examples/Monte Carlo Integration.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problems with Monte Carlo integration\n",
    "\n",
    "By default, we convert \"continuous\" predictions into grid based predictions by using monte carlo integration: evaluate the kernel at random points and average.\n",
    "\n",
    "This seems to introduce a large amount of error...  The following demonstrates this with artificial, random data.  However, I noticed this problem on data from Chicago."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "import open_cp.retrohotspot\n",
    "import open_cp.data\n",
    "import open_cp.predictors\n",
    "import open_cp.evaluation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "n = 1000\n",
    "times = np.random.random(n) * 365\n",
    "times = times * (np.timedelta64(1, \"D\") / np.timedelta64(1, \"s\")) * np.timedelta64(1, \"s\")\n",
    "times = np.sort(np.datetime64(\"2016-01-01\") + times)\n",
    "points = open_cp.data.TimedPoints(times, np.random.random((2,n)) * 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "mask = [[False]*100]*100\n",
    "grid = open_cp.data.MaskedGrid(10, 10, 0, 0, mask)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "predictor = open_cp.retrohotspot.RetroHotSpot()\n",
    "predictor.data = points\n",
    "predictor.weight = open_cp.retrohotspot.TruncatedGaussian(100, 10)\n",
    "cts_pred = predictor.predict(end_time=np.datetime64(\"2016-10-01\"))\n",
    "cts_pred.samples = 5\n",
    "pred = open_cp.predictors.GridPredictionArray.from_continuous_prediction_grid(cts_pred, grid)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PolyCollection at 0x7fc8b43e0b70>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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7b2cfXd+2zLn4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\nAgAAAIAOL0gAAAAA0OEFCQAAAAA6/x8SU/3ZBaz5owAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fc8b6f062e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(ncols=2, figsize=(14,8))\n",
    "\n",
    "ax[0].pcolor(*pred.mesh_data(), pred.intensity_matrix, cmap=\"Greys\")\n",
    "p = open_cp.evaluation.top_slice_prediction(pred, 0.1)\n",
    "ax[1].pcolor(*p.mesh_data(), p.intensity_matrix, cmap=\"Greys\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Repeat: Form the prediction again, top slice the top 10% of risk again, and compare the selected cells (same as looking at the \"mask\" of the top sliced prediction.  They differ by a lot!)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "cts_pred = predictor.predict(end_time=np.datetime64(\"2016-10-01\"))\n",
    "cts_pred.samples = 5\n",
    "pred = open_cp.predictors.GridPredictionArray.from_continuous_prediction_grid(cts_pred, grid)\n",
    "p1 = open_cp.evaluation.top_slice_prediction(pred, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(p.intensity_matrix.mask ^ p.intensity_matrix.mask)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(p1.intensity_matrix.mask ^ p1.intensity_matrix.mask)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "224"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(p.intensity_matrix.mask ^ p1.intensity_matrix.mask)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You might argue that 5 samples is too small..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "cts_pred = predictor.predict(end_time=np.datetime64(\"2016-10-01\"))\n",
    "cts_pred.samples = 25\n",
    "pred = open_cp.predictors.GridPredictionArray.from_continuous_prediction_grid(cts_pred, grid)\n",
    "p = open_cp.evaluation.top_slice_prediction(pred, 0.1)\n",
    "\n",
    "cts_pred = predictor.predict(end_time=np.datetime64(\"2016-10-01\"))\n",
    "cts_pred.samples = 25\n",
    "pred = open_cp.predictors.GridPredictionArray.from_continuous_prediction_grid(cts_pred, grid)\n",
    "p1 = open_cp.evaluation.top_slice_prediction(pred, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "86"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(p.intensity_matrix.mask ^ p1.intensity_matrix.mask)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have added a \"reproducible\" mode, whereby we don't sample at random points, but rather on a regular sub-grid"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "cts_pred = predictor.predict(end_time=np.datetime64(\"2016-10-01\"))\n",
    "cts_pred.samples = -5\n",
    "pred = open_cp.predictors.GridPredictionArray.from_continuous_prediction_grid(cts_pred, grid)\n",
    "p = open_cp.evaluation.top_slice_prediction(pred, 0.1)\n",
    "\n",
    "cts_pred = predictor.predict(end_time=np.datetime64(\"2016-10-01\"))\n",
    "cts_pred.samples = -5\n",
    "pred = open_cp.predictors.GridPredictionArray.from_continuous_prediction_grid(cts_pred, grid)\n",
    "p1 = open_cp.evaluation.top_slice_prediction(pred, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(p.intensity_matrix.mask ^ p1.intensity_matrix.mask)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}