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  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# In which we explore the problems of 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": "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": 5,
       "text": [
        "array([ 0,  2,  4,  6,  8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28])"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we make a function returning an HRF shape for an input vector of times:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Gamma distribution from scipy\n",
      "from scipy.stats import gamma\n",
      "\n",
      "def spm_hrf(times):\n",
      "    \"\"\" Return values for SPM-like HRF at given times \"\"\"\n",
      "    # Make output vector\n",
      "    values = np.zeros(len(times))\n",
      "    # Only evaluate gamma above 0 (undefined at <= 0)\n",
      "    valid_times = times[times > 0]\n",
      "    # Gamma pdf for the peak\n",
      "    peak_values = gamma.pdf(valid_times, 6)\n",
      "    # Gamma pdf for the undershoot\n",
      "    undershoot_values = gamma.pdf(valid_times, 12)\n",
      "    # Combine them, put back into values vector\n",
      "    values[times > 0] = peak_values - 0.35 * undershoot_values\n",
      "    # Scale area under curve to 1\n",
      "    return values / np.sum(values)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Sample the HRF at the given times (to simulate an event starting at time 0), and at times - 2 (simulating an event starting at time 2):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hrf1 = spm_hrf(times)\n",
      "hrf2 = spm_hrf(times - 2) # An HRF with 2 seconds (one TR) delay\n",
      "hrf1 = (hrf1 - hrf1.mean()) # Rescale and mean center\n",
      "hrf2 = (hrf2 - hrf2.mean())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(times, hrf1)\n",
      "plt.plot(times, hrf2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "[<matplotlib.lines.Line2D at 0x106369b50>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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hzx4oKWn8tn//sYF95Ai0bw+ZmZCR4d3q3s/IgIxMJSXzADWZJVRnlFCZVsLh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\n6BtjTIj8f1QKpC90kheyAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x106369910>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The Pearson correlation coefficient between the HRFs for the two events:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.corrcoef(hrf1, hrf2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "array([[ 1.    ,  0.7023],\n",
        "       [ 0.7023,  1.    ]])"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Make a signal that comes from the combination of the two HRFs:"
     ]
    },
    {
     "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": 10,
       "text": [
        "<matplotlib.legend.Legend at 0x1080adbd0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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lFm1ZxENXP1Si17/6Kkyfbhb/dDQv5cWcm+cwcc1ELqZfvNyvX9aH1wohCiRB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\nL4QQ5YgEfSGEKEck6AshRDny/w85XEHgsuG1AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107fc1590>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We're going to make our simulated data from taking the signal (the two HRFs) and adding 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)\n",
      "plt.plot(times, Y, '+')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 11,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108231dd0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x108231f90>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We are going to model this simulated signal in several different ways.  First, we make a model that only has the first HRF as a regressor (plus a column of ones to model the mean of the data):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X_one = np.vstack((hrf1, np.ones_like(hrf1))).T\n",
      "plt.imshow(X_one, interpolation='nearest', cmap='gray')\n",
      "plt.title('Model with first HRF regressor only')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "<matplotlib.text.Text at 0x108534b90>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10847f050>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Next we make a model where we also include the second HRF as a regressor:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X_both = np.vstack((hrf1, hrf2, np.ones_like(hrf1))).T\n",
      "plt.imshow(X_both, interpolation='nearest', cmap='gray')\n",
      "plt.title('Model with both HRF regressors')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "<matplotlib.text.Text at 0x1085c9e50>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x1085a5f90>"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we make a very large number of data vectors, each with the signal (both HRFs) plus a different vector of noise."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "T = len(times)\n",
      "iters = 100000\n",
      "# Make 100000 Y vectors (new noise for each colum)\n",
      "noise_vectors = np.random.normal(size=(T, iters))\n",
      "# add signal to make data vectors\n",
      "Ys = noise_vectors + signal[:, np.newaxis]\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."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Fit X_one to signals + noise\n",
      "B_ones = npl.pinv(X_one).dot(Ys)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Next fit the model with both HRFs as regressors:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Fit X_both to signals + noise\n",
      "B_boths = npl.pinv(X_both).dot(Ys)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "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",
      "which works out to:\n",
      "\n",
      "$$\n",
      "t = \\frac{c^T \\hat\\beta}{\\sqrt{\\hat{\\sigma}^2 c^T (X^T X)^+ c}}\n",
      "$$\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 $X^T X$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "That's the theory.  So, what is the distribution of the estimates we get for the first beta, in the single-HRF model?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(B_ones[0], bins=100)\n",
      "np.std(B_ones[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "1.4859976445167784"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x108694750>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The standard deviation of the estimates is what we observe.  Does this match what we would predict from the t-statistic formula above?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "C_one = np.array([1, 0])[:, None]  # column vector\n",
      "np.sqrt(C_one.T.dot(npl.pinv(X_one.T.dot(X_one)).dot(C_one)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": [
        "array([[ 1.485]])"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Notice that the mean of the estimates, is somewhere above one, even though we only added 1 times the first HRF as the signal:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.mean(B_ones[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": [
        "1.7003184795688511"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is because the single first regresssor has to fit *both* the first HRF in the signal, and as much as possible of the second HRF in the signal, because there is nothing else in the model to fit the second HRF shape."
     ]
    },
    {
     "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_boths[0], bins=100)\n",
      "np.mean(B_boths[0]), np.std(B_boths[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 20,
       "text": [
        "(1.0040164097531259, 2.0898972889301848)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10a595e90>"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Two things have happened now we added the second (correlated) hrf2 regressor.  First, the mean of the parameter for the hrf1 regressor has dropped to 1, because hrf1 is no longer having to model the signal from the second HRF.   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": [
      "C_both = np.array([1, 0, 0])[:, None]  # column vector\n",
      "np.sqrt(C_both.T.dot(npl.pinv(X_both.T.dot(X_both)).dot(C_both)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 21,
       "text": [
        "array([[ 2.0861]])"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The estimate of the parameter for hrf2 has a mean of around 1, like the parameter estimates for hrf1.  This is what we expect because we have 1 x hrf1 and 1 x hrf2 in the signal.  Not surprisingly, the hrf2 parameter estimate has a similar variability to that for the hrf1 parameter estimate:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(B_boths[1], bins=100)\n",
      "np.mean(B_boths[1]), np.std(B_boths[1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 22,
       "text": [
        "(0.99158549385148709, 2.0895047510765408)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10847f5d0>"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "C_both_1 = np.array([0, 1, 0])[:, None]  # column vector\n",
      "np.sqrt(C_both_1.T.dot(npl.pinv(X_both.T.dot(X_both)).dot(C_both_1)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 23,
       "text": [
        "array([[ 2.0865]])"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The parameter estimates for hrf1 and hrf2 are anti-correlated:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Relationship of estimated parameter of hrf1 and hrf2\n",
      "plt.plot(B_boths[0], B_boths[1], '.')\n",
      "np.corrcoef(B_boths[0], B_boths[1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 24,
       "text": [
        "array([[ 1.    , -0.7032],\n",
        "       [-0.7032,  1.    ]])"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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+YrwPIdlgwiYNfhKPqYmTqRG13z5uT9qeHuCNN2Sf2lobK24mMAtF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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10a697e90>"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Orthogonalizing hrf2 with respect to hrf1"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "hrf2 is correlated with hrf1.  That means that we can split up hrf2 into two vectors, one being a multiple of hrf1, and the other being the remaining unique contribution of hrf2.  The sum of the two vectors is the original hrf2 regressor.  Like this:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Regress hrf2 against hrf1 to get best fit of hrf2 using just hrf1\n",
      "y = hrf2\n",
      "X = hrf1[:, np.newaxis]  # hrf1 as column vector\n",
      "B_hrf1_in_hrf2 = npl.pinv(X).dot(y)  # scalar multiple of hrf1 to best fit hrf2\n",
      "hrf1_in_hrf2 = X.dot(B_hrf1_in_hrf2)  # portion of hrf2 that can be explained by hrf1\n",
      "unique_hrf2 = hrf2 - hrf1_in_hrf2  # portion of hrf2 that cannot be explained by hrf1\n",
      "plt.plot(times, hrf1, label='hrf1')\n",
      "plt.plot(times, hrf2, label='hrf2')\n",
      "plt.plot(times, hrf1_in_hrf2, label='hrf1 in hrf2')\n",
      "plt.plot(times, unique_hrf2, label='hrf2 orth wrt hrf1')\n",
      "plt.legend()\n",
      "# hrf1 part of hrf2, plus unique part, equals original hrf2\n",
      "np.allclose(hrf2, hrf1_in_hrf2 + unique_hrf2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 25,
       "text": [
        "True"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Ks0zpPYXvDn7Hk926kVlSwsenTunfiUKhcDnOin4okGn1Pqv8s9q4HVjlZJ+6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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10a697a10>"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "How much of the first regressor did we find in the second regressor?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B_hrf1_in_hrf2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 26,
       "text": [
        "array([ 0.7022])"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "When we have the model with both hrf1 and hrf2, we are effectively multiplying both parts of hrf2 by the same beta parameter, to fit the data.  That is, we are applying the same scaling to the part of hrf2 that is the same shape as hrf1 and the part of hrf2 that cannot be formed from the hrf1 shape."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, what happens if we replace hrf2, by just the part of hrf2, that cannot be explained by hrf1?  Our second regressor is now hrf2 *orthogonalized with respect to* hrf1:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X_both_o = np.vstack((hrf1, unique_hrf2, np.ones_like(hrf1))).T\n",
      "plt.imshow(X_both_o, interpolation='nearest', cmap='gray')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 27,
       "text": [
        "<matplotlib.image.AxesImage at 0x10a5d7790>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFEAAAD7CAYAAAASCHi+AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nF7tHAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10a6e3fd0>"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(times, X_both_o[:,0], times, X_both_o[:,1]) "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 28,
       "text": [
        "[<matplotlib.lines.Line2D at 0x1086b8e10>,\n",
        " <matplotlib.lines.Line2D at 0x1086b8950>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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PH3l0/qOk/DnF6VLK7IcfYNQo+OWX4J/j6ETWCfp83IeW1Vvybr93Nfj97ETW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\nYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10a5b4950>"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "What will happen when we fit this model?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B_boths_o = npl.pinv(X_both_o).dot(Ys)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Distribution of parameter for hrf1 in orth model\n",
      "plt.hist(B_boths_o[0], bins=100)\n",
      "np.mean(B_boths_o[0]), np.std(B_boths_o[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 30,
       "text": [
        "(1.7003184795688511, 1.4859976445167784)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10a4ebe50>"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Predicted variance of hrf1 parameter is the same as for the\n",
      "# model with hrf1 on its own\n",
      "np.sqrt(C_both.T.dot(npl.pinv(X_both_o.T.dot(X_both_o)).dot(C_both)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 31,
       "text": [
        "array([[ 1.485]])"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The parameter for the hrf1 regressor has now returned to the same value and variance as it had when hrf1 was the only regressor in the model (apart from the mean).  For the orthogonalized model, we removed the part of hrf2 that could be explained by hrf1.  Now, the amount of hrf1, that we could find in hrf2, has been added back to the parameter for hrf1, in order to make the fitted $\\hat{y}$ values the same as for the model with both HRFs."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.mean(B_boths[0, :]) + B_hrf1_in_hrf2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 32,
       "text": [
        "array([ 1.7062])"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The hrf1 parameter in the orthogonalized model is the same as for the model that only includes hrf1 - as if the orthogonalized hrf2 was not present.  The parameter for orthogonalized hrf2 is the same as the parameter for hrf2 in the not-orthogonalized model.  We still need the same amount of the *orthogonal part* of the second regressor to explain the signal:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Example parameters from the single model\n",
      "B_ones[:,:5]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 33,
       "text": [
        "array([[ 1.8449,  2.6261, -0.1168,  3.7753,  3.2674],\n",
        "       [-0.1947,  0.0405,  0.0153, -0.3264,  0.295 ]])"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Example parameters from the non-orth model\n",
      "B_boths[:,:5]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 34,
       "text": [
        "array([[ 1.2982, -0.2706, -2.4005,  1.4996,  2.7255],\n",
        "       [ 0.7786,  4.1252,  3.2521,  3.2408,  0.7717],\n",
        "       [-0.1947,  0.0405,  0.0153, -0.3264,  0.295 ]])"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Example parameters from the orth model\n",
      "B_boths_o[:,:5]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 35,
       "text": [
        "array([[ 1.8449,  2.6261, -0.1168,  3.7753,  3.2674],\n",
        "       [ 0.7786,  4.1252,  3.2521,  3.2408,  0.7717],\n",
        "       [-0.1947,  0.0405,  0.0153, -0.3264,  0.295 ]])"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# The parameter for the hrf1 regressor in the orth model\n",
      "# is the same as the parameter for the hrf1 regressor in the\n",
      "# single regressor model\n",
      "plt.plot(B_ones[0], B_boths_o[0], '.')\n",
      "np.allclose(B_ones[0], B_boths_o[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 36,
       "text": [
        "True"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10a4d9550>"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# The parameter for the orthogonalized hrf2 regressor is the same as the\n",
      "# parameter for the non-orthogonalize hrf2 regressor in the \n",
      "# non-orthogonalized model\n",
      "plt.plot(B_boths[1], B_boths_o[1], '.')\n",
      "np.allclose(B_boths[1], B_boths_o[1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 37,
       "text": [
        "True"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10a4d9850>"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "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_boths[0], B_boths_o[0], '.')\n",
      "np.corrcoef(B_boths[0], B_boths_o[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 38,
       "text": [
        "array([[ 1.    ,  0.7121],\n",
        "       [ 0.7121,  1.    ]])"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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JB4r5PHL//V5a9qFDGje8eLGusz/YAwfY8i2KzM7qXdj0NLBwYfg+Bw5k17eT\nkGygmM8j8Thw9dX62Fa3279fMzp37gQuvtjflJdUL64bBdDOQL/7nZZkeOIJL49gaEjv0gBWPCTF\npeBoFhG5HsBdAOoBfMsY87WCRxUhJib88cFupqetR02qn09/2p/BuWuXN2dy7Jj26hwb04Shlhb1\nqT/wQPrm3dluIwQo0DIXkXoA/wTgegArAPyZiCwvxsCiQjyuy/Llert98qTGELNjUHRobtZopHvu\nAR5+WOPDbSlbAFiwAPjkJ9XVtm2bFk47cEDdbi7pSiWn20YIULib5RoArxhjpo0xswD+BcBI4cOq\nLtI1nwD0h2gbUNjEIYYiRof33lO32ec+561bs0ajVmIxrbHz1lv62duGFGEulnS9PNnnk2SiUDH/\nCIDXnedvzK2rKTJZTfaHaHnoIY14INHCvUC/9ZZOgs7OAn/7t953YGgIGBkJbyph+8Pmuo0QoHCf\neVb25ebNmz98nEgkkEgkCnzbyiKT1TQxAVxyCTsFRYm2NuDECa9UbbAAVth3IlNtFRuamOs2Eg0m\nJycxOTmZ9+sLygAVkZUANhtjrp97/iUAH7iToLWQAZqqCJItpHXkiFpttukEqX5sXZX6ei9r8/77\nvc+fhbFIoZQ6A/RpABeJyDIRaQTwpwC+X+Axq4pLL9X0/N27NWrBxfrKz5yhkFc7wdDDvXt14nP7\ndi/U1HWxMQGIlJqCa7OIyDp4oYnfNsb8XWB7pC3zeNwT8d5e4HVnBsH2/iTViYiGG544Eb49FvO3\n+Hv1VYo3KR4lr5pojNkOoOYky8b92qSflhbg8cf9248f1wSRK67QH/6OHbTQq4l43Is+CWvf5gr5\ns89SyEl5YQncPNm2zauIuGAB8NJLwFe+4iV2PPOMN+Fpu8js368uGVL5tLdrbZ2dO1WsBwa8Tj9v\nvqmf7dAQcP75OvGZrrEEIaWAYp4nbkXEz3xGS9u6Au9iOwh95SulGRspnOZmndC0k5iAFsxqbVVh\nb2vTJCEr2jY8FdDXMPKElBqKeZ5Yq82GpI2P+9PzGxq8AlovvKBJRc89V46RknyYnQVuvtnf4u/I\nEeC113T72Jjf+k4Vimibk/T1acVMWu1kvmBziiwIq4thQ8+am9V98rOfeRZ4Z6eK/KOP+kWdVAe2\ncQjgifZ993kT3WGTnWGhiG6rQAvbwZFsYXOKeSAsw9P+wPfv121uM+aTJ4FvfENL3VLIq4fVq1W8\nP/EJfW7Eh66+AAAS8klEQVSt7KkpT8hjMW+y0y3jACSHIlpr3Q1rrHK7hlQwFPMsyKZmRoPjsJqd\nBW64QSdGSeWSSOhdlGXRIhVk2/HHps7bz7irC/jNb9RlAmQu42BT8Fev1ufBLFFCigndLFkQdKmE\nuVt++1stqGSJxbSmdaoYZVJePvYxDSUdG/P33XSjUhYv1s87FtOJz61b/Za3zSMYHk5fM4XZoCQf\ncnWzUMxzwPWB9vdrWJqd3Hr66eSen6QyaGnxil6JaG3xZct0QvLuu7UUrRVa9zNevNib1B4bS45Q\noUiT+aTkSUO1hL3dbmvzRza88UbyvnV1WhmxsdHvTyelQ0TdX1dfrXdIzzyjPuvTp7WmOKBC7oq0\n61KLx9VqT1VAjcWvSCVByzwHjh7VVm9hHYJs1MrAgNbqWLkSeOwxXRdW45yUlqYmFXE30ijMPeJa\n2wAtb1I+6GaZZ6yfdHAQePFFL6W7s1Mt8IEB7SjkZoCSymJkRO+YKNKkkmFoYoBMXYByxUYo7N4N\nXHedrmtt1dC1997TSdDt29nfs5IQ5+fQ2Kh3TpyYJlEj8mJe7N6Jtqfn6Kg+HxkBPv5x/z7t7bqQ\n8uGGii5cqH+7urSeyp49/u9DsS/4hJSDyE+AFrt3oq27YZNI+vs1OkLESwh55x2tlnj8eOHvR3In\nFvO35bv2Wr172rJFU/QB//eBdVVIFIi8ZV6s3onWenvgAU/IGxpUtPfs8Wf2dXYCn/50QcMmWSIB\nj2J9vc5j2HK17e3AN7/pZWeGfR/YLJlEAU6AZkHQGnexFrmtdx2PA7/6FXDLLcl1OUjhuOe7vt4L\nE7WE1cIJixG3jI9r+eJ9+4Ann/SyOwkpN5wAnQfc2hxB7HXq7FntNPTaayoI+/aVbny1hDE6iXn2\nrEYPvfOOty0eB1at8u/f1ZXe2p6a0jurAwc05pyQaoVingVhBZOCDA0Bzz/v3bqzo1DxqKvzRHp4\nGPjkJ73HtihWfb12dPrud4GeHl2XTQcgulhIVKCbJQtsIsnMjGYE2lv5oSHg3HPVUrSNCmy53D17\nWDGxGMTjwI036p2OdYV0dmqjiHff1USgd9/1zvXYmIpyWLJPulLGjDknlUZJk4ZE5E4AnwVwBsA+\nAJ8zxhwL7FM1Yh72Y3exP/w77/TX83Bft3s3cOpUecYfNaxl7c4/dHdrev7x415KviVTwSu37ko6\nPzohlUCpa7M8AuAvjTEfiMhXAXwJwBcLPGbZyBSiZmtxjI9rdufll6vf9sgRWuHF5JJLgOXLvbsd\n6wqpq9OEn+3bPVcKAFx2GXDhhbr/ypXq/47FtPiZO6FJlwqJMgWJuTFmh/P0KQD/vrDhlJdsf+yu\n6LuEdXAPEovRn56J5cu9bjzj42qFu+etsVFF+oMP9LO6917vDumVV7zPYPVq4PXXveNOTNClQqJL\nMZOG/hOA7xXxeCXFikZPj8aSZzNp1tHhTww6ezZzm7iODuDtt4sz5mrmhz9USzpYB35oSNcD4SGh\nXV1a7My+prXV+6y2bfOEvK5O65W7sMohiTIZxVxEdgDoCdn0V8aYbXP7/DWAM8aYibBjbN68+cPH\niUQCiUQin7HOK3bSEkguixrEWnh33qkZoHZKIBbTQlu//GV4e7CGBmaFWm64Qe9kbNmDzk6NWLFW\nNuAPCe3sBD71KRX6sCxOwF9P/g//kDHjpLqYnJzE5ORk3q8vOJpFRG4FsBHAHxpjkqb+qmUCNNuu\nMYB/wnPnztRuE7pUsqO3VzNmbRcn2+HnxRfVR24nQq04p4pAWbtWP4+hIWDXLrpSSHVT6miW6wF8\nHcB1xpjDKfapCjFPJRBhES5uVIRbk8XS3g78wR+oO8V1IbguGBFd3BoiUSXsHFna2oAXXkiOWDk8\n923q7fXH76fj6FENWVy6VN1ZYRFJhFQLpc4A/UcAbQB2iMizIvLNAo9XNqw/NfjjD6u66E6UdnQk\nH+udd9Qqt9X6AGDFCi/xpa5Oxa0ahbypSStF/vSnyXVRUmGFfMECr2yw5cQJ4Atf8DJmOzq0Pyeg\n5zdbIQd0v/PPT66KSEgtUJCYG2MuMsb0GWOG5pbPF2tglYIVmbo6DUc8etRfrOnqq3W7iL/s7cMP\nAydPqvtmdFSF6f33tSl0ukzSSqelBfiHf9BIkeuvz7y/LUXb1qav+e539dy5Am2M50I5fly35Vsc\njeGHpGYxxszrom9R2WzcaMx11xmzbp0xMzP+batWGaNyo0t/v3/fmRljxsb075o1/n0BYxYt0n2D\nx6nmpbdXz83MjDGxWOr9YjFj9u41ZvFib93YmL7WnqvBQT3OunX6fHjY/xmk+2zCcD8PQqqZOe3M\nXmtz2TmfpVLF3BUJV2it2FisyFhh7uhI3tcea80aY5YsCRc2uz6d+FXqUl+f/HzRImNuusmYzk7/\nNhH929VlzPS0/xy6Qh0U3VQifN11qT8bQqJMrmJec7VZ7ITmc89prRVAY8sPHEjd4PfWW9WN8rOf\nAQcP6vrOTmB6OnlCdHQUeOIJdcnU1Xl+8YULvS426Xzl2UTA1NfrsVtaUldzzAd3vPkwPKyZmD/5\niRa9uv/+9PVPMpVPAHKLMiIkSuQ6AVpzlrlr6VlrcXo6u1vzri7vdevXe+uDlueGDepasPu3tc2P\nxbxggTF1dYUfp7lZ7xyC5ybbpaFBz8fMTG6WdE+Pt+/oaPg+GzYY092tdz10nZBaAjla5jVXAtdO\nkA0NaVTGjh06+RYWyWKxXYbq6rzX3nuvtz3YveaRR7Sh88yMhsmtXKn7FXvi89Sp4kTEvPee3km8\n+qreGWQiGMXy/vuaKAXkNgHpJvkEb97sOd+2TcMUd+5kdAohaclF+fNZUGGWeT4TZK612dub+bWN\njX4L/qKL1Hrt7FQr+Kqr/PuELW1txqxdm711LOL356da0vnsFy3yW/12LiEe9x7X1+v/0dSk/5P7\nenteszm/GzfqcQFjLrssva886G8npBYALfPUjI+rT/vEidxe51qb2cQ92/0BzWJ89VW1Xo8d09oh\nzzyj1RathRuPa6IM4K07cUJjubPFmOxKBbS06BxB8C7h0kv1jgPQ//Ott4Af/EDvOF57TR93d+v4\njx1Tq9qtQWMt8VTx+kGmprw5hAsvTN7fnsPBQf3M6C8nJD01JeZhCUDZMDGhNViamrQuiBUhwHMH\nrF/vrR8e1r9DQ5rBaIs/1df7E4mM0QqALS0q4rGY58oB5qcu+rFjGvvd2elff8klOmFp3UWbNvkv\nfPG4F1MPeDH1+Yqte4G8557k82hdV7t3awVFCjkhGcjFjM9nQQW5WVLFMmdDcGLPhiO6k6I9Pd4E\nqJ20s/HUNubaDYO0YXyFLMGwwaBLJTj52tKiE77T0/5j7N2b+v+1sfVr1qjbaGQkedK40HhwhiAS\n4gc5ullqSswLSShxLwQbNqT2PY+N+YVpZMT/nvY4XV3hPu7LLlPBDPqj3WXBAvW72+dLl6p/3fXD\nx2IquE1N3oVjwQIvymbpUv/FxCYChf2/6eLwLYWKcSEXWkKiCMV8npiZUQt11Sq/NQ54gmmFKJ0w\nhWWMDgx4Fq/d325zJx8BYxYu1Of2PTs7k5Nz3ISda69NHmeYdW/3DxunPW53t773unV6QXMt8ULF\nmJmbhPihmM8jYXHYAwPJLgdX+MPcDhs3qksmHtf47ulpv5tiwwZ9bU+PbpuZ0TjskZHwsgDWEg4T\nRFdkbfQI4Fn+1v2TDvv/uHcS3d3+96cYE1JcKOZFIswHbIVxaEgt6dHRcPHauNHvhrEJMWF+9jDX\njPta65/v6dHX2fBBm0ZvLeHgeC+5RPdZtMhL6HGt/b1704tv8Hju+Lq6vGPRLULI/EAxLxJhPuCg\n9Zlq0i9owY+MhK8H1NqdnvbXgLGLFcrg6+rq1GduLfew8bo1U9zCWNlaz2525shIsguHljgh8wvF\nvEik8wGHWdjupJ8rzDYhZuNGb/+BAf9rrUvGTmAODfn952FCH3zf4HitG8RGr+SKOz57B0LxJqR0\n5CrmNVdoK1tSFYcC/IW1AI2VXrHCa3t2993acMEYYOvW5GJcIyOaNGQLSDU1ef1Hgy3UJua6qt56\nqxb6OnTIe9+uLk1ICitmtX+/1g9//PH8emGyBRsh5aXUnYYiS7pMxrD6Lvv3ewlJt9+ufSyPHvWS\njNwkma1b/fVcbLcim2HqHuvii4HLL9f6JAMDwJIluq/ti2nHFxzvV74CXHABcNtt/iSnbLEJRBRy\nQqqEXMz4fBZUiZsll6SXTFEjGzb4fdaZoj2C2+yxwqotBuPWU/0vwfcnhFQXKLWbRUT+B4A7AXQb\nY46EbDeFvkcpcN0gY2Nq5eaC6+YYHfWOZV0hmzZlrt0dPNbMjLo6Ojq07opb0ztdLXD3f3FdMYSQ\n6iFXN0tDgW92HoC1APYXcpxKoNDekdbN4R7LdYXYujCACnG6i8WmTeobj8X0wnDXXeq6aW7W5y0t\nKu7Wzx48Xtj7E0IiTi5mfHABcD+AAQCvAViYYp8Pbxtyrd9RSrKJ1giOP5joYx+HNbvIJUMyVWq8\nu96GDmbKMiWEVCcoVWgigBEAfz/3OCsxr/ZiSsHxu8/Dmha75CKwqYTfXZ9tdyRCSHWSq5injWYR\nkR0i8nzIciOALwG4w909011Aoa6MUmHLsZ53nob32bKswfG7z6+4wr8tSDDaJKx0riXYuShsfabu\nSISQ2iKvCVARuQzATwCcnFvVC+B3AK4xxhwK7GvuuEM1/9Qp4MknE3jooURFi1AwjhxQEd2yxR/L\n7U56Aqnj0jO9R6oJ12waHhNCosHk5CQmJyc/fP7lL385pwnQoiQNichrAK4yVRzN4mI7wnd2ajOH\n+egMn03X+UIjbAgh1Uu5koaqS60DpOpys3dvuLujGKRypbhUi1uKEFJ+mM6PyrGAg24Vuy5b1w0h\nJDqUNM48KlSKBRwWi07XCiEkG1ibBdm5PEpBpVxUCCHVB90sFUS6So2EkNoiVzcLxZwQQioQlsAl\nhJAahGJOCCERgGJOCCERgGJOCCERgGJOCCERgGJewaSrrEgIIS4U8wrGZoRu367CTgghqaCYVzDM\nCCWEZAuThioYZoQSUrswA5QQQiIAM0AJIaQGoZgTQkgEoJgTQkgEoJgTQkgEoJgTQkgEKEjMReTP\nReRlEXlBRL5WrEERQgjJjbzFXEQ+BeBGAAPGmMsA/K+ijSqiTE5OlnsIFQPPhQfPhQfPRf4UYpnf\nBuDvjDGzAGCM+X1xhhRd+EX14Lnw4Lnw4LnIn0LE/CIAnxSRJ0VkUkSGizUoQgghudGQbqOI7ADQ\nE7Lpr+de22WMWSkiVwO4D8BHiz9EQgghmcg7nV9EtgP4qjHm0bnnrwC41hjzdmA/5vITQkge5JLO\nn9Yyz8BDAD4N4FERuRhAY1DIcx0MIYSQ/ChEzL8D4Dsi8jyAMwA2FGdIhBBCcmXeqyYSQgiZf+Yt\nA1RExkTkRRE5KyJXBrZ9SUR+IyL/JiJ/NF9jqEREZLOIvCEiz84t15d7TKVGRK6f++x/IyJ/We7x\nlBMRmRaR5+a+Cz8v93hKiYh8R0QOzt3d23ULRWSHiEyJyCMiUhOV/FOci5y0Yj7T+Z8H8McAHnNX\nisgKAH8KYAWA6wF8U0RqqayAAfC/jTFDc8uPyj2gUiIi9QD+CfrZrwDwZyKyvLyjKisGQGLuu3BN\nuQdTYu6Bfg9cvghghzHmYgA/mXteC4Sdi5y0Yt5E1Bjzb8aYqZBNIwC+Z4yZNcZMA3gFQK19iWt5\nUvgaAK8YY6bnEs7+BfqdqGVq8vtgjPkpgJnA6hsBfHfu8XcBjJZ0UGUixbkAcvhulMMiXgrgDef5\nGwA+UoZxlJM/F5G9IvLtWrmNdPgIgNed57X4+bsYADtF5GkR2VjuwVQA5xhjDs49PgjgnHIOpgLI\nWisKLbS1Q0SeD1n+XY6HitQsbJrzciOAuwH0AxgE8BaAr5d1sKUnUp91EVhljBkCsA7AfxWRPyj3\ngCqFuX6Ttfx9yUkrCglNhDFmbR4v+x2A85znvXPrIkO250VEvgVg2zwPp9IIfv7nwX+nVlMYY96a\n+/t7EXkQ6ob6aXlHVVYOikiPMeaAiJwL4FC5B1QujDEf/u/ZaEWp3Cyu3+f7AG4SkUYR6YfWeKmZ\nWfy5L6jlj6ETxbXE0wAuEpFlItIInQz/fpnHVBZEpEVE2ucetwL4I9Te9yHI9wHcMvf4FmhyYk2S\nq1YUZJlnGMgfA/gGgG4AD4vIs8aYdcaYl0TkPgAvAXgfwOdNbQW7f01EBqG3j68B+C9lHk9JMca8\nLyL/DcCPAdQD+LYx5uUyD6tcnAPgQREB9Ld4rzHmkfIOqXSIyPcAXAegW0ReB/A3AL4K4D4R+c8A\npgH8h/KNsHSEnIs7ACRy0QomDRFCSASopfhuQgiJLBRzQgiJABRzQgiJABRzQgiJABRzQgiJABRz\nQgiJABRzQgiJABRzQgiJAP8fv1xhcT8ZUmsAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10a498d50>"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Relationship of estimated parameters for hrf1 and orthogonalized hrf2\n",
      "# (they should be independent)\n",
      "plt.plot(B_boths_o[0], B_boths_o[1], '+')\n",
      "np.corrcoef(B_boths_o[0], B_boths_o[1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 39,
       "text": [
        "array([[ 1.    , -0.0015],\n",
        "       [-0.0015,  1.    ]])"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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MrT78vDbq015dKMlXiIUOSJFGxCAq1yr3HiLxyOQ4dAg4fpwS1vg4A6HGxjgZ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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10ae674d0>"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Variance inflation factor"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The coefficient of determination, $R^2$, is computed with: \n",
      "$$\n",
      "R^2 = \\frac{SSReg}{SST - C}\n",
      "$$\n",
      "\n",
      "With $C = n \\bar{Y}^2$, and $\\bar{Y}$ the mean of the $Y$s."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n, p = X_both.shape[0], X_both.shape[1]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# R2 And variance inflation factor\n",
      "\n",
      "# function to compute the R2 from Y and X\n",
      "from scipy import linalg as slg\n",
      "\n",
      "def projR(to_proj, X_):\n",
      "    \"\"\" \n",
      "    projects the 2d array `to_proj` onto the residual space of `_X`\n",
      "    \n",
      "    The svd of _X is computed to have a orthonormal basis of the\n",
      "    space of the columns of _X\n",
      "    \"\"\"\n",
      "    u_, _, _ = slg.svd(X_, full_matrices=False)\n",
      "    return (to_proj - u_.dot(u_.T.dot(to_proj)))\n",
      "\n",
      "# try this function with the mean:\n",
      "ones_n1 = np.ones((n,1))\n",
      "y = np.random.normal(3,1,size=(n,1))\n",
      "print(\"this should be close to 3: \", y.mean())\n",
      "print(\"this should be close to 0: \", projR(y, ones_n1).mean())\n",
      "\n",
      "\n",
      "def R2(Y, X):\n",
      "    \"\"\"\n",
      "    Compute the coefficient of determination R2 \n",
      "    Y: numpy array shape (n,1) : the data\n",
      "    X: numpy array shape (n,p) : the design matrix\n",
      "    \"\"\"\n",
      "    (n,p) = X.shape\n",
      "    SST = (Y**2).sum()               # total sum of square\n",
      "    C = (Y.sum())**2 / n             # sum of square of the mean\n",
      "    ones_n1 = np.ones((n,1))\n",
      "    Xdemeaned = projR(X, ones_n1)\n",
      "    # compute the svd to get an orthonormal basis \n",
      "    u_, _, _ = slg.svd(Xdemeaned, full_matrices=False)\n",
      "    \n",
      "    # compute SSReg noticing that Y^t u u^t Y = \\sum (u^t Y)^2\n",
      "    SSReg = ((u_.T.dot(Y))**2).sum()\n",
      "    return SSReg / (SST - C)\n",
      "\n",
      "def SSR(Y, X):\n",
      "    \"compute the sum of square of the regression on X\"\n",
      "    (n_, p_) = X.shape\n",
      "    if n_ < p_:\n",
      "        raise ValueError(\"n must be >= to p\")\n",
      "    u_, _, _ = slg.svd(X, full_matrices=False)\n",
      "    return (((u_.T * Y.T).sum(axis=1))**2).sum()\n",
      "\n",
      "def Proj(X):\n",
      "    \"\"\" This function returns the projector on the \n",
      "        space of the columns of `X`\n",
      "    \"\"\"\n",
      "    (n_, p_) = X.shape\n",
      "    if n_ < p_:\n",
      "        raise ValueError(\"n must be >= to p\")\n",
      "    u_, _, _ = slg.svd(X, full_matrices=False)\n",
      "    return u_.dot(u_.T)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "this should be close to 3:  2.64427014763\n",
        "this should be close to 0:  -3.5527136788e-16\n"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(R2(X_both[:,[0]], X_both[:,1:]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.493287984608\n"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "VIF = 1./(1. - .55)\n",
      "print(VIF)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "2.22222222222\n"
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(R2(X_both_o[:,[0]], X_both_o[:,1:]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.000150782815012\n"
       ]
      }
     ],
     "prompt_number": 44
    }
   ],
   "metadata": {}
  }
 ]
}