GD.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "import numpy as np\n",
    "import sympy as sp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = symbols('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = (x**2) + (6*x) + 17"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [],
   "source": [
    "Fprime = F.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$2 x + 6$$"
      ],
      "text/plain": [
       "2⋅x + 6"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fprime"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$8 x + 11$$"
      ],
      "text/plain": [
       "8⋅x + 11"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fprime+5+6*x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = 1 / (1 + exp(-x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "Fprime = F.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{e^{- x}}{\\left(1 + e^{- x}\\right)^{2}}$$"
      ],
      "text/plain": [
       "    -x    \n",
       "   ℯ      \n",
       "──────────\n",
       "         2\n",
       "⎛     -x⎞ \n",
       "⎝1 + ℯ  ⎠ "
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fprime"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculus in Python:\n",
    "- https://docs.sympy.org/latest/tutorial/calculus.html#derivatives\n",
    "- https://dev.to/erikwhiting88/calculate-derivative-functions-in-python-h58\n",
    "- https://www.youtube.com/watch?v=Dkifb6nytao\n",
    "- https://www.youtube.com/watch?v=EaYEQ30KlWw"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Course Website:\n",
    "https://github.com/lazyprogrammer/machine_learning_examples/tree/master/linear_regression_class"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.array([\n",
    "    [2],\n",
    "    [3],\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2],\n",
       "       [3]])"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = np.array([\n",
    "    [2, 3],\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 3]])"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4, 6],\n",
       "       [6, 9]])"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4, 6],\n",
       "       [6, 9]])"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.matmul(a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "20"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [],
   "source": [
    "total = 0\n",
    "for i in range(len(a)):\n",
    "    total += a[i]*b[i]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "90"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(a*b).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# load the data\n",
    "X = []\n",
    "Y = []\n",
    "for line in open('data_1d.csv'):\n",
    "    x, y = line.split(',')\n",
    "    X.append(float(x))\n",
    "    Y.append(float(y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.array(X)\n",
    "Y = np.array(Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(X, Y)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "352941.7940542859"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.dot(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "352941.7940542859"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(X, X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "53.037609423246906"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5303.76094232469"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "denominator = X.dot(X) - X.mean() * X.sum()\n",
    "# a = ( X.dot(Y) - Y.mean()*X.sum() ) / denominator\n",
    "# b = ( Y.mean() * X.dot(X) - X.mean() * X.dot(Y) ) / denominator\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = len(X)\n",
    "d = ( N * X.dot(X).sum() ) - ( X.sum() ** 2.0 )\n",
    "\n",
    "a = ( (N * Y.dot(X).sum()) - ( X.sum()*Y.sum()) ) / d\n",
    "b = ( (Y.sum() * X.dot(X)) - (X.sum() * Y.dot(X)) ) /d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.9726121674845967"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.864424075660324"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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x/hqQZIpTXnkr8F7gcDPbAnyRTJXNFOBeM4P9ZZTvAb5kZnuBfcAn3f2l0B8sIpl8+kj8/HycGXl2xq+qG8myIOsyoXp7e72/v3+ihyFSVx++4SF+8tRvI59vT7UVrGi96vzZCtYyyszWuntvqfPUvVJkAnT1rSoa5NMd7Vx1/mzSHe1YzmMFeRkPtUAQqaOwNsJTJ09iz96R0cfZXjRa0SrVokAvUqE4m4FAeJBvT7XxpyeneeDJHepFIzWjQC9Sgaiuk7D/pmixzUCGhvfxwJM7+Fnf6bUfrLQs5ehFKhDVdXLp6k2MjHisHZ+0iElqTYFepAJRQXpgcIhjr7h7zLGo1gVaxCS1ptSNtKT8vPppb51eVp48e32c4uQz33YE3774vxekeUCbgEh9KNBLywnLq39nzfOjz4fl2YtdX0xur3htAiITRYFeWk5YXj1fNs8eFoTjXL/m8jP4g0OmFhxXyaRMBAV6aTlxb35GnVfq+lI7PonUmwK9tJyjOtoZiBHso26SRl2f7mgvKJOMW2MvUkuqupGWs2jOLNpTbUXPKXaTNCzIh52fzeUPDA7h7M/9a5cnqTfN6KXlhN0UjVN1E1YTbxB5frEae83qpZ4U6KUllXNT9KVXX+ekL99bcDybi8+mZy5bvn5M0B9vjl+k2hToRYootrK1+8p7OOfEGdy+diC0BUJULl8LpKTelKMXCfGlOx8vCPJtef+1DA4N8501z0emZ8LuBWiBlEwEzehF8oTN4tMxK3Wytg4OaYGUNIxYgd7MbiSzEfh2dz8hOHYYsBzoAjYDH3D3nZbZW/CbwFxgN/BRd3+4+kMXKa2c8sawAP/sV+cyaZIxM0ZzslzZ9IwWSEkjiJu6uQk4K+9YH3Cfux8H3Bc8BjibzKbgxwELgesrH6ZI+copbwwL8puXzGPSpMwm28Xy6vnbcCs9I40mVqB39weB/E2+zwNuDr6/GZifc/wWz1gDdJjZjGoMVqQcUeWNV965cfRxV9+qgiB/7YLugtWti+bMItWWH9IhNcm46JRObfknDa2SHP2R7r4NwN23mdkRwfE08ELOeVuCY9tyLzazhWRm/HR2dlYwDJFwUWWMO3cP88OHt3DZ9x8JfT6soVn2+yvv3MjO3cMAdLSnWHzu8Qrq0vBqcTO2cNpDYTdXd18GLAPo7e2N0+1VJFRUHr5Yq4OoIA/7Z/35AVz5dmlWlZRXvphNyQRftwfHtwDH5Jx3NLC1gtcRiVQsD19Jnnzn7mG1KpDEqCTQrwQuDr6/GPhRzvGPWMYpwK5sikek2kq1GehoT0Veu3nJvMhdn7I/WyQJYgV6M7sV+AUwy8y2mNklwBLgfWb2FPC+4DHA3cCzwNPAt4C/qvqoRQKl2gykJhf+X3zq5Elcu6AboOisX60KJCli5ejd/cKIp84IOdeBT1UyKJG4irUZiFr4lFtLP78nzeKVGxkcGg79GSJJoJWx0vCKLXpaNGdW6LZ++cG/2GYgi889Xnu5SqIp0EtDC9vfNbf8Mb/NQFj5Vqkdn9SqQJLOMpmWidXb2+v9/f0TPQxpQKcuuT/Wbk5RK1tFkszM1rp7b6nz1L1SGlqpm60v7xlWkBcpQakbaWjl3mzN3wxEqRgRBXppIGHBOexm6+RJVhD8r/tQD+e846jRn1Msry/SapS6kYYQtcIV4KrzZ482DQPYOzL2vtLmJfNGgzwUX0Ql0oo0o5eGEBWcL12+PnLTj6e/cjaT87d9onReX6TVaEYvDaFYEA4L8puXzAsN8hC90EkLoKRVKdBLQygnCLen2oo2HNNerSJjKdBLQ4ja2CNMqXz7/J70mLy+NgORVqccvTSEYj1nwpTKt6t3vMh+CvQyYXLLKQ8/aErsIA/Kt4uUQ4FeJkR+rfuO378W+1rl20XKo0AvVRV3RWpYOWWY9lQbf3pymgee3KFVriLjpEAvVVPOitSovVwhs+mwU9g7XkTGZ9yB3sxmActzDh0L/G+gA/hLYEdw/Ap3v3vcI5SmUWpbPwjvMpkvG+Rzu1OKyPiNO9C7+yagG8DM2oAB4IfAx4Br3P3rVRmhNI2oSpjs7D1OkC/1s0SkfNWqoz8DeMbdn6vSz5MmVKwSJj/Ib14yj2sXdNNm4bXzqqoRqZ5qBfoPArfmPP60mT1qZjea2aFVeg1pcHErYbKthOf3pLn6AydqFatIjVW8w5SZHQBsBY539xfN7Ejgt2RSrV8GZrj7X4RctxBYCNDZ2Xnyc8/pj4EkKJaeidoMRL3jRcYn7g5T1ai6ORt42N1fBMh+DQbxLeCusIvcfRmwDDJbCVZhHNLgZvatGhPI8wP8NQu6FeBFaqAagf5CctI2ZjbD3bcFD98PPFaF15AmUOpma26f+f7nXuL2tQPaHESkDioK9GY2DXgf8Imcw18zs24y/11vzntOEqicahrIlFze+tAL7MtLG+aXYopIdVQU6N19N/DGvGMfrmhE0lTKDfJZ+UE+S2WVItWnlbEyLmEBPruiNY42s9Bgr7JKkepTP3opW1iQT3e0xw7y7ak2LnzXMSqrFKkTzegltrAAf+2C7tFNvUsxGFN10/umw1RWKVIHCvQSKrf08YiDp/DiK2PbCJ/fk+YbC7o5dcn9sbpQhvWu0eYgIvWhQC8F8rtQ5gf53IVPcW6eKiUjMrEU6BNsvCtOo3rFH3nwFB76/Jljjh3V0R6atmkzY8RdKRmRBqBAn1BxesNH/SKIyrdvf6VwF6hFc2aNeR3IzOC1GbdI41DVTUIV6w0P+38RDAwOja5YvXT5+qJ18WGlj/N70lx1/mzSHe0YmVy8grxIY9GMPqGicufZ43G38ssqlmfXTVWRxqYZfUJFLTzKHi92EzXbK16zdJFk0Iw+oaJy59lZ+YxDprJ1156C69LBLwLN0kWSQ4E+obJBOuxma1QeXmWQIsmkQJ9g+bPyBzZtLwjyh007gJ27X1cZpEiCKdA3iUp3YQqbxUft+CQiyaJA3wTi1MRHOe+6n/LIll1jjj355bOYmtdQTESSS1U3TaBUTXyUrr5VBUF+85J5CvIiLUYz+iZQqiY+n9I0IpKr4hm9mW02sw1mtt7M+oNjh5nZvWb2VPD10MqH2ppWrBtgklnoc2G18lEVNV19q+i+8h5WrBuo6vhEpPFVa0Z/mrv/NudxH3Cfuy8xs77g8eeq9FotI5ubD9uJKb8UMizAp9qM4X37rx0cGmbRDx4BtAG3SCupVY7+PODm4Pubgfk1ep1Ei2pT0GY2ulJ1xyuvFQT5T7znWNId7WOCfNbwiJfM7YtIslRjRu/APWbmwD+7+zLgSHffBuDu28zsiPyLzGwhsBCgs7OzCsNInqgc/Ih75MKnbC5+2YPPlv1zRSSZqhHoT3X3rUEwv9fMnoxzUfALYRlAb29v3O1Gm162Hn5gcGh0g+x0RF18VK/3g6dOLgjyP/nb0zjmsGklr80+JyKto+LUjbtvDb5uB34IvBN40cxmAARft1f6OkmQ2xoYGM29Z+vi82+ULpozq2ADbYCX9+wd83jzknljgnz22lRb4U3c1CRTmwORFlNRoDezA83s4Oz3wB8DjwErgYuD0y4GflTJ6yRFsdbAYXXxub3ew2xeMi+ybHJ+T5qlf3Yih05LjR7raE+x9IITdSNWpMVUmro5EvihZcr/JgPfc/cfm9mvgO+b2SXA88AFFb5OIkSlUrLCcufze9Jcunx9wfE4dfHqQCkiUGGgd/dngRNDjv8OOKOSn51E2Zx8lPzcuRY+iUg1qAVCHRUL8rl18ftGvCDIv33GGxTkRWRc1AKhjtIRlTC5dfGaxYtItWlGX0dhVTTtqTau/sCJzDz8wIIgf+NHexXkRaRimtHXUdSuT+O92SoiEocCfZ3lVsL8r++vLwjy6hUvItWmQD9BlIsXkXpRoK+RqK3/FOBFpN7Mi5T81Utvb6/39/dP9DCqJn/rP8jcdA1bFasgLyLjZWZr3b231Hma0VcobOYetfVfLgV4EakXlVdWILdJmbO/OVmpVgftqTbt9CQidaNAX4GomXtbxNZ/uedceefGWg5NRGSUAn0FojbwKNbqIGvn7mHN6kWkLhToK1DpBh7a0k9E6kGBvgJRG4MA/OaquWxeMo9rF3RHXq8t/USkHhTox2HFugFOXXI/ly1fH1kyGfToZ35Pmo72VME5oC39RKQ+VF5ZRFjpJFBQI58VVTK5+NzjQ+vqtaWfiNTDuAO9mR0D3AL8ATACLHP3b5rZYuAvgR3BqVe4+92VDrTe8hc9ZUsnp0y20CAftd0fRDcz0+5PIlIPlczo9wKfdfeHg31j15rZvcFz17j71ysf3sSJKp0cGg4/v1S+Xdv6ichEGXegd/dtwLbg+1fM7AkgMZGs1KKnfMq3i0ijqsrNWDPrAnqAh4JDnzazR83sRjM7tBqvUW/FFj2FbR6ifLuINKqKA72ZHQTcDlzq7i8D1wNvBrrJzPivjrhuoZn1m1n/jh07wk6ZUMUWPV11/mzSHe0Ymdx8dhtAEZFGVFH3SjNLAXcBq939GyHPdwF3ufsJxX5Oo3WvDGslnJXuaOdnfafXcTQiIuHidq8c94zeMoXiNwBP5AZ5M5uRc9r7gcfG+xr1ds29vy4a5JWiEZFmVEnVzanAh4ENZpbdD+8K4EIz6wYc2Ax8oqIR1kl+gD/1zW/kgt5jVBIpIk2vkqqbnwJhdyybqmb+4zf38+9PvFhw/OfP/I6Z0w9UmkZEml7Lrowd3jfCcZ//t8jnHfjumufpfdNhmsWLSFNrmUCf284g7u1nJ7NwSoFeRJpZSzQ1y98JKteGxX9ctH2BOkyKSLNL3Iw+7h6ukCmVPHhqikVzZnHZ8vWhM32teBWRZpeoGX3YHq6fu/3RyHYG2dn6/J40F53SWXBnWeWUIpIEiQr0YTP31/aORJ7fMW1/n/i/mz+baxZ0a8WriCROolI35ebT8xcFq8OkiCRRomb0Mw6ZWtb5u6J6DouIJEhiAv36FwbZumtPwfH2VJu28hORltbUqZsV6wb42o+fHBPgD546mYOnTGbbrj1Ft//TjVYRaRVNG+hXrBug7/ZH2ZNzs/WAtkl8+bwTIvPs6lsjIq2oaVM3S1dvGhPkAV7fN8LS1ZtCz5/fk2bRnFkc1dHO1sEhlq7exIp1A/UYqojIhGraGX1UhU3U8ajNvgHN7EUk0Zp2Rh91IzXqeNRm31F/AYiIJEXTBvpFc2aVtXdruX8BiIgkRdMG+vk96bL2bi33LwARkaRo2hw9lLeSddGcWSqxFJGWVLMZvZmdZWabzOxpM+ur1evEVe5fACIiSVGTGb2ZtQH/BLwP2AL8ysxWuvvjtXi9uNTLRkRaUa1m9O8Ennb3Z939deBfgfNq9FoiIlJErQJ9Gngh5/GW4NgoM1toZv1m1r9jx44aDUNERGoV6PP38ADGbuDk7svcvdfde6dPn16jYYiISK0C/RbgmJzHRwNba/RaIiJSRK0C/a+A48xsppkdAHwQWFmj1xIRkSLM87dZqtYPNpsLXAu0ATe6+1eKnLsDeC7v8OHAb2syuMan996a9N5bUyXv/U3uXjL3XbNAXykz63f33okex0TQe9d7bzV677V9703bAkFEROJRoBcRSbhGDvTLJnoAE0jvvTXpvbemmr/3hs3Ri4hIdTTyjF5ERKqgIQN9o3W+rCUzO8bMHjCzJ8xso5l9Jjh+mJnda2ZPBV8Pneix1oKZtZnZOjO7K3g808weCt738mAdRiKZWYeZ3WZmTwaf//9ohc/dzC4L/r/+mJndamZTk/y5m9mNZrbdzB7LORb6OVvGPwSx71EzO6kaY2i4QJ/T+fJs4O3AhWb29okdVU3tBT7r7m8DTgE+FbzfPuA+dz8OuC94nESfAZ7Iefz3wDXB+94JXDIho6qPbwI/dve3AieS+d8h0Z+7maWBvwZ63f0EMutsPkiyP/ebgLPyjkV9zmcDxwX/FgLXV2MADRfoabHOl+6+zd0fDr5/hcx/7Gky7/nm4LSbgfkTM8LaMbOjgXnAt4PHBpwO3Backsj3DWBmbwDeA9wA4O6vu/sgLfC5k2mP3m5mk4FpwDYS/Lm7+4PAS3mHoz7n84BbPGMN0GFmMyodQyMG+pJEEuQYAAACCUlEQVSdL5PKzLqAHuAh4Eh33waZXwbAERM3spq5FvhbYCR4/EZg0N33Bo+T/NkfC+wA/l+Quvq2mR1Iwj93dx8Avg48TybA7wLW0jqfe1bU51yT+NeIgb5k58skMrODgNuBS9395YkeT62Z2TnAdndfm3s45NSkfvaTgZOA6929B3iVhKVpwgS56POAmcBRwIFk0hX5kvq5l1KT/wYaMdC3XOdLM0uRCfLfdfc7gsMvZv9kC75un6jx1cipwLlmtplMeu50MjP8juBPekj2Z78F2OLuDwWPbyMT+JP+uZ8J/Mbdd7j7MHAH8Ie0zueeFfU51yT+NWKgb6nOl0Fe+gbgCXf/Rs5TK4GLg+8vBn5U77HVkrtf7u5Hu3sXmc/4fne/CHgA+LPgtMS97yx3/y/gBTPL7k5/BvA4Cf/cyaRsTjGzacH/97PvuyU+9xxRn/NK4CNB9c0pwK5siqci7t5w/4C5wK+BZ4DPT/R4avxe/4jMn2aPAuuDf3PJ5KvvA54Kvh420WOt4f8G7wXuCr4/Fvgl8DTwA2DKRI+vhu+7G+gPPvsVwKGt8LkDVwJPAo8B/wJMSfLnDtxK5n7EMJkZ+yVRnzOZ1M0/BbFvA5nqpIrHoJWxIiIJ14ipGxERqSIFehGRhFOgFxFJOAV6EZGEU6AXEUk4BXoRkYRToBcRSTgFehGRhPv/ri3ECUu6QxYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "# let's calculate the predicted Y\n",
    "Yhat = a*X + b\n",
    "\n",
    "# let's plot everything together to make sure it worked\n",
    "plt.scatter(X, Y)\n",
    "plt.plot(X, Yhat)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the r-squared is: 0.9911838202977805\n"
     ]
    }
   ],
   "source": [
    "# determine how good the model is by computing the r-squared\n",
    "d1 = ((Y - Yhat) ** 2.0).sum()\n",
    "d2 = ((Y - Y.mean()) ** 2.0).sum()\n",
    "\n",
    "r2 = 1 - ( d1 / d2 )\n",
    "print(\"the r-squared is:\", r2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}