examples/mixture_models/dependent_density_regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Dependent density regression\n",
    "In another [example](dp_mix.ipynb), we showed how to use Dirichlet processes to perform Bayesian nonparametric density estimation.  This example expands on the previous one, illustrating dependent density regression.\n",
    "\n",
    "Just as Dirichlet process mixtures can be thought of as infinite mixture models that select the number of active components as part of inference, dependent density regression can be thought of as infinite [mixtures of experts](https://en.wikipedia.org/wiki/Committee_machine) that select the active experts as part of inference.  Their flexibility and modularity make them powerful tools for performing nonparametric Bayesian Data analysis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Library/Frameworks/Python.framework/Versions/3.5/lib/python3.5/site-packages/h5py/__init__.py:36: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.\n",
      "  from ._conv import register_converters as _register_converters\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running on PyMC3 v3.4.1\n"
     ]
    }
   ],
   "source": [
    "import arviz as az\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import pymc3 as pm\n",
    "import seaborn as sns\n",
    "\n",
    "from IPython.display import HTML\n",
    "from matplotlib import animation as ani\n",
    "from matplotlib import pyplot as plt\n",
    "from theano import shared\n",
    "from theano import tensor as tt\n",
    "\n",
    "print(f\"Running on PyMC3 v{pm.__version__}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%config InlineBackend.figure_format = 'retina'\n",
    "plt.rc(\"animation\", writer=\"ffmpeg\")\n",
    "blue, *_ = sns.color_palette()\n",
    "az.style.use(\"arviz-darkgrid\")\n",
    "SEED = 972915  # from random.org; for reproducibility\n",
    "np.random.seed(SEED)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will use the LIDAR data set from Larry Wasserman's excellent book, [_All of Nonparametric Statistics_](http://www.stat.cmu.edu/~larry/all-of-nonpar/).  We standardize the data set to improve the rate of convergence of our samples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Library/Frameworks/Python.framework/Versions/3.5/lib/python3.5/site-packages/pandas/io/parsers.py:2218: FutureWarning: split() requires a non-empty pattern match.\n",
      "  yield pat.split(line.strip())\n",
      "/Library/Frameworks/Python.framework/Versions/3.5/lib/python3.5/site-packages/pandas/io/parsers.py:2220: FutureWarning: split() requires a non-empty pattern match.\n",
      "  yield pat.split(line.strip())\n"
     ]
    }
   ],
   "source": [
    "DATA_URI = \"http://www.stat.cmu.edu/~larry/all-of-nonpar/=data/lidar.dat\"\n",
    "\n",
    "\n",
    "def standardize(x):\n",
    "    return (x - x.mean()) / x.std()\n",
    "\n",
    "\n",
    "df = pd.read_csv(DATA_URI, sep=\" *\", engine=\"python\").assign(\n",
    "    std_range=lambda df: standardize(df.range), std_logratio=lambda df: standardize(df.logratio)\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>range</th>\n",
       "      <th>logratio</th>\n",
       "      <th>std_logratio</th>\n",
       "      <th>std_range</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>390</td>\n",
       "      <td>-0.050356</td>\n",
       "      <td>0.852467</td>\n",
       "      <td>-1.717725</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>391</td>\n",
       "      <td>-0.060097</td>\n",
       "      <td>0.817981</td>\n",
       "      <td>-1.707299</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>393</td>\n",
       "      <td>-0.041901</td>\n",
       "      <td>0.882398</td>\n",
       "      <td>-1.686447</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>394</td>\n",
       "      <td>-0.050985</td>\n",
       "      <td>0.850240</td>\n",
       "      <td>-1.676020</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>396</td>\n",
       "      <td>-0.059913</td>\n",
       "      <td>0.818631</td>\n",
       "      <td>-1.655168</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   range  logratio  std_logratio  std_range\n",
       "0    390 -0.050356      0.852467  -1.717725\n",
       "1    391 -0.060097      0.817981  -1.707299\n",
       "2    393 -0.041901      0.882398  -1.686447\n",
       "3    394 -0.050985      0.850240  -1.676020\n",
       "4    396 -0.059913      0.818631  -1.655168"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We plot the LIDAR data below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x118213668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8, 6))\n",
    "\n",
    "ax.scatter(df.std_range, df.std_logratio, c=blue)\n",
    "\n",
    "ax.set_xticklabels([])\n",
    "ax.set_xlabel(\"Standardized range\")\n",
    "\n",
    "ax.set_yticklabels([])\n",
    "ax.set_ylabel(\"Standardized log ratio\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This data set has a two interesting properties that make it useful for illustrating dependent density regression.\n",
    "\n",
    "1. The relationship between range and log ratio is nonlinear, but has locally linear components.\n",
    "2. The observation noise is [heteroskedastic](https://en.wikipedia.org/wiki/Heteroscedasticity); that is, the magnitude of the variance varies with the range.\n",
    "\n",
    "The intuitive idea behind dependent density regression is to reduce the problem to many (related) density estimates, conditioned on fixed values of the predictors.  The following animation illustrates this intuition."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, (scatter_ax, hist_ax) = plt.subplots(ncols=2, figsize=(16, 6))\n",
    "\n",
    "scatter_ax.scatter(df.std_range, df.std_logratio, c=blue, zorder=2)\n",
    "\n",
    "scatter_ax.set_xticklabels([])\n",
    "scatter_ax.set_xlabel(\"Standardized range\")\n",
    "\n",
    "scatter_ax.set_yticklabels([])\n",
    "scatter_ax.set_ylabel(\"Standardized log ratio\")\n",
    "\n",
    "bins = np.linspace(df.std_range.min(), df.std_range.max(), 25)\n",
    "\n",
    "hist_ax.hist(df.std_logratio, bins=bins, color=\"k\", lw=0, alpha=0.25, label=\"All data\")\n",
    "\n",
    "hist_ax.set_xticklabels([])\n",
    "hist_ax.set_xlabel(\"Standardized log ratio\")\n",
    "\n",
    "hist_ax.set_yticklabels([])\n",
    "hist_ax.set_ylabel(\"Frequency\")\n",
    "\n",
    "hist_ax.legend(loc=2)\n",
    "\n",
    "endpoints = np.linspace(1.05 * df.std_range.min(), 1.05 * df.std_range.max(), 15)\n",
    "\n",
    "frame_artists = []\n",
    "\n",
    "for low, high in zip(endpoints[:-1], endpoints[2:]):\n",
    "    interval = scatter_ax.axvspan(low, high, color=\"k\", alpha=0.5, lw=0, zorder=1)\n",
    "    *_, bars = hist_ax.hist(\n",
    "        df[df.std_range.between(low, high)].std_logratio, bins=bins, color=\"k\", lw=0, alpha=0.5\n",
    "    )\n",
    "\n",
    "    frame_artists.append((interval,) + tuple(bars))\n",
    "\n",
    "animation = ani.ArtistAnimation(fig, frame_artists, interval=500, repeat_delay=3000, blit=True)\n",
    "plt.close()\n",
    "# prevent the intermediate figure from showing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "HTML(animation.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we slice the data with a window sliding along the x-axis in the left plot, the empirical distribution of the y-values of the points in the window varies in the right plot.  An important aspect of this approach is that the density estimates that correspond to close values of the predictor are similar.\n",
    "\n",
    "In the previous example, we saw that a Dirichlet process estimates a probability density as a mixture model with infinitely many components.  In the case of normal component distributions,\n",
    "\n",
    "$$y \\sim \\sum_{i = 1}^{\\infty} w_i \\cdot N(\\mu_i, \\tau_i^{-1}),$$\n",
    "\n",
    "where the mixture weights, $w_1, w_2, \\ldots$, are generated by a [stick-breaking process](https://en.wikipedia.org/wiki/Dirichlet_process#The_stick-breaking_process).\n",
    "\n",
    "Dependent density regression generalizes this representation of the Dirichlet process mixture model by allowing the mixture weights and component means to vary conditioned on the value of the predictor, $x$.  That is,\n",
    "\n",
    "$$y\\ |\\ x \\sim \\sum_{i = 1}^{\\infty} w_i\\ |\\ x \\cdot N(\\mu_i\\ |\\ x, \\tau_i^{-1}).$$\n",
    "\n",
    "In this example, we will follow Chapter 23 of [_Bayesian Data Analysis_](http://www.stat.columbia.edu/~gelman/book/) and use a probit stick-breaking process to determine the conditional mixture weights, $w_i\\ |\\ x$.  The probit stick-breaking process starts by defining\n",
    "\n",
    "$$v_i\\ |\\ x = \\Phi(\\alpha_i + \\beta_i x),$$\n",
    "\n",
    "where $\\Phi$ is the cumulative distribution function of the standard normal distribution.  We then obtain $w_i\\ |\\ x$ by applying the stick breaking process to $v_i\\ |\\ x$.  That is,\n",
    "\n",
    "$$w_i\\ |\\ x = v_i\\ |\\ x \\cdot \\prod_{j = 1}^{i - 1} (1 - v_j\\ |\\ x).$$\n",
    "\n",
    "For the LIDAR data set, we use independent normal priors $\\alpha_i \\sim N(0, 5^2)$ and $\\beta_i \\sim N(0, 5^2)$.  We now express this this model for the conditional mixture weights using `pymc3`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def norm_cdf(z):\n",
    "    return 0.5 * (1 + tt.erf(z / np.sqrt(2)))\n",
    "\n",
    "\n",
    "def stick_breaking(v):\n",
    "    return v * tt.concatenate(\n",
    "        [tt.ones_like(v[:, :1]), tt.extra_ops.cumprod(1 - v, axis=1)[:, :-1]], axis=1\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING (theano.gof.cmodule): The same cache key is associated to different modules (/Users/jlao/.theano/compiledir_Darwin-17.5.0-x86_64-i386-64bit-i386-3.5.1-64/tmp6mq4zwxw and /Users/jlao/.theano/compiledir_Darwin-17.5.0-x86_64-i386-64bit-i386-3.5.1-64/tmprc22erpn). This is not supposed to happen! You may need to manually delete your cache directory to fix this.\n",
      "WARNING (theano.gof.cmodule): The same cache key is associated to different modules (/Users/jlao/.theano/compiledir_Darwin-17.5.0-x86_64-i386-64bit-i386-3.5.1-64/tmp6mq4zwxw and /Users/jlao/.theano/compiledir_Darwin-17.5.0-x86_64-i386-64bit-i386-3.5.1-64/tmprc22erpn). This is not supposed to happen! You may need to manually delete your cache directory to fix this.\n"
     ]
    }
   ],
   "source": [
    "N, _ = df.shape\n",
    "K = 20\n",
    "\n",
    "std_range = df.std_range.values[:, np.newaxis]\n",
    "std_logratio = df.std_logratio.values[:, np.newaxis]\n",
    "\n",
    "x_lidar = shared(std_range, broadcastable=(False, True))\n",
    "\n",
    "with pm.Model() as model:\n",
    "    alpha = pm.Normal(\"alpha\", 0.0, 5.0, shape=K)\n",
    "    beta = pm.Normal(\"beta\", 0.0, 5.0, shape=K)\n",
    "    v = norm_cdf(alpha + beta * x_lidar)\n",
    "    w = pm.Deterministic(\"w\", stick_breaking(v))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have defined `x_lidar` as a `theano` [`shared`](http://deeplearning.net/software/theano/library/compile/shared.html) variable in order to use `pymc3`'s posterior prediction capabilities later.\n",
    "\n",
    "While the dependent density regression model theoretically has infinitely many components, we must truncate the model to finitely many components (in this case, twenty) in order to express it using `pymc3`.  After sampling from the model, we will verify that truncation did not unduly influence our results.\n",
    "\n",
    "Since the LIDAR data seems to have several linear components, we use the linear models\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "\\mu_i\\ |\\ x\n",
    "    & \\sim \\gamma_i + \\delta_i x \\\\\n",
    "\\gamma_i\n",
    "    & \\sim N(0, 10^2) \\\\\n",
    "\\delta_i\n",
    "    & \\sim N(0, 10^2)\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "for the conditional component means."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "with model:\n",
    "    gamma = pm.Normal(\"gamma\", 0.0, 10.0, shape=K)\n",
    "    delta = pm.Normal(\"delta\", 0.0, 10.0, shape=K)\n",
    "    mu = pm.Deterministic(\"mu\", gamma + delta * x_lidar)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we place the prior $\\tau_i \\sim \\textrm{Gamma}(1, 1)$ on the component precisions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "with model:\n",
    "    tau = pm.Gamma(\"tau\", 1.0, 1.0, shape=K)\n",
    "    obs = pm.NormalMixture(\"obs\", w, mu, tau=tau, observed=std_logratio)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now sample from the dependent density regression model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Sequential sampling (1 chains in 1 job)\n",
      "CompoundStep\n",
      ">Metropolis: [tau]\n",
      ">Metropolis: [delta]\n",
      ">Metropolis: [gamma]\n",
      ">Metropolis: [beta]\n",
      ">Metropolis: [alpha]\n",
      "100%|██████████| 30000/30000 [02:44<00:00, 182.90it/s]\n",
      "Only one chain was sampled, this makes it impossible to run some convergence checks\n"
     ]
    }
   ],
   "source": [
    "SAMPLES = 20000\n",
    "BURN = 10000\n",
    "\n",
    "with model:\n",
    "    step = pm.Metropolis()\n",
    "    trace = pm.sample(SAMPLES, step, chains=1, tune=BURN, random_seed=SEED)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To verify that truncation did not unduly influence our results, we plot the largest posterior expected mixture weight for each component.  (In this model, each point has a mixture weight for each component, so we plot the maximum mixture weight for each component across all data points in order to judge if the component exerts any influence on the posterior.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x119470cf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8, 6))\n",
    "\n",
    "ax.bar(np.arange(K) + 1, trace[\"w\"].mean(axis=0).max(axis=0))\n",
    "\n",
    "ax.set_xlim(1 - 0.5, K + 0.5)\n",
    "ax.set_xticks(np.arange(0, K, 2) + 1)\n",
    "ax.set_xlabel(\"Mixture component\")\n",
    "\n",
    "ax.set_ylabel(\"Largest posterior expected\\nmixture weight\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since only three mixture components have appreciable posterior expected weight for any data point, we can be fairly certain that truncation did not unduly influence our results.  (If most components had appreciable posterior expected weight, truncation may have influenced the results, and we would have increased the number of components and sampled again.)\n",
    "\n",
    "Visually, it is reasonable that the LIDAR data has three linear components, so these posterior expected weights seem to have identified the structure of the data well.  We now sample from the posterior predictive distribution to get a better understand the model's performance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|██████████| 5000/5000 [00:33<00:00, 151.39it/s]\n"
     ]
    }
   ],
   "source": [
    "PP_SAMPLES = 5000\n",
    "\n",
    "lidar_pp_x = np.linspace(std_range.min() - 0.05, std_range.max() + 0.05, 100)\n",
    "x_lidar.set_value(lidar_pp_x[:, np.newaxis])\n",
    "\n",
    "with model:\n",
    "    pp_trace = pm.sample_posterior_predictive(trace, PP_SAMPLES, random_seed=SEED)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below we plot the posterior expected value and the 95% posterior credible interval."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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VlerqaiKRyvo17SCRSASg3rVNvaAfC2CPro0byc3NIyvLSTgcJBgMEQpF18rz8vJxOp1kZmZisVhRVbUhSj0SiZCRYWnVCg+FQhw6dJAdO7Zjt9sZNmw4VmtyjwPAvM1HeGbZbo746sixGrhtyiAuGJGT6qUVCASCVhGiK4iLLMvo9YaULei2ELW2Szl48ACSJKPRyMiyBlAbgtZi6+CKotRHd0ctckVRcDiyyMvLIzc3H71e39Cuqqrs3r2L7du3EYmE0esNhMMhVqxYxpAhQznhhBOx2Wz4/ZUcOVJOIBBAlmUyMx18WRzmz4t2EqhPKXjYV8dDC7YBpE14g8EgxcVHGh4wBIl5++03KS4+3OJ1o1FHIBCKc0Zy3O5crr76moTvB4NBHnroAQ4ePEBGRga//OWvGTCgiKVLP+Ppp6PbFgFuuukWPJ4R3HPPL6mrq2P27HsZOnQY33yznm+/Xc+PfnR9m8fWGlVVlaxevYrp089P6fi///0xrr76h+Tm5nb6WFrj2WefZODAQQwbNpzlyz/nhhtu5tJLz+ODD+Y3Oe6FF56jsDCP8867tNU27713Ng899GjC999//z9cdNGlLbJhdZTO2m4lRFfQ5cSs7XjY7ZkNP39bpefzozaqIjI2jcI0RzUjLX4CgQC7du1k8+bvWpxvMBhwuZxoNMe+2oqicPDgAXbs2I4kyZhMOmprg0DUApckiX/XjSKAoUlbgbDCYwu+g91r0Om0aLU6JCm69au21k8g4CcSiZCVlU1eXj4OhwOLxUpNTTXl5eUcOnSQkpJirFYrubl5uN05WK1Wjh49ite7hX379qIoYXQ6A2eddQ7Dhg0X28cSUFx8OG4C/bYENzbmwIH9Sd+fO/e/mExmnn/+Jfbu3c3jj/+Fv/71Kbzezdx228+ZNu3shmOXLl3M6adP5aSTTuHDD9/njjvu4p133uS++/7Q5nGlwvbt21ixYmnKonvHHXelZRxtoTMrKCUTXIBXX32R88+/qNNFt7PomaMS9Hs2+Yx8Um4lrEYjwqoiGj4ui1qDo6wSJpMp5bZkWSYrK7vh93g36ppd+uanAVAZljl06ACKojaKGo+62KPbxSQOHz7Izp076s84JuQmkwmj0YTP56O4+AihUAhJklFVhYyMDHJycpBlmUDAz8cfz2X4cA9Tp07DYrESCoUoLy+nrKyUQMBfn3XNhMlkxmAwoNVqxbaxNLJr166GfMVFRYPYvXsXAF7vFrZt8zJnzpuMGDGKW2/9GSaTmbq6Ourq6jCZTCxc+AlTp56JwWCI2/asWTPjVvp58snH2bBhPQDnnns+V111DUuXLuatt14ForsfHnjgIV555V9s376N99//DxMmTOIvf3mIuroABoORu+++F0VR+PWv78RmszNx4mRWrVrB7Nn3kpWVzYMP3kdNTQ2RSISbb76VsWPHcd11VzFgwEB0Om2TXNPNqwzdf/+D3HXXz3E4sqiqquLRR//GY489wv79+1AUhZtvvpVTTjmVJUsW8fLLL5CZ6SAUCjFw4CDWrl3D+++/ywMPPEwwGOT++39DcfERhgwZxl133dPk+sSreNSYmKU8a9ZMhg3zsHPnDmprq3nwwT+zZs0XlJeX8fvf38vDDz8Wt61Zs2Y2zMFsNnPVVddw8slj2bLlO1566Z/cd98feOSRP1Jd7aO0tITLL7+Kyy67ojO+VoAQXUEPZUmFpUFwY4RVmSUVFkZZA53en02jUBVpKWA2jUJmpiPpuUajKekxjfeFJzp/wIAi9u7dw+uvv0JmpoOSklgavGNudqDJNrBoHnIZk8mEy+UmJyeH7GwnGRkZaDRaQqEMfL4AGo2mYWuZEOnUGDZsOCtXLmPq1Gls2rSR0tISIpEI48adxpQp08jPL+DRRx/i/fff5bLLrmTVqhW8//5/uOmmW3j66b9x44238Je//ImCgkJ++MOftGi/eaWfceMmcOjQQZ5//iUikQi33noTY8eOY+HC+dxww42cdtoU5s37kJqaGn784xt5//13mTHjcn73u99wxRVXM3HiZNas+ZJ//OMpZs68jfLyMl544TV0Oh2rVq0A4OWXX+DUU8dz1VXXUFJSzG23/ZQ5c97H7/dz/fU3MXz48U3G2LzK0O7duwE455zzOOOMM/nvf/+N3Z7Jb37zOyorj3L77TN56aU3ePLJx/nXv17DZrMze/YdLeYeDNZx660/Jzc3j/vuu4cVKz5veC9RxaREfz8jRozijjvu4rnnnmbhwvlcd931vPTSC/z+9w8lbKvxHFatWsG8eR9y8slj+eijuVxyyWXs37+fc86ZzhlnnEVpaQmzZs0Uoivo+1RF4u+fTvR6czb5jCypsDRxTScT62mOaj4uszYReq0UPa8rkCQJtzuHuro6gsE6cnNzk+4hj6Gqar1VXMqBA/sbEsxHLe1oJHosWYuiqOh0OiwWCx7PCAYNGozL5RIu7ThcdNGl7Nmzi9tu+yknnHAiHs/xaDQaLrpoRoMATJlyBkuWLEaW5QYX7quvvsiVV17Dyy+/wJ133s2//vUce/fuaVEdqHmlH7c7hxNPPAlJktBqtYwadQK7d+/kZz+7kzlzXuOVV15l4MBBTJ06rUk7O3du59VXX+T1118GaFhWycvLbxFwuGfPrgaXtMvlxmzOoKKiHIha882JV2Uoemx0Ljt2bGfDhnV8991GACKRMKWlpdhstoZlotGjx7Ro1+3OJTc3r2H+e/fuaTKfeBWPrNb4runhw6Ov5+TkUFZW1uLaxGur8RzGj5/IM8/8naqqSjZsWMcvfvErysvLmDPnDZYu/QyzOSNpxaP2IERX0CNJZnm2xiafsYmANnVNxxfe2OttEep0YDAYErol4yFJEnp9dJ+1zdZ0nTyeGz0SiRAMBvnqq9WsXr0Sq9XK6NEnkJub1xDQJUQYtmz5jrFjT+PnP7+LLVu+48iRQ6iqyk9+8gP+8Y9/4XbnsGbNV3g8IxrOqagoZ+/ePVx33Q28+earyLKMJEkEAv4W7bes9DOYjz/+gKuv/iHhcJiNGzdwwQUX88EH/+W2225HozHxl7/8ic8/X0JeXj6KEvV2FBUN4pprfsQJJ5zInj27WbfuayBRNaLBfPPNeoYPP56SkmJ8vqqG70y8zzxelSE4llBo4MBBuN1ufvzjG6mrC/Dyy/8iOzub6upqKioqcDgcbNnyXUPQWYySkiOUlpbidDrZsGE9F100o0G4E1U8SkS8cUeXcNSkbcXmIMsyZ555Dv/7v48wZco0NBoNb731GqNHj+Gyy65g7do1rFq1PGH/7UGIrqBH0hHLs72u6VHWQJeLbFej0Wjq14aja+KBgJ+vvvoCRYmuV+v1evLy8hk79tSkN7u+TmFhEf/3f/fyyiv/wmKx8pvf3IckSdxzz3389rezMRiMDBo0mEsvvazhnJdffoGf/OQmAC677EruumsWOTm5DB06vEX7zSv92O2ZrFv3NbfccgOhUIizzjoHjycqjrfffhsGgxGTycSkSacTDAbZuXM7c+a8we2338Fjjz1CMBikri7AHXf8KuGcfvzjG3j44T+wZMki6urquPvu3yYNNopXZeidd95seH/GjMv585//yKxZM6mpqeayy65Ep9Nx5513c9dds7Ba7XHbt9sz+dvfHqWkpJjRo8cwceLkBtGNVzEpVlc4VU488SR+9auf8+STz6XU1kUXXcpVV83grbf+2zCGxx//C4sWLcBisaDRaAgGg20aQzJ6ZZWh7roZtDdSsjfRk+bYVhdxjId3uYm6VJuj8pvBxT1qjumkPfMMh8P4fD58Ph8nnHACEyZMTlhYpCvp6i1D6aStW0/6UvWdZPSlefa5KkOC1GivaPUU2mt5dsQ13ZiedP26aixarRaHw4Hdbsfr3cL27duYOvVMPJ7jU1pjTheJBLIv3agF/QMhun2U9qxrNj+/pwhOW+mMoKiOXr/G7XT0OnbWWNqCLMvk5uYRCASYP/9jvN4tnHXW2S3WjQVtR1T66d8I0e1k0iVWbW23I1tu2nOT7+x5d6S9zgiK6owtS/Gu49xSKwvLrfgVKeVxdfX2qcYYjUaKigZy5MghXn/9Vc4++1yRxEMg6ABCdDuRdFkk7Wm3I1tu2nqT7+x5d0Z7HQ2KSnb9nt7rbLdYqsj4lVhbqc2ro9unOgOXy92QxGPEiJFMmTINs9ncZf0LBH2FtIuuxWJAq+2cDfnRZPk6zObUt1R0JrIsJe176f74YrW00sK4nPbHk7WnXbtOoTLU8rrbdUrSOciylPQmH+/czp53uq4jwDdHdXy638TRkIRdp3CuO8CJmS0DcRJdP5CoimiYV2ZDb9DGPTdGKqKYyrw68ll25t+K2WzA4bBz8OA+/vOfN7nwwos47rjjOq399qDRyGRm9m3x7w9zhP4zz7SLbnV150WJqqpKIBDqtsjTeNGgjd2giagMyR0ac2Uo/jpaZUjm0S3WFlbXJp+RunA0IULjKF6tpHCGvTrpWMxmQ9JApHjnJhtfe+bd2e3FiFrQpgZBrwxpeO+giWBduIWleYZdarEu3JiQKvHhQRPD9NUJXeGJrmNb5xVvLKl+lun4W7Hbs6itreWVV15nzJgTmTTpdIxGY6f3kwr9IZCqP8wR+tY8Xa7EGej6vHs50Q0xHQEuiWgcNdvWfjf5kt3MpBYuyvhjUjHJKudm+VKaY7xAJFAZaop/A++saOF0tRejLW7z5uvC8bYgBVSJ+SVWNtQY47rC41/HlrQ2r56SuKMxZrOZwsJCNm/exO7du7jookvJyRFlEAWC1ujToptobXB/QJfwRtkWQY53E2+J2rAOONRU1+Z+l1RYiL/n9BiNhSP+mCR0ktKm9dD9AR1rq02N+pbYUGOk0Bhq0U5np1Ds7PZa80Ykej22Lvz0XmcCi1ViXbUJtdnnE/s8bi8qBY6JpVFSCakSkWbeh1Tm1RMTd8iyTF5ePlVVVXzwwX+55pofihKFAkEr9GnRTWTZJLtRAgmDeMaZm667JXYpN3brSg3tNBWxY/3OLbUlfBAIq6lFicbG0pGgm00+I0v3W6gM2etHGf8atWYVdtQS68z2UvFGtGZpTnNU80GpjXgPP4lWYmPXu7lY9uatWImw2WyUlBQzf/48Zsy4vMeWVBMIegJ9+q8jkdAku1EmEuoFZVbG5VQ1eT3xml0ioYz/uoqUUJAl1ITjbT6WZGNqTViai1NrYtKcVCyxtghOvPbaI1iteSNSsTRHWQMN23yaE1s5b45RUust5KZjbe069VZRdrnc7N+/n1WrVjJlytTuHo5A0GPpU6Lb/IZlktW4N8pk7tpEohJQJb45qmOY/ti6ZqK1z9bcwfFJJMhRYUjWR2PhaK9rNjVXebTXh3e52ywIHd2zmuo2oubfgeTeCNCl+FGdm+WLe13HZASaeCgAJBRCqkQgknys7Z1jTyU/P5+vv/6K3Nxchg1rmW9YIBD0IdGNd8OSUNCgNllDS04iuyX63sJiI0F7uMlNfUxGgO1+Q8PvIVVKIPQx2ibKjdd2Y30MNdU16bOxWMVzzQ411bGkwsIHpbaEwWSpoTa45dsqCB3ds5pKEFS870CyzxPAr0gpzSOZy7vQGGryerzvQCrJLLozCUZnIMsybrebhQs/ISsrm+zs7O4ekkDQ4+hVorty5XJWrlzOuHHjKSoaiCzL9cKRQVVYA82y5KjIRFAxyhECETkqd61m0kn8fmVI5oMSW0M/VRENG2qMXJgdjQoOBAKsOlTHV+oQIlJLF2/MMoq3phwbcTwLNp5L8jx8ccfY3NJLFLzVfA058dVQG0aVaB08FUFIdc9qovZSWatOFETW2oNOqvOItz7b2IV8qbOKUdZAfcGF5GNty/tdmQSjoxiNRoxGI3PnvscVV1yNxWLp7iEJBD2KXiW6n3++hEWLFrJo0ULMZjNFU6/EP3oGaLRJ7qkS/rowpfOewHnWjWDpyNO31KKfsCozd5/Cm2/8mtLSaLSqecQZOM74CRqbE1QVJBlqy5E2z2dPhZfswadRftx5KPKxy69F4TiphH0RG37JiIRKWJVYVGaiLljHyVlKqw8M8Sy9RGvFiYX/GFpJaXigaK+QxEh1z2os0jvVPa+N16oDuW2RAAAgAElEQVSTjcWmiTR6v+W82ypsySLjE48h+bp6smvUPAtWT177zcx0UFJSzEcffcCMGZd32x5egaAn0utK+z3yyINUVlbi9W5h74k3IVudKZ6soJMghET71lyTj8u16A8UFBSQn1+AXm+gqqqSysqjVFZW4vP5qK2txe+vpba2loqKcjTHTWgQ5oivlIolL1O7eSnmEWeQfcHPkHXGJu2rAR/Kmjlk+XaRmZlJMBikpqaa6upq/P5aDAYj8ow/opodqY66leugcorFz3kuH5t8RuaW2uKKtE0Tadgak4xU9zTHs/YvzI5a9fHXz5u74OMJ87ExJtr+k+o8YnNJdD2kRi745vOKWcKN20nmlWhOsmvR+AEphtls4KsjUreJ8+HDhygsHMCFF16StojmvpRQIRH9YY7Qt+aZrLRfrxPdxvV0E9dNTak1dKhoZerX3xK5e2Mk7qctN2yIzqOsrIyDBw9w8OB+KisrcTiyyMrKYkXWudRKCSyDcJDI6lc5+s1CDAYjGRkZWCwWTCYzdXUBfBc+1MLFnmQQrR6rCVSSc+QLDg84q4lVHkOLwoXO1BJuQFORibdnNdGDQOz6tlyDbirO8YKamovRJp+ReWU2QmpLYU+1+lLyh4dEDzPRWr6xNhaUWQmoTb93sTnE1usTXQsg7oODUVLQy2qDwB5vDbG2Ut+qOKeT/fv3M3LkSM4669y0lAbsSzfqRPSHOULfmmefraebqssyPhImjdJwM4+/D1NKepOD9iVtkCQJp9OJ0+lkzJgTm7y3cFeSXLlaPY4zrud/rru4ycsxMSKS4Lx4ApuCOIcNNvZmnoQ2juCqSoRDH/6Vf+7+kqysbBwOB3q9Ab1ej06nQ6vVoigK4XCYcDiMoiiYzWYG2zOx2+1kZmZSmjGIDRRRreqSRho33vMKxP2swqrMdr+BC7N9rQaRzcivZcFhY7usv9aivBOF4sVcy8lEOzaH24tK2+XODzSLmP7yaEvhjm1/6yrrt6CggE2bNmK12hg/fmJa+hAIehO9WnSTJS1IhcY382g78Y+51FmVNLUiEHcdsjnJ1uEahDPFMTduM5nlpZUUxliSW0+JrDO7VqHK7orbriTJFFxwK0Htr5ACR6la9x7BbSsJhUKEQkHC4TCyLKPRaNFqtciyTE1NNbW18Z9k9Xo97p8+h8basj99qIZ1677GbM5gofY0kON/3lURmVHWAMebqwkE6vD6M1hU7Wqx7vq9fH+bPBPN+0hEMms79mDWmmjH2m9tDTu1/eHxr1NzcU7ntiRJksjPz+eLL1YzZMgwnM4Ul4MEgj5KrxbdZEkLUtma0ziwpbWbnBYI19swjfMYt2X/aKLjIN56ZetjhmQ3cbWJsJ+HL6H1BC33AseEItE6KRIEddGHBNXkQDf5J8y49PK46SybbnGKMFZ7CJd/Lz6fj+rq6D+fz0f5wdWUDz0PNPpjswgFODDvWZ7bvBSAors/SPipqorCQztdRKpKqVj6Bo4zfoLW3nILzod7JUbvmk9NTQ0GgwGTyYTJZMZsNuN25+ByuRK6QhN9TyTUBrdt8y1EjR+uWgvYin2+yfJfFxpDHdwf3v4o9Pag0WgxmYwsW7aU733vclGLV9Cv6dWiC21LWtCY5m7hREklhprqWrweUlX2B3QJk+HHu4kl24MZ+7kliZNgxEh2E29szSUrnCABYVVq2B7UXChSucEnS6PZOIK6KqJlmVLAhW4bJw2J5w3ws6RCc0ywnHUMvvkKSkvPwO/38wlBaokzF1VFkqNiqLW7ybn4TpQ427YA/LKRd9+dQ8bIaTjOvhbZFPVyKP4qyt96nvCOVeTnF1BQUIjb7a7fc+okOzubk+QAy8MFKNKxPx1JCeGpXIumrpb9FisOWeb7eg0ajYzJZCIj41j+7GRLIo0/39byXzd3oyfeH978s4ovzunelpSd7WTv3l3s3LmDIUOGprUvgaAn0+tFN9WkBUZJRZJImP0oUTuJxDLeVpzGNL+JtXcPZmyrSyK3daKsW/Es4kSu5Vi0rQroJLXVZBvJ5pJ4r+wx4j2UNLeOm0b6mikqGhg911fHx2X6lg8BzawnRdYmTKFp00S45U//Yn6VE5Vj7WjMdtyX/BLXzgVUfbuIb75ZR3V1y/X6xlvClOoyyj97id2blzI/7lUBx4nnYptyHZgdSGF/1JJvvE6uRveST8+uaXJNtvsNCa/d7UWlST0pEP0sT8jwN3gZkqV+6WgFp1TIznaxZMkiCgsHYDB0T01sgaC76fWiC4nz/ra1Mku84xOt9bbFdR37va1rdLb6QK9EbPIZCcS5V2pQ22ARN51HSJVaCGLz65J4202ytItNie3HjY2zNRd988hnnazgVySMklofBdySeCk0tZLCeXlBFhzObiK4DedIGgLDp3PnOacAEAj4KSsro7y8jLKyMoxGIzk5ueTk1GCuL4ARGnENR49eQHl5GbW1NSiKgqIoRCIR9uBil3M8qia6f1fVmVHDIaRQNegzkAOVFH/6T+q2Lmft+IlYpk4jGKyjuLiYqrzvx/2axbvG8R6OpucGGKavptAXzx19jFjayvak92wLZrOZiooK1q9fx/jxEzq9fYGgN9AnRDedtCdCurGbsOk2l8Tu4vbmS44nHDpJbXHTbMs8WhPOZPmdE6+xN+dYLWCd1NK93tgabm7FBVQJLVFreEmFpSEoqDnxUmhOc1RzYqbKvw+YU5q/0WiioKCwYZtaPHQ6HS6XC5erZRDY13udqM2uu6TVYTPI3F4U3UJU7JzOp59GM66tWLGs4biC/zcVrb3lOnwsor45jR+ONvmMLCy28O+QOW42sShqw9at2GeW7sCqnJwcvvxyFcOHD8fhyOr09gWCno4Q3VZoW1GDpsFLiQrKQ8t1U2h7KbtkxRk6Mo/2FlUH4lreyQirMuEEe8Vj80u2Hp6soEGiFJpgSPoQEivqEHPHdsT6S2VZwe3O4dprr+Piiy9l06aN2O12XK4cDut1fFLR1FJXVYXKsIY/fashY/tiBlLCxImTyc3Nazgm1YpRAHpZbfHQks7AKq1Wi15vYNmypVxyyfdEUJWg3yFEtxUSFRBoLQlD7Jx465vxkmm01RW+yWdsdU9oe+YRW9NtjXjjfXqvM67l3V6Oud4TC1ci8TTGsfYbM81RzdxSa5zxNl3jjvbTfuuvLaUWbTY7EydObvjdRRBZ9jV5uJCk+v8tTmpHz2DFJ0+x+E8PcPnlVzJt2llIkpRyxahU9kanA6fTya5dO9i61YvHc3za+hEIeiJCdFMgJjBms4Ha2mhpv2TbQmKkckNrTw7dmCUTz2WYzC0dTyibzyO6Dtg+C6e9N2qjpBJGTeheTyZciVzd07PjF4SIEbsOxzJDxS/qEKO91l97Sy02HucoayDuOrqkNeC8+Jeo6i9ZXlXC2ncXcdM5J1MVSbw1rPkYEqfOTF9glSRJuFw5fPbZIgoKCrBYrK2fJBD0EYTotpNULNPWrJz21k9NZMk03iva3nlEHyxSPr0J7V3/jglkooePZMKVLHq9NZrPPdk+ZmjfQ0Uq40vlwStR32p9EQ6t3U1gzGX8+a3ncZ87kIBsanFs44pR4fqAuURem7ZmWWsrRqORqqpKli5dwoUXXizczIJ+gxDdNNKaldPe+qmJb8BdU+w8WfKL1vaFSigY5fhbtxKNvTXhaqtrPhGtPTS01/pLNr5UH7xSeaCRdUZsp/+QfR8+hfviX6DKxyoexdu7HitN2bwmdFcVRXC53Gzb5mXbtuEMH+5Je38CQU9AiG4aaU0s2r93N/V1ws4mtfKBx4LFmmajav8NvbOENRnxg82ipMv6S/XBK9nYmpCRxQm2Or798G/kTL8ZxWhvdc95LN9zVyNJEm63m8WLPyU/P1+4mQX9AiG6aSaZWLRXPDu6TtgRUkl+0TxY7DySr6/2FJo/JHVG9HJrpPrgFX9s8SPPr7/+pyxY8AnvPXEdAwYUce2ts3BYs5LmF+8ujEYTVVVVLFu2lAsuuLj1EwSCXo4Q3W6kveLZkXXMjtKW5Be9kfZa1O0tKt+WB6/m+3ATfXckSeK88y5g8OCBPPPM0zz22J+ZPfs32DSJk5p09rzaQtTNvI3TTisjOzu7U9sWCHoaQnS7kc4MAuoqUg2W6gpXd0+hvQFxkN4Hr5NOOplf/OJXPP74ozz55N+46NYH+NTnTNhXvLX65mvA6UicIUkSOp2WLVu+Y/LkKZ3WrkDQExGi2810l3i2l1SSbHSVq7un0N6AOIjvNo5FFjd+P9G5rbU/aNBgbrnlNp5++gkWv/Rnpt90L8urMlsIdWpr9elLnJGVlc2GDd9w6qmnibzMgj5N7/QBCrqNUdYAF2b76lMRqtg0EU6x+Jv83tZtS72djiaYGGUNMM1RjVaKJeY4liIzWXWoVBk5cjTXX38T27Z5WfXW37m1sJjfDC5uUjQhtbX6KOlYOtDpdIRCQfbs2d3pbQsEPQlh6QraTDwLq7cES6WDzogm74i1nArjxo3H5/MxZ86bvPPOW1x99bVN3m+LkKZr6SAzM5M1a75k2LDhYt+uoM8iLF2BoINErdSmQtRWF3tXpGM866xzmDp1GkuWLKa4+EiT9xILadNEo+lcOrBYrJSWllJcXJyW9gWCnoAQXYGgg8RzubfVxZ5I9DrbqrzookvRarV88slHTV6PCmm8TN4SEipdtXSg1+vYtOnbtLUvEHQ3wr0sEHQCHQ2I66q913a7ndNPn1qffvFSnE4nEB1/on28KvCbwV1jfWZnO9m8eRMTJkzCbE5cflEg6K30K0s3Eglz+PAhDh06SF1dXXcPR0B0m8rTe508vMvN03udnRI41BvpDGs5VaZPvwBZlpg//+Mmr3eVtZ0MjUaDoijs3Lm9y/oUCLqSXmnp+nw+zGYzGk3qyfXLy8upra3l1FPHodfr+eabdZSUlGA0GsnIMBMORwiHw0QiETQaDTabFY2mV16eXkNH9rf2Rbpq+5jD4WDSpNNZsWIZF1xwMVlZ0WLy3ZnprDFZWVmsWfMVI0aMatPfuEDQG+hVqiJJEpMnT2H37l0cPnwYRYmgqnAs0LG+DqqqoNFo0ev1aLUajh49SmFhETNmXI7L5QLglFNO5eDBA3z33UbKy8sxmy2YzSZMJhPV1dXs2bOHSCQMgMlkQlVVamq01NQEUBS1/j0JSYr2ryhq/Sb/aJ+BQPQ4WZZQlKilIMsyqqqiqtEnepPJiMlkQq83oCgKgUCAQMBPIBBAbVbY3Wq1YrPZu+IydxnpjtgVJOb88y9kxYplLFgwjx/84IdA92Y6a4zJZKasbB9r1nzF+PETurRvgSDdSM1v7p2N3x9UtdrOf1pVFIXKykoqKipQFAVJkhq2GQQCAXy+KioqKqiq8jFq1EhGjBiJLKfuTY9EIpSUlHDgwAGKi4+g1WrR63X1Ym7AbDaj1+vR6/VoNFGRramppqLiKLW1NbhcbrKzs8nMzMRmsxEKhfD7/fj9fmpraykvL+PQocMcOXKE6mofkiTjdGaTk5NLbm4Oer2hXqCj/9au/ZojR46Qk5PbInlAtO8a4NgDiCTJ6HS6+gcPLbIsoygKoVCo3qIPYzSa0Ov1TdqKPiSk9zsR475NduLvBVV5cFRl2vrtyjl2J63N84UX/snKlSt47LG/kpnp6MKRtU44HObAgf1cc821FBUVJTxOo5GJRPp29rP+MEfoW/PU6TQJ97ylXXRLSnx95u6WmWnm6NF2FptNQigUQqPRJH0oiEQibN68iWXLPkdVFSwWKz5fFZGIgtVqpbBwQIPgqmr0plVTU43P56OmpgZVVZBlDRkZZiwWK1qtjtLSEvz+WmIWu15vIDs7E1WVG8aiqiqhUIhQKNgg2DE3vFarw2KxYDab27WvMl5hdqBJsYR0EK0Z3PfX9FubZ0lJCffffy9nnnk2V175gy4cWWrU1FRTVxfkmmt+REZGRtxj0vU32ZPoD3OEvjVPl8ua8IbYq9zLfRWdTtfqMRqNhtGjxzBo0HF8+eVqiouPMHnyVAYMKCIrKyup6KmqSjgcjttPbW0tlZVHqaio4MiRw/h8FezbdxBVVVBVFUmSsVgysFisuFwWzOYMLBYLJpOZo0cr2LNnN4cOHazvJ2ppazSa+r6keqEOERV2qd71r0Gr1TLeqPBZbU63ryH2dtpblMDlcnHaaRNYtmwpF154SUJh6y4yMixUVx9hyZJFXHjhJSJhhqBPIES3l2GxWDjrrHPadE50rTm+sJvNZsxmM3l5+YwcOYrMTDPl5dXU1tag0WgxGo1Jb3YTJ04mGAxSUVFOdbWPQKCOmpoaamp8KErUIs/IsGA0GpFlmWAwSG1tDdXV1bjLywns2cvacB416LHIYU63HMVjCtIZX82YZX7sOkSt+b5EsmC0cebWnUznnDOd1atXsmLFMqZPPz+tY20Pbrebbdu28u23Gxgz5sTuHo5A0GGE6ApaIMtymwqK6/V6cnJyycnJbXNfF4VClJeXcfDgQfbt24vPV8PRozXU1dXVB6lFK8eqKvXBZRKyHLWqgXrrGUBFrzcQCgUbgtCMRhMZGRkNgWyKonLo0EEyMkxYLHa02t7/9U8WjDYup/XUnIWFAxg+3MOSJYs4++xze1y0sCRJ5OXls2TJ4vrvWE53D0kg6BC9/64j6NXodLoGwT755FMaXm8ezR0IBAgGg+j1OoxGEwaDEYPBQF1dHT5fFdXVPkpLS7HZbLhcbhyOLCwWS4v+qqoq2bNnO8uWrSASUTAaDQ17tmPiDdFANEmKCr0sa7DbbZhMPS9ZQ2ekjzzrrHP4xz+e5ptv1nHKKad21tA6DZ1Oh81m46OP3ueqq66N+7kKBL0FIbqCHoksyw2u72RYLJY2FT632exMmTKFoUNHsnWrl7KyEnJy8hqizI1GU32wWKg+GK2GAwf2s23bVg4c2F8frBZ1t2u1WrRaLZGIQiQSbrCoW8Ymqmg0WnQ6LTqdDlmW66Pto/8bDIY2RdY3mU8Hiy1s8hlZkXUuRXefz/zaCgw+TY/crmWz2SguPsKnny7gkktm9DiLXCBIFRG93Ab6UnRdIsQcE1Nd7aO8vJy6ujoCAT/V1dX4/bUYDKb6Pdfm+u1ZUVGOur5VgsFQQyR5dbWvIfo79n9ZWRl6vQ6n09Vm8W2+pgsgoWCUwa9ISQOr4p2rIcJFztYDsTb5jCwosxJQo3M1ySrnZqW/pOO+ffs49dRxDcXuxfe179CX5imilwWCTsBisbZprTtVysvLWL9+Pd999y2yLGO12tBoZDSa6P7qZFZd84QWRkklpEr4lejffLIsX/HWgyNoGpKTJIqK3uQzMrfUitooi6xfkfioPndzOoW3oKCAr776Arc7h2HDhqetH4EgXQjRFQi6maysbM4662zGjRvHt99uYP/+fQSDQfx+H8FgiLq6APn5BQkDvxqnj3x6r5NAJLUsX8nWg5NFRS+psDQR3BgRpLRnE5NlGbc7h/nzP8ZsNpOZKYRX0LsQoisQ9BCsVhuTJp3e5DVVVVm/fh1Ll36G2+3GaExeEKItgVWJ1oN1weqkUdHJgrQ6s/5vIoxGI3Z7Ju+88zbB4MUMGiSK3gt6D/2qypBA0NuQJImTTz6FSy6ZQVlZKdXVybcBtaVS0DRHNVqp2evhIBVLX04q3smCtLqqIlFGRgZ5eXnMn/8Jn322iFAo1CX9CgQdRYiuQNALGDJkKFde+QP8/gAVFRUJj4snpImyfMUrJ3hSaDNlaz/BpMR3EcfWdiVaiqsGtUuziel0OoqKBrJp07e89967+HxVXda3QNBehOgKBL2E3Nw8vv/9q6ipqWnYntScttblHWUNcHtRKb8ZXMztRaVM9zixWq3IGz9MKN6jrAEucfowSgrRfc0qJlnhImdVl283kmWZgoJCKirKefvtNzly5HCX9i8QtBWxpisQ9CKcTiejR49m61Yvbnf87EyxwKr2FHbQaLScdtoElix4jRsnT2V1rTNuTueuqv2bKk6nC5/Px5w5bzJ9+gV4PMd395AEgrgIS1cg6GWcfPJYgsFgQmu3o0ycOJlIJIJv4+ImVnBPEtl4WK1WXC4X8+Z9yMqVy4lEIt09JIGgBUJ0BYJehsORxciRoygtTU/5w8LCARQWDmDVqpVpaT+d6PUGCgsH8PXXX/Hqqy+xadNGAoGe/bAg6F8I0RUIeiGnnHIqdXV1abN2J0yYxJ49uxrKNvYmYuu8Op2WxYs/5cUX/8mqVSuoqCjv7qEJBEJ0BYLeSFZWNiNGjKSsLD3W7mmnjUeWZVav7n3WbgyTyUxBQQFZWVmsW/c1r776Mm+88SrffvsNVVWV3T08QT9FBFIJBL2UsWPHsXnzdyiK0u6CCYmw2eyMGnUCq1evYsaMy9vUfqL0kd2FTqcjNzcPgJqaapYu/QxFURkyZChTpkzFZrN329gE/Q9h6QoEvZTs7Gw8Hg9lZWVpaX/ChElUVh5l8+bvUj4nlj4ymulKakgfucmXPJNWV5GRYSE/v4CCggL27dvLa6+9wnffbUybm14gaI4QXYGgFxONZG7btqBUGTPmRMxmc5tczMnSR/YkJEnC7XaTleVgwYJPeO+9dzl6NHHSEYGgsxCiKxD0YjIzHWlrW6fTceqpp7F+/Vr8/tRKrrUl93NPQK83UFQ0kNLSUt5++01KSkq6e0iCPk7P/EsQCAQpYTAYMJlMacs9PHHiZEKhEF9/vSal49uS+7kn4XQ6MRj0/Oc/7wjhFaQVIboCQS8nO9uVtr2ogwYNJjc3j1WrVqR0fFtyP/c0bDZ7g/Cmaw+0QCBEVyDo5bhcbgIBf1raliSJiRMns2PHdo4cOdLq8W3N/dzTsNns6PVCeAXpQ4iuQNDLcblcBIPBtLU/fvwEJEli9erUrN3mRRR6i+DGsNvt6HQ63nrrNZYv/1xULxJ0KkJ0BYJejtVqRZbTV8Q9M9PByJGjWb16Zb/ZWmO323G7c9iwYT0vvfQCS5YsprLyaHcPS9AHEKIrEPRyLBYL6dbCiRMnUVFRgde7Jb0d9SC0Wi25uXnk5uaxefMm3nrrdZFKUtBhhOgKBL0ci8WKJElptUJPPPFkTCZTygFVm3xGnt7r5OFdbp7e6+wxyTHag0ajIScnF41GyyeffJxWV76g7yPSQAoEvRxZlnE4HNTV1WEymdLSh06nY9y48awv1/DUnix8ijZhisdYVqpYkoyqiIYPSm18UGrrEWkh20tWVhYHDhxgxYplnHnm2d09HEEvRVi6AkEfwOVypS2COUbexEuxT78Vn6IjWYrHeFmpQEp6Tm8hLy+P9evXsW3b1u4eiqCXIkRXIOgDuFw5+P3ptR6/0xyHrGsqlvFSPLaWfaonpoVMFVmWycnJYcGCeWJ9V9AuhOgKBH0Au92Oqqppa3+Tz1hfxKAlzUU2lexTPTUtZCoYjUYMBiNz577Pli2bqagoT+u1F/QtxJquQNAHsFptSGnaNRRbo426iFvSXGSnOaqbrOmmck5vIysri8rKShYunI8kRYV48ODjOO64oeTnF2AwGLp7iIIeihBdgaAPYLVa09Z2/DXaKLEUj81r6I7JCLDdb2hk0Uotzunt2O127PZoLd5QKMSOHTv47rtNSJJEUdEgRowYyXHHDUGjie8hEPRPhOgKBH2AqMvTQCgUQqfTdWrbCV3BqsqFTh9Ai2jlDTXGhvSPPa2ofTrQ6XQ4nU4AFEWhtLSYDz/cxujRJ3D22dOR5d7rThd0LmkXXYvFgFbbN570NBqZzExzdw8jrYg59l6KigqorKzCbI66NmVZavi5I9h1CpWhln/Dcl0l43Lgf7e2dCWHVZmllRbG5aiMM6uMy/E1O7tz3K+dNcfOxmIxkZeXw86d28jMtHL22We3W3j76ve1Of1lnmkX3erq9BTY7g4yM80cPZpaXdHeiphj78VstrN79360WkP97wZqazv+93eGXWqxRitFQpQueoGSosuoDNnjnlcZkjul/2R01hzTRXa2mxUrVhEIRDj99ClI7Vh476vf1+b0pXm6XImXe4R7WSDoI7hcboLBtZ3ebswV3NhFPIo9vLXxM9asKcA2eGDcyObGwVL9wcUcD1mWKSwcwNdff4lOp2PChIndPSRBNyNEVyDoI6Sz8MEoa6CJSKqqhWUFhaxcuYJLT7mghSXcOFgqXoaqaDQ0/UZ4CwoKWL16BaFQiEmTJovgqn6MWN0XCPoIVqsVVU1ftaHGxOrs7tmzC4dvR9IauvGin3tzgoz2oNFoKSwcwLp1a5g7932qq3t/9LagfQjRFQj6CBkZFlRV7bLye+PHT0SWNaxatSJpDd1E0c+9OUFGe4i5mg8fPsicOW9w5MiR7h6SoBto9Vvv8XhsHo/nUY/H85HH43nc4/FkdcXABAJB29BoNDgcDoLBrgksslqtnHDCGL74YhWRSDjhcYkSYfT2BBntxe3OQZY1zJnzRr8qlSiIksqj5r+AfcBvgd3AS2kcj0Ag6ABOpzPtOZgbM2nS6VRVVbFhwzcJj5nmqEYrNRXYvpIgo73YbDacThfz5n3IypXLiUQi3T0kQReRiuhme73eJ7xe73qv1/t3wJHuQQkEgvbhduekvdpQY0aPPgGXy80nn8xLmH94lDWQdM23v2IwGCgoKOSrr77k44/n4vd33ecm6D5SEV2Tx+PJBfB4PDmACLsTCHoomZmOLk2+r9FomD79fPbs2cWWLZsTHpdszbc/o9FoGDBgAPv27eOdd94SlYv6AamI7n3ASo/Hsx5YWf+7QCDogVitNhIVJkgXEyZMwm7P5JNPPurSfvsSubm5BINB5s59n0BAPJD0ZVoVXa/Xu9Dr9R4HnOP1eod4vd5FXTAugUDQDqKFD7q2zJxOp+Pcc6fj9W5h584dXdp3XyIrKwufr4qlSxeLUoF9mISi6/F4nqr/f5XH41kJfPjo8EEAACAASURBVODxeFbW/ywQCHogRqMRi8VCXV3XpkY8/fQzyMjI4JNPPu5wW5t8Rp7e6+ThXW6e3utkk8/YCSPsHeTm5rF583d8++2G7h6KIE0ky0j1YP3/PwaCjV4XW4YEgh5Mfn4B+/btxeGwdVmfRqORM888mw8//IADB/ZTUFDYrnb6e/YqSZLIy8tnyZLFuFwu8vLyu3tIgk4mmXtZ8ng8w4FXAT3RsiAm4LmuGJhAIGgf+fkF3RIJe+aZZ2MwGJg/f1672xDZq6Luervdzscfz6Wy8iiBQKDhX01NDWVlZezfv4/t27exa9fO7h6uoI0ks3QnAHcAHuD5+tcUYH66ByUQCNqPw9E9zqiMDAtTppzB4sWfcskl38PlcrW5DZG9KorVaqWkpJhXX30Jk0mH3x9q9K4EqKgqhEJBpkw5g1NPPa1dFYwEXU9C0fV6ve8B73k8ngu9Xm/HF2oEAkGXYLdndlvf55wznaVLP+O///03M2fe2ubzbRql1YpF/QWXyw0kL18YiYRZvvxzgsEgkyadLoS3F5BKlaFyj8fzHKAj+oiV7/V6z0vvsAQCQXvJyMhArzcQCoVaP7iTycx0cP75FzF37nts3vwdI0aMbNP50xzVCSsWxSsPOM7cv6N8NRotAwYU8dVXXxAOh5ky5QxkuX95BXobqXw6zwJLADuwByhN54AEAkHHiAbj5FFTU9Ol/caijjeMvIkBt7/EnC+2Ew4nzskcj0TZqwA+LrPWW8FSQ4DVN0d1nT+RXkaskML69WtZsOATDh06KNJK9mBSsXRLvV7vmx6PZ7rX6/29x+NZmvZRCQSCDlFQUMi6dV+h15u6pL/mUceyxYl20nXM+fILrp10fJvaal67F+Dpvc64AVYLi40MK+y/OZxjxIR3166dbN3qxWAwMHy4h+HDPe2OJBekh1QsXcXj8YwCzB6Px4PYMiQQ9Hiys52oatetg8aLOpZ1RnZkjOiU1IaJAqkqQ8KVGkOSJHJycigsLCQzM5OtW738+99vs3//vu4emqARqXxjfwmMAp4A3gBeSOuIBAJBh8nM7NpgqkSiqLE6+fe/53S4/USBVHZd/wuwSgWdTofL5SIrK4sFC+aJ1JI9iFRE90av1zvH6/Wu8Hq9Y71e79/SPiqBQNAhrFYbGo2my9b2EomiPlTD119/xerVKzuU2jBRecBz3UJMkpGRYcHv97Ny5fLuHoqgnlREd6TH4+m+PQgCgaDNyLJMTk4ufn9tl/SXUBRzggwaNJiXXnqBp5/+OyUlJe1qP1GA1YmZXR+h3dvIycllw4b17N69q7uHIgCk1p4+PR7PHqCAaNSyCqherzfl3GQlJb4+E9OfmWnm6NGuuYl1F2KOfYeNG79m2bJV5OTkdEl/8bb0jLIGiEQifPbZIubOfQ9FUbjggos499zz0ek6HnmcbA9rX6Ez5lhbW0sgEODaa6/DbDZ30sg6l770d+lyWRNumG41etnr9Q7s3OEIBIKuICcnt81bdjpCvKhjiNaMPeec6YwdO4533nmLDz54j+3bt3P77T9Ho2maCCORcAs6htlspqqqiuXLl3L22dNbXHdB15HKliGBQNALyczMpCclKHI4HEy+5k6qz9RSrTHx2FYfFxaqjLZGrbiuKnbQX4U9JyeHLVu2sHfvHsaMOZlhw4Z1W8rQ/owQXYGgj+JwOABQFKVHZClqEFWtjAREjHY+Kg4jITHKGkha7KCzRLE/VzGSJInCwkKCwTq++mo1q1evJC8vjzFjTqKoaCAmU9fs6e7vCNEVCPooWq2WrKxsAoFAj1jHiyeqiqxlQbGeUdZAlxQ76Aph7+no9Qby8vJRVRWfz8eCBZ8gSTBkyFBGjhzNgAFFPeIhra/Squh6PJ4DgBsoAZxAADgC3Ob1ehemd3gCgaAj5OcXsHWrt0eIbiLx9EsGdu3aiU3jbLXYQXPX8PTcAMP0qQcZiSpGx5AkCZvNhs1mQ1EUDhzYz9atXo47bgjnnHOesHzTRCrftM+B0fURyyOA94ALOFbkXiAQ9FDy8vIJBntGdG+ivbxqdTl//euj5BZ/2WLbkYRCSJV4eJebx3e7+KjU1iT/8vsHzWzyGTs8hv5YxagxsiyTlZXNgAFF7N+/j3feeZPSUpFmPx2kIrqFXq/XC+D1encARV6vdzvQdWGRAoGgXXR1ZqpkJNrLOz03yNChQ1n4zz9h+OZdrHIIUDFKCjISfkUGJAKqTISmkWEhVWpTgftEY5jm6N78zbFiEQ/vcvP0XmebHiQ6m1jU+9tvv87Ondu7bRx9lVTWdA95PJ5HgJXAJOCwx+M5FwimdWQCgaDDZGY6UFWV/9/encdHVd8LH/+cWTKTyWSfLCQhbAlHCAk7Isi+CqKgorT6qI+217Z0c2ut9mn73PbW2urtfWr1WmtvvbcqLlBlEwVks9YVFAThAMoW1gRC9n3m+WMSlmRmMklm5pxJvu/Xyxc4Sc75ngD5zm/7fj0ej+69VlvXTNvuHIY4LIt+Tb8FZsorSmjY8By3TRrKxtirqQti2rczU8P+Ymh9XY+dzV3d3BXOWJOSkrHb7axa9QZjx17JuHHjQ3KuWgSXdG8H/gWYC+wGfgGMBL4WvrCEEKFgs9lIS0unpqaauLjgR4Th0nqWtzVhrCpNaPmIAgpYEtMxT76LP637A2kLJkMQ7xM6OzXs7zyxXjubu7K5KxKx2u2xZGfnsGPHJxw+fIjZs68hLS0tJNfuzYJ5i9gItBZwtQJuTdPe1zTtdPjCEkKESl5ePhUVFXqHcUFrwmhdm22bWRWrjay536KpouOSkVbFE7Kp4UDJL5y6srkrUrGazWays3Oor69j2bK/sWPHJ9Krt5uCGek+C5wH1gNTgOfwjn6FEFEgOzsHt9s4G4V8JYy2GmOczIwvY0t1A1hiLryu4MZuglq3csnu5dCM7PTa2Zxgdne4a7utSMealJRMXJyTd9/dxt69e5gwYRL9+vUPeLRo3d7TPP3uYU5X1pMRb+M7k/pzzZDIlCQ1smCSbr6maZNbfv+Gqqr/DGdAQojQcrnSMJnMhimSEUxiSDC7uSrLRkJlLevPNFOr2PBUnWNqSjUTcy5uMvLWJQ5NXF1JfqEwNbnqsqli6Hhzlx6xWq1W+vbtS1VVJatWvY7LlcbEiZPIze3X7u/Vur2n+fX6A9Q1eeM5VVnPr9cfAOj1iTeYpGtXVdWhaVqNqqqxQKeKdjqdNiyWnlHn02w2kZSk/3nHcJJn7DkufU5VHURp6VlD7GZOtLopb/T/M8GqeJidWceBBidby+3UmUw4lUZOf/Aybxx4j0H3P8DAgYMAMJkUHA5bSOKanVnHyhMOGj0Xp7tbYwnVPXwZ6/AQY6tlwxk75Y0mEq3eloXDkzyAzecz6hUreN/opKe7qKio4O23V2MymbBYLBf+GzRoEM/ssl1IuK3qmtw8894RvnbVAJ/X7S3/LoPpMnQr3s1Tu4GhwC80TVsW7A2ky1B0kWfsOS59zj17PmfTpo1kZ+foHFX7TUBe3h8TF3c00+5zzLip3vws53du4J57ljJ0aEHIuwwZsS6zv2c0Sqytu+Pdbjcej4eTJ0/yYvNYn5+rAB/dP9nnx3rSv8vudhl6UVXVdcBA4JCmaWdDGZwQIvwyMvroHcIFHR3bAXjqqKvdum8zJlwz7sZy/DOefPL3TJ06nSVLbqGTk28dxqZ3kg2WUWJVFAVFUS5MMWdnZ+E4Wk8N7UfcGfHhHYVHA79JV1XVZbS+/bz8dTRN+3pYoxJChFRKSgo2m43GxkZDnLfsKGH4W/etclt44IGHWLny72zZsonPPtvBTTctYdSo0bqfQxZeZrOFKUmVrD9vofmSN0R2i4nvTOqvX2AGEWik+0zEohBChJXJZKJ//4EcPXqYlJRUvcPpUKCNQrGxsSxZcisZV17L+9WpvO1IZsMX5UxOrGBCjn6VnLoiVFPERplqbjUiuZmmphLer06h2hMju5cv4Tfpapq2NZKBCCHCa+DAQWjaPr3DCEpHO3r3VNr5RElDifN+3ONIZktDIlu/Uog3NzEtpcYQU6+BhKrAhVHbFY5JU+iv7CchIZHrr79BGii00P/8gBAiItLT0/UOIWgF8XXMS60kwdwMeEgwNzMvtfKy9eC2a76KYgJFodJtZc2ZOD6viPFxZeMIVYELvYp6BMPlSuPcuXO8+eZqGhqkcjBIP10heo2EhEScTid1dXXY7cafhg207tvRWV+3ycLaYzBoQI0h2hr6EqoCF0ZvV5iZmcmJE8d5++03mTt3viH2FOgp0Eaqv+JjIxWApml3hS0iIUTY5OUNZs+eXVGRdAPxt+Z7KXdsIo8+ej/f/vZ3ycrKjlBkwQtVgQu9inp0RlZWNocPH+Kdd9Yza9ZczGb/f3Y9vZJVoLdCLwOvACnAPuAvwC4guv+1CtGL5ebm0tTUqHcY3earRV9bcTRQX1/PY4/9G9u3fxKhyIIXqjaDRm1X2FZWVjaaprF58zt+p5pbK1mdqqzHw8VKVuv29pxS/4E2Ur0NoKrq/Zqm/bbl5fdUVd0QkciEECGXnp4BKIZo9dcdrdPOW8udlDe2jh0uPo9FcTPTVU/2w/+HZ5/9T/785//k3XeHMGPGLAoKCiNaDtPfzuJgzisHI1TXCTdFUcjOzmbv3j2cPn2KOXPm4XK5Lvucp9897LOS1dPvHu4xo91g1nSdqqpOBz7G209XRrpCRKnY2FhcrjRqamqIi4vTO5xuKYivY2yGh5qa+gBHZpK5994H2bRpI5s3v8NTT/2BjIxMZsyYxVVXTez0+mJnj+Z0tLM4VAUujFIooyMmk4ns7BzKyspYtuxvTJs2g4KCwgtvAE9X+q4u5u/1aBRM0r0L+B2QD3wB3BHWiIQQYZWXl89HH30Q9Un3UoGSjtVqZc6ca5g5cxbbt3/CO+9s4KWX/saWLe9wxx13069f/8s+319i7SiB+vq6rvTK7Q2Sk5OJi4tj48b1fPXVl4wYMYq4uDwy4m2c8pFge1Ilqw5rLwOoqjoYb9LdCRzXNC3oespSezm6yDP2HP6e8+TJEyxf/ooh6jB3V1dqL3s8Hnbv3sWLL/6NiopyZs+ey/z512G1Wn3WhbYobualVrYkVF8blpr9nitu8rTvF9wSBT8ZcCZszxgtPB4P586dpa6uDqczltK4PFaesFHffDFt2C0mHp6dH1XTy92qvayq6neBRXg3VD2PN/l+N1TBCSEiy+VKQ1FMhmn1F2mKolBYOJyf/Syf5ctf5q233mTnzs/45je/xZam4X5HpoGO5vgb0Sp4fB4BMdLOYj0pikJqqnddNybGTOOJA4xx2/nc0p/KJnOv273cagkwCzivadr/A64Mb0hCiHCyWq3k5Hj7ovZmDoeD22+/i6VLf0B1dRX/8R+PB0ys/hKl98iO76/zQFTsLDYCi8WCy+Vicn8nt8Tu43bbDv51rIXZg10df3EUCSbpmvD+3Wl9w9Yz5zmE6EXy8/OpqpIf/ACFhUX88IcP4na7cVf6bqLWukbrL4EGSsiBKmsJrz2Vdh7fH8+jh9J56qiL07Yc0tLS2bZtK6++uozz58v0DjFkgtlItQzYBvRTVfVN4I3whiSECLf09EyC2c/RW2RlZfH979/H02teJGHWt1AsFzfutCbWjo7m+KsVHS07i/Xid4NaKhT0tXLu3Flee+0VFi68kbS0NJ2j7b4OR7qapj0J/AtwP/AQ8Fy4gxJChFdqaioxMTaampr0DsUwcnP78Y3Zozm//mk8VWdpOzINdFyoo1rRwr+OakenpKRitVp47bWXOXnyhB4hhlQwG6m+BG7QNO21lv/fBEwPd2BCiPDxtvrrz7Fjx0hJSdE7HMPIy8vnzoYGnn76bpqamkhJSeW9/gPYNWw6xX2upongjwtJwg1OMLWjExISMZvNLF/+CgsWLKR//wGRCi/kgplergGeUlX1/2iathnf+9+FEFFm4MA89u/X8B5M0IcRk9XQoQU8/PDP2LNnN4cPH+Lw4UPYx12BBf+jMSO21osWwdaOjotzYjKZeeONv3PlleMpLCzC6YyPVJghE0zSLQFuBN5QVdUFyHyUED2At9Wffu+hQ90HNpQJPCsr+7ImCY8e8r2WGOi4UG8vgBGsjnonXyo2NpasrCw+/XQ727d/wsiRoygqGk58fEIkQ+6WYJKuomlamaqq84HlQFGYYxJCREBrq7/6+npstshX/OlOsmqbYPNi69lVbQ86gXc2QQcajRm9tZ7Rta2j3dGfh8ViITOzD83NTezc+SmffrqdkSNHM3LkaMO2cbxUMEn3JwCaplWpqnod8P3whiSEiARFURg0KI+9e/eQlhb5BvddTVa+Rsg7qmJpO2r3l8C7MsL2NRoz03yh1KPRW+sZ3aV1tINlNl9Mvp99toNdu3YyYcLVDB1aYOievYH66X5D07TngOtbkq0QoofJze3Hzp2f6XLvrvaB9TVC9jdN7iuBd2WE3fa4UHNFKSVbnmf5+QNkT1iIadBs3MrFH6dSACNyzGYLffpk0dBQz9atm9m+/WNmzZpD3765eofmU6CR7rGWX/dFIhAhROR5W/15dGn115m1vEt1ZtrWVwLv6gj70vO2Z8+W8skVaWhaKTtXPYc1bw8pU+/EHO8iwWKMDWGhYsTNbr7ExNjIycmhurqa119fwbXXLmDgwDy9w2onUNKtVVV1MnAoUsEIISLL4XCQnp5BZWUlCQmR3YzS1T6w/kbI3qJ5l/fU9ZXAgx1hB0o2qaku5sy5hjlzrqGxsZGvvvqS5ct/zpFjR5kyZRr5N94MxPiMP9B1fX1srEO/Iiah3uwWCXFxcZjNZlavXsncufNR1Sv0DukygZLut1t+HYT3b8/HwEigCpga3rCEEJEyZsw41q5dGfGkC13rA+tvhFwUV8fBWluHCTyYEXZnko3VakVVr+BHP3qYlSv/zsaN6zlwYD933/0v7To5Bbou+D56FGOrJT9Gn+q70boz2263k5GRwbp1a2hqaqSgoFDvkC7wm3Q1TfsagKqqa4HrNU1rUlXVDKyNVHBCiPDr338AcXHx1NbWEhsbq3c4HQo0Qp5Dx00cghlhdyXZWK1WbrrpFoYMKeC///sv/OY3v+LWW29n/PgJQV239fdtP7bhjJ38HO8bgkhP9UbzzuyYGBt9+mSxYcPbHD9+nD59+rTs2I8nMTERiyWYfcShF8xd+7T5/MhvcxRChI3ZbObKK8ezefMmsrOzO/4CA2gdIXe112xHI+zuJJuCgmH89Ke/4C9/eZbnn/8LX331JYsXL8FqtXbpuuWN3o/pMdXb1c1uRmG1WsnOzubo0cPs378PjwcUBZKTU7j++kW6FNcI5u3Kc8AeVVVX4G1i/4fwhiSEiLT8fBWr1UJjY6PeoRhCoK5BQX19QiLf//59zJ49l23btvDEE49x7tzZgNf197FEq/f1jkbJ4RCos1K0MJstpKSkkpWVTXa2t+hJdXUVf//7a1RUlEc8nmCS7l5gEvA4cLWmac+HNSIhRMTZbDZGjRpLaWmJ3qEYQiiSjdls5oYbFnPPPUs5deok//Zv/0pezV6/1/V3z1np3lGsHlO90d7IYU+lnaeOui60DNxTaQfA5Uqjvr6BFStei3jbwGCml/+vpmmTgTPhDkYIoZ+hQwv46KMPaG5uxmz2tTu45/G3RtrVndW+jBw5iqysLP70p6d54//9hAlf+yGluVOpaxm1Wls2XPu75/AkDzU1+k31Rmtrwo6m41NTUzl37hwrVrzKokU3kZKSGpG4lI56aqqquhU4B2iAG0DTtIeDvUFJSWWPadqZlOTg/PkavcMIK3nGnqMrz7llyyb27t1DRkZmmKIKLV9rusFuNmr7Qxm8I8twjeQaGup56aUX2FVuJW3eD8By8UhRoPu2PmN34o2Gs7ZdXZ/356mjLj9vUppZmlt64f9LS0tJT89g4cIbQnbvtLR4v4fegxnp/lfIIhFCGNrw4SPYufMz3G43JpPxd6i25W90U1xnbXecKNLHYWJibNxxx138/mA89ZbLz/AGc9+ujr6j8axtKAQ7HR8bG0tjY0MkQgKCS7ovAmMBK96T51lhjUgIoZvk5BQGDx7M0aNHSUvz3VnHyPwl0ktrM7cmnSZP8KUjQ0VRFOotvovyB7UzugtTvdF61ra7jLrzOpik+zrehJsNmIETwLJwBiWE0M+4cVexf//+qBzt+k9c7ZshKHjwtfYV7h/K/pJBvCk8XVOj+awtdH1qvKtlRsMtmO+6S9O0ucCHwGjAHt6QhBB6Sk1NpbCwiJKS6NvJ3JmE6QFdjsP42qXsbqyjdNN/cfDggZDfr7vHn/TUOjXufZOiXJilaN2FHIhRd14HM9Jt3YkRp2laraqqndoY5XTasFh6xk5Is9lEUpLx+zV2hzxjz9Gd55wxYwqHDu0nJsasW+WeYJhMCg7HxV7AszPrWHnCQeNlU8eX12RulWj1HsfZcMZOeaPpwv8PT/IA4esvPNbhIcZWe9l9h5mKWf/l+zz+4WqmT5/BzTffcqE3bNtn7Cxf3xOr4mF2Zl23rhtqvp5za7HvqfGt5U7GZnScisY6PIzNaFul7PJ7eDxN2O0xEfuZEMzu5aVAKtAALASqNE2bGewNZPdydJFn7Dm6+5zvv/8eO3Z8Qp8+xt3GEczu5bYN7uHyXb9G2dlbV1fH6tVvsGnTRhISEliy5NaWxuzd39UbqWfszn18Peejh9Lx3bbRw08GhOYUa3V1NXa7ncWLl4TketDN3cuapj3V+vuWOsyhn/8QQhjO8OEj+fTTHTQ2Nhq6KXhbvjYb5dgbfSYDI+3sbf3BP27ceF544Xn+9KenGTBgIPPnz2fIkMJunZ2OxFnbcHwvjboZqjsCNbH/K/jcZwBwV3jCEUIYhcPhYPz4q3jvvXfbdcuJNv6SjhF39vbr15+HHvop7767jXfe2cAf//gkqakuihZ8g5Ks8VQ2mw151jYc30ujbobqjkAj3Zdbfv028E/gPbxHh8aFOyghhDEUFBSyffsn1NXVYbf3vD2U3dnZG84pW7PZwtSp05k8eSr79u1m9a5THEwZg6nZ0hKf8c7ahmOXdCgrgxlFoNZ+bwOoqnq/pmm/bXn5PVVVN0QkMiGE7mw2G1ddNSGqOhB1RlenLyM1LW0ymRgzZiybnfHt4tR7RN5WuKaCo7UMpT/BvAVxqqo6XVXVeFVV5yBHhoToVbwdiKw0NESuak+kdLWxQaQ7/kTDWdue0JEoEoI5C3AX8DtgMLAHuCOsEQkhDMVmszFmzFg+/PB9Q+9k7oquTl9GOgn6G0Xa3XV4PB4Uxe9mWZ/CMTXeE6eCwyGY3cv7gAURiEUIYVBDhhTw4Yfv09zchNls3HO7XdGV6ctI76r1taHI01TPsTf/yKPVh5k3bwHDh48IKvn6mhpfVZrAqtKEbifKSO2SjubE3uG/HlVVHwZ+hLdIhgJ4NE3rWW93hRABxcXFUVQ0gt27d0VNB6JwivSuWl+jyMkp1VSOHci6dXt55pk/kpPTl0WLbqKgYFjAa/maGm9bl/rSe7alZ9LraC09GhJyMG9ZbwGyNE3r+dUEhBB+FRUN59NPd0RlTeZQ02Mq1ecocsLVXHnlVXz88YesXbuaJ5/8PZMmTeHGG2/2u9u8oynwQBu09D7X3NFaulHOXAcSTNI9BNSGOxAhhLElJiZxxRVDOHToS9LS0vUOR3dG2VVrNpsZP34Co0ePZdWq19m4cT379u3lzjvvZtCgvHaf729q/FL+ErPe55oDraXrHVuwgnm7GgN8rqrqspb/Xgp3UEIIYxo5cjT19fV0VD5WRJ7VauXGG2/mvvsexO128/jjv2HFileprb18zORrl3Fb/tamAyW9YJoQdFeg5g3RsMMbghvpPhb2KIQQUSEtLY3+/Qdw6NBXJCYm4XQ6e/1Us9Hk56v89Ke/YMWKV9mw4W0++OB9rrtuIRMnTsJkMrWbGve6uAEr0Nq0/1GyEpGp3EBr6d7nMX7JyGCS7ufAHC5vYr81nEEJIYxr+vSZfPXVl3z55UFOnDiBx+PG4XCQlJSsd2g9zp5KO1uLnZQ3JnZq3Tg2NpbbbruDSZOm8NprL/Pii//D1sOVOCd+nVrF3q7+dLBr076SXqtITOV2tJYeDSUjg21ivxcoBOq42OpPCNELOZ3xFBWNoKhoBI2NjZw5c5rXX1+BwxFHTEyM3uH1GKHYtNSvX3/uv//HrNl9ks9tQ6lVbD6vFez1Wj9vVWkCvrr/RGIq11+80XJOOJjvkKJp2rcADZgFpIQ3JCFEtLBarWRn5zB16jROnz6ldzg9SqiqXimKwtHEIhTr5X1kmzwmNp/rfA/Zgvi6gGureiqIr2Npbik/GXCGpbmlhku4EFzSbVJV1Q7E4e061LNOxgshum3o0GFkZ/fl7NlSvUNpZ0+lnaeOunj0UDpPHXVFZMNPKIRyY5D/a5k5cEDr9PWk5GPXBfOn9xTwQ2A9cAzvESIhhLjAZDIxY8ZM6uvraWxs1DucC1qnaL0bbJQL06qdSbx6Je1Qjib9fY2n+hy///0TbNq0sVM70gvi65iXWkmCuRnwkGBuZl5qpSFHlkYTzKj1iKZpKwBUVX0NGBnekIQQ0Sg5OYWJEyezbdsW+vbtq3c4QPfPlepZDKI7Va/abo7Ki61nV7W93bVmZnn4oLCQV19dxpEjh7n11tuDXpc3yjnlaBOoif0kYChwr6qq/97ysgn4LhC4zpgQolcqKhrOwYMaJSVncLnSOl2IP9S6O0Xb1aQdinKErZ+/tdxJeWPw1/H1RmFXtZ2iuDoO1traxORm+D1LWbduLWvWrOT48WJuueXr5OcP7lSsIniBRrplQCZgA/q0vObGW4dZCCHaMZvNzJw5hy1bNnHs2DFMJoXExCTi4uJ0iae7jQm6krRDOTouBtQi2QAAHLdJREFUiK9jbIaHmpr6oL/G3xuFg7U2lua2X3M3mUzMn7+A3NxcXnjhf3jiiccoKhrOwoU3kZWlb5n9aKil3FmBmtjvBnarqvpnTdNOqKqaBDRrmlYZufCEENEmOTmFRYtuorKygsOHD7Fz52ccP16MyWQiKSmZ2NjYiMXS3cYEXUnaepcj7OrovrBwOL/85a/ZtGkjb721jl/+8mdMmHA1N910M7Gxnd/l3F1613kOl0DTy6OAvwDjVFW9AXgGKFNV9QFN01ZHKkAhRHSKj0+gsHA4w4YVcfbsWQ4f/orPP9/FuXNn2/WAtVispKenh3w6urtnN7uStPUuR9id0X1MjI25c+dz9dWTWbduLZs3v8PBg/u5556lZGVlhyNcvzr75iVaRsWBppd/B9yhaVqjqqq/Aq4BDgDrAEm6QoigKIqCy+XC5XIxevRYSktLKS0twe12X9gxe/z4Mfbu3Ut2djZmc+Bi/J3VnQ0/XUnake61e6k9lXYa3Are053BlXb0xemMZ/HiJYwYMYo///kZfvObX3H77f+bMWPGBbx3KJNeZ968RNOoOFDSNWuatktV1SwgTtO07QCqqhqrkKUQImooikJaWhppaWmXvT50aAEJCYl88ME/ycrKxmq16hRhe51N2pHutduqbeLx8hBr8jArpWvHefLzB/PIIz/j2Wf/k+ee+xOHDn3FwoU3tvvzCUfS68ybF72n9DsjUNJtPWw3F9gIoKqqFYgPd1BCiN7FZDJx1VUTcTrjeeedDaSnp2G3R27tt6sCje4iPdXprzm9VXF3696JiUncd9+DrFjxGu+8s4E9e3Zz663/i/x8NeC9u5v0OvPmRe8p/c4IlHQ3qqr6HtAXuE5V1UHAH4FXIhKZEKLXKSwswumMY+3a1TidThISEvUOya+ORneRHmGFM/GYzRZuvvlrDB06jGXLXuCJJ37LhAlXc8MNi3E6nWG5d2fevOg5pd9ZgXYvP6aq6iqgvGX38iDgWU3TXo9ceEKI3mbAgEHcfPPXWbNmJadPnyYjI0PvkHwy2pRmJBLPsGGF/Pzn/8ratavZsOFtdu36jK997TYS0q4J+t6dWfsN9s2LXlP6XRHwbYimaXs1TTvR8vsvJeEKISIhPT2dW275OllZWRw7dozm5ma9Q2rHaFOakaqHHBNjY9Gim3j44Z+RmprGn//8DHy+Cgsd3zsUZTl9iaaylNK8QAhhSHFxcVx77fV89NEHfPjhBzgcDpKTkzCbjfFjy2hTmpFeS87J6cuPfvQQ69at5c21f8V16hQp0/43tYrN770DzQ6MzeheCYhoKUtpjL+9Qgjhg9ls5qqrJpKb248vvtjNgQP7aWpqxuFwYLVaaWiopbq6Do/HQ0JCAhZL5H6kGXFKM9KJx2y2cO2111NQUMhf//oc2u/eZMSIkUyaNIUrcobSdjLVaLMDepCkK4QwvOzsHLKzc5gyZTrHjxezf/8+amvrSEyMo6GhGY/Hw759e+nTJ+vCcZZwF0swQtP0cD9jsNcfMGAgjzzyc9atW8M//rGNTz/dgcvlYuLEyUyePPVCGdBwzA5ES1GMVkpn2jl1RUlJZXhvEEFJSQ7On6/RO4ywkmfsOXrDc176jLt372LjxvVkZvbhQH2Cz1GoUdf5AnE4bD5rL/s6lxuKZ7w0iXldXmSjo+s3Njayc+envPvuVjRtH5mZffjBD+4nOTk5YMydrTHdGmt3vwfV1dXY7XYWL17SqXsHkpYW77e0Wu8Z0wsherRhw4qYPXsup06dYnNZnN+1w54i0PpoV7Xd6HRpwg32+larlTFjxnHvvQ9y330/oqzsHE888Rhnz5aGfMNTOL4H4SbTy0KIHmPo0GGYzRZeXHPa58d70tphR+ujXZl29V1gI7j7+jJ4sMoPf/gATz75ex5//DHuvfcBCtIzQjbbEI1rxMaNTAghukBVr8AV67t+sxGLJXSVv2dJMLu7fDQn2GT11FFX0Md8BgwYyL33PkhjYwNPPPEYxcXHgvq6YAT6HhhV2Nd0a2sbPBZLaAuY68VsNtHcbNw/zFCQZ+w5esNz+nvGVTtP8PDrn1PffPHnm1XxcH1WDcOTGtt9vpGZTApud/uf0zvPW1l5wkGj5+IUcOszbjhjp7yx/c/dRGszDwz2fzTn8f3xPr/Ol85+P48fP85jjz1KZWUlM2fOYuHCRZf1Wfb3nIEE+h4EG5d3TdfGrbfe1ql7B2K1mv2u6cpGqk7obRtTeqre8IzQO54z0DOu23uaJ7d+SUl1I06lkemptVG3iQr8b6QC/1PIjx5Kp+16rJeH61wVfqed/TVN8Gp/vQRzM0tzS4N+loqKClavfoN//GMbDoeDBQsWMmnSFMxmc8DnDKS7u5cjvZFKkm4n9PYfYj1Fb3hG6B3PGcwzlpWd4403/k5NTTXp6emGKa4RrK4ko6eOunwezbErbpog4G5fX0lsVWkC/pL4Twac6VRsAMXFx3j11WXs36+Rm9uPb31rKTk5WV1Kut0lu5eFECKEkpNTWLx4CUVFIzh9+gwnT56gsTG6ppg7y19JSEWhw92+BfF1LM0t5ScDzrA017vjONRrpzk5fbn33gf55je/xZkzZ/j1r/+VL774okvXijaSdIUQPZ7T6WTSpCnceefdjBlzJWfPnuX48WKqq6v1Di0s/B3NqXX7HoB1tIEqHHWdFUVh9OixPPTQT3E64/ntb3/Dxo3rCffsq96ia55FCCG6wel0cuWV4ykqGs5XXx3ks892UFxcjNVqISUltV1z9mjmqySkd9q48xWhwll9KzMzk4ceeoS//e15li9/hS+/PMjixbeQkpLa7WsbkSRdIUSvExsbS0FBIUOHDqOkpARN28v27R+Tk9MXk6nnTgB2p150OOs62+2xfO9732flypWsWbOKzz/fyfTpM5k7dz4OhyMs99SLJF0hRK+lKArp6emkp6fT3NzMnj2f06dPlt5hhY0R6kX7YzKZmDNnHmPHjmfVqtfZsOFt3nvvH1x77XVMmTKtx7wZkqQrhBDAuHHj2b9fo66uFrs9Vu9wwsbXiNVITQNSUlK48867mT59JitWvMorr7zE+++/x2233UFubj9dYgqlnvHWQQghusnhcDBlyjTOnOn8EZhoFq7G8t2Vm9uPH/7wAb7xjW9x/nwZjz76S5Yvf4W6Ov1H5d0hI10hhGiRnz+Yzz/P5dy5sz12I09b/poGrC5NANB16llRFMaMGcvQoQW8/vpyNm5cz44d27n77n9h0KA83eLqDhnpCiFEC5PJxNSp06murqG5uenC683NzTQ0NHTqWnsq7Tx11MWjh9I7Vas40vwdF/KgGGLEC95ZiFtvvZ0HHngIk8nEE088xttvr8Ptjr4yp5J0hRDiEi6Xi3HjruTIkSMcP36c48ePU1paQknJGWpra4O6hq8p21WlCYZMwIGOCxmtTV5eXj6PPPIzRowYyeuvL+epp/5AZaX/WtJGJNPLQgjRxqhRY4iPj8fpjCc5OZn4+AQOHNjPW2+tpW/f3A6/3neLPG9hitY1U9B36raVr2NElzJam7zYWAff/Oa32bZtC6+99jK/+tUv+M53vke/fv11jiw4xvpuCiGEAdhsNoYNK6J//wEkJiZhMpnIzx9MVlYWZWVlHX59R4nKSCPI1upVCr4rQenRJq+jqXlFUZgyZRo//vEjmM1mHn/8MT77bEfE4+wKSbpCCBEEk8nE5MnTqKqq7HAtMZhEZaQRZEF8HQtcFSEv9dgVndlN3bdvLg899AjZ2dn86U9Ps3Hj24YvI2mcP3UhhDC4jIxMCguLOHPm9GWvnzt3jsOHD1FRUQH4rlXcltEarfur1xzpKXB/u6m3lDl9joATEhK5774HGTFiFMuXv8qyZS/Q0BD5bkXBkjVdIYTohHHjxrNv3z4aGhowmUycOnWSzMw+TJo0hbfeWktcnKNd5Sevi80G9BhBBiOcpR6D5W8GoKLZdNna8+Vr4/DNb36LlSv/zttvr+PTT7czY8ZspkyZRmyssQqdSNIVQohOcDrjmThxEhs2vE1MTAwTJlzNiBGjsFgslJWV8fHHH5KdnX1ZAjNSxSejSzC7fTZlUPDflrAgvg6TycSiRTdRWDicdevW8MYbK1i/fh1Tp85gxoxZxMXFRegJApOkK4QQnVRQMIyysnMMGVJARkbGhddHjx7DoUNfUlZWRnJy8sXPN8AIUm/+3ni0fT0vtp5d1fZ2TRmaPMG1JczLy+d737uXI0cO89Zba3nzzdVs2fIOc+fOZ+rU6cTExIT1OTsiSVcIITrJYrEwder0dq9brVZmzZrLsmV/w+l09qhWgd3Rujmq7dRwcZ31sgRb0WxmV7Wdorg6DtbaLkvQnW1L2K9ff+65ZynFxcd49cODfJg+nU+K07A31zAro4HChM4VOwkVSbpCCBFCLpeLq6+ewrZtm8nOzukx3XG6w9/mqE+rYvGgtHv9YK2Npbml7a7TlbaE5Yn5NI8ehaXl6+otcaw5baW2poZxmZFPgZJ0hRAixIYPH8GJE8UcPnwYt7v5wut2u53UVJeOkenDf6nJ4D+/q20JfRYqscSw/nQMcWd30r//wA7jDyVJukIIEWImk4n5868DwOPx0NzcTFVVJa+8soyampoe15i9I4E2R/lKvP6mjLuyNu4v4ZviXTz52/9g2rSZLFp0Q6eu2R0y7yGEEGGkKAoWi4WkpGSuuWY+paWllzVT6A18nVu2KG5GOmvDXpDDXwJPsDQzadIUNm/eyHvv/SNk9+uIjHSFECJCcnP7MX78BD788J9B1XDuKQJNDefYG8N6nMpXbWmL4mZacjUFt97OyJGjyc7OCdn9OiJJVwghImjs2HGcOHGcM2dOk56e3uHn7zxvZf2p+Kg/4+tvarj1dYfDRk1N6CtJdbQW3K9ff+z2yHV9kqQrhBARZDabmT17Di+99AJVVVU4nf4bH+yptLPurINGjzE7FEULI52TljVdIYSIMKczngULrqeqqoqKinK/n7elzHkh4bYyUoci0XmSdIUQQgd9+mSxePESmpqaOXfurM/PCVSHWEQn+ZMTQgidpKWlsXjxLcTE2CgpOQNAc3Mz1dVVlJaWEG/yvcvZaB2KRPAk6QohhI4SE5O48cabSU5Oobi4mJKSEuz2WFR1CKOtJw3R41aEjmykEkIIncXFxbFo0U3U1dXidMajKN513Ly8YpTXN/NJUzaVzeao3r0svCTpCiGEAVit1nYNErKzc/judVexYsUb9O3XV+o49wDyJyiEEAZWVFTE6NFjOXHiuN6hiBCQka4QQhjcxIlXU1Z2lkOHviI21kFMTAwxMVZiYmyYze1rGhuNv166vZEkXSGEMDhvQY1r+PLLA5w/f57z589TWVnBqVMncTjiSE5O1jtEv/z10oXeWeAj7EnX6bRhsRj/nVgwzGYTSUk9uzuIPGPP0Rueszc9Y1KSgz59Ui/7WGlpKStXrqSsrJSsrKwLG7CMZGux7166W8udjM242GPIZFJwOGwhuefO81Y2nLFT3mgi0epmVnodw5MafX6ux9OE3R4Tsb9HYU+6VVWhr6Wpl6QkB+fP1+gdRljJM/YcveE5e/szWiwOrr32BrZu3cQXX+yhT5+sdpux9FbemOjnddNltZZDVXvZO7KOvZDoyxvNvHEilob6Jp8j65qaetxuJaR/j9LS4v1+TDZSCSFEFLPZbMyaNZfp02dx8uRJGht9j+j04re1XpgKfPhqWm+k0pmSdIUQIsopikJR0XCmT5/JyZMn8Hh8tYbXh79euuEq8GH00pnGiEIIIUS3FRYWccUVQzh9+pTeoVxQEF/HvNRKEszNgIcEczPzUivDtokq0iPrzpLdy0II0UMoisLUqTM4c+Y05eXnSUxM0jskwHdrvbbHiGZn1pEf0/01XX9N641SOlNGukII0YPY7XauuWYB1dXVNDQYcyNr6zGiimYzoFDRbGblCQd7KrvfTD7SI+vOkpGuEEL0MGlpacycOYd169ZcsptZweNxk5CQSHy8/921keBrs1OjR2FLmTMkydFITevbkqQrhBA9kKpegd1ux+12ExMTg9UaQ0NDPW+99Sbnzp0lJSW144uESSg2O0VrlStJukII0QMpikL//gPavX7zzUtYvXolp06dIjMzU4fIvJuavFPL7V8PRjRXuZI1XSGE6EVa+/dmZ2dTXFyM2x35Xb2+jhFZFU/Qm52MfhY3EEm6QgjRy8TGxjJ//nUUFhZRXFxMXV1tRO/va7PT9Vk1QY9SjX4WNxCZXhZCiF7IYrEwbdoMcnP7sX79W1RVVeFypYXk2sGst7bd7OQtAxnc9bs7Pa0n478tEEIIETaDBuVx663/C5crnWPHjtLQ0NCt6/k6DvTm2fiQHAdqFekqV6EkSVcIIXq5hIRErr9+EZMnT6OiooLi4mIqKsq7dK1IrLca/SxuIDK9LIQQBrJu72mefvcwpyvryYi38eAclcn9wl9Zymw2M3LkKIYNK+To0SPs2PEJxcXHsNnsuFyuoNsGRmq91chncQORpCuEEAaxbu9pfr3+AHVN3qnTU5X1PLJyNw/PyueaIRkRicFqtTJoUB6DBuVRUlLCRx99wMGD+3E640lOTu7w66N5vTUSZHpZCCEM4ul3D19IuK3qGt08/e5hXeJJS0tj/vwF3HLL10lISOTYsaPUdLDbKZrXWyNBkq4QQhjE6UrftZL9vR4pmZl9uOmmm1mwYCHnzp0L2LM3mtdbI0Gml4UQwiAy4m2c8pFgM+JtOkRzOUVRGDhwENOnz2DjxvXk5PTFZPI9bovW9dZIkJGuEEIYxHcm9cduufzHst1q4juT+usTkA8FBYUUFhZx6pRxevZGExnpCiGEQbRultJj93KwFEVh0qSplJSUcPZsKampLr1DiiqSdIUQwkCuGZJx2U7lpCQH588HWaopQmJiYrjmmvn88n/WsONICpVuS1R1+tGTJF0hhBCd9s8T9XzQ3I96tweIrk4/epI1XSGEEJ329LuHqW/2XPZak8fEpnMOGhrq8Xg8fr6yd5ORrhBCiE7zd4ypym2hrq6OkpJSWotY9emTFXRFq55Okq4QQohO83e8KTPezp13foPm5mZqaqrZtm0Lx44dIz09XYcojUeml4UQQnSaz+NNlovHm8xmM/HxCUyePBUg4j17jUqSrhBCiE67ZkgGD8/OJzPehgJkxtt4eHb7GtHx8QlMnz6TM2fOyDovMr0shBCii9oeb/InP38wgwerHD16hPT0yDRuMCoZ6QohhAgrRVGYPHkqiqL0+mlmGekKIYQIO6cznhkzZrNmzSrMZhNms5mYmBis1hgcDofe4UWMJF0hhBARMWhQHosX30J1dTVVVZWUl5dTVlbGqVMnKS83UVvbiN1uIyUltcceMZKkK4QQIiIURSE7O6fd6263G7e7jsOHj7Nr105Onz5JRkamDhGGn6zpCiGE0JXJZMLlcpGXl8/cufOw2exUVlbqHVZYSNIVQghhGA6Hg3nzruX8+fM0NjbqHU7ISdIVQghhKJmZfZg8eQqnTp3UO5SQC/uartNpw2Ixh/s2EWE2m0hK6tm77OQZe47e8JzyjD1H2+ecMmUiZWVnOHr0GJmZ4Vvf9XiasNtjIvY9DnvSraryXRQ7Ghmxr2WoyTP2HL3hOeUZew5fz3nllZM5cuQlDh48BHjweLz/2Ww20tJCU8u5pqYet1sJ6fc4LS3e78dk97IQQghDiouLY8mS22hsbMBisWKxWKivr+Oll16gtraG2NjomwGQNV0hhBCGFRsbS0JCIg6Hg5iYGOLjE5g9ey4lJSW43W69w+s0SbpCCCGiyoABAxk1agwnT0bfRitJukIIIaLO+PETSE5Oprz8vN6hdIokXSGEEFEnJiaGOXPmUVVVFVXneSXpCiGEiEppaWlMmTKdkydPRM36riRdIYQQUauwsIhRo8Zw/PhxPB6P3uF0SJKuEEKIqKUoChMnTiIvLy8qKlhJ0hVCCBHVzGYzM2fOITXVRWlpid7hBCRJVwghRNSz2WzMn78Ai8VCaWmpYaeaJekKIYToEeLjE1i48EYyMzM5frzYkMlXykAKIYToMVJSUrnuukWcPn2a7ds/4uDBg5jNJuLinMTFxWE269uAR5KuEEKIHicjI4N58xZQWlrKkSOHOHr0CCdPnsTtbkZRFFyuNKxWa8TjkqQrhBCix3K5XLhcLkaPHovb7eb8+TK++upLPvroAzweD3FxcRGNR5KuEEKIXsFkMpGSkkpKSipDhw5j166dfPTR+8THJ0YsBkm6Qggheh2Hw8H48VcxbNgwqqurI3ZfSbpCCCF6LaczHqfTf9P5UJMjQ0IIIUSESNIVQgghIkSSrhBCCBEhknSFEEKICJGkK4QQQkSIJF0hhBAiQiTpCiGEEBEiSVcIIYSIEEm6QgghRIRI0hVCCCEiRJKuEEIIESGSdIUQQogIUTwej94xCCGEEL2CjHSFEEKICJGkK4QQQkSIJF0hhBAiQiTpCiGEEBEiSVcIIYSIEEm6QgghRIRI0hVCCCEixKJ3AEJEE1VVHwJmAlbADTygadp2VVULgWRN07Z149pzgSWapt3Zha99GXgGsAO5mqY9G+kYhBAdk6QrRJBUVR0KXAdM1DTNo6rqCOC/geHAjcApoMtJNxQ0TXtLz/sLIQKTpCtE8MqBXOAuVVXf0jTtM1VVx6mqmg3cCTSoqrqj5XOW4h0Ne4BFwDDgx0ADMBB4WdO0f1NVdQjwX0B1y39lAKqqfhe4AYgDSluu8XXgLrzLQj8HrgC+AZwE0lu+7s6W11cCj7bE7QLiNE0boKrq91qu42mJ4Q/+YmilqupU4LGW2J8FajvxfHnA80AjcATor2naVFVVFwP3Ac3APzRNe6hzfxRCRCdZ0xUiSJqmHadlpAu8r6rqPuDaltefB/5d07SPgMHAfE3Trga+AOa0XKIf3hHxeOBHLa/9DviZpmkzgX8CqKpqAlKBmZqmXYn3zfHYls8va7nubuAHLde6HohpE+v7mqZNxZu4K4AbW0bqtwBXA5OAhaqqqr5i8MGuadokTdP+1oXn+7WmadOA91qeLwX4v8CMlmtkq6o6y899hehRJOkKEaSWUVuFpml3aZqWC9wGPNOSRC51BvhvVVX/ChThHRECfK5pWpOmadV4R4vgTWAftfz+PQBN09x4R4zLVFX9C5BzyTW0ll8HAXs0TavXNK3xkmtcGq8TeANvQt2BdzTaD3in5b9UIN9XDD5ol/y+M883hIuJ/N2WX/OANOBNVVW3AENbnkeIHk+SrhDBKwL+qKpq66hyP3Ae7xSpGzCpqpqIdxS3BO/Uby2gtHy+r+4iXwBXtfx+LICqqkXAQk3TbgG+h/ffaes13C2/HgAKVFWNVVXVDIy89KKqqtqAFcBTmqZtbHlZA/YA01pGwc8Du3zF4IO75bqdfb7dl1x7fMuvh4BjwKyWOJ4EPvBzXyF6FFnTFSJImqb9vWX982NVVavwJsMHNU0rV1V1O96p1L14R4vvA01410ez8CYaX+7HO2p8ECgB6oCDQLWqqq2jzpMt17g0lhJVVX+DdxRZgnct9lLfB0YBFlVVv93y2g14R7j/aEnKHwHH/cTgT0Unn+/HwH+pqvoA3jXxxpbY/x3Y2vKG4TDwaoB7CtFjSGs/IUTYqKp6K/ChpmkHVVX9BjBB07S79I5LCL3ISFcIEU7HgJdVVa3BOw1/t87xCKErGekKIYQQESIbqYQQQogIkaQrhBBCRIgkXSGEECJCJOkKIYQQESJJVwghhIgQSbpCCCFEhPx/+FRPHQk/SWcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x19e359dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8, 6))\n",
    "\n",
    "ax.scatter(df.std_range, df.std_logratio, c=blue, zorder=10, label=None)\n",
    "\n",
    "low, high = np.percentile(pp_trace[\"obs\"], [2.5, 97.5], axis=0)\n",
    "ax.fill_between(\n",
    "    lidar_pp_x, low, high, color=\"k\", alpha=0.35, zorder=5, label=\"95% posterior credible interval\"\n",
    ")\n",
    "\n",
    "ax.plot(lidar_pp_x, pp_trace[\"obs\"].mean(axis=0), c=\"k\", zorder=6, label=\"Posterior expected value\")\n",
    "\n",
    "ax.set_xticklabels([])\n",
    "ax.set_xlabel(\"Standardized range\")\n",
    "\n",
    "ax.set_yticklabels([])\n",
    "ax.set_ylabel(\"Standardized log ratio\")\n",
    "\n",
    "ax.legend(loc=1)\n",
    "ax.set_title(\"LIDAR Data\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The model has fit the linear components of the data well, and also accomodated its heteroskedasticity.  This flexibility, along with the ability to modularly specify the conditional mixture weights and conditional component densities, makes dependent density regression an extremely useful nonparametric Bayesian model.\n",
    "\n",
    "To learn more about depdendent density regression and related models, consult [_Bayesian Data Analysis_](http://www.stat.columbia.edu/~gelman/book/), [_Bayesian Nonparametric Data Analysis_](http://www.springer.com/us/book/9783319189673), or [_Bayesian Nonparametrics_](https://www.google.com/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=bayesian+nonparametrics+book).\n",
    "\n",
    "This example first appeared [here](http://austinrochford.com/posts/2017-01-18-ddp-pymc3.html).\n",
    "\n",
    "Author: [Austin Rochford](https://github.com/AustinRochford/)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "arviz  0.8.3\n",
      "pymc3  3.8\n",
      "pandas 1.0.3\n",
      "numpy  1.18.1\n",
      "last updated: Wed Jun 10 2020 \n",
      "\n",
      "CPython 3.6.10\n",
      "IPython 7.13.0\n",
      "watermark 2.0.2\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -n -u -v -iv -w"
   ]
  }
 ],
 "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": "3.8.2"
  },
  "widgets": {
   "state": {},
   "version": "1.1.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}