scipy_sg_filter

scipy_sg.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0.18.1'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import scipy\n",
    "scipy.__version__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numba\n",
    "import numpy as np\n",
    "from scipy import signal\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "np.set_printoptions(precision=3)\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Savitzky-Golay coefficients"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "@numba.jit\n",
    "def savgol_coeffs(window, order):\n",
    "    assert order > 0\n",
    "    \n",
    "    J = np.empty((window, order), dtype=np.float64)\n",
    "    \n",
    "    halfw = window//2\n",
    "    \n",
    "    for i in range(window):\n",
    "        for j in range(order):\n",
    "            J[i,j] = (i-halfw)**j\n",
    "\n",
    "    C = np.dot(np.linalg.inv(np.dot(J.T,J)), J.T)\n",
    "    \n",
    "    return C[0,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# See: http://robert.mellet.pagesperso-orange.fr/rgrs_pol/regrs_06.htm\n",
    "#          window, norm\n",
    "test_cases = [( 5,  35),\n",
    "              ( 7,  21),\n",
    "              ( 9, 231),\n",
    "              (11, 429)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Reference implementation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ -3.  12.  17.  12.  -3.]\n",
      "[-2.  3.  6.  7.  6.  3. -2.]\n",
      "[-21.  14.  39.  54.  59.  54.  39.  14. -21.]\n",
      "[-36.   9.  44.  69.  84.  89.  84.  69.  44.   9. -36.]\n"
     ]
    }
   ],
   "source": [
    "order = 4 # quadratic\n",
    "\n",
    "for window, norm in test_cases:\n",
    "    print(savgol_coeffs(window, order) * norm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Scipy.signal:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ -1.261e-16   8.045e-15   3.500e+01   1.892e-14  -1.821e-15]\n",
      "[  0.455  -2.727   6.818  11.909   6.818  -2.727   0.455]\n",
      "[  8.077 -29.615  16.154  72.692  96.385  72.692  16.154 -29.615   8.077]\n",
      "[  18.  -45.  -10.   60.  120.  143.  120.   60.  -10.  -45.   18.]\n"
     ]
    }
   ],
   "source": [
    "order = 4 # quadratic\n",
    "\n",
    "# scaled\n",
    "for window, norm in test_cases:\n",
    "    print(signal.savgol_coeffs(window, order) * norm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Scipy.signal (order-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ -3.  12.  17.  12.  -3.]\n",
      "[-2.  3.  6.  7.  6.  3. -2.]\n",
      "[-21.  14.  39.  54.  59.  54.  39.  14. -21.]\n",
      "[-36.   9.  44.  69.  84.  89.  84.  69.  44.   9. -36.]\n"
     ]
    }
   ],
   "source": [
    "order = 4 # quadratic\n",
    "\n",
    "for window, norm in test_cases:\n",
    "    print(signal.savgol_coeffs(window, order-1) * norm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### refence implementation, fixed window different orders:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 [ 0.143  0.143  0.143  0.143  0.143  0.143  0.143]\n",
      "2 [ 0.143  0.143  0.143  0.143  0.143  0.143  0.143]\n",
      "3 [-0.095  0.143  0.286  0.333  0.286  0.143 -0.095]\n",
      "4 [-0.095  0.143  0.286  0.333  0.286  0.143 -0.095]\n",
      "5 [ 0.022 -0.13   0.325  0.567  0.325 -0.13   0.022]\n",
      "6 [ 0.022 -0.13   0.325  0.567  0.325 -0.13   0.022]\n"
     ]
    }
   ],
   "source": [
    "window = 7\n",
    "for order in range(1,7):\n",
    "    print(order, savgol_coeffs(window, order))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Scipy.signal, fixed window different orders:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 [ 0.143  0.143  0.143  0.143  0.143  0.143  0.143]\n",
      "2 [-0.095  0.143  0.286  0.333  0.286  0.143 -0.095]\n",
      "3 [-0.095  0.143  0.286  0.333  0.286  0.143 -0.095]\n",
      "4 [ 0.022 -0.13   0.325  0.567  0.325 -0.13   0.022]\n",
      "5 [ 0.022 -0.13   0.325  0.567  0.325 -0.13   0.022]\n",
      "6 [ -2.592e-17   2.718e-16  -9.216e-16   1.000e+00  -4.181e-18  -1.844e-16\n",
      "   2.169e-17]\n"
     ]
    }
   ],
   "source": [
    "window = 7\n",
    "for order in range(1,7):\n",
    "    print(order, signal.savgol_coeffs(window, order))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-3.557, -3.237, -2.434, -2.877, -2.476]),\n",
       " array([-3.557, -3.237, -2.434, -2.877, -2.476]))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_noise = np.linspace(-np.pi, np.pi, 50) + np.random.randn(50)/4\n",
    "y_sg = signal.savgol_filter(y_noise, 7, 6)\n",
    "\n",
    "y_noise[:5], y_sg[:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0xa7351d0>]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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KYZODPcU3ALCqLEt6tohZJSEzdqXUFUqpJqXUEaXUlxNxTCGmWjSmee6wccG0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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xa735048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(y_noise, lw=6, alpha=.5)\n",
    "plt.plot(y_sg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [default]",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}