CJS_MNIST_28x28-CrossEnt.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from sklearn.datasets import fetch_openml\n",
    "from sklearn.model_selection import train_test_split"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "mnist = fetch_openml('mnist_784', version=1, cache=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['data',\n",
       " 'target',\n",
       " 'frame',\n",
       " 'feature_names',\n",
       " 'target_names',\n",
       " 'DESCR',\n",
       " 'details',\n",
       " 'categories',\n",
       " 'url']"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(mnist)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(70000, 784)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mnist.data.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(70000,)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mnist.target.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x1a199a0a10>"
      ]
     },
     "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": [
    "plt.imshow(mnist.data[100].reshape(28,28),cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "images = (0.99 * mnist.data)/ 255"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.99 0.0\n"
     ]
    }
   ],
   "source": [
    "print(np.max(images[0]),np.min(images[0]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sigmoid(x):\n",
    "    return 1/(1+np.exp(-x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "70000\n"
     ]
    }
   ],
   "source": [
    "num = mnist.target.shape[0]\n",
    "print(num)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# one-hot encode the y_true values\n",
    "y_true = np.zeros((num,10))\n",
    "for i in range(num):\n",
    "    y_true[i,int(mnist.target[i])] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "14000\n"
     ]
    }
   ],
   "source": [
    "X_train,X_test,y_train,y_test = \\\n",
    "train_test_split(mnist.data,\n",
    "                 y_true,\n",
    "                 test_size = 0.8, \n",
    "                 random_state = 42)\n",
    "size = X_train.shape[0]\n",
    "print(size)\n",
    "#for i in range(size):\n",
    "#    print(np.argmax(y_train[i]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "X_valid,y_valid = X_test[:14000],y_test[:14000]\n",
    "X_test_final,y_test_final = X_test[14000:],y_test[14000:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(14000, 784)\n",
      "(42000, 784)\n"
     ]
    }
   ],
   "source": [
    "print(X_valid.shape)\n",
    "print(X_test_final.shape)\n",
    "test_size = X_test_final.shape[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/christiansutton/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:2: RuntimeWarning: overflow encountered in exp\n",
      "  \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Epoch 0\n",
      "Error is 1.51; Accuracy is 78.66%\n",
      "Validation error is 1.52 and validation accuracy is 78.20%\n",
      "---------------------------------------------\n",
      "Epoch 1\n",
      "Error is 0.96; Accuracy is 87.56%\n",
      "Validation error is 0.98 and validation accuracy is 87.07%\n",
      "---------------------------------------------\n",
      "Epoch 2\n",
      "Error is 0.82; Accuracy is 88.86%\n",
      "Validation error is 0.85 and validation accuracy is 88.71%\n",
      "---------------------------------------------\n",
      "Epoch 3\n",
      "Error is 0.78; Accuracy is 89.45%\n",
      "Validation error is 0.80 and validation accuracy is 88.88%\n",
      "---------------------------------------------\n",
      "Epoch 4\n",
      "Error is 0.77; Accuracy is 89.69%\n",
      "Validation error is 0.80 and validation accuracy is 89.49%\n",
      "---------------------------------------------\n",
      "Epoch 5\n",
      "Error is 0.73; Accuracy is 90.14%\n",
      "Validation error is 0.76 and validation accuracy is 89.66%\n",
      "---------------------------------------------\n",
      "Epoch 6\n",
      "Error is 0.70; Accuracy is 90.59%\n",
      "Validation error is 0.73 and validation accuracy is 89.91%\n",
      "---------------------------------------------\n",
      "Epoch 7\n",
      "Error is 0.69; Accuracy is 90.75%\n",
      "Validation error is 0.72 and validation accuracy is 90.01%\n",
      "---------------------------------------------\n",
      "Epoch 8\n",
      "Error is 0.70; Accuracy is 90.59%\n",
      "Validation error is 0.73 and validation accuracy is 90.10%\n",
      "---------------------------------------------\n",
      "Epoch 9\n",
      "Error is 0.69; Accuracy is 90.71%\n",
      "Validation error is 0.72 and validation accuracy is 90.26%\n",
      "---------------------------------------------\n",
      "Epoch 10\n",
      "Error is 0.70; Accuracy is 90.84%\n",
      "Validation error is 0.73 and validation accuracy is 90.36%\n",
      "---------------------------------------------\n",
      "Epoch 11\n",
      "Error is 0.65; Accuracy is 91.20%\n",
      "Validation error is 0.68 and validation accuracy is 90.81%\n",
      "---------------------------------------------\n",
      "Epoch 12\n",
      "Error is 0.66; Accuracy is 90.90%\n",
      "Validation error is 0.68 and validation accuracy is 90.71%\n",
      "---------------------------------------------\n",
      "Epoch 13\n",
      "Error is 0.63; Accuracy is 91.24%\n",
      "Validation error is 0.66 and validation accuracy is 90.79%\n",
      "---------------------------------------------\n",
      "Epoch 14\n",
      "Error is 0.67; Accuracy is 90.97%\n",
      "Validation error is 0.70 and validation accuracy is 90.61%\n",
      "---------------------------------------------\n",
      "Epoch 15\n",
      "Error is 0.66; Accuracy is 91.12%\n",
      "Validation error is 0.69 and validation accuracy is 90.49%\n",
      "---------------------------------------------\n",
      "Epoch 16\n",
      "Error is 0.63; Accuracy is 91.32%\n",
      "Validation error is 0.67 and validation accuracy is 90.70%\n",
      "---------------------------------------------\n",
      "Epoch 17\n",
      "Error is 0.62; Accuracy is 91.33%\n",
      "Validation error is 0.66 and validation accuracy is 90.71%\n",
      "---------------------------------------------\n",
      "Epoch 18\n",
      "Error is 0.63; Accuracy is 91.39%\n",
      "Validation error is 0.67 and validation accuracy is 90.69%\n",
      "---------------------------------------------\n",
      "Epoch 19\n",
      "Error is 0.61; Accuracy is 91.52%\n",
      "Validation error is 0.65 and validation accuracy is 90.98%\n",
      "---------------------------------------------\n",
      "Final training accuracy is 91.52%\n"
     ]
    }
   ],
   "source": [
    "input_layer = 784\n",
    "hidden_layer = 256\n",
    "output_layer = 10\n",
    "\n",
    "w1 = np.random.normal(0,\n",
    "                      1/np.sqrt(input_layer),\n",
    "                      size = (hidden_layer,input_layer))\n",
    "\n",
    "w2 = np.random.normal(0,\n",
    "                      1/np.sqrt(hidden_layer),\n",
    "                      size = (output_layer,hidden_layer))\n",
    "\n",
    "b1 = np.ones((hidden_layer,1))\n",
    "\n",
    "b2 = np.ones((output_layer,1))\n",
    "\n",
    "lr = .001\n",
    "epochs = [x for x in range(20)]\n",
    "train_error = []\n",
    "valid_error = []\n",
    "Error = 0\n",
    "for epoch in epochs:\n",
    "    error = 0\n",
    "    cnt = 0\n",
    "    cnt_v = 0\n",
    "    error_v = 0\n",
    "    for i in range(size):\n",
    "        # get the true value\n",
    "        t  = y_train[i].reshape(output_layer,1)\n",
    "        tV = y_valid[i].reshape(output_layer,1)\n",
    "        # set nodes\n",
    "        x0 = X_train[i].reshape(input_layer,1)\n",
    "        x1 = sigmoid(w1.dot(x0)+b1)\n",
    "        x2 = sigmoid(w2.dot(x1)+b2)\n",
    "        # calculate errors\n",
    "        error += np.sum( -t * np.log(x2)-(1-t) * np.log(1-x2))/size\n",
    "        derr = (x2-t)/(x2 * (1-x2))\n",
    "        \n",
    "        x_valid_0 = X_valid[i].reshape(input_layer,1)\n",
    "        x_valid_1 = sigmoid(w1.dot(x_valid_0)+b1)\n",
    "        x_valid_2 = sigmoid(w2.dot(x_valid_1)+b2)\n",
    "        error_v += np.sum( -tV * np.log(x_valid_2)-(1-tV) * np.log(1-x_valid_2))/size\n",
    "        \n",
    "        #calculate deltas\n",
    "        delta_layer2 = (derr * (x2 * (1-x2)))        \n",
    "        delta_layer1 = (((w2.T).dot(delta_layer2)) * (x1 * (1-x1)))\n",
    "        \n",
    "        #clean up notation\n",
    "        w2 = w2 - lr * delta_layer2.dot(x1.T)\n",
    "        b2 = b2 - lr * delta_layer2\n",
    "        w1 = w1 - lr * delta_layer1.dot(x0.T)\n",
    "        b1 = b1 - lr * delta_layer1\n",
    "\n",
    "        #calculate how many are correct, for accuracy\n",
    "        cnt_v += np.argmax(x_valid_2) == np.argmax(tV)\n",
    "        cnt += np.argmax(x2) == np.argmax(t)\n",
    "    train_error.append(error)\n",
    "    valid_error.append(error_v)\n",
    "    if epoch%1 == 0:\n",
    "        print(f'Epoch {epoch}')\n",
    "        print(f'Error is {error:.2f}; Accuracy is {100*cnt/size:.2f}%')\n",
    "        print(f'Validation error is {error_v:.2f} and validation accuracy is {100*cnt_v/size:.2f}%')\n",
    "        print(\"---------------------------------------------\")\n",
    "print(f'Final training accuracy is {100*cnt/size:.2f}%')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cnt = 0\n",
    "print(f'Number of test samples is: {test_size}')\n",
    "for i in range(test_size):\n",
    "    t = y_test_final[i].reshape(output_layer,1)\n",
    "    # set nodes\n",
    "    x0 = X_test_final[i].reshape(input_layer,1)\n",
    "    x1 = sigmoid(w1.dot(x0)+b1)\n",
    "    x2 = sigmoid(w2.dot(x1)+b2)\n",
    "    cnt += np.argmax(x2) == np.argmax(t)\n",
    "print(f'Testing accuracy is {100 * cnt/(test_size):.2f}%')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print('Please input a digit:')\n",
    "i = int(input())\n",
    "blind = X_test_final[i].reshape(28,28)                                 \n",
    "plt.imshow(blind,cmap='gray')\n",
    "x0 = X_test_final[i].reshape(input_layer,1)\n",
    "x1 = sigmoid(w1.dot(x0)+b1)\n",
    "x2 = sigmoid(w2.dot(x1)+b2)\n",
    "print(f'The queried digit is {np.argmax(x2)}.  The correct digit is {np.argmax(y_test_final[i])}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(w2,cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(w1,aspect=10,cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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