uczenie-maszynowe/wyk/03_Regresja_liniowa_2.ipynb

{
 "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": 70,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y : price    x1: isNew    x2: rooms    x3: floor    x4: location x5: sqrMetres\n",
      "476118.0     False        3            1            Centrum      78          \n",
      "459531.0     False        3            2            Sołacz       62          \n",
      "411557.0     False        3            0            Sołacz       15          \n",
      "496416.0     False        4            0            Sołacz       14          \n",
      "406032.0     False        3            0            Sołacz       15          \n",
      "450026.0     False        3            1            Naramowice   80          \n",
      "571229.15    False        2            4            Wilda        39          \n",
      "325000.0     False        3            1            Grunwald     54          \n",
      "268229.0     False        2            1            Grunwald     90          \n"
     ]
    }
   ],
   "source": [
    "import csv\n",
    "\n",
    "reader = csv.reader(open(\"data02_train.tsv\", encoding=\"utf-8\"), delimiter=\"\\t\")\n",
    "for i, row in enumerate(list(reader)[:10]):\n",
    "    if i == 0:\n",
    "        print(\n",
    "            \" \".join(\n",
    "                [\n",
    "                    \"{}: {:8}\".format(\"x\" + str(j) if j > 0 else \"y \", entry)\n",
    "                    for j, entry in enumerate(row)\n",
    "                ]\n",
    "            )\n",
    "        )\n",
    "    else:\n",
    "        print(\" \".join([\"{:12}\".format(entry) for entry in row]))\n"
   ]
  },
  {
   "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": 71,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# Wersje macierzowe funkcji rysowania wykresów punktowych oraz krzywej regresyjnej\n",
    "\n",
    "\n",
    "def hMx(theta, X):\n",
    "    return X * theta\n",
    "\n",
    "\n",
    "def regdotsMx(X, y):\n",
    "    fig = plt.figure(figsize=(16 * 0.6, 9 * 0.6))\n",
    "    ax = fig.add_subplot(111)\n",
    "    fig.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)\n",
    "    ax.scatter([X[:, 1]], [y], c=\"r\", s=50, label=\"Dane\")\n",
    "\n",
    "    ax.set_xlabel(\"Populacja\")\n",
    "    ax.set_ylabel(\"Zysk\")\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 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(\n",
    "        L,\n",
    "        fun(theta, LX),\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": 72,
   "metadata": {
    "slideshow": {
     "slide_type": "notes"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 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",
      "(1339, 4)\n",
      "\n",
      "[[476118.]\n",
      " [459531.]\n",
      " [411557.]\n",
      " [496416.]\n",
      " [406032.]]\n",
      "(1339, 1)\n"
     ]
    }
   ],
   "source": [
    "# Wczytwanie danych z pliku za pomocą numpy – regresja liniowa wielu zmiennych – notacja macierzowa\n",
    "\n",
    "import pandas\n",
    "\n",
    "data = pandas.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",
    "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)\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": 73,
   "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 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",
    "\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))\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": 74,
   "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 dJMx(theta, X, y):\n",
    "    \"\"\"Wersja macierzowa gradientu funckji kosztu\"\"\"\n",
    "    return 1.0 / len(y) * (X.T * (X * theta - y))\n",
    "\n",
    "\n",
    "thetaMx = np.matrix([10, 90, -1, 2.5]).reshape(4, 1)\n",
    "\n",
    "display(\n",
    "    Math(\n",
    "        r\"\\large \\theta = \"\n",
    "        + LatexMatrix(thetaMx)\n",
    "        + r\"\\quad\"\n",
    "        + r\"\\large \\nabla J(\\theta) = \"\n",
    "        + LatexMatrix(dJMx(thetaMx, XMx, yMx))\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": 75,
   "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 GDMx(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",
    "thetaStartMx = np.zeros((n + 1, 1))\n",
    "\n",
    "# Zmieniamy wartości alpha (rozmiar kroku) oraz eps (kryterium stopu)\n",
    "thetaBestMx, history = GDMx(JMx, dJMx, thetaStartMx, XMx, yMx, alpha=0.0001, eps=0.1)\n",
    "\n",
    "######################################################################\n",
    "display(\n",
    "    Math(\n",
    "        r\"\\large\\textrm{Wynik:}\\quad \\theta = \"\n",
    "        + LatexMatrix(thetaBestMx)\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": 76,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
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       "<Figure size 960x540 with 1 Axes>"
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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, 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",
    "        + LatexMatrix(theta_e1)\n",
    "        + r\"\\quad\\theta_{10^{-6}} = \"\n",
    "        + LatexMatrix(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": [
    {
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       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>price</th>\n",
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       "      <th>3</th>\n",
       "      <td>496416.00</td>\n",
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       "      <td>14</td>\n",
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       "      <td>406032.00</td>\n",
       "      <td>3</td>\n",
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       "    <tr>\n",
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       "      <td>450026.00</td>\n",
       "      <td>3</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>571229.15</td>\n",
       "      <td>2</td>\n",
       "      <td>39</td>\n",
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       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>325000.00</td>\n",
       "      <td>3</td>\n",
       "      <td>54</td>\n",
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       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>268229.00</td>\n",
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       "    <tr>\n",
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       "      <td>40</td>\n",
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      "text/plain": [
       "       price  rooms  sqrMetres\n",
       "0  476118.00      3         78\n",
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     },
     "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>"
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     "metadata": {},
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    {
     "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>"
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    "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"
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  {
   "cell_type": "code",
   "execution_count": 86,
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      "text/plain": [
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   "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."
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  },
  {
   "cell_type": "markdown",
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    "slideshow": {
     "slide_type": "slide"
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   },
   "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",
    "$$ \\hat{x_i}^{(j)} := \\frac{x_i^{(j)}}{\\max_j x_i^{(j)}} $$"
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  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
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   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.00 \\leq x_0 \\leq 1.00$"
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      "text/plain": [
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     "data": {
      "text/latex": [
       "$\\displaystyle 0.29 \\leq x_1 \\leq 1.00$"
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       "$\\displaystyle 0.06 \\leq x_2 \\leq 1.00$"
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   "source": [
    "XMx2_scaled = XMx2 / np.amax(XMx2, axis=0)\n",
    "\n",
    "show_mins_and_maxs(XMx2_scaled)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
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    "contour_plot(XMx2_scaled, yMx2)\n"
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   "source": [
    "### Normalizacja średniej"
   ]
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   "metadata": {
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   "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",
    "$$ \\hat{x_i}^{(j)} := \\frac{x_i^{(j)} - \\mu_i}{\\max_j x_i^{(j)}} $$"
   ]
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  {
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   "execution_count": 89,
   "metadata": {
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     "data": {
      "text/latex": [
       "$\\displaystyle 0.00 \\leq x_0 \\leq 0.00$"
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       "$\\displaystyle -0.10 \\leq x_1 \\leq 0.62$"
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   "source": [
    "XMx2_norm = (XMx2 - np.mean(XMx2, axis=0)) / np.amax(XMx2, axis=0)\n",
    "\n",
    "show_mins_and_maxs(XMx2_norm)\n"
   ]
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   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
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   "source": [
    "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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