7 files changed, 2386 insertions, 1 deletions
diff --git a/blog/git_server.md b/blog/git_server.md
index 67191c6..741032c 100644
--- a/blog/git_server.md
+++ b/blog/git_server.md
@@ -1,4 +1,4 @@
-# How to make your own git server/website    
+# How to make a git server/website
 
 ## Basic ssh server
 
diff --git a/generate b/generate
index 7b6492c..56ea15b 100755
--- a/generate
+++ b/generate
@@ -119,3 +119,15 @@ do
         </li>" walks/index.html
     echo "Generated walk at $gpx_path"
 done
+
+for notebook_path in notebooks/*.ipynb
+do
+    notebook_dst_path=$(echo "$notebook_path" | sed 's_.ipynb_.html_')
+    jupyter-nbconvert "$notebook_path"
+
+    sed -i'' "/<!-- NOTEBOOKINDEX -->/ a\
+        <li> \
+        <a href=\"$notebook_dst_path\">$(basename $notebook_path | cut -d . -f 1 | tr '-' ' ')</a> \
+        <a href=\"$notebook_path\">   ($(basename $notebook_path))</a> \
+        </li>" index.html
+done
diff --git a/index.template.html b/index.template.html
index 55a1df7..4a08f1d 100644
--- a/index.template.html
+++ b/index.template.html
@@ -40,3 +40,8 @@
   <!-- <li><a href="https://chat.cacharle.xyz">chat</a></li> -->
   <!-- UTILSINDEX -->
 </ul>
+
+<h2>Notebooks</h2>
+<ul>
+  <!-- NOTEBOOKINDEX -->
+</ul>
diff --git a/notebooks/.ipynb_checkpoints/linear-regression-checkpoint.ipynb b/notebooks/.ipynb_checkpoints/linear-regression-checkpoint.ipynb
new file mode 100644
index 0000000..4d6a6cd
--- /dev/null
+++ b/notebooks/.ipynb_checkpoints/linear-regression-checkpoint.ipynb
@@ -0,0 +1,477 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Linear Regression"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 25,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "%matplotlib inline\n",
+    "\n",
+    "data = np.genfromtxt('./data.csv', delimiter=',')[1:]\n",
+    "km = data[:, 0]\n",
+    "price = data[:, 1]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "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 plot_data():\n",
+    "    plt.scatter(km, price)\n",
+    "    plt.xlabel(\"km\")\n",
+    "    plt.ylabel(\"price\")\n",
+    "plot_data()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "This dataset contains information about cars, their cost and the the number of kilometers they've made.\n",
+    "\n",
+    "We can see just by looking at the data that their is a **linear relationship**, when the distance made diminish, the price rize.\n",
+    "\n",
+    "We can try to trace an approximate line which goes *through* the data."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "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_data()\n",
+    "plt.plot([200_000, 20_000], [3_500, 8_500], color='r')\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We can use this hypothetical line to predict new data, for example predict the price of a car knowing only it's distance traveled.\n",
+    "\n",
+    "The equation of a line is in the form:\n",
+    "\n",
+    "$$\n",
+    "\\boxed {\n",
+    "    y = \\theta_1 x + \\theta_0\n",
+    "}\n",
+    "$$\n",
+    "\n",
+    "Where $\\theta_1$ is the slope of the line and $\\theta_0$ the y-intercept. If we managed to tweak those values correctly we can find the best fitting line for our data (or at least very close to it).\n",
+    "\n",
+    "If we switch to a function, we get $h(x) = \\theta_1 x + \\theta_0$, $h$ is our **hypothesis function**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "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 h(x, theta1, theta0):\n",
+    "    return x * theta1 + theta0\n",
+    "\n",
+    "theta1 = -0.03\n",
+    "theta0 = 9000\n",
+    "\n",
+    "line_xs = [km.min(), km.max()]\n",
+    "line_ys = [h(x, theta1, theta0) for x in line_xs]\n",
+    "plt.plot(line_xs, line_ys, color='r')\n",
+    "\n",
+    "plot_data()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We can adjust our prediction by changing $\\theta_1$ and $\\theta_0$.\n",
+    "\n",
+    "Now, we need a way to know if our line correctly predicts the data or not, so that if not, we can slightly change it to better fit the data.\n",
+    "\n",
+    "Introducing the **error function** $J$ (also called cost function):\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "J(\\theta_1, \\theta_0) &= \\frac{1}{2n} \\sum_{i=1}^{n}(h(x_i) - y_i)^2 \\\\\n",
+    "                      &= \\frac{1}{2n} \\sum_{i=1}^{n}((\\theta_1 x_i + \\theta_0) - y_i)^2\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "It's the sum of the square of the distance between what we predicted and the actual data.\n",
+    "$n$ is the number of entry we got, we divide by it to normalize the result."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def error_function(xs, ys, theta1, theta0):\n",
+    "    return (1 / (2 * len(xs))) * sum(\n",
+    "        [(h(x, theta1, theta0) - y) ** 2\n",
+    "         for x, y in zip(xs, ys)])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "386217.6189791666\n"
+     ]
+    }
+   ],
+   "source": [
+    "xs = km\n",
+    "ys = price\n",
+    "print(error_function(xs, ys, theta1, theta0))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The error value is pretty high, this means that the approximated line is garbage.\n",
+    "\n",
+    "The process of learning will be done by reducing this error function. To make that happen, we need to know *how* we are going to change $\\theta_1$ and $\\theta_0$.\n",
+    "\n",
+    "Here comes partial derivatives:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "    \\theta_1 &:= \\theta_1 - \\alpha \\frac{\\partial}{\\partial \\theta_1} J(\\theta_1, \\theta_0) \\\\\n",
+    "    \\theta_0 &:= \\theta_0 - \\alpha \\frac{\\partial}{\\partial \\theta_0} J(\\theta_1, \\theta_0)\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "First of all $\\alpha$ is the **learning rate** we multiply the change we want to make by it, it describes how *fast* we are changing the hypothesis component.\n",
+    "\n",
+    "The partial derivative of $J$ by regard to $\\theta_1$ for example is how the error function changes if we consider $\\theta_0$ has a constant.\n",
+    "\n",
+    "We compute the *next* $\\theta_0$ by subtracting learning rate * partial derivative from it.\n",
+    "\n",
+    "Lets expand theses partial derivatives:\n",
+    "\n",
+    "$$\n",
+    "\\newcommand{\\partialthetazero}{\\frac{\\partial}{\\partial \\theta_0}}\n",
+    "\\begin{align}\n",
+    "    &  \\partialthetazero \\left[ J(\\theta_1, \\theta_0) \\right] \\\\\n",
+    "    =& \\partialthetazero \\left[ \\frac{1}{2n} \\sum_{i=1}^{n}((\\theta_1 x_i + \\theta_0) - y_i)^2 \\right] \\\\\n",
+    "    =& \\frac{1}{2n} \\sum_{i=1}^{n} \\partialthetazero \\left[ ((\\theta_1 x_i + \\theta_0) - y_i)^2 \\right] \\\\\n",
+    "    =& \\frac{1}{2n} \\sum_{i=1}^{n} 2((\\theta_1 x_i + \\theta_0) - y_i) \\partialthetazero \\left[ ((\\theta_1 x_i + \\theta_0) - y_i) \\right] \\\\\n",
+    "    =& \\boxed{ \\frac{1}{n} \\sum_{i=1}^{n} ((\\theta_1 x_i + \\theta_0) - y_i) }\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "And for $\\theta_1$:\n",
+    "\n",
+    "$$\n",
+    "\\newcommand{\\partialthetaone}{\\frac{\\partial}{\\partial \\theta_1}}\n",
+    "\\begin{align}\n",
+    "    &  \\partialthetaone \\left[ J(\\theta_1, \\theta_0) \\right] \\\\\n",
+    "    =& \\partialthetaone \\left[ \\frac{1}{2n} \\sum_{i=1}^{n}((\\theta_1 x_i + \\theta_0) - y_i)^2 \\right] \\\\\n",
+    "    =& \\frac{1}{2n} \\sum_{i=1}^{n} \\partialthetaone \\left[ ((\\theta_1 x_i + \\theta_0) - y_i)^2 \\right] \\\\\n",
+    "    =& \\frac{1}{2n} \\sum_{i=1}^{n} 2((\\theta_1 x_i + \\theta_0) - y_i) \\partialthetaone \\left[ ((\\theta_1 x_i + \\theta_0) - y_i) \\right] \\\\\n",
+    "    =& \\boxed { \\frac{1}{n} \\sum_{i=1}^{n} ((\\theta_1 x_i + \\theta_0) - y_i) x_i }\n",
+    "\\end{align}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def theta0_partial(xs, ys, theta1, theta0):\n",
+    "    return sum([h(x, theta1, theta0) - y\n",
+    "                for x, y in zip(xs, ys)]) / len(xs)\n",
+    "\n",
+    "def theta1_partial(xs, ys, theta1, theta0):\n",
+    "    return sum([(h(x, theta1, theta0) - y) * x\n",
+    "                for x, y in zip(xs, ys)]) / len(xs)\n",
+    "\n",
+    "\n",
+    "def gradient_descent(xs, ys, theta1, theta0, alpha, iterations):\n",
+    "    for _ in range(iterations):\n",
+    "        next_theta1 = theta1 - alpha * theta1_partial(xs, ys, theta1, theta0)\n",
+    "        next_theta0 = theta0 - alpha * theta0_partial(xs, ys, theta1, theta0)\n",
+    "        theta1 = next_theta1\n",
+    "        theta0 = next_theta0\n",
+    "    return theta1, theta0"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The gradient descent work by repeating the formula above a certain amount of times.\n",
+    "\n",
+    "But before actually running the algorithm, the data need to be normalized. This means that they will be in a range like 0 < x < 1 but they will keep their relative distance to each other.\n",
+    "\n",
+    "If we don't do this, our values will go to infinity or some do some sneaky floating point arithmetic stuff.\n",
+    "\n",
+    "Here is how we normalize:\n",
+    "\n",
+    "$$\n",
+    "X_{normalized} = \\frac{X - X_{min}}{X_{max} - X_{min}}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def normalize(x):\n",
+    "    return (x - x.min()) / (x.max() - x.min())\n",
+    "\n",
+    "def denormalize(x, x_ref):\n",
+    "    return x * (x_ref.max() - x_ref.min()) + x_ref.min()\n",
+    "\n",
+    "X = normalize(xs)\n",
+    "Y = ys"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 33,
+   "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": [
+    "theta1, theta0 = 0, 0\n",
+    "\n",
+    "line_xs = [X.min(), X.max()]\n",
+    "line_ys = [h(x, theta1, theta0) for x in line_xs]\n",
+    "plt.plot(line_xs, line_ys, color='r')\n",
+    "\n",
+    "plt.scatter(X, Y)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 34,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "error before:  20880519.291666664\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(\"error before: \", error_function(X, Y, theta1, theta0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "-4656.591444722055 8008.439832646783\n",
+      "error after:  222822.6225356367\n"
+     ]
+    }
+   ],
+   "source": [
+    "alpha = 0.1\n",
+    "iterations = 20000\n",
+    "\n",
+    "theta1, theta0 = gradient_descent(X, Y, theta1, theta0, alpha, iterations)\n",
+    "\n",
+    "print(theta1, theta0)\n",
+    "print(\"error after: \", error_function(X, Y, theta1, theta0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 36,
+   "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": [
+    "line_xs = [X.min(), X.max()]\n",
+    "line_ys = [h(x, theta1, theta0) for x in line_xs]\n",
+    "plt.plot(line_xs, line_ys, color='r')\n",
+    "\n",
+    "plt.scatter(X, Y)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now that our model is trained, if we want to make a prediction, we have to normalize the prediction the same way that we normalized our trainning data."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "7427.151470348072"
+      ]
+     },
+     "execution_count": 37,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "predict = 50_000\n",
+    "predict = (predict - xs.min()) / (xs.max() - xs.min())\n",
+    "h(predict, theta1, theta0)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "If we plot this point with the original data. We can see that the prediction seems correct."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 38,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAY8AAAD8CAYAAACPWyg8AAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDMuMC4zLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvnQurowAAFY9JREFUeJzt3X+MXeV95/H3N8bQgY0yBlwLj0ntbllLVJFiOgpURJE2bD3AVsUboawjJKwUyVU3u9tWqhu7+SNV0ipkvbvZRrtL4i1ZkYiGsKxjrE22rovzV6UQxh2KgcT1pIHgy69JjIlURl3jfveP+4xzZzq/jn1/nDPzfklX99znnHPvcx4u/sx5zvPcE5mJJElVvGPQFZAkNY/hIUmqzPCQJFVmeEiSKjM8JEmVGR6SpMoMD0lSZYaHJKkyw0OSVNllg67AYq699trcvHnzoKshSY1y/PjxH2Xm+l5+Rq3DY/PmzYyPjw+6GpLUKBHxYq8/w24rSVJlhockqTLDQ5JUmeEhSarM8JAkVVbr0Vaa7dBEi/1HTvLy2Wk2Dg+xZ2wrO7aNDLpaklYhw6MhDk202HfwBNPnzgPQOjvNvoMnAAwQSX1nt1VD7D9y8kJwzJg+d579R04OqEaSVjPDoyFePjtdqVySesnwaIiNw0OVyiWplwyPhtgztpWhtWtmlQ2tXcOesa0DqpGk1cwL5g0xc1Hc0VaS6sDwaJAd20YMC0m1YLeVJKkyw0OSVJnhIUmqzPCQJFW2rPCIiN+JiOci4tmI+GpE/ExEbImIJyNiMiK+FhGXl22vKK8ny/rNHe+zr5SfjIix3hySJKnXlhxtFREjwL8HbszM6Yh4FNgJ3Al8LjMfiYgvAPcBD5TnNzLzFyJiJ/BZ4F9HxI1lv18ENgJ/ERH/LDPPz/Oxl6RJPyDY7bo26dglNddyu60uA4Yi4jLgSuAV4IPAY2X9Q8COsnxXeU1Zf1tERCl/JDP/PjN/AEwC77v0Q5ht5gcEW2enSX76A4KHJlrd/qhL1u26NunYJTXbkuGRmS3gPwI/pB0abwLHgbOZ+XbZ7DQw8+ftCPBS2fftsv01neXz7NM1TfoBwW7XtUnHLqnZlgyPiFhH+6xhC+3upquA23tVoYjYHRHjETE+NTVVef8m/YBgt+vapGOX1GzL6bb6F8APMnMqM88BB4FbgeHSjQWwCZjpG2kB1wOU9e8CftxZPs8+F2TmgcwczczR9evXVz6gJv2AYLfr2qRjl9RsywmPHwK3RMSV5drFbcDzwLeAu8s2u4DHy/Lh8pqy/lhmZinfWUZjbQFuAL7TncP4qSb9gGC369qkY5fUbEuOtsrMJyPiMeCvgLeBCeAA8A3gkYj4w1L2YNnlQeArETEJnKE9worMfK6M1Hq+vM/HejHSqkk/INjtujbp2CU1W7RPCuppdHQ0x8fHB10NSWqUiDiemaO9/AxnmEuSKjM8JEmVGR6SpMoMD0lSZYaHJKkyw0OSVJnhIUmqzPCQJFVmeEiSKjM8JEmVGR6SpMqW/GFE1Z+3npXUb4ZHw83cenbmDoIzt54FDBBJPWN4NMRCZxeL3XrW8JDUK4ZHAyx2duGtZyUNghfMG2CxswtvPStpEAyPfnr4Ydi8Gd7xjvbzww8va7fFzi689aykQbDbql8efhh274a33mq/fvHF9muAe+5ZdNeNw0O05gmQjcND3npW0kB4G9p+2by5HRhz/dzPwQsvLLrr3Gse0D67+MyH3mNISPpH+nEbWs88+uWHP6xW3sGzC0l1Y3j0y7vfPf+Zx7vfvazdd2wbMSwk1YYXzPvlj/4IrrxydtmVV7bLJalhDI9+ueceOHCgfY0jov184MCSF8slqY7stuqne+4xLCStCJ55SJIqMzwkSZUZHpKkygwPSVJlhockqTLDQ5JUmeEhSarM8JAkVWZ4SJIqMzwkSZUtGR4RsTUinu54/CQifjsiro6IoxFxqjyvK9tHRHw+IiYj4pmIuKnjvXaV7U9FxK5eHpgkqXeW/G2rzDwJvBcgItYALeDrwF7gicy8PyL2ltcfB+4AbiiPm4EHgJsj4mrgk8AokMDxiDicmW90/ahUC4cmWt6DRFqhqnZb3QZ8PzNfBO4CHirlDwE7yvJdwJez7dvAcERcB4wBRzPzTAmMo8Dtl3wEqqWZux+2zk6TQOvsNPsOnuDQRGvQVZPUBVXDYyfw1bK8ITNfKcuvAhvK8gjwUsc+p0vZQuWzRMTuiBiPiPGpqamK1VNd7D9yctZtcwGmz51n/5GTA6qRpG5a9k+yR8TlwK8B++auy8yMiK7cDD0zDwAHoH0P82685yDVueuml3V7+ex0pXJJzVLlzOMO4K8y87Xy+rXSHUV5fr2Ut4DrO/bbVMoWKl+x6tx10+u6bRweqlQuqVmqhMdH+GmXFcBhYGbE1C7g8Y7ye8uoq1uAN0v31hFge0SsKyOztpeyFavOXTe9rtuesa0MrV0zq2xo7Rr2jG3tyvtLGqxldVtFxFXArwC/0VF8P/BoRNwHvAh8uJR/E7gTmATeAj4KkJlnIuLTwFNlu09l5plLPoIaq3PXTa/rNtP9VdcuO0mXZlnhkZl/B1wzp+zHtEdfzd02gY8t8D5fAr5UvZrNtHF4iNY8/xjXoeumH3XbsW3EsJBWKGeY91A/u24OTbS49f5jbNn7DW69/9iS1y4G2a1Uta6S6mfZo61UXb+6bmYufs9cw5i5+N1Zh0HVrRt1lVQ/0e5lqqfR0dEcHx8fdDVq79b7j83bBTUyPMRf7v3gAGq0sCbVVWqqiDiemaO9/Ay7rVaAOl+Yn6tJdZW0MMNjBWjSnIom1VXSwgyPFaBJcyqaVFdJC/OC+QrQpDkVTaqrpIV5wVySVph+XDD3zGOVqfMPNUpqDsNjFXGOhaRu8YL5KlLnH2qU1CyGxyriHAtJ3WJ4rCLOsZDULYbHKuIcC0nd4gXzVcQ5FpK6xfBYZbzHhqRusNtKklSZZx4N1K+Jfk4olLQQw6Nh+jXRzwmFkhZjt1XD9GuinxMKJS3G8GiYfk30c0KhpMUYHg3Tr4l+TiiUtBjDo2H6NdHPCYWSFuMF84bp10Q/JxRKWow3g5KkFcabQak2nPMhqZPhoSU550PSXF4w15Kc8yFpLsNDS3LOh6S5DA8tyTkfkuYyPLQk53xImssL5lqScz4kzWV4aFm8iZSkTsvqtoqI4Yh4LCK+FxHfjYhfjoirI+JoRJwqz+vKthERn4+IyYh4JiJu6nifXWX7UxGxq1cHJUnqreWeefwx8GeZeXdEXA5cCfw+8ERm3h8Re4G9wMeBO4AbyuNm4AHg5oi4GvgkMAokcDwiDmfmG109IvWdEwgvju2mJlvyzCMi3gV8AHgQIDP/X2aeBe4CHiqbPQTsKMt3AV/Otm8DwxFxHTAGHM3MMyUwjgK3d/Vo1HczEwhbZ6dJfjqB8NBEa9BVqzXbTU23nG6rLcAU8D8jYiIi/iQirgI2ZOYrZZtXgQ1leQR4qWP/06VsoXI1mBMIL47tpqZbTnhcBtwEPJCZ24C/o91FdUG2f12xK7+wGBG7I2I8Isanpqa68ZbqIScQXhzbTU23nPA4DZzOzCfL68doh8lrpTuK8vx6Wd8Cru/Yf1MpW6h8lsw8kJmjmTm6fv36KseiAXAC4cWx3dR0S4ZHZr4KvBQRMzPCbgOeBw4DMyOmdgGPl+XDwL1l1NUtwJule+sIsD0i1pWRWdtLmRrMCYQXx3ZT0y13tNW/Ax4uI63+Fvgo7eB5NCLuA14EPly2/SZwJzAJvFW2JTPPRMSngafKdp/KzDNdOQoNjBMIL47tpqbzZlCStMJ4MyhJPeEcE10qw0NaZby5l7rBX9WVVhnnmKgbDA9plXGOibrB8JBWGeeYqBsMD2mVcY6JusEL5tIq4xwTdYPhIa1C3txLl8puK0lSZYaHJKkyw0OSVJnhIUmqzPCQJFVmeEiSKjM8JEmVGR6SpMoMD0lSZYaHJKkyw0OSVJm/baUVx1usSr1neGhF8RarUn/YbaUVxVusSv1heGhF8RarUn8YHlpRvMWq1B+Gh1YUb7HaO4cmWtx6/zG27P0Gt95/jEMTrUFXSQPkBXOtKN5itTcciKC5DA+tON5itfsWG4hgW69OdltJWpIDETSX4SFpSQ5E0FyGh6QlORBBc3nNQ9KSHIiguQwPScviQAR1sttKklSZ4SFJqmxZ4RERL0TEiYh4OiLGS9nVEXE0Ik6V53WlPCLi8xExGRHPRMRNHe+zq2x/KiJ29eaQJEm9VuXM459n5nszc7S83gs8kZk3AE+U1wB3ADeUx27gAWiHDfBJ4GbgfcAnZwJHktQsl9JtdRfwUFl+CNjRUf7lbPs2MBwR1wFjwNHMPJOZbwBHgdsv4fMlSQOy3PBI4M8j4nhE7C5lGzLzlbL8KrChLI8AL3Xse7qULVQ+S0TsjojxiBifmppaZvUkSf203KG678/MVkT8LHA0Ir7XuTIzMyKyGxXKzAPAAYDR0dGuvKckqbuWdeaRma3y/DrwddrXLF4r3VGU59fL5i3g+o7dN5WyhcolSQ2zZHhExFUR8c6ZZWA78CxwGJgZMbULeLwsHwbuLaOubgHeLN1bR4DtEbGuXCjfXsokSQ2znG6rDcDXI2Jm+z/NzD+LiKeARyPiPuBF4MNl+28CdwKTwFvARwEy80xEfBp4qmz3qcw807UjkST1TWTW97LC6Ohojo+PD7oaktQoEXG8Y1pFTzjDXJJUmeEhSarM8JAkVWZ4SJIqMzwkSZUZHpKkygwPSVJlhockqTLDQ5JUmeEhSarM8JAkVWZ4SJIqMzwkSZUZHpKkygwPSVJly72HuSRdlEMTLfYfOcnLZ6fZODzEnrGt7Ng2Muhq6RIZHpJ65tBEi30HTzB97jwArbPT7Dt4AsAAaTjDQ6qplfAX+/4jJy8Ex4zpc+fZf+Rk445FsxkeUg2tlL/YXz47XalczeEFc6mGFvuLvUk2Dg9VKldzGB5SDa2Uv9j3jG1laO2aWWVDa9ewZ2zrgGqkbjE8pBpaKX+x79g2wmc+9B5GhocIYGR4iM986D2N6nrT/LzmIdXQnrGts655QHP/Yt+xbcSwWIEMD6mGZv6xbfpoK61chodUU/7FrjrzmockqTLDQ5JUmeEhSarM8JAkVWZ4SJIqMzwkSZUZHpKkygwPSVJlyw6PiFgTERMR8X/K6y0R8WRETEbE1yLi8lJ+RXk9WdZv7niPfaX8ZESMdftgJEn9UeXM47eA73a8/izwucz8BeAN4L5Sfh/wRin/XNmOiLgR2An8InA78N8jYvbPbUqSGmFZ4RERm4B/CfxJeR3AB4HHyiYPATvK8l3lNWX9bWX7u4BHMvPvM/MHwCTwvm4chCSpv5Z75vFfgN8D/qG8vgY4m5lvl9engZkf4RkBXgIo698s218on2cfSVKDLBkeEfGrwOuZebwP9SEidkfEeESMT01N9eMjJUkVLefM41bg1yLiBeAR2t1VfwwMR8TMr/JuAlpluQVcD1DWvwv4cWf5PPtckJkHMnM0M0fXr19f+YAkSb23ZHhk5r7M3JSZm2lf8D6WmfcA3wLuLpvtAh4vy4fLa8r6Y5mZpXxnGY21BbgB+E7XjkSS1DeXcj+PjwOPRMQfAhPAg6X8QeArETEJnKEdOGTmcxHxKPA88Dbwscw8/4/fVpJUd9E+Kain0dHRHB8fH3Q1JKlRIuJ4Zo728jOcYS5Jqszb0Eo9dGii5X3ItSIZHlKPHJpose/gCabPtS/ttc5Os+/gCQADRI1nt5XUI/uPnLwQHDOmz51n/5GTA6qR1D2Gh9QjL5+drlQuNYnhIfXIxuGhSuVSkxgeUo/sGdvK0NrZPxw9tHYNe8a2DqhGUvd4wVzqkZmL4o62UhVNGaFneEg9tGPbSC3/x1c9NWmEnt1WklQTTRqhZ3hIUk00aYSe4SFJNdGkEXqGhyTVRJNG6HnBXJJqokkj9AwPSaqRpozQs9tKklSZ4SFJqszwkCRVZnhIkiozPCRJlUVmDroOC4qIKeDFQdejB64FfjToStSQ7TI/22V+tsv8rgWuysz1vfyQWofHShUR45k5Ouh61I3tMj/bZX62y/z61S52W0mSKjM8JEmVGR6DcWDQFagp22V+tsv8bJf59aVdvOYhSarMMw9JUmWGxyWIiBci4kREPB0R46Xs6og4GhGnyvO6Uh4R8fmImIyIZyLipo732VW2PxURuzrKf6m8/2TZN/p/lEuLiC9FxOsR8WxHWc/bYaHPqIsF2uUPIqJVvjNPR8SdHev2lWM8GRFjHeW3l7LJiNjbUb4lIp4s5V+LiMtL+RXl9WRZv7k/R7w8EXF9RHwrIp6PiOci4rdK+ar+zizSLvX8zmSmj4t8AC8A184p+w/A3rK8F/hsWb4T+L9AALcAT5byq4G/Lc/ryvK6su47Zdso+94x6GNeoB0+ANwEPNvPdljoM+ryWKBd/gD43Xm2vRH4a+AKYAvwfWBNeXwf+Hng8rLNjWWfR4GdZfkLwG+W5X8DfKEs7wS+Nui2mHOs1wE3leV3An9Tjn9Vf2cWaZdafmcG3mBNfjB/eJwEruv4Mpwsy18EPjJ3O+AjwBc7yr9Yyq4DvtdRPmu7uj2Azcz+R7Ln7bDQZ9TpMU+7LPQPwT5gX8frI8Avl8eRuduVfxR/BFxWyi9sN7NvWb6sbBeDbotF2uhx4Ff8zizYLrX8zthtdWkS+POIOB4Ru0vZhsx8pSy/CmwoyyPASx37ni5li5Wfnqe8KfrRDgt9Rt3929L98qWObpOq7XINcDYz355TPuu9yvo3y/a1U7pHtgFP4nfmgjntAjX8zhgel+b9mXkTcAfwsYj4QOfKbMf4qh/O1o92aFBbPwD8U+C9wCvAfxpsdQYnIv4J8L+B387Mn3SuW83fmXnapZbfGcPjEmRmqzy/DnwdeB/wWkRcB1CeXy+bt4DrO3bfVMoWK980T3lT9KMdFvqM2srM1zLzfGb+A/A/aH9noHq7/BgYjojL5pTPeq+y/l1l+9qIiLW0/4F8ODMPluJV/52Zr13q+p0xPC5SRFwVEe+cWQa2A88Ch4GZUR+7aPdbUsrvLSNHbgHeLKfPR4DtEbGunI5up90P+Qrwk4i4pYwUubfjvZqgH+2w0GfU1sw/XMW/ov2dgfax7CyjXrYAN9C+6PsUcEMZJXM57YuZh8tfzd8C7i77z23jmXa5GzhWtq+F8t/xQeC7mfmfO1at6u/MQu1S2+/MoC8KNfVBeyTDX5fHc8AnSvk1wBPAKeAvgKtLeQD/jfYoiBPAaMd7/TowWR4f7SgfLV+U7wP/lZpe9AS+Svt0+hztftT7+tEOC31GXR4LtMtXynE/U/6Hva5j+0+UYzxJx8g62qON/qas+8Sc7+B3Snv9L+CKUv4z5fVkWf/zg26LOe3yftrdRc8AT5fHnav9O7NIu9TyO+MMc0lSZXZbSZIqMzwkSZUZHpKkygwPSVJlhockqTLDQ5JUmeEhSarM8JAkVfb/AV9ouOtM+5SnAAAAAElFTkSuQmCC\n",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plt.scatter(km, price)\n",
+    "plt.scatter([50_000], h(predict, theta1, theta0), color='r')\n",
+    "plt.show()"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.8.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/notebooks/.ipynb_checkpoints/logistic-regression-checkpoint.ipynb b/notebooks/.ipynb_checkpoints/logistic-regression-checkpoint.ipynb
new file mode 100644
index 0000000..8c862d3
--- /dev/null
+++ b/notebooks/.ipynb_checkpoints/logistic-regression-checkpoint.ipynb
@@ -0,0 +1,707 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Logistic Regression\n",
+    "\n",
+    "Logistic regression is a *binary classification algorithm*."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Text(0.5, 1.0, 'Data')"
+      ]
+     },
+     "execution_count": 7,
+     "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": 8,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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z6AtDx9rK5YYbbjjqtssuu4zLLrssAukk1tX4A7zw6RaenLuepIQEHvz2YK49s8dhX72X2KVCF2kmthQe5O63lrF8x34uGpzDr789mE6Z2oY8nqjQReKcc463Fu7goXdWk5yYwLP/OYxLTu3kdSxpAip0kThWWlnDz2d8xT9W7uLs3tk8duVQjcrjWNQVunNO27xGmHYBEJ827inlllcXs62onKkTB/D983qRoLnyuBZVhZ6WlkZRURHZ2dkq9QhxzlFUVHTUfmckts1ZuYufvr2MFimJvHbzKM7sle11JImAqCr0rl27kp+fz969e0/6uQ7tcyVWRTJ/WloaXbt2jci6pGk553hnUzUz5ixmaLcsnrt2mKZYmpGoKvTk5GR69uwZlufKy8sL61HsIy3W80vk+fwBfjV7FTM21DDptM78/junkpp07KNrSfyJqkIXkcYpr/Zx1xtLmbtmD5f0TOaJK0/TfHkzpEIXiXGllTXc+OJClmzfx28mDaZb1VaVeTOlr/6LxLADlTVcP/1Llu3Yz9PXDOO6s3K9jiQe0ghdJEaVlNdw/fQFrC44wLNThnHR4I5eRxKPqdBFYtCByhqufWEB63aV8ty1wxk3MMfrSBIFVOgiMaayxs/NLy1i7a4DTLtuBGMHdPA6kkQJFbpIDKnxB7jj9SUs3FbMU1efrjKXw+hDUZEYEQg4fjHjK+au2cNDk4Zw2dDOXkeSKKNCF4kR//3+OmYu3clPL+jHdWf28DqORCEVukgMmLE4nz/lbeKaM7pzx/l9vI4jUUqFLhLlFm4tZurMrzi7dzYPTRqsHdfJManQRaLYjuJybn11MV3btOSPU4aRnKgfWTk2vTtEolRFtZ/vv7IInz/AC98dQVbLFK8jSZTTZosiUcg5x72zVrBudykv3jCSXu3TvY4kMUAjdJEo9NbCHcxcspM7z+/LmP7a1lzqR4UuEmVW7izhgdmrOK9vO340rq/XcSSGqNBFosiByhp+8NoSslul8ORVp5Go3eBKA2gOXSSKPDBrJTv3V/D2rWeSnZ7qdRyJMRqhi0SJvy3byaxlX3Pn+X0Y3qOt13EkBqnQRaLAzv0V3DdrJcO6Z3HHWH0TVBpHhS7iMX/A8ZO3lhEIOJ686nSS9OUhaSTNoYt47PlPNrNgSzGPXjGU7tktvY4jMUxDAREPbdpbxmMfrOeiwTn8x7AuXseRGKdCF/HIof2bt0hO5DeTh2inW3LSVOgiHnnl860s2raPBy4dRIeMNK/jSBxQoYt4YEdxOb+bs45v9WvP5ZpqkTBRoYtEmHOOe2Z+RWKC8V+Xn6KpFgkbFbpIhM1e/jWfbSziFxP60yWrhddxJI6o0EUiqKSiht+8s4ahXTP5z1E6LqiEl7ZDF4mgx99fR/HBKl68YaR2vCVhpxG6SISsyC/h1S+2cd2ZPTila6bXcSQOqdBFIsAfcNw3awVtW6Xykwv7ex1H4pQKXSQC3l60g+X5Jdx/6UAyWyR7HUfilApdpIkdqKzh0X+u44zctnx7aGev40gc04eiIk3s6Q83UFxezcuXDdI259KkNEIXaUJbCg/y0vytXDm8G0O66INQaVoqdJEm9PC7q0lNSuT/XqQPQqXpqdBFmsjH6/cyd80e7ji/D+0zdHxQaXoqdJEm4A84Hn53DT2yW3LjOblex5FmQoUu0gT+siSfdbtL+cWEAaQmJXodR5oJFbpImFVU+3n8/fWc1i2LiUM6eh1HmhEVukiYTf9sC7sOVDJ14gBtpigRpUIXCaOisir+lLeJ8QNzGNUr2+s40syo0EXC6Ol5Gymv9nHPRG2mKJGnQhcJkx3F5by2YBtXjexGnw4ZXseRZkiFLhImT8xdT4IZPxrXz+so0kyp0EXCYMPuUv66dCffPTuXjplpXseRZkqFLhIGj3+wnlYpSdz2rd5eR5FmTIUucpK+yt/PP1bu4ubzetK2VYrXcaQZU6GLnKRH319Pm5bJfO/cnl5HkWZOhS5yEhZsLuLj9Xv5wZg+ZKTpSETiLRW6SCM553jsg/V0yEjlurN6eB1HRIUu0lifbyriyy3F/GBMb9KStQMu8Z4KXaQRnHM8MXc9HVuncfUZ3b2OIwKo0EUa5dONhSzcuo8fjtXoXKKHCl2kgZxzPPHBejpnpnHlyG5exxH5hgpdpIE+3lDIku37+cHYPjp4hUQVFbpIAxwanXfJasGVIzQ6l+iiQhdpgI83FLJsx35+MLY3KUn68ZHoonekSD055/jD3ODc+XeGd/U6jshRVOgi9TR/UxFLtu/n9jG9NXcuUUmFLlJPf/hwAzmtU7lCc+cSpVToIvXwxebgt0Jv/5a2O5fopUIXqYenPtxA+4xUfStUopoKXeQEFm0tZv6mIm4d3Uujc4lqKnSRE3h63kayW6UwZZT2qCjRTYUuchzLd+znX+v3cvN5vWiRotG5RDcVushxPPPRRjJbJGt/5xITVOgix7Cm4AAfrN7NTef0JD01yes4IiekQhc5hmc+2kh6ahI3nJ3rdRSRelGhi9Rh454y3ltRwPVn9SCzpY4VKrFBhS5Shz/mbSQtKZHvndvT6ygi9VavQjezCWa2zsw2mtk9ddx/g5ntNbNlodPN4Y8qEhl7ygP8bdnXTBnVnez0VK/jiNTbCT/pMbNE4FngAiAfWGhms51zq49Y9C3n3B1NkFEkot7bXEOiGd8f3cvrKCINUp8R+hnARufcZudcNfAmMKlpY4l4o6Ckgk93+rhyZFdyWqd5HUekQeqzLVYXYEet6/nAqDqW+w8zGw2sB37snNtx5AJmdgtwC0BOTg55eXkNDlxfZWVlTfr8TU35vfHamioCzjE0ZW9M5ofY/bc/RPkbL1wb1/4deMM5V2VmtwIvA+cfuZBzbhowDWDEiBFuzJgxYVr90fLy8mjK529qyh95hWVVfPLhPM7ukswVFx/19o0ZsfhvX5vyN159plx2ArV3AN01dNs3nHNFzrmq0NXngeHhiScSOS98uoUqX4BLe2kzRYlN9Sn0hUBfM+tpZinA1cDs2guYWadaV78NrAlfRJGmt7+8mlc/38Ylp3SiYyttzSux6YTvXOecD7gD+CfBon7bObfKzB4ys2+HFrvLzFaZ2XLgLuCGpgos0hRemr+Vsiofd5zfx+soIo1Wrzl059x7wHtH3PZArctTganhjSYSGaWVNUz/dAsXDsphQMfW7FrrdSKRxtHfltLsvfrFNg5U+rjz/L5eRxE5KSp0adbKq308/8kWxvRvzyldM72OI3JSVOjSrL2+YDvFB6u5U3PnEgdU6NJsVdb4mfbxZs7unc3wHm29jiNy0lTo0mz9edEO9pRWacsWiRsqdGmWqnx+/pi3iZG5bTirV7bXcUTCQoUuzdKMxfkUlFRy17i+mJnXcUTCQoUuzU61L8AfP9rE6d2zOLdPO6/jiISNCl2anb8uzWfn/gqNziXuqNClWanxB3jmo42c2jWTMf3aex1HJKxU6NKszFq6kx3FFdx1vkbnEn9U6NJs+EKj80GdWjNuYAev44iEnQpdmo2ZS3eyraicH1/QT6NziUsqdGkWavwBnp63gVO6ZDJeo3OJUyp0aRb+sjifHcUV/PgCzZ1L/FKhS9yr9gV4et5GhnbLYmx/jc4lfqnQJe7NWBzc7vzu8RqdS3xToUtcq/YFePajjZzePUvbnUvcU6FLXHtz4XZ27q/gJ9qyRZoBFbrErfJqH099uJEze7XVPlukWVChS9x6ef42Csuq+NlF/TU6l2ZBhS5xqaSihuf+tYnzB3TQ0Yik2VChS1x6/pPNlFTU8NML+3kdRSRiVOgSdwrLqnjh0y1cemonBnfO9DqOSMSo0CXuPDNvI5U1fn58gUbn0ryo0CWubCs6yGsLtnHVyO70bp/udRyRiFKhS1z573+uIykhgR+P7+t1FJGIU6FL3Fi2Yz/vfFXA90f3okPrNK/jiEScCl3ignOOR95bQ7v0FG4Z3cvrOCKeUKFLXPho3R4WbCnmR+P7kZ6a5HUcEU+o0CXm+fwBHnlvLb3ateLqkd28jiPiGRW6xLzXv9zOhj1l3DNxAMmJektL86V3v8S0/eXVPP7Bes7pk80Fg3K8jiPiKRW6xLQn527gQEUN9186SDvgkmZPhS4xa/3uUl79YhtTRvVgQMfWXscR8ZwKXWKSc47fvLOaVimJ+oq/SIgKXWLSB6t388mGQu4e34+2rVK8jiMSFVToEnPKq308+PfV9M/J4LqzengdRyRq6BsYEnOe+nAjO/dX8OfbztJmiiK16KdBYsr63aU8/8lmrhjelZG5OhKRSG0qdIkZzjnum7WSVqlJ3DNxgNdxRKKOCl1ixswlO/lySzH3TBxAdnqq13FEoo4KXWLC3tIqfvPuak7vnsVVI7S/FpG6qNAlJvx69irKq/z8/j9OJSFB3wgVqYsKXaLenJUFvLuigLvG9aFvTobXcUSilgpdotr+8mrum7WKQZ1ac+u3ensdRySqaTt0iWoPvbOa/eXVvHzTSG1zLnIC+gmRqDVn5S5mLtnJbd/qzeDOmV7HEYl6KnSJSrsPVDJ15lcM6dKau8b19TqOSExQoUvUCQQc//fPy6mo8fPkVaeTkqS3qUh96CdFos6L87fyyYZC7rtkEH06pHsdRyRmqNAlqqwpOMDv5qxl3IAOTBnV3es4IjFFhS5Ro7Syhh+8toTWacn87jun6pByIg2kzRYlKjjnuOcvK9heXM5rN4+infbVItJgGqFLVHhp/lbeXVHAzy7qz5m9sr2OIxKTVOjiucXb9vHwu2sYPzCHW0f38jqOSMxSoYundpVU8oPXFtMpK43HrhyqeXORk6BCF8+UV/u4+ZWFlFX6+J/rR5DZItnrSCIxTR+KiicCAcdP3lrOqq8P8Pz1IxjQsbXXkURinkbo4onHPljHnFW7uPfigYwbmON1HJG4oEKXiHt9wXae/WgTV4/sxvfO7el1HJG4oUKXiHpvRQH3zlrBmP7teWjSEH0IKhJGKnSJmE83FHL3m8sY3r0Nf5oyXDvdEgkz/URJRCzdvo9bXl1Er/ateOG7I2mRkuh1JJG4o0KXJrd42z6uf+FL2qWn8spNZ5DZUpsnijQFFbo0qYVbi7n+hQVkp6fw1q1n0qF1mteRROKWtkOXJvPF5iJuemkhHTPTeOP7Z5KjMhdpUip0aRKLd/uYNvdLurdtyWvfH0WHDJW5SFPTlIuE3cvzt/LM0ioGdmrNm7ecqTIXiRCN0CVsAgHH7+as5f99vJnTOyTy+vfP1NYsIhGkQpewOFBZw0/eWs7cNbu59szunJ9ZqDIXiTBNuchJ27C7lMnPfMZH6/bwwKWD+M2kISToG6AiEacRupyUd776mp/P+IqWKYm8fvMoRuloQyKeUaFLo5RW1vDr2av5y5J8Tu+exZ+mDKdjpj78FPGSCl0abPG2Yu5+axk791Vw5/l9uGtcX5ITNXsn4jUVutRbWZWPx95fx8vzt9I5qwVv33oWI3Lbeh1LREJU6FIvc1fv5oG/raTgQCVTRnXnFxMGkJGmfbKIRBMVuhzXxj2lPPLeWj5cu4f+ORk8/Z/DGN6jjdexRKQOKnSpU1FZFU/O3cDrX26nZXIiUycO4KZze2quXCSKqdDlMEVlVUz7ZDOvfr6NKl+AKaO686NxfclOT/U6moicgApdACgoqeDFz7by6ufbqPT5uezUztw1ri99OqR7HU1E6kmF3swt3b6P6Z9t5b0VBTjnmHRaF344to+KXCQGqdCboZKKGmYv28nbi/JZsbOEjNQkbjonl+vPyqVb25ZexxORRlKhNxNVPj8fry/k78u/5p+rdlHlCzCwU2semjSYy4d1JT1VbwWRWKef4jhWVuXj0w17eX/1bj5YtZvSKh9ZLZO5ckQ3rhrZjSFdMr2OKCJhpEKPI4GAY93uUj7bWMi/1u/li81F1PgdrdOSmDCkI5cO7czZvbO16aFInFKhxzCfP8CaglIWbStm0dZ9fLG5iKKD1QD0at+KG8/pyfkDOjC8RxuVuEgzoEKPETX+AFsKD7JyZwkrdx4Inn9dQnm1H4DOmWl8q197zu7TjrN7Z9M5q4XHiUUk0lToUaa82sfWwnK+LPDx1Ycb2JdqRDgAAAb4SURBVLinjPW7S9m0t4wavwMgLTmBQZ1ac+WIbgzv0YbhPdqowEVEhR5JgYCj6GA1uw9UsvtAJV/vr+DrkuD5juJythdXUFhWVesR6+mS1YJ+OemM6d+B/h3TGdQpk97tW5GkKRQROUK9Ct3MJgB/ABKB551zvz3i/lTgFWA4UARc5ZzbGt6o0aXaF6CsykdpZQ0lFTUcqPBRUlHD/opq9h2sZl95DfsOVlN4sJrC0iqKDlZRWFaNP+AOe57kRKNjZhrd2rRk/MAOdGvbkh7ZLSneuoYrJozRcTlFpN5OWOhmlgg8C1wA5AMLzWy2c251rcW+B+xzzvUxs6uB3wFXNUVgAOcc/oDDF/j3uc8fwB9w1IQuf10WYE3BAXx+R7U/QLUvQE3ovNofoMrnp9oXoMoXoKomeL2yJkBFjZ+KGj+V1X7Kq/2U1/ipqPZxsMrPwWofZZU+yqp8VPkCx83YMiWRNi1TaJeeQqfMNIZ0aU37jFRyWqfRISONnNapdMlqQbv0VBISjj7+Zl7xepW5iDRIfUboZwAbnXObAczsTWASULvQJwG/Dl2eATxjZuacO3w4Wsv63aWc/2geAecIOILngeBlf+iyP1Tchy4HAnxzW718+kn9lgtJMGiRnEiLlERSkxJpmXLolETnrGRapSbRKjWJ9NApIy2JjLRkWqclkdkimcyWyWS1SCGrZTJpySpjEYms+hR6F2BHrev5wKhjLeOc85lZCZANFNZeyMxuAW4BSM/JpX1SJWZgQIJZrctHnDASEhJIOOK+xARINPv39dBtSQY11VW0apFGUkLw9uQEIymB0MlITiB0MpITg5cTDeywo9U7wBc61aEmdCqFCoKnXfX4B62PsrIy8vLywvRskRfL+WM5Oyi/17zMH9EPRZ1z04BpACNGjHBv3T2hydaVl5fHmDFjmuz5m5ryeyeWs4Pye83L/PXZVGIn0K3W9a6h2+pcxsySgEyCH46KiEiE1KfQFwJ9zaynmaUAVwOzj1hmNvDd0OXvAPOON38uIiLhd8Ipl9Cc+B3APwlutjjdObfKzB4CFjnnZgMvAK+a2UagmGDpi4hIBNVrDt059x7w3hG3PVDrciVwRXijiYhIQ+jrhiIicUKFLiISJ1ToIiJxQoUuIhInzKutC81sL7CtCVfRjiO+qRpjlN87sZwdlN9rTZ2/h3OufV13eFboTc3MFjnnRnido7GU3zuxnB2U32te5teUi4hInFChi4jEiXgu9GleBzhJyu+dWM4Oyu81z/LH7Ry6iEhzE88jdBGRZkWFLiISJ+K+0M3sTjNba2arzOz3XudpDDP7qZk5M2vndZb6MrP/Dv27f2VmfzWzLK8z1YeZTTCzdWa20czu8TpPQ5hZNzP7yMxWh97vP/I6U0OZWaKZLTWzd7zO0hhmlmVmM0Lv/TVmdlYk1x/XhW5mYwke73Soc24w8KjHkRrMzLoBFwLbvc7SQB8AQ5xzpwLrgake5zmhWgdEnwgMAq4xs0HepmoQH/BT59wg4EzghzGWH+BHwBqvQ5yEPwBznHMDgKFE+LXEdaEDtwO/dc5VATjn9nicpzGeAH5O8ACnMcM5975z7tDBWL8geKSraPfNAdGdc9XAoQOixwTnXIFzbknocinBMunibar6M7OuwCXA815naQwzywRGEzw+BM65aufc/khmiPdC7wecZ2YLzOxfZjbS60ANYWaTgJ3OueVeZzlJNwH/8DpEPdR1QPSYKcTazCwXOB1Y4G2SBnmS4OAl4HWQRuoJ7AVeDE0bPW9mrSIZIKIHiW4KZjYX6FjHXfcSfH1tCf75ORJ428x6RdPh8U6Q/5cEp1ui0vGyO+f+FlrmXoJTAa9FMltzZmbpwF+Au51zB7zOUx9mdimwxzm32MzGeJ2nkZKAYcCdzrkFZvYH4B7g/kgGiGnOufHHus/Mbgdmhgr8SzMLENxxzt5I5TuRY+U3s1MI/sZfbmYQnLJYYmZnOOd2RTDiMR3v3x7AzG4ALgXGRdMv0eOozwHRo5qZJRMs89ecczO9ztMA5wDfNrOLgTSgtZn9r3PuWo9zNUQ+kO+cO/RX0QyChR4x8T7lMgsYC2Bm/YAUYmQvbs65Fc65Ds65XOdcLsE3y7BoKfMTMbMJBP98/rZzrtzrPPVUnwOiRy0L/uZ/AVjjnHvc6zwN4Zyb6pzrGnqvX03wQPOxVOaEfjZ3mFn/0E3jgNWRzBDzI/QTmA5MN7OVQDXw3RgZKcaDZ4BU4IPQXxhfOOdu8zbS8R3rgOgex2qIc4DrgBVmtix02y9DxwSWyLgTeC00INgM3BjJleur/yIicSLep1xERJoNFbqISJxQoYuIxAkVuohInFChi4jECRW6iEicUKGLiMSJ/w+a+bLGSkCgbAAAAABJRU5ErkJggg==\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": 9,
+   "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": 10,
+   "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": 11,
+   "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": 43,
+   "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": 44,
+   "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",
+    "$$\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": 45,
+   "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": 46,
+   "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": 47,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "weights:   -4.121158543930249 33.10478298104216\n",
+      "train cost: 0.0031212648079922676\n",
+      "test cost:  0.0017122703295056325\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": 48,
+   "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": 49,
+   "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": 50,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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dH3PhQnd/zBw5zMVL1yg8JET2bxDXZf6O+bSNaEtYlTAiO0f67N7mqfGbi6Kpuf12s8qtZ08zEIyJgXHjpE2jV7n6Y7rmwH/7zRT1wECzE+ELL5gC3rCh/MOLDNsau5XwmeFUL1adSe0n+V0xT49fFHQwNwlGR5vdTl9/3ewHExUljWMyLCnJ/OrjGoFf3h+zRg3za5CrP2a+fNZmFT7hxPkTtItoR85sOZkTPoc8QXmsjmQ7flPQwUzNvvaamU/v2dN8nDbNrH4T6XD1x3SNwBctcvfHLF8eevQwBbx5c9ktTXjdpcRLdJ3elf1n9rP4wcWUzl/a6ki25FcF3SU01EzBdOwId99t7jB98kmZV7/GoUNX7kq4b595/Oab4b773PPgpeXNJTLX4J8Gs3D3Qia2n0jDUg2tjmNbflnQwdyfsnq12dDr6afNNbvx4/28Z8GJE2Y/cFcR37bNPH7TTWbkPXSoWUpYsaL89BNZZvza8Xy85mOevfNZ+tTsY3UcW0u3oCulvgLaAke11nekck4z4EMgO3BMa93UmyEzS758Zn36yJFmzfrmzTBrlh/tohofb/pjukbg69ebqZU8eczcd//+poDXqCFLCYUlluxZwmPzHqN1+da8c/c7VsexvXSXLSqlmgBnga9TKuhKqQLACuA+rfU+pVRRrfXR9F44K5ctemL+fDMNHBAAU6dCq1ZWJ8oEFy6YX0su74956ZLpj3nnne6beerWNY8JYaHdJ3dT94u6FMlThFX9V5E/Z36rI9nCDS1b1FovVUqVTeOUHkCU1npf8vnpFnM7at3aTLt07Gimh0eNMu0jHT2zkJhoRt2uEfiyZe7+mHXqmH3BW7Y0d1v59VyTsJu4C3GETg0lSScRHR4txdxD3phDrwhkV0r9CgQDY7TWX6d0olJqIDAQoLQNL6SVL2/uhenXD55/3iyr/uorMwPhCFrD1q1X9sc8dcocu/12s1NZy5bmzkzpjylsKkkn0XNWT7bGbuXHnj9SoVAFqyM5hjcKejagDtASyAWsVEqt0lpvv/pErfV4YDyYKRcvvLbX5c1r+pS67oXZssXMq5cvb3WyVOzZc+VKlH/+MY+XKwedOrn7Y8oG8cIhXlr0EtF/RTP2vrHcfevdVsdxFG8U9APAca11PBCvlFoK1ACuKehOoRQMGQI1a5puSHXrwnffmWkZyx05cmV/zL//No8XK+aeA2/RwhR0IRzmu43fMXL5SAbWHsjj9R63Oo7jeKOgzwE+VkplA4KA+sD/vPC8lmvVyky7hIXB/ffDG2/AsGFZvODj1CmzqZWrgG/aZB7Pn9/cEfX006aAV63q8Al/4e/WHFxD/+j+NCnThI/afISS7+fr5smyxQigGVBYKXUAGIFZnojW+jOt9Val1I/An0AS8KXWelPmRc5a5cqZbUkGDoSXXjLtKidPNpsEZopz52DFCncBj4lx98ds1Mjc4tqihdlGUvpjCh9x8MxB2k9tT/G8xZnZdSZBgbLKKiP8YrdFb9Da7NT43HPmvppZs6BSJS888aVLZnmNq4CvWOHuj1m/vnsKpUED6dIhfNL5S+dpMqkJ245tY0W/FVQrVs3qSLbm97steoNSZnajZk3o2hXq1YNvvjHbCFyXpCSzF/jl/THPnjUvULOm2YOgRQvTrV76Ywofp7Wmf3R/1h5ay6xus6SY3yAp6NepWTMz7RIWZjqjvfIKjBiRxry61rBjx5X9MY8fN8cqVYLevd39MQsVyqK/hRD2MGr5KCI2RfBWi7doX7m91XEcTwp6BpQqZe7ReeQRsxXvunVmtP7f0u4DB9wFfOFCd3/MUqWgXTv3plYlS1r2dxDCatF/RfPiohfpfkd3hjUaZnUcnyAFPYNy5jQ3HdWtC089palX9SyzG4+m6vopZkQOULjwlf0xb7tNVqIIAWw8spEHoh4g5OYQJoROkBUtXiIFPSPi4mDpUtSiRTy6cCHVE/LS+fAM6k8bzKTa2ek0Opcp4tWqyaZWQlwlNj6W0KmhBAcFM6vbLHJlz2V1JJ8hBd0Trv6YrimU339398ds2JBGb7ZkbbWDdH67Fp1XD2fYvfDGHRAotVyIK1xMvEjn6Z05HHeYpX2XUjKfTDt6kxT0lCQkmCufrgLu6o8ZEGDmWIYOdffHzGVGFyWBX+81i1RGjjTz6t99Z7YSF0KYFS1PzHuCpXuXMiVsCvVK1rM6ks+Rgg5mJcqmTe4CvmQJnDljjlWrBg8/7O6PmT/1Xd9y5IDPPzeN7B9/3HycPRuqV8+iv4cQNvbJmk8Yv248L9z1Aj2q9bA6jk/yzxuLtDZ7oLgK+OLFcDR519/bbnPfzNO8ORQtmqGXWLXK7I116hRMmGD2hBHCXy38eyH3fnsvbSq0YXb4bAKUzEdmlNxYBHD48JW7Eu7dax4vUQLuuce9GqVMGa+8XIMGZtamSxfo3t3cwT9qlLkBVAh/suP4DrpM70KVIlWYEjZFinkm8t3ycuKEmTpxFfCtW83jBQuakfeQIWYkXqlSpi0lLF7cvPyzz5pG1OvXm615CxfOlJcTwnZO/3ua0KmhBKgAosOjCc7hnU2Q8o3MR9zFuFSPBwcFc2bYGa+8lpP4TkGPj4fly90FfN06M7WSO7eZ++7b190fMws3tQoKgo8+MvPpgwaZRkGzZpm9tYTwZYlJiXSf2Z2dJ3byS69fKFfQe1s6p1XMPTnuq5xb0C9eND0xXQV81Sqz0VX27KY/5ogRpoDXq2eL/pgPPmiaBoWFmY5v48dDr15WpxIi8wz9ZSjzd87ns/s/o2lZR/SNdzznFfTly+HNN8299+fOmemSOnXgmWfc/TFt2jMuJMTMq3ftarZwiYmB9983P4OE8CWTN0zmg5Uf8FjdxxgUMsjqOH7DeQU9IQH27zeNP139MQsWtDqVx4oUgQULzFL20aNhwwaYNs00HBLCF6zYv4KB3w+kZbmW/O9en+h14xjOK+jNmsHmzVanuCHZspmLpHXqmL7NdepAVJSZHRLCyfaf3k9YZBil85dmWpdpZA+UXz+zkqwfslCPHqafRfbsZvvzCROsTiRExsVfjKf91PacTzhPdHg0N+WS26SzmhR0i9WsaebSmzY1o/VHHjHXe4VwkiSdRJ85fdjwzwYiOkVQpUgVqyP5JSnoNlCoEMyfb+bVP/vMLJM/dMjqVEJ47o0lbzBjywzebfUubSq0yfTXCw5Kez17esd9lX/e+m9j06ebJfPBwTBjhlm0I4Sdzdwyk87TO9O7Rm8mtZ8ke5tnsrRu/ZcRus106WKW1OfJY0bqn31m7o8Swo42/LOB3rN70+CWBnze9nMp5haTgm5Dd9wBa9ZAq1ZmTv2hh8zuvULYyZGzRwiNCOWmXDcxq9sscmbLaXUkvycF3aYKFoS5c+Hll02ruyZNzPJ7IezgQsIFwqaFcezcMaLDoymet7jVkQRS0G0tIMA0oZ41C7ZtM+vVlyyxOpXwd1prHvnhEVbsX8HkDpOpVaKW1ZFEMinoDtChg+l6d9NN5ubYsWNlXl1Y53+r/sfEDRN5pckrdLm9i9VxxGWkoDtE5cqmqLdtC089ZTb7On/e6lTC38zfMZ8hC4bQqUonRjQbYXUccRUp6A6SL5/ZIuD11+Hbb82Sxj17rE4l/MW2Y9sInxlOtaLVmNxhsjSqsCH5P+IwAQHmQuncuaaLXkiI2UFYiMx08vxJQiNCyZktJ3PC55AnyJ47mvo7KegOdf/9Zmlj8eKmg97778u8usgcCUkJdJ3RlT2n9hDVNYoyBbzTplF4nxR0B6tQwdyEFBZmOup1724aNwnhTc/+9Cy//P0Ln7f9nLtKy63LdiYF3eHy5jX7qY8aZbYNuPNO2LXL6lTCV3yx9gvG/j6WZxo8Q99afa2OI9KRbkFXSn2llDqqlNqUznl1lVIJSqnO3osnPKGU2dhr/nw4cMDMq//4o9WphNMt3buUR+c9yr233cu7rd61Oo7wgCcj9EnAfWmdoJQKBN4BfvZCJpFB99xjtuItUwbatIG335Z5dZExe07todO0TtxW8Damdp5KtgDn9cLxR+kWdK31UuBEOqc9AcwEjnojlMi4W281TTPCw+HFF6FzZ4jzzwboIoPiLsQRGhFKQlIC0d2jKZCzgNWRhIdueA5dKVUS6Ah86sG5A5VSMUqpmNjY2Bt9aZGK3LlhyhTTs3TOHKhfH7ZvtzqVcIIknUSvWb3YEruFaZ2nUbFQRasjievgjYuiHwJDtdZJ6Z2otR6vtQ7RWocUKVLECy8tUqMUPPOMaUgdGwt168L331udStjdK4tfYc5fcxh972ha3dbK6jjiOnmjoIcAU5VSe4DOwDilVAcvPK/wgubNYe1as8SxXTt47TVISvdHr/BHERsjeGvZWzxU6yGeqPeE1XFEBtxwQddal9Nal9ValwVmAI9qrWffcDLhNaVLw7JlZv+XV181m32dPm11KmEnaw6uoV90PxqXbswn938ijSocypNlixHASqCSUuqAUqq/UuphpdTDmR9PeEuuXDBxInz8sVneWK8ebNlidSphB4fiDtEhsgPF8hRjZteZBAUGWR1JZFC6a5G01t09fTKtdZ8bSiMylVLw2GNQvbpZ/VK/PkyebO40Ff7p/KXzdIzsyOl/T7Oi/wqK5JFrW04md4r6ocaNYd06uP126NTJLG9MTLQ6lchqWmsGzB3A7wd/59uwb6lerLrVkcQNkoLup0qWNN2PBgwwNyC1bQsnT1qdSmSld397lykbp/Bm8zfpUFnWMfgCKeh+LEcOGD8ePv/cbMEbEgIbN1qdSmSFuX/NZdjCYYTfEc7wxsOtjiO8RAq6YOBAM1o/fx4aNDCbfQnftenoJnpE9aB2idpMCJ0gK1p8iBR0AZhdGteuhVq1oFs3eP55SEiwOpXwtmPnjhEaEUreoLzMCZ9D7uy5rY4kvEgKuvhPiRKwaJFZCfPee3DffXDsmNWphLdcSrxEl+ldOBR3iNndZlMyX0mrIwkvk4IurhAUZNaqT5wIy5ebefX1661OJbzhyflP8uueX/ky9Evq31Lf6jgiE0hBFynq08cU9MREaNjQNKUWzjVuzTg+W/sZQ+8aSs/qPa2OIzKJFHSRqpAQM69evz706gVPPw2XLlmdSlyvhX8v5Mn5T9K2YlveavGW1XFEJpKCLtJUtKjZsfHpp2HMGGjVCo7KrveOsfPETrpM70LlwpWZEjaFwIBAqyOJTOS4NiRJSeZ29QcfhAD5cZQlsmeH//3PjNgHDIA6dSAqymzJK+zr9L+nCY0IRSlFdPdo8uXIZ3Uk28s3Mh9xF1PvCBMcFMyZYWeyMNH1cVxJjIqCfv1g1iyrk/ifBx6A336DwECzfcDEiVYnEqlJTEqkR1QPdpzYwYwuM7i14K1WR3KEtIq5J8et5qiCnpQEQ4aYz4cMkX29rVCrlulb2rix+cH62GNw8aLVqcTVhi0cxrwd8/io9Uc0L9fc6jgiiziqoEdFuddFx8bKKN0qhQubLXiffx7GjYMWLeDwYatTCZev//ia91a8x6Mhj/JwiOxy7U8cU9Bdo/OzZ83XZ8/KKN1K2bLBO+9AZKRZp16nDqxcaXUqsXL/SgbMHUDzss358L4PrY4jsphjCvrlo3MXGaVbr2tXWLXKNKZu2tRs9KW11an80/7T++kY2ZFS+Uoxvct0sgdmtzqSyGKOKOhXj85dZJRuD9WqwZo1cPfd8PDDZrOvCxesTuVfzl06R4fIDpy7dI7o7tEUyl3I6kjCAo4o6LNnw759KR/bt88cF9YqWBDmzjXNMr780ozWDxywOpV/0FrTd05f1h9eT0SnCKoWqWp1JGERR6xDL1PG3NiS1nFhvcBAePNNM5/eu7f5OH06NGlidTLf9ubSN5m2eRrv3v0u91e83+o4jhYcFJzuOnQ7U9qiCc+QkBAdExNjyWuLzLd1K3TsCLt2wejR8Pjjpqep8K6orVF0mtaJXtV7MbnDZNnb3A8opdZqrUNSOuaIKRfhPFWqwOrV0KYNPPmk2ezr/HmrU/mWDf9soNesXjS4pQHj242XYi6koIvMkz+/WYX02mvw9dfQqBHs3Wt1Kt9wNP4o7ae2p2DOgkR1jSJntpxWR9N/yn8AAA6JSURBVBI2IAVdZKqAAHjlFXPBdOdOsx/MokVWp3K2CwkXCIsMIzY+ljnhcygRXMLqSMImpKCLLNG2rVnaWLSo2bFx9GhZr54RWmse/eFRftv/GxPbT6TOzXWsjiRsRAq6yDIVK5qbkDp2hGefhR49ID7e6lTOMmb1GL7a8BUvNX6Jbnd0szqOsBkp6CJLBQebpYxvv222DWjYEP7+2+pUzvDTzp949udn6Vi5I681f83qOMKGpKCLLKcUDBsG8+bB/v1mXv3nn61OZW9/HfuLbjO6cUfRO/i649cEKHnrimvJd4WwzH33ma14S5Uyn48aJfPqKTl5/iTtItoRFBhEdHg0eYPyWh1J2JQUdGGpW2+FFSugWzczau/aFeLs3UMgSyUkJdBtRjf2nNpDVLcoyhSQ26JF6hxx67/wbXnywHffmZZ2Q4bAli1mf54KFaxOZr3nfn6OBX8v4Mt2X9KodCOr44gMyqrWdumO0JVSXymljiqlNqVy/AGl1J9KqY1KqRVKqRo3nEr4HaVg8GAzl37kiCnuP/xgdSprTVg3gTGrx/BU/afoX7u/1XHEDciq1naeTLlMAu5L4/huoKnWuhrwBjDeC7mEn2rZEtauNVMx7drB66/75/bIy/Yu45EfHuGe2+7h/XvetzqOcIh0C7rWeilwIo3jK7TWJ5O/XAXc4qVswk+VKWOaUffqBSNGmHXrp09bnSrr7Dm1h7BpYZQrWI7IzpFkC5CZUeEZb18U7Q/MT+2gUmqgUipGKRUTGxvr5ZcWviRXLpg0CcaONVMv9eubHRx93dmLZ2k/tT2XEi8RHR5NgZwFrI4kHMRrBV0p1RxT0Iemdo7WerzWOkRrHVKkSBFvvbTwUUrBE0/AwoVw8iTUq+fbzUySdBK9ZvVi09FNTOsyjUqFK1kdSTiMVwq6Uqo68CXQXmt93BvPKYRL06ZmXr1qVTP98vLLkJhodSrvG7F4BLO3zeaDez7gntvusTqOcKAbLuhKqdJAFNBLa739xiMJca1bboElS6B/f9MVqV07M2r3FZGbInlz2Zv0q9mPp+o/ZXUc4VCeLFuMAFYClZRSB5RS/ZVSDyulHk4+5RWgEDBOKbVBKSVtiESmyJkTvvgCPvsMfvnFLG3clOJiWmeJORRDnzl9aFS6EePuHyeNKnxQeq3rvNXaTlrQCUdasQI6dTJ3lU6cCF26WJ0oYw7HHabuF3UJDAhkzYA1FM1T1OpIwuakBZ3wOQ0bwrp1UKOG2S5g6FDnzav/m/AvHSI7cOrfU0SHR0sxFzdMCrpwrBIlYPFieOQRePddaN0ajjvkkrzWmgFzB/D7wd/5puM31CguN1iLGycFXThaUBCMGwcTJpiLpiEhsGGD1anS996K9/j2z295vdnrdKzS0eo4wkdIQRc+oV8/WLYMLl0y0zFTplidKHXfb/+eF355ga63d+WlJi9ZHUf4ECnowmfUq2fWq9etCz17wjPPQEKC1amutPnoZnrM7EGtErWY2H6irGgRXiUFXfiUYsXMksannoIPPzQNqY8etTqVcfzccUKnhpInKA9zwueQO3tuqyMJHyMFXfic7NlNMf/mG9OUOiTEdEay0qXES3Se3pmDZw4yq9ssbskne9gJ75OCLnxWz55m18aAAGjUyGz2ZZWnfnyKX/f8yhftvqDBLQ2sCyJ8mhR04dNq1zaj80aNoG9fePxxuHgxazN8uuZTPo35lCENh9CrRq+sfXHhV6SgC59XuDD8+CM89xx88olpovHPP1nz2ot3L+aJ+U9wf4X7GdlyZNa8qPBbUtDFf5KSzG30vtghKFs2eO89iIgwd5jWqWPm1zPTrhO76Dy9MxULVeS7Tt8RGBCYuS8o/J4UdPGfqCiznnvWLKuTZJ7wcFi50mz01aQJjM+kholnLpyhXUQ7AKK7R5MvR77MeSEhLiMFXQBmVD5kiPl8yBDfHKW7VK8Oa9ZAixYwaBAMHAgXLnjv+ROTEukxswfbj29nepfplL+pvPeeXIg0SEEXgBmdHztmPo+N9e1ROsBNN5nWdsOHmy15mzaFgwe989zDFw7nhx0/MLb1WFqUa+GdJxXCA1LQxX+j87Nnzddnz/r+KB0gMBDeegtmzoTNm828+vLlN/ac3/zxDe+ueJeH6zzMo3Uf9U5QITwkBV1cMTp38YdRuktYGKxeDfnyQfPmZiVMRtoErD6wmgFzB9CsbDPGth7r/aBCpEMKup+7enTu4i+jdJeqVc28euvWZq16375w/rzn//2BMwfoENmBm4NvZnqX6WQPzJ55YYVIhRR0Pzd7Nuzbl/KxffvMcX+RP7/5+776KkyeDI0bp/5vc7lzl87Rfmp7zl48y9zucymcu3CmZxUiJdmsDiCsVaYMPP102sf9SUAAjBgBtWpBr15mXn3aNDMVkxKtNf3m9GP94fVEd4/m9qK3Z21gIS4jBd3P1alj/ogrhYbC779Dx45mx8b33jM/+K7e7fatZW8RuTmSUS1H0bZiW2vCCpFMplyESEWlSuZiafv2MHiw2ezr3Dn38VlbZ/Hy4pfpWb0nz9/1vHVBhUgmBV2INAQHw4wZZnljRITphrR7N/x55E96zepFvZL1+KLdF9KoQtiCFHQh0qGUuQHphx9g716o3fgo90wMJX/O/MzuNpuc2XJaHVEIQAq6EB5r3RpWrL7IxY6dOHL2CJ0uzqF43hJWxxLiP1LQhfCQ1poPtj3KucLLufPIJD4aFkK3bteu4RfCKlLQhfDQ2NVjmbB+Ai82fpHfxnfjvffMtgENGsDOnVanE0IKuhAe+XnXzwz+eTAdKnfg9eavo5RpmPHTT6ZZRkgIzJtndUrh76SgC5GO7ce3021GN24vcjvfdPyGAOV+29x9t2lxV64ctG0Lb77pP9slCPuRgi5EGk6eP0m7iHZkC8hGdPdo8gblveacsmVNM+oHHoCXX4ZOneDMmazPKoQUdCFSkZCUQPjMcP4++Tczu86kbIGyqZ6bOzd8/TWMGQNz50L9+rBtW9ZlFQIcWNB9ue+lsJchPw/h510/8+n9n9KkTJN0z1cKnnwSFi6E48ehXj2YMycLggqRLN2CrpT6Sil1VCm1KZXjSik1Vim1Uyn1p1KqtvdjuvlD30thva/Wf8WHqz/kyXpP8lDth67rv23aFNauNVsHdOgAr7wiAxCRNTwZoU8C7kvjeGugQvKfgcCnNx4rZf7U91JYZ/m+5Tz8/cO0urUVH9z7QYaeo1QpWLbM7Kv+xhtms69Tp7wcVIirpFvQtdZLgRNpnNIe+Fobq4ACSqlMuX3O3/peiqy399RewiLDKFugLJGdI8kWkPENSXPmhAkTYNw4+PlnqFvX7AMjRGbxxhx6SWD/ZV8fSH7sGkqpgUqpGKVUTGxs7HW9iL/2vRRZKygwiNolahPdPZqCuQre8PMpBY88AosXmymY4sW9EFKIVGTpRVGt9XitdYjWOqRIkSLX9d/6e99LkTVKBJfgx54/UrlwZa8+7113wfffQ65cXn1aIa7gjYJ+ECh12de3JD/mNdL3Uggh0ueNgh4N9E5e7dIAOK21PuyF5/2P9L0UQoj0pXvFRykVATQDCiulDgAjgOwAWuvPgHlAG2AncA7o6+2Q0vdSCCHSl25B11p3T+e4Bh7zWqIUSN9LIYRIn+PuFBVCCJEyKehCCOEjpKALIYSPkIIuhBA+Qplrmha8sFKxwN4M/ueFgWPpnmUfTsrrpKzgrLxOygrOyuukrHBjectorVO8M9Oygn4jlFIxWusQq3N4ykl5nZQVnJXXSVnBWXmdlBUyL69MuQghhI+Qgi6EED7CqQV9vNUBrpOT8jopKzgrr5OygrPyOikrZFJeR86hCyGEuJZTR+hCCCGuIgVdCCF8hK0Lut0aVKfFg6wPJGfcqJRaoZSqkdUZr8qTZt7LzqurlEpQSnXOqmwpZEg3q1KqmVJqg1Jqs1JqSVbmSyFLet8L+ZVSc5VSfyTn9foOpZ5SSpVSSi1WSm1JzvJUCufY4n3mYVbbvM88yXvZud55n2mtbfsHaALUBjalcrwNMB9QQANgtY2zNgQKJn/e2sqsnuRNPicQWITZIrmzXbMCBYAtQOnkr4va+d8WGA68k/x5EUzP3iCLspYAaid/HgxsB6pedY4t3mceZrXN+8yTvMnHvPY+s/UIXduoQXV60suqtV6htT6Z/OUqTGcny3jwbwvwBDATOJr5iVLnQdYeQJTWel/y+XbPq4FgpZQC8iafm5AV2a4JovVhrfW65M/jgK1c2xPYFu8zT7La6X3m4b8tePF9ZuuC7gGPG1TbTH/MiMe2lFIlgY7Ap1Zn8UBFoKBS6lel1FqlVG+rA6XjY6AKcAjYCDyltba8kaJSqixQC1h91SHbvc/SyHo527zPUsvr7fdZug0uhHcppZpjvtEaWZ0lHR8CQ7XWSWYgaWvZgDpASyAXsFIptUprvd3aWKm6F9gAtABuAxYopZZprc9YFUgplRczSnzayhye8CSrnd5n6eT16vvM6QU90xtUe5NSqjrwJdBaa33c6jzpCAGmJn+TFQbaKKUStNZ27OB6ADiutY4H4pVSS4EamDlLO+oLjNJmAnWnUmo3UBn43YowSqnsmIIzRWsdlcIptnmfeZDVVu8zD/J69X3m9CmXTG9Q7S1KqdJAFNDLxiPH/2ity2mty2qtywIzgEdtWswB5gCNlFLZlFK5gfqY+Uq72of5bQKlVDGgEvC3FUGS5/EnAFu11qNTOc0W7zNPstrpfeZJXm+/z2w9Qlc2aFDtKQ+yvgIUAsYl/zRO0BbuDudBXttIL6vWeqtS6kfgTyAJ+FJrneZyTCvzAm8Ak5RSGzErR4Zqra3a+vUuoBewUSm1Ifmx4UBpsN37zJOsdnqfeZLXq+TWfyGE8BFOn3IRQgiRTAq6EEL4CCnoQgjhI6SgCyGEj5CCLoQQPkIKuhBC+Agp6EII4SP+D6erSXojXA0lAAAAAElFTkSuQmCC\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": 51,
+   "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": 52,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[array([  5.03613532,   6.28604462, -15.81831884]),\n",
+       " array([  3.21796205, -10.90131786,  12.4557529 ]),\n",
+       " array([-9.3864618 ,  6.35954149,  8.46541456])]"
+      ]
+     },
+     "execution_count": 52,
+     "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": 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_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": 54,
+   "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": 55,
+   "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": "3.8.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/notebooks/linear-regression.ipynb b/notebooks/linear-regression.ipynb
new file mode 100644
index 0000000..4d6a6cd
--- /dev/null
+++ b/notebooks/linear-regression.ipynb
@@ -0,0 +1,477 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Linear Regression"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 25,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "%matplotlib inline\n",
+    "\n",
+    "data = np.genfromtxt('./data.csv', delimiter=',')[1:]\n",
+    "km = data[:, 0]\n",
+    "price = data[:, 1]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "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 plot_data():\n",
+    "    plt.scatter(km, price)\n",
+    "    plt.xlabel(\"km\")\n",
+    "    plt.ylabel(\"price\")\n",
+    "plot_data()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "This dataset contains information about cars, their cost and the the number of kilometers they've made.\n",
+    "\n",
+    "We can see just by looking at the data that their is a **linear relationship**, when the distance made diminish, the price rize.\n",
+    "\n",
+    "We can try to trace an approximate line which goes *through* the data."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "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_data()\n",
+    "plt.plot([200_000, 20_000], [3_500, 8_500], color='r')\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We can use this hypothetical line to predict new data, for example predict the price of a car knowing only it's distance traveled.\n",
+    "\n",
+    "The equation of a line is in the form:\n",
+    "\n",
+    "$$\n",
+    "\\boxed {\n",
+    "    y = \\theta_1 x + \\theta_0\n",
+    "}\n",
+    "$$\n",
+    "\n",
+    "Where $\\theta_1$ is the slope of the line and $\\theta_0$ the y-intercept. If we managed to tweak those values correctly we can find the best fitting line for our data (or at least very close to it).\n",
+    "\n",
+    "If we switch to a function, we get $h(x) = \\theta_1 x + \\theta_0$, $h$ is our **hypothesis function**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "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 h(x, theta1, theta0):\n",
+    "    return x * theta1 + theta0\n",
+    "\n",
+    "theta1 = -0.03\n",
+    "theta0 = 9000\n",
+    "\n",
+    "line_xs = [km.min(), km.max()]\n",
+    "line_ys = [h(x, theta1, theta0) for x in line_xs]\n",
+    "plt.plot(line_xs, line_ys, color='r')\n",
+    "\n",
+    "plot_data()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We can adjust our prediction by changing $\\theta_1$ and $\\theta_0$.\n",
+    "\n",
+    "Now, we need a way to know if our line correctly predicts the data or not, so that if not, we can slightly change it to better fit the data.\n",
+    "\n",
+    "Introducing the **error function** $J$ (also called cost function):\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "J(\\theta_1, \\theta_0) &= \\frac{1}{2n} \\sum_{i=1}^{n}(h(x_i) - y_i)^2 \\\\\n",
+    "                      &= \\frac{1}{2n} \\sum_{i=1}^{n}((\\theta_1 x_i + \\theta_0) - y_i)^2\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "It's the sum of the square of the distance between what we predicted and the actual data.\n",
+    "$n$ is the number of entry we got, we divide by it to normalize the result."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def error_function(xs, ys, theta1, theta0):\n",
+    "    return (1 / (2 * len(xs))) * sum(\n",
+    "        [(h(x, theta1, theta0) - y) ** 2\n",
+    "         for x, y in zip(xs, ys)])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "386217.6189791666\n"
+     ]
+    }
+   ],
+   "source": [
+    "xs = km\n",
+    "ys = price\n",
+    "print(error_function(xs, ys, theta1, theta0))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The error value is pretty high, this means that the approximated line is garbage.\n",
+    "\n",
+    "The process of learning will be done by reducing this error function. To make that happen, we need to know *how* we are going to change $\\theta_1$ and $\\theta_0$.\n",
+    "\n",
+    "Here comes partial derivatives:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "    \\theta_1 &:= \\theta_1 - \\alpha \\frac{\\partial}{\\partial \\theta_1} J(\\theta_1, \\theta_0) \\\\\n",
+    "    \\theta_0 &:= \\theta_0 - \\alpha \\frac{\\partial}{\\partial \\theta_0} J(\\theta_1, \\theta_0)\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "First of all $\\alpha$ is the **learning rate** we multiply the change we want to make by it, it describes how *fast* we are changing the hypothesis component.\n",
+    "\n",
+    "The partial derivative of $J$ by regard to $\\theta_1$ for example is how the error function changes if we consider $\\theta_0$ has a constant.\n",
+    "\n",
+    "We compute the *next* $\\theta_0$ by subtracting learning rate * partial derivative from it.\n",
+    "\n",
+    "Lets expand theses partial derivatives:\n",
+    "\n",
+    "$$\n",
+    "\\newcommand{\\partialthetazero}{\\frac{\\partial}{\\partial \\theta_0}}\n",
+    "\\begin{align}\n",
+    "    &  \\partialthetazero \\left[ J(\\theta_1, \\theta_0) \\right] \\\\\n",
+    "    =& \\partialthetazero \\left[ \\frac{1}{2n} \\sum_{i=1}^{n}((\\theta_1 x_i + \\theta_0) - y_i)^2 \\right] \\\\\n",
+    "    =& \\frac{1}{2n} \\sum_{i=1}^{n} \\partialthetazero \\left[ ((\\theta_1 x_i + \\theta_0) - y_i)^2 \\right] \\\\\n",
+    "    =& \\frac{1}{2n} \\sum_{i=1}^{n} 2((\\theta_1 x_i + \\theta_0) - y_i) \\partialthetazero \\left[ ((\\theta_1 x_i + \\theta_0) - y_i) \\right] \\\\\n",
+    "    =& \\boxed{ \\frac{1}{n} \\sum_{i=1}^{n} ((\\theta_1 x_i + \\theta_0) - y_i) }\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "And for $\\theta_1$:\n",
+    "\n",
+    "$$\n",
+    "\\newcommand{\\partialthetaone}{\\frac{\\partial}{\\partial \\theta_1}}\n",
+    "\\begin{align}\n",
+    "    &  \\partialthetaone \\left[ J(\\theta_1, \\theta_0) \\right] \\\\\n",
+    "    =& \\partialthetaone \\left[ \\frac{1}{2n} \\sum_{i=1}^{n}((\\theta_1 x_i + \\theta_0) - y_i)^2 \\right] \\\\\n",
+    "    =& \\frac{1}{2n} \\sum_{i=1}^{n} \\partialthetaone \\left[ ((\\theta_1 x_i + \\theta_0) - y_i)^2 \\right] \\\\\n",
+    "    =& \\frac{1}{2n} \\sum_{i=1}^{n} 2((\\theta_1 x_i + \\theta_0) - y_i) \\partialthetaone \\left[ ((\\theta_1 x_i + \\theta_0) - y_i) \\right] \\\\\n",
+    "    =& \\boxed { \\frac{1}{n} \\sum_{i=1}^{n} ((\\theta_1 x_i + \\theta_0) - y_i) x_i }\n",
+    "\\end{align}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def theta0_partial(xs, ys, theta1, theta0):\n",
+    "    return sum([h(x, theta1, theta0) - y\n",
+    "                for x, y in zip(xs, ys)]) / len(xs)\n",
+    "\n",
+    "def theta1_partial(xs, ys, theta1, theta0):\n",
+    "    return sum([(h(x, theta1, theta0) - y) * x\n",
+    "                for x, y in zip(xs, ys)]) / len(xs)\n",
+    "\n",
+    "\n",
+    "def gradient_descent(xs, ys, theta1, theta0, alpha, iterations):\n",
+    "    for _ in range(iterations):\n",
+    "        next_theta1 = theta1 - alpha * theta1_partial(xs, ys, theta1, theta0)\n",
+    "        next_theta0 = theta0 - alpha * theta0_partial(xs, ys, theta1, theta0)\n",
+    "        theta1 = next_theta1\n",
+    "        theta0 = next_theta0\n",
+    "    return theta1, theta0"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The gradient descent work by repeating the formula above a certain amount of times.\n",
+    "\n",
+    "But before actually running the algorithm, the data need to be normalized. This means that they will be in a range like 0 < x < 1 but they will keep their relative distance to each other.\n",
+    "\n",
+    "If we don't do this, our values will go to infinity or some do some sneaky floating point arithmetic stuff.\n",
+    "\n",
+    "Here is how we normalize:\n",
+    "\n",
+    "$$\n",
+    "X_{normalized} = \\frac{X - X_{min}}{X_{max} - X_{min}}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def normalize(x):\n",
+    "    return (x - x.min()) / (x.max() - x.min())\n",
+    "\n",
+    "def denormalize(x, x_ref):\n",
+    "    return x * (x_ref.max() - x_ref.min()) + x_ref.min()\n",
+    "\n",
+    "X = normalize(xs)\n",
+    "Y = ys"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 33,
+   "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": [
+    "theta1, theta0 = 0, 0\n",
+    "\n",
+    "line_xs = [X.min(), X.max()]\n",
+    "line_ys = [h(x, theta1, theta0) for x in line_xs]\n",
+    "plt.plot(line_xs, line_ys, color='r')\n",
+    "\n",
+    "plt.scatter(X, Y)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 34,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "error before:  20880519.291666664\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(\"error before: \", error_function(X, Y, theta1, theta0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "-4656.591444722055 8008.439832646783\n",
+      "error after:  222822.6225356367\n"
+     ]
+    }
+   ],
+   "source": [
+    "alpha = 0.1\n",
+    "iterations = 20000\n",
+    "\n",
+    "theta1, theta0 = gradient_descent(X, Y, theta1, theta0, alpha, iterations)\n",
+    "\n",
+    "print(theta1, theta0)\n",
+    "print(\"error after: \", error_function(X, Y, theta1, theta0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 36,
+   "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": [
+    "line_xs = [X.min(), X.max()]\n",
+    "line_ys = [h(x, theta1, theta0) for x in line_xs]\n",
+    "plt.plot(line_xs, line_ys, color='r')\n",
+    "\n",
+    "plt.scatter(X, Y)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now that our model is trained, if we want to make a prediction, we have to normalize the prediction the same way that we normalized our trainning data."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "7427.151470348072"
+      ]
+     },
+     "execution_count": 37,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "predict = 50_000\n",
+    "predict = (predict - xs.min()) / (xs.max() - xs.min())\n",
+    "h(predict, theta1, theta0)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "If we plot this point with the original data. We can see that the prediction seems correct."
+   ]
+  },
+  {
+   "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": [
+    "plt.scatter(km, price)\n",
+    "plt.scatter([50_000], h(predict, theta1, theta0), color='r')\n",
+    "plt.show()"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.8.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/notebooks/logistic-regression.ipynb b/notebooks/logistic-regression.ipynb
new file mode 100644
index 0000000..8c862d3
--- /dev/null
+++ b/notebooks/logistic-regression.ipynb
@@ -0,0 +1,707 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Logistic Regression\n",
+    "\n",
+    "Logistic regression is a *binary classification algorithm*."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Text(0.5, 1.0, 'Data')"
+      ]
+     },
+     "execution_count": 7,
+     "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": 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": [
+    "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": 9,
+   "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": 10,
+   "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": 11,
+   "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": 43,
+   "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": 44,
+   "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",
+    "$$\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": 45,
+   "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": 46,
+   "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": 47,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "weights:   -4.121158543930249 33.10478298104216\n",
+      "train cost: 0.0031212648079922676\n",
+      "test cost:  0.0017122703295056325\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": 48,
+   "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": 49,
+   "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": 50,
+   "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": 51,
+   "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": 52,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[array([  5.03613532,   6.28604462, -15.81831884]),\n",
+       " array([  3.21796205, -10.90131786,  12.4557529 ]),\n",
+       " array([-9.3864618 ,  6.35954149,  8.46541456])]"
+      ]
+     },
+     "execution_count": 52,
+     "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": 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_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": 54,
+   "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": 55,
+   "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": "3.8.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}