From dbc0fcfc54630c13a292587e50fa812ee16390a6 Mon Sep 17 00:00:00 2001
From: skorzewski <pawel.skorzewski@amu.edu.pl>
Date: Sat, 19 Dec 2020 13:09:41 +0100
Subject: [PATCH] =?UTF-8?q?Modu=C5=82y=202=20i=203=20-=20szkielet?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 1.ipynb => 1. Jupyter - podstawy.ipynb        |     0
 ... wykresy i prezentacje multimedialne.ipynb |    86 +
 ...przykład materiałów dydaktycznych.ipynb | 42666 ++++++++++++++++
 data01_test.csv                               |    17 +
 data01_train.csv                              |    80 +
 5 files changed, 42849 insertions(+)
 rename 1.ipynb => 1. Jupyter - podstawy.ipynb (100%)
 create mode 100644 2. Jupyter - interaktywne wykresy i prezentacje multimedialne.ipynb
 create mode 100644 3. Jupyter - przykład materiałów dydaktycznych.ipynb
 create mode 100644 data01_test.csv
 create mode 100644 data01_train.csv

diff --git a/1.ipynb b/1. Jupyter - podstawy.ipynb
similarity index 100%
rename from 1.ipynb
rename to 1. Jupyter - podstawy.ipynb
diff --git a/2. Jupyter - interaktywne wykresy i prezentacje multimedialne.ipynb b/2. Jupyter - interaktywne wykresy i prezentacje multimedialne.ipynb
new file mode 100644
index 0000000..851db08
--- /dev/null
+++ b/2. Jupyter - interaktywne wykresy i prezentacje multimedialne.ipynb	
@@ -0,0 +1,86 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Przygotowanie innowacyjnych materiałów szkoleniowych i dokumentacji wewnętrznych w obszarze IT\n",
+    "# 2. Tworzenie materiałów szkoleniowych w Jupyter Notebook - interaktywne wykresy i prezentacje multimedialne"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## 2.1. Umieszczanie wzorów matematycznych"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## 2.2. Tworzenie wykresów"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Tworzenie statycznych wykresów"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Tworzenie interaktywnych wykresów"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## 2.3. Wyświetlanie materiałów Jupyter Notebook w formie prezentacji multimedialnej"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Materiały przygotowane w formacie `.ipynb` można wyświetlać w formie prezentacji multimedialnej.\n",
+    "\n",
+    "Jest to możliwe dzięki narzędziu [**RISE**](https://rise.readthedocs.io).\n",
+    "\n",
+    "RISE (*Reveal.js Ipython Slideshow Extension*) jest rozszerzeniem do Jupytera umożliwiającym wyświetlanie notatników w trybie prezentacji w oparciu o framework [**Reveal.js**](https://revealjs.com/)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "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"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/3. Jupyter - przykład materiałów dydaktycznych.ipynb b/3. Jupyter - przykład materiałów dydaktycznych.ipynb
new file mode 100644
index 0000000..79f636a
--- /dev/null
+++ b/3. Jupyter - przykład materiałów dydaktycznych.ipynb	
@@ -0,0 +1,42666 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Przygotowanie innowacyjnych materiałów szkoleniowych i dokumentacji wewnętrznych w obszarze IT\n",
+    "# 3. Przykład materiałów szkoleniowych w Jupyter Notebook"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Poniższe materiały pochodzą z kursu \"Uczenie maszynowe i big data\"."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Funkcja kosztu"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Zadanie\n",
+    "Znając $x$ – ludność miasta (w dziesiątkach tysięcy mieszkańców),\n",
+    "należy przewidzieć $y$ – dochód firmy transportowej (w dziesiątkach tysięcy dolarów).\n",
+    "\n",
+    "(Dane pochodzą z kursu „Machine Learning”, Andrew Ng, Coursera)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib\n",
+    "import matplotlib.pyplot as pl\n",
+    "import ipywidgets as widgets\n",
+    "\n",
+    "%matplotlib inline\n",
+    "%config InlineBackend.figure_format = 'svg'\n",
+    "\n",
+    "from IPython.display import display, Math, Latex"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Dane"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "6.1101,17.592\n",
+      "\n",
+      "5.5277,9.1302\n",
+      "\n",
+      "8.5186,13.662\n",
+      "\n",
+      "7.0032,11.854\n",
+      "\n",
+      "5.8598,6.8233\n",
+      "\n",
+      "8.3829,11.886\n",
+      "\n",
+      "7.4764,4.3483\n",
+      "\n",
+      "8.5781,12\n",
+      "\n",
+      "6.4862,6.5987\n",
+      "\n",
+      "5.0546,3.8166\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "with open('data01_train.csv') as data:\n",
+    "    for line in data.readlines()[:10]:\n",
+    "        print(line)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Wczytanie danych"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "x = [6.1101, 5.5277, 8.5186, 7.0032, 5.8598, 8.3829, 7.4764, 8.5781, 6.4862, 5.0546]\n",
+      "y = [17.592, 9.1302, 13.662, 11.854, 6.8233, 11.886, 4.3483, 12.0, 6.5987, 3.8166]\n"
+     ]
+    }
+   ],
+   "source": [
+    "import csv\n",
+    "\n",
+    "reader = csv.reader(open('data01_train.csv'), delimiter=',')\n",
+    "\n",
+    "x = list()\n",
+    "y = list()\n",
+    "for xi, yi in reader:\n",
+    "    x.append(float(xi))\n",
+    "    y.append(float(yi)) \n",
+    "    \n",
+    "print('x = {}'.format(x[:10])) \n",
+    "print('y = {}'.format(y[:10]))"
+   ]
+  },
+  {
+   "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",
+    "\n",
+    "Zacznijmy od najprostszego przypadku, kiedy $h(x)$ jest po prostu funkcją liniową. Ogólny wzór funkcji liniowej to\n",
+    "$$ h(x) = a \\, x + b $$\n",
+    "\n",
+    "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).\n",
+    "\n",
+    "Poszukiwaną funkcję $h$ będziemy nazywać **funkcją hipotezy**, a jej współczynniki – **parametrami modelu**.\n",
+    "\n",
+    "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",
+    "\n",
+    "**Parametry modelu** tworzą wektor, który oznaczymy po prostu przez $\\theta$:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "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": {},
+   "source": [
+    "Przyjrzyjmy się teraz, jak wyglądają dane, które mamy modelować:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
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+      ],
+      "text/plain": [
+       "<Figure size 691.2x388.8 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Funkcje rysujące wykres kropkowy oraz prostą regresyjną\n",
+    "\n",
+    "def regdots(x, y):    \n",
+    "    fig = pl.figure(figsize=(16*.6, 9*.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', s=50, label='Dane')\n",
+    "    \n",
+    "    ax.set_xlabel(u'Wielkość miejscowości [dzies. tys. mieszk.]')\n",
+    "    ax.set_ylabel(u'Dochód firmy [dzies. tys. dolarów]')\n",
+    "    ax.margins(.05, .05)\n",
+    "    pl.ylim(min(y) - 1, max(y) + 1)\n",
+    "    pl.xlim(min(x) - 1, max(x) + 1)\n",
+    "    return fig\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(X, Y, linewidth='2',\n",
+    "            label=(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",
+    "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",
+    "\n",
+    "fig = regdots(x,y)\n",
+    "legend(fig)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Hipoteza: funkcja liniowa jednej zmiennej\n",
+    "\n",
+    "def h(theta, x):\n",
+    "    return theta[0] + theta[1] * x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Przygotowanie interaktywnego wykresu\n",
+    "\n",
+    "sliderTheta01 = widgets.FloatSlider(min=-10, max=10, step=0.1, value=0, description=r'$\\theta_0$', width=300)\n",
+    "sliderTheta11 = widgets.FloatSlider(min=-5, max=5, step=0.1, value=0, description=r'$\\theta_1$', width=300)\n",
+    "\n",
+    "def slide1(theta0, theta1):\n",
+    "    fig = regdots(x, y)\n",
+    "    regline(fig, h, [theta0, theta1], x)\n",
+    "    legend(fig)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "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": 8,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "64432dd82e97458f86f45f4ed72c6616",
+       "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__.slide1(theta0, theta1)>"
+      ]
+     },
+     "execution_count": 8,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "widgets.interact_manual(slide1, theta0=sliderTheta01, theta1=sliderTheta11)"
+   ]
+  },
+  {
+   "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, *root-mean-square error, RMSE*):"
+   ]
+  },
+  {
+   "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": "fragment"
+    }
+   },
+   "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))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Oblicz wartość funkcji kosztu i pokaż na wykresie\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(X, Y, linewidth='2',\n",
+    "            label=(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",
+    "sliderTheta02 = widgets.FloatSlider(min=-10, max=10, step=0.1, value=0, description=r'$\\theta_0$', width=300)\n",
+    "sliderTheta12 = widgets.FloatSlider(min=-5, max=5, step=0.1, value=0, description=r'$\\theta_1$', width=300)\n",
+    "\n",
+    "def slide2(theta0, theta1):\n",
+    "    fig = regdots(x, y)\n",
+    "    regline2(fig, h, [theta0, theta1], x, y)\n",
+    "    legend(fig)"
+   ]
+  },
+  {
+   "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": "1d46c4b0e0234d1d9e95f9ef7aee61cc",
+       "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)"
+   ]
+  },
+  {
+   "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 RMSE 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": 11,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Wykres funkcji kosztu dla ustalonego theta_1=1.0\n",
+    "\n",
+    "def costfun(fun, x, y):\n",
+    "    return lambda theta: J(fun, theta, x, y)\n",
+    "\n",
+    "def costplot(hypothesis, x, y, theta1=1.0):\n",
+    "    fig = pl.figure(figsize=(16*.6, 9*.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(X, Y, linewidth='2', label=(r'$J(\\theta_0, {theta1})$'.format(theta1=theta1)))\n",
+    "    return fig\n",
+    "\n",
+    "def slide3(theta1):\n",
+    "    fig = costplot(h, x, y, theta1)\n",
+    "    legend(fig)\n",
+    "\n",
+    "sliderTheta13 = widgets.FloatSlider(min=-5, max=5, step=0.1, value=1.0, description=r'$\\theta_1$', width=300)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "a5409cb710364e6691dd67525cb971f4",
+       "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": 12,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "widgets.interact_manual(slide3, theta1=sliderTheta13)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "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 = pl.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",
+    "    pl.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": 14,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
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+   ],
+   "source": [
+    "costplot3d(h, x, y)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "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:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def costplot2d(hypothesis, x, y, gradient_values=[], nohead=False):\n",
+    "    fig = pl.figure(figsize=(16*.6, 9*.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'$\\theta_1$')\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([[J(hypothesis, [theta0, theta1], x, y) \n",
+    "                    for theta0, theta1 in zip(xRow, yRow)] \n",
+    "                   for xRow, yRow in zip(X, Y)])\n",
+    "    \n",
+    "    N = range(0, 800, 20)\n",
+    "    pl.contour(X, Y, Z, N, cmap='coolwarm', alpha=1)\n",
+    "\n",
+    "    ax.scatter([-3.89578088], [1.19303364], c='r', s=80, marker='x',\n",
+    "               label=r'minimum: $J(-3.90, 1.19) = 4.48$')\n",
+    "    \n",
+    "    if len(gradient_values) > 0:\n",
+    "        prev_theta = gradient_values[0][1]\n",
+    "        ax.scatter([prev_theta[0]], [prev_theta[1]],\n",
+    "                   c='g', s=30, marker='D', zorder=100)\n",
+    "        for cost, theta in gradient_values[1:]:\n",
+    "            dtheta = [theta[0] - prev_theta[0], theta[1] - prev_theta[1]]\n",
+    "            ax.arrow(prev_theta[0], prev_theta[1], dtheta[0], 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",
+    "            prev_theta = theta\n",
+    "    \n",
+    "    return fig"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
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+       "<Figure size 691.2x388.8 with 1 Axes>"
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+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "fig = costplot2d(h, x, y)\n",
+    "legend(fig)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Cechy funkcji kosztu\n",
+    "* $J(\\theta)$ jest funkcją wypukłą\n",
+    "* $J(\\theta)$ posiada tylko jedno minimum lokalne"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Metoda gradientu prostego"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Metoda gradientu prostego\n",
+    "Metoda znajdowania minimów lokalnych."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Idea:\n",
+    " * Zacznijmy od dowolnego $\\theta$.\n",
+    " * Zmieniajmy powoli $\\theta$ tak, aby zmniejszać $J(\\theta)$, aż w końcu znajdziemy minimum."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
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+      ],
+      "text/plain": [
+       "<Figure size 691.2x388.8 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "costplot3d(h, x, y, show_gradient=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
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+   "outputs": [],
+   "source": [
+    "# Przykładowe wartości kolejnych przybliżeń (sztuczne)\n",
+    "\n",
+    "gv = [[_, [3.0, 3.0]], [_, [2.6, 2.4]], [_, [2.2, 2.0]], [_, [1.6, 1.6]], [_, [0.4, 1.2]]]\n",
+    "\n",
+    "# Przygotowanie interaktywnego wykresu\n",
+    "\n",
+    "sliderSteps1 = widgets.IntSlider(min=0, max=3, step=1, value=0, description='kroki', width=300)\n",
+    "\n",
+    "def slide4(steps):\n",
+    "    costplot2d(h, x, y, gradient_values=gv[:steps+1])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
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+      ],
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+       "<Figure size 691.2x388.8 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "<function __main__.slide4(steps)>"
+      ]
+     },
+     "execution_count": 19,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "widgets.interact(slide4, steps=sliderSteps1)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Metoda gradientu prostego\n",
+    "W każdym kroku będziemy aktualizować parametry $\\theta_j$:\n",
+    "\n",
+    "$$ \\theta_j := \\theta_j - \\alpha \\frac{\\partial}{\\partial \\theta_j} J(\\theta) \\quad \\mbox{ dla każdego } j $$"
+   ]
+  },
+  {
+   "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": [
+    "$$ \\begin{array}{rcl}\n",
+    "\\dfrac{\\partial}{\\partial \\theta_j} J(\\theta)\n",
+    " & = & \\dfrac{\\partial}{\\partial \\theta_j} \\dfrac{1}{2m} \\displaystyle\\sum_{i = 1}^{m} \\left( h_{\\theta} \\left( x^{(i)} \\right) - y^{(i)} \\right) ^2 \\\\\n",
+    " & = & 2 \\cdot \\dfrac{1}{2m} \\displaystyle\\sum_{i=1}^m \\left( h_\\theta \\left( x^{(i)} \\right) - y^{(i)} \\right) \\cdot \\dfrac{\\partial}{\\partial\\theta_j} \\left( h_\\theta \\left( x^{(i)} \\right) - y^{(i)} \\right) \\\\\n",
+    " & = & \\dfrac{1}{m}\\displaystyle\\sum_{i=1}^m \\left( h_\\theta \\left( x^{(i)} \\right) - y^{(i)} \\right) \\cdot \\dfrac{\\partial}{\\partial\\theta_j} \\left( \\displaystyle\\sum_{i=0}^n \\theta_i x_i^{(i)} - y^{(i)} \\right)\\\\\n",
+    " & = & \\dfrac{1}{m}\\displaystyle\\sum_{i=1}^m \\left( h_\\theta \\left( x^{(i)} \\right) -y^{(i)} \\right) x_j^{(i)} \\\\\n",
+    "\\end{array} $$"
+   ]
+  },
+  {
+   "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": 20,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Wyświetlanie macierzy w LaTeX-u\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"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "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",
+    "    log = [[current_cost, theta]]  # log przechowuje wartości kosztu i parametrów\n",
+    "    m = len(y)\n",
+    "    while True:\n",
+    "        new_theta = [\n",
+    "            theta[0] - alpha/float(m) * sum(h(theta, x[i]) - y[i]\n",
+    "                                            for i in range(m)),   \n",
+    "            theta[1] - alpha/float(m) * sum((h(theta, x[i]) - y[i]) * x[i]\n",
+    "                                            for i in range(m))]\n",
+    "        theta = new_theta  # jednoczesna aktualizacja - używamy zmiennej tymaczasowej\n",
+    "        try:\n",
+    "            current_cost, prev_cost = cost_fun(h, theta, x, y), current_cost\n",
+    "        except OverflowError:\n",
+    "            break      \n",
+    "        if abs(prev_cost - current_cost) <= eps:\n",
+    "            break     \n",
+    "        log.append([current_cost, theta])\n",
+    "    return theta, log"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\large\\textrm{Wynik:}\\quad \\theta = \\left[\\begin{array}{r}0.0525 \\\\ 0.8015 \\\\ \\end{array}\\right] \\quad J(\\theta) = 6.0273 \\quad \\textrm{po 3604 iteracjach}$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Math object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "best_theta, log = gradient_descent(h, J, [0.0, 0.0], x, y, alpha=0.00001, eps=0.0001)\n",
+    "\n",
+    "display(Math(r'\\large\\textrm{Wynik:}\\quad \\theta = ' + \n",
+    "             LatexMatrix(np.matrix(best_theta).reshape(2,1)) + \n",
+    "            (r' \\quad J(\\theta) = %.4f' % log[-1][0])  \n",
+    "            + r' \\quad \\textrm{po %d iteracjach}' % len(log))) "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Przygotowanie interaktywnego wykresu\n",
+    "\n",
+    "sliderSteps2 = widgets.IntSlider(min=0, max=500, step=1, value=1, description='kroki', width=300)\n",
+    "\n",
+    "def slide5(steps):\n",
+    "    costplot2d(h, x, y, gradient_values=log[:steps+1], nohead=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "74447996dfc8460cb3230139ddb584c4",
+       "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": 24,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "widgets.interact_manual(slide5, steps=sliderSteps2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Współczynnik $\\alpha$ (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": [
+    "## 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 trenujące 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": 25,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "43.01265462001823\n"
+     ]
+    }
+   ],
+   "source": [
+    "example_x = 53.6\n",
+    "predicted_y = h(best_theta, example_x)\n",
+    "print(predicted_y)  ## taki jest przewidywany dochód tej firmy transportowej w 536-tysięcznym mieście"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## 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 wytrenowania 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 juz 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 (RMSE):"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def rmse(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))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "5.283571974307861\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Wczytwanie danych testowych z pliku za pomocą numpy\n",
+    "\n",
+    "test_data = np.loadtxt('data01_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 RMSE na zbiorze testowym (im mniejszy RMSE, tym lepiej!)\n",
+    "evaluation_result = rmse(y_test, y_pred)\n",
+    "\n",
+    "print(evaluation_result)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Otrzymana wartość mówi nam o tym, jak dobry jest stworzony przez nas model.\n",
+    "\n",
+    "W przypadku metryki RMSE im mniejsz wartość, tym lepiej.\n",
+    "\n",
+    "W ten sposób możemy np. porównywać różne modele."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Regresja liniowa wielu zmiennych"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "source": [
+    "Do przewidywania wartości $y$ możemy użyć więcej niż jednej cechy $x$:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Przykład – ceny mieszkań"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "ename": "FileNotFoundError",
+     "evalue": "[Errno 2] No such file or directory: 'data02_train.tsv'",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
+      "\u001b[1;31mFileNotFoundError\u001b[0m                         Traceback (most recent call last)",
+      "\u001b[1;32m<ipython-input-28-b306ec3e56ee>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;32mimport\u001b[0m \u001b[0mcsv\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      2\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 3\u001b[1;33m \u001b[0mreader\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mcsv\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mreader\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mopen\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'data02_train.tsv'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdelimiter\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m'\\t'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      4\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mi\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mrow\u001b[0m \u001b[1;32min\u001b[0m \u001b[0menumerate\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mlist\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mreader\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;36m10\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[0mi\u001b[0m \u001b[1;33m==\u001b[0m \u001b[1;36m0\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
+      "\u001b[1;31mFileNotFoundError\u001b[0m: [Errno 2] No such file or directory: 'data02_train.tsv'"
+     ]
+    }
+   ],
+   "source": [
+    "import csv\n",
+    "\n",
+    "reader = csv.reader(open('data02_train.tsv'), delimiter='\\t')\n",
+    "for i, row in enumerate(list(reader)[:10]):\n",
+    "    if i == 0:\n",
+    "        print(' '.join(['{}: {:8}'.format('x' + str(j) if j > 0 else 'y ', entry)\n",
+    "                        for j, entry in enumerate(row)]))\n",
+    "    else:\n",
+    "        print(' '.join(['{:12}'.format(entry) for entry in row]))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "$$ x^{(2)} = ({\\rm \"False\"}, 3, 2, {\\rm \"Sołacz\"}, 62), \\quad x_3^{(2)} = 2 $$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Hipoteza"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "W naszym przypadku (wybraliśmy 5 cech):\n",
+    "\n",
+    "$$ h_\\theta(x) = \\theta_0 + \\theta_1 x_1 + \\theta_2 x_2 + \\theta_3 x_3 + \\theta_4 x_4 + \\theta_5 x_5 $$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "W ogólności ($n$ cech):\n",
+    "\n",
+    "$$ h_\\theta(x) = \\theta_0 + \\theta_1 x_1 + \\theta_2 x_2 + \\ldots + \\theta_n x_n $$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "Jeżeli zdefiniujemy $x_0 = 1$, będziemy mogli powyższy wzór zapisać w bardziej kompaktowy sposób:\n",
+    "\n",
+    "$$\n",
+    "\\begin{array}{rcl}\n",
+    "h_\\theta(x)\n",
+    " & = & \\theta_0 x_0 + \\theta_1 x_1 + \\theta_2 x_2 + \\ldots + \\theta_n x_n \\\\\n",
+    " & = & \\displaystyle\\sum_{i=0}^{n} \\theta_i x_i \\\\\n",
+    " & = & \\theta^T \\, x \\\\\n",
+    " & = & x^T \\, \\theta \\\\\n",
+    "\\end{array}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Metoda gradientu prostego – notacja macierzowa"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "source": [
+    "Metoda gradientu prostego przyjmie bardzo elegancką formę, jeżeli do jej zapisu użyjemy wektorów i macierzy."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "$$\n",
+    "X=\\left[\\begin{array}{cc}\n",
+    "1 & \\left( \\vec x^{(1)} \\right)^T \\\\\n",
+    "1 & \\left( \\vec x^{(2)} \\right)^T \\\\\n",
+    "\\vdots & \\vdots\\\\\n",
+    "1 & \\left( \\vec x^{(m)} \\right)^T \\\\\n",
+    "\\end{array}\\right] \n",
+    "= \\left[\\begin{array}{cccc}\n",
+    "1 & x_1^{(1)} & \\cdots & x_n^{(1)} \\\\\n",
+    "1 & x_1^{(2)} & \\cdots & x_n^{(2)} \\\\\n",
+    "\\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
+    "1 & x_1^{(m)} & \\cdots & x_n^{(m)} \\\\\n",
+    "\\end{array}\\right]\n",
+    "\\quad\n",
+    "\\vec{y} = \n",
+    "\\left[\\begin{array}{c}\n",
+    "y^{(1)}\\\\\n",
+    "y^{(2)}\\\\\n",
+    "\\vdots\\\\\n",
+    "y^{(m)}\\\\\n",
+    "\\end{array}\\right]\n",
+    "\\quad\n",
+    "\\theta = \\left[\\begin{array}{c}\n",
+    "\\theta_0\\\\\n",
+    "\\theta_1\\\\\n",
+    "\\vdots\\\\\n",
+    "\\theta_n\\\\\n",
+    "\\end{array}\\right]\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Wersje macierzowe funkcji rysowania wykresów punktowych oraz krzywej regresyjnej\n",
+    "\n",
+    "def hMx(theta, X):\n",
+    "    return X * theta\n",
+    "\n",
+    "def regdotsMx(X, y):    \n",
+    "    fig = pl.figure(figsize=(16*.6, 9*.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[:, 1]], [y], c='r', s=50, label='Dane')\n",
+    "    \n",
+    "    ax.set_xlabel('Populacja')\n",
+    "    ax.set_ylabel('Zysk')\n",
+    "    ax.margins(.05, .05)\n",
+    "    pl.ylim(y.min() - 1, y.max() + 1)\n",
+    "    pl.xlim(np.min(X[:, 1]) - 1, np.max(X[:, 1]) + 1)\n",
+    "    return fig\n",
+    "\n",
+    "def reglineMx(fig, fun, theta, X):\n",
+    "    ax = fig.axes[0]\n",
+    "    x0, x1 = np.min(X[:, 1]), np.max(X[:, 1])\n",
+    "    L = [x0, x1]\n",
+    "    LX = np.matrix([1, x0, 1, x1]).reshape(2, 2)\n",
+    "    ax.plot(L, fun(theta, LX), linewidth='2',\n",
+    "            label=(r'$y={theta0:.2}{op}{theta1:.2}x$'.format(\n",
+    "                theta0=float(theta[0][0]),\n",
+    "                theta1=(float(theta[1][0]) if theta[1][0] >= 0 else float(-theta[1][0])),\n",
+    "                op='+' if theta[1][0] >= 0 else '-')))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Wczytwanie danych z pliku za pomocą numpy – regresja liniowa wielu zmiennych – notacja macierzowa\n",
+    "\n",
+    "import pandas\n",
+    "\n",
+    "data = pandas.read_csv('data02_train.tsv', delimiter='\\t', usecols=['price', 'rooms', 'floor', 'sqrMetres'])\n",
+    "m, n_plus_1 = data.values.shape\n",
+    "n = n_plus_1 - 1\n",
+    "Xn = data.values[:, 1:].reshape(m, n)\n",
+    "\n",
+    "# Dodaj kolumnę jedynek do macierzy\n",
+    "XMx = np.matrix(np.concatenate((np.ones((m, 1)), Xn), axis=1)).reshape(m, n_plus_1)\n",
+    "yMx = np.matrix(data.values[:, 0]).reshape(m, 1)\n",
+    "\n",
+    "print(XMx[:5])\n",
+    "print(XMx.shape)\n",
+    "\n",
+    "print()\n",
+    "\n",
+    "print(yMx[:5])\n",
+    "print(yMx.shape)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Funkcja kosztu – notacja macierzowa"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "$$J(\\theta)=\\dfrac{1}{2|\\vec y|}\\left(X\\theta-\\vec{y}\\right)^T\\left(X\\theta-\\vec{y}\\right)$$ \n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "from IPython.display import display, Math, Latex\n",
+    "\n",
+    "def JMx(theta,X,y):\n",
+    "    \"\"\"Wersja macierzowa funkcji kosztu\"\"\"\n",
+    "    m = len(y)\n",
+    "    J = 1.0 / (2.0 * m) * ((X * theta - y) . T * ( X * theta - y))\n",
+    "    return J.item()\n",
+    "\n",
+    "thetaMx = np.matrix([10, 90, -1, 2.5]).reshape(4, 1) \n",
+    "\n",
+    "cost = JMx(thetaMx,XMx,yMx) \n",
+    "display(Math(r'\\Large J(\\theta) = %.4f' % cost))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Gradient – notacja macierzowa"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "$$\\nabla J(\\theta) = \\frac{1}{|\\vec y|} X^T\\left(X\\theta-\\vec y\\right)$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "from IPython.display import display, Math, Latex\n",
+    "\n",
+    "def dJMx(theta,X,y):\n",
+    "    \"\"\"Wersja macierzowa gradientu funckji kosztu\"\"\"\n",
+    "    return 1.0 / len(y) * (X.T * (X * theta - y)) \n",
+    "\n",
+    "thetaMx = np.matrix([10, 90, -1, 2.5]).reshape(4, 1) \n",
+    "\n",
+    "display(Math(r'\\large \\theta = ' + LatexMatrix(thetaMx) + \n",
+    "             r'\\quad' + r'\\large \\nabla J(\\theta) = ' \n",
+    "             + LatexMatrix(dJMx(thetaMx,XMx,yMx))))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Algorytm gradientu prostego – notacja macierzowa"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "$$ \\theta := \\theta - \\alpha \\, \\nabla J(\\theta) $$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Implementacja algorytmu gradientu prostego za pomocą numpy i macierzy\n",
+    "\n",
+    "def GDMx(fJ, fdJ, theta, X, y, alpha, eps):\n",
+    "    current_cost = fJ(theta, X, y)\n",
+    "    log = [[current_cost, theta]]\n",
+    "    while True:\n",
+    "        theta = theta - alpha * fdJ(theta, X, y) # implementacja wzoru\n",
+    "        current_cost, prev_cost = fJ(theta, X, y), current_cost\n",
+    "        if abs(prev_cost - current_cost) <= eps:\n",
+    "            break\n",
+    "        if current_cost > prev_cost:\n",
+    "            print('Długość kroku (alpha) jest zbyt duża!')\n",
+    "            break\n",
+    "        log.append([current_cost, theta])\n",
+    "    return theta, log\n",
+    "\n",
+    "thetaStartMx = np.zeros((n + 1, 1))\n",
+    "\n",
+    "# Zmieniamy wartości alpha (rozmiar kroku) oraz eps (kryterium stopu)\n",
+    "thetaBestMx, log = GDMx(JMx, dJMx, thetaStartMx, \n",
+    "                        XMx, yMx, alpha=0.0001, eps=0.1)\n",
+    "\n",
+    "######################################################################\n",
+    "display(Math(r'\\large\\textrm{Wynik:}\\quad \\theta = ' + \n",
+    "             LatexMatrix(thetaBestMx) + \n",
+    "             (r' \\quad J(\\theta) = %.4f' % log[-1][0])  \n",
+    "             + r' \\quad \\textrm{po %d iteracjach}' % len(log)))  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Metoda gradientu prostego w praktyce"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Kryterium stopu"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "source": [
+    "Algorytm gradientu prostego polega na wykonywaniu określonych kroków w pętli. Pytanie brzmi: kiedy należy zatrzymać wykonywanie tej pętli?\n",
+    "\n",
+    "W każdej kolejnej iteracji wartość funkcji kosztu maleje o coraz mniejszą wartość.\n",
+    "Parametr `eps` określa, jaka wartość graniczna tej różnicy jest dla nas wystarczająca:\n",
+    "\n",
+    " * Im mniejsza wartość `eps`, tym dokładniejszy wynik, ale dłuższy czas działania algorytmu.\n",
+    " * Im większa wartość `eps`, tym krótszy czas działania algorytmu, ale mniej dokładny wynik."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Na wykresie zobaczymy porównanie regresji dla różnych wartości `eps`"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Wczytwanie danych z pliku za pomocą numpy – wersja macierzowa\n",
+    "data = np.loadtxt('data01_train.csv', delimiter=',')\n",
+    "m, n_plus_1 = data.shape\n",
+    "n = n_plus_1 - 1\n",
+    "Xn = data[:, 0:n].reshape(m, n)\n",
+    "\n",
+    "# Dodaj kolumnę jedynek do macierzy\n",
+    "XMx = np.matrix(np.concatenate((np.ones((m, 1)), Xn), axis=1)).reshape(m, n_plus_1)\n",
+    "yMx = np.matrix(data[:, 1]).reshape(m, 1)\n",
+    "\n",
+    "thetaStartMx = np.zeros((2, 1))\n",
+    "\n",
+    "fig = regdotsMx(XMx, yMx)\n",
+    "theta_e1, log1 = GDMx(JMx, dJMx, thetaStartMx, XMx, yMx, alpha=0.01, eps=0.01)  # niebieska linia\n",
+    "reglineMx(fig, hMx, theta_e1, XMx)\n",
+    "theta_e2, log2 = GDMx(JMx, dJMx, thetaStartMx, XMx, yMx, alpha=0.01, eps=0.000001)  # pomarańczowa linia\n",
+    "reglineMx(fig, hMx, theta_e2, XMx)\n",
+    "legend(fig)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "display(Math(r'\\theta_{10^{-2}} = ' + LatexMatrix(theta_e1) +\n",
+    "             r'\\quad\\theta_{10^{-6}} = ' + LatexMatrix(theta_e2)))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Długość kroku ($\\alpha$)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Jak zmienia się koszt w kolejnych krokach w zależności od alfa\n",
+    "\n",
+    "def costchangeplot(logs):\n",
+    "    fig = pl.figure(figsize=(16*.6, 9*.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('krok')\n",
+    "    ax.set_ylabel(r'$J(\\theta)$')\n",
+    "\n",
+    "    X = np.arange(0, 500, 1)\n",
+    "    Y = [logs[step][0] for step in X]\n",
+    "    ax.plot(X, Y, linewidth='2', label=(r'$J(\\theta)$'))\n",
+    "    return fig\n",
+    "\n",
+    "def slide7(alpha):\n",
+    "    best_theta, log = gradient_descent(h, J, [0.0, 0.0], x, y, alpha=alpha, eps=0.0001)\n",
+    "    fig = costchangeplot(log)\n",
+    "    legend(fig)\n",
+    "\n",
+    "sliderAlpha1 = widgets.FloatSlider(min=0.01, max=0.03, step=0.001, value=0.02, description=r'$\\alpha$', width=300)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "widgets.interact_manual(slide7, alpha=sliderAlpha1)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Normalizacja danych"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Normalizacja danych to proces, który polega na dostosowaniu danych wejściowych w taki sposób, żeby ułatwić działanie algorytmowi gradientu prostego.\n",
+    "\n",
+    "Wyjaśnię to na przykladzie."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "Użyjemy danych z „Gratka flats challenge 2017”.\n",
+    "\n",
+    "Rozważmy model $h(x) = \\theta_0 + \\theta_1 x_1 + \\theta_2 x_2$, w którym cena mieszkania prognozowana jest na podstawie liczby pokoi $x_1$ i metrażu $x_2$:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Wczytanie danych przy pomocy biblioteki pandas\n",
+    "import pandas\n",
+    "alldata = pandas.read_csv('data_flats.tsv', header=0, sep='\\t',\n",
+    "                          usecols=['price', 'rooms', 'sqrMetres'])\n",
+    "alldata[:10]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Funkcja, która pokazuje wartości minimalne i maksymalne w macierzy X\n",
+    "\n",
+    "def show_mins_and_maxs(XMx):\n",
+    "    mins = np.amin(XMx, axis=0).tolist()[0]  # wartości minimalne\n",
+    "    maxs = np.amax(XMx, axis=0).tolist()[0]  # wartości maksymalne\n",
+    "    for i, (xmin, xmax) in enumerate(zip(mins, maxs)):\n",
+    "        display(Math(\n",
+    "            r'${:.2F} \\leq x_{} \\leq {:.2F}$'.format(xmin, i, xmax)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Przygotowanie danych\n",
+    "\n",
+    "import numpy as np\n",
+    "\n",
+    "%matplotlib inline\n",
+    "\n",
+    "data2 = np.matrix(alldata[['rooms', 'sqrMetres', 'price']])\n",
+    "\n",
+    "m, n_plus_1 = data2.shape\n",
+    "n = n_plus_1 - 1\n",
+    "Xn = data2[:, 0:n]\n",
+    "\n",
+    "XMx2 = np.matrix(np.concatenate((np.ones((m, 1)), Xn), axis=1)).reshape(m, n_plus_1)\n",
+    "yMx2 = np.matrix(data2[:, -1]).reshape(m, 1) / 1000.0"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "Cechy w danych treningowych przyjmują wartości z zakresu:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "show_mins_and_maxs(XMx2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "Jak widzimy, $x_2$ przyjmuje wartości dużo większe niż $x_1$.\n",
+    "Powoduje to, że wykres funkcji kosztu jest bardzo „spłaszczony” wzdłuż jednej z osi:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def contour_plot(X, y, rescale=10**8):\n",
+    "    theta0_vals = np.linspace(-100000, 100000, 100)\n",
+    "    theta1_vals = np.linspace(-100000, 100000, 100)\n",
+    "\n",
+    "    J_vals = np.zeros(shape=(theta0_vals.size, theta1_vals.size))\n",
+    "    for t1, element in enumerate(theta0_vals):\n",
+    "        for t2, element2 in enumerate(theta1_vals):\n",
+    "            thetaT = np.matrix([1.0, element, element2]).reshape(3,1)\n",
+    "            J_vals[t1, t2] = JMx(thetaT, X, y) / rescale\n",
+    "            \n",
+    "    pl.figure()\n",
+    "    pl.contour(theta0_vals, theta1_vals, J_vals.T, np.logspace(-2, 3, 20))\n",
+    "    pl.xlabel(r'$\\theta_1$')\n",
+    "    pl.ylabel(r'$\\theta_2$')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "contour_plot(XMx2, yMx2, rescale=10**10)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "source": [
+    "Jeżeli funkcja kosztu ma kształt taki, jak na powyższym wykresie, to łatwo sobie wyobrazić, że znalezienie minimum lokalnego przy użyciu metody gradientu prostego musi stanowć nie lada wyzwanie: algorytm szybko znajdzie „rynnę”, ale „zjazd” wzdłuż „rynny” w poszukiwaniu minimum będzie odbywał się bardzo powoli.\n",
+    "\n",
+    "Jak temu zaradzić?\n",
+    "\n",
+    "Spróbujemy przekształcić dane tak, żeby funkcja kosztu miała „ładny”, regularny kształt."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Skalowanie"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Będziemy dążyć do tego, żeby każda z cech przyjmowała wartości w podobnym zakresie.\n",
+    "\n",
+    "W tym celu przeskalujemy wartości każdej z cech, dzieląc je przez wartość maksymalną:\n",
+    "\n",
+    "$$ \\hat{x_i}^{(j)} := \\frac{x_i^{(j)}}{\\max_j x_i^{(j)}} $$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "XMx2_scaled = XMx2 / np.amax(XMx2, axis=0)\n",
+    "\n",
+    "show_mins_and_maxs(XMx2_scaled)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "contour_plot(XMx2_scaled, yMx2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Normalizacja średniej"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Będziemy dążyć do tego, żeby dodatkowo średnia wartość każdej z cech była w okolicach $0$.\n",
+    "\n",
+    "W tym celu oprócz przeskalowania odejmiemy wartość średniej od wartości każdej z cech:\n",
+    "\n",
+    "$$ \\hat{x_i}^{(j)} := \\frac{x_i^{(j)} - \\mu_i}{\\max_j x_i^{(j)}} $$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "XMx2_norm = (XMx2 - np.mean(XMx2, axis=0)) / np.amax(XMx2, axis=0)\n",
+    "\n",
+    "show_mins_and_maxs(XMx2_norm)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "contour_plot(XMx2_norm, yMx2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "notes"
+    }
+   },
+   "source": [
+    "Teraz funkcja kosztu ma wykres o bardzo regularnym kształcie – algorytm gradientu prostego zastosowany w takim przypadku bardzo szybko znajdzie minimum funkcji kosztu."
+   ]
+  }
+ ],
+ "metadata": {
+  "celltoolbar": "Slideshow",
+  "kernelspec": {
+   "display_name": "Python 3",
+   "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"
+  },
+  "livereveal": {
+   "start_slideshow_at": "selected",
+   "theme": "amu"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/data01_test.csv b/data01_test.csv
new file mode 100644
index 0000000..72c4017
--- /dev/null
+++ b/data01_test.csv
@@ -0,0 +1,17 @@
+5.7292,0.47953
+5.1884,0.20421
+6.3557,0.67861
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+5.3054,1.9869
+8.2934,0.14454
+13.394,9.0551
+5.4369,0.61705
diff --git a/data01_train.csv b/data01_train.csv
new file mode 100644
index 0000000..ab04fdf
--- /dev/null
+++ b/data01_train.csv
@@ -0,0 +1,80 @@
+6.1101,17.592
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