{ "metadata": { "name": "", "signature": "sha256:0779da07df52e2ac4c32b2ce932872b7e2479d67813ca1df801c4b6dfbecbb3c" }, "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: population receptive field fitting" ] }, { "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", "import scipy.stats as stats\n", "import scipy.signal as signal\n", "import matplotlib.pyplot as plt\n", "from sympy import *\n", "import math" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's do inline plotting:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%matplotlib inline " ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 12 }, { "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": [ "# define an Impulse Response Function:\n", "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", " t: IRF is defined with respect to 't'\n", " s: scaling factor\n", " n: sets the width\n", " tmax: sets the time to peak \n", " IRF_len: function is evaluated for t = [0:IRF_len]\n", " \n", " Returns\n", " -------\n", " y: IRF evaluated for t = [0:IRF_len]\n", " yprime: IRF first derivative evaluated for t = [0:IRF_len]\n", " \n", " \"\"\"\n", " \n", " # in sympy:\n", " t = Symbol('t')\n", " y = ( (s) * (t**n) * (math.e**((-n*t)/tmax)) )\n", " yprime = y.diff(t)\n", " \n", " # lambdify:\n", " y = lambdify(t, y, \"numpy\")\n", " yprime = lambdify(t, yprime, \"numpy\")\n", " \n", " # evaluate:\n", " y = y(timepoints)\n", " yprime = yprime(timepoints)\n", " \n", " return (y, yprime)\n", "\n", "# create the IRF:\n", "sample_rate = 10\n", "IRF_len = 3.0 # in seconds\n", "timepoints = np.linspace(0,IRF_len,IRF_len*sample_rate)\n", "\n", "IRF, IRF_prime = pupil_IRF(timepoints=timepoints)\n", "IRF = IRF / IRF.std()\n", "IRF_prime = IRF_prime / IRF_prime.std()\n", "\n", "# plot the IRF:\n", "fig = plt.figure()\n", "plt.plot(timepoints, IRF, color='r')\n", "# plt.plot(timepoints, IRF_prime, color='g')\n", "plt.legend(['IRF'])\n", "plt.title('Impulse Response Function')\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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"text": [ "" ] } ], "prompt_number": 13 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's simulate convolved timeseries data based on this IRF." ] }, { "cell_type": "code", "collapsed": false, "input": [ "duration = 60 # in seconds\n", "times_l = np.array([5,15,16,25])\n", "times_r = np.array([35,45,46,55])\n", "input_signal_l = np.zeros(duration * sample_rate)\n", "input_signal_r = np.zeros(duration * sample_rate)\n", "for i in times_l:\n", " input_signal_l[i*sample_rate] = 1.5\n", "for i in times_r:\n", " input_signal_r[i*sample_rate] = 0.5\n", "input_signal = input_signal_l + input_signal_r\n", "\n", "# convolve inputs with IRF: \n", "convolved_signal = (sp.convolve(input_signal, IRF, 'full'))[:-(IRF.shape[0]-1)]\n", "\n", "# let's add some noise:\n", "convolved_signal_noise = convolved_signal + np.random.normal(0,0.1,len(convolved_signal))\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_noise, 'b')\n", "plt.legend(['BOLD time series'], loc=1)\n", "for i in times_l:\n", " plt.axvline(i, color='r', alpha=0.5)\n", "for i in times_r:\n", " plt.axvline(i, color='g', alpha=0.5)\n", "# plt.legend(['\"true\" neural signal', 'pupil time series'], loc=1)\n", "plt.title('BOLD 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": 14, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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eSAQhsNbPaYoJwL8daNCH/wpCa5PS3ApE9LPDYqWUGtCSHRNRAYDnAHQDoAA8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\nbnt6G0RNwyXV1aGPbNRocTj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"text": [ "" ] } ], "prompt_number": 14 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's create a regressor separately for every type of event. In our case, we want to separately model (predict) a BOLD response every time a stimulus was presented on the left side of the screen, and model (predict) a response every time a stimulus was presented on the rights side of the screen: " ] }, { "cell_type": "code", "collapsed": false, "input": [ "duration = 60 # in seconds\n", "times_l = np.array([5,15,16,25])\n", "times_r = np.array([35,45,46,55])\n", "regr_l = np.zeros(duration * sample_rate)\n", "regr_r = np.zeros(duration * sample_rate)\n", "for i in times_l:\n", " regr_l[i*sample_rate] = 1\n", "for i in times_r:\n", " regr_r[i*sample_rate] = 1\n", "\n", "regr_l_convolved = (sp.convolve(regr_l, IRF, 'full'))[:-(IRF.shape[0]-1)]\n", "regr_r_convolved = (sp.convolve(regr_r, IRF, 'full'))[:-(IRF.shape[0]-1)]\n", "\n", "# demean measured data and regressors:\n", "regr_l_convolved = regr_l_convolved - regr_l_convolved.mean()\n", "regr_r_convolved = regr_r_convolved - regr_r_convolved.mean()\n", "convolved_signal_noise = convolved_signal_noise - convolved_signal_noise.mean()\n", "\n", "timepoints = np.linspace(0,duration,duration*sample_rate)\n", "fig = plt.figure()\n", "plt.plot(timepoints, convolved_signal_noise, 'b')\n", "plt.plot(timepoints, regr_l_convolved, color='r', lw=2)\n", "plt.plot(timepoints, regr_r_convolved+0.05, color='g', lw=2)\n", "plt.legend(['BOLD time series', 'regressor left events', 'regressor right events'], loc=1)\n", "plt.title('regressors')\n", "plt.xlabel('time (s)')\n", "plt.ylabel('a.u.')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 15, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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88pT1hs8880x8++23GDVqFA4++GBle2/XUq9c6uUsTzzxxLRlNo2u8VAhlqo0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\nz83MpLKui8pKGovBg2t/16qVW5ensJDGVsSVny5dav8mFKI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"text": [ "" ] } ], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "For every regressor (in the above toy example, we have two regressors) 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 data. Eyebaling the above figure, for the regressor predicting visual events that occured left on the screen, we will need a $b$ > 1; in contrast, for the regressor predicting visual events that occured right on the screen, we will need a $b$ < 1.\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 BOLD timeseries." ] }, { "cell_type": "code", "collapsed": false, "input": [ "# make design matrix:\n", "designMatrix = np.vstack((regr_l_convolved, regr_r_convolved))\n", "\n", "# # plot design matrix:\n", "# fig = plt.figure(figsize=(2,2))\n", "# plt.imshow(designMatrix.T)\n", "# plt.xticks([0,1])\n", "# plt.title('design matrix')\n", "# plt.xlabel('nr samples')\n", "# plt.ylabel('length run')\n", "\n", "# multiple regression:\n", "designMatrix = np.mat(designMatrix).T\n", "betas = ((designMatrix.T * designMatrix).I * designMatrix.T) * np.mat(convolved_signal_noise).T\n", "betas = np.array(betas)\n", "\n", "print 'betas: {}'.format(betas.ravel())\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "betas: [ 1.49740104 0.50932406]\n" ] } ], "prompt_number": 16 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Indeed, we find betas ~1.5 and ~0.5. This is what we used as input to simulate the BOLD timeseries. So this works!! Note that we don't find the exact same values due to the random noise we added to the BOLD time series.\n", "\n", "Let's plot our exaplained signal. We construct this by multiplying our regessors with their respective betas, and add them all up:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "explained_signal = regr_l_convolved*betas[0] + regr_r_convolved*betas[1]\n", "residuals = convolved_signal_noise - explained_signal\n", "\n", "fig = plt.figure()\n", "plt.plot(timepoints, convolved_signal_noise, 'b')\n", "plt.plot(timepoints, explained_signal, 'm', lw=2)\n", "plt.legend(['BOLD timeseries', 'explained signal'], loc=1)\n", "plt.title('predicted signal')\n", "plt.xlabel('time (s)')\n", "plt.ylabel('a.u.')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 17, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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+logPDQWvR6llCbmqhmJ1TmjIMQT+lNlecjM7mHNFGQwdjQmaeuCuqgnlBE+R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\n0Y/7WWepHuY1k2HiafDsc4Jdh8GCRap3XlUb4ju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"text": [ "" ] } ], "prompt_number": 17 }, { "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 BOLD timeseries. Note that this should look like random (white) noise. If not, then there is still some consistent components in the signal that we failed to exaplain. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig = plt.figure()\n", "plt.plot(timepoints, residuals, 'c')\n", "plt.legend(['residuals'], 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": 18, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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SGAizHZPG50oZWrToqeVdF6NybKKk6hEYvLentrR4vxnaihp9fpSUhWE0qfHD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"text": [ "" ] } ], "prompt_number": 18 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, to construct a PRF, let's plot a \"heatmap\" of our betas. The location in the plot of a particular beta should correspond to the location on the screen that that regressor was trying to exaplain: " ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig = plt.figure()\n", "ax = fig.add_subplot(111)\n", "a = ax.imshow(betas.T, interpolation='nearest', vmin=0, vmax=2, cmap='seismic') # cmap='YlOrRd')\n", "ax.set_xticks([0,1])\n", "ax.set_yticks([])\n", "ax.set_xticklabels(['beta left', 'beta right'])\n", "fig.colorbar(a, ticks=[0, 1, 2], orientation='vertical', )" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 19, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAT8AAAD3CAYAAABralPGAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n8P6SAAAAAElFTkSuQmCC\n", "text": [ "" ] } ], "prompt_number": 19 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here you see that this particular voxel indeed was driven more strongly by visual events occuring on the left side of the screen, than by event occuring on the right side of the screen." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, this was just a toy example. In practice we won't create two regressors for two locations in the visual field. Instead, we will create for example 30 by 30 regressors that cover the visual field with high resolution. In that case, the fitted pRF of one voxel might look something like this:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "sizex = 30\n", "sizey = 30\n", "x, y = np.mgrid[-sizex+10:sizex+10+1, -sizey+10:sizey+10+1]\n", "g = np.exp(-0.333*(x**2/float(sizex)+y**2/float(sizey)))\n", "PRF = g/g.sum()\n", "plt.imshow(PRF)\n", "plt.xticks([]); plt.yticks([])\n", "plt.title('\"Real\" pRF')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 20, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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q8Ob0htHjqJzlTVjlOApDbJ/Mte8nICbtoLSB1uM0xCkfz4l3Xm/R+fxh+Dzi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\niz1e3AEnt0dvW4w9t7SYwV2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"text": [ "" ] } ], "prompt_number": 20 }, { "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": [] } ], "metadata": {} } ] }