curvature-and-distance-approximation-example-for-article.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see how the curvature affects the pairwise distances in the case of a balanced binary tree.  In order to make the distance matrices comparable across curvatures, we'll need to ensure that the particular \"ordering\" of the points in the space is the same in each case (recall that there are many choices, and which occurs is a function of the initial conditions)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Throughout, the tree is a balanced binary tree\n",
    "\n",
    "```\n",
    "(((12)(34))((56)(78)))\n",
    "```\n",
    "\n",
    "with all edge lengths equal to $0.25$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "def balanced_binary_tree_distances(tree_height):\n",
    "    \"\"\"\n",
    "    Return the matrix of pairwise distances of the leaves of a balanced binary tree,\n",
    "    where the leaves are enumerated in a depth-first fashion.\n",
    "    \"\"\"\n",
    "    num_leaves = 2 ** tree_height\n",
    "    diss = np.empty((num_leaves, num_leaves), dtype=np.float64)\n",
    "    for i in range(num_leaves):\n",
    "        for j in range(num_leaves):\n",
    "            diss[i,j] = 2 * _lowest_common_ancestor_height(i, j, tree_height)\n",
    "    return diss\n",
    "\n",
    "def _lowest_common_ancestor_height(i, j, tree_height):\n",
    "    \"\"\"\n",
    "    In a balanced binary tree of height `tree_height`, return the height of the lowest\n",
    "    common ancestor of leaves `i` and `j` (which are in range(2 ** tree_height))\n",
    "    \"\"\"\n",
    "    for h in reversed(range(0, tree_height)):\n",
    "        div = 2 ** h\n",
    "        if (i // div) != (j // div):\n",
    "            return h + 1\n",
    "    return h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "base_dist = 1/4\n",
    "true_pairwise_dists = balanced_binary_tree_distances(3)\n",
    "true_pairwise_dists *= base_dist  ## as though the edge lengths were base_dist"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our implementation works with the rates of site difference, so we need to invert the distance correction formula\n",
    "\n",
    "```\n",
    "dist = (-3 / 4) * np.log(1 - (4 / 3) * diffs_per_site)\n",
    "```\n",
    "\n",
    "to get the expected rate of difference, given the distance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def distance_to_rate(dists):\n",
    "    return 0.75 * (1 - np.exp(-4 * dists / 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Normalisations\n",
    "\n",
    "In finite data scenarios, we'd use the variance of the distance estimate to normalise the summands in the objective function.  For simplicity in this example, let's assume that the sequences are of infinite length."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "rates_of_difference = distance_to_rate(true_pairwise_dists)\n",
    "normalisations = np.ones_like(rates_of_difference, dtype=np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "from hyperboloid import Hyperboloid\n",
    "from cythonised.cy_sequence_mds_sg import SequenceMDS\n",
    "\n",
    "num_leaves = true_pairwise_dists.shape[0]\n",
    "init_stddev = 0.1\n",
    "stopping_distance = 1e-8\n",
    "max_rounds = -1\n",
    "lr = 0.001\n",
    "max_step_size = 0.1\n",
    "\n",
    "def fit_smds(rho, dimension, initial_pts=None):\n",
    "    rho_hyperboloid = Hyperboloid(local_dimension=dimension, rho=rho)\n",
    "    if initial_pts is None:\n",
    "        # generate initial points\n",
    "        np.random.seed(0)\n",
    "        initial_pts = rho_hyperboloid.initialisation_points(init_stddev, num_leaves)\n",
    "            \n",
    "    smds = SequenceMDS(rho=rho, lr=lr, max_step_size=max_step_size)\n",
    "    smds.set_points(initial_pts.copy())\n",
    "    converged, failed, checkpoints = smds.fit(rates_of_difference,\n",
    "                                              normalisations,\n",
    "                                              stopping_distance=stopping_distance,\n",
    "                                              max_rounds=max_rounds,\n",
    "                                              checkpoint_interval=1000)    \n",
    "    pw_dists = rho_hyperboloid.pairwise_distances(smds.get_points())\n",
    "    return converged, failed, checkpoints, pw_dists"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "rhos = [0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1., 1.25, 1.5, 1.75]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Dimension 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "fitted_distances_dim2 = {}\n",
    "\n",
    "converged, failed, checkpoints, fitted_distances_dim2[1.] = fit_smds(1., 2)\n",
    "assert converged\n",
    "\n",
    "# keep these points for later plotting\n",
    "converged_points_dim2_rho1 = checkpoints[-1][0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "for rho in rhos[::-1]:\n",
    "    converged, failed, checkpoints, fitted_distances_dim2[rho] = fit_smds(rho, 2)\n",
    "    assert converged"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Dimension 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "fitted_distances_dim3 = {}\n",
    "\n",
    "for rho in rhos[::-1]:\n",
    "    converged, failed, checkpoints, fitted_distances_dim3[rho] = fit_smds(rho, 3)\n",
    "    assert converged"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_points(ax, points):\n",
    "    \"\"\"\n",
    "    Plot the given points, belonging to the 1-hyperboloid of local dimension 2,\n",
    "    on the Poincaré disc.\n",
    "    \"\"\"\n",
    "    rho_hyperboloid = Hyperboloid(local_dimension=2, rho=1.)\n",
    "\n",
    "    circle = plt.Circle((0, 0), 1, fc='none', ec='black', linewidth=3, alpha=0.5)\n",
    "    ax.add_artist(circle)\n",
    "    lim = 1.1\n",
    "    ax.set_xlim(-lim, lim)\n",
    "    ax.set_ylim(-lim, lim)\n",
    "    ax.xaxis.set_visible(False)\n",
    "    ax.yaxis.set_visible(False)\n",
    "    ax.axis('off')\n",
    "    \n",
    "    def get_size(ppt):\n",
    "        euc_dist = np.sqrt(ppt.dot(ppt))\n",
    "        hyp_dist = 2 * np.arctanh(euc_dist)\n",
    "        markersize = 0.7 * max(10 - 3 * hyp_dist, 0.5)\n",
    "        return markersize\n",
    "\n",
    "    for i, pt in enumerate(points):\n",
    "        ppt = rho_hyperboloid.to_poincare_ball_point(pt)\n",
    "        ax.plot(*ppt, marker='o', markersize=get_size(ppt), label='%i' % (i + 1))[0]\n",
    "        ax.annotate(str(i+1), tuple(ppt + 0.02), fontsize=8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(4, 4))\n",
    "plot_points(ax, converged_points_dim2)\n",
    "ax.set_aspect('equal')\n",
    "ax.set_title('Point configuration, dimension 2, $\\\\rho=1$')\n",
    "plt.savefig('article/point-configuration-article.eps')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.\n",
      "The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot distances for dim 2\n",
    "fig, ax2 = plt.subplots(figsize=(4, 4))\n",
    "for j in range(0, num_leaves):\n",
    "    distances = [fitted_distances_dim2[rho][0, j] for rho in rhos]\n",
    "    ax2.plot(rhos, distances, label='$\\mathrm{d}_{1%i}$' % (j + 1))\n",
    "ax2.set_ylim(-0.05, 2.3)\n",
    "ax2.set_xlim(0., 2.3)\n",
    "ax2.set_title('Distances from leaf 1, dimension 2')\n",
    "ax2.set_ylabel('Distance')\n",
    "ax2.set_xlabel('Hyperboloid radius $\\\\rho$')\n",
    "ax2.legend(fontsize=8, handlelength=2)\n",
    "plt.savefig('article/distance-rho-dependence-dimension-2-article.eps')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.\n",
      "The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot distances for dim 3\n",
    "fig, ax3 = plt.subplots(figsize=(4, 4))\n",
    "reordering = [0, 1, 3, 2, 4, 5, 7, 6]\n",
    "for j in range(8):\n",
    "    distances = [fitted_distances_dim3[rho][0, reordering[j]] for rho in rhos]\n",
    "    if j == 4:\n",
    "        linestyle = (0, (3, 2))\n",
    "    elif j == 6:\n",
    "        linestyle = (2, (3, 2))\n",
    "    else:\n",
    "        linestyle = '-'\n",
    "    ax3.plot(rhos, distances, label='$\\mathrm{d}_{1%i}$' % (j + 1), linestyle=linestyle)\n",
    "ax3.set_ylim(-0.05)\n",
    "ax3.set_xlim(0., 2.3)\n",
    "ax3.set_title('Distances from leaf 1, dimension 3')\n",
    "ax3.set_ylabel('Distance')\n",
    "ax3.set_xlabel('Hyperboloid radius $\\\\rho$')\n",
    "ax3.legend(fontsize=8, handlelength=2)\n",
    "plt.savefig('article/distance-rho-dependence-dimension-3-article.eps')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Euclidean, dimension 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [],
   "source": [
    "from cythonised.cy_sequence_mds_sg import SequenceMDS\n",
    "from scipy.spatial.distance import pdist, squareform\n",
    "\n",
    "num_leaves = true_pairwise_dists.shape[0]\n",
    "init_stddev = 0.1\n",
    "stopping_distance = 0.\n",
    "max_rounds = 100000\n",
    "lr = 0.1\n",
    "max_step_size = 0.1\n",
    "\n",
    "def fit_euclidean_smds(dimension, initial_pts=None):\n",
    "    rho_hyperboloid = Hyperboloid(local_dimension=dimension, rho=rho)\n",
    "    if initial_pts is None:\n",
    "        # generate initial points\n",
    "        np.random.seed(1)\n",
    "        initial_pts = np.zeros((num_leaves, dimension + 1), dtype=np.float64)\n",
    "        initial_pts[:,:-1] = np.random.normal(size=(num_leaves, dimension)) * init_stddev\n",
    "            \n",
    "    smds = SequenceMDS(rho=np.inf, lr=lr, max_step_size=max_step_size)\n",
    "    smds.set_points(initial_pts.copy())\n",
    "    converged, failed, checkpoints = smds.fit(rates_of_difference,\n",
    "                                              normalisations,\n",
    "                                              stopping_distance=stopping_distance,\n",
    "                                              max_rounds=max_rounds,\n",
    "                                              checkpoint_interval=1000)    \n",
    "    points = smds.get_points()\n",
    "    pw_dists = squareform(pdist(points))\n",
    "    return converged, failed, checkpoints, pw_dists"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {},
   "outputs": [],
   "source": [
    "converged, failed, checkpoints, pw_dists = fit_euclidean_smds(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x800 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_points(points):\n",
    "    fig, ax = plt.subplots(figsize=(4, 4), dpi=200)\n",
    "    ax.set_aspect('equal')\n",
    "    ax.scatter(points[:,0], points[:,1])\n",
    "    lim = 1.3\n",
    "    ax.set_xlim(-lim, lim)\n",
    "    ax.set_ylim(-lim, lim)\n",
    "    for i, pt in enumerate(points[:,:-1]):\n",
    "        ax.plot(*pt, marker='o', label='%i' % (i + 1))[0]\n",
    "        ax.annotate(str(i+1), tuple(pt + 0.02), fontsize=8)\n",
    "    return ax\n",
    "    \n",
    "points = checkpoints[-1][0].copy()\n",
    "ax = plot_points(points)\n",
    "plt.savefig('article/euclidean-fit-1-article.eps')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x800 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta = np.pi / 3\n",
    "points = checkpoints[-1][0].copy()\n",
    "points[:,:-1] = points[:,:-1].dot(np.array([[np.cos(theta), np.sin(theta)], [-np.sin(theta), np.cos(theta)]]))\n",
    "ax = plot_points(points)\n",
    "plt.savefig('article/euclidean-fit-2-article.eps')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x800 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "points = checkpoints[-1][0].copy()\n",
    "\n",
    "def swap_rows(i, j):\n",
    "    inter = points[i,:].copy()\n",
    "    points[i,:] = points[j,:]\n",
    "    points[j,:] = inter\n",
    "\n",
    "swap_rows(0,3)\n",
    "swap_rows(1,2)\n",
    "ax = plot_points(points)\n",
    "plt.savefig('article/euclidean-fit-3-article.eps')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "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": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}