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   "source": [
    "<center><h1>Improved Graph Laplacian via Geometric Self-Consistency</h1></center>\n",
    "<center>Yu-Chia Chen, Dominique Perrault-Joncas, Marina Meilă, James McQueen. University of Washington</center> <br>\n",
    "\n",
    "<center>Original paper: <a href=https://nips.cc/Conferences/2017/Schedule?showEvent=9223>Improved Graph Laplacian via Geometric Self-Consistency</a> on NIPS 2017</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*The github notebook viewer fails to render some plots in this notebook correctly, please view it on [jupyter nbviewer](https://nbviewer.jupyter.org/github/mmp2/megaman/blob/master/examples/radius_estimation_tutorial.ipynb?flush_cache=true). *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Task\n",
    "1. Problem: Estimate the ``radius`` of heat kernel in manifold embedding\n",
    "1. Formally: Optimize Laplacian w.r.t. parameters (e.g. ``radius``)\n",
    "1. Previous work: \n",
    "    1. asymptotic rates depending on the (unknown) manifold [4]\n",
    "    1. Embedding dependent neighborhood reconstruction [6]\n",
    "1. Challenge: it’s an unsupervised problem! What “target” to choose?\n",
    "\n",
    "## The ``radius`` affects…\n",
    "1. Quality of manifold embedding  via neighborhood selection\n",
    "1. Laplacian-based embedding and clustering via the kernel for computing similarities \n",
    "1. Estimation of other geometric quantities that depend on the Laplacian (e.g Riemannian metric) or not (e.g intrinsic dimension).\n",
    "1. Regression on manifolds via Gaussian Processes or Laplacian regularization.\n",
    "\n",
    "All the reference is the same as the poster."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Radius Estimation on hourglass dataset\n",
    "\n",
    "In this tutorial, we are going to estimate the radius of a noisy hourglass data. The method we used is based on our NIPS 2017 paper \"[Improved Graph Laplacian via Geometric Self-Consistency](https://nips.cc/Conferences/2017/Schedule?showEvent=9223)\" (Perrault-Joncas et. al). Main idea is to find an estimated radius $\\hat{r}_d$ given dimension $d$ that minimize the distorsion. The distorsion is evaluated by the riemannian metrics of local tangent space.\n",
    "\n",
    "\n",
    "Below are some configurations that enables plotly to render Latex properly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
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       "<script>requirejs.config({paths: { 'plotly': ['https://cdn.plot.ly/plotly-latest.min']},});if(!window.Plotly) {{require(['plotly'],function(plotly) {window.Plotly=plotly;});}}</script>"
      ],
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       "<script>requirejs.config({paths: { 'plotly': ['https://cdn.plot.ly/plotly-latest.min']},});if(!window.Plotly) {{require(['plotly'],function(plotly) {window.Plotly=plotly;});}}</script>"
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      "text/html": [
       "<script>var waitForPlotly = setInterval( function() {if( typeof(window.Plotly) !== \"undefined\" ){MathJax.Hub.Config({ SVG: { font: \"STIX-Web\" }, displayAlign: \"center\" });MathJax.Hub.Queue([\"setRenderer\", MathJax.Hub, \"SVG\"]);clearInterval(waitForPlotly);}}, 250 );</script>"
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       "<IPython.core.display.HTML object>"
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   ],
   "source": [
    "import plotly\n",
    "plotly.offline.init_notebook_mode(connected=True)\n",
    "\n",
    "from IPython.core.display import display, HTML\n",
    "display(HTML(\n",
    "    '<script>'\n",
    "        'var waitForPlotly = setInterval( function() {'\n",
    "            'if( typeof(window.Plotly) !== \"undefined\" ){'\n",
    "                'MathJax.Hub.Config({ SVG: { font: \"STIX-Web\" }, displayAlign: \"center\" });'\n",
    "                'MathJax.Hub.Queue([\"setRenderer\", MathJax.Hub, \"SVG\"]);'\n",
    "                'clearInterval(waitForPlotly);'\n",
    "            '}}, 250 );'\n",
    "    '</script>'\n",
    "))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Generate data\n",
    "\n",
    "This dataset used in this tutorial has a shape of hourglass, with ``size = 10000`` and dimension be 13. The first three dimensions of the data is generated by adding gaussian noises onto the noise-free hourglass data, with ``sigma_primary = 0.1``, the variance of the noises added on hourglass data. We made ``addition_dims = 10``, which is the addition noises dimension to make the whole dataset has dimension 13, with ``sigmal_additional = 0.1``, which is the variance of additional dimension.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from plotly.offline import iplot\n",
    "from megaman.datasets import generate_noisy_hourglass\n",
    "\n",
    "data = generate_noisy_hourglass(size=10000, sigma_primary=0.1,\n",
    "                                addition_dims=10, sigma_additional=0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can visualize dataset with the following plots:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from megaman.plotter.scatter_3d import scatter_plot3d_plotly\n",
    "import plotly.graph_objs as go\n",
    "\n",
    "t_data = scatter_plot3d_plotly(data,marker=dict(color='rgb(0, 102, 0)',opacity=0.5))\n",
    "l_data = go.Layout(title='Noisy hourglass scatter plot for first 3 axis.')\n",
    "f_data = go.Figure(data=t_data,layout=l_data)\n",
    "iplot(f_data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Radius estimation\n",
    "\n",
    "To estimate the ``radius``, we need to find the pairwise distance first. \n",
    "\n",
    "To do so, we compute the adjacency matrix using the Geometry modules in megaman."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "rmax=5\n",
    "rmin=0.1\n",
    "\n",
    "from megaman.geometry import Geometry\n",
    "\n",
    "geom = Geometry(adjacency_method='brute',adjacency_kwds=dict(radius=rmax))\n",
    "geom.set_data_matrix(data)\n",
    "dist = geom.compute_adjacency_matrix()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For each data points, the distortion will be estimated. If the size $N$ used in estimating the distortion is large, it will be computationally expensive. We want to choose a sample with size $N'$ such that the average distion is well estimated. In our cases, we choose $N'=1000$. The error will be around $\\frac{1}{\\sqrt{1000}} \\approx 0.03$.\n",
    "\n",
    "In this example, we searched radius from the minimum pairwise distance ``rmin`` to the maximum distance between points ``rmax``. By doing so, the distance matrix will be dense. If the matrix is too large to fit in the memory, smaller maximum radius ``rmax`` can be chosen to make the distance matrix sparse.\n",
    "\n",
    "Based on the discussion above, we run radius estimation with \n",
    "1. sample size=1000 (created by choosing one data point out of every 10 of the original data.)\n",
    "1. radius search from ``rmin=0.1`` to ``rmax=50``, with 50 points in logspace.\n",
    "1. dimension ``d=1``\n",
    "\n",
    "Specify run_parallel=True for searching the radius in parallel."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from megaman.utils.estimate_radius import run_estimate_radius\n",
    "import numpy as np\n",
    "\n",
    "# subsample by 10.\n",
    "sample = np.arange(0,data.shape[0],10)\n",
    "\n",
    "distorion_vs_rad_dim1 = run_estimate_radius(\n",
    "    data, dist, sample=sample, d=1, rmin=rmin, rmax=rmax,\n",
    "    ntry=50, run_parallel=True, search_space='logspace')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run radius estimation same configurations as above except\n",
    "1. dimension  ``d=2``"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "distorion_vs_rad_dim2 = run_estimate_radius(\n",
    "    data, dist, sample=sample, d=2, rmin=0.1, rmax=5,\n",
    "    ntry=50, run_parallel=True, search_space='logspace')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Radius estimation result\n",
    "\n",
    "The estimated radius is the minimizer of the distorsion, denoted as $\\hat{r}_{d=1}$ and $\\hat{r}_{d=2}$. (In the code, it's ``est_rad_dim1`` and ``est_rad_dim2``)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimated radius with d=1 is: 1.0969\n",
      "Estimated radius with d=2 is: 1.0128\n"
     ]
    }
   ],
   "source": [
    "distorsion_dim1 = distorion_vs_rad_dim1[:,1].astype('float64')\n",
    "distorsion_dim2 = distorion_vs_rad_dim2[:,1].astype('float64')\n",
    "rad_search_space = distorion_vs_rad_dim1[:,0].astype('float64')\n",
    "\n",
    "argmin_d1 = np.argmin(distorsion_dim1)\n",
    "argmin_d2 = np.argmin(distorsion_dim2)\n",
    "est_rad_dim1 = rad_search_space[argmin_d1]\n",
    "est_rad_dim2 = rad_search_space[argmin_d2]\n",
    "\n",
    "print ('Estimated radius with d=1 is: {:.4f}'.format(est_rad_dim1))\n",
    "print ('Estimated radius with d=2 is: {:.4f}'.format(est_rad_dim2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot distorsions with different radii"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t_distorsion = [go.Scatter(x=rad_search_space, y=distorsion_dim1, name='Dimension = 1'), \n",
    "                go.Scatter(x=rad_search_space, y=distorsion_dim2, name='Dimension = 2')]\n",
    "\n",
    "l_distorsion = go.Layout(\n",
    "    title='Distorsions versus radii',\n",
    "    xaxis=dict(\n",
    "        title='$\\\\text{Radius } r$',\n",
    "        type='log',\n",
    "        autorange=True\n",
    "    ),\n",
    "    yaxis=dict(\n",
    "        title='Distorsion',\n",
    "        type='log',\n",
    "        autorange=True\n",
    "    ),\n",
    "    annotations=[\n",
    "        dict(\n",
    "            x=np.log10(est_rad_dim1),\n",
    "            y=np.log10(distorsion_dim1[argmin_d1]),\n",
    "            xref='x',\n",
    "            yref='y',\n",
    "            text='$\\\\hat{r}_{d=1}$',\n",
    "            font = dict(size = 30),\n",
    "            showarrow=True,\n",
    "            arrowhead=7,\n",
    "            ax=0,\n",
    "            ay=-30\n",
    "        ),\n",
    "        dict(\n",
    "            x=np.log10(est_rad_dim2),\n",
    "            y=np.log10(distorsion_dim2[argmin_d2]),\n",
    "            xref='x',\n",
    "            yref='y',\n",
    "            text='$\\\\hat{r}_{d=2}$',\n",
    "            font = dict(size = 30),\n",
    "            showarrow=True,\n",
    "            arrowhead=7,\n",
    "            ax=0,\n",
    "            ay=-30\n",
    "        )\n",
    "    ]\n",
    ")\n",
    "f_distorsion = go.Figure(data=t_distorsion,layout=l_distorsion)\n",
    "iplot(f_distorsion)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Application to dimension estimation\n",
    "\n",
    "We followed the method proposed by [Chen et. al (2011)]((http://lcsl.mit.edu/papers/che_lit_mag_ros_2011.pdf) [5] to verify the estimated radius reflect the truth intrinsic dimension of the data. The basic idea is to find the largest gap of singular value of local PCA, which correspond to the dimension of the local structure.\n",
    "\n",
    "We first plot the average singular values versus radii."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from rad_est_utils import find_argmax_dimension, estimate_dimension\n",
    "# rad_est_utils.py is located in https://github.com/mmp2/megaman/tree/master/examples\n",
    "\n",
    "rad_search_space, singular_values = estimate_dimension(data, dist)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The singular gap is the different between two singular values. Since the intrinsic dimension is 2, we are interested in the region where the largest singular gap is the second. The region is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The index which maximize the second singular gap is: [20 21 22 23 24 25 26 27 28 29 30 31 32 33]\n",
      "The start and end index of largest continuous range is 20 and 34, respectively\n"
     ]
    }
   ],
   "source": [
    "singular_gap = -1*np.diff(singular_values,axis=1)\n",
    "second_gap_is_max_range = (np.argmax(singular_gap,axis=1) == 1).nonzero()[0]\n",
    "start_idx, end_idx = second_gap_is_max_range[0], second_gap_is_max_range[-1]+1\n",
    "\n",
    "print ('The index which maximize the second singular gap is: {}'.format(second_gap_is_max_range))\n",
    "print ('The start and end index of largest continuous range is {} and {}, respectively'.format(start_idx, end_idx))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Averaged singular values with different radii\n",
    "\n",
    "Plot the averaged singular values with different radii. The gray shaded area is the continous range in which the largest singular gap is the second, (local structure has dimension equals 2). And the purple shaded area denotes the second singular gap.\n",
    "\n",
    "By hovering the line on this plot, you can see the value of the singular gap."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
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        {
         "name": "1st singular value",
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\"showarrow\": true}], \"yaxis\": {\"title\": \"$\\\\text{Singular value } \\\\sigma$\"}, \"xaxis\": {\"title\": \"$\\\\text{Radius } r $\", \"autorange\": true}}, {\"linkText\": \"Export to plot.ly\", \"showLink\": true})});</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from rad_est_utils import plot_singular_values_versus_radius, generate_layouts\n",
    "\n",
    "t_avg_singular = plot_singular_values_versus_radius(singular_values, rad_search_space, start_idx, end_idx)\n",
    "l_avg_singular = generate_layouts(start_idx, end_idx, est_rad_dim1, est_rad_dim2, rad_search_space)\n",
    "f_avg_singular = go.Figure(data=t_avg_singular,layout=l_avg_singular)\n",
    "iplot(f_avg_singular)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Histogram of estimated dimensions with estimated radius.\n",
    "\n",
    "We first find out the estimated dimensions of each points in the data using the estimated radius $\\hat{r}_{d=1}$ and $\\hat{r}_{d=2}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "dimension_freq_d1 = find_argmax_dimension(data,dist, est_rad_dim1)\n",
    "dimension_freq_d2 = find_argmax_dimension(data,dist, est_rad_dim2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The histogram of estimated dimensions with different optimal radius is shown as below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.plotly.v1+json": {
       "data": [
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2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 2, 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2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], \"type\": \"histogram\", \"name\": \"d=2\"}], {\"title\": \"Dimension histogram\", \"bargroupgap\": 0.1, \"xaxis\": {\"title\": \"Estimated dimension\"}, \"yaxis\": {\"title\": \"Counts\"}, \"bargap\": 0.2}, {\"linkText\": \"Export to plot.ly\", \"showLink\": true})});</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t_hist_dim = [go.Histogram(x=dimension_freq_d1,name='d=1'),\n",
    "            go.Histogram(x=dimension_freq_d2,name='d=2')]\n",
    "\n",
    "l_hist_dim = go.Layout(\n",
    "    title='Dimension histogram',\n",
    "    xaxis=dict(\n",
    "        title='Estimated dimension'\n",
    "    ),\n",
    "    yaxis=dict(\n",
    "        title='Counts'\n",
    "    ),\n",
    "    bargap=0.2,\n",
    "    bargroupgap=0.1\n",
    ")\n",
    "\n",
    "f_hist_dim = go.Figure(data=t_hist_dim,layout=l_hist_dim)\n",
    "\n",
    "iplot(f_hist_dim)\n"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "## Conclusion\n",
    "\n",
    "1. Choosing the correct radius/bound/scale is important in any non-linear dimension reduction task\n",
    "1. The __Geometry Consistency (GC) Algorithm__ required minimal knowledge: maximum radius, minimum radius, (optionally: dimension $d$ of the manifold.)\n",
    "1. The chosen radius can be used in \n",
    "    1. any embedding algorithm\n",
    "    1. semi-supervised learning with Laplacian Regularizer (see our NIPS 2017 paper)\n",
    "    1. estimating dimension $d$ (as shown here)\n",
    "1. The megaman python package is __scalable__, and __efficient__\n",
    "\n",
    "<img src=\"https://raw.githubusercontent.com/mmp2/megaman/master/doc/images/spectra_Halpha.png\" width=\"600\" />\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## __Try it:__\n",
    "\n",
    "<div style=\"float:left;\">All the functions are implemented by the manifold learning package <a href=\"https://github.com/mmp2/megaman\">megaman</a>.　</div><a style=\"float:left;\" href=\"https://anaconda.org/conda-forge/megaman\"><img src=\"https://anaconda.org/conda-forge/megaman/badges/downloads.svg\" /></a>\n",
    "<br>\n",
    "<a href=\"https://plot.ly/python/\">Plotly</a> graphing library is used in this tutorial to create interactive plots.   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Reference\n",
    "\n",
    "[1] R. R. Coifman, S. Lafon. Diffusion maps. Applied and Computational Harmonic Analysis, 2006. <br>\n",
    "[2] D. Perrault-Joncas, M. Meila, Metric learning and manifolds: Preserving the intrinsic geometry , arXiv1305.7255 <br>\n",
    "[3] X. Zhou, M. Belkin. Semi-supervised learning by higher order regularization. AISTAT, 2011 <br>\n",
    "[4] A. Singer. From graph to  manifold laplacian: the convergence rate. Applied and Computational Harmonic Analysis, 2006. <br>\n",
    "[5] G. Chen, A. Little, M. Maggioni, L. Rosasco. Some recent advances in multiscale geometric analysis of point clouds. Wavelets and multiscale analysis. Springer, 2011. <br>\n",
    "[6] L. Chen, A. Buja. Local Multidimensional Scaling for nonlinear dimension reduction, graph drawing and proximity analysis, JASA,2009. <br>"
   ]
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