Linear_regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Intro to Linear Regression and Gradient Descent\n",
    "\n",
    "This is the code for [this video](https://www.youtube.com/watch?v=XdM6ER7zTLk&index=2&list=PL2-dafEMk2A7YdKv4XfKpfbTH5z6rEEj3) on Youtube by Siraj Raval. We're using a small dataset of student test scores and the amount of hours they studied. Intuitively, there must be a relationship right? The more you study, the better your test scores should be. We're going to use linear regression to prove this relationship.\n",
    "\n",
    "The target is to predict the students test score, when given the amount they studied 📖🤓📚."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Overview\n",
    "\n",
    "Linear regression is a very simple machine learning (=ML) technique. In this project we want to find a line which best fits our data and the way we are going to optimize it, is **THE** most popular optimization methode for deep learning called \"Gradient Descent\". Gradient Descent is used everywhere in machine learning and deep learning (=DL) and that's the reason why we do this project.\n",
    "\n",
    "(Actually, linear regession is just ML and not DL, because there aren't any neural networks.)\n",
    "\n",
    "<img style=\"max-width: 66%;\" src=\"Linear_regression_files/gradient_descent_example.gif\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dataset Visualisation\n",
    "\n",
    "Our dataset is stored in a `.csv` file, where the first column contains the amount of hours the students studied and the second column contains their test scores:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7b3b46a9b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Load dependencies\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Load dataset\n",
    "points = np.genfromtxt(\"data.csv\", delimiter=\",\") # a magic function to import .csv files\n",
    "\n",
    "# Display the data using matplotlib\n",
    "fig, ax = plt.subplots()  # creates a figure and an axes object\n",
    "ax.plot(points[:,0], points[:,1], 'ro')\n",
    "ax.axis([0, 100, 0, 150]) # set the limits of the axis\n",
    "ax.set(xlabel='Time Studied [h]', ylabel='Test Score [Points]',\n",
    "       title='Relationship between Test Scores and Time Studied')\n",
    "ax.grid() # show the grid\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A word on `numpy`: 👩🏻‍🏫\n",
    "\n",
    "Numpy is the de facto best python library for matrix multiplication and is therefor used in nearly every machine learning project. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear Regression\n",
    "\n",
    "First we have to define our \"Hyperparameters\". They are some kind of tuning knobs for our ML algorithm: \n",
    "- `learning_rate`: defines \"how fast our model learns\". If the learning rate is to low, our model will be to slow to converge. If it's to high our model will never converge. Typical values range over a few orders of magnitude from 0.0001 up to 1.\n",
    "- `initial_b`: We are fitting our dataseet to the `y = mx + b` formula, where `initial_b` is our first guess for b.\n",
    "- `initial_m`: initial guess for m 😉\n",
    "- `num_iterations`: How often to we want to run our training step on the dataset. If it's to low, the model may not converge. If it's to high the training time will take longer than necessary. 😴\n",
    "\n",
    "Next, we simply pass all our variables to a function called `gradient_descent_runner`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def train():\n",
    "    # Define the Hyperparameters\n",
    "    learning_rate = 0.0001\n",
    "    initial_b = 0 # initial y-intercept guess\n",
    "    initial_m = 0 # initial slope guess\n",
    "    num_iterations = 1000\n",
    "    \n",
    "    # Start the gradient descent algorithm\n",
    "    print(\"Starting gradient descent at b = {0}, m = {1}, error = {2}\".format(initial_b, initial_m, compute_error_for_line_given_points(initial_b, initial_m, points)))\n",
    "    print(\"Running...\")\n",
    "    [b, m] = gradient_descent_runner(points, initial_b, initial_m, learning_rate, num_iterations)\n",
    "    print(\"After {0} iterations b = {1}, m = {2}, error = {3}\".format(num_iterations, b, m, compute_error_for_line_given_points(b, m, points)))\n",
    "    \n",
    "    # return the final result to the global scope\n",
    "    return [b, m]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the `gradient_descent_runner` function we are simply performing the gradient step over and over again. 🔄"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gradient_descent_runner(points, starting_b, starting_m, learning_rate, num_iterations):\n",
    "    b = starting_b\n",
    "    m = starting_m\n",
    "    for i in range(num_iterations):\n",
    "        b, m = step_gradient(b, m, np.array(points), learning_rate)\n",
    "        if i % 100 == 0:\n",
    "            # Print the error every 100th iteration\n",
    "            # The \"error\" indicates how good our model fits the data.\n",
    "            print(\"Iteration {0}: b = {1:.2f}, m = {2:.2f}, error = {3:.5f}\".format(i, b, m, compute_error_for_line_given_points(b, m, points)))\n",
    "    return [b, m]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Who do we measure how good our line fits the data?\n",
    "For this we are using the sum of the squared errors as an indicator:\n",
    "\n",
    "<img style=\"max-width: 50%;\" src=\"Linear_regression_files/sum_of_squared_errors.png\">\n",
    "\n",
    "- $N$ ... number of datapoints in our dataset\n",
    "- $x_{i}$ ... amount student i has studied \n",
    "- $y_{i}$ ... test score of student i\n",
    "- $i$ ... index of the selected student\n",
    "- $m$ and $b$ ... optimization parameters \n",
    "\n",
    "This can be plotted for different `m` and `b` values.\n",
    "\n",
    "<img style=\"max-width: 50%;\" src=\"Linear_regression_files/gradient_descent_error_surface.png\">\n",
    "\n",
    "So basically we are trying to find the values for `b` and `m` where the error is the smallest. Calculating the error for **every** single m and b value is usually very computation intensive and therefor we are using this `Gradient Descent` algorithm.\n",
    "\n",
    "Remember, the gradient of a function gives us the direction of the steepest ascent. (check [this](https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/gradient-and-directional-derivatives/v/why-the-gradient-is-the-direction-of-steepest-ascent) out if you wanna dive into the math for this)\n",
    "\n",
    "<img style=\"max-width: 50%;\" src=\"Linear_regression_files/gradient_descent_illustration.png\">\n",
    "\n",
    "So in short: The gradient of our error function tells us how to update our optimization parameters. The gradient can be calculated as follows:\n",
    "\n",
    "$$\\operatorname{grad}(f)=\\begin{pmatrix}\\frac{\\partial f}{\\partial m}\\\\ \\frac{\\partial f}{\\partial b} \\end{pmatrix}$$\n",
    "\n",
    "\n",
    "<img style=\"max-width: 50%;\" src=\"Linear_regression_files/partial_error_derivatives.png\">\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def step_gradient(b_current, m_current, points, learningRate):\n",
    "    b_gradient = 0\n",
    "    m_gradient = 0\n",
    "    N = float(len(points))\n",
    "    for i in range(0, len(points)):\n",
    "        x = points[i, 0]\n",
    "        y = points[i, 1]\n",
    "        m_gradient += -(2/N) * x * (y - ((m_current * x) + b_current)) #see formula above\n",
    "        b_gradient += -(2/N) * (y - ((m_current * x) + b_current)) \n",
    "    new_b = b_current - (learningRate * b_gradient) \n",
    "    new_m = m_current - (learningRate * m_gradient)\n",
    "    # Note: Minus the gradient we wanna go down. (neg. gradient => steepest descent)\n",
    "    return [new_b, new_m]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's add another function, which will calculate the sum of squared errors. Now we can easily print our error and see how the error is becoming smaller and smaller during training."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# y = mx + b\n",
    "# m is slope, b is y-intercept\n",
    "def compute_error_for_line_given_points(b, m, points):\n",
    "    totalError = 0\n",
    "    for i in range(0, len(points)):\n",
    "        x = points[i, 0]\n",
    "        y = points[i, 1]\n",
    "        totalError += (y - (m * x + b)) ** 2\n",
    "    return totalError / float(len(points))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's train our model 🤖🗜"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Starting gradient descent at b = 0, m = 0, error = 5565.107834483211\n",
      "Running...\n",
      "Iteration 0: b = 0.01, m = 0.74, error = 1484.58656\n",
      "Iteration 100: b = 0.04, m = 1.48, error = 112.64702\n",
      "Iteration 200: b = 0.04, m = 1.48, error = 112.64342\n",
      "Iteration 300: b = 0.05, m = 1.48, error = 112.63982\n",
      "Iteration 400: b = 0.05, m = 1.48, error = 112.63622\n",
      "Iteration 500: b = 0.06, m = 1.48, error = 112.63263\n",
      "Iteration 600: b = 0.07, m = 1.48, error = 112.62905\n",
      "Iteration 700: b = 0.07, m = 1.48, error = 112.62547\n",
      "Iteration 800: b = 0.08, m = 1.48, error = 112.62190\n",
      "Iteration 900: b = 0.08, m = 1.48, error = 112.61834\n",
      "After 1000 iterations b = 0.08893651993741346, m = 1.4777440851894448, error = 112.61481011613473\n"
     ]
    }
   ],
   "source": [
    "[b, m] = train()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Testing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's test our model and predict some test scores."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7b3b5c1160>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# sample studied hours\n",
    "h = np.arange(0, 100, 5)\n",
    "\n",
    "# visualize our model\n",
    "fig, ax = plt.subplots()  # creates a figure and an axes object\n",
    "ax.plot(points[:,0], points[:,1], 'ro')\n",
    "ax.plot(h, m*h + b, 'b')\n",
    "ax.axis([0, 100, 0, 150])\n",
    "ax.set(xlabel='Time Studied [h]', ylabel='Test Score [Points]',\n",
    "       title='Relationship between Test Scores and Time Studied')\n",
    "ax.grid()\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": "3.6.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}