Exercise_04_3_solution.ipynb

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    "# Exercise 4.3 - Solution\n",
    "## Classification\n",
    "In the following tasks, we will repeatedly use some basic functions (e.g., the softmax function or the cross-entropy) of the [Keras](https://keras.io/) Library. To familiarize with them, we will implement the most important of them ourselves in this task.\n",
    "\n",
    "Suppose we want to classify some data (4 samples) into 3 distinct classes: 0, 1, and 2.\n",
    "We have set up a network with a pre-activation output `z` in the last layer.\n",
    "Applying softmax will give the final model output.\n",
    "\n",
    "input X ---> some network --> `z`<br>\n",
    "--> `y_model = softmax(z)`\n",
    "\n",
    "We quantify the agreement between truth (y) and model using categorical cross-entropy.\n",
    "\n",
    "$$J = - \\sum_i (y_i * \\log(y_\\mathrm{model}(x_i))$$\n",
    "\n",
    "In the following you are to implement softmax and categorical cross-entropy\n",
    "and evaluate them values given the values for `z`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Data: 4 samples with the following class labels (input features X irrelevant here)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_cl = np.array([0, 0, 2, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### output of the last network layer before applying softmax"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "z = np.array([\n",
    "    [4,    5,    1],\n",
    "    [-1,  -2,   -3],\n",
    "    [0.1, 0.2, 0.3],\n",
    "    [-1,  17,    1]\n",
    "    ]).astype(np.float32)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Task 1)\n",
    "Write a function that turns any class labels `y_cl` into one-hot encodings `y`.\n",
    "\n",
    "0 --> (1, 0, 0)<br>   \n",
    "1 --> (0, 1, 0)<br>   \n",
    "2 --> (0, 0, 1)<br>\n",
    "\n",
    "Make sure that `np.shape(y) = (4, 3)` for `np.shape(y_cl) = (4)`.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "one-hot encoding of data labels\n",
      "[[1. 0. 0.]\n",
      " [1. 0. 0.]\n",
      " [0. 0. 1.]\n",
      " [0. 1. 0.]]\n"
     ]
    }
   ],
   "source": [
    "def to_onehot(y_cl, num_classes):\n",
    "    y = np.zeros((len(y_cl), num_classes))\n",
    "    y[np.arange(4), y_cl] = 1\n",
    "    return y\n",
    "\n",
    "y = to_onehot(y_cl, num_classes=3)\n",
    "print('one-hot encoding of data labels')\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Task 2)\n",
    "Write a function that returns the softmax of the input `z` along the last axis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "softmax(z)\n",
      "[[2.6538792e-01 7.2139925e-01 1.3212887e-02]\n",
      " [6.6524100e-01 2.4472848e-01 9.0030573e-02]\n",
      " [3.0060962e-01 3.3222499e-01 3.6716542e-01]\n",
      " [1.5229979e-08 9.9999994e-01 1.1253517e-07]]\n"
     ]
    }
   ],
   "source": [
    "def softmax(z):\n",
    "    expz = np.exp(z).T\n",
    "    return (expz / np.sum(expz, axis=0)).T\n",
    "\n",
    "y_model = softmax(z)\n",
    "print('softmax(z)')\n",
    "print(y_model)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Task 3)\n",
    "Compute the categorical cross-entropy between data and model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cross entropy = 0.684028\n"
     ]
    }
   ],
   "source": [
    "crossentropy = -np.mean(np.sum(y * np.log(y_model), axis=1))\n",
    "crossentropy = -np.mean(np.log(y_model[np.arange(4), y_cl]))  # alternative formulation\n",
    "print('cross entropy = %f' % crossentropy)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Task 4)\n",
    "Determine which calsses are predicted by the model (maximum prediction)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "true class labels =  [0 0 2 1]\n",
      "predicted class labels = [1 0 2 1]\n"
     ]
    }
   ],
   "source": [
    "y_model_cl = np.argmax(y_model, axis=1)\n",
    "print('\\ntrue class labels = ', y_cl)\n",
    "print('predicted class labels =', y_model_cl)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Task 5)\n",
    "Estimate how many samples are classified correctly (accuracy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "accuracy = 0.75\n"
     ]
    }
   ],
   "source": [
    "accuracy = np.mean(y_model_cl == y_cl)\n",
    "print('accuracy = %.2f' % accuracy)"
   ]
  }
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