1597 lines
1.0 MiB
1597 lines
1.0 MiB
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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||
"slideshow": {
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"slide_type": "slide"
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||
}
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||
},
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"source": [
|
||
"### Uczenie maszynowe\n",
|
||
"# 2. Regresja liniowa – część 1"
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]
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||
},
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||
{
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||
"cell_type": "markdown",
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||
"metadata": {
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||
"slideshow": {
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||
"slide_type": "slide"
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||
}
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||
},
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"source": [
|
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"## 2.1. Funkcja kosztu"
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||
]
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||
},
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{
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||
"cell_type": "markdown",
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||
"metadata": {
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||
"slideshow": {
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||
"slide_type": "subslide"
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||
}
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||
},
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"source": [
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"### Zadanie\n",
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"Znając $x$ – ludność miasta, należy przewidzieć $y$ – dochód firmy transportowej.\n",
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"\n",
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"(Dane pochodzą z kursu „Machine Learning”, Andrew Ng, Coursera)."
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||
]
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||
},
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||
{
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||
"cell_type": "markdown",
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||
"metadata": {
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||
"slideshow": {
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||
"slide_type": "notes"
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||
}
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||
},
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"source": [
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"**Uwaga**: Ponieważ ten przykład ma być tak prosty, jak to tylko możliwe, ludność miasta podana jest w dziesiątkach tysięcy mieszkańców, a dochód firmy w dziesiątkach tysięcy dolarów. Dzięki temu funkcja kosztu obliczona w dalszej części wykładu będzie osiągać wartości, które łatwo przedstawić na wykresie."
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||
]
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||
},
|
||
{
|
||
"cell_type": "code",
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||
"execution_count": 1,
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||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
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||
}
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||
},
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||
"outputs": [],
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||
"source": [
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||
"import numpy as np\n",
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||
"import matplotlib.pyplot as plt\n",
|
||
"import ipywidgets as widgets\n",
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"\n",
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"%matplotlib inline\n",
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||
"%config InlineBackend.figure_format = \"svg\"\n",
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||
"\n",
|
||
"from IPython.display import display, Math, Latex"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
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||
}
|
||
},
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"source": [
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||
"### Dane"
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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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||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"outputs": [
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||
{
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||
"name": "stdout",
|
||
"output_type": "stream",
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||
"text": [
|
||
" x y\n",
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||
"0 6.1101 17.59200\n",
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||
"1 5.5277 9.13020\n",
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||
"2 8.5186 13.66200\n",
|
||
"3 7.0032 11.85400\n",
|
||
"4 5.8598 6.82330\n",
|
||
".. ... ...\n",
|
||
"75 6.5479 0.29678\n",
|
||
"76 7.5386 3.88450\n",
|
||
"77 5.0365 5.70140\n",
|
||
"78 10.2740 6.75260\n",
|
||
"79 5.1077 2.05760\n",
|
||
"\n",
|
||
"[80 rows x 2 columns]\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"import pandas as pd\n",
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||
"\n",
|
||
"data = pd.read_csv(\"data1_train.csv\", names=[\"x\", \"y\"])\n",
|
||
"print(data)\n"
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||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 3,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"x = data[\"x\"].to_numpy()\n",
|
||
"y = data[\"y\"].to_numpy()\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Hipoteza i parametry modelu"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Jak przewidzieć $y$ na podstawie danego $x$? W celu odpowiedzi na to pytanie będziemy starać się znaleźć taką funkcję $h(x)$, która będzie najlepiej obrazować zależność między $x$ a $y$, tj. $y \\sim h(x)$.\n",
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||
"\n",
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||
"Zacznijmy od najprostszego przypadku, kiedy $h(x)$ jest po prostu funkcją liniową. Ogólny wzór funkcji liniowej to"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"source": [
|
||
"$$ h(x) = a \\, x + b $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Pamiętajmy jednak, że współczynniki $a$ i $b$ nie są w tej chwili dane z góry – naszym zadaniem właśnie będzie znalezienie takich ich wartości, żeby $h(x)$ było „możliwie jak najbliżej” $y$ (co właściwie oznacza to sformułowanie, wyjaśnię potem)."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"source": [
|
||
"Poszukiwaną funkcję $h$ będziemy nazywać **funkcją hipotezy**, a jej współczynniki – **parametrami modelu**."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"W teorii uczenia maszynowego parametry modelu oznacza się na ogół grecką literą $\\theta$ z odpowiednimi indeksami, dlatego powyższy wzór opisujący liniową funkcję hipotezy zapiszemy jako\n",
|
||
"$$ h(x) = \\theta_0 + \\theta_1 x $$\n",
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||
"\n",
|
||
"**Parametry modelu** tworzą wektor, który oznaczymy po prostu przez $\\theta$:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"$$ \\theta = \\left[\\begin{array}{c}\\theta_0\\\\ \\theta_1\\end{array}\\right] $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Żeby podkreślić fakt, że funkcja hipotezy zależy od parametrów modelu, będziemy pisać $h_\\theta$ zamiast $h$:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"source": [
|
||
"$$ h_{\\theta}(x) = \\theta_0 + \\theta_1 x $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Przyjrzyjmy się teraz, jak wyglądają dane, które mamy modelować:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Na poniższym wykresie możesz spróbować ręcznie dopasować parametry modelu $\\theta_0$ i $\\theta_1$ tak, aby jak najlepiej modelowały zależność między $x$ a $y$:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 4,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Funkcje rysujące wykres kropkowy oraz prostą regresyjną\n",
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||
"\n",
|
||
"\n",
|
||
"def regdots(x, y):\n",
|
||
" fig = plt.figure(figsize=(16 * 0.6, 9 * 0.6))\n",
|
||
" ax = fig.add_subplot(111)\n",
|
||
" fig.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)\n",
|
||
" ax.scatter(x, y, c=\"r\", label=\"Dane\")\n",
|
||
"\n",
|
||
" ax.set_xlabel(\"Wielkość miejscowości\")\n",
|
||
" ax.set_ylabel(\"Dochód firmy\")\n",
|
||
" ax.margins(0.05, 0.05)\n",
|
||
" plt.ylim(min(y) - 1, max(y) + 1)\n",
|
||
" plt.xlim(min(x) - 1, max(x) + 1)\n",
|
||
" return fig\n",
|
||
"\n",
|
||
"\n",
|
||
"def regline(fig, fun, theta, x):\n",
|
||
" ax = fig.axes[0]\n",
|
||
" x0, x1 = min(x), max(x)\n",
|
||
" X = [x0, x1]\n",
|
||
" Y = [fun(theta, x) for x in X]\n",
|
||
" ax.plot(\n",
|
||
" X,\n",
|
||
" Y,\n",
|
||
" linewidth=\"2\",\n",
|
||
" label=(\n",
|
||
" r\"$y={theta0}{op}{theta1}x$\".format(\n",
|
||
" theta0=theta[0],\n",
|
||
" theta1=(theta[1] if theta[1] >= 0 else -theta[1]),\n",
|
||
" op=\"+\" if theta[1] >= 0 else \"-\",\n",
|
||
" )\n",
|
||
" ),\n",
|
||
" )\n",
|
||
"\n",
|
||
"\n",
|
||
"def legend(fig):\n",
|
||
" ax = fig.axes[0]\n",
|
||
" handles, labels = ax.get_legend_handles_labels()\n",
|
||
" # try-except block is a fix for a bug in Poly3DCollection\n",
|
||
" try:\n",
|
||
" fig.legend(handles, labels, fontsize=\"15\", loc=\"lower right\")\n",
|
||
" except AttributeError:\n",
|
||
" pass\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 5,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
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||
"data": {
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"text/plain": [
|
||
"<Figure size 960x540 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": [
|
||
"fig = regdots(x, y)\n",
|
||
"legend(fig)\n"
|
||
]
|
||
},
|
||
{
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||
"cell_type": "code",
|
||
"execution_count": 6,
|
||
"metadata": {
|
||
"slideshow": {
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||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Hipoteza: funkcja liniowa jednej zmiennej\n",
|
||
"\n",
|
||
"\n",
|
||
"def h(theta, x):\n",
|
||
" return theta[0] + theta[1] * x\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 7,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Przygotowanie interaktywnego wykresu\n",
|
||
"\n",
|
||
"sliderTheta01 = widgets.FloatSlider(\n",
|
||
" min=-10, max=10, step=0.1, value=0, description=r\"$\\theta_0$\", width=300\n",
|
||
")\n",
|
||
"sliderTheta11 = widgets.FloatSlider(\n",
|
||
" min=-5, max=5, step=0.1, value=0, description=r\"$\\theta_1$\", width=300\n",
|
||
")\n",
|
||
"\n",
|
||
"\n",
|
||
"def slide1(theta0, theta1):\n",
|
||
" fig = regdots(x, y)\n",
|
||
" regline(fig, h, [theta0, theta1], x)\n",
|
||
" legend(fig)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
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||
"execution_count": 8,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
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||
"application/vnd.jupyter.widget-view+json": {
|
||
"model_id": "0672501afb5c4fca818b52c3fdd274fd",
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||
"version_major": 2,
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||
"version_minor": 0
|
||
},
|
||
"text/plain": [
|
||
"interactive(children=(FloatSlider(value=0.0, description='$\\\\theta_0$', max=10.0, min=-10.0), FloatSlider(valu…"
|
||
]
|
||
},
|
||
"metadata": {},
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||
"output_type": "display_data"
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||
},
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||
{
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||
"data": {
|
||
"text/plain": [
|
||
"<function __main__.slide1(theta0, theta1)>"
|
||
]
|
||
},
|
||
"execution_count": 8,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"widgets.interact_manual(slide1, theta0=sliderTheta01, theta1=sliderTheta11)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Skąd wiadomo, że przewidywania modelu (wartości funkcji $h(x)$) zgadzaja się z obserwacjami (wartości $y$)?\n",
|
||
"\n",
|
||
"Aby to zmierzyć wprowadzimy pojęcie funkcji kosztu."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Funkcja kosztu"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Funkcję kosztu zdefiniujemy w taki sposób, żeby odzwierciedlała ona różnicę między przewidywaniami modelu a obserwacjami.\n",
|
||
"\n",
|
||
"Jedną z możliwosci jest zdefiniowanie funkcji kosztu jako wartość **błędu średniokwadratowego** (metoda najmniejszych kwadratów, *mean-square error, MSE*).\n",
|
||
"\n",
|
||
"My zdefiniujemy funkcję kosztu jako *połowę* błędu średniokwadratowego w celu ułatwienia późniejszych obliczeń (obliczenie pochodnej funkcji kosztu w dalszej części wykładu). Możemy tak zrobić, ponieważ $\\frac{1}{2}$ jest stałą, a pomnożenie przez stałą nie wpływa na przebieg zmienności funkcji."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"source": [
|
||
"$$ J(\\theta) \\, = \\, \\frac{1}{2m} \\sum_{i = 1}^{m} \\left( h_{\\theta} \\left( x^{(i)} \\right) - y^{(i)} \\right) ^2 $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"gdzie $m$ jest liczbą wszystkich przykładów (obserwacji), czyli wielkością zbioru danych uczących.\n",
|
||
"\n",
|
||
"W powyższym wzorze sumujemy kwadraty różnic między przewidywaniami modelu ($h_\\theta \\left( x^{(i)} \\right)$) a obserwacjami ($y^{(i)}$) po wszystkich przykładach $i$."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Teraz nasze zadanie sprowadza się do tego, że będziemy szukać takich parametrów $\\theta = \\left[\\begin{array}{c}\\theta_0\\\\ \\theta_1\\end{array}\\right]$, które minimalizują fukcję kosztu $J(\\theta)$:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"source": [
|
||
"$$ \\hat\\theta = \\mathop{\\arg\\min}_{\\theta} J(\\theta) $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"source": [
|
||
"$$ \\theta \\in \\mathbb{R}^2, \\quad J \\colon \\mathbb{R}^2 \\to \\mathbb{R} $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Proszę zwrócić uwagę, że dziedziną funkcji kosztu jest zbiór wszystkich możliwych wartości parametrów $\\theta$."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"$$ J(\\theta_0, \\theta_1) \\, = \\, \\frac{1}{2m} \\sum_{i = 1}^{m} \\left( \\theta_0 + \\theta_1 x^{(i)} - y^{(i)} \\right) ^2 $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 9,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def J(h, theta, x, y):\n",
|
||
" \"\"\"Funkcja kosztu\"\"\"\n",
|
||
" m = len(y)\n",
|
||
" return 1.0 / (2 * m) * sum((h(theta, x[i]) - y[i]) ** 2 for i in range(m))\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 10,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "skip"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Oblicz wartość funkcji kosztu i pokaż na wykresie\n",
|
||
"\n",
|
||
"\n",
|
||
"def regline2(fig, fun, theta, xx, yy):\n",
|
||
" \"\"\"Rysuj regresję liniową\"\"\"\n",
|
||
" ax = fig.axes[0]\n",
|
||
" x0, x1 = min(xx), max(xx)\n",
|
||
" X = [x0, x1]\n",
|
||
" Y = [fun(theta, x) for x in X]\n",
|
||
" cost = J(fun, theta, xx, yy)\n",
|
||
" ax.plot(\n",
|
||
" X,\n",
|
||
" Y,\n",
|
||
" linewidth=\"2\",\n",
|
||
" label=(\n",
|
||
" r\"$y={theta0}{op}{theta1}x, \\; J(\\theta)={cost:.3}$\".format(\n",
|
||
" theta0=theta[0],\n",
|
||
" theta1=(theta[1] if theta[1] >= 0 else -theta[1]),\n",
|
||
" op=\"+\" if theta[1] >= 0 else \"-\",\n",
|
||
" cost=cost,\n",
|
||
" )\n",
|
||
" ),\n",
|
||
" )\n",
|
||
"\n",
|
||
"\n",
|
||
"sliderTheta02 = widgets.FloatSlider(\n",
|
||
" min=-10, max=10, step=0.1, value=0, description=r\"$\\theta_0$\", width=300\n",
|
||
")\n",
|
||
"sliderTheta12 = widgets.FloatSlider(\n",
|
||
" min=-5, max=5, step=0.1, value=0, description=r\"$\\theta_1$\", width=300\n",
|
||
")\n",
|
||
"\n",
|
||
"\n",
|
||
"def slide2(theta0, theta1):\n",
|
||
" fig = regdots(x, y)\n",
|
||
" regline2(fig, h, [theta0, theta1], x, y)\n",
|
||
" legend(fig)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Poniższy interaktywny wykres pokazuje wartość funkcji kosztu $J(\\theta)$. Czy teraz łatwiej jest dobrać parametry modelu?"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 11,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"application/vnd.jupyter.widget-view+json": {
|
||
"model_id": "0e9e869c222b4f928c8b7c358ccdd973",
|
||
"version_major": 2,
|
||
"version_minor": 0
|
||
},
|
||
"text/plain": [
|
||
"interactive(children=(FloatSlider(value=0.0, description='$\\\\theta_0$', max=10.0, min=-10.0), FloatSlider(valu…"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"<function __main__.slide2(theta0, theta1)>"
|
||
]
|
||
},
|
||
"execution_count": 11,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"widgets.interact_manual(slide2, theta0=sliderTheta02, theta1=sliderTheta12)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Funkcja kosztu jako funkcja zmiennej $\\theta$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Funkcja kosztu zdefiniowana jako MSE jest funkcją zmiennej wektorowej $\\theta$, czyli funkcją dwóch zmiennych rzeczywistych: $\\theta_0$ i $\\theta_1$.\n",
|
||
" \n",
|
||
"Zobaczmy, jak wygląda jej wykres."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 12,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Wykres funkcji kosztu dla ustalonego theta_1=1.0\n",
|
||
"\n",
|
||
"\n",
|
||
"def costfun(fun, x, y):\n",
|
||
" return lambda theta: J(fun, theta, x, y)\n",
|
||
"\n",
|
||
"\n",
|
||
"def costplot(hypothesis, x, y, theta1=1.0):\n",
|
||
" fig = plt.figure(figsize=(16 * 0.6, 9 * 0.6))\n",
|
||
" ax = fig.add_subplot(111)\n",
|
||
" fig.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)\n",
|
||
" ax.set_xlabel(r\"$\\theta_0$\")\n",
|
||
" ax.set_ylabel(r\"$J(\\theta)$\")\n",
|
||
" j = costfun(hypothesis, x, y)\n",
|
||
" fun = lambda theta0: j([theta0, theta1])\n",
|
||
" X = np.arange(-10, 10, 0.1)\n",
|
||
" Y = [fun(x) for x in X]\n",
|
||
" ax.plot(\n",
|
||
" X, Y, linewidth=\"2\", label=(r\"$J(\\theta_0, {theta1})$\".format(theta1=theta1))\n",
|
||
" )\n",
|
||
" return fig\n",
|
||
"\n",
|
||
"\n",
|
||
"def slide3(theta1):\n",
|
||
" fig = costplot(h, x, y, theta1)\n",
|
||
" legend(fig)\n",
|
||
"\n",
|
||
"\n",
|
||
"sliderTheta13 = widgets.FloatSlider(\n",
|
||
" min=-5, max=5, step=0.1, value=1.0, description=r\"$\\theta_1$\", width=300\n",
|
||
")\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 13,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"application/vnd.jupyter.widget-view+json": {
|
||
"model_id": "bbfe8cfd8e274970904cd2c9e12fce6f",
|
||
"version_major": 2,
|
||
"version_minor": 0
|
||
},
|
||
"text/plain": [
|
||
"interactive(children=(FloatSlider(value=1.0, description='$\\\\theta_1$', max=5.0, min=-5.0), Button(description…"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"<function __main__.slide3(theta1)>"
|
||
]
|
||
},
|
||
"execution_count": 13,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"widgets.interact_manual(slide3, theta1=sliderTheta13)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 14,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Wykres funkcji kosztu względem theta_0 i theta_1\n",
|
||
"\n",
|
||
"from mpl_toolkits.mplot3d import Axes3D\n",
|
||
"import pylab\n",
|
||
"\n",
|
||
"%matplotlib inline\n",
|
||
"\n",
|
||
"def costplot3d(hypothesis, x, y, show_gradient=False):\n",
|
||
" fig = plt.figure(figsize=(16*.6, 9*.6))\n",
|
||
" ax = fig.add_subplot(111, projection='3d')\n",
|
||
" fig.subplots_adjust(left=0.0, right=1.0, bottom=0.0, top=1.0)\n",
|
||
" ax.set_xlabel(r'$\\theta_0$')\n",
|
||
" ax.set_ylabel(r'$\\theta_1$')\n",
|
||
" ax.set_zlabel(r'$J(\\theta)$')\n",
|
||
" \n",
|
||
" j = lambda theta0, theta1: costfun(hypothesis, x, y)([theta0, theta1])\n",
|
||
" X = np.arange(-10, 10.1, 0.1)\n",
|
||
" Y = np.arange(-1, 4.1, 0.1)\n",
|
||
" X, Y = np.meshgrid(X, Y)\n",
|
||
" Z = np.array([[J(hypothesis, [theta0, theta1], x, y) \n",
|
||
" for theta0, theta1 in zip(xRow, yRow)] \n",
|
||
" for xRow, yRow in zip(X, Y)])\n",
|
||
" \n",
|
||
" ax.plot_surface(X, Y, Z, rstride=2, cstride=8, linewidth=0.5,\n",
|
||
" alpha=0.5, cmap='jet', zorder=0,\n",
|
||
" label=r\"$J(\\theta)$\")\n",
|
||
" ax.view_init(elev=20., azim=-150)\n",
|
||
"\n",
|
||
" ax.set_xlim3d(-10, 10);\n",
|
||
" ax.set_ylim3d(-1, 4);\n",
|
||
" ax.set_zlim3d(-100, 800);\n",
|
||
"\n",
|
||
" N = range(0, 800, 20)\n",
|
||
" plt.contour(X, Y, Z, N, zdir='z', offset=-100, cmap='coolwarm', alpha=1)\n",
|
||
" \n",
|
||
" ax.plot([-3.89578088] * 2,\n",
|
||
" [ 1.19303364] * 2,\n",
|
||
" [-100, 4.47697137598], \n",
|
||
" color='red', alpha=1, linewidth=1.3, zorder=100, linestyle='dashed',\n",
|
||
" label=r'minimum: $J(-3.90, 1.19) = 4.48$')\n",
|
||
" ax.scatter([-3.89578088] * 2,\n",
|
||
" [ 1.19303364] * 2,\n",
|
||
" [-100, 4.47697137598], \n",
|
||
" c='r', s=80, marker='x', alpha=1, linewidth=1.3, zorder=100, \n",
|
||
" label=r'minimum: $J(-3.90, 1.19) = 4.48$')\n",
|
||
" \n",
|
||
" if show_gradient:\n",
|
||
" ax.plot([3.0, 1.1],\n",
|
||
" [3.0, 2.4],\n",
|
||
" [263.0, 125.0], \n",
|
||
" color='green', alpha=1, linewidth=1.3, zorder=100)\n",
|
||
" ax.scatter([3.0],\n",
|
||
" [3.0],\n",
|
||
" [263.0], \n",
|
||
" c='g', s=30, marker='D', alpha=1, linewidth=1.3, zorder=100)\n",
|
||
"\n",
|
||
" ax.margins(0,0,0)\n",
|
||
" fig.tight_layout()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 15,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
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"text/plain": [
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"<Figure size 960x540 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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||
"costplot3d(h, x, y)\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "notes"
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}
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},
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"source": [
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||
"Na powyższym wykresie poszukiwane minimum funkcji kosztu oznaczone jest czerwonym krzyżykiem.\n",
|
||
"\n",
|
||
"Możemy też zobaczyć rzut powyższego trójwymiarowego wykresu na płaszczyznę $(\\theta_0, \\theta_1)$ poniżej:"
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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": 16,
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"metadata": {
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"slideshow": {
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"slide_type": "notes"
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}
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},
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"outputs": [],
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"source": [
|
||
"def costplot2d(hypothesis, x, y, gradient_values=[], nohead=False):\n",
|
||
" fig = plt.figure(figsize=(16 * 0.6, 9 * 0.6))\n",
|
||
" ax = fig.add_subplot(111)\n",
|
||
" fig.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)\n",
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||
" ax.set_xlabel(r\"$\\theta_0$\")\n",
|
||
" ax.set_ylabel(r\"$\\theta_1$\")\n",
|
||
"\n",
|
||
" j = lambda theta0, theta1: costfun(hypothesis, x, y)([theta0, theta1])\n",
|
||
" X = np.arange(-10, 10.1, 0.1)\n",
|
||
" Y = np.arange(-1, 4.1, 0.1)\n",
|
||
" X, Y = np.meshgrid(X, Y)\n",
|
||
" Z = np.array(\n",
|
||
" [\n",
|
||
" [\n",
|
||
" J(hypothesis, [theta0, theta1], x, y)\n",
|
||
" for theta0, theta1 in zip(xRow, yRow)\n",
|
||
" ]\n",
|
||
" for xRow, yRow in zip(X, Y)\n",
|
||
" ]\n",
|
||
" )\n",
|
||
"\n",
|
||
" N = range(0, 800, 20)\n",
|
||
" plt.contour(X, Y, Z, N, cmap=\"coolwarm\", alpha=1)\n",
|
||
"\n",
|
||
" ax.scatter(\n",
|
||
" [-3.89578088],\n",
|
||
" [1.19303364],\n",
|
||
" c=\"r\",\n",
|
||
" s=80,\n",
|
||
" marker=\"x\",\n",
|
||
" label=r\"minimum: $J(-3.90, 1.19) = 4.48$\",\n",
|
||
" )\n",
|
||
"\n",
|
||
" if len(gradient_values) > 0:\n",
|
||
" prev_theta = gradient_values[0][1]\n",
|
||
" ax.scatter(\n",
|
||
" [prev_theta[0]], [prev_theta[1]], c=\"g\", s=30, marker=\"D\", zorder=100\n",
|
||
" )\n",
|
||
" for cost, theta in gradient_values[1:]:\n",
|
||
" dtheta = [theta[0] - prev_theta[0], theta[1] - prev_theta[1]]\n",
|
||
" ax.arrow(\n",
|
||
" prev_theta[0],\n",
|
||
" prev_theta[1],\n",
|
||
" dtheta[0],\n",
|
||
" dtheta[1],\n",
|
||
" color=\"green\",\n",
|
||
" head_width=(0.0 if nohead else 0.1),\n",
|
||
" head_length=(0.0 if nohead else 0.2),\n",
|
||
" zorder=100,\n",
|
||
" )\n",
|
||
" prev_theta = theta\n",
|
||
"\n",
|
||
" return fig\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 17,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
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}
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},
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"outputs": [
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{
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"text/plain": [
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"<Figure size 960x540 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": [
|
||
"fig = costplot2d(h, x, y)\n",
|
||
"legend(fig)\n"
|
||
]
|
||
},
|
||
{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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||
},
|
||
"source": [
|
||
"### Cechy funkcji kosztu\n",
|
||
"Funkcja kosztu $J(\\theta)$ zdefiniowana powyżej jest funkcją wypukłą, dlatego posiada tylko jedno minimum lokalne."
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||
]
|
||
},
|
||
{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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||
},
|
||
"source": [
|
||
"## 2.2. Metoda gradientu prostego"
|
||
]
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||
},
|
||
{
|
||
"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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||
}
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||
},
|
||
"source": [
|
||
"### Metoda gradientu prostego\n",
|
||
"Metoda znajdowania minimów lokalnych."
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||
]
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||
},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
|
||
"Idea:\n",
|
||
" * Zacznijmy od dowolnego $\\theta$.\n",
|
||
" * Zmieniajmy powoli $\\theta$ tak, aby zmniejszać $J(\\theta)$, aż w końcu znajdziemy minimum."
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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": 18,
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
|
||
"<Figure size 960x540 with 1 Axes>"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
}
|
||
],
|
||
"source": [
|
||
"costplot3d(h, x, y, show_gradient=True)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 19,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Przykładowe wartości kolejnych przybliżeń (sztuczne)\n",
|
||
"\n",
|
||
"gv = [\n",
|
||
" [_, [3.0, 3.0]],\n",
|
||
" [_, [2.6, 2.4]],\n",
|
||
" [_, [2.2, 2.0]],\n",
|
||
" [_, [1.6, 1.6]],\n",
|
||
" [_, [0.4, 1.2]],\n",
|
||
"]\n",
|
||
"\n",
|
||
"# Przygotowanie interaktywnego wykresu\n",
|
||
"\n",
|
||
"sliderSteps1 = widgets.IntSlider(\n",
|
||
" min=0, max=3, step=1, value=0, description=\"kroki\", width=300\n",
|
||
")\n",
|
||
"\n",
|
||
"\n",
|
||
"def slide4(steps):\n",
|
||
" costplot2d(h, x, y, gradient_values=gv[: steps + 1])\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 20,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"application/vnd.jupyter.widget-view+json": {
|
||
"model_id": "4b3e33d96ad447e08dace02f0ea543d2",
|
||
"version_major": 2,
|
||
"version_minor": 0
|
||
},
|
||
"text/plain": [
|
||
"interactive(children=(IntSlider(value=0, description='kroki', max=3), Output()), _dom_classes=('widget-interac…"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"<function __main__.slide4(steps)>"
|
||
]
|
||
},
|
||
"execution_count": 20,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"widgets.interact(slide4, steps=sliderSteps1)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Metoda gradientu prostego\n",
|
||
"W każdym kroku będziemy aktualizować parametry $\\theta_j$:\n",
|
||
"\n",
|
||
"$$ \\theta_j := \\theta_j - \\alpha \\frac{\\partial}{\\partial \\theta_j} J(\\theta) $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"source": [
|
||
"Współczynnik $\\alpha$ nazywamy **długością kroku** lub **współczynnikiem szybkości uczenia** (*learning rate*)."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"$$ \\begin{array}{rcl}\n",
|
||
"\\dfrac{\\partial}{\\partial \\theta_j} J(\\theta)\n",
|
||
" & = & \\dfrac{\\partial}{\\partial \\theta_j} \\dfrac{1}{2m} \\displaystyle\\sum_{i = 1}^{m} \\left( h_{\\theta} \\left( x^{(i)} \\right) - y^{(i)} \\right) ^2 \\\\\n",
|
||
" & = & 2 \\cdot \\dfrac{1}{2m} \\displaystyle\\sum_{i=1}^m \\left( \\left( h_\\theta \\left( x^{(i)} \\right) - y^{(i)} \\right) \\cdot \\dfrac{\\partial}{\\partial\\theta_j} \\left( h_\\theta \\left( x^{(i)} \\right) - y^{(i)} \\right) \\right) \\\\\n",
|
||
" & = & \\dfrac{1}{m}\\displaystyle\\sum_{i=1}^m \\left( \\left( h_\\theta \\left( x^{(i)} \\right) - y^{(i)} \\right) \\cdot \\dfrac{\\partial}{\\partial\\theta_j} \\left( \\displaystyle\\sum_{k=0}^n \\theta_k x_k^{(i)} - y^{(i)} \\right) \\right) \\\\\n",
|
||
" & = & \\dfrac{1}{m}\\displaystyle\\sum_{i=1}^m \\left( h_\\theta \\left( x^{(i)} \\right) -y^{(i)} \\right) x_j^{(i)} \\\\\n",
|
||
"\\end{array} $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Czyli dla regresji liniowej jednej zmiennej:\n",
|
||
"\n",
|
||
"$$ h_\\theta(x) = \\theta_0 + \\theta_1x $$\n",
|
||
"\n",
|
||
"w każdym kroku będziemy aktualizować:\n",
|
||
"\n",
|
||
"$$\n",
|
||
"\\begin{array}{rcl}\n",
|
||
"\\theta_0 & := & \\theta_0 - \\alpha \\, \\dfrac{1}{m}\\displaystyle\\sum_{i=1}^m \\left( h_\\theta(x^{(i)})-y^{(i)} \\right) \\\\ \n",
|
||
"\\theta_1 & := & \\theta_1 - \\alpha \\, \\dfrac{1}{m}\\displaystyle\\sum_{i=1}^m \\left( h_\\theta(x^{(i)})-y^{(i)} \\right) x^{(i)}\\\\ \n",
|
||
"\\end{array}\n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"###### Uwaga!\n",
|
||
" * W każdym kroku aktualizujemy *jednocześnie* $\\theta_0$ i $\\theta_1$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"source": [
|
||
" * Kolejne kroki wykonujemy aż uzyskamy zbieżność"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Metoda gradientu prostego – implementacja"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 21,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Wyświetlanie macierzy w LaTeX-u\n",
|
||
"\n",
|
||
"\n",
|
||
"def LatexMatrix(matrix):\n",
|
||
" ltx = r\"\\left[\\begin{array}\"\n",
|
||
" m, n = matrix.shape\n",
|
||
" ltx += \"{\" + (\"r\" * n) + \"}\"\n",
|
||
" for i in range(m):\n",
|
||
" ltx += r\" & \".join([(\"%.4f\" % j.item()) for j in matrix[i]]) + r\" \\\\ \"\n",
|
||
" ltx += r\"\\end{array}\\right]\"\n",
|
||
" return ltx\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 22,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def gradient_descent(h, cost_fun, theta, x, y, alpha, eps):\n",
|
||
" current_cost = cost_fun(h, theta, x, y)\n",
|
||
" history = [\n",
|
||
" [current_cost, theta]\n",
|
||
" ] # zapiszmy wartości kosztu i parametrów, by potem zrobić wykres\n",
|
||
" m = len(y)\n",
|
||
" while True:\n",
|
||
" new_theta = [\n",
|
||
" theta[0] - alpha / float(m) * sum(h(theta, x[i]) - y[i] for i in range(m)),\n",
|
||
" theta[1]\n",
|
||
" - alpha / float(m) * sum((h(theta, x[i]) - y[i]) * x[i] for i in range(m)),\n",
|
||
" ]\n",
|
||
" theta = new_theta # jednoczesna aktualizacja - używamy zmiennej tymczasowej\n",
|
||
" try:\n",
|
||
" prev_cost = current_cost\n",
|
||
" current_cost = cost_fun(h, theta, x, y)\n",
|
||
" except OverflowError:\n",
|
||
" break\n",
|
||
" if abs(prev_cost - current_cost) <= eps:\n",
|
||
" break\n",
|
||
" history.append([current_cost, theta])\n",
|
||
" return theta, history\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": null,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"best_theta, history = gradient_descent(h, J, [0.0, 0.0], x, y, alpha=0.05, eps=0.0001)\n",
|
||
"\n",
|
||
"display(\n",
|
||
" Math(\n",
|
||
" r\"\\large\\textrm{Wynik:}\\quad \\theta = \"\n",
|
||
" + LatexMatrix(np.matrix(best_theta).reshape(2, 1))\n",
|
||
" + (r\" \\quad J(\\theta) = %.4f\" % history[-1][0])\n",
|
||
" + r\" \\quad \\textrm{po %d iteracjach}\" % len(history)\n",
|
||
" )\n",
|
||
")\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 24,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Przygotowanie interaktywnego wykresu\n",
|
||
"\n",
|
||
"sliderSteps2 = widgets.IntSlider(\n",
|
||
" min=0, max=500, step=1, value=1, description=\"kroki\", width=300\n",
|
||
")\n",
|
||
"\n",
|
||
"\n",
|
||
"def slide5(steps):\n",
|
||
" costplot2d(h, x, y, gradient_values=history[: steps + 1], nohead=True)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 25,
|
||
"metadata": {
|
||
"scrolled": true,
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"application/vnd.jupyter.widget-view+json": {
|
||
"model_id": "e2f6797395434b26b93f0392c7b58207",
|
||
"version_major": 2,
|
||
"version_minor": 0
|
||
},
|
||
"text/plain": [
|
||
"interactive(children=(IntSlider(value=1, description='kroki', max=500), Button(description='Run Interact', sty…"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"<function __main__.slide5(steps)>"
|
||
]
|
||
},
|
||
"execution_count": 25,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"widgets.interact_manual(slide5, steps=sliderSteps2)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Współczynnik szybkości uczenia $\\alpha$ (długość kroku)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Tempo zbieżności metody gradientu prostego możemy regulować za pomocą parametru $\\alpha$, pamiętając, że:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"source": [
|
||
" * Jeżeli długość kroku jest zbyt mała, algorytm może działać zbyt wolno."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"source": [
|
||
" * Jeżeli długość kroku jest zbyt duża, algorytm może nie być zbieżny."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"## 2.3. Predykcja wyników"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Zbudowaliśmy model, dzięki któremu wiemy, jaka jest zależność między dochodem firmy transportowej ($y$) a ludnością miasta ($x$).\n",
|
||
"\n",
|
||
"Wróćmy teraz do postawionego na początku wykładu pytania: jak przewidzieć dochód firmy transportowej w mieście o danej wielkości?\n",
|
||
"\n",
|
||
"Odpowiedź polega po prostu na zastosowaniu funkcji $h$ z wyznaczonymi w poprzednim kroku parametrami $\\theta$.\n",
|
||
"\n",
|
||
"Na przykład, jeżeli miasto ma $536\\,000$ ludności, to $x = 53.6$ (bo dane uczące były wyrażone w dziesiątkach tysięcy mieszkańców, a $536\\,000 = 53.6 \\cdot 10\\,000$) i możemy użyć znalezionych parametrów $\\theta$, by wykonać następujące obliczenia:\n",
|
||
"$$ \\hat{y} \\, = \\, h_\\theta(x) \\, = \\, \\theta_0 + \\theta_1 \\, x \\, = \\, 0.0494 + 0.7591 \\cdot 53.6 \\, = \\, 40.7359 $$\n",
|
||
"\n",
|
||
"Czyli używając zdefiniowanych wcześniej funkcji:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 26,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"52.96131370254696\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"example_x = 53.6\n",
|
||
"predicted_y = h(best_theta, example_x)\n",
|
||
"print(\n",
|
||
" predicted_y\n",
|
||
") ## taki jest przewidywany dochód tej firmy transportowej w 536-tysięcznym mieście\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"## 2.4. Ewaluacja modelu"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Jak ocenić jakość stworzonego przez nas modelu?\n",
|
||
"\n",
|
||
" * Trzeba sprawdzić, jak przewidywania modelu zgadzają się z oczekiwaniami!"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Czy możemy w tym celu użyć danych, których użyliśmy do uczenia modelu?\n",
|
||
"**NIE!**\n",
|
||
"\n",
|
||
" * Istotą uczenia maszynowego jest budowanie modeli/algorytmów, które dają dobre przewidywania dla **nieznanych** danych – takich, z którymi algorytm nie miał jeszcze styczności! Nie sztuką jest przewidywać rzeczy, które już sie zna.\n",
|
||
" * Dlatego testowanie/ewaluowanie modelu na zbiorze uczącym mija się z celem i jest nieprzydatne.\n",
|
||
" * Do ewaluacji modelu należy użyć oddzielnego zbioru danych.\n",
|
||
" * **Dane uczące i dane testowe zawsze powinny stanowić oddzielne zbiory!**"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Na wykładzie *5. Dobre praktyki w uczeniu maszynowym* dowiesz się, jak podzielić posiadane dane na zbiór uczący i zbiór testowy.\n",
|
||
"\n",
|
||
"Tutaj, na razie, do ewaluacji użyjemy specjalnie przygotowanego zbioru testowego."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Jako metrykę ewaluacji wykorzystamy znany nam już błąd średniokwadratowy (MSE):"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 27,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "skip"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def mse(expected, predicted):\n",
|
||
" \"\"\"Błąd średniokwadratowy\"\"\"\n",
|
||
" m = len(expected)\n",
|
||
" if len(predicted) != m:\n",
|
||
" raise Exception(\"Wektory mają różne długości!\")\n",
|
||
" return 1.0 / (2 * m) * sum((expected[i] - predicted[i]) ** 2 for i in range(m))\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 28,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"4.36540743711836\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"# Wczytwanie danych testowych z pliku za pomocą numpy\n",
|
||
"\n",
|
||
"test_data = np.loadtxt(\"data1_test.csv\", delimiter=\",\")\n",
|
||
"x_test = test_data[:, 0]\n",
|
||
"y_test = test_data[:, 1]\n",
|
||
"\n",
|
||
"# Obliczenie przewidywań modelu\n",
|
||
"y_pred = h(best_theta, x_test)\n",
|
||
"\n",
|
||
"# Obliczenie MSE na zbiorze testowym (im mniejszy MSE, tym lepiej!)\n",
|
||
"evaluation_result = mse(y_test, y_pred)\n",
|
||
"\n",
|
||
"print(evaluation_result)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Otrzymana wartość mówi nam o tym, jak dobry jest stworzony przez nas model.\n",
|
||
"\n",
|
||
"W przypadku metryki MSE im mniejsza wartość, tym lepiej.\n",
|
||
"\n",
|
||
"W ten sposób możemy np. porównywać różne modele."
|
||
]
|
||
}
|
||
],
|
||
"metadata": {
|
||
"celltoolbar": "Slideshow",
|
||
"kernelspec": {
|
||
"display_name": "Python 3 (ipykernel)",
|
||
"language": "python",
|
||
"name": "python3"
|
||
},
|
||
"language_info": {
|
||
"codemirror_mode": {
|
||
"name": "ipython",
|
||
"version": 3
|
||
},
|
||
"file_extension": ".py",
|
||
"mimetype": "text/x-python",
|
||
"name": "python",
|
||
"nbconvert_exporter": "python",
|
||
"pygments_lexer": "ipython3",
|
||
"version": "3.10.4"
|
||
},
|
||
"livereveal": {
|
||
"start_slideshow_at": "selected",
|
||
"theme": "white"
|
||
},
|
||
"vscode": {
|
||
"interpreter": {
|
||
"hash": "916dbcbb3f70747c44a77c7bcd40155683ae19c65e1c03b4aa3499c5328201f1"
|
||
}
|
||
}
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 4
|
||
}
|