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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Parameter estimates for correlated regressors"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "By JB Poline and Matthew Brett."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Compatibility with Python 3\n",
      "from __future__ import print_function  # print('me') instead of print 'me'\n",
      "from __future__ import division  # 1/2 == 0.5, not 0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Array and plotting libraries\n",
      "import numpy as np\n",
      "import numpy.linalg as npl\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Display plots inside the notebook\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Make numpy print 4 significant digits for prettiness\n",
      "np.set_printoptions(precision=4, suppress=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Set random number seed to make random numbers reproducible\n",
      "np.random.seed(42)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## The linear model again"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We have some vector of data $\\vec{y}$ with values $y_1, y_2, ..., y_N$.\n",
      "\n",
      "We have one or more vectors of regressors $\\vec{x_1}, \\vec{x_2}, ..., \\vec{x_P}$, where $\\vec{x_1}$ has values $x_{1,1}, x_{1,2}, ..., x_{1,N}$, and $\\vec{x_p}$ has values $x_{p,1}, x_{p,2}, ..., x_{p,N}$.\n",
      "\n",
      "Our linear model says that:\n",
      "\n",
      "$$\n",
      "\\vec{y} = \\beta_1 \\vec{x_1} + \\beta_2 \\vec{x_2} +  ... + \\beta_P \\vec{x_P} + \\vec{\\varepsilon}\n",
      "$$\n",
      "\n",
      "Here:\n",
      "\n",
      "* $\\beta_1, \\beta_2, ... \\beta_P$ are scaling coefficients for vectors $\\vec{x_1}, \\vec{x_2}, ..., \\vec{x_P}$ respectively;\n",
      "* $\\vec{\\varepsilon} = \\varepsilon_1, \\varepsilon_2, ... \\varepsilon_N$ are the remaining unexplained errors for each observation.\n",
      "\n",
      "Usually one of vectors $\\vec{x}$ is a vector of constant value 1.  This models the intercept of the regression model.  We will write this special vector as $\\vec{1}$.\n",
      "\n",
      "As we saw in the [introduction to the general linear model](http://perrin.dynevor.org/glm_intro.html), we can express this same linear model as matrices.  We:\n",
      "\n",
      "* assemble the $\\vec{x_p}$ vectors as columns in a design matrix $\\mathbf{X} = [\\vec{x_1}, \\vec{x_2}, ... \\vec{x_P}]$;\n",
      "* assemble the $\\beta_p$ coefficients into a vector $\\vec{\\beta} = \\beta_1, \\beta_2, ..., \\beta_P$.\n",
      "\n",
      "Then matrix multiplication does the rest:\n",
      "\n",
      "$$\n",
      "\\vec{y} = \\mathbf{X} \\cdot \\vec{\\beta} + \\vec{\\varepsilon}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Models with correlated regressors"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Some correlated regressors"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Imagine we have a TR (image) every 2 seconds, for 30 seconds.  Here are the times of the TR onsets, in seconds:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "times = np.arange(0, 30, 2)\n",
      "times"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "array([ 0,  2,  4,  6,  8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28])"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we make a hemodynamic response function (HRF) shape for an event starting at time 0.  Call this `hrf1`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Gamma distribution from scipy\n",
      "from scipy.stats import gamma\n",
      "\n",
      "# Make SPM-like HRF shape with gamma for peak and undershoot\n",
      "hrf1 = gamma.pdf(times, 6) - 0.35 * gamma.pdf(times, 12)\n",
      "# Scale area under curve to 1\n",
      "hrf1 = hrf1 / np.sum(hrf1)\n",
      "# Plot\n",
      "plt.plot(times, hrf1)\n",
      "plt.title('HRF at t=0 (not mean-centered)')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "<matplotlib.text.Text at 0x1080942d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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ZNqV+PPAAnHUWPP44DCvYRS+1pL0dnnoK5swJpbnFi0NCb28vnMhzj4cNC61w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\nhijpi4hkiJK+iEiG/H8Exx2Q0ouNFQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x106592f10>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we make another HRF starting at t=2 (at the beginning of the second TR):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# HRF starting at t=2\n",
      "hrf2 = np.zeros(hrf1.shape)\n",
      "hrf2[1:] = hrf1[0:-1]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For simplicity, we remove the mean from these regressors.  We do this to make the HRF regressors independent of the mean ($\\vec{1}$) regressor, but it may not be clear yet why that is a good idea."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Remove the mean from both HRF regressors\n",
      "hrf1 = (hrf1 - hrf1.mean())\n",
      "hrf2 = (hrf2 - hrf2.mean())\n",
      "plt.plot(times, hrf1, label='hrf1 start t=0')\n",
      "plt.plot(times, hrf2, label='hrf2 start t=2')\n",
      "plt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "<matplotlib.legend.Legend at 0x10647eb10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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/yG3qz+2QzVeA9lQYsikirYG/q+rIsu3+CVwInALsAh5U1ReqiGdn\n+iFu/HiYMwcWLPB21u4nmz/hslmX8U7WOwHTN6HiyJvly50SV1YWDBxY+xmvVak88qZf235knZnF\nj7v+mMQmid59ABMUbHKWCWiqMHYsbNoEb73l7azd2V/PZuzbY1l4w0K6nNLFu8B1UFjofK7ykTfD\nhjmJfsQIb0bevLz8ZV5Z+QodkzvayBsDWNI3QaC4GH7yE+dM/x//8HbW7nNLn+NPH/2Jj3/2Manx\nHi/wX4NvvoGnn4YXX4SzzoIxY5yRN4kuT7wPHT3EjK9mMGXJFI4WH+W6ntcxqscoOrfwcDlTE9Qs\n6ZugUFDgDOE8/3xntIqXHv7gYd5Y8wbZN2T7Wu4oKYF33oEpU5yVKm+80fkrpmNH97FX7FrB1MVT\n+eeKfzI4YzC39b6NwRmDbeSN+R5L+iZo7NsHF1wAN98M99zjXVxV5fY5t7Nu3zrmZM3x/ILmzp0w\nbZrTz7l1a2dU0lVXue/tfKzkGG+sfoMpS6awbu86bjn3Fm4+52ZbwMzUyJK+CSrffusk/okTYdQo\n7+KWlJZw5atX0jSqKS9f8bLrWbuq8PHHzln93LnOonK33eYsi+DW5gObefaLZ3ku5zm6tuzK7X1u\n57LTLyM6Mtp9cBPyLOmboLNiBQwZ4lz8HDrUu7iFRYVcNOMieqf1ZvKwyfUqjRw65BzXlClw9Khz\nVn/99ZBc5bzz2ivVUuZtmMeUxVNYuGkhY3qO4dbet9KtVTd3gU3YsaRvgtLChc7Z87vvOmPXvZJX\nmMeAFwZww1k3cN9599X6fStXwtSpzgSqQYOcZD94sLsx9eBMoHpx2YtMXTKVuOg4bu9zO1k9sqxT\nlKm3hp6cZYwnBg50Rr9ccokza9eLi6EAyc2SmTt6Lhe8cAGp8amM6Tmm2m2PHXNmDk+ZAmvXOtca\nvvoK2rZ1fxxfbPuCpxY/xeurX+eS0y5h+uXT+UHbH9iFWdMoLOmbgHDFFc5F0mHDnPq5V7N22zVv\nd3zWbqvYVgzrPOyE17dscXr8/v3vcMYZcMcdzgziaJcl9cKiQv618l9MWTyFXfm7uK33bay7ax2t\n4lq5C2yMS1beMQHlwQedi6ULFtS9pV9NPv72Yy7/1+XMHT2Xc1J78/778NRT8MEHMHq00+qxe3f3\n+1m/bz1PL3ma6V9Op2+bvtze+3aGdx5u698YX1hN3wQ9Vae0smWLM7u1pjPu0lLnYmteHuzf7/ys\nfL/i42+i3+SbrrcR/8qHtI/vzB13OAm/ql8uhUWF7D+yn7wjeeQV5p1wP+9INY8L8zhSfIQbz7qR\nsb3H0jHZozqVMdWwpG9CQnGxM6NV1Tn7ri6RHzgAzZo5o2nKb0lJNT/+v73P8OK6SYw5K4sD5Ym7\nisSuKMlNk0lqmkRys2SSmyaT3CyZpCaVHjdNOn4/uWkyaQlp1nXKNBhL+iZkFBTA5MnO4mQ1JfSo\nelyVmvHVDL7Z902VSbs8yTeLamYXWk3As6RvjDFhpD5JP7QajBpjjKmRJX1jjAkjlvSNMSaMWNI3\nxpgwYknfGGPCiCV9Y4wJI5b0jTEmjFjSN8aYMGJJ3xhjwki9k76ItBCR90RkrYj8n4gkVbFNOxFZ\nICIrRWSFiNzt7nCNMca44eZM/zfAe6p6GjC/7HFlRcA9qtod6A/cISJdXewzKGVnZzf2IfjKPl9w\ns88XXtwk/R8B08vuTwcur7yBqu5Q1WVl9w8Dq4HWLvYZlEL9Pzr7fMHNPl94cZP0U1R1Z9n9nUBK\nTRuLSDpwNvC5i30aY4xxocaFaUXkPSC1ipd+W/GBqqqIVLtEpojEA/8GflF2xm+MMaYR1HtpZRFZ\nA2Sq6g4RSQMWqOoZVWwXDbwNzFXVJ2uIZ+sqG2NMHdV1aWU3jdFnAz8FJpb9fLPyBuJ0oZgGrKop\n4UPdD9wYY0zduTnTbwG8ArQHcoGrVXW/iLQG/q6qI0XkAmAh8BVQvqP7VfW/ro/cGGNMnQVM5yxj\njDH+a/QZuSIyXETWiMg6ERnX2MfjNRHJFZGvRCRHRBY19vG4JSLPi8hOEVle4bmTTtQLFtV8vgki\nsqXsO8wRkeGNeYz1Vd1kyVD5/mr4fKHy/TUVkc9FZJmIrBKRP5U9X6fvr1HP9EUkEvgaGApsBRYD\no1R1daMdlMdEZCNwrqrua+xj8YKIDAAOA/9Q1R5lz00C9qjqpLJf3MmqWtVkvYBXzecbDxxS1cmN\nenAuiUgqkKqqy8pG1H2BM7/mRkLg+6vh811NCHx/ACISq6oFIhIFfATchzNnqtbfX2Of6fcF1qtq\nrqoWAbOAyxr5mPwQMhepVfVDIK/S0yedqBcsqvl8EALfYTWTJdsQIt9fDZ8PQuD7A1DVgrK7MUAk\nzn+rdfr+GjvptwE2V3i8he++pFChwDwRWSIiNzf2wfikThP1gtRdIvKliEwL1vJHRZUmS4bc91fh\n831W9lRIfH8iEiEiy3C+pwWqupI6fn+NnfTD4Sry+ap6NjACZ+2hAY19QH5Sp14Yat/rVCADOAvY\nDjzeuIfjTlnp4zWcyZKHKr4WCt9fFZNBQ+b7U9VSVT0LaAsMFJFBlV4/6ffX2El/K9CuwuN2OGf7\nIUNVt5f93A28gVPSCjU7y+qplE3U29XIx+MpVd2lZYDnCOLvsGyy5GvAS6paPrcmZL6/Cp9vRvnn\nC6Xvr5yqHgDmAOdSx++vsZP+EqCLiKSLSAxwDc6kr5AgIrEiklB2Pw64CFhe87uCUvlEPahmol4w\nK/sfqdyPCdLvsIbJkiHx/VX3+ULo+2tZXpoSkWbAD4Ec6vj9Nfo4fREZATyJc1Fimqr+qVEPyEMi\nkoFzdg/O7OeXg/3zicg/gQuBljj1wweB/1DFRL3GOkY3qvh844FMnNKAAhuBsRVqqEGjusmSwCJC\n4Pur5vM9AIwiNL6/HjgXaiPKbi+p6qPVTZStNk5jJ31jjDENp7HLO8YYYxqQJX1jjAkjlvSNMSaM\nWNI3xpgwYknfGGPCiCV9Y4wJI5b0jTEmjFjSN8aYMPL/AdocrVSCZgCFAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1080cbe50>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "These `hrf1` and `hrf2` regressors are correlated.  The Pearson correlation coefficient between the HRFs is:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.corrcoef(hrf1, hrf2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "array([[ 1.    ,  0.7023],\n",
        "       [ 0.7023,  1.    ]])"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Some simulated data"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we are going to make some simulated data from the *signal* formed from the correlated regressors, plus some random *noise*.\n",
      "\n",
      "The *signal* comes from the sum of `hrf1` and `hrf2`.  This simulates the occurence of two events, one starting at t=0, one at t=2, both causing an HRF response:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "signal = hrf1 + hrf2\n",
      "plt.plot(hrf1, label='hrf1')\n",
      "plt.plot(hrf2, label='hrf2')\n",
      "plt.plot(signal, label='signal (combined hrfs)')\n",
      "plt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 11,
       "text": [
        "<matplotlib.legend.Legend at 0x108479650>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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rrZdkPrare+fCBZPsf/kFli41wwgLtX+/yUrHjmXO0CpYQnICw74d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\niCR9IYQoRSTpCyFEKSJJXwghSpH/B8tdVw84EZCBAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10826cb10>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The simulated data is this signal combined with some random noise:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "noise = np.random.normal(size=times.shape)\n",
      "Y = signal + noise\n",
      "plt.plot(times, signal, label='signal')\n",
      "plt.plot(times, Y, '+', label='signal + noise')\n",
      "plt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "<matplotlib.legend.Legend at 0x1086cae10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x1084a5690>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We are going apply several linear models to these simulated data.\n",
      "\n",
      "All our models include a regressor of a vector of ones, $\\vec{1}$, modeling the mean.\n",
      "\n",
      "We will call our `hrf1` vector $\\vec{h_1}$.  Call `hrf2` : $\\vec{h_2}$.\n",
      "\n",
      "Our models are:\n",
      "\n",
      "* A model with $\\vec{x}$ vectors $\\vec{h_1}, \\vec{1}$ - single HRF model);\n",
      "* A model with $\\vec{h_1}, \\vec{h_2}, \\vec{1}$ - both HRFs model;\n",
      "* A model with $\\vec{h_1}, \\vec{w}, \\vec{1}$, where $\\vec{w}$ is $\\vec{h_2}$ (`hrf2`) *orthogonalized with respect to* $\\vec{h_1}$ (`hrf1`).  We explain what we mean by this further down the page.\n",
      "\n",
      "First, the model with $\\vec{h_1}, \\vec{1}$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Design matrix for single HRF model\n",
      "X_s = np.vstack((hrf1, np.ones_like(hrf1))).T\n",
      "plt.imshow(X_s, interpolation='nearest', cmap='gray')\n",
      "plt.title('Model with hrf1 regressor')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "<matplotlib.text.Text at 0x108737210>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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H1LLmgFrWHFDLmgNqWXNALWsOqGXNAbWsdXVAG41uVyr7R0lDkkb31U3bJro6oDQe3Q5J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\nWtYcUMva/wP7D4ck8Ctl1QAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10862c590>"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Simulating the effect of noise on parameter estimates"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Remember that the students-t statistic is:\n",
      "\n",
      "$$\n",
      "t = \\frac{c^T \\hat\\beta}{\\sqrt{\\mathrm{var}(c^T \\hat\\beta)}}\n",
      "$$\n",
      "\n",
      "where $c^T$ is a row vector of contrast weights, $\\hat{\\beta}$ is our vector of estimated parameters, and $\\mathrm{var}(c^T \\hat\\beta)$ is the variance of $c^T \\hat\\beta$.\n",
      "\n",
      "On the assumption of zero mean normally distributed independent noise:\n",
      "\n",
      "$$\n",
      "\\mathrm{var}(c^T \\hat\\beta) = \\hat{\\sigma}^2 c^T (X^T X)^+ c\n",
      "$$\n",
      "\n",
      "where $\\hat{\\sigma}^2$ is our estimate of variance in the residuals, and $(X^T X)^+$ is the [pseudo-inverse](https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse) of $X^T X$.\n",
      "\n",
      "Therefore:\n",
      "\n",
      "$$\n",
      "t = \\frac{c^T \\hat\\beta}{\\sqrt{\\hat{\\sigma}^2 c^T (X^T X)^+ c}}\n",
      "$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We will see that this expection of variance correctly predicts that parameter estimates for correlated regressors will have higher variance for a given level of noise (where the level of noise can be captured by $\\hat{\\sigma}^2$)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Put another way, the parameter estimates of correlated regressors are more susceptible to the effect of noise."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can look at the variability of the parameter estimates, by estimating our models on many different simulated data vectors.\n",
      "\n",
      "Each of the data vectors are made of the *signal* (the sum of `hrf1` and `hrf2`) and some *noise*.  We take a new sample of noise for each data vector:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Create array of simulated data vectors in columns\n",
      "n_times = len(times) # number of elements in single data vector\n",
      "n_data_vectors = 100000\n",
      "# Make many noise vectors (new noise for each column)\n",
      "noise_vectors = np.random.normal(size=(n_times, n_data_vectors))\n",
      "# add signal to make data vectors\n",
      "# Use numpy broadcasting to add vector elementwise to 2D array\n",
      "Ys = noise_vectors + signal.reshape(n_times, 1)\n",
      "Ys.shape"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "(15, 100000)"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We first fit the model with only the first HRF regressor to every (signal + noise) sample vector.\n",
      "\n",
      "We will fit our model for each data vector, to make a estimated parameter vector for each data vector.  We can stack these estimated parameter vectors into a 2 by `n_data_vectors` array.\n",
      "\n",
      "Call this array $\\beta^s$ (where the $s$ superscript is for the *single* HRF model)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Fit X_one to signals + noise\n",
      "n_regressors = X_s.shape[1]\n",
      "# beta (parameter estimate) matrix, one column per data vector\n",
      "B_s = np.zeros((n_regressors, n_data_vectors))\n",
      "# Estimate the parameters of the model for each data vector\n",
      "X_pinv = npl.pinv(X_s)\n",
      "for i in range(n_data_vectors):\n",
      "    B_s[:, i] = X_pinv.dot(Ys[:, i])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In fact, because of the way that matrix multiplications works, we can do exactly the same calculation as we did in the loop above, in one matrix multiplication:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B_again = X_pinv.dot(Ys)\n",
      "assert np.allclose(B_s, B_again)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We will use this trick to estimate the parameter matrices for the rest of our models."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us look at the variance of the first parameter estimate (the parameter for the `hrf1` regressor).\n",
      "\n",
      "Here is the variance we observe:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(B_s[0], bins=100)\n",
      "print('Observed B[0] variance for single hrf model:', np.var(B_s[0]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Observed B[0] variance for single hrf model: 2.2014181026\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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3UPskoaqGrrxW2+rZC9zWLN8GPNg3fnOSDUk2AZuBg1V1Eng2ydbmYO+tfftIksZoxVZP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\nyt7FZAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108825d50>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can compare the observed variance of the first parameter estimate with that expected from the formula above:\n",
      "\n",
      "$$\n",
      "\\mathrm{var}(c^T \\hat\\beta) = \\hat{\\sigma}^2 c^T (X^T X)^+ c\n",
      "$$\n",
      "\n",
      "To select only the first regressor, we use a contrast vector of\n",
      "\n",
      "$$\n",
      "c = \\left[\n",
      "\\begin{array}{\\cvec}\n",
      "1 \\\\\n",
      "0 \\\\\n",
      "\\end{array}\n",
      "\\right]\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Our $\\hat{\\sigma}^2$ will be close to 1, because we added noise with variance 1:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Estimate sigma^2 for every data vector\n",
      "predicted = X_s.dot(B_s)\n",
      "residuals = Ys - predicted\n",
      "# Residuals have N-P degrees of freedom\n",
      "N = n_times\n",
      "P = npl.matrix_rank(X_s) # number of independent columns in design\n",
      "sigma_hat_squared = (residuals ** 2).sum(axis=0) / (N - P)\n",
      "print('Mean sigma^2 estimate:', np.mean(sigma_hat_squared))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Mean sigma^2 estimate: 1.01981361247\n"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Because $\\hat{\\sigma}^2 \\approx 1$, $\\mathrm{var}(c^T \\hat\\beta) \\approx c^T (X^T X)^+ c$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "C_s = np.array([[1], [0]]) # column vector\n",
      "# c.T{X.T X}+ c\n",
      "C_s.T.dot(npl.pinv(X_s.T.dot(X_s)).dot(C_s))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": [
        "array([[ 2.2051]])"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Notice that the mean of the parameter estimates for $\\vec{h_1} (`hrf1`), is somewhere above one, even though we only added 1 times the first HRF as the signal:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print('Observed B[0] mean for single hrf model:', np.mean(B_s[0]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Observed B[0] mean for single hrf model: 1.70147967742\n"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is because the single first regresssor has to fit *both* $\\vec{h_1}$ in the signal, and as much as possible of $\\vec{h_2}$ in the signal, because there is nothing else in the model to fit $\\vec{h_2}$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now let us construct the model with both HRFs as regressors:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Design matrix for both HRFs model\n",
      "X_b = np.vstack((hrf1, hrf2, np.ones_like(hrf1))).T\n",
      "plt.imshow(X_b, interpolation='nearest', cmap='gray')\n",
      "plt.title('Model with hrf1, hrf2 regressors')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 21,
       "text": [
        "<matplotlib.text.Text at 0x10d0c4b90>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x106592890>"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We will call the resulting 3 by `n_data_vectors` parameter array : $\\beta^b$ (where the $b$ superscript is for *both* HRF regressors).\n",
      "\n",
      "We will use the matrix multiplication trick above to fit all the data vectors at the same time:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Fit X_both to signals + noise\n",
      "B_b = npl.pinv(X_b).dot(Ys)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "What estimates do we get for the first regressor, when we have both regressors in the model?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(B_b[0], bins=100)\n",
      "print('Observed B[0] mean for two hrf model', np.mean(B_b[0]))\n",
      "print('Observed B[0] variance for two hrf model', np.var(B_b[0]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Observed B[0] mean for two hrf model 1.00299905499\n",
        "Observed B[0] variance for two hrf model 4.35228828591\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10d10cf50>"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Two things have happened now we added the second (correlated) $\\vec{h_2}$ regressor.  First, the mean of the parameter for the $\\vec{h_1}$ regressor has dropped to 1, because $\\beta^b_1 \\vec{h_1}$ is no longer having to model the signal due to $\\vec{h_2}$.   Second, the variability of the estimate has increased.  This is what the bottom half of the t-statistic predicts:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Predicted variance for hrf1 parameter in both HRF model\n",
      "C_b = np.array([[1], [0], [0]])  # column vector\n",
      "C_b.T.dot(npl.pinv(X_b.T.dot(X_b)).dot(C_b))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 24,
       "text": [
        "array([[ 4.3517]])"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The estimate of the parameter for $\\vec{h_2}$ has a mean of around 1, like the parameter estimates for $\\vec{h_1}$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(B_b[1], bins=100)\n",
      "print('Observed B[1] mean for two hrf model', np.mean(B_b[1]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Observed B[1] mean for two hrf model 0.994534456872\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10ad816d0>"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This mean of 1 is what we expect because we have $\\vec{h_1} + \\vec{h_2}$ in the signal.  Not surprisingly, the $\\vec{h_2}$ parameter estimate has a similar variability to that for the $\\vec{h_1}$ parameter estimate:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print('Observed B[1] variance for two hrf model', np.var(B_b[1]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Observed B[1] variance for two hrf model 4.37230501038\n"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The observed variance is very similar to the predicted variance:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "C_b_1 = np.array([0, 1, 0])[:, None]  # column vector\n",
      "C_b_1.T.dot(npl.pinv(X_b.T.dot(X_b)).dot(C_b_1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 27,
       "text": [
        "array([[ 4.3519]])"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The parameter estimates for $\\vec{h_1}$ and $\\vec{h_2}$ are anti-correlated:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Relationship of estimated parameter of hrf1 and hrf2\n",
      "plt.plot(B_b[0], B_b[1], '.')\n",
      "np.corrcoef(B_b[0], B_b[1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 28,
       "text": [
        "array([[ 1.   , -0.703],\n",
        "       [-0.703,  1.   ]])"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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9O7PSwNo15Quja6ocE3b56KOSQOUNvzSx9+3twG23ucPqSDyZNk0GAKXkXnjr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\nCnlCCIkx/x8d/SgvbIb1ewAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10d125190>"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Orthogonalizing hrf2 with respect to hrf1"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\vec{h_2}$ is correlated with $\\vec{h_1}.\n",
      "\n",
      "We can therefore think of $\\vec{h_2}$ as the sum of some scaling of $\\vec{h_1}$ plus an extra part that cannot be explained by $\\vec{h_1}$.\n",
      "\n",
      "$$\n",
      "\\vec{h_2} = p (\\vec{h_1}) + \\vec{w}\n",
      "$$\n",
      "\n",
      "where $p$ is some scalar, and\n",
      "\n",
      "$$\n",
      "\\vec{w} = \\vec{h_2} - p (\\vec{h_1})\n",
      "$$\n",
      "\n",
      "To restate, we can think of $\\vec{h_2}$ as the sum of some scalar amount of $\\vec{h_1}$ plus $\\vec{w}$.\n",
      "\n",
      "We want to chose $p$ such that $\\vec{w} = \\vec{h_2} - p (\\vec{h_1})$ is orthogonal to $\\vec{h_1}$.  In this case $\\vec{w}$ is the part of $\\vec{h_2}$ that cannot be explained by $\\vec{h_1}$.\n",
      "\n",
      "If $\\vec{w}$ is orthogonal to $\\vec{h_1}$ then we call $\\vec{w}$ : $\\vec{h_2}$ *orthogonalized with respect to* $\\vec{h_1}$.\n",
      "\n",
      "Following the same logic as for [projection](http://practical-neuroimaging.github.io/day7.html#key-video-on-projecting-vectors), given $\\vec{w} - p (\\vec{h_1})$ is orthogonal to $\\vec{h_1}$:\n",
      "\n",
      "$$\n",
      "(\\vec{w} - p (\\vec{h_1})) \\cdot \\vec{h_1} = 0 \\implies \\\\\n",
      "\\vec{w} \\cdot \\vec{h_1} - p (\\vec{h_1}) \\cdot \\vec{h_1} = 0 \\implies \\\\\n",
      "\\frac{\\vec{w} \\cdot \\vec{h_1}}{\\vec{h_1} \\cdot \\vec{h_1}} = p\n",
      "$$\n",
      "\n",
      "Put another way, $p (\\vec{h_1})$ such that $\\vec{w} = \\vec{h_2} - p (\\vec{h_1})$ is orthogonal to $\\vec{h_1}$ - is also the projection of $\\vec{h_2}$ onto $\\vec{h_1}$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Project hrf2 onto hrf1\n",
      "p = hrf2.dot(hrf1) / hrf1.dot(hrf1)\n",
      "projection = p * hrf1\n",
      "# Get \\vec{w} \n",
      "w = hrf2 - projection\n",
      "# w and hrf1 are now orthogonal\n",
      "assert np.allclose(w.dot(hrf1), 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Plot the vector parts\n",
      "plt.plot(times, hrf1, label=r'$\\vec{h_1}$')\n",
      "plt.plot(times, hrf2, label=r'$\\vec{h_2}$')\n",
      "plt.plot(times, projection, label=r'$p (\\vec{h_1})$')\n",
      "plt.plot(times, w, label=r'$\\vec{w}$')\n",
      "plt.legend()\n",
      "# hrf1 part of hrf2, plus unique part, equals original hrf2\n",
      "assert np.allclose(hrf2, projection + w)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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K+BUKXbQJ0d+4EeLWn+Lt+PGIhx7SasEawLmCcyxYv4DPZn1GN49u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\n0jczayBO+mZmDcRJ38ysgfwP477l3as1/DYAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10adfbe50>"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "How much of the first regressor did we find in the second regressor?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 31,
       "text": [
        "0.70231917818451162"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us rewrite our original model containing `hrf1, hrf2`:\n",
      "\n",
      "$$\n",
      "\\vec{y} = \\beta^b_1 \\vec{h_1} + \\beta^b_2 \\vec{h_2} + \\beta^b_3 1\n",
      "$$\n",
      "\n",
      "where $\\vec{y}$ is our data vector.\n",
      "\n",
      "Now we know we can also write this as:\n",
      "\n",
      "$$\n",
      "\\vec{y} = \\beta^b_1 \\vec{h_1} + \n",
      "  \\beta^b_2 (p (\\vec{h_1}) + \\vec{w}) + \\beta^b_3 1 \\\\\n",
      "= \\beta^b_1 \\vec{h_1} + \n",
      "  \\beta^b_2 p (\\vec{h_1}) + \\beta^b_2 \\vec{w} + \\beta^b_3 1 \\\\\n",
      "= (\\beta^b_1 + p \\beta^b_2) \\vec{h_1} + \\beta^b_2 \\vec{w} + \\beta^b_3 1\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So, what will happen if we drop $\\vec{h_2}$ from our model and leave only $\\vec{w}$?\n",
      "\n",
      "We have called the parameters from the model with $\\vec{h_1}, \\vec{h_2}$ : $\\beta^b$.  Call the parameters from the model with $\\vec{h_1}, \\vec{w}$ : $\\beta^w$. \n",
      "\n",
      "We can see that we are going to get the exact same fit to the data with these two models if $\\beta^w_2 = \\beta^b_2$ and $\\beta^w_1 = p \\beta^b_2 + \\beta^b_1$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us try this new model and see:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X_w = np.vstack((hrf1, w, np.ones_like(hrf1))).T\n",
      "plt.imshow(X_w, interpolation='nearest', cmap='gray')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 32,
       "text": [
        "<matplotlib.image.AxesImage at 0x10d4d7190>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFEAAAD7CAYAAAASCHi+AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nF7tHAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10d4296d0>"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Fit the model\n",
      "B_w = npl.pinv(X_w).dot(Ys)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us first look at the distribution of $\\beta^w_1$ `== B_w[0]`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Distribution of parameter for hrf1 in orth model\n",
      "plt.hist(B_w[0], bins=100)\n",
      "print('Observed B[0] mean for two hrf orth model',\n",
      "      np.mean(B_w[0]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Observed B[0] mean for two hrf orth model 1.70147967742\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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3UPskoaqGrrxW2+rZC9zWLN8GPNg3fnOSDUk2AZuBg1V1Eng2ydbmYO+tfftIksZoxVZP\nkgeAq4ELknwP+DXgDmBPktuBY8CNAFV1JMke4AjwPLC9r4TfDtwHnAM8XFX7RvtUJEnDGKrVMy62\nemabrZ5p2afH91J7jKvVI2lqmfA1mIlfklrGn16U5lT/9FrbPupnxS/NLS/joOWZ+CWpZWz1aE08\nW1eaPVb8GgFbCtIsMfFLUsvY6tGq2OKRZpcVv9bA9o40i0z8ktQytnqkFvBkLvWz4pdawZlXOs3E\nL0ktY+KXpJYx8Ustk8TpuC1n4pdax15/2zmrR0OzSpTmgxW/zpKzQ6RZZ+KXpJYx8UtSy9jj14rs\n7UvzxYpfQ7KvL82LNSX+JMeSfCvJoSQHm7Hzk+xP8mSSR5Kc17f9ziRPJTma5Nq1Bi9p9Rbn8/uN\nrn3WWvEX0Kmqt1TVO5qxHcD+qroMeLS5TZItwE3AFmAbcFcSv3FIE+MMrbYaReJdWi5cB+xulncD\nNzTL1wMPVNVzVXUMeBp4B5KksRpFxf+VJN9I8v5mbGNVnWqWTwEbm+WLgON9+x4HLl7j40uSztJa\nZ/W8q6r+KMlfBfYnOdq/sqoqyaDvki9bt7Cw8OJyp9Oh0+msMURJmi/dbpdut7vq/TOqH2VIsgv4\nEfB+en3/k0kuBA5U1RuT7ACoqjua7fcBu6rqa333Uf5IxPTpHfwrel29xX+fQWNrXT/N+0xLHKON\n3ffdbEtCVQ19lH7VrZ4kr01ybrP8F4FrgcPAXuC2ZrPbgAeb5b3AzUk2JNkEbAYOrvbxtb6c8SHN\nr7W0ejYC/7FJDK8CPldVjyT5BrAnye3AMeBGgKo6kmQPcAR4HthueT/t+qtEzbPFD3jfku0wslbP\nKNjqmR6n2ztgu2Qa41if2H3/zaazbfV4yQZJL/JH2dvBxK8X2c+X7b128MxZLeHZnNK8M/FLUsuY\n+CWpZezxy96+1DJW/GrY15fawsQvSS1jq0fSspzTP79M/C1lX18rOz2n30s6zBdbPa3mnH0Ny9fJ\nPLHilzQ02z/zwYpf0lnwW+I8sOJvGXv7kqz4W8mKTWozK/4WsMrXenCmz+wy8c+plyd7L7erUVv8\nQRfNGls9c80DcZJezopf0po4xXP2mPjnjP18jd/Lz/AFPwSmmYl/Dizfz/cDQJPgZR5mgYl/hr00\n4XvwVtPGAmRajfXgbpJtSY4meSrJPxvnY88vqylJZ2dsiT/JK4F/A2wDtgC3JLl8XI8/St1ud6yP\nl2TZv5V11zu0EelOOoAhdScdwExa7rU77vfQas1KnGdrnBX/O4Cnq+pYVT0H/Hvg+jE+/siM68Xw\n0gTfPzVz2Gma3fUIax10Jx3AkLqTDmBGLX3twjXXXHMWBczkmPjX7mLge323jzdjrTe4oreVo3m0\ni8XX9uq+zWotxpn4ZzaDffKTn3zJi/LjH//4UC/QMyX0Myf4s63opXnw8m8Ew753lvuw8ENkZRnX\nVKsk7wQWqmpbc3sn8EJVfbJvG7OdJK1CVQ39KTfOxP8q4H8A7wb+N3AQuKWqvjOWACRJwBjn8VfV\n80k+BHwZeCVwj0lfksZvbBW/JGk6TOXVOZN8OMl3knw7ySdX3mNykvyjJC8kOX/SsSwnyb9o/l8+\nnuSLSX5i0jEtmoUT+pJckuRAkiea1+NHJh3TIElemeRQkocmHcuZJDkvyRea1+WR5vjf1Emys/l3\nP5zk80n+wqRjAkhyb5JTSQ73jZ2fZH+SJ5M8kuS8QfcxdYk/yTXAdcCbq+ongX854ZDOKMklwHuA\nZyYdywCPAG+qqr8JPAnsnHA8wEyd0Pcc8CtV9SbgncAvT2mciz4KHGG6p4XdCTxcVZcDbwamruWb\n5FLg/cBbq+oKeu3pmycZU5/P0nvf9NsB7K+qy4BHm9tnNHWJH/gl4BPNSV5U1Z9MOJ5BPg3800kH\nMUhV7a+qF5qbXwPeMMl4+szECX1VdbKqHmuWf0QvSV002aiWl+QNwM8Cv82UXiSn+cb5t6vqXugd\n+6uqH0w4rOU8S+9D/7XNxJTXAicmG1JPVX0V+P6S4euA3c3ybuCGQfcxjYl/M/DTSf4gSTfJ2yYd\n0HKSXA8cr6pvTTqWs/CLwMOTDqIxcyf0NVXgW+h9gE6jfwX8E+CFlTacoE3AnyT5bJI/TPLvkrx2\n0kEtVVX/F/gU8L/ozUL806r6ymSjGmhjVZ1qlk8BGwdtPJGrcybZD7x+mVW/Si+mv1xV70zydmAP\n8NfHGd+iFeLcCVzbv/lYglrGgDg/VlUPNdv8KvDnVfX5sQZ3ZtPciniZJK8DvgB8tKn8p0qSnwP+\nuKoOJelMOp4BXgW8FfhQVX09yWfotSV+bbJhvVSSvwH8Q+BS4AfAf0jyd6vqcxMNbAhVVSudEzWR\nxF9V7znTuiS/BHyx2e7rzYHTv1JV/2dsATbOFGeSn6RXuTzenBn4BuCbSd5RVX88xhCBwf8/AZL8\nPXotgHePJaDhnAAu6bt9Cb2qf+okeTXwe8DvVtWDk47nDP4WcF2SnwVeA/ylJPdX1S9MOK6ljtP7\npvz15vYXWKEfPSFvA/7bYt5J8kV6/4+nNfGfSvL6qjqZ5EJgYB6axlbPg8DfAUhyGbBhEkl/kKr6\ndlVtrKpNVbWJ3ov5rZNI+itJso3e1//rq+rPJh1Pn28Am5NcmmQDcBOwd8IxvUx6n+z3AEeq6jOT\njudMqupjVXVJ83q8GfjPU5j0qaqTwPea9zbAzwBPTDCkMzkKvDPJOc1r4GfoHTSfVnuB25rl2+jl\n0TOaxh9iuRe4t5mq9OfA1L14lzHNbYt/DWwA9jffTv57VW2fbEgzdULfu4CfB76V5FAztrOq9k0w\npmFM82vyw8Dnmg/8/wn8/QnH8zJV9XiS++kVKC8Afwj828lG1ZPkAeBq4IIk36PXJrsD2JPkduAY\ncOPA+/AELklql2ls9UiS1pGJX5JaxsQvSS1j4pekljHxS1LLmPglqWVM/JLUMiZ+SWqZ/w/wMjtV\nyt7FZAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10ace1390>"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Notice that $\\beta^w_1$ now has the same values as for the single HRF model  : $\\beta^s_1$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "assert np.allclose(B_s[0, :], B_w[0, :])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 35
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "It therefore has the same variance, and the predicted variance matches:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print('Observed B[0] variance for two hrf orth model', np.var(B_w[0]))\n",
      "pred_var = C_b.T.dot(npl.pinv(X_w.T.dot(X_w)).dot(C_b))\n",
      "print('Predicted B[0] variance for two hrf orth model', pred_var)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Observed B[0] variance for two hrf orth model 2.2014181026\n",
        "Predicted B[0] variance for two hrf orth model [[ 2.2051]]\n"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The fact that the single hrf and orthogonalized model parameters match may make sense when we remember that adding the $\\vec{w}$ regressor to the model cannot change the parameter for the $\\vec{h_1}$ regressor as $\\vec{w}$ is orthogonal to $\\vec{h_1}$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We predicted above that $\\beta^w_2$ would stay the same as $\\beta^b_2$ from the not-orthogonalized model:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "assert np.allclose(B_b[1, :], B_w[1, :])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 37
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We predicted that $\\beta^w_1$ would become $\\beta^b_1 + p \\beta^b_2$ from the not-orthogonalized model:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "predicted_beta1 = B_b[0, :] + p * B_b[1, :]\n",
      "assert np.allclose(predicted_beta1, B_w[0, :])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 38
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Our predictions were correct.  So let us revise what happened:\n",
      "\n",
      "* We estimated our original model with correlated $\\vec{h_1}, \\vec{h_2}$ to get corresponding estimated parameters $\\beta^b_1, \\beta^b_2$;\n",
      "* we orthogonalized $\\vec{h_2}$ with respect to $\\vec{h_1}$ to give $p$ and $\\vec{w}$;\n",
      "* we replaced $\\vec{h_2}$ with $\\vec{w}$ in the model, and re-estimated, giving new parameters $\\beta^w_1$ for $\\vec{h_1}$, $\\beta^w_2$ for $\\vec{w}$;\n",
      "* $\\beta^w_2 = \\beta^b_2$ - the parameter for the new orthogonalized regressor is unchanged from the non-orthogonalized case;\n",
      "* $\\beta^w_1 = \\beta^b_1 + p \\beta^b_2$ - the parameter for the *unchanged* regressor has increased by $\\beta^b_2$ times the amount of $\\vec{h_2}$ present in $\\vec{h_1}$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here we show some example parameters from the three model fits:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Example parameters from the single hrf model\n",
      "B_s[:,:5]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 39,
       "text": [
        "array([[ 0.0032,  1.2188,  2.289 ,  0.2179,  3.45  ],\n",
        "       [-0.0969,  0.0488,  0.006 , -0.1493, -0.4889]])"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Example parameters from the non-orth two-hrf model\n",
      "B_b[:,:5]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 40,
       "text": [
        "array([[-1.3014,  1.676 ,  0.3588, -0.6153,  0.8819],\n",
        "       [ 1.8574, -0.6509,  2.7482,  1.1865,  3.6566],\n",
        "       [-0.0969,  0.0488,  0.006 , -0.1493, -0.4889]])"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Example parameters from the orth model\n",
      "B_w[:,:5]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 41,
       "text": [
        "array([[ 0.0032,  1.2188,  2.289 ,  0.2179,  3.45  ],\n",
        "       [ 1.8574, -0.6509,  2.7482,  1.1865,  3.6566],\n",
        "       [-0.0969,  0.0488,  0.006 , -0.1493, -0.4889]])"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# The parameter for the hrf1 regressor in the non-orth model\n",
      "# is correlated with the parameter for the hrf1 regressor\n",
      "# in the orth model.\n",
      "plt.plot(B_b[0], B_w[0], '.')\n",
      "plt.title('Orth and non-orth hrf1 parameters correlate')\n",
      "np.corrcoef(B_b[0], B_w[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 42,
       "text": [
        "array([[ 1.    ,  0.7103],\n",
        "       [ 0.7103,  1.    ]])"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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jvk1NwMmTMm/2bODwYeD4cXf9SCS1Nqi1nXs0zlxRShS/DM++PmBoyBXc6dOB+fNFOIPi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\nJ3FxLSV7/IphhcMygLJ5Iqq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       "text": [
        "<matplotlib.figure.Figure at 0x10acf9a50>"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Relationship of estimated parameters for hrf1 and orthogonalized hrf2\n",
      "# (they should be independent)\n",
      "plt.plot(B_w[0], B_w[1], '+')\n",
      "plt.title('hrf1 and orth hrf2 parameters are independent')\n",
      "np.corrcoef(B_w[0], B_w[1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 43,
       "text": [
        "array([[ 1.    ,  0.0013],\n",
        "       [ 0.0013,  1.    ]])"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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LAXv2RF8z2/FRZctUbv/YoaGg0GdyoyxW5AsV5p4eDd1UMqMirywZo6PFHedb5l1dFNtE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\n58QEGwtjuB1weXKMoVht3MgyNTa6+HBxCUVRX+/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       "text": [
        "<matplotlib.figure.Figure at 0x10884aed0>"
       ]
      }
     ],
     "prompt_number": 43
    }
   ],
   "metadata": {}
  }
 ]
}