mirror of
https://gitlab.rlp.net/pgp/pgp1-python-einfuehrung
synced 2024-11-16 13:48:11 +00:00
10789 lines
1,010 KiB
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10789 lines
1,010 KiB
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "5064e2e2",
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"metadata": {},
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"source": [
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"# Fortgeschrittenes Beispiel\n",
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"In diesem Abschnitt wollen wir uns mit einem komplexeren Beispiel beschäftigen, um weitere Methoden von `iminuit` kennzulernen.\n",
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"Hierzu betrachten wir ein Zählexperiment, z.B. ein Teilchendetektor, bei dem ein Energiespektrum aufgenommen wird. Für jedes Energieintervall (Bin) wird die Anzahl der registrierten Ereignisse bestimmt. Hierbei können wir annehmen, dass die Verteilung der gemessenen Anzahl durch eine Poisson-Verteilung beschrieben wird. Dann entspricht der Fehler in jedem Bin gerade $\\sqrt n$. \n",
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"Dieses Spektrum soll aus zwei gauß-förmigen Peaks über einem exponentiellen Untergrund bestehen und wird mit Hilfe eines Zufallszahlengenerator \"erzeugt\"."
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]
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},
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{
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"cell_type": "markdown",
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"id": "100a4fe4-a5c4-4be3-a7f7-13337b97a194",
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"metadata": {},
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"source": [
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"Nun wollen wir die Messdaten mit Hilfe von `iminuit` fitten. Hierzu müssen wir zunächste zwei Module des packages importieren und eine Funktion für die Entladekurve des Kondensators definieren:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "520f4973",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Diese Zelle nur auf JupyterHub des ZDV ausführen um `iminuit` zu installieren!\n",
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"# import sys\n",
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"# import subprocess\n",
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"# subprocess.check_call([\n",
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"# sys.executable, \n",
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"# '-m',\n",
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"# 'pip',\n",
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"# 'install',\n",
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"# '--proxy',\n",
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"# 'http://webproxy.zdv.uni-mainz.de:3128',\n",
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"# 'iminuit'\n",
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"# ])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "2ffe340b-cd0f-45ec-b5b8-42e7a0349d4c",
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"metadata": {},
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"outputs": [],
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"source": [
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"from iminuit import Minuit, cost\n",
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"import matplotlib.pyplot as plt\n",
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"import numpy as np"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "143a2a23-0a62-439f-9d28-9f555ae85589",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"Text(0, 0.5, 'Number of counts per bin')"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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},
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{
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"data": {
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"image/png": "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"text/plain": [
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"<Figure size 640x480 with 1 Axes>"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"rnd_bkd = np.random.exponential(39.7, 5000)\n",
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"rnd_bkd += 40\n",
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"\n",
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"peak1 = np.random.normal(53.3, 2.1, 5000)\n",
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"peak2 = np.random.normal(60.5, 2.78, 12000)\n",
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"data = np.concatenate([rnd_bkd, peak1, peak2])\n",
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"\n",
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"entries, edges = np.histogram(data, bins=120, range=(40, 80))\n",
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"center = edges[:-1] + np.diff(edges)/2\n",
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"\n",
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"plt.errorbar(center, entries, np.sqrt(entries), ls='', marker='.')\n",
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"plt.xlabel('Energy [keV]')\n",
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"plt.ylabel('Number of counts per bin')"
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]
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},
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{
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"cell_type": "markdown",
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"id": "b582615c-9251-409d-bcfc-d19fd579e161",
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"metadata": {},
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"source": [
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"Zunächst wollen wir das Fitmodel in der Form\n",
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"\n",
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"$$f(x) = A_1 \\cdot \\exp \\bigg\\{\\frac{-(x - \\mu_1)^2}{2 \\cdot \\sigma_1^2}\\bigg\\} + A_2 \\cdot \\exp \\bigg\\{\\frac{-(x - \\mu_2)^2}{2 \\cdot \\sigma_2^2}\\bigg\\} + A_3 \\exp\\{-x/\\tau\\}$$\n",
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"\n",
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"definieren. Hier lohnt es sich, erst Funktionen für die einzelnen Komponenten zu definieren und dann das Gesamtmodel. Hierdurch lassen sich später die einzelnen Komponenten besser darstellen."
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]
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},
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{
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|
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"cell_type": "code",
|
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"execution_count": 4,
|
||
|
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"id": "f84d7527-c0d2-475d-966d-5363a8e09369",
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"metadata": {},
|
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"outputs": [],
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"source": [
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|
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"def peak(x, A, mu, sigma):\n",
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" return A*np.exp(-(x-mu)**2/(2*sigma**2))\n",
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"\n",
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"def bkg(x, A, tau):\n",
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" return A*np.exp(-x/tau)\n",
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"\n",
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"def fit_model(x, A_p1, A_p2, mu_p1, mu_p2, sigma_p1, sigma_p2, A_bkg, tau_bkg):\n",
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|
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" return peak(x, A_p1, mu_p1, sigma_p1) + peak(x, A_p2, mu_p2, sigma_p2) + bkg(x, A_bkg, tau_bkg)"
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]
|
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},
|
||
|
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{
|
||
|
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"cell_type": "markdown",
|
||
|
|
"id": "32014861-316c-4692-9d52-48f2fb71321c",
|
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|
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"metadata": {},
|
||
|
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"source": [
|
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|
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"Nun wollen wir wieder die Kostenfunktion und die Minimierungsfunktion definieren. Startwerte können wir anhand unseres Plots ablesen, lediglich $\\tau$ lässt sich auf diese Weise nicht gut bestimmen."
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]
|
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},
|
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|
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{
|
||
|
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"cell_type": "code",
|
||
|
|
"execution_count": 5,
|
||
|
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"id": "a31901cf-a0ce-4db8-a072-a661fbbb7296",
|
||
|
|
"metadata": {},
|
||
|
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"outputs": [],
|
||
|
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"source": [
|
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|
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"ls = cost.LeastSquares(center, entries, np.sqrt(entries), fit_model)\n",
|
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"\n",
|
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|
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"mi = Minuit(ls, \n",
|
||
|
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" A_p1 = 400, \n",
|
||
|
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" A_p2 = 700,\n",
|
||
|
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" mu_p1 = 54,\n",
|
||
|
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" mu_p2 = 60,\n",
|
||
|
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" sigma_p1 = 2,\n",
|
||
|
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" sigma_p2 = 2,\n",
|
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|
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" A_bkg = 100,\n",
|
||
|
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" tau_bkg = 10, \n",
|
||
|
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" )\n",
|
||
|
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"mi.limits['tau_bkg'] = (0, None)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 6,
|
||
|
|
"id": "1e69a046-770f-4c38-9b91-0176bb0686a1",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"<matplotlib.legend.Legend at 0x1a589e4f940>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"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.errorbar(center, entries, np.sqrt(entries), ls='', marker='.')\n",
|
||
|
|
"plt.xlabel('Energy [keV]')\n",
|
||
|
|
"plt.ylabel('Number of counts per bin')\n",
|
||
|
|
"\n",
|
||
|
|
"x = np.arange(40, 80, 0.1)\n",
|
||
|
|
"plt.plot(x, fit_model(x, *mi.values), color='k', label='Initial guess')\n",
|
||
|
|
"plt.legend()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "89f755f4-b780-43a6-a923-49662c4c701a",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Unsere Startparameter sind bereits nicht schlecht, aber weichen noch stark von den Daten ab. Bei komplexeren Daten und Fitmodellen lohnt es sich, den Fit schrittweise durchzuführen. Bevor wir uns den beiden Peaks widmen, welche uns eigentlich interessieren, sollten wir versuchen, den Untergrund etwas besser zu beschreiben. Um den Untergrund besser fitten zu können, sollten wir erst den Fitbereich auf einen Energiebereich limitieren, in welchem der Untergrund dominiert. Dem Plot können wir entnehmen, dass dies für alle Werte unterhalb von 45 keV und oberhalb von 70 keV der Fall ist. Im Allgemeinen können wir Wertebereiche in Python mit Hilfe von „Masken“ selektieren. Eine Maske lässt sich wie folgt erstellen:"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 505,
|
||
|
|
"id": "d53e8386-ea7f-43fa-b4fe-65229308a2ec",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": [
|
||
|
|
"mask_outside_of_peaks = (center < 45) | (center >= 70)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "84cef7a6-13a0-4ba8-ac40-eb86a54411dc",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Die Maske hat hierbei die Selbe länge wie unseren Daten…"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 506,
|
||
|
|
"id": "d1d06116-d726-4163-b414-6ccde6a19027",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"(120, 120)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 506,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"len(mask_outside_of_peaks), len(mask_outside_of_peaks)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "80db0ae0-5cbd-4db9-b184-610d77bf1c58",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"… und beinhaltet Wahrheitswerte `True` und `False`, bzw. 1 und 0, mit welchen wir unsere Daten selektieren können:"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 507,
|
||
|
|
"id": "f24d19d8-3483-45b5-aee9-1d3f8755da22",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"(array([ True, True, True, True, True, True, True, True, True,\n",
|
||
|
|
" True, True, True, True, True, True, False, False, False,\n",
|
||
|
|
" False, False, False, False, False, False, False, False, False,\n",
|
||
|
|
" False, False, False, False, False, False, False, False, False,\n",
|
||
|
|
" False, False, False, False, False, False, False, False, False,\n",
|
||
|
|
" False, False, False, False, False, False, False, False, False,\n",
|
||
|
|
" False, False, False, False, False, False, False, False, False,\n",
|
||
|
|
" False, False, False, False, False, False, False, False, False,\n",
|
||
|
|
" False, False, False, False, False, False, False, False, False,\n",
|
||
|
|
" False, False, False, False, False, False, False, False, False,\n",
|
||
|
|
" True, True, True, True, True, True, True, True, True,\n",
|
||
|
|
" True, True, True, True, True, True, True, True, True,\n",
|
||
|
|
" True, True, True, True, True, True, True, True, True,\n",
|
||
|
|
" True, True, True]),\n",
|
||
|
|
" array([40.16666667, 40.5 , 40.83333333, 41.16666667, 41.5 ,\n",
|
||
|
|
" 41.83333333, 42.16666667, 42.5 , 42.83333333, 43.16666667,\n",
|
||
|
|
" 43.5 , 43.83333333, 44.16666667, 44.5 , 44.83333333,\n",
|
||
|
|
" 70.16666667, 70.5 , 70.83333333, 71.16666667, 71.5 ,\n",
|
||
|
|
" 71.83333333, 72.16666667, 72.5 , 72.83333333, 73.16666667,\n",
|
||
|
|
" 73.5 , 73.83333333, 74.16666667, 74.5 , 74.83333333,\n",
|
||
|
|
" 75.16666667, 75.5 , 75.83333333, 76.16666667, 76.5 ,\n",
|
||
|
|
" 76.83333333, 77.16666667, 77.5 , 77.83333333, 78.16666667,\n",
|
||
|
|
" 78.5 , 78.83333333, 79.16666667, 79.5 , 79.83333333]))"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 507,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"mask_outside_of_peaks, center[mask_outside_of_peaks]"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "5b5c07e7-1865-48f2-bd9e-0540661fd71e",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Unsere Selektion können wir an unsere Kostenfunktion direkt übergeben."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 508,
|
||
|
|
"id": "3034bb22-0b96-498d-9736-ed9bb2189460",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": [
|
||
|
|
"ls.mask = (center < 45) | (center >= 70)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "77a664fd-513e-4c89-ba52-945b6f68512f",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Nun können wir nochmal unsere Funktion und Messwerte für den ausgewählten Bereich plotten…"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 509,
|
||
|
|
"id": "81232354-a7b8-4e2a-9ac0-159ce0a03da4",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"<matplotlib.legend.Legend at 0x268b8259630>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 509,
|
||
|
|
"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.errorbar(center[ls.mask], entries[ls.mask], np.sqrt(entries[ls.mask]), ls='', marker='.', label='Not masked')\n",
|
||
|
|
"plt.errorbar(center[~ls.mask], entries[~ls.mask], np.sqrt(entries[~ls.mask]), ls='', marker='.', label='Masked')\n",
|
||
|
|
"plt.xlabel('Energy [keV]')\n",
|
||
|
|
"plt.ylabel('Number of counts per bin')\n",
|
||
|
|
"\n",
|
||
|
|
"x = np.arange(40, 80, 0.1)\n",
|
||
|
|
"plt.plot(x, fit_model(x, *mi.values), color='k', label='Initial guess')\n",
|
||
|
|
"plt.legend()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "ec675b22",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Außerdem müssen wir noch alle Fitparameter, welche nicht zum Untergrund beitragen, als konstant festhalten"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 510,
|
||
|
|
"id": "4a93a1c2-17df-46c2-b38e-9a509fe16fc7",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stdout",
|
||
|
|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"<FixedView A_p1=True A_p2=True mu_p1=True mu_p2=True sigma_p1=True sigma_p2=True A_bkg=False tau_bkg=False>\n"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"mi.fixed[:] = True\n",
|
||
|
|
"mi.fixed[['tau_bkg', 'A_bkg']] = False\n",
|
||
|
|
"print (mi.fixed)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "c5a8d247-5b71-42ae-9706-d16192374686",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"bevor wir die Minmierung starten und das Resultat darstellen."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 511,
|
||
|
|
"id": "3e90c2ed-c282-47c2-b0fe-3063f9545639",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/html": [
|
||
|
|
"<table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th colspan=\"2\" style=\"text-align:center\" title=\"Minimizer\"> Migrad </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Minimum value of function\"> FCN = 32.8 (χ²/ndof = 0.8) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total number of function and (optional) gradient evaluations\"> Nfcn = 98 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Estimated distance to minimum and goal\"> EDM = 4.43e-05 (Goal: 0.0002) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total run time of algorithms\"> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Valid Minimum </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below EDM threshold (goal x 10) </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> No parameters at limit </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below call limit </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Hesse ok </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Covariance accurate </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th title=\"Variable name\"> Name </th>\n",
|
||
|
|
" <th title=\"Value of parameter\"> Value </th>\n",
|
||
|
|
" <th title=\"Hesse error\"> Hesse Error </th>\n",
|
||
|
|
" <th title=\"Minos lower error\"> Minos Error- </th>\n",
|
||
|
|
" <th title=\"Minos upper error\"> Minos Error+ </th>\n",
|
||
|
|
" <th title=\"Lower limit of the parameter\"> Limit- </th>\n",
|
||
|
|
" <th title=\"Upper limit of the parameter\"> Limit+ </th>\n",
|
||
|
|
" <th title=\"Is the parameter fixed in the fit\"> Fixed </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 0 </th>\n",
|
||
|
|
" <td> A_p1 </td>\n",
|
||
|
|
" <td> 400 </td>\n",
|
||
|
|
" <td> 4 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 1 </th>\n",
|
||
|
|
" <td> A_p2 </td>\n",
|
||
|
|
" <td> 700 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 2 </th>\n",
|
||
|
|
" <td> mu_p1 </td>\n",
|
||
|
|
" <td> 54.0 </td>\n",
|
||
|
|
" <td> 0.5 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 3 </th>\n",
|
||
|
|
" <td> mu_p2 </td>\n",
|
||
|
|
" <td> 60.0 </td>\n",
|
||
|
|
" <td> 0.6 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 4 </th>\n",
|
||
|
|
" <td> sigma_p1 </td>\n",
|
||
|
|
" <td> 2.00 </td>\n",
|
||
|
|
" <td> 0.02 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 5 </th>\n",
|
||
|
|
" <td> sigma_p2 </td>\n",
|
||
|
|
" <td> 2.00 </td>\n",
|
||
|
|
" <td> 0.02 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 6 </th>\n",
|
||
|
|
" <td> A_bkg </td>\n",
|
||
|
|
" <td> 137 </td>\n",
|
||
|
|
" <td> 15 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 7 </th>\n",
|
||
|
|
" <td> tau_bkg </td>\n",
|
||
|
|
" <td> 34.9 </td>\n",
|
||
|
|
" <td> 2.3 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <th> A_bkg </th>\n",
|
||
|
|
" <th> tau_bkg </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_bkg </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 229 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(125,125,250);color:black\"> -33 <strong>(-0.962)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> tau_bkg </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(125,125,250);color:black\"> -33 <strong>(-0.962)</strong> </td>\n",
|
||
|
|
" <td> 5.18 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
|
||
|
|
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
|
||
|
|
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
|
||
|
|
"<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"360pt\" height=\"288pt\" viewBox=\"0 0 360 288\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
|
||
|
|
" <metadata>\n",
|
||
|
|
" <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
|
||
|
|
" <cc:Work>\n",
|
||
|
|
" <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
|
||
|
|
" <dc:date>2024-10-16T21:40:49.875185</dc:date>\n",
|
||
|
|
" <dc:format>image/svg+xml</dc:format>\n",
|
||
|
|
" <dc:creator>\n",
|
||
|
|
" <cc:Agent>\n",
|
||
|
|
" <dc:title>Matplotlib v3.8.2, https://matplotlib.org/</dc:title>\n",
|
||
|
|
" </cc:Agent>\n",
|
||
|
|
" </dc:creator>\n",
|
||
|
|
" </cc:Work>\n",
|
||
|
|
" </rdf:RDF>\n",
|
||
|
|
" </metadata>\n",
|
||
|
|
" <defs>\n",
|
||
|
|
" <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
|
||
|
|
" </defs>\n",
|
||
|
|
" <g id=\"figure_1\">\n",
|
||
|
|
" <g id=\"patch_1\">\n",
|
||
|
|
" <path d=\"M 0 288 \n",
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|
"L 360 288 \n",
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|
"L 360 0 \n",
|
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|
"L 0 0 \n",
|
||
|
|
"z\n",
|
||
|
|
"\" style=\"fill: #ffffff\"/>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"axes_1\">\n",
|
||
|
|
" <g id=\"patch_2\">\n",
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|
" <path d=\"M 29.08774 268.321635 \n",
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|
"L 356.99976 268.321635 \n",
|
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|
|
"L 356.99976 3.00024 \n",
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|
"L 29.08774 3.00024 \n",
|
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|
|
"z\n",
|
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|
|
"\" style=\"fill: #ffffff\"/>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"matplotlib.axis_1\">\n",
|
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|
|
" <g id=\"xtick_1\">\n",
|
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" <g id=\"line2d_1\">\n",
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" <defs>\n",
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" <path id=\"m9404e9db51\" d=\"M 0 0 \n",
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|
|
"L 0 3.5 \n",
|
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|
|
"\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
|
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|
|
" </defs>\n",
|
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|
|
" <g>\n",
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|
|
" <use xlink:href=\"#m9404e9db51\" x=\"42.740303\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
|
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|
|
" </g>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"text_1\">\n",
|
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|
|
" <!-- 40 -->\n",
|
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|
|
" <g transform=\"translate(36.377803 282.920073) scale(0.1 -0.1)\">\n",
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|
|
" <defs>\n",
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|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
|
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|
||
|
|
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|
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|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
"Q 1056 1369 1301 889 \n",
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|
|
"Q 1547 409 2034 409 \n",
|
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|
|
"Q 2525 409 2770 889 \n",
|
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|
|
"Q 3016 1369 3016 2328 \n",
|
||
|
|
"Q 3016 3291 2770 3770 \n",
|
||
|
|
"Q 2525 4250 2034 4250 \n",
|
||
|
|
"z\n",
|
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|
|
"M 2034 4750 \n",
|
||
|
|
"Q 2819 4750 3233 4129 \n",
|
||
|
|
"Q 3647 3509 3647 2328 \n",
|
||
|
|
"Q 3647 1150 3233 529 \n",
|
||
|
|
"Q 2819 -91 2034 -91 \n",
|
||
|
|
"Q 1250 -91 836 529 \n",
|
||
|
|
"Q 422 1150 422 2328 \n",
|
||
|
|
"Q 422 3509 836 4129 \n",
|
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|
|
"Q 1250 4750 2034 4750 \n",
|
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|
"z\n",
|
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|
"\" transform=\"scale(0.015625)\"/>\n",
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" </defs>\n",
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|
" </g>\n",
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|
" </g>\n",
|
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|
" </g>\n",
|
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|
" <g id=\"xtick_2\">\n",
|
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|
|
" <g id=\"line2d_2\">\n",
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" <g>\n",
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" <use xlink:href=\"#m9404e9db51\" x=\"80.316165\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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|
" </g>\n",
|
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|
|
" </g>\n",
|
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|
|
" <g id=\"text_2\">\n",
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|
" <!-- 45 -->\n",
|
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|
|
"L 1269 4134 \n",
|
||
|
|
"L 1269 2991 \n",
|
||
|
|
"Q 1406 3038 1543 3061 \n",
|
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|
|
"Q 1681 3084 1819 3084 \n",
|
||
|
|
"Q 2600 3084 3056 2656 \n",
|
||
|
|
"Q 3513 2228 3513 1497 \n",
|
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|
|
"Q 3513 744 3044 326 \n",
|
||
|
|
"Q 2575 -91 1722 -91 \n",
|
||
|
|
"Q 1428 -91 1123 -41 \n",
|
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|
|
"Q 819 9 494 109 \n",
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|
"L 494 744 \n",
|
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|
|
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|
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|
|
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|
||
|
|
"Q 2250 441 2565 725 \n",
|
||
|
|
"Q 2881 1009 2881 1497 \n",
|
||
|
|
"Q 2881 1984 2565 2268 \n",
|
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|
|
"Q 2250 2553 1709 2553 \n",
|
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|
|
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|
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|
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" </g>\n",
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" </g>\n",
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" <g id=\"xtick_3\">\n",
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" <g>\n",
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" <use xlink:href=\"#m9404e9db51\" x=\"117.892027\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </g>\n",
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" </g>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"xtick_4\">\n",
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" <g id=\"line2d_4\">\n",
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" <g>\n",
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" <use xlink:href=\"#m9404e9db51\" x=\"155.467888\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"line2d_5\">\n",
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" <g>\n",
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" <g id=\"text_5\">\n",
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" <!-- 60 -->\n",
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" <defs>\n",
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" <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
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"Q 1688 2584 1439 2293 \n",
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"Q 1191 2003 1191 1497 \n",
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"Q 1191 994 1439 701 \n",
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|
|
"Q 1688 409 2113 409 \n",
|
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|
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|
|
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|
"z\n",
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"M 3366 4563 \n",
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|
|
"L 3366 3988 \n",
|
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|
|
"Q 3128 4100 2886 4159 \n",
|
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|
|
"Q 2644 4219 2406 4219 \n",
|
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"text/plain": [
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"┌─────────────────────────────────────────────────────────────────────────┐\n",
|
||
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|
"│ Migrad │\n",
|
||
|
|
"├──────────────────────────────────┬──────────────────────────────────────┤\n",
|
||
|
|
"│ FCN = 32.8 (χ²/ndof = 0.8) │ Nfcn = 98 │\n",
|
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|
|
"│ EDM = 4.43e-05 (Goal: 0.0002) │ │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Valid Minimum │ Below EDM threshold (goal x 10) │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ No parameters at limit │ Below call limit │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Hesse ok │ Covariance accurate │\n",
|
||
|
|
"└──────────────────────────────────┴──────────────────────────────────────┘\n",
|
||
|
|
"┌───┬──────────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n",
|
||
|
|
"│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n",
|
||
|
|
"├───┼──────────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n",
|
||
|
|
"│ 0 │ A_p1 │ 400 │ 4 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 1 │ A_p2 │ 700 │ 7 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 2 │ mu_p1 │ 54.0 │ 0.5 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 3 │ mu_p2 │ 60.0 │ 0.6 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 4 │ sigma_p1 │ 2.00 │ 0.02 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 5 │ sigma_p2 │ 2.00 │ 0.02 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 6 │ A_bkg │ 137 │ 15 │ │ │ │ │ │\n",
|
||
|
|
"│ 7 │ tau_bkg │ 34.9 │ 2.3 │ │ │ 0 │ │ │\n",
|
||
|
|
"└───┴──────────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n",
|
||
|
|
"┌──────────┬─────────────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ │ A_p1 A_p2 mu_p1 mu_p2 sigma_p1 sigma_p2 A_bkg tau_bkg │\n",
|
||
|
|
"├──────────┼─────────────────────────────────────────────────────────────────────────┤\n",
|
||
|
|
"│ A_p1 │ 0 0 0 0 0 0 0 0 │\n",
|
||
|
|
"│ A_p2 │ 0 0 0 0 0 0 0 0 │\n",
|
||
|
|
"│ mu_p1 │ 0 0 0 0 0 0 0 0 │\n",
|
||
|
|
"│ mu_p2 │ 0 0 0 0 0 0 0 0 │\n",
|
||
|
|
"│ sigma_p1 │ 0 0 0 0 0 0 0 0 │\n",
|
||
|
|
"│ sigma_p2 │ 0 0 0 0 0 0 0 0 │\n",
|
||
|
|
"│ A_bkg │ 0 0 0 0 0 0 229 -33 │\n",
|
||
|
|
"│ tau_bkg │ 0 0 0 0 0 0 -33 5.18 │\n",
|
||
|
|
"└──────────┴─────────────────────────────────────────────────────────────────────────┘"
|
||
|
|
]
|
||
|
|
},
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|
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"output_type": "execute_result"
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}
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],
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|
|
"source": [
|
||
|
|
"mi.migrad()\n",
|
||
|
|
"mi.hesse()"
|
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|
|
]
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},
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"id": "0b435af3-73ea-42de-9ab7-6a16ae9dbceb",
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"metadata": {},
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"image/png": "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
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 1 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"plt.errorbar(center, entries, np.sqrt(entries), ls='', marker='.')\n",
|
||
|
|
"plt.xlabel('Energy [keV]')\n",
|
||
|
|
"plt.ylabel('Number of counts per bin')\n",
|
||
|
|
"\n",
|
||
|
|
"x = np.arange(40, 80, 0.1)\n",
|
||
|
|
"plt.plot(x, fit_model(x, *mi.values), color='k', label='Initial guess')\n",
|
||
|
|
"plt.legend()\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "6def3e2b-5edf-48bb-99b8-2b7fdaae51c5",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Das Resultat sieht bereits sehr gut aus. Nun können wir uns den eigentlichen Peaks widmen und starten im Folgenden mit dem kleineren der beiden. Zunächst sollten wir den maskierten Bereich entweder neu definieren oder komplett entfernen."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 513,
|
||
|
|
"id": "ebd77c40-6fcd-4881-bc1d-e3ca8ae0bf3b",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": [
|
||
|
|
"ls.mask = None"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "7850ae53-ae2d-49aa-ac7b-dcef60a2dab7",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Außerdem können wir dem Plot entnehmen, dass durch den höheren Untergrund unsere Anfangsstartwerte nicht mehr ganz so gut passen. Diese können wir wie folgt aktualisieren:"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 514,
|
||
|
|
"id": "823e05a0-516c-4d30-8dc7-5381e0e2e617",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": [
|
||
|
|
"mi.values['A_p1'] = 700\n",
|
||
|
|
"mi.values['sigma_p1'] = 3"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "8648bf00-901e-40dc-ada2-9a6b684e8f31",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Nun sollten wir alle Parameter wieder festhalten und nur die Parameter des ersten Peaks freigeben."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 515,
|
||
|
|
"id": "3c83690c-103e-47ff-b18f-13ac763ee87d",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/html": [
|
||
|
|
"<table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th colspan=\"2\" style=\"text-align:center\" title=\"Minimizer\"> Migrad </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Minimum value of function\"> FCN = 1296 (χ²/ndof = 11.1) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total number of function and (optional) gradient evaluations\"> Nfcn = 177 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Estimated distance to minimum and goal\"> EDM = 2.92e-05 (Goal: 0.0002) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total run time of algorithms\"> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Valid Minimum </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below EDM threshold (goal x 10) </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> No parameters at limit </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below call limit </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Hesse ok </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Covariance accurate </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th title=\"Variable name\"> Name </th>\n",
|
||
|
|
" <th title=\"Value of parameter\"> Value </th>\n",
|
||
|
|
" <th title=\"Hesse error\"> Hesse Error </th>\n",
|
||
|
|
" <th title=\"Minos lower error\"> Minos Error- </th>\n",
|
||
|
|
" <th title=\"Minos upper error\"> Minos Error+ </th>\n",
|
||
|
|
" <th title=\"Lower limit of the parameter\"> Limit- </th>\n",
|
||
|
|
" <th title=\"Upper limit of the parameter\"> Limit+ </th>\n",
|
||
|
|
" <th title=\"Is the parameter fixed in the fit\"> Fixed </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 0 </th>\n",
|
||
|
|
" <td> A_p1 </td>\n",
|
||
|
|
" <td> 348 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 1 </th>\n",
|
||
|
|
" <td> A_p2 </td>\n",
|
||
|
|
" <td> 700 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 2 </th>\n",
|
||
|
|
" <td> mu_p1 </td>\n",
|
||
|
|
" <td> 53.51 </td>\n",
|
||
|
|
" <td> 0.04 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 3 </th>\n",
|
||
|
|
" <td> mu_p2 </td>\n",
|
||
|
|
" <td> 60.0 </td>\n",
|
||
|
|
" <td> 0.6 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 4 </th>\n",
|
||
|
|
" <td> sigma_p1 </td>\n",
|
||
|
|
" <td> 2.085 </td>\n",
|
||
|
|
" <td> 0.034 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 5 </th>\n",
|
||
|
|
" <td> sigma_p2 </td>\n",
|
||
|
|
" <td> 2.00 </td>\n",
|
||
|
|
" <td> 0.02 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 6 </th>\n",
|
||
|
|
" <td> A_bkg </td>\n",
|
||
|
|
" <td> 137 </td>\n",
|
||
|
|
" <td> 15 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 7 </th>\n",
|
||
|
|
" <td> tau_bkg </td>\n",
|
||
|
|
" <td> 34.9 </td>\n",
|
||
|
|
" <td> 2.3 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <th> A_bkg </th>\n",
|
||
|
|
" <th> tau_bkg </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <td> 42.2 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(248,248,250);color:black\"> -0.0042 <strong>(-0.016)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(177,177,250);color:black\"> -0.1247 <strong>(-0.558)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(248,248,250);color:black\"> -0.0042 <strong>(-0.016)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td> 0.00167 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,212,212);color:black\"> 0.0004 <strong>(0.252)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(177,177,250);color:black\"> -0.1247 <strong>(-0.558)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,212,212);color:black\"> 0.0004 <strong>(0.252)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td> 0.00118 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_bkg </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> tau_bkg </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.0000 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
|
||
|
|
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
|
||
|
|
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
|
||
|
|
"<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"360pt\" height=\"288pt\" viewBox=\"0 0 360 288\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
|
||
|
|
" <metadata>\n",
|
||
|
|
" <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
|
||
|
|
" <cc:Work>\n",
|
||
|
|
" <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
|
||
|
|
" <dc:date>2024-10-16T21:40:50.416590</dc:date>\n",
|
||
|
|
" <dc:format>image/svg+xml</dc:format>\n",
|
||
|
|
" <dc:creator>\n",
|
||
|
|
" <cc:Agent>\n",
|
||
|
|
" <dc:title>Matplotlib v3.8.2, https://matplotlib.org/</dc:title>\n",
|
||
|
|
" </cc:Agent>\n",
|
||
|
|
" </dc:creator>\n",
|
||
|
|
" </cc:Work>\n",
|
||
|
|
" </rdf:RDF>\n",
|
||
|
|
" </metadata>\n",
|
||
|
|
" <defs>\n",
|
||
|
|
" <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
|
||
|
|
" </defs>\n",
|
||
|
|
" <g id=\"figure_1\">\n",
|
||
|
|
" <g id=\"patch_1\">\n",
|
||
|
|
" <path d=\"M 0 288 \n",
|
||
|
|
"L 360 288 \n",
|
||
|
|
"L 360 0 \n",
|
||
|
|
"L 0 0 \n",
|
||
|
|
"z\n",
|
||
|
|
"\" style=\"fill: #ffffff\"/>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"axes_1\">\n",
|
||
|
|
" <g id=\"patch_2\">\n",
|
||
|
|
" <path d=\"M 29.08774 268.321635 \n",
|
||
|
|
"L 356.99976 268.321635 \n",
|
||
|
|
"L 356.99976 3.00024 \n",
|
||
|
|
"L 29.08774 3.00024 \n",
|
||
|
|
"z\n",
|
||
|
|
"\" style=\"fill: #ffffff\"/>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"matplotlib.axis_1\">\n",
|
||
|
|
" <g id=\"xtick_1\">\n",
|
||
|
|
" <g id=\"line2d_1\">\n",
|
||
|
|
" <defs>\n",
|
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|
|
" <path id=\"m6d5c700e22\" d=\"M 0 0 \n",
|
||
|
|
"L 0 3.5 \n",
|
||
|
|
"\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
|
||
|
|
" </defs>\n",
|
||
|
|
" <g>\n",
|
||
|
|
" <use xlink:href=\"#m6d5c700e22\" x=\"42.740303\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"text_1\">\n",
|
||
|
|
" <!-- 40 -->\n",
|
||
|
|
" <g transform=\"translate(36.377803 282.920073) scale(0.1 -0.1)\">\n",
|
||
|
|
" <defs>\n",
|
||
|
|
" <path id=\"DejaVuSans-34\" d=\"M 2419 4116 \n",
|
||
|
|
"L 825 1625 \n",
|
||
|
|
"L 2419 1625 \n",
|
||
|
|
"L 2419 4116 \n",
|
||
|
|
"z\n",
|
||
|
|
"M 2253 4666 \n",
|
||
|
|
"L 3047 4666 \n",
|
||
|
|
"L 3047 1625 \n",
|
||
|
|
"L 3713 1625 \n",
|
||
|
|
"L 3713 1100 \n",
|
||
|
|
"L 3047 1100 \n",
|
||
|
|
"L 3047 0 \n",
|
||
|
|
"L 2419 0 \n",
|
||
|
|
"L 2419 1100 \n",
|
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|
|
"L 313 1100 \n",
|
||
|
|
"L 313 1709 \n",
|
||
|
|
"L 2253 4666 \n",
|
||
|
|
"z\n",
|
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|
|
"\" transform=\"scale(0.015625)\"/>\n",
|
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|
|
" <path id=\"DejaVuSans-30\" d=\"M 2034 4250 \n",
|
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|
|
"Q 1547 4250 1301 3770 \n",
|
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|
|
"Q 1056 3291 1056 2328 \n",
|
||
|
|
"Q 1056 1369 1301 889 \n",
|
||
|
|
"Q 1547 409 2034 409 \n",
|
||
|
|
"Q 2525 409 2770 889 \n",
|
||
|
|
"Q 3016 1369 3016 2328 \n",
|
||
|
|
"Q 3016 3291 2770 3770 \n",
|
||
|
|
"Q 2525 4250 2034 4250 \n",
|
||
|
|
"z\n",
|
||
|
|
"M 2034 4750 \n",
|
||
|
|
"Q 2819 4750 3233 4129 \n",
|
||
|
|
"Q 3647 3509 3647 2328 \n",
|
||
|
|
"Q 3647 1150 3233 529 \n",
|
||
|
|
"Q 2819 -91 2034 -91 \n",
|
||
|
|
"Q 1250 -91 836 529 \n",
|
||
|
|
"Q 422 1150 422 2328 \n",
|
||
|
|
"Q 422 3509 836 4129 \n",
|
||
|
|
"Q 1250 4750 2034 4750 \n",
|
||
|
|
"z\n",
|
||
|
|
"\" transform=\"scale(0.015625)\"/>\n",
|
||
|
|
" </defs>\n",
|
||
|
|
" <use xlink:href=\"#DejaVuSans-34\"/>\n",
|
||
|
|
" <use xlink:href=\"#DejaVuSans-30\" x=\"63.623047\"/>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"xtick_2\">\n",
|
||
|
|
" <g id=\"line2d_2\">\n",
|
||
|
|
" <g>\n",
|
||
|
|
" <use xlink:href=\"#m6d5c700e22\" x=\"80.316165\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"text_2\">\n",
|
||
|
|
" <!-- 45 -->\n",
|
||
|
|
" <g transform=\"translate(73.953665 282.920073) scale(0.1 -0.1)\">\n",
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|
|
" <defs>\n",
|
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|
|
" <path id=\"DejaVuSans-35\" d=\"M 691 4666 \n",
|
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|
|
"L 3169 4666 \n",
|
||
|
|
"L 3169 4134 \n",
|
||
|
|
"L 1269 4134 \n",
|
||
|
|
"L 1269 2991 \n",
|
||
|
|
"Q 1406 3038 1543 3061 \n",
|
||
|
|
"Q 1681 3084 1819 3084 \n",
|
||
|
|
"Q 2600 3084 3056 2656 \n",
|
||
|
|
"Q 3513 2228 3513 1497 \n",
|
||
|
|
"Q 3513 744 3044 326 \n",
|
||
|
|
"Q 2575 -91 1722 -91 \n",
|
||
|
|
"Q 1428 -91 1123 -41 \n",
|
||
|
|
"Q 819 9 494 109 \n",
|
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|
|
"L 494 744 \n",
|
||
|
|
"Q 775 591 1075 516 \n",
|
||
|
|
"Q 1375 441 1709 441 \n",
|
||
|
|
"Q 2250 441 2565 725 \n",
|
||
|
|
"Q 2881 1009 2881 1497 \n",
|
||
|
|
"Q 2881 1984 2565 2268 \n",
|
||
|
|
"Q 2250 2553 1709 2553 \n",
|
||
|
|
"Q 1456 2553 1204 2497 \n",
|
||
|
|
"Q 953 2441 691 2322 \n",
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|
"L 691 4666 \n",
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|
"z\n",
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|
"\" transform=\"scale(0.015625)\"/>\n",
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" </defs>\n",
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" <use xlink:href=\"#DejaVuSans-35\" x=\"63.623047\"/>\n",
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|
" </g>\n",
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|
|
" </g>\n",
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|
|
" </g>\n",
|
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|
|
" <g id=\"xtick_3\">\n",
|
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|
|
" <g id=\"line2d_3\">\n",
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|
" <g>\n",
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" <use xlink:href=\"#m6d5c700e22\" x=\"117.892027\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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|
" </g>\n",
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|
" </g>\n",
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|
" <g id=\"text_3\">\n",
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|
" <!-- 50 -->\n",
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" <g transform=\"translate(111.529527 282.920073) scale(0.1 -0.1)\">\n",
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" <use xlink:href=\"#DejaVuSans-35\"/>\n",
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" <use xlink:href=\"#DejaVuSans-30\" x=\"63.623047\"/>\n",
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|
|
" </g>\n",
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|
|
" </g>\n",
|
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|
" </g>\n",
|
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|
|
" <g id=\"xtick_4\">\n",
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|
" <g id=\"line2d_4\">\n",
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|
" <g>\n",
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|
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" <use xlink:href=\"#m6d5c700e22\" x=\"155.467888\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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|
|
" </g>\n",
|
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|
|
" </g>\n",
|
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|
|
" <g id=\"text_4\">\n",
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|
|
" <!-- 55 -->\n",
|
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" <g transform=\"translate(149.105388 282.920073) scale(0.1 -0.1)\">\n",
|
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|
|
" <use xlink:href=\"#DejaVuSans-35\"/>\n",
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|
|
" <use xlink:href=\"#DejaVuSans-35\" x=\"63.623047\"/>\n",
|
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|
|
" </g>\n",
|
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|
|
" </g>\n",
|
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|
|
" </g>\n",
|
||
|
|
" <g id=\"xtick_5\">\n",
|
||
|
|
" <g id=\"line2d_5\">\n",
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|
|
" <g>\n",
|
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|
|
" <use xlink:href=\"#m6d5c700e22\" x=\"193.04375\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
|
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|
|
" </g>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"text_5\">\n",
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|
|
" <!-- 60 -->\n",
|
||
|
|
" <g transform=\"translate(186.68125 282.920073) scale(0.1 -0.1)\">\n",
|
||
|
|
" <defs>\n",
|
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"text/plain": [
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"┌─────────────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ Migrad │\n",
|
||
|
|
"├──────────────────────────────────┬──────────────────────────────────────┤\n",
|
||
|
|
"│ FCN = 1296 (χ²/ndof = 11.1) │ Nfcn = 177 │\n",
|
||
|
|
"│ EDM = 2.92e-05 (Goal: 0.0002) │ │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Valid Minimum │ Below EDM threshold (goal x 10) │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ No parameters at limit │ Below call limit │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Hesse ok │ Covariance accurate │\n",
|
||
|
|
"└──────────────────────────────────┴──────────────────────────────────────┘\n",
|
||
|
|
"┌───┬──────────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n",
|
||
|
|
"│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n",
|
||
|
|
"├───┼──────────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n",
|
||
|
|
"│ 0 │ A_p1 │ 348 │ 7 │ │ │ │ │ │\n",
|
||
|
|
"│ 1 │ A_p2 │ 700 │ 7 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 2 │ mu_p1 │ 53.51 │ 0.04 │ │ │ │ │ │\n",
|
||
|
|
"│ 3 │ mu_p2 │ 60.0 │ 0.6 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 4 │ sigma_p1 │ 2.085 │ 0.034 │ │ │ │ │ │\n",
|
||
|
|
"│ 5 │ sigma_p2 │ 2.00 │ 0.02 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 6 │ A_bkg │ 137 │ 15 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 7 │ tau_bkg │ 34.9 │ 2.3 │ │ │ 0 │ │ yes │\n",
|
||
|
|
"└───┴──────────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n",
|
||
|
|
"┌──────────┬─────────────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ │ A_p1 A_p2 mu_p1 mu_p2 sigma_p1 sigma_p2 A_bkg tau_bkg │\n",
|
||
|
|
"├──────────┼─────────────────────────────────────────────────────────────────────────┤\n",
|
||
|
|
"│ A_p1 │ 42.2 0 -0.0042 0 -0.1247 0 0 0 │\n",
|
||
|
|
"│ A_p2 │ 0 0 0.0000 0 0.0000 0 0 0 │\n",
|
||
|
|
"│ mu_p1 │ -0.0042 0.0000 0.00167 0.0000 0.0004 0.0000 0.0000 0.0000 │\n",
|
||
|
|
"│ mu_p2 │ 0 0 0.0000 0 0.0000 0 0 0 │\n",
|
||
|
|
"│ sigma_p1 │ -0.1247 0.0000 0.0004 0.0000 0.00118 0.0000 0.0000 0.0000 │\n",
|
||
|
|
"│ sigma_p2 │ 0 0 0.0000 0 0.0000 0 0 0 │\n",
|
||
|
|
"│ A_bkg │ 0 0 0.0000 0 0.0000 0 0 0 │\n",
|
||
|
|
"│ tau_bkg │ 0 0 0.0000 0 0.0000 0 0 0 │\n",
|
||
|
|
"└──────────┴─────────────────────────────────────────────────────────────────────────┘"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 515,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"mi.fixed[:] = True\n",
|
||
|
|
"mi.fixed[['A_p1', 'mu_p1', 'sigma_p1']] = False\n",
|
||
|
|
"mi.migrad()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "34df75bf-3750-4186-ae12-4f6bb9e49931",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Jetzt wiederholen wir das ganze für den zweiten Peak…"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 516,
|
||
|
|
"id": "264a9891-423c-479a-8906-c048aac2fd2e",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/html": [
|
||
|
|
"<table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th colspan=\"2\" style=\"text-align:center\" title=\"Minimizer\"> Migrad </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Minimum value of function\"> FCN = 137 (χ²/ndof = 1.2) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total number of function and (optional) gradient evaluations\"> Nfcn = 226 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Estimated distance to minimum and goal\"> EDM = 1.24e-06 (Goal: 0.0002) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total run time of algorithms\"> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Valid Minimum </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below EDM threshold (goal x 10) </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> No parameters at limit </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below call limit </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Hesse ok </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Covariance accurate </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th title=\"Variable name\"> Name </th>\n",
|
||
|
|
" <th title=\"Value of parameter\"> Value </th>\n",
|
||
|
|
" <th title=\"Hesse error\"> Hesse Error </th>\n",
|
||
|
|
" <th title=\"Minos lower error\"> Minos Error- </th>\n",
|
||
|
|
" <th title=\"Minos upper error\"> Minos Error+ </th>\n",
|
||
|
|
" <th title=\"Lower limit of the parameter\"> Limit- </th>\n",
|
||
|
|
" <th title=\"Upper limit of the parameter\"> Limit+ </th>\n",
|
||
|
|
" <th title=\"Is the parameter fixed in the fit\"> Fixed </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 0 </th>\n",
|
||
|
|
" <td> A_p1 </td>\n",
|
||
|
|
" <td> 348 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 1 </th>\n",
|
||
|
|
" <td> A_p2 </td>\n",
|
||
|
|
" <td> 584 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 2 </th>\n",
|
||
|
|
" <td> mu_p1 </td>\n",
|
||
|
|
" <td> 53.51 </td>\n",
|
||
|
|
" <td> 0.04 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 3 </th>\n",
|
||
|
|
" <td> mu_p2 </td>\n",
|
||
|
|
" <td> 60.605 </td>\n",
|
||
|
|
" <td> 0.031 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 4 </th>\n",
|
||
|
|
" <td> sigma_p1 </td>\n",
|
||
|
|
" <td> 2.085 </td>\n",
|
||
|
|
" <td> 0.034 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 5 </th>\n",
|
||
|
|
" <td> sigma_p2 </td>\n",
|
||
|
|
" <td> 2.666 </td>\n",
|
||
|
|
" <td> 0.026 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 6 </th>\n",
|
||
|
|
" <td> A_bkg </td>\n",
|
||
|
|
" <td> 137 </td>\n",
|
||
|
|
" <td> 15 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 7 </th>\n",
|
||
|
|
" <td> tau_bkg </td>\n",
|
||
|
|
" <td> 34.9 </td>\n",
|
||
|
|
" <td> 2.3 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> yes </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <th> A_bkg </th>\n",
|
||
|
|
" <th> tau_bkg </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0e-3 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 55.2 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.7e-3 <strong>(0.003)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(177,177,250);color:black\"> -110.5e-3 <strong>(-0.563)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0e-3 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0e-3 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0.7e-3 <strong>(0.003)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0e-3 </td>\n",
|
||
|
|
" <td> 0.000983 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0e-3 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(223,223,250);color:black\"> -0.2e-3 <strong>(-0.210)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0e-3 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0e-3 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0e-3 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(177,177,250);color:black\"> -110.5e-3 <strong>(-0.563)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(223,223,250);color:black\"> -0.2e-3 <strong>(-0.210)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 0.000697 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_bkg </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0e-3 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> tau_bkg </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0e-3 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,250,250);color:black\"> 0 </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
|
||
|
|
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
|
||
|
|
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
|
||
|
|
"<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"360pt\" height=\"288pt\" viewBox=\"0 0 360 288\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
|
||
|
|
" <metadata>\n",
|
||
|
|
" <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
|
||
|
|
" <cc:Work>\n",
|
||
|
|
" <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
|
||
|
|
" <dc:date>2024-10-16T21:40:50.691377</dc:date>\n",
|
||
|
|
" <dc:format>image/svg+xml</dc:format>\n",
|
||
|
|
" <dc:creator>\n",
|
||
|
|
" <cc:Agent>\n",
|
||
|
|
" <dc:title>Matplotlib v3.8.2, https://matplotlib.org/</dc:title>\n",
|
||
|
|
" </cc:Agent>\n",
|
||
|
|
" </dc:creator>\n",
|
||
|
|
" </cc:Work>\n",
|
||
|
|
" </rdf:RDF>\n",
|
||
|
|
" </metadata>\n",
|
||
|
|
" <defs>\n",
|
||
|
|
" <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
|
||
|
|
" </defs>\n",
|
||
|
|
" <g id=\"figure_1\">\n",
|
||
|
|
" <g id=\"patch_1\">\n",
|
||
|
|
" <path d=\"M 0 288 \n",
|
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|
|
"L 360 288 \n",
|
||
|
|
"L 360 0 \n",
|
||
|
|
"L 0 0 \n",
|
||
|
|
"z\n",
|
||
|
|
"\" style=\"fill: #ffffff\"/>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"axes_1\">\n",
|
||
|
|
" <g id=\"patch_2\">\n",
|
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|
|
" <path d=\"M 29.08774 268.321635 \n",
|
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|
|
"L 356.99976 268.321635 \n",
|
||
|
|
"L 356.99976 3.00024 \n",
|
||
|
|
"L 29.08774 3.00024 \n",
|
||
|
|
"z\n",
|
||
|
|
"\" style=\"fill: #ffffff\"/>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"matplotlib.axis_1\">\n",
|
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|
|
" <g id=\"xtick_1\">\n",
|
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|
" <g id=\"line2d_1\">\n",
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" <defs>\n",
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" <path id=\"m8346b2ca2e\" d=\"M 0 0 \n",
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|
|
"L 0 3.5 \n",
|
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|
|
"\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
|
||
|
|
" </defs>\n",
|
||
|
|
" <g>\n",
|
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|
|
" <use xlink:href=\"#m8346b2ca2e\" x=\"42.740303\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
|
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|
|
" </g>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"text_1\">\n",
|
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|
|
" <!-- 40 -->\n",
|
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|
|
" <g transform=\"translate(36.377803 282.920073) scale(0.1 -0.1)\">\n",
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" <defs>\n",
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" <path id=\"DejaVuSans-34\" d=\"M 2419 4116 \n",
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|
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|
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|
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|
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|
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|
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|
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|
|
"Q 1056 1369 1301 889 \n",
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|
|
"Q 1547 409 2034 409 \n",
|
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|
|
"Q 2525 409 2770 889 \n",
|
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|
|
"Q 3016 1369 3016 2328 \n",
|
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|
|
"Q 3016 3291 2770 3770 \n",
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|
"Q 2525 4250 2034 4250 \n",
|
||
|
|
"z\n",
|
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|
|
"M 2034 4750 \n",
|
||
|
|
"Q 2819 4750 3233 4129 \n",
|
||
|
|
"Q 3647 3509 3647 2328 \n",
|
||
|
|
"Q 3647 1150 3233 529 \n",
|
||
|
|
"Q 2819 -91 2034 -91 \n",
|
||
|
|
"Q 1250 -91 836 529 \n",
|
||
|
|
"Q 422 1150 422 2328 \n",
|
||
|
|
"Q 422 3509 836 4129 \n",
|
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|
|
"Q 1250 4750 2034 4750 \n",
|
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|
"z\n",
|
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|
"\" transform=\"scale(0.015625)\"/>\n",
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" </defs>\n",
|
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" <use xlink:href=\"#DejaVuSans-34\"/>\n",
|
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" <use xlink:href=\"#DejaVuSans-30\" x=\"63.623047\"/>\n",
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|
|
" </g>\n",
|
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|
|
" </g>\n",
|
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|
|
" </g>\n",
|
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|
|
" <g id=\"xtick_2\">\n",
|
||
|
|
" <g id=\"line2d_2\">\n",
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|
|
" <g>\n",
|
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|
|
" <use xlink:href=\"#m8346b2ca2e\" x=\"80.316165\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"text_2\">\n",
|
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|
|
" <!-- 45 -->\n",
|
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|
|
" <g transform=\"translate(73.953665 282.920073) scale(0.1 -0.1)\">\n",
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" <defs>\n",
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"L 3169 4666 \n",
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"L 3169 4134 \n",
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|
|
"L 1269 4134 \n",
|
||
|
|
"L 1269 2991 \n",
|
||
|
|
"Q 1406 3038 1543 3061 \n",
|
||
|
|
"Q 1681 3084 1819 3084 \n",
|
||
|
|
"Q 2600 3084 3056 2656 \n",
|
||
|
|
"Q 3513 2228 3513 1497 \n",
|
||
|
|
"Q 3513 744 3044 326 \n",
|
||
|
|
"Q 2575 -91 1722 -91 \n",
|
||
|
|
"Q 1428 -91 1123 -41 \n",
|
||
|
|
"Q 819 9 494 109 \n",
|
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|
|
"L 494 744 \n",
|
||
|
|
"Q 775 591 1075 516 \n",
|
||
|
|
"Q 1375 441 1709 441 \n",
|
||
|
|
"Q 2250 441 2565 725 \n",
|
||
|
|
"Q 2881 1009 2881 1497 \n",
|
||
|
|
"Q 2881 1984 2565 2268 \n",
|
||
|
|
"Q 2250 2553 1709 2553 \n",
|
||
|
|
"Q 1456 2553 1204 2497 \n",
|
||
|
|
"Q 953 2441 691 2322 \n",
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"L 691 4666 \n",
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"z\n",
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"\" transform=\"scale(0.015625)\"/>\n",
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" </defs>\n",
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" <use xlink:href=\"#DejaVuSans-34\"/>\n",
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" <use xlink:href=\"#DejaVuSans-35\" x=\"63.623047\"/>\n",
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" </g>\n",
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" </g>\n",
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|
" </g>\n",
|
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" <g id=\"xtick_3\">\n",
|
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" <g id=\"line2d_3\">\n",
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" <g>\n",
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" <use xlink:href=\"#m8346b2ca2e\" x=\"117.892027\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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|
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" </g>\n",
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|
" </g>\n",
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" <g id=\"text_3\">\n",
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" <!-- 50 -->\n",
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" <g transform=\"translate(111.529527 282.920073) scale(0.1 -0.1)\">\n",
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" <use xlink:href=\"#DejaVuSans-35\"/>\n",
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" <use xlink:href=\"#DejaVuSans-30\" x=\"63.623047\"/>\n",
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|
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" </g>\n",
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|
|
" </g>\n",
|
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|
" </g>\n",
|
||
|
|
" <g id=\"xtick_4\">\n",
|
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|
|
" <g id=\"line2d_4\">\n",
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" <g>\n",
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|
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" <use xlink:href=\"#m8346b2ca2e\" x=\"155.467888\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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|
|
" </g>\n",
|
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|
|
" </g>\n",
|
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|
|
" <g id=\"text_4\">\n",
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|
|
" <!-- 55 -->\n",
|
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" <g transform=\"translate(149.105388 282.920073) scale(0.1 -0.1)\">\n",
|
||
|
|
" <use xlink:href=\"#DejaVuSans-35\"/>\n",
|
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|
|
" <use xlink:href=\"#DejaVuSans-35\" x=\"63.623047\"/>\n",
|
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|
|
" </g>\n",
|
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|
|
" </g>\n",
|
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|
|
" </g>\n",
|
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|
|
" <g id=\"xtick_5\">\n",
|
||
|
|
" <g id=\"line2d_5\">\n",
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" <g>\n",
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|
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" <use xlink:href=\"#m8346b2ca2e\" x=\"193.04375\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
|
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|
|
" </g>\n",
|
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|
|
" </g>\n",
|
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|
|
" <g id=\"text_5\">\n",
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|
|
" <!-- 60 -->\n",
|
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|
|
" <g transform=\"translate(186.68125 282.920073) scale(0.1 -0.1)\">\n",
|
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|
|
" <defs>\n",
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" <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
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|
|
"Q 1688 2584 1439 2293 \n",
|
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|
|
"Q 1191 2003 1191 1497 \n",
|
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|
|
"Q 1191 994 1439 701 \n",
|
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|
|
"Q 1688 409 2113 409 \n",
|
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|
|
"Q 2538 409 2786 701 \n",
|
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|
|
"Q 3034 994 3034 1497 \n",
|
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|
|
"Q 3034 2003 2786 2293 \n",
|
||
|
|
"Q 2538 2584 2113 2584 \n",
|
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|
|
"z\n",
|
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|
|
"M 3366 4563 \n",
|
||
|
|
"L 3366 3988 \n",
|
||
|
|
"Q 3128 4100 2886 4159 \n",
|
||
|
|
"Q 2644 4219 2406 4219 \n",
|
||
|
|
"Q 1781 4219 1451 3797 \n",
|
||
|
|
"Q 1122 3375 1075 2522 \n",
|
||
|
|
"Q 1259 2794 1537 2939 \n",
|
||
|
|
"Q 1816 3084 2150 3084 \n",
|
||
|
|
"Q 2853 3084 3261 2657 \n",
|
||
|
|
"Q 3669 2231 3669 1497 \n",
|
||
|
|
"Q 3669 778 3244 343 \n",
|
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|
|
"Q 2819 -91 2113 -91 \n",
|
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|
|
"Q 1303 -91 875 529 \n",
|
||
|
|
"Q 447 1150 447 2328 \n",
|
||
|
|
"Q 447 3434 972 4092 \n",
|
||
|
|
"Q 1497 4750 2381 4750 \n",
|
||
|
|
"Q 2619 4750 2861 4703 \n",
|
||
|
|
"Q 3103 4656 3366 4563 \n",
|
||
|
|
"z\n",
|
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|
|
"\" transform=\"scale(0.015625)\"/>\n",
|
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|
|
" </defs>\n",
|
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|
" <use xlink:href=\"#DejaVuSans-36\"/>\n",
|
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|
|
" <use xlink:href=\"#DejaVuSans-30\" x=\"63.623047\"/>\n",
|
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|
|
" </g>\n",
|
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|
|
" </g>\n",
|
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|
|
" </g>\n",
|
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|
|
" <g id=\"xtick_6\">\n",
|
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|
|
" <g id=\"line2d_6\">\n",
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|
|
" <g>\n",
|
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"text/plain": [
|
||
|
|
"┌─────────────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ Migrad │\n",
|
||
|
|
"├──────────────────────────────────┬──────────────────────────────────────┤\n",
|
||
|
|
"│ FCN = 137 (χ²/ndof = 1.2) │ Nfcn = 226 │\n",
|
||
|
|
"│ EDM = 1.24e-06 (Goal: 0.0002) │ │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Valid Minimum │ Below EDM threshold (goal x 10) │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ No parameters at limit │ Below call limit │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Hesse ok │ Covariance accurate │\n",
|
||
|
|
"└──────────────────────────────────┴──────────────────────────────────────┘\n",
|
||
|
|
"┌───┬──────────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n",
|
||
|
|
"│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n",
|
||
|
|
"├───┼──────────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n",
|
||
|
|
"│ 0 │ A_p1 │ 348 │ 7 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 1 │ A_p2 │ 584 │ 7 │ │ │ │ │ │\n",
|
||
|
|
"│ 2 │ mu_p1 │ 53.51 │ 0.04 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 3 │ mu_p2 │ 60.605 │ 0.031 │ │ │ │ │ │\n",
|
||
|
|
"│ 4 │ sigma_p1 │ 2.085 │ 0.034 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 5 │ sigma_p2 │ 2.666 │ 0.026 │ │ │ │ │ │\n",
|
||
|
|
"│ 6 │ A_bkg │ 137 │ 15 │ │ │ │ │ yes │\n",
|
||
|
|
"│ 7 │ tau_bkg │ 34.9 │ 2.3 │ │ │ 0 │ │ yes │\n",
|
||
|
|
"└───┴──────────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n",
|
||
|
|
"┌──────────┬─────────────────────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ │ A_p1 A_p2 mu_p1 mu_p2 sigma_p1 sigma_p2 A_bkg tau_bkg │\n",
|
||
|
|
"├──────────┼─────────────────────────────────────────────────────────────────────────────────┤\n",
|
||
|
|
"│ A_p1 │ 0 0 0 0e-3 0 0 0 0 │\n",
|
||
|
|
"│ A_p2 │ 0 55.2 0 0.7e-3 0 -110.5e-3 0 0 │\n",
|
||
|
|
"│ mu_p1 │ 0 0 0 0e-3 0 0 0 0 │\n",
|
||
|
|
"│ mu_p2 │ 0e-3 0.7e-3 0e-3 0.000983 0e-3 -0.2e-3 0e-3 0e-3 │\n",
|
||
|
|
"│ sigma_p1 │ 0 0 0 0e-3 0 0 0 0 │\n",
|
||
|
|
"│ sigma_p2 │ 0 -110.5e-3 0 -0.2e-3 0 0.000697 0 0 │\n",
|
||
|
|
"│ A_bkg │ 0 0 0 0e-3 0 0 0 0 │\n",
|
||
|
|
"│ tau_bkg │ 0 0 0 0e-3 0 0 0 0 │\n",
|
||
|
|
"└──────────┴─────────────────────────────────────────────────────────────────────────────────┘"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 516,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"mi.fixed[:] = True\n",
|
||
|
|
"mi.fixed[['A_p2', 'mu_p2', 'sigma_p2']] = False\n",
|
||
|
|
"mi.migrad()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "32d67543-870f-4bd9-bba4-2d01086c671a",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Zum Schluss geben wir wieder alle Parameter frei und führen einen letzten Fit durch. "
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 517,
|
||
|
|
"id": "72d43004-cd80-418a-996a-f1e7a7133ce9",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/html": [
|
||
|
|
"<table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th colspan=\"2\" style=\"text-align:center\" title=\"Minimizer\"> Migrad </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Minimum value of function\"> FCN = 106.4 (χ²/ndof = 0.9) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total number of function and (optional) gradient evaluations\"> Nfcn = 500 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Estimated distance to minimum and goal\"> EDM = 4.26e-05 (Goal: 0.0002) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total run time of algorithms\"> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Valid Minimum </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below EDM threshold (goal x 10) </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> No parameters at limit </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below call limit </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Hesse ok </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Covariance accurate </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th title=\"Variable name\"> Name </th>\n",
|
||
|
|
" <th title=\"Value of parameter\"> Value </th>\n",
|
||
|
|
" <th title=\"Hesse error\"> Hesse Error </th>\n",
|
||
|
|
" <th title=\"Minos lower error\"> Minos Error- </th>\n",
|
||
|
|
" <th title=\"Minos upper error\"> Minos Error+ </th>\n",
|
||
|
|
" <th title=\"Lower limit of the parameter\"> Limit- </th>\n",
|
||
|
|
" <th title=\"Upper limit of the parameter\"> Limit+ </th>\n",
|
||
|
|
" <th title=\"Is the parameter fixed in the fit\"> Fixed </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 0 </th>\n",
|
||
|
|
" <td> A_p1 </td>\n",
|
||
|
|
" <td> 317 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 1 </th>\n",
|
||
|
|
" <td> A_p2 </td>\n",
|
||
|
|
" <td> 580 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 2 </th>\n",
|
||
|
|
" <td> mu_p1 </td>\n",
|
||
|
|
" <td> 53.24 </td>\n",
|
||
|
|
" <td> 0.07 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 3 </th>\n",
|
||
|
|
" <td> mu_p2 </td>\n",
|
||
|
|
" <td> 60.43 </td>\n",
|
||
|
|
" <td> 0.05 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 4 </th>\n",
|
||
|
|
" <td> sigma_p1 </td>\n",
|
||
|
|
" <td> 1.99 </td>\n",
|
||
|
|
" <td> 0.05 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 5 </th>\n",
|
||
|
|
" <td> sigma_p2 </td>\n",
|
||
|
|
" <td> 2.80 </td>\n",
|
||
|
|
" <td> 0.04 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 6 </th>\n",
|
||
|
|
" <td> A_bkg </td>\n",
|
||
|
|
" <td> 147 </td>\n",
|
||
|
|
" <td> 14 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 7 </th>\n",
|
||
|
|
" <td> tau_bkg </td>\n",
|
||
|
|
" <td> 34.1 </td>\n",
|
||
|
|
" <td> 2.0 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <th> A_bkg </th>\n",
|
||
|
|
" <th> tau_bkg </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <td> 51.5 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,227,227);color:black\"> 10 <strong>(0.153)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,220,220);color:black\"> 0.103 <strong>(0.202)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,210,210);color:black\"> 0.1006 <strong>(0.267)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(223,223,250);color:black\"> -0.0808 <strong>(-0.207)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(208,208,250);color:black\"> -0.0969 <strong>(-0.327)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(246,246,250);color:black\"> -0 <strong>(-0.031)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,245,245);color:black\"> 0 <strong>(0.031)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,227,227);color:black\"> 10 <strong>(0.153)</strong> </td>\n",
|
||
|
|
" <td> 50.6 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,242,242);color:black\"> 0.026 <strong>(0.052)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,234,234);color:black\"> 0.0402 <strong>(0.108)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(248,248,250);color:black\"> -0.0047 <strong>(-0.012)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(191,191,250);color:black\"> -0.1329 <strong>(-0.452)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(247,247,250);color:black\"> -0 <strong>(-0.025)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,247,247);color:black\"> 0 <strong>(0.021)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,220,220);color:black\"> 0.103 <strong>(0.202)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,242,242);color:black\"> 0.026 <strong>(0.052)</strong> </td>\n",
|
||
|
|
" <td> 0.00503 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,142,142);color:black\"> 0.0027 <strong>(0.720)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,151,151);color:black\"> 0.0025 <strong>(0.659)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(163,163,250);color:black\"> -0.0020 <strong>(-0.666)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(243,243,250);color:black\"> -0.057 <strong>(-0.055)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,239,239);color:black\"> 0.010 <strong>(0.072)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,210,210);color:black\"> 0.1006 <strong>(0.267)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,234,234);color:black\"> 0.0402 <strong>(0.108)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,142,142);color:black\"> 0.0027 <strong>(0.720)</strong> </td>\n",
|
||
|
|
" <td> 0.00276 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,156,156);color:black\"> 0.0018 <strong>(0.624)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(162,162,250);color:black\"> -0.0015 <strong>(-0.680)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(241,241,250);color:black\"> -0.0515 <strong>(-0.068)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,241,241);color:black\"> 0.0062 <strong>(0.059)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(223,223,250);color:black\"> -0.0808 <strong>(-0.207)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(248,248,250);color:black\"> -0.0047 <strong>(-0.012)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,151,151);color:black\"> 0.0025 <strong>(0.659)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,156,156);color:black\"> 0.0018 <strong>(0.624)</strong> </td>\n",
|
||
|
|
" <td> 0.00297 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(183,183,250);color:black\"> -0.0012 <strong>(-0.518)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(227,227,250);color:black\"> -0.1413 <strong>(-0.179)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,229,229);color:black\"> 0.0156 <strong>(0.142)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(208,208,250);color:black\"> -0.0969 <strong>(-0.327)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(191,191,250);color:black\"> -0.1329 <strong>(-0.452)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(163,163,250);color:black\"> -0.0020 <strong>(-0.666)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(162,162,250);color:black\"> -0.0015 <strong>(-0.680)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(183,183,250);color:black\"> -0.0012 <strong>(-0.518)</strong> </td>\n",
|
||
|
|
" <td> 0.00171 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,229,229);color:black\"> 0.0818 <strong>(0.137)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(228,228,250);color:black\"> -0.0143 <strong>(-0.172)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_bkg </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(246,246,250);color:black\"> -0 <strong>(-0.031)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(247,247,250);color:black\"> -0 <strong>(-0.025)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(243,243,250);color:black\"> -0.057 <strong>(-0.055)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(241,241,250);color:black\"> -0.0515 <strong>(-0.068)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(227,227,250);color:black\"> -0.1413 <strong>(-0.179)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,229,229);color:black\"> 0.0818 <strong>(0.137)</strong> </td>\n",
|
||
|
|
" <td> 209 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(125,125,250);color:black\"> -28 <strong>(-0.965)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> tau_bkg </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,245,245);color:black\"> 0 <strong>(0.031)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,247,247);color:black\"> 0 <strong>(0.021)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,239,239);color:black\"> 0.010 <strong>(0.072)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,241,241);color:black\"> 0.0062 <strong>(0.059)</strong> </td>\n",
|
||
|
|
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"┌─────────────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ Migrad │\n",
|
||
|
|
"├──────────────────────────────────┬──────────────────────────────────────┤\n",
|
||
|
|
"│ FCN = 106.4 (χ²/ndof = 0.9) │ Nfcn = 500 │\n",
|
||
|
|
"│ EDM = 4.26e-05 (Goal: 0.0002) │ │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Valid Minimum │ Below EDM threshold (goal x 10) │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ No parameters at limit │ Below call limit │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Hesse ok │ Covariance accurate │\n",
|
||
|
|
"└──────────────────────────────────┴──────────────────────────────────────┘\n",
|
||
|
|
"┌───┬──────────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n",
|
||
|
|
"│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n",
|
||
|
|
"├───┼──────────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n",
|
||
|
|
"│ 0 │ A_p1 │ 317 │ 7 │ │ │ │ │ │\n",
|
||
|
|
"│ 1 │ A_p2 │ 580 │ 7 │ │ │ │ │ │\n",
|
||
|
|
"│ 2 │ mu_p1 │ 53.24 │ 0.07 │ │ │ │ │ │\n",
|
||
|
|
"│ 3 │ mu_p2 │ 60.43 │ 0.05 │ │ │ │ │ │\n",
|
||
|
|
"│ 4 │ sigma_p1 │ 1.99 │ 0.05 │ │ │ │ │ │\n",
|
||
|
|
"│ 5 │ sigma_p2 │ 2.80 │ 0.04 │ │ │ │ │ │\n",
|
||
|
|
"│ 6 │ A_bkg │ 147 │ 14 │ │ │ │ │ │\n",
|
||
|
|
"│ 7 │ tau_bkg │ 34.1 │ 2.0 │ │ │ 0 │ │ │\n",
|
||
|
|
"└───┴──────────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n",
|
||
|
|
"┌──────────┬─────────────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ │ A_p1 A_p2 mu_p1 mu_p2 sigma_p1 sigma_p2 A_bkg tau_bkg │\n",
|
||
|
|
"├──────────┼─────────────────────────────────────────────────────────────────────────┤\n",
|
||
|
|
"│ A_p1 │ 51.5 10 0.103 0.1006 -0.0808 -0.0969 -0 0 │\n",
|
||
|
|
"│ A_p2 │ 10 50.6 0.026 0.0402 -0.0047 -0.1329 -0 0 │\n",
|
||
|
|
"│ mu_p1 │ 0.103 0.026 0.00503 0.0027 0.0025 -0.0020 -0.057 0.010 │\n",
|
||
|
|
"│ mu_p2 │ 0.1006 0.0402 0.0027 0.00276 0.0018 -0.0015 -0.0515 0.0062 │\n",
|
||
|
|
"│ sigma_p1 │ -0.0808 -0.0047 0.0025 0.0018 0.00297 -0.0012 -0.1413 0.0156 │\n",
|
||
|
|
"│ sigma_p2 │ -0.0969 -0.1329 -0.0020 -0.0015 -0.0012 0.00171 0.0818 -0.0143 │\n",
|
||
|
|
"│ A_bkg │ -0 -0 -0.057 -0.0515 -0.1413 0.0818 209 -28 │\n",
|
||
|
|
"│ tau_bkg │ 0 0 0.010 0.0062 0.0156 -0.0143 -28 4.03 │\n",
|
||
|
|
"└──────────┴─────────────────────────────────────────────────────────────────────────┘"
|
||
|
|
]
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|
},
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|
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}
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],
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"source": [
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||
|
|
"mi.fixed[:] = False\n",
|
||
|
|
"mi.migrad()"
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]
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},
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{
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"cell_type": "code",
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"metadata": {},
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"image/png": "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
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 1 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"plt.errorbar(center, entries, np.sqrt(entries), ls='', marker='.')\n",
|
||
|
|
"plt.xlabel('Energy [keV]')\n",
|
||
|
|
"plt.ylabel('Number of counts per bin')\n",
|
||
|
|
"\n",
|
||
|
|
"x = np.arange(40, 80, 0.1)\n",
|
||
|
|
"plt.plot(x, fit_model(x, *mi.values), color='k', label='Best fit')\n",
|
||
|
|
"plt.plot(x, peak(x, *mi.values['A_p1', 'mu_p1', 'sigma_p1']), color='gray', ls='--', label='Peak 1')\n",
|
||
|
|
"plt.plot(x, peak(x, *mi.values['A_p2', 'mu_p2', 'sigma_p2']), color='gray', ls='-.', label='Peak 2')\n",
|
||
|
|
"plt.plot(x, bkg(x, *mi.values['A_bkg', 'tau_bkg']), color='gray', label='Background')\n",
|
||
|
|
"plt.legend()\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "7ef19633-0947-4568-b537-a1c69e42b7c2",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Das Ergebnis sieht sehr gut aus. Alle Kacheln sind grün und die Daten scheinen durch die Funktion gut beschrieben zu werden. Natürlich können wir das gesamte Fitverfahren auch etwas kompakter in einer Zelle darstellen:"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 519,
|
||
|
|
"id": "2311f135-8410-4f35-8d58-b9bcef0fed53",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/html": [
|
||
|
|
"<table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th colspan=\"2\" style=\"text-align:center\" title=\"Minimizer\"> Migrad </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Minimum value of function\"> FCN = 106.4 (χ²/ndof = 0.9) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total number of function and (optional) gradient evaluations\"> Nfcn = 530 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Estimated distance to minimum and goal\"> EDM = 1.61e-05 (Goal: 0.0002) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total run time of algorithms\"> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Valid Minimum </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below EDM threshold (goal x 10) </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> No parameters at limit </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below call limit </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Hesse ok </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Covariance accurate </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th title=\"Variable name\"> Name </th>\n",
|
||
|
|
" <th title=\"Value of parameter\"> Value </th>\n",
|
||
|
|
" <th title=\"Hesse error\"> Hesse Error </th>\n",
|
||
|
|
" <th title=\"Minos lower error\"> Minos Error- </th>\n",
|
||
|
|
" <th title=\"Minos upper error\"> Minos Error+ </th>\n",
|
||
|
|
" <th title=\"Lower limit of the parameter\"> Limit- </th>\n",
|
||
|
|
" <th title=\"Upper limit of the parameter\"> Limit+ </th>\n",
|
||
|
|
" <th title=\"Is the parameter fixed in the fit\"> Fixed </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 0 </th>\n",
|
||
|
|
" <td> A_p1 </td>\n",
|
||
|
|
" <td> 317 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 1 </th>\n",
|
||
|
|
" <td> A_p2 </td>\n",
|
||
|
|
" <td> 580 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 2 </th>\n",
|
||
|
|
" <td> mu_p1 </td>\n",
|
||
|
|
" <td> 53.24 </td>\n",
|
||
|
|
" <td> 0.07 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 3 </th>\n",
|
||
|
|
" <td> mu_p2 </td>\n",
|
||
|
|
" <td> 60.43 </td>\n",
|
||
|
|
" <td> 0.05 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 4 </th>\n",
|
||
|
|
" <td> sigma_p1 </td>\n",
|
||
|
|
" <td> 1.99 </td>\n",
|
||
|
|
" <td> 0.05 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 5 </th>\n",
|
||
|
|
" <td> sigma_p2 </td>\n",
|
||
|
|
" <td> 2.80 </td>\n",
|
||
|
|
" <td> 0.04 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 6 </th>\n",
|
||
|
|
" <td> A_bkg </td>\n",
|
||
|
|
" <td> 147 </td>\n",
|
||
|
|
" <td> 14 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 7 </th>\n",
|
||
|
|
" <td> tau_bkg </td>\n",
|
||
|
|
" <td> 34.1 </td>\n",
|
||
|
|
" <td> 2.0 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <th> A_bkg </th>\n",
|
||
|
|
" <th> tau_bkg </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <td> 51.5 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,227,227);color:black\"> 10 <strong>(0.153)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,220,220);color:black\"> 0.103 <strong>(0.202)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,210,210);color:black\"> 0.1006 <strong>(0.267)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(223,223,250);color:black\"> -0.0808 <strong>(-0.207)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(208,208,250);color:black\"> -0.0969 <strong>(-0.327)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(246,246,250);color:black\"> -0 <strong>(-0.031)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,245,245);color:black\"> 0 <strong>(0.031)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,227,227);color:black\"> 10 <strong>(0.153)</strong> </td>\n",
|
||
|
|
" <td> 50.6 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,242,242);color:black\"> 0.026 <strong>(0.052)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,234,234);color:black\"> 0.0402 <strong>(0.108)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(248,248,250);color:black\"> -0.0047 <strong>(-0.012)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(191,191,250);color:black\"> -0.1329 <strong>(-0.452)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(247,247,250);color:black\"> -0 <strong>(-0.025)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,247,247);color:black\"> 0 <strong>(0.021)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,220,220);color:black\"> 0.103 <strong>(0.202)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,242,242);color:black\"> 0.026 <strong>(0.052)</strong> </td>\n",
|
||
|
|
" <td> 0.00503 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,142,142);color:black\"> 0.0027 <strong>(0.720)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,151,151);color:black\"> 0.0025 <strong>(0.659)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(163,163,250);color:black\"> -0.0020 <strong>(-0.666)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(243,243,250);color:black\"> -0.057 <strong>(-0.055)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,239,239);color:black\"> 0.010 <strong>(0.072)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,210,210);color:black\"> 0.1006 <strong>(0.267)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,234,234);color:black\"> 0.0402 <strong>(0.108)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,142,142);color:black\"> 0.0027 <strong>(0.720)</strong> </td>\n",
|
||
|
|
" <td> 0.00276 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,156,156);color:black\"> 0.0018 <strong>(0.623)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(162,162,250);color:black\"> -0.0015 <strong>(-0.680)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(241,241,250);color:black\"> -0.0513 <strong>(-0.068)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,241,241);color:black\"> 0.0062 <strong>(0.059)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(223,223,250);color:black\"> -0.0808 <strong>(-0.207)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(248,248,250);color:black\"> -0.0047 <strong>(-0.012)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,151,151);color:black\"> 0.0025 <strong>(0.659)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,156,156);color:black\"> 0.0018 <strong>(0.623)</strong> </td>\n",
|
||
|
|
" <td> 0.00297 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(183,183,250);color:black\"> -0.0012 <strong>(-0.518)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(227,227,250);color:black\"> -0.1409 <strong>(-0.179)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,229,229);color:black\"> 0.0155 <strong>(0.142)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(208,208,250);color:black\"> -0.0969 <strong>(-0.327)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(191,191,250);color:black\"> -0.1329 <strong>(-0.452)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(163,163,250);color:black\"> -0.0020 <strong>(-0.666)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(162,162,250);color:black\"> -0.0015 <strong>(-0.680)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(183,183,250);color:black\"> -0.0012 <strong>(-0.518)</strong> </td>\n",
|
||
|
|
" <td> 0.00171 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,229,229);color:black\"> 0.0816 <strong>(0.137)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(228,228,250);color:black\"> -0.0142 <strong>(-0.172)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_bkg </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(246,246,250);color:black\"> -0 <strong>(-0.031)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(247,247,250);color:black\"> -0 <strong>(-0.025)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(243,243,250);color:black\"> -0.057 <strong>(-0.055)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(241,241,250);color:black\"> -0.0513 <strong>(-0.068)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(227,227,250);color:black\"> -0.1409 <strong>(-0.179)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,229,229);color:black\"> 0.0816 <strong>(0.137)</strong> </td>\n",
|
||
|
|
" <td> 209 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(125,125,250);color:black\"> -28 <strong>(-0.965)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> tau_bkg </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,245,245);color:black\"> 0 <strong>(0.031)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,247,247);color:black\"> 0 <strong>(0.021)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,239,239);color:black\"> 0.010 <strong>(0.072)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,241,241);color:black\"> 0.0062 <strong>(0.059)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,229,229);color:black\"> 0.0155 <strong>(0.142)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(228,228,250);color:black\"> -0.0142 <strong>(-0.172)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(125,125,250);color:black\"> -28 <strong>(-0.965)</strong> </td>\n",
|
||
|
|
" <td> 4.01 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
|
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|
|
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
|
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|
|
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
|
||
|
|
"<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"360pt\" height=\"288pt\" viewBox=\"0 0 360 288\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
|
||
|
|
" <metadata>\n",
|
||
|
|
" <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
|
||
|
|
" <cc:Work>\n",
|
||
|
|
" <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
|
||
|
|
" <dc:date>2024-10-16T21:40:51.617180</dc:date>\n",
|
||
|
|
" <dc:format>image/svg+xml</dc:format>\n",
|
||
|
|
" <dc:creator>\n",
|
||
|
|
" <cc:Agent>\n",
|
||
|
|
" <dc:title>Matplotlib v3.8.2, https://matplotlib.org/</dc:title>\n",
|
||
|
|
" </cc:Agent>\n",
|
||
|
|
" </dc:creator>\n",
|
||
|
|
" </cc:Work>\n",
|
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|
|
" </rdf:RDF>\n",
|
||
|
|
" </metadata>\n",
|
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|
|
" <defs>\n",
|
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|
|
" <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
|
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|
|
" </defs>\n",
|
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" <g id=\"figure_1\">\n",
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" <g id=\"patch_1\">\n",
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" <path d=\"M 0 288 \n",
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"L 360 288 \n",
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"L 360 0 \n",
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"L 0 0 \n",
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|
"z\n",
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|
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"\" style=\"fill: #ffffff\"/>\n",
|
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|
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" </g>\n",
|
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" <g id=\"axes_1\">\n",
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" <g id=\"patch_2\">\n",
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" <path d=\"M 29.08774 268.321635 \n",
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"z\n",
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"\" style=\"fill: #ffffff\"/>\n",
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" </g>\n",
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" <g id=\"matplotlib.axis_1\">\n",
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" <g id=\"xtick_1\">\n",
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" <g id=\"line2d_1\">\n",
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" <defs>\n",
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" <path id=\"m025334e71f\" d=\"M 0 0 \n",
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"L 0 3.5 \n",
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"\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </defs>\n",
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" <g>\n",
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" <use xlink:href=\"#m025334e71f\" x=\"42.740303\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"text_1\">\n",
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" <!-- 40 -->\n",
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" <g transform=\"translate(36.377803 282.920073) scale(0.1 -0.1)\">\n",
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" <defs>\n",
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"M 2034 4750 \n",
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"Q 2819 4750 3233 4129 \n",
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"Q 3647 3509 3647 2328 \n",
|
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|
|
"Q 3647 1150 3233 529 \n",
|
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|
|
"Q 2819 -91 2034 -91 \n",
|
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|
"Q 1250 -91 836 529 \n",
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"Q 422 1150 422 2328 \n",
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"Q 422 3509 836 4129 \n",
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"Q 1250 4750 2034 4750 \n",
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"z\n",
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"\" transform=\"scale(0.015625)\"/>\n",
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" </g>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"xtick_2\">\n",
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" <g id=\"line2d_2\">\n",
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" <g>\n",
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" <use xlink:href=\"#m025334e71f\" x=\"80.316165\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"text_2\">\n",
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"L 1269 4134 \n",
|
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|
"L 1269 2991 \n",
|
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|
|
"Q 1406 3038 1543 3061 \n",
|
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"Q 1681 3084 1819 3084 \n",
|
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"Q 2600 3084 3056 2656 \n",
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|
"Q 3513 2228 3513 1497 \n",
|
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"Q 3513 744 3044 326 \n",
|
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"Q 2575 -91 1722 -91 \n",
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"Q 2250 441 2565 725 \n",
|
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|
|
"Q 2881 1009 2881 1497 \n",
|
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|
|
"Q 2881 1984 2565 2268 \n",
|
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"Q 2250 2553 1709 2553 \n",
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"Q 1456 2553 1204 2497 \n",
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"Q 953 2441 691 2322 \n",
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"L 691 4666 \n",
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" </g>\n",
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" </g>\n",
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" <g id=\"xtick_3\">\n",
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" <use xlink:href=\"#m025334e71f\" x=\"117.892027\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </g>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"xtick_4\">\n",
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" <g id=\"line2d_4\">\n",
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" <g>\n",
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" <use xlink:href=\"#m025334e71f\" x=\"155.467888\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </g>\n",
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" <g id=\"text_4\">\n",
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" <!-- 55 -->\n",
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" <use xlink:href=\"#DejaVuSans-35\"/>\n",
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" </g>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"xtick_5\">\n",
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" <g id=\"line2d_5\">\n",
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" <g>\n",
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" <use xlink:href=\"#m025334e71f\" x=\"193.04375\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"text_5\">\n",
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" <!-- 60 -->\n",
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" <g transform=\"translate(186.68125 282.920073) scale(0.1 -0.1)\">\n",
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" <defs>\n",
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" <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
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"Q 1688 2584 1439 2293 \n",
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"Q 1191 2003 1191 1497 \n",
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"Q 1191 994 1439 701 \n",
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|
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"Q 1688 409 2113 409 \n",
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"text/plain": [
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||
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"┌─────────────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ Migrad │\n",
|
||
|
|
"├──────────────────────────────────┬──────────────────────────────────────┤\n",
|
||
|
|
"│ FCN = 106.4 (χ²/ndof = 0.9) │ Nfcn = 530 │\n",
|
||
|
|
"│ EDM = 1.61e-05 (Goal: 0.0002) │ │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Valid Minimum │ Below EDM threshold (goal x 10) │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ No parameters at limit │ Below call limit │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Hesse ok │ Covariance accurate │\n",
|
||
|
|
"└──────────────────────────────────┴──────────────────────────────────────┘\n",
|
||
|
|
"┌───┬──────────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n",
|
||
|
|
"│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n",
|
||
|
|
"├───┼──────────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n",
|
||
|
|
"│ 0 │ A_p1 │ 317 │ 7 │ │ │ │ │ │\n",
|
||
|
|
"│ 1 │ A_p2 │ 580 │ 7 │ │ │ │ │ │\n",
|
||
|
|
"│ 2 │ mu_p1 │ 53.24 │ 0.07 │ │ │ │ │ │\n",
|
||
|
|
"│ 3 │ mu_p2 │ 60.43 │ 0.05 │ │ │ │ │ │\n",
|
||
|
|
"│ 4 │ sigma_p1 │ 1.99 │ 0.05 │ │ │ │ │ │\n",
|
||
|
|
"│ 5 │ sigma_p2 │ 2.80 │ 0.04 │ │ │ │ │ │\n",
|
||
|
|
"│ 6 │ A_bkg │ 147 │ 14 │ │ │ │ │ │\n",
|
||
|
|
"│ 7 │ tau_bkg │ 34.1 │ 2.0 │ │ │ 0 │ │ │\n",
|
||
|
|
"└───┴──────────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n",
|
||
|
|
"┌──────────┬─────────────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ │ A_p1 A_p2 mu_p1 mu_p2 sigma_p1 sigma_p2 A_bkg tau_bkg │\n",
|
||
|
|
"├──────────┼─────────────────────────────────────────────────────────────────────────┤\n",
|
||
|
|
"│ A_p1 │ 51.5 10 0.103 0.1006 -0.0808 -0.0969 -0 0 │\n",
|
||
|
|
"│ A_p2 │ 10 50.6 0.026 0.0402 -0.0047 -0.1329 -0 0 │\n",
|
||
|
|
"│ mu_p1 │ 0.103 0.026 0.00503 0.0027 0.0025 -0.0020 -0.057 0.010 │\n",
|
||
|
|
"│ mu_p2 │ 0.1006 0.0402 0.0027 0.00276 0.0018 -0.0015 -0.0513 0.0062 │\n",
|
||
|
|
"│ sigma_p1 │ -0.0808 -0.0047 0.0025 0.0018 0.00297 -0.0012 -0.1409 0.0155 │\n",
|
||
|
|
"│ sigma_p2 │ -0.0969 -0.1329 -0.0020 -0.0015 -0.0012 0.00171 0.0816 -0.0142 │\n",
|
||
|
|
"│ A_bkg │ -0 -0 -0.057 -0.0513 -0.1409 0.0816 209 -28 │\n",
|
||
|
|
"│ tau_bkg │ 0 0 0.010 0.0062 0.0155 -0.0142 -28 4.01 │\n",
|
||
|
|
"└──────────┴─────────────────────────────────────────────────────────────────────────┘"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 519,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"ls = cost.LeastSquares(center, entries, np.sqrt(entries), fit_model)\n",
|
||
|
|
"\n",
|
||
|
|
"mi = Minuit(ls, \n",
|
||
|
|
" A_p1 = 800, \n",
|
||
|
|
" A_p2 = 1400,\n",
|
||
|
|
" mu_p1 = 54,\n",
|
||
|
|
" mu_p2 = 60,\n",
|
||
|
|
" sigma_p1 = 2,\n",
|
||
|
|
" sigma_p2 = 2,\n",
|
||
|
|
" A_bkg = 100,\n",
|
||
|
|
" tau_bkg = 10, \n",
|
||
|
|
" )\n",
|
||
|
|
"mi.limits['tau_bkg'] = (0, None)\n",
|
||
|
|
"mi.fixed[:] = True\n",
|
||
|
|
"ls.mask = (center < 45) | (center >= 70)\n",
|
||
|
|
"mi.fixed[['tau_bkg', 'A_bkg']] = False\n",
|
||
|
|
"mi.migrad()\n",
|
||
|
|
"ls.mask = None\n",
|
||
|
|
"mi.values['A_p1'] = 700\n",
|
||
|
|
"mi.values['sigma_p1'] = 3\n",
|
||
|
|
"mi.fixed[:] = True\n",
|
||
|
|
"mi.fixed[['A_p1', 'mu_p1', 'sigma_p1']] = False\n",
|
||
|
|
"mi.migrad()\n",
|
||
|
|
"mi.fixed[:] = True\n",
|
||
|
|
"mi.fixed[['A_p2', 'mu_p2', 'sigma_p2']] = False\n",
|
||
|
|
"mi.migrad()\n",
|
||
|
|
"mi.fixed[:] = False\n",
|
||
|
|
"mi.migrad()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "b2d4c8e9-da2c-489e-9b2f-de24f042c341",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
" # Wann fittet ein Fit?\n",
|
||
|
|
"Nach dem wir nun unser Model an unsere Daten angepasst haben, stellt sich die Frage: „Spiegelt unser Model unsere Daten gut wider?“. Um diese Frage beantworten zu können, gibt es verschiedene Möglichkeiten, welche wir im Folgenden etwas näher betrachten wollen. \n",
|
||
|
|
"## Fit Residual: \n",
|
||
|
|
"Schauen wir uns zunächst noch einmal an, wie das Chi-Quadrat definiert ist:\n",
|
||
|
|
"$$ \\chi^2 = \\sum_i \\frac{(y_i - \\lambda_i)^2}{\\Delta y_i^2} $$\n",
|
||
|
|
"Wir minimieren den Abstand zwischen einem Messwert und unserem Model und gewichten diesen mit den Unsicherheiten unserer Messwerte. Fitresiduen spiegeln genau dies wider. Sie sind definiert als \n",
|
||
|
|
"$$ \\frac{(y_i - \\lambda_i)}{\\Delta y_i} $$\n",
|
||
|
|
"Für unseren Fit sehen sie wie folgt aus.\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 520,
|
||
|
|
"id": "30cafddc-ea17-4158-82cc-f132dee2c8de",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"Text(0, 0.5, 'Residuals [$\\\\sigma$]')"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 520,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
},
|
||
|
|
{
|
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|
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"data": {
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|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 1 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"residuals = (entries - fit_model(center, *mi.values))/np.sqrt(entries)\n",
|
||
|
|
"\n",
|
||
|
|
"plt.plot(center, residuals, ls='', marker='.')\n",
|
||
|
|
"plt.xlabel('Energy [keV]')\n",
|
||
|
|
"plt.ylabel('Residuals [$\\sigma$]')"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "d0ef61ca-afc5-472d-8e8e-b4726ef2a3dd",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Als einzelner Plot sind sie noch nicht sehr informativ. Hilfreicher ist es bereits, wenn wir die Residuen zusammen mit unseren Daten und Fitmodel darstellen. "
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 521,
|
||
|
|
"id": "d9fbe83b-3146-4d72-89a4-084c29752e24",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stderr",
|
||
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|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"C:\\Users\\Matthias\\AppData\\Local\\Temp\\ipykernel_67644\\53208542.py:7: UserWarning: The figure layout has changed to tight\n",
|
||
|
|
" fig_fit.tight_layout()\n"
|
||
|
|
]
|
||
|
|
},
|
||
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{
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||
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"data": {
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|
|
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAnYAAAHWCAYAAAD6oMSKAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjguMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8g+/7EAAAACXBIWXMAAA9hAAAPYQGoP6dpAADBAUlEQVR4nOzdd3hUZfrw8e/MpPcCSQiBkJAAoUNAiICLgCBNUdaKCygKq4hgl9/r2tjVVVdFWHRBUbBjwxVUqiC9t0BCGukVSE9ImznvH7NzSIUMJJkkc3+45iJzzplz7hPI5J6n3I9GURQFIYQQQgjR5mktHYAQQgghhGgaktgJIYQQQrQTktgJIYQQQrQTktgJIYQQQrQTktgJIYQQQrQTktgJIYQQQrQTktgJIYQQQrQTktgJIYQQQrQTNpYOoDUwGAxkZGTg6uqKRqOxdDhCCCGEsBKKolBUVIS/vz9a7fW3t0liB2RkZNClSxdLhyGEEEIIK5WamkpAQMB1n0cSO8DV1RUwflPd3NwsHI0QQgghrEVhYSFdunRRc5HrJYkdqN2vbm5uktgJIYQQosU11VAwmTwhhBBCCNFOSGInhBBCCNFOSGInhBBCCNFOyBg7IYQQop3T6/VUVlZaOgyrZGtri06na7HrSWInhBBCtFOKopCVlUV+fr6lQ7FqHh4e+Pn5tUitXEnshBBCiHbKlNT5+Pjg5OQkRfhbmKIolJaWkpOTA0CnTp2a/ZqS2AkhhBDtkF6vV5M6b29vS4djtRwdHQHIycnBx8en2btlZfKEEEII0Q6ZxtQ5OTlZOBJh+jdoiXGOktgJIYQQ7Zh0v1peS/4bSGInhBBCCNFOSGInhBBCCNGAV155BV9fXzQaDT/99BOzZ89m2rRplg6rQZLYCSGEEKLVmD17NhqNRn14e3tz6623curUqSa7xiuvvMLAgQOvelx0dDSvvvoqK1euJDMzk4kTJ/L++++zZs0a9ZjRo0ezaNGiJovtekliJ4QQzaCkpET9xVRSUmLpcIRoU2699VYyMzPJzMxk+/bt2NjYMGXKlBaPIyEhAYDbb78dPz8/7O3tcXd3x8PDo8VjaSxJ7IQQQgjRqtjb2+Pn54efnx8DBw7khRdeIDU1lfPnz6vHpKamcvfdd+Ph4YGXlxe33347SUlJ6v6dO3dyww034OzsjIeHByNGjCA5OZk1a9bw6quvcvLkSfXDV/UWOJNXXnmFqVOnAqDVatUJENW7YmfPns0ff/zB+++/r56regyWIHXshBBCCCugKAqVpZZZVszWyfaaZ4YWFxfzxRdfEBISotbjq6ysZMKECURERLB7925sbGz4+9//rnbZarVapk2bxiOPPMLXX39NRUUFhw4dQqPRcM8993D69Gk2bdrEtm3bAHB3d69z3WeeeYZu3brx4IMPkpmZWW9s77//PrGxsfTt25fXXnsNgI4dO17TfTYVSeyEEEIIK1BZWskbLm9Y5NqLixdj52zX6OM3btyIi4sLYBzW0KlTJzZu3IhWa+xoXLduHQaDgY8//lhNGD/99FM8PDzYuXMnQ4YMoaCggClTptC9e3cAwsLC1PO7uLhgY2ODn59fgzG4uLioXa4NHefu7o6dnR1OTk5XPFdLkq5YIYQQQrQqN998MydOnODEiRMcOnSICRMmMHHiRJKTkwE4efIk8fHxuLq64uLigouLC15eXpSVlZGQkICXlxezZ89mwoQJTJ06lffff7/BVrf2RlrshBBCCCtg62TL4uLFFru2OZydnQkJCVGff/zxx7i7u/PRRx/x97//neLiYsLDw/nyyy/rvNbUFfrpp5/yxBNPsGnTJtatW8eLL77I1q1bGT58+PXdTCsniZ0QQghhBTQajVndoa2JRqNBq9Vy6dIlAAYPHsy6devw8fHBzc2twdcNGjSIQYMGsXjxYiIiIvjqq68YPnw4dnZ26PX6JomtKc/VFKQrVgghhBCtSnl5OVlZWWRlZREdHc2CBQsoLi5WZ6nOmDGDDh06cPvtt7N7924SExPZuXMnTzzxBGlpaSQmJrJ48WL2799PcnIyW7ZsIS4uTh1n161bNxITEzlx4gQXLlygvLz8mmPt1q0bBw8eJCkpiQsXLmAwGJrke3CtJLETQohmlp6eXu92qXUnRP02bdpEp06d6NSpE8OGDePw4cN89913jB49GgAnJyd27dpF165dufPOOwkLC2POnDmUlZXh5uaGk5MTZ8+eZfr06fTo0YO5c+cyf/585s2bB8D06dO59dZbufnmm+nYsSNff/31Ncf6zDPPoNPp6N27Nx07diQlJaUpvgXXTKMoimLRCFqBwsJC3N3dKSgouGKTrhBCNNYHH3zA/PnzAWMNrFWrVjFnzpwax5SUlKgz/4qLi3F2dm7xOEX7VVZWRmJiIkFBQTg4OFg6HKt2pX+Lps5BpMVOCCGaWFpaGgsWLFCfGwwG5s2bR1pamgWjEkJYA0nshBCiicXFxdUZZ6PX64mPj7dQREIIayGJnRBCNLHQ0FC1kKqJTqerUb5BCCGagyR2QgjRxAICAli+fLn6XKfTsXLlSgICAiwYlRDCGkhiJ4QQzWDWrFnq11FRUXUmTjREZsqKpiZzJC2vJf8NJLETQggzmZt8de7cuQWiEqImW1vjag+lpaUWjkSY/g1M/ybNSVaeEEKIViA9PZ0ePXpYOgzRjuh0Ojw8PMjJyQGMtd80Go2Fo7IuiqJQWlpKTk4OHh4e6HS6Zr+mJHZCCGEha9euVb8OCwtj1apV3HvvvRaMSLQ3fn5+AGpyJyzDw8ND/bdobpLYCSGEBTRU627UqFEWjEq0NxqNhk6dOuHj40NlZaWlw7FKtra2LdJSZ2LxxC49PZ3nn3+e3377jdLSUkJCQvj0008ZMmQIYGzGfPnll/noo4/Iz89nxIgRfPjhh4SGhqrnyM3NZcGCBWzYsAGtVsv06dN5//331YruQgjR2jRU6y4hIcFCEYn2TKfTtWhyISzHopMn8vLyGDFiBLa2tvz2229ERUXxzjvv4OnpqR7z1ltvsWzZMv7zn/9w8OBBnJ2dmTBhAmVlZeoxM2bM4MyZM2zdupWNGzeya9cu5s6da4lbEkJYKdOECheNCwdXHmT3K7u5mZsZwAAuXbxU5/iGat117969pUIWQrRDFl0r9oUXXmDv3r3s3r273v2KouDv78/TTz/NM888A0BBQQG+vr6sWbOGe++9l+joaHr37s3hw4fVVr5NmzYxadIk0tLS8Pf3v2ocslasEMIc9a3xmnMuh3nd59GPfuio2TKitdEy8KGBjPn7GJw7Xl4Ptvp6sqZad/fee6+sHyuEFWlXa8X+/PPPDBkyhLvuugsfHx8GDRrERx99pO5PTEwkKyuLcePGqdvc3d0ZNmwY+/fvB2D//v14eHioSR3AuHHj0Gq1HDx4sOVuRghhtRK2JPBJ+CcMZCA6dPgO9mXIY0MY8ugQ/Ab6YagycGzVMT7s+yEpe1PU112t1l16enqL3YMQon2w6Bi7c+fO8eGHH/LUU0/xf//3fxw+fJgnnngCOzs7Zs2aRVZWFgC+vr41Xufr66vuy8rKwsfHp8Z+GxsbvLy81GNqKy8vp7y8XH1eWFjYlLclhLAiv6/4neP/dxxFr5BJJhvZSMyumBotbcm7k/nl0V84f+Y8a29ey93f303P23rWOI+p1l19M2UbW9xYCCEs2mJnMBgYPHgwr7/+OoMGDWLu3Lk88sgj/Oc//2nW677xxhu4u7urjy5dujTr9YQQbVvtgsTVk6/bn7+do/qj9L6vNx/zMenUbWULHBXIwwcfptcdvTBUGvju7u9I3JFY57iGZsqmpaU1z40JIdodiyZ2nTp1onfv3jW2hYWFkZJi7Kow1XzJzs6ucUx2dra6z8/Pr059nqqqKnJzcxusGbN48WIKCgrUR2pqapPcjxCi/UtPT6+RfCkobNRspP/L/dGjb/B1ds523PXtXfSa1gt9uZ7v/vwdBSkFNY5paKZsfHx8096EEKLdsmhiN2LECGJiYmpsi42NJTAwEICgoCD8/PzYvn27ur+wsJCDBw8SERE
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 2 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"fig_fit = plt.figure(constrained_layout=True)\n",
|
||
|
|
"gs = fig_fit.add_gridspec(5, 5, hspace=0)\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"main_axis = fig_fit.add_subplot(gs[:4, :])\n",
|
||
|
|
"res_axis = fig_fit.add_subplot(gs[4:, :], sharex=main_axis)\n",
|
||
|
|
"fig_fit.tight_layout()\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"main_axis.errorbar(center, entries, np.sqrt(entries), ls='', marker='.', color='k')\n",
|
||
|
|
"\n",
|
||
|
|
"main_axis.plot(x, peak(x, *mi.values['A_p1', 'mu_p1', 'sigma_p1']), color='gray', ls='--')\n",
|
||
|
|
"main_axis.plot(x, peak(x, *mi.values['A_p2', 'mu_p2', 'sigma_p2']), color='gray', ls='-.')\n",
|
||
|
|
"main_axis.plot(x, bkg(x, *mi.values['A_bkg', 'tau_bkg']), color='gray')\n",
|
||
|
|
"\n",
|
||
|
|
"x = np.arange(40, 80, 0.1)\n",
|
||
|
|
"main_axis.plot(x, fit_model(x, *mi.values), color='purple', label='Best fit')\n",
|
||
|
|
"main_axis.legend()\n",
|
||
|
|
"main_axis.set_ylabel('Number of entries per bin')\n",
|
||
|
|
"main_axis.xaxis.set_tick_params(direction='inout')\n",
|
||
|
|
"main_axis.tick_params(axis='x', labelcolor=(0, 0, 0, 0))\n",
|
||
|
|
"main_axis.set_xlim(40, 80)\n",
|
||
|
|
"\n",
|
||
|
|
"res_axis.set_xlabel('Energy [keV]')\n",
|
||
|
|
"res_axis.set_ylabel('Res [$\\sigma$]')\n",
|
||
|
|
"res_axis.set_ylim(-3, 3)\n",
|
||
|
|
"res_axis.set_yticks([-2, 0, 2])\n",
|
||
|
|
"res_axis.fill_between((40, 80), -1, 1, alpha=0.3, color='purple')\n",
|
||
|
|
"res_axis.fill_between((40, 80), -2, 2, alpha=0.3, color='purple')\n",
|
||
|
|
"res_axis.axhline(0, color='purple')\n",
|
||
|
|
"res_axis.set_xlim(40, 80)\n",
|
||
|
|
"res_axis.plot(center, \n",
|
||
|
|
" residuals,\n",
|
||
|
|
" color='k', marker='.', ls=''\n",
|
||
|
|
" )\n",
|
||
|
|
"plt.tight_layout()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "dbe65a21-572e-4618-bcd8-78f13e945e8a",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Sofern unser Fitmodel unsere Daten gut beschreibt, erwarten wir, dass die Residuen sich Gaußförmig zufällig um den Wert 0 herum verteilen. Dies folgt direkt aus der Annahme, dass sich die Unsicherheiten unserer Messwerte durch eine Gaußverteilung darstellen lassen. Dies können wir direkt überprüfen, sofern wir unsere Residuen in ein Histogramm eintragen. "
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 522,
|
||
|
|
"id": "05e24224-66f7-45ed-99c6-f6d257e2c779",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAjIAAAGzCAYAAAA1yP25AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjguMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8g+/7EAAAACXBIWXMAAA9hAAAPYQGoP6dpAAAukUlEQVR4nO3dfVRVdaLG8ecgcETeQRFQQJIZy3zrYhqjJQqpNGmMTDk194bp1LXQUuyNxmrsRcpetLpmTRnq3Fxa3tDJVppaYI3iKMWoWd5g7Pr+UgrIEQE55/7R9dxhRDxHD+yz8ftZa6/Ffjn7POzlkmft/dt7WxwOh0MAAAAm5GN0AAAAgItFkQEAAKZFkQEAAKZFkQEAAKZFkQEAAKZFkQEAAKZFkQEAAKZFkQEAAKbla3SA1ma323Xw4EEFBwfLYrEYHQcAALjA4XDo5MmTio2NlY/P+c+7tPsic/DgQcXFxRkdAwAAXIR9+/ape/fu513f7otMcHCwpJ8OREhIiMFpAACAK6qrqxUXF+f8O34+7b7InL2cFBISQpEBAMBkLjQshMG+AADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtNr9268BtG8HKmt1wlZvdAy3hAf6q1tYgNExgHaBIgPAtA5U1ir9pWLVNjQaHcUtAX4dtH7GMMoM4AEUGQCmdcJWr9qGRs0bP0BJUUFGx3FJ+dEaTVtephO2eooM4AEUGQCmlxQVpD7dQo2OAcAADPYFAACmRZEBAACmRZEBAACmZWiRWbBggfr166eQkBCFhIQoJSVFH3/8sXN9amqqLBZLk2ny5MkGJgYAAN7E0MG+3bt313PPPaef/exncjgcWrx4sW655RZ99dVXuvrqqyVJd999t5566innZzp16mRUXAAA4GUMLTJjxoxpMv/ss89qwYIFKikpcRaZTp06KTo62oh4AADAy3nNGJnGxkYtW7ZMNptNKSkpzuXvvvuuOnfurD59+igvL0+nTp1qcT91dXWqrq5uMgEAgPbJ8OfI7NixQykpKTp9+rSCgoJUWFio3r17S5LuuOMOJSQkKDY2Vtu3b9cjjzyi3bt364MPPjjv/vLz8zVr1qy2ig8AAAxkeJHp1auXysrKVFVVpRUrVig7O1vFxcXq3bu37rnnHud2ffv2VUxMjNLS0lRRUaGePXs2u7+8vDzl5uY656urqxUXF9fqvwcAAGh7hhcZf39/JSUlSZKSk5O1detWvfLKK3rzzTfP2Xbw4MGSpPLy8vMWGavVKqvV2nqBAQCA1/CaMTJn2e121dXVNbuurKxMkhQTE9OGiQAAgLcy9IxMXl6eMjIyFB8fr5MnT2rp0qUqKirS2rVrVVFRoaVLl+qmm25SZGSktm/frunTp+uGG25Qv379jIwNAAC8hKFF5ujRo7rzzjt16NAhhYaGql+/flq7dq1uvPFG7du3T+vXr9e8efNks9kUFxenrKwszZw508jIAADAixhaZBYuXHjedXFxcSouLm7DNAAAwGy8bowMAACAqygyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtCgyAADAtAwtMgsWLFC/fv0UEhKikJAQpaSk6OOPP3auP336tHJychQZGamgoCBlZWXpyJEjBiYGAADexNAi0717dz333HMqLS3Vtm3bNGLECN1yyy36+uuvJUnTp0/Xhx9+qPfff1/FxcU6ePCgxo0bZ2RkAADgRXyN/PIxY8Y0mX/22We1YMEClZSUqHv37lq4cKGWLl2qESNGSJIKCgp01VVXqaSkRNddd50RkQEAgBfxmjEyjY2NWrZsmWw2m1JSUlRaWqqGhgalp6c7t7nyyisVHx+vzZs3n3c/dXV1qq6ubjIBAID2yfAis2PHDgUFBclqtWry5MkqLCxU7969dfjwYfn7+yssLKzJ9l27dtXhw4fPu7/8/HyFhoY6p7i4uFb+DQAAgFEMLzK9evVSWVmZtmzZonvvvVfZ2dnatWvXRe8vLy9PVVVVzmnfvn0eTAsAALyJoWNkJMnf319JSUmSpOTkZG3dulWvvPKKxo8fr/r6elVWVjY5K3PkyBFFR0efd39Wq1VWq7W1YwMAAC9g+BmZf2a321VXV6fk5GT5+flpw4YNznW7d+/W3r17lZKSYmBCAADgLQw9I5OXl6eMjAzFx8fr5MmTWrp0qYqKirR27VqFhoZq0qRJys3NVUREhEJCQjR16lSlpKRwxxIAAJBkcJE5evSo7rzzTh06dEihoaHq16+f1q5dqxtvvFGSNHfuXPn4+CgrK0t1dXUaNWqUXn/9dSMjAwAAL2JokVm4cGGL6zt27Kj58+dr/vz5bZQIAACYideNkQEAAHAVRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJgWRQYAAJiWoUUmPz9f1157rYKDgxUVFaXMzEzt3r27yTapqamyWCxNpsmTJxuUGAAAeBNDi0xxcbFycnJUUlKidevWqaGhQSNHjpTNZmuy3d13361Dhw45pzlz5hiUGAAAeBNfI798zZo1TeYXLVqkqKgolZaW6oYbbnAu79Spk6Kjo9s6HgAA8HJeNUamqqpKkhQREdFk+bvvvqvOnTurT58+ysvL06lTp867j7q6OlVXVzeZAABA+2ToGZl/ZLfbNW3aNA0ZMkR9+vRxLr/jjjuUkJCg2NhYbd++XY888oh2796tDz74oNn95Ofna9asWW0VGwAAGMhrikxOTo527typL774osnye+65x/lz3759FRMTo7S0NFVUVKhnz57n7CcvL0+5ubnO+erqasXFxbVecAAAYBivKDJTpkzR6tWrtXHjRnXv3r3FbQcPHixJKi8vb7bIWK1WWa3WVskJAAC8i6FFxuFwaOrUqSosLFRRUZESExMv+JmysjJJUkxMTCunAwAA3u6iiozdbld5ebmOHj0qu93eZN0/3m10ITk5OVq6dKlWrVql4OBgHT58WJIUGhqqgIAAVVRUaOnSpbrpppsUGRmp7du3a/r06brhhhvUr1+/i4kOAADaEbeLTElJie644w79z//8jxwOR5N1FotFjY2NLu9rwYIFkn566N0/Kigo0IQJE+Tv76/169dr3rx5stlsiouLU1ZWlmbOnOlubAAA0A65XWQmT56sgQMH6qOPPlJMTIwsFstFf/k/F6F/FhcXp+Li4ovePwAAaN/cLjLfffedVqxYoaSkpNbIAwAA4DK3H4g3ePBglZeXt0YWAAAAt7h9Rmbq1KmaMWOGDh8+rL59+8rPz6/JegbhAgCAtuJ2kcnKypIkTZw40bnMYrHI4XC4PdgXAADgUrhdZPbs2dMaOQAAANzmdpFJSEhojRwAAABuc6nI/PnPf1ZGRob8/Pz05z//ucVtx44d65FgAAAAF+JSkcnMzNThw4cVFRWlzMzM827HGBkAANCWXCoy//gagn9+JQEAAIBR3H6ODAAAgLe4qCKzYcMG3XzzzerZs6d69uypm2++WevXr/d0NgAAgBa5XWR
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 1 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"plt.hist(residuals, bins=10, range=(-3, 3), histtype='step')\n",
|
||
|
|
"plt.xlabel('Residual [$\\sigma$]')\n",
|
||
|
|
"plt.ylabel('#Entries per bin')\n",
|
||
|
|
"plt.show()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "24ce04cc-5234-4326-9a28-7624b9c7d23e",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Bzw. den Anteil an Residuen berechnen, welcher innerhalb der 1 $\\sigma$ Umgebung liegt."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 523,
|
||
|
|
"id": "39009321-41f4-49f4-820a-717be277b1b0",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"0.6833333333333333"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 523,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"np.sum(np.abs(residuals) < 1)/len(residuals)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "08579cdf-3b28-4ea2-9c61-6ae62974af51",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Zeigen unsere Residuen eine Struktur oder ein systematisches Verhalten, deutet dies auf einen ungenauen Fit oder ein falsches Fitmodel hin. Dies ist im Folgenden gezeigt. "
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 524,
|
||
|
|
"id": "850870af-e546-4d95-b9de-8a4e7b61c241",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stderr",
|
||
|
|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"C:\\Users\\Matthias\\AppData\\Local\\Temp\\ipykernel_67644\\2321973434.py:8: UserWarning: The figure layout has changed to tight\n",
|
||
|
|
" fig_fit.tight_layout()\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"image/png": "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
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 2 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"pseudo_data = np.random.normal(0, 2, 5000)\n",
|
||
|
|
"\n",
|
||
|
|
"fig_fit = plt.figure(constrained_layout=True)\n",
|
||
|
|
"gs = fig_fit.add_gridspec(5, 5, hspace=0)\n",
|
||
|
|
"\n",
|
||
|
|
"main_axis = fig_fit.add_subplot(gs[:4, :])\n",
|
||
|
|
"res_axis = fig_fit.add_subplot(gs[4:, :], sharex=main_axis)\n",
|
||
|
|
"fig_fit.tight_layout()\n",
|
||
|
|
"\n",
|
||
|
|
"entries1, edges1, _ = main_axis.hist(pseudo_data, bins=25, range=(-5,5), histtype='step', color='k')\n",
|
||
|
|
"center1 = edges1[:-1] + np.diff(edges1)/2\n",
|
||
|
|
"\n",
|
||
|
|
"residuals1 = (entries1 - peak(center1, 400, 0.2, 2))/np.sqrt(entries1)\n",
|
||
|
|
"\n",
|
||
|
|
"x = np.arange(-5, 5, 0.1)\n",
|
||
|
|
"\n",
|
||
|
|
"main_axis.plot(x, peak(x, 400, 0.2, 2), color='purple')\n",
|
||
|
|
"main_axis.set_ylabel('Number of entries per bin')\n",
|
||
|
|
"main_axis.xaxis.set_tick_params(direction='inout')\n",
|
||
|
|
"main_axis.tick_params(axis='x', labelcolor=(0, 0, 0, 0))\n",
|
||
|
|
"main_axis.set_xlim(-5, 5)\n",
|
||
|
|
"\n",
|
||
|
|
"res_axis.set_xlabel('Energy [keV]')\n",
|
||
|
|
"res_axis.set_ylabel('Res [$\\sigma$]')\n",
|
||
|
|
"res_axis.set_ylim(-3, 3)\n",
|
||
|
|
"res_axis.set_yticks([-2, 0, 2])\n",
|
||
|
|
"res_axis.fill_between((-5, 5), -1, 1, alpha=0.3, color='purple')\n",
|
||
|
|
"res_axis.fill_between((-5, 5), -2, 2, alpha=0.3, color='purple')\n",
|
||
|
|
"res_axis.axhline(0, color='purple')\n",
|
||
|
|
"res_axis.set_xlim(-5, 5)\n",
|
||
|
|
"res_axis.plot(center1, \n",
|
||
|
|
" residuals1,\n",
|
||
|
|
" color='k', marker='.', ls=''\n",
|
||
|
|
" )\n",
|
||
|
|
"plt.tight_layout()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "48e95a88-0742-4221-a716-17dacfc02823",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Zusätzlich zu den Fit-Residuen bietet das $\\chi^2$ selbst einen Weg, um die „goodness-of-fit“ unseres Model bestimmen zu können ...\n",
|
||
|
|
"\n",
|
||
|
|
"### $\\chi^2$:"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "fe1789cf-7ed3-4db3-a0ae-9e563a9dc85e",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Wie gut fittet unsere obige Funktion unsere Messdaten? Sehr gut? Gut? Befriedigend? Oder doch eher schlecht? Wäre es nicht gut, ein Maß für die Güte des Fits zu haben? Wie könnte ein solches Maß aussehen?\n",
|
||
|
|
"\n",
|
||
|
|
"Sie haben das entscheidende Kriterium bereits kennengelernt: bei der Methode der kleinsten Quadrate geht es darum, das $\\chi^2$ zu minimieren. Gucken wir uns hierzu erst noch einmal an, wie sich das $\\chi^2$ berechnet:\n",
|
||
|
|
"\n",
|
||
|
|
"$$ \\chi(\\phi_1 ... \\phi_N)^2 = \\sum_{i = 1}^{N} \\frac{ (y_i - \\lambda(x_i; \\phi))^2}{\\Delta y_i^2}$$\n",
|
||
|
|
"\n",
|
||
|
|
"Bei der Minimierung werden dabei Werte mit geringerer Unsicherheit bevorzugt, d.h. stärker gewichtet (s. Bild unten).\n",
|
||
|
|
"\n",
|
||
|
|
"<figure class=\"image\">\n",
|
||
|
|
"<img src=\"images/MaterialPythonkurs092018/LeastSquare.png\" alt=\"{{ Least Square Beispiel }}\" width=80%>\n",
|
||
|
|
"</figure>\n",
|
||
|
|
"\n",
|
||
|
|
"Damit man für einen gegebenen Datensatz nicht hunderte von verschiedenen Funktionen durchprobieren muss, gibt es für das $\\chi^2$ eine allgemeine Faustregel, welche den berechneten $\\chi^2$-Wert mit der Anzahl unserer Freiheitsgrade vergleicht. Die Anzahl an Freiheitsgrade ist gemeinhin gegeben als *Anzahl der Messwerte - Anzahl der Funktionsparameter* ($m - n$).\n",
|
||
|
|
"\n",
|
||
|
|
"1. Sofern $\\chi^2/\\text{ndof} >> 1$: sollte die Hypothese bzw. die Fitfunktion angezweifelt werden. Sie beschreibt in diesem Fall die Messdaten nur unzureichend. (Bzw. sollte $\\chi^2/\\text{ndof} > 1$ kann dies auch bedeuten, dass die Unsicherheiten unterschätzt sind)\n",
|
||
|
|
"2. Sofern $\\chi^2/\\text{ndof} \\approx 1$: beschreibt die Hypothese bzw. die Fitfunktion die Daten wie erwartet und wird nicht abgelehnt. \n",
|
||
|
|
"3. Falls $\\chi^2/\\text{ndof} << 1$ beschreibt die Hypothese bzw. die Fitfunktion die Daten wesentlich besser als erwartet. In diesem Fall heißt das nicht automatisch, dass unsere Hypothese falsch ist, aber man sollte überprüfen, ob die gemessenen Fehler nicht überschätzt worden sind (oder eine Korrelation zwischen den Messfehlern vorliegt). \n",
|
||
|
|
"\n",
|
||
|
|
"Sofern Sie eine Arbeit schreiben und Ihre **Goodness-of-the-Fit** ($\\chi^2/\\text{ndof}$) angeben wollen, so geben Sie immer beides an, das $\\chi^2$ und die Anzahl an Freiheitsgraden *ndof*. Beide Werte getrennt haben einen größeren Informationsgehalt als der resultierende Quotient (Genaueres lernen Sie z.B. in der Vorlesung *Statistik, Datenanalyse und Simulationen* im Master).\n",
|
||
|
|
"\n",
|
||
|
|
"Sehen wir uns hierzu nochmal unseren Doppelpeakfit etwas genauer an. `iminuit` berechnet hier für uns bereits das reduzierete $\\chi^2$."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 525,
|
||
|
|
"id": "fa85a19a-f066-4567-abb0-6283ae1bc90b",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/html": [
|
||
|
|
"<table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th colspan=\"2\" style=\"text-align:center\" title=\"Minimizer\"> Migrad </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Minimum value of function\"> FCN = 106.4 (χ²/ndof = 0.9) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total number of function and (optional) gradient evaluations\"> Nfcn = 530 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Estimated distance to minimum and goal\"> EDM = 1.61e-05 (Goal: 0.0002) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total run time of algorithms\"> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Valid Minimum </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below EDM threshold (goal x 10) </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> No parameters at limit </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below call limit </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Hesse ok </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Covariance accurate </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th title=\"Variable name\"> Name </th>\n",
|
||
|
|
" <th title=\"Value of parameter\"> Value </th>\n",
|
||
|
|
" <th title=\"Hesse error\"> Hesse Error </th>\n",
|
||
|
|
" <th title=\"Minos lower error\"> Minos Error- </th>\n",
|
||
|
|
" <th title=\"Minos upper error\"> Minos Error+ </th>\n",
|
||
|
|
" <th title=\"Lower limit of the parameter\"> Limit- </th>\n",
|
||
|
|
" <th title=\"Upper limit of the parameter\"> Limit+ </th>\n",
|
||
|
|
" <th title=\"Is the parameter fixed in the fit\"> Fixed </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 0 </th>\n",
|
||
|
|
" <td> A_p1 </td>\n",
|
||
|
|
" <td> 317 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 1 </th>\n",
|
||
|
|
" <td> A_p2 </td>\n",
|
||
|
|
" <td> 580 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 2 </th>\n",
|
||
|
|
" <td> mu_p1 </td>\n",
|
||
|
|
" <td> 53.24 </td>\n",
|
||
|
|
" <td> 0.07 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 3 </th>\n",
|
||
|
|
" <td> mu_p2 </td>\n",
|
||
|
|
" <td> 60.43 </td>\n",
|
||
|
|
" <td> 0.05 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 4 </th>\n",
|
||
|
|
" <td> sigma_p1 </td>\n",
|
||
|
|
" <td> 1.99 </td>\n",
|
||
|
|
" <td> 0.05 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 5 </th>\n",
|
||
|
|
" <td> sigma_p2 </td>\n",
|
||
|
|
" <td> 2.80 </td>\n",
|
||
|
|
" <td> 0.04 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 6 </th>\n",
|
||
|
|
" <td> A_bkg </td>\n",
|
||
|
|
" <td> 147 </td>\n",
|
||
|
|
" <td> 14 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 7 </th>\n",
|
||
|
|
" <td> tau_bkg </td>\n",
|
||
|
|
" <td> 34.1 </td>\n",
|
||
|
|
" <td> 2.0 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <th> A_bkg </th>\n",
|
||
|
|
" <th> tau_bkg </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <td> 51.5 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,227,227);color:black\"> 10 <strong>(0.153)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,220,220);color:black\"> 0.103 <strong>(0.202)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,210,210);color:black\"> 0.1006 <strong>(0.267)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(223,223,250);color:black\"> -0.0808 <strong>(-0.207)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(208,208,250);color:black\"> -0.0969 <strong>(-0.327)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(246,246,250);color:black\"> -0 <strong>(-0.031)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,245,245);color:black\"> 0 <strong>(0.031)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,227,227);color:black\"> 10 <strong>(0.153)</strong> </td>\n",
|
||
|
|
" <td> 50.6 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,242,242);color:black\"> 0.026 <strong>(0.052)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,234,234);color:black\"> 0.0402 <strong>(0.108)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(248,248,250);color:black\"> -0.0047 <strong>(-0.012)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(191,191,250);color:black\"> -0.1329 <strong>(-0.452)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(247,247,250);color:black\"> -0 <strong>(-0.025)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,247,247);color:black\"> 0 <strong>(0.021)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,220,220);color:black\"> 0.103 <strong>(0.202)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,242,242);color:black\"> 0.026 <strong>(0.052)</strong> </td>\n",
|
||
|
|
" <td> 0.00503 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,142,142);color:black\"> 0.0027 <strong>(0.720)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,151,151);color:black\"> 0.0025 <strong>(0.659)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(163,163,250);color:black\"> -0.0020 <strong>(-0.666)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(243,243,250);color:black\"> -0.057 <strong>(-0.055)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,239,239);color:black\"> 0.010 <strong>(0.072)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,210,210);color:black\"> 0.1006 <strong>(0.267)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,234,234);color:black\"> 0.0402 <strong>(0.108)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,142,142);color:black\"> 0.0027 <strong>(0.720)</strong> </td>\n",
|
||
|
|
" <td> 0.00276 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,156,156);color:black\"> 0.0018 <strong>(0.623)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(162,162,250);color:black\"> -0.0015 <strong>(-0.680)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(241,241,250);color:black\"> -0.0513 <strong>(-0.068)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,241,241);color:black\"> 0.0062 <strong>(0.059)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(223,223,250);color:black\"> -0.0808 <strong>(-0.207)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(248,248,250);color:black\"> -0.0047 <strong>(-0.012)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,151,151);color:black\"> 0.0025 <strong>(0.659)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,156,156);color:black\"> 0.0018 <strong>(0.623)</strong> </td>\n",
|
||
|
|
" <td> 0.00297 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(183,183,250);color:black\"> -0.0012 <strong>(-0.518)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(227,227,250);color:black\"> -0.1409 <strong>(-0.179)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,229,229);color:black\"> 0.0155 <strong>(0.142)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(208,208,250);color:black\"> -0.0969 <strong>(-0.327)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(191,191,250);color:black\"> -0.1329 <strong>(-0.452)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(163,163,250);color:black\"> -0.0020 <strong>(-0.666)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(162,162,250);color:black\"> -0.0015 <strong>(-0.680)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(183,183,250);color:black\"> -0.0012 <strong>(-0.518)</strong> </td>\n",
|
||
|
|
" <td> 0.00171 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,229,229);color:black\"> 0.0816 <strong>(0.137)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(228,228,250);color:black\"> -0.0142 <strong>(-0.172)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_bkg </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(246,246,250);color:black\"> -0 <strong>(-0.031)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(247,247,250);color:black\"> -0 <strong>(-0.025)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(243,243,250);color:black\"> -0.057 <strong>(-0.055)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(241,241,250);color:black\"> -0.0513 <strong>(-0.068)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(227,227,250);color:black\"> -0.1409 <strong>(-0.179)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,229,229);color:black\"> 0.0816 <strong>(0.137)</strong> </td>\n",
|
||
|
|
" <td> 209 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(125,125,250);color:black\"> -28 <strong>(-0.965)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> tau_bkg </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,245,245);color:black\"> 0 <strong>(0.031)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,247,247);color:black\"> 0 <strong>(0.021)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,239,239);color:black\"> 0.010 <strong>(0.072)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,241,241);color:black\"> 0.0062 <strong>(0.059)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,229,229);color:black\"> 0.0155 <strong>(0.142)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(228,228,250);color:black\"> -0.0142 <strong>(-0.172)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(125,125,250);color:black\"> -28 <strong>(-0.965)</strong> </td>\n",
|
||
|
|
" <td> 4.01 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
|
||
|
|
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
|
||
|
|
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
|
||
|
|
"<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"360pt\" height=\"288pt\" viewBox=\"0 0 360 288\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
|
||
|
|
" <metadata>\n",
|
||
|
|
" <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
|
||
|
|
" <cc:Work>\n",
|
||
|
|
" <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
|
||
|
|
" <dc:date>2024-10-16T21:40:53.558660</dc:date>\n",
|
||
|
|
" <dc:format>image/svg+xml</dc:format>\n",
|
||
|
|
" <dc:creator>\n",
|
||
|
|
" <cc:Agent>\n",
|
||
|
|
" <dc:title>Matplotlib v3.8.2, https://matplotlib.org/</dc:title>\n",
|
||
|
|
" </cc:Agent>\n",
|
||
|
|
" </dc:creator>\n",
|
||
|
|
" </cc:Work>\n",
|
||
|
|
" </rdf:RDF>\n",
|
||
|
|
" </metadata>\n",
|
||
|
|
" <defs>\n",
|
||
|
|
" <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
|
||
|
|
" </defs>\n",
|
||
|
|
" <g id=\"figure_1\">\n",
|
||
|
|
" <g id=\"patch_1\">\n",
|
||
|
|
" <path d=\"M 0 288 \n",
|
||
|
|
"L 360 288 \n",
|
||
|
|
"L 360 0 \n",
|
||
|
|
"L 0 0 \n",
|
||
|
|
"z\n",
|
||
|
|
"\" style=\"fill: #ffffff\"/>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"axes_1\">\n",
|
||
|
|
" <g id=\"patch_2\">\n",
|
||
|
|
" <path d=\"M 29.08774 268.321635 \n",
|
||
|
|
"L 356.99976 268.321635 \n",
|
||
|
|
"L 356.99976 3.00024 \n",
|
||
|
|
"L 29.08774 3.00024 \n",
|
||
|
|
"z\n",
|
||
|
|
"\" style=\"fill: #ffffff\"/>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"matplotlib.axis_1\">\n",
|
||
|
|
" <g id=\"xtick_1\">\n",
|
||
|
|
" <g id=\"line2d_1\">\n",
|
||
|
|
" <defs>\n",
|
||
|
|
" <path id=\"m848079053e\" d=\"M 0 0 \n",
|
||
|
|
"L 0 3.5 \n",
|
||
|
|
"\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
|
||
|
|
" </defs>\n",
|
||
|
|
" <g>\n",
|
||
|
|
" <use xlink:href=\"#m848079053e\" x=\"42.740303\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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||
|
|
" </g>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <g id=\"text_1\">\n",
|
||
|
|
" <!-- 40 -->\n",
|
||
|
|
" <g transform=\"translate(36.377803 282.920073) scale(0.1 -0.1)\">\n",
|
||
|
|
" <defs>\n",
|
||
|
|
" <path id=\"DejaVuSans-34\" d=\"M 2419 4116 \n",
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|
|
"L 825 1625 \n",
|
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"L 2419 1625 \n",
|
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|
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|
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|
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|
|
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"text/plain": [
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"┌─────────────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ Migrad │\n",
|
||
|
|
"├──────────────────────────────────┬──────────────────────────────────────┤\n",
|
||
|
|
"│ FCN = 106.4 (χ²/ndof = 0.9) │ Nfcn = 530 │\n",
|
||
|
|
"│ EDM = 1.61e-05 (Goal: 0.0002) │ │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Valid Minimum │ Below EDM threshold (goal x 10) │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ No parameters at limit │ Below call limit │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Hesse ok │ Covariance accurate │\n",
|
||
|
|
"└──────────────────────────────────┴──────────────────────────────────────┘\n",
|
||
|
|
"┌───┬──────────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n",
|
||
|
|
"│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n",
|
||
|
|
"├───┼──────────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n",
|
||
|
|
"│ 0 │ A_p1 │ 317 │ 7 │ │ │ │ │ │\n",
|
||
|
|
"│ 1 │ A_p2 │ 580 │ 7 │ │ │ │ │ │\n",
|
||
|
|
"│ 2 │ mu_p1 │ 53.24 │ 0.07 │ │ │ │ │ │\n",
|
||
|
|
"│ 3 │ mu_p2 │ 60.43 │ 0.05 │ │ │ │ │ │\n",
|
||
|
|
"│ 4 │ sigma_p1 │ 1.99 │ 0.05 │ │ │ │ │ │\n",
|
||
|
|
"│ 5 │ sigma_p2 │ 2.80 │ 0.04 │ │ │ │ │ │\n",
|
||
|
|
"│ 6 │ A_bkg │ 147 │ 14 │ │ │ │ │ │\n",
|
||
|
|
"│ 7 │ tau_bkg │ 34.1 │ 2.0 │ │ │ 0 │ │ │\n",
|
||
|
|
"└───┴──────────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n",
|
||
|
|
"┌──────────┬─────────────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ │ A_p1 A_p2 mu_p1 mu_p2 sigma_p1 sigma_p2 A_bkg tau_bkg │\n",
|
||
|
|
"├──────────┼─────────────────────────────────────────────────────────────────────────┤\n",
|
||
|
|
"│ A_p1 │ 51.5 10 0.103 0.1006 -0.0808 -0.0969 -0 0 │\n",
|
||
|
|
"│ A_p2 │ 10 50.6 0.026 0.0402 -0.0047 -0.1329 -0 0 │\n",
|
||
|
|
"│ mu_p1 │ 0.103 0.026 0.00503 0.0027 0.0025 -0.0020 -0.057 0.010 │\n",
|
||
|
|
"│ mu_p2 │ 0.1006 0.0402 0.0027 0.00276 0.0018 -0.0015 -0.0513 0.0062 │\n",
|
||
|
|
"│ sigma_p1 │ -0.0808 -0.0047 0.0025 0.0018 0.00297 -0.0012 -0.1409 0.0155 │\n",
|
||
|
|
"│ sigma_p2 │ -0.0969 -0.1329 -0.0020 -0.0015 -0.0012 0.00171 0.0816 -0.0142 │\n",
|
||
|
|
"│ A_bkg │ -0 -0 -0.057 -0.0513 -0.1409 0.0816 209 -28 │\n",
|
||
|
|
"│ tau_bkg │ 0 0 0.010 0.0062 0.0155 -0.0142 -28 4.01 │\n",
|
||
|
|
"└──────────┴─────────────────────────────────────────────────────────────────────────┘"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 525,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
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|
|
],
|
||
|
|
"source": [
|
||
|
|
"mi"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "9f464246-d333-4143-baf0-aa2a632c5be4",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Eine eigene Abschätzung für das $\\chi^2$ ergibt:"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 526,
|
||
|
|
"id": "b0ad46ce-f541-40bb-898c-154ad5f94787",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stdout",
|
||
|
|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"106.36771764289108 112 0.9497117646686704\n"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"def chi_square_ndof(x_values, y_values, dy_values, fit_model, minuit):\n",
|
||
|
|
" ndof = len(x_values) - len(minuit.values)\n",
|
||
|
|
" chi2 = np.sum((y_values - fit_model(x_values, *minuit.values))**2/dy_values**2)\n",
|
||
|
|
" return chi2, ndof\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"chi_square, ndof = chi_square_ndof(center, entries, np.sqrt(entries), fit_model, mi)\n",
|
||
|
|
"print(chi_square, ndof, chi_square/ndof)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "295031f4-6d18-411c-b5dd-a62ed97da7f1",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"### Hypothesen-Test mittels $\\chi^2$\n",
|
||
|
|
"Wie schon im vorherigen Abschnitt erwähnt, kann man das $\\chi^2$ auch dazu verwenden, die Gültigkeit des gewählten Models zu prüfen.\n",
|
||
|
|
"Hierzu schauen wir uns die $\\chi^2$-Verteilung an. Der einzige freie Parameter ist die Anzahl der Freiheitsgrade. Die Anzahl der Freiheitsgrade ist auch gleichzeitig der Erwartungswert der $\\chi^2$-Verteilung. In unserem Beispiel oben ist die Anzahl der Freiheitsgrade 112 und die entsprechende Verteilung sieht wie folgt aus..."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 527,
|
||
|
|
"id": "8c11bc85-4e25-4d40-8397-257414d48a1f",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": [
|
||
|
|
"from scipy.stats import chi2\n",
|
||
|
|
"# chi_distribution = lambda x, ndof: chi2.pdf(x, ndof)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 528,
|
||
|
|
"id": "76836863-109c-4e7c-989e-04b62ec4ca9d",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
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|
|
"image/png": "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
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 1 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"x = np.arange(40., 180.)\n",
|
||
|
|
"# plt.plot(x, chi_distribution(x, 112))\n",
|
||
|
|
"plt.plot(x,chi2.pdf(x, 112))\n",
|
||
|
|
"x = np.arange(chi_square, 180, 0.1)\n",
|
||
|
|
"plt.fill_between(x, chi2.pdf(x, 112), alpha=0.3)\n",
|
||
|
|
"plt.ylim(0, None)\n",
|
||
|
|
"plt.xlabel('x')\n",
|
||
|
|
"plt.ylabel('$\\chi^2(x)$')\n",
|
||
|
|
"plt.show()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "b30829b3-a9e8-4d93-8895-9fd9f67ab9dc",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Der erste Schritt für den Hypothesen-Test ist die Berechnung des $P$-Werts\n",
|
||
|
|
"$$ P = \\int_{\\chi^2}^{\\infty} f(z,n_d)dz $$\n",
|
||
|
|
"wobei $f(z,n_d)$ die $\\chi^2$-Verteilung und $n_d$ die Anzahl der Freiheitsgrade ist.\n",
|
||
|
|
"Im Bild oben entspricht dies der ausgefüllten Fläche.\n",
|
||
|
|
"\n",
|
||
|
|
"Die praktische Berechnung erfolgt mittels der kumulativen Verteilungsfunktion via\n",
|
||
|
|
"$$ P = 1 - \\chi^2_{CDF}(x, n_d) $$\n",
|
||
|
|
"wobei für $x$ das im Fit bestimmte $\\chi^2$ eingesetzt wird. Die praktische Bedeutung des $P$-Werts ist die Wahrscheinlichkeit bei einer Wiederholung des Experiments in größeres $\\chi^2$ zu erhalten, wenn unser Model die Daten richtig beschreibt und die ermittelten Fitparameter den wahren Werten entsprechen."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 529,
|
||
|
|
"id": "cfa9d88a-eada-49dd-8cb3-73c7dd345c08",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"(0.6323451110506132, 0.884238547608047, 0.48222800598351057)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 529,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"p_value = lambda x, ndof: 1 - chi2.cdf(x, ndof)\n",
|
||
|
|
"p_value(chi_square, ndof), p_value(chi_square*10, ndof*10), p_value(ndof, ndof)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "9cba146a-6309-42d1-92cb-8bdde2da42a2",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Kehren wir zu unserem Doppelpeak-Spektrum zurück und änderen das Fitmodell, indem wir statt eines exponentiellen einen konstanten Untergrund annehmen."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 530,
|
||
|
|
"id": "9b91ee55-ac17-4dd6-9827-48677f772096",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/html": [
|
||
|
|
"<table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th colspan=\"2\" style=\"text-align:center\" title=\"Minimizer\"> Migrad </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Minimum value of function\"> FCN = 369.6 (χ²/ndof = 3.3) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total number of function and (optional) gradient evaluations\"> Nfcn = 415 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:left\" title=\"Estimated distance to minimum and goal\"> EDM = 5.63e-05 (Goal: 0.0002) </td>\n",
|
||
|
|
" <td style=\"text-align:center\" title=\"Total run time of algorithms\"> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Valid Minimum </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below EDM threshold (goal x 10) </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> No parameters at limit </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Below call limit </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Hesse ok </td>\n",
|
||
|
|
" <td style=\"text-align:center;background-color:#92CCA6;color:black\"> Covariance accurate </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th title=\"Variable name\"> Name </th>\n",
|
||
|
|
" <th title=\"Value of parameter\"> Value </th>\n",
|
||
|
|
" <th title=\"Hesse error\"> Hesse Error </th>\n",
|
||
|
|
" <th title=\"Minos lower error\"> Minos Error- </th>\n",
|
||
|
|
" <th title=\"Minos upper error\"> Minos Error+ </th>\n",
|
||
|
|
" <th title=\"Lower limit of the parameter\"> Limit- </th>\n",
|
||
|
|
" <th title=\"Upper limit of the parameter\"> Limit+ </th>\n",
|
||
|
|
" <th title=\"Is the parameter fixed in the fit\"> Fixed </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 0 </th>\n",
|
||
|
|
" <td> A_p1 </td>\n",
|
||
|
|
" <td> 319 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 1 </th>\n",
|
||
|
|
" <td> A_p2 </td>\n",
|
||
|
|
" <td> 583 </td>\n",
|
||
|
|
" <td> 7 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 2 </th>\n",
|
||
|
|
" <td> mu_p1 </td>\n",
|
||
|
|
" <td> 53.31 </td>\n",
|
||
|
|
" <td> 0.08 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 3 </th>\n",
|
||
|
|
" <td> mu_p2 </td>\n",
|
||
|
|
" <td> 60.52 </td>\n",
|
||
|
|
" <td> 0.06 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 4 </th>\n",
|
||
|
|
" <td> sigma_p1 </td>\n",
|
||
|
|
" <td> 2.23 </td>\n",
|
||
|
|
" <td> 0.07 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 5 </th>\n",
|
||
|
|
" <td> sigma_p2 </td>\n",
|
||
|
|
" <td> 2.72 </td>\n",
|
||
|
|
" <td> 0.04 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> 6 </th>\n",
|
||
|
|
" <td> c </td>\n",
|
||
|
|
" <td> 21.4 </td>\n",
|
||
|
|
" <td> 0.6 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> 0 </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" <td> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><table>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <td></td>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <th> c </th>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p1 </th>\n",
|
||
|
|
" <td> 47.8 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,228,228);color:black\"> 10 <strong>(0.148)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,225,225);color:black\"> 0.096 <strong>(0.167)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,216,216);color:black\"> 0.0895 <strong>(0.224)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(219,219,250);color:black\"> -0.108 <strong>(-0.235)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(211,211,250);color:black\"> -0.0881 <strong>(-0.301)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,247,247);color:black\"> 0.1 <strong>(0.023)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> A_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,228,228);color:black\"> 10 <strong>(0.148)</strong> </td>\n",
|
||
|
|
" <td> 52.4 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(242,242,250);color:black\"> -0.036 <strong>(-0.060)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(249,249,250);color:black\"> -0.0034 <strong>(-0.008)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(233,233,250);color:black\"> -0.064 <strong>(-0.132)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(205,205,250);color:black\"> -0.1062 <strong>(-0.347)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,249,249);color:black\"> 0.0 <strong>(0.005)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,225,225);color:black\"> 0.096 <strong>(0.167)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(242,242,250);color:black\"> -0.036 <strong>(-0.060)</strong> </td>\n",
|
||
|
|
" <td> 0.00694 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,132,132);color:black\"> 0.0038 <strong>(0.785)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,139,139);color:black\"> 0.004 <strong>(0.743)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(158,158,250);color:black\"> -0.0025 <strong>(-0.711)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,246,246);color:black\"> 0.002 <strong>(0.030)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> mu_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,216,216);color:black\"> 0.0895 <strong>(0.224)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(249,249,250);color:black\"> -0.0034 <strong>(-0.008)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,132,132);color:black\"> 0.0038 <strong>(0.785)</strong> </td>\n",
|
||
|
|
" <td> 0.00333 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,146,146);color:black\"> 0.0027 <strong>(0.695)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(157,157,250);color:black\"> -0.0017 <strong>(-0.714)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(243,243,250);color:black\"> -0.0018 <strong>(-0.051)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p1 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(219,219,250);color:black\"> -0.108 <strong>(-0.235)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(233,233,250);color:black\"> -0.064 <strong>(-0.132)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,139,139);color:black\"> 0.004 <strong>(0.743)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,146,146);color:black\"> 0.0027 <strong>(0.695)</strong> </td>\n",
|
||
|
|
" <td> 0.00444 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(177,177,250);color:black\"> -0.0016 <strong>(-0.559)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(233,233,250);color:black\"> -0.005 <strong>(-0.132)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> sigma_p2 </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(211,211,250);color:black\"> -0.0881 <strong>(-0.301)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(205,205,250);color:black\"> -0.1062 <strong>(-0.347)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(158,158,250);color:black\"> -0.0025 <strong>(-0.711)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(157,157,250);color:black\"> -0.0017 <strong>(-0.714)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(177,177,250);color:black\"> -0.0016 <strong>(-0.559)</strong> </td>\n",
|
||
|
|
" <td> 0.00179 </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(234,234,250);color:black\"> -0.0033 <strong>(-0.124)</strong> </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" <tr>\n",
|
||
|
|
" <th> c </th>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,247,247);color:black\"> 0.1 <strong>(0.023)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,249,249);color:black\"> 0.0 <strong>(0.005)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(250,246,246);color:black\"> 0.002 <strong>(0.030)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(243,243,250);color:black\"> -0.0018 <strong>(-0.051)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(233,233,250);color:black\"> -0.005 <strong>(-0.132)</strong> </td>\n",
|
||
|
|
" <td style=\"background-color:rgb(234,234,250);color:black\"> -0.0033 <strong>(-0.124)</strong> </td>\n",
|
||
|
|
" <td> 0.39 </td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
"</table><?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
|
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|
|
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
|
||
|
|
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
|
||
|
|
"<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"360pt\" height=\"288pt\" viewBox=\"0 0 360 288\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
|
||
|
|
" <metadata>\n",
|
||
|
|
" <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
|
||
|
|
" <cc:Work>\n",
|
||
|
|
" <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
|
||
|
|
" <dc:date>2024-10-16T21:41:00.276254</dc:date>\n",
|
||
|
|
" <dc:format>image/svg+xml</dc:format>\n",
|
||
|
|
" <dc:creator>\n",
|
||
|
|
" <cc:Agent>\n",
|
||
|
|
" <dc:title>Matplotlib v3.8.2, https://matplotlib.org/</dc:title>\n",
|
||
|
|
" </cc:Agent>\n",
|
||
|
|
" </dc:creator>\n",
|
||
|
|
" </cc:Work>\n",
|
||
|
|
" </rdf:RDF>\n",
|
||
|
|
" </metadata>\n",
|
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|
|
" <defs>\n",
|
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|
|
" <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
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|
|
" </defs>\n",
|
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" <g id=\"figure_1\">\n",
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|
" <g id=\"patch_1\">\n",
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" <path d=\"M 0 288 \n",
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|
"z\n",
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"\" style=\"fill: #ffffff\"/>\n",
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" </g>\n",
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" <g id=\"axes_1\">\n",
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" <g id=\"patch_2\">\n",
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" <path d=\"M 29.08774 268.321635 \n",
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"z\n",
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"\" style=\"fill: #ffffff\"/>\n",
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|
" </g>\n",
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" <g id=\"matplotlib.axis_1\">\n",
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" <g id=\"xtick_1\">\n",
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" <g id=\"line2d_1\">\n",
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" <defs>\n",
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" <path id=\"me6586c5535\" d=\"M 0 0 \n",
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"L 0 3.5 \n",
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"\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </defs>\n",
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" <g>\n",
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" <use xlink:href=\"#me6586c5535\" x=\"42.740303\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </g>\n",
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|
" </g>\n",
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" <g id=\"text_1\">\n",
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" <!-- 40 -->\n",
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" <g transform=\"translate(36.377803 282.920073) scale(0.1 -0.1)\">\n",
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" <defs>\n",
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"z\n",
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"M 2034 4750 \n",
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"Q 2819 4750 3233 4129 \n",
|
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"Q 3647 3509 3647 2328 \n",
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|
"Q 3647 1150 3233 529 \n",
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|
"Q 2819 -91 2034 -91 \n",
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|
"Q 1250 -91 836 529 \n",
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"Q 422 1150 422 2328 \n",
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" </g>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"xtick_2\">\n",
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" <g id=\"line2d_2\">\n",
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" <g>\n",
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" <use xlink:href=\"#me6586c5535\" x=\"80.316165\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"text_2\">\n",
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" <!-- 45 -->\n",
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"L 1269 4134 \n",
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"Q 1406 3038 1543 3061 \n",
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"Q 3513 2228 3513 1497 \n",
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"Q 3513 744 3044 326 \n",
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"Q 2250 441 2565 725 \n",
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|
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"Q 2881 1009 2881 1497 \n",
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|
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"Q 2881 1984 2565 2268 \n",
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"Q 2250 2553 1709 2553 \n",
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" </g>\n",
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" </g>\n",
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" <g id=\"xtick_3\">\n",
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" <g>\n",
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" <use xlink:href=\"#me6586c5535\" x=\"117.892027\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </g>\n",
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" <g id=\"text_3\">\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"xtick_4\">\n",
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" <g id=\"line2d_4\">\n",
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" <g>\n",
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" <use xlink:href=\"#me6586c5535\" x=\"155.467888\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"text_4\">\n",
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" <!-- 55 -->\n",
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" <use xlink:href=\"#DejaVuSans-35\"/>\n",
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" <use xlink:href=\"#DejaVuSans-35\" x=\"63.623047\"/>\n",
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" </g>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"xtick_5\">\n",
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" <g id=\"line2d_5\">\n",
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" <g>\n",
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" <use xlink:href=\"#me6586c5535\" x=\"193.04375\" y=\"268.321635\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
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" </g>\n",
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" </g>\n",
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" <g id=\"text_5\">\n",
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" <!-- 60 -->\n",
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" <g transform=\"translate(186.68125 282.920073) scale(0.1 -0.1)\">\n",
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" <defs>\n",
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" <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
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"Q 1688 2584 1439 2293 \n",
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"Q 1191 2003 1191 1497 \n",
|
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"Q 1191 994 1439 701 \n",
|
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"Q 1688 409 2113 409 \n",
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"Q 2538 409 2786 701 \n",
|
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|
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"Q 3034 994 3034 1497 \n",
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"Q 3034 2003 2786 2293 \n",
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|
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"Q 2538 2584 2113 2584 \n",
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|
"z\n",
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"M 3366 4563 \n",
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"L 3366 3988 \n",
|
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|
|
"Q 3128 4100 2886 4159 \n",
|
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"\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
|
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" </g>\n",
|
||
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|
" </g>\n",
|
||
|
|
" </g>\n",
|
||
|
|
" <defs>\n",
|
||
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|
" <clipPath id=\"pc105aa4edc\">\n",
|
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" <rect x=\"29.08774\" y=\"3.00024\" width=\"327.91202\" height=\"265.321395\"/>\n",
|
||
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|
" </clipPath>\n",
|
||
|
|
" </defs>\n",
|
||
|
|
"</svg>\n"
|
||
|
|
],
|
||
|
|
"text/plain": [
|
||
|
|
"┌─────────────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ Migrad │\n",
|
||
|
|
"├──────────────────────────────────┬──────────────────────────────────────┤\n",
|
||
|
|
"│ FCN = 369.6 (χ²/ndof = 3.3) │ Nfcn = 415 │\n",
|
||
|
|
"│ EDM = 5.63e-05 (Goal: 0.0002) │ │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Valid Minimum │ Below EDM threshold (goal x 10) │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ No parameters at limit │ Below call limit │\n",
|
||
|
|
"├──────────────────────────────────┼──────────────────────────────────────┤\n",
|
||
|
|
"│ Hesse ok │ Covariance accurate │\n",
|
||
|
|
"└──────────────────────────────────┴──────────────────────────────────────┘\n",
|
||
|
|
"┌───┬──────────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n",
|
||
|
|
"│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n",
|
||
|
|
"├───┼──────────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n",
|
||
|
|
"│ 0 │ A_p1 │ 319 │ 7 │ │ │ │ │ │\n",
|
||
|
|
"│ 1 │ A_p2 │ 583 │ 7 │ │ │ │ │ │\n",
|
||
|
|
"│ 2 │ mu_p1 │ 53.31 │ 0.08 │ │ │ │ │ │\n",
|
||
|
|
"│ 3 │ mu_p2 │ 60.52 │ 0.06 │ │ │ │ │ │\n",
|
||
|
|
"│ 4 │ sigma_p1 │ 2.23 │ 0.07 │ │ │ │ │ │\n",
|
||
|
|
"│ 5 │ sigma_p2 │ 2.72 │ 0.04 │ │ │ │ │ │\n",
|
||
|
|
"│ 6 │ c │ 21.4 │ 0.6 │ │ │ 0 │ │ │\n",
|
||
|
|
"└───┴──────────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n",
|
||
|
|
"┌──────────┬────────────────────────────────────────────────────────────────┐\n",
|
||
|
|
"│ │ A_p1 A_p2 mu_p1 mu_p2 sigma_p1 sigma_p2 c │\n",
|
||
|
|
"├──────────┼────────────────────────────────────────────────────────────────┤\n",
|
||
|
|
"│ A_p1 │ 47.8 10 0.096 0.0895 -0.108 -0.0881 0.1 │\n",
|
||
|
|
"│ A_p2 │ 10 52.4 -0.036 -0.0034 -0.064 -0.1062 0.0 │\n",
|
||
|
|
"│ mu_p1 │ 0.096 -0.036 0.00694 0.0038 0.004 -0.0025 0.002 │\n",
|
||
|
|
"│ mu_p2 │ 0.0895 -0.0034 0.0038 0.00333 0.0027 -0.0017 -0.0018 │\n",
|
||
|
|
"│ sigma_p1 │ -0.108 -0.064 0.004 0.0027 0.00444 -0.0016 -0.005 │\n",
|
||
|
|
"│ sigma_p2 │ -0.0881 -0.1062 -0.0025 -0.0017 -0.0016 0.00179 -0.0033 │\n",
|
||
|
|
"│ c │ 0.1 0.0 0.002 -0.0018 -0.005 -0.0033 0.39 │\n",
|
||
|
|
"└──────────┴────────────────────────────────────────────────────────────────┘"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 530,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"def alternative_fit_model(x, A_p1, A_p2, mu_p1, mu_p2, sigma_p1, sigma_p2, c):\n",
|
||
|
|
" return peak(x, A_p1, mu_p1, sigma_p1) + peak(x, A_p2, mu_p2, sigma_p2) + c\n",
|
||
|
|
"\n",
|
||
|
|
"ls = cost.LeastSquares(center, entries, np.sqrt(entries), alternative_fit_model)\n",
|
||
|
|
"\n",
|
||
|
|
"mi = Minuit(ls, \n",
|
||
|
|
" A_p1 = 800, \n",
|
||
|
|
" A_p2 = 1400,\n",
|
||
|
|
" mu_p1 = 54,\n",
|
||
|
|
" mu_p2 = 60,\n",
|
||
|
|
" sigma_p1 = 2,\n",
|
||
|
|
" sigma_p2 = 2,\n",
|
||
|
|
" c = 100, \n",
|
||
|
|
" )\n",
|
||
|
|
"mi.limits['c'] = (0, None)\n",
|
||
|
|
"mi.fixed[:] = True\n",
|
||
|
|
"ls.mask = (center < 45) | (center >= 70)\n",
|
||
|
|
"mi.fixed[['c']] = False\n",
|
||
|
|
"mi.migrad()\n",
|
||
|
|
"ls.mask = None\n",
|
||
|
|
"mi.values['A_p1'] = 700\n",
|
||
|
|
"mi.values['sigma_p1'] = 3\n",
|
||
|
|
"mi.fixed[:] = True\n",
|
||
|
|
"mi.fixed[['A_p1', 'mu_p1', 'sigma_p1']] = False\n",
|
||
|
|
"mi.migrad()\n",
|
||
|
|
"mi.fixed[:] = True\n",
|
||
|
|
"mi.fixed[['A_p2', 'mu_p2', 'sigma_p2']] = False\n",
|
||
|
|
"mi.migrad()\n",
|
||
|
|
"mi.fixed[:] = False\n",
|
||
|
|
"mi.migrad()\n",
|
||
|
|
"mi.hesse()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "c9fbbebc",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Diese Änderung ist gering und der Fit scheint die Daten weiterhin zu beschreiben. Allerdings gibt bei kleinen Energien eine deutlich sichtbare Diskrepanz. Dies zeigt sich auch in einem größeren $\\chi^2$-Wert. Wie wirkt sich dies auf den $P$-Wert aus?"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 478,
|
||
|
|
"id": "4aa0f3d9-1d0b-4b4c-b816-2a0cb9ae9793",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stdout",
|
||
|
|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"329.01941626278426 113 2.911676250113135\n"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"chi_square, ndof = chi_square_ndof(center, entries, np.sqrt(entries), alternative_fit_model, mi)\n",
|
||
|
|
"print(chi_square, ndof, chi_square/ndof)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 479,
|
||
|
|
"id": "607ddd33",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stdout",
|
||
|
|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"329.01941626278426 113\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"0.0"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 479,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"p_value = lambda x, ndof: 1 - chi2.cdf(x, ndof)\n",
|
||
|
|
"print(chi_square, ndof)\n",
|
||
|
|
"p_value(chi_square, ndof)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "bcb62098-1e8b-4c9f-8aa3-048037f0d21e",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Der Fit ist offensichtlich viel schlechter und der $P$-Wert liegt nahe bei null, so dass man dieses Model ausschließen sollte.\n",
|
||
|
|
"\n",
|
||
|
|
"Was aber, wenn die Änderung nicht so dramatisch ist? Ist ein $P$-Wert von 0,4 besser als 0,2? Nein, das kann man so nicht beantworten. Aber für einen Hypothesen-Test sollten man vorher eine Schwelle festlegen für die Akzeptanz oder Ablehnung des Models.\n",
|
||
|
|
"\n",
|
||
|
|
"Wie ein solcher Hypothesen-Test aussehen kann, wollen wir im Folgenden betrachten. Hierbei benutzen wir\n",
|
||
|
|
"1. ein korrektes Model (Normalverteilung),\n",
|
||
|
|
"2. ein korrektes Model mit überschätztem Fehler (10% größer),\n",
|
||
|
|
"3. und ein falsches Model (Lorentzverteilung)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 264,
|
||
|
|
"id": "c3f1f1d4-4b84-45a1-9d23-4cbb8ba32c8c",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": [
|
||
|
|
"def lorentzian( x, x0, a, gam ):\n",
|
||
|
|
" return a * gam**2 / ( gam**2 + ( x - x0 )**2)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "0e3fcfd5",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Den Fit der drei Modelle und die Bestimmung des entsprechenden $P$-Werts wiederholen wir 5000-mal um eine ausreichende Statistik zu erhalten."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 7,
|
||
|
|
"id": "9667c766",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": [
|
||
|
|
"# Diese Zelle nur auf JupyterHub des ZDV ausführen um `tqdm` zu installieren falls es nicht vorhanden sein sollte!\n",
|
||
|
|
"# import sys\n",
|
||
|
|
"# import subprocess\n",
|
||
|
|
"# subprocess.check_call([\n",
|
||
|
|
"# sys.executable, \n",
|
||
|
|
"# '-m',\n",
|
||
|
|
"# 'pip',\n",
|
||
|
|
"# 'install',\n",
|
||
|
|
"# '--proxy',\n",
|
||
|
|
"# 'http://webproxy.zdv.uni-mainz.de:3128',\n",
|
||
|
|
"# 'tqdm'\n",
|
||
|
|
"# ])"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 531,
|
||
|
|
"id": "c3b58808-f155-4194-b02e-e5f649cb86aa",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"application/vnd.jupyter.widget-view+json": {
|
||
|
|
"model_id": "9401e27e0abe463ab485a539ee58e61e",
|
||
|
|
"version_major": 2,
|
||
|
|
"version_minor": 0
|
||
|
|
},
|
||
|
|
"text/plain": [
|
||
|
|
" 0%| | 0/5000 [00:00<?, ?it/s]"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"from tqdm.notebook import tqdm\n",
|
||
|
|
"\n",
|
||
|
|
"res_good_model = []\n",
|
||
|
|
"res_overfitting = []\n",
|
||
|
|
"res_wrong_model = []\n",
|
||
|
|
"\n",
|
||
|
|
"def peak(x, A, mu, sigma):\n",
|
||
|
|
" return A*np.exp(-(x-mu)**2/(2*sigma**2))\n",
|
||
|
|
"\n",
|
||
|
|
"def lorentzian( x, x0, a, gam ):\n",
|
||
|
|
" return a * gam**2 / ( gam**2 + ( x - x0 )**2)\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"for i in tqdm(range(5000)):\n",
|
||
|
|
" \n",
|
||
|
|
" test_data = np.random.normal(0, 2, 5000)\n",
|
||
|
|
" \n",
|
||
|
|
" entries, edges = np.histogram(test_data, bins=25, range=(-4,4))\n",
|
||
|
|
" center = edges[:-1] + np.diff(edges)/2\n",
|
||
|
|
" \n",
|
||
|
|
" ls = cost.LeastSquares(center, entries, np.sqrt(entries), peak)\n",
|
||
|
|
" mi = Minuit(ls, \n",
|
||
|
|
" mu=0.1,\n",
|
||
|
|
" sigma=1.5,\n",
|
||
|
|
" A = 300\n",
|
||
|
|
" )\n",
|
||
|
|
" mi.migrad()\n",
|
||
|
|
" \n",
|
||
|
|
" chi, ndof = chi_square_ndof(center, entries, np.sqrt(entries), peak, mi)\n",
|
||
|
|
" res_good_model.append(p_value(chi, ndof))\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
" ls = cost.LeastSquares(center, entries, np.sqrt(entries)*1.1, peak)\n",
|
||
|
|
" mi = Minuit(ls, \n",
|
||
|
|
" mu=0.1,\n",
|
||
|
|
" sigma=1.5,\n",
|
||
|
|
" A = 300\n",
|
||
|
|
" )\n",
|
||
|
|
" mi.migrad()\n",
|
||
|
|
" \n",
|
||
|
|
" chi, ndof = chi_square_ndof(center, entries, np.sqrt(entries)*1.1, peak, mi)\n",
|
||
|
|
" res_overfitting.append(p_value(chi, ndof))\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
" ls = cost.LeastSquares(center, entries, np.sqrt(entries), lorentzian)\n",
|
||
|
|
" mi = Minuit(ls, \n",
|
||
|
|
" x0=0,\n",
|
||
|
|
" gam=3,\n",
|
||
|
|
" a = 300,\n",
|
||
|
|
" )\n",
|
||
|
|
" mi.migrad()\n",
|
||
|
|
" \n",
|
||
|
|
" chi, ndof = chi_square_ndof(center, entries, np.sqrt(entries), lorentzian, mi)\n",
|
||
|
|
" res_wrong_model.append(p_value(chi, ndof))\n",
|
||
|
|
"\n",
|
||
|
|
"res_wrong_model = np.array(res_wrong_model)\n",
|
||
|
|
"res_good_model = np.array(res_good_model)\n",
|
||
|
|
"res_overfit_model = np.array(res_overfitting)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "7ec4cf79",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Die Schwelle des $P$-Werts für den Hypothesen-Test setzen wir auf 0,1, d.h. Ergebnisse mit eine, $P$-Wert $<$ 0,1 werden verworfen, alle anderen akzeptiert."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 266,
|
||
|
|
"id": "f41e0b38-56b6-4f2a-bc75-075f622a2068",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"image/png": "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
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 2 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"image/png": "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
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 2 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 266,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"fig, axes = plt.subplots()\n",
|
||
|
|
"axes.hist(res_good_model, bins=25, range=(0, 1), histtype='step', color='purple', label='Good model')\n",
|
||
|
|
"axes.hist(res_wrong_model, bins=25, range=(0, 1), histtype='step', color='orange', label='Wrong model')\n",
|
||
|
|
"axes.hist(res_overfitting, bins=25, range=(0, 1), histtype='step', color='firebrick', label='Too large uncertainties (10 %)')\n",
|
||
|
|
"axes.set_xlabel('p-value')\n",
|
||
|
|
"axes.set_ylabel('Number of fits')\n",
|
||
|
|
"axes.legend()\n",
|
||
|
|
"axes.axvline(0.1, color='k')\n",
|
||
|
|
"axes2 = plt.twiny()\n",
|
||
|
|
"axes2.set_xlabel('Red. $\\chi^2$')\n",
|
||
|
|
"axes2.set_xticks([0.2, 0.5, 0.8], ['> 1', '1', '< 1'])\n",
|
||
|
|
"plt.show()\n",
|
||
|
|
"\n",
|
||
|
|
"axes.set_yscale('log')\n",
|
||
|
|
"fig"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "86237b4b",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Wie man sieht, wird das falsche Modell nahezu immer verworfen während das richtige Modell meistens nicht verworfen wird. Das Modell mit dem überschätzten Fehler wird sogar häufiger akzeptiert, so dass man hier keine Unterscheidung vornehmen kann."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 532,
|
||
|
|
"id": "fc58ee5c-308c-4479-9236-751d7f158fe5",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stdout",
|
||
|
|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"Fraction of wrong model fits rejected: 0.9998\n",
|
||
|
|
"Fraction of good model fits rejected: 0.1002\n",
|
||
|
|
"Fraction of overfitting model fits rejected: 0.0250\n"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"print(f'Fraction of wrong model fits rejected: {np.sum(res_wrong_model<0.1)/len(res_wrong_model):.4f}')\n",
|
||
|
|
"print(f'Fraction of good model fits rejected: {np.sum(res_good_model<0.1)/len(res_good_model):.4f}')\n",
|
||
|
|
"print(f'Fraction of overfitting model fits rejected: {np.sum(res_overfit_model<0.1)/len(res_overfit_model):.4f}')"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "392f4ef2",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Wenn man das Limit für den Hypothesen-Test auf 0,05 festlegt, ändern die Ergebnisse wie folgt:"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 533,
|
||
|
|
"id": "d5f5efbe-ef8f-48b0-b27b-166f21cb5a06",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stdout",
|
||
|
|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"Fraction of wrong model fits rejected: 0.9986\n",
|
||
|
|
"Fraction of good model fits rejected: 0.0534\n",
|
||
|
|
"Fraction of overfitting model fits rejected: 0.0114\n"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"print(f'Fraction of wrong model fits rejected: {np.sum(res_wrong_model<0.05)/len(res_wrong_model):.4f}')\n",
|
||
|
|
"print(f'Fraction of good model fits rejected: {np.sum(res_good_model<0.05)/len(res_good_model):.4f}')\n",
|
||
|
|
"print(f'Fraction of overfitting model fits rejected: {np.sum(res_overfit_model<0.05)/len(res_overfit_model):.4f}')"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "de9861f6-7870-4dd8-8366-15e0c7dd5125",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Der Hypothesen-Test kann das Modell nicht ablehnen, statt es zu bestätigen!"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"metadata": {
|
||
|
|
"kernelspec": {
|
||
|
|
"display_name": "jupyter",
|
||
|
|
"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": "3.10.11"
|
||
|
|
}
|
||
|
|
},
|
||
|
|
"nbformat": 4,
|
||
|
|
"nbformat_minor": 5
|
||
|
|
}
|