{
 "metadata": {
  "name": "interpolation-padding"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Interpolation, padding - does the padding x value matter?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.kernel.zmq.pylab.backend_inline].\n",
        "For more information, type 'help(pylab)'.\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import matplotlib.pylab as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import scipy.interpolate as spi"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.arange(5)\n",
      "# Some random numbers for the ys\n",
      "y = np.array([-0.16001414,  0.88690672,  0.32057716,  0.09396015,  0.54394511])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(x, y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10641df90>]"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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fN3c1YtEiIDKSoe9I3boB778vbvZevSq7GqLac2RPvxx7/DZSVga0bQssXSp22iLHURQg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\nT0SkM/8PX/rNTdr8pcgAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1063f8990>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First we try a padding x value at -1, so 1 unit away from the first real value of x == 0"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "padded_x = np.r_[-1, x]\n",
      "padded_y = np.r_[y[0], y]\n",
      "interp = spi.interp1d(padded_x, padded_y, 'cubic')\n",
      "interped = interp([-0.1, 0.5])\n",
      "interped"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "array([-0.28503175,  0.4983204 ])"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we try and padding x value of -0.1, much closer to the first real value of 0.  We use the same padding value of y. The interpolated value for 0.5 is different in this case:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "padded_x_close = np.r_[-0.1, x]\n",
      "interp_closer = spi.interp1d(padded_x_close, padded_y, 'cubic')\n",
      "interped_closer = interp_closer([-0.1, 0.5])\n",
      "interped_closer"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "array([-0.16001414,  0.32943714])"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can see why by plotting the full interpolated time courses:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fine_time_short = np.linspace(-0.1, 4)\n",
      "fine_time_long = np.linspace(-1, 4)\n",
      "plt.plot(fine_time_long, interp(fine_time_long), 'r:', label='Distant padding')\n",
      "plt.plot(padded_x, padded_y, 'go', fillstyle='none') # so we can see the +'s below\n",
      "plt.plot(fine_time_short, interp_closer(fine_time_short), 'b:', label='Close padding')\n",
      "plt.plot(padded_x_close, padded_y, 'k+')\n",
      "plt.xlim([-1.2, 4.2])\n",
      "plt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "<matplotlib.legend.Legend at 0x106929dd0>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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5ROuQqoybmxuLFy+mb9++hISEMGTIEIKDg7UOyyqGDRtG\nt27dOH36NP7+/ixbtkzrkKrc3r17Wb58Obt27SI0NJTQ0FCnGDF04cIFevfuTadOnejSpQsDBw6k\nT58+WodldZbqhq30JCYhhBC2y26v3IUQQpgmyV0IIRyQJHchhHBAktyFEMIBSXIXQggHJMldCCEc\n0P8H3i0ZnbPYHIQAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x106408150>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The nearness of the padding value dictates the sharpness of the curve *after* the next point, because the slope of the curve from the previous segment influences the slope of the next segments.  This is in the nature of the spline interpolation we have used - the slope of the curve has to be the same just before and just after each data point."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}