{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline\n",
    "rc('text', usetex=True)\n",
    "rc('axes', labelsize=15, titlesize=15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import seaborn as sns\n",
    "sns.set_style(\"white\")\n",
    "np.random.seed(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# ABC PMC on a 2D gaussian example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this example we're looking at a dataset that has been drawn from a 2D gaussian distribution. We're going to assume that we don't have a proper likelihood but that we know the covariance matrix $\\Sigma$ of the distribution. Using the ABC PMC algorithm we will approximate the posterior of the distribtion of the mean values."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First we generate a new dataset by drawing random variables from a mulitvariate gaussian around mean=[1.1, 1.5]. This is going to be our observed data set"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.colorbar.Colorbar instance at 0x10c542950>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jakeret/Library/Python/2.7/lib/python/site-packages/matplotlib/collections.py:590: FutureWarning: elementwise comparison failed; returning scalar instead, but in the future will perform elementwise comparison\n",
      "  if self._edgecolors == str('face'):\n"
     ]
    },
    {
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       "AABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x107413ad0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "samples_size = 1000\n",
    "sigma = np.eye(2) * 0.25\n",
    "means = [1.1, 1.5]\n",
    "\n",
    "data = np.random.multivariate_normal(means, sigma, samples_size)\n",
    "matshow(sigma)\n",
    "title(\"covariance matrix sigma\")\n",
    "colorbar()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then we need to define our model/simulation. In this case this is simple: we draw again random variables from a multivariate gaussian distribution using the given mean and the sigma from above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def create_new_sample(theta):\n",
    "    return np.random.multivariate_normal(theta, sigma, samples_size)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we need to define a distance measure. We will use the sum of the absolute differences of the means of the simulated and the observed data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def dist_measure(x, y):\n",
    "    return np.sum(np.abs(np.mean(x, axis=0) - np.mean(y, axis=0)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Verification"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To verify if everything works and to see the effect of the random samples in the simulation we compute the distance for 1000 simulations at the true mean values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "distances = [dist_measure(data, create_new_sample(means)) for _ in range(1000)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10c7317d0>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x107413310>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.distplot(distances, axlabel=\"distances\", )\n",
    "title(\"Variablility of distance from simulations\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Setup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we're going to set up the ABC PMC sampling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import abcpmc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a prior we're going to use a gaussian prior using our best guess about the distribution of the means. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "prior = abcpmc.GaussianPrior(mu=[1.0, 1.0], sigma=np.eye(2) * 0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As threshold $\\epsilon$ we're going to use the $\\alpha^{th}$ percentile of the sorted distances of the particles of the current iteration. The simplest way to do this is to define a constant $\\epsilon$ and iteratively adapt the theshold. As starting value we're going to define a sufficiently high value so that the acceptance ratio is reasonable and we will sample for T iterations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "alpha = 75\n",
    "T = 20\n",
    "eps_start = 1.0\n",
    "eps = abcpmc.ConstEps(T, eps_start)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we create an instance of your sampler. We want to use 5000 particles and the functions we defined above. Additionally we're going to make use of the built-in parallelization and use 7 cores for the sampling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sampler = abcpmc.Sampler(N=5000, Y=data, postfn=create_new_sample, dist=dist_measure, threads=7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Optionally, we can customize the proposal creation. Here we're going to use a \"Optimal Local Covariance Matrix\"-kernel (OLCM) as proposed by (Fillipi et al. 2012). This has shown to yield a high acceptance ratio togheter with a faster decrease of the thresold."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sampler.particle_proposal_cls = abcpmc.OLCMParticleProposal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sampling"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we're ready to sample. All we need to do is to iterate over the yielded values of your sampler instance. The sample function returns a namedtuple per iteration that contains all the information that we're interestend in"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def launch():\n",
    "    eps = abcpmc.ConstEps(T, eps_start)\n",
    "\n",
    "    pools = []\n",
    "    for pool in sampler.sample(prior, eps):\n",
    "        print(\"T: {0}, eps: {1:>.4f}, ratio: {2:>.4f}\".format(pool.t, eps(pool.eps), pool.ratio))\n",
    "\n",
    "        for i, (mean, std) in enumerate(zip(*abcpmc.weighted_avg_and_std(pool.thetas, pool.ws, axis=0))):\n",
    "            print(u\"    theta[{0}]: {1:>.4f} \\u00B1 {2:>.4f}\".format(i, mean,std))\n",
    "\n",
    "        eps.eps = np.percentile(pool.dists, alpha) # reduce eps value\n",
    "        pools.append(pool)\n",
    "    sampler.close()\n",
    "    return pools\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "T: 0, eps: 1.0000, ratio: 0.3907\n",
      "    theta[0]: 1.0648 ± 0.3761\n",
      "    theta[1]: 1.3485 ± 0.3746\n",
      "T: 1, eps: 0.8351, ratio: 0.5106\n",
      "    theta[0]: 1.0694 ± 0.2965\n",
      "    theta[1]: 1.3742 ± 0.2963\n",
      "T: 2, eps: 0.6666, ratio: 0.5363\n",
      "    theta[0]: 1.0802 ± 0.2421\n",
      "    theta[1]: 1.4044 ± 0.2410\n",
      "T: 3, eps: 0.5282, ratio: 0.5230\n",
      "    theta[0]: 1.0808 ± 0.1926\n",
      "    theta[1]: 1.4349 ± 0.1924\n",
      "T: 4, eps: 0.4158, ratio: 0.5268\n",
      "    theta[0]: 1.0863 ± 0.1570\n",
      "    theta[1]: 1.4595 ± 0.1524\n",
      "T: 5, eps: 0.3299, ratio: 0.5270\n",
      "    theta[0]: 1.0855 ± 0.1257\n",
      "    theta[1]: 1.4696 ± 0.1248\n",
      "T: 6, eps: 0.2624, ratio: 0.4958\n",
      "    theta[0]: 1.0926 ± 0.0993\n",
      "    theta[1]: 1.4822 ± 0.0999\n",
      "T: 7, eps: 0.2085, ratio: 0.5031\n",
      "    theta[0]: 1.0904 ± 0.0780\n",
      "    theta[1]: 1.4843 ± 0.0805\n",
      "T: 8, eps: 0.1663, ratio: 0.5006\n",
      "    theta[0]: 1.0906 ± 0.0633\n",
      "    theta[1]: 1.4885 ± 0.0652\n",
      "T: 9, eps: 0.1322, ratio: 0.4851\n",
      "    theta[0]: 1.0905 ± 0.0520\n",
      "    theta[1]: 1.4903 ± 0.0516\n",
      "T: 10, eps: 0.1062, ratio: 0.4743\n",
      "    theta[0]: 1.0919 ± 0.0430\n",
      "    theta[1]: 1.4917 ± 0.0417\n",
      "T: 11, eps: 0.0846, ratio: 0.4652\n",
      "    theta[0]: 1.0915 ± 0.0346\n",
      "    theta[1]: 1.4931 ± 0.0347\n",
      "T: 12, eps: 0.0683, ratio: 0.4381\n",
      "    theta[0]: 1.0920 ± 0.0286\n",
      "    theta[1]: 1.4931 ± 0.0290\n",
      "T: 13, eps: 0.0563, ratio: 0.4155\n",
      "    theta[0]: 1.0923 ± 0.0250\n",
      "    theta[1]: 1.4943 ± 0.0254\n",
      "T: 14, eps: 0.0463, ratio: 0.3605\n",
      "    theta[0]: 1.0920 ± 0.0224\n",
      "    theta[1]: 1.4944 ± 0.0219\n",
      "T: 15, eps: 0.0386, ratio: 0.3187\n",
      "    theta[0]: 1.0924 ± 0.0200\n",
      "    theta[1]: 1.4945 ± 0.0193\n",
      "T: 16, eps: 0.0319, ratio: 0.2739\n",
      "    theta[0]: 1.0919 ± 0.0181\n",
      "    theta[1]: 1.4948 ± 0.0179\n",
      "T: 17, eps: 0.0271, ratio: 0.2259\n",
      "    theta[0]: 1.0917 ± 0.0174\n",
      "    theta[1]: 1.4946 ± 0.0166\n",
      "T: 18, eps: 0.0231, ratio: 0.1805\n",
      "    theta[0]: 1.0917 ± 0.0164\n",
      "    theta[1]: 1.4948 ± 0.0159\n",
      "T: 19, eps: 0.0197, ratio: 0.1425\n",
      "    theta[0]: 1.0922 ± 0.0157\n",
      "    theta[1]: 1.4949 ± 0.0158\n",
      "took 149.579810143\n"
     ]
    }
   ],
   "source": [
    "import time\n",
    "t0 = time.time()\n",
    "pools = launch()\n",
    "print \"took\", (time.time() - t0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Postprocessing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How did the sampled values evolve over the iterations? As the threshold is decreasing we expect the errors to shrink while the means converge to the true means."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c568410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in range(len(means)):\n",
    "    moments = np.array([abcpmc.weighted_avg_and_std(pool.thetas[:,i], pool.ws, axis=0) for pool in pools])\n",
    "    errorbar(range(T), moments[:, 0], moments[:, 1])\n",
    "hlines(means, 0, T, linestyle=\"dotted\", linewidth=0.7)\n",
    "_ = xlim([-.5, T])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How does the distribution of the distances look like after we have approximated the posterior? If we're close to the true posterior we expect to have a high bin count around the values we've found in the earlier distribution plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x10c5682d0>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "hMD/B6Frjj3e+sd9AAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c55ef90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "distances = np.array([pool.dists for pool in pools]).flatten()\n",
    "sns.distplot(distances, axlabel=\"distance\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How did our $\\epsilon$ values behave over the iterations? Using the $\\alpha^{th}$ percentile causes the threshold to decrease relatively fast in the beginning and to plateau later on"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x10d769110>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "rkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c4dcad0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eps_values = np.array([pool.eps for pool in pools])\n",
    "plot(eps_values, label=r\"$\\epsilon$ values\")\n",
    "xlabel(\"Iteration\")\n",
    "ylabel(r\"$\\epsilon$\")\n",
    "legend(loc=\"best\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What about the acceptance ratio? ABC PMC with the OLCM kernel gives us a relatively high acceptance ratio."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x10cc2e210>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": [
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10cc2e890>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "acc_ratios = np.array([pool.ratio for pool in pools])\n",
    "plot(acc_ratios, label=\"Acceptance ratio\")\n",
    "ylim([0, 1])\n",
    "xlabel(\"Iteration\")\n",
    "ylabel(\"Acceptance ratio\")\n",
    "legend(loc=\"best\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline\n",
    "rc('text', usetex=True)\n",
    "rc('axes', labelsize=15, titlesize=15)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally what does our posterior look like? For the visualization we're using triangle.py (https://github.com/dfm/triangle.py)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jakeret/Library/Python/2.7/lib/python/site-packages/matplotlib/collections.py:650: FutureWarning: elementwise comparison failed; returning scalar instead, but in the future will perform elementwise comparison\n",
      "  if self._edgecolors_original != str('face'):\n"
     ]
    },
    {
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       "UDFVFEVpACqmiqIoDeD/B3sC0CPNA3CcAAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c568810>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import triangle\n",
    "samples = np.vstack([pool.thetas for pool in pools])\n",
    "fig = triangle.corner(samples, truths= means)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Omitting the first couple of iterations.."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean: 1.0922 ± 0.0157\n",
      "mean: 1.4949 ± 0.0158\n"
     ]
    },
    {
     "data": {
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       "PDw8zhBPkXp4eHicIf8f/W+ru0KFVsIAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c477d90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "idx = -1\n",
    "samples = pools[idx].thetas\n",
    "fig = triangle.corner(samples, weights=pools[idx].ws, truths= means)\n",
    "\n",
    "for mean, std in zip(*abcpmc.weighted_avg_and_std(samples, pools[idx].ws, axis=0)):\n",
    "    print(u\"mean: {0:>.4f} \\u00B1 {1:>.4f}\".format(mean,std))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}