{
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# [20 POINTS] COS 324: Clustering\n",
        "\n",
        "In this exercise, we'll group datapoints into clusters. The twist is that we'll use a different measure of distance (which is not symmetric). Let $\\Delta = \\{\\mathbf{x}\\in\\mathbb{R}^n: \\mathbf{x}_i\\geq 0, \\sum_{i=1}^n \\mathbf{x}_i = 1\\}$. For a point $\\mathbf{x}\\in\\Delta$ and cluster center $\\mathbf{z}\\in\\Delta$, define the distance between them to be\n",
        "$$d(\\mathbf{x}, \\mathbf{z}) = \\sum_{i=1}^n \\mathbf{x}_i \\log \\frac{\\mathbf{x}_i}{\\mathbf{z}_i}.$$\n",
        "\n",
        "# [14 POINTS] Part A: A Simple Dataset\n",
        "We sample 1000 random datapoints. The variable `X` is a $1000\\times 3$-shaped array."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "n_samples, n_dim = 1000, 3\n",
        "X = np.random.dirichlet(np.ones(n_dim), n_samples)"
      ],
      "outputs": [],
      "execution_count": 1,
      "metadata": {
        "colab": {
          "autoexec": {
            "startup": false,
            "wait_interval": 0
          }
        },
        "colab_type": "code",
        "collapsed": true,
        "id": "ZHlM2VjtqDqn"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "The `plot` function takes in\n",
        "1. the variable `X`, \n",
        "2. a $1000$-dimensional vector `belong` such that `belong[i]` denotes the cluster `X[i]` belongs to.\n",
        "\nClusters are numbered from $0$ to $\\texttt{n_clusters}-1$."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "import matplotlib.pyplot as plt\n",
        "from mpl_toolkits.mplot3d import Axes3D\n",
        "from sklearn.decomposition import PCA\n",
        "\n",
        "def plot(X, belong, ev=30, az=30):\n",
        "    if X.shape[1] > 3:\n",
        "        pca = PCA(n_components=3)\n",
        "        X = pca.fit_transform(X)\n",
        "    color_dict = ['b', 'g', 'r', 'c', 'm', 'y', 'k', 'w']\n",
        "    fig = plt.figure()\n",
        "    ax = fig.add_subplot(111, projection='3d')\n",
        "    for j in range(max(belong)+1):\n",
        "        ax.scatter(X[belong==j,0], \n",
        "                   X[belong==j,1], \n",
        "                   X[belong==j,2], c=color_dict[j])\n",
        "    ax.view_init(elev=ev, azim=az)\n",
        "    plt.show()\n",
        "\n",
        "plot(X, np.zeros(X.shape[0]).astype(int))\n",
        "\n",
        "def lostplot(loss):\n",
        "    plt.plot(loss,'-*')\n",
        "    plt.xlabel('Number of iterations')\n",
        "    plt.ylabel('Loss Value')\n",
        "    plt.grid()\n",
        "    plt.show()"
      ],
      "outputs": [
        {
          "output_type": "display_data",
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            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x106c0cc88>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 2,
      "metadata": {
        "colab": {
          "autoexec": {
            "startup": false,
            "wait_interval": 0
          },
          "base_uri": "https://localhost:8080/",
          "height": 330,
          "output_extras": [
            {
              "item_id": 1
            }
          ]
        },
        "colab_type": "code",
        "executionInfo": {
          "elapsed": 909,
          "status": "ok",
          "timestamp": 1513289333509,
          "user": {
            "displayName": "Karan Singh",
            "photoUrl": "//lh5.googleusercontent.com/-pkqojNWhanc/AAAAAAAAAAI/AAAAAAAAAHk/6viixounHWE/s50-c-k-no/photo.jpg",
            "userId": "114289659728532138225"
          },
          "user_tz": 300
        },
        "id": "Y-AuMiu1EqdP",
        "outputId": "fe0e5e5c-99ae-4e77-b78d-e0c74cdbf38b"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Given vectors $\\mathbf{x},\\mathbf{z}\\in \\Delta$, design a function `d(x,z)` that computes the distance between them."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "# [4 POINTS] for the correct implementation.\n",
        "# 0 for not passing the assertion.\n",
        "# 2-3 POINTS for incorrect code passing the assertion.\n",
        "# ACT 1: Compute the distance between x,z.\n",
        "def d(x, z):\n",
        "    return np.sum(x*np.log(x)-x*np.log(z))\n",
        "\n",
        "x, z = np.array([0.1, 0.2, 0.7]), np.array([0.2, 0.1, 0.7])\n",
        "assert(abs(d(x,z) - 0.069) < 1e-3)"
      ],
      "outputs": [],
      "execution_count": 3,
      "metadata": {
        "colab": {
          "autoexec": {
            "startup": false,
            "wait_interval": 0
          },
          "base_uri": "https://localhost:8080/",
          "height": 34,
          "output_extras": [
            {
              "item_id": 1
            }
          ]
        },
        "colab_type": "code",
        "executionInfo": {
          "elapsed": 234,
          "status": "ok",
          "timestamp": 1513289338339,
          "user": {
            "displayName": "Karan Singh",
            "photoUrl": "//lh5.googleusercontent.com/-pkqojNWhanc/AAAAAAAAAAI/AAAAAAAAAHk/6viixounHWE/s50-c-k-no/photo.jpg",
            "userId": "114289659728532138225"
          },
          "user_tz": 300
        },
        "id": "qw3GdC_CErcu",
        "outputId": "274dff91-b00a-4808-c60d-ab1635784a23"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "The function `cluster` takes in\n",
        "1. a 2d array of datapoints `X`,\n",
        "2. a 2d array of cluster centers `Z`,\n",
        "3. the maximum number of iterations `n_iter`\n",
        "\n",
        "and returns\n",
        "1. a 2d array of cluster centers `Z`,\n",
        "2. a 1d array of cluster assignments `belong`,\n",
        "3. a 1d array of loss (see slide 11) at each iteration."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "# [10 POINTS] Full grade if correct and loss goes down quickly.\n",
        "# 6 for returning all arguments; bad clusters, loss stays the same.\n",
        "# 8 for good looking clusters; loss saturates at a high value.\n",
        "# ACT 2: Design the function `cluster`.\n",
        "def cluster(X, Z, n_iter):\n",
        "    n_clusters, n_samples = Z.shape[0], X.shape[0]\n",
        "    loss = np.zeros(n_iter)\n",
        "    for t in range(n_iter):\n",
        "        dist = [[d(X[i], Z[j]) for j in range(n_clusters)]\n",
        "                   for i in range(n_samples)]\n",
        "        belong = np.argmin(dist, axis=1)\n",
        "        loss[t] = np.min(dist, axis=1).sum()\n",
        "        for j in range(n_clusters):\n",
        "            Z[j] = np.mean(X[belong==j], axis=0)\n",
        "    return Z, belong, loss"
      ],
      "outputs": [],
      "execution_count": 4,
      "metadata": {
        "colab": {
          "autoexec": {
            "startup": false,
            "wait_interval": 0
          }
        },
        "colab_type": "code",
        "collapsed": true,
        "id": "Sb17s7jDEsyy"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n_clusters, n_iter = 4, 20\n",
        "\n",
        "# Initializing the initial centers of clusters.\n",
        "Z = np.random.dirichlet(np.ones(n_dim), n_clusters)\n",
        "\n",
        "Z, belong, loss = cluster(X, Z, n_iter)\n",
        "plot(X, belong)\n",
        "lostplot(loss)"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x106ec72e8>"
            ]
          },
          "metadata": {}
        },
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              "TkSuQmCC\n"
            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x106343588>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 5,
      "metadata": {
        "colab": {
          "autoexec": {
            "startup": false,
            "wait_interval": 0
          },
          "base_uri": "https://localhost:8080/",
          "height": 330,
          "output_extras": [
            {
              "item_id": 1
            }
          ]
        },
        "colab_type": "code",
        "executionInfo": {
          "elapsed": 861,
          "status": "ok",
          "timestamp": 1513289347953,
          "user": {
            "displayName": "Karan Singh",
            "photoUrl": "//lh5.googleusercontent.com/-pkqojNWhanc/AAAAAAAAAAI/AAAAAAAAAHk/6viixounHWE/s50-c-k-no/photo.jpg",
            "userId": "114289659728532138225"
          },
          "user_tz": 300
        },
        "id": "TEUSipYFEtyq",
        "outputId": "a0c3edc4-1cde-4c64-8c06-69dda8693345"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# [6 POINTS] Part B: Real Data\n",
        "In this section, we'll attempt to cluster text documents."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "n_dim = 1000\n",
        "\n",
        "from sklearn.datasets import fetch_20newsgroups\n",
        "data = fetch_20newsgroups().data\n",
        "from sklearn.feature_extraction.text import TfidfVectorizer\n",
        "tfidf = TfidfVectorizer(max_features=n_dim)\n",
        "X = tfidf.fit_transform(data).astype(float).toarray()\n",
        "X += 1\n",
        "X = X/X.sum(axis=1)[:,np.newaxis]"
      ],
      "outputs": [],
      "execution_count": 6,
      "metadata": {
        "colab": {
          "autoexec": {
            "startup": false,
            "wait_interval": 0
          },
          "base_uri": "https://localhost:8080/",
          "height": 34,
          "output_extras": [
            {
              "item_id": 1
            }
          ]
        },
        "colab_type": "code",
        "collapsed": true,
        "executionInfo": {
          "elapsed": 13309,
          "status": "ok",
          "timestamp": 1513289367991,
          "user": {
            "displayName": "Karan Singh",
            "photoUrl": "//lh5.googleusercontent.com/-pkqojNWhanc/AAAAAAAAAAI/AAAAAAAAAHk/6viixounHWE/s50-c-k-no/photo.jpg",
            "userId": "114289659728532138225"
          },
          "user_tz": 300
        },
        "id": "g9SyA7bIEu_B",
        "outputId": "17f09f7b-b04a-4557-8f87-55c5c784ed40"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n_samples, n_clusters, n_iter = X.shape[0], 4, 10\n",
        "\n",
        "# [4 POINTS] 4 for randomly chosen points from dataset.\n",
        "# 2 points if they are just randomly drawn from dirichlet (as in part A).\n",
        "# ACT 3: Find a good way to initialize the cluster centers.\n",
        "# Z is n_clusters x n_dim.\n",
        "Z = X[np.random.choice(n_samples, n_clusters)]\n",
        "\nZ, belong, loss = cluster(X, Z, n_iter)"
      ],
      "outputs": [],
      "execution_count": 7,
      "metadata": {
        "colab": {
          "autoexec": {
            "startup": false,
            "wait_interval": 0
          },
          "base_uri": "https://localhost:8080/",
          "height": 330,
          "output_extras": [
            {
              "item_id": 1
            }
          ]
        },
        "colab_type": "code",
        "executionInfo": {
          "elapsed": 14029,
          "status": "ok",
          "timestamp": 1513289382033,
          "user": {
            "displayName": "Karan Singh",
            "photoUrl": "//lh5.googleusercontent.com/-pkqojNWhanc/AAAAAAAAAAI/AAAAAAAAAHk/6viixounHWE/s50-c-k-no/photo.jpg",
            "userId": "114289659728532138225"
          },
          "user_tz": 300
        },
        "id": "2kR1QEv1Ew1l",
        "outputId": "159e5245-39e2-4e59-fdb7-7ca08098cb67"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Plotting from 3 different angles.\n",
        "plot(X, belong)\n",
        "plot(X, belong, az=120)\n",
        "plot(X, belong, az=90, ev=-30)"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x106343588>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
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            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x10ed04fd0>"
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        },
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              "CrKLi4tLnuAKsouLi0ue4Aqyi4uLS57gCrKLi4tLnvD/AId3L6J3NwfLAAAAAElFTkSuQmCC\n"
            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x11082d8d0>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 8,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "lostplot(loss)"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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              "AElFTkSuQmCC\n"
            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x10ea0a128>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 9,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "# [2 POINTS] iff correct.\n",
        "# ACT 4: Print the number of points in each cluster.\n",
        "for (i, c) in zip(range(n_clusters), np.bincount(belong)):\n",
        "    print(\"Cluster %d has %d points.\"%(i,c))"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Cluster 0 has 1411 points.\n",
            "Cluster 1 has 2700 points.\n",
            "Cluster 2 has 2694 points.\n",
            "Cluster 3 has 4509 points.\n"
          ]
        }
      ],
      "execution_count": 10,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true
      }
    }
  ],
  "metadata": {
    "colab": {
      "default_view": {},
      "name": "C24",
      "provenance": [],
      "version": "0.3.2",
      "views": {}
    },
    "kernel_info": {
      "name": "python3"
    },
    "kernelspec": {
      "name": "python3",
      "language": "python",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python",
      "version": "3.6.1",
      "mimetype": "text/x-python",
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "pygments_lexer": "ipython3",
      "nbconvert_exporter": "python",
      "file_extension": ".py"
    },
    "nteract": {
      "version": "0.4.3"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 1
}