{
 "metadata": {
  "name": "CR\u00dcCIAL P\u0178THON Week 4 - Numpy Broadcasting"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "#CR\u00dcCIAL P\u0178THON Week 4: Numpy Broadcasting"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "from IPython.core.display import Image \nImage(url='http://labrosa.ee.columbia.edu/crucialpython/logo.png', width=600) ",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": "<img src=\"http://labrosa.ee.columbia.edu/crucialpython/logo.png\" width=\"600\"/>",
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 1,
       "text": "<IPython.core.display.Image at 0x104407450>"
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Crucial imports\nimport numpy as np\nimport matplotlib.pyplot as plt",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Let's say we have a (near) periodic signal\nx = np.sin(np.arange(128))\nplt.plot(x)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": "[<matplotlib.lines.Line2D at 0x10483afd0>]"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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NjdBuVVeIXAYgkwulDJQzRvJ+c7nk9OWgT8NJLtSECOiNE8cD8ANAPi/u+fx5tbZhJjBl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\nrkJggg==\n",
       "text": "<matplotlib.figure.Figure at 0x104407590>"
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "To analyze time-varying content, we want to process individual overlapping frames.\n\nWe can use the stride_tricks from last week to get overlapped windows on a linear vector \n\n(from  http://nbviewer.ipython.org/github/craffel/crucialpython/blob/master/week3/stride_tricks.ipynb )"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Build a \"framed\" version of x as successive, overlapped sequences\n# of frame_length points\nfrom numpy.lib import stride_tricks\nframe_length = 16\nhop_length = 4\nnum_frames = 1 + (len(x) - frame_length) / hop_length\nrow_stride = x.itemsize * hop_length\ncol_stride = x.itemsize\nx_framed = stride_tricks.as_strided(x, \n                                    shape=(num_frames, frame_length), \n                                    strides=(row_stride, col_stride))\nplt.imshow(x_framed, interpolation='nearest', cmap='gray')",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": "<matplotlib.image.AxesImage at 0x104907090>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": "<matplotlib.figure.Figure at 0x1044152d0>"
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# If we take the FFT of each row, we can see the short-time fourier transform\nplt.imshow(np.abs(np.fft.rfft(x_framed)), interpolation='nearest')",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": "<matplotlib.image.AxesImage at 0x10493b8d0>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": "<matplotlib.figure.Figure at 0x104418990>"
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Although there's a steady sinusoidal component, we see interference between the \n# window frame and the signal phase.  We need a tapered window applied to each frame.\nwindow = np.hanning(frame_length)\nplt.plot(window)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": "[<matplotlib.lines.Line2D at 0x10496c4d0>]"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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1datp1+vvNP1SAqdOme/0zz4L999vO42IlNYf/mAurF62zP/fZauoF8NxzF2j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\nAABJRU5ErkJggg==\n",
       "text": "<matplotlib.figure.Figure at 0x10494aad0>"
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# But what's the best way to multiply each frame of x_framed by window?\n# Linear algebra way is to multiply by a diagonal matrix\ndiag_window = np.diag(window)\nplt.imshow(diag_window, interpolation='nearest', cmap='gray')",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": "<matplotlib.image.AxesImage at 0x1049fae10>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Co2hoaDCZmZlm9uzZpqKiwupaly9fNnFxcSY/P98UFBSYgoIC8/nnn1td0xhjvv766zE5\nSv/999+bhQsXmnnz5pnXX3/d6lF6Y4ypqqoywWDQ5ObmmrKyMjMwMODr82/cuNG8/PLLJiEhwQQC\nAVNTU2Nu375tVqxYYebOnWvC4bD5/fffra13/PhxM2fOHDNz5syRn5e3337b9/UmTpw48uf73zIy\nMqwfpefUWsAhnGkHOITgAYcQPOAQggccQvCAQwgecAjBAw4heMAh/wWg75amg6UNMAAAAABJRU5E\nrkJggg==\n",
       "text": "<matplotlib.figure.Figure at 0x104976f10>"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Now apply it to each frame using matrix multiplication\nx_framed_windowed = np.dot(x_framed, diag_window)\nplt.imshow(x_framed_windowed, interpolation='nearest', cmap='gray')",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": "<matplotlib.image.AxesImage at 0x104aa0190>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": "<matplotlib.figure.Figure at 0x104a0d910>"
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Matlab way is to construct a matrix of repeating rows of the same size\nwindow_repeated = np.tile(window, (num_frames,1))\nplt.imshow(window_repeated, interpolation='nearest', cmap='gray')\n# then pointwise multiplication applies it to each row\nx_framed_windowed = x_framed * window_repeated",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJgAAAD8CAYAAACLp21tAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nRDEwEsXASBQDI1EMjET9D658GjZYwKZ1AAAAAElFTkSuQmCC\n",
       "text": "<matplotlib.figure.Figure at 0x104aa8690>"
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Numpy broadcasting implicitly (and efficiently) repeats singleton or missing dimensions\n# to make matrices the same size, so tiling is unneeded\nplt.imshow(x_framed*window, interpolation='nearest', cmap='gray')",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": "<matplotlib.image.AxesImage at 0x104b2c6d0>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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RGDBiFAaMGIUBI0ZhwIhRGDBiFHEczKrGPCUlBUuXLkXz5s0BnB/7ue+++2p1YKkSGwDa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\nGDEKA0aMwoARozBgxCgMGDEKA0aM8n+c8mhSWePuNgAAAABJRU5ErkJggg==\n",
       "text": "<matplotlib.figure.Figure at 0x104a12a10>"
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Compare the timings:\n%timeit np.dot(x_framed, np.diag(window))\n%timeit x_framed*np.tile(window, (num_frames,1))\n%timeit x_framed*window\n# The big win is not having to allocate the second num_frames x frame_length array",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "100000 loops, best of 3: 10.7 \u00b5s per loop\n100000 loops, best of 3: 15 \u00b5s per loop"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n100000 loops, best of 3: 4.13 \u00b5s per loop"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n"
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# What about if we had our data in columns?\nx_framed_T = x_framed.T\nplt.imshow(x_framed_T * window, interpolation='nearest', cmap='gray')",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "ename": "ValueError",
       "evalue": "operands could not be broadcast together with shapes (16,29) (16) ",
       "output_type": "pyerr",
       "traceback": [
        "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
        "\u001b[0;32m<ipython-input-12-33858fc02e56>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;31m# What about if we had our data in columns?\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[0mx_framed_T\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx_framed\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mT\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mimshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx_framed_T\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mwindow\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0minterpolation\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'nearest'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcmap\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'gray'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
        "\u001b[0;31mValueError\u001b[0m: operands could not be broadcast together with shapes (16,29) (16) "
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Broadcast works by starting at the last dimensions (the fastest-changing ones in 'C' ordering) \n# and promoting either one if it's one.  \n# So we just have to make our window be frame_length x 1\n# We can do this with slicing and np.newaxis:\nplt.imshow(x_framed_T * window[:,np.newaxis], interpolation='nearest', cmap='gray')\n# Now broadcasting works again",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": "<matplotlib.image.AxesImage at 0x104b0b7d0>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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fFkNERN7x0nQiIk1wYBMRaUL8pqOEZC+Eqqoq5ax0D4ebr9bsi3Tviblz54ryf/vb35Sz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\nsImINMGBTUSkCQ5sIiJN/B9lWB7aedosAgAAAABJRU5ErkJggg==\n",
       "text": "<matplotlib.figure.Figure at 0x104a17910>"
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Broadcasting works across multiple dimensions.  \n# It goes through each dimension from the last, looking for a match \n# or promoting singletons\na = np.random.rand( 3, 4, 5 )\nb = np.random.rand( 3, 5 )\nc = a*b",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "ename": "ValueError",
       "evalue": "operands could not be broadcast together with shapes (3,4,5) (3,5) ",
       "output_type": "pyerr",
       "traceback": [
        "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
        "\u001b[0;32m<ipython-input-14-a039b1ac09b5>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      4\u001b[0m \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[0;34m(\u001b[0m \u001b[0;36m3\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m4\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m5\u001b[0m \u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      5\u001b[0m \u001b[0mb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[0;34m(\u001b[0m \u001b[0;36m3\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m5\u001b[0m \u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 6\u001b[0;31m \u001b[0mc\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
        "\u001b[0;31mValueError\u001b[0m: operands could not be broadcast together with shapes (3,4,5) (3,5) "
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# That didn't work because there was no singleton dimension to promote\n# so we can introduce one, with reshape for instance:\nb2 = np.reshape(b, (3, 1, 5))\nc1 = a*b2\n# or using slicing\nc2 = a * b[:, np.newaxis, :]\nprint np.allclose(c1, c2)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "True\n"
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "For the full description of how broadcasting works, see the SciPy documentation:\nhttp://docs.scipy.org/doc/numpy/user/basics.broadcasting.html"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Remember why we wanted the windowing, to remove artefacts from the STFT?\nplt.subplot(121)\nplt.imshow(np.abs(np.fft.rfft(x_framed)), interpolation='nearest')\nplt.subplot(122)\nplt.imshow(np.abs(np.fft.rfft(x_framed * window)), interpolation='nearest')\n# Windowing drastically reduces framing-related artefacts \n# (at the cost of a little frequency resolution)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": "<matplotlib.image.AxesImage at 0x104c354d0>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": "<matplotlib.figure.Figure at 0x104abdcd0>"
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "",
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}