{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Logistic Regression\n",
    "\n",
    "Logistic regression is a *binary classification algorithm*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Data')"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "from sklearn.datasets import make_blobs\n",
    "from sklearn.model_selection import train_test_split\n",
    "\n",
    "\n",
    "# create linearly separable data\n",
    "sep = False\n",
    "while not sep:\n",
    "    X, Y = make_blobs(n_samples=200, n_features\n",
    "                      =1, centers=2, cluster_std=0.5, center_box=(0, 10))\n",
    "    sep = True\n",
    "    for x_1 in X[Y == 1]:\n",
    "        for x_0 in X[Y == 0]:\n",
    "            if Y[X.argmin()] == 0 and x_0 > x_1:\n",
    "                sep = False\n",
    "            elif x_0 < x_1:\n",
    "                sep = False\n",
    "         \n",
    "X = np.hstack([X, np.ones((X.shape[0], 1))])\n",
    "\n",
    "                \n",
    "X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size=0.33)\n",
    "\n",
    "plt.scatter(X[:, 0], Y, s=10, c='b')\n",
    "plt.xlabel('feature')\n",
    "plt.ylabel('groups')\n",
    "plt.yticks([0, 1])\n",
    "plt.title('Data')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Binary target\n",
    "\n",
    "### Sigmoid function\n",
    "\n",
    "We have to fit a **model** trough our data.  \n",
    "The *sigmoid* (also called *logistic*) function is used:\n",
    "$$\\boxed{\n",
    "\\sigma(x) = \\frac{1}{1 + e^{-x}}\n",
    "}$$\n",
    "\n",
    "It produce an 'S' shape like function that stays between 0 and 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "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": [
    "def sigmoid(x):\n",
    "    return 1 / (1 + np.exp(-x))\n",
    "\n",
    "xs_sig = np.linspace(-7, 7, 100)\n",
    "ys_sig = sigmoid(xs_sig)\n",
    "plt.plot(xs_sig, ys_sig, label=r'$\\sigma(x) = {\\frac{1}{1 + e^{-x}}}$')\n",
    "plt.yticks([0, 0.5, 1])\n",
    "plt.xlim([-7, 7])\n",
    "plt.legend()\n",
    "plt.grid()\n",
    "plt.title('Sigmoid function')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "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": [
    "#  helper function for approximation\n",
    "def sigmoid_mu_s(x, mu, s):\n",
    "    return 1 / (1 + np.exp(-(x + mu) / s))\n",
    "\n",
    "plt.scatter(X[:, 0], Y, s=10, c='b', alpha=0.5)\n",
    "sig_x = np.linspace(X[:, 0].min(), X[:, 0].max(), 100)\n",
    "y_flip = 1 if Y[X[:, 0].argmin()] == 0 else -1\n",
    "sig_y = sigmoid_mu_s(sig_x, -X[:, 0].mean(), y_flip * 0.5)\n",
    "plt.plot(sig_x, sig_y, c='r')\n",
    "plt.yticks([0, 0.5, 1])\n",
    "plt.grid(axis='y')\n",
    "plt.title('Approximation')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hypothesis and Prediction\n",
    "\n",
    "We can tweak the sigmoid function by passing the dot product of $x$ (vector of feature) and $\\theta$ (the vector of weights).\n",
    "\n",
    "$$h(x) = \\sigma(x \\cdot \\theta) = \\sigma(\\theta_0 x_0 + \\theta_1 x_1 + \\cdots + \\theta_n x_n)$$\n",
    "$x_0$ being equal to 1.  \n",
    "$\\theta$ is not a parameter of $h$ because it's considered as a constant.\n",
    "\n",
    "$h$ is our hypothesis function. It return a probability of an $x$ to belong to one of the group.  \n",
    "To predict some new $x$ we input it to $h$ with the weights and see if the returned value if above or below 0.5.\n",
    "\n",
    "$$\n",
    "\\text{prediction} =\n",
    "    \\begin{cases} \n",
    "        1, & h(x) \\ge 0.5 \\\\\n",
    "        0, & h(x) \\lt 0.5\n",
    "    \\end{cases}\n",
    "$$\n",
    "\n",
    "This is equivalent to:\n",
    "\n",
    "$$\n",
    "\\text{prediction} =\n",
    "    \\begin{cases} \n",
    "        1, & x \\cdot \\theta \\ge 0 \\\\\n",
    "        0, & x \\cdot \\theta \\lt 0\n",
    "    \\end{cases}\n",
    "$$\n",
    "\n",
    "Indeed, in the sigmoid funcion:  \n",
    "* $x$ is *above* 0 then $e^{-x}$ tend to be small, $\\frac{1}{1 + \\text{small number}} \\approx 1$.  \n",
    "* $x$ is *below* 0 then $e^{-x}$ tend to be big, $\\frac{1}{1 + \\text{big number}} \\approx 0$. \n",
    "\n",
    "Therefore, a prediction can be visualized by projecting $x$ onto $\\theta$.  \n",
    "It will be more relevant the more the two vectors go in the same direction\n",
    "(or opposite direction for negative projection),\n",
    "for exemple if $x \\cdot \\theta = 0$ then $x$ is perpendicular to $\\theta$,\n",
    "$h(x) = 0.5$ right in the middle of the sigmoid function,\n",
    "the prediction become more irelevant as $x \\cdot \\theta$ tend to 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def hypothesis(x, theta):\n",
    "    # suppose that x_0 = 1 and isn't mentionned\n",
    "    return sigmoid(x.dot(theta))\n",
    "\n",
    "\n",
    "def predict(x, theta):\n",
    "    return int(hypothesis(x, theta) >= 0.5)  #  a bit more clear\n",
    "    # or\n",
    "    return int(x.dot(theta) >= 0)\n",
    "    # there is a nice vector interpretation for this one\n",
    "    # and it probably compute faster"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cost function\n",
    "\n",
    "We have to evaluate our model by introducing a cost function, this one is called the *logloss*\n",
    "\n",
    "For a single $x$:\n",
    "$$\n",
    "j(h(x), y) = \n",
    "    \\begin{cases} \n",
    "        -\\ln(h(x)),     & y = 1 \\\\\n",
    "        -\\ln(1 - h(x)), & y = 0 \n",
    "    \\end{cases}\n",
    "$$\n",
    "It can be rewritten as a single continuous function:\n",
    "$$ j(h(x), y) = -y \\ln(h(x)) - (1 - y) \\ln(1 - h(x))$$\n",
    "\n",
    "Even if it may seem more complicated at first sight, it's just a switch with the values of $y$.  \n",
    "- if $y = 1$ then $- (1 - y) \\ln(1 - h(x)) = 0$ and the $-\\ln(h(x))$ stays.\n",
    "- if $y = 0$ then $-y \\ln(h(x)) = 0$            and the $-\\ln(1 - h(x))$ stays."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "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": [
    "xs_log = np.linspace(0.001, 0.999, 100)\n",
    "ys_0 = [-np.log(1 - x) for x in xs_log]\n",
    "ys_1 = [-np.log(x) for x in xs_log]\n",
    "plt.plot(xs_log, ys_1, label=r'$y = 1, -\\ln(h(x))$')\n",
    "plt.plot(xs_log, ys_0, label=r'$y = 0, -\\ln(1 - h(x))$')\n",
    "plt.ylim([0, 5])\n",
    "plt.xlim([0, 1])\n",
    "plt.xticks([0, 0.5, 1])\n",
    "plt.legend()\n",
    "plt.grid()\n",
    "plt.title('cost function')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "def logloss(x, y, theta):\n",
    "    if y == 1:\n",
    "        return -np.log(hypothesis(x, theta))\n",
    "    elif y == 0:\n",
    "        return -np.log(1 - hypothesis(x, theta))\n",
    "    #  or (less obvious)\n",
    "    return -y * np.log(hypothesis(x, theta)) - (1 - y) * np.log(1 - hypothesis(x, theta))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that if $y$ was suppose to be 0 but is close to 1, the cost will be high.  \n",
    "Same logic if $y$ was suppose to be 1 but is close to 0.\n",
    "\n",
    "Then we generalize the cost function for all $x$:\n",
    "$$J(\\theta) = \\frac{1}{n} \\sum_{i = 1}^{n} j(h(x_i), y_i)$$\n",
    "$n$ is the number of elements in $X$ and $Y$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def cost(xs, ys, theta):\n",
    "    return sum([logloss(x, y, theta) for x, y in zip(xs, ys)]) / len(xs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gradient descent\n",
    "\n",
    "Finally, use gradient descent to minimize the cost function:\n",
    "\n",
    "$$\\theta_i := \\theta_i - \\alpha \\frac{\\partial}{\\partial \\theta_i} J(\\theta)\\ \\text{ for } i = 1, ..., n$$\n",
    "\n",
    "Or with gradient notation:\n",
    "\n",
    "$$ \\theta := \\theta - \\alpha \\nabla J(\\theta)$$\n",
    "\n",
    "\n",
    "$\\alpha$ is the learning rate (how fast we want the step to be).  \n",
    "Since $\\theta_i$ is only used in the dot product, all of them will have similar partial derivative.\n",
    "\n",
    "let $z = x \\cdot \\theta_j$\n",
    "\n",
    "$\\newcommand{\\pt}{\\frac{\\partial}{\\partial \\theta_j}}$\n",
    "$$\n",
    "\\begin{align}\n",
    "      & \\pt J(\\theta) \\\\\n",
    "    = & \\pt \\left[ \\frac{1}{n} \\sum_{i = 1}^{n} j(h(x_i, \\theta), y_i) \\right] \\\\\n",
    "    = & \\pt \\left[ \\frac{1}{n} \\sum_{i = 1}^{n}\n",
    "        (-y_i \\ln(h(x_i, \\theta)) - (1 - y_i) \\ln(1 - h(x_i, \\theta))) \\right] \\\\\n",
    "    = & \\pt \\left[ \\frac{1}{n} \\sum_{i = 1}^{n}\n",
    "        (-y_i \\ln(\\sigma(z)) - (1 - y_i) \\ln(1 - \\sigma(z))) \\right] \\\\\n",
    "    = & \\pt \\left[ \\frac{1}{n} \\sum_{i = 1}^{n}\n",
    "        (-y_i \\ln(\\frac{1}{1 + e^{-z}}) -\n",
    "        (1 - y_i) \\ln(1 - \\frac{1}{1 + e^{-z}})) \\right] \\\\\n",
    "    = & \\frac{1}{n} \\sum_{i = 1}^{n} \\pt \\left[ \n",
    "        -y_i \\ln(\\frac{1}{1 + e^{-z}}) -\n",
    "        (1 - y_i) \\ln(1 - \\frac{1}{1 + e^{-z}}) \\right] \\\\\n",
    "    = & \\frac{1}{n} \\sum_{i = 1}^{n} \\left(\n",
    "        -y_i \\pt \\left[ \\ln(\\frac{1}{1 + e^{-z}}) \\right] -\n",
    "        (1 - y_i) \\pt \\left[ \\ln(1 - \\frac{1}{1 + e^{-z}}) \\right] \\right) \\\\\n",
    "    = & \\frac{1}{n} \\sum_{i = 1}^{n} \\left(\n",
    "        -y_i(1 + e^{-z}) \\pt \\left[ \\frac{1}{1 + e^{-z}} \\right]\n",
    "        - (1 - y_i) \\frac{1}{1 - \\frac{1}{1 + e^{-z}}} \\pt \\left[ 1 - \\frac{1}{1 + e^{-z}} \\right]\n",
    "        \\right) \\\\\n",
    "    = & \\frac{1}{n} \\sum_{i = 1}^{n} \\left(\n",
    "        -y_i(1 + e^{-z}) \\pt \\left[ \\frac{1}{1 + e^{-z}} \\right]\n",
    "        - (1 - y_i) \\frac{1 + e^{-z}}{e^{-z}} (-1) \\pt \\left[ \\frac{1}{1 + e^{-z}} \\right]\n",
    "        \\right) \\\\\n",
    "    = & \\frac{1}{n} \\sum_{i = 1}^{n} \\left(\n",
    "        -y_i(1 + e^{-z}) \\frac{e^{-z}}{(1 + e^{-z})^2} \\pt z\n",
    "        - (1 - y_i) \\frac{1 + e^{-z}}{e^{-z}} (-1) \\frac{e^{-z}}{(1 + e^{-z})^2} \\pt z\n",
    "        \\right) \\\\\n",
    "    = & \\frac{1}{n} \\sum_{i = 1}^{n} \\left(\n",
    "        -y_i \\frac{e^{-z}}{1 + e^{-z}} x_{ij}\n",
    "        + (1 - y_i) \\frac{1}{1 + e^{-z}} x_{ij}\n",
    "        \\right) \\\\\n",
    "    = & \\frac{1}{n} \\sum_{i = 1}^{n} \\left(\n",
    "        \\frac{-y_i e^{-z}}{1 + e^{-z}}\n",
    "        + \\frac{(1 - y_i)}{1 + e^{-z}} \\right) x_{ij} \\\\\n",
    "    = & \\frac{1}{n} \\sum_{i = 1}^{n} \\left(\n",
    "        \\frac{-y_i e^{-z} + 1 - y_i}{1 + e^{-z}} \\right) x_{ij} \\\\\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\boxed{\n",
    "    = \\frac{1}{n} \\sum_{i = 1}^{n} (h(x_i) - y_i) x_{ij}\n",
    "}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gradient(xs, ys, theta):\n",
    "    g = np.array([])\n",
    "    for j in range(len(theta)):\n",
    "        g = np.append(g, [sum([(hypothesis(x, theta) - y) * x[j]\n",
    "                          for x, y in zip(xs, ys)]) / len(xs)],\n",
    "                      axis=0)\n",
    "    return g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gradient_descent(xs, ys, theta, alpha=1, epoch=100):\n",
    "    for _ in range(epoch):\n",
    "        theta = theta - alpha * gradient(xs, ys, theta)     \n",
    "    return theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "weights:   -2.647912220979194 15.448901536265106\n",
      "train cost: 0.0020754129182815793\n",
      "test cost:  0.003588729710123071\n"
     ]
    }
   ],
   "source": [
    "theta_binary = np.random.randn(len(X_train[0]))\n",
    "theta_binary = gradient_descent(X_train, Y_train, theta_binary, 2, 2000)\n",
    "print('weights:  ', *theta_binary)\n",
    "print('train cost:', cost(X_train, Y_train, theta_binary))\n",
    "print('test cost: ', cost(X_test, Y_test, theta_binary))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "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": [
    "plt.scatter(X_train[:, 0], Y_train, s=8, c='b', alpha=0.4, label='train data')\n",
    "plt.scatter(X_test[:, 0], Y_test, s=10, c='g', label='test data')\n",
    "sig_x = np.linspace(X[:, 0].min(), X[:, 0].max(), 100)\n",
    "sig_y = [hypothesis(np.array([x, 1]), theta_binary) for x in sig_x]\n",
    "plt.plot(sig_x, sig_y, c='r', alpha=0.5, label='model')\n",
    "plt.yticks([0, 0.5, 1])\n",
    "plt.ylim([-0.1, 1.1])\n",
    "plt.legend()\n",
    "plt.grid(axis='y')\n",
    "plt.title('Optimized model')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One vs all\n",
    "\n",
    "What if we don't have a binary target but many groups (>2) to classify.  \n",
    "It's in this that one vs all is applied.\n",
    "\n",
    "Here, there is 2 features and 3 categories."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "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": [
    "xs_oa = np.array([[1, 1], [1, 1.3], [1.4, 1.1],\n",
    "               [1.5, 2], [1.6, 1.8], [1.4, 1.8],\n",
    "               [2.4, 1.2], [2.2, 1.5], [2.3, 1.3]])\n",
    "ys_oa = np.array([0, 0, 0,\n",
    "               1, 1, 1,\n",
    "               2, 2, 2])\n",
    "\n",
    "xs_oa = np.hstack([xs_oa, np.ones((xs_oa.shape[0], 1))])\n",
    "\n",
    "xs_plot_spec = [\n",
    "    {'marker': '^', 'c': 'b', 's': 50},\n",
    "    {'marker': 'o', 'c': 'r', 's': 50},\n",
    "    {'marker': 's', 'c': 'g', 's': 50}\n",
    "]\n",
    "\n",
    "def plot_onevsall(xs, ys):\n",
    "    for i, y in enumerate(ys):\n",
    "        plt.scatter(xs[i,0], xs[i,1], **xs_plot_spec[y])\n",
    "        \n",
    "plot_onevsall(xs_oa, ys_oa)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The tricks is to create an hypothesis function for every target.  \n",
    "Each of those will compare his target to all the other one.\n",
    "\n",
    "For example the hypothesis function for *circle* will separate it from *triangle* and *square*"
   ]
  },
  {
   "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": [
    "plot_onevsall(xs_oa, ys_oa)\n",
    "plt.plot([1.0, 2.4], [1.5, 1.9], c='r')\n",
    "plt.plot([1.0, 1.8], [1.7, 1.0], c='b')\n",
    "plt.plot([1.7, 2.3], [1.0, 1.8], c='g')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "def train_against(xs, ys, theta, one, alpha, epoch):\n",
    "    ys_ally = ys.copy()\n",
    "    ys_ally[ys == one] = 0\n",
    "    ys_ally[ys != one] = 1\n",
    "    return gradient_descent(xs, ys_ally, theta, alpha, epoch)\n",
    "\n",
    "def train_thetas(xs, ys, theta, alpha=1, epoch=1000):\n",
    "    thetas = []\n",
    "    for i in np.unique(ys):\n",
    "        thetas.append(train_against(xs, ys, theta, i, alpha, epoch))\n",
    "    return thetas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[array([  5.05076977,   6.31195834, -15.87703677]),\n",
       " array([  3.21813142, -10.8578558 ,  12.38691428]),\n",
       " array([-9.38682431,  6.49524162,  8.28589111])]"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta_oa = np.random.randn(len(xs_oa[0]))\n",
    "thetas_oa = train_thetas(xs_oa, ys_oa, theta_oa, alpha=1, epoch=1000)\n",
    "thetas_oa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "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_onevsall(xs_oa, ys_oa)\n",
    "\n",
    "def plot_boundaries(xs, thetas):\n",
    "    for i, ti in enumerate(thetas):\n",
    "        xs_ti = np.array([min(xs[:, 1]) - 2, max(xs[:, 1]) + 2])\n",
    "        ys_ti = (-1 / ti[1]) * (ti[0] * xs_ti + ti[2])  # from https://utkuufuk.com/2018/05/19/binary-logistic-regression/\n",
    "        plt.plot(xs_ti, ys_ti, linewidth=2, c=xs_plot_spec[i]['c'])\n",
    "\n",
    "plt.xlim([xs_oa[:, 0].min() - 0.3, xs_oa[:, 0].max() + 0.3])\n",
    "plt.ylim([xs_oa[:, 1].min() - 0.3, xs_oa[:, 1].max() + 0.3])\n",
    "plot_boundaries(xs_oa, thetas_oa)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Prediction\n",
    "\n",
    "To make a prediction we get the hypothesis for each group $h^{(i)}(x)$, the predicted group is then $\\max_i h^{(i)}(x)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "def make_prediction(x, thetas):\n",
    "    ps = [hypothesis(x, ti) for ti in thetas]\n",
    "    return ps.index(min(ps))    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "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": [
    "xs_pred = 1.2 * np.random.rand(50, 2) + 1\n",
    "xs_pred = np.hstack([xs_pred, np.ones((xs_pred.shape[0], 1))])\n",
    "\n",
    "ys_pred = np.array([make_prediction(x, thetas_oa) for x in xs_pred])\n",
    "plot_onevsall(xs_pred, ys_pred)\n",
    "plt.xlim([xs_oa[:, 0].min() - 0.3, xs_oa[:, 0].max() + 0.3])\n",
    "plt.ylim([xs_oa[:, 1].min() - 0.3, xs_oa[:, 1].max() + 0.3])\n",
    "plot_boundaries(xs_oa, thetas_oa)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}