Test - Point Cloud Correlation Matrix Testing.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "127ea129",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import random\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "id": "a58acf07",
   "metadata": {},
   "outputs": [],
   "source": [
    "X = [i for i in range(-20,20)]\n",
    "y = [3*x for x in X]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "id": "44b812c4",
   "metadata": {},
   "outputs": [],
   "source": [
    "y_points = list(map(lambda x: x+random.random()*5 * (-1 if random.random()>0.5 else 1), y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "id": "fe958de0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f4309f830a0>]"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plt.plot(X,y)\n",
    "plt.plot(X, y_points, 'ro')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "id": "5032caa6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[(-20, -62.15085162599303), (-19, -54.51403747498398), (-18, -49.000021187095726), (-17, -51.18670163727176), (-16, -43.42391997068769), (-15, -44.259170430663545), (-14, -42.488156944786894), (-13, -41.47193523228339), (-12, -33.257317477299075), (-11, -36.732724022136196), (-10, -30.020179548722403), (-9, -23.23697931732584), (-8, -20.073946884767018), (-7, -24.3647932746398), (-6, -14.206686912455336), (-5, -17.41705585983766), (-4, -12.325088003585503), (-3, -6.986957857072399), (-2, -3.175149915941123), (-1, -6.660403390148619), (0, 1.9768514768874028), (1, 0.2563683975417579), (2, 5.2619427683375255), (3, 7.652441295459932), (4, 12.283475371224782), (5, 13.428432592610914), (6, 22.147629960918504), (7, 17.675419383133825), (8, 23.646729424230706), (9, 30.303070983228338), (10, 27.341282067820913), (11, 35.95610103311485), (12, 33.5855128380639), (13, 41.01248039149766), (14, 42.70484666919732), (15, 47.57152139614551), (16, 52.91956116144942), (17, 55.83972052551522), (18, 51.04324082371767), (19, 61.24507942851913)]\n"
     ]
    }
   ],
   "source": [
    "point_tuples = list(zip(X,y_points))\n",
    "print(point_tuples)\n",
    "dims = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "id": "3cd32273",
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_covariance_matrix(points):\n",
    "    eX = list()\n",
    "    for pt in points:\n",
    "        eX.append(pt)\n",
    "    print(eX)\n",
    "    #print(points)\n",
    "    M = np.zeros((dims))\n",
    "\n",
    "    for x in eX:\n",
    "        M += x\n",
    "\n",
    "    M /= len(eX)\n",
    "    print(M)\n",
    "    for i in range(len(eX)):\n",
    "        eX[i] = eX[i] - M\n",
    "    \n",
    "    dim = dict()\n",
    "    for i in range(dims):\n",
    "        dim[i] = np.zeros((len(eX)))\n",
    "        for j in range(len(eX)):\n",
    "            dim[i][j] = eX[j][i]\n",
    "    def covariance(i, j):\n",
    "        temp_array = dim[i] * dim[j]    \n",
    "        k = np.sum(temp_array)/len(eX)\n",
    "        #print(k, temp_array)\n",
    "        return k\n",
    "    K = np.zeros((dims, dims))\n",
    "\n",
    "    for i in range(dims):    \n",
    "        for j in range(dims):\n",
    "            K[i][j] = covariance(i, j)\n",
    "    \n",
    "    eigenvalues, eigenvectors = np.linalg.eig(K)\n",
    "    \n",
    "    return K, eigenvalues, eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "id": "99815b8a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[(-20, -62.15085162599303), (-19, -54.51403747498398), (-18, -49.000021187095726), (-17, -51.18670163727176), (-16, -43.42391997068769), (-15, -44.259170430663545), (-14, -42.488156944786894), (-13, -41.47193523228339), (-12, -33.257317477299075), (-11, -36.732724022136196), (-10, -30.020179548722403), (-9, -23.23697931732584), (-8, -20.073946884767018), (-7, -24.3647932746398), (-6, -14.206686912455336), (-5, -17.41705585983766), (-4, -12.325088003585503), (-3, -6.986957857072399), (-2, -3.175149915941123), (-1, -6.660403390148619), (0, 1.9768514768874028), (1, 0.2563683975417579), (2, 5.2619427683375255), (3, 7.652441295459932), (4, 12.283475371224782), (5, 13.428432592610914), (6, 22.147629960918504), (7, 17.675419383133825), (8, 23.646729424230706), (9, 30.303070983228338), (10, 27.341282067820913), (11, 35.95610103311485), (12, 33.5855128380639), (13, 41.01248039149766), (14, 42.70484666919732), (15, 47.57152139614551), (16, 52.91956116144942), (17, 55.83972052551522), (18, 51.04324082371767), (19, 61.24507942851913)]\n",
      "[-0.5        -0.82750922]\n"
     ]
    }
   ],
   "source": [
    "cov_matrix, eigenvalues, eigenvectors = get_covariance_matrix(point_tuples)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "id": "8435604c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 133.25      ,  401.74458126],\n",
       "       [ 401.74458126, 1219.1307549 ]])"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cov_matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "id": "3ad7b103",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([7.77198695e-01, 1.35160356e+03])"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigenvalues"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "id": "ba65ca2a",
   "metadata": {},
   "outputs": [],
   "source": [
    "eigenvectors = eigenvectors.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "id": "6210ae4e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.94970102,  0.31315806],\n",
       "       [-0.31315806, -0.94970102]])"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "id": "59897bcf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f4309e6b820>]"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 400x400 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plt.plot(X,y)\n",
    "plt.figure(figsize=(5,5), dpi=80)\n",
    "plt.ylim(ymin = -40, ymax = 40)\n",
    "plt.xlim(xmin= -40, xmax = 40)\n",
    "plt.plot([0, eigenvectors[0][0]*40],[0, eigenvectors[0][1]*40])\n",
    "plt.plot([-eigenvectors[1][0]*40, eigenvectors[1][0]*40],[-eigenvectors[1][1]*40, eigenvectors[1][1]*40])\n",
    "plt.plot(X, y_points, 'ro')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "id": "95c93998",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.0057391610830634e-17"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(eigenvectors[0], eigenvectors[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c5aaebc7",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
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