rotate_error_ellipse.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# rotate_error_ellipse\n",
    "誤差楕円の回転がなぜ、共分散行列の両側から回転行列をかけているのかよくわからない。  \n",
    "なので実際に回してみるコードを書いてみた。  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib.patches import Ellipse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Error_ellipse:\n",
    "    def __init__(self, sigma_x = 1.0, sigma_y = 1.0, cov_xy = 0.0, mu_x = 0.0, mu_y = 0.0):\n",
    "        self.cov = np.array([[sigma_x, cov_xy], [cov_xy, sigma_y]])    # 分散共分散行列\n",
    "        self.mean = np.array([mu_x, mu_y]).T                           # 平均値(楕円の中央)\n",
    "        \n",
    "    def shift(self, delta, angle):\n",
    "        ca = math.cos(angle)\n",
    "        sa = math.sin(angle)\n",
    "        self.rot = np.array([[ca, sa],[-sa, ca]])    # 回転行列\n",
    "        \n",
    "        self.cov = self.rot.dot(self.cov).dot(self.rot.T)    # 回転\n",
    "        self.mean = self.mean + delta                        # 移動\n",
    "        \n",
    "    def draw(self):\n",
    "        eigen = np.linalg.eig(self.cov)    # eigenvalue\n",
    "        \n",
    "        v1 = eigen[0][0] * eigen[1][0]     # eigenvector\n",
    "        v2 = eigen[0][1] * eigen[1][1]\n",
    "        v1_direction = math.atan2(v1[1], v1[0])\n",
    "        \n",
    "        ellipse = Ellipse(self.mean, width=np.linalg.norm(v1), height=np.linalg.norm(v2), angle=v1_direction / 3.14 * 180)\n",
    "        ellipse.set_alpha(0.2)\n",
    "        \n",
    "        fig = plt.figure(0)\n",
    "        sp = fig.add_subplot(111, aspect='equal')\n",
    "        sp.add_artist(ellipse)\n",
    "        sp.set_xlim(-2.0, 2.0)\n",
    "        sp.set_ylim(-2.0, 2.0)\n",
    "        \n",
    "        plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1088f7da0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = Error_ellipse(1.0, 2.0, 0.0)\n",
    "p.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x108c47978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p.shift([0.0, 0.0], math.pi / 4)  # 45度回転\n",
    "p.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "共分散行列を回転させる部分を以下に抜き出してみた。  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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U9arH3cDdPbb/HLi5a30fsC9LX2Y2OJ6ZaWZJDgozS3JQmFmSg8LMkhwUZpbkoDCzJAeF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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108c0c160>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p.cov = p.rot.dot(p.cov).dot(p.rot.T)\n",
    "p.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "試しに左側だけ掛けてみる。  \n",
    "虚数解が出たらしい。傾き22.5度の直線を軸に90度回転しているっぽい。  \n",
    "前後にも飛び出している状況。  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python3.6/site-packages/ipykernel_launcher.py:19: ComplexWarning: Casting complex values to real discards the imaginary part\n"
     ]
    },
    {
     "data": {
      "image/png": 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8LSIezzZsMytS1rMe9wD3LLH+OHDzouWDwMEsfZlZ53hmppklOSjMLMlBYWZJDgozS3JQ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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1088f7518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p.cov = p.rot.dot(p.cov)\n",
    "p.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "当たり前だけど右側も掛けたらちゃんと回転した。  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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JK/5uBI5FxFPNtt9gsRSh78RtVhB5HKPYAzzTsn28+dyyXFLQrPOkKikYEVcV/EnLJQXN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\n/BLpaFcm/QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108c04f60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p.cov = p.cov.dot(p.rot.T)\n",
    "p.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "転置した回転行列を左右入れ替えると逆回転した。  \n",
    "回転行列を転置することと、回転角を逆にするのは同じなので当然。  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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1A8CBLH2ZWf/4ykwzS3JQmFmSg8LMkhwUZpbkoDCzJAeFmSU5KMwsyUFhZkkOCjNLclCY\nWZKDwsySHBRmluSgMLMkB4WZJTkozCzJQWFmSQ4KM0tyUJhZUq9KCv5M0iOSHpY0laVPM+u9TPfM\n5JmSgn+3ira/FRFPZ+zPzPqgFyUFzWzA9WqOIoBvSXpQ0t4e9WlmOelVScHfiIijkp4H3CfpJxHR\ntQJ6J0j2AuzevXuVuzezq6kXJQWJiKOdx+OS7qFd4bxrULj2qFnxXPWPHpLGJK27uAz8Nu1JUDMb\nEFe9pCCwFfiepB8C/wl8IyL+OUu/ZtZbV72kYEQ8Afxqln7MrL98ZaaZJTkozCzJQWFmSQ4KM0ty\nUJhZkoPCzJIcFGaW5KAwsyQHhZklOSjMLMlBYWZJDgozS3JQmFmSg8LMkhwUZpbkoDCzJAeFmSU5\nKMwsyUFhZklZb677N5J+IulHku6RtHGZdjdJelzSIUl3ZenTzHov6xHFfcAvRcSvAD8F/mxpA0ll\n4JPAzcAe4A5JezL2a2Y9lCkoIuJbEdHorH4f2Nml2Y3AoYh4IiIWgC8Dt2Xp18x6K2s188XeA9zd\nZfsO4PCi9SPAK5fbyeKSgsC8pLVYLGgLsBYru6/V1wVr97W9dDWNcqk9KunDQAP44uWMsJvFJQUl\nTUXEZNZ9Fo1f1+BZq69N0tRq2mWuPSrpXcCbgNdHRLdaoUeBXYvWd3a2mdmAyHrW4ybgT4BbI2Jm\nmWYPADdIeqGkGnA7sD9Lv2bWW1nPenwCWAfcJ+lhSZ+GZ9ce7Ux23gncCxwEvhIRj65y//syjq+o\n/LoGz1p9bat6Xer+acHM7Bm+MtPMkhwUZpZU6KBY7SXig0jS2yQ9KqklaeBPu63Vy/QlfU7S8bV2\nPY+kXZK+I+mxzvvwAyu1L3RQsIpLxAfYj4G3APf3eyBZrfHL9D8P3NTvQVwFDeBDEbEHeBXw/pX+\nZoUOilVeIj6QIuJgRDze73HkZM1eph8R9wOn+j2OvEXEsYh4qLN8nvYZyR3LtS90UCzxHuCb/R6E\nddXtMv1l33RWLJKuB14O/GC5Nnl+1+OK9PoS8V5azWsz6ydJ48BXgQ9GxLnl2vU9KHK4RLywUq9t\nDfFl+gNIUpV2SHwxIr62UttCf/RY5SXi1n++TH/ASBLwWeBgRHws1b7QQcEyl4ivBZLeLOkI8Grg\nG5Lu7feYrlTGy/QLTdKXgP8AXirpiKT39ntMOXkN8HbgdZ1/Ww9LumW5xr6E28ySin5EYWYF4KAw\nsyQHhZklOSjMLMlBYWZJDgozS3JQmFnS/wMPNO60eOBcmgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108c04828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p.cov = p.rot.T.dot(p.cov).dot(p.rot)\n",
    "p.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}