1_logistic_regression.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "1.logistic_regression_final.ipynb",
      "version": "0.3.2",
      "provenance": [],
      "collapsed_sections": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "accelerator": "GPU"
  },
  "cells": [
    {
      "cell_type": "code",
      "metadata": {
        "id": "GMkVuv4djG4x",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "import numpy as np\n",
        "import pandas as pd\n",
        "import seaborn as sns\n",
        "from pylab import rcParams\n",
        "import matplotlib.pyplot as plt\n",
        "from collections import OrderedDict\n",
        "from scipy.special import expit\n",
        "import unittest\n",
        "\n",
        "%matplotlib inline\n",
        "\n",
        "sns.set(style='whitegrid', palette='muted', font_scale=1.5)\n",
        "\n",
        "rcParams['figure.figsize'] = 14, 8\n",
        "\n",
        "RANDOM_SEED = 42\n",
        "\n",
        "np.random.seed(RANDOM_SEED)\n",
        "\n",
        "def run_tests():\n",
        "  unittest.main(argv=[''], verbosity=1, exit=False)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "V-I1L92c1Dno",
        "colab_type": "text"
      },
      "source": [
        "# Your data"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "0WFneRgf2NGQ",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "data = OrderedDict(\n",
        "        amount_spent =  [50,  10, 20, 5,  95,  70,  100,  200, 0],\n",
        "        send_discount = [0,   1,  1,  1,  0,   0,   0,    0,   1]\n",
        "    )"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Qg_P_LCv6kFZ",
        "colab_type": "code",
        "outputId": "541955cd-a5c6-4b1f-f974-679e7b39e9d9",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 328
        }
      },
      "source": [
        "df = pd.DataFrame.from_dict(data)\n",
        "df"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/html": [
              "<div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>amount_spent</th>\n",
              "      <th>send_discount</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>50</td>\n",
              "      <td>0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>10</td>\n",
              "      <td>1</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>20</td>\n",
              "      <td>1</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>5</td>\n",
              "      <td>1</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>95</td>\n",
              "      <td>0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>70</td>\n",
              "      <td>0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>100</td>\n",
              "      <td>0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>200</td>\n",
              "      <td>0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>0</td>\n",
              "      <td>1</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   amount_spent  send_discount\n",
              "0            50              0\n",
              "1            10              1\n",
              "2            20              1\n",
              "3             5              1\n",
              "4            95              0\n",
              "5            70              0\n",
              "6           100              0\n",
              "7           200              0\n",
              "8             0              1"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 3
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "qOw7gRk48G2A",
        "colab_type": "code",
        "outputId": "b0954f17-2b85-4cfb-e17a-59999a2427d8",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 506
        }
      },
      "source": [
        "df.plot.scatter(x='amount_spent', y='send_discount', s=108, c=\"blue\");"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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s/Pz8VKdOHQ0YMECnTp3S7t27nV3eNeFf//qX7rrrLt1///1yc3NT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\nAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 1008x576 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OAly_VfNEtiQ",
        "colab_type": "text"
      },
      "source": [
        "# Making decisions with Logistic regression\n",
        "\n",
        "Logistic regression is used for classification problems when the dependant/target variable is binary. That is, its values are true or false. Logistic regression is one of the most popular and widely used algorithms in practice ([see this](https://www.kaggle.com/surveys/2017)).\n",
        "\n",
        "Some examples for problems that can be solved with Logistic regression are:\n",
        "\n",
        "- Email - deciding if it is spam or not\n",
        "- Online transactions - fraudelent or not\n",
        "- Tumor - malignant or bening\n",
        "- Customer upgrade - will the customer buy the premium upgrade or not\n",
        "\n",
        "We want to predict the outcome of a variable $y$, such that:\n",
        "\n",
        "$$y \\in \\{0, 1\\}$$\n",
        "\n",
        "and set $0$: negative class (e.g. email is not spam) or $1$: positive class (e.g. email is spam).\n",
        "\n",
        "## Can't we just use Linear regression?\n",
        "\n",
        "![](https://www.machinelearningplus.com/wp-content/uploads/2017/09/linear_vs_logistic_regression.jpg)\n",
        "\n",
        "The response target variable $y$ of the Linear regression model is not restricted within the $[0, 1]$ interval."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xCclnDlvYxbY",
        "colab_type": "text"
      },
      "source": [
        "# Logistic regression model\n",
        "\n",
        "Given our problem, we want a model that uses 1 variable (predictor) ($x_1$ - amount_spent) to predict whether or not we should send a discount to the customer.\n",
        "\n",
        "$$h_w(x) = w_1x_1 + w_0$$\n",
        "\n",
        "where the coefficients $w_i$ are paramaters of the model. Let the coeffiecient vector $W$ be:\n",
        "\n",
        "$$\n",
        "   W =\n",
        "    \\begin{pmatrix}\n",
        "    w_1 \\\\\n",
        "    w_0 \\\\\n",
        "    \\end{pmatrix}\n",
        "$$\n",
        "\n",
        "Then we can represent $h_w(x)$ in more compact form:\n",
        "\n",
        "$$h_w(x) = w^Tx$$\n",
        "\n",
        "That is the *Linear regression model*.\n",
        "\n",
        "We want to build a model that outputs values that are between $0$ and $1$, so we want to come up with a hypothesis that satisfies $0 \\leq h_w(x) \\leq 1$. For *Logistic regression* we want to modify this and introduce another function $g$:\n",
        "\n",
        "$$h_w(x) = g(w^Tx)$$\n",
        "\n",
        "We're going to define $g$ as:\n",
        "\n",
        "$$g(z) = \\frac{1}{1 + e ^{-z}}$$\n",
        "\n",
        "where $z \\in \\mathbb{R}$. $g$ is also known as the **sigmoid function** or the **logistic function**. So, after substition, we end up with this definition:\n",
        "\n",
        "$$h_w(x) = \\frac{1}{1 + e ^{-(w^Tx)}}$$\n",
        "\n",
        "for our hypothesis.\n",
        "\n",
        "# A closer look at the sigmoid function\n",
        "\n",
        "Recall that the sigmoid function is defined as:\n",
        "\n",
        "$$g(z) = \\frac{1}{1 + e ^{-z}}$$\n",
        "\n",
        "where $z \\in \\mathbb{R}$. Let's translate that to a Python function:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "nIU1rj4OjJf_",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def sigmoid(z):\n",
        "#   return 1 / (1 + np.exp(-z))\n",
        "  return expit(z)\n"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "RW5HT9yhlph2",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "class TestSigmoid(unittest.TestCase):\n",
        "\n",
        "    def test_at_zero(self):\n",
        "      self.assertAlmostEqual(sigmoid(0), 0.5)\n",
        "        \n",
        "    def test_at_negative(self):\n",
        "      self.assertAlmostEqual(sigmoid(-100), 0)\n",
        "        \n",
        "    def test_at_positive(self):\n",
        "      self.assertAlmostEqual(sigmoid(100), 1)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Esx-TA91nmVN",
        "colab_type": "code",
        "outputId": "69c570ba-0053-417d-f747-18c6fae4fe3f",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 102
        }
      },
      "source": [
        "run_tests()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "...\n",
            "----------------------------------------------------------------------\n",
            "Ran 3 tests in 0.006s\n",
            "\n",
            "OK\n"
          ],
          "name": "stderr"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "MVgCLBecoQIn",
        "colab_type": "code",
        "outputId": "60b66394-95ab-47bf-f9f6-a322721f8784",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 506
        }
      },
      "source": [
        "x = np.linspace(-10., 10., num=100)\n",
        "sig = sigmoid(x)\n",
        "\n",
        "plt.plot(x, sig, label=\"sigmoid\")\n",
        "plt.xlabel(\"x\")\n",
        "plt.ylabel(\"y\")\n",
        "plt.legend(prop={'size' : 16})\n",
        "plt.show()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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ariZZLBaz4wAATgNFCQCATtTq8uj9dRWKjrBq/HmJZscBAJwmihIAAJ1o7bYa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\nDhMlIiIiIiIiO/8fljZALwYtWwsAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 1008x576 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1wNwylPKQ-SQ",
        "colab_type": "text"
      },
      "source": [
        "# How can we find the parameters for our model?\n",
        "\n",
        "Let's examine some approaches to find good parameters for our model. But what does good mean in this context?\n",
        "\n",
        "## Loss function\n",
        "\n",
        "We have a model that we can use to make decisions, but we still have to find the parameters $W$. To do that, we need an objective measurement of how good some set of parameters are. For that purpose, we will use a loss (cost) function:\n",
        "\n",
        "$$J(W) = \\frac{1}{m}\\sum^m_{i = 1}Cost(h_w(x^{(i)}), y^{(i)})$$\n",
        "\n",
        "$$\n",
        "    Cost(h_w(x), y) = \n",
        "    \\begin{cases} \n",
        "    -log(h_w(x)) &\\text{if} y = 1\\\\\n",
        "    -log(1 - h_w(x)) &\\text{if} y = 0\n",
        "    \\end{cases}\n",
        "$$\n",
        "\n",
        "Which is also known as the [*Log loss* or *Cross-entropy loss*](https://ml-cheatsheet.readthedocs.io/en/latest/loss_functions.html) function\n",
        "\n",
        "![](https://ml-cheatsheet.readthedocs.io/en/latest/_images/y1andy2_logistic_function.png)\n",
        "\n",
        "We can compress the above function into one:\n",
        "\n",
        "$$J(W) = \\frac{1}{m}(-y \\log{(h_w)} - (1 - y) \\log{(1 - h_w)})$$\n",
        "\n",
        "where\n",
        "\n",
        "$$h_w(x) = g(w^Tx)$$\n",
        "\n",
        "Let's implement it in Python:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "oIeiJSugk4qB",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def loss(h, y):\n",
        "  return (-y * np.log(h) - (1 - y) * np.log(1 - h)).mean()"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Vyzx34TDoBoB",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "class TestLoss(unittest.TestCase):\n",
        "\n",
        "  def test_zero_h_zero_y(self):\n",
        "    self.assertLess(loss(h=0.000001, y=.000001), 0.0001)\n",
        "\n",
        "  def test_one_h_zero_y(self):\n",
        "    self.assertGreater(loss(h=0.9999, y=.000001), 9.0)\n",
        "\n",
        "  def test_zero_h_one_y(self):\n",
        "    self.assertGreater(loss(h=0.000001, y=0.9999), 9.0)\n",
        "\n",
        "  def test_one_h_one_y(self):\n",
        "    self.assertLess(loss(h=0.999999, y=0.999999), 0.0001)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "wd6-FSTPoIsV",
        "colab_type": "code",
        "outputId": "1e84d7ce-fdb3-49fe-bfdd-3a11cb4216a1",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 102
        }
      },
      "source": [
        "run_tests()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            ".......\n",
            "----------------------------------------------------------------------\n",
            "Ran 7 tests in 0.010s\n",
            "\n",
            "OK\n"
          ],
          "name": "stderr"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "AHL3KqF7Rfvf",
        "colab_type": "text"
      },
      "source": [
        "## Approach \\#1 - I'm thinking of a number(s)\n",
        "\n",
        "Let's think of 3 numbers that represent the coefficients $w_0, w_1, w_2$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "sWKZ-4x8SP_b",
        "colab_type": "code",
        "outputId": "af2000b9-d95a-42b7-a2fd-748cdb580f91",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        }
      },
      "source": [
        "X = df['amount_spent'].astype('float').values\n",
        "y = df['send_discount'].astype('float').values\n",
        "\n",
        "def predict(x, w):\n",
        "  return sigmoid(x * w)\n",
        "\n",
        "def print_result(y_hat, y):\n",
        "  print(f'loss: {np.round(loss(y_hat, y), 5)} predicted: {y_hat} actual: {y}')\n",
        "  \n",
        "y_hat = predict(x=X[0], w=.5)\n",
        "print_result(y_hat, y[0])"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "loss: 25.0 predicted: 0.999999999986112 actual: 0.0\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OI1PYdiNTqhu",
        "colab_type": "text"
      },
      "source": [
        "I am pretty lazy this approach seems like too much hard work for me.\n",
        "\n",
        "## Approach \\#2 - Try out many numbers\n",
        "\n",
        "Alright, these days computers are pretty fast, 6+ core laptops are everywhere also your phones can are pretty performant, too! Let's use that power for good™ and try to find these parameters by just trying out a lot of numbers:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "9kTZ4F2qhyHC",
        "colab_type": "code",
        "outputId": "bb11ea57-5ec2-4a9f-ec35-e3825d681762",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 391
        }
      },
      "source": [
        "for w in np.arange(-1, 1, 0.1):\n",
        "  y_hat = predict(x=X[0], w=w)\n",
        "  print(loss(y_hat, y[0]))"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "0.0\n",
            "0.0\n",
            "0.0\n",
            "6.661338147750941e-16\n",
            "9.359180097590508e-14\n",
            "1.3887890837434982e-11\n",
            "2.0611535832696244e-09\n",
            "3.059022736706331e-07\n",
            "4.539889921682063e-05\n",
            "0.006715348489118056\n",
            "0.6931471805599397\n",
            "5.006715348489103\n",
            "10.000045398900186\n",
            "15.000000305680194\n",
            "19.999999966169824\n",
            "24.99999582410784\n",
            "30.001020555434774\n",
            "34.945041100449046\n",
            "inf\n",
            "inf\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:2: RuntimeWarning: divide by zero encountered in log\n",
            "  \n"
          ],
          "name": "stderr"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xvnblS0UzG5E",
        "colab_type": "text"
      },
      "source": [
        "## Approach \\#3 - Gradient descent\n",
        "\n",
        "Gradient descent algorithms (yes, there are a lot of them) provide us with a way to find a minimum of some function $f$. They work by iteratively going in the direction of the descent as defined by the gradient.\n",
        "\n",
        "In Machine Learning, we use gradient descent algorithms to find \"good\" parameters for our models (Logistic Regression, Linear Regression, Neural Networks, etc...).\n",
        "\n",
        "![](https://i.ytimg.com/vi/b4Vyma9wPHo/maxresdefault.jpg)\n",
        "\n",
        "**Somewhat deeper look into how Gradient descent works (Source: PyTorchZeroToAll)**\n",
        "\n",
        "Starting somewhere, we take our first step downhill in the direction specified by the negative gradient. Next, we recalculate the negative gradient and take another step in the direction it specifies. This process continues until we get to a point where we can no longer move downhill - a local minimum.\n",
        "\n",
        "### First derivative of the sigmoid function\n",
        "\n",
        "The first derivative of the sigmoid function is given by the following equation:\n",
        "\n",
        "$$g'(z) = g(z)(1 - g(z))$$\n",
        "\n",
        "Complete derivation can be found [here](https://math.stackexchange.com/a/1225116/499458)."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "O54XG0ApLOUF",
        "colab_type": "text"
      },
      "source": [
        "### First derivative of the cost function\n",
        "\n",
        "Recall that the cost function was given by the following equation:\n",
        "\n",
        "$$J(W) = \\frac{1}{m}(-y \\log{(h_w)} - (1 - y) \\log{(1 - h_w)})$$\n",
        "\n",
        "Given $g'(z) = g(z)(1 - g(z))$\n",
        "\n",
        "Then:\n",
        "\n",
        "$$\\frac{\\partial{J(W)}}{\\partial{W}} =\\frac{1}{m}(y(1 - h_w) - (1 - y)h_w)x = \\frac{1}{m}(y - h_w)x$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "VOIkshOtOr4c",
        "colab_type": "text"
      },
      "source": [
        "### Updating our parameters $W$\n",
        "\n",
        "The parameter updating rule we're going to use is defined by:\n",
        "\n",
        "$$W := W - \\alpha (\\frac{1}{m}(y - h_w)x)$$\n",
        "\n",
        "The parameter $\\alpha$ is known as **learning rate**. High learning rate can converge quickly, but risks overshooting the lowest point. Low learning rate allows for confident moves in the direction of the negative gradient. However, it time-consuming so it will take us a lot of time to get to converge.\n",
        "\n",
        "![](https://cdn-images-1.medium.com/max/1600/0*QwE8M4MupSdqA3M4.png)\n",
        "\n",
        "**Big vs Small learning rate (Source: TowardsDataScoemce)**\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xYcz3d_kTVkP",
        "colab_type": "text"
      },
      "source": [
        "### The gradient descent algorithm\n",
        "\n",
        "```\n",
        "Repeat until convergence {\n",
        "  1. Calculate gradient average\n",
        "  2. Multiply by learning rate\n",
        "  3. Subtract from weights\n",
        "}\n",
        "```"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "TLDOcJfGeV95",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def predict(X, W):\n",
        "  return sigmoid(np.dot(X, W))\n",
        "      \n",
        "def fit(X, y, n_iter=100000, lr=0.01):\n",
        "\n",
        "  W = np.zeros(X.shape[1])\n",
        "\n",
        "  for i in range(n_iter):\n",
        "      z = np.dot(X, W)\n",
        "      h = sigmoid(z)\n",
        "      gradient = np.dot(X.T, (h - y)) / y.size\n",
        "      W -= lr * gradient\n",
        "  return W"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "2tY5RhfnnO9P",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "class TestGradientDescent(unittest.TestCase):\n",
        "\n",
        "    def test_correct_prediction(self):\n",
        "      global X\n",
        "      global y\n",
        "      if len(X.shape) != 2:        \n",
        "        X = X.reshape(X.shape[0], 1)\n",
        "      w, _ = fit(X, y)\n",
        "      y_hat = predict(X, w).round()\n",
        "      self.assertTrue((y_hat == y).all())"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "BV7mMdg5nc0C",
        "colab_type": "code",
        "outputId": "d3eece19-0f4e-464c-b46f-7444bbef475e",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 238
        }
      },
      "source": [
        "run_tests()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "E.......\n",
            "======================================================================\n",
            "ERROR: test_correct_prediction (__main__.TestGradientDescent)\n",
            "----------------------------------------------------------------------\n",
            "Traceback (most recent call last):\n",
            "  File \"<ipython-input-15-59757b6f0a6f>\", line 8, in test_correct_prediction\n",
            "    w, _ = fit(X, y)\n",
            "ValueError: not enough values to unpack (expected 2, got 1)\n",
            "\n",
            "----------------------------------------------------------------------\n",
            "Ran 8 tests in 0.888s\n",
            "\n",
            "FAILED (errors=1)\n"
          ],
          "name": "stderr"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xUWBzDZkpdOZ",
        "colab_type": "text"
      },
      "source": [
        "Well, that's not good, after all that hustling we're nowhere near achieving our goal of finding good parameters for our model. But, what went wrong? Let's start by finding whether our algorithm improves over time. We can use our loss metric for that:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "pLe3grGwqNYx",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def fit(X, y, n_iter=100000, lr=0.01):\n",
        "\n",
        "  W = np.zeros(X.shape[1])\n",
        "  errors = []\n",
        "  for i in range(n_iter):\n",
        "      z = np.dot(X, W)\n",
        "      h = sigmoid(z)\n",
        "      gradient = np.dot(X.T, (h - y)) / y.size\n",
        "      W -= lr * gradient\n",
        "      \n",
        "      if(i % 10000 == 0):       \n",
        "        e = loss(h, y)        \n",
        "        print(f'loss: {e} \\t')\n",
        "        errors.append(e)\n",
        "      \n",
        "  return W, errors"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "MtUNQ0Krqj_q",
        "colab_type": "code",
        "outputId": "94d53d4d-9088-4f0f-ccc5-7410e8536a2c",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 408
        }
      },
      "source": [
        "run_tests()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "loss: 0.6931471805599453 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "stream",
          "text": [
            "F......."
          ],
          "name": "stderr"
        },
        {
          "output_type": "stream",
          "text": [
            "loss: 0.41899283818630056 \t\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "stream",
          "text": [
            "\n",
            "======================================================================\n",
            "FAIL: test_correct_prediction (__main__.TestGradientDescent)\n",
            "----------------------------------------------------------------------\n",
            "Traceback (most recent call last):\n",
            "  File \"<ipython-input-15-59757b6f0a6f>\", line 10, in test_correct_prediction\n",
            "    self.assertTrue((y_hat == y).all())\n",
            "AssertionError: False is not true\n",
            "\n",
            "----------------------------------------------------------------------\n",
            "Ran 8 tests in 0.938s\n",
            "\n",
            "FAILED (failures=1)\n"
          ],
          "name": "stderr"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "gi3ZCFD1znth",
        "colab_type": "code",
        "outputId": "044bc2bb-5969-43cf-a6d1-75e71b4f4f0b",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 681
        }
      },
      "source": [
        "_, errors = fit(X, y)\n",
        "plt.plot(np.arange(len(errors)), errors)\n",
        "plt.xlabel(\"iteration^10000\")\n",
        "plt.ylabel(\"error\")\n",
        "plt.ylim(0, 1)\n",
        "plt.show()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "loss: 0.6931471805599453 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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++x96v8mTJ/9zg9aR+jYPDYPrjlJzzVEOrjvKwXW3das3v0epoqJig7X3jtxd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\n7ueBAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 1008x576 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "l4ej4TD5swd0",
        "colab_type": "text"
      },
      "source": [
        "Good, we found a possible cause for our problem. Our loss doesn't get low enough, in other words, our algorithm gets stuck at some point that is not a good enough minimum for us. How can we fix this? Perhaps, try out different learning rate or initializing our parameter with a different value?"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ODP9jfo7s1i_",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def fit(X, y, n_iter=100000, lr=0.001):\n",
        "\n",
        "  W = np.zeros(X.shape[1])\n",
        "  \n",
        "  errors = []\n",
        "\n",
        "  for i in range(n_iter):\n",
        "      z = np.dot(X, W)\n",
        "      h = sigmoid(z)\n",
        "      gradient = np.dot(X.T, (h - y)) / y.size\n",
        "      W -= lr * gradient\n",
        "           \n",
        "      if(i % 10000 == 0):\n",
        "        e = loss(h, y)\n",
        "        print(f'loss: {e} \\t')\n",
        "        errors.append(e)\n",
        "      \n",
        "  return W, errors"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "X0lW0sYLs3ua",
        "colab_type": "code",
        "outputId": "c87799e9-9ca9-4cfb-9874-817a0e82f1f1",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 408
        }
      },
      "source": [
        "run_tests()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "loss: 0.6931471805599453 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n",
            "loss: 0.41899283818630056 \t\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "stream",
          "text": [
            "F......."
          ],
          "name": "stderr"
        },
        {
          "output_type": "stream",
          "text": [
            "loss: 0.41899283818630056 \t\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "stream",
          "text": [
            "\n",
            "======================================================================\n",
            "FAIL: test_correct_prediction (__main__.TestGradientDescent)\n",
            "----------------------------------------------------------------------\n",
            "Traceback (most recent call last):\n",
            "  File \"<ipython-input-15-59757b6f0a6f>\", line 10, in test_correct_prediction\n",
            "    self.assertTrue((y_hat == y).all())\n",
            "AssertionError: False is not true\n",
            "\n",
            "----------------------------------------------------------------------\n",
            "Ran 8 tests in 0.903s\n",
            "\n",
            "FAILED (failures=1)\n"
          ],
          "name": "stderr"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yAeOAz2VtGM4",
        "colab_type": "text"
      },
      "source": [
        "Hmm, how about adding one more parameter for our model to find/learn?"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "vS0aN8ZHfitp",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def add_intercept(X):\n",
        "  intercept = np.ones((X.shape[0], 1))\n",
        "  return np.concatenate((intercept, X), axis=1)\n",
        "\n",
        "def predict(X, W):\n",
        "  X = add_intercept(X)\n",
        "  return sigmoid(np.dot(X, W))\n",
        "      \n",
        "def fit(X, y, n_iter=100000, lr=0.01):\n",
        "  \n",
        "  X = add_intercept(X)\n",
        "  W = np.zeros(X.shape[1])\n",
        "  \n",
        "  errors = []\n",
        "\n",
        "  for i in range(n_iter):\n",
        "      z = np.dot(X, W)\n",
        "      h = sigmoid(z)\n",
        "      gradient = np.dot(X.T, (h - y)) / y.size\n",
        "      W -= lr * gradient\n",
        "      \n",
        "      if(i % 10000 == 0):       \n",
        "        e = loss(h, y)\n",
        "        errors.append(e)\n",
        "  return W, errors"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "aWb1nxIrtldD",
        "colab_type": "code",
        "outputId": "8e7ba18a-c019-49f0-c408-72c7d5738ae3",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 102
        }
      },
      "source": [
        "run_tests()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "........\n",
            "----------------------------------------------------------------------\n",
            "Ran 8 tests in 0.837s\n",
            "\n",
            "OK\n"
          ],
          "name": "stderr"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "_VlYbo6x1Q04",
        "colab_type": "code",
        "outputId": "7c9fad66-4af6-4278-bf8f-6c939eeae44b",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 511
        }
      },
      "source": [
        "_, errors = fit(X, y)\n",
        "plt.plot(np.arange(len(errors)), errors)\n",
        "plt.xlabel(\"iteration^10000\")\n",
        "plt.ylabel(\"error\")\n",
        "plt.ylim(0, 1)\n",
        "plt.show();"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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hCfEW5WelaF9tpxpaXGbHAQAAACIGRSnGsfodAAAAMBhFKcY5+4oSl98BAAAA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\nREQkjIKSiIiIiIhIGAUlERERERGRMH8DxjAuDXdapDYAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 1008x576 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3lMGWRXJ-YP5",
        "colab_type": "text"
      },
      "source": [
        "### Hiding the complexity of the algorithm\n",
        "\n",
        "Knowing all of the details of the inner workings of the Gradient descent is good, but when solving problems in the wild, we might be hard pressed for time. In those situations, a simple & easy to use interface for fitting a Logistic Regression model might save us a lot of time. So, let's build one!\n",
        "\n",
        "But first, let's write some tests:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ZNiUHh8_iNbx",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "class TestLogisticRegressor(unittest.TestCase):\n",
        "\n",
        "    def test_correct_prediction(self):\n",
        "      global X\n",
        "      global y\n",
        "      X = X.reshape(X.shape[0], 1)\n",
        "      clf = LogisticRegressor()\n",
        "      y_hat = clf.fit(X, y).predict(X)\n",
        "      self.assertTrue((y_hat == y).all())"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "XeDkAWxDiQo4",
        "colab_type": "code",
        "outputId": "58adcf12-bb4f-4601-84a8-3fc722fa0d40",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 238
        }
      },
      "source": [
        "run_tests()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            ".E.......\n",
            "======================================================================\n",
            "ERROR: test_correct_prediction (__main__.TestLogisticRegressor)\n",
            "----------------------------------------------------------------------\n",
            "Traceback (most recent call last):\n",
            "  File \"<ipython-input-25-b3cbfe737800>\", line 7, in test_correct_prediction\n",
            "    clf = LogisticRegressor()\n",
            "NameError: name 'LogisticRegressor' is not defined\n",
            "\n",
            "----------------------------------------------------------------------\n",
            "Ran 9 tests in 0.858s\n",
            "\n",
            "FAILED (errors=1)\n"
          ],
          "name": "stderr"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "oidfenvBGiy8",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "class LogisticRegressor:\n",
        "  \n",
        "  def _add_intercept(self, X):\n",
        "    intercept = np.ones((X.shape[0], 1))\n",
        "    return np.concatenate((intercept, X), axis=1)\n",
        "\n",
        "  def predict_probs(self, X):\n",
        "    X = self._add_intercept(X)\n",
        "    return sigmoid(np.dot(X, self.W))\n",
        "  \n",
        "  def predict(self, X):\n",
        "    return self.predict_probs(X).round()\n",
        "      \n",
        "  def fit(self, X, y, n_iter=100000, lr=0.01):\n",
        "\n",
        "    X = self._add_intercept(X)\n",
        "    self.W = np.zeros(X.shape[1])\n",
        "\n",
        "    for i in range(n_iter):\n",
        "        z = np.dot(X, self.W)\n",
        "        h = sigmoid(z)\n",
        "        gradient = np.dot(X.T, (h - y)) / y.size\n",
        "        self.W -= lr * gradient\n",
        "    return self"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "R3hsL4LCBcTm",
        "colab_type": "code",
        "outputId": "dfeea495-21da-4503-c3f9-248d885027a6",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 102
        }
      },
      "source": [
        "run_tests()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            ".........\n",
            "----------------------------------------------------------------------\n",
            "Ran 9 tests in 1.695s\n",
            "\n",
            "OK\n"
          ],
          "name": "stderr"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SbaBivt46Oxf",
        "colab_type": "text"
      },
      "source": [
        "## Using our Regressor to decide who should receive discount codes\n",
        "\n",
        "Now that you're done with the \"hard\" part let's use the model to predict whether or not we should send discount codes."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "FjUXzun46Sve",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "X_test = np.array([10, 250])\n",
        "X_test = X_test.reshape(X_test.shape[0], 1)\n",
        "y_test = LogisticRegressor().fit(X, y).predict(X_test)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dUz__0al7uQV",
        "colab_type": "code",
        "outputId": "3f929d21-8296-402f-f5ac-77616b3c543a",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        }
      },
      "source": [
        "y_test"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([1., 0.])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 32
        }
      ]
    }
  ]
}