Tuto-GUDHI-simplicial-complexes-from-distance-matrix.ipynb

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    "tags": []
   },
   "source": [
    "# TDA  with Python using the Gudhi Library \n",
    "\n",
    "# Building simplicial complexes from distance matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Authors :** F. Chazal and B. Michel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import pickle as pickle\n",
    "import gudhi as gd  \n",
    "from pylab import *"
   ]
  },
  {
   "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 data.\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 matrix of pairwise distances.\n",
    "\n",
    "## Vietoris-Rips filtration defined from a matrix of distances \n",
    "\n",
    "The [$\\alpha$-Rips complex](https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex) of a metric space $\\mathbb X$ 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",
    "![title](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d0/VR_complex.svg/600px-VR_complex.svg.png)\n",
    "\n",
    " \n",
    "## Protein binding dataset\n",
    "\n",
    "The data we will be studying in this notebook represents configurations of protein binding. This example is borrowed from Kovacev-Nikolic et al. [[1]](https://arxiv.org/pdf/1412.1394.pdf).\n",
    "\n",
    "The paper compares closed and open forms of the maltose-binding protein (MBP), a large biomolecule containing $370$ amino-acid residues. The analysis is not based on geometric distances in $\\mathbb R^3$ but on a metric of *dynamical distances* defined by\n",
    "\n",
    "$$ D_{ij} = 1 - |C_{ij}|, $$\n",
    "\n",
    "where $C$ is the correlation matrix between residues. Correlation matrices between residues can be found at this [link](https://www.researchgate.net/publication/301543862_corr). We are greatful to the authors for sharing data !"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The next statments load the $14$ correlation matrices with pandas:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
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   },
   "outputs": [],
   "source": [
    "path_file = \"./datasets/Corr_ProteinBinding/\"\n",
    "files_list = [\n",
    "    '1anf.corr_1.txt', \n",
    "    '1ez9.corr_1.txt', \n",
    "    '1fqa.corr_2.txt', \n",
    "    '1fqb.corr_3.txt', \n",
    "    '1fqc.corr_2.txt', \n",
    "    '1fqd.corr_3.txt', \n",
    "    '1jw4.corr_4.txt', \n",
    "    '1jw5.corr_5.txt', \n",
    "    '1lls.corr_6.txt', \n",
    "    '1mpd.corr_4.txt', \n",
    "    '1omp.corr_7.txt', \n",
    "    '3hpi.corr_5.txt', \n",
    "    '3mbp.corr_6.txt', \n",
    "    '4mbp.corr_7.txt'\n",
    "]\n",
    "corr_list = [\n",
    "    pd.read_csv(\n",
    "        path_file + u, \n",
    "        header = None, \n",
    "        sep='\\s+'\n",
    "    ) for u in files_list\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The object `corr_list` is a list of $14$ correlation matrices. We can iterate in the list to compute the matrix of distances associated to each configuration:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "dist_list = [1 - np.abs(c) for c in corr_list]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's print out the first lines of the first distance matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
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   "outputs": [
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     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = dist_list[0]\n",
    "D.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vietoris-Rips filtration of Protein binding distance matrix\n",
    "\n",
    "The `RipsComplex()` function creates a [$1$-skeleton](https://en.wikipedia.org/wiki/N-skeleton) from the point cloud (see the [GUDHI documentation](https://gudhi.inria.fr/python/latest/rips_complex_user.html) for details on the syntax)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "skeleton_protein = gd.RipsComplex(\n",
    "    distance_matrix = D.values, \n",
    "    max_edge_length = 0.8\n",
    ") "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `max_edge_length` parameter is the maximal diameter: only the edges of length less vers this value are included in the one skeleton graph. \n",
    "\n",
    "Next, we create the Rips simplicial complex from this one-skeleton graph. This is a filtered Rips complex which filtration function is exacly the diameter of the simplices. We use the `create_simplex_tree()` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "Rips_simplex_tree_protein = skeleton_protein.create_simplex_tree(max_dimension = 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `max_dimension` parameter is the maximum dimension of the simplices included in the the filtration. The object returned by the function is a simplex tree, of dimension 2 in this example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rips_simplex_tree_protein.dimension()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the fonctionalites of the simplex tree object to describe the Rips filtration.\n",
    "For instance we can check that the 370 points of the first distance matrix are all vertices of the Vietoris-Rips filtration:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "370"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rips_simplex_tree_protein.num_vertices()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The number of simplices in a Rips complex increases very fast with the number of points and the dimension. There is more than on million of simplexes in the Rips complex:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1626660"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rips_simplex_tree_protein.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=`0.8. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's compute the list of simplices in the Rips complex with the `get_filtration()` function:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1626660"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rips_filtration = Rips_simplex_tree_protein.get_filtration()\n",
    "rips_list = list(rips_filtration)\n",
    "len(rips_list)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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      "([100], 0.0)\n",
      "([101], 0.0)\n",
      "([102], 0.0)\n",
      "([103], 0.0)\n",
      "([104], 0.0)\n",
      "([105], 0.0)\n",
      "([106], 0.0)\n",
      "([107], 0.0)\n",
      "([108], 0.0)\n",
      "([109], 0.0)\n",
      "([110], 0.0)\n",
      "([111], 0.0)\n",
      "([112], 0.0)\n",
      "([113], 0.0)\n",
      "([114], 0.0)\n",
      "([115], 0.0)\n",
      "([116], 0.0)\n",
      "([117], 0.0)\n",
      "([118], 0.0)\n",
      "([119], 0.0)\n",
      "([120], 0.0)\n",
      "([121], 0.0)\n",
      "([122], 0.0)\n",
      "([123], 0.0)\n",
      "([124], 0.0)\n",
      "([125], 0.0)\n",
      "([126], 0.0)\n",
      "([127], 0.0)\n",
      "([128], 0.0)\n",
      "([129], 0.0)\n",
      "([130], 0.0)\n",
      "([131], 0.0)\n",
      "([132], 0.0)\n",
      "([133], 0.0)\n",
      "([134], 0.0)\n",
      "([135], 0.0)\n",
      "([136], 0.0)\n",
      "([137], 0.0)\n",
      "([138], 0.0)\n",
      "([139], 0.0)\n",
      "([140], 0.0)\n",
      "([141], 0.0)\n",
      "([142], 0.0)\n",
      "([143], 0.0)\n",
      "([144], 0.0)\n",
      "([145], 0.0)\n",
      "([146], 0.0)\n",
      "([147], 0.0)\n",
      "([148], 0.0)\n",
      "([149], 0.0)\n",
      "([150], 0.0)\n",
      "([151], 0.0)\n",
      "([152], 0.0)\n",
      "([153], 0.0)\n",
      "([154], 0.0)\n",
      "([155], 0.0)\n",
      "([156], 0.0)\n",
      "([157], 0.0)\n",
      "([158], 0.0)\n",
      "([159], 0.0)\n",
      "([160], 0.0)\n",
      "([161], 0.0)\n",
      "([162], 0.0)\n",
      "([163], 0.0)\n",
      "([164], 0.0)\n",
      "([165], 0.0)\n",
      "([166], 0.0)\n",
      "([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",
      "([200], 0.0)\n",
      "([201], 0.0)\n",
      "([202], 0.0)\n",
      "([203], 0.0)\n",
      "([204], 0.0)\n",
      "([205], 0.0)\n",
      "([206], 0.0)\n",
      "([207], 0.0)\n",
      "([208], 0.0)\n",
      "([209], 0.0)\n",
      "([210], 0.0)\n",
      "([211], 0.0)\n",
      "([212], 0.0)\n",
      "([213], 0.0)\n",
      "([214], 0.0)\n",
      "([215], 0.0)\n",
      "([216], 0.0)\n",
      "([217], 0.0)\n",
      "([218], 0.0)\n",
      "([219], 0.0)\n",
      "([220], 0.0)\n",
      "([221], 0.0)\n",
      "([222], 0.0)\n",
      "([223], 0.0)\n",
      "([224], 0.0)\n",
      "([225], 0.0)\n",
      "([226], 0.0)\n",
      "([227], 0.0)\n",
      "([228], 0.0)\n",
      "([229], 0.0)\n",
      "([230], 0.0)\n",
      "([231], 0.0)\n",
      "([232], 0.0)\n",
      "([233], 0.0)\n",
      "([234], 0.0)\n",
      "([235], 0.0)\n",
      "([236], 0.0)\n",
      "([237], 0.0)\n",
      "([238], 0.0)\n",
      "([239], 0.0)\n",
      "([240], 0.0)\n",
      "([241], 0.0)\n",
      "([242], 0.0)\n",
      "([243], 0.0)\n",
      "([244], 0.0)\n",
      "([245], 0.0)\n",
      "([246], 0.0)\n",
      "([247], 0.0)\n",
      "([248], 0.0)\n",
      "([249], 0.0)\n",
      "([250], 0.0)\n",
      "([251], 0.0)\n",
      "([252], 0.0)\n",
      "([253], 0.0)\n",
      "([254], 0.0)\n",
      "([255], 0.0)\n",
      "([256], 0.0)\n",
      "([257], 0.0)\n",
      "([258], 0.0)\n",
      "([259], 0.0)\n",
      "([260], 0.0)\n",
      "([261], 0.0)\n",
      "([262], 0.0)\n",
      "([263], 0.0)\n",
      "([264], 0.0)\n",
      "([265], 0.0)\n",
      "([266], 0.0)\n",
      "([267], 0.0)\n",
      "([268], 0.0)\n",
      "([269], 0.0)\n",
      "([270], 0.0)\n",
      "([271], 0.0)\n",
      "([272], 0.0)\n",
      "([273], 0.0)\n",
      "([274], 0.0)\n",
      "([275], 0.0)\n",
      "([276], 0.0)\n",
      "([277], 0.0)\n",
      "([278], 0.0)\n",
      "([279], 0.0)\n",
      "([280], 0.0)\n",
      "([281], 0.0)\n",
      "([282], 0.0)\n",
      "([283], 0.0)\n",
      "([284], 0.0)\n",
      "([285], 0.0)\n",
      "([286], 0.0)\n",
      "([287], 0.0)\n",
      "([288], 0.0)\n",
      "([289], 0.0)\n",
      "([290], 0.0)\n",
      "([291], 0.0)\n",
      "([292], 0.0)\n",
      "([293], 0.0)\n",
      "([294], 0.0)\n",
      "([295], 0.0)\n",
      "([296], 0.0)\n",
      "([297], 0.0)\n",
      "([298], 0.0)\n",
      "([299], 0.0)\n",
      "([300], 0.0)\n",
      "([301], 0.0)\n",
      "([302], 0.0)\n",
      "([303], 0.0)\n",
      "([304], 0.0)\n",
      "([305], 0.0)\n",
      "([306], 0.0)\n",
      "([307], 0.0)\n",
      "([308], 0.0)\n",
      "([309], 0.0)\n",
      "([310], 0.0)\n",
      "([311], 0.0)\n",
      "([312], 0.0)\n",
      "([313], 0.0)\n",
      "([314], 0.0)\n",
      "([315], 0.0)\n",
      "([316], 0.0)\n",
      "([317], 0.0)\n",
      "([318], 0.0)\n",
      "([319], 0.0)\n",
      "([320], 0.0)\n",
      "([321], 0.0)\n",
      "([322], 0.0)\n",
      "([323], 0.0)\n",
      "([324], 0.0)\n",
      "([325], 0.0)\n",
      "([326], 0.0)\n",
      "([327], 0.0)\n",
      "([328], 0.0)\n",
      "([329], 0.0)\n",
      "([330], 0.0)\n",
      "([331], 0.0)\n",
      "([332], 0.0)\n",
      "([333], 0.0)\n",
      "([334], 0.0)\n",
      "([335], 0.0)\n",
      "([336], 0.0)\n",
      "([337], 0.0)\n",
      "([338], 0.0)\n",
      "([339], 0.0)\n",
      "([340], 0.0)\n",
      "([341], 0.0)\n",
      "([342], 0.0)\n",
      "([343], 0.0)\n",
      "([344], 0.0)\n",
      "([345], 0.0)\n",
      "([346], 0.0)\n",
      "([347], 0.0)\n",
      "([348], 0.0)\n",
      "([349], 0.0)\n",
      "([350], 0.0)\n",
      "([351], 0.0)\n",
      "([352], 0.0)\n",
      "([353], 0.0)\n",
      "([354], 0.0)\n",
      "([355], 0.0)\n",
      "([356], 0.0)\n",
      "([357], 0.0)\n",
      "([358], 0.0)\n",
      "([359], 0.0)\n",
      "([360], 0.0)\n",
      "([361], 0.0)\n",
      "([362], 0.0)\n",
      "([363], 0.0)\n",
      "([364], 0.0)\n",
      "([365], 0.0)\n",
      "([366], 0.0)\n",
      "([367], 0.0)\n",
      "([368], 0.0)\n",
      "([369], 0.0)\n",
      "([289, 290], 0.014988169999999967)\n",
      "([45, 46], 0.016141790000000045)\n",
      "([70, 71], 0.016186290000000048)\n",
      "([211, 212], 0.019646410000000003)\n",
      "([48, 49], 0.02002598)\n",
      "([293, 294], 0.020196210000000048)\n",
      "([50, 51], 0.02128167000000003)\n",
      "([360, 361], 0.02139837)\n",
      "([101, 102], 0.021894140000000006)\n",
      "([43, 44], 0.022110460000000054)\n",
      "([135, 136], 0.022791349999999988)\n",
      "([240, 241], 0.023054660000000005)\n",
      "([82, 83], 0.02376383999999998)\n",
      "([355, 356], 0.023989579999999955)\n",
      "([231, 232], 0.024279770000000034)\n",
      "([350, 351], 0.02431161000000004)\n",
      "([238, 239], 0.02514183000000003)\n",
      "([16, 17], 0.025262099999999954)\n",
      "([297, 298], 0.025565999999999978)\n",
      "([287, 288], 0.025724140000000006)\n",
      "([284, 285], 0.026689060000000042)\n",
      "([46, 48], 0.027036240000000045)\n",
      "([10, 38], 0.027239420000000014)\n",
      "([347, 348], 0.027307149999999947)\n",
      "([216, 217], 0.02737215000000004)\n",
      "([10, 11], 0.027430060000000034)\n",
      "([303, 304], 0.02758413000000004)\n",
      "([72, 73], 0.02769299000000003)\n",
      "([362, 363], 0.027711200000000047)\n",
      "([21, 22], 0.028153890000000015)\n"
     ]
    }
   ],
   "source": [
    "for splx in rips_list[0:400] :\n",
    "    print(splx)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The integers represent the points in the metric space: the vertex `[2]` corresponds to the point decribed by the second raw (or the second column) in the distance matrix `D`. \n",
    "\n",
    "The filtration value is the diameter of the simplex, which is zero for the vertices of course. The first edge in the filtration is `[289, 290]`, these two points are the two closest points according to `D`, at distance $0.015$ of each other. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### How to define an Alpha complex from a matrix of distance ?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The [alpha complex filtration](https://en.wikipedia.org/wiki/Alpha_shape#Alpha_complex) of a point cloud in $\\mathbb R^p$ is a filtered simplicial complex constructed from the finite cells of a [Delaunay Triangulation](https://en.wikipedia.org/wiki/Delaunay_triangulation).\n",
    "\n",
    "In our case, the data does not belong to euclideen space $\\mathbb R^p$ and we are not in position to directly compute a Delaunay Triangulation in the metric space, using  the pairwise distances.\n",
    "\n",
    "The aim of [Multidimensional Scaling](https://en.wikipedia.org/wiki/Multidimensional_scaling) (MDS) methods is precisely to find a representation of $n$ points in a space $\\mathbb R^p$ that preserves as well as possible the pairwise distances between the $n$ points in the original metric space. The are several versions of MDS algorithms, the most popular ones are available in the [sckit-learn library](https://scikit-learn.org/stable/index.html), see this [documention](https://scikit-learn.org/stable/modules/generated/sklearn.manifold.MDS.html).\n",
    "\n",
    "Let's compute a (classical) MDS representation of the matrix `D` in $\\mathbb R^3$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(370, 3)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.manifold import MDS\n",
    " \n",
    "embedding = MDS(n_components = 3, dissimilarity = 'precomputed')\n",
    "X_transformed = embedding.fit_transform(D)\n",
    "X_transformed.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can represent this configuration, for instance on the two first axes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "plt.scatter(X_transformed[:, 0], X_transformed[:, 1], label = 'MDS');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course you should keep in mind that MDS provides an embedding of the data in $\\mathbb R^p$ that **approximatively** preserves the distance matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The main advantage of Alpha complexes is that they contain  less simplices than Rips complexes do and so it can be a better option than Rips complexes. As subcomplexes of the Delaunay Triangulation complex, an alpha complex is a geometric simpicial complex.\n",
    "\n",
    "The `AlphaComplex()` function directly computes the simplex tree representing the Alpha complex:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "alpha_complex = gd.AlphaComplex(points = X_transformed)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "st_alpha = alpha_complex.create_simplex_tree()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The point cloud `X_transformed` belongs to $\\mathbb R^3$ and so does the Alpha Complex:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "st_alpha.dimension()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As for the Rips complex, the $370$ points are all vertices of the Alpha complex :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "370\n"
     ]
    }
   ],
   "source": [
    "print(st_alpha.num_vertices())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the number of simplexes in the Alpha complex is much smaller then for the Rips complex we computed before:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9255\n"
     ]
    }
   ],
   "source": [
    "print(st_alpha.num_simplices())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###  References"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "[1] Using persistent homology and dynamical distances to analyze protein binding, V. Kovacev-Nikolic, P. Bubenik, D. Nikolic and G. Heo. Stat Appl Genet Mol Biol 2016 [arxiv link](https://arxiv.org/pdf/1412.1394.pdf)."
   ]
  }
 ],
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