Model Function for Linear Regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "4d3d3e47",
   "metadata": {},
   "source": [
    "# Model Function for Linear Regression\n",
    "\n",
    "<figure>\n",
    " <img src=\"./images/C1_W1_L3_S1_Lecture_b.png\"   style=\"width:600px;height:200px;\">\n",
    "</figure>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "92217e3f",
   "metadata": {},
   "source": [
    "## Goals\n",
    "In this notebook I have:\n",
    "- to implement the model $f_{w,b}$ for linear regression with one variable"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dfee26f3",
   "metadata": {},
   "source": [
    "## Notation\n",
    "Here is a summary of some of the notation you will encounter.  \n",
    "\n",
    "|General <img width=70/> <br />  Notation  <img width=70/> | Description<img width=350/>| Python (if applicable) |\n",
    "|: ------------|: ------------------------------------------------------------||\n",
    "| $a$ | scalar, non bold                                                      ||\n",
    "| $\\mathbf{a}$ | vector, bold                                                      ||\n",
    "| **Regression** |         |    |     |\n",
    "|  $\\mathbf{x}$ | Training Example feature values (in this lab - Size (1000 sqft))  | `x_train` |   \n",
    "|  $\\mathbf{y}$  | Training Example  targets (in this lab Price (1000s of dollars))  | `y_train` \n",
    "|  $x^{(i)}$, $y^{(i)}$ | $i_{th}$Training Example | `x_i`, `y_i`|\n",
    "| m | Number of training examples | `m`|\n",
    "|  $w$  |  parameter: weight                                 | `w`    |\n",
    "|  $b$           |  parameter: bias                                           | `b`    |     \n",
    "| $f_{w,b}(x^{(i)})$ | The result of the model evaluation at $x^{(i)}$ parameterized by $w,b$: $f_{w,b}(x^{(i)}) = wx^{(i)}+b$  | `f_wb` | \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3267063d",
   "metadata": {},
   "source": [
    "## Tools\n",
    "In this notebook I will make use of: \n",
    "- NumPy, a popular library for scientific computing\n",
    "- Matplotlib, a popular library for plotting data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "3a86561e",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a30801dc",
   "metadata": {},
   "source": [
    "# Problem Statement: \n",
    "(2 Training examples given which is the real data we have to use to make future predictions)\n",
    "<img align=\"left\" src=\"./images/C1_W1_L3_S1_trainingdata.png\"    style=\" width:380px; padding: 10px;  \" /> \n",
    "\n",
    "As in the lecture, I will use the motivating example of housing price prediction.  \n",
    "This notebook will use a simple data set with only two data points - a house with 1000 square feet(sqft) sold for \\\\$300,000 and a house with 2000 square feet sold for \\\\$500,000. These two points will constitute our *data or training set*. In this notebook, the units of size are 1000 sqft and the units of price are 1000s of dollars.\n",
    "\n",
    "| Size (1000 sqft)     | Price (1000s of dollars) |\n",
    "| -------------------| ------------------------ |\n",
    "| 1.0               | 300                      |\n",
    "| 2.0               | 500                      |\n",
    "\n",
    "You would like to fit a linear regression model (shown above as the blue straight line) through these two points, so you can then predict price for other houses - say, a house with 1200 sqft.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "03d25af9-31ee-43a5-b3d0-49c6e29ab383",
   "metadata": {},
   "source": [
    "(Note that the feature that we know is always the 'x' and the thing that we have to predict is 'y' which is the price)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c14815ba",
   "metadata": {},
   "source": [
    "Please run the following code cell to create your `x_train` and `y_train` variables. The data is stored in one-dimensional NumPy arrays."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "e979d5a4",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_train = [1. 2.]\n",
      "y_train = [300. 500.]\n"
     ]
    }
   ],
   "source": [
    "# x_train is the input variable (size in 1000 square feet)\n",
    "# y_train is the target (price in 1000s of dollars)\n",
    "x_train = np.array([1.0, 2.0])\n",
    "y_train = np.array([300.0, 500.0])\n",
    "print(f\"x_train = {x_train}\")\n",
    "print(f\"y_train = {y_train}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "55ad3968",
   "metadata": {},
   "source": [
    ">**Note**: The course will frequently utilize the python 'f-string' output formatting described [here](https://docs.python.org/3/tutorial/inputoutput.html) when printing. The content between the curly braces is evaluated when producing the output."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d58a79a",
   "metadata": {},
   "source": [
    "### Number of training examples `m`\n",
    "You will use `m` to denote the number of training examples. Numpy arrays have a `.shape` parameter. `x_train.shape` returns a python tuple with an entry for each dimension. `x_train.shape[0]` is the length of the array and number of examples as shown below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b43ec3bc",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_train.shape: (2,)\n",
      "Number of training examples is: 2\n"
     ]
    }
   ],
   "source": [
    "# m is the number of training examples\n",
    "print(f\"x_train.shape: {x_train.shape}\")\n",
    "m = x_train.shape[0]\n",
    "print(f\"Number of training examples is: {m}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6d4c404e",
   "metadata": {},
   "source": [
    "One can also use the Python `len()` function as shown below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "2cb0fa9a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of training examples is: 2\n"
     ]
    }
   ],
   "source": [
    "# m is the number of training examples\n",
    "m = len(x_train)\n",
    "print(f\"Number of training examples is: {m}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "06bf9969",
   "metadata": {},
   "source": [
    "### Training example `x_i, y_i`\n",
    "\n",
    "You will use (x$^{(i)}$, y$^{(i)}$) to denote the $i^{th}$ training example. Since Python is zero indexed, (x$^{(0)}$, y$^{(0)}$) is (1.0, 300.0) and (x$^{(1)}$, y$^{(1)}$) is (2.0, 500.0). \n",
    "\n",
    "To access a value in a Numpy array, one indexes the array with the desired offset. For example the syntax to access location zero of `x_train` is `x_train[0]`.\n",
    "Run the next code block below to get the $i^{th}$ training example."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "64e36949",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(x^(0), y^(0)) = (1.0, 300.0)\n"
     ]
    }
   ],
   "source": [
    "i = 0 # Change this to 1 to see (x^1, y^1)\n",
    "\n",
    "x_i = x_train[i]\n",
    "y_i = y_train[i]\n",
    "print(f\"(x^({i}), y^({i})) = ({x_i}, {y_i})\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "536a8ae8",
   "metadata": {},
   "source": [
    "### Plotting the data"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "847065f8",
   "metadata": {},
   "source": [
    "You can plot these two points using the `scatter()` function in the `matplotlib` library, as shown in the cell below. \n",
    "- The function arguments `marker` and `c` show the points as red crosses (the default is blue dots).\n",
    "\n",
    "You can use other functions in the `matplotlib` library to set the title and labels to display"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "4fd0995f",
   "metadata": {},
   "outputs": [
    {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the data points\n",
    "plt.scatter(x_train, y_train, marker='x', c='r')\n",
    "# Set the title\n",
    "plt.title(\"Housing Prices\")\n",
    "# Set the y-axis label\n",
    "plt.ylabel('Price (in 1000s of dollars)')\n",
    "# Set the x-axis label\n",
    "plt.xlabel('Size (1000 sqft)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb6fdfb5",
   "metadata": {},
   "source": [
    "## Model function\n",
    "\n",
    "<img align=\"left\" src=\"./images/C1_W1_L3_S1_model.png\"     style=\" width:380px; padding: 10px; \" > As described in lecture, the model function for linear regression (which is a function that maps from `x` to `y`) is represented as \n",
    "\n",
    "$$ f_{w,b}(x^{(i)}) = wx^{(i)} + b \\tag{1}$$\n",
    "\n",
    "The formula above is how you can represent straight lines - different values of $w$ and $b$ give you different straight lines on the plot. <br/> <br/> <br/> <br/> <br/> \n",
    "\n",
    "Let's try to get a better intuition for this through the code blocks below. Let's start with $w = 100$ and $b = 100$. \n",
    "\n",
    "**Note: You can come back to this cell to adjust the model's w and b parameters**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "64e8d41d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w: 100\n",
      "b: 100\n"
     ]
    }
   ],
   "source": [
    "w = 100\n",
    "b = 100\n",
    "print(f\"w: {w}\")\n",
    "print(f\"b: {b}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "66565e5d",
   "metadata": {},
   "source": [
    "Now, let's compute the value of $f_{w,b}(x^{(i)})$ for your two data points. You can explicitly write this out for each data point as - \n",
    "\n",
    "for $x^{(0)}$, `f_wb = w * x[0] + b`\n",
    "\n",
    "for $x^{(1)}$, `f_wb = w * x[1] + b`\n",
    "\n",
    "For a large number of data points, this can get unwieldy and repetitive. So instead, you can calculate the function output in a `for` loop as shown in the `compute_model_output` function below.\n",
    "> **Note**: The argument description `(ndarray (m,))` describes a Numpy n-dimensional array of shape (m,). `(scalar)` describes an argument without dimensions, just a magnitude.  \n",
    "> **Note**: `np.zero(n)` will return a one-dimensional numpy array with $n$ entries   \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "8efc5cdc",
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_model_output(x, w, b):\n",
    "    \"\"\"\n",
    "    Computes the prediction of a linear model\n",
    "    Args: (takes in values of x, w and b)\n",
    "      x (ndarray (m,)): Data, m examples               (x is ofcourse would be an array of training examples, .shape[0] is the length of the array and number of examples )\n",
    "      w,b (scalar)    : model parameters                     \n",
    "    Returns\n",
    "      f_wb (ndarray (m,)): model prediction            (initialized as an empty array with no. of places as m)\n",
    "    \"\"\"\n",
    "    m = x.shape[0]\n",
    "    f_wb = np.zeros(m)\n",
    "    for i in range(m):\n",
    "        f_wb[i] = w * x[i] + b\n",
    "        \n",
    "    return f_wb"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eb90978b",
   "metadata": {},
   "source": [
    "Now let's call the `compute_model_output` function and plot the output.."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "172b3f78",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tmp_f_wb = compute_model_output(x_train, w, b,)\n",
    "\n",
    "# Plot our model prediction\n",
    "plt.plot(x_train, tmp_f_wb, c='b',label='Our Prediction')\n",
    "\n",
    "# Plot the data points\n",
    "plt.scatter(x_train, y_train, marker='x', c='r',label='Actual Values')\n",
    "\n",
    "# Set the title\n",
    "plt.title(\"Housing Prices\")\n",
    "# Set the y-axis label\n",
    "plt.ylabel('Price (in 1000s of dollars)')\n",
    "# Set the x-axis label\n",
    "plt.xlabel('Size (1000 sqft)')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca0bd03c",
   "metadata": {},
   "source": [
    "As you can see, setting $w = 100$ and $b = 100$ does *not* result in a line that fits our data. \n",
    "\n",
    "### Challenge\n",
    "Try experimenting with different values of $w$ and $b$. What should the values be for a line that fits our data?\n",
    "\n",
    "#### Tip:\n",
    "You can use your mouse to click on the green \"Hints\" below to reveal some hints for choosing b and w."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae8efe9d",
   "metadata": {},
   "source": [
    "<details>\n",
    "<summary>\n",
    "    <font size='3', color='darkgreen'><b>Hints</b></font>\n",
    "</summary>\n",
    "    <p>\n",
    "    <ul>\n",
    "        <li>Try $w = 200$ and $b = 100$ </li>\n",
    "    </ul>\n",
    "    </p>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc6d24b5",
   "metadata": {},
   "source": [
    "### Prediction\n",
    "Now that we have a model, we can use it to make our original prediction. Let's predict the price of a house with 1200 sqft. Since the units of $x$ are in 1000's of sqft, $x$ is 1.2.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "89876e4f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "$340 thousand dollars\n"
     ]
    }
   ],
   "source": [
    "w = 200                         \n",
    "b = 100    \n",
    "x_i = 1.2\n",
    "cost_1200sqft = w * x_i + b    \n",
    "\n",
    "print(f\"${cost_1200sqft:.0f} thousand dollars\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3d4976d9",
   "metadata": {},
   "source": [
    "# Lesson Learnt\n",
    "In this notebook I have learned:\n",
    " - Linear regression builds a model which establishes a relationship between features and targets\n",
    "     - In the example above, the feature was house size and the target was house price\n",
    "     - for simple linear regression, the model has two parameters $w$ and $b$ whose values are 'fit' using *training data*.\n",
    "     - once a model's parameters have been determined, the model can be used to make predictions on novel data."
   ]
  }
 ],
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