ML 1.2 - Linear Regression - from scratch.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Internet Resources:  \n",
    "\n",
    "[Python Programming.net - machine learning episodes 1-11](https://pythonprogramming.net/machine-learning-tutorial-python-introduction/)  \n",
    "[Khan Acadamy linear regression (whole playlist)](https://www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/more-on-regression/v/squared-error-of-regression-line)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": false
   },
   "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": [
    "from statistics import mean\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import style\n",
    "\n",
    "#data\n",
    "xs = np.array([1,2,3,4,5,6], dtype=np.float64) \n",
    "ys = np.array([5,4,6,5,6,8], dtype=np.float64)\n",
    "\n",
    "plt.scatter(xs, ys)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Based on given data, we want to find a linear function that fits this data in the \"best possible way\" (to be more precise where the squarred error is at an minimum). With this function we can then also make predictions.\n",
    "The function has the form y = m*x+b. Our task is to find what the best m (=slope) and b (=bias) is to model our data. \n",
    "m and b are given by:\n",
    "\n",
    "$$m=\\frac{\\overline{x}\\cdot\\overline{y}-\\overline{xy}}{(\\overline{x})^2-\\overline{x^2}} \\qquad b=\\overline{y}-m\\overline{x}$$\n",
    "\n",
    "Where the overbar denotes the mean i.e $\\overline x = \\frac{\\sum x_i}{n}$. Khan Acadamys linear regression playlist explains the derivation of these functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "m:  0.5714285714285707\n",
      "b:  3.6666666666666696\n",
      "r_squared 0.6122448979591837\n"
     ]
    }
   ],
   "source": [
    "m = (((mean(xs)*mean(ys)) - mean(xs*ys)) / ((mean(xs)**2) - mean(xs**2)))\n",
    "print(\"m: \", m)\n",
    "\n",
    "b = mean(ys) - m * mean(xs)\n",
    "print(\"b: \", b)\n",
    "\n",
    "# we want to see how well we are doing so we calculate R-squared (a statistical measauremnt for regression models)\n",
    "# R-squared ranges from 0 to 1 where the higher the value, the better the model is\n",
    "def squared_error(ys_orig,ys_line):\n",
    "    return sum((ys_line - ys_orig) * (ys_line - ys_orig))\n",
    "\n",
    "regression_line = [m*x + b for x in xs]\n",
    "y_mean_line = [mean(ys) for y in ys]\n",
    "\n",
    "squared_error_regr = squared_error(ys, regression_line)\n",
    "squared_error_y_mean = squared_error(ys, y_mean_line)\n",
    "    \n",
    "r_squared = 1 - (squared_error_regr/squared_error_y_mean)\n",
    "print(\"r_squared\", r_squared)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "8.238095238095235\n"
     ]
    }
   ],
   "source": [
    "# we now want to use our model to predict the next point given an input of 8\n",
    "predict_x = 8\n",
    "predict_y = m*predict_x + b\n",
    "print(predict_y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plotting\n",
    "style.use('ggplot')\n",
    "plt.scatter(xs,ys,color='#003F72')\n",
    "plt.scatter(predict_x, predict_y, color = \"g\")\n",
    "plt.plot(xs, regression_line)\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}