09-Clustering-IV-GMM-EM.ipynb

{
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "skip"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy as sp\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "import seaborn as sns\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Gaussian Mixture Models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center>\n",
    "\n",
    "<img src=\"figs/L09-MultivariateNormal.png\" width=\"70%\">\n",
    "\n",
    "</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## From Hard to Soft Clustering"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "__So far,__ we have seen how to cluster objects using $k$-means: \n",
    "\n",
    "1. start with an initial set of cluster centers,\n",
    "2. assign each object to its closest cluster center, and \n",
    "3. recompute the centers of the new clusters. \n",
    "4. Repeat 2 $\\rightarrow$ 3 until convergence."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "__Note__ that in $k$-means, every object is assigned to a **single** cluster. \n",
    "\n",
    "This is called __hard__ assignment."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "However, there may be cases were we either __cannot__ use hard assignments or we do not __want__ to do it! \n",
    "\n",
    "In particular, we may believe that the best description of the data is a set of __overlapping__ clusters."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "For example: \n",
    "\n",
    "Imagine that we believe society consists of just __two__ kinds of individuals: poor, or rich.\n",
    "\n",
    "Let's think about how we might model society as a mixture of poor and rich, when viewed in terms of age."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Say we sample 20,000 rich individuals, and 20,000 poor individuals, and get the following histograms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "hide_input": true,
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# original inspiration for this example from\n",
    "# https://www.cs.cmu.edu/~./awm/tutorials/gmm14.pdf\n",
    "from scipy.stats import multivariate_normal\n",
    "np.random.seed(4)\n",
    "df = pd.DataFrame(multivariate_normal.rvs(mean = np.array([37, 45]), cov = np.diag([196, 121]), size = 20000),\n",
    "                   columns = ['poor', 'rich'])\n",
    "df.hist(bins = range(80), sharex = True);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We find that ages of the poor set have mean 37 with standard deviation 14,\n",
    "\n",
    "while the ages of the rich set have mean 45 with standard deviation 11."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "hide_input": false,
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "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": [
    "from scipy.stats import norm\n",
    "plt.figure()\n",
    "x = np.linspace(norm.ppf(0.001, loc = 37, scale = 14), norm.ppf(0.999, loc = 37, scale = 14), 100)\n",
    "plt.plot(x, norm.pdf(x, loc = 37, scale = 14),'b-', lw = 5, alpha = 0.6, label = 'poor')\n",
    "x = np.linspace(norm.ppf(0.001, loc = 45, scale = 11), norm.ppf(0.999, loc = 45, scale = 11), 100)\n",
    "plt.plot(x, norm.pdf(x, loc = 45, scale = 11),'g-', lw = 5, alpha = 0.6, label = 'rich')\n",
    "plt.xlim([15, 70])\n",
    "plt.xlabel('Age', size=14)\n",
    "plt.legend(loc = 'best')\n",
    "plt.title('Age Distributions')\n",
    "plt.ylabel(r'$p(x)$', size=14);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Clearly, viewed along the age dimension, there are two clusters that overlap."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Furthermore, given some particular individual at a given age, say 25, we cannot say for sure which cluster they belong to.  \n",
    "\n",
    "Rather, we will use _probability_ to quantify our uncertainty about the cluster that any single individual belongs to."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Thus, we could say that a given individual (\"John Smith\", age 25) belongs to the _rich_ cluster with some probability, and the _poor_ cluster with some different probability.\n",
    "\n",
    "Naturally we expect the probabilities for John Smith to sum up to 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "This is called __soft assignment,__ and a clustering using this principle is called __soft clustering.__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "More formally, we say that an object can belong to each particular cluster with some probability, such that the sum of the probabilities adds up to 1 for each object. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "For example, assuming that we have two clusters $C_1$ and $C_2$, we can have that an object $x_1$ belongs to $C_1$ with probability $0.3$ and to $C_2$ with probability $0.7$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Note that the distribution over $C_1$ and $C_2$ only refers to object $x_1$.\n",
    "\n",
    "Thus, it is a __conditional__ probability:\n",
    "\n",
    "$$P(C_1 \\,|\\, x_1) = 0.3$$\n",
    "$$P(C_2 \\,|\\, x_1) = 0.7$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "And to return to our previous example:\n",
    "\n",
    "$$P(\\text{rich}\\,|\\,\\text{age 25}) + P(\\text{poor}\\,|\\,\\text{age 25}) = 1 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Mixtures of Gaussians"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We're going to consider a particular model for each cluster: the Gaussian (or Normal) distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$${\\displaystyle f(x\\;|\\;\\mu ,\\sigma ^{2})={\\frac {1}{\\sqrt {2\\sigma ^{2}\\pi }}}\\;exp({-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}}})$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "scrolled": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<center>\n",
    "\n",
    "<img src=\"figs/L09-Normal_Distribution_PDF.png\" width=\"60%\">\n",
    "\n",
    "</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "By <a href=\"//commons.wikimedia.org/wiki/User:Inductiveload\" title=\"User:Inductiveload\">Inductiveload</a> - self-made, Mathematica, Inkscape, Public Domain, <a href=\"https://commons.wikimedia.org/w/index.php?curid=3817954\">https://commons.wikimedia.org/w/index.php?curid=3817954</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "You can see that, for example, this is a reasonable model for the distribution of ages in each of the two population groups (rich and poor)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "In the case of the population example, we have a single feature: age.   \n",
    "\n",
    "How do we use a Gaussian when we have multiple features?\n",
    "\n",
    "The answer is that we use a __multivariate Gaussian.__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<center>\n",
    "\n",
    "<img src=\"figs/L09-MultivariateNormal.png\" width=\"70%\">\n",
    "\n",
    "</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "By <a href=\"//commons.wikimedia.org/wiki/User:Bscan\" title=\"User:Bscan\">Bscan</a> - <span class=\"int-own-work\" lang=\"en\">Own work</span>, <a href=\"http://creativecommons.org/publicdomain/zero/1.0/deed.en\" title=\"Creative Commons Zero, Public Domain Dedication\">CC0</a>, <a href=\"https://commons.wikimedia.org/w/index.php?curid=25235145\">https://commons.wikimedia.org/w/index.php?curid=25235145</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Now each point is a vector in $n$ dimensions: $\\mathbf{x} = (x_1, \\dots, x_n)^T$.  \n",
    "\n",
    "As a reminder, the multivariate Gaussian has a pdf (density) function of:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
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    "variables": {
     "-1": {},
     "\\boldsymbol  \\Sigma ": "<p><strong>SyntaxError</strong>: unexpected character after line continuation character (1186751447.py, line 1)</p>\n",
     "\\boldsymbol  \\mu ": "<p><strong>SyntaxError</strong>: unexpected character after line continuation character (185173116.py, line 1)</p>\n",
     "\\mathbf  x": "<p><strong>SyntaxError</strong>: unexpected character after line continuation character (2390085295.py, line 1)</p>\n",
     "\\mathrm  {T": "<p><strong>SyntaxError</strong>: unexpected character after line continuation character (154575714.py, line 1)</p>\n",
     "\\sqrt  {(2\\pi )^{{n": "<p><strong>SyntaxError</strong>: unexpected character after line continuation character (1998936437.py, line 1)</p>\n"
    }
   },
   "source": [
    "$$\n",
    "\n",
    "f(x_{1},\\ldots ,x_{n})={\\frac  {1}{{\\sqrt  {(2\\pi )^{{n}}\n",
    "\\Vert{\\boldsymbol  \\Sigma }\\Vert}}}}\\exp \\left(-{\\frac  {1}{2}}({{\\mathbf  x}}-{{\\boldsymbol  \\mu }})^{{\\mathrm  {T}}}{{\\boldsymbol  \\Sigma }}^{{-1}}({{\\mathbf  x}}-{{\\boldsymbol  \\mu }})\\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Recall also that the shape of a multivariate Gaussian -- the direction of its axes and the width along each axis -- is determined by the __covariance matrix $\\Sigma$__.\n",
    "\n",
    "The covariance matrix is the multidimensional analog of the variance.   It determines the extent to which vector components are correlated."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "For example, let's say we are looking to classify cars based on their model year and miles per gallon (mpg).  \n",
    "\n",
    "To illustrate a particular model, let us consider the properties of cars produced in the US, Europe, and Asia."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<!-- image credit: https://www.cs.cmu.edu/~./awm/tutorials/gmm14.pdf -->\n",
    "<center>\n",
    "\n",
    "<img src=\"figs/L09-multivariate-example.png\" width=\"90%\">\n",
    "\n",
    "</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "It seems that the data can be described (roughly) as a mixture of __three__ __multivariate__ Gaussian distributions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The general situation is that we assume the data was generated according to a collection of arbitrary Gaussian distributions.\n",
    "\n",
    "This is called a __Gaussian Mixture Model.__\n",
    "\n",
    "Aka a \"GMM\"."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<!-- image credit: http://autonlab.org/_media/tutorials/gmm14.pdf  and\n",
    "https://web.iitd.ac.in/~sumeet/GMM_said_crv10_tutorial.pdf -->\n",
    "\n",
    "\n",
    "<div style = \"float: left; width: 45%; text-align: center;\">\n",
    "    \n",
    "<img src=\"figs/L09-general-GMM.png\" width=\"45%\">\n",
    "Ellipsoid Representation\n",
    "</div>\n",
    "\n",
    "<div style = \"float: left; width: 55%; text-align: center;\">\n",
    "    \n",
    "<img src=\"figs/L09-GMM-density.png\"  style=\"width:55%\">\n",
    "Density\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "A Gaussian Mixture Model is defined by:\n",
    "    \n",
    "$$ w_i, \\mu_i, \\Sigma_i,\\;\\; i = 1,\\dots,k$$\n",
    "\n",
    "Where $w_i$ is the prior probability (weight) of the $i$th Gaussian, such that \n",
    "\n",
    "$$\\sum_i w_i = 1,\\;\\;\\; 0\\leq w_i\\leq 1.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Intuitively, $w_i$ tells us \"what fraction of the data comes from Gaussian $i$.\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Then the probability density at any point $x$ is given by:\n",
    "    \n",
    "$$ p(x) = \\sum_i w_i \\cdot \\mathcal{N}(x\\,|\\, \\mu_i, \\Sigma_i) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Learning the Parameters of a GMM"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This model is all very well, but how do we learn the parameters of such a model, given some data?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "That is, assume we are told there are $k$ clusters.   \n",
    "\n",
    "For each $i$ in $1, \\dots, k$, how do we estimate the \n",
    "* cluster probability $w_i$,\n",
    "* cluster mean $\\mu_i$, and \n",
    "* cluster covariance  $\\Sigma_i$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "There are a variety of ways of finding the best $(w_i, \\mu_i, \\Sigma_i)\\;\\;\\;i = 1,\\dots,k$.\n",
    "\n",
    "We will consider the most popular method:  __Expectation Maximization (EM).__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This is another famous algorithm, in the same \"super-algorithm\" league as $k$-means."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "EM is formulated using a probabilistic model for data.   It can solve a problem like:\n",
    "\n",
    "> Given a set of data points and a parameter $k$, find the $(w_i, \\mu_i, \\Sigma_i)\\;\\;i = 1,\\dots,k$ that __maximizes the likelihood of the data__ assuming a GMM with those parameters."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "(It can also solve lots of other problems involving maximizing likelihood of data under a model.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "However, note that problems of this type are often NP-hard.  \n",
    "\n",
    "EM only guarantees that it will find a __local__ optimum of the objective function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Probabilities We Will Use"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "At a high level, EM for the GMM problem has strong similarities to $k$-means.\n",
    "\n",
    "However, there are two main differences:\n",
    "\n",
    "1. The $k$-means problem posits a **hard** assignment of objects to clusters, while GMM uses __soft__ assignment.\n",
    "2. The parameters of the soft assignment are chosed based on a __probability model__ for the data."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Let's start by reviewing the situation probabilistically.\n",
    "\n",
    "Assume a set data points  $x_1, x_2,\\ldots,x_n$ in  a $d$ dimensional space. \n",
    "\n",
    "Also assume a set of $k$ clusters $C_1, C_2, \\ldots, C_k$.   Each cluster is assumed to follow a Gaussian distribution:\n",
    "\n",
    "$$ C_i \\sim {\\mathcal N}({\\mathbf \\mu_i},{\\mathbf \\Sigma_i}) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "source": [
    "We will be working with conditional probabilities."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "First, \n",
    "\n",
    "$P(x_i\\,|\\,C_j)$ is the probability of seeing data point $x_i$ when sampling from cluster $C_j$.  \n",
    "\n",
    "That is, it is the value of a Gaussian pdf at the point $x_i$, for a Gaussian with parameters $({\\mathbf \\mu_j},{\\mathbf \\Sigma_j})$.\n",
    "\n",
    "Clearly, if I give you the cluster parameters, it is a straightforward thing to compute this conditional probability."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "In our example: $P(\\text{age 25}\\,|\\,\\text{rich})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "hide_input": true,
    "tags": [
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Vq1a88sorPPzww4gIffv2Zfr06QB89tlnPProo+zcuZO//OUvjB8/vtbbX8p6OsbUgpz8HJ5Nf9Y3U/Q1Pa9hULtBYWhVzTujxRncMsCT8AOA11a/xoa9G0Lcovppw4YNTJo0ia+//prOnV1uvoyMDB566CE+/vhjVq1axeOPP/7v43Nycli8eDHvvfceDzzgSfBfqyzoGFPDjhQe4amvnvLdNG14p+Fc3PXiMLSq9gxMHsjVPa72lJdoCc+kP2OLR0Ogc+fODBs27ISyjz/+mPHjx9OqVSsAWrRo8e+6q6++mpiYGHr16sWuXaHNKmFBx5gapKq8tPIl9hze46k7s9WZfP+s70fUws+aMrrbaM7ueLan/EjhEaYun+qbPNTUHL8tCVQ16O9aQkLCCceFkgUdY2rQ/C3zWbVzlae8bZO2TEmbQlxMdN5GFRFu6HuD70y8rXlbmZkxMwytqt8uvPBCZs6cSW5uLgD79kVG1ojo/AswJgw27dvE7MzZnvKG8Q25Z8g9NIpvFIZWhU5cTBw/TPshDy16iNwjuSfUfZr1Kd1adGNI++jMvxvsJn849e7dm9/85jeMGDGC2NhYBgwYwEsvvRTuZlkaHLA0OObU5Rfk88fP/kjesTxP3ZS0KQxIHhD6RoVJVl4Wf/38r54htYS4BH517q9ITkoOcmbdY2lwLA2OMSFXoiU8//XzvgHn4jMurlcBByClWQoTek/wlBcUFfDs8mctY0E9Z0HHmFM0f/N8MvdkesrPaHEG43qMC0OLwm9E5xEMbj/YU56Tn8Nb68K3Gt6EnwUdY07B9gPbeXfDu57yJg2acPvA22t9i+lIVTqxoG2Ttp66z7Z+xupdq8PQqtpRH25R1OQ1WtAxppoKiwt5/uvnKS4pPqFcRLh14K00b9g8TC2LDIlxiUxJm+K7D88rq17hYMHBMLSqZiUmJpKbmxvVgUdVyc3NJTHRm/KoOmz2mjHV9Hbm2+Tk53jKL+56Mb1a9wpDiyJPu6R2TOg1gdfXvH5CeX5BPtNXTeeuwXfV6XVLHTp0IDs7mz17vOuyokliYiIdOnSokeeyoGNMNWTszuDjbz/2lHdo2oGxPcaGoUWR6/zO57N612rW7j5xS+XVu1azaNsizu98fphaduri4+Pp0qVLuJtRp9jwmjFVdPj4YV5e9bKnPC4mjlsH3hq1C0CrS0SY3H+y7x48MzNmsutQaNOwmPAKa9ARkdEiskFENomIJ+ucOE8E6leLyMBAeaKIfCkiq0QkQ0QeLHNOCxGZLyIbA1/r98C6qXFvZLzBgWMHPOXX9LyGdkntwtCiyNc0oSk39r3RU15YXMjLq172TYxqolPYgo6IxAJPAWOAXsBEESk/ED4GSA087gCeDpQXABeoaj+gPzBaREqz3T0ALFDVVGBB4GdjasTqXatZlr3MU96zdU8u6HJBGFpUd/Q7vR/ndT7PU75532YWZi0MfYNMWISzpzME2KSqW1T1ODADKD8YPhZ4RZ2lQDMRSQ78fChwTHzgoWXOKR37eBm4ujYvwtQfRwqP8Nrq1zzljeIbcVP/m+r0DfFQmdBrAq0bt/aUz86cbdmo64lwBp32wPYyP2cHyip1jIjEishKYDcwX1VLP362VdUcgMDXNjXfdFMfzVo3yzfrwHW9r6NZYrOQt6cuSohLYFK/SZ7y48XHmb5qelRPPTZOOIOO38fC8r9xQY9R1WJV7Q90AIaISJ8qvbjIHSKSLiLp0T7d0Zy6zD2ZLN622FPeu01vhnUY5nOGCaZ7y+6MSBnhKV+/d73vv7GJLuEMOtlAxzI/dwB2VPUYVc0DFgKjA0W7RCQZIPB1t9+Lq+pUVU1T1bTWrb3dfWNKFRQVMH31dE95QlwCN/S9wYbVquGantfQomELT/lb695i/9H9YWiRCZVwBp2vgFQR6SIiDYDrgTnljpkDTArMYhsGHFDVHBFpLSLNAESkIXARsL7MOZMD308GvDlKjKmCdze860nVDzC+13jfN05zcolxidzQ9wZP+bGiY7y+5nUbZotiYQs6qloE3APMAzKBmaqaISJTRGRK4LD3gS3AJmAacFegPBn4RERW44LXfFV9L1D3Z+BiEdkIXBz42Zhq2Zq31XcRaPeW3Tmvk3cmlqm83m16c07Hczzlq3etZuXOlaFvkAmJsK5iU9X3cYGlbNkzZb5X4G6f81YDvvniVTUXuLBmW2rqoxIt4bU1r3k+dcfHxnNjvxttWK0GTOg9gbW713rysM1YO4OerXuSGFcz+b5M5LCl06ZeUoWDB2H/fti37z9fjx79zzGZRz9l6eGtADSIh4QEaJAAV595JU2kDapgcefUNIpvxMSzJvJs+ok7b+Ydy2POhjlc1/u6MLXM1BYLOqZeOHQItmyBTZvc16wsKCwMfnyB5LE86R2KygWVxsXt+PSLi1gENG4MZ5wBXbtCt27QuTM0aFCbVxGdBpw+gD5t+nhys3387ccM6zCMTqd1ClPLTG2woGOiUkkJbN4M6emQmQm7qpjea0vDmRTJMU956tEbiMHtkXP4MKxe7R4AMTHQqROcdRYMHgxtvVvJGB8iwsSzJvL7hb+nsPg/nwRUlddWv8Yvz/0lMWJpIqOFBR0TNVRdLyY9HZYvhwPe9GiVsi9uLXvil3vKk4+fR9PiM4KeV1LielBZWfDPf0KHDpCWBoMGQRtbolyhVo1acUX3K5idOfuE8qy8LD7b+hkjU0aGp2GmxlnQMXXekSOwaBEsXOjuy5yKEgrZ1PAfnvIGmkTKsaptPZ2d7R7vvANdusCFF8LAgRBbPzcTPamLu17Msuxl7Mg/cbne7MzZDEweSNOEpmFqmalJFnRMnbV7N3z8MXzxBRQUVP38xERo0cI9mjd3X9cWfMju3BNzgGkJnNtkPK0LGrN//38mHRQVVf61vv0WnnvOvc4FF8C550KjRlVvczSLjYnlB31/wF8//+sJ5ceKjjE7czaT+08OcqapSyzomDonKwvef9/dS6nsGkIRSE52N/5LH61bnzj7LPdILu8vnMvp5e7FnNnqTO4fNvSEY4uLXS9m82b32LQJ8vJO3o79+2HWLHjvPTjnHBg9Gpo1q9w11AfdWnRjeKfhfL7t8xPKv9j+Bed1Po+uzbuGqWWmpljQMXXG7t0wezasWFG542NjoXdvd0+lb9+T9yzeXPfmCTeyAWIkhol9JnrW5MTGutlqnTu7ngvA3r2ubcuXu8BYkYIC+OQTWLwYLroILr0UGjas3HVFu2t6XsPXOV9zpPDICeX/WPMPfnXer2xSQR1nQcdEvPx8+Ne/4NNP3c36iohAr17uBn7//pUfwsrYncHXOV97yi/seiHJScmVeo5WreCSS9xj714XfL780vWIgikshLlz4bPP4Ior4PzzIa6e/1U2adCEsT3G8o81J95b23ZgG4u3La7T21sbCzomghUVwYcfwgcfnPyeTUICDB/ueh1Vzd9aVFLEGxlveMpPSzyNK7pfUbUnC2jVyvVeLrnEDb199BGsWhV8OPDwYXjjDViwAK69FgYMqN8LT8/vfD6Lty1m+4HtJ5S/s/4dBiUPonGDxmFqmTlVFnRMRNq8GaZPh5ycio+riRvzC7YsYNch70Kea3tee8ppWEQgNdU9KjPxYe9eePZZ6NcPJk5011cflQ5r/uXzv5xQfvj4Yd5Z/w4/6PuDMLXMnCoLOiaiHDvm7tt8+mnFkwSaNoWrrnI3409lCnLesTz+tfFfnvJuLboxpP2Q6j+xjzZt4Prr4corXe/t44+Dz4BbtQo2bIBrrnFDbvWx13NGizMY1mEYS7OXnlC+aNsizut8nmUqqKPsjpyJGKtXw+9/79bbBAs4CQkwdiz88Y9w3nmnvuZlduZsCopO7HaUrpCvrYSejRu7IbQ//AGGDQseUI4dg9dfh7/+FXburJWmRLxre3l7m6rKG2vfsO0P6igLOibsCgrg5ZfhqafclGI/MTEwciQ89BBcdpkLPqcqKy/L8ykaYGTKSDo07XDqL3ASLVrAzTfDf/2Xm/wQzObNLkAtWFD5KeLRomlCU64880pP+aZ9m1ie480aYSKfBR0TVtu3u0DyxRfBj0lJgd/8xt3jSEqqmdct/bRcXuMGjbmyu/dNrjZ16AA/+hHceacbNvRTVAQzZ7rAnJ8f0uaF3aiUUZze5HRP+ax1szxT3E3ks6BjwkLVfXL/85+DJ+Ns0AAmTIBf/tK9MdekL7/7ki37t3jKrzrzqrDMjBJxKXIefNDNwgtmzRrX69mwIXRtC7fYmFgm9J7gKd93dB8fbv4wDC0yp8KCjgm5/Hz3iX3mzOA30nv1gt/9zi2cjKnh39KCogLeznzbU94uqV3Y14A0agSTJsH99wef+n3gADz2GLz77snXLUWLPm360KdNH0/5B5s+YP/RIGOyJiJZ0DEhlZXlJgGsWeNf36CBe9P90Y/cWpfaMG/zPPKO5XnKr+t9XcSsdu/RA/7f/4NRo/zrVV0qoEcfrT/DbX7/P8eLjzN7/ewgZ5hIFBl/YaZeWLrUzcQKlqOsQwd372b48NqbIhxsSKbf6f3o2bpn7bxoNTVo4KZY33WXm/HmZ+NG+NOfYNu20LYtHNo2acsFXS7wlC/LXuY7VGoiU1iDjoiMFpENIrJJRB7wqRcReSJQv1pEBgbKO4rIJyKSKSIZInJfmXN+LyLficjKwOOyUF6T8SopcUNpL74YfDjtggvggQfgdO/94ho1O3O25+ZzbEws43uNr90XPgX9+sFvfwvdu/vX79sHf/kLfPVVaNsVDpd3v5wmDZp4yt/MeNOmUNcRYQs6IhILPAWMAXoBE0Wk/MTRMUBq4HEH8HSgvAj4qar2BIYBd5c79zFV7R94vF+b12EqdvgwPP64mzTgp3FjuPtu+N73ID6+dtvy7f5v+fK7Lz3lF3a5kDaNI3uXtebN3X2esWP973EVFrqtE95+O7rv8zSKb8TYHmM95Vv2byF9R3oYWmSqKpw9nSHAJlXdoqrHgRlA+d+mscAr6iwFmolIsqrmqOoKAFXNBzKB9qFsvDm5nBw39LN+vX99p07uE3zfvrXfFlVlZsZMT3lSQhKXd7+89htQA2Ji3BqlH/84+HDbvHnw5JNuYWm0OrfTubRLaucpfzvzbZtCXQeEM+i0B8pm88vGGzhOeoyIpAADgGVliu8JDMe9ICK+2atE5A4RSReR9D179lTzEkwwGze6IZ+9e/3rhw6FX/widLnFlucsDzpF+lTzq4XamWe6e1/BppFnZFR876yui5EYrut9nad839F9fLTlozC0yFRFOIOO363i8oOyFR4jIk2AWcCPVfVgoPhp4AygP5ADPOL34qo6VVXTVDWtdVXTEpsKpafD3/7mtpEuTwTGj3cr8Wt7OK1UYXFh0CnS53Y6NzSNqGEtW7qgnZbmX5+d7dZA7djhX1/X9Wzd03cK9dxNczlYcNDnDBMpwhl0soGOZX7uAJT/Ewl6jIjE4wLOa6r673cUVd2lqsWqWgJMww3jmRBQhfnzYdo0/wkDjRq5qdAXXxzaBJYLvl1A7pFcT/n4XuMjZop0dSQkwG23wbhx/v+e+/e73uY334S+baHg9/9XUFTAnA1zwtQiUxnh/Iv7CkgVkS4i0gC4Hij/2zIHmBSYxTYMOKCqOeIyMT4PZKrqo2VPEJGyO26NA9bW3iWYUiUlbj+Yt97yr2/TBn71q4pzjNWGgwUHmbtxrqe8T5s+9G7TO7SNqQUibsvru+5yU6zLO3rUTeSIxpltyUnJvot5F29bTPbBCnbOM2EVtqCjqkXAPcA83ESAmaqaISJTRGRK4LD3gS3AJlyv5a5A+XDgRuACn6nRfxGRNSKyGhgF3B+iS6q3iopc7+aTT/zru3Z1Q0FtwjBB7J8b/smxohPvqsdITERPka6Ovn3hpz/1z01XVORmtgWbQViXXXnmlTSMP3Gfb1Vl1rpZYWqRORmxue2Qlpam6ek23bI6jh+HZ55xN6/99O8Pt97q/ym8tuXk5/Dgpw961m+MTBnJxLMmhr5BIbB3LzzxRPB8dldeCZdfHl3788zfPJ+31nm72D8a+qOo6M1GMhFZrqpB7iz6q7sD2ibsjh51b3DBAs6oUS5zcjgCDsCszFmegJMYl1jtLajrglatXILUM87wr//nP2HWrOjaImFUl1G0auTNmfTWurco0ShetFRHWdAx1XLokMv7tXGjf/2117oFnzWdrLOy1u9dz5pd3gRvl6VeRlJCDe2PEKEaN3YLSQcM8K+fPx9eey16FpHGxcRxTc9rPOU78nfw+bbPw9AiUxELOqbK8vLg4Yf9832JwE03wSWXhG8Ip0RLeDPjTU95i4YtfHN3RaP4eLjjDjg3yIzwRYvg+eehuDi07aotA5MH0rV5V0/5nA1zPPf0THhZ0DFVkpvrFh7m5HjrYmPdG93ZZ4e+XWUtzV7qO3tpXM9xxMeGaHFQBIiJgRtucNtD+ElPd/fjguXDq0tExHfPnYMFB23PnQhjQcdUWm4uPPKIf5aB+HiXQ23gwNC3q6yCogLeXf+upzylWQqD2w0OQ4vCq3Qx7pVBNkNdvRqefTY6Ak/X5l1Ja+e9p/3h5g9tz50IYkHHVEppwMn1rrEkMdHlA+sdAROFPtryke9eOeN7jUeiacpWFYjAFVe4XVj9rF4dPT2ecT3HERcTd0JZYXEh727wfhAx4WFBx5xURQGncWO3PqRbt9C3q7yDBQeZt3mep3xA8gBSW6aGoUWR5aKL4MYb/e+1rVkDTz/tslXXZa0atWJUF+/Od0uzl7L9wHafM0yoWdAxFdq7100a8As4TZq4gNOpU+jb5WfOhjkUFBWcUBYjMYzrMS5MLYo8557rJnr4BZ61a12Pp64HnstSL6NxgxPTcKsqb617y/bciQAWdExQpT2cffu8dUlJLuC0j5ANJXbk72DxtsWe8pEpI2nbpG0YWhS5hg2rOPA8/XTdHmprFN+Iy1O921Ws37uejD1BFpWZkLGgY3zt3+/W4QQLOD/5CbTzbmkSNrPWeReCNoxvWGf2ygm1YcNcpm+/wJORAVOn1u3p1CNSRtC6sTd7vC0YDT8LOsbj4EF47DH/WWqRGHAy92Sydrc3r+tlqZf5bm1snKFDgweeVavghRfq7gLSYAtGc/JzbMFomFnQMSc4dMgFHL/cXaVDapEUcEq0xDfvVstGLRmV4r2hbE40dCjccot/4ElPh5dfrrspcwacPoAzWnjzAfnd+zOhY0HH/NuRI27zNb+Nv5o0cT2c5GRvXTgty17mvxC0R/1aCHoqhgwJ3uNZutSlzKmLgUdEfLOJB5vlaELDgo4B4Ngx+L//g+0+s0obNXLrcCKphwNwvPg476x/x1Oe0izFd5GgCW7oUJe9wM+iRfDmm3Uz8FS0YNRvPZepfRZ0DIWF8NRTsGWLty4xEe67Dzp29NaFmy0ErVnnngvXX+9ft2ABzKmjG3KO6zmO2JjYE8oKiwt9M1eY2mdBp54rLnYzlfy2NG7QAO69F1JSQt6skzpYcJAPNn3gKe93ej9bCHoKRo1yGcL9vP++y1Bd17Rq1Mr3/t6S7CW2w2gYWNCpx1TdjeLVq711cXEul1okZBrwE2wh6LU9g7xjmkq75JLgudreegs+r4OTvy5LvYxG8Y1OKLMFo+FhQaeeUoUZM2DZMm9dTAxMmQI9eoS+XZURbCHoiJQRthC0hlx+OVx6qX/d9OmwYkVo23OqGjdo7Lt5X+aeTFswGmJhDToiMlpENojIJhF5wKdeROSJQP1qERkYKO8oIp+ISKaIZIjIfWXOaSEi80VkY+Br81BeU10xZw4sXOgtF3HbS591VsibVGnBFoJG846goSYC48bB+ed761Thuedg3brQt+tU2ILRyFDloCMiCSLSRUR6iYj3f7DyzxMLPAWMAXoBE0WkV7nDxgCpgccdwNOB8iLgp6raExgG3F3m3AeABaqaCiwI/GzK+OgjNz7v5wc/gLQInvhlC0FDRwQmToTBPjtCFBe7dDmbN4e+XdVlC0YjQ6WCjogkicgPReQz4ACwCVgL7BSR7SIyTUSqulnJEGCTqm5R1ePADGBsuWPGAq+osxRoJiLJqpqjqisAVDUfyATalznn5cD3LwNXV7FdUW3JEjf91c8118B554W2PVVhC0FDLybGreHp08dbd/w4PPmk/7quSBVswei7G961HUZD5KRBR0TuB7KAW4D5uDf1/kB34Gzgd0AcMF9EPhCRyk4dag+UXRWSzX8CR6WPEZEUYABQeneirarmAAS+tglyXXeISLqIpO/Zs6eSTa7bVq+GV17xr7v00uBj+JFiyXb/2UbX9LzGFoLWothYuPNOSPX5yz5yBB5/3D8LeSQSESb08m4slF+Q7zsb0tS8yvR0zgFGqOpgVf2Dqs5T1TWquklVv1TVF1T1ZqAtMAcYUcnX9ltIUX4aSYXHiEgTYBbwY1U9WMnXdU+iOlVV01Q1rXXrao8S1hmbNrmp0X65tM4/343fR7JjRcd8F4J2ad6FQcmDQt+geqZBAzeb0W8bi7w8l8kiPz/UraqeLs27+C4Ynb95PrlH6kj0rMNOGnRUdYKqrgWo6B6Oqhao6t9V9blKvnY2UHbJYQegfEc96DEiEo8LOK+p6ttljtklIsmBY5KB3ZVsT9T67ju3+NNvn5S0NDduH+lrKedtmsfBAu/nigm9JthC0BBp2BB+9CNo6zNBcPdul9HiWB0Zobqm5zWeHUaLSoqYvX52mFpUf1R1IsEXItK1hl77KyA1MCmhAXA9rqdU1hxgUmAW2zDggKrmiHuXeR7IVNVHfc6ZHPh+MlCvlx3n5rrhjyNHvHU9e7rx+pgInzi/7+g+5m/xrkpMa5fmOz5vak9SkstQ0ayZt27r1rqzF0/LRi25sOuFnvKvvvuKLft9UnOYGlPVt5v3cYFnYNlCETlfRKo0/UNVi4B7gHm4iQAzVTVDRKaIyJQyr7cFN3FhGnBXoHw4cCNwgYisDDwuC9T9GbhYRDYCFwd+rpfy892wx4ED3rrOnd1anLg4b12keWf9OxQWn9hNi4uJY1zPCB8TjFItW7rA06iRt279+rqzJcJlqZeRlJDkKX8z401bMFqLqvSWo6r3ich24BMRmYAbuvoz7s19ZlVfXFXfxwWWsmXPlPlegbt9zluM//0eVDUX8H6EqWcKCtxwx26fwcW2bV16m8TE0LerqrLysliW7V3BemHXC2nVqFUYWmTAJX+95x63DUb5Ydvly12P6PrrI3vYNjEukavOvIrXVr92QvmW/VtYnrPcksbWkioPrKjqw8CfgPdwQ2T5QF9VnVjDbTPVVFTk9rrfutVb16yZ+5Sa5P2AF3FUlZkZ3s8ySQlJjOk2JgwtMmWdcYab1eY3PLtwIcydG/ImVdm5nc6lXZI3ffrbmW97etemZlQp6AQyATwL/Dcu4BQA/1JVyyMRIVTdtGi/1eKNGrmA07Jl6NtVHctzlrN5n3f14VVnXkXD+IZhaJEp76yzYPJk/7p33438PG0xEuO7507ukVw+2vJRGFoU/ara09mIWxNzhaoOB64CHhOR39R4y0y1zJrln08tPt4Nh0TanjjBFBYXMmvdLE95clIy53Y6NwwtMsEMGxY8M/X06f4JZSNJ7za96dPGu/p17qa5HDjmc0PUnJKqBp0bVHWIqs4HUNWPgZHAD0Xk7zXdOFM18+f7p54Xgdtvd8MhdcX8LfPZd3Sfp3xCrwnESIRPt6uHLr4YLrrIW67q1of57dUUScb3Gu/5vSooKvBdG2ZOTZX+elXVk4NEVVfhZpONrKE2mWpYtsylnfdzww3Qr19o23Mq8o7lMXej94ZAnzZ96N2mdxhaZE5GBMaPd1tfl1dY6Ca15OSEvl2VlZyUzIgU77r2JdlL2Jrnc3PUVFuNfGRU1a24wGPCYN06eOkl/7qxY92OkHXJ7MzZHC8+fkJZjMQwobc3fYmJHCLu/k7Pnt660nQ5eXkhb1alXdn9St89d2ZmzLQp1DWoMrnXulTmiVR1f2ARZwRubBy9tm51M9X81kWMHAlj6tgkr6y8LJZmL/WUj+oyitObnB6GFpmqiItz6786d/bW7d8ffKFyJGjcoDFXnundvW7Tvk0sz1kehhZFp8r0dJaIyPMicnawA0SkuYj8EFiHN1O0qSWlqUcKCrx1AwfC974X2eskylNV3lj7hqc82AZcJjIlJrp1YG18Uu3u2AF//7t/SqZIMKLzCJKTkj3ls9bNsinUNaQyQacHsA/4l4jsCWSSflFEnhaRGSKyGrdI9AZc4s0na7PBxjl4EJ54wj/JYvfucMstkZ/eprwvv/vSNwXJVWde5Rn2MJEtKcnlafNbD7ZxY+RmLYiNifXNQr3v6D7mbZ4XhhZFn8ok/MxT1Z/jthT4IbAeaAZ0wW2m9jIwQFWHq6r9r4TAsWNuHxO/HRnat4cf/tBNka5LjhUdY1amd4p0u6R2nN/ZZ/tKE/Fat3Y9noQEb92KFfDGG252W6QJNoX6g00f+M6oNFVT6c/Cqno0MHvtl8DvA19vU9VHSrNQm9pXVATPPuufbaBFC/fp0i8nVqR7f+P7vmsirut9nU2RrsM6d3Yfgupa1oIJvb1T8wuLC303ETRVU9WMBBfjNlVbEXjkishzIuK7UZqpWRVlG2jc2AUcv+y/kW734d2+q7/7n96fnq19pkKZOqVnT7jpJv+6SM1acHqT07mgywWe8uU7lrNh74YwtCh6VPUj5JPAp0Aq0BmX6TkVWFG6h42pPSfLNpBcR/8HZmbMpLik+ISyuJg4myIdRYYOdet4/Lz6amRmLbii+xU0TWjqKZ+xdgYlGoE3pOqIqgadTsCvVHWLqmar6tuqOgL4BBeQTC2pKNvAHXdA15ra5SjE1uxaw5pdazzll3a71LJIR5lgWQtKSiIza0HD+Ia+22fsyN/Bp1mfhqFF0aGqQWcNbvfO8v4IXHrqzTF+Kso2cOON0LdvaNtTU4pKinyzSDdv2JzR3UaHoUWmto0f73o95RUWuskxkZa14OwOZ9OluXep4pwNc8gvqCP7c0eYqgad14FpItKtXHlrYH/NNMmUdbJsA8PrcB6I+Zvns/uwd8OfCb0m0CC2QRhaZGqbCEya5J+14PBht3h0fwS9k4gI1/e53lN+pPCI5WWrpqoGnUeBM4B1IjJLRP5LRP4fblfPn9Z46+q5aMs2UFbukVz+tfFfnvLuLbszMHmgzxkmWpwsa8ETT7gAFClSmqUwvJP3093ibYtta+tqqGrQaQ1cAvwaOApcD/wW6Ab8LrBY9NciYsvHT9GuXe6PL1qyDZQ3M2OmZ4V3jMRwfZ/rkbp8YaZSSrMWtG7trduxA556Co4f99aFy9U9rvbdw+n1Na/bpIIqqmqW6VxV/UhVH1bVG1S1D9AEOBt4DNgLXAa8WvNNrT/y8twww6FD3rq6mm2grDW71rBy50pP+aguo2jftH3oG2TCIikp+C62mzfDtGlQXOytC4emCU0Ze6Y3w9f2A9ttUkEVnfJbl6oWqGq6qj6nqveo6rmq2qwy54rIaBHZICKbROQBn3oRkScC9atFZGCZuhdEZLeIrC13zu9F5DsRWRl4XHaq1xhKpdl4c3O9dXU120BZhcWFzFg7w1N+WuJpXHXmVWFokQmn1q1d4ElM9NatXu02gYuUrAUjUkbQ8TRvPuN31r/DwYKDYWhR3RS2z8siEgs8BYwBegETRaRXucPG4NYBpQJ3AE+XqXsJCDbF6TFV7R94vF+jDa9Fx4+7GTw7dnjrWrZ0f5x1MdtAWR9s+oC9R/Z6yif0mkBinM87j4l6HTu6D1Nxcd66JUvg7bdD3yY/MRLD98/6vqf8WNExy1RQBeEcpBkCbAqs+TkOzMCboXos8Io6S4FmpYtQVfUzXCLSqFBS4oYTNm/21iUlwY9/DKedFvJm1ajdh3fzwaYPPOU9WvUgrV1aGFpkIkWPHnDrrf73KT/80H+NWjh0bd7Vd7v0ZdnL+Cb3mzC0qO4JZ9Bpj0upUyo7UFbVY/zcExiOe0FEmvsdICJ3iEi6iKTv8cucGUKqwfeST0gInia+LlFV/rHmHxSVFJ1QHhsTy8SzJtrkAcPAgfB9b0cCcOvUliwJbXuCGddzHI0bNPaUv77mdc/vt/EKZ9Dxe5cpP3pbmWPKexo3rbs/kAM84neQqk5V1TRVTWvtN4UmRFThzTfhiy+8dXFxcNdd/lNL65ovv/uSdXu8SeMu7nqxbc5m/u388+FK7z5qgMs7uGpVaNvjp0mDJozr4c1UkJOfw7xNlmj/ZMIZdLKBsnflOgDl72ZU5pgTqOouVS1W1RLc+iGfXdsjx9y5sGCBt1zEzVLr0SP0bapph48f5s11b3rKWzRswWWpdWqehwmByy9369DKK02XsyEC8m2e2+lc30wF7298n12HdoWhRXVHOIPOV0CqiHQRkQa4NT9zyh0zB5gUmMU2DDigqhUmyiiXeHQcELHbLixc6LLs+pk4EQYNCmlzas2szFm+KUMmnjWRhDifzVZMvSbi1qGl+dzmKypya3j8tvYIJRHhhr43eLY/KCop4rU1r6GRMuUuAoUt6KhqEXAPMA/IBGaqaoaITBGRKYHD3ge2AJtwvZa7Ss8XkX8AS4AzRSRbRG4NVP1FRNYEdjQdBdwfmiuqmi+/hBnemcOAS28zYkRo21Nbvsn9hs+3eXPXD0weSN+2dTRpnKl1MTFw883Qq/x8VtyC6ccfD3+etg5NO3DxGRd7yjfs3cCS7Ai5ARWBxCIypKWlaXp6esheb80at0+8X3qbiy5ySRGj4b56UUkR//3pf3uGGxLjEnlw1IM0S2wWnoaZOqOgAP72N/8M1M2bw89/7pYThMvx4uM8uPBBzzKAxg0a8+DIB0lK8Fn5GkVEZLmqVmnqaR1e1143ffON2/nTL+Ccc070BByAuRvn+o5vX9PzGgs4plISEtxeUe3aeev273cB6YB3w9mQaRDbwHftTrD7mMaCTkh9+61b/FlY6K3r399tUxAtASf7YDbvb/Suy+3avCvndz4/DC0ydVXjxm6dWiuf7ZV273aBJ5wJQnu36c3g9oM95cuyl7F2d8TeUg4bCzohkp0dPIFnjx5w2211O59aWSVawiurXvEkQoyRGG7oe4OtyTFVdtppcP/9/gukd+xw93iOHg19u0pd1/s6GsV704W8uvpVjhaGsWERKEre5iLbzp3u09iRI966lJS6n0+tvPmb57M1zzu96NJul1pCT1NtrVq5Hk9j77pMtm51owh+H+pCoWlCU8b38u7Hvf/ofmZlzgpDiyKXBZ1atncvPPYY5PtsMtihA/zoR/7JDuuqnYd2MmdD+ZnvkJyUzOWpl4ehRSaatGsX/G9m0yZ4+mk3rToczul4Dj1be3enW7R1Eev3rg9DiyKTBZ1alJfnAk5enreubVuXwNPvU1tdVaIlvLzyZU8qEBFhUr9JxMdGUXfOhE1Kiptc4Dc6kJnpFpCGY0sEEeHGvjf6rj17ZdUrFBSFqRsWYSzo1JIDB+CRR1xPp7yWLd34dNOmoW9Xbfrk2098d1K8qOtFdG3eNQwtMtEqNTV4ZupVq+C558ITeFo2ask1Pa/xlOceyWX2+tmhb1AEsqBTCw4ehEcfdTNryiu9IdrcNw1p3bXr0C7fP6o2jdvYPjmmVvTuHXwCzooV8MIL/ksTatuIziNIbZnqKf/k20/YsDcCcviEmQWdGpaf7wLOzp3euiZNXMAJY37RWlGiJbzw9Que7acBJvWbRIPYBmFolakPBgyAm27yX2qQng4vvRT6wFPRcPJLK1+q97PZLOjUoEOH3D0cv/QcjRq5mTfJyd66um7uxrlk5WV5ykemjPT9xGdMTRo61K1x87NsmctOHerEK20at+HqHld7yvcd3cfMjJmhbUyEsaBTQw4fdtOiv/vOW9ewoQs4Hb073dZ5W/O28t4373nK2zRu4zu2bUxtGD4cfvAD/7olS8Kz7fUFXS7w/dD1xfYvWLlzZWgbE0Es6NSA0h7O9u3eusREF3CiYU+c8gqLC3lx5YueRaAiws0DbrYM0iakzj/fZWf38/nnoQ88MRLDTf1v8v07eHX1q76Z1+sDCzqnqPQejl/ASUhw06JTUkLerJCYvX42OfnescQx3cbYbDUTFiNHwnXX+dd9/nno7/G0atSK7/X+nqc8vyCf6aun18stECzonIKDB920aL8htYQEt4ita5S+92bszmDBFu/ucx1P68jl3W0RqAmfCy+Ea6/1r1u6FF58MbSB55yO5/hu47Fq5yoWbVsUuoZECAs61VS6Dsdv0kCDBm7xWrduoW9XKBwsOMiLK1/0lMfFxHHLgFuIi/FZPGFMCF1yCYzz7igNuL2snn8+dOt4RIQb+91IkwZNPHUzM2ayI7/CzZCjjgWdati/Hx5+2H9adGkPp3v30LcrFFSVF79+0Xc8+uoeV9MuyScHvTFhMHp08B5PerpbQBqqlDlNE5pyQ98bPOWFxYVMWz7Nd7lBtLKgU0V797qA47fwMzHR3cNJjeJZwvO3zGfdnnWe8p6te3JR14vC0CJjgrvkkuD3eFascHtb+W01UhsGJA/gvM7necp35O+oV3vvWNCpgpwc+Otf/VPbNGzoFn6ecUbo2xUqWXlZzM70Zh1ISkji5v4325YFJiJdeGHwWW2rV8P//R8cOxaatlzX+zqSk7yL9T7N+pSvc74OTSPCLKxBR0RGi8gGEdkkIg/41IuIPBGoXy0iA8vUvSAiu0VkbblzWojIfBHZGPhaIwlntm51AccveWejRi7gROssNYBjRcd4bsVznunRADf1v4nTEn02OjEmQowcCTd4R7cA2LDBLXkIxUZwDWIbcNvA23zve76y6hX2Hd1X+40Is7AFHRGJBZ4CxgC9gIki0qvcYWOA1MDjDuDpMnUvAaN9nvoBYIGqpgILAj+fko0b3bRov1/Kxo3hJz+JznU4pVSVl1a+xJ7Dezx1F3W9iD5t+oShVcZUzXnnwaRJ/ilzsrLcsHkotr7u0LQDE3pP8JQfKTzCs+nPerK0R5tw9nSGAJtUdYuqHgdmAGPLHTMWeEWdpUAzEUkGUNXPAL+PBWOBlwPfvwxcfSqNXLvW7Uro1/1u1gx+/vPozDRQ1kdbPvLt+ndu1plxPYNMETImAg0fDrfe6p8kdMcON5qRm1v77RjReQT9Tu/nKc/Ky+LNjOi+vxPOoNMeKLukMjtQVtVjymurqjkAga9t/A4SkTtEJF1E0vfs8X6CB5e36amn/G80tmrlAk405lIra2PuRt7OfNtTnhCXEHSYwJhINngw3HWX/348e/bAX/7itpevTSLC5H6Tad7QO/q/MGshy7KX1W4DwiicQcfvrnP55bmVOaZaVHWqqqapalrrcmmfVWHevOCp0du1cwGnVauaaEnkOnDsAFOXT/W9jzO532TaNPaN58ZEvLPOcksbEnwyNeXluR7PN9/UbhsaN2jMHYPuIDYm1lM3ffV0vjvos+o8CoQz6GQDZQemOgDlV0lV5pjydpUOwQW++kxuDk4VZs6Et70f7gF37+ZnP3NDa9GsuKSYaSumcbDgoKfuoq4XMajdoDC0ypia0727ux/rt3vvsWNuWD09vXbb0LV5Vyb08t7fKSwu5Jn0ZzhWFKJpdSEUzqDzFZAqIl1EpAFwPTCn3DFzgEmBWWzDgAOlQ2cVmANMDnw/GXi3sg0qLIRp0+Djj/3rK/oljTZvrnuTjbkbPeXdWnSz7NEmaqSkuA+Rp/lMviwqcgtIF3izPdWokSkjGdJ+iKd89+HdPL/ied+RhrosbEFHVYuAe4B5QCYwU1UzRGSKiEwJHPY+sAXYBEwD7io9X0T+ASwBzhSRbBG5NVD1Z+BiEdkIXBz4+aSOHIEnnoDly/3rBw503fHExCpeaB20aOsiPvn2E09504Sm3D7odt/hAGPqqnbt4Je/hLZtvXWlIx+zZtVehmoR4Ya+N/hm81i9azXvrq/05+Y6QepjltPy+vdP06uvTvfNowYwapRb1ew34yXafJP7DY8teczz6SpGYrj/7Pvp3jJK8/uYeu/wYXjySdiyxb9+4EC4+WaXW7E27Dq0i4cWPURBUYGn7pYBtzC0w9DaeeFTICLLVTWtKufUg7fRk9u50z9xJ8A118D3vlc/As7eI3t5Jv0Z3+78tb2utYBjolrjxm6Rdz/vTGbApc155BGXXb42tG3SllsG3OJb98qqV/h2/7e188IhVg/eSk/OL9tsTIz7VHPppf6LyaLNsaJjPPXlUxw+7l0BO7zTcC7scmEYWmVMaDVoAFOmuIWkfrKy4E9/qr0p1f1P78/YHuWXK0JRSRF//+rv7D+6v3ZeOIQs6PhITHRbEwwbFu6WhEZxSTFTl0/1TbHerUU3vn/W9y2vmqk3YmLc1tdXXeVfv3+/W8uzZk3tvP6YbmMY3H6wp/xgwUGe/PLJOj+jzYJOOS1awC9+Ab17h7sloaGqvLr6VTJ2Z3jqWjRswZS0KbYA1NQ7InD55XDbbRDn8+tfUOAWjn/4Yc1PMChdONq5mTe3VvbBbJ5Jf6ZOp8qxoFNGly7wq19B+5PlPIgi733zHl9s/8JTnhCXwN1D7iYpISkMrTImMgwe7JZJJPn8Gai6WW3PP++CUE2Kj43nrsF3+SbSzdyTyfRVdXeraws6AWlp8NOfQtOm4W5J6Czauoj3vnnPUy4i3DrgVjo07RCGVhkTWc44Ax54IHjKq6++csNtfluenIpmic24e/DdNIj1Tpdbmr2UdzfUzanUFnRwC8Nuu80/F1O0Wr1rNa+ved237vtnfd83GaEx9VWrVm4tT6/yefADsrPhoYcgM7NmX7dzs87cmXYnMeJ9q567cS4LsxbW7AuGgAUdXEqb+nSffP3e9Tyb/qzv1OgxqWM4v/P5YWiVMZGtYUO49164KMgGuUeOuNQ5c+fW7H2ePm36+G51DTBj7QyWZi+tuRcLAQs69cyuqY/RpvcQnrzy7/zp1tcZvPA/qW6GdRjG2DO90zWNMU5MDEyYALfc4j8yogrvvON2I83Pr7nXHd5pOFd0v8Ln9dxeVytyVtTci9UyCzr1yJ5pj9P8R7+gxe58RKHlnkPc+NQiBi/cSM/WPbmx3402NdqYShg61A23tWzpX5+RAX/8o9sAsqZc0f0Khnca7ilXVaYtn8aaXbU0h7uGWRocIC0tTdNrO51smO3I30Fitx602O39+HWgbTMSsnNIjKsHieWMqUGHDrkkwevX+9eLuPU+Y8bUzBB+iZYwdflU300V42LiuHfovfRo1ePUX6iSLA2O8ZV9MJtHlzxK8z3+/f2muw9YwDGmGpo0gfvuc5lL/KjCu++6ez15eaf+ejESw20Db6N3G+9CwqKSIp788kky99TwbIYaZkEnymXlZfHIF4+QX5DPvlZNfI+RTp1C3CpjokdMjMvReO+9wbc9ycyE//7v4FnsqyIuJo4paVN8cyEWFhfy5JdPsnrX6lN/oVpiQSeKbd63mceWPMaRwiMAzL5xMAUJ5ZZXN2rk5noaY05Jnz7w29+6dT1+Dh+GqVPdjsRHj57aazWIbcDdQ+6ma/OunrqikiKe/urpiJ1cYEEnSm3Yu4HHlz1+Qp6mr0amMv3u89jfpikq4rZBnTrVJZoyxpyy5s3dpnBjxgQ/Ztky1+s51e2wE+MSuXfovb7pckrv/SzLXnZqL1ILbCIB0TeRYPmO5bzw9Qu++ZnaNG7DT87+Cc0bNg9Dy4ypPzIy4KWXgm+FIAIjRsC4cae2OeSRwiP837L/Y8t+/42AJvSewEVdgywuOkU2kcCwYMsCpq2Y5htwkpOS+dk5P7OAY0wI9O4Nv/sdDBjgX68KCxfC738Pa9dW/3UaxTfivmH3Bd3v6s2MN5mZMTNicrVZT4fo6OmoKm+te4uPtnzkW9/xtI7cN/Q+S+BpTIipwtKlMGMGHKtgV4IhQ9wOxX7JRSvjePFxnv7qadbtWedbP6jdIG7ufzPxsTWX76s6PR0LOtT9oHO8+DgvrXyJ5Tv8p8Z0bd6Ve4feS6P4RiFumTGmVG4uvPhixQtGmzSB8ePdXl7VWddTVFLEtOXTWLlzpW99astUpqRNoUkD/5msVVXnhtdEZLSIbBCRTSLygE+9iMgTgfrVIjLwZOeKyO9F5DsRWRl4XBaq6wmH3CO5/OXzvwQNOP1O78f9Z99vAceYMGvZ0m2TMGGC26HUz6FD7j7QX/8K27dX/TXiYuK4M+1ORqSM8K3fmLuRPy36E9sPVOPJa0jYejoiEgt8A1wMZANfARNVdV2ZYy4D7gUuA4YCj6vq0IrOFZHfA4dU9eHKtqWu9nS+yf2GZ9Of5dDxQ771I1JGcH2f630z1BpjwmfvXnj11YqzUovA+efD2LHB1/8Eo6rM2zyP2ZmzfevjY+O5qf9NpLWrUifFp411q6czBNikqltU9TgwAyifbXIs8Io6S4FmIpJcyXOjlqryybef8NiSx4IGnHE9xzGxz0QLOMZEoFatXCaDyZPdUjk/qvDpp27tz2efQYk3KXxQIsLobqO5ZcAtxMbEeuoLiwuZtnwab2e+7ZttvjaF8x2pPVC2j5cdKKvMMSc7957AcNwLIuI7VUtE7hCRdBFJ37NnT3WvIeSOFB5h2oppzFg7w/eXJT42nlsH3srobqMteacxEUwEzjkHHnzQbSIZzOHD8Npr7rhVq6q2bcLQDkO5b+h9NG7g31Wat2kejy55lP1H91ex9dUXzqDj945Y/p8z2DEVnfs0cAbQH8gBHvF7cVWdqqppqprWunXrSjU43Lbs38IfP/tj0Ps3zRs25+fn/Jwh7YeEuGXGmOpq2hRuvx3uvz/47qQAO3fC3/8ODz8MW/yX5Pg6s9WZ/Pq8X9O+afnP9M7G3I384bM/hCx1TjiDTjbQsczPHYAdlTwm6LmquktVi1W1BJiGG4qr01SVeZvm8dfP/0rukVzfY1JbpvKb837juzrZGBP5evSA//ovN3stISH4cZs2wf/+LzzzDOwo/44ZRKtGrfjl8F8yMHmgb/3h44d56sunmJkx03eNX00K50SCONxkgAuB73CTAb6vqhlljrkcuIf/TCR4QlWHVHSuiCSrak7g/PuBoap6fUVtieSJBLsO7eLlVS+zed/moMeMSBnBdb2vIy4mLugxxpi6Iy8P3n7bpcw5mYED4YoroL1/R+YEqsrcTXOZs2FO0MWiyUnJ3NT/JlKapZz0+ercOp3A7LS/AbHAC6r6kIhMAVDVZ8TdlHgSGA0cAW5W1fRg5wbKp+OG1hTIAu4sDULBRGLQKdESPv72Y95Z/w6FxYW+xyTGJXJjvxtPeQaKMSYyZWXBW29VbjO4qgSfDXs38PzXz3Pg2AHfehHhkjMu4cruV1a4mLTOBZ1IEWlBZ0f+Dqavmh40lxJASrMUbh90O60atQphy4wxoaYKa9a4nk9OhR+fnbPOgksugdTUiheY5hfk89LKl1i7O3gOntObnM6N/W6kW4tuvvUWdKopUoLO0cKjvPfNe3z87ccVTmO85IxLGNtjrA2nGVOPlJTAF1/AP/9ZuQ3hOnd2wWfgQLfnjx9VZcG3C3g7822KS4qDPtfQDkO5tue1nJZ42gnlFnSqKdxBR1X58rsveWvdWxwsCJKSFnczcHL/yUET+xljol9hISxeDHPnwgH/0bETtGwJI0e66dlNgmS/yT6YzcsrX2bbgW1BnycxLpErz7ySUSmj/r32x4JONYUz6Kzfu57ZmbPJysuq8LhRXUYxrsc4EuIqmNZijKk3SoPPBx9UrucTFweDBrntFLp29Q69FZcUM2/zPN775r0Kez1tm7Tl6h5XM+D0AcTExFjQqY5wBJ3tB7bzdubbQTPClmrTuA2T+k0itWVqiFpmjKlLCgvh88/hww9dUtHK6NABzjsPBg/2ptjZkb+Dl1a+xNa8rRU+R0qzFH59/q8t6FRHKIPO1rytfLDpg5NuJRsfG89lqZdxcdeLazQVuTEmOpWUwIoVLvhsrThe/FtsLPTtC2ef7fb/iQvcJi7REhZvW8w769/h8PHDQc+fetXUKgcduxMdIhtzNzJ301wydmec9NgByQOY0GsCLRu1DEHLjDHRICbGpdMZNMgtIP3wQ1h9kiQDxcXw9dfu0aSJ6/kMGgTdusVwfufzGZQ8iNnrZ7N42+Ia2wTOgk4tKiopYvmO5SzMWljh9OdSHU/ryLU9r6Vn654haJ0xJhqJuOnSqamwezcsWuSG3w4H77AAbluFTz5xj9NOc7PeBg5szPf73MB5nc5jVuYsNuzdcOrts+G1mh9e2390P4u2LWLR1kUVzkYr1apRK67ucTVp7dIsSacxpsYVFrqht08/hc3Bk5v4atrUrf3p00eh9Tre/3b2v/fjseG1MCosLmTlzpV8sf0LMvdmVqor2iyxGaO7jea8zufZmhtjTK2Jj4ehQ93ju+9gyRKXYufgyT8Tc/Cg6yl9/rkQG9ub1NRe9O2SzubYf1arLdbTofo9neKSYjbkbmD5juWk70jnWFEFG6CX0apRK0Z3G83ZHc+2YGOMCYuSEli3DpYuhZUrXW+oKpQSpk2NtZ5ObTtefJz1e9fzdc7XrNy5kiOFRyp9bsfTOnLpGZcyqN0g21zNGBNWMTHQp497HD3q9upZvtwFoqJKJJqWam5SYEHnJFSVPUf2sHb3WtbuXsuGvRuqlPo7RmIY1G4Qo1JG0bV5V7tnY4yJOA0bwrBh7nH0qJv1tnw5ZGRULgBVhQWdckqDzMbcjWzI3cA3ud9Ua1e9Vo1acXbHszmv03mefEXGGBOpGjb8z/2fggJYv94lHF2zpnKZD07Ggg5wrOgYczfOZcv+LXyb9y35BfnVep6EuAQGJQ/inI7n0K1FN+vVGGPqtIQE6NfPPVQhO9sFn3XrqrZ7aVk2kQBondpar3n0mmqdmxCXwFltzmJg8kD6tOljudGMMfVCQQEkJlY94af1dKqhaUJTerfpTb+2/ejdpjcNYhuEu0nGGBNSFW2pXRELOpUQGxNLSrMUerfuTZ82feh0WicbOjPGmGqwoOMjLiaOzs06071ld85seSZdm3e1YTNjjKkBYQ06IjIaeByIBZ5T1T+Xq5dA/WXAEeAmVV1R0bki0gJ4A0gBsoDrVLXC6WfxMfEMaT+Ers270qV5Fzo07WCLNo0xphaEbYWiiMQCTwFjgF7ARBHpVe6wMUBq4HEH8HQlzn0AWKCqqcCCwM8VapfUjlsH3sqoLqNIaZZiAccYY2pJOJfFDwE2qeoWVT0OzADGljtmLPCKOkuBZiKSfJJzxwIvB75/Gbi6lq/DGGNMJYUz6LQHtpf5OTtQVpljKjq3rarmAAS+tvF7cRG5Q0TSRSR9z5491b4IY4wxlRfOoOM3/av8oqFgx1Tm3Aqp6lRVTVPVtNatW1flVGOMMdUUzqCTDXQs83MHYEclj6no3F2BITgCX3fXYJuNMcacgnAGna+AVBHpIiINgOuBOeWOmQNMEmcYcCAwZFbRuXOAyYHvJwPv1vaFGGOMqZywTdNS1SIRuQeYh5v2/IKqZojIlED9M8D7uOnSm3BTpm+u6NzAU/8ZmCkitwLbgAkhvCxjjDEVsNxr1Px21cYYUx+IVD33mu0kZowxJmQs6BhjjAkZCzrGGGNCxu7pACKyB9gahpduBewNw+uGk11z/WDXXD+cqapJVTnBkowBqhqW1aEikl7Vm3B1nV1z/WDXXD+ISJVnYNnwmjHGmJCxoGOMMSZkLOiE19RwNyAM7JrrB7vm+qHK12wTCYwxxoSM9XSMMcaEjAUdY4wxIWNBJ0REpKOIfCIimSKSISL3BcpbiMh8EdkY+No83G2tKSKSKCJfisiqwDU/GCiP2msGt526iHwtIu8Ffo72680SkTUisrJ0Cm09uOZmIvKWiKwP/E2fHc3XLCJnBv5/Sx8HReTH1blmCzqhUwT8VFV7AsOAu0WkF/AAsEBVU4EFgZ+jRQFwgar2A/oDowNbVETzNQPcB2SW+TnarxdglKr2L7NOJdqv+XHgA1XtAfTD/X9H7TWr6obA/29/YBAu6/9sqnPNqmqPMDxw+/xcDGwAkgNlycCGcLetlq63EbACGBrN14zbUHABcAHwXqAsaq83cE1ZQKtyZVF7zUBT4FsCE7HqwzWXu85LgM+re83W0wkDEUkBBgDLgLbqNqYj8LVNGJtW4wJDTStxO7jOV9Vov+a/Ab8ASsqURfP1gtsq/kMRWS4idwTKovmauwJ7gBcDw6jPiUhjovuay7oe+Efg+ypfswWdEBORJsAs4MeqejDc7altqlqsrkveARgiIn3C3KRaIyJXALtVdXm42xJiw1V1IDAGN2x8frgbVMvigIHA06o6ADhMFA2lVSSwU/NVwJvVfQ4LOiEkIvG4gPOaqr4dKN4lIsmB+mRcjyDqqGoesBAYTfRe83DgKhHJAmYAF4jIq0Tv9QKgqjsCX3fjxvmHEN3XnA1kB3rtAG/hglA0X3OpMcAKVd0V+LnK12xBJ0RERIDngUxVfbRM1RxgcuD7ybh7PVFBRFqLSLPA9w2Bi4D1ROk1q+qvVLWDqqbghiA+VtUbiNLrBRCRxiKSVPo9brx/LVF8zaq6E9guImcGii4E1hHF11zGRP4ztAbVuGbLSBAiInIusAhYw3/G+3+Nu68zE+gEbAMmqOq+sDSyholIX+BlIBb3AWemqv63iLQkSq+5lIiMBH6mqldE8/WKSFdc7wbcsNPrqvpQNF8zgIj0B54DGgBbgJsJ/I4TvdfcCNgOdFXVA4GyKv8/W9AxxhgTMja8ZowxJmQs6BhjjAkZCzrGGGNCxoKOMcaYkLGgY4wxJmQs6BhjjAkZCzrGhIGIDBCRYhH5PNxtMSaULOgYEx63A38H+ohIz3A3xphQsaBjTIgFUgJ9H5iGy9t1a7n6oSKyQkSOBbIYXyYiGshyUHpMLxH5l4jki8huEfmHiJweyuswpjos6BgTeuOBraq6GpgOTAokgy3NQv4eLkfdINw2CX8te3IgseJnuBxnQ3A57ZoAc0TE/qZNRLNfUGNC7zZcsAH4FLcL41WBn3+Ay1V3q6pmqOp84KFy5/8QWKWqv1TVzEDwmgQMBtIwJoJZ0DEmhESkG24LhNcB1CU/fA0XiAB6AGtV9WiZ05ZxokHA+SJyqPSBS8QIcEatNd6YGhAX7gYYU8/chuvJbHO7XQAgACLSMfD9ybLwxgD/An7mU7fLp8yYiGFBx5gQEZE43J4jv8LdtylrOi49fibuHk/DMr2dIeWOXQFch7svVFiLTTamxtnwmjGhcznQCpimqmvLPnA7jd6C2yCrGJgWmKF2EW7fJfhPD+gp4DTgjcBMt64icpGITC3dUM2YSGVBx5jQuRX4RFVzfereBDoDw4Argd7A17iZa78PHHMM/r099HDcZoAfABm4QFQQeBgTsWwTN2MinIiMxe3O2UZV94a7PcacCrunY0yEEZHJuC2QtwN9gL8B/7SAY6KBBR1jIk9b4EEgGdiJm6n2y7C2yJgaYsNrxhhjQsYmEhhjjAkZCzrGGGNCxoKOMcaYkLGgY4wxJmQs6BhjjAmZ/w/efSy8KmlsXAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "x = np.linspace(norm.ppf(0.001, loc = 37, scale = 14), norm.ppf(0.999, loc = 37, scale = 14), 100)\n",
    "plt.plot(x, norm.pdf(x, loc = 37, scale = 14),'b-', lw = 5, alpha = 0.6, label = 'poor')\n",
    "x = np.linspace(norm.ppf(0.001, loc = 45, scale = 11), norm.ppf(0.999, loc = 45, scale = 11), 100)\n",
    "plt.plot(x, norm.pdf(x, loc = 45, scale = 11),'g-', lw = 5, alpha = 0.6, label = 'rich')\n",
    "plt.plot(25, norm.pdf(25, loc = 45, scale = 11), 'ro')\n",
    "plt.xlim([15, 70])\n",
    "plt.xlabel('Age', size=14)\n",
    "plt.legend(loc = 'best')\n",
    "plt.title('Age Distributions')\n",
    "plt.ylabel(r'$p(x)$', size=14);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "We will also work with $P(C_j\\,|\\,x_i)$.\n",
    "\n",
    "This is the probability that a data point at $x_i$ was drawn from cluster $C_j$.   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "That is, the data point could have been drawn from any of the $k$ clusters -- what is the probability it was drawn from $C_j$ in particular?\n",
    "\n",
    "In our example: $P(\\text{rich}\\,|\\,\\text{age 25})$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "How can we compute $P(C_j\\,|\\,x_i)$?\n",
    "\n",
    "Note that the reverse conditional probability is easy to compute -- so this is a job for __Bayes' Rule!__\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$ P(C_j\\,|\\,x_i)=\\frac{P(x_i\\,|\\,C_j)}{P(x_i)}P(C_j)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "hide_input": true,
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "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.figure()\n",
    "x = np.linspace(norm.ppf(0.001, loc = 37, scale = 14), norm.ppf(0.999, loc = 37, scale = 14), 100)\n",
    "plt.plot(x, norm.pdf(x, loc = 37, scale = 14),'b-', lw = 5, alpha = 0.6, label = 'poor')\n",
    "plt.plot(25, norm.pdf(25, loc = 37, scale = 14), 'ko', markersize = 8)\n",
    "x = np.linspace(norm.ppf(0.001, loc = 45, scale = 11), norm.ppf(0.999, loc = 45, scale = 11), 100)\n",
    "plt.plot(x, norm.pdf(x, loc = 45, scale = 11),'g-', lw = 5, alpha = 0.6, label = 'rich')\n",
    "plt.plot(25, norm.pdf(25, loc = 45, scale = 11), 'ro', markersize = 8)\n",
    "plt.xlim([15, 70])\n",
    "plt.xlabel('Age', size=14)\n",
    "plt.legend(loc = 'best')\n",
    "plt.title('Age Distributions')\n",
    "plt.ylabel(r'$p(x)$', size=14);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ P(\\text{rich}\\,|\\,\\text{age 25}) = \\frac{\\text{red}}{\\text{red} \\cdot P(\\text{rich}) + \\text{black} \\cdot P(\\text{poor})} \\cdot P(\\text{rich}) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Finally, we will also need to estimate the parameters of each Gaussian, given some data.\n",
    "\n",
    "This is also an easy problem.   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "For example, the best estimate for the mean of a Gaussian, given some data, is the __average__ of the data points.\n",
    "\n",
    "Likewise, there is a simple formula for the covariance."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "These are called __Maximum Likelihood Estimates__ (MLE) of the parameters.\n",
    "\n",
    "Often we use $\\mathbf \\theta$ to denote \"all the parameters of the model.\"   \n",
    "\n",
    "In this case $\\mathbf \\theta_i = (w_i, {\\mathbf \\mu_i},{\\mathbf \\Sigma_i})$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Expectation Maximization for GMM -- The Algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "OK, now we have all the necessary pieces.   Here is the algorithm."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "__Initialization__: \n",
    "\n",
    "Start with an initial set of clusters  $C_1^1, C_2^1, \\ldots, C_k^1$ and the initial probabilities that a random point belongs to each cluster $P(C_1), P(C_2), \\ldots, P(C_k)$.\n",
    "\n",
    "The result will be sensitive to this choice, so a good (and fast) initialization procedure is $k$-means."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<!-- Source http://bengio.abracadoudou.com/lectures/gmm.pdf -->\n",
    "\n",
    "<center>\n",
    "\n",
    "<img src=\"figs/L09-EM-E-step.png\" width=\"60%\">\n",
    "\n",
    "</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "*Step 1* (__Expectation__): For each point $x_i$, compute the probability that it belongs to each cluster $C_j$:\n",
    "\n",
    "$$ P(C_j\\,|\\,x_i)=\\frac{P(x_i\\,|\\,C_j)}{P(x_i)}\\,P(C_j)$$\n",
    "    \n",
    "(Thank you, Rev. Bayes!)\n",
    "\n",
    "This is called the _posterior_ probability of $C_j$ given $x_i$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We know how to compute everything on the right.\n",
    "\n",
    "Note that: $P(x_i) = \\sum_{j=1}^k P(x_i\\,|\\,C_j)\\,P(C_j)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<!-- Source http://bengio.abracadoudou.com/lectures/gmm.pdf -->\n",
    "\n",
    "<center>\n",
    "\n",
    "<img src=\"figs/L09-EM-M-step.png\" width=\"60%\">\n",
    "\n",
    "</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "*Step 2* __(Maximization)__: Using the cluster membership probabilities computed in the previous step, compute new clusters (parameters) and cluster probabilities.\n",
    "\n",
    "This is easy, using maximum likelihood estimates of the parameters $\\theta$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$w_j = P(C_j) = \\frac{1}{n}\\sum_{i=1}^n P(C_j\\,|\\,x_i)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Likewise, compute new parameters for the clusters $C_1, C_2, \\ldots, C_n$ using MLE."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "__Repeat__ Steps 1 and 2 until stabilization."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Let's pause for a minute and compare GMM/EM with $k$-means."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "GMM/EM:\n",
    "\n",
    "1. Initialize randomly or using some rule\n",
    "2. Compute the probability that each point belongs in each cluster\n",
    "3. Update the clusters (weights, means and variances).\n",
    "4. Repeat 2-3 until convergence."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$k$-means:\n",
    "\n",
    "1. Initialize randomly or using some rule\n",
    "2. Assign each point to a single cluster\n",
    "3. Update the clusters (means).\n",
    "4. Repeat 2-3 until convergence."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "From a practical standpoint, the main difference is that in GMM, data points do not belong to a __single__ cluster, but have some probability of belonging to __each__ cluster.\n",
    "\n",
    "In other words, GMM uses soft assignment.\n",
    "\n",
    "For that reason, GMM is also sometimes called __soft $k$-means.__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "However, there is also an important conceptual difference. \n",
    "\n",
    "The GMM starts by making an __explicit assumption__ about how the data were generated.  \n",
    "\n",
    "It says: \"the data came from a collection of multivariate Gaussians.\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Note that we made no such assumption when we came up with the $k$-means problem.   In that case, we simply defined an objective function and declared that it was a good one."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Nonetheless, it appears that we were making a sort of Gaussian assumption when we formulated the $k$-means objective function.   However, __we didn't explicitly state it.__\n",
    "\n",
    "The point is that because the GMM makes its assumptions explicit, we can\n",
    "\n",
    "* examine them and think about whether they are valid\n",
    "* replace them with different assumptions if we wish"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This sort of assumption is called a \"prior\" on the data.\n",
    "\n",
    "Philosophically, __exposing__ this assumption is the heart of Bayesian statistics."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "For example, it is perfectly possible to replace the Gaussian assumption with some other probility distribution.   As long as we can estimate the parameters of such distributions from data (eg, have MLEs), we can use EM in that case as well."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Instantiating EM with the Gaussian Model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "If we model each cluster as a multi-dimensional Gaussian, then we can instatiate every part of\n",
    "the algorithm. \n",
    "\n",
    "This is the GMM (Gaussian Mixture Model) algorithm implemented in *sklearn.mixture* module."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "In that case $C_i$ is represented by $(\\mu_i, \\Sigma_i)$ and in EM Step 1 we compute:\n",
    "\n",
    "$$ P(x_i|C_j) = \\frac{1}{\\sqrt{(2\\pi)^d |\\Sigma_j|}} exp (-\\frac{1}{2}(x_i-\\mu_j)^T\\Sigma_j^{-1}(x_i-\\mu_j))$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "In EM Step 2, we estimate the parameters of the Gaussian using the appropriate MLEs:\n",
    "\n",
    "$$\\mu_j'=\\frac{\\sum_{i=1}^n P(C_j|x_i) x_i}{\\sum_{i=1}^n P(C_j|x_i)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "and\n",
    "\n",
    "$$\\Sigma_j = \\frac{\\sum_{i=1}^n P(C_j|x_i) (x_i-\\mu_j')(x_i-\\mu_j')^T}{\\sum_{i=1}^n P(C_j|x_i)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "A final statement about EM generally.   EM is a versatile algorithm that can be used in many other settings.  What is the main idea behind it?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Notice that the problem definition only required that we find the clusters, $C_i$, meaning that we were to find the $(\\mu_i, \\Sigma_i)$.\n",
    "\n",
    "However, the EM algorithm posited that we should find as well the $P(C_j|x_i)$, that is, the probability that each point is a member of each cluster."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This is the true heart of what EM does.   \n",
    "\n",
    "The idea is called \"data augmentation.\"\n",
    "\n",
    "By __adding parameters__ to the problem, it actually finds a way to make the problem solvable!\n",
    "\n",
    "These parameters don't show up in the solution.  They are sometimes called \"hidden parameters.\"\n",
    "\n",
    "So the basic strategy for using EM is: think up some __additional__ information which, if you had it, would make the problem solvable.\n",
    "\n",
    "Figure out how to estimate the additional information from a solved problem, and put the two steps into a loop."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Here is an example using **GMM**.\n",
    "\n",
    "We're going to create two clusters, one spherical, and one highly skewed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The covariance matrix of our skewed cluster will be:\n",
      " [[2.9  0.67]\n",
      " [0.67 0.17]]\n"
     ]
    }
   ],
   "source": [
    "# Number of samples of larger component\n",
    "n_samples = 1000\n",
    "\n",
    "# C is a transfomation that will make a heavily skewed 2-D Gaussian\n",
    "C = np.array([[0.1, -0.1], [1.7, .4]])\n",
    "\n",
    "print(f'The covariance matrix of our skewed cluster will be:\\n {C.T@C}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "# make the sample deterministic\n",
    "np.random.seed(0)\n",
    "\n",
    "# now we construct a data matrix that has n_samples from the skewed distribution,\n",
    "# and n_samples/2 from a symmetric distribution offset to position (-4, 2)\n",
    "X = np.r_[(np.random.randn(n_samples, 2) @ C),\n",
    "          .7 * np.random.randn(n_samples//2, 2) + np.array([-4, 2])]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "hide_input": true,
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(X[:, 0], X[:, 1], s = 10, alpha = 0.8)\n",
    "plt.axis('equal')\n",
    "plt.axis('off');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "# Fit a mixture of Gaussians with EM using two components\n",
    "import sklearn.mixture\n",
    "gmm = sklearn.mixture.GaussianMixture(n_components=2, \n",
    "                                      covariance_type='full', \n",
    "                                      init_params = 'kmeans')\n",
    "y_pred = gmm.fit_predict(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "cell_style": "split",
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "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": [
    "colors = ['bg'[p] for p in y_pred]\n",
    "plt.title('Clustering via GMM')\n",
    "plt.axis('off')\n",
    "plt.axis('equal')\n",
    "plt.scatter(X[:, 0], X[:, 1], color = colors, s = 10, alpha = 0.8);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "cell_style": "split",
    "hide_input": true,
    "scrolled": true,
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cluster 0:\n",
      " weight: 0.667\n",
      " mean: [-0.01895709 -0.00177815]\n",
      " cov: \n",
      "[[2.79024354 0.64760422]\n",
      " [0.64760422 0.16476598]]\n",
      "\n",
      "Cluster 1:\n",
      " weight: 0.333\n",
      " mean: [-4.05403279  1.9822596 ]\n",
      " cov: \n",
      "[[0.464702   0.02385764]\n",
      " [0.02385764 0.42700883]]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "for clus in range(2):\n",
    "    print(f'Cluster {clus}:')\n",
    "    print(f' weight: {gmm.weights_[clus]:0.3f}')\n",
    "    print(f' mean: {gmm.means_[clus]}')\n",
    "    print(f' cov: \\n{gmm.covariances_[clus]}\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "cell_style": "split",
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/png": 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dLZ5Nqry3M/A7VKHsD1bYG6ywiQ2GmYSiJ9rT19S6918L7IgW9HYFXtHbvIIg2PBE9IJiaScgcZoIvBJ4K/A2JGzf6iFm9iiUAGGoxsKewCussNuBf3nNV9bvkKzhc4EdgEfTvi8Drk6bHJ7msRpZ2ecBp1hhx6BIh22A+5HI347ig9cgy7nKmsrPjiIpgiAYZDYZ0U0W5KnIZfBNr3kuQHMvEqVJ6Hrc4jVfbIV9GfgscIkVdgVwTtruLcAJSPBuQdEGlsYajR7zVwBzrbDjkJX5fOA24Epk4T4XRT/MBJ5IY2XuRwK5Vfp9CbA1EubT0Q3isfT7q5A1O89rvrDulL8HHArsBdwAXNK/KxYEwYZg2IpueowfBzzmtZ5XA62wCSg8awoSyGcArwfwmt9hhZ2OohH2At5ghf0HieULkeidAvwTWcPvRVbo0Sgh4t/A7mhBLC9kLUMRCSencUcicf5AGnc5EuYtkRB/pzLd3wF/BF6NxDdbqAegeN75SKxf4jX/LvC/ynmOQ5lyD6bz/ARwn9d8UU/XJwiC5jEsfbpW2AvQAtF1wKfSSn5PzEACvQAlNTyz7v1r0eP+PCRyX0CWpqOQrm2AC4HPIws1Fyp/AYrFvRlZnjel98cg0dsc3djmI9F9LvCjdExHVvZZXvPqo/+nkD92LBLrLYCzgbvT+yNoUCoyRURciW4uf09fP0B+5i16uT5BEDSJYSm6SITWIAE9Hj1i98QDyH86DbkRrsxvWGGHAN9CAtmJxLQVJSisQlblKCSu+6NH/clIEDcHViIL+ZfAcWnYBUg8f4EWuaaha/0nFN71EWT1HuU1f6QyF0MLXjl1uCUd54o0x2vS3C5FacVVXoBihZchod4CWdRbId9zEARDgOHqXliJVuvzYtHqnjb2mq+ywl6GXAJLgF8BWGFPR3UU8s1nZxTC9SWv+e1W2OuRwGWr15Ewn5vm8HEk0pMpF6tmAM+vhH+9GtVp+BcS/N+mY92S55G2Gwm8BAl9lQ5gpde8neQS6YZFaQ4j0z4t6GbRiSz4IAiGAMMyOcIK2wOVSJwCfCb5Nvuz/xS00HQAihQYRfnY/hCyEF+AFqzmUC5qObJin4JE92Jk5e6IbgAdaZzD0MLbdOB8r/l30nGvo2zjMwl4hdf8tvTeF5Hotqb9QFbr773mJ/fhnFqQBX0M6lrRhmJ5O9M4F3rNf9DXaxQEwYZhWFq6XvPbgYPWY4jXoVCrR9BjeNUn3JJ+fzrq+DANuQja0usHe80XWmGzkO92JQr3OiDtewnwQeQHXgF80Aq72ms+Gwn6jLTd2PTzbem4RyJ3SQcS5a+l+f2yLyeUMs4+kb4AsMI+i9wvK4CzrbB/e81v7ct4QRBsGIarT3d9GVH5Oftx25GoTkYlF3cDnobEFnStrvWa35l+/yyqJrYtEuhXIp/vqciKbUPWryOBBUUy3JD22Rz4oRX2rvTezZQJGYtRfPBP1rOo+fbI9bIyzSMW1IJgkNkoRNcKG22FvcQKO7qPqbjfQ5EDE9EC2zIkvvORQK1CC11TKd0N/0UWcmZPJNDjkIBO9pr/O4WvFUjYJ6EQsLsAvOYPoQWwsejajwfOtMK2Q77mB1F23CsHqIj5Bel7roB2/QCMGQTBejAsfbpV0or/RcjqBCUafAoJ5l1e8we62a8FieYKlLzQgRaqno3Ecmx6rw25D95dLShjhd1E2RW4E3iD1/x7lffzIt0+yIr+JvIHvxrF9eYL/zgS9G3Ta/OQ1bs3cBnw2fUpVmOFbY9cHbdGN4ogGHw2BtGdjERqEVrQGo/E63EUy3pC8gH3Ns40ZAlujyIA2lDL9RZghtd8jRW2JbIen4LCuranjGo4rbqgZ4V9GyVVVP3mq9PYLWkfS8cYjeJwO1G88FJk+Y5L4/6hn5clCIIhyrBcSKtjKXIRzEKCCzqvqcjCPI6uabZrSa6IU9K+DyMhnJ1+b0WLaG1ALS1KfQS5FXJq7lLkilgD/Kdu+M3o6jsGRUmsQWKbC5ZbOtYM5OZ4glL0x7NuIZsgCIYxw150veadVtiJyI/6Ckp/6Zj01VN92vOBN6LrsBo92q9EC1kTkfU8Hng38HLK1jpTkXVqSFwXo0W0qs/0cyjsbHzltSywUFq6E9LvS1GG3RFp/BZU2LxbKze5SD6Aav/eArzPa76sh/MNgmCQGbaim2J1z0m/fsxrfqoV9myU4JB9JjfRjWglK/dEZFWCBHoNEq+8SDYdCeODadxWuoaXORLrUcBHrbA1KJNtNKrV8C6UUjwOWcRLkWjnmOAR6Wsuulk8G7lFViHL+yhgKyvsJ8gS/rrX/KuV4x+Jbho5rng2qkoWBMEQZVhGLyQL70JgFxTWdaEVNgIJYhsSwk7Ukfc+K+w3qQVP3v/dyB0woW7oEUjIRwD3UArjNpTinPHKPqNRuNf7kSvjBpSBdi4S1DlImJcj90FO8SXNs40yNXkyEubJyOo+D/l524B3WGHPqMxhauWcO1EBnSAIhjDDUnSRgE1BPtB2tLD1Z1Rj4QnkFgCJIeiR/WRYW0D8bcgl8Bhlam9uDPmNNP5UZNlm4YWuNWmzxbsm7T8Hif22SKBnpTF2SK8/glwgP0nH6qgc+05Uw/cWVOshh6H9E3gxsnJnIGu46q74LQoFG5/GvMIKO8MKe01fu1gEQdBcNoh7IZVd3AaYvZ7B/Y3GnoEKh89FYjsFPY5PQWJ5MRLb11J2ZKgmBlTdA0tRtbKvIotyRdpuFRLKLdP3ViTuHXS1jjtQNMOR6HxzKvC26TjtlPG67/CaX2aFPYjq4G5J6eN9EarD+0+U2TYfifUsdGOZns6tHfmQdVI1X5Aqru2c9rk0jesoVO3dvV/RIAiayYBbulbYU1E1rcuA36WSgwM19hjg5yiz6yAkgC1I7CYiEfu21/w9qFh4LvzyAPCzNMxs4P9RFql5L0oJ3gKJXF4gOwAtwo1GwplTgTN5IezQNI+ONI/5lK6HFuSbPdRrfll67eVIQOtTj08CrkIiPyWNt5LS2u5M230nha5l9kMLiMemc5qHrPjn9XApgyAYJDaEpft69M+/ANgOWXEXD9DYM5Cl14pEaCQSxLzg1QF8zgr7IxLRh5HA/bTSZfcI4Dmohu5ZaZ+Xp/luhkLBTgfekM5jDWVR8Onp97ygtgiFj41BYjsNha9l/2orsl5zxbEXoM4T9Te7ThSva6hWwp7oRnFW+hkkvK3pvC61wo5EN4z3IHFemc5lWtr+1327pEEQNJMNIboL6BoatWQAx34YCdQ4JLbZEszVvVpQpa5DkPU5B1l9b0qZWTcjQc2NHkcDZ6QxH0eP8kvT75+mFLrtkP93KWUdhTYUrnUKSsrIJRXHIB9rjnrYEzgw+ZI/T1lQJ9OJ3BlbAT9EXSNO95p7EtZxwNdRi6A1yBLeHDgT+abHp2N2oOI4s5GAD9SNLgiCAWRDiO43gKei7gw/ZGAtrlMpw6/WoO4Iu9G19CKUhWY2R6I0Elmzz0eLWx2UroFzkEU7Ewna/5CPNz/WZ0FvpXQ9tKcxv12ZmyNBrYaErUw/fwQ1nBxNV9qQSO6U3lsFvBQtuOU2RMtTXd9pKBZ4QTrO1mm+eX6jgMu95r/Lg6cU6bNQWNkc4OTu0qKDIGgOwyoN2Ar7KRLZ5Whx6gNo9f5iypq2LchqdSREm1WGyJZrrij2OBK+NiR8y5DVmMU1Z42togwN64l8MbP/tR0J82vSXFopBTlv45Xv7Wn/mV7zLm3fk2/8k2me30xz/w6yvFtRYsTFadsWFA3xcuBgSt/2VV7zF/ZyDkEQbECGZHKEFXYA8CUkch/1mmdr+U9oVX46Et3zKRe0HkQLUdciYX4CWXiTKZMQcp2EDuSDnU1Zl7clvZ6Ly2Tf7TxKP3KvU0/jZ0t7AbKmcz2ITLaU8z6ZVcgds86xvOYLgNO6HKywV6Immzd5ze+uvHUC8D7kRhlBGer21D6cQxAEG5AhJ7rpkfgbyIJbA3zBCvtTSm/9OhLLryDBbEERBw8gt8OeXvM3JEvvViQ62ZVQFb3VKJzq0+kYuZHk8vQ1HgltG3IRTKcrWcTqayusQYLajsLPRiCh60Aim32y2YrOZMt3PnLHPEwf8JrfgmJ763l6+p4X9/LC4y/6Mm4QBBuOoZocMR6JU45zHQVruyNcisSkjdJNsD3y386ywnZEIrM5pX81P753pi8H/oaEdUw65hpkJR+GHssfRLHA2V+c43TbKB/X68l1Gx5L402ndBu0o4WuB1m3D5ojkf4m8M7eWsr3gV+mebYg98t1qNzlh9Zz3CAI1pMhZ+mmVftzgI+ll76THq3z+6utsPdTRhfchJIDliNh28trfp8V9g9Uu6ADRR0sRe6KzZAFfCtJzPPQwLK074MoS+z56b0n0r6/QI/zW6KIhuzayPtPTNtticR9JWUG2VLkZ90b+aIPpRTutWP0VDvXCtsJFWDfCviC1/zr6cngBLRQ9xfg+17zm6ywlyCL91av+f3djVkZe/90Tn/xmj/e4P0tkH9613QdPlDXOj4Igj4wJC1dr/lFqPjLIV7zcxu8fzkSv93QYtLS9FYHZRnHTsqog+XAR9EKfvbtTkcimWsfdCJLE695G1r8mofcGI8i6/rNqKdZtprXTimNkRfFoKx09lcU3rUXiqk9HXUBXk3p8jDkTnmuFTaR7vkRStqYCZxvhT0XZcOdg2KPz0JJEnjN/+s1v7yPgvsKFGlyHvBrK6zenUKa+27I53wsincOgqCfDKroWmG7WWFvt8KOTBbbWrzmj1c7NdTjNe9MVuF3kah9EzjRa36PFXYY8ELKDLIJ6NF6FjrnVroWEm9DxcJvr4w/Hy3mTUZhZp9IbdA/iGJpc4uf7K7IC1bV1zqRZX0muhm8Gfl43wZ8hrKFPGn/Q4ErrbCnNLhWY1HMb2tl/gehgj8j0I3H0M2ov5yAnhIWoQXKZzbYZhxdbzZjGmwTBEEvDJp7IaUL/xRZeB3IEv1Rf8ep+Hmr5Kys3HRyDBKNhygXlqAsOvPz+jbuVtjWqPTjSiRmT7PCrkSVzUCRCfchQb4FWcUXowW6F1MmiOTt801lNbK870rfJ1MurLWhWNyrrLDP1JVxPLUy77z9CmStt6dxqtXK+sNtyO2RnwJmN9jmK8Cz0nH+Bfz+SRwnCDZ5BtOnuxdlSNZE5D/ts+gmyziv0v+nbvHpN6jx5E5IRH6M4lXHoYWskUgsRyIR/GSDQ+yP/LFLkB/zA0jo5iMRXwUs4J+M5koOYg1bAEewP1fzIh5FC3l5ES/H7OaUZVDRm9Hp9ZxWnNOOlwDvssK+gaz0pyF/a67v0Jq2eXfabwkKn7vRa17fwaIvfCZdh6cBF9eFnwHgNf9fqlc8DXh0ffq2BcGmzKAlR1hhu6KiOCORMH682tixD/sXqMkjqHD41WiFvgPVtf03ahz5IBK396Eoh0uRNfg+ykpe7ZS1cF+IhPV+VCRnK8q0YagmMdzCSn7FeDoqNy9jNcewmH2YTNdkihwjnMPXOimz3qp0UPqQX4QWrSZSVkqblI6ffbVj03FO8ppf0+A6TUrnPTu6SgTB4DOoGWlW2H4owuBO4LK+Wk9W2Dj0iLsovTSFMgyrJf28T2rl82rg/yh9kLeix/95wB3ptZy9tghZqJ2oEtlNqFh6LsOYBbMDuIfPszNLGzwtjGI+H2YMpeskZ8JV43OrkQ/UvbYadZ1Yg0o+PpHO8XLk0liJrO+3Ut60fuk1P77uOs1EN5kJKJTtZakNfBAEg8Sghox5zW9CwtZf2tBC1jgkUsuQK2Bxen8yMDJ1WTifsm8ayOXwFhRyNhqJVo7FzYVqHAnzt9AC3WmUMbcdKHmhraHganbTkPBnv262jqshavUJG9W048WoPu9rUGW1LdBC2YHAy5Dozkc+3VxOcl8rrKXuxnVCmvc8dOM4DtWVCIJgkBhycbp9IbVDfz3yxTrwYSSSr0XndC2qu/t01u2m257GaE+9xz5IWRZya0rrsxU1mjTKNN629PtWwNZMxlm8jrUKk3Ek5jnd11HCxLaUwr0cCeZkSldDtdbDvkh481iTgN2RcN+DLPKHkethArKGT7TCrkBdMnLsct6/E1hghY1EC2KrgRsGIBEjCIJ+MKwK3vREWlh7HmUK8VZoMW1LSgu2DQnpq73mj1thv0Nxrrn4TCY/5udFLijFOC+ILec2xnAFI7sEfo0EjgH2wJHFOomuViyoWear0QLYDigy4Y1p3iuBN6HkkL0rI6+kbGz5BBLUNyHr9zVIeLOVPCn9vAbFCe8JXIMiRL5FWW/iIq/5x3u6rkEQDCxDMjmiv6R+YB9Aq/CbUboZZiIBvQ51WNgCOKKScZWTIuqtVav7ueoKyD7XUeyJcwztTEy1GCZTFdzV6Yu6fdcAo73mC7zma7zm/0Pi+gRwNwpFyzWIOyv75Sply1GtibO95tciER2d9p+PWhitovR3/8hr/kyv+ZlIsA9K12cpEusgCJrIsHQvNOBNKPFgBFpwykVlslDtAEz2mq+o2+8D6FF7Iuu2V++g6/WpLoDNQyK3E3uymj15HD3yv4SySHkrEsT309VabgE6rbBTkTU7Bvm1x6af25CFfgey0nODzEfSPB9DlvvKNOYiFH+8XRr/f2m7MUhcb6ucwyLkB56UrlWjeNwgCDYgG4V7wQr7MgqvWk5p6WVrMydKzEnbbIdW/eegVjpnpm33pWvVsHvQolu2NkcgMZyE4oDHIQs1RzaMRjVuZ1BatCvQI/1nkIiPSsd6iNIKBwnop5HA/ghFWLwGLfg9A1m+j6I03IfSeY4BDvCaL7TCtkIuinYUdbEX8h9f7TV/sO5a7YVuNstRmF4UNQ+CJrKxiO6zkNjk9udzKRfFRiPBWwy8E1mXhkRrMyR4TlmRLFu0q9J7UyqvLUmvX45SdqdSinq1y4Sn7UairLZPp313ocz4qk/XPQLFFh+AYoWPTMfNLYksHetBJOiTgGfXFzsPgmBoM+zdC1bYFOReyB0g5lDG7C5CVp8jgf0gslBzK/MJSJQbNYpckfabn7aZkL4mAq9L21Ut42rSQwuyauehlkI/A16JrEtD1nh1n3ZU4OdzyHUwK207L431RmTh7ouK27QA3wvBDYLhx7AXXdQR91C0MNSBLNMVwI2U9XAnIrHdPL1fPe9qvd1cJ7cFCexsWJvksAaJZLZkc9GZepZQRg+ckor2fMgK+y1K0piGBLna0uehNL/JlNby5DROGyrWc57X/Cep0/For/kj1YOmwu0vQ9b0r1OB8yAIhhgbg+huSVlcfDHwNbQwdSPqm3YUZReIx5BwjkQW7PZ0rWnbQlklbCl6lN+Dsmda9sXkJIccWparirWfdBsTP3kVLdstYcTDE/nN626xZd/bm5+lY06pjJVZjSzcScg90lIZO5eHPBLY2wo7pFFdhMQbKWOOT7LCjvGa39OH6xcEQRPZGET3G5QRCHNQ0fMccnWXFXYoarN+CrImZyPx25rS9zqGMokgi2gbSjDIiROt6bXRlNZuW3p9DXDz269n/LlXscf41F5y26WM+NqvmbymhVN+sOfagubVeru50HlHmt9DyE+cY26zBU46v51QWFkjDqbM1JuEFt1CdINgiDHs43TTY/TBwPHA0RXBze8vRMVw5iLL9a/I4m1DYV9jWbdfWQuK6b0ZCWB2QYxGAvsYEsrsGvgvcM8nr2abLLiZ8Wvgk1fTSpkWnI9hyJpehnzEO1JmmT1M2dG4WsbxH42uQSp8/hQkyk9Lc7ut0bZBEAwuG4OlmzvlLqh/PSVN7IkiG0DuhP0p024n0dXyzOSEiT2Q2G1GWVZxBWX9hT+jBTAHnju+bW0kQxe2W7x2/Cy62W/7BHJVbIus9O2RD/oNKKxtK8rqZNc3aqOTOAXdJB5FtRb+A8y1wp6Piqb/MSVhBEEwyAxJ0bXCZiCrck5/awOkylpnpF/3RRbk1pSlEA1Zp09FIpoXz7KFmy3LxelrRNp+B7RItpSyDdCRyCfbCfDYBFZvvaxLOUcAHh/PKuTSgNJ6/QnqdzYFKNK4c4F/eM0vtcK+QNkMcyQ9MyZtMz3N7bmoutiR6fcFVtj+fWndEwTBhmXIuRessNcAf0KdCc6pb+PTy76twCXI1XAKqkswH4lrtVnkGFSD4QlyxTBt8yAK1VqS9huPHv8PBc5GoV8PpbFmIpHbClnE251zMHNW1N3GVrbSeeaRa/23uY/aUuC1XvMrkTjOTuN1Arkn3K1IMMci8T2r7lx3scJOTHWJL0I3iJZ0fgtRUfhcb2IzUu+0IAgGl0G1dNPj/0FIKG5KVu2HKUsWvgq1iXm0j0NOQX7RBUiwpiHRegJ1yv0OsnBXIgF9KYrjvQpVKDsA+C0q53g2uj5nec0XWGE/SO/NpGtR87WLXV/fn50XjoFzr5JL4YHJ8LHnsfriPddGTOS43g5gWytsNnJR7EeZyLGHFXYf8s0+nvab7TW/tXLd9kLdMPIi3kko2+5KdEMxFOO7deWYkfIbBEOAweyR1oJE8MD00ndR54dFaAU/P1qvTNtPQW3GFwLXdFPwfAGyEJ+FrNA1KAJhLqplsBCFVm2OFp7O9ppfkvb9Y91YR1fmOgPVps01eyfT9SkhW+MtP9oDfrRHl3FyllveJkcl7IpKMD6z8vrENL+r0GfzKGU9iSqnoHTmVegGdaTX/Fwr7IXAIajQ+SrKAua/Aq4gCIJBZzDb9WyLyg1mv+k4FFOb6+AuQgK0AgnnqWixCOBbXvPzuhl3HGoDtDtyETwNie1yJOCj0OP9lkiMLvGaX93DPPdH3SPGIiu3vhDO2k1ZtxtEzlAbQdeYXktzMrqmGTvKYHs4nWtOZf5SblKZzu9OVOOBNJ9Xec1/2WDuo4Ax9REdQRAMHk2zdK2wCciv2IlCn6Yga2xSmsf/gI8gK/LO9PpByMIci1wFdyMRehlwnhU2GhhZ7f3lNV9hhf0cWbJTKFN5c32FJcjS3QIVPn++FXaK1/yv3Uz9VPSInxfIVqVzyO4CQ4VwNqNsRlmlmuwAXQvojEM3gnHptZUokmFqOt73USTCtZXxcvPKByn7pTW0Yr3mOTU6CIIhQlNEN7kSLkbhW2ORyDyKBGUlWqz6ePrKIVKgcKx2ZJlOR6K2GrjBCjsc+XtHWmFf85qfXznkd9L3Z6UxxlD2PTuNMlRsCyTIe6P43UY8gAR1KhLYcWk+bcjnOgY1j/wHikaYkPar1tDN5Pq989N5jYO1Mbz/TfMZiazy8cDp6b2lqVPGSBSFcTVa3FuIkkGGf9WiINhEaIp7Iflj/4lEYhf0iH8H8mEe6zW/M223CxLnzZDr4W4kkvkR/W/ptS8i4cn1ZycCh6Q6B/XHnoHiXg9HPt89kZugapE+D1nXn0a+0q/kx/X0OP8IElNP+y2jTKpw4GNe809ZYU9PY+yLFu92zdNA4vptVLrx60hAb0Pivy9yEyyhtKBnIFHO8cELkdiDCvy0oxvQrfWia4XtgGJ+b/GaLyYIgiFDs0S3BS0ObY98qaDV9FHAoVWxTD28JiCf7gj0eL8L8DOv+d8r292Ytludvj/Xa75OlEMSzespY1m3oHycz7wX+YAPQZZlK3CU1/y+yrH2QuI/Kh0zW8+Oohr+gHzSfwXOQ0I7B1nopG1PQTcA0I3jSiS6RyPXx2/SsfcCfkrp0siVyHKN4F94zV9Xf65prs9FxdNBQn2013x+o22DYFPCjKmoMesz0NrROe40WpDfoDTFvZBaoZ+IFsYmoroAk4HP1lunXvN2JBYgC+9bNOY9wAVpvM82EtzEZCSyiyhr31YbRnaitN4XI1fHyjTmdBQFAKqJ+wtUcrE9bVMVxK2AdyDB3h2J6J2ojXqRjtuJfLRL0/iHIHfKp7zml1cnbIWtoezPlhM2RqTrMQHY2QoblXy29Zya9lmSzuFgtLAYBJs670XRUktQiOh16Im5qTRtIS2J4jkDON51VtgewIgk1N3xKIqFfQHyCS+mtI7HIOvyhyhK4jx0Tf5LpXaB1/xeK+wQFO2wDxLcHPplSNizb3Zn5HrYHIn03sC/kJ92CnKd5Pm+Dlm0/62b83xU0HwXJLaPpv0mp+M9Fd1wXt/gfGcjd8nY9Ps6LpcgGA4ky3R34D535q7HONuiJ9SplA1bQf9PjbYfi0I5n3Dnrid73G7nM5Q6RyRf5BlIFL/YQ62B/o7bCpyPoh4eQZbnBfVhZ1bYTkgsb/Gar7bCtkDC5ujueCby526N/MObocW0SZRVx2aixbdcVewKVEdhOfLPTkMf+jwk9MeiVutrgD95zTvSXLZG7ojVyH3xfeSPXk3ZNugZXvPcKy2fwziU2LE7akr5/fW4dEEw4CQRfBUygC52X9t0oLrN1uh/J6+lnOzOP3sYcxQych5xX7v2gRlvRO2pQC7OZ5FKDADHV7dN249B8e07IaPqI+785EmdaHdzHSqim2JK/4IeiQ2422v+wgEcf0/KLK4O1Ib95m62HYOE8JuodgOUIrsE3SH/jXxD85GQnodKKf4/5G4Axdxuh/5otktjPICs4aXIWt0VPfIYcLnX/L3dzGkcSo/OLpIH0eLh0PgAg6APmDEOLZJvif4vfu2+tlZKdbuTkfttAXpCvMKdd3cz5mSUYToTGTInuHO3GQbcRVk+dTJaUB8FzHFfN5zSjP1RWv1S9LT4uDvPW59zrmco1V6YggRlYfraJS3ADQhe89tQ2u9HgJfWC64VNsYKe70Vdjby9fwW3RUXI2HdDNV1cJS9dhqKwJiWtv8e+sBXIV9wro3bks7nIcpGmYtQ6Nc3KVuiLwZeboXtnYr21M9/BfBy5J+9DN00QnCDIY8Zu5lxrBkzkAU5DT3pLUbFmRrxCHIFjEX/Q3N6OMRRyAAag4y2NwO44+kYY9LXGmCeO/9tJLiJ3AJrXDr2gKfPDyVLtwVV3tobCdvvvOanV943VF9gH1Sq8O8NB9K2W6APc67X/MY+Hv+raDFtEnIL3Imy2Vaiu96NXvNXNdhvRMUl8Ay04LYSPRbdhJIdpiH3wnQUSkZ6bTdkvU5Bf1g5zbgVrbJ+zWu+ui/zD4KhiBlHolDJkWgtYxFKXOpEVuwdwNnu3Fi3n6Fu2Mei/6Nz0v5HI2PmN+48bsabUMftbSlban3DnXemcfYBPo8E9Cx3/tCHOb8QuTnnpn2e6GWXfjFkRBfWPkIfhXyjv6sukFlhx1OGYq1Bqa/r9AGzwqaheN7t00vnes2LBtvtiCpxPYAqmt2KxH4qci2sRkJ5GXJ7/DxZm/Xj7AHU0B/R+cAx6OYwG31wuS3QXMouFwBXec3fYoU9FXgfevTZD/1hzUKPQ39D0Q//qPfdBsFQwYxtUKGqMcDn3bkzvT4SPd5vT9diT/9Bceg5Nt2BD7nzs27GH4MWwt+FYtrzusYHkKBPRYZLR3rvTHcuGODTHDCGlOj2RKovewyyBKcCH/eaX9hgu6NRDF6uk7sa2KxaICclTPwWWbUdKKFhN+AlaezxSNhXo3jdhlZ1WqC7KW0/CsUA348s3WPRH9SM9PsnkK/3L2mfP9TdVGYh8d8K/fGuQg7/x1BCyHHVdOe+YIWNQL6zed2ElwXBemPG79E6RSf6m3+WO6vM2AO54vI6TeZhynZV85GV+ghwuDuP1I09Eq2f7EhZw8SRe+Ja5DLMNadXp3GOd+f2DXGuA8FQ8un2xrXog5uOrMGGi2DIr5o7+maruL4mb242OY8krMD70ar/PUg4/4UEb1IPcxqb3l+KPvSR6fhTUUzyn9FK6PXAc5AAvxq4rT7MzWs+BzWWXEZZd9fQH/H2lBZyn7DCJgG/Ru6LP1lh2/Vn/yDoDrMy1DS5AXZCboNF6P/hZDM+g1LwGxVbyusgbai41WYoPPIqM8ancbc047foCXEnJKxQVuybRlkmYGQa68fAKUNZcGEYiW5Ky30LyuQ60Wu+zoW1wg5AvtBlyC+6DPhw9rlWyM0dn4I6QrQAbV7z7wHvRh9kbnTZXU0GvOZLkfthImV6cG6vvg9lEfRc62EF5R9Mo/EuRdloP0NCviifGrpB9IeXIJ/0YmTtvrGf+wcbMWbsZ8bJZuzUj31GmnEB8D8zbjbjS6gG9eUojHJrtAD1NeQKOBuJZQddCz49gSzSH6H/i470+vbAYWm7d6N49MWwtlP32qmk37dM3x8H/uLOm9zpdq1nqDAk2/V0h9f8KhRr1x1vR+d0L7I2P+w1/3GDcWZbYdegCmbzUEzrc5FlejPyQ+2HhK+3zhXvR1ENLcj98dz0u6EV1ZGU5SsnoaSL//Rwjo8Dr7PCpiK3x9OAC7sLb+uB1XRtbBk+4Y0QM3ZDgfy3uzd++kthWq9HN/uLkLHxVfQ3udqMlyCr8Rno73grlKL+gxQBkHkBcETab29kIJyGjJMOZHCMpnQBTEMLyh2U/0fz0P/VLpT/B9nfa8DnzLibMmN0OaU1m5MaoHyaHYmeOD/R68UaIgwr0e0Dj6FzGoU+6J7CPVrQHXcxcsLnGgknoCSER5C1ejJaAGtI8hXflH69Ib+eHu93TGN9Hd3VN0MFatpTtMa47vy0qdnmm3uYf29cjhYKj0DFhv5vPcYKhiBma5+KRgDtZpzq3vDJ7LNocdfRWsON6P9kcySS11AWZdoMxYAfADxsxg0oqejB9L5T1rXOHVF2Qm651ZRZk7mvX06Xz2nsOXxrZ8pIpTXp/cfTMQ5EN4XnoEXlNSiaaEoafxUyqkAi/gN3/tbHyzbobGyi+0kknrugGNiewsW+jT7UCcixf016fVTddr01hWxIKhx+aqP3UsTCxcB0K+xK4B0NXCDrRVo4e6sVZhHPOzxIi0YjGmVoNdh2BBKm7ZEluADdYBuJ7gHIt9qO/t7nInHNgjgDPd7nAvu5W/bT0c07lyu9HT2prRNHjsSxhXXrmmQrNqffLkdJQW9F8euT0z6jKvvc6c5sM56D3AynUfp0r0znMw6J740oqmnYsFGJbrIOT4G1GW4npOLpl3nN59Vte6MVdhjKFLvTa748vfVj4Dj0RzQb1WUYaD6E7tQLUcfe59K1UPmAEYI7PDDj+UhER5nxOffun64SB1CGFrYiK3CdEMrE7WgBdwRaV9gGWaa7UVqvIygFcgJ6CtwXrUtkdk/HXEoZsZOFclRlnLXtq9L37BYYibI0b0RPcR0oeuFWJOajgF+58w8Ad9rN+C6y0megBKP3pnMwd4Zl2dJhEzLWX6ywr6EPCxSLe0Rfw6ZSqNVUZD0cih6Dfu41H5DsFCvs/6HqX4uRn/cNXvM/p/d2S+/9x2t+TfejBBsTZtxMuQg7HjjInW5rj5jxTOSyyhE9twOH1PlgMWM/4JfIlVAVwVa6Fm3yyvcVqM7IdshA6I4spqvpKs5VliCfaxb8U1FI52tQpcCpKMLmne4s6OZcR6FFs8d6yCQbNgyb6IUnwfPQ6v989IFt29cdveYdXvMn0MLcb1DDzNuTW2Ag+BTyRW2GHpeuA7DCnoaqjr0f+JYVduwAHS8YHkxGK/bb0biCXJV/okWxFiS4pzYQ3FNQlueWdG2Q2koZAfAIXReoQC6AAi2s9VRvtgWJdLaW62lHpVkfQSGYjwKHpnn+Fwn1IrSI1624u9PmztyNQXBh4xbdGynrOSyGrkHXfeRt6I+zHfm/Gvpo+4vX/B4Ud7ur1/wdlcSNZyJrZx76Yz8CINVj+JMVdpMV9qKGgwZDDjNGmTHajKlmPNtsbSGkRpyJFqgMidMbzdaNETfj6akoy0zgS8htcANwqBlHmHG9GX8z4xjkC13OuoJYjcgZi54Eq2yBLONjKIW1kahm63glEs9qaNgaFNVwCPK/jk9j5UXnvBDWlrbt6dpsVGxUPt06TkfWwiTg+9U02lTH4QyUzXITUHjNGy1e3I/iePN1qq972y1W2H5ooe4Or3l9e/dc2H2HVP3sFq/53SgPvQNZwMDamMMLKBtSnm+FXRcdfoc2ZhyHFnjyotJqoMOME9z5V4Nd/oL8qCPR6v4U4O1mXIP8r+NQofwzSNEKlBbrYiR82YXQgpJyqotjjcidT3asvGZ1P2fBrbog8r5rkH83F4Z6LnrCnIuMneORqLchn+0vgB+k/f+MCkPNSuN8tZs5bnRstKKbFsa+0s3bR6Dg7ZUoROxRZDVkn+orkeC+BiU/7Ji+X9jbcZOgfwp4J/rHmG+FvcNr/qu67fZFi3QjgDVW2Ku85rdZYa8DXogWF3LHh0nonzaH3IylcaZPMARIkQWfRn9fY5EVdzu6mZ4AfDRlcp2KYl//iqzLSWgRKz9dfRx9zh1o0XUWpYiOpmQKuilXn1zr/7ergpmtUaO0OLs9nfQ9C3QHcC7yz7YC33Und235efrCjM+l7Vch0b3cnYvMONCMI9L1yAk8j/bkv97YGJaimyITOtYjzGor9Me0Cv1TzEzjzkDRC+PS+9t4zQ+ywjZH1Y2Os8J+6TWv94FVORhZ2K2UCRGHoq4TVY5EroT5aCHkMJQe/DdYJ+bwU8DH0s8/8po/RjAcmIz8qa0oYmAV5aP8C1CRmHaUiDMBZTZmRlbG6EQ32xxhUG+5WoPX6ql/v4OuzVm7oy3NJd9ARgOvAI6or5NQx0XIeJiObjanmrEaJTFkH/BYd9ZJXtrYGXY+XSvsNBQofbsVdvCTHOYPyG+aG1tenF5/CvqDWIDu5M+ywsYii/NjwOfoveXQFpSt1nOQ+F8abPfvtE0uunN3g22wwsaj9Mpr0cLexxptFzQPM8yMF5vxKTOOqn/fnQ4kqFsjYetE4jsN2NWMFmS1jqcUvs3qx6nQQmkI9Cau3VH1yRplNEF3vtpO9Hc5F1nZ49I81qDC+3earbXY1x3AuQ3dWNagpKUZwGfScRek8Z/zJM9lWDOsLF0rbCsUZrIUWYmfA/bv7zhe80essCPR6uzsSnPMO9EdfTr6g/st8ulORxZpK/pD+mAPw1+NrJkOZBl8EoXs1HMFZYPKq1EUQyPORQsanajg+c2w6TyKDSapsMvY+pYuwEnIB+lowetS5E/dBfnhv4EWxkbQdSHKURjj99B6whaUN+ncELVKvVCu1+lUfs6uAlBi0DZ1xzT095tDy76HznkzSkNtNYqJvwwZEI1YkMbKWZfj0pjT0/emN4UcCgyrON3UN+zPlIVlVnrNnznAx5iJHoseQuFi4ykLjRsqydhjeq4VNhFZA3Prux0/iflchx4xczPNk73m/1ifMYPeMWNPJDYT0aLUh1BoUwsK65tKVx9pddGpnXVdAY78s0uRkFVX6x2t/jeydqu+2PUlx+XmRTBDi1kr0N9rDiO7Ez397Z3O4Rxktf+YMoPsf8jV8HJ37ujugGbUUCq9oz5/V5Di0IGr6sPcNgWGlegCWGHvQItUbcDbmpFAkBpWvhbdub9dyV7b4Fhh70WhPzmU6AX9rasb9Ex6RJ4GLHVndUoo+DJl9+hJKC52X2StbU33Qlhd7a9f8QeFcGX/bHUfQzf6qtU50MxB5zIxHXN+ms9xKNpnZ+SLvQhZoTORQC9HGXDtqJ7HF9B1+DHw0Z6EM13bXdKvd22KIlvPsBNdIHeYWLMpFOZOhXEOR49kv69PZw76ToqTPRMJxhdT88IRqCDR85G4/BLdYKcii/Qu9IQxAd3oJyFrzxn4NZGBsGqrkQmZ7Cr4N7Ja90augTbgd8Dbq2KYfM73IOu7Ez3lPStHGKQaEWMauF2CPjAsRTcIngxm/Ao9Rq/tcICs1wuRRTsdWZqdlC6ChSi29DS6proO5GP/QNCO1hJGo0Wr7EboROL6K+BEd2WYmbEjuvncmV+rYsY5qE06aCH4DWGlDgwhusEmgxn/o+zyMRUF778f+fAXIFEdkzdHC5YXopCuIxgckc2+12qPsVwvIb+fU3FXo0f+w9ACXSsS4x8Bb3Onz01Ok7V7IFo7+as77b3sEvSREN1goyI1STR3HjRjJrJkF6FogWehUK2pyPpbjUQ2x1S3U1qIrZXvg0m2VPPNYE16bSGa/wxKV8cytECVayY4Srx4SSNrNhgchl2cbhB0hxlvR/HM15jxd7Rw9HPU8POVKA57OhLYscg3m0XXkFU3In1vSa9zIj9gNrPooIXZzOLEtZmsG5wc611NUR+BRHhF+lqOrN/HkHth1/T7uLTtu0JwhxZh6QbDnvQo/CzUyimvymcLECRcq5FboV+Gxon8gG/xZsazYu1ryxnHm/gmP+KkAZh9r6xAIjqh8toTKPRqaxRl8RZUG9fSdo+mfSYDh7kztxkTDfpGiG4wbDFjBoql3RktIE2j7FaQ3QK9+GF/AHwEGYkzUS5LKaazmcUs7l9nrzlszw7MWb8TWJecmpvdHC3Iqs3xs3nx7q/uHJp3MmM0CulahJIv3pC2vQF1xw1LdwgRohsMCczYG5W2vA2JxzuRlfc5lC56FloEOwNVY9sMJS08BwlTC/Jz5lCvaquYbvgBamCwovLaONTpScLbQQstDRbtOzFGDIyWdSL3QQuyUMekCc2iTMVdharhHYB6lb3MnbsaDZbiYnM88d9jAWzoEaIbNB0z3oTKFD6GQrEmoRX23J9uOhLOlchim4CiC5ahiIIVSGyrlbYaxadmVezGpTALGlixajs2B9jglm4O6XoM+Z3fiBIXDkM3ma2Q1bsKeLY7EaO9ERCiGzQNM8aiHldnIbFZgIrNX4tCt+ahlfeeLNSc8dUf32w3MbX5iX2dmZL1uh8+3RXIuuzLXLLfuRP5mo8DbkuFcjQDYxfgfJSYcI47v+7D2MEwIEQ3aApmPBXFkOZmiDm+9O/p58NoehzsLHqzdEHC+yk+wkwe4AFm8mE+WS+4jmojr0F+iRGUtWc7KMO9QNb7CFTzwFHq8QEDdELBMCBEN1hvzJgKtOTH3+RXPAs4EbgXrbBfSOk2qBaBOQP4LH2zEgeY3n26dTSymHMjx52Re+Q3yDodgSzVFagc5yjkez4fCe9p6b3T3Ll+IM4mGB6E6AZYYU9BizR3ISFY4jV/sMs2xjPQMn8H8Al3tS4y4zWUorIAta3/Kep6sBKFNY2m7AibRasDxZh+Hgn0ICUh9Bi9UC+yj1HG+YLOYQXwIvfUXFQW/aHoOvzRHTdjM2AvYI47s9N2LYBHau2mR4juJk7qcPwL9Ag8DUUIrAA+6DW/FNYWOLkB1jZKfALFxRoqvr4StV0ZhR6xVyIBnkZZ/LpewDxt+3DarhqHOphkt8AIdFMYR5mk8Cia81R0E3Hg1+4cB+RW4dsCD7t3SWgIgrUMdopjMPgcRBmmtC0SzDZUjevStM2ulPVWFyFBGokEKKfS5gaMnaiGwWjKaARo3J+rFZmXQ6VwTK43m9vJjEcRE39CiRe5S3MN3Zxy1hdmbIGy3zYHFprxcnfWq5ZysHESacDBXZRpo6AQpTHQJTzp7UiIRyBRecidNvQc/gSy+rI1mFNoq4JbpdpyZn3az/SX7h7pctvw/JXDzPK8xgIfceeL7lyMfLYLkNXfStnq6eXAdkikt6Ibp3AQhKW7CZO6ZCxEiQgvApxVEw9l5bTV3HvEJwHM2B/VLKiyixkfQRaf0Vzx7AtrWLeBY25B0ygc7QG0+DWeMhwti/RSVEQGAHcWmfFi1EXiQXduSW8tp2soWxSaDxoSPt1NDCusFVmhx6PFrjGoluxD3Paa27js+y9B/tVVaIHso8hVsC1lJ4QOlBk1E1nGoxgaopv/mLP1mnuU5fdyMe/RlOcCqq17FvAm5F+eXtnnVncO7O3AZoxBkQmHom7Ob3fvEhYRBEBYupsUVtgBwLeRqE5Hj86gTrUT2P7qAxi1eARtk3Ou/zeQxTaGri6BVvQo3ULXrLANSf1CXC5duDDNZwKlFfsAevz/O3AUZUrwQ8C7UWrxrLRtjqXdC3guSqH9CrJ6O4FL+jQ5LZyd9iTOK9jECNHdtDgXfebjqI8W6LStaB/ndI7Kgf1ZXMfWD5Lopa7BgJIbN06hFN42FE1wFuraXKPsdLB5mt9TUMbbrLT9O925woy7kW92BmVtg4XuLAauNuPlwEtQ2NfPN+ypBZsa4V7YSLDCTkHCsxB4o9f8lsp7M5DF9w3KG23XRdROcxbs3MZX7wRaB9J6zREBjehLyxtHgnkTEs+FyMqeA3zcncvyhmZMQlEXb0ahaKPT9zOB+e48UNl2J+ATlEV23ubOE/07tSDoPyG6QwwrbBKwJ2rfPqeP++yEKm/lNi73AOchgWoHvkvXlt9dcWDZlp38+DLnoYOyQA6Uj3Ypsqr7M162Pkcif/NKVMJxCrp53AK8xZ1F9Tum5IRfo4Wt8cC/chxtEAwFwr0whLDCNkPdaLcA3Ap7C0qj/QB6zP+41/yBBrs+m7IfVitKVPgWEuF2evO7zn8K/OmjxsP7VBedeiKHh/VFSNf0MMYaJKyZXAhmNooYuBvVZPgX8Bn33iMC3PmvGZ9HVczmomsXBEOGEN2hxXOQRboYlfh7E7AHsEt6/1grbKbXvH5VfAGyDkdTxprm8KXqIlhj5hzs3H+IYS29y61oYd0GiZnOymsdSFTrF8A6kLBOpLTA16AbzBHAT1BTxH2BV7tzR59mlXDnAuCC/uwTBM0iRHdo8Wj6PhaJ1cPAsUikOlH7lUOtsD3RotGdyF95NnqcnojEK/f6qoZFNSrUotd2vdS44yRYNRXaRgAtRuNY1+q+KyhFfhRlWm81FtYp02XzzWAhqsj1TeBLyBJ3ZNW+EzgECfECVKj8VOA9vVy3IBg2hOgOMimM6yTkh70Axc6ejAT1E0h0d0CCtQKtuL8N+St3RAKVa9C2pq9HURgYrLuIlS3UNvLnP3KlsfU/VtE+ZixLtmlj2cwcezsPuTqy2yKLaRbieUh4l1K6EXLd1xlp373RDWQkCtnaq+KLPWKd62HMohRyg0ilDTYuYiFtELHCdgCuRMI1CRWVeZPX/CErbBzyZ45DjQe3RnGnR6BEBWhczwAkqIuQUOeqWNVMq+x+WAMs4c8fnMp1HxpBx0ijY3QHtDxBmQo8G6X6PhslDoxFQtiOrNbpKK71PhRRcBfwZ8q2ORNRtwdQv64berwmqr71PtQy/Z/AByPJINiYCNEdRKyww1GywjaUwngHEtvvIgsW1MrmSspA/W27jrROemvOvqp/ksmZWvcj8VwGnMJnHv0mK6bvCi0ONiK9fzxwNBLqa1C21fj03ldQScaRqODNq9y5FcCMmcAfUZZXC6Vv+r5qyFYQbKqEe2FwuZVKpSokiFugzKinU1b0eg1yQWxJ46SEahPGamJDlWpdgJxNNhk4iRVb3oiqiOUxWlD7nP3Tfkelr/8DdkcFaD+EkhCuz4KbeBhZvU9N+/7SnWt7uxBBsKkQVcYGEa/5PBTa1U7ZZnsa8EoUk7oVEsa8sJW7xlbpRI/191K2QMgW7dpDpa92tOCW/aWPAc9HnRtWp/3a0nYHIvdBXtB6EWq1k10KB7vzjUrBFx3IWZPm/2GUcntmvy9MEGzEhKW7AbDCRgAFejxvB/4KfMVrfm+DzT8JnIKErBMJ4ivTzytRTduF0G2A/wrgepSN9jLgdUiot6b8fBsVqQG5NS5x524zbkWWbQuyYC9FkQSGLNdc6jHH53bb2tudJagfWhAEdYSlu2E4BvUHm4UKqZwMXGKFrVPHIMXc/pKuq/Q5RGwpigY4G/gVpcVaZRXwOa/5bK/555Ev9k6UXLCM0nIFiWkLigNeiKIczkjvTUaW74L03nkoseDsNOaVwO/QAtls5NMNgqCfhKW7YZhMmZjQiYRvEhK9RotJH0SC93Jkiea6riuAm73mS4F3WGEL07a5QPgy4Fte89vzQF7zu4GjksDPpoyl7UjbL0F+44eBU9PYoJC1PdIx24BH3Lmrbp6nm9GaXAhBEDwJwtIdYFKo133IilyeXs7ZVg1jTr3mS9AiVRa85Siq4Riv+ezKpl9GkQTL0jY3AF/sZsyVKLHgcSToS5C74jGUHvt6r/k1lV1OQ5bsP4FTU8WtdccNwQ2C9SJCxgYQK2wL4DK0GLYKFQAfn96+Molro/12RaUGt0ECvQL4f17zdRahrDBD1msn0O61nj/A1B1iWyTm/4cW6H4LvMNr3tHPUwyCYD0J98LAcjTKxJqPfJ/7ec1rPe1ghe2LLMwp6MkjN308oNH2SWT73Gk2Fch5IB3rAGBcd+IfBMGGZ5MQXSusBfkxF3jN257kGEYqRtOg4ExmEfKf5iIv87rZrsrHkdjmWNvsyx3Z005PBq+5MtCCIBg0NnqfbqpPewXwF+BPVth2T2KMVuA7KFvsUSvss0nI67kcFXNpR+6C7/Rh+FwEZiWpsi3yw17a005BEAxPNgVL92iU3bUAZXS9GbV46Q8HoOSAiciCfRtwHRLZtSQf6UfTV1/5CEr5NdTZ4Q5kIf+hn3MMgmAYMOxFNz32t3rNuwvWz+6EHNS/Ku03BvUMOwj4PVDkhaW0IHY+sDOyVm9G4V/5kX8EsA91otuPOW+J6hfsCPw/YL90Dk/K9REEwfBhWItualPzfWCGFfZT4INe8866zX4JHAkcjmodfCO9fioqm7gUJS9sZoVNRW6I3ZEYL0NprK9CyQYHUhbhXp+V/7NRb65lKFX2eq/5zesxXhAEw4RhLbrAx5DLYAFKk/0F8LfqBsl6fLMVZnXhVVuk720orOsEFHVwICpl2J7eG4MiC76OIhPyGP9Yj3lvXjf+5PUYKwiCYcRwX0hrpWtqbLc3kQbxrD9E6bDjUbTAKmR5GvKrdiIf7t2oju0PUaeDfwKfQmmxT5YvIEt5EkrZ/ft6jBUEwTBiWCdHpKSCi5Al+nvgjP4E/KdGkLOQ0F6EfLYdqIzigyhE7K4N4Wu1wqYja/t/PfijgyDYyBjWogtghY1E1uri3rKzehlnFipO8y+v+T0DM7sgCIKuDHvRDYIgGE4Md59uEATBsCJENwiCoImE6AZBEDSREN0gCIImEqIbBEHQREJ0gyAImkiIbhAEQRMJ0Q2CIGgiIbpBEARNJEQ3CIKgiYToBkEQNJEQ3SAIgiYSohsEQdBEQnSDIAiaSIhuEARBEwnRDYIgaCIhukEQBE0kRDcIgqCJhOgGQRA0kRDdIAiCJhKiGwRB0ERCdIMgCJpIiG4QBEETCdENgiBoIiG6QRAETSRENwiCoImE6AZBEDSREN0gCIImEqIbBEHQREJ0gyAImkiIbhAEQRMJ0Q2CIGgiIbpBEARNJEQ3CIKgiYToBkEQNJEQ3SAIgiYSohsEQdBEQnSDIAiaSIhuEARBEwnRDYIgaCIhukEQBE0kRDcIgqCJhOgGQRA0kRDdIAiCJhKiGwRB0ET+PwSe0ZHmDcb5AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import sklearn.cluster\n",
    "kmeans = sklearn.cluster.KMeans(init = 'k-means++', n_clusters = 2, n_init = 100)\n",
    "y_pred_kmeans = kmeans.fit_predict(X)\n",
    "colors = ['bg'[p] for p in y_pred_kmeans]\n",
    "plt.title('Clustering via $k$-means\\n$k$-means centers: red, GMM centers: black')\n",
    "plt.axis('off')\n",
    "plt.axis('equal')\n",
    "plt.scatter(X[:, 0], X[:, 1], color = colors, s = 10, alpha = 0.8)\n",
    "plt.plot(kmeans.cluster_centers_[:,0], kmeans.cluster_centers_[:,1], 'ro')\n",
    "plt.plot(gmm.means_[:,0], gmm.means_[:,1], 'ko');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "cell_style": "split",
    "hide_input": true,
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cluster 0:\n",
      " center: [0.28341737 0.06741478]\n",
      "\n",
      "Cluster 1:\n",
      " center: [-3.88381453  1.56532945]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "for clus in range(2):\n",
    "    print(f'Cluster {clus}:')\n",
    "    print(f' center: {kmeans.cluster_centers_[clus]}\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Now, let's construct __overlapping__ clusters.  What will happen?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "X = np.r_[np.random.randn(n_samples, 2) @ C,\n",
    "          0.7 * np.random.randn(n_samples, 2) ]\n",
    "gmm = sklearn.mixture.GaussianMixture(n_components=2, covariance_type='full')\n",
    "y_pred_over = gmm.fit_predict(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "cell_style": "split",
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "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": [
    "colors = ['bgrky'[p] for p in y_pred_over]\n",
    "plt.title('GMM for overlapping clusters\\nNote they have nearly the same center')\n",
    "plt.scatter(X[:, 0], X[:, 1], color = colors, s = 10, alpha = 0.8)\n",
    "plt.axis('equal')\n",
    "plt.axis('off')\n",
    "plt.plot(gmm.means_[:,0], gmm.means_[:,1], 'ro');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "cell_style": "split",
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cluster 0:\n",
      " weight: 0.507\n",
      " mean: [0.00881243 0.00217253]\n",
      "\n",
      "Cluster 1:\n",
      " weight: 0.493\n",
      " mean: [ 0.00346227 -0.00250677]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "for clus in range(2):\n",
    "    print(f'Cluster {clus}:')\n",
    "    print(f' weight: {gmm.weights_[clus]:0.3f}')\n",
    "    print(f' mean: {gmm.means_[clus]}\\n')\n",
    "    # print(f' cov: \\n{gmm.covariances_[clus]}\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## How many parameters are estimated?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Most of the parameters in the model are contained in the covariance matrices.\n",
    "\n",
    "In the most general case, for $k$ clusters of points in $n$ dimensions, there are $k$ covariance matrices each of size $n \\times n$.   \n",
    "\n",
    "So we need $kn^2$ parameters to specify this model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "It can happen that you may not have enough data to estimate so many parameters.\n",
    "\n",
    "Also, it can happen that you believe that clusters should have some constraints on their shapes.\n",
    "\n",
    "Here is where the GMM assumptions become __really__ useful."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Let's say you believe all the clusters should have the same shape, but the shape can be arbitrary. \n",
    "\n",
    "Then you only need to estimate __one__ covariance matrix - just $n^2$ parameters.\n",
    "\n",
    "This is specified by the GMM parameter `covariance_type='tied'`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "X = np.r_[np.dot(np.random.randn(n_samples, 2), C),\n",
    "          0.7 * np.random.randn(n_samples, 2) ]\n",
    "gmm = sklearn.mixture.GaussianMixture(n_components=2, covariance_type='tied')\n",
    "y_pred = gmm.fit_predict(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "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": [
    "colors = ['bgrky'[p] for p in y_pred]\n",
    "plt.scatter(X[:, 0], X[:, 1], color=colors, s=10, alpha=0.8)\n",
    "plt.title('Covariance type = tied')\n",
    "plt.axis('equal')\n",
    "plt.axis('off')\n",
    "plt.plot(gmm.means_[:,0],gmm.means_[:,1], 'ok');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Perhaps you believe in even more restricted shapes: all clusters should have their axes aligned with the coordinate axes.\n",
    "\n",
    "That is, clusters are not skewed.\n",
    "\n",
    "\n",
    "Then you only need to estimate the diagonals of the covariance matrices - just $kn$ parameters.\n",
    "\n",
    "This is specified by the GMM parameter `covariance_type='diag'`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "X = np.r_[np.dot(np.random.randn(n_samples, 2), C),\n",
    "          0.7 * np.random.randn(n_samples, 2)]\n",
    "gmm = sklearn.mixture.GaussianMixture(n_components=4, covariance_type='diag')\n",
    "y_pred = gmm.fit_predict(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "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": [
    "colors = ['bgrky'[p] for p in y_pred]\n",
    "plt.scatter(X[:, 0], X[:, 1], color = colors, s = 10, alpha = 0.8)\n",
    "plt.axis('equal')\n",
    "plt.axis('off')\n",
    "plt.plot(gmm.means_[:,0], gmm.means_[:,1], 'oc');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Finally, if you believe that all clusters should be round, then you only need to estimate the $k$ variances.  \n",
    "\n",
    "This is specified by the GMM parameter `covariance_type='spherical'`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.r_[np.dot(np.random.randn(n_samples, 2), C),\n",
    "          0.7 * np.random.randn(n_samples, 2)]\n",
    "gmm = sklearn.mixture.GaussianMixture(n_components=2, covariance_type='spherical')\n",
    "y_pred = gmm.fit_predict(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "hide_input": true,
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "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": [
    "colors = ['bgrky'[p] for p in y_pred]\n",
    "plt.scatter(X[:, 0], X[:, 1], color = colors, s = 10, alpha = 0.8)\n",
    "plt.axis('equal')\n",
    "plt.axis('off')\n",
    "plt.plot(gmm.means_[:,0], gmm.means_[:,1], 'oc');"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.10"
  },
  "rise": {
   "scroll": true,
   "theme": "beige",
   "transition": "fade"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}