byoyo2018-giris-1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Yapay Öğrenmeye Giriş I\n",
    "\n",
    "Ali Taylan Cemgil\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Parametrik Regresyon, Parametrik Fonksyon Oturtma Problemi (Parametric Regression, Function Fitting)\n",
    "\n",
    "\n",
    "Verilen girdi ve çıktı ikilileri $x, y$ için parametrik bir fonksyon $f$ oturtma problemi. \n",
    "\n",
    "Parametre $w$ değerlerini öyle bir seçelim ki \n",
    "$$\n",
    "y \\approx f(x; w)\n",
    "$$\n",
    "\n",
    "$x$: Girdi (Input)\n",
    "\n",
    "$y$: Çıktı (Output)\n",
    "\n",
    "$w$: Parametre (Weight, ağırlık)\n",
    "\n",
    "$e$: Hata\n",
    "\n",
    "Örnek 1: \n",
    "$$\n",
    "e = y - f(x)\n",
    "$$\n",
    "\n",
    "Örnek 2:\n",
    "$$\n",
    "e = \\frac{y}{f(x)}-1\n",
    "$$\n",
    "\n",
    "$E$, $D$: Hata fonksyonu (Error function), Iraksay (Divergence)\n",
    "\n",
    "\n",
    "\n",
    "# Doğrusal Regresyon (Linear Regression)\n",
    "\n",
    "Oturtulacak $f$ fonksyonun **model parametreleri** $w$ cinsinden doğrusal olduğu durum (Girdiler $x$ cinsinden doğrusal olması gerekmez).  \n",
    "\n",
    "## Tanım: Doğrusallık\n",
    "Bir $g$ fonksyonu doğrusaldır demek, herhangi skalar $a$ ve $b$ içn\n",
    "$$\n",
    "g(aw_1 + b w_2) = a g(w_1) + b g(w_2)\n",
    "$$\n",
    "olması demektir.\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Örnek: Doğru oturtmak (Line Fitting)\n",
    "\n",
    "* Girdi-Çıktı ikilileri\n",
    "$$\n",
    "(x_i, y_i)\n",
    "$$\n",
    "$i=1\\dots N$ \n",
    "\n",
    "* Model\n",
    "$$\n",
    "y_i \\approx f(x; w_1, w_0) = w_0 + w_1 x \n",
    "$$\n",
    "\n",
    "\n",
    ">  $x$ : Girdi \n",
    "\n",
    ">  $w_1$: Eğim\n",
    "\n",
    ">  $w_0$: Kesişme\n",
    "\n",
    "$f_i \\equiv f(x_i; w_1, w_0)$\n",
    "\n",
    "## Örnek 2: Parabol Oturtma\n",
    "\n",
    "* Girdi-Çıktı ikilileri\n",
    "$$\n",
    "(x_i, y_i)\n",
    "$$\n",
    "$i=1\\dots N$ \n",
    "\n",
    "* Model\n",
    "$$\n",
    "y_i \\approx f(x_i; w_2, w_1, w_0) = w_0 + w_1 x_i + w_2 x_i^2\n",
    "$$\n",
    "\n",
    "\n",
    ">  $x$ : Girdi \n",
    "\n",
    ">  $w_2$: Karesel terimin katsayısı \n",
    "\n",
    ">  $w_1$: Doğrusal terimin katsayısı\n",
    "\n",
    ">  $w_0$: Sabit terim katsayısı\n",
    "\n",
    "$f_i \\equiv f(x_i; w_2, w_1, w_0)$\n",
    "\n",
    "Bir parabol $x$'in doğrusal fonksyonu değil ama $w_2, w_1, w_0$ parametrelerinin doğrusal fonksyonu.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10deb0048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "%matplotlib inline\n",
    "from __future__ import print_function\n",
    "from ipywidgets import interact, interactive, fixed\n",
    "import ipywidgets as widgets\n",
    "import matplotlib.pylab as plt\n",
    "from IPython.display import clear_output, display, HTML\n",
    "\n",
    "x = np.array([8.0 , 6.1 , 11.,  7.,   9.,   12. , 4.,   2.,   10,    5,    3])\n",
    "y = np.array([6.04, 4.95, 5.58, 6.81, 6.33, 7.96, 5.24, 2.26, 8.84, 2.82, 3.68])\n",
    "\n",
    "def plot_fit(w1, w0):\n",
    "    f = w0 + w1*x\n",
    "\n",
    "    plt.figure(figsize=(4,3))\n",
    "    plt.plot(x,y,'sk')\n",
    "    plt.plot(x,f,'o-r')\n",
    "    #plt.axis('equal')\n",
    "    plt.xlim((0,15))\n",
    "    plt.ylim((0,10))\n",
    "    for i in range(len(x)):\n",
    "        plt.plot((x[i],x[i]),(f[i],y[i]),'b')\n",
    "#    plt.show()\n",
    "#    plt.figure(figsize=(4,1))\n",
    "    plt.bar(x,(f-y)**2/2)\n",
    "    plt.title('Toplam kare hata = '+str(np.sum((f-y)**2/2)))\n",
    "    plt.ylim((0,10))\n",
    "    plt.xlim((0,15))\n",
    "    plt.show()\n",
    "    \n",
    "plot_fit(0.0,3.79)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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       "model_id": "364bf28c721942f6a81c114b512e6cde",
       "version_major": 2,
       "version_minor": 0
      },
      "text/html": [
       "<p>Failed to display Jupyter Widget of type <code>interactive</code>.</p>\n",
       "<p>\n",
       "  If you're reading this message in Jupyter Notebook or JupyterLab, it may mean\n",
       "  that the widgets JavaScript is still loading. If this message persists, it\n",
       "  likely means that the widgets JavaScript library is either not installed or\n",
       "  not enabled. See the <a href=\"https://ipywidgets.readthedocs.io/en/stable/user_install.html\">Jupyter\n",
       "  Widgets Documentation</a> for setup instructions.\n",
       "</p>\n",
       "<p>\n",
       "  If you're reading this message in another notebook frontend (for example, a static\n",
       "  rendering on GitHub or <a href=\"https://nbviewer.jupyter.org/\">NBViewer</a>),\n",
       "  it may mean that your frontend doesn't currently support widgets.\n",
       "</p>\n"
      ],
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.0, description='w1', max=2.0, min=-2.0, step=0.01), FloatSlider(value=0.0, description='w0', max=5.0, min=-5.0, step=0.01), Output()), _dom_classes=('widget-interact',))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(plot_fit, w1=(-2, 2, 0.01), w0=(-5, 5, 0.01));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rasgele Arama"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "999 6.88573142353\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 2.01760086,  0.49685693])"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.array([8.0 , 6.1 , 11.,  7.,   9.,   12. , 4.,   2.,   10,    5,    3])\n",
    "y = np.array([6.04, 4.95, 5.58, 6.81, 6.33, 7.96, 5.24, 2.26, 8.84, 2.82, 3.68])\n",
    "\n",
    "\n",
    "def hata(y, x, w):\n",
    "    N = len(y)\n",
    "    f = x*w[1]+w[0]\n",
    "    e = y-f\n",
    "    return np.sum(e*e)/2\n",
    "\n",
    "\n",
    "w = np.array([0, 0])\n",
    "E = hata(y, x, w)\n",
    "\n",
    "for e in range(1000):\n",
    "    g = 0.1*np.random.randn(2)   \n",
    "    w_temp = w + g\n",
    "    E_temp = hata(y, x, w_temp)\n",
    "    if E_temp<E:\n",
    "        E = E_temp\n",
    "        w = w_temp\n",
    "        #print(e, E)\n",
    "print(e, E)\n",
    "w"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Gerçek veri: Türkiyedeki araç sayıları"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Year</th>\n",
       "      <th>Car</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1966</td>\n",
       "      <td>91469</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1967</td>\n",
       "      <td>112367</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1968</td>\n",
       "      <td>125375</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1969</td>\n",
       "      <td>137345</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1970</td>\n",
       "      <td>137771</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1971</td>\n",
       "      <td>153676</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>1972</td>\n",
       "      <td>187272</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>1973</td>\n",
       "      <td>240360</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>1974</td>\n",
       "      <td>313160</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>1975</td>\n",
       "      <td>403546</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>1976</td>\n",
       "      <td>488894</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>1977</td>\n",
       "      <td>560424</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>1978</td>\n",
       "      <td>624438</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>1979</td>\n",
       "      <td>688687</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>1980</td>\n",
       "      <td>742252</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>1981</td>\n",
       "      <td>776432</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>1982</td>\n",
       "      <td>811465</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>1983</td>\n",
       "      <td>856350</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>1984</td>\n",
       "      <td>919577</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>1985</td>\n",
       "      <td>983444</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>1986</td>\n",
       "      <td>1087234</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>1987</td>\n",
       "      <td>1193021</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>1988</td>\n",
       "      <td>1310257</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>1989</td>\n",
       "      <td>1434830</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>1990</td>\n",
       "      <td>1649879</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>1991</td>\n",
       "      <td>1864344</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>1992</td>\n",
       "      <td>2181388</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>1993</td>\n",
       "      <td>2619852</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>1994</td>\n",
       "      <td>2861640</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>1995</td>\n",
       "      <td>3058511</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>1996</td>\n",
       "      <td>3274156</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>31</th>\n",
       "      <td>1997</td>\n",
       "      <td>3570105</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>32</th>\n",
       "      <td>1998</td>\n",
       "      <td>3838288</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>33</th>\n",
       "      <td>1999</td>\n",
       "      <td>4072326</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>34</th>\n",
       "      <td>2000</td>\n",
       "      <td>4422180</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>35</th>\n",
       "      <td>2001</td>\n",
       "      <td>4534803</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>36</th>\n",
       "      <td>2002</td>\n",
       "      <td>4600140</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>37</th>\n",
       "      <td>2003</td>\n",
       "      <td>4700343</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>38</th>\n",
       "      <td>2004</td>\n",
       "      <td>5400440</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>39</th>\n",
       "      <td>2005</td>\n",
       "      <td>5772745</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>40</th>\n",
       "      <td>2006</td>\n",
       "      <td>6140992</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>41</th>\n",
       "      <td>2007</td>\n",
       "      <td>6472156</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>42</th>\n",
       "      <td>2008</td>\n",
       "      <td>6796629</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>43</th>\n",
       "      <td>2009</td>\n",
       "      <td>7093964</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>44</th>\n",
       "      <td>2010</td>\n",
       "      <td>7544871</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>45</th>\n",
       "      <td>2011</td>\n",
       "      <td>8113111</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>46</th>\n",
       "      <td>2012</td>\n",
       "      <td>8648875</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>47</th>\n",
       "      <td>2013</td>\n",
       "      <td>9283923</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>48</th>\n",
       "      <td>2014</td>\n",
       "      <td>9857915</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49</th>\n",
       "      <td>2015</td>\n",
       "      <td>10589337</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>2016</td>\n",
       "      <td>11317998</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>51</th>\n",
       "      <td>2017</td>\n",
       "      <td>12035978</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    Year       Car\n",
       "0   1966     91469\n",
       "1   1967    112367\n",
       "2   1968    125375\n",
       "3   1969    137345\n",
       "4   1970    137771\n",
       "5   1971    153676\n",
       "6   1972    187272\n",
       "7   1973    240360\n",
       "8   1974    313160\n",
       "9   1975    403546\n",
       "10  1976    488894\n",
       "11  1977    560424\n",
       "12  1978    624438\n",
       "13  1979    688687\n",
       "14  1980    742252\n",
       "15  1981    776432\n",
       "16  1982    811465\n",
       "17  1983    856350\n",
       "18  1984    919577\n",
       "19  1985    983444\n",
       "20  1986   1087234\n",
       "21  1987   1193021\n",
       "22  1988   1310257\n",
       "23  1989   1434830\n",
       "24  1990   1649879\n",
       "25  1991   1864344\n",
       "26  1992   2181388\n",
       "27  1993   2619852\n",
       "28  1994   2861640\n",
       "29  1995   3058511\n",
       "30  1996   3274156\n",
       "31  1997   3570105\n",
       "32  1998   3838288\n",
       "33  1999   4072326\n",
       "34  2000   4422180\n",
       "35  2001   4534803\n",
       "36  2002   4600140\n",
       "37  2003   4700343\n",
       "38  2004   5400440\n",
       "39  2005   5772745\n",
       "40  2006   6140992\n",
       "41  2007   6472156\n",
       "42  2008   6796629\n",
       "43  2009   7093964\n",
       "44  2010   7544871\n",
       "45  2011   8113111\n",
       "46  2012   8648875\n",
       "47  2013   9283923\n",
       "48  2014   9857915\n",
       "49  2015  10589337\n",
       "50  2016  11317998\n",
       "51  2017  12035978"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import scipy as sc\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pylab as plt\n",
    "\n",
    "df_arac = pd.read_csv(u'data/arac.csv',sep=';')\n",
    "df_arac[['Year','Car']]\n",
    "#df_arac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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VWl93b3P3R4CO9O8pl/oWkgJBblaT/GMC+CiwT8A5HwNuTj1vJLkFZ79NqWPl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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x117b11550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "BaseYear = 1995\n",
    "x = np.matrix(df_arac.Year[0:]).T-BaseYear\n",
    "y = np.matrix(df_arac.Car[0:]).T/1000000.\n",
    "\n",
    "plt.plot(x+BaseYear, y, 'o-')\n",
    "plt.xlabel('Yil')\n",
    "plt.ylabel('Araba (Milyon)')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x115d24f60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "from __future__ import print_function\n",
    "from ipywidgets import interact, interactive, fixed\n",
    "import ipywidgets as widgets\n",
    "import matplotlib.pylab as plt\n",
    "from IPython.display import clear_output, display, HTML\n",
    "\n",
    "\n",
    "w_0 = 0.27150786\n",
    "w_1 = 0.37332256\n",
    "\n",
    "BaseYear = 1995\n",
    "x = np.matrix(df_arac.Year[0:]).T-BaseYear\n",
    "y = np.matrix(df_arac.Car[0:]).T/1000000.\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "f = w_1*x + w_0\n",
    "plt.plot(x+BaseYear, y, 'o-')\n",
    "ln, = plt.plot(x+BaseYear, f, 'r')\n",
    "\n",
    "\n",
    "plt.xlabel('Years')\n",
    "plt.ylabel('Number of Cars (Millions)')\n",
    "ax.set_ylim((-2,13))\n",
    "plt.close(fig)\n",
    "\n",
    "def set_line(w_1, w_0):\n",
    "\n",
    "    f = w_1*x + w_0\n",
    "    e = y - f\n",
    "\n",
    "    ln.set_ydata(f)\n",
    "    ax.set_title('Total Error = {} '.format(np.asscalar(e.T*e/2)))\n",
    "    display(fig)\n",
    "\n",
    "set_line(0.32,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "0dc1b6b7aa7a48df90342ab0e057075f",
       "version_major": 2,
       "version_minor": 0
      },
      "text/html": [
       "<p>Failed to display Jupyter Widget of type <code>interactive</code>.</p>\n",
       "<p>\n",
       "  If you're reading this message in Jupyter Notebook or JupyterLab, it may mean\n",
       "  that the widgets JavaScript is still loading. If this message persists, it\n",
       "  likely means that the widgets JavaScript library is either not installed or\n",
       "  not enabled. See the <a href=\"https://ipywidgets.readthedocs.io/en/stable/user_install.html\">Jupyter\n",
       "  Widgets Documentation</a> for setup instructions.\n",
       "</p>\n",
       "<p>\n",
       "  If you're reading this message in another notebook frontend (for example, a static\n",
       "  rendering on GitHub or <a href=\"https://nbviewer.jupyter.org/\">NBViewer</a>),\n",
       "  it may mean that your frontend doesn't currently support widgets.\n",
       "</p>\n"
      ],
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.0, description='w_1', max=2.0, min=-2.0, step=0.01), FloatSlider(value=0.0, description='w_0', max=5.0, min=-5.0, step=0.01), Output()), _dom_classes=('widget-interact',))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(set_line, w_1=(-2, 2, 0.01), w_0=(-5, 5, 0.01));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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XeVytygPvLLPK4wTyezDzB4Apqro3rnnrYp/jMalu1CjX9HTdOr8jyS7vvOMm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\nbm6s4NKSAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x118945ef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w_0 = 0.27150786\n",
    "w_1 = 0.37332256\n",
    "w_2 = 0.1\n",
    "\n",
    "BaseYear = 1995\n",
    "x = np.array(df_arac.Year[0:]).T-BaseYear\n",
    "y = np.array(df_arac.Car[0:]).T/1000000.\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "f = w_2*x**2 + w_1*x + w_0\n",
    "plt.plot(x+BaseYear, y, 'o-')\n",
    "ln, = plt.plot(x+BaseYear, f, 'r')\n",
    "\n",
    "\n",
    "plt.xlabel('Yıl')\n",
    "plt.ylabel('Araba Sayısı (Milyon)')\n",
    "ax.set_ylim((-2,13))\n",
    "plt.close(fig)\n",
    "\n",
    "def set_line(w_2, w_1, w_0):\n",
    "    f = w_2*x**2 + w_1*x + w_0\n",
    "    e = y - f\n",
    "    ln.set_ydata(f)\n",
    "    ax.set_title('Ortalama Kare Hata = {} '.format(np.sum(e*e/len(e))))\n",
    "    display(fig)\n",
    "\n",
    "set_line(w_2, w_1, w_0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "3728522841be43cabe5d562d38ca7a9f",
       "version_major": 2,
       "version_minor": 0
      },
      "text/html": [
       "<p>Failed to display Jupyter Widget of type <code>interactive</code>.</p>\n",
       "<p>\n",
       "  If you're reading this message in Jupyter Notebook or JupyterLab, it may mean\n",
       "  that the widgets JavaScript is still loading. If this message persists, it\n",
       "  likely means that the widgets JavaScript library is either not installed or\n",
       "  not enabled. See the <a href=\"https://ipywidgets.readthedocs.io/en/stable/user_install.html\">Jupyter\n",
       "  Widgets Documentation</a> for setup instructions.\n",
       "</p>\n",
       "<p>\n",
       "  If you're reading this message in another notebook frontend (for example, a static\n",
       "  rendering on GitHub or <a href=\"https://nbviewer.jupyter.org/\">NBViewer</a>),\n",
       "  it may mean that your frontend doesn't currently support widgets.\n",
       "</p>\n"
      ],
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.0, description='w_2', max=0.1, min=-0.1, step=0.001), FloatSlider(value=0.0, description='w_1', max=2.0, min=-2.0, step=0.01), FloatSlider(value=0.0, description='w_0', max=5.0, min=-5.0, step=0.01), Output()), _dom_classes=('widget-interact',))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.set_line>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interact(set_line, w_2=(-0.1,0.1,0.001), w_1=(-2, 2, 0.01), w_0=(-5, 5, 0.01))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Örnek 1, devam: Modeli Öğrenmek\n",
    "\n",
    "* Öğrenmek: parametre kestirimi $w = [w_0, w_1]$\n",
    "\n",
    "* Genelde model veriyi hatasız açıklayamayacağı için her veri noktası için bir hata tanımlıyoruz:\n",
    "\n",
    "$$e_i = y_i - f(x_i; w)$$\n",
    "\n",
    "* Toplam kare hata \n",
    "\n",
    "$$\n",
    "E(w) = \\frac{1}{2} \\sum_i (y_i - f(x_i; w))^2 = \\frac{1}{2} \\sum_i e_i^2\n",
    "$$\n",
    "\n",
    "* Toplam kare hatayı $w_0$ ve $w_1$ parametrelerini değiştirerek azaltmaya çalışabiliriz.\n",
    "\n",
    "* Hata yüzeyi "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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oA7w+jjs9BtqnKtalsvMkoDzRpxmhuj7jttnV2xyrrNwZETcnD/WEdus4JiPFtUYorrC+\nUWATivPCsAnCyXNi3BOE01HfotFfOxBz12YToMWuz6a+2c3ZjP7uz5FiE3jTAJwd9fWNBssR37IB\n2NRPnteM8prw2trK8ng2AMdb9Ybv1jd10vS5ZV1TPyl3Dw52wq+PKym2iunyA7UMkfG0/bzltEK0\nPxDlShqNimnlbs+ekWI1DcO+sie7RNsjxfKxsTHqGRm2R4ijYtfp9LheNSlv5HLr44nVZOTX7DLd\nGLV3mXZHh/uqqrXrdDqd5dyNum+jmht+3V27G915BOHaGzQUbZ+p+qbiI0kXRcT/HLI/q6Jv1NeM\ntJpQnBeIzaisDMMmEOcd+9tuRLg5rs8kbUaEZX82+6ajDL9pIM7uCm0HYrObqrsLs+943QxLinpU\nEYq81q7+gKrPB60kjesgCEvjKgPO+aIZDpOR36h/FLhsEHbnT/q5g9FgWV+GXTfkZupyt2mrbrpr\ntNz9ORN6uRt1EpCVF5a7xwa7xwyb4Nu3rw7NMhTLY4OTY3oZkBs95e7xw31VNRuG1eyxw32VNWoC\nrwnCnJ6e2DMNwSPymOK8UCwfePAZLBRtjyT9nqR/K+l2SZ+w/Z6IuH6oPq2KmeNZTf2c9pOzA7Nt\nXW6u+5Iqh0auj3/Uu3ly8FSFxmouG5Ciaq6TywBRs/upDpXKoU1L1TjkrdDY9fGXahwaZTBWlScj\nXEkaj+vp0ajvOGE1czJMPV11TqRZPBosA7Q7veiEmd5tXR47rBsUb0xxHHHmTVtyNNhVeXZeK+im\n73Hzswml1nTfiTE9xwa7gdcXjN0TaLqjvtljhbOPjVHVOjHlyI1qEkDliK+cbkJtehwwgzDPMG0f\nE6wmxwSbchN6zQkzzSiwOWmm72zRvhN6jihGihujZW8PjQeLIUeKT5N0Y0TcLEm23yrpbEmHfSjW\nZ+C1R4GVpXFVh9yis0Sb+n0hjceVtkbLnzU6OclmK7Q1rvJs0emlEd3LJpoRZHmiTHn26P2b45lL\nJiLaZ4+WwTe5lGJr0Zmi00smFl8qsbOzRbtB2djuTNF5I9elzxYty61LJLY5W7QTdi4umSiPg03P\nCq1m5jUB1wRb06478ptepzfnkomRZy6pmARfnvxy5EY1uXSiG3TTXZjFpRMLzg7daPWpfXZoX/h1\nR33dYOQyCpSGDMWTJN1WTN8u6dsH6svKcR5X2ShObdvcGuvIjap1PV55ScWi6wb77h5ThmV5ofxm\nFNcQhupjhRGt6wmnITm9ML68eH5zHLp/a9o2Qq1LLcpwjFyH5lrC0HS6nBea3ulmcRhO6yS12kua\nKdc/p+2b6dK8vbb9793se9mdNw256XS33Byf6gZef+hN9ww01wE280f5z1QTAM28JvBcBlour7Wb\n0CqOnU2vKdxoQq1ov6/VpsrnVjqiqorAy/ArAq55ze6lEH3T076quDyifdYqcLBW/kQb2+dJOk+S\nTnn0owfuzbDqU9Bn67sXrjeh14TmdneTKUNznIFRzm9CsrlNWnmXmfpn+y4zTf2BIiib26ptjetd\nts3yN4tQrG+/puK1p+2j6WcReJO7ySha9U2gjTvhV95bVJoNxu6t1Mp58/QNMvpuqdYNwfK2avWZ\nkdPyZESXZ0Xas/cHHVVWJU1GPtNQqZfdBF9595im/eQ2akXANbde694tZuTpLddGrlTegq28m0zz\nGk3/uneLaS6d6N5BZt5dY7jdGoYyZCjeIemUYvrkrGuJiAslXShJp5++fwf/rx8+mt1c+5ZoOwmX\nDKwoAnRchGTZbtE9Rft+hmaDNaT266lc7vQepONJyDVltUaJk+mY3l9UmrYbd9azLtc/u6EotQ8P\n9oXi3G3eScPy87sMvtZ0sRu0Ugbg5DnOuml9OfKzivuQysVlA5qEZqUioMpllCFbBPC8e4lud2/R\nUbGc1n1Ei2BjlyTW1ZCh+AlJj7N9muowPEfSfxywP4eF5kNttGSILtIdsU1Dq9jNqenuylbINYfw\nOvVq2i8xrU6dNFs/7WtRnrMu8/QeH2zN72/vzry++ibwtpuusrIplyHl+rBjHdTlqFKz3yQBYLHB\nQjEiNl1/T+MHVF+S8YaI+PRQ/cHOTY5nLfiaIgBYJ4MeU4yI90l635B9AACgwUU4AAAkQhEAgEQo\nAgCQCEUAABKhCABAIhQBAEiEIgAAiVAEACARigAAJEIRAIBEKAIAkAhFAAASoQgAQCIUAQBIhCIA\nAIlQBAAgEYoAACRCEQCARCgCAJAIRQAAEqEIAEAiFAEASIQiAACJUAQAIBGKAAAkQhEAgEQoAgCQ\nCEUAABKhCABAIhQBAEiDhKLtX7f9GdvX2n6X7eOG6AcAAKWhRoofkvTEiPg2SX8r6RUD9QMAgIlB\nQjEiPhgRmzl5maSTh+gHAAClVTim+KOS3j90JwAA2NitBdu+RNIJPbMuiIg/yzYXSNqUdPGC5Zwn\n6TxJOuXRj96FngIAUNu1UIyIZy+ab/tcSc+TdEZExILlXCjpQkk6/fT9c9sBAPBA7VooLmL7TEkv\nl/Q9EfHPQ/QBAICuoY4p/q6kYyV9yPbVtv9goH4AADAxyEgxIr5piNcFAGCRVTj7FACAlUAoAgCQ\nCEUAABKhCABAIhQBAEiEIgAAiVAEACARigAAJEIRAIBEKAIAkAhFAAASoQgAQCIUAQBIhCIAAIlQ\nBAAgEYoAACRCEQCARCgCAJAIRQAAEqEIAEAiFAEASIQiAACJUAQAIBGKAAAkQhEAgEQoAgCQCEUA\nABKhCABAIhQBAEiEIgAAadBQtP1ztsP28UP2AwAAacBQtH2KpO+X9IWh+gAAQGnIkeLrJL1cUgzY\nBwAAJgYJRdtnS7ojIq4Z4vUBAOizsVsLtn2JpBN6Zl0g6ZdU7zpdZjnnSTovJ7921D5fd2h6OJjj\nJX156E48QOu+Duvef4l1WBWsw2p4/KFakCP2du+l7W+V9GFJ/5xVJ0v6oqSnRcRd2zz3iojYv8td\n3FWsw/DWvf8S67AqWIfVcCjXYddGivNExKck/atm2vatkvZHxLr/pwIAWHNcpwgAQNrzkWJXRJy6\ng+YX7lY/9hDrMLx177/EOqwK1mE1HLJ12PNjigAArCp2nwIAkAhFAADSSoei7VfavsP21fk4a067\nM21/1vaNts/f634uYvvXbX/G9rW232X7uDntbrX9qVzPK/a6nz39WbhNXfvtnH+t7acM0c95bJ9i\n+6O2r7f9adsv7Wnzvbb/ofj9+tUh+rrIdr8Xa/A+PL7Yvlfbvtf2yzptVu59sP0G23fb0+uibT/c\n9odsfy5/fv2c567E59GcdVibz6M5/d/9TIiIlX1IeqWkn9+mzUjSTZIeK+kISddIesLQfS/69/2S\nNrL8KkmvmtPuVknHD93fZbeppLMkvV+SJT1d0uVD97vTvxMlPSXLx0r62551+F5J7x26r9usx8Lf\ni1V/H3p+r+6S9JhVfx8kfbekp0i6rqh7taTzs3x+39/yKn0ezVmHtfk8mtP/Xc+ElR4pLulpkm6M\niJsj4n5Jb5V09sB9moiID0bEZk5epvpmBatumW16tqQ3Ru0yScfZPnGvOzpPRNwZEVdl+auSbpB0\n0rC92hUr/T50nCHppoj4/NAd2U5EfEzS33eqz5b0x1n+Y0kv6Hnqynwe9a3DOn0ezXkPlvGA3oN1\nCMWfyqH+G+bsrjhJ0m3F9O1a3Q+/H1X9X32fkHSJ7Svz1nZDWmabrs12t32qpCdLurxn9nfm79f7\nbX/LnnZsOdv9XqzN+yDpHElvmTNv1d8HSXpkRNyZ5bskPbKnzTq9H+vyedS1q5kweCjavsT2dT2P\nsyX9vuoh8JMk3SnpNwbt7BzbrEPT5gJJm5IunrOY74qIJ0l6jqT/avu796DrD3q2j5H0Dkkvi4h7\nO7OvkvToiPg2Sb8j6d173b8lPCh+L2wfIen5kt7WM3sd3oeWqPfTre31bGv8ebTrmbAKF+8/e5l2\ntv9Q0nt7Zt0h6ZRi+uSs2zPbrYPtcyU9T9IZ+cfUt4w78ufdtt+lehfAxw5xV5e1zDYdfLtvx/Y+\n1YF4cUS8szu/DMmIeJ/t/2X7+FihWw4u8Xux8u9Deo6kqyLiS90Z6/A+pC/ZPjEi7sxd1Hf3tFn5\n92MNP48myt+f3cqEwUeKi3SOjbxQUt83ZHxC0uNsn5b/jZ4j6T170b9l2D5T9fdGPj8i/nlOm6Nt\nH9uUVR8MH/LbQJbZpu+R9KI8+/Hpkv6h2LU0ONuW9HpJN0TEa+e0OSHbyfbTVP89/N3e9XKxJX8v\nVvp9KPwHzdl1uurvQ+E9kl6c5RdL+rOeNnwe7aI9yYQhzy5a4uyjP5H0KUnX5kqdmPWPkvS+ot1Z\nqs8uvEnSBUP3u7MON6rev311Pv6guw6qdwdck49Pr8I69G1TST8p6SezbEm/l/M/pfqm7oNv76L/\n36V699a1xbY/q7MOL8ntfY3qkw6+c+h+d9ah9/dind6H7OPRqkPuYUXdSr8PqgP8TkkHVB+T+jFJ\n36D6G34+J+kSSQ/Ptiv5eTRnHdbm82hO/3c9E7jNGwAAaaV3nwIAsJcIRQAAEqEIAEAiFAEASIQi\nAACJUATWlO3T85sMbnT9TRkeuk/AuiMUgfX1+5J+QtLj8nHmsN0B1h+hCKwo279g+6ez/DrbH8ny\ns2x/WNJDI+KyqC82fqP6v7UBwA4QisDqulTSM7O8X9IxeT/XZ0r6oOq7fDRW+dsYgLVBKAKr60pJ\np9t+qKSvSfq46nB8ZpYBHGKDf0sGgH4RccD2LZLOlfR/Vd/v8fskfZPq+2+WXxC7ct/GAKwjRorA\nartU0s+r/tqeS1XfSPuTUX8Txr22n55nnb5I/d/aAGAHCEVgtV0q6URJH4/6u+TuyzpJ+i+SLlL9\nzQc3af63qANYEt+SAQBAYqQIAEAiFAEASIQiAACJUAQAIBGKAAAkQhEAgEQoAgCQCEUAABKhCABA\nIhQBAEiEIgAAiVAEACARigAAJEIRAIBEKAIAkAhFAAASoQgAQCIUAQBIhCIAAIlQBAAgEYoAACRC\nEQCARCgCAJAIRQAAEqEIAEAiFAEASBtDd2AnRg99TMTmffWEu3PdU+w2ck913/N6nuuZF1zyNfuf\n324+7znNj8Xz57bprsLcde15au/6tpftcqLndee9RZ7XgXnPeyCv2Xru/HWat7qe02jmN2u75897\n/5daRv/KLbOM/jaztUv9SZTTc9t4iTZF3ba/knPeNc9/rxet3bw/t8XLmt3+2/x1zC5j3koss7wH\n8rq9T1rydQ/itbf72Fj4V7jd3/D8p+iqq678QEScuYOnzrVWoRib/6Ijv/mcesKWXB2CciVVnpa9\nRHm75W2zDNvTDyZPP6TKenv6V7/M/GXqt3vNbv32z9PCclUtatMsR63yssteXO5fdlmuel6z6lue\n+uureevs6R/93GV26nPTtvo0r69NfVU8t7Jay5uWp8+tinWtWsueXZ/K04+upZbtYtlqL6e7nVuv\nM2/91Xlfiu3Z2ra99XUfytd02SeV70N7PV08b359sR5qXrTdplkH9bz+ntT39Olg6128ULe++89a\n630r/mvp1nvyov31y65z+fyj9vl4HSLsPgUAIBGKAAAkQhEAgEQoAgCQCEUAABKhCABAIhQBAEiE\nIgAAiVAEACARigAAJEIRAIBEKAIAkAhFAAASoQgAQCIUAQBIhCIAAIlQBAAgEYoAACRHxNB9WJrt\n6yTdN3Q/1tDxkr48dCfWENvt4LDdDg7b7eA9JCKeeCgWtHEoFrKH7ouI/UN3Yt3YvoLttnNst4PD\ndjs4bLeDZ/uKQ7Usdp8CAJAIRQAA0rqF4oVDd2BNsd0ODtvt4LDdDg7b7eAdsm23VifaAACwm9Zt\npAgAwK5Zi1C0fabtz9q+0fb5Q/dnldg+xfZHbV9v+9O2X5r1D7f9Idufy59fXzznFbktP2v7B4br\n/fBsj2x/0vZ7c5rttg3bx9l+u+3P2L7B9new3bZn+2fyb/Q622+x/RC2Wz/bb7B9d16G19TteFvZ\nPt32p3Leb9v2ti8eESv9kDSSdJOkx0o6QtI1kp4wdL9W5SHpRElPyfKxkv5W0hMkvVrS+Vl/vqRX\nZfkJuQ0YtSpZAAADB0lEQVSPlHRabtvR0Osx4Pb7WUlvlvTenGa7bb/N/ljSj2f5CEnHsd223WYn\nSbpF0lE5/b8lnct2m7u9vlvSUyRdV9TteFtJ+htJT5dkSe+X9JztXnsdRopPk3RjRNwcEfdLequk\nswfu08qIiDsj4qosf1XSDar/AM9W/eGl/PmCLJ8t6a0R8bWIuEXSjaq38WHH9smSnivpoqKa7baA\n7Yep/sB6vSRFxP0R8RWx3ZaxIeko2xuSvk7SF8V26xURH5P0953qHW0r2ydKemhEXBZ1Qr6xeM5c\n6xCKJ0m6rZi+PevQYftUSU+WdLmkR0bEnTnrLkmPzDLbc+o3Jb1c0rioY7stdpqkeyT9Ue52vsj2\n0WK7LRQRd0h6jaQvSLpT0j9ExAfFdtuJnW6rk7LcrV9oHUIRS7B9jKR3SHpZRNxbzsv/kjjNuGD7\neZLujogr57Vhu/XaUL1b6/cj4smS/kn1rqwJttusPP51tup/Kh4l6WjbP1S2Ybstbze31TqE4h2S\nTimmT846JNv7VAfixRHxzqz+Uu4+UP68O+vZnrVnSHq+7VtV75J/lu03ie22ndsl3R4Rl+f021WH\nJNttsWdLuiUi7omIA5LeKek7xXbbiZ1uqzuy3K1faB1C8ROSHmf7NNtHSDpH0nsG7tPKyLOpXi/p\nhoh4bTHrPZJenOUXS/qzov4c20faPk3S41QfjD6sRMQrIuLkiDhV9e/URyLih8R2Wygi7pJ0m+3H\nZ9UZkq4X2207X5D0dNtfl3+zZ6g+/s92W96OtlXuar3X9tNzm7+oeM58Q59ltOSZSGepPqvyJkkX\nDN2fVXpI+i7VuxGulXR1Ps6S9A2SPizpc5IukfTw4jkX5Lb8rJY4G+vB/pD0vZqefcp22357PUnS\nFfk7925JX892W2q7/Zqkz0i6TtKfqD5bku3Wv63eovrY6wHVeyd+7GC2laT9ub1vkvS7yhvWLHpw\nRxsAANI67D4FAGBPEIoAACRCEQCARCgCAJAIRQAAEqEIAEAiFAEASIQiAADp/wODpAJA68YRdgAA\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x117cd6e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from itertools import product\n",
    "\n",
    "BaseYear = 1995\n",
    "x = np.matrix(df_arac.Year[0:]).T-BaseYear\n",
    "y = np.matrix(df_arac.Car[0:]).T/1000000.\n",
    "\n",
    "# Setup the vandermonde matrix\n",
    "N = len(x)\n",
    "A = np.hstack((np.ones((N,1)), x))\n",
    "\n",
    "left = -5\n",
    "right = 15\n",
    "bottom = -4\n",
    "top = 6\n",
    "step = 0.05\n",
    "W0 = np.arange(left,right, step)\n",
    "W1 = np.arange(bottom,top, step)\n",
    "\n",
    "ErrSurf = np.zeros((len(W1),len(W0)))\n",
    "\n",
    "for i,j in product(range(len(W1)), range(len(W0))):\n",
    "    e = y - A*np.matrix([W0[j], W1[i]]).T\n",
    "    ErrSurf[i,j] = e.T*e/2\n",
    "\n",
    "plt.figure(figsize=(7,7))\n",
    "plt.imshow(ErrSurf, interpolation='nearest', \n",
    "           vmin=0, vmax=1000,origin='lower',\n",
    "           extent=(left,right,bottom,top),cmap='Blues_r')\n",
    "plt.xlabel('w0')\n",
    "plt.ylabel('w1')\n",
    "plt.title('Error Surface')\n",
    "plt.colorbar(orientation='horizontal')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Modeli Nasıl Kestirebiliriz?\n",
    "\n",
    "## Fikir: En küçük kare hata \n",
    "(Gauss 1795, Legendre 1805)\n",
    "\n",
    "* Toplam hatanın $w_0$ ve $w_1$'e göre türevini hesapla, sıfıra eşitle ve çıkan denklemleri çöz\n",
    "\n",
    "\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "y_0 \\\\ y_1 \\\\ \\vdots \\\\ y_{N-1} \n",
    "\\end{array}\n",
    "\\right)\n",
    "\\approx\n",
    "\\left(\n",
    "\\begin{array}{cc}\n",
    "1 & x_0 \\\\ 1 & x_1 \\\\ \\vdots \\\\ 1 & x_{N-1}  \n",
    "\\end{array}\n",
    "\\right) \n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    " w_0 \\\\ w_1  \n",
    "\\end{array}\n",
    "\\right)\n",
    "\\end{eqnarray}\n",
    "\n",
    "\\begin{eqnarray}\n",
    "y \\approx A w\n",
    "\\end{eqnarray}\n",
    "\n",
    "> $A = A(x)$: Model Matrisi\n",
    "\n",
    "> $w$: Model Parametreleri\n",
    "\n",
    "> $y$: Gözlemler\n",
    "\n",
    "* Hata vektörü: $$e = y - Aw$$\n",
    "\n",
    "\\begin{eqnarray}\n",
    "E(w) & = & \\frac{1}{2}e^\\top e = \\frac{1}{2}(y - Aw)^\\top (y - Aw)\\\\\n",
    "& = & \\frac{1}{2}y^\\top y - \\frac{1}{2} y^\\top Aw - \\frac{1}{2} w^\\top A^\\top y + \\frac{1}{2} w^\\top A^\\top Aw \\\\\n",
    "& = & \\frac{1}{2} y^\\top y - y^\\top Aw + \\frac{1}{2} w^\\top A^\\top Aw \\\\\n",
    "\\end{eqnarray}\n",
    "\n",
    "### Gradyan\n",
    "https://tr.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/partial-derivative-and-gradient-articles/a/the-gradient\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\frac{d E}{d w } & = & \\left(\\begin{array}{c}\n",
    " \\partial E/\\partial w_0 \\\\ \\partial E/\\partial w_1 \\\\ \\vdots \\\\  \\partial E/\\partial w_{K-1}\n",
    "\\end{array}\\right)\n",
    "\\end{eqnarray}\n",
    "  \n",
    "Toplam hatanın gradyanı\n",
    "\\begin{eqnarray}\n",
    "\\frac{d}{d w }E(w) & = & \\frac{d}{d w }(\\frac{1}{2} y^\\top y) &+ \\frac{d}{d w }(- y^\\top Aw) &+ \\frac{d}{d w }(\\frac{1}{2} w^\\top A^\\top Aw) \\\\\n",
    "& = & 0 &- A^\\top y &+ A^\\top A w \\\\\n",
    "& = & - A^\\top (y - Aw) \\\\\n",
    "& = & - A^\\top e \\\\\n",
    "& \\equiv & \\nabla E(w)\n",
    "\\end{eqnarray}\n",
    "\n",
    "### Yapay zekaya gönül veren herkesin bilmesi gereken eşitlikler\n",
    "* Vektör iç çarpımının gradyeni\n",
    "\\begin{eqnarray}\n",
    "\\frac{d}{d w }(h^\\top w) & = & h\n",
    "\\end{eqnarray}\n",
    "\n",
    "* Karesel bir ifadenin gradyeni\n",
    "\\begin{eqnarray}\n",
    "\\frac{d}{d w }(w^\\top K w) & = & (K+K^\\top) w\n",
    "\\end{eqnarray}\n",
    "\n",
    "\n",
    "### En küçük kare hata çözümü doğrusal modellerde doğrusal denklemlerin çözümü ile bulunabiliyor\n",
    "\n",
    "\n",
    "\\begin{eqnarray}\n",
    "w^* & = & \\arg\\min_{w} E(w)\n",
    "\\end{eqnarray}\n",
    "\n",
    "* Eniyileme Şartı (gradyan sıfır olmalı )\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\nabla E(w^*) & = & 0\n",
    "\\end{eqnarray}\n",
    "\n",
    "\\begin{eqnarray}\n",
    "0 & = & - A^\\top y + A^\\top A w^*  \\\\\n",
    "A^\\top y & = &  A^\\top A w^* \\\\\n",
    "w^* & = & (A^\\top A)^{-1} A^\\top y \n",
    "\\end{eqnarray}\n",
    "\n",
    "* Geometrik (Projeksyon) yorumu:\n",
    "\n",
    "\\begin{eqnarray}\n",
    "f & = A w^*  = A (A^\\top A)^{-1} A^\\top y \n",
    "\\end{eqnarray}\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Parametreler: \n",
      "w0 =  [[ 4.13258253]] \n",
      "w1 =  [[ 0.20987778]]\n",
      "Toplam Kare Hata: [[ 37.19722385]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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YF1hW14LNbCcze8/MZpjZdDO7qq7nlAZm/Hg45hg49NDQcXznnaGzuG/flIYB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\nIhQIIiICKBBERCRCgSAiIgD8P0oAG/fp/vL9AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x119bfd710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Solving the Normal Equations\n",
    "\n",
    "# Setup the Design matrix\n",
    "N = len(x)\n",
    "A = np.hstack((np.ones((N,1)), x))\n",
    "\n",
    "#plt.imshow(A, interpolation='nearest')\n",
    "# Solve the least squares problem\n",
    "w_ls,E,rank,sigma = np.linalg.lstsq(A, y)\n",
    "\n",
    "print('Parametreler: \\nw0 = ', w_ls[0],'\\nw1 = ', w_ls[1] )\n",
    "print('Toplam Kare Hata:', E/2)\n",
    "\n",
    "f = np.asscalar(w_ls[1])*x + np.asscalar(w_ls[0])\n",
    "plt.plot(x+BaseYear, y, 'o-')\n",
    "plt.plot(x+BaseYear, f, 'r')\n",
    "\n",
    "\n",
    "plt.xlabel('Yıl')\n",
    "plt.ylabel('Araba sayısı (Milyon)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Polinomlar \n",
    "\n",
    "\n",
    "### Parabol\n",
    "\\begin{eqnarray}\n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "y_0 \\\\ y_1 \\\\ \\vdots \\\\ y_{N-1} \n",
    "\\end{array}\n",
    "\\right)\n",
    "\\approx\n",
    "\\left(\n",
    "\\begin{array}{ccc}\n",
    "1 & x_0 & x_0^2 \\\\ 1 & x_1 & x_1^2 \\\\ \\vdots \\\\ 1 & x_{N-1} & x_{N-1}^2   \n",
    "\\end{array}\n",
    "\\right) \n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    " w_0 \\\\ w_1  \\\\ w_2\n",
    "\\end{array}\n",
    "\\right)\n",
    "\\end{eqnarray}\n",
    "\n",
    "### $K$ derecesinde polinom\n",
    "\\begin{eqnarray}\n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "y_0 \\\\ y_1 \\\\ \\vdots \\\\ y_{N-1} \n",
    "\\end{array}\n",
    "\\right)\n",
    "\\approx\n",
    "\\left(\n",
    "\\begin{array}{ccccc}\n",
    "1 & x_0 & x_0^2 & \\dots & x_0^K \\\\ 1 & x_1 & x_1^2 & \\dots & x_1^K\\\\ \\vdots \\\\ 1 & x_{N-1} & x_{N-1}^2 & \\dots & x_{N-1}^K  \n",
    "\\end{array}\n",
    "\\right) \n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    " w_0 \\\\ w_1  \\\\ w_2 \\\\ \\vdots \\\\ w_K\n",
    "\\end{array}\n",
    "\\right)\n",
    "\\end{eqnarray}\n",
    "\n",
    "\n",
    "\\begin{eqnarray}\n",
    "y \\approx A w\n",
    "\\end{eqnarray}\n",
    "\n",
    "> $A = A(x)$: Model matrisi \n",
    "\n",
    "> $w$: Model Parametreleri\n",
    "\n",
    "> $y$: Gözlemler\n",
    "\n",
    "Polinom oturtmada ortaya çıkan özel yapılı matrislere  __Vandermonde__ matrisleri de denmektedir."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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NzEaAKo8wLgPmNpTdABwdEX8B/A44v5/lj4+IWbm/NWtmZtWqLGFExI3A4w1lP46Ibenh\nzcAhVT2/mZkNrzrPYbwb+FGLugB+ImmZpAX9rUTSAkk9kno2bNgw7EGamVmhloQh6VPANuDKFk1e\nExGzgHnAByS9rtW6IuKSiOiOiO6urq4KojUzM6ghYUg6GzgFeHtERLM2EdGb/q4HrgFmty1AMzNr\nqq0JQ9Jc4BPAGyPi6RZtJkjau28eOAlY0aytmZm1T5XDahcDvwZmSFoj6T3AxcDewA1pyOzC1Hay\npKVp0UnATZJuB34L/DAirqsqTjMzyzO2qhVHxJlNii9t0XYtMD/NrwKOqSouMzMbHF/pbWZmWZww\nzMwsixOGmZllccIwM7MsThhmZpbFCcPMzLI4YZiZWRYnDDMzy+KEYWZmWSq70tvMbKCuva2XL15/\nH2s3bmHyxPF8/OQZnHbslLrDssQJw8xGhGtv6+X879/JlmeeBaB34xbO//6dAE4aI4S7pMxsRPji\n9fc9nyz6bHnmWb54/X01RWSNnDDMbERYu3HLgMqt/ZwwzGxEmDxx/IDKrf2cMMxsRPj4yTMYv9uY\n7crG7zaGj588o6aIrJFPepvZiNB3YtujpEYuJwwzGzFOO3aKE8QI5i4pMzPLUuVvei+StF7SilLZ\n/pJukHR/+rtfi2XnSrpP0kpJ51UVo5mZ5avyCOMyYG5D2XnATyNiOvDT9Hg7ksYAXwfmAUcBZ0o6\nqsI4zcyG3bW39XLcRT/j8PN+yHEX/Yxrb+utO6Qh22nCkPTBVkcC/YmIG4HHG4pPBS5P85cDpzVZ\ndDawMiJWRcRW4Kq0nJnZqNB31Xrvxi0EL1y1PtqTRs4RxiTgFknfS11FGsLzTYqIR9L8urTuRlOA\nh0uP16SypiQtkNQjqWfDhg1DCM3MbHh06lXrO00YEfF3wHTgUuBs4H5Jn5N0xFCeOCICiKGsI63n\nkojojojurq6uoa7OzGzIOvWq9axzGOnDfV2atgH7AVdL+sIAn+9RSQcDpL/rm7TpBQ4tPT4klZmZ\njQqdetV6zjmMD0taBnwB+HfgpRHxPuDlwJsH+HxLgLPS/FnAD5q0uQWYLulwSeOAM9JyZmajQqde\ntZ5z4d7+wOkR8WC5MCKek3RKq4UkLQbmAAdKWgN8BrgI+J6k9wAPAm9NbScD346I+RGxTdK5wPXA\nGGBRRNw18E0zM6tHp161rqK3qTN0d3dHT09P3WGYmY0akpZFRHdOW1/pbWZmWZwwzMwsixOGmZll\nccIwM7Msvr25WYWuva2340bK2K7LCcOsIn33E+q7RUTf/YQAJw0bldwlZVaRTr2fkO26nDDMKtKp\n9xOyXZcThllFOvV+QrbrcsIwq0in3U+oE38QyAbGJ73NKtJJ9xPyCXwDJwyzSp127JSO+EDt7wR+\nJ2yf5XHCMBvl2nGth0/gG/gchtmo1q7fjvYJfAMnDLNRrV3XenTaCXwbHHdJmY1i7eoq6qQT+DZ4\nThhmo9jkiePpbZIcqugq6pQT+DZ47pIyG8XcVTRwvp5k8NqeMCTNkLS8ND0p6SMNbeZIeqLU5tPt\njtNsNDjt2Cl8/vSXMmXieARMmTiez5/+Uh8JtNCuQQKdqu1dUhFxHzALQNIYoBe4pknTX0bEKe2M\nzWw0cldRPl9PMjR1n8N4A/BARDxYaxQf+QgsX15rCGZWvS+t+kPryusOaF8gw23WLPjqVyt/mroT\nxhnA4hZ1r5Z0B8URyN9GxF3NGklaACwAmDp1aiVB2gse2/wnHnp8C1u3Pcu4sWOYuv94Dtxr97rD\nMssybuwYtm57tmm57Zwiop4nlsYBa4GZEfFoQ90+wHMRsVnSfOBrETF9Z+vs7u6Onp6eagK2He4n\nBMUJVveZ22jh9/COJC2LiO6ctnWOkpoH3NqYLAAi4smI2JzmlwK7STqw3QHa9vyDQDbaeZDA0NTZ\nJXUmLbqjJB0EPBoRIWk2RWLrp/PR2qHT7ifk39veNXmQwODVkjAkTQBOBP66VHYOQEQsBN4CvE/S\nNmALcEbU1Xdmz2vnRWJV8+26zQauli6piHgqIg6IiCdKZQtTsiAiLo6ImRFxTES8MiJ+VUectr1O\nukjM3WtmA1f3KCkbRTrpfkKd1r1m1g5OGDYgndL/20nda2bt4ntJ2S6pk7rXzNrFRxi2S+qk7jWz\ndnHCsF1Wp3SvmbWLu6TMzCyLE4aZmWVxwjAzsyxOGGZmlsUJw8zMsjhhmJlZFg+r7SC++6qZVckJ\no0P47qtmVjV3SXUI333VzKrmhNEhfPdVM6uaE0aHaHWXVd991cyGixNGh/DdV82saj7p3SF891Uz\nq1pdv+m9GtgEPAtsi4juhnoBXwPmA08DZ0fEre2Oc7Tx3VfNrEp1HmEcHxGPtaibB0xP0yuAb6a/\nZmZWk5HaJXUq8J2ICOBmSRMlHRwRj9QdmFXPFyCajUx1nfQO4CeSlkla0KR+CvBw6fGaVLYDSQsk\n9Ujq2bBhQwWhWjv1XYDYu3ELwQsXIF57W2/doZnt8upKGK+JiFkUXU8fkPS6wa4oIi6JiO6I6O7q\n6hq+CK0WvgDRbOSqJWFERG/6ux64Bpjd0KQXOLT0+JBUZh3OFyCajVxtTxiSJkjau28eOAlY0dBs\nCfBOFV4JPOHzF7sGX4BoNnLVcYQxCbhJ0u3Ab4EfRsR1ks6RdE5qsxRYBawEvgW8v4Y4rQa+ANFs\n5Gr7KKmIWAUc06R8YWk+gA+0M64qedRPPl+AaDZyjdRhtR3Dtx0fOF+AaDYy+V5SFfOoHzPrFE4Y\nFfOoHzPrFE4YFfOoHzPrFE4YFfOoHzPrFD7pXTGP+jGzTuGE0QYe9WNmncBdUmZmlsUJw8zMsjhh\nmJlZFicMMzPL4oRhZmZZnDDMzCyLE4aZmWVxwjAzsyxOGGZmlsUJw8zMsjhhmJlZlrYnDEmHSvq5\npLsl3SXpw03azJH0hKTlafp0u+M0M7Pt1XHzwW3A30TErZL2BpZJuiEi7m5o98uIOKWG+MzMrIm2\nH2FExCMRcWua3wTcA/hWrmZmI1yt5zAkTQOOBX7TpPrVku6Q9CNJM9samJmZ7aC238OQtBfwL8BH\nIuLJhupbgakRsVnSfOBaYHqL9SwAFgBMnTq1wojNzHZttRxhSNqNIllcGRHfb6yPiCcjYnOaXwrs\nJunAZuuKiEsiojsiuru6uiqN28xsV1bHKCkBlwL3RMSXW7Q5KLVD0myKOP/QvijNzKxRHV1SxwHv\nAO6UtDyVXQBMBYiIhcBbgPdJ2gZsAc6IiKghVjMzS9qeMCLiJkA7aXMxcHF7IjIzsxy+0tvMzLI4\nYZiZWRYnDDMzy+KEYWZmWZwwzMwsixOGmZllccIwM7MsThhmZpbFCcPMzLI4YZiZWRYnDDMzy+KE\nYWZmWZwwzMwsixOGmZllccIwM7MsThhmZpbFCcPMzLI4YZiZWRYnDDMzy1JLwpA0V9J9klZKOq9J\nvST9z1R/h6SX1RGnmZm9oO0JQ9IY4OvAPOAo4ExJRzU0mwdMT9MC4JttDdLMzHZQxxHGbGBlRKyK\niK3AVcCpDW1OBb4ThZuBiZIObnegZmb2grE1POcU4OHS4zXAKzLaTAEeaVyZpAUURyEAmyXdN3yh\nDrsDgcfqDmKYeFtGnk7ZDvC2tNNhuQ3rSBjDKiIuAS6pO44cknoiorvuOIaDt2Xk6ZTtAG/LSFVH\nl1QvcGjp8SGpbKBtzMysjepIGLcA0yUdLmkccAawpKHNEuCdabTUK4EnImKH7igzM2uftndJRcQ2\nSecC1wNjgEURcZekc1L9QmApMB9YCTwNvKvdcVZkVHSdZfK2jDydsh3gbRmRFBF1x2BmZqOAr/Q2\nM7MsThhmZpbFCaNNJI2RdJukf607lqGQNFHS1ZLulXSPpFfVHdNgSfqopLskrZC0WNIedceUS9Ii\nSeslrSiV7S/pBkn3p7/71Rljrhbb8sX0HrtD0jWSJtYZY65m21Kq+xtJIenAOmIbDk4Y7fNh4J66\ngxgGXwOui4gjgWMYpdskaQrwIaA7Io6mGIBxRr1RDchlwNyGsvOAn0bEdOCn6fFocBk7bssNwNER\n8RfA74Dz2x3UIF3GjtuCpEOBk4CH2h3QcHLCaANJhwB/BXy77liGQtK+wOuASwEiYmtEbKw3qiEZ\nC4yXNBbYE1hbczzZIuJG4PGG4lOBy9P85cBpbQ1qkJptS0T8OCK2pYc3U1yLNeK1eF0AvgJ8AhjV\no4ycMNrjqxRvlufqDmSIDgc2AP87da99W9KEuoMajIjoBf6R4hvfIxTX+vy43qiGbFLpeqV1wKQ6\ngxlG7wZ+VHcQgyXpVKA3Im6vO5ahcsKomKRTgPURsazuWIbBWOBlwDcj4ljgKUZPt8d2Uv/+qRRJ\ncDIwQdJ/qzeq4RPFePlR/W0WQNKngG3AlXXHMhiS9gQuAD5ddyzDwQmjescBb5S0muLOvK+XdEW9\nIQ3aGmBNRPwmPb6aIoGMRicAv4+IDRHxDPB94NU1xzRUj/bd1Tn9XV9zPEMi6WzgFODtMXovGDuC\n4kvJ7ekz4BDgVkkH1RrVIDlhVCwizo+IQyJiGsVJ1Z9FxKj8JhsR64CHJc1IRW8A7q4xpKF4CHil\npD0liWJbRuUJ/JIlwFlp/izgBzXGMiSS5lJ0474xIp6uO57Biog7I+JFETEtfQasAV6W/pdGHScM\nG6gPAldKugOYBXyu5ngGJR0lXQ3cCtxJ8b8wam7hIGkx8GtghqQ1kt4DXAScKOl+iiOoi+qMMVeL\nbbkY2Bu4QdJySQtrDTJTi23pGL41iJmZZfERhpmZZXHCMDOzLE4YZmaWxQnDzMyyOGGYmVkWJwwz\nM8vihGFmZlmcMMwqIuk/p99z2EPShPTbG0fXHZfZYPnCPbMKSfoHYA9gPMV9uD5fc0hmg+aEYVYh\nSeOAW4A/Aq+OiGdrDsls0NwlZVatA4C9KO6LNGp+AtasGR9hmFVI0hKK29ofDhwcEefWHJLZoI2t\nOwCzTiXpncAzEfFdSWOAX0l6fUT8rO7YzAbDRxhmZpbF5zDMzCyLE4aZmWVxwjAzsyxOGGZmlsUJ\nw8zMsjhhmJlZFicMMzPL8v8BE3TtUu6vYpAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x117bacc88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.array([10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5])\n",
    "N = len(x)\n",
    "x = x.reshape((N,1))\n",
    "y = np.array([8.04, 6.95, 7.58, 8.81, 8.33, 9.96, 7.24, 4.26, 10.84, 4.82, 5.68]).reshape((N,1))\n",
    "#y = np.array([9.14, 8.14, 8.74, 8.77, 9.26, 8.10, 6.13, 3.10, 9.13, 7.26, 4.74]).reshape((N,1))\n",
    "#y = np.array([7.46, 6.77, 12.74, 7.11, 7.81, 8.84, 6.08, 5.39, 8.15, 6.42, 5.73]).reshape((N,1))\n",
    "\n",
    "def fit_and_plot_poly(degree):\n",
    "\n",
    "    #A = np.hstack((np.power(x,0), np.power(x,1), np.power(x,2)))\n",
    "    A = np.hstack((np.power(x,i) for i in range(degree+1)))\n",
    "    # Setup the vandermonde matrix\n",
    "    xx = np.matrix(np.linspace(np.asscalar(min(x))-1,np.asscalar(max(x))+1,300)).T\n",
    "    A2 = np.hstack((np.power(xx,i) for i in range(degree+1)))\n",
    "\n",
    "    #plt.imshow(A, interpolation='nearest')\n",
    "    # Solve the least squares problem\n",
    "    w_ls,E,rank,sigma = np.linalg.lstsq(A, y)\n",
    "    f = A2*w_ls\n",
    "    plt.plot(x, y, 'o')\n",
    "    plt.plot(xx, f, 'r')\n",
    "\n",
    "    plt.xlabel('x')\n",
    "    plt.ylabel('y')\n",
    "\n",
    "    plt.gca().set_ylim((0,20))\n",
    "    #plt.gca().set_xlim((1950,2025))\n",
    "    \n",
    "    if E:\n",
    "        plt.title('Mertebe = '+str(degree)+' Hata='+str(E[0]))\n",
    "    else:\n",
    "        plt.title('Mertebe = '+str(degree)+' Hata= 0')\n",
    "        \n",
    "    plt.show()\n",
    "\n",
    "fit_and_plot_poly(0)"
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    "interact(fit_and_plot_poly, degree=(0,10))"
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   "source": [
    "Overfit: Aşırı uyum"
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