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   "source": [
    "# Exercise 4.3\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": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Task 3)\n",
    "Compute the categorical cross-entropy between data and model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Task 4)\n",
    "Determine which calsses are predicted by the model (maximum prediction)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Task 5)\n",
    "Estimate how many samples are classified correctly (accuracy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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