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+ "version": 3
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+ "file_extension": ".py",
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+++ b/lecture_1.css
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+div.myheader {
+ position: absolute;
+ margin: 30px;
+ left: 8%;
+ background: blue;
+ font-size: xx-large;
+}
+
+div.myfooter {
+ position: absolute;
+ background: red;
+ font-size: 120%;
+ right: 10%;
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+
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diff --git a/lecture_1.ipynb b/lecture_1.ipynb
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--- /dev/null
+++ b/lecture_1.ipynb
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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "87c9d786-6d5c-4366-9701-5fa127727caa",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Lecture 1\n",
+ "\n",
+ "---\n",
+ "\n",
+ "## Basic statistics \n",
+ "\n",
+ "<br>\n",
+ "<br>\n",
+ "\n",
+ " Hartmut Stadie\n",
+ "\n",
+ "hartmut.stadie@uni-hamburg.de"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9eae9199-9401-40c4-b212-ae57f1ccab38",
+ "metadata": {
+ "slideshow": {
+ "slide_type": "slide"
+ }
+ },
+ "source": [
+ "## Samples\n",
+ "\n",
+ "---\n",
+ "\n",
+ "Sample: $X = x_1, x_2,\\dots, x_N$ \n",
+ "\n",
+ "Expectation value: $<f(x)> = \\frac{1}{N}\\sum_i^N f(x_i)$\n",
+ "\n",
+ "Describing samples: minimum, maximum, frequency/histogram, means, variance, standard deviation,....\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "b44e356e-b829-4879-b5fc-9706fffe873d",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "array([[5., 0.],\n",
+ " [2., 1.],\n",
+ " [0., 1.],\n",
+ " [0., 3.],\n",
+ " [0., 1.],\n",
+ " [2., 2.],\n",
+ " [4., 0.],\n",
+ " [2., 1.],\n",
+ " [1., 3.]])"
+ ]
+ },
+ "execution_count": 1,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "import numpy as np\n",
+ "data = np.loadtxt('./exercises/09_data.txt')\n",
+ "\n",
+ "data[0:9]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "id": "2aeacb94-518d-464f-a7ab-5282b97bc225",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "array([5., 2., 0., 0., 0., 2., 4., 2., 1.])"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "data[0:9,0]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "id": "1829fa78-7d7b-4bac-9f24-5129dca63629",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(np.float64(0.0), np.float64(6.0))"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "np.min(data), np.max(data)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "caa12e94-0b72-4f60-875d-5a306a09d036",
+ "metadata": {
+ "slideshow": {
+ "slide_type": "slide"
+ }
+ },
+ "source": [
+ "### Histograms"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "id": "871f915d-09f8-4d14-a79a-8fbd2f16ab75",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(array([74., 96., 0., 67., 0., 43., 13., 0., 10., 3.]),\n",
+ " array([0. , 0.6, 1.2, 1.8, 2.4, 3. , 3.6, 4.2, 4.8, 5.4, 6. ]),\n",
+ " <BarContainer object of 10 artists>)"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ },
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "import matplotlib.pyplot as plt\n",
+ "\n",
+ "plt.hist(data[:, 0])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "id": "f8263279-48b5-4421-be05-604ddbfd8d6f",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Text(0.5, 0, 'k')"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ },
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "plt.hist(data[:, 0], bins=np.arange(-0.25,6.25,0.5))\n",
+ "plt.xlabel(\"k\")"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "id": "5056a717-6561-41ec-be39-1757984863a9",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/plain": [
+ "Text(0.5, 0, 'l')"
+ ]
+ },
+ "execution_count": 6,
+ "metadata": {},
+ "output_type": "execute_result"
+ },
+ {
+ "data": {
+ "image/png": "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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "plt.hist(data[:, 0], bins=np.arange(-0.25,6.26,0.5))\n",
+ "plt.xlabel(\"k\")\n",
+ "#plt.savefig(\"hist.pdf\")\n",
+ "plt.show()\n",
+ "plt.hist(data[:, 1], bins=np.arange(-0.25,6.26,0.5))\n",
+ "plt.xlabel(\"l\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4183331e-8a9f-4af5-829f-3fcb9b2abb31",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Cumulated Distribution"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "id": "4ecfb198-c621-4ee4-9d24-9f9a8cb8bec7",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "median 1.0\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "plt.hist(data[:, 0], bins=100, cumulative=True, density = True, label=\"kumuliert\")\n",
+ "plt.xlabel(\"k\")\n",
+ "#plt.savefig(\"hist2.pdf\")\n",
+ "print(\"median\", np.median(data[:, 0]))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "d00cadf3-0cfa-4f59-9962-4b9e6b707e0b",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ceb5ca27-a96a-4ee3-967b-5f02efe66540",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Means\n",
+ "\n",
+ "---\n",
+ "\n",
+ "different means:\n",
+ " - arithmetric mean: $ \\overline{x} = \\frac{1}{n}\\sum\\limits_{i=1}^n x_i (= \\mu)$\n",
+ " - geometric mean: $ \\overline{{x}}_\\mathrm {geom} = \\sqrt[n]{\\prod\\limits_{i=1}^{n}{x_i}}$\n",
+ " - quadratic mean: $ \\overline{{x}}_\\mathrm{quadr} = \\sqrt {\\frac {1}{n} \\sum\\limits_{i=1}^{n}x_i^2} = \\sqrt{\\overline{x^2}}$\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "68cea392-8d4f-48d7-aacc-0f10d46af3af",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Exercise: Compute mean and variance of $X$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "id": "76d92d18-1d77-40d2-a910-592183635d3b",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "fragment"
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "mean [1.56535948 1.26470588]\n"
+ ]
+ }
+ ],
+ "source": [
+ "print(\"mean\", np.mean(data, axis=0))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "id": "5f0f34b4-5bbe-439b-8b4e-fbb05794b790",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "variance [1.85357128 1.27306805]\n"
+ ]
+ }
+ ],
+ "source": [
+ "print(\"variance\", np.var(data, axis=0))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "id": "70a1920f-beda-4154-ad77-22a9ffe2e39f",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "standard deviation: [1.36145925 1.12830317]\n"
+ ]
+ }
+ ],
+ "source": [
+ "print(\"standard deviation:\", np.std(data, axis=0))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b8ab42a5-b7af-4942-9a7a-91b73e0ce625",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Probability Density Functions\n",
+ "\n",
+ "\n",
+ "Sie $x$ eine reelle Zahl, die das Ergebnis eines Zufallsexperiments\n",
+ "beschreibt:\n",
+ "\n",
+ "Wahrscheinlichkeitsdichte $f(x)$: (probability density function (pdf)) \n",
+ "\n",
+ "- Wahrscheinlichkeit, dass x im Intervall $[x, x + dx]$ liegt:\n",
+ " $f(x)\\,dx$ \n",
+ "\n",
+ "- Normierung: $$\\int_S f(x)\\,dx = 1$$\n",
+ "\n",
+ "Kumulierte Dichte $F(x)$: (cumulative density function (cdf);\n",
+ "Mathematik: Verteilungsfunktion) \n",
+ "Wahrscheinlichkeit, dass x kleiner x ist:\n",
+ "$$F(x) = \\int_{-\\infty}^x f(x^\\prime)\\,dx^\\prime$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "13563524-0626-4a69-a6ab-f87d7f19a016",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Example\n",
+ "\n",
+ "$$P(a \\le x \\le b) = \\int_a^b f(x)\\,dx = F(b) - F(a)$$\n",
+ "\n",
+ "<img src=\"./figures/08/lognormal_pdf.png\"\n",
+ "style=\"width:55%\" /> <img src=\"./figures/08/lognormal_cdf.png\"\n",
+ "style=\"width:41.4%\" /> CC0, via Wikimedia Commons\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "14e4b782-9573-4e87-89d0-901dade25538",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Quantiles \n",
+ "\n",
+ "\n",
+ "Quantile $x_\\alpha$ Wert der Zufallsvariable $x$ mit\n",
+ "$$F(x_\\alpha) = \\alpha$$ Also: $$x_\\alpha = F^{-1}(\\alpha)$$\n",
+ "\n",
+ "Median: $x_{\\sfrac{1}{2}}$ \n",
+ "$$F(x_{\\sfrac{1}{2}}) = 0{,}5$$ $$x_{\\sfrac{1}{2}} = F^{-1}(0{,}5)$$\n",
+ "\n",
+ "<img src=\"./figures/09/Normalverteilung.png\" alt=\"image\" />"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8e74c511-b730-4781-a751-e1895983cc4c",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Mehrdimensionale Wahrscheinlichkeitsdichten \n",
+ "\n",
+ "Beispiel: Es werden zwei Größen auf einmal gemessen mit Zufallsvektor:\n",
+ "$x,y$. \n",
+ "Ereignis A: $x$ innerhalb $[x, x + dx]$, y beliebig \n",
+ "Ereignis B: $y$ innerhalb $[y, y + dy]$, x beliebig\n",
+ "$$P(A \\cap B) = \\text{W. für $x$ in $[x, x + dx]$ und $y$ in $[y, y + dy]$} = f(x, y)\\,dxdy$$\n",
+ "\n",
+ "\n",
+ "Randverteilung: $$f_x(x) = \\int_{-\\infty}^\\infty f(x,y)\\,dy$$\n",
+ "$$f_y(y) = \\int_{-\\infty}^\\infty f(x,y)\\,dx$$\n",
+ "\n",
+ "<img src=\"./figures/09/W_top.png\" alt=\"image\" />"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7624d391-9fd5-4567-bf84-632eead1bb20",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Randverteilungen \n",
+ "\n",
+ "<img src=\"./figures/09/W.png\" style=\"width:47.0%\"\n",
+ "alt=\"image\" />\n",
+ "<img src=\"./figures/09/top.png\" style=\"width:47.0%\"\n",
+ "alt=\"image\" />\n",
+ "\n",
+ "\n",
+ "Bedingte Verteilung: $$g(x|y) = \\frac{f(x,y)}{f_y(y)}$$\n",
+ "$$h(y|x) = \\frac{f(x,y)}{f_x(x)}$$\n",
+ "\n",
+ "<img src=\"./figures/09/top_cond.png\" style=\"width:96.0%\"\n",
+ "alt=\"image\" />"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ba8fae9b-636b-4b7e-a36d-3be582c000f5",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Satz von Bayes \n",
+ "\n",
+ "$g(x|y) = \\frac{f(x,y)}{f_y(y)}$ und\n",
+ "$h(y|x) = \\frac{f(x,y)}{f_x(x)}$\n",
+ "\n",
+ "Satz: $$g(x|y) = \\frac{h(y|x) f_x(x)}{f_y(y)}$$\n",
+ "\n",
+ "Mit $f(x,y) = h(y|x) f_x(x) = g(x|y) f_y(y)$:\n",
+ "$$f_x(x) = \\int_{-\\infty}^\\infty g(x|y) f_y(y)\\,dy$$\n",
+ "$$f_y(y) = \\int_{-\\infty}^\\infty h(y|x) f_x(x)\\,dy$$\n",
+ "\n",
+ "<img src=\"./figures/08/bayes.gif\" style=\"width:80.0%\" />\n",
+ "höchstwahrscheinlich nicht Bayes"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fdd11743-1e7b-48b6-9e9b-41bee7d4f23b",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Funktionen von Zufallsvariablen\n",
+ "\n",
+ "### Funktionen von Zufallsvariablen \n",
+ "\n",
+ "Sei $x$ eine Zufallsvariable, $f(x)$ ihre Wahrscheinlichkeitsdichte und\n",
+ "$a(x)$ eine stetige Funktion: \n",
+ "\n",
+ "Was ist die Wahrscheinlichkeitsdichte $g(a)$? gleiche Wahrscheinlichkeit\n",
+ "für $x$ in $[x, x+dx]$ und $a$ in $[a, a+da]$:\n",
+ "$$g(a) da = \\int_{dS} f(x)\\,dx$$ Wenn die Umkehrfunktion $x(a)$\n",
+ "existiert:\n",
+ "$$g(a) da = \\left| \\int_{x(a)}^{x(a +da)} f(x^\\prime)\\,dx^\\prime \\right| = \\int_{x(a)}^{x(a) + |\\frac{dx}{da}|da} f(x^\\prime)\\,dx^\\prime$$\n",
+ "oder $$g(a) = f(x(a)) \\left|\\frac{dx}{da}\\right|$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "525847ad-2ddf-468c-883f-35bc853bac8c",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Beispiel: \n",
+ "\n",
+ "Beispiel 1: $a(x) = \\sqrt{x}$, $x(a) = a^2$ Für $x$ gleichverteilt\n",
+ "zwischen 0 und 1, also $u(x) = 1$, ist die Wahrscheinlichkeitsdichte\n",
+ "$g(a)$:\n",
+ "$$g(a) = u(x(a)) \\left|\\frac{dx}{da}\\right| = 1 \\cdot \\left|\\frac{da^2}{da}\\right| = 2a \\text{ (linear verteilt)}$$\n",
+ "\n",
+ "Beispiel 1: $a(x) = F^{-1}(x)$, $x(a) = F(a)$ Für $x$ gleich verteilt\n",
+ "zwischen 0 und 1, also $u(x) = 1$, ist die Wahrscheinlichkeitsdichte\n",
+ "$g(a)$:\n",
+ "$$g(a) = u(x(a)) \\left|\\frac{dx}{da}\\right| = 1 \\cdot \\left|\\frac{dF(a)}{da}\\right| = f(a) \\text{ (qed).}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0c669bab-6c81-45fb-a506-8ef709cd4687",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Funktionen von Zufallsvektoren \n",
+ "\n",
+ "Sei $\\vec x$ ein Zufallsvektor, $f(\\vec x)$ seine\n",
+ "Wahrscheinlichkeitsdichte und $\\vec a(\\vec x)$ eine stetige Funktion: \n",
+ "\n",
+ "Was ist die Wahrscheinlichkeitsdichte $g(\\vec a)$?\n",
+ "$$g(\\vec a) = f(\\vec x) \\left| J \\right| \\text{mit } J = \n",
+ "\\begin{array}{rrrr} \n",
+ "\\frac{\\partial x_1}{\\partial a_1} & \\frac{\\partial x_1}{\\partial a_2} & \\dots & \\frac{\\partial x_1}{\\partial a_m} \\\\[6pt]\n",
+ "\\frac{\\partial x_2}{\\partial a_1} & \\frac{\\partial x_2}{\\partial a_2} & \\dots & \\frac{\\partial x_2}{\\partial a_m} \\\\[6pt]\n",
+ "\\vdots & \\vdots & \\ddots & \\vdots \\\\[6pt]\n",
+ "\\frac{\\partial x_n}{\\partial a_1} & \\frac{\\partial x_n}{\\partial a_2} & \\dots & \\frac{\\partial x_n}{\\partial a_m} \\\\[6pt]\n",
+ "\\end{array}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4b7f6667-c019-4b23-83fd-a1758d748762",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Grundbegriffe\n",
+ "\n",
+ "### Grundbegriffe \n",
+ "\n",
+ "Diskrete Zufallsvariable Mittelwert:\n",
+ "$$<r> = \\bar r = \\sum _{i=1}^N r_i P(r_i)$$\n",
+ "\n",
+ "Kontinuierliche Zufallsvariable Wahrscheinlichkeitsdichte $f(x)$ mit\n",
+ "\n",
+ "- $P(a \\leq x \\leq b) = \\int_a^b f(x)\\,dx$\n",
+ "\n",
+ "- $f(x) \\geq 0$\n",
+ "\n",
+ "- $\\int_{-\\infty}^{\\infty} f(x)\\,dx = 1$\n",
+ "\n",
+ "Mittelwert:\n",
+ "$$<x> = \\bar x = \\int_{-\\infty}^{\\infty} x \\, f(x)\\,dx = \\mu_x$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2907584f-bb17-463e-b084-749f2011bd4c",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Erwartungswerte und Momente\n",
+ "\n",
+ "### Erwartungswert \n",
+ "\n",
+ "Definition Erwartungswert der Funktion $h(x)$ f\"ur die\n",
+ "Wahrscheinlichkeitsdichte $f(x)$:\n",
+ "$$E[h] = \\int_{-\\infty}^{\\infty} h(x) \\, f(x)\\,dx$$\n",
+ "\n",
+ "Spezialfall $h(x) = x$\n",
+ "$$E[x] = \\int_{-\\infty}^{\\infty} x \\, f(x)\\,dx = <x>$$\n",
+ "\n",
+ "Erwartungswert ist ein linearer Operator\n",
+ "$$E[a\\cdot g(x) + b \\cdot h(x)] = a\\cdot E[g(x)] + b\\cdot E[h(x)]$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5cd273e7-3129-454b-a31c-4e91c4316bf4",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Varianz und Standardabweichung \n",
+ "\n",
+ "Varianz $V[x]$\n",
+ "\n",
+ "- ein Maß für die Breite einer Wahrscheinlichkeitsdichte\n",
+ "\n",
+ "- zweites zentrales Moment\n",
+ "\n",
+ "- Definition\n",
+ " $$V[x] = E[(x - \\mu_x)^2] = \\int_{-\\infty}^{\\infty} (x-\\mu_x)^2 \\, f(x)\\,dx$$\n",
+ "\n",
+ "- n\"utzliche Formeln: \n",
+ " $V[x] = E[x^2] - <x>^2$ und \n",
+ " $V[ax] = a^2 V[x]$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "30d01c8b-50a5-4d90-8e0b-89a68f52f9bd",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Varianz und Standardabweichung \n",
+ "\n",
+ "Standardabweichung $\\sigma$\n",
+ "\n",
+ "- ein Maß für die Größe der statistischen Schwankungen der\n",
+ " Zufallsvariablen um den Mittelwert\n",
+ "\n",
+ "- in der Physik oft “der Fehler”\n",
+ "\n",
+ "- Definition $$\\sigma = \\sqrt{V[x]}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ae67472e-f9b2-46fa-9679-15553d31aaaf",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Kovarianz \n",
+ "\n",
+ "Kovarianz $V_{xy}$ für zwei Zufallsvariablen $x$ und $y$:\n",
+ "$$V_{xy} = E[(x - \\mu_x)(y - \\mu_y)] = E[xy] - \\mu_x \\mu_y$$\n",
+ "$$V_{xy} = \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} xy\\, f(x, y)\\,dx \\,dy - \\mu_x\\mu_y$$\n",
+ "\n",
+ "Kovarianz $V_{ab} = \\text{cov}[a, b]$ seien $a$ und $b$ Funktionen des\n",
+ "Zufallsvektors $\\vec x$:\n",
+ "$$\\text{cov}[a, b] = E[(a - \\mu_a)(b - \\mu_b)] = E[ab] - \\mu_a \\mu_b$$\n",
+ "$$\\text{cov}[a, b] = \\int_{-\\infty}^{\\infty} \\dots \\int_{-\\infty}^{\\infty} a(x) b(x)\\, f(\\vec x)\\,dx_1 \\dots \\,dx_n - \\mu_a\\mu_b$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "576b4570-8a06-4224-b867-31459483e6bb",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Kovarianzmatrix \n",
+ "\n",
+ "$$C = \\left( \n",
+ " \\begin{array}{rr} \n",
+ " V_{xx} & V_{xy} \\\\ \n",
+ " V_{yx} & V_{yy}\\\\ \n",
+ " \\end{array} \n",
+ " \\right)$$\n",
+ "\n",
+ "Anmerkungen:\n",
+ "\n",
+ "- auch Fehlermatrix genannt\n",
+ "\n",
+ "- $V_{xy} = V_{yx}$, Matrix symmetrisch\n",
+ "\n",
+ "- $V_{ii} > 0$ Matrix positiv (semi)definit\n",
+ "\n",
+ "- Korrelationsmatrix: $$C^\\prime = \\left( \n",
+ " \\begin{array}{rr} \n",
+ " V_{xx}/V_{xx} & V_{xy}/\\sqrt{V_{xx}V_{yy}} \\\\ \n",
+ " V_{xy}/\\sqrt{V_{xx}V_{yy}} & V_{yy}/V_{yy}\\\\ \n",
+ " \\end{array} \n",
+ " \\right) = \\left( \n",
+ " \\begin{array}{rr} \n",
+ " 1 & \\rho_{xy} \\\\ \n",
+ " \\rho_{xy} & 1\\\\ \n",
+ " \\end{array} \n",
+ " \\right)$$\n",
+ "\n",
+ "- Korrelationskoeffizient:\n",
+ " $\\rho_{xy} = \\frac{V_{xy}}{\\sqrt{V_{xx}V_{yy}}}$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ba08bfd1-4c17-4a1d-bdf6-9c517edffdd8",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Wahrscheinlichkeitsdichten\n",
+ "\n",
+ "## Diskrete Verteilungen\n",
+ "\n",
+ "### Binomialverteilung \n",
+ "\n",
+ "Binomialverteilung Ist $p$ die Wahrscheinlichkeit f\"ur das Auftreten\n",
+ "eines Ereignisses, so ist die Wahrscheinlichkeit, dass es bei $n$\n",
+ "Versuchen $k$-mal auftritt, gegeben durch die Binomialverteilung:\n",
+ "$$P(k) = {n \\choose k} p^k(1-p)^{n-k} \\text{, } k = 0,1,2...n$$\n",
+ "\n",
+ "Erwartungswert und Varianz\n",
+ "$$<k> = E[k] = \\sum \\limits_{k = 0}^{n} k P(k) = np$$\n",
+ "$$V[k] = \\sigma^2 = np(1-p)$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c1dbdd5b-c1f4-40b3-bad6-d6639a9a635c",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Beispiel \n",
+ "\n",
+ "Werfen von fünf Münzen $n = 5$, $p = 0.5$ \n",
+ "\n",
+ "| k | 0 | 1 | 2 | 3 | 4 | 5 |\n",
+ "|:-----|:----:|:----:|:-----:|:-----:|:----:|:----:|\n",
+ "| P(k) | 1/32 | 5/32 | 10/32 | 10/32 | 5/32 | 1/32 |\n",
+ "\n",
+ "<img src=\"./figures/08/binom5.pdf\" style=\"width:75.0%\" />\n",
+ "\n",
+ "### Beispiel II \n",
+ "\n",
+ "Fehler in der Effizienzbestimmung eines Selektionsschittes Es soll die\n",
+ "Effizienz eines Selektionschnittes und ihr Fehler bestimmt werden, wenn\n",
+ "in einer Stichprobe von $n$ Datenpunkten $k$ Punkte diesen Schnitt\n",
+ "überleben. \n",
+ "Die Zufallsvariable ist die gefundene Effizienz $h_k = \\frac{k}{n}$. \n",
+ "Wie groß ist der Fehler? \n",
+ "Die Zahlen $k$ folgen einer Binomialverteilung mit der\n",
+ "Wahrscheinlichkeit $p_k = E[h_k] = E[\\frac{k}{n}]$: $$\\begin{aligned}\n",
+ " \\sigma(h_k) &= &\\sqrt{V[\\frac{k}{n}]} = \\sqrt{\\frac{1}{n^2} V[k]} = \\sqrt{\\frac{1}{n^2}\\cdot np_k(1-p_k)}\\\\\n",
+ " &=& \\sqrt{\\frac{p_k(1-p_k)}{n}}\\\\\n",
+ " \n",
+ "\\end{aligned}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "60849c4a-dad2-4acd-9780-f9560da2ed9b",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Poisson-Verteilung \n",
+ "\n",
+ "Poisson-Verteilung Die Possionverteilung gibt die Wahrscheinlichkeit an,\n",
+ "genau $k$ Ereignisse zu erhalten, wenn die Zahl der Versuche $n$ sehr\n",
+ "groß und die Wahrscheinlichkeit $p$ sehr klein ist. Mit $\\mu = np$\n",
+ "$$P(k) = \\frac{\\mu^ke^{-\\mu}}{k!}$$\n",
+ "\n",
+ "Erwartungswert und Varianz\n",
+ "$$E[k] = \\sum \\limits_{k = 1}^{\\infty} k \\frac{e^{-\\mu}\\mu^k}{k!}\n",
+ " = \\mu \\sum \\limits_{k = 1}^{\\infty} k \\frac{e^{-\\mu}\\mu^{k-1}}{(k-1)! k}\n",
+ " = \\mu \\sum \\limits_{s = 0}^{\\infty} \\frac{e^{-\\mu}\\mu^{s}}{s!} = \\mu$$\n",
+ "$$V[k] = \\sigma^2 = \\mu$$\n",
+ "\n",
+ "### Poisson- und Binomialverteilung \n",
+ "\n",
+ "Binomialverteilung mit $n= 1000$ und $p = 0.01$ \n",
+ "Poisson-Verteilung mit $\\mu = 10$(schraffiert) \n",
+ "\n",
+ "<img src=\"./figures/08//bp.jpg\" style=\"width:85.0%\"\n",
+ "alt=\"image\" />"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7e80723a-ac12-4af1-a43a-70593ef791b8",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Beispiel aus vielen alten Statistikbüchern \n",
+ "\n",
+ "Tod durch Pferdetritte in der preußischen Armee\n",
+ "\n",
+ "In der preußischen Armee wurde f\"ur jedes Jahr und jedes Armeekorps die\n",
+ "Anzahl der Todesfälle durch Huftritte registriert. Für 20 Jahre\n",
+ "(1875–1894) und 14 Armeekorps ergibt sich:\n",
+ "\n",
+ "| Anzahl des Todesf\"alle $k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 |\n",
+ "|:------------------------------------------|----:|----:|----:|----:|----:|----:|----:|\n",
+ "| Zahl der Korps-Jahre mit $k$ Todesf\"allen | 144 | 91 | 32 | 11 | 2 | 0 | 0 |\n",
+ "\n",
+ "<img src=\"./figures/08/poisson70.pdf\" style=\"width:55.0%\" />\n",
+ "\n",
+ "Poisson-Verteilung f\"ur $\\mu = \\frac{196}{280} = 0.70$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b9772cfd-74fb-4a8b-9c0c-fd2c3554a986",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# What is meant with error/uncertainty on a measured quantity?"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "42e65c7a-4636-4319-b21a-acc95140c2de",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "fragment"
+ },
+ "tags": []
+ },
+ "source": [
+ "If we quote $a = 1 \\pm 0.5$, we usually mean that the probability for the *true* value of $a$ is Gaussian $G(a, \\mu, \\sigma)$ distributed with $\\mu = 1$ and $\\sigma = 0.5$ "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4b8a5b72-aa82-4acf-9499-8736ed6246f8",
+ "metadata": {
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# How often can/should the measurement be outside one sigma?"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1f03bf12-f17b-409d-8932-4e3e24023445",
+ "metadata": {
+ "slideshow": {
+ "slide_type": "fragment"
+ },
+ "tags": []
+ },
+ "source": [
+ "Let's use pseudo-experiments/Monte Carlo:\n",
+ "\n",
+ " * generate 10.000 Gaussian distributed measurements\n",
+ " * count how ofter they differ by more than one sigma\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "40541a16-abc8-4f5b-b504-71951aa891f5",
+ "metadata": {
+ "slideshow": {
+ "slide_type": "subslide"
+ },
+ "tags": []
+ },
+ "source": [
+ "Relatively easy with *scipy* and *numpy*:\n",
+ " * use [scipy.stats](https://docs.scipy.org/doc/scipy/reference/stats.html)\n",
+ " * use [scipy.stats.norm](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html) class\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "id": "5e55929e-7028-4ae2-9e05-29812e933733",
+ "metadata": {
+ "slideshow": {
+ "slide_type": "fragment"
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[ 2.04228058 2.0372654 0.16987033 ... 0.95302626 1.32747258\n",
+ " -0.02868072]\n",
+ "[ True True True ... False False True]\n",
+ "fraction outside one sigma: 0.3203\n"
+ ]
+ }
+ ],
+ "source": [
+ "import scipy.stats as stats\n",
+ "import numpy as np\n",
+ "\n",
+ "pseudo_a = stats.norm.rvs(1, 0.5, 10000)\n",
+ "print(pseudo_a)\n",
+ "is_outside = abs(pseudo_a - 1) > 0.5\n",
+ "print(is_outside)\n",
+ "print(\"fraction outside one sigma:\", sum(is_outside)/len(pseudo_a)) "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8abb14b4-80fd-494a-89a0-310bceb277dc",
+ "metadata": {
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Why is it a Gaussian"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "85185cef-6b18-4c03-8040-437f1fd40b9e",
+ "metadata": {
+ "slideshow": {
+ "slide_type": "fragment"
+ },
+ "tags": []
+ },
+ "source": [
+ "Central limit theorem:\n",
+ "\n",
+ "\"let $X_{1},X_{2},\\dots ,X_{n}$ denote a statistical sample of size $n$ from a population with expected value (average) $\\mu$ and finite positive variance $\\sigma ^{2}$, and let $\\bar {X_{n}}$ denote the sample mean (which is itself a random variable). Then the limit as $n\\to \\infty$ of the distribution of $\\frac {({\\bar {X}}_{n}-\\mu )}{\\frac {\\sigma }{\\sqrt {n}}}$, is a normal distribution with mean 0 and variance 1.\""
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ac354d95-cede-4215-8138-b0d7c6ae9a5e",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.11.10"
+ },
+ "rise": {
+ "autolaunch": true,
+ "overlay": "<div class='myheader'><h2>my company</h2></div><div class='myfooter'><h2>the date</h2></div>"
+ },
+ "toc": {
+ "base_numbering": 1
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/lecture_2.ipynb b/lecture_2.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..c87a639d59245872c34e433a461d5118dfa397eb
--- /dev/null
+++ b/lecture_2.ipynb
@@ -0,0 +1,646 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "74976eec-e9a9-46b1-824d-01dad15478bc",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Lecture 1\n",
+ "\n",
+ "---\n",
+ "\n",
+ "## Basic statistics \n",
+ "\n",
+ "<br>\n",
+ "<br>\n",
+ "\n",
+ " Hartmut Stadie\n",
+ "\n",
+ "hartmut.stadie@uni-hamburg.de"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "3e407505-a77f-4abf-9312-bbf63543e206",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b3097b69-91ec-413e-935e-e310c96d5c25",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Wahrscheinlichkeitsdichten\n",
+ "\n",
+ "## Spezielle Wahrscheinlichkeitsdichten\n",
+ "\n",
+ "### Normalverteilung \n",
+ "\n",
+ "Normal- oder Gauß-Verteilung\n",
+ "$$f(x) = \\frac{1}{\\sqrt{2\\pi}\\sigma}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}$$\n",
+ "\n",
+ "Erwartungswert und Varianz $$<x> = E[x] = \\mu$$ $$V[x] = \\sigma^2$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f7050597-71f8-45a4-a4ac-64567d17827a",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Standardisierte Gauß-Verteilung\n",
+ "\n",
+ "oder Normalverteilung: $\\mu = 0$ und $\\sigma = 1$\n",
+ "\n",
+ "<img src=\"./figures/10/gauss_alpha.png\" style=\"width:60.0%\"\n",
+ "alt=\"image\" />\n",
+ "\n",
+ "Wahrscheinlichkeit einiger Intervalle\n",
+ "\n",
+ "| | | |\n",
+ "|:----------------------|---------------------------------:|------------:|\n",
+ "| $|x-\\mu| \\ge \\sigma$ | (x außerhalb $\\pm 1\\sigma$) ist: | $31{,}74$ % |\n",
+ "| $|x-\\mu| \\ge 2\\sigma$ | (x außerhalb $\\pm 2\\sigma$) ist: | $4{,}55$ % |\n",
+ "| $|x-\\mu| \\ge 3\\sigma$ | (x außerhalb $\\pm 3\\sigma$) ist: | $0{,}27$ % |"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b5910862-3a33-46be-91e0-868a3093cd48",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Mehrdimensionale Normalverteilung \n",
+ "\n",
+ "$$f_{X}(\\vec x )= \\frac {1}{\\sqrt {(2\\pi )^{p}\\det({C)}}}\\exp \\left(-{\\frac {1}{2}}(\\vec x- \\vec\\mu)^{\\top}C^{-1}(\\vec x - \\vec \\mu)\\right)$$\n",
+ "\n",
+ "<img src=\"./figures/10/Multivariate_Gaussian.png\"\n",
+ "style=\"width:90.0%\" alt=\"image\" />"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bf784648-04e4-4d6b-a6d0-e85f4210ee1d",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### 2-D Normalverteilung \n",
+ "\n",
+ "$$\\vec\\mu = (\\bar x, \\bar y) \\text{ und } C = \\left( \n",
+ " \\begin{array}{rr} \n",
+ " \\sigma_x ^2 & \\rho \\sigma_x \\sigma_y \\\\ \n",
+ " \\rho \\sigma_x \\sigma_y & \\sigma_y^2\\\\ \n",
+ " \\end{array} \\right)$$\n",
+ "\n",
+ "<img src=\"./figures/10/error_elipse.png\" alt=\"image\" />\n",
+ "\n",
+ "### Poisson- Binomial- und Gauß-Verteilung \n",
+ "\n",
+ "Binomialverteilung mit $n= 1000$ und $p = 0.01$ \n",
+ "Poisson-Verteilung mit $\\mu = 10$(schraffiert) \n",
+ "\n",
+ "<img src=\"./figures/08/bpg.jpg\" style=\"width:85.0%\"\n",
+ "alt=\"image\" />"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a3b105a2-4332-46f7-a423-df435d4b06e8",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Gleichverteilung \n",
+ "\n",
+ "Gleichverteilung Die Wahrscheinlichkeitsdichte f(x) ist konstant\n",
+ "$\\frac{1}{b-a}$ für $a \\leq x \\leq b$ und null außerhalb.\n",
+ "\n",
+ "Erwartungswert und Varianz\n",
+ "$$<x> = E[x] = \\int_a^b \\frac{x}{b-a}\\, dx = \\frac{1}{2(b-a)} [b^2-a^2] = \\frac{a + b}{2}$$\n",
+ "$$\\begin{aligned}\n",
+ " V[x] & = &\\sigma^2 = E[(x-<x>)^2] = E[x^2] - <x>^2 \\\\\n",
+ " & = & \\int_a^b \\frac{x^2}{b-a}\\, dx - <x>^2 = \\frac{b^3 - a^3}{3(b-a)} - \\frac{(a+b)^2}{4}\\\\\n",
+ " & = & \\frac{b^2+ab+a^2}{3}-\\frac{a^2 + 2ab + b^2}{4} = \\frac{(b-a)^2}{12} \\\\\n",
+ " \n",
+ "\\end{aligned}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2acc164e-255b-43f4-93ee-d569d0bb9e50",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Beispiel: Mehrere Streifendetektoren \n",
+ "\n",
+ "<img src=\"./figures/10/cms_strip.jpg\" style=\"width:40.0%\"\n",
+ "alt=\"image\" />\n",
+ "<img src=\"./figures/10/strip.png\" style=\"width:40.0%\"\n",
+ "alt=\"image\" /> \n",
+ "Eine Messung: Signal im Streifen $i$, $p(x)$ Gleichverteilung zwischen\n",
+ "$b_i$ und $a_i$ \n",
+ "\n",
+ "Wie sieht die Verteilung der Kombination mehrerer Ortsmessungen aus?"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e4d5dad2-983e-4f8f-b305-fb0a787282d6",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Zentraler Grenzwertsatz \n",
+ "\n",
+ "Zentraler Grenzwertsatz: Die Wahrscheinlichkeitsdichte der Summe\n",
+ "$\\sum_{i=0}^{n} x_i$ einer Stichprobe aus $n$ unabhängigen\n",
+ "Zufallsvariablen $x_i$ mit einer beliebigen Wahrscheinlichkeitsdichte\n",
+ "mit Mittelwert $<x>$ und Varianz $\\sigma^2$ geht in der Grenze\n",
+ "$n \\to \\infty$ gegen eine Gau\"s-Wahrscheinlichkeitsdichte mit Mittelwert\n",
+ "$\\mu = n \\cdot <x>$ und Varianz $V[w] = n \\cdot \\sigma^2$.\n",
+ "\n",
+ "<embed src=\"./figures/08/Summe-von-Gleichverteilungen3.pdf\"\n",
+ "style=\"width:60.0%\" /> "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "69556eab-a67e-4251-a4b3-0290013aa344",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Schätzer\n",
+ "\n",
+ "## Fehlerrechnung\n",
+ "\n",
+ "### Fehlerrechnung: Beispiel I \n",
+ "\n",
+ "Widerstandsmessung: $U = 24$ V und $I = 0{,}6 \\pm 0{,}1$ A \n",
+ "Annahme: $I$ gaußverteilt $g(I)$ mit $\\mu_I = 0{,}6$ und\n",
+ "$\\sigma_I = 0{,}1$ \n",
+ "Was ist $p(R)$? \n",
+ "$R = U/I$, $I = U/R$ und $|dI/dR| = U/R^2$ \n",
+ "$$\\begin{aligned}\n",
+ " p(R) & = & U/R^2 g(I(R)) = \\frac{U}{R^2 \\sqrt{2\\pi}\\sigma_I}e^{-\\frac{(U/R-\\mu_I)^2}{2\\sigma_I^2}} \\\\\n",
+ " & = & \\frac{U}{R^2 \\sqrt{2\\pi}\\sigma_I}e^{-\\frac{(R-U/\\mu_I)^2}{2\\sigma_I^2*R^2/\\mu_I^2}} \\\\\n",
+ " \n",
+ "\\end{aligned}$$ (show in Jupyter)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "06ff9896-15bc-4404-a789-bbe7557350eb",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Fehlerrechnung: Beispiel II \n",
+ "\n",
+ "Spannungsmessung: $R = 40$ $\\Omega$ und $I = 0{,}6 \\pm 0{,}1$ A \n",
+ "Annahme: $I$ gaußverteilt $g(I)$ mit $\\mu_I = 0{,}6$ und\n",
+ "$\\sigma_I = 0{,}1$ \n",
+ "Was ist $p(U)$? \n",
+ "$U = RI$, $I = U/R$ und $|dI/dU| = 1/R$ \n",
+ "$$\\begin{aligned}\n",
+ " p(U) & = &1/R g(I(R)) = \\frac{1}{R \\sqrt{2\\pi}\\sigma_I}e^{-\\frac{(U/R-\\mu_I)^2}{2\\sigma_I^2}} \\\\\n",
+ " & = & \\frac{1}{\\sqrt{2\\pi}R\\sigma_I}e^{-\\frac{(U-R\\mu_I)^2}{2R^2\\sigma_I^2}} \\\\\n",
+ " & = & g(U) \\text{ mit $\\mu_U = R\\mu_I$ und $\\sigma_U = R\\sigma_I$}\\\\\n",
+ " \n",
+ "\\end{aligned}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0fe93c64-f06b-49ec-addb-22b12cf06ec6",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Fehlerrechung: Allgemein \n",
+ "\n",
+ "Gaußsche Fehlerfortpflanzung: Annahme: $x$ gaußverteilt $g(x)$ mit\n",
+ "$\\mu_x$ und $\\sigma_x$ \n",
+ "Was ist $p(y(x))$? \n",
+ "$y(x) \\approx f(\\mu_x) + \\frac{dy}{dx}\\big|_{x=\\mu_x}(x-\\mu_x)$ \n",
+ "$x - \\mu_x \\approx (y-y(\\mu_x))(\\frac{dy}{dx}\\big|_{\\mu_x})^{-1}$,\n",
+ "$|dx/dy| = |\\frac{dy}{dx}(\\mu_x)|^{-1}$ $$\\begin{aligned}\n",
+ " p(y) & \\approx & |\\frac{dy}{dx}(\\mu_x)|^{-1} g(x(y)) = \\frac{1}{\\frac{dy}{dx}\\big|_{\\mu_x}\\sqrt{2\\pi}\\sigma_x}e^{-\\frac{(x(y)-\\mu_x)^2}{2\\sigma_x^2}} \\\\\n",
+ " & = & \\frac{1}{\\frac{dy}{dx}\\big|_{\\mu_x}\\sqrt{2\\pi}\\sigma_x}e^{-\\frac{(y-y(\\mu_x))^2}{2(\\frac{dy}{dx}\\big|_{\\mu_x})^2\\sigma_x^2}}\\\\\n",
+ " & = & g(y) \\text{ mit $\\mu_y = y(\\mu_x)$ und $\\sigma_y = \\frac{dy}{dx}\\Big|_{\\mu_x}\\sigma_x$}\\\\\n",
+ " \n",
+ "\\end{aligned}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "85f35e43-f5cd-4ccf-acd2-cbb5e963ce00",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Fehlerrechung: Mehrere Dimensionen \n",
+ "\n",
+ "Gaußsche Fehlerfortpflanzung: Annahme: $\\vec x$ gaußverteilt $g(\\vec x)$\n",
+ "mit $\\vec \\mu$ und Kovarianz $C$ \n",
+ "Was ist $p(\\vec y(\\vec x))$? \n",
+ "$y_i(\\vec x) \\approx y_i(\\vec \\mu) + \\frac{dy_i}{dx_j}\\big|_{\\vec \\mu} (x_j- \\mu_j)$;\n",
+ "$$\\begin{aligned}\n",
+ " E([y_i y_j]) \\approx & y_i(\\vec \\mu)y_j(\\vec \\mu) + y_i(\\vec \\mu)\\frac{dy_j}{dx_k}\\big|_{\\vec \\mu} E[x_k - \\mu_k] + y_j(\\vec \\mu)\\frac{dy_i}{dx_k}\\big|_{\\vec \\mu}E[x_k- \\mu_k] \\\\\n",
+ " & + \\frac{dy_i}{dx_k}\\big|_{\\vec \\mu}\\frac{dy_j}{dx_l}\\big|_{\\vec \\mu}E[(x_k- \\mu_k) (x_l - \\mu_l)]\\\\\n",
+ " = & y_i(\\vec \\mu)y_j(\\vec \\mu) + \\frac{dy_i}{dx_k}\\big|_{\\vec \\mu}\\frac{dy_j}{dx_l}\\big|_{\\vec \\mu}C_{kl} \\text{ und }V[y_iy_j] = E[y_iy_j] - E[y_i]E[y_j]\n",
+ " \n",
+ "\\end{aligned}$$ mit $A_{ij} = [ \\frac{dy_i}{dx_j}\\big|_{\\vec \\mu}]$ ist\n",
+ "$\\vec y(\\vec x)$ gaußverteilt mit\n",
+ "$$\\vec \\mu_y = \\vec y(\\vec \\mu_x) \\text{ und Kovarianz } C_{yy} = A C_{xx} A^T$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8507d724-f9f8-4d41-a3b0-3ddaf0846dde",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Stichproben\n",
+ "\n",
+ "### Schätzer: Mittelwert aus Stichprobe\n",
+ "\n",
+ "Beispiel: Schätze $I$ aus drei Messungen der Stromstärke $I_1$, $I_2$,\n",
+ "$I_3$ mit gleichem $\\sigma_I = 0.1$ A.\n",
+ "\n",
+ "Annahme: Messungen gauß-verteilt um I. \n",
+ "Schätzer für $\\mu$: Mittelwert: $$\\hat I = 1/3(I_1+I_2+I_3)$$ Fehler für\n",
+ "unkorrellierte Messungen (keine syst. Fehler): \n",
+ "$$C = \\left( \n",
+ " \\begin{array}{rrr} \n",
+ " \\sigma_I^2 &0 & 0 \\\\ \n",
+ " 0 & \\sigma_I^2 & 0\\\\\n",
+ " 0 & 0 & \\sigma_I^2 \n",
+ " \\end{array} \\right) \\text{ und } A = \\left( \\begin{array}{rrr} \n",
+ " d\\hat I/dI_1 & d\\hat I/dI_2 & d\\hat I/dI_3\n",
+ " \\end{array} \\right)$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6463365e-354d-4e98-8f63-6903e2dbd6e6",
+ "metadata": {
+ "editable": true,
+ "jp-MarkdownHeadingCollapsed": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Beispiel: Fehlerfortpflanzung \n",
+ "\n",
+ "Fehler für unabhängige Messungen (keine syst. Fehler): \n",
+ "$$C = \\left( \n",
+ " \\begin{array}{rrr} \n",
+ " \\sigma_I^2 &0 & 0 \\\\ \n",
+ " 0 & \\sigma_I^2 & 0\\\\\n",
+ " 0 & 0 & \\sigma_I^2 \n",
+ " \\end{array} \\right) \\text{ und } A = \\left( \\begin{array}{rrr} \n",
+ " 1/3 & 1/3 & 1/3\n",
+ " \\end{array} \\right)$$ Fehlerfortpflanzung: $$\\begin{aligned}\n",
+ "C_{\\hat I} & = & ACA^T = \\left( \n",
+ " \\begin{array}{rrr} \n",
+ " 1/3 & 1/3 & 1/3 \n",
+ " \\end{array} \n",
+ "\\right) \n",
+ " \\left( \n",
+ " \\begin{array}{rrr} \n",
+ " \\sigma_I^2 &0 & 0 \\\\ \n",
+ " 0 & \\sigma_I^2 & 0\\\\\n",
+ " 0 & 0 & \\sigma_I^2 \n",
+ " \\end{array} \\right) \n",
+ " \\left( \\begin{array}{r} \n",
+ " 1/3 \\\\ 1/3 \\\\ 1/3 \n",
+ " \\end{array} \n",
+ "\\right)\\\\\n",
+ "& = & \\left( \n",
+ " \\begin{array}{rrr} \n",
+ " \\sigma_I^2/3 & \\sigma_I^2/3 & \\sigma_I^2/3 \n",
+ " \\end{array} \n",
+ "\\right) \n",
+ " \\left( \\begin{array}{r} \n",
+ " 1/3 \\\\ 1/3 \\\\ 1/3 \n",
+ " \\end{array} \n",
+ "\\right) = \\sigma_I^2/9 + \\sigma_I^2/9 + \\sigma_I^2/9 = \\sigma_I^2/3 \\\\\n",
+ "\\hat \\sigma_I & = & \\sigma_I/\\sqrt{3}\n",
+ "\\end{aligned}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a038c48f-7143-480d-b0ba-817f0e5211c1",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Schätzer \n",
+ "\n",
+ "Schätzer $\\hat a$ für $a$ aus Stichprobe $x_1,\\dots x_n$\n",
+ "\n",
+ "Anforderungen:\n",
+ "\n",
+ "- erwartungstreu: \n",
+ " $E[\\hat a]= a$\n",
+ "\n",
+ "- konsistent: \n",
+ " $\\lim_{n\\to \\infty } \\hat a = a$\n",
+ "\n",
+ "- effizient: $V[\\hat a]$ möglichst klein\n",
+ "\n",
+ "Aufgabe: Schätze Mittelwert $\\mu$ und Varianz $\\sigma^2$ einer\n",
+ "Wahrscheinlichkeitsdichtefunktion $p(x)$ aus einer Stichprobe\n",
+ "$x_1,\\dots,x_n$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d32672c8-da0b-468b-bf73-df40aef7245a",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Schätzer für Mittelwert \n",
+ "\n",
+ "Schätzer für den Mittelwert $\\mu$:\n",
+ "$$\\hat \\mu = \\bar x = \\frac{1}{n}\\sum_{i=1}^n x_i$$\n",
+ "\n",
+ "Tests:\n",
+ "\n",
+ "- erwartungstreu:\n",
+ " $$E[\\hat \\mu]= E[ \\frac{1}{n}\\sum_{i=1}^n x_i ] = \\frac{1}{n} \\sum_{i=1}^n E[x_i] = \\frac{1}{n} \\sum_{i=1}^n \\mu = \\mu$$\n",
+ "\n",
+ "- konsistent:\n",
+ " $$\\lim_{n\\to \\infty } \\hat \\mu = \\lim_{n\\to \\infty } \\frac{1}{n}\\sum_{i=1}^n x_i = \\int x p(x)\\,dx = \\mu$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4021d6cd-150e-4a6d-bdc4-ed049fd983d3",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Schätzer für Mittelwert \n",
+ "\n",
+ "Schätzer für den Mittelwert $\\mu$:\n",
+ "$$\\hat \\mu = \\bar x = \\frac{1}{n}\\sum_{i=1}^n x_i$$\n",
+ "\n",
+ "Test:\n",
+ "\n",
+ "- effizient: $V[\\hat a]$ möglichst klein\n",
+ " $$V[\\hat \\mu] = V[ \\frac{1}{n}\\sum_{i=1}^n x_i ] = \\frac{1}{n^2} V[ \\sum_{i=1}^n x_i ] = \\frac{1}{n^2} \\sum_{i=1}^n\\sum_{j=1}^n Cov(x_i, x_j)$$\n",
+ " $$V[\\hat \\mu] = \\frac{1}{n^2} \\sum_{i=1}^n Cov(x_i, x_i)= \\frac{n\\sigma^2}{n^2} = \\frac{\\sigma^2}{n}$$\n",
+ " (oder über zentralen Grenzwertsatz)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4dfd3266-a5a2-4db7-a96b-ef21118bc85e",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Schätzer für Varianz \n",
+ "\n",
+ "Schätzer für die Varianz $\\sigma^2$:\n",
+ "$$\\hat \\sigma^2 = V[x] = \\frac{1}{n}\\sum_{i=1}^n (x_i - <x>)^2$$\n",
+ "\n",
+ "Tests:\n",
+ "\n",
+ "- erwartungstreu:\n",
+ " $$E[\\hat {\\sigma^2}]= E[ \\frac{1}{n}\\sum_{i=1}^n (x_i^2 - \\bar x^2)] = \\frac{1}{n} \\sum_{i=1}^n E[x_i^2-\\bar x^2] = E[x^2] - E[\\bar x^2]$$\n",
+ " $$E[\\hat{\\sigma^2}]= E[x^2] - E[x]^2 + E[x]^2 - E[\\bar x^2] = E[x^2] - E[x]^2 - (E[\\bar x^2] - E[\\bar x]^2)$$\n",
+ " $$E[\\hat {\\sigma^2}] = V(x) - V(\\bar x) = \\sigma^2 - \\frac{\\sigma^2}{n} = \\frac{n-1}{n}\\sigma^2 \\ne \\sigma^2$$\n",
+ "\n",
+ "- konsistent:\n",
+ " $\\lim_{n\\to \\infty } \\hat {\\sigma^2} = \\lim_{n\\to \\infty } \\frac{n-1}{n}\\sigma^2 =\\sigma^2$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1621af0b-8216-4959-9cc9-8c41cdb8cc79",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Stimmt das Modell? \n",
+ "\n",
+ "Der angewandte Schätzer basiert immer auf Annahmen über die analysierte\n",
+ "Stichprobe. \n",
+ "Der Fehler auf den Schätzwert sagt nichts darüber aus, ob die Annahmen\n",
+ "stimmen."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8fdd0fc7-af9c-492c-9c7f-5a195ee58c50",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Nochmal die Stromstärke \n",
+ "\n",
+ "Stromstärke aus zwei Messreihen, $\\sigma_I = 0.1$ A: \n",
+ "Reihe 1: $[0.64441099 0.63895571 0.73984042 0.62145699 0.66971489]$ \n",
+ "Reihe 2: $[0.93179978 0.46326547 0.41350898 0.12281948 0.61426579]$ \n",
+ "Schätzer: $I_1 = 0.66 \\pm 0.04$, $I_2 = 0.51 \\pm \\pm 0.04$ \n",
+ "Passen die Daten zur Erwartung? \n",
+ "\n",
+ "Residuum: $$R_i = \\frac{x_i - \\mu}{\\sigma}$$ ist normal verteilt, wenn\n",
+ "$\\mu$ und $\\sigma$ stimmen.\n",
+ "\n",
+ "(Betrachte Summe der Residuenquadrate $\\sum R_i^2$)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0c34b270-4e04-40be-82b1-f305eadf82a9",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### $\\chi^2$-Verteilung\n",
+ "\n",
+ "0.5\n",
+ "\n",
+ "$$\\chi^2 = \\sum_i \\frac{x_i - \\mu_i}{\\sigma_i}$$ Die $p(\\chi^2, n)$ ist\n",
+ "die Wahrscheinlichkeitsdichte für die Summe der Quadrate von $n$\n",
+ "normalverteilten Zufallszahlen. \n",
+ "Mittelwert: $<\\chi^2> = n$ Zahl der Freiheitsgrade.\n",
+ "\n",
+ "0.5 <embed src=\"./figures/10/chi2.pdf\" />\n",
+ "\n",
+ "$p$-Wert (Wahrscheinlichkeit für größeres $\\chi^2$):\n",
+ "$$p = 1- \\int_0^{\\chi 2} p(\\chi^2, n)$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3dbe4a35-771e-4129-9529-76f3d38fddae",
+ "metadata": {
+ "editable": true,
+ "jp-MarkdownHeadingCollapsed": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Zusammenfassung und Ausblick\n",
+ "\n",
+ "Zusammenfassung\n",
+ "\n",
+ "- Wahrscheinlichkeitsdichten\n",
+ "\n",
+ "- Fehlerrechnung (lineare Näherung)\n",
+ "\n",
+ "- Korrelationen nicht vernachlässigen\n",
+ "\n",
+ "- Schätzer\n",
+ "\n",
+ "- p-Wert\n",
+ "\n",
+ "- Literatur: \n",
+ "\n",
+ " - Glen Cowan, Statistical Data Analysis,\n",
+ " [pdf](https://www.sherrytowers.com/cowan_statistical_data_analysis.pdf)\n",
+ "\n",
+ " - Roger John Barlow, Statistics: A Guide to the Use of Statistical\n",
+ " Methods in the Physical Sciences,\n",
+ " [Skript](https://arxiv.org/pdf/1905.12362.pdf)\n",
+ "\n",
+ " - Volker Blobel, Erich Lohrmann, Statistische und numerische\n",
+ " Methoden der Datenanalyse,\n",
+ " [pdf](https://www.desy.de/~sschmitt/blobel/eBuch.pdf)\n",
+ "\n",
+ "# Bibliography\n",
+ "\n",
+ "Bibliography"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.11.10"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/lecture_3.ipynb b/lecture_3.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..e7074cfac4704735143d129a2102447413a3bdc7
--- /dev/null
+++ b/lecture_3.ipynb
@@ -0,0 +1,1142 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "977b88f5-3cb7-445b-adac-f44be4d69c90",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Wieder einmal Bundesligatore..."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "id": "641ec6cc-6f1a-4399-b652-7e3da333782a",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "subslide"
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "(array([24., 45., 66., 67., 58., 25., 18., 1., 2., 0.]), array([-0.5, 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5]), <BarContainer object of 10 artists>)\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "\n",
+ "summe = np.loadtxt(\"../summe.txt\")\n",
+ "\n",
+ "\n",
+ "freq = plt.hist(summe, bins=10, range=(-0.5,9.5))\n",
+ "plt.xlabel(\"k\")\n",
+ "print(freq)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e6bcf92b-386d-492b-9f09-e29118e93505",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "subslide"
+ },
+ "tags": []
+ },
+ "source": [
+ "18-mal fielen sechs Tore. Wie groß ist der Fehler auf die 18? Wie groß ist das 68,27\\%-Konfidenzintervall für das 6-Tore-Bin?"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8c4a1016-7fc9-4eb6-b515-b51c0d16fed8",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "subslide"
+ },
+ "tags": []
+ },
+ "source": [
+ "naiv: $k=18$, Poisson mit $\\hat \\mu = 18$ und $\\sigma = \\sqrt{\\hat \\mu} = \\sqrt{18}$: $18\\pm 4.12$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b26cd06e-7700-4011-b5d8-6bf56e22489f",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Konfidenz: \n",
+ "$P(x_- \\le x \\le x_+) = \\int_{x_-}^{x^+} P(x) dx$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1903f299-4d16-41b0-bb48-5f17a07a9006",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "subslide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Aber was ist hier $x$? $\\mu$ oder $k$???"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b54b11f9-e897-4d2a-a5df-961c16cdbb3f",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "Konfidenz für $P(k=18) = 1$ (Messung!) \n",
+ "\n",
+ "Suchen Konfidenzintervall in $\\mu$.\n",
+ "\n",
+ "\n",
+ "Also: $P(\\mu_- \\le \\mu \\le\\mu_+) = \\frac{\\int_{\\mu_-}^{\\mu_+} P(k, \\mu) d\\mu}{\\int_{0}^{\\infty} P(k, \\mu) d\\mu}$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "id": "5052bdb2-6e4b-4ea6-b08e-ba22f2a4a3d2",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "subslide"
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(1.000000000000001, 4.6105475479254554e-11)"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ },
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "import scipy\n",
+ "\n",
+ "mus = np.linspace(0,50,1000)\n",
+ "plt.plot(mus, scipy.stats.poisson.pmf(18,mus))\n",
+ "plt.xlabel(r\"$\\mu$\")\n",
+ "scipy.integrate.quad(lambda x: scipy.stats.poisson.pmf(18,x), 0, 100)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "71c0df91-c1e0-4fbf-a94b-fd88f839ef07",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "subslide"
+ },
+ "tags": []
+ },
+ "source": [
+ "Also: $P(\\mu_- \\le \\mu \\le\\mu_+) = \\int_{\\mu_-}^{\\mu_+} P(k, \\mu) d\\mu$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d278b736-1f45-4b6c-8e1a-c851bb618aeb",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "source": [
+ "Test der naiven Antwort:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "7c461ddf-0cb8-49b0-adc6-a7ae6f11212b",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "scipy.integrate.quad(lambda x: scipy.stats.poisson.pmf(18,x), 18-np.sqrt(18), 18+np.sqrt(18))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "88f176b2-deb2-4212-b4ba-d86b7964e545",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Example from Publication:\n",
+ "\"Search for flavor changing neutral currents in decays of top quarks\" (D0)\n",
+ "\n",
+ "Physics Letters B 701 (2011), pp. 313-320\n",
+ "[https://arxiv.org/abs/1103.4574]\n",
+ "\n",
+ "![wrong](./figures/12/ht_bq_1jets_comb_5pc.png \"Example\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5135b8e6-711b-440e-8507-f049d017c137",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "source": [
+ "Hm, ok!\n",
+ "\n",
+ "Nächstes Bin: 1-mal fielen sieben Tore. \n",
+ "\n",
+ "Naives Konfidenzintzervall: $1\\pm 1$\n",
+ "\n",
+ "Test:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e6102fec-1275-439a-b6b4-8cb1ee4960fc",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "mus = np.linspace(0,10,1000)\n",
+ "plt.plot(mus, scipy.stats.poisson.pmf(1,mus))\n",
+ "plt.xlabel(r\"$\\mu$\")\n",
+ "scipy.integrate.quad(lambda x: scipy.stats.poisson.pmf(1,x), 1-np.sqrt(1), 1+np.sqrt(1))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3139be1a-8a4d-4ae9-9fa7-aa99d7efffe6",
+ "metadata": {
+ "editable": true,
+ "jupyter": {
+ "source_hidden": true
+ },
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "source": [
+ "zu klein!\n",
+ "\n",
+ "Suche Intervall: $[\\mu_-, \\mu_+]$\n",
+ "\n",
+ "hier:\n",
+ "$\\mu_- = 0$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "630c83e0-2be4-41dd-a124-2ac672a2c507",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def intervall(mu_plus):\n",
+ " return scipy.integrate.quad(lambda x: scipy.stats.poisson.pmf(1,x), 0, mu_plus)[0]\n",
+ "\n",
+ "mus = np.linspace(0,10,100)\n",
+ "plt.plot(mus, np.vectorize(intervall)(mus))\n",
+ "plt.grid()\n",
+ "plt.xlabel(r\"$\\mu$\")\n",
+ "\n",
+ "scipy.optimize.brentq(lambda x: intervall(x)-0.6827, 0,10)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1c4ad7b4-bbdb-4f9d-a795-0e2dae39a181",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "Korrektes Intervall: $[0; 2{,}36]$\n",
+ "\n",
+ "\n",
+ "## Was ist denn nun das richtige Intervall für das 6-Tore-Bin?\n",
+ "\n",
+ "Welches?\n",
+ "\n",
+ "\n",
+ "z.B. symmetrisch in der Konfidenz:\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "5fffd76e-14da-4854-8af0-5262e0c3c9a3",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "def intervall_minus(mu):\n",
+ " return scipy.integrate.quad(lambda x: scipy.stats.poisson.pmf(18,x), mu, 18)[0]\n",
+ "\n",
+ "def intervall_plus(mu):\n",
+ " return scipy.integrate.quad(lambda x: scipy.stats.poisson.pmf(18,x), 18, mu)[0]\n",
+ "\n",
+ "print(\"naiv:\", 18-np.sqrt(18), 18+np.sqrt(18))\n",
+ "mu_minus = scipy.optimize.brentq(lambda x: intervall_minus(x)-0.6827/2, 0,18)\n",
+ "mu_plus = scipy.optimize.brentq(lambda x: intervall_plus(x)-0.6827/2, 18,40)\n",
+ "print(\"symmetrisch in P:\", mu_minus, mu_plus) \n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "52f8f412-cb94-4c32-95b6-03f178cbcdb6",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "scipy.integrate.quad(lambda x: scipy.stats.poisson.pmf(18,x), mu_minus, mu_plus)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f7ef04e8-1dc9-4e8b-b597-39d277bde591",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Konfidenz-Intervalle: Poisson"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e0a6084d-dd5d-4ffc-8849-a61a8327fb4e",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "def intervall_minus(k, mu):\n",
+ " return scipy.integrate.quad(lambda x: scipy.stats.poisson.pmf(k,x), mu, k)[0]\n",
+ "\n",
+ "def intervall_plus(k, mu):\n",
+ " return scipy.integrate.quad(lambda x: scipy.stats.poisson.pmf(k,x), k, mu)[0]\n",
+ "\n",
+ "def conv_limits(k, c):\n",
+ " c_low = intervall_minus(k, 0)\n",
+ " c_up = c / 2\n",
+ " if c_low < c/2:\n",
+ " mu_minus = 0\n",
+ " else:\n",
+ " c_low = c / 2\n",
+ " mu_minus = scipy.optimize.brentq(lambda x: intervall_minus(k, x)-c_low, 0,k)\n",
+ " c_up = c - c_low\n",
+ " mu_plus = scipy.optimize.brentq(lambda x: intervall_plus(k, x)-c_up, k, 2*k + 20)\n",
+ " return mu_minus, mu_plus\n",
+ "\n",
+ "conv_limits(3, 0.6827)\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "382e5a13-5f5f-44e5-90dd-7272fbdb171a",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "ks = np.arange(0, 20)\n",
+ "\n",
+ "limits = np.zeros((len(ks), 2))\n",
+ "for i,k in enumerate(ks):\n",
+ " limits[i] = conv_limits(k, 0.6827)\n",
+ "#print(limits)\n",
+ "plt.plot(ks, limits[:,0], 'b_')\n",
+ "plt.plot(ks, limits[:,1], 'r_')\n",
+ "plt.plot(ks, ks-np.sqrt(ks), 'b.')\n",
+ "plt.plot(ks, ks+np.sqrt(ks), 'r.')\n",
+ "plt.xlabel(\"$k$\")\n",
+ "plt.ylabel(r\"$\\mu$\")\n",
+ "plt.grid()\n",
+ "\n",
+ "print(limits[2])\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "61d94d51-fc1c-4ddb-9cbc-c416a28158f8",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Konfidenzintervall: Gauß"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "cb5826ae-29df-41c3-ad8f-dde61b5d90ba",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def intervall_minus(x, mu, sigma):\n",
+ " return scipy.integrate.quad(lambda y: scipy.stats.norm.pdf(x,y, sigma), mu, x)[0]\n",
+ "\n",
+ "def intervall_plus(x, mu, sigma):\n",
+ " return scipy.integrate.quad(lambda y: scipy.stats.norm.pdf(x, y, sigma), x, mu)[0]\n",
+ "\n",
+ "def conv_limits(x, c, sigma):\n",
+ " c_low = c / 2\n",
+ " mu_minus = scipy.optimize.brentq(lambda mu: intervall_minus(x, mu, sigma)-c_low, x - 10*sigma,x)\n",
+ " c_up = c - c_low\n",
+ " mu_plus = scipy.optimize.brentq(lambda mu: intervall_plus(x, mu, sigma)-c_up, x, x + 10*sigma)\n",
+ " return mu_minus, mu_plus\n",
+ "\n",
+ "conv_limits(3, 0.6827, 1)\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e87da118-0c11-4345-a656-096de6f980e9",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "xs = np.linspace(0,20,100)\n",
+ "sigma = 1\n",
+ "limits = np.zeros((len(xs), 2))\n",
+ "for i,k in enumerate(xs):\n",
+ " limits[i] = conv_limits(k, 0.6827, sigma)\n",
+ "#print(limits)\n",
+ "plt.plot(xs, limits[:,0], 'b')\n",
+ "plt.plot(xs, limits[:,1], 'r')\n",
+ "plt.plot(xs, xs-sigma, 'b.')\n",
+ "plt.plot(xs, xs+sigma, 'r.')\n",
+ "plt.xlabel(\"$x$\")\n",
+ "plt.ylabel(r\"$\\mu$\")\n",
+ "plt.grid()\n",
+ "\n",
+ "print(limits[2])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "202144ae-1057-41fc-8757-7f6f40c75868",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "print(scipy.integrate.quad(lambda y: scipy.stats.norm.pdf(3, y, 1), 2, 3)[0], scipy.integrate.quad(lambda x: scipy.stats.norm.pdf(x, 3, 1), 2,3)[0])"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "24359dd4-7289-4235-872e-1138d5fe0b0b",
+ "metadata": {},
+ "source": [
+ "Für Gauß $G(x, \\mu, \\sigma) = G(\\mu, x, \\sigma)$: $\\int_{\\mu_-}^{\\mu_+} G(x, \\mu, \\sigma) d\\mu = \\int_{\\mu_-}^{\\mu_+} G(\\mu, x, \\sigma) d\\mu$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "10f482d3-6667-4250-aefa-5df896e46936",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Konfidenzregionen\n",
+ "\n",
+ "Erst einmal Konfidenzen für Gauß-Verteilung zum Intervall $[\\mu - z\\sigma, \\mu - z\\sigma]$:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "580b83dc-3cb0-40df-bc77-8ad48d90fce5",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def conv_gaus(z):\n",
+ " return scipy.stats.norm.cdf(z) - scipy.stats.norm.cdf(-z)\n",
+ "\n",
+ "for z in [1,2,3,4,5]:\n",
+ " print(z, conv_gaus(z))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3904ad68-2b4d-4207-a7e8-35def0372238",
+ "metadata": {},
+ "source": [
+ "$z$ für 68\\%, 90\\%, 95\\% und 99\\%: "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "5a71ee06-db32-4349-a3da-ae0e394fbd74",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "for c in [0.68, 0.90, 0.95, 0.99]:\n",
+ " print(c, scipy.optimize.brentq(lambda z: conv_gaus(z)-c,0, 10))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2d5f1dab-cebd-45fe-934e-36cd06ba2782",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Regionen in zwei oder drei Dimensionen:\n",
+ "\n",
+ "$\\vec x^T = (x_1, x_2, \\dots, x_n)$ \n",
+ "\n",
+ "$x_i$ seien unabhängige Zufallsvariablen und Gauß verteilt:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "73c11f8d-c8e4-4dc7-ad37-4ca4a33c55ea",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def conv_gaus_nd(z, n):\n",
+ " return np.power(scipy.stats.norm.cdf(z) - scipy.stats.norm.cdf(-z),n)\n",
+ "\n",
+ "for z in [1,2,3,4,5]:\n",
+ " print(z, conv_gaus_nd(z, 1), conv_gaus_nd(z, 2), conv_gaus_nd(z, 3))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8dd1036c-edf9-40c4-be16-56025f49134b",
+ "metadata": {},
+ "source": [
+ "$z$ für 68\\%, 90\\%, 95\\% und 99\\%: "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "7f14d38e-1e41-460e-9876-2afb62bed08b",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "for c in [0.68, 0.90, 0.95, 0.99]:\n",
+ " print(c, scipy.optimize.brentq(lambda z: conv_gaus_nd(z, 1)-c,0, 10), scipy.optimize.brentq(lambda z: conv_gaus_nd(z, 2)-c,0, 10), scipy.optimize.brentq(lambda z: conv_gaus_nd(z,3)-c,0, 10))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9aca662e-bdd6-408e-8f28-852614e51577",
+ "metadata": {},
+ "source": [
+ "1,2,3-$\\sigma$-Äquivalente in zwei und drei Dimensionen:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "702daac8-81e8-406c-b70f-c3146c58a493",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "for k in [1, 2, 3]:\n",
+ " c = conv_gaus_nd(k, 1)\n",
+ " print(c, scipy.optimize.brentq(lambda z: conv_gaus_nd(z, 2)-c,0, 10), scipy.optimize.brentq(lambda z: conv_gaus_nd(z,3)-c,0, 10))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bba9c7fc-9e11-4441-90e0-f1d186a1726e",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Hypothesentest\n",
+ "\n",
+ "\n",
+ "Hypothese: \"Die $k_i$ Tore pro Bundesligaspiel $i$ sind Poisson-verteilt mit einem gemeinsamen $\\mu = <k>$.\"\n",
+ "\n",
+ "Benötigt für den Test eine alternative Hypothese: \"Die Tore pro Bundesligaspiel $k_i$ sind Poisson verteilt mit $\\mu_i = ki$.\"\n",
+ "\n",
+ "Fehler 1. und 2. Art\n",
+ "* Fehler 1. Art: Die Hypothese stimmt, wird aber verworfen.\n",
+ " \n",
+ " Irrtumswahrscheinlichkeit: $\\alpha$ (Signifikanzniveau, significance)\n",
+ " \n",
+ " Spezifität: $1-\\alpha$ (Effizienz) \n",
+ "\n",
+ "\n",
+ "* Fehler 2. Art: Die Hypothese wird angenommen, ist aber falsch (falsch positiv).\n",
+ "\n",
+ " Wahrscheinlichkeit für Fehler: $\\beta$\n",
+ " \n",
+ " Trennschärfe, Sensitivität: $1-\\beta$ (power)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4e67fff2-211b-4eef-95e2-44e3b363098c",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Beispiel:\n",
+ "Gauß verteilte Zufallsvariable $x$ ($\\sigma = 1$)\n",
+ "\n",
+ "Hypothese: $\\mu = 0$\n",
+ "\n",
+ "Für welchen Bereich $x_- < x < x_+$ sollte man die Hypothese annehmen für Fehler 1. Art $\\alpha = 5\\%$?"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "fb7d561b-51c8-4857-b0ea-81dec8facbda",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "85e25147-c902-4434-b3dc-908486408c5b",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "x = np.linspace(-5,5,100)\n",
+ "xout1 = np.linspace(-5, -1.9599639845401602)\n",
+ "xout2 = np.linspace(1.9599639845401602, 5)\n",
+ "plt.plot(x, scipy.stats.norm.pdf(x))\n",
+ "plt.fill_between(xout1, scipy.stats.norm.pdf(xout1), np.zeros_like(xout1),color=\"c\")\n",
+ "plt.fill_between(xout2, scipy.stats.norm.pdf(xout2), np.zeros_like(xout2),color=\"c\")\n",
+ "\n",
+ "plt.xlabel(\"$x$\")\n",
+ "\n",
+ "plt.grid()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6cbb0d56-981d-4dfa-9a7a-0d6d4a51a357",
+ "metadata": {},
+ "source": [
+ "Fehler 2. Art:?"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cb79afd6-ba74-448b-9096-05cedeb842a6",
+ "metadata": {},
+ "source": [
+ "Beispiel: wahres $\\mu = 3$:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "603a31e1-1425-4eb6-9b12-c44ba31d06a6",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "x = np.linspace(-5,8,200)\n",
+ "xin = np.linspace(-1.9599639845401602, 1.9599639845401602)\n",
+ "xout1 = np.linspace(-5, -1.9599639845401602)\n",
+ "xout2 = np.linspace(1.9599639845401602, 5)\n",
+ "plt.plot(x, scipy.stats.norm.pdf(x))\n",
+ "plt.fill_between(xout1, scipy.stats.norm.pdf(xout1), np.zeros_like(xout1),color=\"c\")\n",
+ "plt.fill_between(xout2, scipy.stats.norm.pdf(xout2), np.zeros_like(xout2),color=\"c\")\n",
+ "plt.plot(x, scipy.stats.norm.pdf(x, 3))\n",
+ "plt.fill_between(xin, scipy.stats.norm.pdf(xin, 3), np.zeros_like(xin),color=\"orange\")\n",
+ "\n",
+ "plt.xlabel(\"$x$\")\n",
+ "\n",
+ "plt.grid()\n",
+ "print(\"Fehler 2. Art: beta:\", scipy.stats.norm.cdf(1.9599639845401602, 3) - scipy.stats.norm.cdf(-1.9599639845401602, 3))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f6a454b7-fce4-4dca-ad3c-9e220956706b",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Neyman-Pearson Test\n",
+ "\n",
+ "Hypothese: $H$\n",
+ "\n",
+ "Alternative Hypothese: $A$\n",
+ "\n",
+ "Suche Kriterium, das $\\alpha$ und $\\beta$ minimiert. $H$ wird für $x$ im Bereich $V$ verworfen:\n",
+ "\n",
+ "$\\int_{V} P_{H}(x) dx = \\alpha$ (klein)\n",
+ "\n",
+ "$\\int_{V} P_{A}(x) dx = 1 - \\beta$ (groß)\n",
+ "\n",
+ "In $V$ sind die $x$-Werte, für die $\\frac{P_{A}(x)}{P_{H}(x)}$ groß ist.\n",
+ "\n",
+ "Neyman-Pearson-Kriterium: $\\frac{P_{A}(x)}{P_{H}(x)} > c$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "39da5505-d3e1-4be6-a1b5-ecef10c58bce",
+ "metadata": {},
+ "source": []
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b636ff7e-5ba1-4fc5-bb40-fb29b63ca88c",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Endlich: Passt das Modell?\n",
+ "\n",
+ "Hypothese: \"Die $k_i$ Tore pro Bundesligaspiel $i$ sind Poisson-verteilt mit einem gemeinsamen $\\mu = <k>$.\"\n",
+ "\n",
+ "Benötigt für den Test eine alternative Hypothese: \"Die Tore pro Bundesligaspiel $k_i$ sind Poisson verteilt mit $\\mu_i = ki$.\"\n",
+ "\n",
+ "Wie gut unser Modell einer Poissonverteilung für alle Spiele zu den Daten passt, lässt sich durch den Vergleich der log-Likelihood unseres Modells $-lnL(\\mu; \\vec k)$ zur log-Likelihood eines saturierten Modells (je Spiel ein eigener $\\mu$-Parameter mit $\\mu_i = k_i$), also $-ln\\hat L(\\vec k; \\vec k)$, abschätzen.\n",
+ "\n",
+ "$P_{H} (\\vec k) = \\prod_{i} P(k_i,\\mu)$; $P_{A} (\\vec k) = \\prod_{i} P(k_i, k_i)$\n",
+ "\n",
+ "Neyman-Pearson: Likelihoodquotient: $\\frac{P_{A}(x)}{P_{H}(x)} = \\frac{\\hat L(\\vec k; \\vec k)}{L(\\mu; \\vec k)} > c$\n",
+ "\n",
+ "$d = \\ln \\frac{P_{A}(\\vec k)}{P_{H}(\\vec k)} = -\\ln \\frac{P_{H}(\\vec k)}{P_{A}(\\vec k)} = -\\ln L(\\mu; \\vec k)-(-\\ln\\hat L(\\vec k; \\vec k))$\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "75234801-341f-4474-a2bf-93aafe9add77",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "mu = np.mean(summe)\n",
+ "\n",
+ "print(\"mu:\", mu)\n",
+ "\n",
+ "print(\"P(H):\", np.prod(scipy.stats.poisson.pmf(summe,mu)))\n",
+ "print(\"P(A):\", np.prod(scipy.stats.poisson.pmf(summe,summe)))\n",
+ "print(\"-ln P(H):\", -np.sum(scipy.stats.poisson.logpmf(summe,mu)))\n",
+ "print(\"-ln P(A):\", -np.sum(scipy.stats.poisson.logpmf(summe,summe)))\n",
+ "print(\"d:\", -np.sum(scipy.stats.poisson.logpmf(summe,mu)) + np.sum(scipy.stats.poisson.logpmf(summe,summe)))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e840a509-4f5b-48ef-9594-eb1dc0bbdc4a",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "Ist das gut?\n",
+ "\n",
+ "Wie sieht die Verteilung von d aus, wenn das Modell stimmt und $\\mu=<k>$?"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "90e27f87-36fd-4b11-add1-c82a4eeb79dd",
+ "metadata": {
+ "editable": true,
+ "jupyter": {
+ "source_hidden": true
+ },
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "#simuliere 1000 Spielzeiten\n",
+ "muobs = np.mean(summe)\n",
+ "\n",
+ "tore = scipy.stats.poisson.rvs(muobs,size=(1000,306))\n",
+ "\n",
+ "d = np.zeros(len(tore))\n",
+ "mu = np.zeros(len(tore))\n",
+ "for i in range(len(tore)):\n",
+ " mu[i] = np.mean(tore[i,:])\n",
+ " d[i] = np.sum(-scipy.stats.poisson.logpmf(tore[i],mu[i]) + scipy.stats.poisson.logpmf(tore[i],tore[i]))\n",
+ " \n",
+ "plt.hist(mu, bins=50)\n",
+ "muobs = np.mean(summe)\n",
+ "plt.plot([muobs, muobs], [0, 100], linestyle = 'dotted')\n",
+ "plt.grid()\n",
+ "plt.xlabel(\"$\\hat \\mu$\")\n",
+ "plt.show()\n",
+ "dobs = -np.sum(scipy.stats.poisson.logpmf(summe,muobs)) + np.sum(scipy.stats.poisson.logpmf(summe,summe))\n",
+ "plt.hist(d, bins=50)\n",
+ "plt.plot([dobs, dobs], [0, 100], linestyle = 'dotted')\n",
+ "\n",
+ "plt.grid()\n",
+ "plt.xlabel(\"$-\\ln(P(H)/P(A))$\")\n",
+ "plt.show()\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f6db4bd4-de67-48e1-add9-4fdc05dbbcf7",
+ "metadata": {},
+ "source": [
+ "$p$-Wert \n",
+ "\n",
+ "Anteil der erwarteter Werte, die höher als der Daten-Wert sind."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "6647f14e-2282-4080-9139-c332db4af23a",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "print(\"p:\", np.sum(d > dobs)/len(d))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c835358c-1291-431c-b0ab-792a4b940d68",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "## Geht es auch ohne MC?\n",
+ "\n",
+ "Einfacherer Fall: Gauß-Verteilungen:\n",
+ "\n",
+ "$d = -\\ln L(\\mu; \\vec x)-(-\\ln\\hat L(\\vec x; \\vec x) = -\\ln \\prod_{i} \\frac{G(x_i,\\mu, \\sigma)}{G(x_i, x_i, \\sigma)}=-\\ln \\prod_{i} \\frac{\\exp(-\\frac{1}{2}(\\frac{x_i-\\mu}{\\sigma})^2)}{\\exp(-\\frac{1}{2}(\\frac{x_i-x_i}{\\sigma})^2)} = -\\ln \\exp(-\\frac{1}{2}(\\frac{x_i-\\mu}{\\sigma})^2) = \\frac{1}{2} \\sum_i\\left(\\frac{x_i - \\mu}{\\sigma}\\right)^2$\n",
+ "\n",
+ "$\\chi^2 = -2 \\ln\\frac{P(H}{P(A)}$ (Wilks Theorem)\n",
+ "\n",
+ "$-2 \\ln\\frac{P(H}{P(A)}$ sollte gemäß $\\chi^2$-Verteilung verteilt sein, mit n Freiheitsgraden n = Zahl der Messpunkte - Zahl der Modellparameter"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "9d667a3b-6a89-419a-8be4-79cb7de96151",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "plt.hist(2*d, bins=50, density=True)\n",
+ "plt.plot([2*dobs, 2*dobs], [0, 0.02], linestyle = 'dotted')\n",
+ "ds = np.linspace(200, 425, 100)\n",
+ "plt.plot(ds,scipy.stats.chi2.pdf(ds, 305))\n",
+ "\n",
+ "plt.grid()\n",
+ "plt.xlabel(\"$-2\\ln(P(H)/P(A))$\")\n",
+ "plt.show()\n",
+ "\n",
+ "\n",
+ "print(\"p-Wert über Chi2:\", scipy.stats.chi2.sf(2*dobs, 305))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "eb7b32c7-a9a7-44de-aa70-ce9744218a35",
+ "metadata": {},
+ "source": [
+ "Stimmt die Gaußsche Näherung in unserem Fall?"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "cda0957a-414f-4bea-ab33-e61218a4a4f5",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "plt.hist( scipy.stats.chi2.sf(2*d, 305), bins=50)\n",
+ "plt.plot([ scipy.stats.chi2.sf(2*dobs, 305), scipy.stats.chi2.sf(2*dobs, 305)], [0, 200], linestyle = 'dotted')\n",
+ "\n",
+ "plt.grid()\n",
+ "plt.xlabel(\"p\")\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "557559e1-8e4a-412a-9486-a2fb9b97de90",
+ "metadata": {},
+ "source": [
+ "Für große $\\mu$? Handball??"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "472aac93-e2dd-424a-ba0e-51c10d409f8e",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "#simuliere 5000 Spielzeiten\n",
+ "muobs = 35\n",
+ "\n",
+ "tore = scipy.stats.poisson.rvs(muobs,size=(5000,306))\n",
+ "\n",
+ "d = np.zeros(len(tore))\n",
+ "mu = np.zeros(len(tore))\n",
+ "for i in range(len(tore)):\n",
+ " mu[i] = np.mean(tore[i,:])\n",
+ " d[i] = np.sum(-scipy.stats.poisson.logpmf(tore[i],mu[i]) + scipy.stats.poisson.logpmf(tore[i],tore[i]))\n",
+ " \n",
+ "plt.hist(mu, bins=50)\n",
+ "plt.grid()\n",
+ "plt.xlabel(\"$\\hat \\mu$\")\n",
+ "plt.show()\n",
+ "\n",
+ "plt.hist(2 * d, bins=50, density=True)\n",
+ "ds = np.linspace(200, 425, 100)\n",
+ "plt.plot(ds,scipy.stats.chi2.pdf(ds, 305))\n",
+ "plt.grid()\n",
+ "plt.xlabel(\"$-2\\ln(P(H)/P(A))$\")\n",
+ "plt.show()\n",
+ "\n",
+ "plt.hist( scipy.stats.chi2.sf(2*d, 305), bins=50)\n",
+ "plt.grid()\n",
+ "plt.xlabel(\"p\")\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "450edccb-2c13-4400-9613-4a1f3e069ce6",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# $\\chi^2$-Test\n",
+ "\n",
+ "\n",
+ "Haben wir schon die ganze Zeit gemacht...\n",
+ "\n",
+ "Wie sind die Konfidenzintervalle $n$ Freiheitsgrade/Dimensionen?\n",
+ "\n",
+ "$$\\chi^2 = (\\overrightarrow{y\\ } - \\vec{f})^{T} V(\\vec{y}- \\vec{f})$$\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "2fd5c302-732b-4c6b-b224-5b9f79df57cc",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "def conv_chi2_nd(z, n):\n",
+ " return scipy.stats.chi2.cdf(z,n)\n",
+ "\n",
+ "\n",
+ "for z in [1,2,3,4,5]:\n",
+ " print(z, conv_chi2_nd(z, 1), conv_chi2_nd(z, 2), conv_chi2_nd(z, 3))\n",
+ "\n",
+ "\n",
+ "print(\"Kritische chi2-Werte\")\n",
+ "for k in [1, 2, 3]:\n",
+ " c = conv_gaus_nd(k, 1)\n",
+ " print(c, scipy.optimize.brentq(lambda z: conv_chi2_nd(z, 1)-c,0, 30), scipy.optimize.brentq(lambda z: conv_chi2_nd(z, 2)-c,0, 30), scipy.optimize.brentq(lambda z: conv_chi2_nd(z,3)-c,0, 30))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "23626cec-4198-44f4-a1c8-81e36992268f",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "for c in [0.68, 0.90, 0.95, 0.99]:\n",
+ " print(c, scipy.optimize.brentq(lambda z: conv_chi2_nd(z, 1)-c,0, 30), scipy.optimize.brentq(lambda z: conv_chi2_nd(z, 2)-c,0, 30),scipy.optimize.brentq(lambda z: conv_chi2_nd(z,3)-c,0, 30))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "d79940bc-7c64-42c8-a564-898995f51ddc",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.11.10"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/lecture_4.ipynb b/lecture_4.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..c2f9b5c19893176b53d487e803036c536d8d1175
--- /dev/null
+++ b/lecture_4.ipynb
@@ -0,0 +1,591 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "60c7547b-61ea-4709-bb5d-30ad3c94c340",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d2193805-8359-4380-82da-4b1cb5358290",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Lecture 1\n",
+ "\n",
+ "---\n",
+ "\n",
+ "## Basic statistics \n",
+ "\n",
+ "<br>\n",
+ "<br>\n",
+ "\n",
+ " Hartmut Stadie\n",
+ "\n",
+ "hartmut.stadie@uni-hamburg.de"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "668ad170-00ca-4d67-ba06-71aebd7092a3",
+ "metadata": {
+ "editable": true,
+ "jp-MarkdownHeadingCollapsed": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Parameterschätzung\n",
+ "\n",
+ "## Einführung\n",
+ "\n",
+ "### Schätzer \n",
+ "\n",
+ "Schätzer $\\hat a$ für $a$ aus Stichprobe $x_1,\\dots x_n$\n",
+ "\n",
+ "Anforderungen:\n",
+ "\n",
+ "- erwartungstreu: \n",
+ " $E[\\hat a]= a$\n",
+ "\n",
+ "- konsistent: \n",
+ " $\\lim_{n\\to \\infty } \\hat a = a$\n",
+ "\n",
+ "- effizient: $V[\\hat a]$ möglichst klein\n",
+ "\n",
+ "Schätzer für den Mittelwert:\n",
+ "$$\\hat \\mu = \\bar x = \\frac{1}{n}\\sum_1^n x_i \\text{ mit } V[\\hat \\mu] = \\frac{\\sigma_x^2}{N}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7c7c6ecd-1611-4787-8985-da0881e90d9f",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Methode der kleinsten Quadrate\n",
+ "\n",
+ "## Herleitung\n",
+ "\n",
+ "### Methode der kleinsten Quadrate \n",
+ "\n",
+ "$y(x) = mx + a$: Finde $\\hat m$ und $\\hat a$!"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "73e6e46a-4e41-4e66-857b-2946946498d1",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "skip"
+ },
+ "tags": []
+ },
+ "source": [
+ "<img src=\"./figures/11/line.png\" style=\"width:90.0%\" alt=\"image\" />"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "55bf0cad-8a50-4831-8a37-daf9bcb39b71",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "#hideme\n",
+ "import numpy as np\n",
+ "import scipy.stats as stats\n",
+ "import matplotlib.pyplot as plt \n",
+ "\n",
+ "def f(x):\n",
+ " return 2*x + 1\n",
+ "\n",
+ "n = 10\n",
+ "xs = np.linspace(0,4,n)\n",
+ "sigma_y=0.4\n",
+ "ys = stats.multivariate_normal.rvs(f(xs), np.eye(n)*sigma_y**2, 1, random_state=42)\n",
+ " \n",
+ "x_axis = np.linspace(0,4,100)\n",
+ "plt.errorbar(xs,ys,yerr=sigma_y,fmt=\".\")\n",
+ "plt.plot(x_axis, f(x_axis),'--')\n",
+ "plt.xlabel(\"x\")\n",
+ "plt.ylabel(\"y\")\n",
+ "plt.savefig(\"line.png\")\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "64e67452-e6bd-442c-a174-e12bdb18dba0",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "source": [
+ "### Methode der kleinsten Quadrate \n",
+ "\n",
+ "$$\\chi^2 = \\sum_i \\left(\\frac{y_i - \\hat y(x)}{\\sigma_i}\\right)^2$$\n",
+ "quantifiziert die Übereinstimmung von Modell zu Daten \n",
+ "$\\rightarrow$ $\\hat m$ und $\\hat a$ sollten $\\chi^2$ minimieren.\n",
+ "<img src=\"./figures/11/line.png\" style=\"width:80.0%\"\n",
+ "alt=\"image\" />"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bdc0949d-791e-4e29-9adc-169578e62821",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Methode der kleinsten Quadrate II \n",
+ "\n",
+ "Minimiere\n",
+ "$\\chi^2 = \\sum_i \\left(\\frac{y_i - \\hat y(x)}{\\sigma_i}\\right)^2 = \\sum_i \\frac{(y_i - m x_i - a)^2}{\\sigma_i^2}$:\n",
+ "\n",
+ "Erste Ableitung ist Null: \n",
+ "\n",
+ "$$\\begin{aligned}\n",
+ " \\frac{d\\chi^2}{dm} &=& -2\\sum_i x_i\\frac {y_i -\\hat m x_i - \\hat a}{\\sigma_i^2} = 0\\\\\n",
+ " \\frac{d\\chi^2}{da} &=& -2\\sum_i \\frac{y_i - \\hat m x_i - \\hat a}{\\sigma_i^2} = 0 \\\\\n",
+ " \\sum_i\\frac{x_iy_i}{\\sigma_i^2} - \\hat m \\sum_i\\frac{x_i^2}{\\sigma_i^2}- \\hat a \\sum_i \\frac{x_i}{\\sigma_i^2} &=& 0 \\\\\n",
+ " \\sum_i\\frac{y_i}{\\sigma_i^2} - \\hat m \\sum_i\\frac{x_i}{\\sigma_i^2}- \\hat a \\sum_i \\frac{1}{\\sigma_i^2} &=& 0 \n",
+ "\\end{aligned}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "322c67fc-ddd1-4830-a5f5-6f118ef54c4c",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Methode der kleinsten Quadrate III \n",
+ "\n",
+ "Minimiere\n",
+ "$\\chi^2 = \\sum_i \\left(\\frac{y_i - \\hat y(x)}{\\sigma_i}\\right)^2 = \\sum_i \\frac{(y_i - m x_i - a)^2}{\\sigma_i^2}$: \n",
+ "$$\\begin{aligned}\n",
+ " \\sum_i\\frac{x_iy_i}{\\sigma_i^2} - \\hat m \\sum_i\\frac{x_i^2}{\\sigma_i^2}- \\hat a \\sum_i \\frac{x_i}{\\sigma_i^2} &=& 0 \\\\\n",
+ " \\sum_i\\frac{y_i}{\\sigma_i^2} - \\hat m \\sum_i\\frac{x_i}{\\sigma_i^2}- \\hat a \\sum_i \\frac{1}{\\sigma_i^2} &=& 0 \n",
+ "\\end{aligned}$$ mit\n",
+ "$\\frac{1}{\\sum_i 1/\\sigma_i^2} \\sum_i \\frac{f}{\\sigma_i^2} = \\langle f \\rangle$: \n",
+ "$$\\begin{aligned}\n",
+ " \\langle xy \\rangle -\\langle x^2 \\rangle \\hat m& - \\langle x \\rangle \\hat a&= 0\\\\\n",
+ " \\langle y \\rangle - \\langle x \\rangle \\hat m& - \\hat a& = 0 \n",
+ "\\end{aligned}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fb25687c-4410-4281-b540-39369732fb26",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Methode der kleinsten Quadrate IV \n",
+ "\n",
+ "$$\\begin{aligned}\n",
+ " \\hat m&=&\\frac{\\langle xy \\rangle - \\langle y \\rangle\\langle x \\rangle}{\\langle x^2 \\rangle - \\langle x \\rangle^2} = \\frac{1}{\\sum_i 1/\\sigma_i^2} \\sum_i \\frac{x_i - \\langle x \\rangle}{\\sigma_i^2(\\langle x^2 \\rangle - \\langle x \\rangle^2)}y_i\\\\\n",
+ " \\hat a &=& \\frac{ \\langle y \\rangle \\langle x^2 \\rangle- \\langle y \\rangle \\langle x \\rangle^2- \\langle x \\rangle \\langle xy \\rangle+ \\langle y \\rangle \\langle x \\rangle^2}{ \\langle x^2 \\rangle- \\langle x \\rangle^2}\\\\\n",
+ " &=& \\frac{ \\langle y \\rangle \\langle x^2 \\rangle - \\langle x \\rangle \\langle xy \\rangle}{ \\langle x^2 \\rangle - \\langle x \\rangle^2} = \\frac{1}{\\sum_i 1/\\sigma_i^2} \\sum_i \\frac{\\langle x^2 \\rangle - \\langle x \\rangle x_i}{\\sigma_i^2(\\langle x^2 \\rangle - \\langle x \\rangle^2)}y_i\n",
+ "\\end{aligned}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b08ade05-e5eb-4a7e-9315-31a45c51aadb",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "source": [
+ "## Fehler\n",
+ "\n",
+ "### Fehler \n",
+ "\n",
+ "$$\\begin{aligned}\n",
+ "V(\\hat m) = \\sum_i \\left(\\frac{d\\hat m}{y_i}\\sigma_i\\right)^2\\text{; }\\frac{d\\hat m}{y_i} & = & \\frac{1}{\\sum_i 1/\\sigma_i^2} \\frac{x_i - \\langle x \\rangle}{\\sigma_i^2(\\langle x^2 \\rangle - \\langle x \\rangle^2)} \\\\\n",
+ "V(\\hat a) = \\sum_i \\left(\\frac{d\\hat a}{y_i}\\sigma_i\\right)^2\\text{; }\\frac{d\\hat a}{y_i} & = & \\frac{1}{\\sum_i 1/\\sigma_i^2} \\frac{\\langle x^2 \\rangle - \\langle x \\rangle x_i}{\\sigma_i^2(\\langle x^2 \\rangle - \\langle x \\rangle^2)}\n",
+ "\\end{aligned}$$ $$\\begin{aligned}\n",
+ "V(\\hat m) &=& \\left(\\frac{1}{\\sum_i 1/\\sigma_i^2}\\right)^2 \\sum_i \\left(\\frac{x_i - \\langle x \\rangle}{\\sigma_i^2(\\langle x^2 \\rangle - \\langle x \\rangle^2)}\\right)^2 \\sigma_i^2 \\\\\n",
+ "&=& \\frac{1}{\\sum_i 1/\\sigma_i^2} \\frac{\\langle x^2 \\rangle - 2\\langle x \\rangle \\langle x \\rangle + \\langle x \\rangle^2}{(\\langle x^2 \\rangle - \\langle x \\rangle^2)^2} \n",
+ "= \\frac{1}{\\sum_i 1/\\sigma_i^2} \\frac{1}{\\langle x^2 \\rangle - \\langle x \\rangle^2} \\\\\n",
+ "V(\\hat a) &=& \\frac{1}{\\sum_i 1/\\sigma_i^2} \\frac{\\langle x^2 \\rangle^2 - 2\\langle x^2 \\rangle\\langle x \\rangle^2 + \\langle x^2 \\rangle\\langle x \\rangle^2}{(\\langle x^2 \\rangle - \\langle x \\rangle^2)^2}\n",
+ "= \\frac{1}{\\sum_i 1/\\sigma_i^2} \\frac{\\langle x^2 \\rangle}{\\langle x^2 \\rangle - \\langle x \\rangle^2}\n",
+ "\\end{aligned}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d8828d04-0af8-4dc3-a846-24acbc9a0f8e",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "source": [
+ "### Korrelation \n",
+ "\n",
+ "$$\\begin{aligned}\n",
+ "V(\\hat m) &=& \\frac{1}{\\sum_i 1/\\sigma_i^2} \\frac{1}{\\langle x^2 \\rangle - \\langle x \\rangle^2} \\\\\n",
+ "V(\\hat a) &=& \\frac{1}{\\sum_i 1/\\sigma_i^2} \\frac{\\langle x^2 \\rangle}{\\langle x^2 \\rangle - \\langle x \\rangle^2}\\\\\n",
+ "\\text{cov}(\\hat m, \\hat a) &=&= \\frac{1}{\\sum_i 1/\\sigma_i^2} \\frac{\\langle (x-\\langle x \\rangle)(\\langle x^2 \\rangle - \\langle x \\rangle x)\\rangle}{(\\langle x^2 \\rangle - \\langle x \\rangle^2)^2}\\\\\n",
+ "&=& \\frac{1}{\\sum_i 1/\\sigma_i^2} \\frac{\\langle x^2 \\rangle \\langle x \\rangle - \\langle x \\rangle \\langle x^2 \\rangle - \\langle x \\rangle \\langle x^2 \\rangle + \\langle x \\rangle^2\\langle x \\rangle}{(\\langle x^2 \\rangle - \\langle x \\rangle^2)^2}\\\\\n",
+ "&=& - \\frac{1}{\\sum_i 1/\\sigma_i^2} \\frac{\\langle x \\rangle}{\\langle x^2 \\rangle - \\langle x \\rangle^2}\n",
+ "\\end{aligned}$$\n",
+ "\n",
+ "### Beispiel in Jupyter "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "424bdd1f-53bf-422b-bc4a-0702231b976d",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "source": [
+ "### Minimales $\\chi^2$ \n",
+ "\n",
+ "$$\\begin{aligned}\n",
+ " \\chi^2 &=& \\sum_i \\frac{(y_i - \\hat m x_i - \\hat a)^2}{\\sigma_i^2} = \\sum_i \\frac{\\left[y_i - \\frac{\\langle xy \\rangle - \\langle y \\rangle\\langle x \\rangle}{\\langle x^2 \\rangle - \\langle x \\rangle^2} x_i - \\frac{ \\langle y \\rangle \\langle x^2 \\rangle - \\langle x \\rangle \\langle xy \\rangle}{ \\langle x^2 \\rangle - \\langle x \\rangle^2} \\right]^2}{\\sigma_i^2}\\\\\n",
+ " & = & \\sum_i \\frac{\\left[(\\langle x^2 \\rangle - \\langle x \\rangle^2)y_i - (\\langle xy \\rangle - \\langle y \\rangle\\langle x \\rangle)x_i - \\langle y \\rangle \\langle x^2 \\rangle + \\langle x \\rangle \\langle xy \\rangle\\right]^2}{\\sigma_i^2 ( \\langle x^2 \\rangle - \\langle x \\rangle^2)^2} \\\\\n",
+ " &=& \\dots\\\\\n",
+ "& =& (\\sum_i \\frac{1}{\\sigma_i^2}) V(y) ( 1- \\rho^2_{xy})\n",
+ "\\end{aligned}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e89f415e-8836-4b97-893c-a7335c3a21e5",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "source": [
+ "### Beispiel in Jupyter \n",
+ "\n",
+ "## In Python"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7311c0ff-0ce0-4427-a50e-b6698100e454",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "source": [
+ "### Mit Python I\n",
+ "\n",
+ "Mit scipy.optimize:\n",
+ "\n",
+ "```\n",
+ "import scipy.optimize as opti\n",
+ "def fitf(x, m , a):\n",
+ " return m*x + a\n",
+ "pfit, Vfit = opti.curve_fit(fitf , xs, ys, \n",
+ " sigma=[sigma_y]*len(ys),absolue_sigma=True)\n",
+ "print(pfit, Vfit)\n",
+ "```\n",
+ "\n",
+ "Vorsicht! Falsche Unsicherheit ohne `absolute_sigma=True`"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a5fec52e-bd3a-4437-bb43-620de44939b2",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "source": [
+ "### Mit Python II\n",
+ "\n",
+ "Mit scipy.optimize:\n",
+ "\n",
+ "```\n",
+ "def chi2(x, y, sy, a, m):\n",
+ " my = m * x + a\n",
+ " r = (y - my)/sy\n",
+ " return np.sum(r**2)\n",
+ " \n",
+ "res = opti.minimize( lambda p: chi2(xs, ys, sigma_y, p[1], p[0]),x0=np.zeros(2))\n",
+ "print(res.x, res.hess_inv*2)\n",
+ "```"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "84b00301-c0d3-4595-858e-4f784d69c876",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Inverse Hesse-Matrix und $\\chi^2$ \n",
+ "\n",
+ "$\\Delta \\chi2$ und Kovarianz Ellipse um Minimum gemäß Kovarianzmatrix\n",
+ "genau bei $\\Delta \\chi^2 = 1$. \n",
+ "$$1 = \\delta \\chi^2 = (\\vec a -\\hat \\vec a)^T V^{-1} (\\vec a-\\hat \\vec a)$$\n",
+ "Mit\n",
+ "$\\chi^2(\\vec a) = \\chi^2(\\hat \\vec a) + (\\vec a -\\hat \\vec a)^T V^{-1} (\\vec a-\\hat \\vec a)$\n",
+ "und\n",
+ "$H_{ij} = \\frac{\\partial^2 \\chi^2(\\vec a)}{\\partial a_i \\partial a_j}$ \n",
+ "$$H_{ij} = \\frac{\\partial^2 (a_k -\\hat a_k) V^{-1}_{kl} (a_l -\\hat a_l)}{\\partial a_i \\partial a_j} = \\frac{\\partial( \\delta_{ik}V^{-1}_{kl} (a_l -\\hat a_l) + (a_k -\\hat a_k) V^{-1}_{kl} \\delta_{il})}{\\partial a_j}$$\n",
+ "$$H_{ij} = \\delta_{ik}V^{-1}_{kl}\\delta_{lj} + \\delta_{jk}V^{-1}_{kl}\\delta_{il} = 2V^{-1}_{ij} \\text{ und } V_{ij} = 2 * H^{-1}_{ij}$$\n",
+ "\n",
+ "Vorsicht! Manche Algorithmen in `minimize` berechnen keine inverse\n",
+ "Hesse-Matrix."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "632583d4-1e97-498b-b8f3-91e259baa24d",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Maximum-Likelihood \n",
+ "\n",
+ "Maximum-Likelihood (ML) Daten: $x_1,...,x_N$ \n",
+ "Wahrscheinlichkeit der Daten für Modell mit Parametern $a$:\n",
+ "$$P(x_1,...,x_N; a) = \\prod_i P(x_i ; a)$$\n",
+ "\n",
+ "Likelihoodfunktion: $$L(a) = \\prod_i P(x_i ; a)$$\n",
+ "\n",
+ "ML-Schätzer $\\hat a$: Maximum von $L(a)$:\n",
+ "$$\\left.\\frac{dL}{da}\\right|_{a = \\hat a} = 0$$ (praktischer:\n",
+ "Log-Likelihood: $-\\ln L = \\sum_i -\\ln P(x_i; a)$)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "dd0afbf7-8504-46f8-b3da-4fb33487a635",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "### Beispiel\n",
+ "\n",
+ "\n",
+ "$y(x) = mx + a$: Finde $\\hat m$ und $\\hat a$ Daten: $y_1,...,y_N$ und\n",
+ "Modell: $$P(y_i; m, a) = G(y_i; \\mu = m x_i + a, \\sigma=\\sigma_i)$$\n",
+ "$$L(m, a) = \\prod_i G(y_i; \\mu = m x_i + a, \\sigma=\\sigma_i)$$\n",
+ "\n",
+ "0.5 <img src=\"./figures/11/line.png\" alt=\"image\" />\n",
+ "\n",
+ "<img src=\"./figures/11/like_a.png\" style=\"width:49.0%\"\n",
+ "alt=\"image\" />\n",
+ "<img src=\"./figures/11/loglike_a.png\" style=\"width:49.0%\"\n",
+ "alt=\"image\" />\n",
+ "\n",
+ "ML-Schätzer für Poisson $\\mu$ $$\\begin{aligned}\n",
+ " L(\\mu) & = & \\prod_i^N P(k_i; \\mu) = \\prod_i^N \\frac{\\mu^{k_i}e^{-\\mu}}{k_i!}\\\\\n",
+ " \\ln L(\\mu) & = & \\sum_{i=1}^N \\left( \\ln \\mu^{k_i} + \\ln e^{-\\mu} - \\ln k_i!\\right)\\\\\n",
+ " & = & \\sum_{i=1}^N \\left( k_i \\ln \\mu -\\mu - \\ln k_i!\\right)\\\\\n",
+ " 0 \\stackrel{!}{=} \\frac{d \\ln L(\\mu)}{d\\mu} \\Big|_{\\hat \\mu}& = & \\sum_{i=1}^N \\left( \\frac{k_i}{\\hat \\mu} - 1\\right) = \\sum_{i=1}^N \\frac{k_i}{\\hat\\mu} - N\\\\\n",
+ " N & = & \\frac{1}{\\hat\\mu} \\sum_{i=1}^N k_i \\rightarrow \\hat\\mu = \\frac{1} {N} \\sum_{i=1}^N k_i\n",
+ " \n",
+ "\\end{aligned}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d9571970-772e-4e95-8474-82e0afb1dd68",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": ""
+ },
+ "tags": []
+ },
+ "source": [
+ "Varianz des ML-Schätzers\n",
+ "\n",
+ "Rao-Cramér-Frechet-Ungleichung: Schätzer $\\hat a$ mit Bias (Verzerrung)\n",
+ "$b$\n",
+ "$$V(\\hat a) \\geq \\frac{\\left(1+ \\frac{\\partial b}{\\partial a} \\right)^2}{E\\left[-\\frac{\\partial^2 \\ln L}{\\partial a^2}\\right]}$$\n",
+ "Fisher-Information:\n",
+ "$$I(\\hat a) = E\\left [-\\frac{\\partial^2 \\ln L}{\\partial a^2}\\right]$$\n",
+ "\n",
+ "ML-Schätzer für Poisson $V(\\hat \\mu)$ $$\\begin{aligned}\n",
+ "V(\\hat \\mu) & \\geq &\\frac{\\left(1+ \\frac{\\partial b}{\\partial \\mu} \\right)^2}{E\\left[-\\frac{\\partial^2 \\ln L}{\\mu^2}\\right]} \\\\\n",
+ " & = & \\frac{1}{E\\left[-\\frac{\\partial(\\sum_{i=1}^N \\frac{k_i}{\\mu} - N)}{\\partial \\mu^2}\\right]} \\\\\n",
+ " & = & \\frac{1}{E\\left[-\\sum_{i=1}^N \\frac{-k_i}{\\hat \\mu^2}\\right]} = \\frac{1}{E\\left[\\sum_{i=1}^N \\frac{k_i}{\\hat \\mu^2}\\right]} \\\\\n",
+ " & = & \\frac{1}{\\frac{1}{\\hat \\mu^2}E\\left[\\sum_{i=1}^N k_i \\right]} = \\frac{1}{\\frac{1}{\\hat \\mu^2}E\\left[N \\hat \\mu \\right]}\\\\\n",
+ " & = & \\frac{\\hat \\mu}{N}\n",
+ "\\end{aligned}$$\n",
+ "\n",
+ "Varianz für mehrere Parameter $\\vec \\theta$\n",
+ "\n",
+ "Für effizienten und erwartungstreuen Schätzer:\n",
+ "$$\\left(V^{-1}\\right)_{ij} = E\\left[ -\\frac{\\partial^2 \\ln L(\\theta)}{\\partial \\theta_i \\partial \\theta_j}\\right]$$\n",
+ "\n",
+ "Näherung für große Datensätze:\n",
+ "$$\\left(\\hat V^{-1}\\right)_{ij} = -\\frac{\\partial^2 \\ln L(\\theta)}{\\partial \\theta_i \\partial \\theta_j}\\Big|_{\\theta=\\hat \\theta} =$$\n",
+ "\n",
+ "Graphisch:\n",
+ "$$\\ln L(\\theta) \\approx \\ln L(\\hat \\theta) + \\frac{\\partial \\ln L}{\\partial \\theta}\\Big|_{\\hat \\theta}(\\theta - \\hat \\theta) + \\frac{1}{2} \\frac{\\partial^2 \\ln L}{\\partial \\theta^2}(\\theta - \\hat \\theta)^2$$\n",
+ "$$\\ln L(\\hat \\theta + \\sigma_\\theta) \\approx \\ln L(\\hat \\theta) + \\frac{1}{2} \\frac{\\partial^2 \\ln L}{\\partial \\theta^2}(\\sigma_\\theta)^2 = \\ln L(\\hat \\theta) - \\frac{1}{2}$$\n",
+ "\n",
+ "Zusammenhang ML und $\\chi^2$\n",
+ "\n",
+ "Likelihood-Quotient:\n",
+ "$$\\lambda = -2 \\ln \\frac{L(\\hat \\theta)}{L(\\hat \\theta^\\prime_\\text{saturiert})}$$\n",
+ "\n",
+ "Mit Normalverteilung: $$\\begin{aligned}\n",
+ "\\lambda &=& -2 \\ln \\frac{L(\\hat \\theta)}{L(\\hat \\theta^\\prime_\\text{saturiert})} = -2 \\ln \\frac{\\prod_i G(x_i; \\hat \\mu, \\sigma_i)}{\\prod_i G(x_i; x_i, \\sigma_i)}\\\\\n",
+ "& = & -2 \\ln \\frac{\\frac{1}{\\sqrt{2\\pi}\\sigma_i}exp\\left(\\frac{(x_i-\\hat \\mu)^2}{2\\sigma_i^2}\\right)}{\\frac{1}{\\sqrt{2\\pi}\\sigma_i}exp\\left(\\frac{(x_i-x_i)^2}{2\\sigma_i^2}\\right)} = -2\\ln exp\\left(\\frac{(x_i-\\hat \\mu)^2}{2\\sigma_i^2}\\right) \\\\\n",
+ "& = & -2 \\frac{(x_i-\\hat \\mu)^2}{2\\sigma_i^2} = \\chi^2 \\text{; also } \\ln L(\\theta) = - \\chi^2(\\theta) / 2 \n",
+ "\\end{aligned}$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7ae85d99-d6fa-409b-abb2-144cc24570db",
+ "metadata": {
+ "editable": true,
+ "slideshow": {
+ "slide_type": "slide"
+ },
+ "tags": []
+ },
+ "source": [
+ "# Zusammenfassung und Ausblick\n",
+ "\n",
+ "## Zusammenfassung und Ausblick\n",
+ "\n",
+ "Zusammenfassung\n",
+ "\n",
+ "- Methode der kleinsten Quadrate ($\\chi^2$)\n",
+ "\n",
+ "- Maximum-Likelihood\n",
+ "\n",
+ "- Zusammenhang $\\chi^2$-ML\n",
+ "\n",
+ "- Minimierung\n",
+ "\n",
+ "- Literatur: \n",
+ "\n",
+ " - Glen Cowan, Statistical Data Analysis,\n",
+ " [pdf](https://www.sherrytowers.com/cowan_statistical_data_analysis.pdf)\n",
+ "\n",
+ " - Roger John Barlow, Statistics: A Guide to the Use of Statistical\n",
+ " Methods in the Physical Sciences,\n",
+ " [Skript](https://arxiv.org/pdf/1905.12362.pdf)\n",
+ "\n",
+ " - Volker Blobel, Erich Lohrmann, Statistische und numerische\n",
+ " Methoden der Datenanalyse,\n",
+ " [pdf](https://www.desy.de/~sschmitt/blobel/eBuch.pdf)\n",
+ "\n",
+ "# Bibliography\n",
+ "\n",
+ "Bibliography"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.11.10"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/rise.css b/rise.css
new file mode 100644
index 0000000000000000000000000000000000000000..2f4475493dc1e689adda1897017a16ae4c0fdf81
--- /dev/null
+++ b/rise.css
@@ -0,0 +1,73 @@
+/* this goes with my global default setting
+ * that has rise use reveal's theme sky
+ */
+/*.reveal {
+ font-family: "Quicksand", sans-serif;
+}*/
+
+.reveal h1, .reveal h2, .reveal h3, .reveal h4, .reveal h5, .reveal h6 {
+ text-transform: initial; /* sky.css says uppercase */
+ letter-spacing: initial ; /* sky.css says -0.08em */
+}
+
+body.rise-enabled .reveal ol, body.rise-enabled .reveal dl, body.rise-enabled .reveal ul {
+ margin-left: 0.1em;
+ margin-top: 0.2em;
+}
+
+.reveal .rendered_html h1:first-child,
+.reveal .rendered_html h2:first-child,
+.reveal .rendered_html h3:first-child,
+.reveal .rendered_html h4:first-child,
+.reveal .rendered_html h5:first-child {
+ margin-top: 0.2em;
+}
+
+h1.plan, h2.plan, h3.plan {
+ text-align: center;
+ padding-bottom: 30px;
+}
+
+ul.plan>li>span.plan-bold {
+ font-size: 110%;
+ padding: 4px;
+ font-weight: bold;
+ background-color: #eee;
+}
+
+ul.plan>li>ul.subplan>li>span.plan-bold {
+ font-weight: bold;
+}
+
+.plan-strike {
+ opacity: 0.4;
+/* text-decoration: line-through; */
+}
+
+div.plan-container {
+ display: grid;
+ grid-template-columns: 50% 50%;
+}
+
+/* something big and obvious again just to outline
+ that this file is actually loaded */
+
+div.cell.code_cell.rendered, div.input_area {
+ border-width: 10px;
+}
+
+/* this is only to check that rise.css properly gets
+ * ignored when quitting reveal mode */
+div.text_cell_render.rendered_html {
+ color: #5050b0;
+}
+
+/*
+ * this is to void xarray's html output to show the fallback textual representation
+ * see also
+ * xarray.md and
+ * https://github.com/damianavila/RISE/issues/594
+ */
+.reveal pre.xr-text-repr-fallback {
+ display: none;
+}
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