code/bonus/softmax-regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Sebastian Raschka](http://sebastianraschka.com), 2016\n",
    "\n",
    "https://github.com/rasbt/python-machine-learning-book"
   ]
  },
  {
   "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": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sebastian Raschka \n",
      "last updated: 2016-06-05 \n",
      "\n",
      "CPython 3.5.1\n",
      "IPython 4.2.0\n",
      "\n",
      "matplotlib 1.5.1\n",
      "numpy 1.11.0\n",
      "scipy 0.17.0\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -a 'Sebastian Raschka' -u -d -v -p matplotlib,numpy,scipy"
   ]
  },
  {
   "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": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bonus Material - Softmax Regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Softmax Regression* (synonyms: *Multinomial Logistic*, *Maximum Entropy Classifier*, or just *Multi-class Logistic Regression*) is a generalization of logistic regression that we can use for multi-class classification (under the assumption that the classes are  mutually exclusive). In contrast, we use the (standard) *Logistic Regression* model in binary classification tasks."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below is a schematic of a *Logistic Regression* model that we discussed in Chapter 3."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](./images/logistic_regression_schematic_2.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In *Softmax Regression* (SMR), we replace the sigmoid logistic function by the so-called *softmax* function $\\phi_{softmax}(\\cdot)$.\n",
    "\n",
    "$$P(y=j \\mid z^{(i)}) = \\phi_{softmax}(z^{(i)}) = \\frac{e^{z^{(i)}}}{\\sum_{j=0}^{k} e^{z_{k}^{(i)}}},$$\n",
    "\n",
    "where we define the net input *z* as \n",
    "\n",
    "$$z = w_1x_1 + ... + w_mx_m  + b= \\sum_{l=0}^{m} w_l x_l + b= \\mathbf{w}^T\\mathbf{x} + b.$$ \n",
    "\n",
    "(**w** is the weight vector, $\\mathbf{x}$ is the feature vector of 1 training sample, and $b$ is the bias unit.)   \n",
    "Now, this softmax function computes the probability that this training sample $\\mathbf{x}^{(i)}$ belongs to class $j$ given the weight and net input $z^{(i)}$. So, we compute the probability $p(y = j \\mid \\mathbf{x^{(i)}; w}_j)$ for each class label in  $j = 1, \\ldots, k.$. Note the normalization term in the denominator which causes these class probabilities to sum up to one."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](./images/softmax_schematic_1.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To illustrate the concept of softmax, let us walk through a concrete example. Let's assume we have a training set consisting of 4 samples from 3 different classes (0, 1, and 2)\n",
    "\n",
    "- $x_0 \\rightarrow \\text{class }0$\n",
    "- $x_1 \\rightarrow \\text{class }1$\n",
    "- $x_2 \\rightarrow \\text{class }2$\n",
    "- $x_3 \\rightarrow \\text{class }2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "y = np.array([0, 1, 2, 2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we want to encode the class labels into a format that we can more easily work with; we apply one-hot encoding:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "one-hot encoding:\n",
      " [[ 1.  0.  0.]\n",
      " [ 0.  1.  0.]\n",
      " [ 0.  0.  1.]\n",
      " [ 0.  0.  1.]]\n"
     ]
    }
   ],
   "source": [
    "y_enc = (np.arange(np.max(y) + 1) == y[:, None]).astype(float)\n",
    "print('one-hot encoding:\\n', y_enc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A sample that belongs to class 0 (the first row) has a 1 in the first cell, a sample that belongs to class 2 has a 1 in the second cell of its row, and so forth."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, let us define the feature matrix of our 4 training samples. Here, we assume that our dataset consists of 2 features; thus, we create a 4x2 dimensional matrix of our samples and features.\n",
    "Similarly, we create a 2x3 dimensional weight matrix (one row per feature and one column for each class)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Inputs X:\n",
      " [[ 0.1  0.5]\n",
      " [ 1.1  2.3]\n",
      " [-1.1 -2.3]\n",
      " [-1.5 -2.5]]\n",
      "\n",
      "Weights W:\n",
      " [[ 0.1  0.2  0.3]\n",
      " [ 0.1  0.2  0.3]]\n",
      "\n",
      "bias:\n",
      " [ 0.01  0.1   0.1 ]\n"
     ]
    }
   ],
   "source": [
    "X = np.array([[0.1, 0.5],\n",
    "              [1.1, 2.3],\n",
    "              [-1.1, -2.3],\n",
    "              [-1.5, -2.5]])\n",
    "\n",
    "W = np.array([[0.1, 0.2, 0.3],\n",
    "              [0.1, 0.2, 0.3]])\n",
    "\n",
    "bias = np.array([0.01, 0.1, 0.1])\n",
    "\n",
    "print('Inputs X:\\n', X)\n",
    "print('\\nWeights W:\\n', W)\n",
    "print('\\nbias:\\n', bias)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To compute the net input, we multiply the 4x2 matrix feature matrix `X` with the 2x3 (n_features x n_classes) weight matrix `W`, which yields a 4x3 output matrix (n_samples x n_classes) to which we then add the bias unit: \n",
    "\n",
    "$$\\mathbf{Z} = \\mathbf{X}\\mathbf{W} + \\mathbf{b}.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Inputs X:\n",
      " [[ 0.1  0.5]\n",
      " [ 1.1  2.3]\n",
      " [-1.1 -2.3]\n",
      " [-1.5 -2.5]]\n",
      "\n",
      "Weights W:\n",
      " [[ 0.1  0.2  0.3]\n",
      " [ 0.1  0.2  0.3]]\n",
      "\n",
      "bias:\n",
      " [ 0.01  0.1   0.1 ]\n"
     ]
    }
   ],
   "source": [
    "X = np.array([[0.1, 0.5],\n",
    "              [1.1, 2.3],\n",
    "              [-1.1, -2.3],\n",
    "              [-1.5, -2.5]])\n",
    "\n",
    "W = np.array([[0.1, 0.2, 0.3],\n",
    "              [0.1, 0.2, 0.3]])\n",
    "\n",
    "bias = np.array([0.01, 0.1, 0.1])\n",
    "\n",
    "print('Inputs X:\\n', X)\n",
    "print('\\nWeights W:\\n', W)\n",
    "print('\\nbias:\\n', bias)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "net input:\n",
      " [[ 0.07  0.22  0.28]\n",
      " [ 0.35  0.78  1.12]\n",
      " [-0.33 -0.58 -0.92]\n",
      " [-0.39 -0.7  -1.1 ]]\n"
     ]
    }
   ],
   "source": [
    "def net_input(X, W, b):\n",
    "    return (X.dot(W) + b)\n",
    "\n",
    "net_in = net_input(X, W, bias)\n",
    "print('net input:\\n', net_in)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, it's time to compute the softmax activation that we discussed earlier:\n",
    "\n",
    "$$P(y=j \\mid z^{(i)}) = \\phi_{softmax}(z^{(i)}) = \\frac{e^{z^{(i)}}}{\\sum_{j=0}^{k} e^{z_{k}^{(i)}}}.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "softmax:\n",
      " [[ 0.29450637  0.34216758  0.36332605]\n",
      " [ 0.21290077  0.32728332  0.45981591]\n",
      " [ 0.42860913  0.33380113  0.23758974]\n",
      " [ 0.44941979  0.32962558  0.22095463]]\n"
     ]
    }
   ],
   "source": [
    "def softmax(z):\n",
    "    return (np.exp(z.T) / np.sum(np.exp(z), axis=1)).T\n",
    "\n",
    "smax = softmax(net_in)\n",
    "print('softmax:\\n', smax)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, the values for each sample (row) nicely sum up to 1 now. E.g., we can say that the first sample   \n",
    "`[ 0.29450637  0.34216758  0.36332605]` has a 29.45% probability to belong to class 0.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, in order to turn these probabilities back into class labels, we could simply take the argmax-index position of each row:\n",
    "\n",
    "[[ 0.29450637  0.34216758  **0.36332605**] -> 2   \n",
    "[ 0.21290077  0.32728332  **0.45981591**]  -> 2  \n",
    "[ **0.42860913**  0.33380113  0.23758974]  -> 0  \n",
    "[ **0.44941979**  0.32962558  0.22095463]] -> 0  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "predicted class labels:  [2 2 0 0]\n"
     ]
    }
   ],
   "source": [
    "def to_classlabel(z):\n",
    "    return z.argmax(axis=1)\n",
    "\n",
    "print('predicted class labels: ', to_classlabel(smax))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, our predictions are terribly wrong, since the correct class labels are `[0, 1, 2, 2]`. Now, in order to train our logistic model (e.g., via an optimization algorithm such as gradient descent), we need to define a cost function $J(\\cdot)$ that we want to minimize:\n",
    "\n",
    "$$J(\\mathbf{W}; \\mathbf{b}) = \\frac{1}{n} \\sum_{i=1}^{n} H(T_i, O_i),$$\n",
    "\n",
    "which is the average of all cross-entropies over our $n$ training samples. The cross-entropy  function is defined as\n",
    "\n",
    "$$H(T_i, O_i) = -\\sum_m T_i \\cdot log(O_i).$$\n",
    "\n",
    "Here the $T$ stands for \"target\" (i.e., the *true* class labels) and the $O$ stands for output -- the computed *probability* via softmax; **not** the predicted class label."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cross Entropy: [ 1.22245465  1.11692907  1.43720989  1.50979788]\n"
     ]
    }
   ],
   "source": [
    "def cross_entropy(output, y_target):\n",
    "    return - np.sum(np.log(output) * (y_target), axis=1)\n",
    "\n",
    "xent = cross_entropy(smax, y_enc)\n",
    "print('Cross Entropy:', xent)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cost:  1.32159787159\n"
     ]
    }
   ],
   "source": [
    "def cost(output, y_target):\n",
    "    return np.mean(cross_entropy(output, y_target))\n",
    "\n",
    "J_cost = cost(smax, y_enc)\n",
    "print('Cost: ', J_cost)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to learn our softmax model -- determining the weight coefficients -- via gradient descent, we then need to compute the derivative \n",
    "\n",
    "$$\\nabla \\mathbf{w}_j \\, J(\\mathbf{W}; \\mathbf{b}).$$\n",
    "\n",
    "I don't want to walk through the tedious details here, but this cost derivative turns out to be simply:\n",
    "\n",
    "$$\\nabla \\mathbf{w}_j \\, J(\\mathbf{W}; \\mathbf{b}) = \\frac{1}{n} \\sum^{n}_{i=0} \\big[\\mathbf{x}^{(i)}\\ \\big(O_i - T_i \\big) \\big]$$\n",
    "\n",
    "We can then use the cost derivate to update the weights in opposite direction of the cost gradient with learning rate $\\eta$:\n",
    "\n",
    "$$\\mathbf{w}_j := \\mathbf{w}_j - \\eta \\nabla \\mathbf{w}_j \\, J(\\mathbf{W}; \\mathbf{b})$$ \n",
    "\n",
    "for each class $$j \\in \\{0, 1, ..., k\\}$$\n",
    "\n",
    "(note that $\\mathbf{w}_j$ is the weight vector for the class $y=j$), and we update the bias units\n",
    "\n",
    "\n",
    "$$\\mathbf{b}_j := \\mathbf{b}_j   - \\eta \\bigg[ \\frac{1}{n} \\sum^{n}_{i=0} \\big(O_i - T_i  \\big) \\bigg].$$ \n",
    " \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a penalty against complexity, an approach to reduce the variance of our model and decrease the degree of overfitting by adding additional bias, we can further add a regularization term such as the L2 term with the regularization parameter $\\lambda$:\n",
    "    \n",
    "L2:        $\\frac{\\lambda}{2} ||\\mathbf{w}||_{2}^{2}$, \n",
    "\n",
    "where \n",
    "\n",
    "$$||\\mathbf{w}||_{2}^{2} = \\sum^{m}_{l=0} \\sum^{k}_{j=0} w_{i, j}$$\n",
    "\n",
    "so that our cost function becomes\n",
    "\n",
    "$$J(\\mathbf{W}; \\mathbf{b}) = \\frac{1}{n} \\sum_{i=1}^{n} H(T_i, O_i) + \\frac{\\lambda}{2} ||\\mathbf{w}||_{2}^{2}$$\n",
    "\n",
    "and we define the \"regularized\" weight update as\n",
    "\n",
    "$$\\mathbf{w}_j := \\mathbf{w}_j -  \\eta \\big[\\nabla \\mathbf{w}_j \\, J(\\mathbf{W}) + \\lambda \\mathbf{w}_j \\big].$$\n",
    "\n",
    "(Please note that we don't regularize the bias term.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# SoftmaxRegression Code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bringing the concepts together, we could come up with an implementation as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Sebastian Raschka 2016\n",
    "# Implementation of the mulitnomial logistic regression algorithm for\n",
    "# classification.\n",
    "\n",
    "# Author: Sebastian Raschka <sebastianraschka.com>\n",
    "#\n",
    "# License: BSD 3 clause\n",
    "\n",
    "import numpy as np\n",
    "from time import time\n",
    "#from .._base import _BaseClassifier\n",
    "#from .._base import _BaseMultiClass\n",
    "\n",
    "\n",
    "class SoftmaxRegression(object):\n",
    "\n",
    "    \"\"\"Softmax regression classifier.\n",
    "\n",
    "    Parameters\n",
    "    ------------\n",
    "    eta : float (default: 0.01)\n",
    "        Learning rate (between 0.0 and 1.0)\n",
    "    epochs : int (default: 50)\n",
    "        Passes over the training dataset.\n",
    "        Prior to each epoch, the dataset is shuffled\n",
    "        if `minibatches > 1` to prevent cycles in stochastic gradient descent.\n",
    "    l2 : float\n",
    "        Regularization parameter for L2 regularization.\n",
    "        No regularization if l2=0.0.\n",
    "    minibatches : int (default: 1)\n",
    "        The number of minibatches for gradient-based optimization.\n",
    "        If 1: Gradient Descent learning\n",
    "        If len(y): Stochastic Gradient Descent (SGD) online learning\n",
    "        If 1 < minibatches < len(y): SGD Minibatch learning\n",
    "    n_classes : int (default: None)\n",
    "        A positive integer to declare the number of class labels\n",
    "        if not all class labels are present in a partial training set.\n",
    "        Gets the number of class labels automatically if None.\n",
    "    random_seed : int (default: None)\n",
    "        Set random state for shuffling and initializing the weights.\n",
    "\n",
    "    Attributes\n",
    "    -----------\n",
    "    w_ : 2d-array, shape={n_features, 1}\n",
    "      Model weights after fitting.\n",
    "    b_ : 1d-array, shape={1,}\n",
    "      Bias unit after fitting.\n",
    "    cost_ : list\n",
    "        List of floats, the average cross_entropy for each epoch.\n",
    "\n",
    "    \"\"\"\n",
    "    def __init__(self, eta=0.01, epochs=50,\n",
    "                 l2=0.0,\n",
    "                 minibatches=1,\n",
    "                 n_classes=None,\n",
    "                 random_seed=None):\n",
    "\n",
    "        self.eta = eta\n",
    "        self.epochs = epochs\n",
    "        self.l2 = l2\n",
    "        self.minibatches = minibatches\n",
    "        self.n_classes = n_classes\n",
    "        self.random_seed = random_seed\n",
    "\n",
    "    def _fit(self, X, y, init_params=True):\n",
    "        if init_params:\n",
    "            if self.n_classes is None:\n",
    "                self.n_classes = np.max(y) + 1\n",
    "            self._n_features = X.shape[1]\n",
    "\n",
    "            self.b_, self.w_ = self._init_params(\n",
    "                weights_shape=(self._n_features, self.n_classes),\n",
    "                bias_shape=(self.n_classes,),\n",
    "                random_seed=self.random_seed)\n",
    "            self.cost_ = []\n",
    "\n",
    "        y_enc = self._one_hot(y=y, n_labels=self.n_classes, dtype=np.float)\n",
    "\n",
    "        for i in range(self.epochs):\n",
    "            for idx in self._yield_minibatches_idx(\n",
    "                    n_batches=self.minibatches,\n",
    "                    data_ary=y,\n",
    "                    shuffle=True):\n",
    "                # givens:\n",
    "                # w_ -> n_feat x n_classes\n",
    "                # b_  -> n_classes\n",
    "\n",
    "                # net_input, softmax and diff -> n_samples x n_classes:\n",
    "                net = self._net_input(X[idx], self.w_, self.b_)\n",
    "                softm = self._softmax(net)\n",
    "                diff = softm - y_enc[idx]\n",
    "                mse = np.mean(diff, axis=0)\n",
    "\n",
    "                # gradient -> n_features x n_classes\n",
    "                grad = np.dot(X[idx].T, diff)\n",
    "                \n",
    "                # update in opp. direction of the cost gradient\n",
    "                self.w_ -= (self.eta * grad +\n",
    "                            self.eta * self.l2 * self.w_)\n",
    "                self.b_ -= (self.eta * np.sum(diff, axis=0))\n",
    "\n",
    "            # compute cost of the whole epoch\n",
    "            net = self._net_input(X, self.w_, self.b_)\n",
    "            softm = self._softmax(net)\n",
    "            cross_ent = self._cross_entropy(output=softm, y_target=y_enc)\n",
    "            cost = self._cost(cross_ent)\n",
    "            self.cost_.append(cost)\n",
    "        return self\n",
    "\n",
    "    def fit(self, X, y, init_params=True):\n",
    "        \"\"\"Learn model from training data.\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",
    "        y : array-like, shape = [n_samples]\n",
    "            Target values.\n",
    "        init_params : bool (default: True)\n",
    "            Re-initializes model parametersprior to fitting.\n",
    "            Set False to continue training with weights from\n",
    "            a previous model fitting.\n",
    "\n",
    "        Returns\n",
    "        -------\n",
    "        self : object\n",
    "\n",
    "        \"\"\"\n",
    "        if self.random_seed is not None:\n",
    "            np.random.seed(self.random_seed)\n",
    "        self._fit(X=X, y=y, init_params=init_params)\n",
    "        self._is_fitted = True\n",
    "        return self\n",
    "    \n",
    "    def _predict(self, X):\n",
    "        probas = self.predict_proba(X)\n",
    "        return self._to_classlabels(probas)\n",
    " \n",
    "    def predict(self, X):\n",
    "        \"\"\"Predict targets from 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",
    "        target_values : array-like, shape = [n_samples]\n",
    "          Predicted target values.\n",
    "\n",
    "        \"\"\"\n",
    "        if not self._is_fitted:\n",
    "            raise AttributeError('Model is not fitted, yet.')\n",
    "        return self._predict(X)\n",
    "\n",
    "    def predict_proba(self, X):\n",
    "        \"\"\"Predict class probabilities of X from the net input.\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 probabilties : array-like, shape= [n_samples, n_classes]\n",
    "\n",
    "        \"\"\"\n",
    "        net = self._net_input(X, self.w_, self.b_)\n",
    "        softm = self._softmax(net)\n",
    "        return softm\n",
    "\n",
    "    def _net_input(self, X, W, b):\n",
    "        return (X.dot(W) + b)\n",
    "\n",
    "    def _softmax(self, z):\n",
    "        return (np.exp(z.T) / np.sum(np.exp(z), axis=1)).T\n",
    "\n",
    "    def _cross_entropy(self, output, y_target):\n",
    "        return - np.sum(np.log(output) * (y_target), axis=1)\n",
    "\n",
    "    def _cost(self, cross_entropy):\n",
    "        L2_term = self.l2 * np.sum(self.w_ ** 2)\n",
    "        cross_entropy = cross_entropy + L2_term\n",
    "        return 0.5 * np.mean(cross_entropy)\n",
    "\n",
    "    def _to_classlabels(self, z):\n",
    "        return z.argmax(axis=1)\n",
    "    \n",
    "    def _init_params(self, weights_shape, bias_shape=(1,), dtype='float64',\n",
    "                     scale=0.01, random_seed=None):\n",
    "        \"\"\"Initialize weight coefficients.\"\"\"\n",
    "        if random_seed:\n",
    "            np.random.seed(random_seed)\n",
    "        w = np.random.normal(loc=0.0, scale=scale, size=weights_shape)\n",
    "        b = np.zeros(shape=bias_shape)\n",
    "        return b.astype(dtype), w.astype(dtype)\n",
    "    \n",
    "    def _one_hot(self, y, n_labels, dtype):\n",
    "        \"\"\"Returns a matrix where each sample in y is represented\n",
    "           as a row, and each column represents the class label in\n",
    "           the one-hot encoding scheme.\n",
    "\n",
    "        Example:\n",
    "\n",
    "            y = np.array([0, 1, 2, 3, 4, 2])\n",
    "            mc = _BaseMultiClass()\n",
    "            mc._one_hot(y=y, n_labels=5, dtype='float')\n",
    "\n",
    "            np.array([[1., 0., 0., 0., 0.],\n",
    "                      [0., 1., 0., 0., 0.],\n",
    "                      [0., 0., 1., 0., 0.],\n",
    "                      [0., 0., 0., 1., 0.],\n",
    "                      [0., 0., 0., 0., 1.],\n",
    "                      [0., 0., 1., 0., 0.]])\n",
    "\n",
    "        \"\"\"\n",
    "        mat = np.zeros((len(y), n_labels))\n",
    "        for i, val in enumerate(y):\n",
    "            mat[i, val] = 1\n",
    "        return mat.astype(dtype)    \n",
    "    \n",
    "    def _yield_minibatches_idx(self, n_batches, data_ary, shuffle=True):\n",
    "            indices = np.arange(data_ary.shape[0])\n",
    "\n",
    "            if shuffle:\n",
    "                indices = np.random.permutation(indices)\n",
    "            if n_batches > 1:\n",
    "                remainder = data_ary.shape[0] % n_batches\n",
    "\n",
    "                if remainder:\n",
    "                    minis = np.array_split(indices[:-remainder], n_batches)\n",
    "                    minis[-1] = np.concatenate((minis[-1],\n",
    "                                                indices[-remainder:]),\n",
    "                                               axis=0)\n",
    "                else:\n",
    "                    minis = np.array_split(indices, n_batches)\n",
    "\n",
    "            else:\n",
    "                minis = (indices,)\n",
    "\n",
    "            for idx_batch in minis:\n",
    "                yield idx_batch\n",
    "    \n",
    "    def _shuffle_arrays(self, arrays):\n",
    "        \"\"\"Shuffle arrays in unison.\"\"\"\n",
    "        r = np.random.permutation(len(arrays[0]))\n",
    "        return [ary[r] for ary in arrays]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 1 - Gradient Descent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c307668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Js6ciISISI3d4771QLCZMgHnzoGfPUDBOOw1atkw64bapSIiI1KJPPw37e48f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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c307080>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from mlxtend.data import iris_data\n",
    "from mlxtend.plotting import plot_decision_regions\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Loading Data\n",
    "\n",
    "X, y = iris_data()\n",
    "X = X[:, [0, 3]] # sepal length and petal width\n",
    "\n",
    "# standardize\n",
    "X[:,0] = (X[:,0] - X[:,0].mean()) / X[:,0].std()\n",
    "X[:,1] = (X[:,1] - X[:,1].mean()) / X[:,1].std()\n",
    "\n",
    "lr = SoftmaxRegression(eta=0.01, epochs=10, minibatches=1, random_seed=0)\n",
    "lr.fit(X, y)\n",
    "\n",
    "plot_decision_regions(X, y, clf=lr)\n",
    "plt.title('Softmax Regression - Gradient Descent')\n",
    "plt.show()\n",
    "\n",
    "plt.plot(range(len(lr.cost_)), lr.cost_)\n",
    "plt.xlabel('Iterations')\n",
    "plt.ylabel('Cost')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Continue training for another 800 epochs by calling the `fit` method with `init_params=False`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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vBwcemJ3yTi6PfjGh4C15bd265WzatBZ4hH375gMNbNq0hnXrljNq1IQObau6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\nhP4/At6xV/9yk2QAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c31dfd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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HuoDPRcQzko7pp7Z9gduycywAn87/cc1qw925m5lZLm7CMjOzXBwgZmaWiwPE\nzMxycYCYmVkuDhAzM8vFAWJmZrk4QMzMLBcHiJmZ5fL/AVuYxX3dt3eTAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c31df98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lr.epochs = 800\n",
    "\n",
    "lr.fit(X, y, init_params=False)\n",
    "\n",
    "plot_decision_regions(X, y, clf=lr)\n",
    "plt.title('Softmax Regression - Stochastic Gradient Descent')\n",
    "plt.show()\n",
    "\n",
    "plt.plot(range(len(lr.cost_)), lr.cost_)\n",
    "plt.xlabel('Iterations')\n",
    "plt.ylabel('Cost')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### Predicting Class Labels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Last 3 Class Labels: [2 2 2]\n"
     ]
    }
   ],
   "source": [
    "y_pred = lr.predict(X)\n",
    "print('Last 3 Class Labels: %s' % y_pred[-3:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Predicting Class Probabilities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Last 3 Class Labels:\n",
      " [[  1.22579350e-09   1.22074818e-02   9.87792517e-01]\n",
      " [  1.84962350e-12   3.99742930e-04   9.99600257e-01]\n",
      " [  3.10919973e-07   1.37047172e-01   8.62952517e-01]]\n"
     ]
    }
   ],
   "source": [
    "y_pred = lr.predict_proba(X)\n",
    "print('Last 3 Class Labels:\\n %s' % y_pred[-3:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 2 - Stochastic Gradient Descent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ffd0c50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11141ca58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from mlxtend.data import iris_data\n",
    "from mlxtend.plotting import plot_decision_regions\n",
    "from mlxtend.classifier import SoftmaxRegression\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Loading Data\n",
    "\n",
    "X, y = iris_data()\n",
    "X = X[:, [0, 3]] # sepal length and petal width\n",
    "\n",
    "# standardize\n",
    "X[:,0] = (X[:,0] - X[:,0].mean()) / X[:,0].std()\n",
    "X[:,1] = (X[:,1] - X[:,1].mean()) / X[:,1].std()\n",
    "\n",
    "lr = SoftmaxRegression(eta=0.05, epochs=200, minibatches=len(y), random_seed=0)\n",
    "lr.fit(X, y)\n",
    "\n",
    "plot_decision_regions(X, y, clf=lr)\n",
    "plt.title('Softmax Regression - Stochastic Gradient Descent')\n",
    "plt.show()\n",
    "\n",
    "plt.plot(range(len(lr.cost_)), lr.cost_)\n",
    "plt.xlabel('Iterations')\n",
    "plt.ylabel('Cost')\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": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}