{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bayesian filtering in the frequency domain\n",
    "---------------------------------------------------\n",
    "\n",
    "This notebook contains an exploration of the characteristic function as a way to represent a probability distribution, yielding an efficient implementation of a Bayes filter.\n",
    "\n",
    "Copyright 2015 Allen Downey\n",
    "\n",
    "MIT License: http://opensource.org/licenses/MIT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from __future__ import print_function, division\n",
    "\n",
    "import thinkstats2\n",
    "import thinkplot\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from scipy import stats\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "sns.set(style=\"white\", palette=\"muted\", color_codes=True)\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose you are tracking a rotating part and want to estimate its angle, in degrees, as a function of time, given noisy measurements of its position and velocity.\n",
    "\n",
    "I'll represent possible positions using a vector of 360 values in degrees."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "n = 360\n",
    "xs = np.arange(n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I'll represent distributions using a Numpy array of probabilities.  The following function takes a Numpy array and normalizes it so the probabilities add to 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def normalize(dist):\n",
    "    dist /= sum(dist)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following function creates a discrete approximation of a Gaussian distribution with the given parameters, evaluated at the given positions, `xs`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def gaussian(xs, mu, sigma):\n",
    "    dist = stats.norm.pdf(xs, loc=180, scale=sigma)\n",
    "    dist = np.roll(dist, mu-180)\n",
    "    normalize(dist)\n",
    "    return dist"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose that initially we believe that the position of the part is 180 degrees, with uncertainty represented by a Gaussian distribution with $\\sigma=4$.\n",
    "\n",
    "Here's what that looks like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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imuqZ0fPXPH+RNoW/FMJQs30q6vmLJBT+knvNZkgrHHx3L9D+l4EqfxGFvxRAe0XPrKme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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f20c04edb50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pos = gaussian(xs, mu=180, sigma=4)\n",
    "plt.plot(xs, pos);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And suppose that we believe the part is rotating at an angular velocity of 15 degrees per time unit, with uncertainty represented by a Gaussian distribution with $\\sigma=3$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207ce5bf90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "move = gaussian(xs, mu=15, sigma=3)\n",
    "plt.plot(xs, move);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###The predict step\n",
    "\n",
    "What should we believe about the position of the part after one time unit has elapsed?\n",
    "\n",
    "A simple way to estimate the answer is to draw samples from the distributions of position and velocity, and add them together.\n",
    "\n",
    "The following function draws a sample from a distribution (again, represented by a Numpy array of probabilities).  I'm using a Pandas series because it provides a function that computes weighted samples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def sample_dist(xs, dist, n=1000):\n",
    "    series = pd.Series(xs)\n",
    "    return series.sample(n=n, weights=dist, replace=True).values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a quick check, the sample from the position distribution has the mean and standard deviation we expect."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(179.989, 3.9825719076998474)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pos_sample = sample_dist(xs, pos)\n",
    "pos_sample.mean(), pos_sample.std()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And so does the sample from the distribution of velocities:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(14.922000000000001, 3.1240864264613424)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "move_sample = sample_dist(xs, move)\n",
    "move_sample.mean(), move_sample.std()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When we add them together, we get a sample from the distribution of positions after one time unit.\n",
    "\n",
    "The mean is the sum of the means, and the standard deviation is the hypoteneuse of a triangle with the other two standard deviations.  In this case, it's a 3-4-5 triangle:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(194.911, 5.0055048696410243)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sample = pos_sample + move_sample\n",
    "sample.mean(), sample.std()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Based on the samples, we can estimate the distribution of the sum.\n",
    "\n",
    "To compute the distribution of the sum exactly, we can iterate through all possible values from both distributions, computing the sum of each pair and the product of their probabilities:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def add_dist(xs, dist1, dist2):\n",
    "    res = np.zeros_like(dist1)\n",
    "    for x1, p1 in zip(xs, dist1):\n",
    "        for x2, p2 in zip(xs, dist2):\n",
    "            x = (x1 + x2) % 360\n",
    "            res[x] = res[x] + p1 * p2\n",
    "    return res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This algorithm is slow (taking time proportional to $N^2$, where $N$ is the length of `xs`), but it works:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "new_pos = add_dist(xs, pos, move)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's what the result looks like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Ag/TZuWzSzn48n4k/BpJFpDryXu912aff1wFqvGp5G/A38c9/C9ywtM2ZNUX0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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207cf34f10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(xs, new_pos);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we can check the mean and standard deviation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def mean_dist(xs, dist):\n",
    "    return sum(xs * dist)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The mean of the sum is the sum of the means:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "195.00000000000006"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mu = mean_dist(xs, new_pos)\n",
    "mu"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And the standard deviation of the sum is the hypoteneuse of the standard deviations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def std_dist(xs, dist, mu):\n",
    "    return np.sqrt(sum((xs - mu)**2 * dist))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which should be 5:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.0"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigma = std_dist(xs, new_pos, mu)\n",
    "sigma"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What we just computed is the convolution of the two distributions.\n",
    "\n",
    "Now we get to the fun part.  The characteristic function is an alternative way to represent a distribution.  It is the Fourier transform of the probability density function, or for discrete distributions, the DFT of the probability mass function.\n",
    "\n",
    "I'll use Numpy's implementation of FFT.  The result is an array of complex, so I'll just plot the magnitude and ignore the phase."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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aGj9BOI/lXbjnIn3Iu12W53pQE68J1tn7qMpeAD5rrX0fcCvwP3n1HhFh3nB+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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207ccb8c10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from numpy.fft import fft\n",
    "\n",
    "char_pos = fft(pos)\n",
    "plt.plot(xs, np.abs(char_pos));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Fourier transform of a Gaussian is also a Gaussian, which we can see more clearly if we rotate the characteristic function before plotting it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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CruS7yhizB7gMyAP3WGtfOcVzvgxcbq29rvwlStiNLYV89UfyoK0NJNxWHckbY64Bdlpr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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207cbac950>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(xs, np.roll(np.abs(char_pos), 180));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also compute the characteristic function of the velocity distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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aa6cBrLVHjDGbjTE7cMP9zcA7S/x+vwCuxX2hyK21WBGRCIrjBvwv1vOLnVKT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\niEiIKeRFREJMIS8iEmIKeRGREPv/IRUkKL0SDtAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207cad5dd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "char_move = fft(move)\n",
    "plt.plot(xs, abs(char_move));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You might notice that the narrower (more certain) the distribution is in the space domain, the wider (less certain) it is in the frequency domain.  As it turns out, the product of the two standard deviations is constant.\n",
    "\n",
    "We can see that more clearly by plotting the distribution side-by-side in the space and frequency domains:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def plot_dist_and_char(dist):\n",
    "    f, (ax1, ax2) = plt.subplots(1, 2, figsize=(7, 4))\n",
    "    ax1.plot(xs, dist)\n",
    "    ax1.set_xlabel('space domain')\n",
    "    char = fft(dist)\n",
    "    ax2.plot(xs, np.roll(abs(char), 180))\n",
    "    ax2.set_xlabel('frequency domain')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following function plots Gaussian distributions with given parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def plot_gaussian_dist_and_char(mu=180, sigma=3):\n",
    "    dist = gaussian(xs, mu, sigma)\n",
    "    plot_dist_and_char(dist)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a simple example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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vYmRm5wInu/tZwOXAjWWX3AS81d1fA6wDLl7hIkZlX5q5rWBwA9jcv4Z9g5MU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\nRKTtKLiJiEjbUXATEZG2o+AmIiJt5/8DGRKkEDYupUQAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207ca10e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_gaussian_dist_and_char(mu=180, sigma=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can make sliders to control `mu` and `sigma`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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vYmRm5wInu/tZwOXAjWWX3AS81d1fA6wDLl7hIkZlX5q5rWBwA9jcv4Z9g5MU\nNd+06RTcIpQu79NREtzSx1r6J1rnA7cCuPujwEYz6ys5/wp3H0ge7wWOWuHyRWXf4Ul6u7P0dK3M\n0lupzf2djE0UODKuCafNpuAWITVLtqWtwL6S53uBbekTdz8MYGbbgIuAO1a0dJHZPzjJ5vVrVvx9\nNyd9fPsGJ1b8vWU2BbcITVWZCjCl4NYuMoS+t2lmtgX4GnCFux9sSqkiMDaRZ3gsv+JNkjDTDJo2\ni0rzaEBJhNQs2ZZ2EbK31DHA7vSJmfUTsrUPuvudK1y2qKSBZfMKjpRMTWduGlTSdMrcIpRPmvMr\nN0s2o0RSBzuBtwKY2WnAgLuPlJz/BHCDu+9sRuFisq8JIyVTaVPofmVuTafMLUIaLdl+3P1BM3vY\nzO4nDPe/0szeBQwC3wDeAZxsZu9NXvLX7v5nTSpuS2vGHLfUdLOkMremU3CLkCZxtyd3v7rs0CMl\nj7tXsiwxm8ncmjegRJlb86lZMkL5SpmbgpsIMBNYmtHn1tudo2dNln2HNVqy2RTcIjRVaSqAmiVF\nAKYDSzP63CAEVY2WbD4FtwhptKTI/J49OElXZ5Z1a1d2Anfq6A1rOHwkz9iEVjFvJgW3CFXqc1Oz\npAgUi0X2HBhn21FryGQytV/QAFuPCn19ew6oabKZag4oqbGY64XAxwiju+5w92uT49cBr0nu/3F3\nv7UBZV+10qbHbMkXU03iFoGhI3mOjBemA0wzbDuqCwjB7cStPU0rx2pXNXNbwGKunwYuA84GLjKz\nHcnq5acmr7kY+FT9i726VRwtqWZJEXYn2dLWJMA0QxpYdx8Yb1oZpHaz5LyLuZrZScABdx9Itt64\nA7gAuAf4heT1g0CvmTWnfaBNpRO1OyrMc9MkblnN9hwMAaWZmZuaJVtDrWbJrcDDJc/3JsceS/7e\nW3LuOeD57p4H0pUVLge+rn2n6mu6WbJCn5tGS8pqlgaUbS3SLCnNs9hJ3NUysFnnzOzNwHuA1y+2\nUFKdRkuKVLbnQJq5Na9Zsrc7R19Pjt0H1SzZTLWaJast5jpQdm57cgwzewNwNXCxuw/Vp6iS0gol\nIpWlfW7P29i8zA1C5vjsgQkKqo9NUyu4zbuYq7s/CfSb2Qlm1gFcAuw0s/XA9cCl7n6ocUVfvdIR\nkaXNkuljjZaU1WzPgQk29XfS1dncWU5bj+piYqrIweGpppZjNav6CXD3B4F0MddPkSzmamZvSS65\nAvgSYRCLT2qsAAAS+ElEQVTJl939MeAXgU3AV8zs7uTPcY37EVafqakQwDo7ZoJbZ9IsOTml4Car\n01S+yN5DE00dTJKaHjG5X02TzVKzz63aYq7ufi9wVtn1NwE31aV0UtFEpeDWqeAmq9vu/eMUinDM\npub1t6WOTcrwzL5xXvwzfU0uzeqkFUoiNJmM91/TMfPrSx9Pai6ArFJP7R0D4Pgtzd9A4fjnhTI8\n+exYk0uyeim4RWhyskLmljxOz4msNk89lwa35mduaYB98tnRJpdk9VJwi9DE1PyZW3pOZLV5+rnQ\nv3VcC2Ruvd05tmzoVObWRApuEUr71dZ0lE7ihmxmpj9OZLV56rkxujozbNnQ/AElACc8r4cDQ1MM\nHdGIyWZQcItQmp11lmRumUyGzo4sk8rcZBXKF4o8s3eMYzd3z9rnsJlOUL9bUym4RajSaMn0uUZL\nymr0zN5xxieLPP+Y1lmFfzq4Pafg1gwKbhGanO5zmx3c1nRk1Ocmq9Lju44AtGRwe2KPglszKLhF\naHKqSCYze21JCINKlLnJavT4rjAq8eRWCm5busll4bGBI80uyqqk4BahiakinbnMnJ2GOzsyGlAi\nq9Jju0bJZOCkba0T3NZ0ZjlpWw+P7x5VX3gTKLhFaGKqMGsaQKqzI6tmSVl18vkiP37mCNuP7qKn\nK1f7BSvohdvXMjlV5KdqmlxxCm4RmpwsTi+3VWqNBpTIKvTTPaOMThQ49YTWW+bKjlsLgD89UuNK\nqTcFtwhN5itnbms6s0zli9pmQ1aVf30iBI5TT+xtcknm2nF8KNMPnlBwW2kKbhGamCrOmQYAMzsD\nqN9NVpMfPDEMwItOaL3gduzmLjb1d/Ivjw9TLKperiQFtwiNjhfoWTP3V9fdFY6NTajfTVaHfL7I\n9x4f5uj1nWxrga1uymUyGV52Uh+DI1M8ocncK0rBLTJT+SLjkwXWVug4T48dGcuvdLFEmuLRp0cY\nHs1zhvXPGT3cKl5+cugLfPhHQ00uyeqi4BaZ0fEQuNZ2zw1uvcmxI+MKbrI6fMdDwDjd+ptckvmF\nwAsP/nCw2UVZVRTcIjOSZGVru+b+6tJjI8rcZBUoFovc+8hBujozvPz5rTdSMrWhr5NTT+zl354a\n4cDQZLOLs2oouEXmyHjoT6uUua2dztzU5ybt70fPHGHX/gletWN9y81vK/eaUzdQLMK3/+Vgs4uy\naii4RSbtT+ut1iypzE1WgW88dACA816+scklqe11L99IRy7D339nv0ZNrhAFt8ioWVIEBkem+OZ3\nD7B145qW7m9Lre/t4OxT1/P0c+P8v8eGm12cVUHBLTIjC8jcFNyk3X31nueYmCry5rOPbpn922p5\n62u3AHDLN/coe1sBCm6R2X84dEgfta5zzrn0WHqNSDsa2DfObffvZcuGTt74yk3NLs6CnXzsWl79\non5++OQId39PfW+N1lHrAjO7ATgTKAJXuftDJecuBD4G5IE73P3aWq+R5XnuUAhcWzbOnbCaHkuv\nkbgspa6tNuOTBf7Hl59gKl/kvW86lq7OuL6f/8Ylx/LdHw/z2dsHOOX4Xo7Z1NXsIrWtqp8MMzsX\nONndzwIuB24su+TTwGXA2cBFZrZjAa+RZXju0AQAWzbMzdx6u3P0dmfZm1wj8VhKXVvhIjbd6Hie\na//qp/x4YJTXv+IoznnJhmYXadG2HtXFFf/+WIZG8/zezY+z58B4s4vUtmp97TkfuBXA3R8FNppZ\nH4CZnQQccPcBdy8CdwAXVHuNLE+xWOSpZ8fo68nR11M56d56VBe79o8zManpAJFZSl1bFQqFIv/0\nw0F+609+xEM/GuL0F67jfW/Z3uxiLdkbTt/EL5//PHbtn+A3b3T+7oG9WjKvAWo1S24FHi55vjc5\n9ljy996Sc88Bzwc2V3jNNuDH1d5oaDTP4MgUAHP7WmcfKD9f4/KKh8o7dGvdY+7ra79prXLOuUeN\n65/eO8aegxOc/eL1c94r9bLn9/H4rlHu/H8HeNWO9XRE0tneLJ0dmVaZI7WUulbV0GieQ8NTpJ+k\n9PM2/bkqzjxO60PpufSvYtnB8vsUi7OvL70fJdfMur7sfqWHRsbyHBqZYt/gJI/vOsL3Hh9m3+Ak\n2Qz83NlH8543HjNnF/rYvPP129h21Bo+938G+OztA/zlP+7mpSet45Tj1rJl4xo29XfSsybLms4s\nXZ0ZctmwOfH0T51h+nGm5GD6uJn/Oh0dmYrLA654ORZ5fbV/s/nOZagYbqblAK647gE6euLpHG6m\nn93ezTPPPFPx3Iu3jvG3Y3v55F89t8KlilM2C//tF0/khcm+W0u1Z8+e9GG9avVS6tqsMrRLnVrb\nleVV1s/Fpx/FsZuL7Nk90Owi1cWO58E1v9TPN797gPv/9SD3fGcP93yn2aVavmwGfufnj+elJy2v\nwW65dapWcNtF+NaYOgbYnTweKDu3Pbl+osprKtkGsPue369dWgHgA99odgnayxV/X9fbbQMeX8Lr\nFlvXqv0P33Z1yoG/bnYhZMHe/w91vd2S6lSt4LYTuAa4ycxOAwbcfQTA3Z80s34zO4FQ0S4BfgU4\ner7XzOM7wDmEiqwJWhKrHKESLvW791Lq2nxUp6QdLKtOZWpNJjSzjwOvJVSSK4HTgEF3v83MzgH+\nR3LpV939k5Ve4+6PLKVwIqvJUuqaiFRWM7iJiIjEJq4ZkCIiIgug4CYiIm1HwU1ERNrOYue51V29\n1qE0s9cBXwF+kBz6PnA98FeEIL4beIe7L3htKjN7KWHViE+6+2fM7DjgL8vvZ2ZvB64CCsBN7v75\nRd73C4TBA/uTS65z979fwn2vA15D+L1+HHioTuUtv++bl1teM1sLfAHYAnQDf0D4nS2rvPPc923L\nLW/Ze/QQPmcfBe5abpnrTXVKdYo61qkq965bvWpEnWpq5taAdSjvdvfzkj9XEX4Bf+zuryWs9PCe\nRZRtLfAJ4BvMTEL/aPn9zKwX+D3CckivA95vZvPunjjPfYvAfysp+98v4b7nAacm/5YXE9YivKYO\n5a1032WXF7gU+Gd3fx3wC8AN9SjvPPetR3lLfQjYlzxe9meinlSnVKcaUKfmu3c961Xd61SzmyXr\nvQ5l+coN5wJfSx7fDly4iHuNE36hz9a43yuB77j7kLuPAfcTFrddyH1Ly1te9jMXed97CB86gEGg\nt07lLb/vWsL8k2WV193/xt3/KHl6PPA04UO7rPLOc1+WW96UmZ0CnAJ8PTlUj3/jelKdmr/sqlNL\n/Hw2sl41qk41u1my0np6NdehnEcReJGZ/R1wFCH697p7uv9Leu8Fcfc8kDez0sOV7ldp3b9532ee\n+wK8z8w+kLz+fUu8bzpZ/nLCB+UNdSpv6X3vIMzDWlZ5U2b2AGE1jn8H3Lnc8s5z3w/Uq7yEZrkr\ngV9Lni/7M1FnqlMzVKfqWKcq3Lte9aohdarZmVu5WutQVvNj4Pfd/c3Au4C/YPaaZPVeS7TaWpqL\n9ZfAf3X3C4DvAb/P3H+HBd3XzN5M+JC8b4GvX8x930P4ENatvEnTzJuBW+pZ3pL7/hXwxXqU18ze\nCdzj7k/N85p6fibqRXVKdaou5S27d13qVSPrVLODW7X19BbF3Xe5+1eSxz8B9hCaZNLdAI9N3m85\nhivcr/xnqLXu3xzufpe7fz95+jXgJUu5r5m9Afgg8EZ3P1yv8ib3vRq4OGkWWHZ5zewVyWAC3P1f\nCK0IQ2bWvZzyznPfH9Tj3xd4E/A2M3sQeC+hn2DZZa4z1SlUp+pZp6rcux71qmF1qtnBbSfwVgBb\n2DqU8zKzXzGzjySPtxDWuLw5vT/w88BSlsjNMPMt4c4K9/u/wBlmtj7p2zgLuHeB903L/lUze0ny\n9FzgkcXe18zWE9L7S9z9UL3KW3LfS9P71qO8hLUPP5Dc73mE/ow7k3IuubwV7tsHfK4O5cXdf8nd\nX+nurwb+nDC44pt1KHM9qU6hOlXnOlXp3nWpV42sU01ffsvqtA5l8gP/NaFvIEcYJfQ9QurcDTwB\n/FrS3r2Q+70K+DPC0NcpwnDXiwnDYWfdz8x+HvjPhJT8Rnf/0iLuewD4COHb4TAwlNx33yLv+xvJ\nfX6UHCoC7yZ8YJZT3vL7QvgP7reXWd5uQjPXcUAPoUnjYSr8vupw3xHgj5ZT3grv8xHgp4Rgsqwy\n15vqlOoUdaxTVe5d13pV7zrV9OAmIiJSb81ulhQREak7BTcREWk7Cm4iItJ2FNxERKTtKLiJiEjb\nUXATEZG2o+C2ypjZhWZ2dwPv/zIzW+5K9LJKmdnfmNlDZnZMs8tST2bWYWaFBr/H3WbWzKXeWkqz\nF06WNpMszfPbzS6HROsywsK5480uSGzc/bxml6GVKLg1WPINNF3AtAf4nLvfbGbfIqwe8GLC6tZ/\n6O5ftrD9w03ABNAPfMjdd1rYzO9mwgoBAFe7+z0W9ob6MGHpoUng1939ibIyvAW4FniGktXhzeyF\nwGeT13YQ9ma638JGj3uBHcCphDXwLgVeCtzn7r9pYX+lLxJWr+gFvuru11nY4PIP3P2c5Gf8R8JS\nOS8EPuLuf72cf09pX2b254TWpH8wsw8Df0rYbPPf3P1aM/tDwmepB/i2u/+XJFP5HPAKwvqD+4Fn\n3P1DSabU4e4FM3s3cIG7v8PCxqZ/BHQmf97n7t+b7/OaLD12M6E+5gkLHV9NWHH/C0nZPwv8i7v/\nz5KfxwgLDI8A3yo53kuo49uT9/+iu382KePFyWWnJa/tImxbkwEudPcjZvZRwjYwecL6ir/q7lPJ\nz9tJ2PdsE2FdxhcQ9uRbdV841SzZeL9IqJznEdZf602OF4Gcu78B+DngU0lFfR7wYXe/kLDr7MeS\n638XeNLdzyas0P7eJOB9Fvg5D5sI/gmh0pb7Y+Dn3f1iwi62xZLjn0nKdgUhWKW2uPulhGV2/gT4\nTcKeSu82s37COoNfS157DvBBm7tvWJHwLfwSwtYe/2WB/2ayCrn7e5OHFxD2CzuFsCvBtWb2NuAY\nd3+du58JnGxmlybXvgw4nVCPXkz4jJcrMvO5vwX4D8ln90rCclrpNZU+rx8H/o+7n0P4IvkOQkB9\nD4QmR+CNhGBU6iPAnyd18/slx38bOODu5xL23/uvZvYzyblXJPd/ffJe30jq/DjwejPLEYLlOUl5\nNgAXVfh5X05Yl/EM4NeS9SxXFWVujXcHcLuZ3UzYD+qzJed2Arj742ZWJASMPcD1ybezNYRvYBAC\ny58m1z8GvNPMXklYKfvW8CWRHGUV28w2AT3u7smhuwgZWHrPtyX3/IGZ9SfXF4EHkmsGCMH5cHK/\n/cB6QmZ3drJO3gThG+ZRFX7+byV/PzXPeZH5HHD3tKXhPODVJf3F/cCJhDpyn7sXgaka/ckZMzua\nkJV93mb2f1tX0lf1reTv0s/rK0m+NLr7PYTNRjGzTWb2AuBk4FvuPlT2fi9m5svpXSXHX0nIBHH3\nMTN7iJCpFYGH3H3SzAYIycd9yWueAfqTNRYLwLfNbIrwBWBzhZ/13uTfZMzM9iU/y2CVf5u2o+DW\nYO7uZvYiQtb2NuB3gNckpyvtjfUnwC3u/gUzezFhJ1pIMr2y248DT9Voa88wO+CV/s4r7b2UHitd\nDHeq7Lps8nN0Jt8qMbO9VFb6WnV2y2JMlDweA25y90+UXmBm/6nsNfMtlptuUzMOjFeqM0mwq/R5\nrVT3IDQtvpvQvPjnFc6X1r3S1xeZXRdK692suubupXU3a2ZnE/aWe4W7j5rZVyq8L8yuv+l7rCpq\nlmwwM/tl4Ax3/yahCeT4pGkhQ2iSSPu+pgjZ0Bbgh8nLf4mZSvkASXu8mZ1kZncCDmw2s1OT4681\ns18vK8J+wi7FJyfPLyw5908l9/xZYJ+7H2BhFWEL8G/Ja/89sJawirdII9wHXJbUHczsw8ln+gfA\nWWaWMbM1zG6iOwwcnzw+DygmLRBPmNkbk/u80Mx+r8Z7l9a9c5I+aQjN+JcBp7p7pe1Xfkjov4O5\n9e4Nyf16CU2RD7HwevdEEthOAF6N6l1FCm6N90Pgk0ln9V3Af0+2CCkCHWZ2G/BV4LeSZoRPAF80\ns53A/cABM7seuJGwUeQ9hLb9P3D3MeBXgb9I7n8NJR3XAMk9fwe4zcy+Bhxh5lvibwG/bmZ3Jfd/\nR3K8tH+i9HHp888T+t/uJjTL3JL8Kb+esteKVFP+WQPA3f83oT48YGYPEJrwHyc07TthcNbfAf/K\nTJD478BOM/s6YSuV1DuBq83s24TtdnbWKMvvAa9Lrr+WmSbKg4S+tL+Z5/UfBX7TzP6B0BQ6mRz/\nY0JT6LcJe5dd42En6kp1rbw8O4F+M7s/KddHCP3dL2D+OrsqacubJkmCwh+4+101LxaRBUn2BOtw\n91rZWD3eayMh4J6dBDppIcrcRKTdNPwbu5m9B/g28EEFttakzE1ERNqOMjcREWk7Cm4iItJ2FNxE\nRKTtKLiJiEjbUXATEZG2o+AmIiJt5/8DGRKkEDYupUQAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207cf101d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from IPython.html.widgets import interact, fixed\n",
    "from IPython.html import widgets\n",
    "\n",
    "slider1 = widgets.IntSliderWidget(min=0, max=360, value=180)\n",
    "slider2 = widgets.FloatSliderWidget(min=0, max=100, value=3)\n",
    "interact(plot_gaussian_dist_and_char, mu=slider1, sigma=slider2);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you increase `sigma`, the distribution gets wider in the space domain and narrower in the frequency domain.\n",
    "\n",
    "As you vary `mu`, the location changes in the space domain, and the phase changes in the frequency domain, but the magnitudes are unchanged.\n",
    "\n",
    "But enough of that; I still haven't explained why characteristic functions are useful.  Here it is: If the characteristic function of X is $\\phi_X$ and the characteristic function of Y is $\\phi_Y$, the characteristic function of the sum X+Y is the elementwise product of $\\phi_X$ and $\\phi_Y$.\n",
    "\n",
    "So the characteristic function of the new position (after one time step) is the product of the two characteristic functions we just computed:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Mfr+WxVU6eZ14FZGtqK6sUSd/OjN7FHjCOfcI/ljmJufc9c65f77kpdUxTi2pkxeRWmjn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\nppAXEWlhCnkRkRb2/wEmrEfI+EgTnAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207ca73d10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "char_new_pos = char_pos * char_move\n",
    "plt.plot(xs, abs(char_new_pos));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we compute the inverse FFT of the characteristic function, we get the PMF of the new position:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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iIoECQUREgHRXGS04M9tHNN+hAuxx98OLXCVgei2mR4EfhE3fB+4HvkAUrqeA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\nvi7yISluAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207cb98350>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from numpy.fft import ifft\n",
    "\n",
    "new_pos = ifft(char_new_pos).real\n",
    "plt.plot(xs, new_pos);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can check the mean and standard deviation of the result:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def mean_std_dist(xs, dist):\n",
    "    xbar = mean_dist(xs, dist)\n",
    "    s = std_dist(xs, dist, xbar)\n",
    "    return xbar, s"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Yup, that's what we expected (forgiving some floating-point errors):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(195.00000000000006, 4.9999999999995675)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mean_std_dist(xs, new_pos)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can encapsulate this process in a function that computes the convolution of two distributions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def fft_convolve(dist1, dist2):\n",
    "    prod = fft(dist1) * fft(dist2)\n",
    "    dist = ifft(prod).real\n",
    "    return dist"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since FFT is $N \\log N$, and elementwise multiplication is linear, the whole function is $N \\log N$, which is better than the $N^2$ algorithm we started with.\n",
    "\n",
    "The results from the function are the same:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(195.00000000000006, 4.9999999999995675)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_pos = fft_convolve(pos, move)\n",
    "mean_std_dist(xs, new_pos)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###The update step\n",
    "\n",
    "Now suppose that after the move we measure the position of the rotating part with a noisy instrument.  If the measurement is 197 and the standard deviation of measurement error is 4, the following distribution shows the likelihood of the observed measurement for each possible, actual, position of the part:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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iIoECQUREgHRXGS04M9tHNN+hAuxx98OLXCVgei2mR4EfhE3fB+4HvkAUrqeA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\nvi7yISluAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207cc24e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "likelihood = gaussian(xs, mu=197, sigma=4)\n",
    "plt.plot(xs, new_pos);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can take our belief about the position of the part and update it using the observed measurement.  By Bayes's theorem, we compute the product of the prior distribution and the likelihood, then renormalize to get the posterior distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ALfG6G4AbgGeA+l+/RDrA6NhE3foBQKlUolIuqYYgwSkaCAPAydT7E/GyZN2J1LrjwNL4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\n/wN08IYmwlLQUQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207ce47e90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "new_pos = new_pos * likelihood\n",
    "normalize(new_pos)\n",
    "plt.plot(xs, new_pos);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The prior mean was 195 and the measurement was 197, so the posterior mean is in between, at 196.2 (closer to the measurement because the measurement error is 4 and the standard deviation of the prior is 5).\n",
    "\n",
    "The posterior standard deviation is 3.1, so the measurement decreased our uncertainty about the location."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(196.21951219512201, 3.123475237772122)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mean_std_dist(xs, new_pos)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can encapsulate the prediction step in a function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def predict(xs, pos, move):\n",
    "    new_pos = fft_convolve(pos, move)\n",
    "    return new_pos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And likewise the update function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def update(xs, pos, likelihood):\n",
    "    new_pos = pos * likelihood\n",
    "    normalize(new_pos)\n",
    "    return new_pos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following function takes a prior distribution, velocity, and a measurement, and performs one predict-update step.\n",
    "\n",
    "(The uncertainty of the velocity and measurement are hard-coded in this function, but could be parameters.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def predict_update(xs, pos1, velocity, measure):\n",
    "    # predict\n",
    "    move = gaussian(xs, velocity, 3)\n",
    "    pos2 = predict(xs, pos1, move)\n",
    "    \n",
    "    #update\n",
    "    likelihood = gaussian(xs, measure, 4)\n",
    "    pos3 = update(xs, pos2, likelihood)\n",
    "    \n",
    "    #plot\n",
    "    plt.plot(xs, pos1, label='pos1')\n",
    "    plt.plot(xs, pos2, label='pos2')\n",
    "    plt.plot(xs, pos3, label='pos3')\n",
    "    plt.legend()\n",
    "    \n",
    "    return pos3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the figure below, `pos1` is the initial belief about the position, `pos2` is the belief after the predict step, and `pos3` is the posterior belief after the measurement.\n",
    "\n",
    "The taller the distribution, the narrower it is, indicating more certainty about position.  In general, the predict step makes us less certain, and the update makes us more certain."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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rQ+Gy2rpDdeueB1Y55yrOuVpn6wbgW2FgiMy60WzQZTSU6axXtJo4TiXfz2iu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\nIgIoEEREJKRAEBERAP4/QICaRNNxnfIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207cf94050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pos1 = gaussian(xs, 180, 4)\n",
    "pos3 = predict_update(xs, pos1, velocity=15, measure=197)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far I've been using Gaussian distributions for everything, but in that case we could skip all the computation and get the results analytically.\n",
    "\n",
    "The implementation I showed generalizes to arbitrary distribitions.  For example, suppose our initial beliefs are multimodal, for example, if we can't tell whether the part has been rotated 90 degrees."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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kk9QRxYpLJprkJhdKRVajpMtKNvIl0U8oi1Jn6qMQxDIZYWj1SvnrXFJDRpJlNEHaRivp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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207c996d90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pos1 = (gaussian(xs, 0, 4) + gaussian(xs, 90, 4) + \n",
    "        gaussian(xs, 180, 4) + gaussian(xs, 270, 4))\n",
    "normalize(pos1)\n",
    "plt.plot(xs, pos1);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can do the same predict-update step:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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atRgYjsqrRAnBnWdAiE9N0dFUics1c7hMe1MFhjG7GqRvqo9QPER3Wl65DBdr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\n2CQgCCGEACQgCCGEsElAEEIIAcD/D5Cfejv9wmLTAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207c8e9890>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pos3 = predict_update(xs, pos1, velocity=15, measure=151)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After the predict step, our belief is still multimodal.\n",
    "\n",
    "Then I chose a measurement, 151, that's halfway between two modes.  The result is bimodal (with the other two modes practically eliminated).\n",
    "\n",
    "If we perform one more step:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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B8QeBHmNM5XUHRGoYqnsMwf/1SuYH6Yp24WYyRDo6cJzqv3aRYI6Cm0k3qbUi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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f207ccb7ed0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pos5 = predict_update(xs, pos3, velocity=15, measure=185)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the posterior is unimodal again (and very close to Gaussian).  In general, if the likelihood function is Gaussian, the result will converge to Gaussian over time, regardless of the initial distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}