{ "metadata": { "name": "", "signature": "sha256:f2442a2dcc1c0a94b4d057bac3d91828e23c6bb34d4bd87cc53897877ef2ae44" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "General Linear Model" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Application: dealing with individual IRF differences" ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "A tutorial by Jan Willem de Gee (jwdegee@gmail.com)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is Python, so let's start with importing some modules:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import os\n", "import numpy as np\n", "import scipy as sp\n", "from scipy.linalg import sqrtm, inv\n", "import scipy.stats as stats\n", "import scipy.signal as signal\n", "import matplotlib.pyplot as plt\n", "import sympy\n", "import math" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's do inline plotting:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%matplotlib inline " ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's first create and plot the true pupil response to a transient input. This is called an \"Inpulse Response Function\" (IRF)." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def pupil_IRF(timepoints, s=1.0/(10**26), n=10.1, tmax=0.93):\n", " \n", " \"\"\" pupil_IRF defines the IRF (response to a transient input) of the pupil.\n", " \n", " Parameters\n", " ----------\n", " timepoints: timepoints to evulate function\n", " s: scaling factor\n", " n: sets the width\n", " tmax: sets the time to peak \n", " \n", " Returns\n", " -------\n", " y: IRF evaluated for 'timepoints'\n", " y_dt: IRF first derivative evaluated for 'timepoints'\n", " \n", " \"\"\"\n", " \n", " # in sympy:\n", " t = sympy.Symbol('t')\n", " y = ( (s) * (t**n) * (math.e**((-n*t)/tmax)) )\n", " yprime = y.diff(t)\n", " \n", " # lambdify:\n", " y = sympy.lambdify(t, y, \"numpy\")\n", " y_dt = sympy.lambdify(t, yprime, \"numpy\")\n", " \n", " # evaluate:\n", " y = y(timepoints)\n", " y_dt = y_dt(timepoints)\n", " \n", " # normalize:\n", " y = y/np.std(y)\n", " y_dt = y_dt/np.std(y_dt)\n", " \n", " return (y, y_dt)\n", "\n", "\n", "# create the IRF:\n", "tmax_shifted = 0.93-0.3\n", "sample_rate = 20\n", "IRF_len = 3.0 # in seconds\n", "timepoints = np.linspace(0,IRF_len,IRF_len*sample_rate)\n", "IRF, IRF_prime = pupil_IRF(timepoints=timepoints)\n", "IRF_shifted, IRF_prime_shifted = pupil_IRF(timepoints=timepoints, tmax=tmax_shifted)\n", "\n", "# IRF_len = 20.0 # in seconds\n", "# timepoints = np.linspace(0,IRF_len,IRF_len*sample_rate)\n", "# IRF, IRF_prime = HRF(timepoints=timepoints)\n", "\n", "# plot the IRF:\n", "fig = plt.figure()\n", "plt.plot(timepoints, IRF_shifted, color='b')\n", "plt.plot(timepoints, IRF, color='r', ls='--')\n", "plt.plot(timepoints, IRF_prime, color='g', ls='--')\n", "\n", "plt.legend(['individual IRF', 'canonical IRF', 'canonical IRF dt',])\n", "plt.title('Impulse Response Function')\n", "plt.xlabel('time (s)')\n", "plt.ylabel('a.u.')\n" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 3, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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Nm7SMNGInxpJ8MrmYjf7UubO9Nfann7wTjtdFRdlk8H//53QkSlU6AdGGICKx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\nGU3Dmg355YDvZnMbOdI22l9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"text": [ "" ] } ], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's simulate convolved timeseries data based on this individual IRF." ] }, { "cell_type": "code", "collapsed": false, "input": [ "true_beta = 1\n", "duration = 15 # in seconds\n", "times = np.array([2,2.5,5,6,8.75,9,10,10.5,])\n", "input_signal = np.zeros(duration * sample_rate)\n", "for i in times:\n", " input_signal[i*sample_rate] = true_beta\n", "\n", "# convolve inputs with IRF: \n", "convolved_signal = (sp.convolve(input_signal, IRF, 'full'))[:-(IRF.shape[0]-1)]\n", "convolved_signal_shifted = (sp.convolve(input_signal, IRF_shifted, 'full'))[:-(IRF.shape[0]-1)]\n", "\n", "# # let's add some noise:\n", "# convolved_signal = convolved_signal + np.random.normal(0,0.2,len(convolved_signal))\n", "# convolved_signal_shifted = convolved_signal_shifted + np.random.normal(0,0.2,len(convolved_signal_shifted))\n", " \n", "# plot simulated convolved signal with noise:\n", "timepoints = np.linspace(0,duration,duration*sample_rate)\n", "fig = plt.figure()\n", "plt.plot(timepoints, convolved_signal_shifted, 'b')\n", "plt.plot(timepoints, convolved_signal, 'r', ls='--')\n", "plt.ylim(-1,8)\n", "plt.legend(['pupil time series', 'expected timeseries\\nbased on canonical IRF',], loc=1)\n", "for i in times:\n", " plt.axvline(i, color='k', alpha=0.5)\n", "plt.title('pupil time series, with measurement noise')\n", "plt.xlabel('time (s)')\n", "plt.ylabel('a.u.')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 4, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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ZNp0FrLZKHo3GapzNE9nZ2RQVFQFw+PBhbrzxRnJyckhJSWH48OGUl5cjpaRp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\nGzWqvSuOG457j2K6dlVmp6uvDnkw2FqxY4ft2W3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"text": [ "" ] } ], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's create regressors (predictors) we want to put in our design matrix. Model 1 will be based on only the canonical IRF. Model 2 will be based on both the canonical IRF and it's temporal derivative." ] }, { "cell_type": "code", "collapsed": false, "input": [ "input_signal = np.zeros(duration * sample_rate)\n", "for i in times:\n", " input_signal[i*sample_rate] = 1\n", "\n", "# convolve inputs with IRF: \n", "regr_convolved = (sp.convolve(input_signal, IRF, 'full'))[:-(IRF.shape[0]-1)]\n", "regr_convolved_dt = (sp.convolve(input_signal, IRF_prime, 'full'))[:-(IRF_prime.shape[0]-1)]\n", "\n", "# z-score measured data and regressors:\n", "regr_convolved = (regr_convolved - regr_convolved.mean()) / regr_convolved.std()\n", "regr_convolved_dt = (regr_convolved_dt - regr_convolved_dt.mean()) / regr_convolved_dt.std()\n", "convolved_signal_shifted = (convolved_signal_shifted - convolved_signal_shifted.mean()) / convolved_signal_shifted.std()\n", "\n", "fig = plt.figure()\n", "plt.plot(timepoints, regr_convolved, 'r',)\n", "plt.ylim(-4,6)\n", "plt.legend(['regressor'], loc=1)\n", "plt.title('model 1')\n", "plt.xlabel('time (s)')\n", "plt.ylabel('a.u.')\n", "\n", "fig = plt.figure()\n", "plt.plot(timepoints, regr_convolved, 'r',)\n", "plt.plot(timepoints, regr_convolved_dt, 'g')\n", "plt.ylim(-3,6)\n", "plt.legend(['regressor', 'regressor dt'], loc=1)\n", "plt.title('model 2')\n", "plt.xlabel('time (s)')\n", "plt.ylabel('a.u.')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 5, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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HLp6sAYDRKaPYw2Gazj8nsAM4LYIwrETq7dyzhmaxf+tay/c8dqzlgvARV7Nu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\nBjlJSfD3v8MvfgF/+5vVwCYjw6qfNWE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"text": [ "" ] } ], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "For every regressor (in the above model 2, we have two regressors, one based on the canonical IRF, and one based on it's temporal derivative) we want to find an associated scalar value (the \"beta\", $b$) that we can use to scale that particular regressor with, such that it best describes the measured timeseries.\n", "\n", "In the GLM, with a procedure called \"multiple regression\" we look for betas that minimimize the sum of squares of errors across all $k$ regressors in our design matrix at the same time.\n", "\n", "To do so, we set up the following equation (for a derivation, see all the way below):\n", "\n", "$ b = (X'X)^{-1} X'y $\n", "\n", "In which,\n", "\n", "$b$ is a vector containing the betas (size: number of regressors; in the above toy example: 2);\n", "\n", "$X$ is the design matrix (size: length BOLD time series x number of regressors);\n", "\n", "$y$ is the measured timeseries." ] }, { "cell_type": "code", "collapsed": false, "input": [ "# make design matrix:\n", "designMatrix_0 = np.mat(regr_convolved).T\n", "designMatrix_1 = np.mat(np.vstack((regr_convolved, regr_convolved_dt))).T\n", "\n", "# multiple regression, yielding the betas:\n", "betas_0 = np.array(((designMatrix_0.T * designMatrix_0).I * designMatrix_0.T) * np.mat(convolved_signal_shifted).T).ravel()\n", "betas_1 = np.array(((designMatrix_1.T * designMatrix_1).I * designMatrix_1.T) * np.mat(convolved_signal_shifted).T).ravel()\n", "betas_1_combined = np.sign(betas_1[::2])*np.sqrt((betas_1[::2]**2) + (betas_1[1::2]**2))\n", "\n", "# explained signal:\n", "explained_signal_0 = regr_convolved*betas_0[0]\n", "explained_signal_1 = regr_convolved*betas_1[0] + regr_convolved_dt*betas_1[1]\n", "\n", "# model fit:\n", "r_0, p_0 = sp.stats.pearsonr(convolved_signal_shifted, explained_signal_0)\n", "r_1, p_1 = sp.stats.pearsonr(convolved_signal_shifted, explained_signal_1)\n", "\n", "print 'model 0:'\n", "print '---------'\n", "print 'beta: {}\\nR2: {}'.format(round(betas_0,3), round(r_0,3))\n", "print\n", "print 'model 1:'\n", "print '---------'\n", "print 'beta: {} ({} ; {})\\nR2: {}'.format(round(betas_1_combined,3), round(betas_1[0],3), round(betas_1[1],3), round(r_1,3))\n", "print\n", "print '---------'\n", "print 'true beta: {}'.format(true_beta)\n", "print '---------'\n", "print" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "model 0:\n", "---------\n", "beta: 0.694\n", "R2: 0.694\n", "\n", "model 1:\n", "---------\n", "beta: 0.972 (0.694 ; 0.681)\n", "R2: 0.972\n", "\n", "---------\n", "true beta: 1\n", "---------\n", "\n" ] } ], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Indeed, with model 2 we find a beta of about 1. This is what we used as input to simulate the \"measured timeseries\". So this works!! \n", "\n", "Let's plot our explained signal. We construct this by multiplying our regessors with their respective betas, and add them all up:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# plotting:\n", "fig = plt.figure()\n", "plt.plot(timepoints, convolved_signal_shifted, 'b')\n", "plt.ylim(-3,6)\n", "plt.plot(timepoints, explained_signal_0, 'm', lw=1)\n", "plt.legend(['pupil timeseries', 'explained signal model 1'], loc=2)\n", "plt.title('predicted signal model 1')\n", "plt.xlabel('time (s)')\n", "plt.ylabel('a.u.')\n", "\n", "fig = plt.figure()\n", "plt.plot(timepoints, convolved_signal_shifted, 'b')\n", "plt.plot(timepoints, explained_signal_1, 'm', lw=1)\n", "plt.ylim(-3,6)\n", "plt.legend(['pupil timeseries', 'explained signal model 2'], loc=2)\n", "plt.title('predicted signal model 2')\n", "plt.xlabel('time (s)')\n", "plt.ylabel('a.u.')\n" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 12, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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h0CA44ACYMwdeftnpFCknFRZaQSCsnbN1+HDI+aGepvqmgKbnyy8hNRVGjYKf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\nN8CKi63A5S7Pr1xbSWRSJP2TO98J3Ak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3Au46aTUkxkcCgRAxoq4OknvdJOTunyOoaonH0+yR5xKIcZFAIESMqKsDS6cL\ny+z9cwR77SbiZ8bjtkuuQIROAoEQMUBro2hItQxfWVxXB5YciwQCMS4SCISIAZ2doLTGY3djyRkc\nCGw2I1CYMi14Gj0RSqGIZRIIhIgBdXUwN9uD2WbGnGgeNE0pmD0bPMnxuB2SIxChkw5lQsQAux1K\nM9xYkizDTs/Oht7EeMkRiHGRQCBEDHA4ID/ZjSV9+ECQkwOdcRY8DgkEInRSNCREDHA4INviwTJr\n5BxBG1I0JMZHAoEQMaCxEWbEuYmfFT/s9OxsaOqTymIxPhIIhIgBDgek65FzBDk5YHdKjkCMjwQC\nIWKAwwHJfaPnCGq7pbJYjE8kn1BWoJR6Qym1USn1uVLqvyOVFiGiXWMjWHtHryPY1WZUFsvjXUWo\nIpkj8AA/0lovAo4ErlRKLYxgeqJKSwusWgVPPRXplIho4HBAXNfIOYKcHNjTaAYTeLu9k56exkb4\n2c/g2WcnfVNiCkQsEGit7VrrT3zvu4DNQG6k0hNtrr8e1qyBG280eo2K6a2xEVTb6DmChgawzJqa\nJqQ/+Qm8/jrccsukb0pMgaioI1BKFQFLgXWRTUl00Bqeew4eegg8Hvjww0inSESSxwMdHeBtchOf\nPXyOwGoFiwVMmZNfT9DXBy++CKtXG+Mfbdw4qZsTUyDigUAplQI8BVzlyxlMe598AsnJsGABXHAB\nPP54pFMkIqmpCWZneEGDOdk84nzZ2eC1TX7LobVrIT8fiorgW9+S8/NAENGexUqpeOAfwKNa62eG\nm2fVqlX+9xUVFVRUVExJ2iLphRfgK18x3h93HKxcGdn0iMhqbISSDDfxSfGDHlo/VE4OuKyT35dg\n6Pn5+99P6uZEECorK6msrBz38hELBMo4ox8ENmmtfzPSfIGBYLpYtw4uucR4v3gxfPop9PeDKeL5\nNxEJDgcU2DxY0oavHxiQnQ09nsnPEaxbBz/9qfF+yRIjBysia+hN8s033xzS8pG8tHwROB84Xim1\n3vc6JYLpiRrr18PSpdDv7sfa7iQjTVNdHelUiUhxOCDXOnKLoQHZ2dCu4ie1slhr48K/dKnxOS8P\nvF5jUDwRuyLZaugdrbVJa71Ea73U93o5UumJBPtf7fvdvTkc0N0Nc+ZoNp65kQ+Xfsg1bOHTTyOU\nyEniaZGOT8FqbIRZo4wzNCAnB1r6J7doqKYGEhN9Yxu92ca2y7dyekGr5ApinBQ2TDGt4dJL4aHL\nmtj+g+1vcVJTAAAgAElEQVSsP2Y97sZ9wWD9eiO77fi/Blx7XRxZdSRzm5vZ8oYzgqkOj/5+Y98f\nu6mT9/Leo+m5pgmvs68PTj0Vfv3rA7eZrcMBWabgcgQO98SKhtxuOPFEuPlm4/811EButa+9j03n\nbSIuM44zq7bw2cfyrORYNq5AoJR6IdwJmS4+/xxefklje2QH3psOIuNLGez55R7/9PXrYeliza5b\nd1H22zLis+JxnzSbxJf2RjDV4fHpp/DKyxrzbZswfzmb6huq0f0Tu3qvWwfbt8PvfgcffRSmhEYZ\nhwNS+8fOERjDTEwsR/Cf/xhNQh97DN57b//p69fDsmVQc1sNmV/OpPSuUlS+FedzDePepoi88eYI\nLgtrKqaRl1+Gc09wkZbo5Z+70im8oZD6B+r9uYL16+FobyOWmRbSjkkDYNbZM8mpbYlkssPilVfg\nW1/qJS3By7tfmAca2v/TPqF1vvwynHkmnH46/PvfYUpolGlshGTP2DmCnBzY3TGxHMGLLxrH89xz\n4V//2n/6+vWwdEEf9Q/WM+dncwBI+VYeOZsbx71NEXnjCgRa67pwJ2S6ePllOCG7nZQj0njmX4qE\nvERmnTOL2l/XAvDJx5rsNTUUXlfobyo475QU0l0u3DFerv7yy1Axq4O4xamseVWRfnw6HWs7JrTO\nl14yioZOqugn+65P6Fg3sfVFI4cDEp3B5QiqW40OZeMdb2jgeJ5xBrzxd9egYkswAkHZzgbSl6dj\nLbICUHJGGvkd7WjvAVo2Nw2MGQiUUtXDvHZOReIONB6Pkd0u6u4g76RU4uNhwwYovLaQuvvraKl2\nU7rbgTXNRNZXs/zLpWWZqIq3UfNC7F7kXC54/30o7Oyg6NRU1q2DhKWpEwoEXV2weTMcdRQs3F5H\nQruLjeduoq+zL4wpj7zGRjB3BldHUOswY0ow4e0Ifbyh1lajMvjww+Hg0j5W1XzAe8XraHzauNt3\nOKC7S9O7ei95/53nXy633EILCdj/I/1BY1UwOYLDA17HAr8FHpvMRB2odu82fqzOjzpIOzqV44+H\nN9+ExDmJ5P0gj8+O/Ijv9e9g7m/L9us45MhOo+6ViRWjRNKuXZCbCz0ftjPr+DTmzYO9aRMLBFVV\nUFwMcXGaht/s5vH55XhmWmn9d2v4Eh4FHA5glHGGBgwMM2HOit/vTj4YVVVQWgpmMzgebaChIIPd\nPziEbd/dhmuvi/Xr4RtzWlHxivTl6f7llILajDRqnovd83O6GzMQaK2bAl61vs5fX5mCtB1wqqqg\ntFjT9VkXtkNtLF9uBAKA4lXFVJ85nzfOXEraUWn7Lds3N5Wej2M3R1BVBaVF/XR/3k3KshQWLYLN\nrYlot6a3tnf86ywF1x4XaMg43EZTQTptb7aFOfWR43JBb4/G2+whfsboOQIw6gl0mgVPQ+jFiAPH\nE6DuvjosZ+Xyyp408q7MY8vFW/h0bR9fte8k/8f5+92odBWn0fZu7J6f010wRUOHKqWW+V6HKaWu\nAEYe8ESMaOdOOCjbRfyMeMzJZo47Dt56a1+zx8ruTOZWJA27bNqSZEy7e6YwteG1cyccPMuFJceC\n2Wpm0SLYtFlhO9RG1yfjK1LYuRNKSqDz405/cNlkSaf9rQPnzrSxEYpm9GG2mTFZxs7AZ2eDJ3li\nOQJPq4feXb0cfmk6b74JhdcVghmW3fIu8XOTybkoZ79lkw5KxlvVHfI2RXQIpmjo7oDXHcChwNmT\nmagD1c6dMC/ZibXUqGQrLISUFKNJaX8/vPoqfOlLwy9bsCwB5fLGbEesofteXm6MWmmdb8W5dXx9\nJAYuXF0fd2FbZmPRInivxYZzuxNPW2wep6EcDihOd2PJHr1YaEB2NjgTxjcC6cDx7P6sm+SDk5k3\nX+H1ws4aE4ueX8x/pR7J4tULhh3vKPuwJBKanPT3SX+CWBRM0VCF1vp43+skrfVlWuutU5G4A01V\nFRSYnVjLrP7vzjzTGL1x/Xqw2WDu3OGXLS5R2BOS6Nkcm7mCqirIU/v2fdEiIxAkzU+iZ8v49mng\nwjWQIygvhw2bTaQsS6Hzg85wJj9iHA7IT3YTP3PsYiEwioY648b3TAJ/YP2si5RDUlAKvvENY7jp\n996DtOIE8guHH/SuaIGZjngLzh2x3/FxOhpvh7JDw52Q6WDnTsjq3XdXDHDRRfDXv8Lf/gannTby\nssXFUNWXTPem2Mx+79wJmQH7XlQEzc1AQRI9WycWCLo+7iJlaYp/nfHzkunZFJsBc6jGRshN9ISU\nI2jVEysa6vq0i+TFyYBxfv7lL/Dkk2Ofn7tUMj0bD4zjPt2Mt0PZFWFNxTSgtfFDs7YMDgQHHWSM\n7f7008aPbiQ5OVDlTaLt09j7oWltBILEgH03mWD+fNijxhcI+vqgthYKsvroa+8jcU6if53t6bEb\nMIdyOGBm/MgPpBkqOxsaPaHnCHp7jaBTUGAUDaUckgLAYYdBaiq89hqcf/7IyxcWwjZXMp0bDozj\nPt0ENQy1UioTmAsk+L56dNJSdIBqazOa2fXtGVw0BEa3/rGGmFYKenOSaV0fe00jm5qMgco8uwbv\ne1kZVLdbyHf242n1EJ8R3MUOjGEQZswAb10viUWJ/nLrsjKwJySRtMkR9v2IBIcDDib4OoKcHPis\nN/TexXV1MHs2mJSme2M3yQcZOQKl4OOPjb+jiY+H9owkmj5spiSkLYtoEEyrocuAN4GXgZuBV4Cb\nJjldBxy7HXJna3qrekksTRw0LdjnDCSUWHHtjL0yWP++7+wlsWTfvpeVwY4qhXWeNeRcgd1uXLh6\nq41AMGidniS6N3aPu3dtNGlshNS+0CqL93SGXllcX28cT3eDG3OymbjUffeIYwWBAabCJLq3x975\nKYIrGroKOAKo0Vofj/Fs4QOnfd4UsduNZoCYIT49+DvfQJkLE6HRFXMtM+x2KMnyYEoyEWfbd4Ep\nLTWKy5LmJdFbFVpfArvduPvt3dVLYvG+QFBaCpvtxkVzKh7iPtkcDrC6PFhygg8E1W2hFw35j2dN\nL4lzEsdeYBi2+Yn0j7NPiIisYAJBr9baCaCUStRabwHmT26yDjx2OxSnuEjITxh75hHMKTPhslpw\n1brCmLLJZ7dDUfL++15WBjt2QGJJIs4Qczr+C9eQHEFpKVTtVCSVJx0Q9QSNjWDpCi1HUNUYj6fJ\nE9LIrgPH01XjImHO+M7R3AXx6D6NpzX2A/B0E0wg2KOUygCeAV5VSj0L7ArHxpVSDymlGpRSG8Kx\nvmhmt0NBgovEgvHdbYHRMqM5IZHenbF112W3Q75l/30fyBFYS6wh79PAhctZ7cRaPLjeYccOSJqb\ndEA0ZXQ4wNQWfGWx1QpxVhOmZDN9bcGPuRSOHEFRsaItKZHe6tg6P0Vw/Qi+rrVu1VqvAm4EHgDO\nCNP2HwamxeMp7XbINk8sR1BcDHv7rTH3Q7PbYZZp/33PzYX2dtCzJ5Aj2DU4R5Cba1TMmwsScVYd\nAIGgQdPfEnyOAIxcAemhVRiHIxAUF4NdWUP+X4rIC6n5qNa6Umv9rNY6LE/H1lq/DcReM5hxsNsh\ns2/igWBHT+xd4Ox2yPDsv+8mk7FPDaZx5giytVE0FFBHYDIZw060JVtDrneINt3dYO3vw2QxYbYG\nP6pLTg54baFVGPuLhna7JhQIdrliL8cq5FGVU8Zuh5ReFwkF4w8E6enGRbN9a2z90Ox2SHEOv+9l\nZVDdkYC70Y23N/ihk+12yE71or2auIzBraBLS6FeWWO+aKixEUozg+9MNiA3F5yJoVUYB+YIEgrH\nWUeQC7vcVrq2xfZxn44kEEwRux0SOyaWI1AKTHmJdI5zbJ5IsdshYYR9Ly2FHdWKxMJEXDXBV4Lb\n7TADFwl5CfuNfVNWBjucVpxVzphuQupwwBybO+gWQwPy8qDTHFrv4nAUDZlM4J2VSNvm2LpREUF2\nKIukVatW+d9XVFRQUVERsbRMhN0OJlvvhAIBQHJZIn3vxtYPzW4HlTr8vpeVGc8yPtnXcihp/vCj\nrwbS2lhnmseNK3f44LJhQzxHmBWeJg+WmaFdSKNFYyPkJQVfUTwgPx9a+oPPEfT3Q0MDZFn72DVM\nDisUicVWeqWOYMpVVlZSWVk57uVjKhDEqr4+aGnW9HVPLEcAkL3Agn7NS19H36BOP9HK5YLODk1f\nj3vEHME//gHWecHXE3R1+XJHrS4suftf5MvK4J//hMRSoz4lVgOBwwHZltAqisHIEWxxx+N2BNdJ\nr7XVGAWXZhcJufvnsEKRsTAR1hl9XUxxUuAwVYbeJN98880hLR/R/5RS6nHgP8A8pdQepdQlkUzP\nZGlshILMPpRJTfjiXVyi6EqJnSZ6DgeUzPBgspowJ+1f4VlWZjQhDaUvwUAxhrvOTcIIOYKqKrCW\nxnaFscMBWabQA0F+PtR2B19ZHHg8LbMnFjQLS319XfbEVl+X6S6it5Ra629GcvtTxW6HeRkuEuIm\nlhsAo2VGfZwVZ7WTlMUpYUjd5PLve/zw+15YaAxvEFdgpeM/wT3hyt/Cpc41qOnogDlzjLFzLMWx\nXWHc2Agl/R4s2baQlsvLg5oQehfX1/sCQf3EA0FgX5fA/h0iuknebQqEo1fxgOJi2N0XO030RupV\nPCA+3hjxsiUh+ByB/8I1Qo4gPt64K+5OtcZcU9tADgcku0OvLJ49G6rbgq8s9gfWetewxzMUA31d\npC9BbJFAMAXsdshPMFq4TFRREWzrjJ0LnN0OeZbR9720FGo8Rh1BMK18AnMEw9URgG8UUnPsHKfh\nNDZCQk/olcUWC6h0Cy578EVDs2eHL0dQ1RM7NyrCIIFgCtjtkD1Mz9rxSE6GjuRE2rfExg/NbodZ\navR9LyuDKnscymK08glmnaPlCMAILtWu2Ot8F8jhgLiO0OsIAFIL4vC2edDe4ANrOOoIZsyAvdpK\nR4w1cZ7uJBBMgYYGyJhgr+JA8QXWmBnut6EB0ofpVRyotNQYHyjYMYcGehW76lwjXrjKymBLUwLe\ndi99XcGPuRNNHA0aQhxeYkBugYn+pLignnEdzqIhf1+XLbFxfgqDBIIpYPQqHr755Hikzk9E23tD\nGl0yUux2SHaOvu+hthyy2yEnuQ9zknnYlkiwbxTSxJLYLKbQGrobvSjL8K2txlJUBG5rcBXG/hxB\nGIqGAJLKrHj2xN4xn84kEEyB0XrWjkfBXDPuxHhcddHfRM9uB8sY+z6eHMFM08j1A7BvFFJraWy2\nHOrshJlxbiwh1g8MKC6GzrjgKozDWTQEkD0vHu3ReNpkOOpYIYFgCtjtoJrDFwiKi6HdGht3uvZ6\njWocvUd1SQns2gWWouBzBOl9I9cPBK4zoTg2K4wdDihODb3F0ICiImjRFjwNweUIZqb0GeM2pU28\nRXlxiaIzhvq6CAkEU6Ktvg88/RPquh+ouNg3qFoMNNHrsHtR5tE70lmtMHMmdKaMnSPweo3WNElO\nF5a8kS+SVqtRcenKis0K48ZGKEgZX/0A+IaEdllwN4yeI3C7oaMDbG4jNzCRXsWB226Mi80APF1J\nIJhkTqdRRp6YP7Gu+4FKSmCnM/pzBF1dkNXvIjGInFBZGeztHztH0NxsjMLa7xg9RwBGkVOjJTZ7\nFzsckJswsRzBrk4Lrr2jFx86HEYQ9tjHPp7BKimBGnf0n59iHwkEk6yhwdezNkzFQmD8yKucVrqi\nvIme3Q5z04Pb93nzYFtbAm67m373yM9krq83HrwyWh+CwHXudMXmnanD4XuYzziHLU9Lg454Cx27\nRs8RDHTOc9WP3AIrVHPnwpZOK90xWDczXUkgmGT+nrVh6Ew2wGwGc370D/cbyr6Xl8OmrSYS8hPo\nrRl5v+rrjXHvR+tDMGDhQtjgSMS11zVqcIlGDQ2Q6Z3YDURCbgKdYwSCujrf8QxTiyEwiuXcWYm0\nbYru81PsI4Fgkvl7FYcxRwCQXm7FXRPdd1z+XsVB7PvChbBp09gthwYCQTA5gvJy2LjVRELu6MEl\nGtntkNo7sWdc24osuMdoWVZf7+tVHERgDWnb82MzJzZdSSCYZMH0rB2PwiUW6InuzlJ2O8wkuH0v\nL4fNm8fuS1BXF/yFy7/O0tirMK6vh8TOiZ03OYss6ObgcgThLBoCmL04ERqN4ahF9JNAMMns9vD2\nKh6wYKGiPSm6m+jZ7WP3Kh6Qn288o5fZo+cI6uogN0fjbhi7IrWgANrbwVwQexXGDfUac9voLaPG\nUro0HpPLi9c58iNA/UVDde4xc1ihmH+wiZ5EGY46VkggmGT+5/WGOxAsgDod+kPfp5LRqzi4fVfK\nKB5qjBs9R1BfD7nJHuLS4jBZRj99B9bZlhx7xRRdtW5MtjjMiaH3Kh6wsFzRbrbgto+cKwisI0iY\nHb5zdMECaDBH9/kp9pFAMMmC6Vk7HgsWGKM8dkXxmEN2O1jag9/38nKjNdRYOYLsEFrTlJdDbX9s\n9S7WGrTDReI4WwwNmD8fGrwJ9NaOHAgG6giCqXMJRXm50cS5J4aO+3QW6SeUnaKU2qKU2q6UuiaS\naZksTXVeTM4+4meOb6iAkSQngyvLSv1H0ftDa9nrRXn6icsMriPdkiXwsd3IEYw0HHV9PWT0Bf/s\n56VLYUNzbOUIOjqMeiVr4cQCQWoqdFks1H8+cvFMXR3kpHvRbk1cevieU5WVBW1JVuwfx85xn84i\nFgiUUmbgD8ApQDnwTaXUwkilZ7K49roxz7KgTOHpTBbINj+6m+h56lzE5QTfW3XZMlj7eTzKpOhr\n2b8SfOCh9cldwecIli2Dd6uNupRYGKQPjH0stY2/D8EgMyzs/WT4QODxQEsLpHnD16s4UFJpIk0b\novf8FPtEMkdwBLBDa71La+0BngBOj2B6wk5rsDQ6SS6bnEf25R1mpW9PdN5x9fdDQrOT5LnB7/uS\nJbBhAyQUD19P0NxsPGTd2xB8cdOSJfDhxjjMqWbc9cE9sSvS6uthTrwTa+nEzxvrnESatwwfCOx2\no1dxX0N4WwwNyF5qxVUdneenGCySgSAP2BPwudb33QGjvR0KzT2kLEyalPUvOD6RxPbobKLX0gIl\nCT0kzw/+YmazGS19PDOGrycYaDrq2hP83XJqqvEMXz07doqH7HbI7XdiDSGIjiRncSLdI9S5+Jvi\nhrEzWaCy46xYmkYu5hPRI5IPrw/q7Fi1apX/fUVFBRUVFZOUnPCz26EsMTw/6OEsPdLM69qCs6qX\n5PmTE2zGy26H0gQnSfNCS9eyZdDYnkjeMJWMu3YZw2u49oTW0WrZMmjfY1QYpx+XHlJ6IqG+HuY6\nw3PezD0mgY8fHD5HUF1tDBDnqpv4A2mGs+S4eD7zmnDVu0mchPWLfSorK6msrBz38pEMBHuBgoDP\nBRi5gkECA0Gs2bMHCk1OrHMzJmX9M2eCw5rElld6ODTKAsGePVCgnFjnZoW03JFHwqZ/JjN3U8t+\n03bsMAaScz0bWiusI4+Eqp1WFmzrCSktkVK7q5/F3b1YS8IRCBLZ6e6ltRUyhpyGO3YY4wJNVo5g\nzhx4KT6Jqtd7WHS+BILJNPQm+eabbw5p+UgWDX0IzFVKFSmlLMA5wLMRTE/YVVdDjqdn0nIEAObi\nJHa8Hn0XuOpqmOUJ/a52+XJ4vTqZ7s+795u2YweUlfgeURlCR6vly+HduhS6N+y/zmjUsqkXnZkw\nZj+JYFjzLKTQx4fv7t+pbMcOY9RXd314h5cYoBTo/CS2rIm+81MMFrFAoLXuA74PvAJsAv6mtd4c\nqfRMhurt/ST3uMJyZzeSWYcl0fZp9F3gdm3rJ8npIrEotLFyDj4YNnQk0bPNSb9ncN3Hjh1QNsNN\nXFpoHa0OOQQ+7Uim45PoO07D6d3hxBKmc0aZFO60BD59df/ioYFAMNqznycqc0kSjR9KIIh2Ee1H\noLV+SWs9X2tdprW+I5JpmQzNG3vpD9Od3UgWnpKEeW8P0VYf17LRSf/MRExxoe272QxHHGvGk56A\nc0hnuR07oCC+l8Q5oQUXsxnmHpeIu6UPT2t0Pz5RazDVO0lfFL6bh8TCRDa9sX+FcWCOIJydyQKV\nnZhE/67YCMDTmfQsnkR6ayeW8pRJ3cbcLyWR29fDhg1RFgm2d5Ewzn0/8UTYaxlcPOR2w969kN7p\nxBpCSyT/Ok9SNNuGL3KKJi0tUNbfSdYXwnfeZB+SSNvWXrq69n3X2Wl0XPMP4BfG4SUClZ+STI6r\nhx07JmX1IkwkEEyitLoOso5JndRtWGZaMCWZefmR6Oq4k1HfwczjxrfvZ5wBHzQm0/nJvivXrl1G\nM1D3zp6QWyIBnH66UTzUGeXFQzt3Qrmpg9Qjw3fepMxNZGlOL2++ue+7qiqj4l27vHh7vEH3/g6V\ntSiBFLOX51fHRh+O6UoCwSTp7IQydwe5J05uIABIPDiFTc90jT3jFGlthXl9HeScML59LyyEzjwb\nu1/t9H/3+efG+ErObU6s80LPERQVQdu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"text": [ "" ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's also plot our residual error, the part of the signal that we were not able to explain. We construct this my subtracting our explained signal from the measured timeseries. Note that this should look like random (white) noise. If not, then there is still some components in the signal that we failed to explain. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "# residuals:\n", "residuals_0 = convolved_signal_shifted - explained_signal_0\n", "residuals_1 = convolved_signal_shifted - explained_signal_1\n", "\n", "fig = plt.figure()\n", "plt.plot(timepoints, residuals_0, 'c')\n", "plt.ylim(-3,6)\n", "plt.legend(['residuals model 1'], loc=1)\n", "plt.title('residuals')\n", "plt.xlabel('time (s)')\n", "plt.ylabel('a.u.')\n", "\n", "fig = plt.figure()\n", "plt.plot(timepoints, residuals_1, 'c')\n", "plt.ylim(-3,6)\n", "plt.legend(['residuals model 2'], loc=1)\n", "plt.title('residuals')\n", "plt.xlabel('time (s)')\n", "plt.ylabel('a.u.')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 13, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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3hwKBnJ2WA+Dt7m4WJpEIUqkaSqSr6lDzvN68X2kv31kiyKB0lQjmlpbGXLcg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\nJae6j35IRPYCpwFPi8hqJ+L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"text": [ "" ] } ], "prompt_number": 13 }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 9 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Derivation of the multiple regression equation:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In raw form the regression equation is:\n", "\n", "$ y = a + b_1X_1 + b_2X_2 + ... + B_kX_k + e $\n", "\n", "This says that y, our dependent variable, is composed of a linear part and error. The linear part is composed of an intercept, a, and k independent variables, $X_1 ... X_k$ along with their associated regression weights $b_1 ... b_k$.\n", "\n", "In matrix terms, the same equation can be written:\n", "\n", "$ y = X b + e $\n", "\n", "If we solve for the $b$ weights, we find that\n", "\n", "$ b = (X'X)^{-1} X'y $\n", "\n", "To give you an idea why it looks like that, first remember the regression equation:\n", "\n", "$ y = X b + e $\n", "\n", "Let's assume that error will equal zero on average. Now we can simply drop the error term, in order to sketch a proof:\n", "\n", "$ y = Xb $\n", "\n", "Now we want to solve for $b$, so we need to get rid of $X$. First we will make $X$ into a square, symmetric matrix by premultiplying both sides of the equation by $X'$:\n", "\n", "$ X'y = X'Xb $\n", "\n", "And now we have a square, symmetric matrix that with any luck has an inverse, which we will call $(X'X)^{-1}$. Multiply both sides by this inverse, and we have:\n", "\n", "$ (X'X)^{-1} X'y = (X'X)^{-1} (X'X)b $\n", "\n", "It turns out that a matrix multiplied by its inverse is the identity matrix $(A^{-1}A=I)$:\n", "\n", "$ (X'X)^{-1} X'y=Ib $\n", "\n", "and a matrix multiplied by the identity matrix is itself $(AI = IA = A)$:\n", "\n", "$ (X'X)^{-1} X'y=b $\n", "\n", "which is the desired result." ] }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 258 }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [] } ], "metadata": {} } ] }