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  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Example of running PCA on a 4D image"
     ]
    },
    {
     "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": [
      "# - show figures inside the notebook\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - import common modules\n",
      "import numpy as np  # the Python array package\n",
      "import matplotlib.pyplot as plt  # the Python plotting package"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - import numpy.linalg with a shorter name\n",
      "import numpy.linalg as npl"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - set gray colormap and nearest neighbor interpolation by default\n",
      "plt.rcParams['image.cmap'] = 'gray'\n",
      "plt.rcParams['image.interpolation'] = 'nearest'"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Load the image 'ds114_sub009_t2r1.nii' with nibabel"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Get the data array from the image\n",
      "import nibabel as nib\n",
      "img = nib.load('ds107_sub012_t1r2.nii')\n",
      "data = img.get_data()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Reshape to volumes by voxels"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "vol_shape = data.shape[:-1]\n",
      "n_vols = data.shape[-1]\n",
      "n_voxels = np.prod(vol_shape)\n",
      "X = data.reshape(n_voxels, n_vols).T"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Subtract the mean across columns (row means):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X = X - X.mean(axis=1).reshape(n_vols, 1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Calculate the unscaled covariance (covariance not divided by N-1):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "unscaled_cov = X.dot(X.T)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Get the principal components (array `V`) using SVD:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "U, S, V = npl.svd(unscaled_cov)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Plot the first principal component:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(V[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 11,
       "text": [
        "[<matplotlib.lines.Line2D at 0x11952af90>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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ARON6GSLRnubataFVF9u0CS7u559vq0Iee6x5HEFwS26BcE/D+3CSrbzGW2/Z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\nQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1078c8a90>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Get the projection coefficients (lengths) for each principal component:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c_values = V.dot(X)\n",
      "c_values.shape"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "(166, 143360)"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Reshape the (number of volumes) by (voxels) array to x by y by z by number of volumes:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c_images = c_values.T.reshape(data.shape)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Show the projection coefficients for the first principal component:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.imshow(c_images[:, :, 15, 0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "<matplotlib.image.AxesImage at 0x124c24710>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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NyNpV9T7rTr0/rls0ZhG5R3VcgvWw1pndiaq/uSwaC9Dw3aoLmUTh19rW7ArT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\njOz0SdIwstMnScPITp8kDSM7fZI0jP8PvkOYkJlUkYEAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x11955d790>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Show second principal component:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(V[1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 15,
       "text": [
        "[<matplotlib.lines.Line2D at 0x124e5a150>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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XAw/Mrrt4ccwuOmgQ3HADbLllcTXPmwf77APz58MWW2Tf/+ADuOkmeOqpWLZm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\nAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x124be8190>"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Show projection coefficients for second principal component:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.imshow(c_images[:, :, 15, 1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "<matplotlib.image.AxesImage at 0x124f79290>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x124e65bd0>"
       ]
      }
     ],
     "prompt_number": 16
    }
   ],
   "metadata": {}
  }
 ]
}