Tuto-GUDHI-simplicial-complexes-from-data-points.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# TDA  with Python using the Gudhi Library  \n",
    "\n",
    "# Building simplicial complexes from a point cloud : Rips and Alpha complexes "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Authors:** F. Chazal and B. Michel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pickle as pickle\n",
    "import gudhi as gd  \n",
    "from pylab import *\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "TDA typically aims at extracting topological signatures from a point cloud in $\\mathbb R^d$ or in a general metric space. [Simplicial complexes](https://en.wikipedia.org/wiki/Simplicial_complex) are used in computational geometry to infer topological signatures from a point cloud.\n",
    "\n",
    "This tutorial explains how to build [Vietoris-Rips complexes](https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex)  and [alpha complexes](https://en.wikipedia.org/wiki/Alpha_shape#Alpha_complex) from a data point in $\\mathbb R ^d$.\n",
    "\n",
    "![title](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d0/VR_complex.svg/600px-VR_complex.svg.png)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The walk of 3 persons A, B and C, has been recorded using the accelerometer sensor of a smartphone in their pocket, giving rise to 3 multivariate time series in $\\mathbb R ^d$: each time series represents the 3 coordinates of the acceleration of the corresponding walker in a coordinate system attached to the sensor. \n",
    "\n",
    "Notice that the smartphone was carried in a possibly different position for each person and thus these time series cannot be compared coordinates by coordinates.\n",
    "\n",
    "Using a sliding window, each serie have been splitted in a list of 100 times series made of 200 consecutive points, that are stored in `data_A`, `data_B` and `data_C`.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We load the data with the pickle module :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = open(\"./datasets/data_acc\",\"rb\")\n",
    "data = pickle.load(f)    \n",
    "f.close()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A'\n",
      " 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A'\n",
      " 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A'\n",
      " 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A'\n",
      " 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A'\n",
      " 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'A' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B'\n",
      " 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B'\n",
      " 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B'\n",
      " 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B'\n",
      " 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B'\n",
      " 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B' 'B'\n",
      " 'B' 'B' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C'\n",
      " 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C'\n",
      " 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C'\n",
      " 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C'\n",
      " 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C'\n",
      " 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C' 'C']\n"
     ]
    }
   ],
   "source": [
    "data_A = data[0]\n",
    "data_B = data[1] \n",
    "data_C = data[2]\n",
    "label  = data[3]\n",
    "print(label)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The object <code>data_A</code> is a list of 100 time series of the 3d acceleration for Walker A, let's have a look at the dimensions of <code>data_A</code>:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(100, 200, 3)\n"
     ]
    }
   ],
   "source": [
    "print(np.shape(data_A))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now represent the trajectory of accelerations for the first trajectory of walker A :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "data_A_sample = data_A[0]\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111, projection = '3d')\n",
    "ax.scatter(data_A_sample[:, 0], data_A_sample[:, 1], data_A_sample[:, 2]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vietoris-Rips complex\n",
    "\n",
    "\n",
    "The [$\\alpha$-Rips complex](https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex) of a point cloud $\\mathbb X$ in $(\\mathbb R^d,\\| \\|_2)$  is an [abstract simplicial complex](https://en.wikipedia.org/wiki/Abstract_simplicial_complex) that can be defined  by forming a simplex for every finite subset of $\\mathbb X$ that has diameter at most $\\alpha$. \n",
    "\n",
    "Definition of diameter here?\n",
    "\n",
    "Vietoris-Rips complexes can be defined for any metric space from the  matrix of pairwise distances (see this [notebook](https://github.com/GUDHI/TDA-tutorial/blob/master/Tuto-GUDHI-simplicial-complexes-from-distance-matrix.ipynb)).\n",
    "\n",
    "In order to efficiently compute an $\\alpha$-Rips complex, one can start by building a topological graph with:\n",
    "\n",
    "- as many vertices as there are points; and,\n",
    "- as edges only pairs of points whose distance is smaller than or equal to $\\alpha$.\n",
    "\n",
    "In other words, it corresponds to the [1-skeleton](https://en.wikipedia.org/wiki/N-skeleton) of the point cloud, with upper-bounded edge length. In GUDHI, this is performed via the `RipsComplex()` function (see the [Documentation](https://gudhi.inria.fr/python/latest/rips_complex_user.html) for details on the syntax):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "skeleton = gd.RipsComplex(points = data_A_sample, max_edge_length = 0.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `max_edge_length` parameter is the maximal diameter: only edges of length smaller than or equal to this value are included in the 1-skeleton.\n",
    "\n",
    "From the $\\alpha$-*truncated* 1-skeleton, it is then possible to add higher dimensional simplices to the simplicial complex, subject to the condition that all their faces are already in the complex. Their filtration value is then defined as the maximum filtration value of its faces which matches their diameter by design. This process necessarily yields the desired $\\alpha$-Rips complex since filtration values can never exceeds $\\alpha$ in this way. \n",
    "\n",
    "In practice, one should define a maximal dimension of simplicies to be added to the Rips complex for computational reasons. In GUDHI, this is achieved through the `create_simplex_tree()` method of the `RipsComplex` class which takes the argument `max_dimension` to limit the tree:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "Rips_simplex_tree_sample = skeleton.create_simplex_tree(max_dimension = 3) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `max_dimension` parameter is the maximum dimension of the simplices included in the filtration. The result is a simplex tree, of dimension 3 in this example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rips_simplex_tree_sample.dimension()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the methods of the simplex tree object to describe the Rips filtration. For instance, we can check that the 200 points of `data_A_sample` are all vertices of the Rips filtration:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "200"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rips_simplex_tree_sample.num_vertices()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The number of simplices in the Rips complex is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "27851"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rips_simplex_tree_sample.num_simplices()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this is actually the number of simplices in the \"last\" Rips complex of the filtration, namely with parameter $\\alpha=$ `max_edge_length=`0.2. \n",
    "\n",
    "Notice that the number of simplices in a Rips complex increases very fast with the number of points and the dimension. AND THE $\\alpha$ ??"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's compute the list of simplices in the Rips complex with the `get_filtration() ` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "rips_generator = Rips_simplex_tree_sample.get_filtration()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Let's print the 300 first elements in the list:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "([0], 0.0)\n",
      "([1], 0.0)\n",
      "([2], 0.0)\n",
      "([3], 0.0)\n",
      "([4], 0.0)\n",
      "([5], 0.0)\n",
      "([6], 0.0)\n",
      "([7], 0.0)\n",
      "([8], 0.0)\n",
      "([9], 0.0)\n",
      "([10], 0.0)\n",
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      "([167], 0.0)\n",
      "([168], 0.0)\n",
      "([169], 0.0)\n",
      "([170], 0.0)\n",
      "([171], 0.0)\n",
      "([172], 0.0)\n",
      "([173], 0.0)\n",
      "([174], 0.0)\n",
      "([175], 0.0)\n",
      "([176], 0.0)\n",
      "([177], 0.0)\n",
      "([178], 0.0)\n",
      "([179], 0.0)\n",
      "([180], 0.0)\n",
      "([181], 0.0)\n",
      "([182], 0.0)\n",
      "([183], 0.0)\n",
      "([184], 0.0)\n",
      "([185], 0.0)\n",
      "([186], 0.0)\n",
      "([187], 0.0)\n",
      "([188], 0.0)\n",
      "([189], 0.0)\n",
      "([190], 0.0)\n",
      "([191], 0.0)\n",
      "([192], 0.0)\n",
      "([193], 0.0)\n",
      "([194], 0.0)\n",
      "([195], 0.0)\n",
      "([196], 0.0)\n",
      "([197], 0.0)\n",
      "([198], 0.0)\n",
      "([199], 0.0)\n",
      "([6, 34], 0.010037972604067026)\n",
      "([95, 120], 0.011934283430520687)\n",
      "([24, 108], 0.017975277911620735)\n",
      "([151, 177], 0.018713766643837386)\n",
      "([79, 106], 0.018944080262710026)\n",
      "([42, 43], 0.02139398300924822)\n",
      "([81, 136], 0.02153918522135876)\n",
      "([4, 191], 0.022836281768273937)\n",
      "([156, 186], 0.02404079478719446)\n",
      "([52, 190], 0.02528062297491887)\n",
      "([47, 185], 0.02543660160084278)\n",
      "([4, 53], 0.02656518191166776)\n",
      "([137, 165], 0.026689955732447364)\n",
      "([12, 68], 0.027999307509293882)\n",
      "([23, 51], 0.028305883664001743)\n",
      "([10, 38], 0.030244785153807872)\n",
      "([54, 165], 0.030334711041972974)\n",
      "([163, 190], 0.030841760585284376)\n",
      "([53, 191], 0.03269553417517446)\n",
      "([4, 53, 191], 0.03269553417517446)\n",
      "([48, 155], 0.03270191445466141)\n",
      "([97, 154], 0.033380721157578334)\n",
      "([42, 181], 0.03371579898207958)\n",
      "([21, 181], 0.034473350301356065)\n",
      "([16, 99], 0.035402079628179935)\n",
      "([43, 181], 0.03604829516079782)\n",
      "([42, 43, 181], 0.03604829516079782)\n",
      "([40, 174], 0.03635244907843206)\n",
      "([122, 149], 0.036790327465245506)\n",
      "([108, 190], 0.037527917767976426)\n",
      "([96, 178], 0.038341942830274006)\n",
      "([52, 108], 0.03848452715053156)\n",
      "([52, 108, 190], 0.03848452715053156)\n",
      "([107, 190], 0.03952421033999288)\n",
      "([125, 180], 0.04042591259328599)\n",
      "([10, 94], 0.04086899075093489)\n",
      "([39, 178], 0.041456725944049144)\n",
      "([79, 162], 0.04152965660826004)\n",
      "([24, 164], 0.041682370973830186)\n",
      "([80, 136], 0.042696594676859156)\n",
      "([6, 10], 0.04296810997705157)\n",
      "([65, 68], 0.043114058785969134)\n",
      "([49, 104], 0.044036842087052615)\n",
      "([85, 198], 0.04441595016432722)\n",
      "([92, 151], 0.044417584119355306)\n",
      "([9, 123], 0.04474070645396651)\n",
      "([25, 80], 0.044775941184971184)\n",
      "([24, 190], 0.04503018556479643)\n",
      "([24, 108, 190], 0.04503018556479643)\n",
      "([108, 164], 0.04509993323720115)\n",
      "([24, 108, 164], 0.04509993323720115)\n",
      "([120, 177], 0.0452854613645483)\n",
      "([21, 43], 0.04536807662883676)\n",
      "([21, 43, 181], 0.04536807662883676)\n",
      "([54, 137], 0.045461176546587496)\n",
      "([54, 137, 165], 0.045461176546587496)\n",
      "([52, 163], 0.04621308427274681)\n",
      "([52, 163, 190], 0.04621308427274681)\n",
      "([95, 177], 0.04646166650046034)\n",
      "([95, 120, 177], 0.04646166650046034)\n",
      "([37, 123], 0.04669949572532886)\n",
      "([43, 71], 0.04670615440174885)\n",
      "([106, 162], 0.046711459461249945)\n",
      "([79, 106, 162], 0.046711459461249945)\n",
      "([23, 53], 0.046858285510675696)\n",
      "([37, 150], 0.04704462724902802)\n",
      "([39, 92], 0.04705525816526781)\n",
      "([8, 147], 0.04728024972438278)\n",
      "([120, 175], 0.04745268912295697)\n",
      "([52, 107], 0.04796088461444387)\n",
      "([52, 107, 190], 0.04796088461444387)\n",
      "([159, 186], 0.049566945306322815)\n",
      "([25, 136], 0.049588833581765135)\n",
      "([25, 80, 136], 0.049588833581765135)\n",
      "([10, 34], 0.049595510048793735)\n",
      "([6, 10, 34], 0.049595510048793735)\n",
      "([23, 134], 0.04963560183980849)\n",
      "([55, 166], 0.05056851213947273)\n",
      "([4, 136], 0.050702254644147744)\n",
      "([80, 81], 0.05071354281649033)\n",
      "([80, 81, 136], 0.05071354281649033)\n",
      "([100, 156], 0.051819557360131926)\n",
      "([135, 136], 0.052141185592581234)\n",
      "([107, 163], 0.05222368637888371)\n",
      "([52, 107, 163], 0.05222368637888371)\n",
      "([107, 163, 190], 0.05222368637888371)\n",
      "([52, 107, 163, 190], 0.05222368637888371)\n",
      "([50, 161], 0.052446656995465385)\n",
      "([70, 125], 0.05251374992323431)\n",
      "([79, 134], 0.052579455550623565)\n",
      "([49, 187], 0.052584575552152)\n",
      "([106, 134], 0.05290928140695164)\n",
      "([79, 106, 134], 0.05290928140695164)\n",
      "([24, 52], 0.05308009651460706)\n",
      "([24, 52, 108], 0.05308009651460706)\n",
      "([24, 52, 190], 0.05308009651460706)\n",
      "([24, 52, 108, 190], 0.05308009651460706)\n",
      "([120, 178], 0.05312372783417972)\n",
      "([63, 90], 0.05346525304344867)\n",
      "([95, 175], 0.053922095267153705)\n"
     ]
    }
   ],
   "source": [
    "rips_list = list(rips_generator)\n",
    "for splx in rips_list[0:300] :\n",
    "    print(splx)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "The filtration value is the diameter of the simplex, which is zero for the vertices of course. The first edge in the filtration is `[6, 34]`, these two points are the two closest points in `data_A_sample`, at distance $0.0100$ of each other. If you scroll down, you will see that the first triangle is `[4, 53, 191]`, for the filtration value $0.0327$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### Alpha complex "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "The [alpha complex](https://en.wikipedia.org/wiki/Alpha_shape#Alpha_complex) is a simplicial complex built from the finite cells of a [Delaunay triangulation](https://en.wikipedia.org/wiki/Delaunay_triangulation). Alpha complexes contain less simplices than Rips complexes and so they can be a better option. They are sub-complexes of the Delaunay complex and, as such, they are geometric simpicial complexes.\n",
    "\n",
    "The `AlphaComplex()` function directly computes the simplex tree representing the alpha complex:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "alpha_complex = gd.AlphaComplex(points = data_A_sample)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "st_alpha = alpha_complex.create_simplex_tree(max_alpha_square = 0.2**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Talk about `max_alpha_square` ?\n",
    "\n",
    "Talk about `default_filtration_value` ?\n",
    "\n",
    "The point cloud `data_A_sample` belongs to $\\mathbb R^3$ and so does the Alpha Complex:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "st_alpha.dimension()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As for the Rips complex, the 200 points of `data_A_sample` are all vertices of the Alpha complex :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "200"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "st_alpha.num_vertices()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the number of simplices in the Alpha complex is smaller than for the Rips complex:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3334"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "st_alpha.num_simplices()"
   ]
  }
 ],
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