python_example.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Reconstruction of plant--pollinator networks from observational data \n",
    "\n",
    "A tutorial on how to use our [Stan](http://mc-stan.org) model to infer the structure and properties of real ecosystems from noisy, error-prone measurements.\n",
    "Based on the article \n",
    "\"*[Reconstruction of plant–pollinator networks from observational data](https://doi.org/10.1038/s41467-021-24149-x)*\"<br/> [J.-G. Young](http://jgyoung.ca), [F. S. Valdovinos](https://www.fsvaldovinos.com/) and [M.E.J. Newman](http://www-personal.umich.edu/~mejn/).\n",
    "\n",
    "\n",
    "*Author*: J.-G. Young <jean-gabriel.young@uvm.edu> <br/>\n",
    "*Date*: July 2019"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Important Note:** In this notebook, I assume that you have installed `pystan` (version 2.19 or earlier) and will interact with the model in `python3` (check the [README](README.md) file otherwise). `Stan` also has a [R interface](https://cran.r-project.org/web/packages/rstan/index.html), for which we do not provide helper functions.\n",
    "That said, all what we show here should easily be transferable to the `R` version of `Stan`!\n",
    "The only purpose of our python code is to manipulate the samples once they have been generated by `Stan`.\n",
    "\n",
    "For a pystan3 version of this code, check the base repo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Loading data and modules"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, let's import some required modules: 'pystan `numpy` and `pyplot` to handle the model the data and visualize our results.\n",
    "We'll also import the custom helper modules `utilities` (load / save / compress / sort data matrices) and `plant_pol_inference` (interact with model, draw samples, compute estimates)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline  \n",
    "import utilities as ut\n",
    "import plant_pol_inference as pp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll use the data matrix shown in the paper as an example. \n",
    "It comes from the Supplementary information of [this paper by Kaiser-Bunbury et al. [Nature **542**, 2017]](https://www.nature.com/articles/nature21071). It records the number of interactions between 8 plant species and 21 pollinator species. This is for a single ecosystem, as observed over the course of one month."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = np.loadtxt('example_matrix.txt').astype('int')\n",
    "M = ut.sort(M, M)\n",
    "n_p = M.shape[0]\n",
    "n_a = M.shape[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's visualize it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x144 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(6,2))\n",
    "plt.pcolormesh(M, cmap=plt.cm.Blues)\n",
    "cb= plt.colorbar(fraction=0.04, pad=0.02, aspect=10)\n",
    "cb.ax.get_yaxis().labelpad = 15\n",
    "cb.set_label('Number of interactions', rotation=270)\n",
    "plt.yticks(np.array(range(n_p)) + 0.5, range(1, n_p + 1))\n",
    "plt.xticks(np.array(range(n_a)) + 0.5, range(1, n_a + 1))\n",
    "plt.xlabel('Pollinator identifier');\n",
    "plt.ylabel('Plant identifier');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pystan model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In `pystan`, we write the statistical model in a separate file.\n",
    "In our case, it looks like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "data {\n",
      "  // Dimensions of the data matrix, and matrix itself.\n",
      "  int<lower=1> n_p;\n",
      "  int<lower=1> n_a;\n",
      "  int<lower=0> M[n_p, n_a];\n",
      "}\n",
      "transformed data {\n",
      "  // Pre-compute the marginals of M to save computation in the model loop.\n",
      "  int M_rows[n_p] = rep_array(0, n_p);\n",
      "  int M_cols[n_a] = rep_array(0, n_a);\n",
      "  int M_tot = 0;\n",
      "  for (i in 1:n_p) {\n",
      "    for (j in 1:n_a) {\n",
      "      M_rows[i] += M[i, j];\n",
      "      M_cols[j] += M[i, j];\n",
      "      M_tot += M[i, j];\n",
      "    }\n",
      "  }\n",
      "}\n",
      "parameters {\n",
      "  real<lower=0> C;\n",
      "  real<lower=0> r;\n",
      "  simplex[n_p] sigma;\n",
      "  simplex[n_a] tau;\n",
      "  real<lower=0, upper=1> rho;\n",
      "}\n",
      "model {\n",
      "  // Global sums and parameters\n",
      "  target += M_tot * log(C) - C;\n",
      "  // Weighted marginals of the data matrix \n",
      "  for (i in 1:n_p) {\n",
      "    target += M_rows[i] * log(sigma[i]);\n",
      "  }\n",
      "  for (j in 1:n_a) {\n",
      "    target += M_cols[j] * log(tau[j]);\n",
      "  }\n",
      "  // Pairwise loop\n",
      "  for (i in 1:n_p) {\n",
      "    for (j in 1:n_a) {\n",
      "      real nu_ij_0 = log(1 - rho);\n",
      "      real nu_ij_1 = log(rho) + M[i,j] * log(1 + r) - C * r * sigma[i] * tau[j];\n",
      "      if (nu_ij_0 > nu_ij_1)\n",
      "        target += nu_ij_0 + log1p_exp(nu_ij_1 - nu_ij_0);\n",
      "      else\n",
      "        target += nu_ij_1 + log1p_exp(nu_ij_0 - nu_ij_1);\n",
      "    }\n",
      "  }\n",
      "} \n",
      "generated quantities {\n",
      "  // Posterior edge probability matrix\n",
      "  real<lower=0> Q[n_p, n_a];\n",
      "  for (i in 1:n_p) {\n",
      "    for (j in 1:n_a) {\n",
      "      real nu_ij_0 = log(1 - rho);\n",
      "      real nu_ij_1 = log(rho) + M[i,j] * log(1 + r) - C * r * sigma[i] * tau[j];\n",
      "      if (nu_ij_1 > 0) \n",
      "        Q[i, j] = 1 / (1+ exp(nu_ij_0 - nu_ij_1));\n",
      "      else\n",
      "        Q[i, j] = exp(nu_ij_1) / (exp(nu_ij_0) + exp(nu_ij_1));\n",
      "    }\n",
      "  }\n",
      "}\n"
     ]
    }
   ],
   "source": [
    "with open('model.stan', 'r') as f:\n",
    "    print(f.read())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `data` block contains the model parameters, in our case the dimension of the observation matrix, and the matrix itself.\n",
    "\n",
    "The `transformed data` block allows us to calculate quantities that we'll use later. Here, we calculate the row and column sums of M, as well as its total sum.\n",
    "\n",
    "The `parameter` block contains... well.. the parameters of the model.\n",
    "These parameters are the ones that will be sampled by `Stan`, and returned to us  once we run the program.\n",
    "`sigma` and `tau` are of the \"simplex\" type---just a synonym for normalized distributions.\n",
    "\n",
    "The `model` block calculates the log-posterior of the model. We accumulate this value in the variable `target`.\n",
    "\n",
    "The `generated quantities` block contains other quantities, functions of the sampled parameters, that will be returned to us once we run the program. In this case, we generate the matrix Q of edge probabilities given the parameters."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Compiling the model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To  start using the model, we first need to \"compile\" it.\n",
    "During compilation, `pystan` builds an executable whose purpose is to generate samples from our model, given an input.\n",
    "Compilation takes a bit of time, but we only need to compile the model once. We are free to re-use it as many times as we like afterwards.\n",
    "\n",
    "Our module `plant_pol_inference` (imported as `pp`) provides utilities to handle this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "model.bin [Skipping --- already compiled]\n"
     ]
    }
   ],
   "source": [
    "model = pp.load_model()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sampling"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With the model compiled, we are ready to generate our first samples.\n",
    "\n",
    "Our `plant_pol_inference` module also provide utilities that handle this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING:pystan:n_eff / iter below 0.001 indicates that the effective sample size has likely been overestimated\n",
      "WARNING:pystan:Rhat above 1.1 or below 0.9 indicates that the chains very likely have not mixed\n"
     ]
    }
   ],
   "source": [
    "samples = pp.generate_sample(M, model, warmup=2000, num_samples=500)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The warnings above indicate that at least one chain has gotten stuck in a minima.\n",
    "\n",
    "We can see this by plotting the value of the posterior distribution, evaluated for each samples.\n",
    "\n",
    "Note that in this plot, vertical lines separate Markov chains: All samples between two of these lines are generated sequentially, by an independent chain."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'log posterior')"
      ]
     },
     "execution_count": 7,
     "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.scatter(range(2000), samples['lp__'], s=10, c='k')\n",
    "for sep in [500, 1000, 1500]:\n",
    "    plt.axvline(sep, c='k', ls='--', lw=1)\n",
    "plt.xlabel('Sample ID')\n",
    "plt.ylabel('log posterior')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this particular case, two chains have gotten stuck in an optima with lower posterior probability."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll re-run the sampling until we get a clean sample:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING:pystan:n_eff / iter below 0.001 indicates that the effective sample size has likely been overestimated\n",
      "WARNING:pystan:Rhat above 1.1 or below 0.9 indicates that the chains very likely have not mixed\n",
      "WARNING:pystan:n_eff / iter below 0.001 indicates that the effective sample size has likely been overestimated\n",
      "WARNING:pystan:Rhat above 1.1 or below 0.9 indicates that the chains very likely have not mixed\n"
     ]
    }
   ],
   "source": [
    "while not pp.test_samples(samples, tol=0.05):\n",
    "    samples = pp.generate_sample(M, model, warmup=2000, num_samples=500)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After a few attempt we find a nice set of samples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'log posterior')"
      ]
     },
     "execution_count": 9,
     "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.scatter(range(2000), samples['lp__'], s=10, c='k')\n",
    "for sep in [500, 1000, 1500]:\n",
    "    plt.axvline(sep, c='k', ls='--', lw=1)\n",
    "plt.xlabel('Sample ID')\n",
    "plt.ylabel('log posterior')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the summary of our succesful sampling procedure:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING:pystan:Truncated summary with the 'fit.__repr__' method. For the full summary use 'print(fit)'\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "Warning: Shown data is truncated to 100 parameters\n",
       "For the full summary use 'print(fit)'\n",
       "\n",
       "Inference for Stan model: plant_pol_e215d9f28a31a5a7790444e1b0d7f7ff.\n",
       "4 chains, each with iter=2500; warmup=2000; thin=1; \n",
       "post-warmup draws per chain=500, total post-warmup draws=2000.\n",
       "\n",
       "           mean se_mean     sd   2.5%    25%    50%    75%  97.5%  n_eff   Rhat\n",
       "C         18.52    0.46   7.21    7.2  12.95  17.86  22.88  34.98    247   1.03\n",
       "r         51.14     0.7  22.34  23.42  35.75  46.04  61.27 108.63   1014    1.0\n",
       "sigma[1]   0.03  8.5e-4   0.04 2.4e-3 9.2e-3   0.02   0.04   0.13   1797    1.0\n",
       "sigma[2]   0.02  4.4e-4   0.01 8.9e-3   0.02   0.02   0.02   0.04    904    1.0\n",
       "sigma[3]    0.1  1.6e-3   0.05   0.04   0.07   0.09   0.13   0.24   1131    1.0\n",
       "sigma[4]   0.16  1.2e-3   0.05   0.08   0.13   0.16   0.19   0.27   1716    1.0\n",
       "sigma[5]   0.06  3.6e-4   0.02   0.03   0.05   0.06   0.07   0.09   1788    1.0\n",
       "sigma[6]   0.25  1.7e-3   0.06   0.14   0.21   0.25   0.29   0.38   1282    1.0\n",
       "sigma[7]   0.19  1.1e-3   0.04   0.12   0.17   0.19   0.22   0.28   1470    1.0\n",
       "sigma[8]   0.18  7.8e-4   0.03   0.12   0.15   0.17   0.19   0.24   1631    1.0\n",
       "tau[1]     0.19  2.8e-3   0.04   0.11   0.16   0.19   0.22   0.27    225   1.03\n",
       "tau[2]     0.22  2.9e-3   0.05   0.13   0.19   0.22   0.25   0.31    267   1.03\n",
       "tau[3]     0.05  6.9e-4   0.02   0.02   0.04   0.05   0.06   0.08    526   1.01\n",
       "tau[4]     0.06  8.3e-3   0.08 6.2e-3   0.01   0.02   0.09   0.27     89    1.1\n",
       "tau[5]     0.05  8.7e-4   0.03   0.01   0.03   0.05   0.07   0.12   1130   1.01\n",
       "tau[6]     0.03  5.4e-4   0.01 8.9e-3   0.02   0.03   0.04   0.07    765   1.01\n",
       "tau[7]     0.04  4.8e-4   0.02   0.01   0.02   0.03   0.05   0.08   1374    1.0\n",
       "tau[8]     0.05  8.2e-4   0.03   0.01   0.03   0.05   0.07   0.12   1268   1.01\n",
       "tau[9]     0.02  4.4e-4   0.01 4.6e-3   0.01   0.02   0.03   0.05   1079    1.0\n",
       "tau[10]    0.02  4.0e-4   0.02 4.7e-3   0.01   0.02   0.03   0.06   1682    1.0\n",
       "tau[11]    0.02  1.0e-3   0.02 4.2e-3   0.01   0.02   0.03    0.1    551   1.01\n",
       "tau[12]    0.02  8.0e-4   0.02 2.4e-3 8.2e-3   0.01   0.02   0.09    772   1.01\n",
       "tau[13]    0.03  4.8e-4   0.02 3.5e-3   0.01   0.02   0.03   0.08   1711    1.0\n",
       "tau[14]    0.02  1.1e-3   0.03 2.3e-3 6.4e-3   0.01   0.02   0.11    650   1.01\n",
       "tau[15]    0.02  5.5e-4   0.02 2.7e-3 8.1e-3   0.01   0.02   0.07   1157    1.0\n",
       "tau[16]    0.02  7.8e-4   0.03 1.3e-3 5.9e-3   0.01   0.03   0.11   1358    1.0\n",
       "tau[17]    0.02  4.9e-4   0.02 2.0e-3 8.5e-3   0.02   0.03   0.09   2189    1.0\n",
       "tau[18]    0.03  8.6e-4   0.03 1.1e-3 5.5e-3   0.01   0.04   0.11   1138    1.0\n",
       "tau[19]    0.03  7.3e-4   0.03 2.1e-3   0.01   0.02   0.05   0.11   1674    1.0\n",
       "tau[20]    0.03  9.2e-4   0.03 1.1e-3 5.3e-3   0.01   0.04   0.11   1094    1.0\n",
       "tau[21]    0.02  6.5e-4   0.03 1.4e-3 7.2e-3   0.02   0.03   0.09   1652    1.0\n",
       "rho        0.25  3.0e-3   0.06   0.14   0.21   0.25   0.29   0.37    370   1.02\n",
       "Q[1,1]     0.45    0.01   0.37 4.7e-9   0.02    0.5   0.83   0.95   1335    1.0\n",
       "Q[2,1]      1.0  2.0e-3   0.04    1.0    1.0    1.0    1.0    1.0    500   1.01\n",
       "Q[3,1]   2.8e-4  3.8e-5 1.6e-39.3e-208.1e-10 6.4e-7 2.0e-5 2.7e-3   1694    1.0\n",
       "Q[4,1]   5.3e-3  1.2e-3   0.043.3e-191.4e-10 9.4e-8 1.6e-5   0.03   1344    1.0\n",
       "Q[5,1]      1.0  7.8e-7 3.5e-5    1.0    1.0    1.0    1.0    1.0   2009    1.0\n",
       "Q[6,1]   1.5e-6  1.3e-6 5.9e-54.9e-321.7e-193.1e-159.5e-12 5.5e-7   2009    1.0\n",
       "Q[7,1]      1.0     nan    0.0    1.0    1.0    1.0    1.0    1.0    nan    nan\n",
       "Q[8,1]      1.0     nan    0.0    1.0    1.0    1.0    1.0    1.0    nan    nan\n",
       "Q[1,2]     0.04  1.5e-3   0.064.1e-13 1.8e-4   0.01   0.07   0.21   1611    1.0\n",
       "Q[2,2]     0.99  2.2e-3   0.06   0.95    1.0    1.0    1.0    1.0    712    1.0\n",
       "Q[3,2]   1.6e-4  2.8e-5 1.2e-34.9e-232.8e-11 5.2e-8 4.7e-6 1.2e-3   1898    1.0\n",
       "Q[4,2]   1.8e-6  8.4e-7 3.8e-52.8e-263.1e-167.1e-135.0e-10 2.6e-6   2011    1.0\n",
       "Q[5,2]      1.0 3.0e-121.2e-10    1.0    1.0    1.0    1.0    1.0   1529    1.0\n",
       "Q[6,2]   7.1e-8  3.8e-8 1.7e-61.7e-386.8e-239.7e-187.3e-14 5.0e-8   2013    1.0\n",
       "Q[7,2]   7.7e-6  5.7e-6 2.6e-41.1e-216.4e-154.2e-127.4e-10 2.4e-6   2008    1.0\n",
       "Q[8,2]      1.0     nan    0.0    1.0    1.0    1.0    1.0    1.0    nan    nan\n",
       "Q[1,3]     0.13  2.8e-3   0.09 7.0e-4   0.05   0.13    0.2    0.3   1002    1.0\n",
       "Q[2,3]     0.14  2.2e-3   0.06   0.04    0.1   0.13   0.17   0.25    635   1.01\n",
       "Q[3,3]     0.36  9.5e-3    0.3 4.4e-5   0.06    0.3   0.63   0.89    991    1.0\n",
       "Q[4,3]      1.0  9.1e-8 4.1e-6    1.0    1.0    1.0    1.0    1.0   2022    1.0\n",
       "Q[5,3]     0.04  9.5e-4   0.03 3.4e-3   0.02   0.03   0.06   0.13   1258    1.0\n",
       "Q[6,3]      1.0 8.5e-11 3.8e-9    1.0    1.0    1.0    1.0    1.0   2015    1.0\n",
       "Q[7,3]   1.4e-3  1.1e-4 3.9e-3 1.8e-7 2.5e-5 2.0e-4 1.0e-3   0.01   1290    1.0\n",
       "Q[8,3]      1.0  8.6e-6 3.7e-4    1.0    1.0    1.0    1.0    1.0   1827    1.0\n",
       "Q[1,4]     0.16  8.6e-3   0.11 4.0e-8   0.06   0.18   0.25   0.34    153   1.05\n",
       "Q[2,4]     0.16    0.01    0.1 1.9e-4   0.06   0.19   0.24   0.33    102   1.08\n",
       "Q[3,4]     0.57    0.04   0.392.2e-16   0.01   0.77   0.89   0.96    102   1.08\n",
       "Q[4,4]     0.48    0.03   0.361.9e-23 1.4e-4    0.6   0.81   0.94    128   1.06\n",
       "Q[5,4]     0.11  7.5e-3   0.09 1.9e-9 5.0e-3   0.12   0.18   0.27    131   1.06\n",
       "Q[6,4]     0.65    0.05   0.447.1e-36 5.2e-6   0.95   0.99    1.0     93   1.09\n",
       "Q[7,4]     0.73    0.05   0.443.7e-24 8.9e-3    1.0    1.0    1.0     79   1.11\n",
       "Q[8,4]     0.04  2.6e-3   0.051.0e-24 7.4e-7   0.02   0.06   0.16    308   1.03\n",
       "Q[1,5]     0.13  2.6e-3   0.09 1.9e-4   0.05   0.13    0.2   0.31   1178    1.0\n",
       "Q[2,5]     0.84  3.6e-3   0.12   0.52   0.81   0.87   0.92   0.96   1061   1.01\n",
       "Q[3,5]      1.0  1.7e-3   0.05   0.99    1.0    1.0    1.0    1.0    787    1.0\n",
       "Q[4,5]     0.01  6.9e-4   0.02 1.1e-9 3.3e-5 8.3e-4 7.9e-3   0.09   1229    1.0\n",
       "Q[5,5]     0.05  1.4e-3   0.05 4.1e-4   0.01   0.03   0.07   0.19   1247    1.0\n",
       "Q[6,5]     0.08  5.6e-3   0.183.6e-13 6.3e-6 8.3e-4   0.04    0.7   1034    1.0\n",
       "Q[7,5]   5.8e-3  4.7e-4   0.021.7e-10 4.8e-6 1.9e-4 2.8e-3   0.05   1282    1.0\n",
       "Q[8,5]   6.9e-3  5.2e-4   0.02 1.6e-9 1.7e-5 4.2e-4 4.1e-3   0.07   1226    1.0\n",
       "Q[1,6]     0.17  3.1e-3   0.09 4.5e-3    0.1   0.17   0.23   0.33    774    1.0\n",
       "Q[2,6]     0.18  2.5e-3   0.06   0.07   0.13   0.17   0.22   0.31    615   1.01\n",
       "Q[3,6]     0.06  1.7e-3   0.06 6.3e-5   0.01   0.05   0.09    0.2   1214   1.01\n",
       "Q[4,6]     0.03  7.6e-4   0.03 2.1e-5 3.1e-3   0.01   0.03   0.12   1842    1.0\n",
       "Q[5,6]      0.1  1.7e-3   0.06   0.01   0.05   0.09   0.13   0.23   1069   1.01\n",
       "Q[6,6]      1.0  1.6e-6 7.2e-5    1.0    1.0    1.0    1.0    1.0   1989    1.0\n",
       "Q[7,6]     0.02  6.5e-4   0.03 7.7e-6 1.1e-3 5.8e-3   0.02   0.09   1660    1.0\n",
       "Q[8,6]     0.02  6.6e-4   0.03 3.4e-5 2.1e-3 8.9e-3   0.03   0.09   1704    1.0\n",
       "Q[1,7]     0.16  2.9e-3   0.09 2.6e-3   0.09   0.16   0.22   0.32    941    1.0\n",
       "Q[2,7]     0.16  2.4e-3   0.06   0.05   0.12   0.16    0.2    0.3    644   1.01\n",
       "Q[3,7]     0.05  1.4e-3   0.05 3.4e-5 6.8e-3   0.03   0.07    0.2   1427    1.0\n",
       "Q[4,7]      1.0  3.8e-6 1.6e-4    1.0    1.0    1.0    1.0    1.0   1771    1.0\n",
       "Q[5,7]     0.08  1.6e-3   0.06 4.2e-3   0.04   0.07   0.11   0.22   1257    1.0\n",
       "Q[6,7]   6.9e-3  4.2e-4   0.02 4.2e-9 3.2e-5 6.2e-4 4.9e-3   0.05   1878    1.0\n",
       "Q[7,7]     0.01  6.1e-4   0.02 3.0e-7 2.4e-4 2.4e-3   0.01   0.08   1682    1.0\n",
       "Q[8,7]     0.01  6.0e-4   0.03 1.9e-6 5.7e-4 3.7e-3   0.02   0.08   1760    1.0\n",
       "Q[1,8]     0.13  2.5e-3   0.09 8.5e-5   0.05   0.14   0.21   0.31   1271    1.0\n",
       "Q[2,8]     0.14  1.9e-3   0.06   0.02   0.09   0.13   0.18   0.28   1180    1.0\n",
       "Q[3,8]      1.0  3.4e-6 1.5e-4    1.0    1.0    1.0    1.0    1.0   2033    1.0\n",
       "Q[4,8]     0.01  6.3e-4   0.023.1e-10 3.2e-5 1.2e-3 9.5e-3   0.09   1346    1.0\n",
       "Q[5,8]     0.05  1.4e-3   0.05 3.6e-4   0.01   0.04   0.08   0.18   1301    1.0\n",
       "Q[6,8]   3.9e-3  3.4e-4   0.011.7e-15 1.6e-7 3.6e-5 1.2e-3   0.04   1429    1.0\n",
       "Q[7,8]   5.5e-3  3.9e-4   0.011.0e-10 7.4e-6 3.0e-4 3.4e-3   0.05   1290    1.0\n",
       "Q[8,8]   6.8e-3  4.4e-4   0.025.3e-10 2.2e-5 5.6e-4 5.3e-3   0.06   1242    1.0\n",
       "Q[1,9]     0.18  3.0e-3   0.08   0.01   0.13   0.19   0.24   0.34    765   1.01\n",
       "Q[2,9]     0.19  2.8e-3   0.06   0.08   0.15   0.19   0.23   0.33    501   1.01\n",
       "Q[3,9]     0.09  2.0e-3   0.07 5.8e-4   0.03   0.08   0.13   0.25   1268   1.01\n",
       "lp__      76.27    0.15   4.06  67.41  73.62  76.64  79.21  83.15    693    1.0\n",
       "\n",
       "Samples were drawn using NUTS at Wed Jul 31 18:12:51 2019.\n",
       "For each parameter, n_eff is a crude measure of effective sample size,\n",
       "and Rhat is the potential scale reduction factor on split chains (at \n",
       "convergence, Rhat=1)."
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "samples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It shows that we can safely ignore the `Rhat` warning, because, as we see, the warning is triggered by the fact that all the samples agree on some entries of the matrix Q.\n",
    "\n",
    "For example, Q[7,1], is always equal to 1. That is, there is such a strong evidence for the link (7,1) that all samples place an edge between these two nodes. `stan` does not see any variances and  throws a warning. In our case this is by design. \n",
    "\n",
    "To see this, let's inspect the entry of the data matrix corresponding to this node pair:\n",
    "\n",
    "(*Note*: `stan` indices start from 1 not 0. Hence Q[7,1] is associated with M[6,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "30"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M[6,0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Indeed, we see that there were 30 observations for this entry---no wonder the samples all place an edge between these nodes!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now have well-behaved samples, let's move on to model checking and estimation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Posterior-predictive check"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We provide a simple function that calculates the posterior predictive matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "M_tilde = pp.get_posterior_predictive_matrix(samples)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's compare it with the input"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x144 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x144 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(6,2))\n",
    "plt.title('Input matrix')\n",
    "plt.pcolormesh(M, cmap=plt.cm.Blues)\n",
    "cb= plt.colorbar(fraction=0.04, pad=0.02, aspect=10)\n",
    "cb.ax.get_yaxis().labelpad = 15\n",
    "cb.set_label('Number of interactions', rotation=270)\n",
    "plt.yticks(np.array(range(n_p)) + 0.5, range(1, n_p + 1))\n",
    "plt.xticks(np.array(range(n_a)) + 0.5, range(1, n_a + 1))\n",
    "plt.xlabel('Pollinator identifier');\n",
    "plt.ylabel('Plant identifier');\n",
    "plt.figure(figsize=(6,2))\n",
    "plt.title('Posterior predictive')\n",
    "plt.pcolormesh(M_tilde, cmap=plt.cm.Blues)\n",
    "cb= plt.colorbar(fraction=0.04, pad=0.02, aspect=10)\n",
    "cb.ax.get_yaxis().labelpad = 15\n",
    "cb.set_label('Number of interactions', rotation=270)\n",
    "plt.yticks(np.array(range(n_p)) + 0.5, range(1, n_p + 1))\n",
    "plt.xticks(np.array(range(n_a)) + 0.5, range(1, n_a + 1))\n",
    "plt.xlabel('Pollinator identifier');\n",
    "plt.ylabel('Plant identifier');\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "They are quite close!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "More systematically we can check the distribution of the differences between the posterior predictive and the input:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Count')"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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jiO8neSxwdZKHARu6UcWpI/RzA4MAeeSJ+gO4nz63MpgCqWdep6ITSnIN8MHjH0uY5CjwCODpwEOA/wFOr6r3r7CfXwFey+B5uvfbX1V9J8mzj+8zyY3Ay6rq8Er6VztOf3RCVfXipZ5zWlWbu2nLLwG3AU8E5kfo5xYGo6Jty/TH8X0m2Qg80EBZHZz+aFSPBA4C51fVSOc0qurxI/T5tFH6VjuOVDSSqtoO/CFw11ruU8PznIqkphypSGrKUJHUlKEiqSlDRVJThoqkpgwVSU0ZKpKaMlQkNWWoSGrq/wH3XxU05UcbwQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(4,4))\n",
    "plt.hist((M-M_tilde).flatten(), density=False, bins=20, rwidth=0.9);\n",
    "\n",
    "plt.xlabel(r'$M_{ij}-\\langle \\tilde{M}_{ij}\\rangle$')\n",
    "plt.ylabel('Count')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or compare the posterior predictive value to the input directly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-4.9, 38)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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cKwJxWlu2AZfkemIzW0z4om2J9elwzjWeyOQh6SYz+6GkR6H2ao5mdkaikTnnilq2ksfdwbN35nLO1RKZPMxsafCy0sxuzvxM0qXA00kG5pwrbnFaW8aFbBvfwHE450pMtjqPscA3ge6SHsn4qA3wQdKBOeeKW7Y6j+dJ9QBtx769QHcALyUZlHOu+GWr81gPrJd0DrDRzP4JIKkM6ESqj4ZzromKU+dxP59OPwippRgeSCYc51ypiJM8WpjZv9Jvgte5zOvhnNsPxUkeWyRVdwiTdCbwfnIhOedKQZyxLROB2ZJuJdXd/G3gW4lG5ZwrenHGtvwdGBjMyyEz25F8WM65Yhen5IGkEUAPoGVqpD2Y2U8TjMs5V+TiTEN4O/AN4AekbltGA10Tjss5V+TiVJh+2cy+BWwLJkMeBHRONiznXLGLkzz+GTzvknQosBvonlxIzrlSECd5PCqpLTAdWEaqZ+m9dR0kqbOkJyX9TdIrwUhcJFVIelzSa8Fz6OzpzrniljV5SGoGLDKz7Wb2B1J1HUea2TUxzp1e9OkoYCDwfUlfAiYF5/wisCh475wrMVmTRzDD+Q0Z7z82s1jr1prZJjNbFrzeAaQXfToTmBXsNgs4K4+4nXMFFue2ZYGk/1C6jTYPNRZ9OiRYTS69qtzB+Z7XOVc4cfp5XEZqnZY9kqpINdeamR0U5wI1F32Km4MkTQAmAHTp0iXWMc65xlNnycPM2phZMzM7wMwOCt7HTRy1Fn0C3kuv3RI8b464rq8Y51wRi9NJTJLOlXR18L6zpDpXeYta9Al4hE+nNhwHPJx72M65QotT53EbqY5h3wze7wR+FeO49KJPJ0paETxOBaYCJ0l6DTgpeO+cKzFx6jyONbN+kpZDahEoSXXO55Fl0SeAr+YQo3OuCMUpeeyW1Jxg4SdJ7dl3ZjHnXBMUJ3ncAjwEHCzpF8Bi4PpEo3LOFb0483nMlrSU1K2GgLPM7G+JR+acK2p1Jg9Jd5vZecDqkG3OuSYqzm1Lj8w3Qf1H/2TCcc6VisjkIWmypB1Ab0n/kLQjeL8Z75vhXJMXmTzM7HozawNMz+hZ2sbMPmdmkxsxRudcEYpTYTpZUkdSw/FbZGx/JsnAnHPFLU6F6VRgDPAqqdXiINXnw5OHc01YnB6mZwNHmNnHSQfjnCsdcVpb3gAOSDoQ51xpiVPy2AWskLQIqC59mNkliUXlnCt6cZLHI8HDOeeqxWltmVXXPs65picyeUi638y+LmkVwYjaTGbWO9HInHNFLVvJ49Lg+bTGCMQ5V1oik0fGDOfrGy8c51ypiNNUmxdJd0raLOnljG1TJG2oMS2hc64EJZY8gLuA4SHbZ5hZZfCYl+D1nXMJyjaqdlHwPC2fEwdjX7bmGZdzrshlK3l0kHQCcIakvpL6ZT7qcc2LJb0U3NZELnItaYKkJZKWbNmypR6Xc84lQWa1WmFTH0ijgG8DxwFLanxsZnZinSdPLTP5mJn1DN4fArxPqun3Z0AHM7ugrvMMGDDAliypGYJzLmmSlprZgLDPsrW2zAHmSLrazH7WEIGY2XsZQf0WeKwhzuuca3xxepj+TNIZwPHBpqfMLK9fekkd0k3ApEbrvpxtf+dc8Yozn8f1wDHA7GDTpZIG1zWbmKR7ga8A7SS9A1wLfEVSJanblnXARfmH7pwrpDgD40YAlWa2F0DSLGA5kDV5mNnYkM135Byhc64oxe3n0Tbj9WeTCMQ5V1rilDyuB5ZLepLUok/HU0epwzm3/4tTYXqvpKeAo0kljyvN7N2kA3POFbc4JY/0IDmfEMg5Vy3JsS3Ouf2YJw/nXF6yJg9JzTKH1O8PKioqkFTrUVFRUejQnCspWes8zGyvpJWSupjZW40VVJK2bdtG2HgeSQWIxrnSFafCtAPwiqS/AB+lN5rZGYlF5ZwrenGSx3WJR+GcKzlx+nk8Lakr8EUzWyjpQKB58qE554pZna0tkr4DzAF+E2zqCMxNMqhC8YpU5+KLc9vyfVKjal8EMLPXJB2caFQJKi8vj6wc9YpU5+KLkzw+NrN/pX+JJLUgZBGoUrF1a/i0qp4knMtNnE5iT0v6CVAm6STgAeDRZMNyzhW7OCWPSaTmMl1FavKeecD/JBlU0uYu38D0+WvYuL2KQ9uWccWwIwodknMlJ05ry95gAqAXSd2urLGoWZMzSLqT1FKVmzMmQK4Afg90IzWT2NfNbFve0edh7vINTH5wFVW79wCwYXsVkx9cVf2ZJxXn4okzDeEI4Hbg76SG5HeXdJGZ/amOQ+8CbgX+N2PbJGCRmU2VNCl4f2U+gedr+vw11YkjrWr3Hpq3bM3Z/TpVb1sHnD0ZWh/kcx85FybObcsNwFAzex1A0heAPwJZk4eZPRMsvZDpTFLzmgLMAp6ikZPHxu1Vods7XXpf6PaObcuSDMe5khWnwnRzOnEE3gA253m9QzIW0N4ERDb5JrXo06E5JoOoZONcU5dtucmRkkaSGtcyT9J4SeNItbT8NenAzGymmQ0wswHt27dvsPNeMewIyg7Yt4Ns2QHNKT/wgND9c002zjUV2W5bTs94/R5wQvB6CxC5TGQd3kuv3SKpA/mXYPJ2Vt+OAKEVo5kVqZBKKl5p6ly4bCvGnZ/A9R4BxgFTg+eHE7hGnc7q27E6idQUllQGT31in21RxzrXlESuVVu9g9Qd+AGp5tXqZFPXkPzMRZ9IlVyuJTUm5n6gC/AWMNrMwrt8ZijUWrU1m3UhVRq5fmQvTyCuSchrrdoMc0kt1vQosDfuRSMWfQL4atxzFFpUs+70+Ws8ebgmL07y+KeZ3ZJ4JEUoqqXFW2Cci5c8bpZ0LbAA+Di90cyWJRZVkXjnlrF8UrWj1vYWZW1g6j8KEJFzxSNO8ugFnAecyKe3LRa83699UrWDI6/6U606j9U/P6WAUTlXHOJ0Ejsb+HczO8HMhgaP/T5xpF0/shcd25YhUr1Nrx/Zq9AhOVcU4pQ8VpJa6LrR+2QUg2zNus41ZXGSxyHAakl/Zd86D5893bkmLE7yuDbxKEpQ2Mxj5eXl3LloVa2OZl5ycfujWLOnN0YgxSif+U6j5grxBOL2N3FmT98h6R/B45+S9khqEu2UW7duxcxqPbKJ6lTm3P4mTsmjTeZ7SWeRmk3dxeSdytz+KE6dxz7MbG4wC1iTlsuUhT6s3+2P4kxDODLjbTNgACW89EJDiarbKDuguQ/rd01CnFG1v8t4+wmp6T1/a2aN1u+jUKNqoxxw4EGh3dZzUV5eHrmGjHPFol6jahOa16Okdbrk3tCi1/ppp9WqUE231kRtd65URSYPSddkOc7M7GcJxFMSDm1bxoYcK0FrTijkXKnL1lT7UcgDUgtANeqM58Umah7UbDZsr8LYt37EuVKWbRrCG9KvJbUBLgXOB+4jtRxD3iStA3YAe4BPou6pilXUPKhn/zze8TX7gjhXirLWeQQrvF0GnENqnZV+DbjC21Aze7+BztXoGmLAXNx6D69cdcUoW53HdGAkMBPoZWY7Gy2qEpWtO/v6aafV2uaVq66UZSt5/JjUKNqrgP/M+A8sUhWmB9XjugYskGTAb8xsZj3OVTTCSgdRkyj7hEKu1GWr84gzUVC+BpvZRkkHA49LWm1mz2TuIGkCMAGgS5cuCYaSrHzqR7xlxpWCOjuJJR6ANAXYaWb/HbVPsXUSawiSIm9Pul75WPW2dCml0P9OrmnK1kksydJFVDCtgtYbJLUCTgZebuw4SkX6dkdSrEdFRUWBI3ZNRc4D4xrAIcBDwV/ZFsA9ZvZ/BYijoHKtXK05EXN6H69cdYXS6MnDzN4A+jT2dYvNnYtW1apIFeEjDtdPOy2nviFhI359MiLX0ApR8nCEr0Zn1E4gdfVcDeOzmbnG0Oh1Hi4laoIgg5yWehg89Qm6T/ojg6c+wdzlGwCfzcw1Di95FEjU4LqObct4blLtZXFqzhOSlj5HZgkjrM5kPaDJ8WILawkC7+nq9uXJo0CuGHZEaOexsH4d5eXlkZ3KwhJFZlNv5n5Rlatfvn7Rvn1Q+nXioWXv1O6b0q9TTt/R7d88eRRIVOexsHqJsL/23Sf9MbJyNWw2s2zCSi9R9SbOpXnyKKD6DK7LNqfI9SN71XvEb1i9iXOZPHmUqKjbHohOSjWbcPNRs+t8tuTnTcb7N08eJSqfMTM1b0Vy8fbNYwB4fvJXgdREtmdPhujeKftu/3T/kD0jKmjru69X8Car4GNb4tgfx7YkpaKigm3bGmrKlX01a9mazpfet8+2sIpYSP2S16y4TVfuhlXQ5jLOp2ZlbjqhhZ2jFP5/F7N6TYDsSkvUX9qoClZINQ/XvLWoecvx/OSv1koc+QqroI0jXWqKKkH5aOTG5cmjici1X0nNepO4fURylU9FbNQx+SYllx/vYdpERE3anMtf6Fwnfc6na31D8dah5HnyaCLO6tuR60f2qtX1PZfWj7Djc9m/vvJJRmHTFjRr1iz2FAcNsW9jX68hY8v6sy2FCiWvMC28qIrYqErJ3Coro1ps6rtv7akMoiZXyqW3bbqCNk7Fb/rcUecIqxCOmn6hvrFF9RrOVgHtFaau3qIqYqPmaI0q1UT1/cilT0jNfYce2Z4/LN0QmiSiOrtF9XmJ09s2275R4jSTR3XQa6jYonoN59uE78nD1UvY1ALpUbxhv/xhHdhqJqC6phEIO8eArhUN3uelrl/msH3DkmDcc+Qin9iiEmnUOQZPfYLPfP6w/lHn9eTh6iVqaoGo7WFyTUBhcu3qn1SF6v40JqiuUkhBKkwlDZe0RtLrkiYVIgbXMKK6uefS/b0hElCUXFuI6qs+Y4LSsRWylSoXjV5hKqk5sBY4CXgH+Csw1sxejTrGK0yLV651HmEGT30ipz4ocSXZ2zYpxdYr9jOfP4yPN70W2uxSiNuWY4DXg7lMkXQfcCYQmTxc8cplaoEoucxtkotcK3n/o39Hnly9JVZlLtT+ztPnr4lMgunP8/0Z5VLRHBZbtkrpmtt3/esTtu3aDcCmWT+MjKkQJY9RwHAzuzB4fx5wrJldXGO/zEWf+q9fv75R43SNq7FH4CZxvYYohRWDzO+xadYPi6rkERZIrQwWLEE5E1K3LUkH5QqrIRYOL/T1GqIUVgwyv8emLPsVouQxCJhiZsOC95MBzOz6qGO8zsO5wiiqFeNIVZB+UVJ3SZ8BxgCPFCAO51w9FGLRp08kXQzMB5oDd5rZK40dh3OufgrSSczM5gHzCnFt51zD8FG1zrm8ePJwzuXFk4dzLi+ePJxzefHk4ZzLiycP51xePHk45/LiycM5lxdPHs65vHjycM7lxZOHcy4vnjycc3nx5OGcy4snD+dcXjx5OOfy4snDOZcXTx7Oubx48nDO5cWTh3MuL42+9EI+JG0BfNUn5xpfVzNrH/ZBSSQP51zx8dsW51xePHk45/LiycM5lxdPHkVI0h5JKyS9LOkBSQcG23fW45zjJR3acFGCpCmSLg9e3yVpVPD6fyR9Kc9zfkXSlxsyzpjXnSepbWNft5R58ihOVWZWaWY9gX8BExvgnOOBBk0eUczsQjN7Nc/DvwLklDwk1XvlQzM71cy21zivJPnvSAT/wRS/Z4HDMjdIai1pkaRlklZJOjPY3k3S3yT9VtIrkhZIKgtKBAOA2UGJpizsQpLWSZom6S/B47Bge9fgei8Fz12yBSzpKUkDgtc7Jf1C0kpJL0g6JNh+uqQXJS2XtFDSIZK6kUqUPwriHBJ17aCkc6OkJ4FpNa4/XtLDkv5P0hpJ12Z8NlfS0uDnM6HGd2+X8TO8DVgGdA6u9XLws/5R3f9kTYSZ+aPIHsDO4LkF8DDw3ZDtBwWv2wGvAwK6AZ8AlcFn9wPnBq+fAgbUcd11wH8Gr78FPBa8fhQYF7y+AJgbvJ4CXB68vgsYVfNagAGnB6//H3BV8LqcT7sKXAjcUPOcdVz7LuAxoHnI9xgPbAI+B5QBL2fEUxE8p7d/LuO7twt+hnuBgcH2/sDjGeduW+j/H8Xy8JJHcSqTtAJYArwF3FHjcwH/JeklYCHQETgk+OxNM1sRvF5K6pchF/dmPA8KXg8C7gle3w0cl8P5/kXql7xmPJ2A+ZJWAe+Of58AAAG/SURBVFcAPSKOz3btB8xsT8Rxj5vZB2ZWBTyYcdwlklYCLwCdgS+GHLvezF4IXr8B/LukX0oaDvwj4npNTr3vFV0iqsysMsvn5wDtgf5mtlvSOqBl8NnHGfvtIfUXdh+SjgV+k7FpuZmdH7zO7DUY1YMwl56Fuy34kx3Ek/4/90vgRjN7RNJXSJU44si89kcx9wOw4DpfAwaZ2S5JT/Hpzy1T9XnNbJukPsAw4PvA10mVgJo8L3mUps8Cm4PEMRToGuOYHUAbADN70VIVsunH+Rn7fSPj+c/B6+eBMcHrc4DF9f4Gqe+wIXg9LizOel77JEkVQf3OWcBzwTW3BYnjSGBgXSeR1A5oZmZ/AK4G+sW8/n7PSx6laTbwqKQlwApgdYxj7gJul1RF6i9vVcR+/ybpRVJ/WMYG2y4B7pR0BbAFOD/i2FxMAR6QtIHULUT3YPujwJygEvgH9bj2YlK3OYcB95jZkuAWaWJwu7cmuG5dOgK/y2h1mRzz+vs9H9viqgW3PwPM7P1Cx1IfksaT+h4XFzqW/Znftjjn8uIlD+dcXrzk4ZzLiycP51xePHk45/LiycM5lxdPHs65vPx/DZMazFjtQ+IAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(4,4))\n",
    "sort_order = np.argsort(-M.flatten())\n",
    "\n",
    "plt.scatter(range(1, len(sort_order)+1), M.flatten()[sort_order], marker='s', facecolors='none', edgecolors='k', label='Data')\n",
    "plt.scatter(range(1, len(sort_order)+1), M_tilde.flatten()[sort_order], label='Posterior-predictive', zorder=-10)\n",
    "\n",
    "plt.xlim(0,40)\n",
    "plt.xlabel('Plant–pollinator pairs')\n",
    "plt.xticks([])\n",
    "plt.ylabel(r'Number of interactions')\n",
    "plt.legend(frameon=False, markerfirst=False, loc=1)\n",
    "plt.ylim(-4.9,38)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All these point of view confirm one thing: The model fits well.\n",
    "\n",
    "Which means we're ready to calculate some estimates."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Estimates"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We provide a simple function to get the network estimate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "edge_prob = pp.estimate_network(samples)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x144 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(6,2))\n",
    "\n",
    "plt.pcolormesh(edge_prob, cmap=plt.cm.Blues, vmin=0, vmax=1)\n",
    "cb= plt.colorbar(fraction=0.04, pad=0.02, aspect=10)\n",
    "cb.ax.get_yaxis().labelpad = 15\n",
    "cb.set_label('Edge probability', rotation=270)\n",
    "plt.yticks(np.array(range(n_p)) + 0.5, range(1, n_p + 1))\n",
    "plt.xticks(np.array(range(n_a)) + 0.5, range(1, n_a + 1))\n",
    "plt.xlabel('Pollinator identifier');\n",
    "plt.ylabel('Plant identifier');\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The parameters themselves are contained in the `samples` object:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.21464613, 0.14904356, 0.36833021, ..., 0.2382869 , 0.25922686,\n",
       "       0.24231355])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "samples['rho']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we can, for example, visualize posterior distributions as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Count')"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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/riSvYPbk5iSH6bx9rqrjq2qqqqaALwHvXMiBAhMwUqmnmLqf5M+73z+T5HnADPAc4JdJzmf2LPojYyv8EAzTZ+Cvgd8ALu7+ne2uCb2rFYbu8+uBtyb5BfB/wJsGTtxOnCH7PHGcpi+pqUk4/JE0QQwVSU0ZKpKaMlQkNWWoSGrKUJHUlKEiqSlDRWOR5J1Jbk9yf5J3jbsetePkN41cktcDfwqcxez9LD8EnldVu8damJpY8NP0tSj9BfD2qvoFsKObdu+oeZHwL1IjleTpwEv33gSZZAXw0+4hRVoEDBWN2gnAc5O8IMnTgL8BPjXmmtSQoaJRezlwOXAFcBvwQM2+H1uLhOdUNGprgH+uqi+OuxD1w5GKRm0NcOu4i1B/vKQsqSlHKpKaMlQkNWWoSGrKUJHUlKEiqSlDRVJThoqkpv4flM8f1qZVJ9sAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(4,4))\n",
    "plt.hist(samples['rho'], density=False, bins=20, rwidth=0.9);\n",
    "\n",
    "plt.xlabel(r'$\\rho$')\n",
    "plt.ylabel('Count')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And find posterior averages with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Posterior average of the connectance: 0.249840802684265\n"
     ]
    }
   ],
   "source": [
    "print('Posterior average of the connectance:', np.mean(samples['rho'], axis=0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The same thing goes for, say, the plant abundances (now a dimension `samples * n_p` object---hence why we average along the axis 0, the `samples` axis):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "mean_sigma = np.mean(samples['sigma'], axis=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Effective abundance')"
      ]
     },
     "execution_count": 22,
     "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.bar([i + 1 for i in range(n_p)], mean_sigma, color='orange')\n",
    "plt.xlabel('Plant ID')\n",
    "plt.ylabel('Effective abundance')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can do the same with the pollinators:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "mean_tau = np.mean(samples['tau'], axis=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Effective abundance')"
      ]
     },
     "execution_count": 24,
     "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.bar([i + 1 for i in range(n_a)], mean_tau, color='orange')\n",
    "plt.xlabel('Pollinator ID')\n",
    "plt.ylabel('Effective abundance')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The last thing we'll calculate is the posterior distribution of **some arbitrary function** of the incidence matrix.\n",
    "\n",
    "To make this example simple, we'll consider the number of non-zero entries, i.e.,\n",
    "\n",
    "$$f(B) = \\sum_{ij} B_{ij}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The python definition of this function is simple:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "def edge_sum(B):\n",
    "    return np.sum(B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, we provide a simpler interface to make the calculation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "posterior_values_edge_sum = pp.get_network_property_distribution(samples, edge_sum)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We find, in this case:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Count')"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "plt.figure(figsize=(4,4))\n",
    "plt.hist(posterior_values_edge_sum, density=False, bins=20, rwidth=0.9);\n",
    "\n",
    "plt.xlabel(r'$\\sum_{ij} B_{ij}$')\n",
    "plt.ylabel('Count')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Further information: \n",
    "\n",
    "\"*[Reconstruction of plant–pollinator networks from observational data](https://doi.org/10.1038/s41467-021-24149-x)*\"<br/>\n",
    "[J.-G. Young](http://jgyoung.ca), [F. S. Valdovinos](https://www.fsvaldovinos.com/) and [M.E.J. Newman](http://www-personal.umich.edu/~mejn/)<br/>\n",
    "Nature Communications **12**, 3911 (2021)\n",
    "\n",
    "\n",
    "Or get in touch at <jean-gabriel.young@uvm.edu> and on [github](https://github.com/jg-you/plant-pol-inference)!"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.10.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}