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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Low Rank Process with Non Gaussian Likelihood\n",
      "\n",
      "# Gaussian Process Summer School, Melbourne, Australia\n",
      "### 25th-27th February 2015\n",
      "### James Hensman and Neil Lawrence\n",
      "\n",
      "In this lab we are going to consider approaches to dealing with non-Gaussian likelihoods."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt\n",
      "import GPy\n",
      "import pods\n",
      "from IPython.display import display"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Sparse GP Classification\n",
      "\n",
      "In this section we'll combine expectation propagation with the low rank approximation to build a simple image classificationa application. For this toy example we'll classify whether or not the subject of the image is wearing glasses.\n",
      "correspond to whether the subject of the image is wearing glasses.\n",
      "Set up the ipython environment and download the data:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "data = pods.datasets.olivetti_glasses()\n",
      "Xtrain = data['X']\n",
      "ytrain = data['Y']"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here\u2019s a simple way to visualise the data. Each pixel in the image will become an input\n",
      "to the GP."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.imshow(Xtrain[120].reshape(64,64,order='F'),\n",
      "interpolation='nearest',cmap=plt.cm.gray)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "<matplotlib.image.AxesImage at 0x10f903210>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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lp8kuSZKx5KBPkp4xcfF+KFJ18bpzoiuX6XEcaEFjxrOIpvHyGRW7XKx5Z9bh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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10f8b9790>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now fetch the class labels"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Divide the data into a training/testing set:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Next we choose some inducing inputs. Here we've chosen inducing inputs by applying k-means clustering to the training data. Think about whether this is a good scheme for choosing the inputs? Can you\n",
      "devise a better one?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy import cluster\n",
      "M = 8\n",
      "#Z, distortion = cluster.vq.kmeans(Xtrain,M)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Xtrain_std = (Xtrain - Xtrain.mean(0)[None,:] )/ (Xtrain.std(0)[None,:])\n",
      "Z = np.random.permutation(Xtrain_std)[:M].copy()\n",
      "print Xtrain.mean()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "132.132182617\n"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Finally, we\u2019re ready to build the classifier object."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "k = GPy.kern.RBF(Xtrain.shape[1],lengthscale=20) + GPy.kern.White(Xtrain.shape[1],0.001)\n",
      "model = GPy.models.SparseGPClassification(Xtrain_std, ytrain, kernel=k, Z=Z)\n",
      "display(model)\n",
      "model.optimize()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "clang: warning: argument unused during compilation: '-fopenmp'\n",
        "In file included from /Users/neil/.cache/scipy/python27_compiled/sc_1790bf65208b11355ffcfd4b65a5f1091.cpp:11:\n",
        "In file included from /Users/neil/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/scipy/weave/blitz/blitz/array.h:26:\n",
        "In file included from /Users/neil/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/scipy/weave/blitz/blitz/array-impl.h:37:\n",
        "/Users/neil/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/scipy/weave/blitz/blitz/range.h:120:34: warning: '&&' within '||' [-Wlogical-op-parentheses]\n",
        "        return ((first_ < last_) && (stride_ == 1) || (first_ == last_));\n",
        "                ~~~~~~~~~~~~~~~~~^~~~~~~~~~~~~~~~~ ~~\n",
        "/Users/neil/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/scipy/weave/blitz/blitz/range.h:120:34: note: place parentheses around the '&&' expression to silence this warning\n",
        "        return ((first_ < last_) && (stride_ == 1) || (first_ == last_));\n",
        "                                 ^\n",
        "                (                                 )\n",
        "In file included from /Users/neil/.cache/scipy/python27_compiled/sc_1790bf65208b11355ffcfd4b65a5f1091.cpp:23:\n",
        "In file included from /Users/neil/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/numpy/core/include/numpy/arrayobject.h:4:\n",
        "In file included from /Users/neil/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/numpy/core/include/numpy/ndarrayobject.h:17:\n",
        "In file included from /Users/neil/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/numpy/core/include/numpy/ndarraytypes.h:1761:\n",
        "/Users/neil/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/numpy/core/include/numpy/npy_1_7_deprecated_api.h:15:2: warning: \"Using deprecated NumPy API, disable it by \"          \"#defining NPY_NO_DEPRECATED_API NPY_1_7_API_VERSION\" [-W#warnings]\n",
        "#warning \"Using deprecated NumPy API, disable it by \" \\\n",
        " ^\n",
        "/Users/neil/.cache/scipy/python27_compiled/sc_1790bf65208b11355ffcfd4b65a5f1091.cpp:24:10: fatal error: 'omp.h' file not found\n",
        "#include <omp.h>\n",
        "         ^\n",
        "2 warnings and 1 error generated.\n",
        "\n",
        " Weave compilation failed. Falling back to (slower) numpy implementation\n",
        "\n"
       ]
      },
      {
       "html": [
        "<style type=\"text/css\">\n",
        ".pd{\n",
        "    font-family:\"Courier New\", Courier, monospace !important;\n",
        "}\n",
        "</style>\n",
        "\n",
        "<p class=pd>\n",
        "<b>Model</b>: SparseGPClassification<br>\n",
        "<b>Log-likelihood</b>: -329.515082269<br>\n",
        "<b>Number of Parameters</b>: 32771<br>\n",
        "</p>\n",
        "<style type=\"text/css\">\n",
        ".tg  {border-collapse:collapse;border-spacing:0;border-color:#999;}\n",
        ".tg td{font-family:Arial, sans-serif;font-size:14px;padding:2px 3px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:#999;color:#444;background-color:#F7FDFA;}\n",
        ".tg th{font-family:Arial, sans-serif;font-size:14px;font-weight:normal;padding:2px 3px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:#999;color:#fff;background-color:#26ADE4;}\n",
        ".tg .tg-left{font-family:\"Courier New\", Courier, monospace !important;;text-align:left}\n",
        ".tg .tg-right{font-family:\"Courier New\", Courier, monospace !important;;text-align:right}\n",
        "</style>\n",
        "<table class=\"tg\">\n",
        "<tr>\n",
        "  <th><b>SparseGPClassification.</b></th>\n",
        "  <th><b>Value</b></th>\n",
        "  <th><b>Constraint</b></th>\n",
        "  <th><b>Prior</b></th>\n",
        "  <th><b>Tied to</b></th>\n",
        "</tr>\n",
        "<tr><td class=tg-left>inducing inputs        </td><td class=tg-right>(8, 4096)</td><td class=tg-left>          </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>add.rbf.variance       </td><td class=tg-right>      1.0</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>add.rbf.lengthscale    </td><td class=tg-right>     20.0</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "<tr><td class=tg-left>add.white.variance     </td><td class=tg-right>    0.001</td><td class=tg-left>   +ve    </td><td class=tg-left>     </td><td class=tg-left>       </td></tr>\n",
        "</table>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<GPy.models.sparse_gp_classification.SparseGPClassification at 0x11045df50>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Exercise 1\n",
      "Look at the following figure. What is being shown? Why does it look\n",
      "like this?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure()\n",
      "plt.imshow(model.Z[0].gradient.reshape(64,64,order='F'),\n",
      "           interpolation='nearest',\n",
      "           cmap=plt.cm.gray)\n",
      "plt.colorbar()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 20,
       "text": [
        "<matplotlib.colorbar.Colorbar instance at 0x118b6d4d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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3m5kkSS/om8HM3VcCmN/k+9dQi1OUJEkf0zeDWSdoTJHVdYGn6lFkDEVfTVeV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\ncjBLkqQvyMEsSZK+IAezJEn6gv8PlmKxNHpnqigAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x11310b350>"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "raw",
     "metadata": {},
     "source": [
      "# Exercise 1  answer here"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Exercise 2\n",
      "Write some code to evaluate the model\u2019s performance, using the held-out data that we separated earlier. How is the error rate? Is that better than random guessing? \n",
      "\n",
      "*Hint:* \n",
      "```python\n",
      "GPy.util.classification.conf_matrix(prob_estimate,ytest)\n",
      "```"
     ]
    },
    {
     "cell_type": "raw",
     "metadata": {},
     "source": [
      "# Exercise 2 answer here"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Exercise 3\n",
      "\n",
      "Write another simple for loop to optimize the model. How low can you get the error rate to go? What kind of kernel do you think might be appropriate for this classification task?"
     ]
    },
    {
     "cell_type": "raw",
     "metadata": {},
     "source": [
      "# Exercise 3 answer here"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}