{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Uczenie maszynowe\n",
    "# 3. Regresja liniowa – część 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 3.1. 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": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "       y:price  x1:isNew  x2:rooms  x3:floor   x4:location  x5:sqrMetres\n",
      "1     476118.0     False         3         1       Centrum            78\n",
      "2     459531.0     False         3         2        Sołacz            62\n",
      "3     411557.0     False         3         0        Sołacz            15\n",
      "4     496416.0     False         4         0        Sołacz            14\n",
      "5     406032.0     False         3         0        Sołacz            15\n",
      "...        ...       ...       ...       ...           ...           ...\n",
      "1335  349000.0     False         4         0  Szczepankowo            29\n",
      "1336  399000.0     False         5         0  Szczepankowo            68\n",
      "1337  234000.0      True         2         7         Wilda            50\n",
      "1338  210000.0      True         2         1         Wilda            65\n",
      "1339  279000.0      True         2         2        Łazarz            36\n",
      "\n",
      "[1339 rows x 6 columns]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "data = pd.read_csv(\"data02_train.tsv\", sep=\"\\t\")\n",
    "data.rename(columns={col: f\"x{i}:{col}\" if i > 0 else f\"y:{col}\" for i, col in enumerate(data.columns)}, inplace=True)\n",
    "data.index = np.arange(1, len(data) + 1)\n",
    "print(data)"
   ]
  },
  {
   "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": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# Wersje macierzowe funkcji rysowania wykresów punktowych oraz krzywej regresyjnej\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "\n",
    "def h(theta, x):\n",
    "    return x * theta\n",
    "\n",
    "\n",
    "def regdots(x, y, xlabel=\"populacja\", ylabel=\"zysk\"):\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[:, 1]], [y], c=\"r\", s=50, label=\"Dane\")\n",
    "\n",
    "    ax.set_xlabel(xlabel)\n",
    "    ax.set_ylabel(ylabel)\n",
    "    ax.margins(0.05, 0.05)\n",
    "    plt.ylim(y.min() - 1, y.max() + 1)\n",
    "    plt.xlim(np.min(x[:, 1]) - 1, np.max(x[:, 1]) + 1)\n",
    "    return fig\n",
    "\n",
    "\n",
    "def regline(fig, fun, theta, x):\n",
    "    ax = fig.axes[0]\n",
    "    x_min = np.min(x[:, 1])\n",
    "    x_max = np.max(x[:, 1])\n",
    "    x_range = [x_min, x_max]\n",
    "    x_matrix = np.matrix([1, x_min, 1, x_max]).reshape(2, 2)\n",
    "    ax.plot(\n",
    "        x_range,\n",
    "        fun(theta, x_matrix),\n",
    "        linewidth=\"2\",\n",
    "        label=(\n",
    "            r\"$y={theta0:.2}{op}{theta1:.2}x$\".format(\n",
    "                theta0=float(theta[0][0]),\n",
    "                theta1=(\n",
    "                    float(theta[1][0]) if theta[1][0] >= 0 else float(-theta[1][0])\n",
    "                ),\n",
    "                op=\"+\" if theta[1][0] >= 0 else \"-\",\n",
    "            )\n",
    "        ),\n",
    "    )\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x[:5]=matrix([[ 1.,  3.,  1., 78.],\n",
      "        [ 1.,  3.,  2., 62.],\n",
      "        [ 1.,  3.,  0., 15.],\n",
      "        [ 1.,  4.,  0., 14.],\n",
      "        [ 1.,  3.,  0., 15.]])\n",
      "x.shape=(1339, 4)\n",
      "y[:5]=matrix([[476118.],\n",
      "        [459531.],\n",
      "        [411557.],\n",
      "        [496416.],\n",
      "        [406032.]])\n",
      "y.shape=(1339, 1)\n"
     ]
    }
   ],
   "source": [
    "# Wczytwanie danych z pliku – regresja liniowa wielu zmiennych – notacja macierzowa\n",
    "\n",
    "import pandas as pd\n",
    "\n",
    "data = pd.read_csv(\n",
    "    \"data02_train.tsv\", delimiter=\"\\t\", usecols=[\"price\", \"rooms\", \"floor\", \"sqrMetres\"]\n",
    ")\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",
    "x = np.matrix(np.concatenate((np.ones((m, 1)), xn), axis=1)).reshape(m, n_plus_1)\n",
    "y = np.matrix(data.values[:, 0]).reshape(m, 1)\n",
    "\n",
    "print(f\"{x[:5]=}\")\n",
    "print(f\"{x.shape=}\")\n",
    "print(f\"{y[:5]=}\")\n",
    "print(f\"{y.shape=}\")\n"
   ]
  },
  {
   "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": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\Large J(\\theta) = 85104141370.9717$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from IPython.display import display, Math, Latex\n",
    "\n",
    "\n",
    "def J(theta, X, y):\n",
    "    \"\"\"Wersja macierzowa funkcji kosztu\"\"\"\n",
    "    m = len(y)\n",
    "    cost = 1.0 / (2.0 * m) * ((X * theta - y).T * (X * theta - y))\n",
    "    return cost.item()\n",
    "\n",
    "\n",
    "theta = np.matrix([10, 90, -1, 2.5]).reshape(4, 1)\n",
    "\n",
    "cost = J(theta, x, y)\n",
    "display(Math(r\"\\Large J(\\theta) = %.4f\" % cost))\n"
   ]
  },
  {
   "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": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "outputs": [],
   "source": [
    "# Wyświetlanie macierzy w LaTeX-u\n",
    "\n",
    "\n",
    "def latex_matrix(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": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\large \\theta = \\left[\\begin{array}{r}10.0000 \\\\ 90.0000 \\\\ -1.0000 \\\\ 2.5000 \\\\ \\end{array}\\right]\\quad\\large \\nabla J(\\theta) = \\left[\\begin{array}{r}-373492.7442 \\\\ -1075656.5086 \\\\ -989554.4921 \\\\ -23806475.6561 \\\\ \\end{array}\\right]$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from IPython.display import display, Math, Latex\n",
    "\n",
    "\n",
    "def dJ(theta, X, y):\n",
    "    \"\"\"Wersja macierzowa gradientu funckji kosztu\"\"\"\n",
    "    return 1.0 / len(y) * (X.T * (X * theta - y))\n",
    "\n",
    "\n",
    "theta = np.matrix([10, 90, -1, 2.5]).reshape(4, 1)\n",
    "\n",
    "display(\n",
    "    Math(\n",
    "        r\"\\large \\theta = \"\n",
    "        + latex_matrix(theta)\n",
    "        + r\"\\quad\"\n",
    "        + r\"\\large \\nabla J(\\theta) = \"\n",
    "        + latex_matrix(dJ(theta, x, y))\n",
    "    )\n",
    ")\n"
   ]
  },
  {
   "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": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\large\\textrm{Wynik:}\\quad \\theta = \\left[\\begin{array}{r}17446.2135 \\\\ 86476.7960 \\\\ -1374.8950 \\\\ 2165.0689 \\\\ \\end{array}\\right] \\quad J(\\theta) = 10324864803.1591 \\quad \\textrm{po 374575 iteracjach}$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Implementacja algorytmu gradientu prostego za pomocą numpy i macierzy\n",
    "\n",
    "\n",
    "def gradient_descent(fJ, fdJ, theta, X, y, alpha, eps):\n",
    "    current_cost = fJ(theta, X, y)\n",
    "    history = [[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",
    "        history.append([current_cost, theta])\n",
    "    return theta, history\n",
    "\n",
    "\n",
    "theta_start = np.zeros((n + 1, 1))\n",
    "\n",
    "# Zmieniamy wartości alpha (rozmiar kroku) oraz eps (kryterium stopu)\n",
    "theta_best, history = gradient_descent(J, dJ, theta_start, x, y, alpha=0.0001, eps=0.1)\n",
    "\n",
    "display(\n",
    "    Math(\n",
    "        r\"\\large\\textrm{Wynik:}\\quad \\theta = \"\n",
    "        + latex_matrix(theta_best)\n",
    "        + (r\" \\quad J(\\theta) = %.4f\" % history[-1][0])\n",
    "        + r\" \\quad \\textrm{po %d iteracjach}\" % len(history)\n",
    "    )\n",
    ")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 3.2. 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": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "x and y must be the same size",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "Cell \u001b[0;32mIn [18], line 3\u001b[0m\n\u001b[1;32m      1\u001b[0m theta_start \u001b[39m=\u001b[39m np\u001b[39m.\u001b[39mzeros((\u001b[39m2\u001b[39m, \u001b[39m1\u001b[39m))\n\u001b[0;32m----> 3\u001b[0m fig \u001b[39m=\u001b[39m regdots(x[\u001b[39m1\u001b[39m], y)\n",
      "Cell \u001b[0;32mIn [6], line 15\u001b[0m, in \u001b[0;36mregdots\u001b[0;34m(x, y, xlabel, ylabel)\u001b[0m\n\u001b[1;32m     13\u001b[0m ax \u001b[39m=\u001b[39m fig\u001b[39m.\u001b[39madd_subplot(\u001b[39m111\u001b[39m)\n\u001b[1;32m     14\u001b[0m fig\u001b[39m.\u001b[39msubplots_adjust(left\u001b[39m=\u001b[39m\u001b[39m0.1\u001b[39m, right\u001b[39m=\u001b[39m\u001b[39m0.9\u001b[39m, bottom\u001b[39m=\u001b[39m\u001b[39m0.1\u001b[39m, top\u001b[39m=\u001b[39m\u001b[39m0.9\u001b[39m)\n\u001b[0;32m---> 15\u001b[0m ax\u001b[39m.\u001b[39;49mscatter([x[:, \u001b[39m1\u001b[39;49m]], [y], c\u001b[39m=\u001b[39;49m\u001b[39m\"\u001b[39;49m\u001b[39mr\u001b[39;49m\u001b[39m\"\u001b[39;49m, s\u001b[39m=\u001b[39;49m\u001b[39m50\u001b[39;49m, label\u001b[39m=\u001b[39;49m\u001b[39m\"\u001b[39;49m\u001b[39mDane\u001b[39;49m\u001b[39m\"\u001b[39;49m)\n\u001b[1;32m     17\u001b[0m ax\u001b[39m.\u001b[39mset_xlabel(xlabel)\n\u001b[1;32m     18\u001b[0m ax\u001b[39m.\u001b[39mset_ylabel(ylabel)\n",
      "File \u001b[0;32m~/.local/lib/python3.10/site-packages/matplotlib/__init__.py:1423\u001b[0m, in \u001b[0;36m_preprocess_data.<locals>.inner\u001b[0;34m(ax, data, *args, **kwargs)\u001b[0m\n\u001b[1;32m   1420\u001b[0m \u001b[39m@functools\u001b[39m\u001b[39m.\u001b[39mwraps(func)\n\u001b[1;32m   1421\u001b[0m \u001b[39mdef\u001b[39;00m \u001b[39minner\u001b[39m(ax, \u001b[39m*\u001b[39margs, data\u001b[39m=\u001b[39m\u001b[39mNone\u001b[39;00m, \u001b[39m*\u001b[39m\u001b[39m*\u001b[39mkwargs):\n\u001b[1;32m   1422\u001b[0m     \u001b[39mif\u001b[39;00m data \u001b[39mis\u001b[39;00m \u001b[39mNone\u001b[39;00m:\n\u001b[0;32m-> 1423\u001b[0m         \u001b[39mreturn\u001b[39;00m func(ax, \u001b[39m*\u001b[39;49m\u001b[39mmap\u001b[39;49m(sanitize_sequence, args), \u001b[39m*\u001b[39;49m\u001b[39m*\u001b[39;49mkwargs)\n\u001b[1;32m   1425\u001b[0m     bound \u001b[39m=\u001b[39m new_sig\u001b[39m.\u001b[39mbind(ax, \u001b[39m*\u001b[39margs, \u001b[39m*\u001b[39m\u001b[39m*\u001b[39mkwargs)\n\u001b[1;32m   1426\u001b[0m     auto_label \u001b[39m=\u001b[39m (bound\u001b[39m.\u001b[39marguments\u001b[39m.\u001b[39mget(label_namer)\n\u001b[1;32m   1427\u001b[0m                   \u001b[39mor\u001b[39;00m bound\u001b[39m.\u001b[39mkwargs\u001b[39m.\u001b[39mget(label_namer))\n",
      "File \u001b[0;32m~/.local/lib/python3.10/site-packages/matplotlib/axes/_axes.py:4512\u001b[0m, in \u001b[0;36mAxes.scatter\u001b[0;34m(self, x, y, s, c, marker, cmap, norm, vmin, vmax, alpha, linewidths, edgecolors, plotnonfinite, **kwargs)\u001b[0m\n\u001b[1;32m   4510\u001b[0m y \u001b[39m=\u001b[39m np\u001b[39m.\u001b[39mma\u001b[39m.\u001b[39mravel(y)\n\u001b[1;32m   4511\u001b[0m \u001b[39mif\u001b[39;00m x\u001b[39m.\u001b[39msize \u001b[39m!=\u001b[39m y\u001b[39m.\u001b[39msize:\n\u001b[0;32m-> 4512\u001b[0m     \u001b[39mraise\u001b[39;00m \u001b[39mValueError\u001b[39;00m(\u001b[39m\"\u001b[39m\u001b[39mx and y must be the same size\u001b[39m\u001b[39m\"\u001b[39m)\n\u001b[1;32m   4514\u001b[0m \u001b[39mif\u001b[39;00m s \u001b[39mis\u001b[39;00m \u001b[39mNone\u001b[39;00m:\n\u001b[1;32m   4515\u001b[0m     s \u001b[39m=\u001b[39m (\u001b[39m20\u001b[39m \u001b[39mif\u001b[39;00m mpl\u001b[39m.\u001b[39mrcParams[\u001b[39m'\u001b[39m\u001b[39m_internal.classic_mode\u001b[39m\u001b[39m'\u001b[39m] \u001b[39melse\u001b[39;00m\n\u001b[1;32m   4516\u001b[0m          mpl\u001b[39m.\u001b[39mrcParams[\u001b[39m'\u001b[39m\u001b[39mlines.markersize\u001b[39m\u001b[39m'\u001b[39m] \u001b[39m*\u001b[39m\u001b[39m*\u001b[39m \u001b[39m2.0\u001b[39m)\n",
      "\u001b[0;31mValueError\u001b[0m: x and y must be the same size"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 960x540 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta_start = np.zeros((2, 1))\n",
    "\n",
    "fig = regdots(x[1], y)\n",
    "# theta_e1, history1 = GDMx(\n",
    "#     JMx, dJMx, thetaStartMx, XMx, yMx, alpha=0.01, eps=0.01\n",
    "# )  # niebieska linia\n",
    "# reglineMx(fig, hMx, theta_e1, XMx)\n",
    "# theta_e2, history2 = GDMx(\n",
    "#     JMx, dJMx, thetaStartMx, XMx, yMx, alpha=0.01, eps=0.000001\n",
    "# )  # pomarańczowa linia\n",
    "# reglineMx(fig, hMx, theta_e2, XMx)\n",
    "# legend(fig)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\theta_{10^{-2}} = \\left[\\begin{array}{r}0.0531 \\\\ 0.8365 \\\\ \\end{array}\\right]\\quad\\theta_{10^{-6}} = \\left[\\begin{array}{r}-3.4895 \\\\ 1.1786 \\\\ \\end{array}\\right]$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(\n",
    "    Math(\n",
    "        r\"\\theta_{10^{-2}} = \"\n",
    "        + latex_matrix(theta_e1)\n",
    "        + r\"\\quad\\theta_{10^{-6}} = \"\n",
    "        + latex_matrix(theta_e2)\n",
    "    )\n",
    ")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Długość kroku ($\\alpha$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "outputs": [],
   "source": [
    "# Jak zmienia się koszt w kolejnych krokach w zależności od alfa\n",
    "\n",
    "\n",
    "def costchangeplot(history):\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(\"krok\")\n",
    "    ax.set_ylabel(r\"$J(\\theta)$\")\n",
    "\n",
    "    X = np.arange(0, 500, 1)\n",
    "    Y = [history[step][0] for step in X]\n",
    "    ax.plot(X, Y, linewidth=\"2\", label=(r\"$J(\\theta)$\"))\n",
    "    return fig\n",
    "\n",
    "\n",
    "def slide7(alpha):\n",
    "    best_theta, history = gradient_descent(\n",
    "        h, J, [0.0, 0.0], x, y, alpha=alpha, eps=0.0001\n",
    "    )\n",
    "    fig = costchangeplot(history)\n",
    "    legend(fig)\n",
    "\n",
    "\n",
    "sliderAlpha1 = widgets.FloatSlider(\n",
    "    min=0.01, max=0.03, step=0.001, value=0.02, description=r\"$\\alpha$\", width=300\n",
    ")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "52b0d91e39104f4facbb7f57819aae0c",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.02, description='$\\\\alpha$', max=0.03, min=0.01, step=0.001), Button…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.slide7(alpha)>"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "widgets.interact_manual(slide7, alpha=sliderAlpha1)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 3.3. Normalizacja danych"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "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": 81,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>price</th>\n",
       "      <th>rooms</th>\n",
       "      <th>sqrMetres</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>476118.00</td>\n",
       "      <td>3</td>\n",
       "      <td>78</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>459531.00</td>\n",
       "      <td>3</td>\n",
       "      <td>62</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>411557.00</td>\n",
       "      <td>3</td>\n",
       "      <td>15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>496416.00</td>\n",
       "      <td>4</td>\n",
       "      <td>14</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>406032.00</td>\n",
       "      <td>3</td>\n",
       "      <td>15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>450026.00</td>\n",
       "      <td>3</td>\n",
       "      <td>80</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>571229.15</td>\n",
       "      <td>2</td>\n",
       "      <td>39</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>325000.00</td>\n",
       "      <td>3</td>\n",
       "      <td>54</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>268229.00</td>\n",
       "      <td>2</td>\n",
       "      <td>90</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>604836.00</td>\n",
       "      <td>4</td>\n",
       "      <td>40</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       price  rooms  sqrMetres\n",
       "0  476118.00      3         78\n",
       "1  459531.00      3         62\n",
       "2  411557.00      3         15\n",
       "3  496416.00      4         14\n",
       "4  406032.00      3         15\n",
       "5  450026.00      3         80\n",
       "6  571229.15      2         39\n",
       "7  325000.00      3         54\n",
       "8  268229.00      2         90\n",
       "9  604836.00      4         40"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Wczytanie danych przy pomocy biblioteki pandas\n",
    "import pandas\n",
    "\n",
    "alldata = pandas.read_csv(\n",
    "    \"data_flats.tsv\", header=0, sep=\"\\t\", usecols=[\"price\", \"rooms\", \"sqrMetres\"]\n",
    ")\n",
    "alldata[:10]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "outputs": [],
   "source": [
    "# Funkcja, która pokazuje wartości minimalne i maksymalne w macierzy X\n",
    "\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(r\"${:.2F} \\leq x_{} \\leq {:.2F}$\".format(xmin, i, xmax)))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "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": 84,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.00 \\leq x_0 \\leq 1.00$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2.00 \\leq x_1 \\leq 7.00$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 12.00 \\leq x_2 \\leq 196.00$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_mins_and_maxs(XMx2)\n"
   ]
  },
  {
   "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": 85,
   "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",
    "    plt.figure()\n",
    "    plt.contour(theta0_vals, theta1_vals, J_vals.T, np.logspace(-2, 3, 20))\n",
    "    plt.xlabel(r\"$\\theta_1$\")\n",
    "    plt.ylabel(r\"$\\theta_2$\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
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      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "contour_plot(XMx2, yMx2, rescale=10**10)\n"
   ]
  },
  {
   "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": 87,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.00 \\leq x_0 \\leq 1.00$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0.29 \\leq x_1 \\leq 1.00$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0.06 \\leq x_2 \\leq 1.00$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "XMx2_scaled = XMx2 / np.amax(XMx2, axis=0)\n",
    "\n",
    "show_mins_and_maxs(XMx2_scaled)\n"
   ]
  },
  {
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      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "contour_plot(XMx2_scaled, yMx2)\n"
   ]
  },
  {
   "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": 89,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0.00 \\leq x_0 \\leq 0.00$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -0.10 \\leq x_1 \\leq 0.62$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -0.23 \\leq x_2 \\leq 0.70$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "XMx2_norm = (XMx2 - np.mean(XMx2, axis=0)) / np.amax(XMx2, axis=0)\n",
    "\n",
    "show_mins_and_maxs(XMx2_norm)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
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      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
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    "contour_plot(XMx2_norm, yMx2)\n"
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    "Teraz funkcja kosztu ma wykres o bardzo regularnym kształcie – algorytm gradientu prostego zastosowany w takim przypadku bardzo szybko znajdzie minimum funkcji kosztu."
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