wahrscheinlichkeitsdichte dieser Korrelationskoeffizienten. +\part Ist die Korrelation der urspr\"unglichen Daten signifikant? +\part Variiere den Parameter $a$ und \"uberpr\"ufe auf gleiche Weise die Signifikanz. +\end{parts} + + +\end{questions} + +\end{document} \ No newline at end of file diff --git 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+ax.set_ylim( -4.0, 4.0) +ax.annotate('Median', + xy=(3.9, 0.1), xycoords='data', + xytext=(3.5, -2.5), textcoords='data', ha='right', + arrowprops=dict(arrowstyle="->", relpos=(0.8,1.0), + connectionstyle="angle3,angleA=-110,angleB=60") ) +ax.annotate('1. quartile', + xy=(5.8, -0.7), xycoords='data', + xytext=(5.5, -3.5), textcoords='data', ha='right', + arrowprops=dict(arrowstyle="->", relpos=(0.5,1.0), + connectionstyle="angle3,angleA=30,angleB=70") ) +ax.annotate('3. quartile', + xy=(6.1, 0.6), xycoords='data', + xytext=(6.5, 3.0), textcoords='data', ha='left', + arrowprops=dict(arrowstyle="->", relpos=(0.0,0.0), + connectionstyle="angle3,angleA=30,angleB=70") ) +ax.annotate('minimum', + xy=(6.1, -2.3), xycoords='data', + xytext=(7.2, -3.3), textcoords='data', ha='left', + arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5), + connectionstyle="angle3,angleA=10,angleB=100") ) +ax.annotate('maximum', + xy=(5.9, 2.8), xycoords='data', + xytext=(4.9, 3.5), textcoords='data', ha='right', + arrowprops=dict(arrowstyle="->", relpos=(1.0,0.5), + connectionstyle="angle3,angleA=0,angleB=120") ) +ax.boxplot( x, whis=100.0 ) +plt.tight_layout() +plt.savefig('boxwhisker.pdf') +plt.show() + diff --git a/statistics/lecture/boxwhiskerdata.npy b/statistics/lecture/boxwhiskerdata.npy new file mode 100644 index 0000000000000000000000000000000000000000..6751afa775ea24f0c9cf7a38661516374ed7f54a GIT binary patch literal 3280 zcmXX|c{mh`+Z{`VY;}b;q(#=7rIhVejnpkm_DU&BCCXBbgv$-3EH^5ZB?;Fqlsi=J zJCQBQGLxk##@I3jGt3xSe|^vMo&V1H@0{nn?{gklJ6hSE5F#8Rgs6FXUG)f5Gg47A zIlos;OGWLxU*NUCb64E_0zJL{Pd{|d_o~+~{p!VY{$9WMPJ>-qD!RMWwN%1X{vQ%f ze}sF_M$TX(>(4un^Il+;vT=SevOwH4Y;UM%8(d$@FU-Et1}EUr(4&jO6bX6l^LNY# zQSyqh*?tBSo_)By`?dxH{`1ysE+y2#bBDE7+1x>h@a~z8(q>>yN`Y#ePa8O2p5$kI z=)$C2@m68aP9T%ckldb+z@&8cRSac9?;f8xkmY0V@+k55<$l<zku1aVZ3Ftw>gdtd zMMx>SUiD0`2AU<+L_~g6V}wj<%wbI)UL2~(i2Zv2pUS<jUnuB9^6@mW!|Yx(Uc>Up z*f9lnYr1QRdw3{GO-=Yqbr@P5Mb<s@nZ^@m9?%NwMnUzaOJ&5(WW4p7`m~EU0EK)x 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b/statistics/lecture/descriptivestatistics.tex @@ -82,6 +82,8 @@ \arabic{theexercise}:} \stepcounter{theexercise}\newline \newcommand{\exercisesource}{#1}}% {\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\equal{\showlisting}{yes}}{\medskip\lstinputlisting{\exercisesource}}{}}\medskip} +\graphicspath{{figures/}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -269,6 +271,24 @@ spricht von einer Wahrscheinlichkeitsdichte. \tr{Why?}{Warum?} \end{exercise} +\begin{exercise}[boxwhisker.m] + \tr{Generate eine $40 \times 10$ matrix of random numbers and + illustrate their distribution in a box-whicker plot + (\code{boxplot()} function). How to interpret the plot?} + {Erzeuge ein $40 \times 10$ Matrix + von Zufallszahlen und illustriere ihre Verteilungen in einem + Box-Whisker Plot (\code{boxplot()} Funktion, lies die Hilfe!). Wie ist der + Box-Whisker Plot zu interpretieren? Was hat es mit den Ausreissern auf sich? + Wie kann man erreichen, dass die Whisker den kleinsten und den gr\"o{\ss}ten + Datenwert anzeigen? Warum sind die unterschiedlichen Box-Whiskers nicht alle gleich, + obwohl sie aus der selben Verteilung gezogen worden sind?} +\end{exercise} + +\begin{figure}[t] + \includegraphics[width=1\textwidth]{boxwhisker} + \caption{\label{boxwhiskerfig} Box-whisker plots illustrate distributions.} +\end{figure} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Data types} @@ -390,6 +410,71 @@ spricht von einer Wahrscheinlichkeitsdichte. \end{itemize} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\chapter{\tr{Bootstrap Methods}{Bootstrap Methoden}} + +Beim Bootstrap erzeugt man sich die Verteilung von Statistiken durch Resampling +aus der Stichprobe. Das hat mehrere Vorteile: +\begin{itemize} +\item Weniger Annahmen (z.B. muss eine Stichprobe nicht Normalverteilt sein). +\item H\"ohere Genauigkeit als klassische Methoden. +\item Allgemeing\"ultigkeit: Bootstrap Methoden sind sich sehr + \"ahnlich f\"ur viele verschiedene Statistiken und ben\"otigen nicht + f\"ur jede Statistik eine andere Formel. +\end{itemize} + +\begin{figure}[t] + \includegraphics[width=0.8\textwidth]{2012-10-29_16-26-05_771}\\[2ex] + \includegraphics[width=0.8\textwidth]{2012-10-29_16-41-39_523}\\[2ex] + \includegraphics[width=0.8\textwidth]{2012-10-29_16-29-35_312} + \caption{\tr{Why can we only measure a sample of the + population?}{Warum k\"onnen wir nur eine Stichprobe der + Grundgesamtheit messen?}} +\end{figure} + +\begin{figure}[t] + \includegraphics[height=0.2\textheight]{srs1}\\[2ex] + \includegraphics[height=0.2\textheight]{srs2}\\[2ex] + \includegraphics[height=0.2\textheight]{srs3} + \caption{Bootstrap der Stichprobenvertielung (a) Von der + Grundgesamtheit (population) mit unbekanntem Parameter + (z.B. Mittelwert $\mu$) zieht man Stichproben (SRS: simple random + samples). Die Statistik (hier Bestimmung von $\bar x$) kann f\"ur + jede Stichprobe berechnet werden. Die erhaltenen Werte entstammen + der Stichprobenverteilung. Meisten wird aber nur eine Stichprobe + gezogen! (b) Mit bestimmten Annahmen und Theorien kann man auf + die Stichprobenverteilung schlie{\ss}en ohne sie gemessen zu + haben. (c) Alternativ k\"onnen aus der einen Stichprobe viele + Bootstrap-Stichproben generiert werden (resampling) und so + Eigenschaften der Stichprobenverteilung empirisch bestimmt + werden. Aus Hesterberg et al. 2003, Bootstrap Methods and + Permuation Tests} +\end{figure} + +\section{Bootstrap des Standardfehlers} + +Beim Bootstrap erzeugen wir durch resampling neue Stichproben und +benutzen diese um die Stichprobenverteilung einer Statistik zu +berechnen. Die Bootstrap Stichproben haben jeweils den gleichen Umfang +wie die urspr\"unglich gemessene Stichprobe und werden durch Ziehen +mit Zur\"ucklegen gewonnen. Jeder Wert der urspr\"unglichen Stichprobe +kann also einmal, mehrmals oder gar nicht in einer Bootstrap +Stichprobe vorkommen. + +\begin{exercise}[bootstrapsem.m] + Ziehe 1000 normalverteilte Zufallszahlen und berechne deren Mittelwert, + Standardabweichung und Standardfehler ($\sigma/\sqrt{n}$). + + Resample die Daten 1000 mal (Ziehen mit Zur\"ucklegen) und berechne jeweils + den Mittelwert. + + Plotte ein Histogramm dieser Mittelwerte, sowie deren Mittelwert und + die Standardabweichung. + + Was hat das mit dem Standardfehler zu tun? +\end{exercise} + \end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% diff --git a/statistics/lecture/figures/2012-10-29_16-26-05_771.jpg b/statistics/lecture/figures/2012-10-29_16-26-05_771.jpg new file mode 100755 index 0000000000000000000000000000000000000000..a997cdd2e755e21cef5a620742ea3a7888744ec6 GIT binary patch literal 740827 zcmdSAbyOTr*FQJ}f=h6h;O@aC5Zoo`;1Vpjy9Eud!QEX4m*6_MyJWDz9Rkbud7k&( 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zh%+|{xCrQXA$bgk#!8@DdLBT=x$gqGqUjJ=X*h5y;d6PT1VR$%{e@F=+;>4Gq48G& zh94U4!B)V5NeLRYs5J$P5eFt=i)h(V4CYwQo=bplaoPj(3uo=6KuqX<WnjePl!3Q{ zE|cTUZEP3<TP}wVfu)KAr(oaU%mXl*GiE6WCiK36<(ebsNFkp;4U<w}tn~Qcx*J-j z0}vN-@D1(CoPL39(CZ@Q+9SX?4u&aeT|hUW?I2JOL<CMf35Q=wfj4sI24t7sOR(f} za7!uzXBORuNXnfrNI#sl5_4=&AdF++IeX8+w$9;YQY}0HTuK~%4rSc2fWpNY3xFSI zt-y@V84IYooV5a%2u(|X<8gR0cwIPr5Y%(p)*=RJP0PDnoR$Jg)3^nfU2||t#^d;5 zQN`xy8sOcceGW*9Ir0H$IGlN)4}&`v5Gbj=1iuD{mH@?}?NXrbb9{?X#^tLr5IgC8 zgS)CU_b4XHFBlH28aHbO5q=n22Oe+8yVI_JGoe2(AixjISboxBe5uS2&;{r-$XL?A j0GW8$&=zIa7zpPuApFiRDu%ezVbI3)>b39UVf_C9tDM6g literal 0 HcmV?d00001 From 69b9afbc80a757e2363382b62608b9c24b1340c6 Mon Sep 17 00:00:00 2001 From: Jan Benda <jan.benda@uni-tuebingen.de> Date: Tue, 20 Oct 2015 18:26:43 +0200 Subject: [PATCH 4/4] Added correlation plots --- statistics/code/diehistograms.m | 6 +- statistics/lecture/correlation.py | 34 ++++ statistics/lecture/descriptivestatistics.tex | 165 ++++++++++++++++++- statistics/lecture/nonlincorrelation.py | 42 +++++ 4 files changed, 242 insertions(+), 5 deletions(-) create mode 100644 statistics/lecture/correlation.py create mode 100644 statistics/lecture/nonlincorrelation.py diff --git a/statistics/code/diehistograms.m b/statistics/code/diehistograms.m index 8af7d68..d04ca98 100644 --- a/statistics/code/diehistograms.m +++ b/statistics/code/diehistograms.m @@ -3,16 +3,16 @@ nrolls = [ 20, 100, 1000 ]; for i = [1:length(nrolls)] d = rollthedie( nrolls(i) ); % plain hist: - % hist( d ) + %hist( d ) % check bin counts of plain hist: % h = hist( d ) % force 6 bins: - % hist( d, 6 ) + %hist( d, 6 ) % set the right bin centers: - bins = 1:6; + %bins = 1:6; %hist( d, bins ) % normalize histogram and compare to expectation: diff --git a/statistics/lecture/correlation.py b/statistics/lecture/correlation.py new file mode 100644 index 0000000..db317c7 --- /dev/null +++ b/statistics/lecture/correlation.py @@ -0,0 +1,34 @@ +import numpy as np +import matplotlib.pyplot as plt + +plt.xkcd() +fig = plt.figure( figsize=(6,5) ) +n = 200 +for k, r in enumerate( [ 1.0, 0.6, 0.0, -0.9 ] ) : + print r + x = np.random.randn( n ) + y = r*x + np.sqrt(1.0-r*r)*np.random.randn( n ) + ax = fig.add_subplot( 2, 2, k+1 ) + ax.spines['right'].set_visible(False) + ax.spines['top'].set_visible(False) + ax.yaxis.set_ticks_position('left') + ax.xaxis.set_ticks_position('bottom') + ax.text( -2, 2.5, 'r=%.1f' % r ) + if k == 0 : + ax.text( 2.8, -2, 'positively\ncorrelated', ha='right' ) + elif k == 1 : + ax.text( 2.8, -2.5, 'weakly\ncorrelated', ha='right' ) + elif k == 2 : + ax.text( 2.8, -2.5, 'not\ncorrelated', ha='right' ) + elif k == 3 : + ax.text( -2.5, -2, 'negatively\ncorrelated', ha='left' ) + ax.set_xlabel('x') + ax.set_ylabel('y') + ax.set_xlim( -3.0, 3.0) + ax.set_ylim( -3.0, 3.0) + ax.scatter( x, y ) + +plt.tight_layout() +plt.savefig('correlation.pdf') +plt.show() + diff --git a/statistics/lecture/descriptivestatistics.tex b/statistics/lecture/descriptivestatistics.tex index 21e7493..ac99345 100644 --- a/statistics/lecture/descriptivestatistics.tex +++ b/statistics/lecture/descriptivestatistics.tex @@ -19,10 +19,141 @@ \usepackage[left=25mm,right=25mm,top=20mm,bottom=30mm]{geometry} \setcounter{tocdepth}{1} -%%%% graphics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[sf,bf,it,big,clearempty]{titlesec} +\setcounter{secnumdepth}{-1} + + +%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro + + +%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{graphicx} \usepackage{xcolor} -\newcommand{\texpicture}[1]{{\sffamily\small\input{#1.tex}}} +\pagecolor{white} + +\newcommand{\ruler}{\par\noindent\setlength{\unitlength}{1mm}\begin{picture}(0,6)% + \put(0,4){\line(1,0){170}}% + \multiput(0,2)(10,0){18}{\line(0,1){4}}% + \multiput(0,3)(1,0){170}{\line(0,1){2}}% + \put(0,0){\makebox(0,0){{\tiny 0}}}% + \put(10,0){\makebox(0,0){{\tiny 1}}}% + \put(20,0){\makebox(0,0){{\tiny 2}}}% + \put(30,0){\makebox(0,0){{\tiny 3}}}% + \put(40,0){\makebox(0,0){{\tiny 4}}}% + \put(50,0){\makebox(0,0){{\tiny 5}}}% + \put(60,0){\makebox(0,0){{\tiny 6}}}% + \put(70,0){\makebox(0,0){{\tiny 7}}}% + \put(80,0){\makebox(0,0){{\tiny 8}}}% + \put(90,0){\makebox(0,0){{\tiny 9}}}% + \put(100,0){\makebox(0,0){{\tiny 10}}}% + \put(110,0){\makebox(0,0){{\tiny 11}}}% + \put(120,0){\makebox(0,0){{\tiny 12}}}% + \put(130,0){\makebox(0,0){{\tiny 13}}}% + \put(140,0){\makebox(0,0){{\tiny 14}}}% + \put(150,0){\makebox(0,0){{\tiny 15}}}% + \put(160,0){\makebox(0,0){{\tiny 16}}}% + \put(170,0){\makebox(0,0){{\tiny 17}}}% + \end{picture}\par} + +% figures: +\setlength{\fboxsep}{0pt} +\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{#1.tex}}} +%\newcommand{\texpicture}[1]{\fbox{\sffamily\footnotesize\input{#1.tex}}} +%\newcommand{\texpicture}[1]{\setlength{\fboxsep}{2mm}\fbox{#1}} +%\newcommand{\texpicture}[1]{} +\newcommand{\figlabel}[1]{\textsf{\textbf{\large \uppercase{#1}}}} + +% maximum number of floats: +\setcounter{topnumber}{2} +\setcounter{bottomnumber}{0} +\setcounter{totalnumber}{2} + +% float placement fractions: +\renewcommand{\textfraction}{0.2} +\renewcommand{\topfraction}{0.8} +\renewcommand{\bottomfraction}{0.0} +\renewcommand{\floatpagefraction}{0.5} + +% spacing for floats: +\setlength{\floatsep}{12pt plus 2pt minus 2pt} +\setlength{\textfloatsep}{20pt plus 4pt minus 2pt} +\setlength{\intextsep}{12pt plus 2pt minus 2pt} + +% spacing for a floating page: +\makeatletter + \setlength{\@fptop}{0pt} + \setlength{\@fpsep}{8pt plus 2.0fil} + \setlength{\@fpbot}{0pt plus 1.0fil} +\makeatother + +% rules for floats: +\newcommand{\topfigrule}{\vspace*{10pt}{\hrule height0.4pt}\vspace*{-10.4pt}} +\newcommand{\bottomfigrule}{\vspace*{-10.4pt}{\hrule height0.4pt}\vspace*{10pt}} + +% captions: +\usepackage[format=plain,singlelinecheck=off,labelfont=bf,font={small,sf}]{caption} + +% put caption on separate float: +\newcommand{\breakfloat}{\end{figure}\begin{figure}[t]} + +% references to panels of a figure within the caption: +\newcommand{\figitem}[1]{\textsf{\bfseries\uppercase{#1}}} +% references to figures: +\newcommand{\panel}[1]{\textsf{\uppercase{#1}}} +\newcommand{\fref}[1]{\textup{\ref{#1}}} +\newcommand{\subfref}[2]{\textup{\ref{#1}}\,\panel{#2}} +% references to figures in normal text: +\newcommand{\fig}{Fig.} +\newcommand{\Fig}{Figure} +\newcommand{\figs}{Figs.} +\newcommand{\Figs}{Figures} +\newcommand{\figref}[1]{\fig~\fref{#1}} +\newcommand{\Figref}[1]{\Fig~\fref{#1}} +\newcommand{\figsref}[1]{\figs~\fref{#1}} +\newcommand{\Figsref}[1]{\Figs~\fref{#1}} +\newcommand{\subfigref}[2]{\fig~\subfref{#1}{#2}} +\newcommand{\Subfigref}[2]{\Fig~\subfref{#1}{#2}} +\newcommand{\subfigsref}[2]{\figs~\subfref{#1}{#2}} +\newcommand{\Subfigsref}[2]{\Figs~\subfref{#1}{#2}} +% references to figures within bracketed text: +\newcommand{\figb}{Fig.} +\newcommand{\figsb}{Figs.} +\newcommand{\figrefb}[1]{\figb~\fref{#1}} +\newcommand{\figsrefb}[1]{\figsb~\fref{#1}} +\newcommand{\subfigrefb}[2]{\figb~\subfref{#1}{#2}} +\newcommand{\subfigsrefb}[2]{\figsb~\subfref{#1}{#2}} + +% references to tables: +\newcommand{\tref}[1]{\textup{\ref{#1}}} +% references to tables in normal text: +\newcommand{\tab}{Tab.} +\newcommand{\Tab}{Table} +\newcommand{\tabs}{Tabs.} +\newcommand{\Tabs}{Tables} +\newcommand{\tabref}[1]{\tab~\tref{#1}} +\newcommand{\Tabref}[1]{\Tab~\tref{#1}} +\newcommand{\tabsref}[1]{\tabs~\tref{#1}} +\newcommand{\Tabsref}[1]{\Tabs~\tref{#1}} +% references to tables within bracketed text: +\newcommand{\tabb}{Tab.} +\newcommand{\tabsb}{Tab.} +\newcommand{\tabrefb}[1]{\tabb~\tref{#1}} +\newcommand{\tabsrefb}[1]{\tabsb~\tref{#1}} + + +%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newcommand{\eqref}[1]{(\ref{#1})} +\newcommand{\eqn}{Eq.} +\newcommand{\Eqn}{Eq.} +\newcommand{\eqns}{Eqs.} +\newcommand{\Eqns}{Eqs.} +\newcommand{\eqnref}[1]{\eqn~\eqref{#1}} +\newcommand{\Eqnref}[1]{\Eqn~\eqref{#1}} +\newcommand{\eqnsref}[1]{\eqns~\eqref{#1}} +\newcommand{\Eqnsref}[1]{\Eqns~\eqref{#1}} + %%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{listings} @@ -290,6 +421,36 @@ spricht von einer Wahrscheinlichkeitsdichte. \end{figure} +\subsection{Korrelation} + +\begin{figure}[t] + \includegraphics[width=1\textwidth]{correlation} + \caption{\label{correlationfig} Korrelationen zwischen zwei Datens\"atzen $x$ und $y$.} +\end{figure} + +Bisher haben wir Eigenschaften einer einzelnen Me{\ss}gr\"o{\ss}e angeschaut. +Bei mehreren Me{\ss}gr\"o{\ss}en, kann nach Abh\"angigkeiten gefragt werden. +Der Korrelationskoeffizient +\[ r_{x,y} = \frac{Cov(x,y)}{\sigma_x \sigma_y} = \frac{\langle + (x-\langle x \rangle)(y-\langle y \rangle) \rangle}{\sqrt{\langle + (x-\langle x \rangle)^2} \rangle \sqrt{\langle (y-\langle y + \rangle)^2} \rangle} \] quantifiziert einfache lineare +Zusammenh\"ange. Perfekt korrelierte Variablen ergeben einen +Korrelationskoeffizienten von $+1$, antikorrelierte Daten einen +Korrelationskoeffizienten von $-1$ und nicht korrelierte Daten einen +Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}). + +\begin{figure}[t] + \includegraphics[width=1\textwidth]{nonlincorrelation} + \caption{\label{nonlincorrelationfig} Nichtlineare Zusammenh\"ange + werden durch den Korrelationskoeffizienten nicht erfasst! Sowohl + die quadratische Abh\"angigkeit (links) als auch eine + Rauschkorrelation (rechts), bei der die Streuung der $y$-Werte von + $x$ abh\"angen, ergeben Korrelationskeffizienten nahe Null. + $\xi$ sind normalverteilte Zufallszahlen.} +\end{figure} + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Data types} diff --git a/statistics/lecture/nonlincorrelation.py b/statistics/lecture/nonlincorrelation.py new file mode 100644 index 0000000..c0ca723 --- /dev/null +++ b/statistics/lecture/nonlincorrelation.py @@ -0,0 +1,42 @@ +import numpy as np +import matplotlib.pyplot as plt + +plt.xkcd() +fig = plt.figure( figsize=(6,3) ) +n = 200 +x = np.random.randn( n ) +y = np.random.randn( n ) + +z = x*x+0.2*y +r =np.corrcoef(x,z)[0,1] +ax = fig.add_subplot( 1, 2, 1 ) +ax.spines['right'].set_visible(False) +ax.spines['top'].set_visible(False) +ax.yaxis.set_ticks_position('left') +ax.xaxis.set_ticks_position('bottom') +ax.text( 0, 4.0, 'r=%.1f' % r, ha='center' ) +ax.text( 0, 5.5, r'$y = x^2+\xi/5$', ha='center' ) +ax.set_xlabel('x') +ax.set_ylabel('y') +ax.set_xlim( -3.0, 3.0) +ax.set_ylim( -0.5, 6.0) +ax.scatter( x, z ) + +z = 0.5*x*y +r =np.corrcoef(x,z)[0,1] +ax = fig.add_subplot( 1, 2, 2 ) +ax.spines['right'].set_visible(False) +ax.spines['top'].set_visible(False) +ax.yaxis.set_ticks_position('left') +ax.xaxis.set_ticks_position('bottom') +ax.text( 0, 1.5, 'r=%.1f' % r, ha='center' ) +ax.text( 0, 2.5, r'$y = x \cdot \xi/2$', ha='center' ) +ax.set_xlabel('x') +ax.set_ylabel('y') +ax.set_xlim( -3.0, 3.0) +ax.set_ylim( -3.0, 3.0) +ax.scatter( x, z ) + +plt.tight_layout() +plt.savefig('nonlincorrelation.pdf') +plt.show()