Regression.ipynb

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    "# Regression and Goodness of the fit\n",
    "\n",
    "\n",
    "**Course: \"Introduction to data analysis methods for HEP\"**\n",
    "\n",
    "*Author: Jose A. Hernando*, January 2017\n",
    "\n",
    "*Particle Physics Deparment. Universidade de Santiago de Compostela, Spain.*"
   ]
  },
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   "source": [
    "These is a series of lectures on Data Analysis for High Energy Physics. Emphasis is put on Hypothesis testing and Confidence Intervals. They are based on the excellent lectures on statistics by Prosper [1], Cowan [2] and Cranmer[3] given at CERN Academic Training.\n",
    "\n",
    "The most common data analysis packages in HEP are ROOT, RooFit and TMVA [9]. But we will use the Python scientic toolkits, Matplotlib, Numpy, Scipy, Scikit-learn [10] that come with the Anaconda Python distribution [10]. \n",
    "\n",
    "Structure of the course:\n",
    "* General Concepts: probability density functions and likelihoods. Frequentist and bayesian statistics\n",
    "* Hypothesis testings: test statistic. Continuous hypothesis. CLs method.\n",
    "* Confidence Interval: Classical and F.C. intervals.\n",
    "* MVA: Classification and Regression\n",
    "* Regression: Likelihood estimation and goodness of the fit.\n",
    "\n",
    "\n",
    "Structure of the chapter:\n",
    "* Introduction\n",
    "* Estimator of the mean and variance\n",
    "* Maximun Likelihood\n",
    "* Least Squares\n",
    "* Goodness of the fit: chi squares, kolmogoriv-smirnov test\n",
    "* Goodness of the fit, mutivariate case: nearest neighbors, energy and kernel methods"
   ]
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   "source": [
    "# general imports\n",
    "%matplotlib inline\n",
    "\n",
    "# numpy and matplotlib\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.stats as stats\n",
    "\n",
    "# possible styles: ggplot (simplicity), bmh (scientify data), \n",
    "matplotlib.style.use('ggplot')\n",
    "\n",
    "# ploting images\n",
    "from IPython.display import display, Image"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " Last Execution  Mon Feb 13 20:51:50 2017\n"
     ]
    }
   ],
   "source": [
    "import time\n",
    "\n",
    "print ' Last Execution ', time.asctime()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction\n",
    "\n",
    "**regression** is the task of estimating the parameters of a model using data.\n",
    "The most common regression methods are the Least Squares (LS) and the Maximum Likelihood (ML).\n",
    "\n",
    "**goodness of the fit** is the task to quantify the agreement between a dataset and a model or two datasets. A typical example is to check if the model obtained by regression is consistent with the original dataset. Another example is to check if two datasets are consistent. The most common goodness of the fit used are the $\\chi^2$ and the Kolmogorov-Smirnov. The later is only valid in 1D problems. Extending goodness of the fit to multivariate datasets is not simple, and not settled issue.\n",
    "\n",
    "\n",
    "## Estimators\n",
    "\n",
    "An estimator is a method to estimate the value of a parameter, $\\hat{\\mu}$, using the data $x$, where the true parameter is $\\mu$. An estimator is optimal if:\n",
    "\n",
    "* there is not bias in the average of $\\hat{\\mu}$, that is,  $ E[\\hat{\\mu}]-\\mu = 0,$. The mean of the distribution of $\\hat{\\mu}$ is the true value, $\\mu$. \n",
    "* it has the smallest variance: $V[\\hat{\\theta}]$. \n",
    "\n",
    "### Estimators of the mean and the variance\n",
    "\n",
    "\n",
    "The estimator of the mean of a dataset with $n$ samples of a measurement $x$, distributed with a given pdf, $f(x |\\mu)$, is:\n",
    "$$\n",
    "\\hat{\\mu} = \\frac{1}{n} \\sum_{i=1}^n x_i\n",
    "$$\n",
    "and its variance\n",
    "$$\n",
    "V[\\hat{\\mu}] = \\frac{\\sigma^2}{n}\n",
    "$$\n",
    "were $\\sigma^2$ is the variance of the x, $V[x]$.\n",
    "\n",
    "The estimator of the variance, $\\sigma^2$, $V[x]$, of $n$ samples of $x$ measurements, distributed with a pdf $f(x | \\mu)$ is:\n",
    "$$\n",
    "\\hat{\\sigma}^2 = \\frac{1}{n-1} \\sum_{i=1}^n (x_i - \\hat{\\mu})^2\n",
    "$$\n",
    "and its variance:\n",
    "$$\n",
    "V[\\hat{\\sigma}^2] = \\frac{1}{n} \\left( \\mu_4 - \\frac{n-3}{n-1} \\mu_2 \\right),\n",
    "$$\n",
    "where:\n",
    "$$\n",
    "\\mu_n = \\int (x-\\mu)^k \\, f(x |\\mu) \\mathrm{d}x\n",
    "$$\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example** Generate $m$ datasets, each one of $n$ samples of $x$, expected to be normal distributed. Obtain the distribution of its mean and variance. Check that the estimators are not biased. Verify that the estimator of the variance is not bias using a small value of $n$ samples."
   ]
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   "execution_count": 3,
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " E[mu] =  1.00, mu-true  1.00\n",
      " E[var] =  4.01, sigma-true  2.00 \n"
     ]
    },
    {
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3a/Pmzdq+fbumT58u6ejZmVAopDlz5mjdunWRtgsWLDjp75k9e7Zmz56tnTt36j/+4z90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\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x103cbd3d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mu, sigma = 1., 2.\n",
    "m, n = 100000, 4\n",
    "xs = [stats.norm.rvs(mu, sigma, size=10) for i in range(m)]\n",
    "xhats = np.array([np.mean(xis) for xis in xs])\n",
    "\n",
    "def var(xs, xhat):\n",
    "    dx = (xs-xhat)\n",
    "    n = 1.*len(xs)\n",
    "    return np.sum(dx*dx)/(n-1)\n",
    "\n",
    "xsigs = np.array([var(xis, xihat) for xis, xihat in zip(xs, xhats)])\n",
    "print ' E[mu] = {:5.2f}, mu-true {:5.2f}'.format(np.mean(xhats), mu)\n",
    "print ' E[var] = {:5.2f}, sigma-true {:5.2f} '.format(np.mean(xsigs), sigma)\n",
    "\n",
    "fig, axs = plt.subplots(2, 1, figsize=(8, 6))\n",
    "\n",
    "ax0, ax1 = axs.ravel()\n",
    "\n",
    "ax0.hist(xhats, bins=50, normed=True, color='blue', alpha=0.5)\n",
    "ax0.set_xlabel(r'$\\hat{\\mu}$', fontsize=14)\n",
    "ax0.set_ylabel('pdf', fontsize=14)\n",
    "\n",
    "ax1.hist(xsigs, bins=50, normed=True, color='blue', alpha=0.5)\n",
    "ax1.set_xlabel(r'$\\hat{\\sigma}^2$', fontsize=14)\n",
    "ax1.set_ylabel('pdf', fontsize=14)\n",
    "\n",
    "fig.tight_layout()\n",
    "fig.savefig('figs/re_estimators.png');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Maximum likelihood estimator (MLE)\n",
    "\n",
    "The likelihood is the evaluation of the probability at the $x$ data. If $x$ is a list of independent measurements, the likelihood is the product of the individual likelihoods (also called **joint likelihood**).\n",
    "\n",
    "$$\n",
    "\\mathcal{L}(\\mu) = f(x | \\mu) = \\Pi_{i=1}^n f(x_i|\\mu)\n",
    "$$\n",
    "\n",
    "As this is a small number, we take the logarithm, and called it the log-likelihood function: \n",
    "\n",
    "$$\n",
    "\\log f(x | \\mu).\n",
    "$$\n",
    "\n",
    "Given a set of measurements $\\{x\\}$, the maximum likelihood estimator (MLE) maximizes the likelihood function. The MLE are very good, but not always optimal! Be aware that they can be bias!\n",
    "\n",
    "The second degree Taylor's expansion of the likelihood around $\\hat{\\mu}$ is:\n",
    "$$\n",
    "\\log \\mathcal{L} (\\mu) \\simeq \\log \\mathcal{L}(\\hat{\\mu}) + \\frac{\\partial \\log \\mathcal{L}(\\mu}{\\partial \\mu} (\\mu-\\hat{\\mu})+ \\frac{1}{2!}  \\frac{\\partial^2 \\log \\mathcal{L}(\\mu)}{\\partial \\mu^2} (\\mu-\\hat{\\mu})^2\n",
    "$$\n",
    "\n",
    "From this, we can associate the variance to:\n",
    "$$\n",
    "\\sigma^2_{\\hat{\\mu}} = - \\left[\\frac{\\partial^2 \\log \\mathcal{L}(\\mu)}{\\partial \\mu^2}\\right]^{-1}_{\\hat{\\mu}}\n",
    "$$\n",
    "\n",
    "Or: \n",
    "\n",
    "$$\n",
    "\\log \\mathcal{L} (\\hat{\\mu} \\pm \\sigma_{\\hat{\\mu}} )= \\log \\mathcal{L}(\\hat{\\mu}) - \\frac{1}{2}.\n",
    "$$\n",
    "\n",
    "If we use $-2 \\log \\mathcal{L}(\\mu)$, The standar desviation is now related with the increase of one unit in $-2 \\log \\mathcal{L}(\\mu)$ respect the value at the best estimate, $\\hat{\\mu}$:\n",
    "\n",
    "$$\n",
    "-2 \\log \\frac{\\mathcal{L}(\\hat{\\mu}\\pm \\sigma_{\\hat{\\mu}})}{ \\mathcal{L}(\\hat{\\mu})} = 1\n",
    "$$\n",
    "\n",
    "Be aware that LL is *not a criteria for the goodness of the fit!*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Consider the decay of a particle with a lifetime, $\\tau = 1$. Generate a dataset with the measurements $x_i, \\, i=1, \\dots, n$ of $n$ decays. Calculate $\\hat{\\tau}$ using the MLE. Use the scan in $\\log \\mathcal{L} (\\tau)$ to estimate the variance of $\\tau$. Generate $m$ datasets with $n$ decays each one, and obtain the distribution of $\\hat{\\tau}$, its mean and variance. \n",
    "\n",
    "First notice, that the MLE for an exponential is just the mean:\n",
    "\n",
    "$$\n",
    "f(x \\ | \\tau) = \\frac{1}{\\tau^n} \\Pi_{i=1}^n e^{-x_i/\\tau}\n",
    "$$\n",
    "\n",
    "$$\n",
    "-2 \\log \\mathcal{L}(\\tau) = 2 n \\log(\\tau) + 2 \\sum_{i=1}^n \\frac{x_i}{\\tau}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\hat{\\tau} = \\frac{1}{n} \\sum_{i=1}^n x_i\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "E[tau] =  0.99\n",
      "E[sigma_tau] =  0.39\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x115f627d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import scipy.optimize as optimize\n",
    "\n",
    "m, n = 10000, 10\n",
    "tau = 1.\n",
    "\n",
    "xs = [stats.expon.rvs(scale=tau, size=n) for i in range(m)]\n",
    "\n",
    "def ll(xs, tau):\n",
    "    return 2.*np.sum(xs)/tau + 2.*len(xs)*np.log(tau) \n",
    "\n",
    "def muhat(xs):\n",
    "    return np.mean(xs)\n",
    "\n",
    "def tmu(xs, mu):\n",
    "    mhat = muhat(xs)\n",
    "    return ll(xs, mu) - ll(xs, mhat)\n",
    "\n",
    "def fun_root(xs, delta=0.):\n",
    "    def fun(mu):\n",
    "        return tmu(xs, mu) - delta \n",
    "    return fun\n",
    "\n",
    "def sigmas(xs, nsize=3.):\n",
    "    muhat0 = muhat(xs)\n",
    "    xfun = fun_root(xs, delta=1.)\n",
    "    # xl = optimize.brentq(xfun, 0., muhat0)\n",
    "    xu = optimize.brentq(xfun, muhat0, nsize*muhat0)\n",
    "    return xu-muhat0\n",
    "\n",
    "\n",
    "taus = np.array([np.mean(xis) for xis in xs])\n",
    "sigs = np.array([sigmas(xis) for xis in xs])\n",
    "\n",
    "fig, axs = plt.subplots(2, 1, figsize=(8, 6))\n",
    "\n",
    "ax0, ax1 = axs.ravel()\n",
    "\n",
    "ax0.hist(taus, bins=50, normed=True, color='blue', alpha=0.5);\n",
    "ax0.set_xlabel(r'$\\tau$', fontsize=14)\n",
    "\n",
    "ax1.hist(sigs, bins=50, normed=True, color='blue', alpha=0.5);\n",
    "ax1.set_xlabel(r'$\\sigma_\\tau$', fontsize=15)\n",
    "\n",
    "print 'E[tau] = {:5.2f}'.format(np.mean(taus))\n",
    "print 'E[sigma_tau] = {:5.2f}'.format(np.mean(sigs))\n",
    "\n",
    "fig.tight_layout();\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Binned Maximum Likelihood\n",
    "\n",
    "When data is a histogram (with bins $x_i$ and contents $n_i$, with $i=1, \\dots, n$), the maximum likelihood is said to be binned. There is some loss of information passing from unbin to bin ML, but if the number of samples is very large, the binned ML reduces the computational time.\n",
    "\n",
    "In the case of the histogram:\n",
    "$$\n",
    "- 2 \\log \\mathcal{L}(\\mu) = -2 \\sum_{i=1}^n n_i \\log f(x_i | \\mu)\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example:** Fill a histogram with $n$ samples of a measurement $x$ distributed with an exponential with $\\tau=1$. Obtain $\\hat{\\tau}$ with a binned ML."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "best estimate tau 0.966\n"
     ]
    },
    {
     "data": {
      "image/png": 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6uH1dXV1wf6fTic1mw2az4XQ6u70e2Cfw3Ov10tzcTGJiIjabjcrKyi77TJ06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\n/QIx0OQsOavATgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113730490>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tau = 1.\n",
    "n = 1000000\n",
    "xss = stats.expon.rvs(size=n, scale=tau)\n",
    "\n",
    "ns, xedges = np.histogram(xss, bins=40, range=(0., 5.))\n",
    "xs = 0.5*(xedges[1:]+xedges[:-1])\n",
    "\n",
    "def ll(xtau, xs, ns):\n",
    "    return -2*np.sum(ns*np.log(stats.expon.pdf(xs, scale=xtau)))\n",
    "\n",
    "def llfun(xs, ns):\n",
    "    def fun(xtau):\n",
    "        return ll(xtau, xs, ns)\n",
    "    return fun\n",
    "\n",
    "def muhat(xtau0, xs, ns):\n",
    "    fun = llfun(xs, ns)\n",
    "    result = optimize.minimize(fun, x0=xtau0, method='Nelder-Mead')\n",
    "    return result.x[0]\n",
    "\n",
    "fig, axs = plt.subplots(2, 1, figsize=(6, 6))\n",
    "ax0, ax1 = axs.ravel()\n",
    "\n",
    "ax0.hist(xss, bins=40, normed=True, range=(0., 5.), color='blue', alpha=0.5);\n",
    "ax0.set_xlabel(r'x', fontsize=14)\n",
    "\n",
    "taus = np.linspace(0.2, 3., 100)\n",
    "lls = np.array([ll(itau, xs, ns) for itau in taus])\n",
    "ax1.plot(taus, lls, color='red', lw=2)\n",
    "ax1.set_xlabel(r'$\\tau$', fontsize=14)\n",
    "ax1.set_ylabel(r'$\\mathcal{L}(\\tau)$', fontsize=14)\n",
    "\n",
    "fig.tight_layout()\n",
    "\n",
    "tauhat = muhat(2., xs, ns)\n",
    "print 'best estimate tau {:5.3f}'.format(tauhat)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Extended Maximum Likelihood\n",
    "\n",
    "Consider the case of $n$ samples of $x$ measurements, distributed with a pdf $f(x | \\mu)$, and $n$ is itself poisson distributed $f(n | \\nu)$. The complete likelihood is:\n",
    "\n",
    "$$\n",
    "\\mathcal{L}(\\nu,\\mu) = \\frac{\\nu^n}{n!} e^{-\\nu} \\Pi_{i=1}^n f(x_i|\\mu)\n",
    "$$\n",
    "\n",
    "And it is called **Extended Maximum Likelihood function**. In the cases where $\\nu$ depends on $\\mu$, we must use EML. If $\\nu$ is independent of $\\mu$, then $\\hat{\\nu} = n$, and we recover the ML result.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Signal and background samples combined\n",
    "\n",
    "Consider two types of samples, background and signal, each one has a given pdf in one variable $x$, $f_b(x | \\mu), f_s(x | \\mu')$. We measure $n$ samples of $x$, we do not know how many are signal or background, but we expect them distributed with a poisson with means $\\nu_s, \\, \\nu_b$. The pdf of the variable $x$, $f(x| \\nu_s, \\nu_b, \\mu, \\mu')$ is the product of two pdfs: \n",
    "\n",
    "The pdf for $x$:\n",
    "$$\n",
    "f(x|\\nu_s,\\nu_s, \\mu, \\mu') = \\frac{\\nu_s}{\\nu_s+\\nu_b} f_s(x | \\mu) + \\frac{\\nu_b}{\\nu_s+\\nu_b} f_b(x | \\mu')\n",
    "$$\n",
    "And pdf of the number of events, that is poisson distributed:\n",
    "$$\n",
    "p(n|\\nu_s,\\nu_b) = \\frac{(\\nu_s+\\nu_b)^n}{n!} e^{-(\\nu_s+\\nu_b)}\n",
    "$$\n",
    "\n",
    "The complete pdf is:\n",
    "$$\n",
    "f(n, x | \\nu_s,\\nu_b, \\mu, \\mu') = p(n|\\nu_s,\\nu_b) \\, f(x|\\nu_S,\\nu_B, \\mu, \\mu')\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Consider a dataset with $n$ samples of a measurement $x$. The samples can be either background or signal. For the background $x$ follows and exponential and for the signal a gaussian. Use EML to compute the number of signal and background events in the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " true: signal = 30, bkg = 95\n",
      " minimize:  True\n",
      " best : signal = 31.04, bkg = 93.96 \n"
     ]
    },
    {
     "data": {
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LiY+PD3BUQggR/KxW6RMohOha0NcEAtxzzz28+OKLeDwehg8fzoMPPhjokIQQIuhJn0Ah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\nIiIiIg1iCCQiIiLSoP8P5NN+riuErCUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x116188cd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "import scipy.optimize as optimize\n",
    "\n",
    "nus, nub = 20, 100\n",
    "tau = 1\n",
    "mu, sigma = 5, 1\n",
    "\n",
    "# code\n",
    "def ll(xs, mus):\n",
    "    nus, nub = float(mus[0]), float(mus[1])\n",
    "    nu = nus+nub\n",
    "    if (float(nu)<=0.):\n",
    "        return 1.e33\n",
    "    q = nus/nu\n",
    "    n = len(xs)\n",
    "    pn = stats.poisson.pmf(n, nu)\n",
    "    pb = stats.expon.pdf(xs, scale=tau)\n",
    "    ps = stats.norm.pdf(xs, scale=sigma, loc=mu)\n",
    "    xll = -2.*np.sum(np.log((1-q)*pb+q*ps))-2*np.log(pn)\n",
    "    return xll\n",
    "\n",
    "def llfun(xnus, xnub):\n",
    "    return ll(xs, (xnus, xnub))\n",
    "\n",
    "def minllfun(xx):\n",
    "    return ll(xs, xx)\n",
    "\n",
    "# generate samples\n",
    "ns = stats.poisson.rvs(nus, size=1)[0]\n",
    "nb = stats.poisson.rvs(nub, size=1)[0]\n",
    "bxs = stats.expon.rvs(scale=tau, size=nb)\n",
    "sxs = stats.norm.rvs(scale=sigma, loc=mu, size=ns)\n",
    "print ' true: signal = {:d}, bkg = {:d}'.format(ns, nb)\n",
    "\n",
    "# generate combined sample and shuffle it!\n",
    "xs = np.array(list(sxs)+list(bxs))\n",
    "np.random.shuffle(xs)\n",
    "\n",
    "# get the best estimates of nus and nub\n",
    "result = optimize.minimize(minllfun, x0=(100., 100.), \n",
    "                           method='Nelder-Mead')\n",
    "print ' minimize: ', result.success\n",
    "nushat, nubhat = result.x[0], result.x[1]\n",
    "print ' best : signal = {:5.2f}, bkg = {:5.2f} '.format(nushat, nubhat)\n",
    "\n",
    "\n",
    "# plots\n",
    "fig, axs = plt.subplots(2, 2, figsize=(9, 6))\n",
    "ax0, ax1, ax2, ax3 = axs.ravel()\n",
    "\n",
    "ax0.hist(bxs, bins=50, color='blue', alpha=0.5, label='bkg')\n",
    "ax0.hist(sxs, bins=50, color='red', alpha=0.5, label='sig')\n",
    "ax0.set_xlabel(r'$x$', fontsize=14)\n",
    "ax1.set_ylabel('samples', fontsize=14)\n",
    "\n",
    "bins = max(nus, nub)\n",
    "xnus = np.linspace(1., 2.*bins, 100)\n",
    "llss = np.array([ll(xs, (nui, nub)) for nui in xnus])\n",
    "llbs = np.array([ll(xs, (nus, nui)) for nui in xnus])\n",
    "ax1.plot(xnus, llss, color='red', lw=2, label='signal')\n",
    "ax1.plot(xnus, llbs, color='blue', lw=2, label='bkg');\n",
    "ax1.set_xlabel('n', fontsize=14)\n",
    "ax1.set_ylabel(r'$\\mathcal{L}(n)$', fontsize=14)\n",
    "\n",
    "xnus = np.linspace(1., 2.*bins, 100)\n",
    "ynus = np.linspace(1., 2.*bins, 100)\n",
    "mxnus, mynus = np.meshgrid(xnus, ynus)\n",
    "zz = [[llfun(inus, inub) for inus in xnus] for inub in ynus]\n",
    "zz = np.array(zz)\n",
    "c0 = ax2.contourf(mxnus, mynus, zz, 20, alpha=0.8, \n",
    "                 cmp=plt.cm.jet)\n",
    "ax2.set_xlabel(r'$\\nu_s$', fontsize=14)\n",
    "ax2.set_ylabel(r'$\\nu_b$', fontsize=14)\n",
    "ax2.plot(nushat, nubhat, color='black', \n",
    "         marker='*', markersize=8)\n",
    "ax2.set_title('likelihood')\n",
    "fig.colorbar(c0, ax=ax2)\n",
    "\n",
    "fig.tight_layout();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Least squared method\n",
    "\n",
    "Consider a dataset of $n$ pair samples $\\{(x_i, \\, y_i)\\}$, where $i=1,\\dots,n$. And that a relation between $y$ and $x$ exists and it is given by a model $y=\\lambda(x|\\mu)$, that depends on a parameter $\\mu$.  If the $y_i$ measurements are gaussian distributed with known variances $V[y_i] = \\sigma^2_i$. The likelihood function is:\n",
    "\n",
    "$$\n",
    "\\mathcal{L} (\\mu) = \\Pi_{i=1}^n \\frac{1}{\\sqrt{2\\pi}\\sigma_i} e^{- \\frac{(y_i-\\lambda(x_i|\\mu))^2}{2\\sigma^2_i}}\n",
    "$$\n",
    "\n",
    "If we take the logarithm and remove the constant terms, those ones that do not depend on $\\mu$, we get:\n",
    "\n",
    "$$\n",
    "-  2 \\log \\mathcal{L}(\\mu) = \\sum_{i=1}^n \\frac{(y_i - \\lambda(x_i|\\mu))^2} { \\sigma^2_i}\n",
    "$$\n",
    "that is the $\\chi^2$.\n",
    "\n",
    "Maximizing $\\log \\mathcal{L} (\\mu)$ is now the same as minimizing  the $\\chi^2 (\\mu)$.\n",
    "\n",
    "$$\n",
    "\\chi^2(\\mu) = - 2 \\log \\mathcal{L} (\\mu)\n",
    "$$\n",
    "\n",
    "The best estimate, $\\hat{\\mu}$, is the minimum of the $\\chi^2$. The variance of $\\hat{\\mu}$ is:\n",
    "\n",
    "$$\n",
    "\\sigma^2_{\\hat{\\mu}} \\simeq 2 \\left[ \\frac{\\partial^2 \\chi^2}{\\partial \\mu^2}\\right]^{-1}_{\\hat{\\mu}}\n",
    "$$\n",
    "\n",
    "or where the expansion of $\\chi^2$ around the minimum gets increased by one.\n",
    "\n",
    "$$\n",
    "\\chi^2(\\hat \\mu \\pm \\sigma_{\\hat \\mu}) = \\chi^2(\\hat \\mu) + 1\n",
    "$$\n",
    "\n",
    "### $\\chi^2$ as goodness of the fit\n",
    "\n",
    "The least squares method provides a criteria to estimate the goodness of the fit. The variable $\\chi^2$ follows a $\\chi^2$ distribution with $n$ degrees of freedom (the number of measurements minus the number of parameters), $\\chi^2(t|n)$. Therefore it can be used as a test statistics to compute a $p$-value!\n",
    "\n",
    "$$\n",
    "p = \\int_{\\chi^2}^{\\infty} \\chi^2(t|n) \\, \\mathrm{d}t\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "**Exercise:** Consider a linear dependence $y=ax+b$, where $a,b$ are parameters ($a=1, \\, b=0$). Generate an dataset with $n=4$ pairs of measurements $\\{(x_i,y_i)\\}$, with $i=1,\\dots,n$; and a variance on $y$ of $\\sigma^2_y = 0.1$. Estimate the $a,b$ parameters using LS. Compute the $\\chi^2$ and the $p$-value. Generate now $m$, large number, datasets, and get he $\\chi^2$ distribution, check that it follows a $\\chi^2_n(t|n=2)$  distribution. Get also the pull distributions for $a, b$. Are they gaussian distributed with mean zero and sigma one?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def ls_chi2(result):\n",
    "    \"\"\" it computes the chi2 and cov matrix from the result of a least_squares scipy fit\n",
    "    return the chi2/ndf\n",
    "    result is updated with the information of the chi2, and the cov matrix!\n",
    "    \"\"\"\n",
    "    chi2 = np.sum(result.fun**2)\n",
    "    ndf = float(len(result.fun)-len(result.x))\n",
    "    chi2ndf = chi2/ndf\n",
    "    cov = (np.linalg.inv(np.dot(result.jac.T, result.jac)))*chi2ndf\n",
    "    result.chi2 = chi2\n",
    "    result.ndf = ndf\n",
    "    result.cov = cov\n",
    "    return chi2/ndf\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " pull a  0.01398351849  sigma  1.14161763261\n",
      " pull b  0.00086643575087  sigma  1.1516032296\n"
     ]
    },
    {
     "data": {
      "image/png": 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zf1Css9uV2L5dfNnDqFH8o4coHgVVBF533XUoLS3F0KFDMW7cONH3HnvssZi8\niJxiV0sLMGiQePh34cIGLFsWW9fBERERXUxQRaBSqcSYMWMCfo/X7lE0effdXli4MFXU9vXXNbj0\nUq9EiYiIiKQhmxlDiMKt/bP/AA7/EhFR/JLFjCFE4XTqlMKvAPz9750sAImIKK6xJ5Bi2oYNvbFy\nZR9Rm8VyHMnJgkSJiEhO+NgYimcsAilmte/9S0wU8P33xyVKQ0RyxMfGUDzjcDDFHJtN6VcAbtrk\nYAFIRER0AfYEUkx55JE+eOMN8dy/R47YkMDfdCIiIhF+NFLMaN/7d+WVrfjkE7tEaYiIiOSNw8EU\n9b79NsGvAHzvPTsLQCIiootgTyBFtTlz9Pj0U42ozWq1QaGQKBAREVGUYE8gRSWv9+zw74UF4IwZ\nzTh2jAUgERFRMNgTSFHnX/9S49ZbLxW1ff75CWRleSRKREREFH1YBFJUyc/vi++/F//acuYPIiKi\nruNwMEUFt/vs8O+FBeC99zayACRJVVRU4MEHH8QDDzyArVu3drhedXU1br/9duzZsyeC6YiILo49\ngSR7772nwYIFelFbeXkN+vb1SpSICPB6vSgpKcHy5cuRmpqKZcuWITc3FxkZGX7r/elPf8Lw4cMl\nSkpEFBiLQJK19o9+ATj8S/JQXV2NAQMGwPCf+cXy8/NRWlrqVwR++OGHGDt2LKqrq6WISUTUIQ4H\nkyw5HP5Tvz355CkWgCQbDocDaWlpvmW9Xg+Hw+G3TmlpKa677rpIxyMi6hR7Akl25s9Pxd//3kvU\n9u23x5GSIkiUiKh7Nm/ejNmzZ/uWBaHj32Gz2Qyz2exbLiwshFarDXkmjUYBtTpR1JaY2BqwTaVS\nQa1Wd7rehW3BrNPVts5yhPo9NRoFtNokUZtarQ7Lz6Mr5JBBLjnkkEFOObZs2eL72mQywWQyBfU6\nFoEkKxz+pWih1+tx8uRJ37LD4YBeL7529dChQ3jhhRcgCAIaGhpQXl6OhIQEjB492m97gU7cDQ0N\nIc/tcqnhdouL0dZWT8A2j8cDt9vd6XoXtgWzTlfbOssR6vd0uVrQ0OAWtWm12rD8PLpCDhnkkkMO\nGeSSQ6vVorCwsFuvZRFIsvDddyqMH99P1LZ48WksWtQoUSKii8vKykJNTQ3sdjtSU1Oxe/duPPDA\nA6J11q1b5/t6/fr1uOaaawIWgEREUmARSJK79loDLBbxcMx339mg0XTwAiIZUCqVKCoqwsqVKyEI\nAiZNmgTfaSgeAAAgAElEQVSj0YgdO3ZAoVCgoKBA6ohERBfFIpAkxeFfimYjRoxAcXGxqG3KlCkB\n112wYEEkIhERBY13B5Mk9uxR+xWAL71UzwKQiIgoQtgTSBEXqPfv6FEblPyThCjk7PYE2O3n/+dq\nbFRJmIaI5IRFIEWMIABGI4d/iSLJbldi+/bzjzsZNcojYRoikhP2vVBE/OUvCX4F4N/+dpIFIBER\nkUTYE0hhx5s/iIiI5Ic9gRQ2LS3+BeAVV7SyACQiIpIBWRWBXq8XS5cuxapVq6SOQj1UXJyCQYPE\nBeDnn5/A//6vPeht1NfXY8+ePXA6naGOR0TUoZYWJSor1b5/djsHzSg2yeo3e9u2bcjIyEBzc7PU\nUagHAg3/nj7dgIaG4C9I37hxI0pKSmCz2ZCeno6ioiLMmzcvlDGJiAJyOpUoKzv/APvp01swaJCE\ngYjCRDY9gXV1dSgvL8fkyZOljkLd5HAo/ArAn/2sucvDv/X19SgpKYHVaoXX64XVakVJSQnq6+tD\nGZeIiCiuyaYn8I033sCcOXPQ1NQkdRTqhvvv1+Gdd5JFbf/3fzVIS/N2eVtVVVWw2cSFo81mg8Vi\nQV5eXo9yEhF1VUuLEl9/rYDLpfa1GQxeGAxtEqYi6jlZFIFlZWXo06cPMjMzYTabIQiC1JGoC0J9\n929OTg7S09NhtVp9benp6cjOzu72NomIusvpVGLXrkS43ec/m6ZPb4HBIGEoohCQRRF48OBB7Nu3\nD+Xl5XC73Whubsa6deuwcOFC0Xpmsxlms9m3XFhYCK1WG+m4YaVWq6Nmnw4dUmDEiBRR28MPt+Cx\nx9wAxPvQlf3SarVYsGABNmzYgGPHjiEjIwPz58/Hj370o1BFD4lo+ll1Razu15YtW3xfm0wmmEwm\nCdMQEUlPFkXgHXfcgTvuuAMAUFlZiffff9+vAAQCn7gbGhoikjFStFptVOzTddcZYDYnitqqq23o\n1QsIFL+r+zV37lzceOONsFgsyM7Ohk6nk91xiZafVVfF4n5ptVoUFhZKHYOISFZkUQRSdInUw59T\nU1N5DSAREVGYyK4IvOqqq3DVVVdJHYMC2LcvETfdJL4I5rnn6nHbbXykDxERUbSRXRFI0qmvr0dV\nVRVycnKg0+lE3wvU+3f0qA1K2TxkiIiIiLqCH+EE4OzDmadNm4Zbb70VU6dOxcaNGwEAgtDx8C8L\nQCIioujFj3Hq8OHMb73lgdEoLgDffvsk5/4lIiKKARwOpoAPZ7Zaj+Khh8TrsfgjIiKKHewJJN/D\nmc9KBCB+WLfR2MYCkIiIKMawJ5Cg0+lQVFSEF19sQH39s6LvffZZLbKzOTUSEdGFWlqUqKxUi9o4\nlRxFGxaBBAD4/e9XoLVVIWpj7x8RUWBOpxJlZeIH5nMqOYo2HA6Oc01NCmRkpIsKwKlTm1kAEhER\nxTj2BMax//7vZCxbJn4eYGXlcfTpI3TwCiIi6giHiCnasAiMU5Ga+o2IKF5wiJiiDYeD48zJk0q/\nAvCll+pZABIREcUZ9gTGkaef1mLdOq2o7dAhG5KSJApEREREkmERGCfa9/717+/Bv/99QqI0RERE\nJDUOB8e4w4dVfgXgn/98kgUgERFRnGNPYAxbsECH995LFrUdPWqDkqU/ERFR3GMRGIMEATAaxb1/\n+fkt2LKlTqJEREREJDcsAmNMeXkirr9e/DyCjz+uhcnE51QRERHReSwCY8jPfnYpKirEDyrlo1+I\niIgoEBaBMcDjAQYOFA//3nHHGaxZc0qiREREBHAWEZI3FoFR7pNPkjB3bpqobe/eE8jI8EiUiIiI\nzuEsIiRnLAJDoL6+HlVVVcjJyYFOp+v8BSGSnd0fZ86Ib/Xl8C8REREFgw8L6aGNGzdi2rRpuPXW\nWzF16lRs3Lgx7O/Z3KxARka6qAB8+OHTLACJiIgoaOwJ7IH6+nqUlJTAarUCAKxWK0pKSjBr1iyk\npqaG5T3/9KdkLFki7m08cOA4UlOFsLwfEXWsoqICmzdvhiAImDhxImbOnCn6/q5du/Dee+8BADQa\nDe6++24MHDhQiqhERH5YBPZAVVUVbDZx75vNZoPFYkFeXl7I36/9zB8Ah3+JpOL1elFSUoLly5cj\nNTUVy5YtQ25uLjIyMnzr9O3bF0888QSSk5NRUVGBV199FU899ZSEqYmIzuNwcA/k5OQgPV1cmKWn\npyM7Ozuk7+NwKP0KwOefr8eBA2bs2bMHTqczpO9HRJ2rrq7GgAEDYDAYkJCQgPz8fJSWlorWyc7O\nRnLy2Vl7Bg8eDIfDIUVUIqKAWAT2gE6nQ1FREYxGI5RKJYxGI4qKikJ6c8izz2oxdGh/Udt339ng\ndBZH/FpEIjrP4XAgLe38nfl6vf6iRd4///lPjBgxIhLRiIiCwuHgHpo3bx5mzZoFi8WC7OzskBaA\n7Xv/9HoP9u8/Icm1iETUfQcOHMDOnTvxu9/9rsN1zGYzzGazb7mwsBBarbbH763RKKBWn39ESWJi\nq2j5Ym0qlQpqtbpLr+3K9kOVIxzv2X773TkWHbVpNApotUnoKrVaHZLfiZ6SQw45ZJBTji1btvi+\nNplMMJlMQb2ORWAIpKamhvQawCNHVBg3rp+o7U9/qsOECS0AIn8tIhH50+v1OHnypG/Z4XBAr9f7\nrXfkyBFs3LgRjzzyCFJSUjrcXqATd0NDQ49zulxquN3nbxxrbfWIli/W5vF44Ha7u/Tarmw/VDnC\n8Z7tt9+dY9FRm8vVgoYGN7pKq9WG5Heip+SQQw4Z5JJDq9WisLCwW6/lcLDMzJun8SsAf/jB5isA\ngchdi0hEHcvKykJNTQ3sdjva2tqwe/dujB49WrTOyZMn8eyzz2LhwoXo379/B1siIpIGewJlQhAA\no1Fc2OXlteDdd+v81j13LWJJSQlsNhvS09NDfi0iEV2cUqlEUVERVq5cCUEQMGnSJBiNRuzYsQMK\nhQIFBQX461//isbGRpSUlEAQBKhUKjz99NNhy2S3J8BuF/9t39ioCtv7UfdwKjmSC1kUgXV1dVi3\nbh1OnToFhUKByZMnY8aMGVLHipj9+xMxbZp4DqEPP7Rj6NDWDl8TzmsRiSg4I0aMQHFxsahtypQp\nvq/vvfde3HvvvRHLY7crsX27+FqzUaM4haTccCo5kgtZFIEqlQpz585FZmYmXC4Xli5diuHDh4ue\ntxWrbrrpUuzbJ/6L0Gq1QaHo/LWhvhaRiIiI4ocsrgnU6XTIzMwEcPap+hkZGTH/PC2P5+zdvxcW\ngIWFTTh9uiGoApCIiIioJ2TRE3ih2tpaHDlyBIMHD5Y6Stjs3JmE2bPTRG3/+tcJDBzoASD9reZE\nREQU+2RVBLpcLjz33HO48847odFo/L4frudoRdJll6Wgvl7c1Xf6dAOAs7MKyOWZQ6EWi/sVi/sE\nxO5+dfc5WkREsUo2RaDH48Gzzz6L8ePHIzc3N+A64XqOViQ0NwNZWeK7fx966DR+/etGXLgLcnjm\nUDjE4n7F4j4BsblfPXmOFhFRrJJNEbhhwwYYjcaYvCv47bd74de/Fs/ksX//cej1QgevICIiIgov\nWRSBBw8exBdffIGBAwfi4YcfhkKhwO233x4T82y2n/oNAI4dswVYk4iIiChyZFEEDhkyBG+//bbU\nMUKqvl6Bq68eIGpbu9aJ229vkigRERER0XmyKAJjzQsvpGDNmktEbdXVx9GrF4d/iYiISB5YBIZY\n++FfrdaLgwdrJEpDRETRoP1UcpxGjiKBRWCIWK0qjBnTT9T23/9dh8mTWyRKRERE0aL9VHKcRo4i\ngUVgCDz8cB/8z//0FrX98IMNKs7bTkRERDIli2njollhYZqoABw50o1jx1gAEhERkbyxJ7CbamuV\nGDmyv6jtH/+wY8SIVokSEREREQWPRWA3tH/489ixLfjrX+ugUFzkRUREREFqf6MIAAwcKCAlRaJA\nFJNYBHaBIADXXmtAdfX5i3dfe82BGTNcEqYiIqJY0/5GEQC4+WYFi0AKKRaBQTpyRIVx48R3/x44\ncBypqXz2HxERhV9Tk+DXO8hHyVBPsAgMwiuv9MaTT/bxLd9wQzNeeaVewkRERBRvnE4F9uxJErXx\nUTLUEywCL0IQgGHD+sHhOH+r79tvn8RPfuKWMBURERFRz7EIvIjy8kRRAWixHEdyMod/iYhIHgLd\nQMIhYgoWi8CL+PLL16DV9kFj43FkZLyBN98swrx586SORUREBCDwDSQcIqZg8WHRHaivr8cf/7gB\nDQ0LIQhPwWq1oqSkBPX1vBaQiIiIoh97AjtQVVUFm80marPZbLBYLMjLy5MoFRHRWXZ7Aux28d/x\njY2cqog4REzBYxHYgZycHKSnp8Nqtfra0tPTkZ2dLWEqIqKz7HYltm8X3yk6apRHojQkJxwipmBx\nOLgDOp0ORUVFMBqNUCqVMBqNKCoqgk6nkzoaERERUY+xJ/Ai5s2bh1mzZsFisSA7O5sFIBEREcUM\nFoGdSE1N5TWAREREFHM4HExEREQUh9gTSEREFOMC3TGclAS0tJxf5h3E8YdFIBERUYwLdMfwqFEe\nlJWdf6zQpEmtfo8dYmEY21gEEhERER8tE4d4TSARERFRHGIRSERERBSHOBxMREREAQW6oWTgQAEp\nKRIFopBiEUhEREQBBbpO8OabFSwCYwSLQCIiIgpaU5Pg1zvIu4ijE4tAIiKZq69XQRDOL6tUHa9L\nFG5OpwJ79iSJ2gLdRWy3J4geOROoUGy/TkfrUXjIpgisqKjA5s2bIQgCJk6ciJkzZ0odiYjoooI5\nb23atAkVFRVISkrCfffdh8zMzC6/z7ZtGtTWKnzLgwZ5MXgwPyRJPgJdO9jYqMIXX5wvMwI9h7D9\nOgAfSxNJsigCvV4vSkpKsHz5cqSmpmLZsmXIzc1FRkaG1NFkob6+HlVVVcjJyYFOp5M6DhEhuPNW\neXk5Tpw4gRdffBEWiwWvvfYannrqqS6/lyAAXq9CtEwkJx09jLqr61BkyeIRMdXV1RgwYAAMBgMS\nEhKQn5+P0tJSqWPJwsaNGzFt2jTceuutmDp1KjZu3Ch1JCJCcOet0tJSTJgwAQAwePBgNDU1wel0\nShGXiMiPLHoCHQ4H0tLSfMt6vR7V1dUSJpKH+vp6lJSUwGq1AgCsVitKSkowa9YspKamSpyOKL4F\nc94KtI7D4WCPPlEXtb92UKNRQBDUormPAf/5kNsvA7w28UIKQZB+YOGrr77C119/jXvuuQcA8L//\n+7+orq7G//t//0+0ntlshtls9i0XFhZGNCcRRa8tW7b4vjaZTDCZTD3aXjDnrWeeeQY333wzcnJy\nAABPPvkkZs+ejUGDBvltj+c3Iuqu7p7fZDEcrNfrcfLkSd+yw+GAXq/3W89kMqGwsND378KdjhWx\nuE9AbO5XLO4TEJv7tWXLFtG5o6cFIBDceUuv16Ours63XFdXF/DcBsjz/CaHDIA8cjDDeXLIIYcM\ngDxy9OT8JosiMCsrCzU1NbDb7Whra8Pu3bsxevRoqWMREXUomPPW6NGj8fnnnwMAqqqq0Lt3bw4F\nE5FsyOKaQKVSiaKiIqxcuRKCIGDSpEkwGo1SxyIi6lBH560dO3ZAoVCgoKAAo0aNQnl5Oe6//35o\nNBrMnz9f6thERD6yKAIBYMSIESguLu7Sa0IxpCM3sbhPQGzuVyzuExCb+xWufQp03poyZYpouaio\nqFvblsPPQQ4ZAHnkYIbz5JBDDhkAeeToSQZZ3BhCRERERJEli2sCiYiIiCiyWAQSERERxSEWgURE\nRERxSDY3hnRFMJO2R5u6ujqsW7cOp06dgkKhwOTJkzFjxgypY4WE1+vFsmXLoNfrsXTpUqnjhERT\nUxNeeeUVHD16FAqFAvPnz8fgwYOljtUjH3zwAT777DMoFAoMHDgQCxYsQEJC9J0iNmzYgLKyMvTp\n0wdr164FADQ2NuKFF16A3W5H3759sWjRIiQnJ0ucNLhz2aZNm1BRUYGkpCTcd999yMzMjHiOyspK\nrF69Gv369QMA5OXl4ZZbbglphkA/t/bCfSw6yxCJ4xDsZ0G4j0UwOcJ9PFpbW7FixQq0tbXB4/Fg\n7NixmDVrlt964TwWwWSIxO8F0PlnabeOgxBlPB6PsHDhQqG2tlZobW0VFi9eLFitVqlj9Vh9fb1w\n+PBhQRAEobm5WfjVr34VE/slCILw/vvvC8XFxcIzzzwjdZSQWbdunfDpp58KgiAIbW1twpkzZyRO\n1DN1dXXCfffdJ7S2tgqCIAjPPfecsHPnTolTdc8333wjHD58WHjooYd8bX/84x+FrVu3CoIgCO++\n+67w5ptvShXPJ5hzWVlZmfD73/9eEARBqKqqEh555BFJcpjN5rD//xvo53ahSByLzjJE4jgE81kQ\niWMRTI5IHA+XyyUIwtnf00ceeUSwWCyi70fiWHSWIRLHQRAu/lna3eMQdcPBwUzaHo10Op2vatdo\nNMjIyIDD4ZA2VAjU1dWhvLwckydPljpKyDQ1NeHgwYOYOHEiAEClUsmiV6mnvF4vXC4XPB4PWlpa\nonZ+6iFDhqB3796itn379mHChAkAgGuvvVYW54xgzmWlpaW+3IMHD0ZTUxOcTmfEcwCAEOYHSQT6\nuV0oEseiswxA+I9DMJ8FkTgWwX4mhft4JCUlATjbI+fxePy+H4lj0VkGIPzHobPP0u4eh6gb6wlm\n0vZoV1tbiyNHjkT98CIAvPHGG5gzZw6ampqkjhIytbW10Gq1WL9+PY4cOYJBgwbhrrvuglqtljpa\nt+n1elx//fVYsGABkpKSMGzYMAwbNkzqWCFz6tQp30wdOp0Op06dkjhRcOeyQOs4HI6QzjoS7DnV\nYrFgyZIl0Ov1mDNnTsQf6B+JYxGMSB6Hjj4LIn0sLvaZFO7j4fV68Zvf/AYnTpzA1KlTkZWVJfp+\nJI5FZxmA8B+Hzj5Lu3scoq4nMNa5XC4899xzuPPOO6HRaKSO0yPnrq3JzMyEIAhh/0spUrxeLw4f\nPoypU6di1apVSEpKwtatW6WO1SNnzpzBvn37sH79erz66qtwuVzYtWuX1LHCRqFQSB0hqgwaNAjr\n16/HmjVrMG3aNKxZs0bqSJKI5HGQy2fBxXJE4ngolUqsXr0aGzZsgMVigdVqDfl79DRDuI9DOD9L\no64IDGbS9mjl8Xjw7LPPYvz48cjNzZU6To8dPHgQ+/btw8KFC1FcXAyz2Yx169ZJHavH9Ho90tLS\ncMUVVwAAxo4di0OHDkmcqmf279+Pvn37IiUlBUqlEmPGjMG3334rdayQ0el0vqERp9OJPn36SJwo\nuHOZXq9HXV2db7muri7k57tgcmg0Gt+Q2MiRI9HW1obGxsaQ5ggmZ7iPRWcidRw6+yyI1LHoLEck\nfy+Sk5NhMplQUVEhao/k70VHGcJ9HIL5LO3ucYi6IjCYSduj1YYNG2A0GmPmruA77rgDGzZswLp1\n6/Dggw/i6quvxsKFC6WO1WM6nQ5paWmw2WwAzhZQ0T7X9aWXXgqLxQK32w1BELB//35kZGRIHavb\n2v+1fM0112Dnzp0AgJ07d8rinBHMuWz06NH4/PPPAQBVVVXo3bt3yIf8gslx4bVF54aKU1JSQpoD\n8P+5XSgSx6KzDJE6Dp19FkTqWHSWI9zH4/Tp077hT7fbjf379yM9PV20TriPRTAZwn0cgvks7e5x\niMpp4yoqKvCHP/zBN2l7LDwi5uDBg1ixYgUGDhwIhUIBhUKB22+/HSNGjJA6WkhUVlbi/fffj5lH\nxHz//fd49dVX0dbWhn79+mHBggVRf3PIX/7yF3z55ZdQqVTIzMzEvffeG5WPiCkuLkZlZSUaGhrQ\np08fFBYWIjc3F88//zxOnjwJg8GARYsWdXoDQCQEOpft2LEDCoUCBQUFAICSkhJUVFRAo9Fg/vz5\nGDRoUMRzfPjhh9ixYwdUKhXUajXmzp0b8muWA/3c2traInosOssQiePQ0WeB3W6P6LEIJke4j8cP\nP/yAl19+GV6vF4IgYNy4cfj5z38e0f9HgskQid+Lcy78LA3FcYjKIpCIiIiIeibqhoOJiIiIqOdY\nBBIRERHFIRaBRERERHGIRSARERFRHGIRSERERBSHWAQSERERxSEWgURERERxKPqeBEtERBTHBEHA\nRx99BLfbDQC48cYbJU5E0Yo9gURERFGkrKwMeXl5uPHGG2GxWHD48GGpI1GUYhFIREQURU6cOIFd\nu3YBAPr164e6ujqJE1G04nAwERFRFLnuuuvQ1tYGADh69Ciuv/56iRNRtGIRSDGhqqoKVqsVP/zw\nAwYPHoympiZUVFRg7ty56Nu3r9TxiIhCJiEhAQkJCfj2229x1VVXQafTSR2JohSHgynqNTc3w2az\nYdKkSRg6dCi2bduGKVOmQKPRQK1WSx2PiCjkmpubYTabcdNNN0kdhaKYQhAEQeoQRD3R2toKpVIJ\nlUqFt956C7169eKJkYhi2ieffIKJEydCEAR88803GDp0qNSRKAqxJ5CiXmJiIlQqFQDg66+/9p0M\nm5qapIxFRNRlzc3N2LZtG1avXo1//etfou/t3bsXtbW1+PLLL/Hmm29i3rx5mDdvHlJTUyVKS9GO\n1wRS1Pv3v/+N48ePY8yYMfjhhx+QmZkJAPj8888xffp0acMREXWBRqPBjBkzkJWVhVWrViE3NxcJ\nCQn45ptvkJKSgr59+6Jv374YN26c1FEpBrAIpKjX0NCA2tpa/Pvf/8Ztt92Gbdu2ISkpCfn5+VJH\nIyLqEoVCAQDIzs5GamoqysrKkJ6ejoaGBuTl5UmcjmINrwkkIiKSoXfeeQfffvstxowZg0mTJkkd\nh2IQrwkkIiKSoauvvhoHDx7ExIkTpY5CMYpFIBERkcy0tbXhm2++gVarhcVikToOxSgWgURERDKz\nfft2TJ06Fbm5ufjiiy+kjkMxikUgERGRjHz88cfIz8+HRqPBhAkT8OWXX6K1tRUej0fqaBRjeGMI\nERGRTJSXl0Ov1+Oyyy7ztb355ptoa2uD0WhEQUGBhOko1rAIJCIiIopDHA4mIiIiikMsAomIiIji\nEItAIiIiojjEIpCIiIgoDrEIJCIiIopDLAKJiIiI4hCLQCIiIqI4xCKQiIiIKA6xCCQiIiKKQywC\niYiIiOJQQrjfoLW1FStWrEBbWxs8Hg/Gjh2LWbNmidaprKzE6tWr0a9fPwBAXl4ebrnllnBHIyLq\nkYqKCmzevBmCIGDixImYOXOm3zpmsxlvvPEGPB4PLrnkEqxYsUKCpEREAQgR4HK5BEEQBI/HIzzy\nyCOCxWIRfd9sNgvPPPNMl7d74MCBkOTrCTlkEAR55JBDBkGQRw5mOE8OOcKRwePxCAsXLhRqa2uF\n1tZWYfHixYLVahWtc+bMGWHRokVCXV2dIAiCcOrUqaC3L4fj1hXRllcQoi9ztOUVhOjLHG15BaFn\nmSMyHJyUlATgbK+gx+PpqBjt8nbNZnOPcoWCHDIA8sghhwyAPHIww3lyyBGODNXV1RgwYAAMBgMS\nEhKQn5+P0tJS0Tq7du3CmDFjoNfrAQCXXHJJ0NuXw3HrimjLC0Rf5mjLC0Rf5mjLC/Qsc9iHgwHA\n6/XiN7/5DU6cOIGpU6ciKyvLbx2LxYIlS5ZAr9djzpw5MBqNkYhGRNQtDocDaWlpvmW9Xo/q6mrR\nOjabDR6PB0888QRcLhemT5+O8ePHRzoqEVFAESkClUolVq9ejaamJqxZswZWq1VU5A0aNAjr169H\nUlISysvLsWbNGhQXF0ciGhFR2Hi9Xhw+fBjLly9HS0sLHnvsMWRnZ6N///5SRyMigkLozjhsD/z1\nr3+FRqPB9ddf3+E69913H1atWoWUlBRRu9lsFnV7FhYWhi0nEcWWLVu2+L42mUwwmUw92l5VVRX+\n8pe/4NFHHwUAbN26FQBEN4ds3boVra2tvpvhXnnlFYwYMQJjx4712x7Pb0TUXd09v4W9J/D06dNI\nSEhAcnIy3G439u/fj5tuukm0jtPphE6nAwDfcEr7AhAIvGM2my1MyYOj1WrR0NAgaQa55JBDBrnk\nYAZ55UhPTw95UZWVlYWamhrY7XakpqZi9+7deOCBB0Tr5ObmYtOmTfB6vWhtbYXFYunwD2A5nt+6\nQg4/566KtszRlheIvszRlhfo2fkt7EWg0+nEyy+/DK/XC0EQMG7cOIwaNQo7duyAQqFAQUEBvvrq\nK+zYsQMqlQpqtRoPPvhguGMREfWIUqlEUVERVq5cCUEQMGnSJBiNRtG5LSMjA8OHD8fixYuhVCpR\nUFDA652JSDYiPhwcalL/pSyXvxrkkEMOGeSSgxnklSM9PV3S9+8uqc9vXSGHn3NXRVvmaMlrtyfA\nbj/78BGNRgOXywWDwQuDoU3iZJ2LlmN8oZ6c3yJyYwgRERHFB7tdie3bzz4aTq1OhNstYPr0FhgM\nEgcjP5w2joiIiCgOsSeQKMTs9gQcOqSAy6UWtUfLcAgREcUHFoFEIWa3K/HPf54dArkQh0OIiEhO\nOBxMREREFIdYBBIRERHFIRaBRERERHGI1wQSERFRWLW0KFFZyZvl5IZFIBEREYWV06lEWVmiqI03\ny0mPw8FEREREcYhFIBEREVEc4nAwERERdcuF8wSf09iokigNdRWLQKJuCnTyA3gCJKL4ceE8weeM\nGuWRKA11FYtAom4KdPIDeAIkIqLowGsCiYiIiOIQewKJIiTQc7LO4fOyiIgo0lgEEkVIoOdkncPn\nZRERUaRxOJiIiIgoDrEIJCIiIopDLAKJiIiI4hCLQCIiIqI4xCKQiIiIKA6xCCQiIiKKQywCiYiI\niOJQ2J8T2NraihUrVqCtrQ0ejwdjx47FrFmz/NbbtGkTKioqkJSUhPvuuw+ZmZnhjkZEREQUt8Je\nBCYmJmLFihVISkqC1+vF448/jpEjRyIrK8u3Tnl5OU6cOIEXX3wRFosFr732Gp566qlwRyMiIiKK\nW9bMhx4AACAASURBVBEZDk5KSgJwtlfQ4/H4fb+0tBQTJkwAAAwePBhNTU1wOp2RiEZEREQUlyIy\nbZzX68VvfvMbnDhxAlOnThX1AgKAw+FAWlqab1mv18PhcECn00UiHhEREVHciUhPoFKpxOrVq7Fh\nwwZYLBZYrdZIvC0RERERdSAiPYHnJCcnw2QyoaKiAkaj0deu1+tRV1fnW66rq4Ner/d7vdlshtls\n9i0XFhZCq9WGN3Qn1Gq15BnkkkMOGSKZQ6NRQK1O9GtPTGyFSqWCWq32aw+0/rltabVJIc0Xbz+P\nzmzZssX3tclkgslkkjANEZH0wl4Enj59GgkJCUhOTobb7cb+/ftx0003idYZPXo0PvroI4wbNw5V\nVVXo3bt3wKHgQCfuhoaGsObvjFarlTyDXHLIIUMkc7hcarjdgl97a6sHHo8Hbrfbrz3Q+me31YKG\nBnfA73VXvP08OstQWFgY8u1WVFRg8+bNEAQBEydOxMyZM0Xfr6ysxOrVq9GvXz8AQF5eHm655ZaQ\n5yAi6o6wF4FOpxMvv/wyvF4vBEHAuHHjMGrUKOzYsQMKhQIFBQUYNWoUysvLcf/990Oj0WD+/Pnh\njkVE1CNerxclJSVYvnw5UlNTsWzZMuTm5iIjI0O03pVXXomlS5dKlJKIqGNhLwIHDhyIVatW+bVP\nmTJFtFxUVBTuKEREIVNdXY0BAwbAYDAAAPLz81FaWupXBApC4N5fIiKpccYQIqJu6OipBu1ZLBYs\nWbIETz/9NG+KIyJZieiNIURE8WTQoEFYv349kpKSUF5ejjVr1qC4uDjgunK88a0r5HIDUFdEW2Y5\n5g10g9yFN8Gdu0ku0I1x4bghrqfkeIyD0d0b31gEEhF1g16vx8mTJ33LDofD76kGGo3G9/XIkSPx\n+uuvo7GxESkpKX7bk+ONb10hhxuAuiraMssxb6Ab5C68CU6tVsPtdge8MS4cN8T1lByPcWd6cuMb\nh4OJiLohKysLNTU1sNvtaGtrw+7duzF69GjROhfOfFRdXQ0AAQtAIiIpsCeQiKgblEolioqKsHLl\nSgiCgEmTJsFoNIqefPDVV19hx44dviGxBx98UOrYREQ+LAKJiLppxIgRftf4Xfjkg2nTpmHatGmR\njkVEFBQWgUQXYbcnwG4PfNVEY6MqwmmIiGJHS4sSlZXimZUMBi8MhjaJEsUfFoFEF2G3K7F9e+C7\n10aN8kQ4DRFR7HA6lSgrE98xPH16C/7z6E2KAN4YQkRERBSHWAQSERERxSEWgURERERxiEUgERER\nURxiEUhEREQUh1gEEhEREcUhFoFEREREcYhFIBEREVEcYhFIREREFIdYBBIRERHFIRaBRERERHGI\nRSARERFRHEqQOgARAS0tSlRWqv3aDQYvDIY2CRIREVGsYxFIJANOpxJlZYl+7dOnt8BgkCAQERHF\nPA4HExEREcUhFoFEREREcSjsw8F1dXVYt24dTp06BYVCgcmTJ2PGjBmidSorK7F69Wr069cPAJCX\nl4dbbrkl3NGIiIiI4lbYi0CVSoW5c+ciMzMTLpcLS5cuxfDhw5GRkSFa78orr8TSpUvDHYeIiIiI\nEIHhYJ1Oh8zMTACARqNBRkYGHA6H33qCIIQ7ChERERH9R0TvDq6trcWRI0cwePBgv+9ZLBYsWbIE\ner0ec+bMgdFojGQ0IiIiorgSsSLQ5XLhueeew5133gmNRiP63qBBg7B+/XokJSWhvLwca9asQXFx\nsd82zGYzzGazb7mwsBBarTbs2S9GrVZLnkEuOeSQIdQ5NBoF1Gr/R7cAQGJia8DvJSa2QqVSQa1W\nB7X+xb6n0Sig1SZ1I3ls/jx6YsuWLb6vTSYTTCaThGmIiKQXkSLQ4/Hg2Wefxfjx45Gbm+v3/QuL\nwpEjR+L1119HY2MjUlJSROsFOnE3NDSEJ3SQtFqt5BnkkkMOGUKdw+VSw+0OfKlCa6sn4PdaWz3w\neDxwu91BrX+x77lcLWhocAd4Redi8efRkwyFhYWSZiCKZnZ7Aux2/yvIGhtVEqShUIlIEbhhwwYY\njUa/u4LPcTqd0P1/9u4/uI36zv/4ayVb0TexQixXQIibpjSh6bltgpsE5lLKJXEY0h+DGTqaAuX4\nYXoQAg3clYKhBGichjakabhc0oOaunO0gK/TQrkjHVym/PIMYLBdinwp6KDkSjCRrTqxCf4l7fcP\nGjWq5Fi2V1qt9/mYyYx39Vn5pWTz9tv76zNnjiQpGo1KUkYDCAAA7BGLebR3b+ZZierqhA1pYJW8\nN4H79u3Ts88+q/nz5+ub3/ymDMPQhRdeqFgsJsMwVFNTo+eff14tLS2pU2jXX399vmMBAAC4Wt6b\nwMWLF+vhhx8+7phzzz1X5557br6jAAAA4C+YMQQAAMCFaAIBAABciCYQACaps7NT119/vTZu3KhH\nHnlkzHHRaFQXXnihXnjhhQKmA4DjowkEgElIJpNqbGzUrbfequ3bt6u1tVVvv/121nE/+9nPtGTJ\nEhtSAsDYaAIBYBKi0ajmzp2rUCikkpISrVy5Um1tbRnjfv3rX+vMM8/U7NmzbUgJAGOjCQSASYjH\n46qoqEgtB4PBjHnR4/G42tradM455xQ6HgCMq6BzBwPFiqfhIx+ampp08cUXp5ZNM/uMMVJxTos5\nEcUyPeBEOC2znXnHmkIz25SXx647+vzf8cYd+30mO1WmFZy2Txw12WkxaQIB8TR8TFwwGFRPT09q\nOR6PKxgMpo1544039IMf/ECmaaq/v18dHR0qKSnRsmXLMt6vGKfFnIhimB5wopyW2c68Y02hmW3K\ny2PX+Xw+DQ8Pjzvur99n8lNlWsFp+4Q0tWkxaQIBYBIWLlyo7u5uxWIxlZeXq7W1VRs3bkwbs2vX\nrtTXu3fv1mc+85msDSAA2IEmEAAmwePxqK6uTg0NDTJNU6tXr1ZlZaVaWlpSU2ICQDGjCQSASVq6\ndKl27tyZtm7t2rVZx15zzTWFiAQAOePuYAAAABeiCQQAAHAhmkAAAAAXogkEAABwIZpAAAAAF6IJ\nBAAAcCGaQAAAABeiCQQAAHAhmkAAAAAXogkEAABwIZpAAAAAF6IJBAAAcCGaQAAAABcqyfc36O3t\n1a5du3To0CEZhqE1a9bo85//fMa4+++/X52dnZoxY4Y2bNigBQsW5DsaAACAa+W9CfR6vbr00ku1\nYMECDQ4O6qabbtKSJUs0b9681JiOjg69++67uueee/T666/rvvvu05YtW/IdDQAAwLXyfjp4zpw5\nqaN6fr9f8+bNUzweTxvT1tams88+W5K0aNEiHTlyRH19ffmOBgAA4Fp5PxJ4rIMHD+qtt97SokWL\n0tbH43FVVFSkloPBoOLxuObMmVPIeAAAwEZDQx51dfky1odCSYVCozYkmt4K1gQODg7q+9//vi67\n7DL5/f5JvUckElEkEkkth8NhBQIBqyJOis/nsz1DseQohgyTzeH3G/L5SjPWl5aOZF1/vNdKS0fk\n9Xrl8/lyGn+81/x+Q4HAjFw+QgYn/3vkQ3Nzc+rrqqoqVVVV2ZgGQDZ9fR61t2fWwnXrhhQK2RBo\nmitIE5hIJLR9+3Z97nOf0/LlyzNeDwaD6u3tTS339vYqGAxmjMtWuPv7+60PPAGBQMD2DMWSoxgy\nTDbH4KBPw8NmxvqRkUTW9cd7bWQkoUQioeHh4Sm/1+DgkPr7h7NsMT4n/3vkI0M4HLY1AwAUm4I8\nImbPnj2qrKzMelewJC1btkxPP/20JOm1117TrFmzOBUMAACQR3k/Erhv3z49++yzmj9/vr75zW/K\nMAxdeOGFisViMgxDNTU1qq6uVkdHh6677jr5/X6tX78+37EAAABcLe9N4OLFi/Xwww+PO66uri7f\nUQAAAPAXzBgCAADgQjSBAAAALkQTCAAA4EI0gQAAAC5U0BlDAABAcYvFShSLpR8jGhjw2pQG+UQT\nCAAAUmIxj/buTZ+pqLo6YVMa5BOngwEAAFyII4FAERtrMnWJCdWLQWdnp5qammSaplatWqXa2tq0\n11966SU9/PDDMgxDXq9Xl156qRYvXmxTWgBIRxMIFLGxJlOXmFDdbslkUo2Njdq0aZPKy8tVX1+v\n5cuXa968eakxn/rUp7Rs2TJJ0v79+7Vjxw7t2LHDrsgAkIbTwQAwCdFoVHPnzlUoFFJJSYlWrlyp\ntra2tDEzZvz1uqrBwUEZhlHomAAwJo4EAsAkxONxVVRUpJaDwaCi0WjGuBdffFEPPvigDh8+rJtv\nvrmQEQHguGgCASCPVqxYoRUrVmjfvn166KGHdNttt9kdCQAk0QQCwKQEg0H19PSkluPxuILB4Jjj\nFy9erIMHD2pgYEBlZWUZr0ciEUUikdRyOBxWIBCwNnQe+Xw+R+WVnJe5UHn9fkM+X/q1yKWlIxnr\nxlp/7Dqv1yufzzfuuPG+j99vKBCYkbHeak7bJ45qbm5OfV1VVaWqqqqctqMJBIBJWLhwobq7uxWL\nxVReXq7W1lZt3LgxbUx3d7dOPvlkSdIbb7yh0dHRrA2glL1w9/f35yd8HgQCAUfllZyXuVB5Bwd9\nGh4209aNjCQy1o21/th1Pp9Pw8PD444b7/sMDg6pv394Up9nIpy2T0gfZA6Hw5PaliYQACbB4/Go\nrq5ODQ0NMk1Tq1evVmVlpVpaWmQYhmpqavTCCy/omWeeUUlJiXw+n2644Qa7YwNACk0gAEzS0qVL\ntXPnzrR1a9euTX193nnn6bzzzit0LADICY+IAQAAcCGaQAAAABeiCQQAAHAhmkAAAAAXogkEAABw\nIZpAAAAAF6IJBAAAcCGaQAAAABfK+8Oi9+zZo/b2dp1wwgm6++67M17v6urS9773PZ100kmSPphs\n/YILLsh3LLhULFaiWCzzd5+BAa8NaQAAsE/em8BVq1Zp3bp12rVr15hjPvGJT+imm27KdxRAsZhH\ne/dmTkJeXZ2wIQ0AAPbJ++ngxYsXa9asWccdY5qZk0UDAAAgf4pi7uDXX39dN954o4LBoC655BJV\nVlbaHQkAAGBas70JPPXUU7V7927NmDFDHR0d2rZtW8aE7EdFIhFFIpHUcjgcViAQKFTUrHw+n+0Z\niiVHMWQYL4ffb8jnK81YX1o6MqH1423j9Xrl8/ksea+xtvH7DQUCmae2j3LCv0chNTc3p76uqqpS\nVVWVjWkAwH62N4F+vz/19emnn64f/ehHGhgYUFlZWcbYbIW7v78/7xmPJxAI2J6hWHIUQ4bxcgwO\n+jQ8nHn5wchIYkLrx9smkUhoeHjYkvcaa5vBwSH19w9nfU1yxr9HITOEw2FbMwBAsSnII2JM0xzz\nur++vr7U19FoVJKyNoAAAACwTt6PBO7cuVNdXV3q7+/X+vXrFQ6HNTo6KsMwVFNTo+eff14tLS2p\n02fXX399viMBAAC4Xt6bwI0bNx739XPPPVfnnntuvmMAAADgGMwYAgAA4EI0gQAAAC5EEwgAAOBC\nNIEAAAAuRBMIAADgQjSBAAAALkQTCAAA4EI0gQAAAC5EEwgAAOBCNIEAAAAulPdp4wAAAKZiaMij\nri5f2rpQKKlQaNSmRNMDTSAAAChqfX0etbeXpq1bt25IoZBNgaYJTgcDAAC4EEcCAWCSOjs71dTU\nJNM0tWrVKtXW1qa9/txzz+nRRx+VJPn9fn3ta1/T/Pnz7YgKABk4EggAk5BMJtXY2Khbb71V27dv\nV2trq95+++20MSeeeKLuvPNObdu2TRdccIH+/d//3aa0AJCJJhAAJiEajWru3LkKhUIqKSnRypUr\n1dbWljbmtNNO08yZMyVJixYtUjwetyMqAGRFEwgAkxCPx1VRUZFaDgaDx23ynnzySS1durQQ0QAg\nJ1wTCDhUtkcmSDw2oRi9+uqreuqpp/Ttb397zDGRSESRSCS1HA6HFQgEChHPEj6fz1F5JedlLlRe\nv9+Qz5d+J25p6UjGurHWH7vO6/XK5/ONO24y38fvNxQIzMj9g+XAafvEUc3Nzamvq6qqVFVVldN2\nNIGAQ2V7ZILEYxMKJRgMqqenJ7Ucj8cVDAYzxr311lu69957dcstt6isrGzM98tWuPv7+60LnGeB\nQMBReSXnZc5H3lisRLFY+knBgQGvhofNtHUjI4mMdWOtP3adz+fT8PDwuOMm830GB4fU3z88ziec\nGKftE9IHmcPh8KS2pQkEgElYuHChuru7FYvFVF5ertbWVm3cuDFtTE9Pj7Zv365rr71WJ598sk1J\ngbHFYh7t3Zt+NK26OmFTGhQaTSAATILH41FdXZ0aGhpkmqZWr16tyspKtbS0yDAM1dTU6Oc//7kG\nBgbU2Ngo0zTl9Xq1detWu6MDgCSaQACYtKVLl2rnzp1p69auXZv6+uqrr9bVV19d6FgAkBPuDgYA\nAHAhmkAAAAAXyvvp4D179qi9vV0nnHCC7r777qxj7r//fnV2dmrGjBnasGGDFixYkO9YAAAArpb3\nI4GrVq3SrbfeOubrHR0devfdd3XPPffon/7pn3TfffflOxIAAIDr5b0JXLx4sWbNmjXm621tbTr7\n7LMlfTCt0pEjR9TX15fvWAAAAK5m+zWBE516CQAAAFPnqEfEFOO0SsUyxUwx5CiGDJIUi3l14MDs\nrK8ND5vy+TKnWpvIVEW5bHN0qiQr3mui2xydSqlY/j2KJcdkp1UCgOnK9iYwGAyqt7c3tdzb25t1\n6iWpOKdVKpYpZoohRzFkkKQDB2brl7/MnHZI+uBJ+MPDmdMMTWSqoly2SSQyv89k32ui2xydSqlY\n/j2KIcdUplUCgOmqIKeDTdOUaWb/QbZs2TI9/fTTkqTXXntNs2bN0pw5cwoRCwAAwLXyfiRw586d\n6urqUn9/v9avX69wOKzR0dHUtErV1dXq6OjQddddJ7/fr/Xr1+c7EgAAgOvlvQn82wnVs6mrq8t3\nDAAAABzD9ruDAQAAUHg0gQAAAC5EEwgAAOBCNIEAAAAuRBMIAADgQjSBAAAALkQTCAAA4EI0gQAA\nAC5EEwgAAOBCeZ8xBAAAwGpDQx51dfnS1oVCSYVCozYlch6aQAAA4Dh9fR61t5emrVu3bkihkE2B\nHIjTwQAAAC7EkUAAAFwgFitRLJZ+7GdgwGtTGhQDmkAAAFwgFvNo794ZaeuqqxM2pUExoAkEppmj\nF0v7/YYGB/960TQXTAMAjkUTCEwzRy+W9vlKNTxsptZzwTQA4FjcGAIAAOBCNIEAAAAuxOlgOFa2\nO90kpZ0CBfKps7NTTU1NMk1Tq1atUm1tbdrrBw4c0O7du/Xmm2/qwgsv1Be/+EWbkgJAJppAOFa2\nO90k6YwzDEk0gsivZDKpxsZGbdq0SeXl5aqvr9fy5cs1b9681JiysjJdccUVevHFF21MCgDZcToY\nACYhGo1q7ty5CoVCKikp0cqVK9XW1pY2Zvbs2Tr11FPl9fIsNgDFhyYQACYhHo+roqIitRwMBhWP\nx21MBAATw+lgACgCkUhEkUgktRwOhxUIBGxMNDE+n89ReSXnZZ5qXr/fkM+XPtduaenIpNflMtbr\n9crn8+X9+xz7GQOBzMuEcuW0feKo5ubm1NdVVVWqqqrKabuCNIHjXTzd1dWl733vezrppJMkSStW\nrNAFF1xQiGgAMCnBYFA9PT2p5Xg8rmAwOOn3y1a4+/v7J/1+hRYIBByVV3Je5qnmHRz0Zdw4NzKS\nmPS6XMb6fD4NDw/n/fv89TMOqb9/OGP7XDltn5A+yBwOhye1bd6bwFwunpakT3ziE7rpppvyHQcA\nLLFw4UJ1d3crFoupvLxcra2t2rhx45jjTZOblQAUl7w3gcdePC0pdfH03zaBFEgATuLxeFRXV6eG\nhgaZpqnVq1ersrJSLS0tMgxDNTU16uvrU319vd5//30ZhqHHH39cO3bskN/vtzs+AOS/Ccx28XQ0\nGs0Y9/rrr+vGG29UMBjUJZdcosrKynxHA4ApWbp0qXbu3Jm2bu3atamv58yZoz179hQ6FgDkpChu\nDDn11FO1e/duzZgxQx0dHdq2bVtGYQUAAIB18t4E5nLx9LGnRk4//XT96Ec/0sDAgMrKytLGFePd\nc8VyJ1Ex5Ch0hmx3ukmSYSTk82V/LttE7jw73vrxtjl6R5wV7zXZbf42w1TvmpusYtg3pcnfPQcA\n01Xem8BcLp7u6+vTnDlzJCl1qvhvG0CpOO+eK5Y7iYohR6EzZLvTTZJM09DwcCLrNhO58+x468fb\nJpFIaHh4OKfx+fj+w8Nm6q68o6Z619xkFcu+Odm75wBgusp7E5jLxdPPP/+8WlpaUkcurr/++nzH\nAgAAcLWCXBM43sXT5557rs4999xCRAFca2jIo64uX9bXQqGkQqHRAicCAGtlq3PUt7EVxY0hAPKv\nr8+j9vbs1xeuWzekvzzFCQAcK1udo76NjbmDAQAAXIgjgQAATCOxWIliscxjPAMD2Z+aAPeiCQQA\nYBqJxTzauzfzcVDV1dmfmgD34nQwAACAC3EkEEWPUxsAAFiPJhBFj1MbAABYj9PBAAAALkQTCAAA\n4EI0gQAAAC5EEwgAAOBCNIEAAAAuRBMIAADgQjSBAAAALsRzAgFoaMijri5fxvpQKKlQaNSGRACA\nfKMJBKC+Po/a20sz1q9bN6RQyIZAAIC8owlEURhrajiJ6eEAYCzZaic1E7miCURRGGtqOInp4QBg\nLNlqJzUzHZe7jI0mEAAATFtc7jI2mkAAAOA62Y4Qzp9vqqzMpkA2oAkEAACuk+0I4fnnGzSBACBx\nLQ0ATGc0gSiose4C5m624sS1NEDxeOcdU/v3p/9SRu3EVNAEoqDGuguYu9mcZawjhBJHCYF8OXiQ\nO4FhrYI0gZ2dnWpqapJpmlq1apVqa2szxtx///3q7OzUjBkztGHDBi1YsKAQ0QBMwlhHCCV3HSWk\ntgFwsrw3gclkUo2Njdq0aZPKy8tVX1+v5cuXa968eakxHR0devfdd3XPPffo9ddf13333actW7bk\nOxryhAc/ww2obQCcLu9NYDQa1dy5cxX6y6GBlStXqq2tLa1QtrW16eyzz5YkLVq0SEeOHFFfX5/m\nzJmT73jIAx78DDegtiGfsv0yPTxs2pTGPY4cMTMudZkxQxoaSh83XS57yXsTGI/HVVFRkVoOBoOK\nRqPjjonH4xRKwIH+9npBv9/Q4KAvayE9yokFldoGq4w19duzz6b/iD7jDEMSjWA+9fUZeuGFzOsu\n29vTz2JNl8tepuWNIcmkodHR7B/N6x2V1+ve/0RjnaqdzA/ov32voz/sOeXrbn97vaDPV6rhYTNr\nIT1q9eqRCe2X062hdJqRkczrQUdGqKvHyraPjrXfZmv4OGtS3LLdHDeRelUsdcowTTOv/3Nfe+01\n/ed//qduvfVWSdIjjzwiSWkXUN9777365Cc/qb//+7+XJF1//fW64447Mn5bjkQiikQiqeVwOJzP\n6ACmkebm5tTXVVVVqqqqmtL7WVnbJOobgMmbbH3LfvW+hRYuXKju7m7FYjGNjo6qtbVVy5YtSxuz\nbNkyPf3005I+KKyzZs3KWiSrqqoUDodTf4790HYphgxSceQohgxSceQgw18VQ47m5ua02jHVBlCy\ntrZJxVnfJsJpeSXnZXZaXsl5mZ2WV5pafcv76WCPx6O6ujo1NDTINE2tXr1alZWVamlpkWEYqqmp\nUXV1tTo6OnTdddfJ7/dr/fr1+Y4FAFNCbQPgdAW5JnDp0qXauXNn2rq1a9emLdfV1RUiCgBYhtoG\nwMm8d9xxxx12h5iKE0880e4IRZFBKo4cxZBBKo4cZPirYshRDBkmymmZnZZXcl5mp+WVnJfZaXml\nyWfO+40hAAAAKD55vzEEAAAAxYcmEAAAwIUc8bDoYpikfbwMzz33nB599FFJkt/v19e+9jXNnz+/\noBmOikajuu2223T99dfrjDPOsDRDrjkikYh+8pOfKJFIaPbs2br99tsLmuHIkSP613/9V/X09CiZ\nTOpLX/qS/uEf/sHSDHv27FF7e7tOOOEE3X333VnH5Hu/zCVHIfbNXP4upPzum7lkyPd+aaU//vGP\nuu+++zQyMiKv16srr7xSH/vYx+yONa69e/fqiSeekMfjUXV1tS6++GK7I+Xkscce0wMPPKDGxkaV\nlZXZHWdMDzzwgF5++WWVlJTopJNO0jXXXKOZM2faHStDrj+vikVvb6927dqlQ4cOyTAMrVmzRp//\n/OftjjWuZDKp+vp6BYNB3XTTTRN/A7PIJRIJ89prrzUPHjxojoyMmN/4xjfMP/3pT2lj2tvbze98\n5zumaZrma6+9Zt5yyy0Fz/CHP/zBfO+990zTNM2Ojg5bMhwdd+edd5pbt241n3/+eUsz5Jrjvffe\nM2+44Qazt7fXNE3TPHToUMEz/OIXvzB/+tOfpr7/5Zdfbo6Ojlqa43/+53/MN9980/yXf/mXrK/n\ne7/MNUe+981cMphm/vfN8TLke7+0WkNDg9nZ2Wma5gf70h133GFzovG9+uqr5ubNm1P/14r97/io\nnp4es6GhwbzmmmvM/v5+u+Mc1+9+9zszkUiYpmmaDzzwQKrOFZNcf14Vkz//+c/mm2++aZqmab7/\n/vvm17/+9aLPbJqm+dhjj5k7d+4077rrrkltX/Sng4+dpL2kpCQ1SfuxxpqkvZAZTjvttNRvY4sW\nLVI8Hrfs++eaQZJ+/etf68wzz9Ts2bMt/f4TyfHcc8/pjDPOUDAYlCTLs+SSwTAMvf/++5KkwcFB\nBQIBeb3WTme3ePFizZo1a8zX871f5poj3/tmLhmk/O+b42XI935pNcMwdOTIEUnSe++9p/LycpsT\nje+JJ55QbW1t6v9asf8dH/WTn/xEl1xyid0xcvLpT39aHs8HP7oXLVqk3t5emxNlyvXnVTGZM2dO\n6kyN3+/XvHnz8lIrrdTb26uOjg6tWbNm0u9R9E3gWBOwT3RMvjMc68knn9TSpUst+/65ZojHQHqb\nwAAAIABJREFU42pra9M555xj6feeaI4DBw5oYGBAd955p+rr6/XMM88UPMO5556rP/3pT7rqqqt0\n44036rLLLrM0Qy7yvV9ORj72zVwUYt8cT773S6tdeuml+o//+A+tX79eP/3pT3XRRRfZHWlc77zz\njrq6unTrrbfqzjvv1P/+7//aHWlcL730kioqKiy/RKIQfvvb3+r000+3O0aGYqx9E3Hw4EG99dZb\nWrRokd1RjuvoLy+GYUz6PRxxTaCTvPrqq3rqqaf07W9/u+Dfu6mpKe36G9Omp/8kk0m9+eab2rRp\nk4aGhvStb31Lp512mk4++eSCZejs7NRHP/pR3X777eru7lZDQ4Puvvtu+f3+gmUoNm7fN4thv/xb\nmzdv1qFDh1LLpmnKMAx95Stf0e9//3tddtllWrFihZ5//nnt2bNHt912m21Zjzpe5kQioffee09b\ntmxRNBrVjh07tGvXLhvTfuB4mX/5y1/qW9/6Vtprdjte3qNTE/7iF7+Q1+vVZz/7WbtiTkuDg4P6\n/ve/r8suu6yof14cvf55wYIFikQik95vi74JDAaD6unpSS3H4/HU6Zxjxxx7SLy3tzdjTL4zSNJb\nb72le++9V7fccovlFxbnkuGNN97QD37wA5mmqf7+fnV0dKikpCRjPtN85wgGgwoEAvL5fPL5fPrE\nJz6hP/7xj5b9sM0lw1NPPZW6EPnkk0/WiSeeqLfffrugF9bne7+ciHzum7koxL45nnzvl5NxvKZu\n165duvzyyyVJZ555pvbs2VOoWMd1vMwtLS2pG34WLlwowzDU39+vQCBQqHhZjZV5//79OnjwoG68\n8UaZpql4PK6bb75Z3/nOd3TCCScUOOVfjdfsP/XUU+ro6NCmTZsKlGhicv2ZWWwSiYS2b9+uz33u\nc1q+fLndcY5r3759eumll9TR0aHh4WG9//772rVrl6699toJvU/Rnw62epL2fGXo6enR9u3bde21\n1+blh0ouGXbt2qVdu3bp3/7t33TmmWfqyiuvtPyHbC45li9frn379imZTGpoaEivv/66KisrC5rh\nQx/6kH7/+99Lkvr6+vTOO+/opJNOsizDUaZpjvkbWL73y1xz5HvfzCVDIfbN8TLke7+0WjAYVFdX\nlyTp97//vU455RSbE41v+fLlevXVVyV9cPo9kUjY3gAez/z583Xfffel9s1gMKjvfve7tjaA4+ns\n7NSvfvUrffOb31RpaandcbLKpUYXoz179qiystIRdwVfdNFF2rNnj3bt2qXrr79en/zkJyfcAEoO\nmTGks7NTP/7xj1OTtNfW1qZN0i5JjY2N6uzsTE3SfuqppxY0ww9/+EO9+OKLCoVCMk1TXq9XW7du\nLWiGY+3evVuf+cxn8vaImPFy/OpXv9JTTz0lj8ejNWvWaN26dQXN8Oc//1m7d+/Wn//8Z0lSbW2t\n5adNdu7cqa6uLvX39+uEE05QOBzW6OhoQffLXHIUYt/M5e/iqHztm7lkyPd+aaU//OEP+vGPf6xk\nMqnS0lJdeeWV+uhHP2p3rOMaHR3Vnj179Mc//lGlpaX6x3/8R/3d3/2d3bFydu211+quu+4q6kfE\nfP3rX9fo6GiquV60aJGuvPJKm1Nlylaji9m+fft0++23a/78+TIMQ4Zh6MILL7TlGuqJ6urq0mOP\nPTapR8Q4ogkEAACAtYr+dDAAAACsRxMIAADgQjSBAAAALkQTCAAA4EI0gQAAAC5EEwgAAOBCNIEA\nABSpO++8U/fff7/dMTBN0QQCAAC4EE0gAACAC5XYHQCwQmdnp375y19q//79MgxDH/vYx3TZZZdp\n3rx5dkcDgClJJpNqampKzUW+Zs0affWrX7U5FaYDjgRiWhgaGtIXvvAF3XXXXbrjjjs0a9Ysffe7\n31UikbA7GgBMybPPPivTNLVlyxZdddVV+s1vfqP//u//tjsWpgGOBGJaOOOMM9KWr776al122WWK\nRqP6+Mc/blMqAJi68vJyXX755ZKkU045RQcOHNB//dd/6Qtf+ILNyeB0NIGYFt5991099NBDikaj\nOnz4sEzTlGma6unpoQkE4GiLFi1KWz7ttNP08MMPa3BwUH6/36ZUmA5oAjEt3HXXXfrQhz6kq666\nSsFgUB6PRzfccINGR0ftjgYAQFHimkA43sDAgA4cOKDzzz9fn/zkJ3XKKafoyJEjSiaTdkcDgCmL\nRqNpy6+99pqCwSBHATFlHAmE482aNUuBQEC/+c1vFAwGFY/H9cADD8jr9dodDQCmLB6Pq6mpSeec\nc47279+vxx57TBdccIHdsTANcCQQjmcYhm644Qbt379f3/jGN3T//ffrK1/5ikpLS+2OBgBTdtZZ\nZymZTOrWW2/VvffeqzVr1nBTCCxhmKZp2h0CAAAAhcWRQAAAABeiCQQAAHAhmkAAAAAXogkEAABw\nIZpAAAAAF6IJBAAAcCGaQAAAABeiCQQAAHAhmkAAAAAXogkEAABwIZpAAAAAF6IJBAAAcCGaQAAA\nABeiCQQAAHAhmkAAAAAXogkEAABwIZpAAAAAF7KsCezs7NT111+vjRs36pFHHhlzXDQa1YUXXqgX\nXnhhwtv+rUgkMqXMhea0vBKZC8FpeSXnZZ5qXurb+JyWV3JeZqfllZyX2Wl5palltqQJTCaTamxs\n1K233qrt27ertbVVb7/9dtZxP/vZz7RkyZIJb5uN0/6xnJZXInMhOC2v5LzMU8lLfcuN0/JKzsvs\ntLyS8zI7La9UBE1gNBrV3LlzFQqFVFJSopUrV6qtrS1j3K9//WudeeaZmj179oS3BQA7UN8ATFeW\nNIHxeFwVFRWp5WAwqHg8njGmra1N55xzzoS3BQC7UN8ATFclhfpGTU1Nuvjii6f0HpFIJO2wZzgc\nnmqsgnJaXonMheC0vJLzMofDYTU3N6eWq6qqVFVVZdn7U9+cl1dyXman5ZWcl9lpeaWp1TdLmsBg\nMKienp7UcjweVzAYTBvzxhtv6Ac/+IFM01R/f786Ojrk9Xpz2vaobB/swIEDVnyEgggEAurv77c7\nxoSQOf+clldyXuZTTjll0sWd+pYbp+0TkvMyOy2v5LzMTssrTa2+WdIELly4UN3d3YrFYiovL1dr\na6s2btyYNmbXrl2pr3fv3q3PfOYzWrZsmZLJ5LjbAoBdqG8ApitLmkCPx6O6ujo1NDTINE2tXr1a\nlZWVamlpkWEYqqmpmfC2AFAMqG8ApivDNE3T7hBTwemS/CJz/jktr+S8zKeccordESaF+pZfTsvs\ntLyS8zI7La80tfrGjCEAAAAuRBMIAADgQjSBAAAALlSw5wQChfLOO6b27/elrQuFkgqFRm1KBAAT\nE4uVKBZLP04zf76psjKbAmFaognEtHPwoEd7985IW7du3ZBCIZsCAcAExWKZdez88w2aQFiK08EA\nAAAuxJFAuMLQkEddXb6M9ZwmBgC4FU0gXKGvz6P29tKM9ZwmBgC4FU0gHCvbhdOSNDzs6OefAwBQ\nEDSBcKxsF05L0hlnGJJoBAEAOB5uDAEAAHAhmkAAAAAX4nQwAAAOlu36aJ58gFzQBAIA4GDZro/m\nyQfIBaeDAQAAXIgmEAAAwIVoAgEAAFyIJhAAAMCFaAIBAABcyLK7gzs7O9XU1CTTNLVq1SrV1tam\nvf7SSy/p4YcflmEY8nq9uvTSS7V48WJJ0oYNGzRz5szUa1u3brUqFgBMGfUN+TLW9JcDA14b0sBt\nLGkCk8mkGhsbtWnTJpWXl6u+vl7Lly/XvHnzUmM+9alPadmyZZKk/fv3a8eOHdqxY4ckyTAM3X77\n7SorK7MiDgBYhvqGfBpr+svq6oQNaeA2lpwOjkajmjt3rkKhkEpKSrRy5Uq1tbWljZkx4687+eDg\noAzDSC2bpinTZK5XAMWH+gZgurLkSGA8HldFRUVqORgMKhqNZox78cUX9eCDD+rw4cO6+eabU+sN\nw1BDQ4M8Ho/WrFmjmpoaK2IBwJRR3wBMVwWdMWTFihVasWKF9u3bp4ceeki33XabJGnz5s0qLy/X\n4cOHtXnzZlVWVqaupwEAJ6C+Id+OHDHV1eXLWM/1g5gsS5rAYDConp6e1HI8HlcwGBxz/OLFi3Xw\n4EENDAyorKxM5eXlkqTZs2drxYoVikajWYtkJBJRJBJJLYfDYQUCASs+QkH4fD5H5ZWKO7Pfb8jn\nK81YbxgJ+XzpRbG0dCTrWL/fUCCQeT1OIRXz3/FYnJi5ubk59XVVVZWqqqpy2o76lhsn7hPFkHms\nOpatZh06lFB7e2be00/PHFsMtU0qjr/jiXBa3qMmW98saQIXLlyo7u5uxWIxlZeXq7W1VRs3bkwb\n093drZNPPlmS9MYbb2h0dFRlZWUaGhqSaZry+/0aHBzUK6+8oi9/+ctZv0+2D9bf32/FRyiIQCDg\nqLxScWceHPRpeDjzWivTNDQ8nH5R9chIIuvYwcEh9fcP5y1jLor573gsTsscCAQUDocntS31LTdO\n2yek4sg8Vh3LVrOy1baxxhZDbZOK4+94IpyWV5pafbOkCfR4PKqrq1NDQ4NM09Tq1atVWVmplpYW\nGYahmpoavfDCC3rmmWdUUlIin8+nG264QZJ06NAhbdu2TYZhKJFI6KyzztKSJUusiAUAU0Z9AzBd\nGabDb1s7cOCA3RFy5tTfMIo1c1eXL+ujFc44w9ALL6Tv1tXVCbW3Z143s27dkP7u7zgSOFFOy3zK\nKafYHWFSqG/5VQyZx6pj2WpWtto21tjVq0c0Y0YyY2wolFQoNDrF1Lkrhr/jiXBaXmlq9a2gN4YA\nAID86+vzqL0981rDdeuGFArZEAhFiWnjAAAAXIgjgSh6TKsEAID1aAJR9JhWCQAA63E6GAAAwIVo\nAgEAAFyIJhAAAMCFuCYQrjY05MmYi7PQz9ECAMAONIFwtWzP0uI5WgAAN+B0MAAAgAvRBAIAALgQ\nTSAAAIAL0QQCAAC4EE0gAACAC9EEAgAAuBBNIAAAgAvxnEAUlVisRLFY+u8mAwNem9IAADB90QSi\nqMRiHu3dOyNtXXV1wqY0AABMX5wOBgAAcCGaQAAAABey7HRwZ2enmpqaZJqmVq1apdra2rTXX3rp\nJT388MMyDENer1eXXnqpFi9enNO2AGAn6huA6ciSJjCZTKqxsVGbNm1SeXm56uvrtXz5cs2bNy81\n5lOf+pSWLVsmSdq/f7927NihHTt25LQtANiF+gZgurLkdHA0GtXcuXMVCoVUUlKilStXqq2tLW3M\njBl/vdh/cHBQhmHkvC0A2IX6BmC6suRIYDweV0VFRWo5GAwqGo1mjHvxxRf14IMP6vDhw7r55psn\ntC0A2IH6BqvwCCwUm4I+ImbFihVasWKF9u3bp4ceeki33XbbhLaPRCKKRCKp5XA4rEAgYHXMvPH5\nfI7KKxU+s99vyOcrTVtXWjqSse546w0jIZ/Pm9PYbOv9fkOBwIyMsfnCflEYzc3Nqa+rqqpUVVVl\n6ftT35y3TxQ68xtvGHryyfR6c/rpudembLVtrLFj1Tzq2/E5Le9Rk61vljSBwWBQPT09qeV4PK5g\nMDjm+MWLF+vgwYMaGBiY0LbZPlh/f/8U0xdOIBBwVF6p8JkHB30aHjbT1o2MJDLWHW+9aRoaHk7k\nNDbb+sHBIfX3D08m/qSwX+RfIBBQOBye1LbUt9w4bZ+QnFffstW2scaO9b7Ut+NzWl5pavXNkmsC\nFy5cqO7ubsViMY2Ojqq1tTV1kfRR3d3dqa/feOMNjY6OqqysLKdtAcAu1DcA05UlRwI9Ho/q6urU\n0NAg0zS1evVqVVZWqqWlRYZhqKamRi+88IKeeeYZlZSUyOfz6YYbbjjutgBQDKhvAKYry64JXLp0\nqXbu3Jm2bu3atamvzzvvPJ133nk5bwsAxYL6BmA6YsYQAAAAF6IJBAAAcCGaQAAAABeiCQQAAHAh\nmkAAAAAXogkEAABwIZpAAAAAF6IJBAAAcCGaQAAAABeiCQQAAHAhmkAAAAAXogkEAABwIZpAAAAA\nF6IJBAAAcCGaQAAAABeiCQQAAHAhmkAAAAAXogkEAABwIZpAAAAAFyqxOwBQbIaGPOrq8mWsD4WS\nCoVGbUgEAID1LGsCOzs71dTUJNM0tWrVKtXW1qa9/txzz+nRRx+VJPn9fl155ZX6yEc+IknasGGD\nZs6cKcMw5PV6tXXrVqtiARPW1+dRe3tpxvp164YUCtkQCLajvmG64JdcHMuSJjCZTKqxsVGbNm1S\neXm56uvrtXz5cs2bNy815sQTT9Sdd96pmTNnqrOzU/fee6+2bNkiSTIMQ7fffrvKysqsiAMAlqG+\nYTrhl1wcy5JrAqPRqObOnatQKKSSkhKtXLlSbW1taWNOO+00zZw5U5K0aNEixePx1Gumaco0TSui\nAIClqG8ApitLjgTG43FVVFSkloPBoKLR6Jjjn3zySS1dujS1bBiGGhoa5PF4tGbNGtXU1FgRCwCm\njPoGYLoq+I0hr776qp566il9+9vfTq3bvHmzysvLdfjwYW3evFmVlZVavHhxxraRSESRSCS1HA6H\nFQgECpLbCj6fz1F5pcJn9vsN+XzppypKS0cy1h1vvWEk5PN5J/0eY431+w0FAjNy+hwTwX5RGM3N\nzamvq6qqVFVVZfn3oL45J6/kvPqWrbaNNXaidZP69gGn5T1qsvXNkiYwGAyqp6cntRyPxxUMBjPG\nvfXWW7r33nt1yy23pF0fU15eLkmaPXu2VqxYoWg0mrVIZvtg/f39VnyEgggEAo7KKxU+8+CgT8PD\n6afORkYSGeuOt940DQ0PJyb9HmONHRwcUn//cE6fYyLYL/IvEAgoHA5PalvqW26ctk9Izqtv2Wrb\nWGMnWjepbx9wWl5pavXNkmsCFy5cqO7ubsViMY2Ojqq1tVXLli1LG9PT06Pt27fr2muv1cknn5xa\nPzQ0pMHBQUnS4OCgXnnlFX34wx+2IhYATBn1DcB0ZcmRQI/Ho7q6OjU0NMg0Ta1evVqVlZVqaWmR\nYRiqqanRz3/+cw0MDKixsVGmaaYelXDo0CFt27ZNhmEokUjorLPO0pIlS6yIBQBTRn0DMF1Zdk3g\n0qVLtXPnzrR1a9euTX199dVX6+qrr87Y7sQTT9S2bdusigEAlqO+AZiOmDEEtojFShSLZV6NMDCQ\nedEzAACwHk0gbBGLebR3b+adaNXVmRc9A4CT8EsunIImEAAAC/FLLpzCkruDAQAA4Cw0gQAAAC5E\nEwgAAOBCNIEAAAAuRBMIAADgQjSBAAAALkQTCAAA4EI0gQAAAC5EEwgAAOBCNIEAAAAuRBMIAADg\nQjSBAAAALlRidwBMf7FYiWKx9N83Bga8NqUBAAASTSAKIBbzaO/eGWnrqqsTNqUBAAASp4MBAABc\niSYQAADAhWgCAQAAXMiyawI7OzvV1NQk0zS1atUq1dbWpr3+3HPP6dFHH5Uk+f1+XXnllfrIRz6S\n07ZAMRga8qiry5e2LhRKKhQatSkRCoX6BmA6sqQJTCaTamxs1KZNm1ReXq76+notX75c8+bNS405\n8cQTdeedd2rmzJnq7OzUvffeqy1btuS0LVAM+vo8am8vTVu3bt2QQiGbAqEgqG84Hp5+ACezpAmM\nRqOaO3euQn/5abhy5Uq1tbWlFbrTTjst9fWiRYsUj8dz3hYA7EJ9w/Hw9AM4mSXXBMbjcVVUVKSW\ng8Fgqghm8+STT2rp0qWT2hYACon6BmC6KvhzAl999VU99dRT+va3vz3hbSORiCKRSGo5HA4rEAhY\nGS+vfD6fo/JK1mT2+w35fOmnUUtLRzLWjbV+ImMlyTAS8vm8OY2d6vfz+w0FAjMyxk6EW/eLQmtu\nbk59XVVVpaqqKsu/B/XNOXkl59W3bLXNivc9+jmmWsuycdp+4bS8R022vlnSBAaDQfX09KSW4/G4\ngsFgxri33npL9957r2655RaVlZVNaFsp+wfr7++34iMURCAQcFReyZrMg4M+DQ+baetGRhIZ68Za\nP5GxkmSahoaHEzmNner3GxwcUn//cMbYiXDrflFIgUBA4XB4UttS33LjtH1Ccl59y1bbrHjfDz7H\n1GtZNk7bL5yWV5pafbPkdPDChQvV3d2tWCym0dFRtba2atmyZWljenp6tH37dl177bU6+eSTJ7Qt\nANiF+gY3OPr0g2P/xGJMKjbdWfIv7PF4VFdXp4aGBpmmqdWrV6uyslItLS0yDEM1NTX6+c9/roGB\nATU2Nso0TXm9Xm3dunXMbQGgGFDf4AY8/cCdLGvzly5dqp07d6atW7t2berrq6++WldffXXO2wJA\nsaC+AZiOmDEEAADAhWgCAQAAXIgmEAAAwIVoAgEAAFyIJhAAAMCFaAIBAABciCYQAADAhWgCAQAA\nXIgmEAAAwIVoAgEAAFyIJhAAAMCFaAIBAABciCYQAADAhWgCAQAAXIgmEAAAwIVoAgEAAFyIJhAA\nAMCFaAIBAABciCYQAADAhUqseqPOzk41NTXJNE2tWrVKtbW1aa8fOHBAu3fv1ptvvqkLL7xQX/zi\nF1OvbdiwQTNnzpRhGPJ6vdq6datVsQBgyqhvcKOhIY+6unwZ60OhpEKhURsSwWqWNIHJZFKNjY3a\ntGmTysvLVV9fr+XLl2vevHmpMWVlZbriiiv04osvZmxvGIZuv/12lZWVWREHACxDfYNb9fV51N5e\nmrF+3bohhUI2BILlLGkCo9Go5s6dq9Bf9oqVK1eqra0trUjOnj1bs2fP1ssvv5yxvWmaMk3TiigA\nYCnqGyQpFitRLJZ5BdXAgNeGNIA1LGkC4/G4KioqUsvBYFDRaDTn7Q3DUENDgzwej9asWaOamhor\nYgHAlFHfIEmxmEd7987IWF9dnbAhDWANy64JnIrNmzervLxchw8f1ubNm1VZWanFixfbHQsApoz6\nBqBYWdIEBoNB9fT0pJbj8biCwWDO25eXl0v64JTKihUrFI1GsxbJSCSiSCSSWg6HwwoEAlNIXlg+\nn89ReSVrMvv9hny+9OtKSktHMtaNtX4iYyXJMBLy+bw5jZ3q9/P7DQUCmUcHJsKt+0WhNTc3p76u\nqqpSVVVVTttR33LjxH1iIpmz1TFp6jVkImOz1TYr3nei7zGRuue0/cJpeY+abH2zpAlcuHChuru7\nFYvFVF5ertbWVm3cuHHM8cdeHzM0NCTTNOX3+zU4OKhXXnlFX/7yl7Nul+2D9ff3W/ERCiIQCDgq\nr2RN5sFBn4aH06+JGhlJZKwba/1ExkqSaRoaHk7kNHaq329wcEj9/cMZYyfCrftFIQUCAYXD4Ult\nS33LjdP2CWlimbPVMWnqNWQiY7PVNived6LvMZG657T9wml5panVN0uaQI/Ho7q6OjU0NMg0Ta1e\nvVqVlZVqaWmRYRiqqalRX1+f6uvr9f7778swDD3++OPasWOHDh8+rG3btskwDCUSCZ111llasmSJ\nFbEAYMqobwCmK8uuCVy6dKl27tyZtm7t2rWpr+fMmaM9e/ZkbOf3+7Vt2zarYgCA5ahvAKYjZgwB\nAABwIZpAAAAAF6IJBAAAcKGieE4g4FTMrQkAcCqaQGAKmFsTAOBUnA4GAABwIY4EwjJMsA4AgHPQ\nBMIyTLAOAIBzcDoYAADAhWgCAQAAXIgmEAAAwIVoAgEAAFyIJhAAAMCFaAIBAABciCYQAADAhWgC\nAQAAXIiHRWNSss0OwswgAAA4B00gJiXb7CDMDPJXQ0MedXX5MtaHQkmFQqM2JAKQC6a/hJvQBAJ5\n0NfnUXt7acb6deuGFArZEAhATpj+Em7CNYEAAAAuZNmRwM7OTjU1Nck0Ta1atUq1tbVprx84cEC7\nd+/Wm2++qQsvvFBf/OIXc94WAOxEfQMwHVlyJDCZTKqxsVG33nqrtm/frtbWVr399ttpY8rKynTF\nFVfoS1/60oS3BQC7UN8ATFeWNIHRaFRz585VKBRSSUmJVq5cqba2trQxs2fP1qmnniqv1zvhbQHA\nLtQ3IN3RG9+O/ROLcYuBE1nyrxaPx1VRUZFaDgaDikajed8WAPKN+gaky3bjGze9OZOjWvdIJKJI\nJJJaDofDCgQCNiaaGJ/P56i80tiZ/X5DPl96ESgtHclYN9b6fI2VJMNIyOfz5jS20Nn8fkOBQPqd\nh9Npvyhmzc3Nqa+rqqpUVVVlY5pM1LfCy5Y5W22TiqOGZKttVryvFe+RrbZJztsvnJb3qMnWN0ua\nwGAwqJ6entRyPB5XMBi0fNtsH6y/v38Sie0RCAQclVcaO/PgoE/Dw2baupGRRMa6sdbna6wkmaah\n4eFETmMLnW1wcEj9/cNp66bTflGsAoGAwuHwpLalvuXGafuElD1zttomFUcNyVbbrHhfK94jW22T\nnLdfOC2vNLX6Zsk1gQsXLlR3d7disZhGR0fV2tqqZcuWjTneNM1JbwsAhUR9AzBdWXIk0OPxqK6u\nTg0NDTJNU6tXr1ZlZaVaWlpkGIZqamrU19en+vp6vf/++zIMQ48//rh27Nghv9+fdVsAKAbUNwDT\nlWXXBC5dulQ7d+5MW7d27drU13PmzNGePXty3hYAigX1DcB0xIwhAAAALkQTCAAA4EI0gQAAAC5E\nEwgAAOBCNIEAAAAuRBMIAADgQjSBAAAALuSouYMBAEDxGRryqKvLl7F+/nxTZWU2BEJOaAIBAMCU\n9PV51N5emrH+/PMNmsAiRhOI43rnHVP792f+djcw4LUhDQAAsApNII7r4EGP9u6dkbG+ujphQxrn\ny3bKhNMlAAA70AQCBZTtlAmnSwAAduDuYAAAABfiSCAAwJWyXfPM9c5wE5pAAIArZbvmmeud4Sac\nDgYAAHAhmkAAAAAXogkEAABwIZpAAAAAF6IJBAAAcCHL7g7u7OxUU1OTTNPUqlWrVFtbmzHm/vvv\nV2dnp2bMmKFrrrlGH/3oRyVJGzZs0MyZM2UYhrxer7Zu3WpVLACYMuobgOnIkiYwmUz91vRcAAAR\nFElEQVSqsbFRmzZtUnl5uerr67V8+XLNmzcvNaajo0Pvvvuu7rnnHr3++uv60Y9+pC1btkiSDMPQ\n7bffrjKmTQBQZKhvAKYrS04HR6NRzZ07V6FQSCUlJVq5cqXa2trSxrS1tenss8+WJC1atEhHjhxR\nX1+fJMk0TZmmaUUUALAU9Q3AdGXJkcB4PK6KiorUcjAYVDQaHXdMPB7XnDlzZBiGGhoa5PF4tGbN\nGtXU1FgRCwCmjPoGYLoqihlDNm/erPLych0+fFibN29WZWWlFi9enDEuEokoEomklsPhsAKBQCGj\nTonP53NUXknyer3y+TIPGJeWjsjnKx13XaHHSpJhJOTzeXMaW+hs2dYPDY3qjTdmZ4w98cSk5s41\nMtYXAyfuy83Nzamvq6qqVFVVVZDvS32z3zvvmDp4MLOODQwYefk/na+x2WqbFe+bz8/h9SaLdr/I\nppj34+OZbH2zpAkMBoPq6elJLcfjcQWDwYwxvb29qeXe3t7UmPLycknS7NmztWLFCkWj0axFMtsH\n6+/vt+IjFEQgEHBUXklKJGZreHg0Y/3ISELDw+a46wo9VpJM09DwcCKnsYXOlm19b6+hF17IHLtu\n3bDKyoYz1hcDp+3LgUBA4XB4UttS33JTzPvE/v2+jOnhJOmMM6Th4fT/Y1b8n87X2Gy1zYr3zefn\nSCSMot0vsinm/XgsU6lvllwTuHDhQnV3dysWi2l0dFStra1atmxZ2phly5bp6aefliS99tprmjVr\nlubMmaOhoSENDg5KkgYHB/XKK6/owx/+sBWxMAGxWIm6unwZfw4f5lomuBv1DZi8I0fMjJ8rsVhR\nnISELDoS6PF4VFdXp4aGBpmmqdWrV6uyslItLS0yDEM1NTWqrq5WR0eHrrvuOvn9fq1fv16SdOjQ\nIW3btk2GYSiRSOiss87SkiVLrIiFCYjFMidSl6QzzjAk0QjCvahvwOT19Rl64YX0ny3r1g0pFLIp\nENJY1o4vXbpUO3fuTFu3du3atOW6urqM7U488URt27bNqhgAYDnqG4DpiBlDAAAAXIgmEAAAwIVo\nAgEAAFyIW3QAANNGLFaiWCz9+MbAQOaz9WCfoSGPurp8GetDoaRCocxHkiF/aAIBANNGticdVFdn\nPlsP9unr86i9PfPB0tw1XHg0gS7Eb8rOkO23ZX5TBgBYhSbQhfhN2Rmy/bbMb8oAAKtwYwgAAIAL\n0QQCAAC4EE0gAACAC3FNIOAgPFoB+EC2G9wkbnIDJoImcBqjSE4/PFoB+EC2G9wkbnIDJoImcBqj\nSAIAnIIzHYVHEwgAAGzHmY7CowkEABQ1HnAP5AdN4DRBkQQwXfGAeyA/aAId5ng3ezz7bPo/J0XS\nPZhiDsB0RX3LH5pAh+FmD2TDFHMApivqW/7QBAIAigKPtUKuuJPYGjSBwDQ1VpGcMUMaGsocT/FE\nPmRr7Mba1zjTgVyNdSfx6tUjOe9vsLAJ7OzsVFNTk0zT1KpVq1RbW5sx5v7771dnZ6dmzJihDRs2\naMGCBTlvC2BixiqS1dUJtbdnHlmheI6N+jZ52Rq7bPuaxBE/TB2njifGkiYwmUyqsbFRmzZtUnl5\nuerr67V8+XLNmzcvNaajo0Pvvvuu7rnnHr3++uu67777tGXLlpy2ne7GOgWS7YgNRRL5kq14jvXD\nev58U2VlhUpmL+qb9Y73CwpgNU4dj82SJjAajWru3LkK/aXVXrlypdra2tIKXVtbm84++2xJ0qJF\ni3TkyBH19fXp4MGD425bzMZq4Caycx3vFMjfHrGhSKKQxvphff75hmuaQDfXt4l45x1T+/dn/qDl\nF1fYbSIPoR5rP56ul9FY0gTG43FVVFSkloPBoKLR6Lhj4vF4TtsW2vvvl8g0jbR1hiH9v/83kjF2\nrAYu2xGUQMBUfz9FEs535IiZ8yMbsv2i5KSCOr3qm6GuLr+SyfS1FRWjKikxc/53yrZ+eNjQk09y\nPR+cI9sRwuPtx7leRuOk+maYpmlO9U2ef/55/e53v9NVV10lSXrmmWcUjUZ1xRVXpMbcddddOv/8\n8/Xxj39ckrR582ZdfPHFOnjw4LjbHhWJRBSJRFLL4XB4qtEBuERzc3Pq66qqKlVVVeW0HfUNQLGb\nbH3LPI85CcFgUD09PanleDyuYDCYMaa3tze13Nvbq2AwmNO2R1VVVSkcDqf+HPuhncBpeSUyF4LT\n8krOy9zc3JxWO3ItkBL1LVdOyys5L7PT8krOy+y0vNLU6pslTeDChQvV3d2tWCym0dFRtba2atmy\nZWljli1bpqefflqS9Nprr2nWrFmaM2dOTtsCgF2obwCmK0uuCfR4PKqrq1NDQ4NM09Tq1atVWVmp\nlpYWGYahmpoaVVdXq6OjQ9ddd538fr/Wr19/3G0BoBhQ3wBMV5ZcE2iXSCQyocOednNaXonMheC0\nvJLzMjstr+S8zE7LKzkvs9PySs7L7LS80tQyO7oJBAAAwORYck0gAAAAnIUmEAAAwIVoAgEAAFzI\nkruD7bZ371498cQT8ng8qq6u1sUXX2x3pHE99thjeuCBB9TY2KiyIp9/64EHHtDLL7+skpISnXTS\nSbrmmms0c+ZMu2Nl6OzsVFNTk0zT1KpVq1RbW2t3pOPq7e3Vrl27dOjQIRmGoTVr1ujzn/+83bHG\nlUwmVV9fr2AwqJtuusnuOOM6cuSIfvjDH+r//u//ZBiG1q9fr0WLFtkdKydOrG0S9S0fnFTfnFrb\nJPfVN8c3gZFIRC+//LLuvvtueb1eHT582O5I4+rt7dUrr7yiD33oQ3ZHycmnP/1pXXTRRfJ4PPrp\nT3+qRx55RBdddJHdsdIkk0k1NjZq06ZNKi8vV319vZYvX17Uc7R6vV5deumlWrBggQYHB3XTTTdp\nyZIlRZ1Zkh5//HHNmzdP77//vt1RcvLjH/9Yp59+uv75n/9ZiURCQ9nmcypCTqxtEvUtH5xW35xa\n2yT31TfHnw5+4oknVFtbK6/3gzn9Zs+ebXOi8f3kJz/RJZdcYneMnH3605+Wx/PBrrJo0aK0mRGK\nRTQa1dy5cxUKhVRSUqKVK1eqra3N7ljHNWfOHC1YsECS5Pf7NW/ePMXjcXtDjaO3t1cdHR1as2aN\n3VFycuTIEe3bt0+rVq2S9MEPp2I8ypONE2ubRH3LB6fVNyfWNsmd9c3xRwLfeecddXV16cEHH5TP\n59NXv/pVfexjH7M71pheeuklVVRUaP78+XZHmZTf/va3Wrlypd0xMsTjcVVUVKSWg8GgotGojYkm\n5uDBg3rrrbeK/jTl0R/wR44csTtKTg4ePKhAIKDdu3frrbfe0qmnnqrLL79cPp9v/I1t5rTaJlHf\n8sXJ9c0ptU1yZ31zRBO4efNmHTp0KLVsmqYMw9BXvvIVJRIJvffee9qyZYui0ah27NihXbt22Zj2\n+Hl/+ctf6lvf+lbaa8XgeJmPTnP1i1/8Ql6vV5/97GftijktDQ4O6vvf/74uu+wy+f1+u+OMqb29\nXSeccIIWLFigSCRSNPvu8SSTSb355v9v7+5Cmvr/OIC/t34TQVdssxZr7CK1i5LoZs2LCknoIiHo\ncWYPsAyUWYSlsogeBiobYTcJQg9joPRAJZoVUSOEFWFESsmSlJJRIx9YUFYqc+d/IZ1/9stlv3Lb\n8bxfVzue89XPUXj78fv9zvMGxcXFyMzMhNfrRUtLC3bs2JHo0gBIL9sA5hvNnFSyDZBvvkmiCTx+\n/Pi05+7fvw+LxQJg8hmfCoUCnz59glqtjld5/zJdvcFgEIODg6isrIQgCAiHw3A4HKitrcWCBQvi\nXOVUsb7HANDe3o7Ozk6cOHEiThX9Hq1Wi+HhYfE4HA5Dq9UmsKKZmZiYQF1dHdatWwez2ZzocmLq\n6enB06dP0dnZifHxcXz9+hX19fU4cOBAokubllarhU6nE2fQcnNz0dLSkuCq/k9q2QYw3xJBivkm\npWwD5JtvkmgCYzGbzeju7sby5csRCoUwMTGR8JCcjslkwvnz58XjsrIyuN3upH/3XFdXF27evAmn\n0wmVSpXocn4qKysL79+/x9DQEDQaDR49eoRDhw4luqxfamhogNFolMQ754qKisQN84FAAG1tbUkd\nkMDk3iSdTodQKASDwYAXL15I5tm9Uso2gPk2m6SYb1LKNkC++Sb5JjAvLw8NDQ04cuQIVCpV0v/Q\nvqdQKBJdwox4PB5EIhFUV1cDmNw8vX///gRXNZVSqURxcTGqq6shCALWr1+f9L/se3p64Pf7YTKZ\nUFVVBYVCgZ07d2LVqlWJLm1OsdlsOHv2LCKRiPgvQKRAytkGMN/+JqnlG7Mtfv403/jsYCIiIiIZ\nkvy/iCEiIiKi38cmkIiIiEiG2AQSERERyRCbQCIiIiIZYhNIREREJENsAomIiIhkiE0gyV4gEIDV\nasXIyMhPj4mI/oTT6YTH40nI17Zarejo6Jj2mOSNTSARERGRDLEJJCIiIpIhNoGUNLq6unDy5EnY\nbDbs27cPNTU1ePfu3bTXf1u2ffbsGaqqqrBr1y44HA68fv1avKa9vR179+796Tgu9xJRvESjUXi9\nXthsNthsNjQ1NcW83mq14u7du3C5XNizZw/Kysrg9/vF80NDQ7BarVPy7ts4LvfSTLEJpKQxNjaG\ngoICuFwunDp1CmlpaXC73ZiYmIg5rrGxEbt374bb7caiRYvgdrsxPj4unpfKM0yJaO7y+/0QBAE1\nNTUoKSmBz+fD7du3Y465du0azGYzTp8+jfz8fNTX1/+r6SP6E2wCKWlYLBasXr0aer0eJpMJpaWl\nGBwcRF9fX8xx27Ztw8qVK2E0GmG32zE+Po6HDx/GqWoiol/TaDSw2WwwGAzIzc3Fpk2bcOvWrZhj\nLBYL8vPzsXjxYmzZsgU5OTm4c+dOnComOfgn0QUQfTMwMIArV66gr68PHz9+hCAIEAQBw8PDaG5u\nxsuXLwEACxcuRF1dnTguOztbfJ2amgqTyYS3b9/GvX4ioul8n1MAsGzZMly9ehU3btxAa2srgMlV\nizNnzkCn04nX/Dims7MzPgWTLLAJpKThcrmQkZGBkpISaLVaKJVKlJeXIxKJoLS0VFzinTdv3ow/\np1KphCAIUz4WiUT+at1ERP9VXl4e1qxZIx5rNJoZjfvZNpdfbZ0h+hGXgykpjIyMIBQKYfPmzcjJ\nyYHBYMCXL18QjUYBTAajXq+HXq9HRkbGlLG9vb3i69HRUQSDQRiNRgDA/PnzMTY2htHRUfGa/v7+\n2b8hIqLv/Lit5dWrV9BqtdDpdGK26fV6KJXKKdd8r7e3F0uWLAEwmW0A8OHDB/H8mzdvZqt8mqM4\nE0hJIS0tDWq1Gj6fD1qtFuFwGE1NTTOa9WtuboZarYZGo8H169ehUqnEv6yzsrKQmpqKS5cuYePG\njejv78e9e/dm+3aIiKYIh8Pwer3YsGEDgsEg2trasHXr1phjnjx5gszMTKxYsQKPHz9Gd3c3amtr\nAQApKSnIzs5Ga2sr9Ho9Pn/+jMuXL8fjVmgO4UwgJQWFQoHy8nIEg0FUVFTA4/GgsLAQKpXql2OL\niorQ2NgIh8OBgYEBOBwOpKSkAADS09Nx8OBBPH/+HJWVlXjw4AEKCwtn+3aIiKZYu3YtotEojh07\nhnPnziE/Px8FBQUxx2zfvh0dHR2oqKiAz+eD3W7H0qVLxfN2ux0AcPToUVy4cIHZRr9NIfy4YYpI\nIgKBAJxOJy5evIj09PREl0NE9NdYrVYcPnwYFosl0aXQHMaZQCIiIiIZYhNIREREJENcDiYiIiKS\nIc4EEhEREckQm0AiIiIiGWITSERERCRDbAKJiIiIZIhNIBEREZEM/Q9huZz+L7tOFgAAAABJRU5E\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x116000090>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import scipy.optimize as optimize\n",
    "\n",
    "a, b = 1., 0.\n",
    "sigma = 1.\n",
    "m, n = 10000, 10\n",
    "xsize = 10.\n",
    "nbins = 40\n",
    "\n",
    "def experiment(n):\n",
    "    xs = xsize*stats.uniform.rvs(size=n)\n",
    "    ys = a*xs+b+stats.norm.rvs(scale=sigma, size=n)\n",
    "    return (xs, ys)\n",
    "\n",
    "xs = [experiment(n) for i in range(m)]\n",
    "\n",
    "def resfun(xs, ys, errors):\n",
    "    def fun(pars):\n",
    "        dx = (ys-pars[0]*xs+pars[1])/errors\n",
    "        return dx\n",
    "    return fun\n",
    "\n",
    "def ls(xi, x0):\n",
    "    xs, ys = xi[0], xi[1]\n",
    "    lsfun = resfun(xs, ys, sigma)\n",
    "    result = optimize.least_squares(lsfun, x0=x0)\n",
    "    chi2 = ls_chi2(result)\n",
    "    a, sigma_a = result.x[0], np.sqrt(result.cov[0, 0])\n",
    "    b, sigma_b = result.x[1], np.sqrt(result.cov[1, 1])\n",
    "    return (chi2, a, b, sigma_a, sigma_b)\n",
    "\n",
    "x0 = [a, b]\n",
    "lsvals = [ls(xi, x0) for xi in xs]\n",
    "\n",
    "fig, axs = plt.subplots(3, 2, figsize=(9 , 12))\n",
    "ax0, ax1, ax2, ax3, ax4, ax5 = axs.ravel()\n",
    "\n",
    "xxs, ys = xs[0][0], xs[0][1]\n",
    "ax0.plot(xxs, ys, color='black', marker='o', markersize=5, \n",
    "         ls='none', label='data')\n",
    "a0, b0 = lsvals[0][1], lsvals[0][2]\n",
    "ax0.plot(xxs, a0*xxs+b0, color='blue', lw=2, label='ls fit')\n",
    "ax0.legend(loc=2)\n",
    "ax0.set_xlabel(r'$x$', fontsize=14)\n",
    "ax0.set_ylabel(r'$y$', fontsize=14)\n",
    "\n",
    "chi2 = [xi[0] for xi in lsvals]\n",
    "ax1.hist(chi2, bins=50, normed=True, color='blue', alpha=0.5)\n",
    "ax1.set_xlabel(r'$\\chi^2$', fontsize=14)\n",
    "\n",
    "xas = np.array([xi[1] for xi in lsvals])\n",
    "xsas = np.array([xi[3] for xi in lsvals])\n",
    "ax2.hist(xas, bins=nbins, normed=True, color='blue', alpha=0.5)\n",
    "ax2.set_xlabel('a', fontsize=14)\n",
    "\n",
    "xbs = np.array([xi[2] for xi in lsvals])\n",
    "xsbs = np.array([xi[4] for xi in lsvals])\n",
    "ax3.hist(xbs, bins=nbins, normed=True, color='blue', alpha=0.5)\n",
    "ax3.set_xlabel('b', fontsize=14)\n",
    "\n",
    "aps = (xas-a)/xsas\n",
    "print ' pull a ', np.mean(aps), ' sigma ', np.std(aps)\n",
    "ax4.hist(aps, bins=nbins, normed=True, range=(-5., 5.), \n",
    "         color='blue', alpha=0.5)\n",
    "ax4.set_xlabel('a-pull', fontsize=14)\n",
    "\n",
    "bps = (xbs-b)/xsbs\n",
    "print ' pull b ', np.mean(bps), ' sigma ', np.std(bps)\n",
    "ax5.hist(bps, bins=nbins, normed=True, range=(-5., 5.),\n",
    "         color='blue', alpha=0.5)\n",
    "ax5.set_xlabel('b-pull', fontsize=14);\n",
    "\n",
    "fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Goodness of the fit\n",
    "\n",
    "How I can know is two samples come from the same pdf? Are they identical? \n",
    "\n",
    "To check the goodness of the fit we require a model and a test dataset (or two datasets). As we do for hypothesis testing, we need to define a test-statistics that quantifies the agreement of the test with the model. \n",
    "\n",
    "### p-values\n",
    "\n",
    "(See chapter on Hypothesis testing)\n",
    "\n",
    "The p-value measures the probability of obtaining a measurement equal or less compatible than the current measurement with the model, $H_0$. First, we define a test statistics, $t(x)$, which cuantifies the agreement of the measurement, $x$, with the model. Usually low values of $t(x)$ indicates good compatibility with $H_0$ and larger values bad compatibility. To compute the p-value we need to know the pdf of the test statistics: $f(t|H_0)$ \n",
    "\n",
    "From the test-statistic of our observation, $t_0$, we determine the p-value of the test, computing the integral:\n",
    "$$\n",
    "\\alpha = \\int_{t_0}^{\\infty} f(t | H_0) \\mathrm{d}t\n",
    "$$\n",
    "where $\\alpha$, is the p-value.\n",
    "\n",
    "If our dataset follows a hypothesis $H_0$, the p-values are uniform distributed between [0, 1]. If the p-value of our observation is low, i.e. 1%, then can conclude that we have a bad agreement with the model. In general if $p < 0.05 \\, (0.1)$ we exclude that the test dataset comes from $H_0$ at 95% (90%) CL. (see confidence level chapter)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Generate an experiment with $n$ samples normal distributed. Compute its $p$-value. What happens if you were considering that the model was a gaussian with mean 3 and sigma 1?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\chi^2$ test\n",
    "\n",
    "Consider the case of a model $H_0$, that predicts a set of measurements ${\\bf x}'$, with a covariance matrix ${\\bf V}$. The actual measurements are ${\\bf x}$. The $\\chi^2$ quantifies the agreement between the measurments and the predictions. In matricial form:\n",
    "\n",
    "$$\n",
    "\\chi^2 = ({\\bf x} - {\\bf x}') \\, {\\bf V} \\, ({\\bf x} - {\\bf x}')^t\n",
    "$$\n",
    "\n",
    "The $\\chi^2$ is distributed with a $\\chi^2(t | n)$ distribution of $n$ degrees of freedom, where $n$ is the number of measurements minus the number of parameters of our model that we have estimated from the data. \n",
    "\n",
    "We can use that we know the pdf of $\\chi^2$ to compute its p-value:\n",
    "\n",
    "$$\n",
    "\\alpha = \\int_{\\chi^2}^{\\infty} \\chi^2(t | n) \\mathrm{d} t\n",
    "$$\n",
    "\n",
    "This is the test we use to quantify the goodness of the a least squares fit! (see section)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Generate $n$ measurements according with a polinomial model $y = f(x)$ with equal error, $\\sigma$, for each measurement. Compute the $\\chi^2$ distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Kolmogorov-Smirnov test\n",
    "\n",
    "\n",
    "The KS test cuantifies the largest desviation of the cumulative of the test dataset with respect the cumulative of the model. The test only works on one dimension!\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Generate a datasets with $n$ samples of $x$, that is normal distributed. Do a KS test with respect the normal distribution. Repeat the test with respect a gaussian model of mean 3 and sigma 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "m, n = 10000, 100\n",
    "mu0, mu1 = 0., 0.\n",
    "sig0, sig1 = 1., 1.\n",
    "\n",
    "h0 = stats.norm(loc=mu0, scale=sig0)\n",
    "h1 = stats.norm(loc=mu1, scale=sig1)\n",
    "\n",
    "def experiment(n):\n",
    "    return h1.rvs(size=n)\n",
    "\n",
    "sets = [experiment(n) for i in range(m)]\n",
    "\n",
    "ksvals = [stats.kstest(iset, h0.cdf) for iset in sets]\n",
    "\n",
    "kss = np.array([val[0] for val in ksvals])\n",
    "pss = np.array([val[1] for val in ksvals])\n",
    "\n",
    "fig, axs = plt.subplots(2, 1, figsize=(9, 6))\n",
    "\n",
    "ax0, ax1 = axs.ravel()\n",
    "\n",
    "ax0.hist(kss, bins=50, normed=True, color='blue', alpha=0.5)\n",
    "ax0.set_xlabel('ks-test')\n",
    "\n",
    "ax1.hist(pss, bins=50, normed=True, color='blue', alpha=0.5)\n",
    "ax1.set_xlabel('p-value')\n",
    "\n",
    "fig.tight_layout();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## n-dimensional goodness of the fit tests\n",
    "\n",
    "Testing the goodness of the fit for n-dimensional model or dataset is complex. The KS test is only valid in 1D and the $\\chi^2$ test is not always possible. Moving to many dimensions, one faces the *curse of dimensionality*! In general in $n$ dimensions, we have two datasets and we are left with the question if they come from the same pdf, model!\n",
    "\n",
    "There is no unique or best method in a n-dimensional goodness of the test. The following methods, k nearest neighbor, energy test are used in HEP. (see ref [8])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### k nearest neighbour\n",
    "\n",
    "Consider the case where we have two datasets. If both they come from the same model, mixing them, will generate another dataset from the same model. \n",
    "\n",
    "The k-nearest neighbour is based on a test-statistic that measures the average fraction of neighbor points of the same dataset in a mix of the two datasets. First, we need to define the distance between two points, ${\\bf x}_i, \\ {\\bf x}_j$, in the $n$ dimensional space. The euclidian distance is a natural possibility, but it is not the only one.\n",
    "\n",
    "$$\n",
    "| {\\bf x}_i - {\\bf x}_j |^2 = \\sum_{k=i}^n \\left( \\frac{x^k_i-x^k_j}{w^k}\\right)^2 \n",
    "$$\n",
    "\n",
    "where $k$ is the index in the dimension of the space. For the euclidian distance with all coordinates equal, we take the weight $w^k =1$.\n",
    "\n",
    "The test statistics, $t$, is then computed from the average number of points from the same dataset with respect any point in the pooled sample.\n",
    "\n",
    "$$\n",
    "t = \\frac{1}{n_a+n_b} \\sum_{i=1}^{n_a+n_b} \\frac{1}{n_k} \\sum_{k=1}^{n_k} I(i,k)\n",
    "$$\n",
    "where $n_a, n_b$ are the sample sizes of the test and the model datasets, $n_k$ are the number of neighbors considered (i.e 5), and I(i, k) is a function, that gives 0, if the kth neighbour of the $i$ point is of the same dataset, (i.e if $i$ belongs to the sample $a$, and its kth neighbour belongs to sample $b$, then $I(i, k)$ is zero); and gives 1 if both points belongs to the same dataset.\n",
    "\n",
    "In the case that both datasets are from the same model, and have the same size, one expects that $t$ is distributed around $0.5$. If the datasets come from different models, $t$ should be greater, towards 1.\n",
    "\n",
    "To compute the p-value we need to obtain the distribution of the test statistics $f(t)$, we can do that even not knowing the model, via permutations (see section)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Generate two datasets of $n samples$ of $(x, y)$ distributed with a normal 2d gaussian. Compute the k nearest neighbor goodness of the fit."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Energy test\n",
    "\n",
    "The energy test is based on the potential of a charged particles in electromagnetics. Consider that the pdfs of the samples are $f_a({\\bf x}), \\, f_b({\\bf x})$. The Energy test statistics is defined as:\n",
    "\n",
    "$$\n",
    "t = \\frac{1}{2} \\int (f_a({\\bf x})-f_b({\\bf x'})) \\; \\Phi(|{\\bf x} - {\\bf x'}|) \\; (f_a({\\bf x})-f_b({\\bf x'})) \\mathrm{d}x \\mathrm{d}x'\n",
    "$$\n",
    "\n",
    "that is minimal when $f_a({\\bf x}) = f_b({\\bf x})$\n",
    "\n",
    "We have to define an energy interaction, by analogy with the electromagnetism, we can take:\n",
    "$$\n",
    "\\Phi(|{\\bf x} - {\\bf x'}|) = \\frac{1}{|{\\bf x} - {\\bf x'}|}\n",
    "$$\n",
    "but other choices are also possible, i.e:\n",
    "$$\n",
    "\\Phi(|{\\bf x} - {\\bf x'}|) = e^{-({\\bf x} - {\\bf x'})^2}\n",
    "$$\n",
    "\n",
    "\n",
    "\n",
    "There is no need to know the pdfs $f_a({\\bf x}), \\, f_b({\\bf x'})$, we can re-write the test-statistics:\n",
    "\n",
    "$$\n",
    "t = \\frac{1}{n_a (n_a-1)} \\sum_{i>j}^{n_a} \\Phi(\\Delta {\\bf x}_{ij}) + \\frac{1}{n_b (n_b-1)} \\sum_{i>j}^{n_b} \\Phi(\\Delta {\\bf x}_{ij}) - \\frac{1}{n_a n_b} \\sum_{i=1}^{n_a} \\sum_{j=1}^{n_b} \\Phi(\\Delta {\\bf x}_{ij})\n",
    "$$\n",
    "\n",
    "The Energy test is a powerful goodness of the test, but it can be expensive on computation time."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Compute the energy test for a 1D sample normal distributed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "m, n = 10000, 5\n",
    "\n",
    "mu0, mu1 = 0., 3.\n",
    "\n",
    "def ene(xi, xj):\n",
    "    # return 1./abs(xi-xj)\n",
    "    return np.exp(-(xi-xj)*(xi-xj))\n",
    "\n",
    "def total_ene(xas, xbs):\n",
    "    ee = 0.\n",
    "    na, nb = len(xas), len(xbs)\n",
    "    eas, ebs, eabs = 0., 0., 0.\n",
    "    for i in range(na):\n",
    "        for j in range(i+1, na):\n",
    "            eas += ene(xas[i], xas[j])\n",
    "    for i in range(nb):\n",
    "        for j in range(i+1, nb):\n",
    "            ebs += ene(xbs[i], xbs[j])\n",
    "    for i in range(na):\n",
    "        for j in range(nb):\n",
    "            eabs += ene(xas[i], xbs[j])\n",
    "    ee = eas/(na*(na-1))+ebs/(nb*(nb-1))-eabs/(na*nb)\n",
    "    return ee\n",
    "\n",
    "def experiment():\n",
    "    xas = stats.norm.rvs(loc=mu0, size=n)\n",
    "    xbs = stats.norm.rvs(loc=mu1, size=n)\n",
    "    return (xas, xbs)\n",
    "\n",
    "dsets = [experiment() for i in range(m)]\n",
    "\n",
    "enes0 = np.array([total_ene(dsets[i][0], dsets[m-1-i][0]) for i in range(m)])\n",
    "enes1 = np.array([total_ene(iset[0], iset[1]) for iset in dsets])\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "# emin = np.min(np.min(enes0), np.min(enes1))\n",
    "# emax = np.max(np.max(enes0), np.max(enes1))\n",
    "ax.hist(enes0, bins=50, normed=True, range=(np.min(enes0), np.max(enes0)), color='blue', alpha=0.5, label='H0')\n",
    "ax.hist(enes1, bins=50, normed=True, range=(np.min(enes1), np.max(enes1)), color='red', alpha=0.5, label='H1' )\n",
    "ax.set_xlabel('energy', fontsize=14)\n",
    "ax.set_ylabel('pdf', fontsize=14)\n",
    "ax.legend(fontsize=14);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### pdf of the test statistics from permutations\n",
    "\n",
    "To compute the p-value we need to know the pdf of the test statistics, $f(t|H_0)$. For the $\\chi^2$ test, it is just a $\\chi^2$ distribution with $n$ degrees of freedom. But in many other cases, in particular in the n-dimensional cases, we have to do MC datasets to obtain the pdf. \n",
    "\n",
    "One possibility is to use permutations of the samples of the datasets. In the case that both datasets, $A, \\,B$, are identical, any mixture of them is algo a dataset of the same model. We then can generate $m$ datasets doing permutations, by selecting randomly samples from both datasets, compute the test statistics, and computing its pdf distribution. \n",
    "\n",
    "Notice that this works because we are considering that both datasets are identical!, that is the p-value definition: what is the fraction of datasets with equal or worse test-statistics than the observed one if both datasets are identical."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Generate two datasets with $n$ samples of a 2d normal distribution. Compute a test-statistics and get its distribution using permutations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bibliography\n",
    "\n",
    "[1] \"Practical Statistic for LHC physicist,\" H. B. Prosper, CERN Academic Training Lectures (2015). https://indico.cern.ch/event/358542/ https://arxiv.org/pdf/1504.00945.pdf\n",
    "\n",
    "[2] \"Statistic for HEP,\" G. Cowan. CERN Academic Training Lectures (2012). http://indico.cern.ch/event/173726/\n",
    "\n",
    "[3] \"Statistics for Particle Physics,\" K. Cranmer, CERN Academic Training Lectures (2009). \n",
    "https://indico.cern.ch/event/48425/\n",
    "\n",
    "[4] \"Unified approach to the classical statistical analysis of small signals, \"G. J. Feldman and R. D. Cousins, Phys. Rev. D57 (1998) 3873. http://journals.aps.org/prd/abstract/10.1103/PhysRevD.57.3873\n",
    "\n",
    "[5] “Asymptotic formulae for likelihood-based tests of new physics,” Glen Cowan, Kyle Cranmer, Eilam Gross, Ofer Vitells. Eur. Phys. J. C71 1554 (2011). https://arxiv.org/abs/1007.1727\n",
    "\n",
    "[6] \"Incorporating systematic uncertainties into an upper limit,\" R.D. Cousins and V.L. Highland. Nucl. Instrum. Meth. A320, 331 (1992). http://www.sciencedirect.com/science/article/pii/0168900292907945\n",
    "\n",
    "[7] \"Confidence Level Computation for Combining Searches with Small Statistics,\" T. Junk, Nucl. Instrum. Meth. A434, 435 (1999). https://arxiv.org/abs/hep-ex/9902006  \n",
    "\n",
    "[8] \"How good are your fits? Unbinned multivariate goodness-of-fit tests in high energy physics,\" M. Willians, https://arxiv.org/abs/1006.3019\n",
    "\n",
    "[9] ROOT https://root.cern.ch, TMVA http://tmva.sourceforge.net, RooFit https://root.cern.ch/roofit\n",
    "\n",
    "[10] Anaconda https://anaconda.org, SciPy https://www.scipy.org, NumPy http://www.numpy.org, http://scikit-learn.org/stable/, http://matplotlib.org\n"
   ]
  }
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