Classification_MNIST_NaiveBayes.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Name: Ibrahim Odumas Odufowora <br />\n",
    "Course: Machine Learning <br />\n",
    "Project Name: Classification with MNIST <br />\n",
    "\n",
    "If the Latex formula are not showing correctly, please use chrome browser."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problems Identification:\n",
    "This project involves the implementation of efficient and effective Naive Bayes and Logistic Regression classifiers on MNIST data set. The MNIST data comprises of digital images of several digits ranging from 0 to 9. Each image is 28 x 28 pixels. Thus, the data set has 10 levels of classes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These are the needed packages:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<script>requirejs.config({paths: { 'plotly': ['https://cdn.plot.ly/plotly-latest.min']},});if(!window.Plotly) {{require(['plotly'],function(plotly) {window.Plotly=plotly;});}}</script>"
      ],
      "text/vnd.plotly.v1+html": [
       "<script>requirejs.config({paths: { 'plotly': ['https://cdn.plot.ly/plotly-latest.min']},});if(!window.Plotly) {{require(['plotly'],function(plotly) {window.Plotly=plotly;});}}</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from mnist import MNIST\n",
    "from time import time\n",
    "import pandas as pd\n",
    "import os\n",
    "import matplotlib.pyplot as plt\n",
    "from IPython.display import display, HTML\n",
    "from itertools import chain\n",
    "from sklearn.metrics import confusion_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Load the dataset, having downloaded it from Yann. The train data set comprises of 60,000 samples and the test data set has 10,000 samples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "__mndata__ = MNIST(os.getcwd() +'\\\\image')\n",
    "__train_image__, __train_label__ = __mndata__.load_training() #train & train_label\n",
    "__test_image__, __test_label__ = __mndata__.load_testing() #test & test_label"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The train and test images were reshaped to 784 dimension long, which translate to one feature for each pixel in the image."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Visualization:\n",
    "Visualizing a random 28 by 28 image."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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BBAAwAoEEADACgQQAMAKBBAAwAoEEADACgQQAMAKBBAAwQlNvFwA0Runp6bbH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\nEQgkAIAR/g8oMXFNBI2kMAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2ae1e5ccf98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(np.reshape(train[random.randint(1,1000)], [28,28]), cmap='Greys_r'); plt.title('Visualizing a random sample'); plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Implementation functions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Utility:\n",
    "    \"\"\"This class warehouses the data and other general functions\"\"\"\n",
    "    def __init__(self):\n",
    "        self.train = np.asarray([np.reshape(x, (784)) for x in __train_image__]).astype('float64') #change train_image to numpy array #784 means 28 by 28 pixel\n",
    "        self.train_label = np.asarray(__train_label__)\n",
    "        self.test = np.asarray([np.reshape(x, (784)) for x in __test_image__]).astype('float64') #change test_image to numpy array\n",
    "        self.test_label = np.asarray(__test_label__)\n",
    "        self.n_class = list(range(10)) #class labels\n",
    "        \n",
    "    def confusionMatrix(self, actual, predict, print_cfm=True, print_err_digit=True):\n",
    "        cfm = pd.DataFrame(confusion_matrix(actual, predict))\n",
    "        err_all = round((1 - sum(np.diagonal(cfm)) / len(predict)) * 100, 4)\n",
    "        error_digit = []\n",
    "        for i in self.n_class:\n",
    "            error_digit.append(round(1 - cfm.iloc[i,i] / sum(cfm.iloc[i,:]), 4) * 100)\n",
    "\n",
    "        tab_error = pd.DataFrame(columns=['digit', 'error per digit in %'])\n",
    "        tab_error['digit'] = self.n_class\n",
    "        tab_error['error per digit in %'] = error_digit\n",
    "\n",
    "        if print_cfm:\n",
    "            print('Testing Confusion Matrix: Actual vs. Prediction')\n",
    "            display(cfm)\n",
    "\n",
    "        if print_err_digit:\n",
    "            print('% error per digit')\n",
    "            display(tab_error)\n",
    "            print('\\nThe overall testing error is {}%'.format(err_all))\n",
    "\n",
    "        return cfm, tab_error, err_all\n",
    "\n",
    "\n",
    "    def nFoldCV_NB(self, train, train_label, smoothings, kFolds):\n",
    "        stra_all = self.folds_stratify(nSample=len(train), kFolds=kFolds)\n",
    "        ave_test_err = []\n",
    "        for s, sVal in enumerate(smoothings):\n",
    "            test_err = []\n",
    "            for k in range(kFolds):\n",
    "                stra = stra_all.copy()\n",
    "                te = train[stra[k]]\n",
    "                te_lb = train_label[stra[k]]\n",
    "                del stra[k]  # del test list\n",
    "                tr = train[list(chain.from_iterable(stra))]\n",
    "                tr_lb = train_label[list(chain.from_iterable(stra))]\n",
    "                nb = NaiveBayes(train=tr, train_lb=tr_lb, test=te, test_lb=te_lb, smoothing=sVal)\n",
    "                nb.predict()\n",
    "                conf = self.confusionMatrix(te_lb, nb.pred, False, False)\n",
    "                test_err.append(conf[2])\n",
    "\n",
    "            ave_test_err.append(np.mean(test_err))\n",
    "\n",
    "        return ave_test_err\n",
    "\n",
    "    def folds_stratify(self, nSample, kFolds):\n",
    "        \"\"\"This return kfold stratification\"\"\"\n",
    "        foldSize = round(nSample / kFolds, 0)\n",
    "        randomList = list(range(0, nSample))\n",
    "        random.shuffle(randomList)\n",
    "        stratify = []\n",
    "\n",
    "        for k in range(0, kFolds):\n",
    "            strt = int(k * foldSize)\n",
    "            end = int((k + 1) * foldSize)\n",
    "            if k == (kFolds - 1):\n",
    "                end = nSample\n",
    "            stratify.append(list(randomList)[strt:end])\n",
    "\n",
    "        return stratify\n",
    "    \n",
    "    def CV_plot(k_error, k_list, title):\n",
    "        k_error = np.round(k_error, 4)\n",
    "        best_k = k_list[np.argmin(k_error)]\n",
    "        err_best_k = np.min(k_error)\n",
    "        plt.plot(k_list, k_error, '-gD', color='black')\n",
    "        ax = subplot(111)\n",
    "        ax.set_xticks(k_list, k_list)\n",
    "        plt.ylabel('% Error')\n",
    "        plt.xlabel(title)\n",
    "        plt.title('Plot of ' + title + ' vs. average 5 foldCV error')\n",
    "        plt.show()\n",
    "        print('The best value of ' + title + ' is {} with an error of {}%'.format(best_k, err_best_k))\n",
    "        print('\\n' + title + ' = {} will be used to model the entire training set and prediction on testing set:'.format(best_k))\n",
    "        return best_k, err_best_k\n",
    "        \n",
    "    def images_plot(imageData):\n",
    "        classes = [\"P(x|c=0)\", \"P(x|c=1)\", \"P(x|c=2)\", \"P(x|c=3)\", \"P(x|c=4)\", \"P(x|c=5)\", \"P(x|c=6)\", \"P(x|c=7)\",\n",
    "                   \"P(x|c=8)\", \"P(x|c=9)\"]\n",
    "\n",
    "        num_classes = len(classes)\n",
    "        plt.subplots(figsize=(15, 2))\n",
    "        for y, cls in enumerate(classes):\n",
    "            plt_idx = y + 1\n",
    "            plt.subplot(1, num_classes, plt_idx)\n",
    "            if prob:\n",
    "                plt.imshow(imageData[y].reshape((28, 28)))\n",
    "            else:\n",
    "                plt.imshow(imageData[y].reshape(785, 1)[1:785].reshape((28, 28)))\n",
    "            plt.axis(\"off\")\n",
    "            plt.title(cls)\n",
    "\n",
    "        plt.tight_layout()\n",
    "        plt.show()\n",
    "        \n",
    "#End of Class Utility\n",
    "\n",
    "class NaiveBayes:\n",
    "    \"\"\"This class warehouses the Naive bayes classifier\"\"\"\n",
    "    def __init__(self, train, train_lb, test, test_lb, smoothing, jupyter=True, plot_dis=False):\n",
    "        self.n_class = np.unique(train_lb)\n",
    "        self.tr = train\n",
    "        self.te = test\n",
    "        self.tr_lb = train_lb\n",
    "        self.te_lb = test_lb\n",
    "        self.plot_dis = plot_dis\n",
    "        self.jupyter = jupyter\n",
    "        self.smoothing = smoothing\n",
    "        if jupyter:\n",
    "            self.plotly = iplot\n",
    "        else:\n",
    "            self.plotly = plot\n",
    "\n",
    "    def mean_std_prior(self):\n",
    "        \"\"\"Mean and Std per feature per class, and prior per class\"\"\"\n",
    "        self.mean, self.std, self.priors, self.count = [], [], [], []\n",
    "        for i, val in enumerate(self.n_class):\n",
    "            sep = [self.tr_lb == val] #separated\n",
    "            self.count.append(len(self.tr_lb[sep]))\n",
    "            self.priors.append(len(self.tr_lb[sep]) / len(self.tr_lb))\n",
    "            self.mean.append(np.mean(self.tr[sep], axis=0))\n",
    "            self.std.append(np.std(self.tr[sep], axis=0))\n",
    "\n",
    "        if self.plot_dis:\n",
    "            bar_data = Bar(x=list(range(len(self.count))), y=self.count)\n",
    "            data_plt = Data([bar_data])\n",
    "            layout = Layout(yaxis=YAxis(title='counts'), xaxis=XAxis(title='classes', dtick=1),\n",
    "                            title='Class distribution in Train set')\n",
    "            fig = Figure(data=data_plt, layout=layout)\n",
    "            self.plotly(fig)\n",
    "\n",
    "    def predict(self):\n",
    "        \"\"\"prediction: by summing the natural log of all the conditional probability given the class.\"\"\"\n",
    "        str_time = time()\n",
    "        self.mean_std_prior()\n",
    "        self.pred = []\n",
    "        self.likelihood = []\n",
    "        self.logsum_chk = []\n",
    "        for n in range(len(self.te_lb)):\n",
    "            classifier = []\n",
    "            sample = self.te[n] #test sample\n",
    "            likelih = []\n",
    "            for i, val in enumerate(self.n_class):\n",
    "                mean = self.mean[i]\n",
    "                var = np.square(self.std[i]) + self.smoothing\n",
    "                prob = 1 / np.sqrt(2 * np.pi * var) * np.exp(-np.square(sample - mean)/(2 * var))\n",
    "                result = np.sum(np.log(prob)) #, np.log(self.priors[i])) #not needed, we assume equal prior\n",
    "                classifier.append(result)\n",
    "                likelih.append(prob)\n",
    "\n",
    "            self.pred.append(np.argmax(classifier))\n",
    "            self.likelihood.append(likelih)\n",
    "            self.logsum_chk.append(classifier)\n",
    "\n",
    "        self.end_time = time() - str_time\n",
    "\n",
    "#End of Class Naive Bayes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 1: \n",
    "I implemented a Naive Bayes classifier and presented the testing error for each digit in a table. \n",
    "\n",
    "Below are the implemention explanations:\n",
    "\n",
    "Assuming that the probability model for each pixel is Gaussian and that the probability of each class — i.e., digit—is equal. That is, P(c = 0) = P(c = 1) = : : : = P(c = 9), thus, there is no need to model the prior probabilities.\n",
    "\n",
    "$$P(c \\ | \\ X) \\approx \\prod_{i=1}^{784}P(x_{i} \\ | \\ c)$$\n",
    "\n",
    "In order to manage underflow error, the function below will be modeled:\n",
    "\n",
    "$$ ln \\ P(c \\ | \\ X) \\approx \\sum_{i=1}^{784} ln \\ P(x_{i} \\ | \\ c) $$\n",
    "\n",
    "$$ where \\ P(x_{i}|c) = N(\\mu_{i},\\sigma_{i}^{2}\\ | \\ c) \\    is \\ modeled \\ as \\ a \\ Gaussian \\ from \\ the \\ training \\ data. $$\n",
    "\n",
    "$$ Thus, \\ \\mu_{i}|c \\ and \\ \\sigma_{i}^{2}|c  \\ were \\ determined \\ using \\ the \\ training \\ set.$$\n",
    "\n",
    "\n",
    "In order to avoid zero variance while computing the parameters above, smoothing will be applied. This involves adding a reasonable value to all the variances. The maximum variance given a class is about 13083, based on this, smoothing values ranging from 500 to 2000 with an increment of 100 will be tested through 5 fold cross validation by using the training set. The best smoothing value will then be applied to the entire training to get the model parameters. Finally, the parameters will be used for predicting the class label of the testing set. Also, I tried to apply zero - one normalization, but it gave a poor accuracy rate. Thus, feature scaling is not applied."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "util = Utility()\n",
    "smoothings_nb = list(range(500, 2100, 100))\n",
    "kfold_nb = util.nFoldCV_NB(util.train, util.train_label, smoothings_nb, kFolds=5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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ff/wRt2/fBvDP6ewTJ05g3LhxBt8zRdmxYwcqV66M/v37G7zWzZs3h5eXV6HXumnTpsVO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\nnrF1/QlNWRC6kzUHok8//RTLly8HAKxfvx4tW7ZUuCIiogo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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1f0aee35470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The best value of smoothing is 1000 with an error of 19.86%\n",
      "\n",
      "smoothing = 1000 will be used to model the entire training set and prediction on testing set:\n"
     ]
    }
   ],
   "source": [
    "sm_plot = Utility.CV_plot(kfold_nb, smoothings_nb, 'smoothing')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Testing Confusion Matrix: Actual vs. Prediction\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <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",
       "      <th>5</th>\n",
       "      <th>6</th>\n",
       "      <th>7</th>\n",
       "      <th>8</th>\n",
       "      <th>9</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>902</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>11</td>\n",
       "      <td>20</td>\n",
       "      <td>1</td>\n",
       "      <td>37</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0</td>\n",
       "      <td>1096</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>6</td>\n",
       "      <td>0</td>\n",
       "      <td>28</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>15</td>\n",
       "      <td>36</td>\n",
       "      <td>778</td>\n",
       "      <td>31</td>\n",
       "      <td>9</td>\n",
       "      <td>5</td>\n",
       "      <td>59</td>\n",
       "      <td>14</td>\n",
       "      <td>79</td>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>6</td>\n",
       "      <td>34</td>\n",
       "      <td>26</td>\n",
       "      <td>814</td>\n",
       "      <td>0</td>\n",
       "      <td>18</td>\n",
       "      <td>15</td>\n",
       "      <td>11</td>\n",
       "      <td>44</td>\n",
       "      <td>42</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>4</td>\n",
       "      <td>6</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>640</td>\n",
       "      <td>4</td>\n",
       "      <td>22</td>\n",
       "      <td>0</td>\n",
       "      <td>19</td>\n",
       "      <td>282</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>19</td>\n",
       "      <td>28</td>\n",
       "      <td>7</td>\n",
       "      <td>94</td>\n",
       "      <td>23</td>\n",
       "      <td>574</td>\n",
       "      <td>24</td>\n",
       "      <td>11</td>\n",
       "      <td>64</td>\n",
       "      <td>48</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>12</td>\n",
       "      <td>22</td>\n",
       "      <td>12</td>\n",
       "      <td>1</td>\n",
       "      <td>8</td>\n",
       "      <td>24</td>\n",
       "      <td>865</td>\n",
       "      <td>0</td>\n",
       "      <td>13</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>1</td>\n",
       "      <td>38</td>\n",
       "      <td>8</td>\n",
       "      <td>4</td>\n",
       "      <td>20</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>837</td>\n",
       "      <td>25</td>\n",
       "      <td>94</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>8</td>\n",
       "      <td>67</td>\n",
       "      <td>8</td>\n",
       "      <td>36</td>\n",
       "      <td>11</td>\n",
       "      <td>17</td>\n",
       "      <td>12</td>\n",
       "      <td>4</td>\n",
       "      <td>742</td>\n",
       "      <td>69</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>6</td>\n",
       "      <td>16</td>\n",
       "      <td>3</td>\n",
       "      <td>9</td>\n",
       "      <td>38</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>15</td>\n",
       "      <td>20</td>\n",
       "      <td>901</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     0     1    2    3    4    5    6    7    8    9\n",
       "0  902     0    0    4    1   11   20    1   37    4\n",
       "1    0  1096    1    4    0    0    6    0   28    0\n",
       "2   15    36  778   31    9    5   59   14   79    6\n",
       "3    6    34   26  814    0   18   15   11   44   42\n",
       "4    4     6    5    0  640    4   22    0   19  282\n",
       "5   19    28    7   94   23  574   24   11   64   48\n",
       "6   12    22   12    1    8   24  865    0   13    1\n",
       "7    1    38    8    4   20    0    1  837   25   94\n",
       "8    8    67    8   36   11   17   12    4  742   69\n",
       "9    6    16    3    9   38    1    0   15   20  901"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "% error per digit\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
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       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>digit</th>\n",
       "      <th>error per digit in %</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0</td>\n",
       "      <td>7.96</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>3.44</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>2</td>\n",
       "      <td>24.61</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>3</td>\n",
       "      <td>19.41</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>4</td>\n",
       "      <td>34.83</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>5</td>\n",
       "      <td>35.65</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>6</td>\n",
       "      <td>9.71</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>7</td>\n",
       "      <td>18.58</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>8</td>\n",
       "      <td>23.82</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>9</td>\n",
       "      <td>10.70</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   digit  error per digit in %\n",
       "0      0                  7.96\n",
       "1      1                  3.44\n",
       "2      2                 24.61\n",
       "3      3                 19.41\n",
       "4      4                 34.83\n",
       "5      5                 35.65\n",
       "6      6                  9.71\n",
       "7      7                 18.58\n",
       "8      8                 23.82\n",
       "9      9                 10.70"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "The overall testing error is 18.51%\n"
     ]
    }
   ],
   "source": [
    "util = Utility()\n",
    "best_s = smoothings_nb[np.argmin(kfold_nb)]\n",
    "nb = NaiveBayes(train=util.train, train_lb=util.train_label, test=util.test, test_lb=util.test_label, smoothing=best_s)\n",
    "nb.predict()\n",
    "conf_matix = util.confusionMatrix(util.test_label, nb.pred)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the information above the overall error is 18.51% (which is about 81.49% accuracy). The digits 0 and 1 tend to have minimal error rate compare to the other digits, this can be attributed to the fact that the digits 0 and 1 have low variabilities. Generally, people tend to write digits 0 and 1 in the right way.\n",
    "\n",
    "Digits 4 & 5 have the high error rates, visualizing some of them show that they have high variations. Digit 4 is mostly misclassified has 9, some people tend to write 4 in a way that it looks like 9. Also, digit 2 is mostly misclassified as 8.\n",
    "\n",
    "Digit 9 is mostly misclassified as either 4 or 8, likewise, digits 4 and 8 are also occasionally misclassified as 9.\n",
    "\n",
    "Generally, Naive Bayes did poorly on the MNIST dataset, this could be attributed to the independent assumption which is likely not to be correct. Query time is faster compared to KNN, however, KNN provided better performance on the MNIST dataset. Naive Bayes doesn't perform well when there are repeated attributes or when attributes are not equally important, which are the case in the MNIST dataset."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Visualizing the 784 conditional probabilities per class (as 28 x 28 images) for a test sample with actual label 1 which is rightly classified:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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jydv+v89C+aS/wbzZ5Z7duG6aG++O174/PltFCCGEEEIIIYQQQgghhBwH8CU6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\nVVE+hq73nfY/rNaVHZ85ddVO/hv0KXMdw7wQ+gE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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1f0aee2dc88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Utility.images_plot(nb.likelihood[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is really fascinating, the classifier actually modeled the image that corresponds to the class, i.e P(x|c=0) modeled an image of 0, etc. While the actual label is presented on the modeled image. Thus, in the case above, the actual label really match the C=1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Visualizing the 784 conditional probabilities per class (as 28 x 28 images) for a test sample with actual label 4 which is wrongly classified as 9:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ueonSjgou+VtZsJYZP9x7FtS9sH4+lOOb7ZKS7gFcxhIfxnWf0VG7FCKYwWUQ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\ns4u7atuxvejXFt2pvKIc7zSjvNK0L1Gx1X6/8gz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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1f0aee556a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Utility.images_plot(nb.likelihood[9997])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this instance, the actual label doesn't fit well on P(x|c=4), it seems more fitted on P(x|c=9), hence the classier prediction it as 9. On the confusion matrix, digit 4 is mostly classified as 9. Some people tend to write 4 in a way that it looks like 9."
   ]
  }
 ],
 "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": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}