convolution.ipynb

{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.rcParams['image.cmap'] = 'gray'\n",
      "plt.rcParams['image.interpolation'] = 'nearest'"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The Nipy HRF routines"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import nipy.modalities.fmri.hrf as hrfs"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Remember tab completion"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "t = np.arange(0, 25, 0.1)\n",
      "hrf = hrfs.glovert(t)\n",
      "plt.plot(t, hrf)\n",
      "plt.xlabel('time (seconds)')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "<matplotlib.text.Text at 0x1079aedd0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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/9bmTn2/WzNR6mJqPXI4dyyUoCO67r/F/l0vjDCoqKnA4HGzYsAGAe++9l9TU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\nlAxERAQlAxERAf4/PiOhc9iXYSQAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1079ae210>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "What is convolution?\n",
      "\n",
      "Wikipedia knows: http://en.wikipedia.org/wiki/Convolution"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In practice:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "dt = 1/8. # time interval in seconds\n",
      "t = np.arange(0, 40, dt) # times\n",
      "experiment = np.zeros_like(t)\n",
      "experiment[t == 5] = 1 # Event at 5 seconds\n",
      "plt.plot(experiment)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "[<matplotlib.lines.Line2D at 0x1079cce10>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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bVVpaqqamJiUSiZRtEomE1q9frzlz5qilpUWnT5/O3bRZdP2tCm70TtZC/4+kkOcv5Nkl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\nHgCMIvAAYNT/AVvTk6hrtnAIAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107d26950>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We make an HRF at a corresponding time resolution:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hrf_t = np.arange(0, 24, dt)\n",
      "hrf = hrfs.glovert(hrf_t)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Numpy has a convolve routine:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hrf_experiment = np.convolve(experiment, hrf)\n",
      "plt.plot(hrf_experiment)\n",
      "hrf_experiment.shape"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "(511,)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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2228bhmEYW7ZsMebMmdOpzOHDh43c3NzQ+4ULFxrr1q3rVO6dd94xrrrqqi6v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\nAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107915610>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "What happened?  More slowly:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def dumb_convolve(data, kernel):\n",
      "    data_len = len(data)\n",
      "    kernel_len = len(kernel)\n",
      "    out_len = data_len + kernel_len - 1\n",
      "    out = np.zeros((out_len,))\n",
      "    for i, value in enumerate(data):\n",
      "        out[i:i+kernel_len] += value * kernel\n",
      "    return out"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(dumb_convolve(experiment, hrf))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "[<matplotlib.lines.Line2D at 0x107b19b90>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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2228bhmEYW7ZsMebMmdOpzOHDh43c3NzQ+4ULFxrr1q3rVO6dd94xrrrqqi6v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\nAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1079ad810>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's add another event:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "experiment[t == 15] = 1\n",
      "plt.plot(dumb_convolve(experiment, hrf))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 11,
       "text": [
        "[<matplotlib.lines.Line2D at 0x107b39e10>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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EiRP9nuvx7rtiR7KeHkDv7/n7fkODqC385jfAyJFy/kWUNWSI2LCnoUHUHvyl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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107b20e50>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Rephrased - the convolution adds a scaled copy of the kernel to the time course, where the scaling comes from the data value at each time point."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "What about a block design?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "block = np.zeros((400,))\n",
      "block[50:150] = 1\n",
      "plt.plot(np.convolve(block, hrf))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "[<matplotlib.lines.Line2D at 0x107b6e8d0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXYAAAEACAYAAACnJV25AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAHBRJREFUeJzt3XlwlEX+BvBnIBBUDjnkcEM4Qu5AMmAOg8SY1ZhCQ3Cx\nSnHFlaCLUbxI7bI/XRVlV6Uo5UjpLrtrELWQdZWSQ00Q2SGAkiAknOHIIQEEDCiQhAQC6d8fvYmA\nOeZ43+mZnudT9VbCzOR9v03Jk7bffrstQggBIiLSRifVBRARkbEY7EREmmGwExFphsFORKQZBjsR\nkWYY7EREmrEr2DMzMzFgwACMHDmy5bXZs2cjICAAVqsVVqsVeXl5phVJRET2syvYp06d+ovgtlgs\nmDlzJoqLi1FcXIy0tDRTCiQiIsfYFezjxo1D7969f/E6n20iIvI8Lo2x5+TkICEhAXPnzkVNTY1R\nNRERkQucDvasrCxUVlYiPz8f5eXlWLx4sZF1ERGRs4SdKisrRVRUVKvvlZSUiMTExFbfCwoKEgB4\n8ODBg4cDR1BQkL3x/AtO99iPHTsGALh48SKWLVuG8ePHt/q58vJyCCG0PV566SXlNbB9bBvbp99R\nXl7ubDzDz54PTZ48GRs2bMDJkycxePBgvPzyy7DZbCgpKUHXrl2RlJSErKwsp4sgIiLj2BXsH374\n4S9ey8zMNLwYIiJyHZ88dVFycrLqEkylc/t0bhvA9vkyixBCmHoBiwUmX4KISDuuZCd77EREmmGw\nExFphsFORKQZBjsRkWYY7EREmmGwExFphsFORKQZBjsRkWYY7EREmmGwExFphsFORKQZBjsRkWYY\n7EREmmGwExFphsFORKQZBjsRkWYY7EREmmGwExFphsFORKQZBjsRkWYY7EREmmGwExFphsFORKQZ\nBjsRkWYY7EREmmGwu9Hp08DzzwO33AKkpgKLFgF1daqrIiLdMNjdZO9eICoKqK4GXnkFePJJYMMG\nICYGKC5WXR0R6cQihBCmXsBigcmX8Hjffw/ExQGvvw48+OCV7/373zLkV6yQPXkiIsC17GSwm0wI\nICMDiI4G5sxp/TNr1wJTpgAbNwIhIe6tj4g8kyvZyaEYk61cCVRUAC+80PZnUlNl6E+YIMfhiYhc\nwR67iZqaAKsV+MtfgPT0jj+flSWD/cMPza+NiDwbe+weavVqwM8PuPtu+z7/xhvyRury5ebWRUR6\nY7Cb6O23gWefBSwW+z5/7bXABx8ATz0FHD9ubm1EpC8OxZikslLOhDl8GOjWzbGfnTULOHpUhjwR\n+SYOxXigJUvk1EZHQx0AXnwR2LQJ+Oor4+siIv0x2E0gBPDRR8Dkyc79/HXXATk5wOOPA+fPG1sb\nEemPwW6C3buB+nogNtb5c6SnA+HhwJtvGlcXEfkGjrGb4KWXgNpaOcvFFRUVcpx+xw7gV78ypjYi\n8g4cY/cwn30mHzZy1fDhwPTp8mYqEZG92GM32IkTQGioXOyrSxfXz1dbC4SFyTH7xETXz0dE3oE9\ndg+ydi2QkmJMqANA9+7A3LlybntTkzHnJCK9MdgNlpcHpKUZe84HHgD8/eUUSiKijnAoxkBCAAEB\ncp31ESOMPfe2bXJpgn37gF69jD03EXkeDsV4iMpKOVwSFGT8uceMkcH+yivGn5uI9MJgN9DGjUBS\nkv1rwzjqr38F3ntP9tqJiNpiV7BnZmZiwIABGDlyZMtrNTU1yMjIQGBgICZOnIja2lrTivQWGzcC\n48aZd/7+/YHnngOeeUYO+xARtcauYJ86dSry8vKueO1vf/sbAgMDcfDgQQQEBODvf/+7KQV6E7OD\nHQBmzAAOHQLWrDH3OkTkvewK9nHjxqF3795XvFZUVIRp06bB398fmZmZKCwsNKVAb3HiBPDDD3LD\najN16QIsWCCXA+Y6MkTUGqfH2Ldu3YqwsDAAQFhYGIqKigwryhtt2iQfIOrc2fxr3XknEBEhA56I\n6Gp+zv6gI9NwZs+e3fJ9cnIykpOTnb2sxyoqAhIS3He9N9+U13voIWDQIPddl4jMYbPZYLPZDDmX\n08EeGxuL0tJSWK1WlJaWIradpQwvD3ZdbdsGZGe773ojRgCPPAL86U/A0qXuuy4RmePqTu/LL7/s\n9LmcHoqJj49Hbm4u6uvrkZubiwR3dlc9jBAy2MeMce91n38eWLcO2LLFvdclIs9mV7BPnjwZiYmJ\nOHDgAAYPHowlS5YgKysLVVVVCA0NxdGjR/HYY4+ZXavHKi8HevaU0xHdqUcP4LXXuI4MEV2JSwoY\nYPly4D//AT75xP3XbmqSN22nTwemTnX/9YnIHFxSQLFvvwVuuknNtTt1AhYtksMyZ8+qqYGIPAuD\n3QAqxtcvFxcnp0DOmaOuBiLyHByKcVFTE9C7t9zGrm9fdXUcPy4fjvr6ayAkRF0dRGQMDsUoVFEB\nXH+92lAHgIED5RZ6zz6rtg4iUo/B7qJdu4DL1kZT6umngdJS2WsnIt/FYHfR7t2eE+xdu8qpj4sW\nqa6EiFRisLvIk3rsgJzyuHYtcPSo6kqISBUGu4s8Ldh79QLuvRf44APVlRCRKpwV44KGBjkj5swZ\nOQziKTZtkg8s7d5t3m5ORGQuzopRpLRULsblSaEOAGPHAvX1QHGx6kqISAUGuwt27TJ/Yw1nWCzA\nffcBH3+suhIiUoHB7gJPmhFztYwMYOVK1VUQkQoMdhd42o3Ty8XFAT/+CJSVqa6EiNyNwe4CTw72\nTp2A9HRg9WrVlRCRuzHYnfTTT3I1xSFDVFfSttRUuREHEfkWBruTdu8GIiM9ezrhbbcBGzcCFy6o\nroSI3InB7qTSUiAiQnUV7evbFwgOBgoLVVdCRO7EYHfS/v1AaKjqKjp2++3AV1+proKI3InB7iRv\nCfakJDkcQ0S+g8HupAMHvGNDi5tvBoqKgIsXVVdCRO7CYHfChQtAVRUQFKS6ko716QMMHgzs3Km6\nEiJyFwa7EyoqZFh62hoxbRk7Fti8WXUVROQuDHYneMv4erPERO6qRORLGOxO2L/fO8bXm8XGAtu2\nqa6CiNyFwe6EAwe8q8ceFiZ3VDp7VnUlROQODHYneNtQjJ+fXNOmpER1JUTkDgx2J3hbsAPAmDHA\n9u2qqyAid2CwO+inn+SWeAMHqq7EMaNHc5ydyFcw2B3U/GCSJy/+1ZrRo9ljJ/IVDHYHeeMwDCBX\noqysBOrqVFdCRGZjsDvIW4O9a1e5GiWfQCXSH4PdQd42h/1yMTGcGUPkCxjsDvLWHjsAREUBe/ao\nroKIzMZgd0BTE1Be7r099shIufMTEemNwe6Aqiq5K9F116muxDlRUTLYhVBdCRGZicHuAG8ehgHk\n3PumJuCHH1RXQkRmYrA7wJtvnAJy7j3H2Yn0x2B3gLct/tWayEgGO5HuGOwO8PahGIA3UIl8AYPd\nAToEO4diiPRnEcLcORIWiwUmX8It6uqAfv2A2lqgc2fV1TivuhoIDpaLmXnbejdEvsSV7GSP3U5l\nZXLzam8OdQC44QagSxfg+HHVlRCRWRjsdtJhGKZZSIi8EUxEemKw20mnYA8Nle0hIj0x2O3k7XPY\nL8dgJ9Ibg91OOvXYORRDpDcGux2E0OPhpGbssRPpzc/VEwwdOhQ9e/ZE586d0aVLFxQVFRlRl0c5\ncUJuVNGnj+pKjBEUBBw6BFy4INtFRHpxOdgtFgtsNhv66JJ6rdBpfB0A/P2BgACgogIIC1NdDREZ\nzZChGB0eQGqPTuPrzUJDOc5OpCuXg91isSAlJQUTJ07EqlWrjKjJ4+g0vt6M4+xE+nJ5KGbz5s0Y\nNGgQSktLkZ6ejri4OAwcOPCKz8yePbvl++TkZCQnJ7t6Wbfavx+45RbVVRgrJATYtk11FUTUzGaz\nwWazGXIuQ9eKmTlzJsLDw/Hoo4/+fAEN1ooJCQE+/RSIiFBdiXHWrwdmzwYKClRXQkStUbZWzLlz\n51BTUwMAqK6uRn5+PtLS0lw5pce5cEFuiRcUpLoSY3EohkhfLg3FnDhxAvfccw8AoG/fvsjOzsbg\nwYMNKcxTVFTIGST+/qorMdaNN8oVK0+fBq6/XnU1RGQkl4J92LBhKCkpMaoWj6TjjVNALtkbEiJ7\n7fHxqqshIiPxydMO6DaH/XLBwcDBg6qrICKjMdg7sG8fEB6uugpzMNiJ9MRg70Bpqb5PZzLYifTE\nYG+HELLHrnOwl5WproKIjMZgb0d1tbzJeMMNqisxR3OP3csfMyCiqzDY29HcW9d10+d+/WSonzql\nuhIiMhKDvR06D8MA8hcWx9mJ9MNgb4fON06bMdiJ9MNgb4fuPXaAwU6kIwZ7OxjsROSNGOxtOHcO\nOH4cGDZMdSXmGjGCwU6kGwZ7Gw4elCs6+rm8Yr1n45RHIv0w2NvgCzdOAaBvX/nLq7padSVEZBQG\next8YXy9GcfZifTCYG+Dr/TYAQY7kW4Y7G3YvRuIilJdhXsw2In0wmBvxfnzcuckXZfrvRqDnUgv\nDPZW7Nsnpznqth1eWxjsRHphsLdi1y5g5EjVVbhP8/K9nPJIpAcGeyt8Ldivvx7o1k0+kEVE3o/B\n3gpfC3aAT6AS6YTB3gpfDHbupkSkDwb7VX76CTh9Ghg6VHUl7sUbqET6YLBfZfduIDIS6ORjfzMM\ndiJ9+Fh8daykBBg1SnUV7sdgJ9IHg/0q334LxMaqrsL9OOWRSB8M9qts3eqbwd6zJ9C9O/D996or\nISJXMdgvU1MDHDokx9h9EYdjiPTAYL9McbGc5tili+pK1GCwE+mBwX4ZXx2GacZgJ9IDg/0y334L\n3HST6irUGTGCDykR6YDBfhlfD3b22In0YBHC3AluFosFJl/CECdPys2rf/wR6NxZdTVq1NQAAwYA\ntbW+94AWkadxJTv5z/d/Nm0CEhN9N9QBoEcPoFcv4OhR1ZUQkSsY7P9TUAAkJamuQj0OxxB5Pwb7\n/zDYJQY7kfdjsAM4e1Zuh+fLN06bMdiJvB+DHcBXX8nxdV/Z47Q9DHYi78dgB/DFF8D48aqr8AwM\ndiLv5/PTHYUABg8G1q8HQkJUV6PeuXNA375yyqMvzxAiUo3THV2wY4ccggkOVl2JZ7j2WhnsR46o\nroSInOXzwb58OXDvvYDForoSz8GNrYm8m08He1MTsGwZ8Nvfqq7Es3Ccnci7+XSwb9won7T0xa3w\n2sNgJ/JuPh3sixYBjz6qugrPw2An8m5+qgtQ5eBB+bTpe++prsTzMNiJvJvPTne8/34gPBx46SXV\nlXie+nqgd2855dHPZ3/1E6mldLpjQUEBwsPDERwcjJycHFdP5xaffy7XXv/jH1VX4pmuuQbo3x+o\nqlJdCRE5w+Vgf/rpp7F48WKsW7cOb731Fk6ePGlEXabZuxeYOhX45z9lgFHrgoO5mxKRt3Ip2M+c\nOQMASEpKwpAhQ5CamorCwkJDCjPapUvAhx8CKSnAG28At92muiLPxnF2Iu/l0gjq1q1bERYW1vLn\niIgIbNmyBXfdddcVn3vxRflVCHk0f+/Ka478TFWVHHoZMgRYuRKIj3el1b6BwX6lujqgulruMlVX\nJ4/a2p+/b2wELl6UHYirv3rgLSbyIGFh8p6fkdxya6ygYHbLk53DhiVj+PBkAD8/7WmxXPm9K6+1\n9v6vfw0sWCC3viP7jBgh18/xFdXVwM6dcvipvFweVVXy9R9+kOHcrx/QsyfQvTtw3XXy6N5dLsPg\n7y/X1vHzk18v/57bDJI9bDYbbDabIedyaVbMmTNnkJycjOLiYgDAk08+ibS0tCt67J46K4bat3cv\nMHEicOCA6kqMJwRQWgrk5QGbN8v/mzt7FoiOlr/QgoLkERgobyL37y9DnMtOkDu5kp0uT3e0Wq1Y\nuHAhAgMDkZaWhk2bNqFfv36GFEfqNDTIp3Jra4EuXVRXY4ySEuDdd4EVK2QvOi0NuPVWIDYWGD6c\nPWvyLK5kp8tDMQsWLMD06dPR2NiIp5566opQJ+/VrRsQEABUVAChoaqrcd6FC8AHHwALFwJnzgAP\nPQTk58txTfbASVc++4ASdSwjQwbhpEmqK3HcpUvAO+8Ac+bIB9H+8Ad5r4W9cvIWSnvspK+oKGD3\nbu8L9s2bgSeekDc6V6yQQy1EvoT9F2pTc7B7i/PngVmz5Pr6//d/wIYNDHXyTQx2alNkJLBnj+oq\n7FNVBdx8M7Bvn9wV6777OIZOvovBTm0KDQUqK2VP2JN98w2QkCA3TPn0Uzk9kciXMdipTf7+wLBh\nwP79qitp2+rV8ibvv/4FZGezl04EMNipA548zr5iBfDII8BnnwHjx6uuhshzcFYMtctTx9lXrgQe\nf1w+PWq1qq6GyLOwx07tiooCdu1SXcWVvvlG9tTXrGGoE7WGwU7tiomRj+J7ioMHgXvuAZYuBW66\nSXU1RJ6JwU7tGj5cLpDlCfun1NbKG6Uvv8wxdaL2MNipXRaLHO743wKeygghh18SE4Hp09XWQuTp\nGOzUIasV2L5dbQ05OXIJYS/ZVpdIKc6KoQ6NHi3ni6uyc6dczKuwkPvUEtmDPXbq0OjR6nrs588D\nDz4IzJsnx/uJqGNctpc6dOmS3HTj6FH51Z1mzZJDMCtW8KlS8i2uZCd77NShzp2BUaPcP+1x2zY5\nrfEf/2CoEzmCwU52GTMG2LrVfddragJmzABefRW44Qb3XZdIBwx2ssvNN8snPt1l6VI5xfHhh913\nTSJdcIyd7HLoEBAfDxw7Zv6wyOnTck/SNWv4dCn5Lo6xk+kCAwE/P7m5tdlefFE+YcpQJ3IO57GT\nXSwW+dTn5s1AUJB519m5E1i+HNi717xrEOmOPXay29ixMtjNIoS8YfrKK0C/fuZdh0h3DHay29ix\nwNdfm3f+ZcuAujrg0UfNuwaRL+DNU7JbYyPQp4+8kdqnj7HnPnsWCA8HPv5YzsAh8nW8eUpu0aUL\ncMstwH//a/y558wBUlMZ6kRGYLCTQ+64A/jyS2PPuXcv8O67wOuvG3teIl/FYCeHpKYCa9cadz4h\ngKeeAl54ARgwwLjzEvkyBjs5JDISaGgAysqMOd/HHwMnTsiNqYnIGAx2cojFAtx1F7BypevnqqsD\nsrOBt96SDz8RkTEY7OSw3/xGLqPrqldfBcaNA5KSXD8XEf2M0x3JYefPAwMHypuegwY5d459++QM\nm507gRtvNLY+Ih1wuiO5lb8/cPfdwEcfOffzQgCPPSZvmDLUiYzHYCenPPwwkJsrQ9pR778P1NTI\n5QOIyHgciiGnNDXJxcA++UTuiWqvkyeBqCguyUvUEQ7FkNt16iTXdFm40P6fEQKYPl1uTs1QJzIP\ne+zktNOnZa992zZg6NCOP79kCTB/vtxiz9/f9PKIvJor2clgJ5c895zcVWnJkvY/t307cOedwPr1\nwMiR7qmNyJsx2EmZmhr5NOp77wHJya1/5vvv5bZ6CxYAkya5tTwir8UxdlKmRw/g7beBKVOAw4d/\n+f7hw0BKipwBw1Ancg8GO7ns7ruBZ56RPfaNG+VN0sZGuXFGbCzw+98Ds2aprpLId3CFDjJEdra8\ngfrgg/LJ1Pp6YNQoufRAYqLq6oh8C8fYyVBCyOGX7t2N32WJyJfw5ikRkWZ485SIiFow2ImINMNg\nJyLSjNPBPnv2bAQEBMBqtcJqtSIvL8/IuoiIyElOB7vFYsHMmTNRXFyM4uJipKWlGVmX17DZbKpL\nMJXO7dO5bQDb58tcGorhbBf9/+PSuX06tw1g+3yZS8Gek5ODhIQEzJ07FzU1NUbVRERELmg32O+4\n4w6MHDnyF8eqVauQlZWFyspK5Ofno7y8HIsXL3ZXzURE1B5hgJKSEpGYmNjqe0FBQQIADx48ePBw\n4AgKCnI6k51eK+bYsWMYNGgQLl68iGXLlmH8+PGtfq6srMzZSxARkROcHmOfNWsWRo0ahYSEBDQ2\nNiIrK8vIuoiIyEmmrxVDRETuZdqTpwUFBQgPD0dwcDBycnLMuoypMjMzMWDAAIy8bC+3mpoaZGRk\nIDAwEBMnTkRtbW3Le4sWLUJwcDAiIiKwadMmFSU75PDhw7jtttsQGRmJ5ORkLFu2DIA+bWxoaEB8\nfDxiYmKQkJCA+fPnA9CnfQBw6dIlWK1WpKenA9CrbUOHDsWoUaNgtVoRFxcHQK/21dXV4Xe/+x1C\nQkIQERGBwsJC49rn9Oh8B2JiYsSGDRvEd999J0JDQ0V1dbVZlzJNQUGB2L59u4iKimp5be7cuWLG\njBmioaFBPPHEE2LevHlCCCFOnDghQkNDxaFDh4TNZhNWq1VV2XY7duyYKC4uFkIIUV1dLYYNGybO\nnj2rVRvr6uqEEEI0NDSIyMhIceDAAa3a98Ybb4gHHnhApKenCyH0+u9z6NCh4tSpU1e8plP7srOz\nxZ///GdRX18vGhsbxenTpw1rnyk99jNnzgAAkpKSMGTIEKSmpqKwsNCMS5lq3Lhx6N279xWvFRUV\nYdq0afD390dmZmZLuwoLC5GWlobAwEDceuutEEJ4/Nz+gQMHIiYmBgDQr18/REZGYuvWrVq18dpr\nrwUA1NbW4uLFi/D399emfUeOHMHnn3+ORx55pOVhQV3a1kxcNVKsU/vWrVuH5557Dt26dYOfnx96\n9eplWPtMCfatW7ciLCys5c8RERHYsmWLGZdyu8vbFhYWhqKiIgDyLz48PLzlc6GhoS3veYOysjLs\n2bMHcXFxWrWxqakJ0dHRGDBgAGbMmIHAwEBt2vfss89i3rx56NTp53/GurQNkMuWpKSkYOLEiVi1\nahUAfdp35MgRNDQ0ICsrC/Hx8Zg7dy7q6+sNax9Xd3TQ1T2I9lgsFhMrMU5NTQ3uu+8+zJ8/H927\nd9eqjZ06dcKOHTtQVlaGt99+G8XFxVq0b82aNejfvz+sVusV7dGhbc02b96MHTt24LXXXsPMmTNx\n/PhxbdrX0NCAAwcOYNKkSbDZbNizZw8++ugjw9pnSrDHxsZi3759LX/es2cPEhISzLiU28XGxqK0\ntBQAUFpaitjYWABAfHw89u7d2/K5ffv2tbznyRobGzFp0iRMmTIFGRkZAPRrIyBvxI0fPx6FhYVa\ntO/rr7/GqlWrMGzYMEyePBnr16/HlClTtGhbs0GDBgEAwsPDMWHCBKxevVqb9o0YMQKhoaFIT0/H\nNddcg8mTJyMvL8+w9pkS7L169QIgZ8Z89913+PLLLxEfH2/GpdwuPj4eubm5qK+vR25ubssvrLi4\nOOTn56Oqqgo2mw2dOnVCjx49FFfbPiEEpk2bhqioKDzzzDMtr+vSxpMnT+L06dMAgFOnTmHt2rXI\nyMjQon2vvvoqDh8+jMrKSixfvhwpKSl4//33tWgbAJw7d65lDLm6uhr5+flIS0vTpn0AEBwcjMLC\nQjQ1NeGzzz7D7bffblz7zLjbK4QQNptNhIWFiaCgILFw4UKzLmOq+++/XwwaNEh07dpVBAQEiNzc\nXHH27FkxYcIEMXjwYJGRkSFqampaPr9gwQIRFBQkwsPDRUFBgcLK7bNx40ZhsVhEdHS0iImJETEx\nMeKLL77Qpo07d+4UVqtVjBo1SqSmpoqlS5cKIYQ27Wtms9laZsXo0raKigoRHR0toqOjRUpKinjn\nnXeEEPq0Twgh9u/fL+Lj40V0dLTIzs4WtbW1hrWPDygREWmGN0+JiDTDYCci0gyDnYhIMwx2IiLN\nMNiJiDTDYCci0gyDnYhIMwx2IiLN/D+jZu92h+FSowAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107b4ac50>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Convolution is a little easier to see at coarser detail:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "dt = 1. # time interval in seconds\n",
      "t_low = np.arange(0, 40, 1) # dt == 1\n",
      "experiment_low = np.zeros_like(t_low)\n",
      "experiment_low[5] = 1\n",
      "experiment_low[15] = 1\n",
      "hrf_low = hrfs.glovert(np.arange(24))\n",
      "plt.plot(hrf_low)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "[<matplotlib.lines.Line2D at 0x107b9cdd0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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vsyXmbt3O3mJiGt+vfzw8PDhLlx7/SQsXLiQ7O5u6ujpycnKIjIwkPz8fgOzs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\noGQgIiIoGYiICPD/AfAM0SWXsmyhAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107b5c210>"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Ugly, but, now lets redo the dumb convolution.  This time, instead of summing up all the components, (the copies of the HRF corresponding to each time point in the experiment), we'll keep the individual components. There will be one component for each time point in the experiment."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "data_len = len(experiment_low)\n",
      "kernel_len = len(hrf_low)\n",
      "out_len = data_len + kernel_len - 1\n",
      "# Let's stack the summed time courses\n",
      "components = np.zeros((data_len, out_len))\n",
      "for i, value in enumerate(experiment_low):\n",
      "   components[i, i:i+kernel_len] = value * hrf_low\n",
      "# Then sum to get the result\n",
      "out = np.sum(components, axis=0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This gives the right convolution:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(out)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 15,
       "text": [
        "[<matplotlib.lines.Line2D at 0x107bcbd50>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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5yN3PTq5AlkHJbqIBA8SsnNpaZa7nDkkSLYNRozw73zGATETGxTCQQckwALQbN7hwQbRG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\ngWFARERgGBAREYD/B4o0G3OtfRoUAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107b7ca90>"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.imshow(components)\n",
      "plt.plot((0, data_len), (0, data_len), 'r')\n",
      "plt.axis('tight')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "(-0.5, 62.5, 40.0, -0.5)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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UjIVNNA4+SUnBZosFWyQJlq4upeOQSrGwicbJ7kmT8Hp6OmolCebubqXjkAqx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\n+P98SAK0xreWcAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107b7e810>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Another way to calculate convolution - using a backwards kernel vector"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Looking at the components matrix give us another way to calculate the convolution.\n",
      "\n",
      "Notice that each component starts at the red line.  The sums to get the convolution result are over the columns.\n",
      "\n",
      "What is this sum made up of?  First we need some notation."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Call our data vector $d$ ($d$ is `experiment_low` in the case above).  $d$ has $D$ elements $d_1, d_2 ... d_D$ ($D$ is `data_len` above). Call our kernel vector $k$ ($k$ is `hrf_low` above). $k$ has $K$ elements $k_1, k_2, ... k_K$ ($K$ is `kernel_len` above).\n",
      "\n",
      "Let's call $\\mathbf{C}$ the $D$ x $M$ matrix of $D$ components (rows) and $M = D+K-1$ out values (columns).  $\\mathbf{C}[i, j]$ is the value in row $i$ and column $j$.\n",
      "\n",
      "Consider the sum of column $j$.  If $v$ is the output vector from the convolution, $v_j$ is the sum of column $j$ of $\\mathbf{C}$.\n",
      "\n",
      "All values of $\\mathbf{C}[i>j, j]$ are zero because the components start at diagonal cells $\\mathbf{C}[i, j]$ where $i = j$.\n",
      "\n",
      "At the diagonal we have the data value at this time point times the first value in the kernel: $\\mathbf{C}[j, j] = d_j * k_1$.\n",
      "\n",
      "Working upwards from the diagonal, $\\mathbf{C}[j-1, j]$ is the second element of the vector resulting from multiplying the kernel with data value $d_{j-1}$ so $\\mathbf{C}[j-1, j] = d_{j-1} * k_2$.  The kernel goes to $0$ after $K$, so the whole column sum is given by:\n",
      "\n",
      "$$\n",
      "\\sum_{i=1}^M{C(i, j)} = d_j * k_1 + d_{j-1} * k_2 + ... + d_{j-(K-1)} * k_K\n",
      "$$\n",
      "\n",
      "(if we run out of data ($d_i$ where $i < 1$ or $i > D$) assume the value is zero)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This formula gives another way to do the convolution. We step through each value in the data, and starting at that point, go back through the data and the kernel, summing up the data times the kernel values. This is the direct implementation of the column sum formula above. Like this:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def ugly_convolve(data, kernel):\n",
      "    # Starts the same as the dumb convolve\n",
      "    data_len = len(data)\n",
      "    kernel_len = len(kernel)\n",
      "    out_len = data_len + kernel_len - 1\n",
      "    out = np.zeros((out_len,))\n",
      "    # Different here though\n",
      "    for i in range(out_len):\n",
      "        # Sum up the value of the data * kernel going backwards through the data\n",
      "        val = 0\n",
      "        # Go back over the kernel and data\n",
      "        for k in range(kernel_len):\n",
      "            # Go back from current point in data\n",
      "            data_ind = i - k\n",
      "            # If we've run out of data, assume 0\n",
      "            if data_ind < 0 or data_ind > data_len - 1:\n",
      "                d_val = 0\n",
      "            else:\n",
      "                d_val = data[data_ind]\n",
      "            # Multiply the corresponding kernel value with data\n",
      "            val = val + d_val * kernel[k]\n",
      "        out[i] = val\n",
      "    return out"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ugly_out = ugly_convolve(experiment_low, hrf_low)\n",
      "plt.plot(ugly_out)\n",
      "assert np.all(ugly_out == out)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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bm1gb8u23Ykqw45/HjokWXnW1eN5DRISYOkzGk5SkjzCw2Wyw2Wweny8rDJKS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\ngWFARERgGBAREYD/B4o0G3OtfRoUAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107b0a350>"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Another way to think of convolution - filter matrices"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can also do convolution with a *filter matrix* using the elements of the column sum formula.  Here it is again:\n",
      "\n",
      "$$\n",
      "\\sum_{i=1}^M{C(i, j)} = d_j * k_1 + d_{j-1} * k_2 + ... + d_{j-(K-1)} * k_K\n",
      "$$\n",
      "\n",
      "We can split this formula into the *data* part and the *kernel* part.  We'll make a matrix representing the kernel part by constructing a matrix with the $k_1, k_2 .. k_K$ values going upwards along the columns from the diagonal, for each column. For our HRF kernel, we could call this the upward HRF matrix.\n",
      "\n",
      "Then we'll make a matrix for the data part of the formula; the $d_j, d_{j-1} ...$ from the equation above. \n",
      "\n",
      "Multiplying the elements of the matrix for the kernel part and the matrix for the data part reconstructs the components matrix above.\n",
      "\n",
      "We first make the upwards HRFs like this:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "up_hrfs_rows = kernel_len - 1 + data_len + kernel_len - 1\n",
      "up_hrfs = np.zeros((up_hrfs_rows, out_len))\n",
      "backwards_hrf = hrf_low[::-1]\n",
      "for i in range(0, kernel_len -1 + data_len):\n",
      "    up_hrfs[i:i+kernel_len, i] = backwards_hrf\n",
      "plt.imshow(up_hrfs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": [
        "<matplotlib.image.AxesImage at 0x1082fb3d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMEAAAD/CAYAAABIMrJBAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1082d1f10>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can chop off the top of and bottom of these upwards HRFs because they'll be multiplying outside the data range. Once we've done this we've got something the same shape as `components` above:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "up_hrfs_trimmed = up_hrfs[kernel_len-1:kernel_len-1+data_len, :]\n",
      "plt.imshow(up_hrfs_trimmed)\n",
      "plt.plot((0, data_len), (0, data_len), 'r')\n",
      "plt.axis('tight')\n",
      "up_hrfs_trimmed.shape"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 20,
       "text": [
        "(40, 63)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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zVqxYAavVCgDYtm0bCgsLsWHDBrjd7oAWSqRG7WFhWC6K+NGFC/jhxYtKl0PT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\nNhGRRjCwiYg0goFNRKQRDGwiIo34fwaMZNCngpVmAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108a01610>"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Think of each column as being an HRF flipped to go upwards rather than downwards.  For example, here are columns 28 through 31:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, axes = plt.subplots(4, 1)\n",
      "for i in range(4):\n",
      "    axes[i].plot(up_hrfs_trimmed[:, 28+i])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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JEBFhbDW/zd/+Jo3B119r8w5GUX8Magzq4kC+dOkSAQEB5ObmYm5uTqdOnTh+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\netZWAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108a0d050>"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can make another matrix to give the data corresponding to each upwards HRF. These are the $d_j, d_{j-1} ... $ values from the column formala above.  Of course the $d$ values are just the $d$ vector repeated for each column. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "data = np.tile(experiment_low[:, None], (1, out_len))\n",
      "plt.imshow(data)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 22,
       "text": [
        "<matplotlib.image.AxesImage at 0x107bfec10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x108075a10>"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If we do an element-wise multiply the of the data matrix with the upward HRFs, we get the components back:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.imshow(data * up_hrfs_trimmed)\n",
      "plt.plot((0, data_len), (0, data_len), 'r')\n",
      "plt.axis('tight')\n",
      "assert np.all(data * up_hrfs_trimmed == components)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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UjIVNNA4+SUnBZosFWyQJlq4upeOQSrGwicbJ7kmT8Hp6OmolCebubqXjkAqx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\n+P98SAK0xreWcAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1082e28d0>"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "upwards_out = np.sum(data * up_hrfs_trimmed, 0)\n",
      "assert np.all(out == upwards_out)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can do the same thing by taking the transposed upwards HRF matrix and matrix multiplying by the data vector.  The transposed upwards HRF matrix is now called a filter matrix because the matrix multiplication applies the HRF convolution 'filter' to the data:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "filter_matrix = up_hrfs_trimmed.T\n",
      "plt.imshow(filter_matrix)\n",
      "filtered_out = filter_matrix.dot(experiment_low)\n",
      "assert np.all(filtered_out == out)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x108a0ab90>"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In the transposed (filter) matrix, the HRFs are now going backwards instead of upwards:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(filter_matrix[30, :])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 26,
       "text": [
        "[<matplotlib.lines.Line2D at 0x109a2f450>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x108a06450>"
       ]
      }
     ],
     "prompt_number": 26
    }
   ],
   "metadata": {}
  }
 ]
}