python-exercises/sympy.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "from __future__ import division, print_function\n",
    "from sympy import Symbol, diff, solve, lambdify, simplify\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Model parameters: We look for a line y = b1*x + b2.\n",
    "b1 = Symbol('b1')\n",
    "b2 = Symbol('b2')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Data points\n",
    "data = [(1,14), (2, 13), (3, 12), (4, 10), (5,9), (7,8), (9,5)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Function to minimize: S = 185*b1**2 + 62*b1*b2 - 524*b1 + 7*b2**2 - 142*b2 + 779\n"
     ]
    }
   ],
   "source": [
    "# S is the function to minimize:\n",
    "#\n",
    "# For each data point the vertical error/residual is x*b1 + b2 - y. We want to\n",
    "# minimize the sum of the squared residuals (least squares).\n",
    "S = sum((p[0] * b1 + b2 - p[1]) ** 2 for p in data)\n",
    "S = simplify(S)\n",
    "print(\"Function to minimize: S = {}\".format(S))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "S is minimal for b1 = -367/334, b2 = 5013/334\n"
     ]
    }
   ],
   "source": [
    "# Minimize S by setting its partial derivatives to zero.\n",
    "d1 = diff(S, b1)\n",
    "d2 = diff(S, b2)\n",
    "solutions = solve([d1, d2], [b1, b2])\n",
    "print(\"S is minimal for b1 = {}, b2 = {}\".format(solutions[b1], solutions[b2]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Fitted line: y = -367*x/334 + 5013/334\n"
     ]
    }
   ],
   "source": [
    "# Construct fitted line from the solutions\n",
    "x = Symbol('x')\n",
    "fitted_line = solutions[b1] * x + solutions[b2]\n",
    "print(\"Fitted line: y = {}\".format(fitted_line))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Construct something we can plot with matplotlib\n",
    "fitted_line_func = lambdify(x, fitted_line, modules=['numpy'])\n",
    "x_plot = np.linspace(min(p[0] for p in data),\n",
    "                     max(p[0] for p in data), 100)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
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96ipf6D17wrBh0Lw5zJ8fOplIsanQRQ5WowY89RS8+qp/COnss6F3b9i+PXQykSKp0EUO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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fdb7f6c8b70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot data points and fitted line\n",
    "plt.scatter([p[0] for p in data], [p[1] for p in data], marker=\"+\")\n",
    "plt.plot(x_plot, fitted_line_func(x_plot), 'r')\n",
    "plt.show()"
   ]
  },
  {
   "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",
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