code/bonus/logistic_regression.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Sebastian Raschka](http://sebastianraschka.com), 2015\n",
    "\n",
    "https://github.com/rasbt/python-machine-learning-book"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Python Machine Learning Essentials - Code Examples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bonus Material - A Simple Logistic Regression Implementation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the optional watermark extension is a small IPython notebook plugin that I developed to make the code reproducible. You can just skip the following line(s)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sebastian Raschka \n",
      "Last updated: 12/22/2015 \n",
      "\n",
      "CPython 3.5.1\n",
      "IPython 4.0.1\n",
      "\n",
      "numpy 1.10.2\n",
      "pandas 0.17.1\n",
      "matplotlib 1.5.0\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -a 'Sebastian Raschka' -u -d -v -p numpy,pandas,matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# to install watermark just uncomment the following line:\n",
    "#%install_ext https://raw.githubusercontent.com/rasbt/watermark/master/watermark.py"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Overview"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Please see Chapter 3 for more details on logistic regression.\n",
    "\n",
    "![](./images/logistic_regression_schematic.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Implementing logistic regression in Python"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following implementation is similar to the Adaline implementation in [Chapter 2](../ch02/ch02.ipynb) except that we replace the sum of squared errors cost function with the logistic cost function\n",
    "\n",
    "$$J(\\mathbf{w}) = \\sum_{i=1}^{m} - y^{(i)} log \\bigg( \\phi\\big(z^{(i)}\\big) \\bigg) - \\big(1 - y^{(i)}\\big) log\\bigg(1-\\phi\\big(z^{(i)}\\big)\\bigg).$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class LogisticRegression(object):\n",
    "    \"\"\"LogisticRegression classifier.\n",
    "\n",
    "    Parameters\n",
    "    ------------\n",
    "    eta : float\n",
    "        Learning rate (between 0.0 and 1.0)\n",
    "    n_iter : int\n",
    "        Passes over the training dataset.\n",
    "\n",
    "    Attributes\n",
    "    -----------\n",
    "    w_ : 1d-array\n",
    "        Weights after fitting.\n",
    "    cost_ : list\n",
    "        Cost in every epoch.\n",
    "\n",
    "    \"\"\"\n",
    "    def __init__(self, eta=0.01, n_iter=50):\n",
    "        self.eta = eta\n",
    "        self.n_iter = n_iter\n",
    "\n",
    "    def fit(self, X, y):\n",
    "        \"\"\" Fit training data.\n",
    "\n",
    "        Parameters\n",
    "        ----------\n",
    "        X : {array-like}, shape = [n_samples, n_features]\n",
    "            Training vectors, where n_samples is the number of samples and\n",
    "            n_features is the number of features.\n",
    "        y : array-like, shape = [n_samples]\n",
    "            Target values.\n",
    "\n",
    "        Returns\n",
    "        -------\n",
    "        self : object\n",
    "\n",
    "        \"\"\"\n",
    "        self.w_ = np.zeros(1 + X.shape[1])\n",
    "        self.cost_ = []       \n",
    "        for i in range(self.n_iter):\n",
    "            y_val = self.activation(X)\n",
    "            errors = (y - y_val)\n",
    "            neg_grad = X.T.dot(errors)\n",
    "            self.w_[1:] += self.eta * neg_grad\n",
    "            self.w_[0] += self.eta * errors.sum()\n",
    "            self.cost_.append(self._logit_cost(y, self.activation(X)))\n",
    "        return self\n",
    "\n",
    "    def _logit_cost(self, y, y_val):\n",
    "        logit = -y.dot(np.log(y_val)) - ((1 - y).dot(np.log(1 - y_val)))\n",
    "        return logit\n",
    "    \n",
    "    def _sigmoid(self, z):\n",
    "        return 1.0 / (1.0 + np.exp(-z))\n",
    "    \n",
    "    def net_input(self, X):\n",
    "        \"\"\"Calculate net input\"\"\"\n",
    "        return np.dot(X, self.w_[1:]) + self.w_[0]\n",
    "\n",
    "    def activation(self, X):\n",
    "        \"\"\" Activate the logistic neuron\"\"\"\n",
    "        z = self.net_input(X)\n",
    "        return self._sigmoid(z)\n",
    "    \n",
    "    def predict_proba(self, X):\n",
    "        \"\"\"\n",
    "        Predict class probabilities for X.\n",
    "        \n",
    "        Parameters\n",
    "        ----------\n",
    "        X : {array-like, sparse matrix}, shape = [n_samples, n_features]\n",
    "            Training vectors, where n_samples is the number of samples and\n",
    "            n_features is the number of features.\n",
    "        \n",
    "        Returns\n",
    "        ----------\n",
    "          Class 1 probability : float\n",
    "        \n",
    "        \"\"\"\n",
    "        return activation(X)\n",
    "\n",
    "    def predict(self, X):\n",
    "        \"\"\"\n",
    "        Predict class labels for X.\n",
    "        \n",
    "        Parameters\n",
    "        ----------\n",
    "        X : {array-like, sparse matrix}, shape = [n_samples, n_features]\n",
    "            Training vectors, where n_samples is the number of samples and\n",
    "            n_features is the number of features.\n",
    "        \n",
    "        Returns\n",
    "        ----------\n",
    "        class : int\n",
    "            Predicted class label.\n",
    "        \n",
    "        \"\"\"\n",
    "        # equivalent to np.where(self.activation(X) >= 0.5, 1, 0)\n",
    "        return np.where(self.net_input(X) >= 0.0, 1, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Reading-in the Iris data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>145</th>\n",
       "      <td>6.7</td>\n",
       "      <td>3.0</td>\n",
       "      <td>5.2</td>\n",
       "      <td>2.3</td>\n",
       "      <td>Iris-virginica</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>146</th>\n",
       "      <td>6.3</td>\n",
       "      <td>2.5</td>\n",
       "      <td>5.0</td>\n",
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       "      <td>Iris-virginica</td>\n",
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       "    <tr>\n",
       "      <th>147</th>\n",
       "      <td>6.5</td>\n",
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       "      <td>Iris-virginica</td>\n",
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       "    <tr>\n",
       "      <th>148</th>\n",
       "      <td>6.2</td>\n",
       "      <td>3.4</td>\n",
       "      <td>5.4</td>\n",
       "      <td>2.3</td>\n",
       "      <td>Iris-virginica</td>\n",
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       "      <th>149</th>\n",
       "      <td>5.9</td>\n",
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       "       0    1    2    3               4\n",
       "145  6.7  3.0  5.2  2.3  Iris-virginica\n",
       "146  6.3  2.5  5.0  1.9  Iris-virginica\n",
       "147  6.5  3.0  5.2  2.0  Iris-virginica\n",
       "148  6.2  3.4  5.4  2.3  Iris-virginica\n",
       "149  5.9  3.0  5.1  1.8  Iris-virginica"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "\n",
    "df = pd.read_csv('https://archive.ics.uci.edu/ml/'\n",
    "        'machine-learning-databases/iris/iris.data', header=None)\n",
    "df.tail()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "# select setosa and versicolor\n",
    "y = df.iloc[0:100, 4].values\n",
    "y = np.where(y == 'Iris-setosa', 1, 0)\n",
    "\n",
    "# extract sepal length and petal length\n",
    "X = df.iloc[0:100, [0, 2]].values\n",
    "\n",
    "# standardize features\n",
    "X_std = np.copy(X)\n",
    "X_std[:,0] = (X[:,0] - X[:,0].mean()) / X[:,0].std()\n",
    "X_std[:,1] = (X[:,1] - X[:,1].mean()) / X[:,1].std()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A function for plotting decision regions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from matplotlib.colors import ListedColormap\n",
    "\n",
    "def plot_decision_regions(X, y, classifier, resolution=0.02):\n",
    "\n",
    "    # setup marker generator and color map\n",
    "    markers = ('s', 'x', 'o', '^', 'v')\n",
    "    colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan')\n",
    "    cmap = ListedColormap(colors[:len(np.unique(y))])\n",
    "\n",
    "    # plot the decision surface\n",
    "    x1_min, x1_max = X[:, 0].min() - 1, X[:, 0].max() + 1\n",
    "    x2_min, x2_max = X[:, 1].min() - 1, X[:, 1].max() + 1\n",
    "    xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, resolution),\n",
    "                         np.arange(x2_min, x2_max, resolution))\n",
    "    Z = classifier.predict(np.array([xx1.ravel(), xx2.ravel()]).T)\n",
    "    Z = Z.reshape(xx1.shape)\n",
    "    plt.contourf(xx1, xx2, Z, alpha=0.4, cmap=cmap)\n",
    "    plt.xlim(xx1.min(), xx1.max())\n",
    "    plt.ylim(xx2.min(), xx2.max())\n",
    "\n",
    "    # plot class samples\n",
    "    for idx, cl in enumerate(np.unique(y)):\n",
    "        plt.scatter(x=X[y == cl, 0], y=X[y == cl, 1],\n",
    "                    alpha=0.8, c=cmap(idx),\n",
    "                    marker=markers[idx], label=cl)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
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snnxy8/DKX96woTO4KsNs993TzMUxYxxk1hwcUjkOKRsInn++engtXgxLlsDS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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1088a1b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "lr = LogisticRegression(n_iter=500, eta=0.2).fit(X_std, y)\n",
    "plt.plot(range(1, len(lr.cost_) + 1), np.log10(lr.cost_))\n",
    "plt.xlabel('Epochs')\n",
    "plt.ylabel('Cost')\n",
    "plt.title('Logistic Regression - Learning rate 0.01')\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Meri2Ybm7v1fsIAvEoCQlUkYTz8qfwEB9JUrxdSRJPe7u+5csshSUpESyQ8Ot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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x107f1add8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_decision_regions(X_std, y, classifier=lr)\n",
    "plt.title('Logistic Regression - Gradient Descent')\n",
    "plt.xlabel('sepal length [standardized]')\n",
    "plt.ylabel('petal length [standardized]')\n",
    "plt.legend(loc='upper left')\n",
    "plt.tight_layout()"
   ]
  }
 ],
 "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": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}