add beta distribution graphs

7 files changed, 351 insertions, 13 deletions

diff --git a/buch/papers/dreieck/images/Makefile b/buch/papers/dreieck/images/Makefile
index 3907d13..c979599 100644
--- a/buch/papers/dreieck/images/Makefile
+++ b/buch/papers/dreieck/images/Makefile
@@ -3,8 +3,16 @@
 #
 # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
+all:	order.pdf beta.pdf
+
 order.pdf:	order.tex orderpath.tex
 	pdflatex order.tex
 
 orderpath.tex:	order.m
 	octave order.m
+
+beta.pdf:	beta.tex betapaths.tex
+	pdflatex beta.tex
+
+betapaths.tex:	betadist.m
+	octave betadist.m
diff --git a/buch/papers/dreieck/images/beta.pdf b/buch/papers/dreieck/images/beta.pdf
new file mode 100644
index 0000000..c3ab4f6
--- /dev/null
+++ b/buch/papers/dreieck/images/beta.pdf
diff --git a/buch/papers/dreieck/images/beta.tex b/buch/papers/dreieck/images/beta.tex
new file mode 100644
index 0000000..50509ee
--- /dev/null
+++ b/buch/papers/dreieck/images/beta.tex
@@ -0,0 +1,214 @@
+%
+% beta.tex -- display some symmetric beta distributions
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math,calc}
+\input{betapaths.tex}
+\begin{document}
+\def\skala{12}
+\definecolor{colorone}{rgb}{1.0,0.6,0.0}
+\definecolor{colortwo}{rgb}{1.0,0.0,0.0}
+\definecolor{colorthree}{rgb}{0.6,0.0,0.6}
+\definecolor{colorfour}{rgb}{0.6,0.0,1.0}
+\definecolor{colorfive}{rgb}{0.0,0.0,1.0}
+\definecolor{colorsix}{rgb}{0.4,0.6,1.0}
+\definecolor{colorseven}{rgb}{0.0,0.0,0.0}
+\definecolor{coloreight}{rgb}{0.0,0.8,0.8}
+\definecolor{colornine}{rgb}{0.0,0.8,0.2}
+\definecolor{colorten}{rgb}{0.2,0.4,0.0}
+\definecolor{coloreleven}{rgb}{1.0,0.8,0.4}
+
+\def\achsen{
+	\foreach \x in {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9}{
+		\draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala});
+		\node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$};
+	}
+	\foreach \y in {1,2,3,4}{
+		\draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy});
+		\node at ({-0.1/\skala},{\y*\dy}) [left] {$\y$};
+	}
+	\def\x{1}
+	\draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala});
+	\node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$};
+	\def\x{0}
+	\node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$};
+
+	\draw[->] ({-0.1/\skala},0) -- ({1*\dx+0.4/\skala},0)
+		coordinate[label={$x$}];
+	\draw[->] (0,{-0.1/\skala}) -- (0,{\betamax*\dy+0.4/\skala},0)
+		coordinate[label={right:$\beta(a,b,x)$}];
+}
+
+\def\farbcoord#1#2{
+	({\dx*(0.7+((#1-1)/4)*0.27)},{\dx*(0.15+((#2-1)/4)*0.27)})
+}
+\def\farbviereck{
+	\foreach \x in {1,2,3,4,5}{
+		\draw[color=gray!30] \farbcoord{\x}{1} -- \farbcoord{\x}{5};
+		\draw[color=gray!30] \farbcoord{1}{\x} -- \farbcoord{5}{\x};
+	}
+	\draw[->] \farbcoord{1}{1} -- \farbcoord{5.4}{1}
+		coordinate[label={$a$}];
+	\draw[->] \farbcoord{1}{1} -- \farbcoord{1}{5.4}
+		coordinate[label={left: $b$}];
+	\foreach \x in {1,2,3,4,5}{
+		\node[color=gray] at \farbcoord{5}{\x} [right] {\tiny $b=\x$};
+		\fill[color=white,opacity=0.7] 
+			\farbcoord{(\x-0.1)}{4.3}
+			rectangle
+			\farbcoord{(\x+0.1)}{5};
+		\node[color=gray] at \farbcoord{\x}{5} [left,rotate=90]
+			{\tiny $a=\x$};
+	}
+}
+\def\farbpunkt#1#2#3{
+	\fill[color=#3] \farbcoord{#1}{#2} circle[radius={0.1/\skala}];
+}
+
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\dx{1}
+\def\dy{0.1}
+\def\opa{0.1}
+
+\def\betamax{4.2}
+
+\fill[color=colorone,opacity=\opa] (0,0) -- \betaaa -- (\dx,0) -- cycle;
+\fill[color=colortwo,opacity=\opa] (0,0) -- \betabb -- (\dx,0) -- cycle;
+\fill[color=colorthree,opacity=\opa] (0,0) -- \betacc -- (\dx,0) -- cycle;
+\fill[color=colorfour,opacity=\opa] (0,0) -- \betadd -- (\dx,0) -- cycle;
+\fill[color=colorfive,opacity=\opa] (0,0) -- \betaee -- (\dx,0) -- cycle;
+\fill[color=colorsix,opacity=\opa] (0,0) -- \betaff -- (\dx,0) -- cycle;
+\fill[color=colorseven,opacity=\opa] (0,0) -- \betagg -- (\dx,0) -- cycle;
+\fill[color=coloreight,opacity=\opa] (0,0) -- \betahh -- (\dx,0) -- cycle;
+\fill[color=colornine,opacity=\opa] (0,0) -- \betaii -- (\dx,0) -- cycle;
+\fill[color=colorten,opacity=\opa] (0,0) -- \betajj -- (\dx,0) -- cycle;
+\fill[color=coloreleven,opacity=\opa] (0,0) -- \betakk -- (\dx,0) -- cycle;
+
+\draw[color=colorone]    \betaaa;
+\draw[color=colortwo]    \betabb;
+\draw[color=colorthree]  \betacc;
+\draw[color=colorfour]   \betadd;
+\draw[color=colorfive]   \betaee;
+\draw[color=colorsix]    \betaff;
+\draw[color=colorseven]  \betagg;
+\draw[color=coloreight]  \betahh;
+\draw[color=colornine]   \betaii;
+\draw[color=colorten]    \betajj;
+\draw[color=coloreleven] \betakk;
+
+\achsen
+
+\farbviereck
+
+\farbpunkt{\alphaeleven}{\betaeleven}{coloreleven}
+\farbpunkt{\alphaten}{\betaten}{colorten}
+\farbpunkt{\alphanine}{\betanine}{colornine}
+\farbpunkt{\alphaeight}{\betaeight}{coloreight}
+\farbpunkt{\alphaseven}{\betaseven}{colorseven}
+\farbpunkt{\alphasix}{\betasix}{colorsix}
+\farbpunkt{\alphafive}{\betafive}{colorfive}
+\farbpunkt{\alphafour}{\betafour}{colorfour}
+\farbpunkt{\alphathree}{\betathree}{colorthree}
+\farbpunkt{\alphatwo}{\betatwo}{colortwo}
+\farbpunkt{\alphaone}{\betaone}{colorone}
+
+
+\def\betamax{4.9}
+
+\begin{scope}[yshift=-0.6cm]
+\fill[color=colorone,opacity=\opa] (0,0) -- \betaaa -- (\dx,0) -- cycle;
+\fill[color=colortwo,opacity=\opa] (0,0) -- \betaab -- (\dx,0) -- cycle;
+\fill[color=colorthree,opacity=\opa] (0,0) -- \betaac -- (\dx,0) -- cycle;
+\fill[color=colorfour,opacity=\opa] (0,0) -- \betaad -- (\dx,0) -- cycle;
+\fill[color=colorfive,opacity=\opa] (0,0) -- \betaae -- (\dx,0) -- cycle;
+\fill[color=colorsix,opacity=\opa] (0,0) -- \betaaf -- (\dx,0) -- cycle;
+\fill[color=colorseven,opacity=\opa] (0,0) -- \betaag -- (\dx,0) -- cycle;
+\fill[color=coloreight,opacity=\opa] (0,0) -- \betaah -- (\dx,0) -- cycle;
+\fill[color=colornine,opacity=\opa] (0,0) -- \betaai -- (\dx,0) -- cycle;
+\fill[color=colorten,opacity=\opa] (0,0) -- \betaaj -- (\dx,0) -- cycle;
+\fill[color=coloreleven,opacity=\opa] (0,0) -- \betaak -- (\dx,0) -- cycle;
+
+\draw[color=colorone]    \betaaa;
+\draw[color=colortwo]    \betaab;
+\draw[color=colorthree]  \betaac;
+\draw[color=colorfour]   \betaad;
+\draw[color=colorfive]   \betaae;
+\draw[color=colorsix]    \betaaf;
+\draw[color=colorseven]  \betaag;
+\draw[color=coloreight]  \betaah;
+\draw[color=colornine]   \betaai;
+\draw[color=colorten]    \betaaj;
+\draw[color=coloreleven] \betaak;
+
+\achsen
+
+\farbviereck
+
+\farbpunkt{\alphaone}{\betaeleven}{coloreleven}
+\farbpunkt{\alphaone}{\betaten}{colorten}
+\farbpunkt{\alphaone}{\betanine}{colornine}
+\farbpunkt{\alphaone}{\betaeight}{coloreight}
+\farbpunkt{\alphaone}{\betaseven}{colorseven}
+\farbpunkt{\alphaone}{\betasix}{colorsix}
+\farbpunkt{\alphaone}{\betafive}{colorfive}
+\farbpunkt{\alphaone}{\betafour}{colorfour}
+\farbpunkt{\alphaone}{\betathree}{colorthree}
+\farbpunkt{\alphaone}{\betatwo}{colortwo}
+\farbpunkt{\alphaone}{\betaone}{colorone}
+
+\end{scope}
+
+\begin{scope}[yshift=-1.2cm]
+\fill[color=colorone,opacity=\opa]    (0,0) -- \betaak -- (\dx,0) -- cycle;
+\fill[color=colortwo,opacity=\opa]    (0,0) -- \betabk -- (\dx,0) -- cycle;
+\fill[color=colorthree,opacity=\opa]  (0,0) -- \betack -- (\dx,0) -- cycle;
+\fill[color=colorfour,opacity=\opa]   (0,0) -- \betadk -- (\dx,0) -- cycle;
+\fill[color=colorfive,opacity=\opa]   (0,0) -- \betaek -- (\dx,0) -- cycle;
+\fill[color=colorsix,opacity=\opa]    (0,0) -- \betafk -- (\dx,0) -- cycle;
+\fill[color=colorseven,opacity=\opa]  (0,0) -- \betagk -- (\dx,0) -- cycle;
+\fill[color=coloreight,opacity=\opa]  (0,0) -- \betahk -- (\dx,0) -- cycle;
+\fill[color=colornine,opacity=\opa]   (0,0) -- \betaik -- (\dx,0) -- cycle;
+\fill[color=colorten,opacity=\opa]    (0,0) -- \betajk -- (\dx,0) -- cycle;
+\fill[color=coloreleven,opacity=\opa] (0,0) -- \betakk -- (\dx,0) -- cycle;
+
+\draw[color=colorone]    \betaak;
+\draw[color=colortwo]    \betabk;
+\draw[color=colorthree]  \betack;
+\draw[color=colorfour]   \betadk;
+\draw[color=colorfive]   \betaek;
+\draw[color=colorsix]    \betafk;
+\draw[color=colorseven]  \betagk;
+\draw[color=coloreight]  \betahk;
+\draw[color=colornine]   \betaik;
+\draw[color=colorten]    \betajk;
+\draw[color=coloreleven] \betakk;
+
+\achsen
+
+\farbviereck
+
+\farbpunkt{\alphaeleven}{\betaeleven}{coloreleven}
+\farbpunkt{\alphaten}{\betaeleven}{colorten}
+\farbpunkt{\alphanine}{\betaeleven}{colornine}
+\farbpunkt{\alphaeight}{\betaeleven}{coloreight}
+\farbpunkt{\alphaseven}{\betaeleven}{colorseven}
+\farbpunkt{\alphasix}{\betaeleven}{colorsix}
+\farbpunkt{\alphafive}{\betaeleven}{colorfive}
+\farbpunkt{\alphafour}{\betaeleven}{colorfour}
+\farbpunkt{\alphathree}{\betaeleven}{colorthree}
+\farbpunkt{\alphatwo}{\betaeleven}{colortwo}
+\farbpunkt{\alphaone}{\betaeleven}{colorone}
+
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/papers/dreieck/images/betadist.m b/buch/papers/dreieck/images/betadist.m
new file mode 100644
index 0000000..9ff78ed
--- /dev/null
+++ b/buch/papers/dreieck/images/betadist.m
@@ -0,0 +1,50 @@
+#
+# betadist.m
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+global N;
+N = 201;
+global n;
+n = 11;
+
+t = (0:n-1) / (n-1)
+alpha = 1 + 4 * t.^2
+
+#alpha = [ 1, 1.03, 1.05, 1.1, 1.25, 1.5, 2, 2.5, 3, 4, 5 ];
+beta = alpha;
+names = [ "one"; "two"; "three"; "four"; "five"; "six"; "seven"; "eight";
+	  "nine"; "ten"; "eleven" ]
+
+function retval = Beta(a, b, x)
+	retval = x^(a-1) * (1-x)^(b-1) / beta(a, b);
+end
+
+function plotbeta(fn, a, b, name)
+	global N;
+	fprintf(fn, "\\def\\beta%s{\n", name);
+	fprintf(fn, "\t({%.4f*\\dx},{%.4f*\\dy})", 0, Beta(a, b, 0));
+	for x = (1:N-1)/(N-1)
+		X = (1-cos(pi * x))/2;
+		fprintf(fn, "\n\t--({%.4f*\\dx},{%.4f*\\dy})",
+			X, Beta(a, b, X));
+	end
+	fprintf(fn, "\n}\n");
+end
+
+fn = fopen("betapaths.tex", "w");
+
+for i = (1:n)
+	fprintf(fn, "\\def\\alpha%s{%f}\n", names(i,:), alpha(i));
+	fprintf(fn, "\\def\\beta%s{%f}\n", names(i,:), beta(i));
+end
+
+for i = (1:n)
+	for j = (1:n)
+		printf("working on %d,%d:\n", i, j);
+		plotbeta(fn, alpha(i), beta(j),
+			char(['a' + i - 1, 'a' + j - 1]));
+	end
+end
+
+fclose(fn);
diff --git a/buch/papers/dreieck/images/order.m b/buch/papers/dreieck/images/order.m
index d37a258..762f458 100644
--- a/buch/papers/dreieck/images/order.m
+++ b/buch/papers/dreieck/images/order.m
@@ -26,6 +26,10 @@ function retval = orderd(p, n, k)
 	end
 end
 
+function retval = orders(p, n, k)
+	retval = k * nchoosek(n, k) * p^(k-1) * (1-p)^(n-k);
+end
+
 function orderpath(fn, k, name)
 	fprintf(fn, "\\def\\order%s{\n\t(0,0)", name);
 	global N;
@@ -51,29 +55,65 @@ function orderdpath(fn, k, name)
 	fprintf(fn, "\n}\n");
 end
 
+function orderspath(fn, k, name)
+	fprintf(fn, "\\def\\orders%s{\n\t(0,0)", name);
+	global N;
+	global subdivisions;
+	for i = (1:subdivisions-1)
+		p = i/subdivisions;
+		fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})",
+			p, orders(p, N, k));
+	end
+	fprintf(fn, "\n\t-- ({1*\\dx},0)");
+	fprintf(fn, "\n}\n");
+end
+
 fn = fopen("orderpath.tex", "w");
+
 orderpath(fn,   0, "zero");
 orderdpath(fn,  0, "zero");
+orderspath(fn,  0, "zero");
+
 orderpath(fn,   1, "one");
 orderdpath(fn,  1, "one");
+orderspath(fn,  1, "one");
+
 orderpath(fn,   2, "two");
 orderdpath(fn,  2, "two");
+orderspath(fn,  2, "two");
+
 orderpath(fn,   3, "three");
 orderdpath(fn,  3, "three");
+orderspath(fn,  3, "three");
+
 orderpath(fn,   4, "four");
 orderdpath(fn,  4, "four");
+orderspath(fn,  4, "four");
+
 orderpath(fn,   5, "five");
 orderdpath(fn,  5, "five");
+orderspath(fn,  5, "five");
+
 orderpath(fn,   6, "six");
 orderdpath(fn,  6, "six");
+orderspath(fn,  6, "six");
+
 orderpath(fn,   7, "seven");
 orderdpath(fn,  7, "seven");
+orderspath(fn,  7, "seven");
+
 orderpath(fn,   8, "eight");
 orderdpath(fn,  8, "eight");
+orderspath(fn,  8, "eight");
+
 orderpath(fn,   9, "nine");
 orderdpath(fn,  9, "nine");
+orderspath(fn,  9, "nine");
+
 orderpath(fn,  10, "ten");
 orderdpath(fn, 10, "ten");
+orderspath(fn, 10, "ten");
+
 fclose(fn);
 
 
diff --git a/buch/papers/dreieck/images/order.pdf b/buch/papers/dreieck/images/order.pdf
index 6d9c8c0..98a5fbe 100644
--- a/buch/papers/dreieck/images/order.pdf
+++ b/buch/papers/dreieck/images/order.pdf
diff --git a/buch/papers/dreieck/images/order.tex b/buch/papers/dreieck/images/order.tex
index 083f014..9a2511c 100644
--- a/buch/papers/dreieck/images/order.tex
+++ b/buch/papers/dreieck/images/order.tex
@@ -13,10 +13,25 @@
 \begin{document}
 \def\skala{8}
 \definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\n{10}
+\def\E#1#2{
+	\draw[color=#2]
+		({\dx*#1/(\n+1)},{-0.1/\skala}) -- ({\dx*#1/(\n+1)},{4.4*\dy});
+	\node[color=#2] at ({\dx*#1/(\n+1)},{3.2*\dy})
+		[rotate=90,above right] {$k=#1$};
+}
+\def\var#1#2{
+	\pgfmathparse{\dx*sqrt(#1*(\n-#1+1)/((\n+1)*(\n+1)*(\n+2)))}
+	\xdef\var{\pgfmathresult}
+	\fill[color=#2,opacity=0.5]
+		({\dx*#1/(\n+1)-\var},0) rectangle ({\dx*#1/(\n+1)+\var},{4.4*\dy});
+}
+
 \input{orderpath.tex}
 \begin{tikzpicture}[>=latex,thick,scale=\skala]
 
-\def\dx{1}
+\def\dx{1.6}
 \def\dy{0.5}
 
 \def\pfad#1#2{
@@ -24,7 +39,7 @@
 	--
 	#1
 	--
-	({1+0.1/\skala},0.5);
+	({1*\dx+0.1/\skala},0.5);
 }
 
 \pfad{\orderzero}{darkgreen!20}
@@ -39,11 +54,11 @@
 \pfad{\orderten}{darkgreen!20}
 \pfad{\orderseven}{darkgreen}
 
-\draw[->] ({-0.1/\skala},0) -- (1.1,0) coordinate[label={$x$}];
-\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={left:$F(X)$}];
+\draw[->] ({-0.1/\skala},0) -- ({1.03*\dx},0) coordinate[label={$x$}];
+\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={right:$F(X)$}];
 \foreach \x in {0,0.2,0.4,0.6,0.8,1}{
-	\draw (\x,{-0.1/\skala}) -- (\x,{0.1/\skala});
-	\node at (\x,{-0.1/\skala}) [below] {$\x$};
+	\draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala});
+	\node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$};
 }
 \foreach \y in {0.5,1}{
 	\draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy});
@@ -55,17 +70,25 @@
 \begin{scope}[yshift=-0.7cm]
 \def\dy{0.125}
 
+\foreach \k in {1,2,3,4,5,6,8,9,10}{
+	\E{\k}{blue!30}
+}
+\def\k{7}
+\var{\k}{orange!40}
+\node[color=blue] at ({\dx*\k/(\n+1)},{4.3*\dy}) [above] {$E(X_{7:n})$};
+
 \def\pfad#1#2{
 	\draw[color=#2,line width=1.4pt] ({-0.1/\skala},0) 
 		--
 		#1
 		--
-		({1+0.1/\skala},0.0);
+		({1*\dx+0.1/\skala},0.0);
 }
 
 \begin{scope}
 \clip ({-0.1/\skala},{-0.1/\skala})
-	rectangle ({1+0.1/\skala},{0.56+0.1/\skala});
+	rectangle ({1*\dx+0.1/\skala},{0.56+0.1/\skala});
+
 \pfad{\orderdzero}{red!20}
 \pfad{\orderdone}{red!20}
 \pfad{\orderdtwo}{red!20}
@@ -76,21 +99,24 @@
 \pfad{\orderdeight}{red!20}
 \pfad{\orderdnine}{red!20}
 \pfad{\orderdten}{red!20}
+\E{\k}{blue}
 \pfad{\orderdseven}{red}
+
 \end{scope}
 
-\draw[->] ({-0.1/\skala},0) -- (1.1,0) coordinate[label={$x$}];
-\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={left:$\varphi(X)$}];
+\draw[->] ({-0.1/\skala},0) -- ({1.03*\dx},0) coordinate[label={$x$}];
+\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={right:$\varphi(X)$}];
 \foreach \x in {0,0.2,0.4,0.6,0.8,1}{
-	\draw (\x,{-0.1/\skala}) -- (\x,{0.1/\skala});
-	\node at (\x,{-0.1/\skala}) [below] {$\x$};
+	\draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala});
+	\node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$};
 }
 \foreach \y in {1,2,3,4}{
 	\draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy});
 	\node at ({-0.1/\skala},{\y*\dy}) [left] {$\y$};
 }
 
-\node[color=red] at (0.67,{2.7*\dy}) [above] {$k=7$};
+\node[color=red] at ({0.67*\dx},{2.7*\dy}) [above] {$k=7$};
+
 
 \end{scope}