Logistic_Regression.ipynb

{
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   "cell_type": "markdown",
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    "# Classification with Logistic Regression\n",
    "\n",
    "## A companion notebook implementing logistic regression from scratch in `python`\n",
    "\n",
    "**Ingo Scholtes**  \n",
    "*September 11th 2020*  \n",
    "\n",
    "This tutorial accompanies a short lecture on classification with logistic regression. It showcases a simple machine learning workflow, including feature extraction, model fitting, and validation in a test data set. We implement a simple binary logistic regression classifier from scratch, including gradient ascent to perform a maximum likelihood estimation of the parameters. We further show how we can use the professional implementation of logistic regression in the popular machine learning package `sklearn`. \n",
    "\n",
    "We first need to import the following python packages:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/ingo/opt/anaconda3/lib/python3.7/site-packages/statsmodels/tools/_testing.py:19: FutureWarning: pandas.util.testing is deprecated. Use the functions in the public API at pandas.testing instead.\n",
      "  import pandas.util.testing as tm\n"
     ]
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "# The package numpy contains mathematical functions that we will need for our tutorial\n",
    "import numpy as np\n",
    "\n",
    "import seaborn as sns\n",
    "\n",
    "# matplotplib allows us to generate plots in python. The parameter inline, which we set in the first line of this cell, ensures that matplotlib plots are properyl displayed within this jupyter notebook.\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# the mplot3d module of the matplotlib toolkits enables us to generate 3D plots\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "\n",
    "# we need the cm module since we want to use color maps to color matplotlib surface plots\n",
    "from matplotlib import cm\n",
    "\n",
    "# The machine learning package sklearn contains some exemplary data sets that we will use to test our implementation of \n",
    "# logistic regression. We specifically use the digits data set, which contains pixel images of hand-written digits.\n",
    "from sklearn import datasets\n",
    "\n",
    "# the pandas package simplifies the handling of panel data\n",
    "import pandas as pd\n",
    "\n",
    "# the special module of the package scipy contains an implementation of the logistic function\n",
    "from scipy.special import expit\n",
    "\n",
    "plt.style.use('default')\n",
    "sns.set_style(\"whitegrid\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To keep the code in the tutorial as simple as possible we further define some helper functions. To understand logistic regression, it is not neccessary to understand the technical details of those functions. In the comments below, we nevertheless briefly explain what each of those methods does:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# This function displays an 8x8 grayscale pixel image stored in a two-dimensional numpy array\n",
    "def plotImage(img):\n",
    "    font = {'color':  'red',\n",
    "        'weight': 'bold',\n",
    "        'size': 16,\n",
    "        }    \n",
    "    plt.imshow(img, cmap=plt.cm.gray_r, interpolation='none')    \n",
    "    plt.xticks(range(8))\n",
    "    plt.yticks(range(8))  \n",
    "    plt.grid()\n",
    "    plt.tick_params(axis='both', which='major', labelsize=18)\n",
    "    plt.show()\n",
    "\n",
    "\n",
    "# This function displays a rectangular grid with multiple pixel images showing different digit. \n",
    "# It colors a customizable fraction of the images according to the digit contained in the images\n",
    "def plotDigitPanel(digits, indices, color_frac):\n",
    "    for index, image in enumerate(digits.images[indices]):\n",
    "        plt.subplot(int(np.sqrt(n)), int(np.sqrt(n)), index+1)\n",
    "        plt.axis('off')\n",
    "        if index < int(color_frac * n):            \n",
    "            alpha = 1.0\n",
    "            if digits.target[indices[index]] == 5:\n",
    "                plt.imshow(image, cmap=plt.cm.Blues, interpolation='nearest', alpha=alpha)\n",
    "            elif digits.target[indices[index]] == 6:\n",
    "                plt.imshow(image, cmap=plt.cm.Reds, interpolation='nearest', alpha=alpha)            \n",
    "        else:\n",
    "                plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest', alpha=0.5)\n",
    "    plt.gca().set_axis_off()\n",
    "    plt.subplots_adjust(top = 1, bottom = 0, right = 1, left = 0, hspace = 0.1, wspace = 0.1)\n",
    "    plt.margins(0,0)\n",
    "    plt.gca().xaxis.set_major_locator(plt.NullLocator())\n",
    "    plt.gca().yaxis.set_major_locator(plt.NullLocator())\n",
    "    plt.show()\n",
    "\n",
    "\n",
    "# This function plots features of images as defined in the lecture. A configurable fraction of those features \n",
    "# will be colored according to their respective classes. If highlight is not None, the feature with the given \n",
    "# index will be highlighted. \n",
    "def plotData(data, classes, color_frac=1.0, highlight=None, show_validation=True, file=None):\n",
    "    h = 0.02    \n",
    "    alpha = 0.7\n",
    "    s = 60\n",
    "\n",
    "    x_min, x_max = 0, 7\n",
    "\n",
    "    if color_frac>0:\n",
    "        y = range(n, 0, -1)\n",
    "        y_min, y_max = 0, n+1\n",
    "    else:\n",
    "        y = range(n, 0, -1)\n",
    "        y_min, y_max = 0, n+1\n",
    "\n",
    "    colored = int(n*color_frac)\n",
    "    if not highlight == None:\n",
    "        plt.axhline(data[colored+highlight][1])\n",
    "        plt.axvline(data[colored+highlight][0])\n",
    "        plt.scatter(data[:, 0][:colored], [y[:colored]], c=classes[:colored], cmap = plt.cm.coolwarm, alpha=0.3, s=s, edgecolors='k')    \n",
    "        plt.scatter(data[:, 0][colored:], y[colored:], s=s, edgecolors='k', c='black', alpha=0.3, marker='x')\n",
    "        plt.scatter(data[:, 0][colored+x], y[colored+highlight], s=s, edgecolors='k', c='black', marker='x')\n",
    "    else:\n",
    "        plt.scatter(data[:, 0][:colored], y[:colored], c=classes[:colored], cmap = plt.cm.coolwarm, s=s, edgecolors='k', alpha=alpha)\n",
    "        if show_validation:\n",
    "            plt.scatter(data[:, 0][colored:], y[colored:], s=s, edgecolors='k', c='black', alpha=0.5, marker='x')\n",
    "    plt.xlim(x_min, x_max)\n",
    "    plt.ylim(y_min, y_max)\n",
    "    plt.xlabel(r'feature $X$', fontsize=18)\n",
    "    plt.ylabel(r'image', fontsize=18)\n",
    "    plt.tick_params(axis='both', labelsize=14)\n",
    "    frame1 = plt.gca()\n",
    "    frame1.axes.xaxis.set_ticks([0, 1, 2, 3, 4, 5, 6, 7])\n",
    "    frame1.axes.yaxis.set_ticks([1, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100])\n",
    "    frame1.axes.yaxis.grid()\n",
    "    plt.show()"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Data Import and Feature Extraction\n",
    "\n",
    "Our is to develop a classifier that can recognize hand-written digits contained in a set of 8x8 pixel images. The machine learning package `sklearn` contains an example data set, which we can use to illustrate this simple classification problem. We can load this data set as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "digits = datasets.load_digits()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's have a look at the structure of the data. The variable `digits` is a `dictionary`, which contains multiple entries. The entry `digits.DESCR` contains a detailed explanation of the data set:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      ".. _digits_dataset:\n",
      "\n",
      "Optical recognition of handwritten digits dataset\n",
      "--------------------------------------------------\n",
      "\n",
      "**Data Set Characteristics:**\n",
      "\n",
      "    :Number of Instances: 5620\n",
      "    :Number of Attributes: 64\n",
      "    :Attribute Information: 8x8 image of integer pixels in the range 0..16.\n",
      "    :Missing Attribute Values: None\n",
      "    :Creator: E. Alpaydin (alpaydin '@' boun.edu.tr)\n",
      "    :Date: July; 1998\n",
      "\n",
      "This is a copy of the test set of the UCI ML hand-written digits datasets\n",
      "https://archive.ics.uci.edu/ml/datasets/Optical+Recognition+of+Handwritten+Digits\n",
      "\n",
      "The data set contains images of hand-written digits: 10 classes where\n",
      "each class refers to a digit.\n",
      "\n",
      "Preprocessing programs made available by NIST were used to extract\n",
      "normalized bitmaps of handwritten digits from a preprinted form. From a\n",
      "total of 43 people, 30 contributed to the training set and different 13\n",
      "to the test set. 32x32 bitmaps are divided into nonoverlapping blocks of\n",
      "4x4 and the number of on pixels are counted in each block. This generates\n",
      "an input matrix of 8x8 where each element is an integer in the range\n",
      "0..16. This reduces dimensionality and gives invariance to small\n",
      "distortions.\n",
      "\n",
      "For info on NIST preprocessing routines, see M. D. Garris, J. L. Blue, G.\n",
      "T. Candela, D. L. Dimmick, J. Geist, P. J. Grother, S. A. Janet, and C.\n",
      "L. Wilson, NIST Form-Based Handprint Recognition System, NISTIR 5469,\n",
      "1994.\n",
      "\n",
      ".. topic:: References\n",
      "\n",
      "  - C. Kaynak (1995) Methods of Combining Multiple Classifiers and Their\n",
      "    Applications to Handwritten Digit Recognition, MSc Thesis, Institute of\n",
      "    Graduate Studies in Science and Engineering, Bogazici University.\n",
      "  - E. Alpaydin, C. Kaynak (1998) Cascading Classifiers, Kybernetika.\n",
      "  - Ken Tang and Ponnuthurai N. Suganthan and Xi Yao and A. Kai Qin.\n",
      "    Linear dimensionalityreduction using relevance weighted LDA. School of\n",
      "    Electrical and Electronic Engineering Nanyang Technological University.\n",
      "    2005.\n",
      "  - Claudio Gentile. A New Approximate Maximal Margin Classification\n",
      "    Algorithm. NIPS. 2000.\n"
     ]
    }
   ],
   "source": [
    "print(digits.DESCR)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For our tutorial, the dictionary entries `digits.images` and `digits.target` are particularly relevant. `digits.images` is a list of two-dimensional 8x8 arrays, which contain the pixel values of each image. The pixel values from 0 to 16 represent different grayscales, with 0 being white and 16 being black. We can, for instance, print the two-dimensional array of pixel values of the image with index 15:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.  5. 12. 13. 16. 16.  2.  0.]\n",
      " [ 0. 11. 16. 15.  8.  4.  0.  0.]\n",
      " [ 0.  8. 14. 11.  1.  0.  0.  0.]\n",
      " [ 0.  8. 16. 16. 14.  0.  0.  0.]\n",
      " [ 0.  1.  6.  6. 16.  0.  0.  0.]\n",
      " [ 0.  0.  0.  5. 16.  3.  0.  0.]\n",
      " [ 0.  1.  5. 15. 13.  0.  0.  0.]\n",
      " [ 0.  4. 15. 16.  2.  0.  0.  0.]]\n"
     ]
    }
   ],
   "source": [
    "print(digits.images[15])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the array displayed above, we can guess that this image shows a five. To check this, we can use the list `digits.target`, which contains the actual digit displayed in the image, i.e. the ground truth that we want to predict with our classifier."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5\n"
     ]
    }
   ],
   "source": [
    "print(digits.target[15])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To plot the gray scale image contained in the pixel array above, we can use the helper function `plotImage`. We simply pass the two-dimensional array of pixel values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "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": [
    "plotImage(digits.images[15])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now know how we can access our test data and how we can display them. Let us now start with the actual classification, i.e. we are interested in a function which, for each image, predicts which digit is displayed in the image. We can then use the ground-truth digit stored in `digits.target` to validate our prediction.\n",
    "\n",
    "Each of our images contains 64 gray-scale values, i.e. we can see our set of images as a set of points in a 64-dimensional space. Real images will have million pixels, which highlights the problem that we are typically confronted with high-dimensional data. A first step before applying machine learning technique is thus to reduce the dimensionality of our problem. This is important for multiple reasons: First, dealing with high dimensional data is computationally demanding. Second, there is the so-called *curse of dimensionality* which states that the number of points required in a high-dimensional feature space grows exponentially with the dimensionality of the space. Hence, even millions of images in a feature space that has millions of dimensions is a very sparse data set! This comes with a high risk of overfitting, so it is a good idea to  reduce the dimensionality of our problem in the first step. \n",
    "\n",
    "We call this process of mapping a high-dimensional data set to a feature space with smaller dimensions *dimensionality reduction* or *feature extraction*. In our case, we want to extract a small number of (ideally one) numerical features that capture patterns in our images that are relevant for the classification. We could think of many different approaches: counting the number of white pixels, the average brightness of pixels, the borders based on the contrast of pixels. etc. To simplify matters, we use a very simple and rather naive approach: We simply compute the weighted average pixel coordinate in the two dimensions, and then simply pick one of those dimensions. This allows us to embed our image with 64 pixel values into a one-dimensional feature space. The idea is that this average coordinate tells us how far up/down left/right the digit is, and we can use variations between the images that are due to the shape of the different digits displayed to perform the classification. \n",
    "\n",
    "The following function computes the two average coordinates in x- and y direction of our image, using the pixel value as a weight in the weighted average:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def imageTo2D(digit):\n",
    "    m = np.reshape(digit, (8,8))\n",
    "    mass = 0\n",
    "    for i in range(8):\n",
    "        for j in range(8):\n",
    "            mass += m[i,j] * np.array([i,j])\n",
    "    return mass/np.sum(m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us apply this to our example image:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "tags": []
   },
   "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"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2.97878788 2.93333333]\n"
     ]
    }
   ],
   "source": [
    "plotImage(digits.images[15])\n",
    "d = imageTo2D(digits.images[15])\n",
    "print(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can compare this with the image displaying a **6**. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "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"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[4.16507937 3.89206349]\n"
     ]
    }
   ],
   "source": [
    "plotImage(digits.images[16])\n",
    "d = imageTo2D(digits.images[16])\n",
    "print(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The fact that the shape of this digit is different is captured in the difference in the weighted average of the coordinate. Our idea is now to utilize those variations to distinguish images showing a five from those showing a six. This yields a binary classification problem that we can address using logistic regression.\n",
    "\n",
    "Let us first select 100 random images showing a five or a six. We then store the image data as well as ground truth classes in the variables `data` and `classes`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 100\n",
    "\n",
    "indices = np.where( (digits.target==5) | (digits.target==6))[0]\n",
    "indices = np.random.choice(indices, n)\n",
    "\n",
    "data = np.array([ imageTo2D(d) for d in digits.images[indices]])\n",
    "classes = digits.target[indices]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now use our helper function `plotDigitPanel` to visually display our 100 images. We further color all images based on the ground truth classes, i.e. the digits displayed in the image. Those images showing a **5** are colored in blue, while those showing a six are colored in red. Note that this coloring is only for illustrative purposes, i.e. we do not change the pixels of the underlying data! "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 100 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotDigitPanel(digits, indices, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To simplify some later tasks (like the use of logistic regression in `sklearn`), we additionally store our data in a `pandas` dataframe. Here we also introduce a new column `target` which assumes a value of 1 if the digit is six, and a zero if the digit is five. This highlights that we will be using the digit six as the positive class of our binary classification problem. We also store the average x, and y coordinates of the images (our two one-dimensional features) in two columns of the dataframe."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "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>x</th>\n",
       "      <th>y</th>\n",
       "      <th>digit</th>\n",
       "      <th>target</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>4.328767</td>\n",
       "      <td>3.684932</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>4.324324</td>\n",
       "      <td>3.096525</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3.272414</td>\n",
       "      <td>3.458621</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>3.376384</td>\n",
       "      <td>3.328413</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>2.871972</td>\n",
       "      <td>3.207612</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>95</th>\n",
       "      <td>3.206790</td>\n",
       "      <td>3.049383</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>4.237805</td>\n",
       "      <td>3.207317</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>4.242424</td>\n",
       "      <td>3.724242</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>98</th>\n",
       "      <td>4.293629</td>\n",
       "      <td>3.332410</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99</th>\n",
       "      <td>4.168498</td>\n",
       "      <td>3.758242</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>100 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "           x         y  digit  target\n",
       "0   4.328767  3.684932      6       1\n",
       "1   4.324324  3.096525      6       1\n",
       "2   3.272414  3.458621      5       0\n",
       "3   3.376384  3.328413      5       0\n",
       "4   2.871972  3.207612      5       0\n",
       "..       ...       ...    ...     ...\n",
       "95  3.206790  3.049383      5       0\n",
       "96  4.237805  3.207317      6       1\n",
       "97  4.242424  3.724242      6       1\n",
       "98  4.293629  3.332410      6       1\n",
       "99  4.168498  3.758242      6       1\n",
       "\n",
       "[100 rows x 4 columns]"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.DataFrame({'x': data[:,0], 'y': data[:,1], 'digit': classes, 'target': (classes==6).astype(int)})\n",
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now use the helper function `plot_data` to generate a scatter plot of the feature values. Here we will only use the x-dimension from above, which turns out to be sufficient for our classification. Every image thus corresponds to a point on the real line, and we will plot all of the points of the 100 images above, coloring each point according to the digit displayed in the corresponding image."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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     },
     "metadata": {
      "needs_background": "light"
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     "output_type": "display_data"
    }
   ],
   "source": [
    "plotData(data, classes)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that the two classes of images have different average x coordinates, where the images showing a five tend to have a smaller average x-coordinate than those images showing a six. We can further illustrate this by a plot of the histograms of the coordinates of the two classes, with a kernel density estimate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.distplot(df.query('digit==5')[['x']], color='blue')\n",
    "sns.distplot(df.query('digit==6')[['x']], color='darkred')\n",
    "plt.xlabel(r'feature $x$', fontsize=18);\n",
    "plt.ylabel(r'count', fontsize=18);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Supervised Statistical Classification\n",
    "\n",
    "Now that we have embedded the 64 pixel values of our images into a one-dimensional feature space $\\mathbb{F}=\\mathbb{R}$ we can formally the classification problem. There are a number of different classification problems, which generally fall into two categories: **unsupervised** and **supervised classification**.\n",
    "\n",
    "**Unsupervised** methods detect patterns in unlabeled data and allow to infer which of the images belong to the same class. In the example above, we could use clustering algorithms to detect that there are two different digits shown in the images, and we could infer which images are showing the same digits, even though we may not know what these digits actually are. Such unsupervised methods do not require labeled examples, because they are based on patterns in the distribution of features.\n",
    "\n",
    "\n",
    "**Supervised** methods utilize labeled training data, in which we have access to the correct classes, i.e. a subset of the images for which we know which digit is being displayed. Our goal is to use such labeled examples to learn a classifier function, that can then be generalized to unseen examples. Examples include Support Vector Machines, Decision Trees, or logistic regression, which we explain in more detail below. \n",
    "\n",
    "In a nutstell, we will use a set of labeled training examples consisting both the features and the classes to learn a classifier function $C: \\mathbb{R} \\rightarrow \\{5,6\\}$. In the following, we will use 70 % of the data as training data, i.e. we use 30 of the 100 images to validate the classification. For simplicity, we use the first 70 images in our pandas data frame as training data, and the remaining 70 images as test data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\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>x</th>\n",
       "      <th>y</th>\n",
       "      <th>digit</th>\n",
       "      <th>target</th>\n",
       "      <th>train</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>4.328767</td>\n",
       "      <td>3.684932</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>True</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>4.324324</td>\n",
       "      <td>3.096525</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>True</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3.272414</td>\n",
       "      <td>3.458621</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>True</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>3.376384</td>\n",
       "      <td>3.328413</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>True</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>2.871972</td>\n",
       "      <td>3.207612</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>True</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>95</th>\n",
       "      <td>3.206790</td>\n",
       "      <td>3.049383</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>4.237805</td>\n",
       "      <td>3.207317</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>4.242424</td>\n",
       "      <td>3.724242</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>98</th>\n",
       "      <td>4.293629</td>\n",
       "      <td>3.332410</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99</th>\n",
       "      <td>4.168498</td>\n",
       "      <td>3.758242</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>100 rows × 5 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "           x         y  digit  target  train\n",
       "0   4.328767  3.684932      6       1   True\n",
       "1   4.324324  3.096525      6       1   True\n",
       "2   3.272414  3.458621      5       0   True\n",
       "3   3.376384  3.328413      5       0   True\n",
       "4   2.871972  3.207612      5       0   True\n",
       "..       ...       ...    ...     ...    ...\n",
       "95  3.206790  3.049383      5       0  False\n",
       "96  4.237805  3.207317      6       1  False\n",
       "97  4.242424  3.724242      6       1  False\n",
       "98  4.293629  3.332410      6       1  False\n",
       "99  4.168498  3.758242      6       1  False\n",
       "\n",
       "[100 rows x 5 columns]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "training_frac = 0.7\n",
    "training_samples = int(training_frac*n)\n",
    "\n",
    "df['train'] = df.index < training_frac*n\n",
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's use our helper function to color the first 70 images, which highlights which of the images are used to learn the classifier function, and which images are used as a test set:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 100 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plotDigitPanel(digits, indices, training_frac)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can further use the function `plotData` to show which data points in our feature space belong to the training set: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "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": [
    "plotData(data, classes, color_frac=training_frac)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Logistic Regression\n",
    "\n",
    "Let us now explain the logistic regression classifier. First of all, logistic regression is a statistical classification technique, i.e. we are interested in a conditional probability $P(C=c|x)$ which for each point $x$ in the feature space assigns a probability that an image with this feature belongs to class $c$. \n",
    "\n",
    "In its most elementary form, logistic regression is a binary classification algorithm, which requires dichomotous classes such as {False, True}, {0, 1}, or {Negative, Positive}. In our case we will use the two classes {5, 6} where we treat the class 6 as the Positive class.\n",
    "\n",
    "As explained in the lecture, starting from a linear model for the log-odds of the two classes, logistic regression uses the logistic function $\\sigma$ to express the probability of the positive class as: \n",
    "\n",
    "$P(C=6|x) = \\sigma(\\beta_0 + \\beta_1 x)$\n",
    "\n",
    "where the intercept $\\beta_0$ and the slope $\\beta_1$ are the coefficients of a linear model. The following helper function plots a logistic model with given coefficients along with a set of training data points. Following our coding as positive/negative class, instances of class six are encoded as 1 while instances of class five are coded as zero: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_logistic_model(df, intercept, slope):\n",
    "    plt.scatter(x=df.query('digit==5')[['x']], y=df.query('digit==5')['target'], alpha=0.5, color='blue', label='class 5',  s=60)\n",
    "    plt.scatter(x=df.query('digit==6')[['x']], y=df.query('digit==6')['target'], alpha=0.5, color='darkred', label='class 6',  s=60)\n",
    "    xx = np.linspace(1, 6, 50)\n",
    "    plt.ylim(-0.1,1.1)\n",
    "    plt.axhline(y=0.5, color='black')    \n",
    "    frame1 = plt.gca()\n",
    "    frame1.axes.set_yticks([0, 0.5, 1])\n",
    "\n",
    "    boundary = -intercept/slope\n",
    "    if slope>0:\n",
    "        frame1.axes.fill_between(np.linspace(boundary, 7), 0, 1, alpha=0.2, facecolor='darkred')\n",
    "        frame1.axes.fill_between(np.linspace(0, boundary), 0, 1, alpha=0.2, facecolor='blue')\n",
    "    else:\n",
    "        frame1.axes.fill_between(np.linspace(boundary, 7), 0, 1, alpha=0.2, facecolor='blue')\n",
    "        frame1.axes.fill_between(np.linspace(0, boundary), 0, 1, alpha=0.2, facecolor='darkred')\n",
    "\n",
    "    plt.plot(xx, intercept + slope*xx, c='green', lw=2, label=r'$\\beta_0 + x \\beta_1$')\n",
    "    plt.plot(xx, expit(intercept + slope*xx), c='red', lw=2, label=r'$\\sigma(\\beta_0 + \\beta_1 x)$')\n",
    "    plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us plot this for a specific set of parameters:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "scrolled": true
   },
   "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": [
    "plot_logistic_model(df[df['train']==True], intercept = -12.5, slope = 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the plot above, we have highlighted the decision boundary between class five (blue) and class six (red), which is at the point where the logistic model crosses the probability of 0.5 for class six. The model above is reasonable in so far as it assigns a higher probability of class six to those points in the feature space that are larger, which matches the pattern in our feature space. Let's see what happens if we invert the sign of the coefficients:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "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": [
    "plot_logistic_model(df[df['train']==True], intercept = 12.5, slope = -3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have inverted the curve, and now the model assigns smaller probability of class six to larger values of x, which is exactly the opposite of the pattern in the feature space, so this is not a reasonable model. \n",
    "\n",
    "We can use likelihood estimation to assess the explanatory power or plausibility of a logistic model with a given value for the coefficients in our training data. In the lecture we have seen that we can easily compute this based on the classes and features in our training set. In the following, for reasons of numerical stability, we will calculate the log-likelihood, i.e. we have made a logarithmic transformation of the likelihood function given in the lecture. Since the logarithm is a monotonic transformation, this does not affect the maximum, i.e. it does not matter if we maximize the likelihood or the log-likelihood function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "def log_likelihood(intercept, slope, data):\n",
    "    log_l = 0\n",
    "    for index, row in data.iterrows():\n",
    "        x_i = row['x']\n",
    "        y_i = row['target']\n",
    "        if y_i == 1: # we fit the probability for class 1\n",
    "            log_l += np.log(expit(intercept+slope*x_i))\n",
    "        else:\n",
    "            log_l += np.log(1-expit(intercept+slope*x_i))\n",
    "    return log_l"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now see whether we can use the likelihood function to assess the plausibility of the different coefficients above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-128.32636808059826\n"
     ]
    }
   ],
   "source": [
    "print(log_likelihood(intercept=12.5, slope=-3, data=df[df['train']==True]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the parameters that capture the opposite of the pattern in our feature space, the log-likelihood is very small.  If we invert the sign of the coefficients, we should get a much more plausible model with a larger likelihood:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-33.63646641523374\n"
     ]
    }
   ],
   "source": [
    "print(log_likelihood(intercept=-12.5, slope=3, data=df[df['train']==True]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Indeed, we get a value for the model likelihood that is more than 90 orders of magnitude larger! \n",
    "\n",
    "## Maximum Likelihood Estmation via Gradient Ascent\n",
    "\n",
    "But how do we find the parameters that maximise the likelihood? In the lecture, I have mentioned that there is no simple analytical solution for the maximum likelihood estimate of the logistic regression model, but we can use the partial derivatives of the log-likelihood function to implement a gradient ascent algorithm. \n",
    "\n",
    "This algorithm effectively uses the smoothness of the likehood manifold in the parameter space of the logistic regression model, moving up the surface until a maximum is reached.\n",
    "\n",
    "Let us first plot the likelihood manifold. Since we have two parameters, we get a surface in a three-dimensional space."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "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": [
    "plt.clf()\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "\n",
    "beta0_mesh = np.linspace(-10, 10, 20)\n",
    "beta1_mesh = np.linspace(-4, 4, 20)\n",
    "beta0_mesh, beta1_mesh = np.meshgrid(beta0_mesh, beta1_mesh)\n",
    "ax.plot_surface(beta0_mesh, beta1_mesh, log_likelihood(beta0_mesh, beta1_mesh, df), cmap=cm.coolwarm, alpha=0.55, rstride=1, cstride=1, linewidth=0.2, shade=True)\n",
    "\n",
    "ax.set_xlabel(r'$\\beta_0$')\n",
    "ax.set_ylabel(r'$\\beta_1$')\n",
    "ax.set_zlabel(r'$\\log\\mathcal{L}$')\n",
    "ax.view_init(30, 115)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us now implement the gradient ascend algorithm to estimate the optimal parameters heuristically. For this, we need the two partial derivatives of the log-likelihood function. They are easy to calculate for the log-transformed likelihood function as follows:\n",
    "\n",
    "$\\log \\mathcal{L}(\\beta_0, \\beta_1)  = \\sum_{i, c_i=6} \\log P(C=6|x_i) + \\sum_{i, c_i=5} \\log \\left(1-P(C=6|x_i))\\right) = \\sum_{i, c_i=6} \\log \\sigma (\\beta_0 +\\beta_1 x_i) + \\sum_{i, c_i=5} \\log \\left(1-\\sigma (\\beta_0 +\\beta_1 x_i)\\right) $\n",
    "\n",
    "The partial derivates of $\\mathcal{L}(\\beta_0, \\beta_1)$ in the direction $\\beta_0$ and $\\beta_1$ can then be given as: \n",
    "\n",
    "$ \\frac{\\partial \\log \\mathcal{L}(\\beta_0, \\beta_1)}{\\partial \\beta_0} = \\sum_i y_i - \\sigma(\\beta_0 + \\beta_1 x_i)  $  \n",
    "          $ \\frac{\\partial \\log \\mathcal{L}(\\beta_0, \\beta_1)}{\\partial \\beta_1} = \\sum_i x_i \\left[y_i - \\sigma(\\beta_0 + \\beta_1 x_i)\\right]$\n",
    "          \n",
    "where we define $y_i=1$ for all data points with $c_i=6$ and $y_i=0$ for all data points with $c_i=5$.\n",
    "\n",
    "Let us implement those partial derivatives as functions, so we can use them in our gradient ascent algorithm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "def partial_beta_0(beta_0, beta_1, data):\n",
    "    r = 0\n",
    "    for index, row in data.iterrows():\n",
    "        x_i = row['x']\n",
    "        y_i = row['target']\n",
    "        r += y_i - expit(beta_0+beta_1*x_i)\n",
    "    return r\n",
    "\n",
    "def partial_beta_1(beta_0, beta_1, data):\n",
    "    r = 0\n",
    "    for index, row in data.iterrows():\n",
    "        x_i = row['x']\n",
    "        y_i = row['target']\n",
    "        r += x_i * (y_i - expit(beta_0+beta_1*x_i))\n",
    "    return r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now use these partial derivatives to gradually move up the likelihood manifold, until we either reach a maximum, or the change of the iterative move is smaller than a given threshold. In each step, we move a distance of gamma along the partial derivatives at the current point in the parameter space and we perform a maximum of 1000 iterations before we stop."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "def estimate_gradient_ascent(data, initial_params=(0,0), gamma = 0.01, tol = 1e-12, max_iter=1000):\n",
    "    fitted_beta_0 = initial_params[0]\n",
    "    fitted_beta_1 = initial_params[1]\n",
    "    L = log_likelihood(fitted_beta_0, fitted_beta_1, data)\n",
    "\n",
    "    path_beta_0 = []\n",
    "    path_beta_1 = []\n",
    "    path_loglikelihood = []\n",
    "\n",
    "    for i in range(max_iter):\n",
    "        path_beta_0.append(fitted_beta_0)\n",
    "        path_beta_1.append(fitted_beta_1)        \n",
    "        path_loglikelihood.append(L)\n",
    "\n",
    "        last_beta_0 = fitted_beta_0\n",
    "        last_beta_1 = fitted_beta_1\n",
    "        last_L = L\n",
    "\n",
    "        # move along the gradient for both parameters\n",
    "        fitted_beta_0 = last_beta_0 + gamma * partial_beta_0(last_beta_0, last_beta_1, data)\n",
    "        fitted_beta_1 = last_beta_1 + gamma * partial_beta_1(last_beta_0, last_beta_1, data)\n",
    "\n",
    "        step_beta_0 = fitted_beta_0 - last_beta_0\n",
    "        step_beta_1 = fitted_beta_1 - last_beta_1\n",
    "\n",
    "        L = log_likelihood(fitted_beta_0, fitted_beta_1, data)\n",
    "\n",
    "        if abs(step_beta_0) + abs(step_beta_1) <= tol:\n",
    "            break\n",
    "    return fitted_beta_0, fitted_beta_1, path_beta_0, path_beta_1, path_loglikelihood"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us start at a point that we know has very small likelihood, i.e. a point where both the slope and the intercept are negative."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-16.87404463422563\n",
      "4.696337271032865\n"
     ]
    }
   ],
   "source": [
    "beta_0, beta_1, path_beta_0, path_beta_1, path_loglikelihood = estimate_gradient_ascent(df[df['train']==True], (-1, -1))\n",
    "\n",
    "print(beta_0)\n",
    "print(beta_1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conveniently, the implementation of the gradient ascent algorithm above also returns the path in the likelihood landscape that the gradient ascent has taken. We can plot this path in the 3D plot from before, to visualize the optimization process:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "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": [
    "plt.clf()\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "\n",
    "min_b0 = min(path_beta_0)\n",
    "max_b0 = max(path_beta_0)\n",
    "min_b1 = min(path_beta_1)\n",
    "max_b1 = max(path_beta_1)\n",
    "beta0_mesh = np.linspace(min(min_b0, -20), max(max_b0, 0), 20)\n",
    "beta1_mesh = np.linspace(min(min_b1, 0), max(max_b1, 5), 20)\n",
    "beta0_mesh, beta1_mesh = np.meshgrid(beta0_mesh, beta1_mesh)\n",
    "ax.plot_surface(beta0_mesh, beta1_mesh, log_likelihood(beta0_mesh, beta1_mesh, df[df['train']==True]), cmap=cm.coolwarm, alpha=0.55, rstride=1, cstride=1, linewidth=0.2, shade=True)\n",
    "\n",
    "#ax.scatter3D(path_beta_0, path_beta_1, path_loglikelihood, color='blue')\n",
    "ax.plot3D(path_beta_0, path_beta_1, path_loglikelihood, color='blue', alpha=0.5, linewidth=3)\n",
    "\n",
    "ax.set_xlabel(r'$\\beta_0$')\n",
    "ax.set_ylabel(r'$\\beta_1$')\n",
    "ax.set_zlabel(r'$\\log\\mathcal{L}$')\n",
    "ax.view_init(30, 115)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The resulting log-likelihood is much larger than the two points that we have tested above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-10.930023384905128"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log_likelihood(beta_0, beta_1, df[df['train']==True])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now plot the fitted logistic function as well as the fitted linear log-odds function. The decision boundary is given by the point at which the logistic model crosses 0.5, which coincides with the point at which the linear model of the log-odds function crosses zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "scrolled": true
   },
   "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": [
    "plot_logistic_model(df[df['train']==True], beta_0, beta_1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For all data points right of that boundary we predict the digit six (class 1), while for all points left of that boundary we predict the digit five (class 0). We implement this in terms of a predict function and apply it to some example images:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "def predict(beta_0, beta_1, x):\n",
    "    if expit(beta_0 + x * beta_1)>0.5:\n",
    "        return 6\n",
    "    else:\n",
    "        return 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our first example shows a five:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "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": [
    "plotImage(digits.images[indices][45])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, y = imageTo2D(digits.images[indices][45])\n",
    "predict(beta_0, beta_1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our second example shows a six, so let us see whether we correctly classify this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "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": [
    "plotImage(digits.images[indices][46])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, y = imageTo2D(digits.images[indices][46])\n",
    "predict(beta_0, beta_1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "At least in those two examples, our classifier actually works. We can now test it a bit more systematically, by applying it to the 30 test images that we have not used to fit our logistic regression model. This gives us a first idea how well it generalizes to unseen data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th>79</th>\n",
       "      <td>4.379928</td>\n",
       "      <td>3.655914</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
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       "      <th>81</th>\n",
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       "      <td>0</td>\n",
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       "      <td>6</td>\n",
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       "    <tr>\n",
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       "      <td>5</td>\n",
       "      <td>False</td>\n",
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       "    <tr>\n",
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       "      <td>5</td>\n",
       "      <td>0</td>\n",
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       "      <td>5</td>\n",
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       "      <td>0</td>\n",
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       "      <td>5</td>\n",
       "      <td>False</td>\n",
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       "      <td>False</td>\n",
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       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>88</th>\n",
       "      <td>4.202952</td>\n",
       "      <td>3.568266</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>6</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>89</th>\n",
       "      <td>3.781870</td>\n",
       "      <td>3.300283</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>6</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>90</th>\n",
       "      <td>4.178914</td>\n",
       "      <td>3.654952</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>6</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>91</th>\n",
       "      <td>4.379928</td>\n",
       "      <td>3.655914</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>6</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>92</th>\n",
       "      <td>4.168498</td>\n",
       "      <td>3.758242</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>6</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>93</th>\n",
       "      <td>4.079027</td>\n",
       "      <td>3.243161</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>6</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>94</th>\n",
       "      <td>3.864935</td>\n",
       "      <td>3.615584</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>6</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>95</th>\n",
       "      <td>3.206790</td>\n",
       "      <td>3.049383</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "      <td>5</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>4.237805</td>\n",
       "      <td>3.207317</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>6</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>4.242424</td>\n",
       "      <td>3.724242</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>6</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>98</th>\n",
       "      <td>4.293629</td>\n",
       "      <td>3.332410</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>6</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99</th>\n",
       "      <td>4.168498</td>\n",
       "      <td>3.758242</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>6</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           x         y  digit  target  train  predicted  error\n",
       "70  4.217143  3.357143      6       1  False          6  False\n",
       "71  4.129825  3.270175      6       1  False          6  False\n",
       "72  4.119454  3.658703      6       1  False          6  False\n",
       "73  3.862434  3.605820      6       1  False          6  False\n",
       "74  4.202454  3.245399      6       1  False          6  False\n",
       "75  3.469091  3.541818      5       0  False          5  False\n",
       "76  3.254613  3.059041      5       0  False          5  False\n",
       "77  3.533141  3.521614      5       0  False          5  False\n",
       "78  3.927954  3.478386      6       1  False          6  False\n",
       "79  4.379928  3.655914      6       1  False          6  False\n",
       "80  3.758017  3.183673      6       1  False          6  False\n",
       "81  2.889262  3.647651      5       0  False          5  False\n",
       "82  4.152027  3.344595      6       1  False          6  False\n",
       "83  3.026230  3.121311      5       0  False          5  False\n",
       "84  3.573477  3.566308      5       0  False          5  False\n",
       "85  3.308123  3.585434      5       0  False          5  False\n",
       "86  3.160448  2.791045      5       0  False          5  False\n",
       "87  3.405405  3.699324      5       0  False          5  False\n",
       "88  4.202952  3.568266      6       1  False          6  False\n",
       "89  3.781870  3.300283      6       1  False          6  False\n",
       "90  4.178914  3.654952      6       1  False          6  False\n",
       "91  4.379928  3.655914      6       1  False          6  False\n",
       "92  4.168498  3.758242      6       1  False          6  False\n",
       "93  4.079027  3.243161      6       1  False          6  False\n",
       "94  3.864935  3.615584      6       1  False          6  False\n",
       "95  3.206790  3.049383      5       0  False          5  False\n",
       "96  4.237805  3.207317      6       1  False          6  False\n",
       "97  4.242424  3.724242      6       1  False          6  False\n",
       "98  4.293629  3.332410      6       1  False          6  False\n",
       "99  4.168498  3.758242      6       1  False          6  False"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test_data = df[df['train']==False].copy()\n",
    "\n",
    "xx = test_data['x']\n",
    "test_data['predicted'] = [ predict(beta_0, beta_1, x) for x in xx]\n",
    "test_data['error'] = test_data['predicted']!=test_data['digit']\n",
    "test_data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this particular example, our simple classifier correctly classifies all 30 test images. We can plot the classification visually, by coloring the predicted digits in red (predicted six) and blue (predicted five):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 100 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#%% plot predicted classes for validation set\n",
    "training_samples = int(training_frac*n)\n",
    "prediction = [predict(beta_0, beta_1, s[0]) for s in data[training_samples:]]\n",
    "\n",
    "for index, image in enumerate(digits.images[indices]):\n",
    "    plt.subplot(int(np.sqrt(n)), int(np.sqrt(n)), index+1)\n",
    "    plt.axis('off')\n",
    "    if index < int(training_frac * n):\n",
    "        plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest', alpha=0.3)\n",
    "    else:\n",
    "        x = int(training_frac*n)\n",
    "        if prediction[index-x] == 5:\n",
    "            plt.imshow(image, cmap=plt.cm.Blues, interpolation='nearest', alpha=1)\n",
    "        elif prediction[index-x] == 6:\n",
    "            plt.imshow(image, cmap=plt.cm.Reds, interpolation='nearest', alpha=1)        \n",
    "plt.gca().set_axis_off()\n",
    "plt.subplots_adjust(top = 1, bottom = 0, right = 1, left = 0, \n",
    "        hspace = 0.1, wspace = 0.1)\n",
    "plt.margins(0,0)\n",
    "plt.gca().xaxis.set_major_locator(plt.NullLocator())\n",
    "plt.gca().yaxis.set_major_locator(plt.NullLocator())\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Logistic regression in scikit-learn\n",
    "\n",
    "Now that we have implemented the logistic regression model from scratch, we can also use the implementation of logistic regression in scikit-learn, which is provided in the class `LogisticRegression`. Among others, it takes parameters that influence the heuristic algorithm to estimate the parameters:\n",
    "\n",
    "In the following, we plot the fitted coefficients of the logistic model and we output the wrongly classified images in our test data set:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.linear_model import LogisticRegression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-14.326443668648944\n",
      "3.9647458122757087\n"
     ]
    },
    {
     "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": [
    "logreg = LogisticRegression(solver='lbfgs')\n",
    "training_data = df[df['train']==True]\n",
    "test_data = df[df['train']==False].copy()\n",
    "\n",
    "# Create an instance of Logistic Regression Classifier and fit the data.\n",
    "logreg.fit(training_data[['x']], training_data['target'])\n",
    "\n",
    "print(logreg.intercept_[0])\n",
    "print(logreg.coef_[0,0])\n",
    "\n",
    "plot_logistic_model(df[df['train']==True], logreg.intercept_[0], logreg.coef_[0,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x</th>\n",
       "      <th>y</th>\n",
       "      <th>digit</th>\n",
       "      <th>target</th>\n",
       "      <th>train</th>\n",
       "      <th>predicted</th>\n",
       "      <th>error</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>70</th>\n",
       "      <td>4.217143</td>\n",
       "      <td>3.357143</td>\n",
       "      <td>6</td>\n",
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       "      <th>71</th>\n",
       "      <td>4.129825</td>\n",
       "      <td>3.270175</td>\n",
       "      <td>6</td>\n",
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       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>72</th>\n",
       "      <td>4.119454</td>\n",
       "      <td>3.658703</td>\n",
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       "      <td>1</td>\n",
       "      <td>False</td>\n",
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       "    <tr>\n",
       "      <th>73</th>\n",
       "      <td>3.862434</td>\n",
       "      <td>3.605820</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>74</th>\n",
       "      <td>4.202454</td>\n",
       "      <td>3.245399</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
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       "    <tr>\n",
       "      <th>75</th>\n",
       "      <td>3.469091</td>\n",
       "      <td>3.541818</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "      <td>0</td>\n",
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       "      <td>0</td>\n",
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       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>77</th>\n",
       "      <td>3.533141</td>\n",
       "      <td>3.521614</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
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       "    <tr>\n",
       "      <th>78</th>\n",
       "      <td>3.927954</td>\n",
       "      <td>3.478386</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>79</th>\n",
       "      <td>4.379928</td>\n",
       "      <td>3.655914</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>80</th>\n",
       "      <td>3.758017</td>\n",
       "      <td>3.183673</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>81</th>\n",
       "      <td>2.889262</td>\n",
       "      <td>3.647651</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>82</th>\n",
       "      <td>4.152027</td>\n",
       "      <td>3.344595</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>83</th>\n",
       "      <td>3.026230</td>\n",
       "      <td>3.121311</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
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       "    <tr>\n",
       "      <th>84</th>\n",
       "      <td>3.573477</td>\n",
       "      <td>3.566308</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>85</th>\n",
       "      <td>3.308123</td>\n",
       "      <td>3.585434</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
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       "    <tr>\n",
       "      <th>86</th>\n",
       "      <td>3.160448</td>\n",
       "      <td>2.791045</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>87</th>\n",
       "      <td>3.405405</td>\n",
       "      <td>3.699324</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
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       "    <tr>\n",
       "      <th>88</th>\n",
       "      <td>4.202952</td>\n",
       "      <td>3.568266</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>89</th>\n",
       "      <td>3.781870</td>\n",
       "      <td>3.300283</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>90</th>\n",
       "      <td>4.178914</td>\n",
       "      <td>3.654952</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>91</th>\n",
       "      <td>4.379928</td>\n",
       "      <td>3.655914</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>92</th>\n",
       "      <td>4.168498</td>\n",
       "      <td>3.758242</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>93</th>\n",
       "      <td>4.079027</td>\n",
       "      <td>3.243161</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>94</th>\n",
       "      <td>3.864935</td>\n",
       "      <td>3.615584</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>95</th>\n",
       "      <td>3.206790</td>\n",
       "      <td>3.049383</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "      <td>0</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>4.237805</td>\n",
       "      <td>3.207317</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>4.242424</td>\n",
       "      <td>3.724242</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>98</th>\n",
       "      <td>4.293629</td>\n",
       "      <td>3.332410</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99</th>\n",
       "      <td>4.168498</td>\n",
       "      <td>3.758242</td>\n",
       "      <td>6</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "      <td>1</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           x         y  digit  target  train  predicted  error\n",
       "70  4.217143  3.357143      6       1  False          1  False\n",
       "71  4.129825  3.270175      6       1  False          1  False\n",
       "72  4.119454  3.658703      6       1  False          1  False\n",
       "73  3.862434  3.605820      6       1  False          1  False\n",
       "74  4.202454  3.245399      6       1  False          1  False\n",
       "75  3.469091  3.541818      5       0  False          0  False\n",
       "76  3.254613  3.059041      5       0  False          0  False\n",
       "77  3.533141  3.521614      5       0  False          0  False\n",
       "78  3.927954  3.478386      6       1  False          1  False\n",
       "79  4.379928  3.655914      6       1  False          1  False\n",
       "80  3.758017  3.183673      6       1  False          1  False\n",
       "81  2.889262  3.647651      5       0  False          0  False\n",
       "82  4.152027  3.344595      6       1  False          1  False\n",
       "83  3.026230  3.121311      5       0  False          0  False\n",
       "84  3.573477  3.566308      5       0  False          0  False\n",
       "85  3.308123  3.585434      5       0  False          0  False\n",
       "86  3.160448  2.791045      5       0  False          0  False\n",
       "87  3.405405  3.699324      5       0  False          0  False\n",
       "88  4.202952  3.568266      6       1  False          1  False\n",
       "89  3.781870  3.300283      6       1  False          1  False\n",
       "90  4.178914  3.654952      6       1  False          1  False\n",
       "91  4.379928  3.655914      6       1  False          1  False\n",
       "92  4.168498  3.758242      6       1  False          1  False\n",
       "93  4.079027  3.243161      6       1  False          1  False\n",
       "94  3.864935  3.615584      6       1  False          1  False\n",
       "95  3.206790  3.049383      5       0  False          0  False\n",
       "96  4.237805  3.207317      6       1  False          1  False\n",
       "97  4.242424  3.724242      6       1  False          1  False\n",
       "98  4.293629  3.332410      6       1  False          1  False\n",
       "99  4.168498  3.758242      6       1  False          1  False"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test_data = df[df['train']==False].copy()\n",
    "test_data['predicted'] = logreg.predict(test_data[['x']])\n",
    "test_data['error'] = test_data['predicted']!=test_data['target']\n",
    "test_data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Just like our own implementation, the implementation of logistic regression in the `sklearn` module correctly classifies all images in the test data set!\n",
    "\n",
    "While there is much more to say about topics like how to use logistic regression for non-binary classification, how  it is related to neural networks, or how we can apply regularization to logistic regression, we close this introductory tutorial here, and I must refer you to my MSc course on data science."
   ]
  }
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