From b6d3c5c91bd9714de3bca098eb4c1648f83231e0 Mon Sep 17 00:00:00 2001
From: svoboj68 <svoboj68@fjfi.cvut.cz>
Date: Tue, 15 Feb 2022 17:18:42 +0100
Subject: [PATCH 1/4] WIP: pridan ntb s teorii k chybam
---
README.md | 5 +-
chyby.ipynb | 267 ++++++++++++++++++++++++++++++++++++++++++++++++++++
2 files changed, 271 insertions(+), 1 deletion(-)
create mode 100644 chyby.ipynb
diff --git a/README.md b/README.md
index 4f932b6..04f44c2 100644
--- a/README.md
+++ b/README.md
@@ -3,7 +3,7 @@ Výukový nastroj pro použití programovacího jazyka python při zpracování
Využívá interaktviní prostředí pro python [jupyter notebook](https://jupyter.readthedocs.io/en/latest/install/notebook-classic.html).
Bývá součástí distribucí python jako například [annaconda](https://www.anaconda.com/products/individual).
-Obsah:
+Tutorialy:
- základy práce s proměnnými v pythonu [prace_s_daty](prace_s_daty.ipynb)
- indexování
- načítání z txt
@@ -21,6 +21,9 @@ Obsah:
- odvození vzorce pro chybu nepřímého měření pomoci symbolického počítání [symbolics](symbolics.ipynb)
- vykreslení dvourozměrných dat [2D_grafy](2D_grafy.ipynb)
+Teorie:
+ - Chyby a rozdelovaci funkce [chyby](chyby.ipynb)
+
Příklady:
- [Extrapolace](Extrapolace.ipynb)
- [Hobit](Hobit.ipynb)
diff --git a/chyby.ipynb b/chyby.ipynb
new file mode 100644
index 0000000..a5c0682
--- /dev/null
+++ b/chyby.ipynb
@@ -0,0 +1,267 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# NEDOKONCENO, muze obsahovat chyby"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Chyby statisticke, systematicke a rozdelovaci funkce"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "pri mereni se prirozene vykytuji chyby, tyto chyby se nedaji zcela ostranit a jejich vliv na mereni muze byt ruzny. Chyby delime na statisticke, systematicke a hrube, podle jejich puvodu pripadne vlivu na mereni\n",
+ "statisticke jsou nahodile nahodne se vyskytujici\n",
+ "systematicke jsou dane pouzitymi pristroji nebo metodou\n",
+ "hrube chyby jsou zpusobeny lidskym faktorem pripadne selhanim mericich pristroju\n",
+ "kazda "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import matplotlib.pyplot as plt\n",
+ "import numpy as np"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# TODO remove\n",
+ "np.random.seed(931031)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Rozdelovaci funkce\n",
+ "pokud se vyskytuje nekonecne mnoho malych nahodnych vlivu -> pravdepodobnost namereni urcite hodnoty se ridi normalnim rozdelenim.\n",
+ "\n",
+ "pravdepodobnost namereni hodnoty $p(x)$\n",
+ "$$ p(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}} \\mathrm{e}^{ -\\frac{1}{2} \\left( \\frac{x - \\mu}{ \\sigma} \\right)^2 }$$\n",
+ "kde $\\mu$ je prumer a $\\sigma$ je rozptyl"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "mu = 100\n",
+ "sigma = 10"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "x = np.linspace(-4*sigma, 4*sigma, 101)\n",
+ "y = 1 / (sigma * (2 * np.pi)**0.5 ) * np.exp(-0.5 * x/sigma * x/sigma)\n",
+ "\n",
+ "plt.figure(dpi=100)\n",
+ "plt.plot(x+mu, y, label=r'$ \\frac{1}{10 \\sqrt{2\\pi}} \\mathrm{e}^{ -\\frac{1}{2} \\left( \\frac{x - 100}{10} \\right)^2} $')\n",
+ "plt.axvline(mu, c='k', lw=1, ls='--')\n",
+ "plt.xlabel('x [-]')\n",
+ "plt.ylabel('p(x) [-]')\n",
+ "plt.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# generovani 10000 vzorku s rozdelenim vyse\n",
+ "n = 10000\n",
+ "d1 = np.random.normal(mu, sigma, n)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# chybove intervaly\n",
+ "interval1 = (d1 < mu + sigma) * (d1 > mu - sigma)\n",
+ "interval2 = (d1 < mu + 2 * sigma) * (d1 > mu - 2 * sigma)\n",
+ "interval3 = (d1 < mu + 3 * sigma) * (d1 > mu - 3 * sigma)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# pravdepodobnosti s jakymi se vyskytuji vygenerovane hodnoty v chybovych intervalech\n",
+ "interval1.sum() / n, interval2.sum() / n, interval3.sum() / n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "67% hodnot v intervalu $\\pm\\sigma$, 99,7% hodnot v intervalu $\\pm 3 \\sigma$, s velkou pravdepodobnosti hodnoty mimo tento interval jsou dusledkem hrube chyby, lze je tedy vyradit ze zpracovani mereni"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "plt.figure()\n",
+ "plt.hist(d1, bins=np.linspace(60, 140, 17), density=True)\n",
+ "plt.plot(x+mu, y, label='rozdelovaci fuknce')\n",
+ "plt.axvline(mu, c='k', label='$\\mu$')\n",
+ "plt.axvline(mu-sigma, ls='--', c='k')\n",
+ "plt.axvline(mu+sigma, ls='--', c='k', label='$\\mu\\pm\\sigma$')\n",
+ "plt.axvline(mu-3*sigma, ls='--', c='r')\n",
+ "plt.axvline(mu+3*sigma, ls='--', c='r', label='$\\mu\\pm 3\\sigma$')\n",
+ "plt.legend()\n",
+ "plt.xlabel('x [-]')\n",
+ "plt.ylabel('Pravdepodobnost [-]')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Statisticka chyba\n",
+ " - nelze ji zcela odstranit\n",
+ " - dobre popsana teorii\n",
+ " - jeji vliv lze zmensit opakovanim mereni"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# 10, 20 a 100 mereni s rozdelenim stejnym jako u prikladu vyse\n",
+ "d10 = np.random.normal(mu, sigma, 10)\n",
+ "d20 = np.random.normal(mu, sigma, 20)\n",
+ "d100 = np.random.normal(mu, sigma, 100)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "plt.hist([d10, d20, d100], density=True)\n",
+ "plt.plot(x+mu, y, c='k', label='rozdelovaci fuknce')\n",
+ "\n",
+ "# vyplneni legendy\n",
+ "plt.plot([], ls='', marker='s', c='C0', label='10')\n",
+ "plt.plot([], ls='', marker='s', c='C1', label='20')\n",
+ "plt.plot([], ls='', marker='s', c='C2', label='100')\n",
+ "\n",
+ "plt.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "plt.plot(d10, 'x')\n",
+ "plt.axhline(mu, c='k', label='prumer rozdelovaci funkce')\n",
+ "plt.axhline(d10.mean(), ls='--', label='prumer vzorku')\n",
+ "plt.xlabel('Index [-]')\n",
+ "plt.xticks(range(10))\n",
+ "plt.ylabel('x [-]')\n",
+ "plt.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "mask = np.ones_like(d10, dtype=bool)\n",
+ "mask[1] = 0\n",
+ "d10[mask].std() / (d10[mask].size - 1)**.5 , d10[mask].mean() - mu"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Systematicka chyba"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# spatne zkalibrovany pristroj posune merene hodnoty o konstantu\n",
+ "posun = d1 + 10"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "plt.hist([d1, posun], bins=25)\n",
+ "plt.show()"
+ ]
+ }
+ ],
+ "metadata": {
+ "interpreter": {
+ "hash": "b3ba2566441a7c06988d0923437866b63cedc61552a5af99d1f4fb67d367b25f"
+ },
+ "kernelspec": {
+ "display_name": "Python 3.8.3 64-bit ('base': conda)",
+ "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": "3.8.3"
+ },
+ "orig_nbformat": 4
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
--
GitLab
From 0799965be0c0305613986ed656682841bc7a573a Mon Sep 17 00:00:00 2001
From: svoboj68 <svoboj68@fjfi.cvut.cz>
Date: Mon, 4 Apr 2022 09:09:55 +0200
Subject: [PATCH 2/4] Pridany zakladni definice a notebook
---
definice.py | 30 +++++++++++++
spousteni.ipynb | 112 ++++++++++++++++++++++++++++++++++++++++++++++++
2 files changed, 142 insertions(+)
create mode 100644 definice.py
create mode 100644 spousteni.ipynb
diff --git a/definice.py b/definice.py
new file mode 100644
index 0000000..622aa9d
--- /dev/null
+++ b/definice.py
@@ -0,0 +1,30 @@
+import numpy as np
+
+
+def sem(pole):
+ """standard error of mea"""
+ return pole.std() / np.sqrt(pole.size-1)
+
+
+def sem_annotated(pole: np.ndarray) -> float:
+ """
+ Spocita chybu prumeru
+
+ Funguje uplne stejne jako sem, akorat jsou zaneseny typy parametru a vystupu
+
+ Parameters
+ ----------
+ pole : np.ndarray
+ hodnoty
+
+ Returns
+ -------
+ float
+ chyba prumeru
+ """
+ return pole.std() / np.sqrt(pole.size-1)
+
+
+def foo():
+ print('bar')
+ return
diff --git a/spousteni.ipynb b/spousteni.ipynb
new file mode 100644
index 0000000..f5e6da9
--- /dev/null
+++ b/spousteni.ipynb
@@ -0,0 +1,112 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Spouštění skriptů uvnitř ipython notebooku"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import os"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "print(os.getcwd()) # vypise pracovni adresar"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# pokud je pracovni slozkou prakpy, lze soubor definice.py spustit primo prikazem\n",
+ "%run definice.py\n",
+ "# pokud ne, je potreba zadat cestu k souboru, napr.\n",
+ "# %run c:\\Users\\student\\prakpy\\definice.py # windows\n",
+ "# %run /home/student/prakpy/definice.py # unix"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "%run /cesta/k/souboru/definice.py"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "foo()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "np.random.seed(15022022)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "a = np.random.normal(10, 2, size=10)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "a.mean(), sem(a), sem_annotated(a)"
+ ]
+ }
+ ],
+ "metadata": {
+ "interpreter": {
+ "hash": "b3ba2566441a7c06988d0923437866b63cedc61552a5af99d1f4fb67d367b25f"
+ },
+ "kernelspec": {
+ "display_name": "Python 3.8.3 ('base')",
+ "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": "3.8.3"
+ },
+ "orig_nbformat": 4
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
--
GitLab
From 1408d7a5b07d7d75d8c970703867b8a7b3c341d7 Mon Sep 17 00:00:00 2001
From: svoboj68 <svoboj68@fjfi.cvut.cz>
Date: Tue, 7 Jun 2022 18:26:23 +0200
Subject: [PATCH 3/4] Pridana numericka integrace
---
README.md | 4 ++
numerika.ipynb | 127 +++++++++++++++++++++++++++++++++++++++++++++++++
2 files changed, 131 insertions(+)
create mode 100644 numerika.ipynb
diff --git a/README.md b/README.md
index 04f44c2..99cc64e 100644
--- a/README.md
+++ b/README.md
@@ -4,6 +4,7 @@ Využívá interaktviní prostředí pro python [jupyter notebook](https://jupyt
Bývá součástí distribucí python jako například [annaconda](https://www.anaconda.com/products/individual).
Tutorialy:
+**********
- základy práce s proměnnými v pythonu [prace_s_daty](prace_s_daty.ipynb)
- indexování
- načítání z txt
@@ -20,11 +21,14 @@ Tutorialy:
- prokládání dat [fity](fity.ipynb)
- odvození vzorce pro chybu nepřímého měření pomoci symbolického počítání [symbolics](symbolics.ipynb)
- vykreslení dvourozměrných dat [2D_grafy](2D_grafy.ipynb)
+ - numerická integrace dat [numerika](numerika.ipynb)
Teorie:
+******
- Chyby a rozdelovaci funkce [chyby](chyby.ipynb)
Příklady:
+*********
- [Extrapolace](Extrapolace.ipynb)
- [Hobit](Hobit.ipynb)
diff --git a/numerika.ipynb b/numerika.ipynb
new file mode 100644
index 0000000..37fa57c
--- /dev/null
+++ b/numerika.ipynb
@@ -0,0 +1,127 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Numerická integrace\n",
+ "Numerická integrace dat (rozpracováno)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import matplotlib.pyplot as plt\n",
+ "import numpy as np\n",
+ "from scipy.integrate import trapz, cumtrapz"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "x = np.linspace(0, np.pi, 200)\n",
+ "y = np.sin(x)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "plt.figure()\n",
+ "plt.plot(x, y, '.', ms=1, label='sin')\n",
+ "plt.legend()\n",
+ "plt.xlabel('x [-]')\n",
+ "plt.ylabel('y [-]')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "pocatek = 0\n",
+ "y_integral = cumtrapz(y, x, initial=0) # postupná integrace a počáteční podmínka\n",
+ "y_integrated = trapz(y, x) # plocha pod křivkou\n",
+ "\n",
+ "plt.figure()\n",
+ "plt.plot(x, y_integral, '.', ms=1, label=r'$\\int\\sin(x) \\,\\mathrm{d} x$')\n",
+ "plt.plot(x, -np.cos(x), label=r'$-\\cos (x)$') # referenční funkce, analyticky určený integrál\n",
+ "plt.text(2, 0, r'$\\int_0^\\pi sin(x) \\,\\mathrm{{d}} x = {:.3f}$'.format(y_integrated)) # vloženi textu do grafu\n",
+ "plt.legend()\n",
+ "plt.xlabel('x [-]')\n",
+ "plt.ylabel('y [-]')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "delta = 0.1 # konstantní posunuti dat (systematická chyba)\n",
+ "y_offset = np.sin(x) + delta\n",
+ "\n",
+ "plt.figure()\n",
+ "plt.plot(x, y, '.', ms=1, label=r'$\\sin(x)$')\n",
+ "plt.plot(x, y_offset, '.', ms=1, label=r'$\\sin(x) + \\Delta$')\n",
+ "plt.legend()\n",
+ "plt.xlabel('x [-]')\n",
+ "plt.ylabel('y [-]')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "y_int_offset = cumtrapz(y_offset, x, initial=pocatek) # integrace posunutých dat\n",
+ "\n",
+ "plt.figure()\n",
+ "plt.plot(x, y_integral, '.', ms=1, label=r'$\\int \\sin(x) \\,\\mathrm{d}x$')\n",
+ "plt.plot(x, y_int_offset, '.', ms=1, label=r'$\\int ( \\sin(x) + \\Delta ) \\,\\mathrm{d}x$')\n",
+ "plt.legend()\n",
+ "plt.xlabel('x [-]')\n",
+ "plt.ylabel('y [-]')\n",
+ "plt.show()"
+ ]
+ }
+ ],
+ "metadata": {
+ "interpreter": {
+ "hash": "ad2bdc8ecc057115af97d19610ffacc2b4e99fae6737bb82f5d7fb13d2f2c186"
+ },
+ "kernelspec": {
+ "display_name": "Python 3.8.3 ('base')",
+ "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": "3.8.3"
+ },
+ "orig_nbformat": 4
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
--
GitLab
From 924702b3a7443b856b7f30458436ae0d6d45b5a8 Mon Sep 17 00:00:00 2001
From: svoboj68 <svoboj68@fjfi.cvut.cz>
Date: Thu, 18 Aug 2022 15:16:14 +0200
Subject: [PATCH 4/4] Pridana numericka derivace a komentare
---
numerika.ipynb | 108 ++++++++++++++++++++++++++++++++++++++++++++++---
1 file changed, 103 insertions(+), 5 deletions(-)
diff --git a/numerika.ipynb b/numerika.ipynb
index 37fa57c..412be62 100644
--- a/numerika.ipynb
+++ b/numerika.ipynb
@@ -4,8 +4,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "# Numerická integrace\n",
- "Numerická integrace dat (rozpracováno)"
+ "# Numerické zpracování signálů"
]
},
{
@@ -19,6 +18,16 @@
"from scipy.integrate import trapz, cumtrapz"
]
},
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Vytvoříme rozsah od 0 do $\\pi$ a jako signál zvolíme funkci sinus, u které je triviální určit primitivní funkci a derivaci.\n",
+ "\n",
+ " - $\\int \\sin(x) \\mathrm{d}x = - \\cos(x) + c $\n",
+ " - $\\frac{\\mathrm{d}}{\\mathrm{d}x} \\sin(x) = \\cos(x)$"
+ ]
+ },
{
"cell_type": "code",
"execution_count": null,
@@ -43,6 +52,13 @@
"plt.show()"
]
},
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Integrace"
+ ]
+ },
{
"cell_type": "code",
"execution_count": null,
@@ -50,13 +66,14 @@
"outputs": [],
"source": [
"pocatek = 0\n",
- "y_integral = cumtrapz(y, x, initial=0) # postupná integrace a počáteční podmínka\n",
- "y_integrated = trapz(y, x) # plocha pod křivkou\n",
+ "y_integral = cumtrapz(y, x, initial=pocatek) # postupná (kumulativní) numerická integrace a počáteční podmínka\n",
+ "y_integrated = trapz(y, x) # určitý numerický integrál ~ plocha pod křivkou\n",
"\n",
"plt.figure()\n",
"plt.plot(x, y_integral, '.', ms=1, label=r'$\\int\\sin(x) \\,\\mathrm{d} x$')\n",
"plt.plot(x, -np.cos(x), label=r'$-\\cos (x)$') # referenční funkce, analyticky určený integrál\n",
"plt.text(2, 0, r'$\\int_0^\\pi sin(x) \\,\\mathrm{{d}} x = {:.3f}$'.format(y_integrated)) # vloženi textu do grafu\n",
+ "plt.axhline(y_integrated, lw=1, color='k')\n",
"plt.legend()\n",
"plt.xlabel('x [-]')\n",
"plt.ylabel('y [-]')\n",
@@ -69,7 +86,30 @@
"metadata": {},
"outputs": [],
"source": [
- "delta = 0.1 # konstantní posunuti dat (systematická chyba)\n",
+ "print('numerická integrace:', y_integrated)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$\\int_0^\\pi \\sin(x) \\,\\mathrm{{d}} x = [\\cos(x)]_0^\\pi = \\cos(0) - \\cos(\\pi) = 1 - (-1) = 2$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Posunutí dat"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "delta = 0.1 # konstantní posunuti dat (systematická chyba, offset na osciloskopu)\n",
"y_offset = np.sin(x) + delta\n",
"\n",
"plt.figure()\n",
@@ -97,6 +137,64 @@
"plt.ylabel('y [-]')\n",
"plt.show()"
]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Derivace"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "dx = np.diff(x) # rozdíly mezi hodnotami nezávisle proměnné\n",
+ "dydx = np.gradient(y, x) # derivace\n",
+ "\n",
+ "plt.figure()\n",
+ "plt.plot(x, dydx, '.', ms=1, label=r'$\\frac{\\mathrm{d} y}{\\mathrm{d} x}$')\n",
+ "plt.plot(x, np.cos(x), label=r'$\\cos (x)$', alpha=.5)\n",
+ "plt.legend()\n",
+ "plt.xlabel('x [-]')\n",
+ "plt.ylabel('y [-]')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "print(x.size, dx.size) # počet členů se derivací zmenší"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Posunutí dat"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "dydx_offset = np.gradient(y_offset, x)\n",
+ "\n",
+ "plt.figure()\n",
+ "plt.plot(x, dydx_offset, '.', ms=1, label=r'$\\frac{\\mathrm{d}}{\\mathrm{d} x} (y + \\Delta)$')\n",
+ "plt.plot(x, np.cos(x), label=r'$\\cos (x)$', alpha=.5)\n",
+ "plt.legend()\n",
+ "plt.xlabel('x [-]')\n",
+ "plt.ylabel('y [-]')\n",
+ "plt.show()"
+ ]
}
],
"metadata": {
--
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