{ "cells": [ { "cell_type": "markdown", "metadata": { "deletable": false }, "source": [ "# [Introduction to Data Science: A Comp-Math-Stat Approach](http://datascience-intro.github.io/1MS041-2020/)\n", "## 1MS041, 2020 \n", "©2020 Raazesh Sainudiin, Benny Avelin. [Attribution 4.0 International (CC BY 4.0)](https://creativecommons.org/licenses/by/4.0/)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 10. Convergence of Limits of Random Variables, Confidence Set Estimation and Testing\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Inference and Estimation: The Big Picture\n", "\n", "- [Limits](#Limits)\n", " - Limits of Sequences of Real Numbers\n", " - Limits of Functions\n", " - Limit of a Sequence of Random Variables\n", "- [Convergence in Distribution](#Convergence-in-Distribution)\n", "- [Convergence in Probability](#Convergence-in-Probability)\n", "- [Some Basic Limit Laws in Statistics](#Some-Basic-Limit-Laws-in-Statistics)\n", "- [Weak Law of Large Numbers](#Weak-Law-of-Large-Numbers)\n", "- [Central Limit Theorem](#Central-Limit-Theorem)\n", "- [Asymptotic Normality of the Maximum Likelihood Estimator](#Properties-of-the-MLE)\n", "- [Set Estimators - Confidence Intervals and Sets from Maximum Likelihood Estimators](#Confidence-Interval-and-Set-Estimation-from-MLE)\n", "- [Parametric Hypothesis Test - From Confidence Interval to Wald test](#Hypothesis-Testing)\n", " \n", "\n", "### Inference and Estimation: The Big Picture\n", "\n", "The Models and their maximum likelihood estimators we discussed earlier fit into our Big Picture, which is about inference and estimation and especially inference and estimation problems where computational techniques are helpful. \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "
 Point estimationSet estimationHypothesis Testing
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Parametric

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MLE of finitely many parameters
done

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Asymptotically Normal Confidence Intervals
about to see ...

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Wald Test from Confidence Interval
about to see ...

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Non-parametric
(infinite-dimensional parameter space)

\n", "		coming up ... coming up ... coming up ...
\n", "\n", "But before we move on we have to discuss what makes it all work: the idea of limits - where do you get to if you just keep going?\n", "\n", "## Limits\n", "\n", "We talked about the likelihood function and maximum likelihood estimators for making point estimates of model parameters. For example for the $Bernoulli(\\theta^*)$ RV (a $Bernoulli$ RV with true but possibly unknown parameter $\\theta^*$, we found that the likelihood function was $L_n(\\theta) = \\theta^{t_n}(1-\\theta)^{(n-t_n)}$ where $t_n = \\displaystyle\\sum_{i=1}^n x_i$. We also found the maxmimum likelihood estimator (MLE) for the $Bernoulli$ model, $\\widehat{\\theta}_n = \\frac{1}{n}\\displaystyle\\sum_{i=1}^n x_i$. \n", "\n", "We demonstrated these ideas using samples simulated from a $Bernoulli$ process with a secret $\\theta^*$. We had an interactive plot of the likelihood function where we could increase $n$, the number of simulated samples or the amount of data we had to base our estimate on, and see the effect on the shape of the likelihood function. The animation belows shows the changing likelihood function for the Bernoulli process with unknown $\\theta^*$ as $n$ (the amount of data) increases.\n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", " \n", " \n", "
Likelihood function for Bernoulli process, as $n$ goes from 1 to 1000 in a continuous loop.
\n", "\n", "For large $n$, you can probably make your own guess about the true value of $\\theta^*$ even without knowing $t_n$. As the animation progresses, we can see the likelihood function 'homing in' on $\\theta = 0.3$. \n", "\n", "We can see this in another way, by just looking at the sample mean as $n$ increases. An easy way to do this is with running means: generate a very large sample and then calculate the mean first over just the first observation in the sample, then the first two, first three, etc etc (running means were discussed in an earlier worksheet if you want to go back and review them in detail in your own time). Here we just define a function so that we can easily generate sequences of running means for our $Bernoulli$ process with the unknown $\\theta^*$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Preparation: Let's just evaluate the next cell and focus on concepts.\n", "\n", "You can see what they are as you need to." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "def likelihoodBernoulli(theta, n, tStatistic):\n", " '''Bernoulli likelihood function.\n", " theta in [0,1] is the theta to evaluate the likelihood at.\n", " n is the number of observations.\n", " tStatistic is the sum of the n Bernoulli observations.\n", " return a value for the likelihood of theta given the n observations and tStatistic.'''\n", " retValue = 0 # default return value\n", " if (theta >= 0 and theta <= 1): # check on theta\n", " mpfrTheta = RR(theta) # make sure we use a Sage mpfr \n", " retValue = (mpfrTheta^tStatistic)*(1-mpfrTheta)^(n-tStatistic)\n", " return retValue\n", " \n", "def bernoulliFInverse(u, theta):\n", " '''A function to evaluate the inverse CDF of a bernoulli.\n", " \n", " Param u is the value to evaluate the inverse CDF at.\n", " Param theta is the distribution parameters.\n", " Returns inverse CDF under theta evaluated at u'''\n", " \n", " return floor(u + theta)\n", " \n", "def bernoulliSample(n, theta, simSeed=None):\n", " '''A function to simulate samples from a bernoulli distribution.\n", " \n", " Param n is the number of samples to simulate.\n", " Param theta is the bernoulli distribution parameter.\n", " Param simSeed is a seed for the random number generator, defaulting to 30.\n", " Returns a simulated Bernoulli sample as a list.'''\n", " \n", " set_random_seed(simSeed)\n", " us = [random() for i in range(n)]\n", " set_random_seed(None)\n", " return [bernoulliFInverse(u, theta) for u in us] # use bernoulliFInverse in a list comprehension\n", " \n", "def bernoulliSampleSecretTheta(n, theta=0.30, simSeed=30):\n", " '''A function to simulate samples from a bernoulli distribution.\n", " \n", " Param n is the number of samples to simulate.\n", " Param theta is the bernoulli distribution parameter.\n", " Param simSeed is a seed for the random number generator, defaulting to 30.\n", " Returns a simulated Bernoulli sample as a list.'''\n", " \n", " set_random_seed(simSeed)\n", " us = [random() for i in range(n)]\n", " set_random_seed(None)\n", " return [bernoulliFInverse(u, theta) for u in us] # use bernoulliFInverse in a list comprehension\n", "\n", "def bernoulliRunningMeans(n, myTheta, mySeed = None):\n", " '''Function to give a list of n running means from bernoulli with specified theta.\n", " \n", " Param n is the number of running means to generate.\n", " Param myTheta is the theta for the Bernoulli distribution\n", " Param mySeed is a value for the seed of the random number generator, defaulting to None.'''\n", " \n", " sample = bernoulliSample(n, theta=myTheta, simSeed = mySeed)\n", " from pylab import cumsum # we can import in the middle of code\n", " csSample = list(cumsum(sample))\n", " samplesizes = range(1, n+1,1)\n", " return [RR(csSample[i])/samplesizes[i] for i in range(n)]\n", " \n", "#return a plot object for BernoulliLikelihood using the secret theta bernoulli generator\n", "def plotBernoulliLikelihoodSecretTheta(n):\n", " '''Return a plot object for BernoulliLikelihood using the secret theta bernoulli generator.\n", " \n", " Param n is the number of simulated samples to generate and do likelihood plot for.'''\n", " \n", " thisBSample = bernoulliSampleSecretTheta(n) # make sample\n", " tn = sum(thisBSample) # summary statistic\n", " from pylab import arange\n", " ths = arange(0,1,0.01) # get some values to plot against\n", " liks = [likelihoodBernoulli(t,n,tn) for t in ths] # use the likelihood function to generate likelihoods\n", " redshade = 1*n/1000 # fancy colours\n", " blueshade = 1 - redshade\n", " return line(zip(ths, liks), rgbcolor = (redshade, 0, blueshade))\n", " \n", "def cauchyFInverse(u):\n", " '''A function to evaluate the inverse CDF of a standard Cauchy distribution.\n", " \n", " Param u is the value to evaluate the inverse CDF at.'''\n", " \n", " return RR(tan(pi*(u-0.5)))\n", " \n", "def cauchySample(n):\n", " '''A function to simulate samples from a standard Cauchy distribution.\n", " \n", " Param n is the number of samples to simulate.'''\n", " \n", " us = [random() for i in range(n)]\n", " return [cauchyFInverse(u) for u in us]\n", "\n", "def cauchyRunningMeans(n):\n", " '''Function to give a list of n running means from standardCauchy.\n", " \n", " Param n is the number of running means to generate.'''\n", " \n", " sample = cauchySample(n)\n", " from pylab import cumsum\n", " csSample = list(cumsum(sample))\n", " samplesizes = range(1, n+1,1)\n", " return [RR(csSample[i])/samplesizes[i] for i in range(n)]\n", "\n", "def twoRunningMeansPlot(nToPlot, iters):\n", " '''Function to return a graphics array containing plots of running means for Bernoulli and Standard Cauchy.\n", " \n", " Param nToPlot is the number of running means to simulate for each iteration.\n", " Param iters is the number of iterations or sequences of running means or lines on each plot to draw.\n", " Returns a graphics array object containing both plots with titles.'''\n", " xvalues = range(1, nToPlot+1,1)\n", " for i in range(iters):\n", " shade = 0.5*(iters - 1 - i)/iters # to get different colours for the lines\n", " bRunningMeans = bernoulliSecretThetaRunningMeans(nToPlot)\n", " cRunningMeans = cauchyRunningMeans(nToPlot)\n", " bPts = zip(xvalues, bRunningMeans)\n", " cPts = zip(xvalues, cRunningMeans)\n", " if (i < 1):\n", " p1 = line(bPts, rgbcolor = (shade, 0, 1))\n", " p2 = line(cPts, rgbcolor = (1-shade, 0, shade))\n", " cauchyTitleMax = max(cRunningMeans) # for placement of cauchy title\n", " else:\n", " p1 += line(bPts, rgbcolor = (shade, 0, 1))\n", " p2 += line(cPts, rgbcolor = (1-shade, 0, shade))\n", " if max(cRunningMeans) > cauchyTitleMax: cauchyTitleMax = max(cRunningMeans)\n", " titleText1 = \"Bernoulli running means\" # make title text\n", " t1 = text(titleText1, (nToGenerate/2,1), rgbcolor='blue',fontsize=10) \n", " titleText2 = \"Standard Cauchy running means\" # make title text\n", " t2 = text(titleText2, (nToGenerate/2,ceil(cauchyTitleMax)+1), rgbcolor='red',fontsize=10)\n", " return graphics_array((p1+t1,p2+t2))\n", "\n", "def pmfPointMassPlot(theta):\n", " '''Returns a pmf plot for a point mass function with parameter theta.'''\n", " \n", " ptsize = 10\n", " linethick = 2\n", " fudgefactor = 0.07 # to fudge the bottom line drawing\n", " pmf = points((theta,1), rgbcolor=\"blue\", pointsize=ptsize)\n", " pmf += line([(theta,0),(theta,1)], rgbcolor=\"blue\", linestyle=':')\n", " pmf += points((theta,0), rgbcolor = \"white\", faceted = true, pointsize=ptsize)\n", " pmf += line([(min(theta-2,-2),0),(theta-0.05,0)], rgbcolor=\"blue\",thickness=linethick)\n", " pmf += line([(theta+.05,0),(theta+2,0)], rgbcolor=\"blue\",thickness=linethick)\n", " pmf+= text(\"Point mass f\", (theta,1.1), rgbcolor='blue',fontsize=10)\n", " pmf.axes_color('grey') \n", " return pmf\n", " \n", "def cdfPointMassPlot(theta):\n", " '''Returns a cdf plot for a point mass function with parameter theta.'''\n", " \n", " ptsize = 10\n", " linethick = 2\n", " fudgefactor = 0.07 # to fudge the bottom line drawing\n", " cdf = line([(min(theta-2,-2),0),(theta-0.05,0)], rgbcolor=\"blue\",thickness=linethick) # padding\n", " cdf += points((theta,1), rgbcolor=\"blue\", pointsize=ptsize)\n", " cdf += line([(theta,0),(theta,1)], rgbcolor=\"blue\", linestyle=':')\n", " cdf += line([(theta,1),(theta+2,1)], rgbcolor=\"blue\", thickness=linethick) # padding\n", " cdf += points((theta,0), rgbcolor = \"white\", faceted = true, pointsize=ptsize)\n", " cdf+= text(\"Point mass F\", (theta,1.1), rgbcolor='blue',fontsize=10)\n", " cdf.axes_color('grey') \n", " return cdf\n", " \n", "def uniformFInverse(u, theta1, theta2):\n", " '''A function to evaluate the inverse CDF of a uniform(theta1, theta2) distribution.\n", " \n", " u, u should be 0 <= u <= 1, is the value to evaluate the inverse CDF at.\n", " theta1, theta2, theta2 > theta1, are the uniform distribution parameters.'''\n", " \n", " return theta1 + (theta2 - theta1)*u\n", "\n", "def uniformSample(n, theta1, theta2):\n", " '''A function to simulate samples from a uniform distribution.\n", " \n", " n > 0 is the number of samples to simulate.\n", " theta1, theta2 (theta2 > theta1) are the uniform distribution parameters.'''\n", " \n", " us = [random() for i in range(n)]\n", " \n", " return [uniformFInverse(u, theta1, theta2) for u in us]\n", "\n", "def exponentialFInverse(u, lam):\n", " '''A function to evaluate the inverse CDF of a exponential distribution.\n", " \n", " u is the value to evaluate the inverse CDF at.\n", " lam is the exponential distribution parameter.'''\n", " \n", " # log without a base is the natural logarithm\n", " return (-1.0/lam)*log(1 - u)\n", " \n", "def exponentialSample(n, lam):\n", " '''A function to simulate samples from an exponential distribution.\n", " \n", " n is the number of samples to simulate.\n", " lam is the exponential distribution parameter.'''\n", " \n", " us = [random() for i in range(n)]\n", " \n", " return [exponentialFInverse(u, lam) for u in us]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To get back to our running means of Bernoullin RVs:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "def bernoulliSecretThetaRunningMeans(n, mySeed = None):\n", " '''Function to give a list of n running means from Bernoulli with unknown theta.\n", " \n", " Param n is the number of running means to generate.\n", " Param mySeed is a value for the seed of the random number generator, defaulting to None\n", " Note: the unknown theta parameter for the Bernoulli process is defined in bernoulliSampleSecretTheta\n", " Return a list of n running means.'''\n", " \n", " sample = bernoulliSampleSecretTheta(n, simSeed = mySeed)\n", " from pylab import cumsum # we can import in the middle of code\n", " csSample = list(cumsum(sample))\n", " samplesizes = range(1, n+1,1)\n", " return [RR(csSample[i])/samplesizes[i] for i in range(n)]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can use this function to look at say 5 different sequences of running means (they will be different, because for each iteration, we will simulate a different sample of $Bernoulli$ observations). " ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 5 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "nToGenerate = 1500\n", "iterations = 5\n", "xvalues = range(1, nToGenerate+1,1)\n", "for i in range(iterations):\n", " redshade = 0.5*(iterations - 1 - i)/iterations # to get different colours for the lines\n", " bRunningMeans = bernoulliSecretThetaRunningMeans(nToGenerate)\n", " pts = zip(xvalues,bRunningMeans)\n", " if (i == 0):\n", " p = line(pts, rgbcolor = (redshade,0,1))\n", " else:\n", " p += line(pts, rgbcolor = (redshade,0,1))\n", "show(p, figsize=[5,3], axes_labels=['n','sample mean'])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "What we notice is how the different lines **converge** on a sample mean of close to 0.3. \n", "\n", "Is life always this easy? Unfortunately no. In the plot below we show the well-behaved running means for the $Bernoulli$ and beside them the running means for simulated standard $Cauchy$ random variables. They are all over the place, and each time you re-evaluate the cell you'll get different all-over-the-place behaviour. " ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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F7P/vf69q1jx2rBj7ZOVKUU7PmuWdsqK2hAQxgvacOeLG7MqVIpn0B3feKc7Fww+L2L74oqrFoqcHEHvySeDDD8V52LtXfEeWLROPgQNEc3CHA3jjDeDwYTHY2VtvtWwfR46IBHvLFnHD+qefxHH16ePZYyHylZgzzxSsXXudnZ1dp5a7OpbZrUBeniiYPv5YDNp05IgYyfPFF0VB6da1q0gIMzPFiMmAKFiWL69qrjx1atPNm2qbOlUkt7fdJgaHWrVKjDRZXUKCSJZXrxY/pk88UffubH169RKDWN1+u+hLtmOHqP1tqhb5+edFgTlihCgw0tJE8vXee6Kpb0mJGP1bp6sqLL79VgyMVt2IEaJJ/L//LRLDTz+t2xz/qaeAzz4T/6aniyZWL77Y9LHVNnOmSPqdTnHj42yMHSsKvY0bxUXM9Ok1m3MrpXdv8QzzO+4QI9Tv3Cn+Nho9f5HQnIvEhARxntLTxffqppua1zKhuiefBL75Rnwv9u4Vg/fxIqFd8ctEm6gpP/4ouvbExooyzj36t7vrlMkkBpLs3FkM9Nmnj7j5W14OnOs1oMkkrgPy80Wf4+uuE62Bqg/ie+utouyaNk2MSh0f3/jAoJ6i1YqyfN8+0R1u3jzxCDB/EB8vBoNdvlzc3F+8uGrUcb3es/u69FJRnq1ZIz6jkSPFQKTuVnaDBonpefNELfQnn4inxbSEySTO87XXihrxO+4QI5/feadnj4XIV+Lj4xETE4M1a9ZUzrPZbNiwYQOSk5MVjIzOWWCgKCxffRW48ELxw/fEEyI5rV54vfKK+OGMixO1yoBYJzRU1DZPnix+YKv3o23u/r/7TiSzgweLH/9582ou849/iAJ7yhQRa16euEvdHEuXiphHjxbbuOMOUVvZmNBQUat+882ioBw8WNRaf/aZeKSH2SxqI99/X1xgJCaKJLf2DYKwMHEDY9UqUev82Wc1H5UFiIuTL78UifqgQSLZ/f335h1bdTfeKPoxT50q7vifjUcfFd+Byy8XtbVXXVVzMBIlffihqLG48EJxI+H220Wf57M91oY05yLxvffEzabBg0VNyl13Nf2dqk2nE+d7wABxTGq16LNN7YYky83tJXn2SkpKcPDgQQDA4MGDMX/+fIwZMwZhYWHo3Llzk+sXFRXBbDbDYrHwTjlRG/Lcc6ImmS1NiXyjqfJ43rx5mDt3LpYuXYoePXrg+eefx/r167F//34EBQU1ax8ss4m85MQJ0eogJaXlNztao5MnxQ2UtWubHkyOyA9pfLGT7du3Y0y16rz77rsPADB9+nS839Rol0TUZixaJGqYw8NFv/aXXlLm+eNE7VVT5fFDDz2E8vJyzJo1CwUFBRgxYgR++umnZifZROQFdrvoV/3II6KJVltNstetE032+/cXx/vQQ+LGwoUXKh0Z0VnxSY32ueLdcaK24d57RV/p/HzRYuvvfxetqjQ+ueVHRL7AMpvIw9avF/3PevYUfbCqP7qqLVm9Grj/ftEfPihIdFdYsKDp0dWJ/BQTbSIiIvIYltlEREQcDI2IiIiIiIjIo5hoExEREREREXkQE20iIiIiIiIiD2KiTURERERERORBTLSJiIiIiIiIPIiJNhEREREREZEHMdEmIiIiIiIi8iAm2kREREREREQexESbiIiIiIiIyIOYaBMRERERERF5EBNtIiIiOmcLFy5EYmIikpKSAAA7zaNRceC4wlEREREpg4k2ERERnbPZs2cjLS0NKSkplfMs3/7qs/3vuXsJfr/0KZ/tj4iIqDEapQMgIiKitkl2OHy2r6Ovf+ezfRERETXFr2u0azdDIyIiotZDdrp8vk9bXpHP90lERFSbXyfa9TVDIyIiotZBdjh9tq+Q4T0BAIXbDvhsn0RERA3x60SbiIiIWjEf1mibuscAAAq3M9EmIiLlMdEmIiIir/BlH21IEgDAsuOQ7/ZJRETUACbaRERE5BW+bDruxkSbiIj8ARNtIiIi8g4FBkOrOJkLa3ahz/dLRERUHRNtIiIi8gpf12gbOoYDAIr+POrT/RIREdXGRJuIiIi8wqd9tCEGRFMZdSjefcyn+yUiIqqNiTYRERF5ha9rtCW1CkF9O6No1xGf7peIiKg2JtpERETkFUoMhhY8MJ6JNhERKY6JNhEREXmHQol2SdoJuOy+bbZORERUHRNtIiIi8gpFarQHdIXL5kDpX6d9vm8iIiI3JtpERETkFb4eDA0Agvp2BgAU7+GAaEREpBwm2kREROQVStRo6yKCYeoeg4It+3y+byIiIjcm2kREROQVSiTaABB6Xm/kb2aiTUREymGiTURERN6hVKKd3BtFOw/DWWZVZP9ERERMtImIiOicLVy4EImJiUhKSqqcp2SNtuxw4tjbPyqyfyIiIibaREREdM5mz56NtLQ0pKSkVM5TKtF2D4iWdt+7iuyfiIiIiTYRERF5hRKjjgOASquBeVgCAMBZzubjRETke0y0iYiIyCuUqtEGgL6v3Q4AyPt1r2IxEBFR+8VEm4iIiLzDl4m2LNeYDD2vNwxxEchetcN3MRAREZ3BRJuIiIi8QskabUmSEHnJYOT9vEuxGIiIqP3y60S7vhFMiYiIqHWQncol2gAQMXYAivceR0VmgaJxEBFR++PXiXZ9I5gSERFR66BkjTYAhI/pDwDYcc1cReMgIqL2x68TbSIiImrFfJ1oS1KNSUNsGADAXljq2ziIiKjdY6JNREREXqF0jTYA9HxmKqwZ+ZBdLqVDISKidoSJNhEREXmFPyTaYcl9YC8sRcm+k0qHQkRE7QgTbSIiIvIKf0i0Q0b0BFQq5P+WrnQoRETUjjDRJiIiIq+QHQ6lQ4Am0IiA7jEoSWeNNhER+Q4TbSIiIvIOP6jRBoCg/l1w6pP1SodBRETtCBNtIiIi8gp/aDoOiObjtmwLLH8cUjoUIiJqJ5hoExERkVf4S6Id/6/LAQA5a1MVjoSIiNoLJtpERETkFf6SaKuNeuijQ7Dv4Q8gy7LS4RARUTvARJuIiIi8wh8GQ3OLv/dKAOBjvoiIyCeYaBMREZF3+EmNNgB0nT0RklaD3J93KR0KERG1A0y0iYiI6JwtXLgQiYmJSEpKUjqUemkCjQgb1Qe5P7GfNhEReR8TbSIiIjpns2fPRlpaGlJSUpQOpUGRlw5G7ro/4aywKR0KERG1cUy0iYiIqF2IujwJztIK5G3Yo3QoRETUxjHRJiIionYhqG9nGLtEIft7/611JyKitoGJNhEREbULkiQh6vJhyPo+hY/5IiIir2KiTURERF7jbwlt9OVJKD+ajZK0E0qHQkREbRgTbSIiIvIa2e4/z9IGgPCL+kMdYEDWd9uUDoWIiNowJtpERETkNbLVRyN8N7PmXG3QIWxUH+x79EMvB0RERO2ZXyfa/v5MTiIiImqcbLMrHUId6iAjAKD0cKbCkRARUVvl14l2a3gmJxERETXM5YfPrO6/6B8AgKyvtyocCRERtVV+nWgTERFR6+azpuMtoI8KQdSkYcj8lv20iYjIO5hoExERkdf4Y402IEYfL/gtHbbcIqVDISKiNoiJNhEREXmNP9ZoA0DsdaMAScKJ939WOhQiImqDmGgTERGR1/hrjbYuIhjB/bsg/cGlyFmbqnQ4RETUxjDRJiIiIq/xZY22JEktWn7g+3cDAI4v+ckb4RARUTvGRJuIiIi8xl9rtAEguH9XdLz5IpTuP6V0KERE1MYw0SYiIiKvkf040QaAmGvOQ9GuIyjafVTpUIiIqA1hok1ERERe4/LTwdDcoi9Pgi7SjOPvrFY6FCIiakOYaBMREdE5W7hwIRITE5GUlFRjvr/XaKu0GjiKynD0zZXIXbdL6XCIiKiNYKJNRERE52z27NlIS0tDSkpKjfn+XqMNAKGj+gAAtl78BGwFJQpHQ0REbQETbSIiIvIaf6/RBoCkr/9d+ff2K/6jYCRERNRWMNEmIiIir6k4cFzpEJqkCTJhom05tGFByN+UBnsha7WJiOjcMNEmIiIiryn8eoPSITSLSqvB6L1vQqXT4NjbHBiNiIjODRNtIiIi8prQ68cpHUKzGWJC0XHaWBx57Tu4bHalwyEiolaMiTYRERF5jVxuVTqEFul235WwZuQj85vflQ6FiIhaMSbaRERE5DWy06l0CC0S1CcOKr0W1sxCpUMhIqJW7KwS7UWLFiE+Ph4GgwFDhw7Fxo0bG13+k08+wcCBA2EymRAbG4tbbrkFeXl5ZxUwERERtSKy0gG0nKTVQHa0rhsERETkX1qcaC9btgz33HMPHnvsMezcuRMXXHABLrvsMhw/Xv+oops2bcK0adNw2223Ye/evfjyyy+RkpKCmTNnnnPwRERE5Ofk1pdpq7RquOwOpcMgIqJWrMWJ9vz583Hbbbdh5syZ6NOnDxYsWIC4uDgsXry43uW3bt2Krl274q677kJ8fDzOP/983Hnnndi+ffs5B09ERER+TJJ8ty8P5vOSVg3ZzhptIiI6ey1KtG02G3bs2IFLLrmkxvxLLrkEmzdvrned5ORknDx5EqtWrYIsy8jKysJXX32FSZMmNbgfq9WKoqKiGi8iIiJqfeRWWaOtYY02ERGdkxYl2rm5uXA6nYiOjq4xPzo6GpmZmfWuk5ycjE8++QRTpkyBTqdDTEwMQkJC8MYbbzS4n7lz58JsNle+4uLiWhImERER+QNJap1Nx/Va2HJ5k5+IiM7eWQ2GJtVqCibLcp15bmlpabjrrrvw5JNPYseOHfjxxx9x5MgR/OMf/2hw+48++igsFkvl68SJE2cTJhERESmplSbaEeMH4djCVbDlFysdChERtVItSrQjIiKgVqvr1F5nZ2fXqeV2mzt3LkaNGoUHH3wQAwYMwKWXXopFixbhvffeQ0ZGRr3r6PV6BAcH13gRERER+ULCI9cCADYl3a9wJERE1Fq1KNHW6XQYOnQo1qxZU2P+mjVrkJycXO86ZWVlUKlq7katVgNonf22iIiI2qo5c+ZAkqQar5iYmLPfoIRW+XgvU9doRIwbiLLDmSj684jS4RARUSvU4qbj9913H/773//ivffeQ3p6Ou69914cP368sin4o48+imnTplUuP3nyZCxfvhyLFy/G4cOH8dtvv+Guu+7C8OHD0aFDB88dCREREZ2zvn37IiMjo/K1e/fus96W1EqbjgPA8FVPwdglCvuf+ETpUIiIqBXStHSFKVOmIC8vD8888wwyMjLQr18/rFq1Cl26dAEAZGRk1Him9owZM1BcXIw333wT999/P0JCQjB27FjMmzfPc0dBREREHqHRaM6tFrs6SWq1rddUWg16PXczUm+ej7xf9yD8wn5Kh0RERK2IJLeCErCoqAhmsxkWi4X9tYmIiLxkzpw5eOmll2A2m6HX6zFixAg8//zz6NatW4PrWK1WWK3WyumioiLExcVhPQYhSG9E6JTxiP/gaa/H/scNL8GWW4SRa5/12DZllwubhj8ASa3CqC0vQlKd1RiyRETUDrHEICIiIgDAiBEj8OGHH2L16tVYsmQJMjMzkZycjLy8vAbXafSRnK246TgASCoVEl+5FYXb/sKpTzcoHQ4REbUiTLSJiIgIAHDZZZfh2muvRf/+/TFu3DisXLkSAPDBBx80uE6Tj+RsxYk2AISP7oeYa87Dvkc/gr2wROlwiIiolWCiTURERPUKCAhA//79ceDAgQaXafSRnL4edVzyzmYTX7kVjuJy/HnHQu/sgIiI2hwm2kRERFQvq9WK9PR0xMbGntX6kuSlzNfHTF2j0eOJ65G5fAsf90VERM3CRJuIiIgAAA888AA2bNiAI0eO4Pfff8d1112HoqIiTJ8+vVnruxxOrImpesRna++jXV38XZNh6BiOQy+tUDoUIiJqBVr8eC8iIiJqm06ePIkbb7wRubm5iIyMxMiRI7F169bKR3g2xVFcDme5tca8VvBwk2ZRaTUIHzsAJeknml6YiIjaPSbaREREBAD4/PPPPbvBNtJ03E12ulD4+1+wZhdCHxWidDhEROTH2HSciIiIvMJVYYX9RJbSYXiMs6QcAHDyg3UKR0JERP6OiTYRERF5h8MJR2Gx0lF4zKAP7gEApD/0vrKBEBGR32OiTUS1DmuAAAAgAElEQVRERF5h6N0VanOg0mF4jCbIVPl3YUrDjzwjIiJiok1EREQeUbtLtirACFdZhTLBeMlE+woED+6G35Ifgr2wROlwiIjITzHRJiIiIq9QmQxtLtFWadQY8M5syA4nNg65V+lwiIjITzHRJiIiIq9oi4k2AIQM64HuD1+LsiNZOLF0rdLhEBGRH+LjvYiIiMg7VBLg8s1ztH39vO7ec6fBnl+MXTPfhHlodwQPiPfp/omIyL+xRpuIiIg8o3YnbUkCfJwA+4okSejz4gzA5UJJ+kmlwyEiIj/j14n2woULkZiYiKSkJKVDISIioqbUk1Q7rXYFAvENSSsaBspOl8KREBGRv/HrRHv27NlIS0tDSkqK0qEQERFRCxUdzoU1rwSWg9lKh+IVklpcRjHRJiKi2vw60SYiIqJWpFbTcafNCQCw5pUqEY3XMdEmIqKGMNEmIiKic7Zw4UKMGD6ixjyVRlxm2Eva3sjjQPVE26lwJERE5G+YaBMREdE5mz17Nn7f9nuNeSqd6MNc0VZrtFWs0SYiovox0SYiIiKvUOu1kCCjIqdY6VC8RlKrmGgTEVEdTLSJiIjIKyStGgBgL7UpHIn3MNEmIqL6MNEmIiIir2gXTauZaBMRUT2YaBMREZFn1Bp13D3dlhNRV7kNaff8FwW/71c6FCIi8iNMtImIiMg7zuTdLkfbTbR7PH49AOC3kQ/i6OJVsOZYFI6IiIj8ARNtIiIi8gxZrjktSZAgt+ka7V7P3owJxcvQYcoF2DPrLayJ+jvKT+YqHRYRESmMiTYRERF5VVtOtAFAE2jE4M8eQK//3AwA2HrxE7DlFSkcFRERKYmJNhEREXmHr/to1+4j7kOSJKHHY9djzIG3UPrXKazrdgec5VbF4iEiImUx0SYiIiKPUGk1NWecyXtlp1x34TYqIKEDEl+5FY6iMqzvPQty7eb0RETULjDRJiIiIo9Qm/SozK4BQFJBggxXG286Xlu3+65CtweuRvnxHJTsO9ni9e1FZSg/nuOFyIiIyFeYaBMREZHnVGu+7f5TdjgVCkY5CQ9fC0gSTr7/c4vXXW2+AT93uQ0nP/7FC5EREZEvMNEmIiIiL3H30W5/zad1EcHo8fj1OPTicnwvXYHyU3nNXlcTbAIApP79VY5gTkTUSjHRJiIiIo+pMR7Zmb/tpe1zULCec25E1KRhAIBNw+6Do7isWesF9esM87AE6KLM2DbxafbzJiJqhZhoExERkXecyborcksUDkQZkkqF4d8/iVFbXoSzzIo/bnwZrlrN6MuOZOL3S59C2dGsqpkyENy/C3o9PRXFu49hpepKfC9dgfzf0nx8BEREdLaYaBMREZGXSJAgt9tE2y10ZG8MWfYgslduxyrt1bCkHq58L/2RD5Hz005sSrofp5dtxPfSFSjYsg+QJHS8+SKEJveuXHbz+Y/gyJvf87FhREStABNtIiIi8pzqbcfP/FmR44NE28+bV0dNGIqER68DAGwcfA+OvfMjAEAfHQIAsOUW4eALX1UuX5FRAE2gEckbX8DIX57D+OyP0OHGC7H3X+/gB9PfsOfuJXDZHb4/ECIiaha/TrQXLlyIxMREJCUlKR0KERERtZQkLjNshc3rm9zW9X5+GiYUfY7APnHYfecibJv0DEoPZiCgV0cEJsahKPUI9NEh6PHUDeg6eyIA0fw84qL+0EeaMeTTBzBsxb8BAEdf/w6rdNdgx5QXa9SQExGRf/DrRHv27NlIS0tDSkqK0qEQERFRI9w3x+3OqlpWubQMEgBtYbZygfkZTZAJo/e+iYFL70b2qu3I+WEHZLsT5//+MmKvS4YxPhq95kxF9KT6KxlirhqJSy2fo8OUCwAAGV9swsbB92D/nE99eRhERNQEv060iYiIqHVw3xzXarSV85zZ4tFUoXK+UmH5JUmSEDfjYlywcwHMwxIQe10yNIFGDP3yEYz6bV6T62uDTRjy+YO4XP4WYw69g9jrknH45a9hL2zffeGJiPwJE20iIiLyGJVOU/m3pFaLfyHDabUrFZLfMg/qhgtS5qPPvBmV8yRVyy7NArrFoOecqXCW27Ax6f46o5oTEZEymGgTERGRx0j6qhptVEsabUUVCkTTPgT17YzE+bei7GAGfu50C4688T2fvU1EpDAm2kREROQxkrraqONqcZkhQYa9hI+k8qZud1+B/m/NgjWrEHvvegcrVVdyVHIiIgUx0SYiIiKPkao93kuqlmg7mGh7XZc7J2CibXnl9NqOt2D/nE9hL+Ko70REvsZEm4iIiDyneh9jVVUf7Yo8DtTlCyqtBpfL3+LC1NcQdmFfHHj6c6w234C0+99Fyf6TKDuWDdnJftxERN6maXoRIiIiouaRVNWajmvEZYYaTpSdtigUUfsUPDAew756BOUnc7Hrltdw4oN1ODz/GwCANiQAyZtfRFCfOIWjJCJqu1ijTURERJ5Tvel4oAkAoIMNpacKlIqoXTN2isDINc9i3MmlSHz1NmiCjJCdLmwcci+2jn8CZUcylQ6RiKhNYqJNREREHlO9RlsVHgoA0EoOlJ4qVCokAqA26NDtnisxoWgZxmd+iB6PX4/ctbuwrtsdKD+Vp3R4RERtDhNtIiIi8pxqibZ2SD8AQGlkPBNtP6I26dHjsesx5MuHAQA/d7oF2yY/i0Mvr8DRxas4eBoRkQewjzYRERF5TI1Rx7Va2KGBOiQIJZlFCkZF9elw3SgEpS3EscU/oGDrfqQ/uBQAsGfWW+gyayIiLx2MqMuGQqXl5SIRUUu1il9OWVY6AiIiImqW6oOhnaEN0qMsg4Oh+aOgPnHo9/odAICKzALkrP4DJ95bi6xvfsexRasAACqDDkOWPYioCUOg0mmVDJeIqNVoFU3Hx0UqHQERERE1hyTVvbRwJ9oy75z7NUNMKOKmX4zkDXNx8fF3MWL10+gwdTRcFTZsv/I57L17SeWy6Q+/jx8C/oa99yxBxvLNsFtKFYyciMj/tIoabYdd6QiIiIioWaR6arRNejjKiuEst0Nj0nlx13X3TWdHUqkQeclgRF4yGIM+uAe/nfcgjr31I2y5RRjw7l2w7DwMZ5kVWd+n4Mhr30FSqxB6fiJirzkPQX07I3zsAH4eRNSutYpEm4iIiFoHqZ6m4yqVqMl2VHg30SbvUGnU6Pn0VOz552JkfLUZGV9tBgBEXjYUI1Y9hbKjWcj+YQcyl2+pUesdedlQ5P28C9FXjUTsNechatIwaAKNSh0GEZFPMdEmIiIiz6mVaOthReT3H2AygPIFvWF45l5l4qJzEj1xGKKO/hcFm9NxetkmnPp0A8yDuwEATF2j0fWfE9H1nxNhyyvC3ruXoHjvceRvTIPL5kDumlRkfLEJkCSoDFoED4xHYK+OCOwTh/AL+yJkeA9IarXCR0hE5FlMtImIiMhjajcXllDVL7vi2ZcBJtqtliRJCBuViLBRiZUDqNWmCw/G4I/vBwDIsgxHcTk0QUZYdhxE0Z9HUbBlP8qPZaMo9QgyvtqMfaUV0MeEosOU89H9ketgiBHPXi8/kQNrViFChvVAwdZ9OLLgO4SdScrNQ7pDUrWKYYaIqB1jok1ERESeU71GW5Yhg/102ytJkqANNgEAQob1QMiwHuh86/jK951WO3LXpiJ71Q4cfXMljrz2HcLH9Mfgj+/DljGPoexQJoIHd0PRzsMAgIz/bYbscEIbGoiwC/vCPLQ7Qkf0Qtj5iVAZdXDZHFDra46KXnE6D8VpJ6AJNEBSqxHQswO05gDfnQQiareYaBMREdE5W7hwIRYuXIg7D8UqHQq1Emq9FtGTkhA9KQndH7oGpz/dgH2PfYy1HW+pXMZ6Oh8AEHnpYAxb8W8UbjuAvPW7kfvLbhx+5Rs4LKVQBxjgLLcBLhfCxw5A2Kg+6Pn0VEiShL33viuarVcTmBiHiHGDED1pGMIu7Au1QYwbUH4yF/kb9yJsVCKMnfnIGyI6N5LcCp61MVgqQirMsFgsCA4OVjocIiIiasAP/f+BiXvexnoMQocXn0TJQ88gBtmV78eUHYRk9PyAWDv+9gIcReUYsfppj2+bfKf8eA7yN6Xh0IvLEXHJICS+eEuDy8qyjLxf/sTpZZtw/J3VAABdpBm2HAv0sWGIuHgATn28HsED45G4YCY0AQbkb0pD8d7jyFm9ExUnc6E26RF+UX9kr9peY9shI3shetIw9Hh8ilePl4jaLr+u0XbfHddjq9KhEBERUTMU7znW6PuOIyegTezpo2iotTF2jkTHqaPRceroJpeVJAkRYwciYuxA9Hx6Kux5xTAlxCJ/wx7k/rwL2T/sAABEjBuIiIv6AwBCknoAEEl68d7jyPlhB3JW76zcZp8XZ0AfHYI9dy1B4db96DJ7EnShgV44UiJq61ijTURERB7zhXQZpuDHBmu0oVYj6uR2qGOiPLpf1mhTfVw2OyStpslnessuV40B1koPnsaGfv+C7HDCPCwBI9c+y0eTEVGL+HWNNhEREbVux9C5ZqLtdCJ/7PWITFuvWEzUfqh02qYXAuqMYh6Q0AHnb3sZx99di6Ovf4c10dMgO5yImjwcAd1jEDy4G6InD4cmwFBjvUMvr4A+yoyQ4T1h6BTO5JyoHWOiTURERF5TBHOdeY70AwpEQtQywQPi0e+12xFz9UhkfbsNpz5Zj8z/bYYhLgKHXlwOSa1CxPhBiLttPIp3H8Ph+d/AWVJeYxsacwCC+nWG9XQ+Anp1REDPjgjs2QGSWgV1kBHmQd0Q0KsjVBo+R5yorWGiTURERETUgIiL+iPiov7oO/+2ynkl+07i9JebcPzt1fjjb/Mq53d/6BrE3TIOFZkFsJ7OF88M//MobNkWOMusyP1pJ44t/gGy3VG5jkqvRVC/LgjoEYucH/+AIS4CuohgBPaJQ8TFAxA1KanOY8uIyP8x0SYiIiKfc1mKoDJz3BVqnQJ7d0LPJ25Aj8enoCTtBPI3pSFkeA+YB3evfL8hLocTzjIrZKcLxX8ehSX1MIp3HUHpwQzYC0thLyyFSq9F/sY0HFu0CiqDDqauUYBKgmx3IvGVWxE9ebivDpWIzhITbSIiIvKKxoZbPfbsR4j41w0I6hLuu4CIPEySJAT17Yygvp2bvY5Ko4Yq2AQACB/dD+Gj+9W7nLPMioyvfkPZsWzY80tgzSjA6WUbkXLFfzA+60Poo0I8cgzkH6xZBTg4bzls2YUIGd4T8XdNVjokOkdMtImIiMgzLBaMxi/NWvTwK8ux5pU/cYf8tpeDImqd1CY9Ok0bW2Ne7JTzseOauVgTPQ2mbjHQhgWi7HAWAMDQKRzmwd0QMX4QzIO7wRgXAZVBB5WWl/utQc5PqTjy6jfQx4Yhe+V2OMusUBm0UAcYoIsIhi2rEJJWDVN8NLShgdAEGaEJNkETZITKoIMkSXCUVqB0/ykE9ukEtVGv9CG1e/yfR0RERJ4hyzDCWu9bJVFdEaiuADIyAQBGiEGjyrOLYIxiE3J/IDudAABJzYG5/FXs1edh3KmlyPjfFlScyhO1n8MSoI8NQ8XpfGSv2o6TH6yrXF7SahAyvAdG/vQM1CYmXv5MdrkAAIM/vhd7/vUODr20AvaCkjNvNv40ZpVBB114ECpO5VXO03cIgybICF14EAJ6doQ9rxj6DmHQRQRDFxYIbVgQdFFmRI4bCEmjhrPMCme5DZJGDV1oIJwV4m8O1Hf2mGgTERGRZzTyrGKH1QX9ZSNg/fwbAIAaIqnL3nYUXS4f4JPwqHEFk2fA+sM66MYkQ9O/D4xTJkM7cmidR1+RsgwdwhH/r8vrfU92uWDLLULx3uOwZhbAsuMQDr/yNSoy8hHQPdbHkZLsckF2yc1LVl0imQ6/qD8u2rtQrC/LgMsFe0EpbPnFMMSGoiKzEI6iMvEqLoejqAy2vGLYC0rgKCyFo9SKwF4d4Sgph7OkAhUn81Cy7yQcljKUn8iBPb8EtvySyhHyVToNXA4XcCbRBwBj50iUH8+pnNbHhCJkeA+YusXAFB8NQ8dwaEMCoAkJgDYkACqdBpogI7QhgR48e60fE20iIiLyjEYSbaelGFmfr4e7V6khWA8UAeWZRb6JjZpk35UGAHDs2Q/bL5tR9vq7gNEATc9uUIWHQjcmGYarJkDbr7fCkbaMLMuoWL4KqpBgaIcOgCqk7iPn2gpJpYI+KqSy/7ahQzgOv/I1ZKeriTXJG/644SVkfPkb9B3CoDbq4LCUwVFagY43XoiON43GyQ9/gSbYBEOncBx9Y6VYqdrvqCRJgFotaqEjRMufwCCTR2Jz2ezIWb0TltTD0EUEQ23SQxsSgPJjOSjZdxLWzALookIg2+ywF5XDVW5D9g87UH40Gy6rvd5tupuyu6x2mLrHQB8dAn1MKEKTeyOob2foY0Jhy7Zg18w3RJN3vRaF2w5AH2WGsUskzMMS0PPJG+AstaIiswCG2FBow4LEeWiFmGgTERGRZ7TgYihubA/sS4tGzvZj6D3zfC8GRc2lCg+F4dqJML/+LOSKCti2pcK+LRWO3elwZeWi5KlXUPLES9CNHQXtoL6ARgN1p1iYZk336+bm9pRUFF53R+W0KjYaqsgwBL/8JPTjLwQAuIpLAKezzSXhklq0RmCi7XvZP+5Axpe/QaXXovPtl8CWWwSVVoNji1bhxHtrceK9tQBEX3xJo4ajqAySVuOzpFKl0yJ68vAWj2Avu1yw55fAbimFw1IGe2EJinYfg0qrQcWpPMhOF1RaNcpP5sGWY0H2qh04/s7qOtuJGDcQkloFY+cIhCb3Qfb3KchduwuHXvhfjeUkrQb66BCEXZCImKtGAioJrgob7PklyFyxFWqTHhUZ+XCWVKD0UCZUOg2MXSIRMXYATN1jRO27OQAaswnakADowoOgiwj2yW8WE20iIiLyjNoXiLUmq/cydJ3MQNyES3D0m11eD4uayeUCVOJDkwwG6C8cCf2FI6veLrSg8Jb74MrJQ8V3a+A8cAQAUHTf01DHdYDh6gkIeukJv2tqLpeWAQBCv1kKubgEjn0HUfKf15B/yY3QTxwL66qqPs36qyZAFWiCKjIcqo4xcKQfhH3nHqi7dAKcTkiBAdAO6ovAB/+p1OG0CBNt5bgHqRu9980azfa7/t8kZH2fAl1EMKIuHVJZU20vKoNK5/+pmaRS1ahhB4CIsQMbXF52uVB+LBt2SxnKj2YDKgmG2DCEJPWos6xl5yFY/jgE2e6ENjwIkkoFa2YBjr39I05/9itOf/ZrnXXMQ7rDGBeBgJ4dAQlQaTUoTjuB/N/SceLdNXDZHHXWgSRBazbBGB8NY1wE1IFGdLvninpjOhf+/2kSERFR69DMmhht8jDYN29H7M1G7DmWh8J9mQjpHePl4KhJLlejSbIqxIywFe9WTss2G+wpu2BP3YuKb1ajdP47MN56A7R9e/ki2hbT9O0JTfeuAAD9pReh/KOv4DyZAVVkOFz5hYBOC7m4BLY9+wAZcGVmVybp6shwOA4ehfPQUVSs+KHVJNo4k2iDibZPWLMKsHvWW9BHmVFxKh8qvbZO3/iA7rHodvcVddbVBnumSbi/kVQqmOLF77t5ULdGlzUP7l75LPrqusyaCGdpBWSXLGrMdRpoAgxN7luWZThLK2C3lMFRWAp7YQlsecWwZhbAXlCK0gOnUXEiF3ZLGZwVtrM7wEYw0SYiIiLPqJZoa2BHz4euwQFMrLOY4erLYN+WigjkAgC+6PMUH/PlB2SnE6iWaDvKbcj94zgik7pCXU9Nm6TTQTcqCbpRSdBPHIucbuchb/S1CJg9A4FP3ANJI9ZxZmRBZQ5G8ePzIJmDoR2YCO3wQVB3aP7NFXvaX5C0Gqi7dWl5k896RmzWnT8cuvMbbzIryzLkomJIeh0kg7ioL/r3XFR8/m3L9q8g1mj7luWPw8hcvgUBvTrCejofQQO6Kh1SmyBJEjSBxrNeTxNoBDqGeyGyxjHRJiIiIs+olmjrzzzm6wJsRCmCAEiwQQcA0CYnQQoMgGSrehRY+pKN6HP7BT4Nl2pxuYBqSezBT7fh15kfQW3UImp4PGJH90DCTSMQ0jO6zqqa+M4I+fJtlL2xFCXPvAr9JaOh7tYZhTfOhm3Dlnp3J4WYoRs1DKZZ06EbOQSqsNAGQ8vtO0asYzJC07834HRBCguBYeJYSEGBkIICALUa2v69oelRq9bMnWi3sO+rJEmQzK370XNMtH3LPUhY8sYXoI9sW/39qeWYaBMREdE5W7hwIZa88QbcPeicEAlbMIpRiiD0+PEN7JnwAE4jFlH9e0MutKD4of/gik0b8O35L2HjHR+j48W9EdwtUrmDaMVcdickjarFAylV5JfixKrdSH9nI/odzcOhRRuQl2JEcPdIZPx6AACQ9OyVyPztIHbNW40/nlmJ+GuHIP7awQhNjEXpyUJ0uLg3NAYtjNddDv1FyciK7I/C6fdAMgfB8cduqKIiEPjo/0EyGWGcORXOA4dh/2MPKr79CRXLf4B15c8AAE1iT2gG94MqwAhnTj40CV2h7hgD3ThxA0Y/6WLoxiTDsSsN9p17Yf9pA2y//g5UVNQ4pujCdMg2O9SRZ2qwzjLRrpckNflMY39SlWg7FY7EOxwl5SjYuh+SRg21SQ99ZDAcReXI+3UvtGYTtGFB0AQbEdCzIwwxDd/I8RR3f2CVXuv1fZH/O6tEe9GiRXjppZeQkZGBvn37YsGCBbjggobvQlutVjzzzDP4+OOPkZmZiU6dOuGxxx7DrbfeetaBExERkf+YPXs2Zk2fjuKgIACAGZaGF3ZVJSoxoxIw9rOZWHfjf/Hny2tQlmnBefP/hqCuEd4OucrJk0BcnPj72DGgc2ff7dsDMn79C9+NfgVxE/th3Bd3QBugb3Id2elEyoNfIPXV9ZXz+mlkyDKgDwtA5qaDsBWVI25CXwy4fzwG3D8e1sIy7HntZxz9ZhfWTX23xvYMkUEwxZpRkHYa4265E6rjRyEdPw5nh67QLX0bJUFGhA/uDEgSNL0SoOmVAMMNV2LDjKUo3rIbMcWHYMwpgfH7zTBZMgEAVogB9NzpcZ65G/Z8no+IoSMQ88hNMCdEIWpEPOSKCshl5ahY/gMstz+IrJA+AIDIw1ugie9cNQpfK31E0LlwJ9pFu4/Bmi3+T6pNeqiNOqiNeqiDjNAGG0W/V7221T1K6egb32Pfvz9q1rKGjuEwdAyHKT4aKr0Gkk4LlU4DlU4DdYABgX06QRtsgux0wVlhg9YcAFuOBQW//wWXzSFGqw4PgqO0AsZOEYi7dVydMQ3KDmUAQKsY1Iy8r8XfgmXLluGee+7BokWLMGrUKLz99tu47LLLkJaWhs4NFEzXX389srKy8O677yIhIQHZ2dlwOOoZAY6IiIhaLbmktPLvSOQ0vJzLheA3n0PR/c8AABJuSMLeN35B2uINAICjK1Jxw8H/QKXXILBTC2qhzjY/OH686u+JE4E9e85yQ8o49Pl2AMCJVXvwQei90IWYYIoJRrcpwzDksbp95AGg9NUliHv1WcQBqIhLQNi0SSh77nv0f2Qcgp69s9519CEmDH1qMoY+NRklJwuQvfUwJJUEe4kVJcfyUXQkFwV7T2PN0lMAtADODGp06RuV21Bp1XDZnTD3ikbRoRzIDtGk2XTDFbA4XbAVlaNgbwbKTuYBAKJHxiMqYxdceQU4+MVxVDi0yNl+DOlv/wp9WACm582HZDBAMhhgvGUK1N27oOKL71D21kewpPyFgKCwqq9FAwlk/p5T+G70K9CFGBHSMxpB3SNhCAtA+OA4mGLN0Jh0COgQAo1JB0mSILtccDldUKnrDhwny2cGa9L4x+POtCEBAIDddyxs3vLhQdCFiccf6SKDoYs0V/6tjwqBNjQQklZdmaD+eftCWLMLYYgNgz4mBPrYMESM6S8eGaVWQW3QQm1s+sbP2XKUVECl02D03jfhLLXCmmOBLduC4MHdYIqPhr2gBBWn82HZfhAVp/NRfjQL5SdyIdudcNkccNkdkG0OlOw/BdlRf62/NiwIhg5hKD2UAdnurFxu/1OfwjyoG9QmHSBJUOk0OPWJ+A1jok3AWSTa8+fPx2233YaZM2cCABYsWIDVq1dj8eLFmDt3bp3lf/zxR2zYsAGHDx9GWFgYAKBr167nFjURERH5n2q1O6LpuLgglVCzf2j6oKmIvqi36BN8RtxlfZG1+VDl9OcJjwMAdGYjZhQuaHEoZZkWyC4ZplhzozV0pacLYf8rEyEAcPvtwJIlwOuvA/36AUOHAmb/7WcpyzIkSYIuxIjAzmGYsOpfOLU2HUWHcrD3jV+Qv/sUIMvo8feRMEYFQaXTQJJEM3PHsVOV2zGcOIiy514DAKhio5q178BOoQi8bmid+aNenwLLwRyUnSqEPjwAgZ3DUHqiAMXH8lCeXYyyDAuyNh9C0YFsyE5R1TzTtggqbc3E1Gm1w2lzQhdUNbJwcrXj3jVvNbY9ugKfdH4Exsgg5P4hbpb0vuMCaPMj0RXAqSn34zRi0bmnAaEAVo1fADkiEhf+9+8ITexQud3UuT/Cml+KyKQuAIDT6/bD8ldWvf2ae2E/OiEf32j+CUmtgiZAB2NUMPQhRjitDlgOZMNZYRejIhu10IcHQBugR/jgOHS6JBE9bhrRrPPrKYYO4bj4+LuwF5RAHWiEJkAPZ7kNzjIrSg+cRuG2AwgZ3gOSSgVbXjFK9p+EbHfCllsEW04Rincfgy3HAmtOEZwl5fXuI2hAV4SP7gdrRj4ylm3EqY9+qfG+KT4a2ohgaAINCOrbGYZOEdCFB8GaWQCoJBg7R0Jt0iM0uQ/0kU0/39hZYcOmpPuh0mlg+eMQjF2iEJDQod5l1bFh4lFSQxMa3abLZoejuBwuuxOy0wVNkBGOojLIDieMXaJq/IbITicOPPclyo9mwZZXDFtuMSS1Ci6HE2EXJKLLP3OcC1oAACAASURBVC7zu0fckTIkWW5+RxObzQaTyYQvv/wSV199deX8u+++G6mpqdiwYUOddWbNmoW//voLw4YNw0cffYSAgABcccUVePbZZ2E01j96nNVqhdVaNUDKaDOQCjMsFguCg1v3oBRERERtlSs3DyWRETADyIEGERCt17IQjZAfP8eeCQ9ULmtCKUI0ZYi1HwMAFKRn4MvEOQ1u+8YjzzXcnLygAKl/fxlWuwojbugG3Horfu92I3YdFs3Yr/r9ERQfzcOhZdtxdPlOxF83BOO/FLW2K8e9CsfP63ElvgWWLQOmTKm57f37gZ49z+6EeEDZx/+D9esfITuckPQ6OE+cBmx22Hf8CQBQd+8KW1Y+Sux6xNw4BgH/dwugVqGwTIsdo+6GA1rkIhw21KxVHKDagxBXPpwLFyPxzgsAhwNySSmksNBW0XS4Ir8Uf72/Gdb8UpTnlOD0uv0oOpiNsAGdYM+zYKh1C4xleVCXlUA603Y8beJ9OLTqL5hizeg0oS9C+8QgY8MBHF+5G3ET+uKyH+6q3L7L7oS1oBRlGRaUZVhg+SsbaoMGqg8/gDF1K0pffQuywwVbcQUqcophs5RDdp6pzdZrENa/IxylVljzS5G15TAyNx4EAPSccR46XtwbXa4YCF1wy0dRVpKz3Ap7YemZ2mA7XHYnIMsI7N2pMrG0F5YgZ+0uSCoJznIbbFmFKD+eA0epFRWn8lB2MAO23CLYC0rq34kkQRNsgqlbNHThQQgd2QuO4nJow4KgDQmASq9F4faDOPHfnwAAHaaORlhyb3SdPclXp4Go2VpUo52bmwun04no6JqjTUZHRyMzM7PedQ4fPoxNmzbBYDBgxYoVyM3NxaxZs5Cfn4/33nuv3nXmzp2Lp59+unJ6UGP9vIiIiMg/VKvF0aGpLmJSjRrt0D6xGPXmDTi1Nh1Hv94FAOh16yjsf+83AMBn8Y9hhmVBzeRk2TLghhsAAIPc88T1N0b8P3v3HSdFeT9w/DMz29v1fnSQdoAIioAKVozYYm9IJD9jASwYNVGjsbckmkSssRtLDNiNFFEE6R1EOhztetm9vds+z++PgYXjjoODq/K8X6993e7MMzPPw+4d+53neb7Plg8IcSrr6M2nQ56qdeWt/13G4gc+5afJswlX1pC5dxLvgAFGYL1kCaxcCc88A4sWtUqgrXt9BD/5H94bJsW3KU5HfF3nvay/Op3gB1+T4C8i8NZ/CLz1n/i+AfuVExYrSjhEeYd+cOqp2JbtgtIoWWf3NnoQNQ3F2nxDfJuaLdlJ/0lnH7KciEbRS8sR/mrO7N4F/7CnKV60jQ1vzouX8XRPJ3dUn1rHqWYNe7oHe7qHlAEd6HCusb1qxwICO1bTsZEZ8gPFPn648T22f7WGDW8ZWdidOYmkDu6E2WlFj+nkntWbnr8d3qo3OkKVNZSv3kXWqT3q7NPs1kMOAzcnusi+bPhB9+uRGKpZQ8RiBAsqMHkcqBYTgW3F+NftjA/93vzsJ/iWb6F0pvG3wJqVTKTCjx4Mg6pizU7m1CV/w5aVfHQNlqRm1Kge7d27d5OTk8O8efMYOnRofPvjjz/Ou+++y7p16+occ8455zBnzhwKCwtJ2DP8aurUqVx22WVUV1fX26ste7QlSZIkqf3RK734kxJJALzA3v+x6+/RriERL1liV63jizufTMgb4DtOp/8jl9FjzBC+Out5fJuNOd/drj6R5H459Pq/U7CnH+Q7wSWXwNSpAIQHDcWydD7FpKGmJLNt1E2sfH8Nsf36GjIp4EK+4Lshf2DjwnIyhnVjwD3n0PnigQCI665D+ctfIKPuslbNpeqBp/E//g8AnJN+h/vZPxlzjCMR46euxwPjRX/8hF3//o6TCz6DaBT30/ejOOxE127AdtVF6MWlRJatJjJ3MeE5C+PXsJwzgpRp77dYm9qSWChC2cqd2NLceLocfuK9qgefpeat/5CxffERXVeP6RR8vx7vxmK8G4spX70L38ZiqrYZc9KdOYkomootzUXOWb2xJNhBCKKBCGanFXuGm8RemXi6p2NLdTV5UL7g7v+y6i8zSOiZQdaI4xAxHbPbhmbRcHVKwZroIBqMkDm8G56uaXWG/B9ob5ihKAolS/P5ZPATqBYTnq6p2NJc2NLcWBLsWDw2FFVBj+pGMG7RsKc66XbpQMI1EcxuO5rVhMlhwZbWvhK2SceuRvVop6amomland7r4uLiOr3ce2VlZZGTkxMPsgF69+6NEIKdO3fSo0fdO2ZWqxVrrbuqvsZUU5IkSZKk1nCYX357LXmX7YMvBfZ8Edd1Ck37EqpagFFMJ+Gax3F0TuWqTY+x9JEvWfrQF2z+YDGbP1jM4vs+5frBp2BbMhe2bmXJ7z8k5g8w5BtjRNzynlcycMN/sCw1eg/TKYGyElLfn8TgPdcRZjNKJBK/btHCrUACRfM2M/3ilxirWLGKEMp777F1RSUdX7gHbeVyI+A+cIh5E9O9VaiZ6STP/BBTz2775nxaLHXKCiEImt3xYfj1sV92vnHe6hoCb34EZhPWc09vlrq3B5rVWBu8pamaSs6Zvck5s3et7ZHqEKv+OgN/fjmxUATvhmI2f7iEiD+IoihoNjMRf4hw5b4RDapZo9+dZ5J5SndETKCYVFSTkYDM0z19T34CGjdfeM/vcGKvTEoWbUOzmgj7glT+XFBv8bTBnfBtKcXTPY20QZ1w5CRSOGcjpUu3E60JEw1EUDQFbb/kYP0nnUUsFCFQ4idYUkXN7koiviBCN24oWJMchL0BqraWsuTBL+tc057uxtkhifO+uR1bquvw2yZJLaxRgbbFYmHQoEHMmDGj1hztGTNmcNFFF9V7zPDhw/n444/x+/24XMYvw4YNG1BVldzc3KOouiRJkiRJbUpDX+j3C8Kdg3qjJbjA6yVWVY16QHzu/OMEqp98AW/3E3Hs6fEe9OD5nPDAeeR/uZrvxrxBxBekeEk+HS65BKWeJKvb0wah5WTQ37EF3nwT3G6YNAmmTYMtW4wq7Q2y+/SBtWux9+zIyNd/h6drGt/f8BZvTxsLwBV8RJc1X8LI/b70X3FFkywXFVmxBt9dj2A961SU5EQUsxnr2acSWb4GxeXA3LfnYZ3ncHv4VKfDmMctHZlmWkfb7LQy6MHzD1kuGghTsiSfQHEVc258l5XPTGflM9MbPMbksGByWlE0BVVTUTQVT7c0et90Kp7u6ThzErEmOQiV17Du1Tm4u6Yy6tNb65xHCEG0JoxvUzHFi7ZRvGCrMRc7HKNqaxkli7ZhTXaiR2J0uWQgKcfnYnJY4r3UeiSGLdVFz98Mq6eWtfm2lvJh1/sByDmzFyc8OJpoTZj1b83HkeFhzT9mMf2Sl8gc3p1tn67AmuIkoXs6JqeFpD5Z2FJdxEJR9HAMPRrD0zUNR06iMa9cF4T23LCweGw4shNxZCXg21xC5c8FOLITSTm+A6qmEqqoJlQZoHRpPr7NJThzEkFR0KxG1nWz04IlwUgyZ3ZaMbutmF02/DvKqSn0GQnxkhzGTQ+TRumy7bg7JWP22DHZ6q61HSipQjWpWBLsMqHaL0Cjs45PmjSJMWPGMHjwYIYOHcqrr77K9u3bufnmmwH44x//yK5du3jnnXcAuOaaa3j00Ue54YYbePjhhyktLeXuu+9m3LhxB02GJkmSJElSO3SIYM8xuA81S9YC4Bo5CD7bRdWsxSScMThexvXnu3A/NInqJ18AjKXA9NJyVJcTxWGn84UDuMH7d15VbjKSXOl1M0MbB0J5xxPgrX/s2/bii/ue+3zw+utGpnGXC3Sdi/b7YnveN7dTU+DF7LYSmT+GzeeMJXtwNlpRAZYdm6kcNwmtS0fMg/tjO+/MRv5D7RP65nvCs34ksmAZIhiq1R7L6YcOSIy2imNyjehjkcluic+f7nrpCYSrgkSrQ6gmDT2mI6IxItVhSpfmEw1EqNpaSiwQwZJkrA8tYjrFC7dSNH8Lu69aX+810k7qXO92RVGMDOoDOpAyoAO9GzlPvTE8XVK5etsTRHxBPD3S40Fph3PziNaEKVqwBd/mUirWFhD2BsjO7UnF2gLKVu5EDzd+CWHNZiYWjNTaZs/0EChsvlG1JqcVs8t4CF0QC0aoKfDW2g/GdAKLx2aMWDBrqCZt33OzhtllpdeNp5I9ovWSNkr1a3SgfeWVV1JWVsYjjzxCQUEBeXl5fP3113TqZCyJUFBQwPb91qN0uVzMmDGDiRMnMnjwYFJSUrjiiit47LHHmq4VkiRJkiS1vkMEe70WvR1/bu+eSxAIz5xLJN3IDp74n5exX35BrWMKtQ7x5xkVa1ETjaloZ338O7j8ayp+LiI0ZyNivwB19/frKZq3GUdWA0tzeTxw5537XqsqQtcJfTUTEY5gHTWS0N0P4J36NeazR/ItZ3HuyL54/vIQFiDw1kfsv3C3qc9xJH39LqZOBx+tF9u+i5K+p+N+5n4c/3cNwc+mUfXHJ1GSE8ks+wmA6JZ8wt/PR3HYsZ5/1sHrL7WOZurRPlIWt63WEmh7JR536HwCoYpqvJtKCBT5CJVXEyz1kzqwI5mn1Z3W2RrcnVLq3W5yWPj1wj/Wuy8WjhIqq8bkshq9zmYNhKBkcT7+nRU4shLiPcaqScO7sYhoTRj/jgoW3/8ZebefgbtTMoGiKuOEqoK7UzKWBDtZI43RJaqmGsvPhaJEa8KEfUGi/hARf5CIP0RNoQ89FCX7jJ7okRg7p63F1TEZPaYT9Ydwdkgi4gsSqqgm4g8R8Yfi67MLAWmDOlK5vggR1TE5LQSKq4j4Q4iojh41RgbEn4djbP9yNapZk4F2G3REq6nfeuut3Hpr3SElAG+99Vadbb169WLGjBlHcilJkiRJktqRhkKQ/Yc3q8mJAOiTX6Z88ssARNdtiu9PnvY+5aOuqXV8UVIfkqe9j/WcEVgSHQigcn0hM0/7C0lU0HGUMe/1h/97FwDN2rivOeVnXkH4+/l1tkc+/4ZRmDH/Zd/Q8czgVkR1Df5nXyYyfynh2fOpfu5VEp5/5KDn997+IMJfje/W+/Ddel98u+M3V8Sfm7p2wtS1U6PqLTu0pSNhTXKSfqKztavRpDSLqe4NNkUhfUgX0ofUnZPv6ZYWf97v9sMfmWJy1M2VcDDNnQvgixF/ia9JL7UtcvC/JEmSJElNLoCV4tMuO+h+y0Wj6mzbf+6w9ZwRZIldZIldpK79HsfEcQDoFZUAZI88jqS+WST3y4kfE90z9NOZm0T6kC4Mn3x1o+qsl5QDYD7JyDZuH3s5yTM/xHbz9ewkh8KM/tj/dJfRvrIaNn21Ht8ZF5E4/UPMI4ehF5bUOWf1C29SoORQlHk8oU+/QU1LIfmbf+N+6j7UtBTcf3sIz18falQ9pVakKA3fTZKkFqZoKno01trVkOpxRD3akiRJkiRJDREo4Ks66H7N7WI3WaRRgpkoWC2oSYn1ljX37oHn6fuo+ecbBN75L/YrL0I1abhyksDj4by/3cH8sx+iYPZGAPRojMRemVgTHY2qs5qTia1XN5L++1qt7ZYzTqFq14v89MUqfI/OYATwXs697D90/CRlMzln111juOqBZ4w6FZVgH3Mp9usuxXrOCKyjRuK6d3yj6ndQsktbko5ZiqYiYgfJVSG1KhloS5IkSZLUNA6Yu6pWlNUtU1kJiYlYOmaiOu2UVyeRQQmWSw+RcdlmzEMNfT0LEYuhaPvW7z0we68eiR1yfd96xWKg1T1OURTO/Xw8W/67FMvUD+GDHzjj3XGUrtzJ7lnrKV22HSEgOnN2rXWDhRAIfzWeF5/AecvYxteH2usQezcWEaqoIfWEjkQDEfRwFNViIlIVOqJzS0egjc3RliRFU+TQ8TZKBtqSJEmSJDULLX9L7Q333APPPms813UG+ucQ2rabNV0upOvF9S8TupeiKNh/cwWBt/5DoakjjluuJ2FPwJMxvBuWRDtRr7FkzxEH2rpeb6C9V9fLBoFvJXwA3a85ie7XnQxA6bLt5A/6tbEeuJqL69G7cT9wB6K6BmKxeAK3xpj9f++w/YtVoCgEipov87EkSe2b7NFuu2SgLUmSJElSM6hnKHNFxb7nqgrjxmF+YTIA5R9NJ+nyhrNsJ7zxN4L//Qrhr6bmpXewmb1YLhiFoig4sxOprKxh7Ss/ULZ8BykDDp79+6BiesNrgR+EYlJZSx8cdkgMFOL/07MoJhOm/kZyNiWp4UBb6DolS7eDEPHESfmfryJYUkWfW0ew9b/LSBnYgVgoSu45fdCsJqzJTsxuG3okRvnKnbg6JTe+vVLjKcgebalNUU1au5yjrUdjKKpy2OuFR0oqiJZWEvP6qdxQTLQmBIoKJhNmi4KrQxKmJDeiqppoZRXm9CS05AQ0jxPFYka1W1Es5lpJOZtbuwi0Kw5dRJIkSZKkNqROKKIokJlZe9sbb6C+8QYOeqFP+QTs/zSCcVvd5YqMUyhkeNcR/PAzKq+dAJEowl+NAtgzPFSu3c3cm/8NQNYRLHVjDElvfKDt6ZaGpUdn5mx0cM4t3bC+9Heq/vhkfL+a6AGMZG3zbvuQYKmf3jedRsHsDYiYzpp/fkcsYCRy6zvxdEoWbyNYUsUJD45m8MMXcsrka+q9btxVJza6ztJRkIG21IYomoIeaVufyXBVkFgwQtW2Mkpf/A+xolJQFNS0VNSTBlH49jTURQvQiGEhwt7/MQQKoBi99IBQVIQAVY9hEpGGLnl4TBr2vt2I2l2IaIxYTRhdF0TdSXSbfCdpJ3Y9+mvsf7kmPVszkTOPJEmSJKn1CSGoqqqd4CwUChEKGf9Ti+qa+HYfYEVgAqrQUbfnY33sMWPnzTfDkCFwg5FlPId1xjFBYNo0GDmydnKvA5J9idFnEL70V/imvIM5FkP1+Yi4NVyDs+n62wvocHp3YtNnUrk1G//fX0dNS8Y18beHbF9VOIQWi6L6GhiqHQjsaaCvVu/3r5bczZsJt/PZSz+TPXIiFd8vpzPbUBDUPDYLc/ISTHYTP782F4CfP1mEyWElWhNCs5npefNQ1r78A0v+OQ2AvAmnk3V5f3wN1UVqcVWhEAE9hl2+L9JREEKg+6rRElxHfa4aPcSu2etY/I9vSOiRjsVjQ+iChB4ZaDYTeiRGxB/C5LTWyWfRHH66920qX/4AgYKKjp0gADFUNHR4+/34slfKaUOJuN2gacSCUfRwhGhVAAUFRQUFYfR6awpKNIbWuztaciK2zARcHZJQhIBoFN+2UvSYQrTST+WWcny7/WjhAFF/kOptxbhzElBiMSzeIpSVO4C9QT1YCWElzK6p/bD2vKrBtrnd7kb1iCtCtP3bclmKj0IS8Hq9eDye1q6OJEmSJB2TfD4fCQmNn28sSZIkSe1dY2PRdtGjLUmSJElS63O73Xi93lrbavVo+2ugd086Amtw48BEChX4cONhT0/4smXQrdu+EyxbxubTx1GNk45sJ5EDegpV1UhSBrBpE6SlARBetAxx9jmYhp/Moxt2cFNJkKjdTfJjk6i662EAtB5dsV91Ef5Hn8N8Qj/cf32I8Oz5+P/8V5I+fwvriKGM7z2QP+3e1xPvuONGPA///uD/CO++CxMmkADs2LHjsL50CSHwbizGv72cm+68la9Wf3fIY+pz4oknsnjx4hY5zufz0aFDh8NuY1NcszWOa2w7q559icDL75K+eV6L1/VIjz2a97Kl63qkx7XG57Vq7nI2jb6j1jbXsAG4Tjuesre+QnXsmROsqgQ370SEwgCYUhIpqSwnKabiGHAceiiM7q8hUlIZL9MYprRkVLsFFIjVhEFArNILe+Zta/16E9ldQqisDJvDjogJ1FCg1jkEUEkiFSQhUNBREajEUImhEcXEgXk3VJOGq1MKZpcFIQTqnmk3iTvX4IqUMzD/83bzedUDQYadeioLljR8rNvtbtR5ZaAtSZIkSdJhURSlwS8+epU/Hia7UXCi4AEshLEBvP46DBxY+6CRI+n47gtsG/MgJfSgFJ2BmQUohYV7TrpfNt3u3ePzY8NOl/GF8MfF3E0yoEKgGu56FDcqnsmP47z1NwCUzVlE+Pv5RE6/AgVwo6L960M8F4wiSzdeJ055DfOg/pg6HSKJmt1u/FsAHo/nsL8IJgxOgMFQ9kDgiEfnaZp2RMce6XHQuDY2xTVbo41w+O1UbDZMiorH42mVurb0e9kadW3pNuo1QbqFTYi5q6j4z0yC67aBSUMEQoS3F+I65Xi6ffIXAMK7iolVVhHz+ql5YQouNLp89CSqw0bhY68T/GkbgZ+3Y4/GSDzrZLRENwiB9uszcQzsSeIlZ6AoCn369GHt2rV16uKfu4JIQSmqw4aW4IoPLdcDQRSTiUjnDHqlZLJm5SpcLieWDpko5vrDuZjPj2IyoTqMnBfGNZcBICJRfLMWE95eSLS4nNI3v8C9eScdqKr3XHEmDVNqIprbCUIgosVoisuogxAE1m5BBELYB/Y8qt8RaOHPq8cDpqP7G1IfGWhLkiRJktTklPgMOIhiRjg0lHHj6i1r798j/lygEp6/FOvOLTBhAqxcCWefDTNmGPu//ZZCrx2XFsBywHlKBp5Nl9F9cP7+ZtSEfV+YPP94lPCchSgeN7FN2/A//Ddi23dROWYidxYa8wftl5zXZG1vyPjx41v82KO55pFq6bq2VBv3n5/ZGnVt6ffyWPi8Fv39Ax7dZo/3TrvPOBFzbjoiEiW4Pp/KT7/n5yFjiRaVE84vqHVsKMlJ8hVnA5B4/qmHfc2DtdF1yvENHufz+SgkjLlzFtZDBIWap/b87/2vqZhNJIwaGn+ded84Yl4/IhxBhCPowTB6VTWxqhq++PeHnHPSMBCgh8LxzN97A3y9qgY9GEIPhHAOycNxYh9cwwc02M7m0tZ+t9rFHO1MxUeRnKMtSZIkSW2a7qvCl+AhCdiFAxs2kinHjwunQ6BU+w96rG/GAsK7Ssi/4WHytn6OtXN2fJ8QAsxmlJgxFLKATNbRm2H8iG51UJXbhx19RxENxRjyzcOHrGeBvRsEg/HX1gvPIfmzNw+vkW+8Ab/9LSpQ+Qv+XrJ3Pv4v/btXY9vpf+IfVD/3Ghklq5usDnpZOSKmI7w+9PJKQtNnoxeVYhk5FPtl5x/1+ZvqvdS9PmKbthErLkUvLUcvLIFIBK1TLuYTB2A6rludY0Q4TPjHxYjqGkQ4Qix/J+HpPxArKkEvKEbrmEPqwi+PpnnAvjaWb9uJIyoI5Rfgn7uC0OadxMq86DVBYtUBoqWVmFISsHbLBV2gh8J4P5sNQM/5b2Ltko05IyV+3vD2QgqffQcRDIOqYEpJJGH0cGJVNVg6ZmLOTsOU2LjhxE3Rzl/y7+UvqY3toke7rLUrIEmSJEnSIemRfWu5VuPAxt5h34e+p+85+2R83y6qd9/Gxz7CFxvIYJYAkEUhhUMuhoU/oofClG2uoGDzEtLOGVjv8QdKWzmd0LdzMR/fF/NJA1E07bCO299DDz6I1Wpt9HHthdVq5aGHHvpFtxEa304hBHppOZE16zDn9TrsY0R5BbGCYvSCYmIFRcS2bEf4qsBiofrpyXWOUVxOIotXNkmgbbVaeeS++1F/XIJ/5Vr0ohKsF41Cy8lETU4kunEr0dXrwG5D374LTCYii1egOB2YenUn/ONiQjPn1ro5VR/XI78ntqMA4fWhuF2IQJDw7AXou2r3AitJCVgv/hWh7dOILFpOYOZcRDCECEcIb9pBtLSCmK8avdyL7q1C9/kRJSVou3eiRCKEO3RBWO1ofi+aRUNJTUZUellv70FF51MpR0VHJYoJS3YqZo+dWP+BWLvlUr30Z0JbdqFaLShWC4rFCIWSrjoH18n96rTJ0jGTjv+856jfg6ZyLPxe/pLa2C56tBXFB7JHW5IkSZLatMD67UR6dSIBqEBBJ2lPj7YTp1NB8Tc8/8/37SI2nnVrvEc7Uuln0XmPUDHfWP7LjY/TlB9QTj+d8BPPIk4eSgQzO+jIZrqTfGofhv3wVPM28s03Ydw4iMVqLe8lHRv8T/yDqvufBkDr1hlTr27oxWUobieRlWsRNQHQBWqC2wg2QyH04jII105ypSQlggKK0wGBIK4HbsfUvzeqx43WKZeqPz1DzcvvYsrrhalnV1x/vgtzXi/08goQgtiuQkRNwEieFYmgl5QT+m4eBIMoDiOPgAhHEKEwekkZ4Rk/HF4DTSYwmyBgBNVKYgKKw465f29s1/4aU6/uaJlphIq9eGcuIvLzZmLfz8FSuAOiUURMRxfGxBELEXQggoUKEuOTSfaulWwjQDKVh1UtoSigqCj6vpt5uqohUEFREGYTwuZAtVkwWVTE9p21jtd6dMHUvQuOG68htn0Xoe/mYT6+r1FnFBSzCb2sAvt1l2A56fBu2NVbTyEQlV4UlxM0DeGvRlT5EVXV6FV+VLcLU6/uR3z+WFGJcc7qGoTPj15SBqqKXlSCiETBpGE7/yz0Ci+K24WiKCgJ7lpTaUQsRmxnAaImgLl3jwau9suhV3oRwRBEIojqAJhNKE4HWmY6sGfUlBAoTfw3vV30aEuSJEmS1Pbp+2XMVRE4//og3HVHA0c0bOllT8eD7F5Pj6X7PZfCoEEwaxbas08RBUx2C+7unWE1LRv4tv1+CqkZOCaOw3zyCUSWrCI07XtCX30LgOX0YTgnjkNNSgBFQa/wIqprUMwm1Iw01OwMtKx01Czjp2KzNXgd21UXIQJBRE2A4MdfEpzyNaa8XkTXrGvwONOAPsZn02QyemytFrBYwGTC/ejdOO+6CVFdQ3TdJvTiMkQ0ipaVjta5gxGM2W0oioLu9aG4nAcd7bH9svuoXrgGFAXFasE1/BwsHTKwdM7C1iEDEdNRNBXFbMLhtJOSkoCW5EH3VRPctAPN7UCxWQh8Ox9TcgJqogvFpGFKTcLevweq3Wpc324DWEQQkgAAIABJREFUqxVFUYzRBLsKEKEwWnYGyp7EhPURkYhx0yAcpvK6iYTnLyX0v1mE/jcrXib02TS0TrnoVUbSML24FFHlx3LSQPSycmK7i4yANhgiun4z+o7diOoaql96F9XjQnE5jSC6JoCS4AFNRS8qhUjkkO+RmpSA8NfgevAORFU1Iho13i+bDawWYhu2EN2cj767EIRATU9F91YRfP+TBs8NHLhug0FV9z323PTRenUn/efZhzxfexBZ+ROxLdvRq2sQpeV75pqHEZU+oj9tIPTNQVZ6sFiMf5NolKSpr2G74JwmrVebDrQnT57M5MmTgQWtXRVJkiRJkg5lvwzhW7KGkrd/NlylnvKHkH7eIEq/XclJ3/yZ1DP6GxvPPBOWLUOb8jEaEMCJJTMJj5ZK3t9vPLr6S9IhqG4X1jNOwXrGKbjuuZW9A0P3T5LWFKwjhmIdYSSrqnrgaQJTvsbUpwfWc0diHjIQLScTxeM2ElJZLKjJiUZgehg3m5TEBPSkFPw/7QBFIbZ9A/rMFcR81cR8fmJVNei+amx53UgddyHoguCmHZS+PIXwzmIiu0uIllSQ9eCNZD98U6Pbtn/Cr8TRh59ATFEUtNzsQxcEFLPZeGK1kvTxqwDopeXGMlQeF7rPj5ro2VcOqLj0RgIffU7oq2+NnuIDqB2yUUwmCIWwjr4INTMd1e1EcdiNIe6l5aipyWidchA+P4rDjuJ2orhdqG4XIhYj8PoHoKpEN2whsmQlFRfecNA2qOmpmPr0MIL3zfmoCR6047rivP23mAf1R/G4QdcRlT7MA/NQHHbC85egl5Yb+4IhRDSKvrsIoevG3+eYjmK34bv3cQiGDvvfvi0TQlA69ML4KAysVhS7bc9NJjNaRhrm4SfinHADSqIHxaQZS6Bt24GorgFNM7Kp9+3Z5HVr04H2+PHjGT9+/J6h45IkSZIktWn7BRvRhLSD7jscpd+uZO1dbwCQPuqEfTueeQZOPhkuvZQoGtXODMwJTk6b/vcjrrYkHammDrAPVPHxTPzVJqwTJ1CzuwS9qobY10uJVdUQK/chIlEsnTKx9+2GluDCdcrxWI/raKzf3EDdfup1Wfy56rSjuh1oHieax4XmdhApKMX71VyKnn671nFpt16OOSsFc3Yaib8+vdna3RzU1OT4cy2t7vxf5x8nYOrdHcxmtKx0TH17oiS4UWxWlAQPWlpKnWMayzpyWPx5ZPXPRhCeloJi0hChsDEVIBRCRGNonXIb/fmyDB18WOUiy9cQnlt/Tox2JxSCQBDPS09iv+5SFKej2X8vD1ebDrQlSZIkSWo/VJeDvTMoxREkGNurYuF6llz13MEL/PrXhKfPouyc61BVC1ob+VIlSY1V8ton+KYvILBiA6FNO7D26IilUyaBlRtRHbZay0mZczPQEl1oLocRGCe4iJZ5qV68lspPZ6P7a2qdu9Prf0K1WVFsFlSrhZi/Br2qhvCOIgCSrh5Fl3ceNnppD1CzaiMFj7yG86S+OIf2p3rhGpxD8nCfeuTzl9s6y+ABWAYPaLHrmfv1rvVaacnkX4pyxNNfRChEeO4i1Mx0FKvFmC5hMqGXlBnzxssqiK7fbOQr0FRETQA1KRHFZjV61a0W49p7lhIT4TCEI8QKixHllYg9PdP2cVdBKIxeWo7WvbPRQx2NGYnz/NUEv5yJvrsoPixcTU9FdTmb7J+oSYh2ALwCEF6vt7WrIkmSJEnSQdRs2im8RopxsbD/ZSLyj8lCgKjCKXS355DHe2cuFEsYJL7mV+ILLog/6hNavELsJlvsJluUnXddo+r5xBNPiMGDBwuXyyXS0tLERRddJNatW1erjK7r4qGHHhJZWVnCZrOJESNGiDVr1gjxxhtG2pxoVASDQTFhwgSRkpIiHA6HuOCCC8SOHTtqnae8vFxcd911wuPxCI/HI6677jpRUVHRqPo2hSeeeEIA4vbbb49vO2gb99Me2rhz505x7bXXiuTkZGG328WAAQPEkiVL4vvbcjuXMEgsYZBYf+Yt8ec/D71BbLxokth6w59F4fPvi8DuYvGne/8gOnfuLGw2m+jSpYt4+OGHRSwWq9XGJ++4R1yd0FXcYMqJn6uhx5O9T2/V93L27Nni/PPPF1lZWQIQn3zySa39Lfm+5efni/PPP184HA6RkpIiJk6cKEKhULO2MRwOi3vuuUfk5eUJh8MhsrKyxJgxY8SuXbuavY3P2tLFipReh9VG/8vviNLTLhGlIy4VxX1Gxv/uHvJh6igK3MeJAmd3sVvrUH8ZaxdR4OkpClLzxM/ubuILU6ZYS+bhXyN7oCjMGCCKe48QkQ2bm+19PFIy0JYkSZIkqUnUbN4VD7QXHH+FiPzzxSMOtL/rfasQwviyXZ/9A23vvY83qp6jRo0Sb775plizZo1YsWKFGD16tOjYsaPw+/3xMk899ZRwu91iypQpYvXq1eLKK68UWVlZoubFF+OB9s033yxycnLEjBkzxLJly8Tpp58uBgwYIKLRaPw85557rsjLyxPz5s0T8+bNE3l5eeL8889vVH2P1qJFi0Tnzp1F//79awXaB2ujz+eLl2nrbSwvLxedOnUSv/nNb8TChQvF1q1bxcyZM8WmTZvaRTuXaCeJ4pc+jr+u7/P+2GOPiZSUFPHll1+KrVu3io8//li4XC7x/PPPH7SN40ddLE5N6SgqduwWwU07RHh3iZg47kaRm50jZnz1P7F02ixx+siRrfpefv311+L+++8XU6ZMqTfQbqn3LRqNiry8PHH66aeLZcuWiRkzZojs7GwxYcKEZm1jZWWlOOuss8RHH30k1q1bJ+bPny+GDBkiBg0aVOsczdHG1RePFT+YDq+NpWddKXaTLcqvGS8q/u/3wnf/U6L6X++L4PfzRHDGbPHj/Y+Kty6+Wsx88HHRH7P46pV/Cb2mps55Ppk6VQwYMEB0ycoWz//lr7U+67XauGSJuPLk4eLM3nkiXFgsIus3ifDSVWLc8JHi3B69xaIpn4oF/5smBvbp22Lv45GSgbYkSZIkSU0isLs0Hmgv7ndJPND24xS6J+GQx+8faOf/a1qDZcNLVsYD7fCqtUdV7+LiYgGI2bNnCyGMYCczM1M89dRT8TLBYFAkJCSI766/XggQlWVlwmw2iw8//DBeZteuXUJVVfHNN98IIYRYu3atcdNhwYJ4mfnz5wugTg96c6mqqhI9evQQM2bMECNGjIgH2g218eWXXxZCGIFAW2/jvffeK0455ZSD7m/r7VyiniiKX5nSYJnRo0eLcePG1dp2ySWXiOuuu65dtPFwHBiEtmSbvv76a6Gqaq2e5A8++EBYrdYmjT3qu5lwoEWLFglA5OfnN2sbK2//k9hNtnhASxLF4/8oKn5zh6gYM1FU/OYOUX7VLSKycUv8XKVnXiHKr7rlqNq4c+dOkZOTI9asWSM6deoknnvuufi+9vY+NoZcAFKSJEmSpCahWCzx52Gr64CdjTtX7tgzGy5g2zef0dSjS+NOfgCv1wtAcrKRLGnr1q0UFhZyzjn7lnqxWq2MGDGCTZs2AbB06VIikUitMtnZ2eTl5TFv3jwA5s+fT0JCAkOGDImXOfnkk0lISIiXaW7jx49n9OjRnHXWWbW2N9TGvXVrD238/PPPGTx4MJdffjnp6ekMHDiQ1157Lb6/LbZTRKJEisoIbsivlan/YE455RS+/fZbNmzYAMDKlSuZO3cu5513Xptt49FqyTbNnz+fvLw8srP3ZTQfNWoUoVCIpUuXNms7D+T1elEUhcTERKD52mge1B+y0hkdM1P95QyiP20gsupnous3E/zwM0p6nILv948QmvY9ennlUS2dqOs6Y8aM4e6776Zv37519v8S38e9ZDI0SZIkSZKaxn5JyWLmhtcJPhTV1HAyNXPfnqiZ6ejFpYdck7ghQggmTZrEKaecQl5eHgCFhYUAZGRk1CqbkZGBd/NmAIqKirBYLCQlJdUps/f4wsJC0tPT61wzPT09XqY5ffjhhyxbtozFixfX2ddQG/Pz8+Nl2nobt2zZwksvvcSkSZO47777WLRoEbfddhtWq5Xrr7++xdsphCC0cXutZbBiFVUUPPovbL06Ey3zEi3z1gqwNbejwTbee++9eL1eevXqhaZpxGIxHn/8ca6++up43VqyjS2hJdtUWFhY5zpJSUlYLJYWbXcwGOQPf/gD11xzDR6PJ1635mijY8xlOMZcRmerlbeefD7+WQKovP42ous2U/3XV6j+6ysAWE4dwpF6+umnMZlM3HbbbfXu/6W9j/uTgbYkSZIkSU1j/+zfmung+w6hw+9GHVa51BXT0csqDvu89ZkwYQKrVq1i7ty5dfYduESMEOKQ7RBC1DquvmVmDizTHHbs2MHtt9/O9OnTsTVwI6K+Nh6qbm2ljWD0lg0ePJgnnngCgIEDB/LTTz/x0ksvcf311x+0joeqX6w6gCWiowDR0krUSIzkqErpvz4luGkHenWAaFE5k7e48Pz5U9Z9uZFIYRnhbQUH7aV2DuuPtWsOpvRkzBnJaAkuNLcT+4AeDbbxo48+4r333uP999+nb9++rFixgjvuuIPs7GzGjh17xG2sr0xrvpf1aak2tXa7I5EIV111Fbqu8+KLLx6yfHO2MfGdfwAQKyhCL6tAcTrQOuYcdlv2t3TpUv7+97+zbNmyRv9btsf38UAy0JYkSZIkqekpymEF17GaELFAiHCpjwVn/YkUwN2342FdQstIQ8tIO3TBg5g4cSKff/45P/zwA7m5ufHtmZmZgNFDkpWVFd9eXFzMkD09TRkZGYTDYSoqKmr1xBQXFzNs2LD4eYqKiupct6SkpE7PS1NbunQpxcXFDBo0KL4tFovxww8/8MILL7B+/Xqg/jburVtmZmabbiNAVlYWffr0qbWtd+/eTJkyJV4/OHg7Y14/ueuLuSDsYevj/8IaiVH17WL8c1fwMKD/sIyVr57F2cDZuMi/6QlM6UmYUhMxZ6SwVPVzwsknk5aThWv4ACwdjDZ7zh2GOTsN1bZnOoUQ9S6jdTjuvvtu/vCHP3DVVVcB0K9fP/Lz83nyyScZO3bsIdu499+hrb+X+2vJNmVmZrJw4cJa+ysqKohEIi3S7kgkwhVXXMHWrVuZNWtWvDd7b91aq41aVgZa1tG1f86cORQXF9Ox476/6bFYjLvuuovnn3+ebdu2/WLex/rIOdqSJEmSJB21yZMnM2TIyfHXIho9oETdoLtkxnL+57yc6anX8X2vW+PbPcd3ba5qGnUTggkTJjB16lRmzZpFly6153h36dKFzMxMZsyYEd8WDoeZPXs23bt3B2DQoEGYzeZaZQoKClizZk38y+HQoUPxer0sWrQoXmbhwoV4vd54meZy5plnsnr1alasWBF/DB48mGuvvZYVK1bQtWvXg7Zxb93aehsBhg8fHr9psNeGDRvo1KkTcPD3csH3PzCiV3+2XvcnnI+8x910pPSx1yl99RP0YBj7xafxNkXU3HI+Xac8g+WxG7mfLcS+eZoBBdPou/ojKh4fyx8jG+n06v10ff9xcp+5nfSJV5E+8SpsPTqiOe0ommY8jjDIBqipqUE9YI6spmnoe3rOG/q8tqf3cn8t2aahQ4eyZs0aCgr2rVk+ffp0rFZrrRtVzWFvkL1x40ZmzpxJSkpKrf3tvY1jxoxh1apVtf4OZWdnc/fddzNt2rRfRBsb1IKJ146YzDouSZIkSW1fyFsTzzo+u+cVIvLCS3uyjjuEnpRcq6zvp/xaa2V/wQViGueIJQwSwa27DnKFpnHLLbeIhIQE8f3334uCgoL4o2a/JWmeeuopkZCQIKZOnSpWr14trr766nqX98rNzRUzZ84Uy5YtE2eccUa9y+70799fzJ8/X8yfP1/069evxZf32mv/rONCHLyNBy6f1JbbuGjRImEymcTjjz8uNm7cKP79738Lh8Mh3nvvPSGEkb36lf+7U9xg6yi+O+f/xLLzJ4hvnUNqrSe98cI723Q7x44dK3JycuLLe02dOlWkpqaKe+65J16mPb6XVVVVYvny5WL58uUCEH/729/E8uXL4xm3W6pNe5eFOvPMM8WyZcvEzJkzRW5ubpMsC9VQGyORiLjwwgtFbm6uWLFiRa2/Rfuv/dye21ifA7OOt4c2HikZaEuSJEmS1CRqBdp9rmow0N720td1Au0Zpl+1SKDNnjoe+HjzzTfjZXRdFw899JDIzMwUVqtVnHbaaWL16tVCvPFGPNAOBAJiwoQJIjk5WdjtdnH++eeL7du317pWWVmZuPbaa4Xb7RZut1tce+21oqKiolnbdzAHBtoHbeN+2kMbv/jiC5GXlyccFqv4XWae+N+IMWLDqAm1gumFphPFfE4QU5Q88UHCiWLZlXeLyi/niOqlP4tYINim2+nz+cTtt98uOnbsKGw2m+jatau4//77awVj7fG9/O677+r9PRw7dmyLtyk/P1+MHj1a2O12kZycLCZMmCCCwWCztnHr1q0H/Vv03Xff/SLaWJ/6Au223sYjpQghRHP3mh8tRfEBCXi93lrzFiRJkiRJajvCvgDBBAcJwOy8axl2yymYxt9CNQ4cyTaUsrJ42UUXPErxl4tRNBURM4bA9nvsckof+Dt5Wz/H2jn7IFdpZW++CePGQTQKWsOZ0aWWVfn5bDZfdFetbcljzsN5Ul/Sbr0c5SiWKJIkSWosmQxNkiRJkqSmpx0Q1ByQGC1UWEHHG88h9ezjWXbFMwyf9wxajY9SgLbcB9BK2WulhololMDKjQAMDM1HtZhbuUaSJB3rZKAtSZIkSVLTO0TvoXfpZlLPHED25aeQpQ9HURR83y5q8BhJOpj8Gx+n7K0vsPXqLINsSZLaBDmGRpIkSZKkpqdq1M40vu95sLAChKB6k5EdtrXWOJWaj9B1alZtpPTNz4mWVjbrtfJvMoLspCvPpvfKD5r1WpIkSYdL9mhLkiRJktQ89g+g1X3Pl/z6CQB6P3V9S9dIagIxfw2BFRuIllZSvfgnXMMGoIcjBNdsxv/jSnwzF6G57MS8fqP83+4k485rm60+5R9OByDrod/J3mxJktoMGWhLkiRJktT0DuylNhsBkIjFqFxgrHvs7J7d8DFtWVueR36UhBDEKnyEdxRh7ZaL5nLE9+mhMOuHjSOwelOd47REN3p1APdpA3GfPhjXqcez4fSb2TnpuSMKtP0/riC8owh7/x7o/hqELggsX0/JK1NRLCaSLj2DmuXr0X3VdPjH77H37nLok0qSJLUQGWhLkiRJktT01H0ZuRWIZ3yuXFI3QJPajuC6bWy+7B6CP22Jb+u363/4v19KwWOvE/x5KwCpv/s1abdeTvWin1BUBc+5wzBnp9WZBmAfcBzB9fm1tgkhIBqjZtVGquevwj9vFXpVDUlXnk1g9SaCa7fi/XLOIetas3w9tl6dSRl7PgkXjmiC1ktSOxSLQVlZ3UckYqyQYJajPFqLDLQlSZIkSWpySmZavds3Pz0FgFEV77dkdaQGCCGonPItxS/8B//sZahOO53f+jM7bv8LMa+f1Tm/ipdNuuJs0ideiXP4ABRFwTHguAbPnXTZGez+08sUPf8+NUt/pnrBGkKbdsT3K2YTIhIFwPvlHEyZKei+aixdc3Cc0Iu0my9FtVlQHTZi/gCOgT3RXA70YAjFYpZLdknHpgULYNEi2LwZ3n0XKirqL3f77ZCdDTt2QHo6zJ0LXeTIj5YiA21JkiRJkpqc4nTtGwputYCiENhRQuEnC0g86TjMia7WreAxTg+F8c9ZTslL/8X/4yqiRcYa55l/vIG0Wy/DkptB4q9HUv7BNExpSThP6oslN6PR11HMxlfNnXf+DUvXHOx9uuIaPgDHoF44BvXGcUIvFKsF/+yl2Pp0xZyefFjnVW3WRtdFktqtWAwKC2H3biOoHjUKLBbo2BGGDYMrrzQC6ORkSEkBjwduvtmY4pKdbRz79tuQny8D7RbUpgPtyZMnM3nyZGBBa1dFkiRJkqQjpDiNOb4/DrsXgOQRfVuzOse8gifeYPeDr0AshiktiZRxF+Ia2g/3yMFoCftugGgeF2k3XXpU10q5/nxM6cm4Rw7C2iXnoOXcIwcf1XUkqd2rqYGffzZ6q+fMgZ07weeDykrYtQt0vXb5WbNg+PCDn+/tt/c937Sp9mupRbTpQHv8+PGMHz8eRfG1dlUkSZIkSTpCissJhLCmJxLcWUq3uy5u7SodkyLF5fi+mUfh02/jPKkvuc/ehmNgL1SHrdmuac5KJfWGC5vt/JL0i1BZafROV1UZr7t3N3qqPR5wOIxe6W7djJ/JyZCRAXZ769ZZOqQ2HWhLkiRJktSOHJAIS1TXAKC6HFAdJv1XJ+BduglrRlJr1O6YVDVnOcXPvU/lJ9/Ftzn2BNmu4ce3Ys0k6RiwfDnMnm0M7c7K2rc9GjXmTb/8sjFveskSCIeN+dYXXQRud+vVWWoyMtCWJEmSJKlZqGkpAGjdu8DK5Wx87D+tXKO2bW9SsspPZ5N0+ZkkXjSy0eeIVvgoeXkKpa99ijkzher5qwBQrBaSr/sVmXePwdazc9NWXJKOZbpuzIHevh22bIFgEC6/HN56C267zSjz1FNwySXg9RqB9a5dxrxrMHqnJ00yeqyvuQaaI8Hf3pugv+BlCdsiGWhLkiRJktQstFQjsZWWktS+1shuQUIIgj9tJrhhOwWP/IvAyg0AxPw1hx1oh7btRjGbKH/va3b94YX4dnu/bnR+52GSrx6FYpJf+STpqAkBW7cagfLu3TBtmtELHY3WLvfb3xo/c3KMskVF8OOP4HLB8cfD738Pxx0HXbsaycnk7+cvknxXJUmSJElqXnt6UTIuPpmYP9DKlWl9oc072ThqAqHNOzFnpRIpKAXAelxHMh/4LeXv/Q/N5ahznH/eSlSXA0f/Hghdp3LKLEpf/wzftPm1yvVZ+zG2Xp3rrGktSdJhiESMYd35+bBtm/HYvt1IVLZundErvVdWFlx2GVxxhRE0Z2UZw8GzsqB/fxgyxJh3bTaDrflyIUhtkwy0JUmSJElqXkIQDYQp+nQBKSP7tXZtWk3MX8O2MQ/i/WY+IhgCwHPuUBSrhZQx5+Ec2h9FUfDPWV5riGd4dwm7//QSZW98DkCPmS+y47a/EFy7BWu3XNxnnYTzpL6k/t/FDWb2liQJY9706tVGJu5Nm4y1qBctMpbLcjiMTN/7Z/jOyjISlXXpAhdfDHl5xjDvtDRjKa0Db2g9+GDt13K+9TFLBtqSJEmSJB21yZMn8+o/XuYvnAgsNjbuNy8w6jUSo1kzE1ungq0o5q9h513PUfrqJwCkjb+cnKdvQ7VbUeqZj6moKmJPoL3rvskUv/AfFLMJxWxCRKJsPOtWrMd1JOuhG8l68MZ6zyFJUj02bYITTjCeZ2VBjx6QlARnn230ZI8YAbm50KkTdO4MHTr8Mnqi5RztVtGuAm1fmZHlXpIkSZKktmX8+PHceP1veclzC/FAey8hEDGjhyh9dDtfL/kwh2PHqqpZN+Q3BH/eWmt7ty+eI/H8Uw95Db06wI67nqP4b//GddoJdP34KWqWr2fr1ffT4blJJF8/Wg4Nl1qPEFBQYCT00jRj2an2YO+w7wULjGHdktSM2lWgvfBzyL29tWshSZIkSVJjqTYLRCDn2pGtXZUWUfraJ/EgW/U46bv6IywdMw/r2EhhGcG1WwBIuvwsunzwOIqmkTBqKMeXz2q2OkvSQZWVwU8/wZo1MG8eTJ8OJSX79nftagzBbuvCYeOny9W69ZCOCe0q0JYkSZIkqR0SAqELUs8a8IvqhRVCUL1wDa6Ta887D+UXsPvBV3Cc0Ivjvn8Fze1s1HkTLjgVVIVuU57BdlynpqyyJB0+XTeC60cegf/+19hmMkHv3vC730G/fkZv9uWXG8tataZg0Ph5qGHekYjx02xu3vq0Nb+gv7vtSbsKtOVnRJIkSZLaJ6HrWFIPc/5XG55HGPVWYQIqv5qDXh1i67UP0PPH13ENG0C0rJKapevYcs39qG4H3b98vtFBNkDuUxPJfWpi01dekuoTDBo91T/9ZGTI3roVVq6EFSuMnmyTyZjD/Nxzxpxmi6X28c88A0880fz1jEaN7N8rVxrJzAoKjPqtX2/U/YQTjKRmDTlWA+292vDf1l+idhVoy8+GJEmSJLVDe+ZoW9ISWrsmR0wIgfeLH6i4/a90AYr/8RGuUwYCUL14LdtvfTq+Brbn3GF0efcRTKnHXuI3qY2LxWDnTpg/3xgCPm+eEbjuXQdaVY0kYAMGwE03wRlnwNChRjbug1GUpv2Svm6dEURv2QJr1xqvN20ybgDsDZTT04154ampMGiQ8XrmTKP8jz/CnDng98Pbbxs3EnbtMm4c/O9/xvHHaqAttah2FWhLkiRJktQO7Qm0rentM9DWQ2G2XnUflZ9+T/KebVWzFlM1awkAO+/4a7xs9hPjybx3rMwELrUt27bBmDGwZMm+YdbduxtB9G9/a/QGd+liZB1ubJZtVa29HNbh+OEH+Owzoz4rV8JttxlB8po1Rg81GAF8167GUPXRo43e9O7doWdPY7mt/T3xhBFo9+1buz6ffFK7XPfuMHYsZB5evgRJOhrtKtAu3NTaNZAkSZIkqbHE3h7tdhhol737FduufwiApCvPpvOo42DcuNqFFIXcv91J+vgrUMzt6quV9EsWDsPGjUbv7lNPQX4+/PWv0KuX0WOd00RrriuKMeS8pmZfz7cQRk/06tXw6adG4Lt1qzHUe/+kaQkJRibwyZPh+ONh1Ci49lo491xjDvjhBv133WWsV92jB5x6KgQC8MILxs0Dm81YsmvAgGM3CZqcf9sq2tX/Bl++AOP/2dq1kCRJkiSpPgf7LicixrBUc0Lj5yu3lrL3vmbbmAfjr7t/+TwJo08xhqICx5fMRGgapiS57qhCF+UAAAAgAElEQVTUCgoKYNkyo/d3/Xqjh3fYMKNXuKgINmyAKVP2DQkHI9icNKnp69Kli/GzUye46CIj2H3lFSOw3qtzZyMATkuD2bON9aqffdbofd4/QD9SVitM3C+vgdMJf/7z0Z3zl0jOw21R7SrQliRJkiSp/RHRGACq3XKIkm1DzaqN8SDbnJ1G3w1T0Zz2WmW0BJec5ym1jIICo3c4Px++/dYYIl1YaOyz243e271UFZKTjfnLf/iDkcQsM9MIhk3N9LX/ggvg5JONtalff92o07Bh8M9/wuDBxvzphnpUjzbIlqQ2SgbakiRJkiQ1KxHTEShohwi028LSXyIWY/vvHsfSMZO+Gz9BtchgWmpFY8fCO+8YzxXFmIN83XVGYHvCCUZP8bZtxlzkkSON4dctnR9A04zkakLA7t1GYK9pLVsHSWqDZKAtSZIkSVKz2tujrdmtrVyTQyt8+m2qF/3Ecd+9LINsqfWtXGkMx37ySWNOtaeeqQpdujTPkPDGUpSmm/ctNa02cBOzTRDCyCdQUGDcFCoo2Pf8hhsgL69JLycDbUmSJEmSmpX5f19gBirMbbuXK/DTZnbf/yIZ91yPe8Sg1q6OdKwLBqGyEo47zsi8LUlSXXuD502bYMcOY/m6HTuMALqwEKqrjVEee4Pq6urax7tcxlSLc8+VgbYkSZIkSe2Tp1/n1q5Cg0JbdgGQcec1rVwT6Zj21VdG0r0vvzTmX3fu3No1kn4pfgnJ0ISAkhIjiP7972Hu3Np5CsxmI/FfVpaRGDA93UjUN2SIsS07u/bPZsxE36YD7cmTJzN58mRgQWtXRZIkSZKkxtozXDGS24XSnQEyTG13bWkhBKWvTMWcnYYpLam1qyMda3w+I8nZmDFGFu6+feGhh4z1o/v2be3aSVLLiUaNXunt241HQYExJWHbNmNpuKlTjZEee513Hlx/vRE09+xpZLZv6TwFB9GmA+3x48czfvx4FMXX2lWRJEmSJOkIKX4fEZxtdp5geHcJa7pehAiF6fzuIygykZPU3Hw+oxfuzTeN5a6++Wbfvt69jfWn2+jvi9QOtcZnSQhj3fRNm4zl5nTdCIbPPBOKi43AecsW47FtmzH8e8cOWLOm7vBugJQUY3SHzQb/+pcxpWLAgPrzFrQRbTrQlv6fvfuOr7I+/z/+OifjZIcRIGxBRIVQByICWheifhHcYq240GqbOtuqaKsWtWj7s67iqqtaVFQc1eJAq4iCyFQQ2XuETXZOxrl/f1zn5GRCxklOTvJ+Ph7ncc65zz2u+5yT3Of6TBERkcjnzt5PMe1wuVte4uArKGJp97PLn7e/4LQwRiOt1sqVNjL4woWwYIElFmBTWw0cCLfeCtdea1NzdemiJFsih9cLX35pCfKyZfDTT1aQtG0bZGcffPvOnS2BTkmBAQPgggssge7dG3r2tBrtzp1bdEJdGyXaIiIi0qTcThnFxLa45MEpK+OHnqMB6HjNWHo//6eDTzHWws5BWiCfz5p/B/p+zpplU28lJMDxx1sikZEBJSX2OC0trOFKG9KYPto7d8LWrVZotGYN7N5tTbk//NBeDxQa9eoF7dtDv37BW+fO1tz7P/+xwqSuXS2R7tPn4H2k+/VreMxhpkRbREREmlwxHlwtpN+cr7iEFceNp3DpGgC6//Um0v9wRZijkog1d67NG3388VZTPWyYDdT04IPw0Uc2WJPHYwM4JSSEO1ppi+paQOjzWfK8caPd1q2zZt+rVsEPPwTXS00N1kRfdBGcdhpcf/2B+0Z37drmRs+PuETbVwZudZ0SERFpUaZMmcJzTz7NNQyp8fViYps5otptvGZSeZLdKfNiJdnSMDt22PzWjz9uz7t3txq/gLvvtsT6mmvgF79Qki0tz5o18OOPsHo1zJ5tTcBzKoyN1bOn9YUeMcKS6bFj4eijrcZaDiriEu0ZD8I594Q7ChEREakoMzOTX105gaeSf13j6y0l0S7LyWPv1I8ASDxhED2f+EOYI5KIsnmzjQ4+axZMm2Y1hcceC4sWwQknwMUXw89/Dl99ZYlJp07hjlgkaOpU+O47+77m5tp3GSA2FoYPh9tus/sOHaxpt7o1NErEJdo7V4c7AhEREanRAZontpREe9Nv/wpA/1nPkfzzY8McjUSM6dOtVrqkxL7nAwfCxIkwYYLVZFc1blzzxyhSm0Brildesft27awZ91VXwZ//bAPweTxhC6+1iqhEO5rWMc+6iIhIW3P4c7eGOwTyv1vG3ldnkHb9BUqy5eBKS23wpjfegLfeslq/N9+06Yk6dAh3dCJ117Gj9b1u317NvptRRCXapQBKtEVERFo+x6Foxz7i/E+7Xzu6HtuGPhxfYRHrr7gXgF7/uD30B5DW4fvv4YUX4MknKy//+9/hxhshOqJ+OosE9e0b7gjanIj7b5G7M9wRiIiISF0UrMsiDijDTVQYp8Vadfqvyf3ffADaXzwSl5IlqSg722qsn37a+q4GTJ4MJ58Mxx0HMTHhi09EIlLLmGejHn76LNwRiIiISJ2UWdW0l/D1/StauaE8yQboes+1odmx+rJFtv37bTCzCy6waYquu84GfrrCPwL966/DnXfaVF1KskWkAVSkKyIiIk3CKfMBNhBaOCY2yvnffFafbqOgd7jsLGJ6dCY+o18YIpEWwXFgxgy4/36YN8+WDR4MDz1kCXfv3rbsn/+0/tgiIo2gRFtERESahM9nibYvKjxJy5r/uxmALn8YT4+/3hyWGKQF2LHD+l2/8AKsW2fLLrwQJk2CAQOqr68kW0RCQIm2iIiINA1/jXY4po3J+WwejreYdhedriS7rXIc+PRTuOQS8HrhoovgwQdtruuoqHBHJyKtnBJtERERaRJlRSX2IC7uwCuGmOM4rD4jE4A+Ux9o1mNLC7B7N9x7Lzz1lD0/9lh4+23o0ye8cYlIm6JEW0RERJpEyb48AFwJ8c163KIf1wJwyMv34Y7VQFatmtcLH3wAPXrAq68Gk2uAQw+1/tiXXgphHPVeRNomJdoiIiLSJDydUwFIGnJEsx43f8FPAKSee3KzHleaUWEhPPccPPoobNwYXB4VBb/6Fdx9N3TvHr74RKTNa9D0Xk899RR9+vQhLi6OwYMHM3v27Dpt98033xAdHc3RRx/dkMOKiIhIJPHPgBWdnla39UNU6bjx6j+TcPxAotslh2aH0nJs22ZTcCUkwC23wPDhVmv95JOwbx+UllqttpJsEQmzetdoT5s2jVtuuYWnnnqKESNG8Oyzz3L22WezfPlyevXqVet22dnZXHHFFZx++uns2LGjwQEfMqTBm4qIiEgzcvxzTbuSmy/hzZvzPQAJxxzeNAdQE+Tw+PFH+Mc/4KWXrLn40KHw/POQkRHuyEREalTvGu2///3vTJgwgWuvvZYjjzySxx57jJ49e/L0008fcLvrr7+eyy67jGHDhjU4WBEREYkgZWV2n5jYbIfc9/bnAPR45NZmO6Y0gU2bwOezGupLL7WE+t134Z57YP9++PZbJdki0qLVK9EuLi5m4cKFjBo1qtLyUaNGMWfOnFq3e+mll1i7di333ntvnY7j9XrJyckpv1XkK6tPxCIiItJcXFVqe92F+fYgJaXZYsibvQSAqMTmHYBNQmTrVpvjundvm+f6mGPgzTfh4Yct+b7rLkhNDXeUIiIHVa9Ee/fu3ZSVldGlS5dKy7t06UJWVlaN26xevZo777yTqVOnEh1dt5bqkydPJjU1tfxWkRJtERGRyOAq8CfazZQYleUXUvjjWro98OtmOZ6E0ObNNs91377w/vu27M9/hthY+O47uP12eywiEiEaNBha1RJrx3GqLQMoKyvjsssu489//jP9+/ev8/4nTpxIdnZ2+a2in1Y3JGIRERFpSlOmTOHYY4+ttKy8RruZEu2sh17GKfTS4RdnNsvxJAQcB6ZPh8MPt/ubbrIBz2JjYdgwmDULjjsu3FGKiNRbvQZDS0tLIyoqqlrt9c6dO6vVcgPk5uayYMECFi9ezG9/+1sAfD4fjuMQHR3Np59+ymmnnVZtO4/Hg8fjqbAk2Hw8r6A+EYuIiEhzyMzM5Pqrr2NK4vXly1yF/ot2MzQddxyHrAdeAMDTt0eTH09CYNs2uOACmDcPzjkH/v3vYKFMYSG4G1QfJCLSItTrP1hsbCyDBw9m5syZlZbPnDmT4cOHV1s/JSWFpUuXsmTJkvLbDTfcwOGHH86SJUsYOnRo46IXERGRFqvoEP/82Yc30QjgFRSv3wrAIa9OavJjSQP5fLBgAWRnW5/rjAxYuRI+/hj+85/KLR+UZItIhKv39F633XYb48eP57jjjmPYsGE899xzbNq0iRtuuAGwZt9bt27llVdewe12k1FlRMjOnTsTFxdXbbmIiIi0LoVHHMvHiRdxVs+eTX6skh17AUg4uu5d1aQZZGXB734Hc+fC+vXB5dHR8KtfwR//CF27hi8+EZEmUu9Ee9y4cezZs4dJkyaxfft2MjIymDFjBr179wZg+/btbNq0KeSBioiISGRxfD6IimqWYwWm9Ypq13xzdstB/O9/8ItfwM6dlZenpcGXX8LAgWEJS0SkOdQ70Qb4zW9+w29+85saX3v55ZcPuO19993Hfffd15DDioiISCRxgBoGSz3oZo5T7/V3/n0qALE9qo8ZI2GwaVNwFPG//c1qsM86y/rrR0U16HshIhJJGpRoi4iIiByM4/Phcjd9QlW0fB0A3f6S2eTHKlfPwoA2Zf16OOEESEqyAc6OOCLcEYmINLuIHGmitCTcEYiIiMhB+Zxmqbnc+8anuJMT6XLrZU1+LDmIDz+EoUOt1nr+fCXZItJmRVSiHSg7fuKasIYhIiIiNajW5NtxmrxG23Ec9vzrQzr8YhTuOM/BN2iJfD647DJ48MFwR9JwBQXw61/DmDFw/PGweDHUMPWriEhbEVFNx3cB3YGFH4U7EhERETkYx+dAEyfa+XN/oGTzDtqPG9Wkx2kyjgOXXgpvvWXP7747vPE0xJIldg6bNsHTT8P116sPtoi0eRGVaOf57wefHdYwREREpC4cB1cTJ1wrR0wAIPmkY5r0OE3m//2/YJJdcR7pSLB7N1xzDcycaU3EFy1SU3EREb+ISrQDDdKKi8IahoiIiNSkStNxq9Fuul5qvkL7QZBy9nBcMRH1kwb27YMOHezxxIk2Kve//hXemA5m3z4b4Cwmxp7fdBN88AFcdx08/jjEx4c3PhGRFiSi+mjv89+XKNEWERFp8RyfD5qwQjv7ozkA9Hz0tqY7SFWNraHfsQN69w4m2dHR8MADjY+rKe3bByNGWMynngoTJsCsWfDVVzB4MDz3nJJsEZEqIirR3u2/V422iIhIy1NtxiuHJm06vinzYWJ6diHu8EOa7Bghd8891pcZ4OOPobCwSWv9G23ZMhg+HOZYoQbffAMvvginnGJzZL/zTljDExFpqVrwf/aaLQLcUeGOQkRERA7G5y3BHRfTJPsuy86jNGsPKaNOaJL9h1xpKVx8sdX+AgwaBGeeaTXaLdXHH8OQIRbjjz/CE0/A7bfba3fdBV98Ab16hTdGEZEWKuIS7a3AkcPDHYWIiEjrc9VVV+FyuSrdTjihHolslSrtssJiouKbZsqtvG+WAJB+x5VNsv+Q2r7d+jW//bYl11lZsHBh9fWqNQkIE58P/vpXOOccq7n+7jsYMABuvBEefhjKymwqsijVfIiI1KYFF6PWrsQb7ghERERap7POOouXXnqp/HlsbGyD91VW4MUd3/DtD2TDlfcB4OnXs0n2HzKOA2ecYY8zM+HJJ2vu591SpsPKz4crr4Tp022QtkmTqte6t+Sm7iIiLUREJtqfPAeXP9ByrkkiIiKthcfjIT09vWEbV6mQ9RV6iUqoR412HS/sJTv3Urp7v3+TFvxjYM0aOOwwe/zBB1ZD3JKtXAm//CWsWGG17xdeGO6IREQiVkQWSebsho3Lwh2FiIhI6/Pll1/SuXNn+vfvz3XXXcfOnTsPuL7X6yUnJ6f8VpE1HQ99jXbe19ZsvOeUO0K+75Dx+eCyy+zxiBEtP8meN8+ah2dn22jiSrJFRBolIhNtH7BhQbijEBERaV3OPvtspk6dyv/+9z8eeeQR5s+fz2mnnYbXW3ufrcmTJ5Oamkpqaiq9qgyMVVZQzxrtOsqbvZjY3l3p/JuLQ77vkDn3XJg/H555xhLXluztt+GEE6xwYNEiOPbYcEckIhLxWnSiPWXKFAYMGFBt+X+Bj15s/nhERERai6lTp5KUlFR+mz17NuPGjWP06NFkZGQwZswYPvroI1atWsV///vfWvczceJEsrOzyc7OZtPGTZVea6oa7Z2PvY7n0B4h32/ILF4MH34Ixx0H119f9z7NzT0Yms8HP/+5jYYOMHcuJCc3bwwiIq1Ui+6jnZmZSWZmJi5XTrXX5n8NZaUQ1aLPQEREpGUaO3YsQ4cOLX/evXv3aut07dqV3r17s3r16lr34/F48His1tpbVnkUahsMLbQ12t61WwBIHDYopPsNiZdftnmyi4ogLs4S15aqpAR+8QuYPdsGP3vxRQ1yJiISQhGbppYBC6bB0F+GOxIREZHIk5ycTPJBai/37NnD5s2b6dq1a4OO4SssDnnT8TVjbgWg068uCOl+G+2bb+Dqq4PPzzmn5c6R/frrwf7j06bBJZeENx4RkVYoYosu84DignBHISIi0jrk5eXx+9//nrlz57Jhwwa+/PJLxowZQ1paGueff36d9hGTEk/nIYcA1gq6rMAb8qbjTkkpuN3E9mrgyOhN5f77Kz9/4on6bd9co6f/9a/BJPv995Vki4g0kYhNtB0gewfk7Al3JCIiIpEvKiqKpUuXcu6559K/f3+uvPJK+vfvz9y5cw9a8x3gjnIz+rNby583uI92LX2Vy7Lz8K7bSq9nJtZ/n01p1iz45BN44w3IyrLpsfr0CXdUlTmOzYt9xx1w111QWgpjx4Y7KhGRVquFtmk6OAd452l47k/wn2YeO0RERKS1iY+P55NPPgnpPus9j/ZB5H65AHw+UkYeH7J9NljFwoB16+z+wgutuXiXLuGJqTY+nw3K9vzz8MgjcNtt4Y5IRKTVi9hE2wfs3hbuKERERKQ2Nhha6JqO53z2HbF9u+PpU33gtrAqK7P7qKgDr3cwTTHquOPAzTfDCy/YYG1XXhn6Y4iISDURm2hXvBR99hKMvLrWVUVERKSZOT4fvuLS0NZof/Zdy6jNrqqszPpYN1c/67rauhV6+KdBe+YZJdkiIs0oYvtoV+ya/cQ1dl9WEpZQREREpApfoRcgZIOhFW/ZQdGKDaSMHHrwlZtSTcl0WVnja7ND7bvvgkn2E09Y03EREWk2EZtob6NyrfbKL+A3sTB/WrgiEhERkQBfkZV+R4VoHu3cLxYAkHzqcSHZX0i1pES7qAj69oXAHOkLFsCNN4Y3JhGRNihiE22onGgvmG73z19at223/wQl3pCHJCIiIgQTbbcnJiT7y5vzA3FH9iE6rV1I9hdSoUi0Q9HsvLgYLr4Y1q+35+vXw+DBjd+viIjUW0Qn2p9VePzpFLvvcviBt9m3Ba53wX0D4LVfN1loIiIibZrjHyDMFR2anxr5c5eSOGxQSPYVcj5f+Gu0S0ttfuxPP7Vk++234ZBDwhuTiEgbFtGJdsUK6X3AbmDJSnjrDti6tOZtfvgg+HjOS5C9vfb9z3oNPn8Z8vbDvh2wfzcsmweF+bD+JzitA2Rtatw5lBTDtjWN24eIiEhL45T5AHBFNz4BLcsvpPCH1SQN/1mj9xVyjgMffQS9e4dmXw31xBMwfTq8+abdLryw8fGIiEiDRXSiDfCXL4KPA2OhvfpXmPgzKMqrvv6S9yo/v/vQmve7azP89Zfw6NVwfns4Ix1GdoKrToCTkuDiAZCzD87pDcX+jN9xrFC7Jrl7rWXZrE9h905btnIZjPbADYfBWBf8fTxckW6PA7ePn4M/jYSi/Dq/JdX88Ubo7oJP/xOcgaSoCIYfCrM/P/C2jmMt0QLX/gVz4bZrYO+e6r8HHKdpZiYREZHIU55oRzX+p0bhkpXgOCQMPrLR+wq5mTPhs8/gL38JXwwzZsDtt0NmJpx7bvjiEBGRchE7vVfAs3fWvHw/8PrNcPULwWXPXwbLP7XHf/ga/nYilBRaU/JnKySIhXlwdq/aj5kDFANRQBHQJw5e/gC+ex7mvQ9jbobrHguuv2Iu/GE4rAWy/ct6psPmrOA6UcDCf0MS0BnIB7oAT15vBQiXJME/18PWldBvMLijITG19i5de3fDziw4vUIru6truPZeOrL28zyQaS8dfJ1hJ9v98SdBbja8+CT06gPHDYfLrgu+XhPHgQ/fhofvhvWrK+/z4ivhuBFwyKE1t9TLy4XP/wu3XW0FCpl3wLbNUFgA27dYwcht98GgYyGtC3w3G7ZshFgPHD4QDukHScmQmwNZW6H3oeDxtIyWgSIikSKUiXbe7CW4kxOJz6ildDyc/vY36wd9zjnhOf6iRdZU/Jxz4PHHwxODiIhUE/GJ9j/mwZhaXnv3RUu013wDfYbCt69b0hoH9BtRed3r/Qnrhf+E+6+reX+xWCIcaG1ecTaxq8bYfo8APngchv8SHrwOXp4HZw+3xLyiikk2gL+imTz/DWBHlW1G94GOgAvw+c/jHz9CrwHBdR57AP72p+qxT5wMkyfWfF71dceDlgAfzNxZle8BNq232ztTIS4eTjkTPnm/7jXhc2dV3l9dTHm4+rLrGtGizu2Gux6CgnxIbQ8vPgEb18G5l1ohQL8j7ByHngTRdfwLC7QGcEd8GxMREVPeRzsUifbXS0gaNghXXf+pNpcVK6w2+7XXwjOHdk4OjBkDAwdaDCoNFhFpMVrYFevgRo2ycT4qSgeq5K18iI1K/sYz8NmvrcbY32KbLv77Z3xwgxsKsJpmB3jyOij0vx4NlPof/+o++NW9cM/N8OMTweNEYwl4AVa7vcS//yuPt9eHxlVOskedDD26w4uvQQzw1PRg0vfS+zXXOgdsJ5jkBwwbCN2Ah9+E8ZdU3+a2e+F391kSd+2N9jsg2mM13p26VF43Nwd+XGI1zq8+A5deA/EJNcdy0121xxkwdxZ8v8Bqsxd9C4++DMkp8OUn8ODtlpx+/N6B9/Hvj+DkUcEEdMUyeOgu+OkHq4WuzRlj4Ka74ajj4L3XIa0zJKdaTXWx12rLP3zLassTEmHc1Xb+SxfaeDK7suAX18I3/4PEJJg3O7hvnw8euL36Md9/w25VJafAUUNg41pI72417n36WTwzP7DPIiAuDjp0gmNPgIFHW416STEcngH/dyF07V75t1xersUXjt93IiIH4pRaCWpj+2g7jkPe3KV0vmlcKMIKrX/9C5KT4fzzw3P8K6+0ZPubbyChlgu2iIiERcQl2vHx1Ze5gL/MgrtOhkQgmeDUXzf92hLaLlhtcDbgRMM/b4YzroXrPoBJYyxJ3ltlvzOyoIM/GS0ttX7OAaPGwFcfwAAs+XUIJuhVa6Lxx5QILJ0FS4GuwMNvw1HDYGuF2tyKjxfNg38/C3c/bIndmy/X/J5so3KS3RUrfHADnq3W17uqS++Fy+6rEmMKpMdDZgyM/pMl2Y4DXzwJ0262dY65AC7+O6z+Ctr3hMNOAnctv6GGnRxsHl5WCgtmwKBT4KRT4M93Q4EDpdFw+lgrhE9OqXk/FR2RAS//p/Iyx7FE03GgpARiYyu/fuHl1feTcQzc+eDBj1cTnw9mf2YJ8qrltv8+h1nz9PffsOT5xNNh8waYMd0S+K/9feE3b7D7n34I7q93X4v/iEGwdBFs3WT7+vCtyse99xa7j4mx8+ycbt0DOqRZUt63vyXvySlWmBCfYK916QZrfoStGyA+1o7TfxCsmA+HHw99B1iz+nbtaz7XtQshOQ3S+zTs/RKRtmHKlClMmTKFMn9Ndqiajpds2UHZ3mwSjj2i0TGGlM8HU6faSN9xcY3fX31LTFeuhPfegxdf1OjiIiItkMtxWv7wVS5XDpAKZJOWlsLu3ZVfvwZ4wYH138OFp0GPo+CDCoOkdfPfbwM8QEaFZWBJ8iaC/adLgI5J8POx8OS/reb08//C3b+1188YAy++BzdEBWvSo4H1BGvNq+pah/O88Aa47l547Hfw51dqbgHm84G3EOITLfnfsR2O9/cnjwEaMh7r3+fBbUPtcQJWQ7+/wuvxWCFCLNYyIBlL4gM/CTodCjvXQmw8FBVCdCfI9kJ+zoGP6wX2+B8POgHGXgXvTIKc/TDgdOhzKCR3gi9ehJhYOGU8bFsN29fALf8CHHhlInz9ZvV99zgCjhoJE/4O0f4pXBd9Aq/dA+uWQFpPaJ8Ohx0HA4+HI0+BohxIPwJKiiB3F6R2s8+gIAe+et2O+/U0GyjPZYfH4wGvfzC85I5w0qXwv5ft99IhR1m82f4vRWw89BoIIy6yWuojh9tc7tk7Ycd6SEmDXhmwbjHs3ghfv29JcEIKlEXDwpWwfCVElUBuGaR3hCgHUlKh0IGdO6GgALwl9vnUMi5frdongVMGcdFAmf2tlBRYt4Z4oEtXKCyy8QGS24GvBGJdEBsTbB1RUGYFWfu2Qde+MHAE9M2Aw4+Bzr2tZcK+LBtrID7FBixM6wOxCVYw4Cuzgpvo6Lo3u2+N8rNh83LYvdnGjCjKs7/9uESIjQNckNQOElKhuNDWj46BDt2gUy/7H5G11r67OzfYPj0J9l2L8UB8sr3/aT0gJg4Ksm3AxlIv5OyB/TugMMfWa9fFvttlpVaY5YmHpA4QFW2vlxbb9zi1E7RLt88wLhEcnw3iWFoC+fts+/xs2LMVcKB9V4hPAm+B7a9jd7updUbky8nJITU1lQ8OvQbW7uLn3z9Oys/qVlKXO2shq065noGr3iHuMLu47f9wNmvH3MqgjR8S2yMHT14AACAASURBVCu9KUOvm6lT4fLLrcn4yJEwezaceGLj9ztpEjzzDGzbdvB1CwogMdFGOv/hB0ipQ0m1iIg0q4j7KVtQUH3ZbxfZfZd+sGovLP6i+jqB0gQvljhsw5KRXGxws6r25MG7r9mtorsessG1/vuAJVtdga4D4OZPoX13eOlqeO5lS9zbe6C/12rSv8cSnwwsgTn8Znitypgl05+xG8DHr8HH2yCtKzx1Lgw8C177TXDdMn/cA34GW/wZlctlP3hXfA4/fgxn3glXdrcfs6n+jt+B/t3bsSbtcQSTbLAm8FXf4kBNfaAJfCAJL/HfCtfa8+hCO7e4XfbFCvxerliXUUDlJD5g6bd2C9j6QeXXU4BNd1nLgzjg2r7B/ToEm/8HbFsB362Af/6j8n4CCeimtcBa+PwbWx5InKOwpDK5Qvw1Kf8+VZhjLncPzJgSfP7TN5W3KS6ENQvsVld7qzSP71vxSZbFyw67bxcDHbtYIub4P6wSHxQUQ1Ii9M+Azv1gWy7s2gXdesPKb61WvLQUduba51lUbMn0/mIocVsCt8MLGyr2W9hVJdBVNQS/BqjQzSMK8LjB5bPvb5F/WVkNmzpYk/6OHW2k/LhYq6nveQikJEFpPuzaCi4HolyQ7IGt26FgjzW7T+4EXh/s2WyFUV4fxHeAnofDoUdaoVGsCzq0g+wscLktcezWH8pKggUkaxdDQpIlmLs3Q2mRJYxxSZZQ7t8FqR0huT3k59pUgEX5EB0LnmSIcVuBU0mRrZeQAq4omyIwZxfgtoTU8dnxS4ohp8D+L/mwvyMHiHFBlBtKy6w7S2FgOXYuMf51o22XFPtvPqDMbdu7ymyFYgfy/G96nH9bx78fN5Dvts8/ymOFNk4JFJYF/z5c/vXK/Nt4CRbqxFbYT5R/uRv7vxBbIb4y/y2Gyn9nyR2sQCBnj33vSrz22cTEWqFC+3To0B1SOtryQGuaojzI22c3x7F9xMRaAU7+flsvKho6dLVCtl2bYPcWK6woyLGk35NghRKFuVYY4CuD9L5WCJCYCp5E274o3woz3FGwdzvsz7LCo6ho+xvP3gWJ7axgbehYOHV8DV/wNsLxleGink3HayhpKfx+FVHtkonp2aWGDcLoiy+sCdUJJzT/sW+7ze7fektJtohICxVxifZh/eD0n0OsG+Y8AV/5l99yM/zjiZp/tMcCmys8XwIMBrY24PjjLgkOnAZwy0w4ssLI3Ve/BBdMhvXz4OhzYcaD8P4f4Zgq+1n5uMXgw34Q76V6/+vTuwV/2LqqNJcOWPcDzHUHa4b7E2w6/8mT1mS+NA+WY4lNUjzkFVbeRyfsB+/db8Dbj1nyde+nkHES/OIo2FRTElWDUv+tPjORZWCf2U7/dkX+5Un+fQWe51BzgUh91VbL6/hvJf7jBgak8wCdEiGxENw++z3zp/nQqa/90C7YD948KC6Al6+Cdd/aPlZghSsxWCLiwhJ4L1YokIp9Nnn+Y3iBtCjILwNvInhj4Mi+1r99TzYMOQKS9kKPDtAlHbbkwra90KM9DDkRliyGzfNg/xbIj4Meh1mCGF8MXYshNgq2zoFVFQowVvrvu3ksMexRAElpVqtZlGuJBlhwxdhnkdbFkqHtu2F/rtWOOh7IL4SyeHDHg9sD7TpDXgGU5UB2ocW7Iw+KfBDnAk80eKIgNtpqNEv8yXdxhffLlw9rK3yZXIDz/cE/408/Dj6uWLPvWguu+cHWGG7sH2AgBQgkqmX+70IgGfQR/L8SjX03ywjupzjbngfijgK8+VC6z/73BBLU6IJgAhr4vgUKG1z+fQb+hqoJBNQQFb/0Ne68hvUDmbpf4Fx9BAuZAu9tIP667r4iN1b44gaiXUA2JObbY0+pdZMoBdwucOVB9DpwVlthh9sFce5gTNHR0C4FCktgRw7sK7TXSh0r8EiKhuISoAySPFb4FOWzbX1FkFcKJS4rxCj2QUEpuDdafKU++1uOB3wuK6jw4c8JXbbM50BCjBWIFJfaMU/Z2cYT7VKfXb8a2XS88PvVxP/sMFwtrbnD3/4GEyY0f/ObGTPg2WfhqadgyJDmPbaIiNRZxCXaq36AXT/Awp9gnX9QsmOPrXndnlGwuQw21PBaSQ3Lhp1sA3hdMAy8c61Z+EL/a8cDf3wbPv1bcP2YuMpJdkBquiXZAP93t902LIDnLoaCfZaYBZKYQG1PNyo3Z/8yFlZV+KF7DLVPer6nwuOD5cRVk2wIVk5mXhpcdv2og+wIawbc8zC4+Ddw7gT4fDp06Aw3nX3g7d5bCz36Vl5WVgLzp8E7t1tT4rVzKr++A9iCJajeKCiqknTEAIdhNXS1/RRzgH5nQOY0mLcYTjkFNm+GiRPh9ddrOD/8hRb5lhxHAzF58MZAyCkNJhXxsf75xsuqJ/IV3+6iCo+zqVwDD7DNf07pydCtC3y1yFoFbtoNm76u5aSgUq1x+YGWBp8mJFgBQVQceNP8A6jFQ1IC9D3Umv86DnToAK4UKCyEvDw79u5dsGyJzaW+e5cVKMQWWSLjC2SKAbkVHlf5IqZEQ792cNQRsHkP7Nhhf4PZhZDSAVLa26BuCW4o3W6fYfsESE2xWs3SMthTCHFlkJdv8cWUWNIfnQx7syEpFuLaQ3G0nWu7jpBfDL36WiK1aytkbYTdu60JfEks7CqyfDLaDV4HvP7PIDYqWBAQHWUxxCdAiQP78vxJp8/uo6OtUqtLgk0nV1wCKbFWy1pS4QsRKMgpJvgdjSGYKCYnQYf2VsvuLoVOna2GdP1ayM23ZK+k2H/MCvsMFAyktLMWDT4gOdFale7daev36W/vced0SEyAzp2tFn3NKli70loKbdpon337dtZKwO2/+XywOwuSkuy7Ex0N2Tn2mTj+c3FViMfBCkITPeCKgbwie99iYqGgyLZLiofUBPCWQn4JOC5730p9kOWtuUAs0OqECses2u/J2R0sRImtsF4hkOcNbrOnAJyC4PaOf5uKN0/gM/N/zoFCskBhSlSFEw78by72+pN+//o7ttRwIm1IqPpoF3y/ipQzh4UipNAqLoY//rH5j/vEEzBiBNxwQ/MfW0RE6qxFJ9qBgVUg2KY4yX8fFWs/hGqTAmTeD3fWMjr2D1iz72hgWAa88UP1FmvXu+AcbNCvfZvhuYuCr13wMJxZw8jTtel5DNzzk42X8sq/4JN3IXYjrNsAX+2Hvt1gUBd4f7F/gyrzgS2u8PjG6+Dm22F/tv2YnDwBxl0O0/4QzG+82I/4ij9vLrzB+kuedRkcfaL98H04E9566sCxf11gU3FBcOCxgNWrYe5c6zMe3RlGnA3p6fDKK3DGGbbO009bMvvYY3DqqVYr6vPZfgL7ioqBEy6Hob+0H/2lpZYgxsRY7VVUNOzZY/vKz4dbb4VOnWxQsCcetvfiwUesiXGveEj2wqFFcN698M9Z8MWX/oBnwh0daj/XESNsANelS+3HfsVkuBSrkQ1kAS5sgDt3cbC2PcH/3nciOCBdYF0INqWtuM9AjWqZ/7ErC+Jz4cIjYO8O63pQhtV+V2zWn4L9qC912Wdd6kCnrjDoaOjcA3Ztt+bURTE22FlUlPWhz/XXwCUmwJb19lqpz9//OsoSMG+xP8Fw7JzAjhEDxKfC8YdBx1Qo80F6T+vH27kbjD4XBmZYk/rYWNi71z7jDh0sSQx8jxrCWwSbV0POPutP7PggY7g1345LtKbGxUWwaxt06Qnd+1T+rjqOfZdy9sHGlfa42Osf8G0ZrFwMa3+ErWuhqIDyKvaYWH8Naj4c9jPoMwASU6BjOvTub+9ZYgrk7rMm5D372fGKvbB7p22/chlkbYH2Haz/dPs069sc47EkPzYWcvZa4VWMxxLmhGR7XpBn793eHTYAX1Q09Opn3/28HOvD3aGTFVRsWQdrfrBjdEy384iNs219+fDTh7B7u+0PrAl7QFICZBxhf8t7/INPpPeE9F72Xhd7rXWLy+WPyWffncRkO7bbDQW59vfd50hbx3HsfSnwNxFxR0G7NIvL7bb+3lFR1qx+02rbfucWcMdZS46d2/3N2/3V5w7giYN2nSxp37PL/mi6dbem+j7s/0+nTnbcnH12jLRudszs/ZbQl5TZ/xm325aXltr/xg7trGAlymVNv4u99n45QFyC1VTHx1lhgbfE3z+92LoH+EqCTeMDX7tRIei2G8lCkWj7Corwrt5Mwu1XhCqsxsv1lyqedx5063bgdUPt229h5kxLtltaDb+IiFQScYOhdSGFKGB1HtycBM/XsH73ZPDlwsKt9gMsoHNnGD8WHvFvFA1MvAkmPV7DTrDmn3s2QIfecHNy5deeree7NnIkfP55/bb5080w6TE491z4Ty1NxwPuvBMeeqjm19avrzwgaXGxf2Al/yCpjmM/hD1xwYHDauPzwWGHwbp1cMwxsHjxgddvadLSoHt3+5Ht88GwYVazfcUVltRXlZtrtd59+9rAZ9nZVjscKCT4fg4cmmHJ1JqlsH+3NdnuO9DuC/IsqetzpA0gFiio8BZZMrRuOUx/2vZxwpmwdK6t/8U7VhvWd4DV+PYbBHt2QKzHama7HwpDTrMWBNs32mvR0fZ40SxLMKJjLKnZtMqSDZfLlgUSwf27rcl4xy623HEs2SgptuPm5UBaOnTsCkkplgxl74Fl8ywxLSu1pHDpXHsvA9p3gq69IXuvJb7FRfY4JhaOPA6697XH+3bZcZNS/IPoxVjSVlZqzwH2bIfc/Rbrjs2Vj3MwcQn2/uzJgk7dbR95VZsRVHD4MfY59TjUksvk9hajt9CSwWKvJePrl9t7tHOL9csOFbf74OcXFWV91g+k7wBLRPdk2XtQUgxJqfb3nd7b3n9PnL3P7TrZue7fDVvX2a2owD7DqGh7vmsr1jy6DA45wr5LCclWyx8Ta99xb6HFlZhs79nWdZbgu932vY9Psv15Cy0pdbutj3VBrn3ue3dCt0Ns3bRudqzYOCvIiI61711cgiX8+f5kPCbWPquiQiuA2bkl+Lm43HYOiSm2bP9u27ZDF/s+FhUE/x46plsBQEJSsAACbJ3oGCu0crmChRK5/n7gCcm2TUys/Y32ONTiz8+1z6kw35YdObgx34rIFBgM7Z20S4jZXcjpG18gvlenOm2b+9UiVp38q/LB0PIX/sSK48ZzxLyXSTw+o4kjr6Pp0+Gii2DNGjj00NDt9/77rTS5tsHQHMcu5j6fXdjb8oiRIiIRIOIS7W7YoB8/7bfRlqsW6KYA2f4zeu95uP86q5ha4l+2ZgUceWSw2e+P39uAYnVR4rX+1mffBdPetWUTJljB8o031r5dfr41u6yrnBzYsAEGDQoucxyrJezbF7ZX7cxdByeeCF/X0Px44kTo3x+uvjq47C9/gRUr7Jrfyz+ieXGxJZz9+lXfR2KinSPA//5n7+/48TYga8CiRTYLyaRJ9Y+9ogkTLMG/6y57nwDatbO4nnkGBgyALVsgK8vGqSkuhowMuOCC6tN+SWiUlsL2DbB1vSX1e3fY43b+AaviEiwB8RZajfGWNZbwtu9kSVTOXv8Abr5gzafHX/PdobMVFnRMt2S5x6GW/Kd1s4HH1iyzRDE/xxJHl8sGEKwYS/vOViuZlGo13QlJdp/czmJzu+15TD2/H45jtedlpZZcJbezc9m2wRKtQKJZVGBJZvtOlnB64u1/iePY7+WkFHsfklKCA6Lt22WJmq/MaphdLjuPqGg7XtYm2z4pxZK8glwrCOjQRb+9JfwCifb0dhcRu9/LyG0vE9f1AE2JKihPtFdOJ65/b/b860M2XHUfR+d+RVRSC5onurCw5vlGG+P++63fdU0XeZ/PRofcvx9++1t48snQHltEREIuIhLteFcORVUS7e93WDPUinMvdwUm/gX+dRf88jaY/YE1R7znRRhbIZHs7rJRxwNycysnwhs2wNtvwy9/CV1rmJerW7fq18H+/W1Ky4Df/94S8JIKncGffhr++U+49FL4wx9smddrP8ob8+N42jTb5+efw2mn2fXY7W+p99//wjnnNHzftcnNtST94YetiXdDlZZaTZjPZ/vs3Ln6OlWbq4uISMsVSLTfTjwfT34po3b9m9i0uo2MXTXR3nLnk+x7/RMGbfywiaNuAQ6UaHfubNNFgP1I6d27WUMTEZH6a9wIJWHk9cLhFa7bgUGwbpxoz6f+3WqJAMZcVX37ivncyAoDms2dC336WCLcrRvMmmXJaqCp8PPP13wNXLXKmhUD3HQTPPJI5SQb4KqrYOHCYJIN1hy5sTVQ48ZZMnraafbcXeFTHT3aCt4vuAAuucT6VC9aZIntihVWQw5w8cWW9K5eDSedZM2ra3LPPf6atCQrUG9Mkg127h6PVQzUlGSDkmwRkUjklFo/B1dMPab3qqJ43VY8/XqGKqTINGmSJdnR0fDdd0qyRUQiRMQ2Miz2z1+cgg1E1d6/vGL9/Lof7b5qorbFB1nb4NHHbXaOefNg2TJrYjx8eOV1Tzml8vPrrqv83OeDBQvg+OOtCXNNzj47mKyHQ1ycdSmr6vDDYe3aysv69YOvvqq+roiISH04JTbcY73m0a6iZNc+YrqmhS6oSLN2Ldx7L6Sm2uiS7oitHxERaXMi9j/2f9+2+yRsWqxADrv+p4Nv63JB1+7w178Glw0aVDkRnjz5wPvYvj3YpPm446q//pe/2OuOY1NeqlZWRETaEsc/up+7ETXaTnEJbk8bHWCjsNBKv9u3t8FHlGSLiESUiP2vPXli5eeZd8AH39rIxBWdduGB91NYw7zSYKN4+3zw6quwdavNj+s4Ng7J5s02hUyAyz/X7D/+Ya+Xllr/ZRERkbbOFdPwxnNOcSmu2IhtfNc4gTm6H3qofiOqiohIixCxiXZVEyfDsUPhXw9XXn7CqANvFxdno1UHXHttsG+1ywWXX259tT3+SbtTU6FHj+r7cbkgM9Nej2p44b2IiEir4Ypy42pEky6nuARX7EHmnWwtXK5g/7dVq6z0fuJE+NWvwhuXiIg0SKtItH82ONg0e/Oayq/Nm3nw7ceNsz7WmzfbqOCaHkdERKTxGtM/G6yfd5tJtAMcx0ruu3eHP/0p3NGIiEgDRVSinVrL8oFHBx8Hmo5P8Le4evC1g++3XTsbEK2mmmoRERFpmMY0Gwd/H+22lmi/+SZ89pnVaId6rm4REWk2EZVo13S5ueVP8MA/qi//9f2wwIHoNnZ9FhERaSkaMxAaQFl+Ia64NjQYWk4O3HornH8+/N//hTsaERFphIhvJH14hvWz3r4R5nxsyzKGhjcmERERaVzTcV+Rl9KsPcT2Sj/4yq1FYaGNLv744+GOREREGimiarRr8un7dn/T2TD5Bns87MzwxSMiIiLG1Yga7dKd+wDazjzaq1bZ/b33Qs+e4Y1FREQaLSIT7c4VHhcX231BbnDZuuXNGo6IiEibN2XKFAYMGMCQIUPKl7kb0Ue7dL9d2KPbpzQ6tohw/vk2b/bNN4c7EhERCYGITLQrThRyo3++6hhPcNnXHzZrOCIiIm1eZmYmy5cvZ/78+eXLGlOjXbY3G4Co9smNji0inH8+rF4NsW2oT7qISCsW8Yl2nH+EtC1rg8smv9ms4YiIiEgN3I3oo128MQuA2B5dQhWOiIhIs4n4RPud52DHFmjfyZ4/+yX8fEw4ohIREZGKGlOj7V27hZiuabgT4kIYkYiISPOIyES7ojcfg9E9Yd8uez745PDGIyIiIqYxfbS9a7fgObRHCKMRERFpPhGbaMcDsVSu3RYREZGWo9412q7gVd27dguefhp9W0REIlPEJtrtgI7hDkJERERq1fga7e4hjEZERKT5tOhEOzBVSFUdNS6KiIhIi+dq4GBoZdl5lO7er6bjIiISsVp0oh2YKqSiaGD8H2pe/65nmz4mERERqZuGDobmXbMZQIm2iIhErIa36QqTBKBdWvXl97wIY65q7mhERESkNg1tOl60ciMAnsN6hTIcERGRZtOia7RrktYRjq1hZPG4hEpjqIiIiEiYNbRGu3hjFlHtkolunxLiiERERJpHxCXaD0yFhKTqy5Vki4iItCzuhibam7KI7d01xNGIiIg0n4hrOh4dDfEVEu3ZefDfV+G0C8MXk4iIiFTX0MHQijduJ25AnxBHIyIi0nwirkbbHQWxnuDz+ES46AaIati1XERERJqIq4F9tFWjLSIikS7iEm2XS83ERUREIkFDm447JaXE9koPcTQiIiLNJ+KajgeS7FcXQGrH8MYiIiIitWto03FANdoiIhLRIi7RDjhycLgjEBERkQNp6KjjgGq0RUQkokVc03HHCXcEIiIiUhfuxtRoK9EWEZEIFnGJtvpni4iIRIbGNB2P6dIhhJGIiIg0r4hLtFGiLSIi0uJMmTKFAQMGMGTIkPJljWk67tJ0IiIiEsEiLtH2+cIdgYiIiFSVmZnJ8uXLmT9/fvmyhjYdjzvikBBFJSIiEh4RlWjHAIOODXcUIiIiUhcNbToe07NLiCMRERFpXhGVaHcEEpPCHYWIiIjURUMT7Vgl2iIiEuEiKtE+9ucaDE1ERCRS1LePti83H4DYHkq0RUQkskVUoq3+2SIiIpHDFVW/nxklWXsAiOneqSnCERERaTYNSrSfeuop+vTpQ1xcHIMHD2b27Nm1rvvOO+9wxhln0KlTJ1JSUhg2bBiffPJJg4J1lGiLiIhEjPom2mW5BQBEJSc0RTgiIiLNpt6J9rRp07jlllu4++67Wbx4MSeddBJnn302mzZtqnH9r776ijPOOIMZM2awcOFCTj31VMaMGcPixYvrHaxqtEVERCJHfafoih/YF4C4gYc2RTgiIiLNxuU4jlOfDYYOHcqxxx7L008/Xb7syCOP5LzzzmPy5Ml12sfAgQMZN24c99xzT53Wj3flUEQqvzw+m3/PS6lPuCIiItKMcnJySE1NZRpncfyUGznkN/9Xr+1L9+UQ3V7XehERiWz1qtEuLi5m4cKFjBo1qtLyUaNGMWfOnDrtw+fzkZubS4cOHWpdx+v1kpOTU34LUNNxERGRyFHfpuOAkmwREWkV6nUF3L17N2VlZXTpUnk00C5dupCVlVWnfTzyyCPk5+dzySWX1LrO5MmTSU1NLb8FHDm4PtGKiIhIWLk1VYiIiLRNDRoMzVVlji3Hcaotq8nrr7/Offfdx7Rp0+jcuXOt602cOJHs7OzyW/C4DYlWREREwqEhNdoiIiKtQXR9Vk5LSyMqKqpa7fXOnTur1XJXNW3aNCZMmMBbb73FyJEjD7iux+PB4/FUWGLNx+vXm1xERETCSYm2iIi0VfW6AsbGxjJ48GBmzpxZafnMmTMZPnx4rdu9/vrrXHXVVbz22muMHj26YZECJ5/X4E1FRESkmSnRFhGRtqpeNdoAt912G+PHj+e4445j2LBhPPfcc2zatIkbbrgBsGbfW7du5ZVXXgEsyb7iiit4/PHHOeGEE8prw+Pj4yv1v66Lo2rP5UVERKSFUaItIiJtVb0T7XHjxrFnzx4mTZrE9u3bycjIYMaMGfTu3RuA7du3V5pT+9lnn6W0tJTMzEwyMzPLl1955ZW8/PLLjT8DERERaZHqO4+2iIhIa1HvebTDITCPdnZ2NikpmvZDRESkpao4j/aJb/2RbheNCHdIIiIizU5tukRERKRJqOm4iIi0VboCioiISKNNmTKFAQMGMGTIkPJlSrRFRKSt0hVQREREGi0zM5Ply5czf/788mVKtEVEpK3SFVBERESahBJtERFpq3QFFBERkSahRFtERNoqXQFFRESkSbjc+pkhIiJtk66AIiIi0jRUoy0iIm2UroAiIiLSJNR0XERE2ipdAUVERKRJKNEWEZG2SldAERERaRJKtEVEpK3SFVBERESahBJtERFpq3QFFBERkSahUcdFRKSt0hVQREREmoRqtEVEpK2KiCugE+4AREREpP6UaIuISBulK6CIiIg0CdVoi4hIW9Wir4BTpkxhwIAB4Q5DREREGkCJtoiItFUt+gqYmZnJ8uXLwx2GiIhIxHvnnXc488wzSUtLw+VysWTJkmrreL1ebrzxRtLS0khMTGTs2LFs2bKlwcdUoi0iIm2VroAiIiJtQH5+PiNGjOChhx6qdZ1bbrmFd999lzfeeIOvv/6avLw8zjnnHMrKyhp0TJfb1dBwRUREIlp0uAOoC12mRUREGmf8+PEAbNiwocbXs7OzeeGFF3j11VcZOXIkAP/+97/p2bMnn332GWeeeWa9j+mKimpwvCIiIpFMNdoiIiLCwoULKSkpYdSoUeXLunXrRkZGBnPmzKl1O6/XS05OTqVbOTUdFxGRNkpXQBERESErK4vY2Fjat29faXmXLl3IysqqdbvJkyeTmppafuvZs2f5a+qjLSIibZWugCIiIq3M1KlTSUpKKr/Nnj27wftyHAeXq/ZOXBMnTiQ7O7v8tnnz5vLXlGiLiEhbFRF9tEVERKTuxo4dy9ChQ8ufd+/e/aDbpKenU1xczL59+yrVau/cuZPhw4fXup3H48Hj8dT4mhJtERFpq3QFFBERaWWSk5Pp169f+S0+Pv6g2wwePJiYmBhmzpxZvmz79u0sW7bsgIn2gWjUcRERaatUoy0iItIG7N27l02bNrFt2zYAVq5cCVhNdnp6OqmpqUyYMIHf/e53dOzYkQ4dOvD73/+eQYMGlY9CXm9uleeLiEjbpCugiIhIG/Cf//yHY445htGjRwNw6aWXcswxx/DMM8+Ur/Poo49y3nnncckllzBixAgSEhL44IMPiGrgNF2q0RYRkbbK5TiOE+4gDibelUMRqWRnZ5OSkhLucERERKQWOTk5pKamMo2zuDD/PaISau6/LSIi0pqpRltERESahmq0RUSkjVKiLSIiIk1CTcdFRKStUqItIiIiTUODoYmISBulK6CIiIg0CdVoi4hIWxURiXaLH61NbKIgpwAADg1JREFUREREqnMp0RYRkbYpIhJtERERiTwuJdoiItJGKdEWERERERERCSEl2iIiIiIiIiIhpERbREREREREJIRadKI9ZcoUBgwYgIZDExERERERkUjRohPtzMxMli9fDmgwFREREREREYkMLTrRFhERkcgQaIU2ZMiQcIciIiISdi7HcVp8u+w4Vw5eUsnOziYlJSXc4YiIiEgtcnJySE1NZRpncYnzUbjDERERCQvVaIuIiIiIiIiEkBJtERERERERkRBSoi0iIiIiIiISQkq0RUREREREREJIibaIiIiIiIhICCnRFhEREREREQkhJdoiIiIiIiIiIaREW0RERERERCSElGiLiIiIiIiIhJASbREREREREZEQUqItIiIiIiIiEkJKtEVERERERERCSIm2iIiIiIiISAgp0RYREREREREJISXaIiIiIiIiIiGkRFtEREREREQkhJRoi4iIiIiIiISQEm0RERFptClTpjBgwACGDBkS7lBERETCrkUn2oGLtoMT7lBERETkADIzM1m+fDnz588PdygiIiJh16IT7cBF24Ur3KGIiIiIiIiI1EmLTrRFREREREREIo0SbREREREREZEQUqItIiIiIiIiEkJKtEVERERERERCSIm2iIiIiIiISAgp0RYREREREREJISXaIiIiIiIiIiGkRFtEREREREQkhCIi0XbCHYCIiIiIiIhIHUVEoi0iIiIiIiISKZRoi4iIiIiIiISQEm0RERERERGREFKiLSIiIiIiIhJCSrRFREREREREQkiJtoiIiIiIiEgIKdEWERERERERCSEl2iIiItJoU6ZMYcCAAQwZMiTcoYiIiISdy3EcJ9xBHIzHlUMxqWRnZ5OSkhLucERERKQWOTk5pKamMo2zuMT5KNzhiIiIhEV0uANwHIfc3NxKy7xeL16vt/x5sf8+JyenGSMTERFpWsnJybhcrnCHISIiIiEW9kQ7NzeX1NTUOq3bs2fPJo5GRESk+bTmllrpF58Y7hBERETCJuxNx+tSo719+3aOP/54li9fTvfu3eu1/yFDhjB//vwGxdbQbRuyXU5ODj179mTz5s0N+tHVnLE2ZrvGnGdzx9rQbcNxjo3ZVt/X2un7GvrjNXbb1vZ9bY012oGm4625EEFERORgwl6j7XK56nwhTk5OrvdFOyoqqsEX+oZu25hjpqSkNOsxw3GO0LDzDEeszf1Z6vvasrYL0Pc19MfT91VERERas1Y/6nhmZmazb9uYYzZUc8faFs6xsds29/H0WYZ+u8bQ97Vptm0rn6WIiIhEtrA3Ha+LLVu2lDf769GjR7jDaRJtpaldWzhPnWPr0RbOU+cooab3W0REJEJqtD0eT6X71sjj8XDvvfe26nOEtnGeOsfWoy2cp85RREREJPQiokZbpeMiIiKRQddsERGRCKnRFhEREREREYkUSrRFREREREREQkiJtoiIiIiIiEgIRUQfbcdxyM3NJTk5GZfLFe5wREREpBa6ZouIiERIjbbL5SIlJaVFX7AnT57MkCFDSE5OpnPnzpx33nmsXLmy0jqO43DffffRrVs34uPjOeWUU/jxxx8rreP1ernxxhtJS0sjMTGRsWPHsmXLlkrr7Nu3j/Hjx5Oamkpqairjx49n//79TX6OVU2ePBmXy8Utt9xSvqy1nOPWrVu5/PLL6dixIwkJCRx99NEsXLiw/PVIP8/S0lL++Mc/0qdPH+Lj4+nbty+TJk3C5/NF9Dl+9dVXjBkzhm7duuFyuXjvvfcqvd6c57Rp0ybGjBlDYmIiaWlp3HTTTRQXFzfpOZaUlHDHHXcwaNAgEhMT6datG1dccQXbtm2LqHM82HlWdf311+NyuXjsscci7jxbo0i4ZouIiDQ5R0LizDPPdF566SVn2bJlzpIlS5zRo0c7vXr1cvLy8srXeeihh5zk5GRn+vTpztKlS51x48Y5Xbt2dXJycsrXueGGG5zu3bs7M2fOdBYtWuSceuqpzlFHHeWUlpaWr3PWWWc5GRkZzpw5c5w5c+Y4GRkZzjnnnNOs5/vdd985hxxyiPOzn/3Mufnmm1vVOe7du9fp3bu3c9VVVznz5s1z1q9f73z22WfOmjVrWs15PvDAA07Hjh2dDz/80Fm/fr3z1ltvOUlJSc5jjz0W0ec4Y8YM5+6773amT5/uAM67775b6fXmOqfS0lInIyPDOfXUU51FixY5M2fOdLp16+b89re/bdJz3L9/vzNy5Ehn2rRpzooVK5y5c+c6Q4cOdQYPHlxpHy39HA92nhW9++67zlFHHeV069bNefTRRyPuPEVERKR1UqLdRHbu3OkAzqxZsxzHcRyfz+ekp6c7Dz30UPk6RUVFTmpqqvPMM884jmM/kmNiYpw33nijfJ2tW7c6brfb+fjjjx3HcZzly5c7gPPtt9+WrzN37lwHcFasWNEcp+bk5uY6hx12mDNz5kzn5JNPLk+0W8s53nHHHc6JJ55Y6+ut4TxHjx7tXHPNNZWWXXDBBc7ll1/uOE7rOMeqyVlzntOMGTMct9vtbN26tXyd119/3fF4PE52dnaTnWNNvvvuOwdwNm7cGJHn6Di1n+eWLVuc7t27O8uWLXN69+5dKdGOxPMUERGR1iMimo5HouzsbAA6dOgAwPr168nKymLUqFHl63g8Hk4++WTmzJkDwMKFCyn5/+3dT0jTfRwH8HePy2lgS4u51rD0ZDkD2aAs6ZBBkNGt0MbcOVpaxkrq0Mn+XII6JNShi0VddrBTulqSMFq4LWcXhWR22FrEWIGpo32ew4M/nv2xnodnLn6/5/2CXX7fj/t+38h+8OG3fb+ZTE6N2WyG1WpVagKBAAwGA/bt26fU7N+/HwaDQalZb2fPnkVXVxeOHDmSc10rGUdHR2G323Hy5EkYjUa0tbXhwYMHyrgWcnZ0dODFixeYnZ0FALx79w6Tk5M4duyYZjLmK2emQCAAq9UKs9ms1Bw9ehTLy8s5P0Eoh3Q6jQ0bNmDLli0AtJMxm83C6XTC4/GgpaWlYFwrOYmIiEiddL97AVokIhgYGEBHRwesVisAIJFIAADq6+tzauvr6xGLxZSayspK1NbWFtSs/n0ikYDRaCyY02g0KjXr6cmTJwiFQnj79m3BmFYyfvjwAcPDwxgYGMCVK1cQDAbR19cHvV6P3t5eTeS8fPky0uk0mpubUVFRgR8/fmBoaAg9PT3K2lbXm79+tWTMV85MiUSiYJ7a2lpUVlaWNffS0hIGBwdx+vRpbN68WVmbFjLeunULOp0OfX19Rce1kpOIiIjUiY32OnC73Ziensbk5GTBWP7mMCLyyw1j8muK1f+T9/mvPn78iP7+foyNjaGqqmrNOjVnBP56Uma323H9+nUAQFtbG96/f4/h4WH09vauuUY15Xz69ClGRkbw+PFjtLS0IBKJ4Pz58zCbzXC5XGuuT00Z11KuTL87dyaTQXd3N7LZLO7du/fLejVlnJqawp07dxAKhf71XGrKSUREROrFr46X2Llz5zA6Ogq/3w+LxaJcN5lMAFDwBCSZTCpPS0wmE1ZWVpBKpX5a8+nTp4J5P3/+XPDUpdSmpqaQTCZhs9mg0+mg0+kwMTGBu3fvQqfTKfOrOSMAbN++HXv27Mm5tnv3biwsLCjrA9Sd0+PxYHBwEN3d3WhtbYXT6cSFCxdw48YNZW2AujPmK2cmk8lUME8qlUImkylL7kwmg1OnTmF+fh7j4+PK0+zVtak94+vXr5FMJtHQ0KDci2KxGC5evIhdu3Yp61N7TiIiIlIvNtolIiJwu93wer14+fIlGhsbc8YbGxthMpkwPj6uXFtZWcHExAQOHDgAALDZbNi4cWNOTTwex8zMjFLT3t6OdDqNYDCo1Lx58wbpdFqpWS+dnZ2IRqOIRCLKy263w+FwIBKJoKmpSfUZAeDgwYMFR7PNzs5i586dALTxv1xcXMQff+R+/CsqKpTjvbSQMV85M7W3t2NmZgbxeFypGRsbg16vh81mW9ecq0323NwcfD4ftm7dmjOuhYxOpxPT09M59yKz2QyPx4Pnz59rJicRERGpWPn2XdO2M2fOiMFgkFevXkk8Hldei4uLSs3NmzfFYDCI1+uVaDQqPT09RY8Wslgs4vP5JBQKyeHDh4seR7N3714JBAISCASktbW17Md7rfr7ruMi2sgYDAZFp9PJ0NCQzM3NyaNHj2TTpk0yMjKimZwul0t27NihHO/l9Xpl27ZtcunSJVVn/Pbtm4TDYQmHwwJAbt++LeFwWNlxu1yZVo+E6uzslFAoJD6fTywWS0mOhPpZxkwmIydOnBCLxSKRSCTnXrS8vKyajL/KWUz+ruNqyUlERETaxEa7RAAUfT18+FCpyWazcu3aNTGZTKLX6+XQoUMSjUZz3uf79+/idrulrq5Oqqur5fjx47KwsJBT8+XLF3E4HFJTUyM1NTXicDgklUqVI2aB/EZbKxmfPXsmVqtV9Hq9NDc3y/3793PG1Z7z69ev0t/fLw0NDVJVVSVNTU1y9erVnGZMjRn9fn/Rz6HL5Sp7plgsJl1dXVJdXS11dXXidrtlaWlpXTPOz8+veS/y+/2qyfirnMUUa7TVkJOIiIi0aYOISDmenBMRERERERH9H/A32kREREREREQlxEabiIiIiIiIqITYaBMRERERERGVEBttIiIiIiIiohJio01ERERERERUQmy0iYiIiIiIiEqIjTYRERERERFRCbHRJiIiIiIiIiohNtpEREREREREJcRGm4iIiIiIiKiE2GgTERERERERlRAbbSIiIiIiIqIS+hMLmHGmYQyYWwAAAABJRU5ErkJggg==\n", "text/plain": [ "Graphics Array of size 1 x 2" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "nToGenerate = 15000\n", "iterations = 5\n", "g = twoRunningMeansPlot(nToGenerate, iterations) # uses above function to make plot\n", "show(g,figsize=[10,5])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We talked about the Cauchy in more detail in an earlier notebook. If you cannot recall the detail and are interested, go back to that in your own time. The message here is that although with the Bernoulli process, the sample means converge as the number of observations increases, with the Cauchy they do not. \n", "\n", "\n", "\n", "# Limits of a Sequence of Real Numbers\n", "\n", "A sequence of real numbers $x_1, x_2, x_3, \\ldots $ (which we can also write as $\\{ x_i\\}_{i=1}^\\infty$) is said to converge to a limit $a \\in \\mathbb{R}$,\n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} x_i = a$$\n", "\n", "if for every natural number $m \\in \\mathbb{N}$, a natural number $N_m \\in \\mathbb{N}$ exists such that for every $j \\geq N_m$, $\\left|x_j - a\\right| \\leq \\frac{1}{m}$\n", "\n", "What is this saying? $\\left|x_j - a\\right|$ is measuring the closeness of the $j$th value in the sequence to $a$. If we pick bigger and bigger $m$, $\\frac{1}{m}$ will get smaller and smaller. The definition of the limit is saying that if $a$ is the limit of the sequence then we can get the sequence to become as close as we want ('arbitrarily close') to $a$, and to stay that close, by going far enough into the sequence ('for every $j \\geq N_m$, $\\left|x_j - a\\right| \\leq \\frac{1}{m}$')\n", "\n", "($\\mathbb{N}$, the natural numbers, are just the 'counting numbers' $\\{1, 2, 3, \\ldots\\}$.)\n", "\n", " \n", "\n", "Take a trivial example, the sequence $\\{x_i\\}_{i=1}^\\infty = 17, 17, 17, \\ldots$\n", "\n", "Clearly, $\\underset{i \\rightarrow \\infty}{\\lim} x_i = 17$, but let's do this formally:\n", "\n", "For every $m \\in \\mathbb{N}$, take $N_m =1$, then\n", "\n", "$\\forall$ $j \\geq N_m=1, \\left|x_j -17\\right| = \\left|17 - 17\\right| = 0 \\leq \\frac{1}{m}$, as required.\n", "\n", "($\\forall$ is mathspeak for 'for all' or 'for every')\n", "\n", "\n", "\n", "What about $\\{x_i\\}_{i=1}^\\infty = \\displaystyle\\frac{1}{1}, \\frac{1}{2}, \\frac{1}{3}, \\ldots$, i.e., $x_i = \\frac{1}{i}$?\n", "\n", "$\\underset{i \\rightarrow \\infty}{\\lim} x_i = \\underset{i \\rightarrow \\infty}{\\lim}\\frac{1}{i} = 0$\n", "\n", "For every $m \\in \\mathbb{N}$, take $N_m = m$, then $\\forall$ $j \\geq m$, $\\left|x_j - 0\\right| \\leq \\left |\\frac{1}{m} - 0\\right| = \\frac{1}{m}$\n", "\n", "### YouTry\n", "\n", "Think about $\\{x_i\\}_{i=1}^\\infty = \\frac{1}{1^p}, \\frac{1}{2^p}, \\frac{1}{3^p}, \\ldots$ with $p > 0$. The limit$\\underset{i \\rightarrow \\infty}{\\lim} \\displaystyle\\frac{1}{i^p} = 0$, provided $p > 0$.\n", "\n", "You can draw the plot of this very easily using the Sage symbolic expressions we have already met (`f.subs(...)` allows us to substitute a particular value for one of the symbolic variables in the symbolic function `f`, in this case a value to use for $p$)." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "var('i, p')\n", "f = 1/(i^p)\n", "# make and show plot, note we can use f in the label\n", "plot(f.subs(p=1), (x, 0.1, 3), axes_labels=('i',f)).show(figsize=[6,3]) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "What about $\\{x_i\\}_{i=1}^\\infty = 1^{\\frac{1}{1}}, 2^{\\frac{1}{2}}, 3^{\\frac{1}{3}}, \\ldots$. The limit$\\underset{i \\rightarrow \\infty}{\\lim} i^{\\frac{1}{i}} = 1$.\n", "\n", "This one is not as easy to see intuitively, but again we can plot it with SageMath." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "var('i')\n", "f = i^(1/i)\n", "n=500\n", "p=plot(f.subs(p=1), (x, 0, n), axes_labels=('i',f)) # main plot\n", "p+=line([(0,1),(n,1)],linestyle=':') # add a dotted line at height 1\n", "p.show(figsize=[6,3]) # show the plot" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, $\\{x_i\\}_{i=1}^\\infty = p^{\\frac{1}{1}}, p^{\\frac{1}{2}}, p^{\\frac{1}{3}}, \\ldots$, with $p > 0$. The limit$\\underset{i \\rightarrow \\infty}{\\lim} p^{\\frac{1}{i}} = 1$ provided $p > 0$.\n", "\n", "You can cut and paste (with suitable adaptations) to try to plot this one as well ..." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "x" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(end of You Try)\n", "\n", "---\n", "\n", "*back to the real stuff ...*\n", "\n", "# Limits of Functions\n", "\n", "We say that a function $f(x): \\mathbb{R} \\rightarrow \\mathbb{R}$ has a limit $L \\in \\mathbb{R}$ as $x$ approaches $a$:\n", "\n", "$$\\underset{x \\rightarrow a}{\\lim} f(x) = L$$\n", "\n", "provided $f(x)$ is arbitrarily close to $L$ for all ($\\forall$) values of $x$ that are sufficiently close to but not equal to $a$.\n", "\n", "For example\n", "\n", "Consider the function $f(x) = (1+x)^{\\frac{1}{x}}$\n", "\n", "$\\underset{x \\rightarrow 0}{\\lim} f(x) = \\underset{x \\rightarrow 0}{\\lim} (1+x)^{\\frac{1}{x}} = e \\approx 2.71828\\cdots$\n", "\n", "even though $f(0) = (1+0)^{\\frac{1}{0}}$ is undefined!" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "# x is defined as a symbolic variable by default by Sage so we do not need var('x')\n", "f = (1+x)^(1/x)\n", "# uncomment and try evaluating next line\n", "#f.subs(x=0) # this will give you an error message" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can get some idea of what is going on with two plots on different scales" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics Array of size 1 x 2" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "f = (1+x)^(1/x)\n", "n1=5\n", "p1=plot(f.subs(p=1), (x, 0.001, n1), axes_labels=('x',f)) # main plot\n", "t1 = text(\"Large scale plot\", (n1/2,e), rgbcolor='blue',fontsize=10) \n", "n2=0.1\n", "p2=plot(f.subs(p=1), (x, 0.0000001, n2), axes_labels=('x',f)) # main plot\n", "p2+=line([(0,e),(n2,e)],linestyle=':') # add a dotted line at height e\n", "t2 = text(\"Small scale plot\", (n2/2,e+.01), rgbcolor='blue',fontsize=10) \n", "show(graphics_array((p1+t1,p2+t2)),figsize=[6,3]) # show the plot" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "all this has been laying the groundwork for the topic of real interest to us ...\n", "\n", "# Limit of a Sequence of Random Variables\n", "\n", "We want to be able to say things like $\\underset{i \\rightarrow \\infty}{\\lim} X_i = X$ in some sensible way. $X_i$ are some random variables, $X$ is some 'limiting random variable', but what do we mean by 'limiting random variable'?\n", "\n", "To help us, lets introduce a very very simple random variable, one that puts all its mass in one place. " ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics Array of size 1 x 2" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "theta = 2.0\n", "show(graphics_array((pmfPointMassPlot(theta),cdfPointMassPlot(theta))),\\\n", " figsize=[8,2]) # show the plots" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is known as the $Point\\,Mass(\\theta)$ random variable, $\\theta \\in \\mathbb(R)$: the density $f(x)$ is 1 if $x=\\theta$ and 0 everywhere else\n", "\n", "$$\n", "f(x;\\theta) =\n", "\\begin{cases}\n", "0 & \\text{ if } x \\neq \\theta \\\\\n", "1 & \\text{ if } x = \\theta\n", "\\end{cases}\n", "$$\n", "\n", "$$\n", "F(x;\\theta) =\n", "\\begin{cases}\n", "0 & \\text{ if } x < \\theta \\\\\n", "1 & \\text{ if } x \\geq \\theta\n", "\\end{cases}\n", "$$\n", "\n", "So, if we had some sequence $\\{\\theta_i\\}_{i=1}^\\infty$ and $\\underset{i \\rightarrow \\infty}{\\lim} \\theta_i = \\theta$\n", "\n", "and we had a sequence of random variables $X_i \\sim Point\\,Mass(\\theta_i)$, $i = 1, 2, 3, \\ldots$\n", "\n", "then we could talk about a limiting random variable as $X \\sim Point\\,Mass(\\theta)$:\n", "\n", "i.e., we could talk about $\\underset{i \\rightarrow \\infty}{\\lim} X_i = X$" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 202 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# mock up a picture of a sequence of point mass rvs converging on theta = 0\n", "ptsize = 20\n", "i = 1\n", "theta_i = 1/i\n", "p = points((theta_i,1), rgbcolor=\"blue\", pointsize=ptsize)\n", "p += line([(theta_i,0),(theta_i,1)], rgbcolor=\"blue\", linestyle=':')\n", "while theta_i > 0.01:\n", " i+=1\n", " theta_i = 1/i\n", " p += points((theta_i,1), rgbcolor=\"blue\", pointsize=ptsize)\n", " p += line([(theta_i,0),(theta_i,1)], rgbcolor=\"blue\", linestyle=':')\n", "p += points((0,1), rgbcolor=\"red\", pointsize=ptsize)\n", "p += line([(0,0),(0,1)], rgbcolor=\"red\", linestyle=':')\n", "p.show(xmin=-1, xmax = 2, ymin=0, ymax = 1.1, axes=false, gridlines=[None,[0]], \\\n", " figsize=[7,2])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we want to generalise this notion of a limit to other random variables (that are not necessarily $Point\\,Mass(\\theta_i)$ RVs)\n", "\n", "What about one many of you will be familiar with - the 'bell-shaped curve' \n", "\n", "## The $Gaussian(\\mu, \\sigma^2)$ or $Normal(\\mu, \\sigma^2)$ RV?\n", "\n", "The probability density function (PDF) $f(x)$ is given by\n", "\n", "$$\n", "f(x ;\\mu, \\sigma) = \\displaystyle\\frac{1}{\\sigma\\sqrt{2\\pi}}\\exp\\left(\\frac{-1}{2\\sigma^2}(x-\\mu)^2\\right)\n", "$$\n", "\n", "The two parameters, $\\mu \\in \\mathbb{R} := (-\\infty,\\infty)$ and $\\sigma \\in (0,\\infty)$, are sometimes referred to as the location and scale parameters.\n", "\n", "To see why this is, use the interactive plot below to have a look at what happens to the shape of the density function $f(x)$ when you change $\\mu$ or increase or decrease $\\sigma$:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "da67126df9d24c6f9634d0e068ca5d7c", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Interactive function with 2 widgets\n", " my_mu: EvalText(value='0', description='mu',…" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "@interact\n", "def _(my_mu=input_box(0, label='mu') ,my_sigma=input_box(1,label='sigma')):\n", " '''Interactive function to plot the normal pdf and ecdf.'''\n", " \n", " if my_sigma > 0:\n", " html('

Normal('+str(my_mu)+','+str(my_sigma)+'2)

')\n", " var('mu sigma')\n", " f = (1/(sigma*sqrt(2.0*pi)))*exp(-1.0/(2*sigma^2)*(x - mu)^2)\n", " p1=plot(f.subs(mu=my_mu,sigma=my_sigma), \\\n", " (x, my_mu - 3*my_sigma - 2, my_mu + 3*my_sigma + 2),\\\n", " axes_labels=('x','f(x)'))\n", " show(p1,figsize=[8,3])\n", " else:\n", " print( \"sigma must be greater than 0\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Consider the sequence of random variables $X_1, X_2, X_3, \\ldots$, where\n", "\n", "- $X_1 \\sim Normal(0, 1)$\n", "- $X_2 \\sim Normal(0, \\frac{1}{2})$\n", "- $X_3 \\sim Normal(0, \\frac{1}{3})$\n", "- $X_4 \\sim Normal(0, \\frac{1}{4})$\n", "- $\\vdots$\n", "- $X_i \\sim Normal(0, \\frac{1}{i})$\n", "- $\\vdots$\n", "\n", "We can use the animation below to see how the PDF $f_{i}(x)$ looks as we move through the sequence of $X_i$ (the animation only goes to $i = 25$, $\\sigma = 0.04$ but you get the picture ...)\n", "\n", "\n", " \n", " \n", " \n", "\n", " \n", " \n", "
Normal curve animation, looping through $\\sigma = \\frac{1}{i}$ for $i = 1, \\dots, 25$
\n", "\n", "We can see that the probability mass of $X_i \\sim Normal(0, \\frac{1}{i})$ increasingly concentrates about 0 as $i \\rightarrow \\infty$ and $\\frac{1}{i} \\rightarrow 0$\n", "\n", "Does this mean that $\\underset{i \\rightarrow \\infty}{\\lim} X_i = Point\\,Mass(0)$?\n", "\n", "No, because for any $i$, however large, $P(X_i = 0) = 0$ because $X_i$ is a continuous RV (for any continous RV $X$, for any $x \\in \\mathbb{R}$, $P(X=x) = 0$).\n", "\n", "So, we need to refine our notions of convergence when we are dealing with random variables\n", "\n", "# Convergence in Distribution\n", "\n", "Let $X_1, X_2, \\ldots$ be a sequence of random variables and let $X$ be another random variable. Let $F_i$ denote the distribution function (DF) of $X_i$ and let $F$ denote the distribution function of $X$.\n", "\n", "Now, if for any real number $t$ at which $F$ is continuous,\n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} F_i(t) = F(t)$$\n", "\n", "(in the sense of the convergence or limits of functions we talked about earlier)\n", "\n", "Then we can say that the sequence or RVs $X_i$, $i = 1, 2, \\ldots$ **converges to $X$ in distribution** and write $X_i \\overset{d}{\\rightarrow} X$.\n", "\n", "An equivalent way of defining convergence in distribution is to go right back to the meaning of the probabilty space 'under the hood' of a random variable, a random variable $X$ as a mapping from the sample space $\\Omega$ to the real line ($X: \\Omega \\rightarrow \\mathbb{R}$), and the sample points or outcomes in the sample space, the $\\omega \\in \\Omega$. For $\\omega \\in \\Omega$, $X(\\omega)$ is the mapping of $\\omega$ to the real line $\\mathbb{R}$. We could look at the set of $\\omega$ such that $X(\\omega) \\leq t$, i.e. the set of $\\omega$ that map to some value on the real line less than or equal to $t$, $\\{\\omega: X(\\omega) \\leq t \\}$. \n", "\n", "Saying that for any $t \\in \\mathbb{R}$, $\\underset{i \\rightarrow \\infty}{\\lim} F_i(t) = F(t)$ is the equivalent of saying that for any $t \\in \\mathbb{R}$, \n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} P\\left(\\{\\omega:X_i(\\omega) \\leq t \\}\\right) = P\\left(\\{\\omega: X(\\omega) \\leq t\\right)$$\n", "\n", "Armed with this, we can go back to our sequence of $Normal$ random variables $X_1, X_2, X_3, \\ldots$, where\n", "\n", "- $X_1 \\sim Normal(0, 1)$\n", "- $X_2 \\sim Normal(0, \\frac{1}{2})$\n", "- $X_3 \\sim Normal(0, \\frac{1}{3})$\n", "- $X_4 \\sim Normal(0, \\frac{1}{4})$\n", "- $\\vdots$\n", "- $X_i \\sim Normal(0, \\frac{1}{i})$\n", "- $\\vdots$\n", "\n", "and let $X \\sim Point\\,Mass(0)$,\n", "\n", "and say that the $X_i$ **converge in distribution** to the $x \\sim Point\\,Mass$ RV $X$,\n", "\n", "$$X_i \\overset{d}{\\rightarrow} X$$\n", "\n", "What we are saying with convergence in distribution, informally, is that as $i$ increases, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by the limiting random variable. In this case, as $i$ increases, the $Point\\,Mass(0)$ is becoming a better and better model for the next outcome of a random experiment with outcomes $\\sim Normal(0,\\frac{1}{i})$." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 14 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# mock up a picture of a sequence of converging normal distributions\n", "my_mu = 0\n", "upper = my_mu + 5; lower = -upper; # limits for plot\n", "var('mu sigma')\n", "stop_i = 12\n", "html('

N(0,1) to N(0, 1/'+str(stop_i)+')

')\n", "f = (1/(sigma*sqrt(2.0*pi)))*exp(-1.0/(2*sigma^2)*(x - mu)^2)\n", "p=plot(f.subs(mu=my_mu,sigma=1.0), (x, lower, upper), rgbcolor = (0,0,1))\n", "for i in range(2, stop_i, 1): # just do a few of them\n", " shade = 1-11/i # make them different colours\n", " p+=plot(f.subs(mu=my_mu,sigma=1/i), (x, lower, upper), rgbcolor = (1-shade, 0, shade))\n", "textOffset = -0.2 # offset for placement of text - may need adjusting \n", "p+=text(\"0\",(0,textOffset),fontsize = 10, rgbcolor='grey') \n", "p+=text(str(upper.n(digits=2)),(upper,textOffset),fontsize = 10, rgbcolor='grey') \n", "p+=text(str(lower.n(digits=2)),(lower,textOffset),fontsize = 10, rgbcolor='grey') \n", "p.show(axes=false, gridlines=[None,[0]], figsize=[7,3])" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 15 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# mock up a picture of a sequence of converging normal distributions\n", "my_mu = 0\n", "upper = my_mu + 5; lower = -upper; # limits for plot\n", "var('mu sigma')\n", "stop_i = 12\n", "html('

N(0,1) to N(0, 1/'+str(stop_i)+')

')\n", "f = (1/2)*(1+erf((x - mu)/(sqrt(2)*sigma)))\n", "p=plot(f.subs(mu=my_mu,sigma=1.0), (x, lower, upper), rgbcolor = (0,0,1))\n", "for i in range(2, stop_i, 1): # just do a few of them\n", " shade = 1-11/i # make them different colours\n", " p+=plot(f.subs(mu=my_mu,sigma=1/i), (x, lower, upper), rgbcolor = (1-shade, 0, shade))\n", "textOffset = -0.2 # offset for placement of text - may need adjusting \n", "p+=text(\"0\",(0,textOffset),fontsize = 10, rgbcolor='grey') \n", "p+=text(str(upper.n(digits=2)),(upper,textOffset),fontsize = 10, rgbcolor='grey') \n", "p+=text(str(lower.n(digits=2)),(lower,textOffset),fontsize = 10, rgbcolor='grey') \n", "p.show(axes=false, gridlines=[None,[0]], figsize=[7,3])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### There is an interesting point to note about this convergence: \n", "\n", "We have said that the $X_i \\sim Normal(0,\\frac{1}{i})$ with distribution functions $F_i$ converge in distribution to $X \\sim Point\\,Mass(0)$ with distribution function $F$, which means that we must be able to show that for any real number $t$ at which $F$ is continuous,\n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} F_i(t) = F(t)$$\n", "\n", "Note that for any of the $X_i \\sim Normal(0, \\frac{1}{i})$, $F_i(0) = \\frac{1}{2}$, and also note that for $X \\sim Point,Mass(0)$, $F(0) = 1$, so clearly $F_i(0) \\neq F(0)$. \n", "\n", "What has gone wrong? \n", "\n", "Nothing: we said that we had to be able to show that $\\underset{i \\rightarrow \\infty}{\\lim} F_i(t) = F(t)$ for any $t \\in \\mathbb{R}$ at which $F$ is continuous, but the $Point\\,Mass(0)$ distribution function $F$ is not continous at 0!" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics Array of size 1 x 2" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "theta = 0.0\n", "# show the plots\n", "show(graphics_array((pmfPointMassPlot(theta),cdfPointMassPlot(theta))),figsize=[8,2]) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Convergence in Probability\n", "\n", "Let $X_1, X_2, \\ldots$ be a sequence of random variables and let $X$ be another random variable. Let $F_i$ denote the distribution function (DF) of$X_i$ and let $F$ denote the distribution function of $X$.\n", "\n", "Now, if for any real number $\\varepsilon > 0$,\n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} P\\left(|X_i - X| > \\varepsilon\\right) = 0$$\n", "\n", "Then we can say that the sequence $X_i$, $i = 1, 2, \\ldots$ **converges to $X$ in probability** and write $X_i \\overset{P}{\\rightarrow} X$.\n", "\n", "Or, going back again to the probability space 'under the hood' of a random variable, we could look the way the $X_i$ maps each outcome $\\omega \\in \\Omega$, $X_i(\\omega)$, which is some point on the real line, and compare this to mapping $X(\\omega)$. \n", "\n", "Saying that for any $\\varepsilon \\in \\mathbb{R}$, $\\underset{i \\rightarrow \\infty}{\\lim} P\\left(|X_i - X| > \\varepsilon\\right) = 0$ is the equivalent of saying that for any $\\varepsilon \\in \\mathbb{R}$, \n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} P\\left(\\{\\omega:|X_i(\\omega) - X(\\omega)| > \\varepsilon \\}\\right) = 0$$\n", "\n", "Informally, we are saying $X$ is a limit in probabilty if, by going far enough into the sequence $X_i$, we can ensure that the mappings $X_i(\\omega)$ and $X(\\omega)$ will be arbitrarily close to each other on the real line for all $\\omega \\in \\Omega$.\n", "\n", "**Note** that convergence in distribution is implied by convergence in probability: convergence in distribution is the weakest form of convergence; any sequence of RV's that converges in probability to some RV $X$ also converges in distribution to $X$ (but not necessarily vice versa). " ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 25 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# mock up a picture of a sequence of converging normal distributions\n", "my_mu = 0\n", "var('mu sigma')\n", "upper = 0.2; lower = -upper\n", "i = 20 # start part way into the sequence\n", "lim = 100 # how far to go\n", "stop_i = 12\n", "html('

N(0,1/'+str(i)+') to N(0, 1/'+str(lim)+')

')\n", "f = (1/(sigma*sqrt(2.0*pi)))*exp(-1.0/(2*sigma^2)*(x - mu)^2)\n", "p=plot(f.subs(mu=my_mu,sigma=1.0/i), (x, lower, upper), rgbcolor = (0,0,1))\n", "for j in range(i, lim+1, 4): # just do a few of them\n", " shade = 1-(j-i)/(lim-i) # make them different colours\n", " p+=plot(f.subs(mu=my_mu,sigma=1/j), (x, lower,upper), rgbcolor = (1-shade, 0, shade))\n", "textOffset = -1.5 # offset for placement of text - may need adjusting \n", "p+=text(\"0\",(0,textOffset),fontsize = 10, rgbcolor='grey') \n", "p+=text(str(upper.n(digits=2)),(upper,textOffset),fontsize = 10, rgbcolor='grey') \n", "p+=text(str(lower.n(digits=2)),(lower,textOffset),fontsize = 10, rgbcolor='grey') \n", "p.show(axes=false, gridlines=[None,[0]], figsize=[7,3])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For our sequence of $Normal$ random variables $X_1, X_2, X_3, \\ldots$, where\n", "\n", "- $X_1 \\sim Normal(0, 1)$\n", "- $X_2 \\sim Normal(0, \\frac{1}{2})$\n", "- $X_3 \\sim Normal(0, \\frac{1}{3})$\n", "- $X_4 \\sim Normal(0, \\frac{1}{4})$\n", "- $\\vdots$\n", "- $X_i \\sim Normal(0, \\frac{1}{i})$\n", "- $\\vdots$\n", "\n", "and $X \\sim Point\\,Mass(0)$,\n", "\n", "It can be shown that the $X_i$ converge in probability to $X \\sim Point\\,Mass(0)$ RV $X$,\n", "\n", "$$X_i \\overset{P}{\\rightarrow} X$$\n", "\n", "Since we are going to be using Markovs inequality later, we might as well take a look at it here and prove it.\n", "\n", "### Markovs inequality\n", "\n", "Let $x$ be a nonnegative random variable. Then for $a > 0$,\n", "$$\n", " P(X \\geq a) \\leq \\frac{E[X]}{a}\n", "$$\n", "\n", "#### Proof\n", "\n", "Let us begin by assuming that $X$ is a continuous random variable and let us write\n", "$$\n", " P(X \\geq a) = \\int_{a}^\\infty f(x) dx = \\frac{1}{a} \\int_{a}^\\infty a f(x) dx \\leq \\frac{1}{a} \\int_{a}^\\infty x f(x) dx \\leq \\frac{1}{a} E[x]\n", "$$\n", "\n", "#### Convergence in probability of our sequence\n", "Lets now use it! Remember we need to prove\n", "\n", "$$\\underset{i \\rightarrow \\infty}{\\lim} P\\left(|X_i - X| > \\varepsilon\\right) = 0$$\n", "\n", "but $X = 0$ since its a $Point\\,Mass(0)$ r.v. so by Markovs inequality we get\n", "\n", "$$\n", " P\\left(|X_i - X| \\geq \\varepsilon\\right) = P\\left(|X_i - X|^2 \\geq \\varepsilon^2 \\right) \\leq \\frac{E[|X_i-X|^2]}{\\varepsilon^2} = \\text{Var}[X_i]\\frac{1}{\\varepsilon^2} \\leq \\frac{1}{i}\\frac{1}{\\varepsilon^2}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Some Basic Limit Laws in Statistics\n", "\n", "Intuition behind Law of Large Numbers and Central Limit Theorem\n", "\n", "Take a look at the Khan academy videos on the Law of Large Numbers and the Central Limit Theorem. This will give you a working idea of these theorems. In the sequel, we will strive for a deeper understanding of these theorems on the basis of the two notions of convergence of sequences of random variables we just saw.\n", " \n", "\n", "## Weak Law of Large Numbers\n", "\n", "Remember that a statistic is a random variable, so a sample mean is a random variable. If we are given a sequence of independent and identically distributed RVs, $X_1,X_2,\\ldots \\overset{IID}{\\sim} X_1$, then we can also think of a sequence of random variables $\\overline{X}_1, \\overline{X}_2, \\ldots, \\overline{X}_n, \\ldots$ ($n$ being the sample size). \n", "\n", "Since $X_1, X_2, \\ldots$ are $IID$, they all have the same expection, say $E(X_1)$ by convention.\n", "\n", "If $E(X_1)$ exists, then the sample mean $\\overline{X}_n$ converges in probability to $E(X_1)$ (i.e., to the expectatation of any one of the individual RVs):\n", "\n", "$$\n", "\\text{If} \\quad X_1,X_2,\\ldots \\overset{IID}{\\sim} X_1 \\ \\text{and if } \\ E(X_1) \\ \\text{exists, then } \\ \\overline{X}_n \\overset{P}{\\rightarrow} E(X_1) \\ .\n", "$$\n", "\n", "Going back to our definition of convergence in probability, we see that this means that for any real number $\\varepsilon > 0$, $\\underset{n \\rightarrow \\infty}{\\lim} P\\left(|\\overline{X}_n - E(X_1)| > \\varepsilon\\right) = 0$\n", "\n", "Informally, this means that means that, by taking larger and larger samples we can make the probability that the average of the observations is more than $\\varepsilon$ away from the expected value get smaller and smaller.\n", "\n", "Proof of this is beyond the scope of this course, but we have already seen it in action when we looked at the $Bernoulli$ running means. Have another look, this time with only one sequence of running means. You can increase $n$, the sample size, and change $\\theta$. Note that the seed for the random number generator is also under your control. This means that you can get replicable samples: in particular, in this interact, when you increase the sample size it looks as though you are just adding more to an existing sample rather than starting from scratch with a new one. " ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "347a97c419934a3ab3d97757233c72cc", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Interactive function with 3 widgets\n", " nToGen: TransformIntSlider(value=100, descri…" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "@interact\n", "def _(nToGen=slider(1,1500,1,100,label='n'),my_theta=input_box(0.3,label='theta'),rSeed=input_box(1234,label='random seed')):\n", " '''Interactive function to plot running mean for a Bernoulli with specified n, theta and random number seed.'''\n", " \n", " if my_theta >= 0 and my_theta <= 1:\n", " html('

Bernoulli('+str(my_theta.n(digits=2))+')

')\n", " xvalues = range(1, nToGen+1,1)\n", " bRunningMeans = bernoulliRunningMeans(nToGen, myTheta=my_theta, mySeed=rSeed)\n", " pts = zip(xvalues, bRunningMeans)\n", " p = line(pts, rgbcolor = (0,0,1))\n", " p+=line([(0,my_theta),(nToGen,my_theta)],linestyle=':',rgbcolor='grey')\n", " show(p, figsize=[5,3], axes_labels=['n','sample mean'],ymax=1)\n", " else:\n", " print ('Theta must be between 0 and 1')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Central Limit Theorem\n", "\n", "You have probably all heard of the Central Limit Theorem before, but now we can relate it to our definition of convergence in distribution. \n", "\n", "Let $X_1,X_2,\\ldots \\overset{IID}{\\sim} X_1$ and suppose $E(X_1)$ and $V(X_1)$ both exist,\n", "\n", "then\n", "\n", "$$\n", "\\overline{X}_n = \\frac{1}{n} \\sum_{i=1}^n X_i \\overset{d}{\\rightarrow} X \\sim Normal \\left(E(X_1),\\frac{V(X_1)}{n} \\right)\n", "$$\n", "\n", "And remember $Z \\sim Normal(0,1)$?\n", "\n", "Consider $Z_n := \\displaystyle\\frac{\\overline{X}_n-E(\\overline{X}_n)}{\\sqrt{V(\\overline{X}_n)}} = \\displaystyle\\frac{\\sqrt{n} \\left( \\overline{X}_n -E(X_1) \\right)}{\\sqrt{V(X_1)}}$\n", "\n", "If $\\overline{X}_n = \\displaystyle\\frac{1}{n} \\displaystyle\\sum_{i=1}^n X_i \\overset{d}{\\rightarrow} X \\sim Normal \\left(E(X_1),\\frac{V(X_1)}{n} \\right)$, then $\\overline{X}_n -E(X_1) \\overset{d}{\\rightarrow} X-E(X_1) \\sim Normal \\left( 0,\\frac{V(X_1)}{n} \\right)$\n", "\n", "and $\\sqrt{n} \\left( \\overline{X}_n -E(X_1) \\right) \\overset{d}{\\rightarrow} \\sqrt{n} \\left( X-E(X_1) \\right) \\sim Normal \\left( 0,V(X_1) \\right)$\n", "\n", "so $Z_n := \\displaystyle \\frac{\\overline{X}_n-E(\\overline{X}_n)}{\\sqrt{V(\\overline{X}_n)}} = \\displaystyle\\frac{\\sqrt{n} \\left( \\overline{X}_n -E(X_1) \\right)}{\\sqrt{V(X_1)}} \\overset{d}{\\rightarrow} Z \\sim Normal \\left( 0,1 \\right)$\n", "\n", "Thus, for sufficiently large $n$ (say $n>30$), probability statements about $\\overline{X}_n$ can be approximated using the $Normal$ distribution. \n", "\n", "The beauty of the CLT, as you have probably seen from other courses, is that $\\overline{X}_n \\overset{d}{\\rightarrow} Normal \\left( E(X_1), \\frac{V(X_1)}{n} \\right)$ does not require the $X_i$ to be normally distributed. \n", "\n", "We can try this with our $Bernoulli$ RV generator. First, a small number of samples:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[7/10, 2/5, 3/5, 4/5, 3/5]" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "theta, n, samples = 0.6, 10, 5 # concise way to set some variable values\n", "sampleMeans=[] # empty list\n", "for i in range(0, samples, 1): # loop \n", " thisMean = QQ(sum(bernoulliSample(n, theta)))/n # get a sample and find the mean\n", " sampleMeans.append(thisMean) # add mean to the list of means\n", "sampleMeans # disclose the sample means" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can use the interactive plot to increase the number of samples and make a histogram of the sample means. According to the CLT, for lots of reasonably-sized samples we should get a nice symmetric bell-curve-ish histogram centred on $\\theta$. You can adjust the number of bins in the histogram as well as the number of samples, sample size, and $\\theta$. " ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "f9e4807b9c30430e97cdca69156c3f51", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Interactive function with 4 widgets\n", " replicates: TransformIntSlider(value=100, de…" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import pylab\n", "@interact\n", "def _(replicates=slider(1,3000,1,100,label='replicates'), \\\n", " nToGen=slider(1,1500,1,100,label='sample size n'),\\\n", " my_theta=input_box(0.3,label='theta'),Bins=5):\n", " '''Interactive function to plot distribution of replicates of sample means for n IID Bernoulli trials.'''\n", " \n", " if my_theta >= 0 and my_theta <= 1 and replicates > 0:\n", " sampleMeans=[] # empty list\n", " for i in range(0, replicates, 1): \n", " thisMean = RR(sum(bernoulliSample(nToGen, my_theta)))/nToGen\n", " sampleMeans.append(thisMean)\n", " pylab.clf() # clear current figure\n", " n, bins, patches = pylab.hist(sampleMeans, Bins, density=true) \n", " pylab.ylabel('normalised count')\n", " pylab.title('Normalised histogram for Bernoulli sample means')\n", " pylab.savefig('myHist') # to actually display the figure\n", " pylab.show()\n", " #show(p, figsize=[5,3], axes_labels=['n','sample mean'],ymax=1)\n", " else:\n", " print ('Theta must be between 0 and 1, and samples > 0')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Increase the sample size and the numbe rof bins in the above interact and see if the histograms of the sample means are looking more and more normal as the CLT would have us believe." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "But although the $X_i$ do not have to be $\\sim Normal$ for $\\overline{X}_n = \\overset{d}{\\rightarrow} X \\sim Normal\\left(E(X_1),\\frac{V(X_1)}{n} \\right)$, remember that we said \"Let $X_1,X_2,\\ldots \\overset{IID}{\\sim} X_1$ and suppose $E(X_1)$ and $V(X_1)$ both exist\", then,\n", "\n", "$$\n", "\\overline{X}_n = \\frac{1}{n} \\sum_{i=1}^n X_i \\overset{d}{\\rightarrow} X \\sim Normal \\left(E(X_1),\\frac{V(X_1)}{n} \\right)\n", "$$\n", "\n", "This is where is all goes horribly wrong for the standard $Cauchy$ distribution (any $Cauchy$ distribution in fact): neither the expectation nor the variance exist for this distribution. The Central Limit Theorem cannot be applied here. In fact, if $X_1,X_2,\\ldots \\overset{IID}{\\sim}$ standard $Cauchy$, then $\\overline{X}_n = \\displaystyle \\frac{1}{n} \\sum_{i=1}^n X_i \\sim$ standard $Cauchy$.\n", "\n", "### YouTry\n", "\n", "Try looking at samples from two other RVs where the expectation and variance do exist, the $Uniform$ and the $Exponential$:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "67196e1715f14e4d9430c6784e6cfe66", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Interactive function with 5 widgets\n", " replicates: EvalText(value='100', descriptio…" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import pylab\n", "@interact\n", "def _(replicates=input_box(100,label='replicates'), \\\n", " nToGen=slider(1,1500,1,100,label='sample size n'),\\\n", " my_theta1=input_box(2,label='theta1'),\\\n", " my_theta2=input_box(4,label='theta1'),Bins=5):\n", " '''Interactive function to plot distribution of \n", " sample means for n IID Uniform(theta1, theta2) trials.'''\n", " \n", " if (my_theta1 < my_theta2) and replicates > 0:\n", " sampleMeans=[] # empty list\n", " for i in range(0, replicates, 1):\n", " \n", " thisMean = RR(sum(uniformSample(nToGen, my_theta1, my_theta2)))/nToGen\n", " sampleMeans.append(thisMean)\n", " pylab.clf() # clear current figure\n", " n, bins, patches = pylab.hist(sampleMeans, Bins, density=true) \n", " pylab.ylabel('normalised count')\n", " pylab.title('Normalised histogram for Uniform sample means')\n", " pylab.savefig('myHist') # to actually display the figure\n", " pylab.show()\n", " #show(p, figsize=[5,3], axes_labels=['n','sample mean'],ymax=1)\n", " else:\n", " print ('theta1 must be less than theta2, and samples > 0')" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "746dca82f4bf4d3ba949aea898903c16", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Interactive function with 4 widgets\n", " replicates: EvalText(value='100', descriptio…" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import pylab\n", "@interact\n", "def _(replicates=input_box(100,label='replicates'), \\\n", " nToGen=slider(1,1500,1,100,label='sample size n'),\\\n", " my_lambda=input_box(0.1,label='lambda'),Bins=5):\n", " '''Interactive function to plot distribution of \\\n", " sample means for an Exponential(lambda) process.'''\n", " \n", " if my_lambda > 0 and replicates > 0:\n", " sampleMeans=[] # empty list\n", " for i in range(0, replicates, 1): \n", " thisMean = RR(sum(exponentialSample(nToGen, my_lambda)))/nToGen\n", " sampleMeans.append(thisMean)\n", " pylab.clf() # clear current figure\n", " n, bins, patches = pylab.hist(sampleMeans, Bins, density=true) \n", " pylab.ylabel('normalised count')\n", " pylab.title('Normalised histogram for Exponential sample means')\n", " pylab.savefig('myHist') # to actually display the figure\n", " pylab.show()\n", " #show(p, figsize=[5,3], axes_labels=['n','sample mean'],ymax=1)\n", " else:\n", " print ('lambda must be greater than 0, and samples > 0')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Properties of the MLE\n", "\n", "The LLN (law of large numbers) and CLT (central limit theorem) are statements about the limiting distribution of the sample mean of IID random variables whose expectation and variance exists. How does this apply to the MLE (maximum likelihood estimator)?\n", "\n", "Consider the following generic parametric model for our data or observations:\n", "\n", "$$\n", "X_1,X_2,\\ldots,X_n \\overset{IID}{\\sim} F(x; \\theta^*) \\ \\text{ or } \\ f(x; \\theta^*)\n", "$$\n", "\n", "We do not know the true parameter $\\theta^*$ under the model for our data. Our task is to estimate the unknown parameter $\\theta^*$ using the MLE:\n", "\n", "$$\\widehat{\\Theta}_n = argmax_{\\theta \\in \\mathbf{\\Theta}} l(\\theta)$$\n", "\n", "The amazing think about the MLE is its following properties:\n", "\n", "### 1. The MLE is *asymptotically consistent*\n", "\n", "$$\\boxed{\\widehat{\\Theta}_n \\overset{P}{\\rightarrow} \\theta^*}$$\n", "\n", "So when the number of observations (sample size $n$) goes to infinity, our MLE converges in probability to the true parameter $\\theta^* \\in \\mathbf{\\Theta}$.\n", "\n", "Interestingly, one can work out the details and find that the MLE $\\widehat{\\Theta}_n$, which is also a random variable based on $n$ IID samples that takes values in the parameter space $\\mathbf{\\Theta}$, is also normally distributed for large sample sizes.\n", "\n", "### 2. The MLE is *equivariant*\n", "\n", "$$\\boxed{\\text{If } \\ \\widehat{\\Theta}_n \\ \\text{ is the MLE of } \\ \\theta^* \\ \\text{ then } \\ g(\\widehat{\\Theta}_n) \\ \\text{ is the MLE of } \\ g(\\theta^*)}$$\n", "\n", "This is a very useful property, since any function $g : \\mathbf{\\Theta} \\to \\mathbb{R}$ of interest is at our disposal by merely applying $g$ to the the MLE. Often $g$ is some sort of reward that depends on the unknown parameter $\\theta^*$.\n", "\n", "### 3. The MLE is *asymptotically normal* \n", "\n", "$$\\boxed{ \\frac{\\left(\\widehat{\\Theta}_n - \\theta^*\\right)}{\\widehat{se}_n} \\overset{d}{\\rightarrow} Normal(0,1) }\n", "\\quad \\text{ or equivalently, } \\quad\n", "\\boxed{ \\widehat{\\Theta}_n \\overset{d}{\\rightarrow} Normal( \\theta^*, \\widehat{se}_n^2) }\n", "$$\n", "\n", "where, $\\widehat{se}_n$ is the *estimated standard error* of the MLE:\n", "\n", "$$\\boxed{ \\widehat{se}_n \\ \\text{ is an estimate of the } \\ \\sqrt{V\\left(\\widehat{\\Theta}_n \\right)}}$$\n", "\n", "We can compute $\\widehat{se}_n$ with the following formula:\n", "\n", "$$\\boxed{\\widehat{se}_n = \\sqrt{\\frac{1}{ \\left. n E \\left(-\\frac{\\partial^2 \\log f(X;\\theta)}{\\partial \\theta^2} \\right) \\right\\vert_{\\theta=\\widehat{\\theta}_n} } }}$$\n", "\n", "where, the expectation is called the *Fisher information* of one sample or $I_1$:\n", "\n", "$$\\boxed{ I_1 := E \\left(-\\frac{\\partial^2 \\log f(X;\\theta)}{\\partial \\theta^2} \\right) = \n", "\\begin{cases}\n", "\\displaystyle{\\int{\\left(-\\frac{\\partial^2 \\log f(x;\\theta)}{\\partial \\theta^2} \\right) f(x; \\theta)} dx} & \\text{ for continuous RV } X\\\\\n", "\\displaystyle{\\sum_x{\\left(-\\frac{\\partial^2 \\log f(x;\\theta)}{\\partial \\theta^2} \\right) f(x; \\theta)}}& \\text{ for discrete RV } X\n", "\\end{cases}\n", "}\n", "$$\n", "\n", "Other two properties (not needed for this course) include:\n", "\n", "- *asymptotic efficiency*, i.e., among a class of well-behaved estimators, the MLE has the smallest variance at least for large samples, and\n", "- *approximately Bayes*, i.e., the MLE is approximately the *Bayes estimator* (some of you may see Bayesian methods of estimation in advanced courses in statistical machine learning or in latest AI methods)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Confidence Interval and Set Estimation from MLE\n", "\n", "An immediate implication of the asymptotic normality of the MLE, which informally states that the distribution of the MLE can be approximated by a Normal random variable, is to obtain confidence intervals for the unkown parameter $\\theta^*$.\n", "\n", "Recall that in set estimation, as opposed to point estimation, we estimate the unknown parameter using a random set based on the data (typically intervals in 1D) that \"traps\" the true parameter $\\theta^*$ with a very high probability, say $0.95$. We typically express such probality in terms of $1-\\alpha$, so the $95\\%$ confidence interval is seen as a $1-\\alpha$ confidence interval with $\\alpha=0.05$. From the the asymptotic normality of the MLE, we get the following confidence interval for the unknown $\\theta^*$:\n", "\n", "\n", "$$\n", "\\boxed{\\text{If } \\quad \n", "\\displaystyle{C_n := \\left( \\widehat{\\Theta}_n - z_{\\alpha/2} \\widehat{se}_n, \\, \\widehat{\\Theta}_n + z_{\\alpha/2} \\widehat{se}_n \\right)} \\quad \\text{ then } \\quad P \\left( \\{ \\theta^* \\in C_n \\} ; \\theta^* \\right) \\underset{n \\to \\infty}{\\longrightarrow} 1-\\alpha , \\quad \\text{ where } z_{\\alpha/2} = \\Phi^{[-1]}(1-\\alpha/2).\n", "}\n", "$$\n", "\n", "Recall that $P \\left( \\{ \\theta^* \\in C_n \\} ; \\theta^* \\right)$ is simply the probability of the event that $\\theta^*$ will be in $C_n$, the $1-\\alpha$ confidence interval, given the data is distributed according to the model with true parameter $\\theta^*$.\n", "\n", "NOTE: $\\Phi^{[-1]}(1-\\alpha/2)$ is merely the inverse distribution function (CDF) of the standard normal RV. \n", "\n", "$$\n", "\\text{For } \\alpha=0.05, z_{\\alpha/2}=1.96 \\approxeq 2, \\text{ so: } \\quad \\boxed{\\widehat{\\Theta}_n \\pm 2 \\widehat{se}_n} \\quad \\text{ is an approximate 95% confidence interval.}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example of Confidence Interval for IID $Bernoulli(\\theta)$ Trials\n", "\n", "We already know that the MLE for the model with $n$ IID $Bernoulli(\\theta)$ Trials is the sample mean, i.e.,\n", "\n", "$$X_1,X_2,\\ldots, X_n \\overset{IID}{\\sim} Bernoulli(\\theta^*) \\implies \\widehat{\\Theta}_n = \\overline{X}_n$$\n", "\n", "Our task now is to obtain the $1-\\alpha$ confidence interval based on this MLE.\n", "\n", "To get the confidence interval we need to obtain $\\widehat{se}_n$ by computing the following:\n", "\n", "$$\n", "\\begin{array}{cc}\n", "\\widehat{se}_n &=& \\displaystyle{\\sqrt{\\frac{1}{ \\left. n E \\left(-\\frac{\\partial^2 \\log f(X;\\theta)}{\\partial \\theta^2} \\right) \\right\\vert_{\\theta=\\widehat{\\theta}_n} } }}\n", "\\end{array}\n", "$$\n", "$I_1 := E \\left(-\\frac{\\partial^2 \\log f(X;\\theta)}{\\partial \\theta^2} \\right)$ is called the Fisher Information of one sample.\n", "Since our IID samples are from a discrete distribution with \n", "\n", "$$\n", "\\begin{array}{cc}\n", "f(x; \\theta) = \\theta^x (1-\\theta)^{1-x} \n", "&\\implies& \\displaystyle{\\log \\left( f(x;\\theta) \\right) = x \\log(\\theta) +(1-x) \\log(1-\\theta)}\\\\\n", "&\\implies& \\displaystyle{\\frac{\\partial}{\\partial \\theta} \\left(\\log \\left( f(x;\\theta) \\right)\\right)} \n", "= \\displaystyle{\\frac{x}{\\theta} -\\frac{1-x}{1-\\theta}} \\\\\n", "&\\implies& \\displaystyle{\\frac{\\partial^2}{\\partial \\theta^2} \\left(\\log \\left( f(x;\\theta) \\right)\\right)} \n", "= \\displaystyle{-\\frac{x}{\\theta^2} - \\frac{1-x}{(1-\\theta)^2}}\\\\\n", "&\\implies& \\displaystyle{E \\left( - \\frac{\\partial^2}{\\partial \\theta^2} \\left(\\log \\left( f(x;\\theta) \\right)\\right) \\right)} \n", "= \\displaystyle{\\sum_{x\\in\\{0,1\\}} \\left( \\frac{x}{\\theta^2} + \\frac{1-x}{(1-\\theta)^2} \\right) f(x; \\theta) = \\frac{\\theta}{\\theta^2} + \\frac{1-\\theta}{(1-\\theta)^2} = \\frac{1}{\\theta(1-\\theta)}}\n", "\\end{array}\n", "$$\n", "\n", "Note that we have implicitly assumed that the $x$ values are only $0$ or $1$ by ignoring the indicator term $\\mathbf{1}_{\\{0,1\\}}(x)$ in $f(x;\\theta)$. But this is okay as we are carefully doing the sums over just $x \\in \\{0,1\\}$.\n", "\n", "Now, by using the formula for $\\widehat{se}_n$, we can obtain:\n", "\n", "$$\n", "\\begin{array}{cc}\n", "\\widehat{se}_n \n", "&=& \\displaystyle{\\sqrt{\\frac{1}{ \\left. n E \\left(-\\frac{\\partial^2 \\log f(X;\\theta)}{\\partial \\theta^2} \\right) \\right\\vert_{\\theta=\\widehat{\\theta}_n} } }}\\\\\n", "&=& \\displaystyle{\\sqrt{\\frac{1}{ \\left. n \\frac{1}{\\theta(1-\\theta)} \\right\\vert_{\\theta=\\widehat{\\theta}_n} } }}\\\\\n", "&=& \\displaystyle{\\sqrt{\\frac{\\widehat{\\theta}_n(1-\\widehat{\\theta}_n)}{n}}}\n", "\\end{array}\n", "$$\n", "\n", "Finally, we can complete our task by obtaining the 95% confidence interval for $\\theta^*$ as follows:\n", "\n", "$$\n", "\\displaystyle{ \\widehat{\\theta}_n \\pm 2 \\widehat{se}_n = \\widehat{\\theta}_n \\pm 2 \\sqrt{\\frac{\\widehat{\\theta}_n(1-\\widehat{\\theta}_n)}{n}} = \\overline{x}_n \\pm 2 \\sqrt{\\frac{\\overline{x}_n(1-\\overline{x}_n)}{n}} }\n", "$$" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "Graphics object consisting of 42 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "nToGenerate = 100\n", "replicates = 20\n", "xvalues = range(1, nToGenerate+1,1)\n", "for i in range(replicates):\n", " redshade = 0.5*(replicates - 1 - i)/replicates # to get different colours for the lines\n", " bRunningMeans = bernoulliSecretThetaRunningMeans(nToGenerate)\n", " pts = zip(xvalues,bRunningMeans)\n", " if (i == 0):\n", " p = line(pts, rgbcolor = (redshade,0,1))\n", " else:\n", " p += line(pts, rgbcolor = (redshade,0,1))\n", " mle=bRunningMeans[nToGenerate-1]\n", " se95Correction=2.0*sqrt(mle*(1-mle)/nToGenerate)\n", " lower95CI = mle-se95Correction\n", " upper95CI = mle+se95Correction\n", " p += line([(nToGenerate+i,lower95CI),(nToGenerate+i,upper95CI)], rgbcolor = (redshade,0,1), thickness=0.5)\n", "p += line([(1,0.3),(nToGenerate+replicates,0.3)], rgbcolor='black', thickness='2')\n", "p += text('sample mean up to n='+str(nToGenerate)+' and their 95% confidence intervals',(nToGenerate/1.5,1),fontsize=16)\n", "show(p, figsize=[10,6])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Sample Exam Problem 5\n", "\n", "Obtain the 95% Confidence Interval for the $\\lambda^*$ from the experiment based on $n$ IID $Exponential(\\lambda)$ trials.\n", "\n", "Write down your answer by returning the right answer in the function `SampleExamProblem5` in the next cell.\n", "Your function call `SampleExamProblem5(sampleWaitingTimes)` on the Orbiter waiting times data should return the 95% confidence interval for the unknown parameter $\\lambda^*$." ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "ename": "NameError", "evalue": "name 'XXX' is not defined", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 15\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 16\u001b[0m \u001b[0;31m# do NOT change anything below\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 17\u001b[0;31m \u001b[0mlowerCISampleExamProblem5\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mupperCISampleExamProblem5\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mSampleExamProblem5\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msampleWaitingTimes\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 18\u001b[0m \u001b[0mprint\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m\"The 95% CI for lambda in the Orbiter Waiting time experiment = \"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 19\u001b[0m \u001b[0mprint\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mlowerCISampleExamProblem5\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mupperCISampleExamProblem5\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m\u001b[0m in \u001b[0;36mSampleExamProblem5\u001b[0;34m(exponentialSamples)\u001b[0m\n\u001b[1;32m 7\u001b[0m '''return the 95% confidence interval as a 2-tuple for the unknown rate parameter lambda* \n\u001b[1;32m 8\u001b[0m from n IID Exponential(lambda*) trials in the input numpy array called exponentialSamples'''\n\u001b[0;32m----> 9\u001b[0;31m \u001b[0mXXX\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 10\u001b[0m \u001b[0mXXX\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0mXXX\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mNameError\u001b[0m: name 'XXX' is not defined" ] } ], "source": [ "# Sample Exam Problem 5\n", "# Only replace the XXX below, do not change the function naemes or parameters\n", "import numpy as np\n", "sampleWaitingTimes = np.array([8,3,7,18,18,3,7,9,9,25,0,0,25,6,10,0,10,8,16,9,1,5,16,6,4,1,3,21,0,28,3,8,6,6,11,\\\n", " 8,10,15,0,8,7,11,10,9,12,13,8,10,11,8,7,11,5,9,11,14,13,5,8,9,12,10,13,6,11,13,0,\\\n", " 0,11,1,9,5,14,16,2,10,21,1,14,2,10,24,6,1,14,14,0,14,4,11,15,0,10,2,13,2,22,10,5,\\\n", " 6,13,1,13,10,11,4,7,9,12,8,16,15,14,5,10,12,9,8,0,5,13,13,6,8,4,13,15,7,11,6,23,1])\n", "\n", "def SampleExamProblem5(exponentialSamples):\n", " '''return the 95% confidence interval as a 2-tuple for the unknown rate parameter lambda* \n", " from n IID Exponential(lambda*) trials in the input numpy array called exponentialSamples'''\n", " XXX\n", " XXX\n", " XXX\n", " lower95CI=XXX\n", " upper95CI=XXX\n", " return (lower95CI,upper95CI)\n", "\n", "# do NOT change anything below\n", "lowerCISampleExamProblem5,upperCISampleExamProblem5 = SampleExamProblem5(sampleWaitingTimes)\n", "print (\"The 95% CI for lambda in the Orbiter Waiting time experiment = \")\n", "print (lowerCISampleExamProblem5,upperCISampleExamProblem5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Sample Exam Problem 5 Solution\n", "\n", "We can obtain the 95% Confidence Interval for the $\\lambda^*$ for the experiment based on $n$ IID $Exponential(\\lambda)$ trials, by hand or using SageMath symbolic computations (typically both).\n", "\n", "Let $X_1,X_2,\\ldots,X_n \\overset{IID}{\\sim} Exponential(\\lambda^*)$. \n", "\n", "We saw that the ML estimator of $\\lambda^* \\in (0,\\infty)$ is $\\widehat{\\Lambda}_n = 1/\\, \\overline{X}_n$ and its ML estimate is $\\widehat{\\lambda}_n=1/\\, \\overline{x}_n$, where $x_1,x_2,\\ldots,x_n$ are our observed data.\n", "\n", "Let us obtain $I_1$, the Fisher Information of one sample, for this experiment to find the standard error:\n", "\n", "$$\n", "\\widehat{\\mathsf{se}}_n(\\widehat{\\Lambda}_n) = \\frac{1}{\\sqrt{n \\left. I_1 \\right\\vert_{\\lambda=\\widehat{\\lambda}_n}}}\n", "$$\n", "\n", "and construct an approximate $95\\%$ confidence interval for $\\lambda^*$ using the asymptotic normality of its ML estimator $\\widehat{\\Lambda}_n$.\n", "\n", "Since the probability density function $f(x;\\lambda)=\\lambda e^{-\\lambda x}$, for $x\\in [0,\\infty)$, we have,\n", "\n", "$$\n", "\\begin{align}\n", "I_1 &= - E \\left( \\frac{\\partial^2 \\log f(X;\\lambda)}{\\partial^2 \\lambda} \\right) = - \\int_{x \\in [0,\\infty)} \\left( \\frac{\\partial^2 \\log \\left( \\lambda e^{-\\lambda x} \\right)}{\\partial^2 \\lambda} \\right) \\lambda e^{-\\lambda x} \\ dx\n", "\\end{align}\n", "$$\n", "\n", "Let us compute the above integrand next.\n", "$$\n", "\\begin{align}\n", "\\frac{\\partial^2 \\log \\left( \\lambda e^{-\\lambda x} \\right)}{\\partial^2 \\lambda}\n", "&:=\n", "\\frac{\\partial}{\\partial \\lambda} \\left( \\frac{\\partial}{\\partial \\lambda} \\left( \\log \\left( \\lambda e^{-\\lambda x} \\right) \\right) \\right)\n", "= \\frac{\\partial}{\\partial \\lambda} \\left( \\frac{\\partial}{\\partial \\lambda} \\left( \\log(\\lambda) + \\log(e^{-\\lambda x} \\right) \\right) \\\\\n", "&= \\frac{\\partial}{\\partial \\lambda} \\left( \\frac{\\partial}{\\partial \\lambda} \\left( \\log(\\lambda) -\\lambda x \\right) \\right)\n", "= \\frac{\\partial}{\\partial \\lambda} \\left( {\\lambda}^{-1} - x \\right) = - \\lambda^{-2} - 0 = -\\frac{1}{\\lambda^2}\n", "\\end{align}\n", "$$\n", "Now, let us evaluate the integral by recalling that the expectation of the constant $1$ is 1 for any RV $X$ governed by some parameter, say $\\theta$. For instance when $X$ is a continuous RV, $E_{\\theta}(1) = \\int_{x \\in \\mathbb{X}} 1 \\ f(x;\\theta) = \\int_{x \\in \\mathbb{X}} \\ f(x;\\theta) = 1$. Therefore, the Fisher Information of one sample is\n", "$$\n", "\\begin{align}\n", "I_1(\\theta) = - \\int_{x \\in \\mathbb{X} = [0,\\infty)} \\left( \\frac{\\partial^2 \\log \\left( \\lambda e^{-\\lambda x} \\right)}{\\partial^2 \\lambda} \\right) \\lambda e^{-\\lambda x} \\ dx\n", " &= - \\int_{0}^{\\infty} \\left(-\\frac{1}{\\lambda^2} \\right) \\lambda e^{-\\lambda x} \\ dx \\\\\n", "& = - \\left(-\\frac{1}{\\lambda^2} \\right) \\int_{0}^{\\infty} \\lambda e^{-\\lambda x} \\ dx = \\frac{1}{\\lambda^2} \\ 1 = \\frac{1}{\\lambda^2}\n", "\\end{align}\n", "$$\n", "Now, we can compute the desired estimated standard error, by substituting in the ML estimate $\\widehat{\\lambda}_n = 1/(\\overline{x}_n) := 1 / \\left( \\sum_{i=1}^n x_i \\right)$ of $\\lambda^*$, as follows:\n", "$$\n", "\\widehat{\\mathsf{se}}_n(\\widehat{\\Lambda}_n) \n", "= \\frac{1}{\\sqrt{n \\left. I_1 \\right\\vert_{\\lambda=\\widehat{\\lambda}_n}}}\n", "= \\frac{1}{\\sqrt{n \\frac{1}{\\widehat{\\lambda}_n^2} }}\n", "= \\frac{\\widehat{\\lambda}_n}{\\sqrt{n}}\n", "= \\frac{1}{\\sqrt{n} \\ \\overline{x}_n}\n", "$$\n", "Using $\\widehat{\\mathsf{se}}_n(\\widehat{\\lambda}_n)$ we can construct an approximate $95\\%$ confidence interval $C_n$ for $\\lambda^*$, due to the asymptotic normality of the ML estimator of $\\lambda^*$, as follows:\n", "$$\n", "C_n\n", "= \\widehat{\\lambda}_n \\pm 2 \\frac{\\widehat{\\lambda}_n}{\\sqrt{n}}\n", "= \\frac{1}{\\overline{x}_n} \\pm 2 \\frac{1}{\\sqrt{n} \\ \\overline{x}_n} .\n", "$$\n", "Let us compute the ML estimate and the $95\\%$ confidence interval for the rate parameter for the waiting times at the Orbiter bus-stop. The sample mean $\\overline{x}_{132}=9.0758$ and the ML estimate is:\n", "$$\\widehat{\\lambda}_{132}=1/\\,\\overline{x}_{132}=1/9.0758=0.1102 ,$$\n", "and the $95\\%$ confidence interval is:\n", "$$\n", "C_n\n", "= \\widehat{\\lambda}_{132} \\pm 2 \\frac{\\widehat{\\lambda}_{132}}{\\sqrt{132}}\n", "= \\frac{1}{\\overline{x}_{132}} \\pm 2 \\frac{1}{\\sqrt{132} \\, \\overline{x}_{132}} = 0.1102 \\pm 2 \\cdot 0.0096 = [0.091, 0.129] .\n", "$$\n" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "logfx = log(lam*e^(-lam*x))\n", "d2logfx = -1/lam^2\n", "FisherInformation1 = lam^(-2)\n", "StdErr = lam/sqrt(n)\n", "EstStdErr = 1/(sqrt(n)*sampMean)\n" ] }, { "data": { "text/plain": [ "(1/sampMean - 2/(sqrt(n)*sampMean), 1/sampMean + 2/(sqrt(n)*sampMean))" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Sample Exam Problem 5 Solution\n", "# solution is straightforward by following these steps symbolically\n", "# or you can do it by hand with pen/paper or do both to be safe\n", "\n", "## STEP 1 - define the variables you need\n", "lam,x,n = var('lam','x','n')\n", "\n", "## STEP 2 - get symbolic expression for the likelihood of one sample\n", "logfx = log(lam*exp(-lam*x)).full_simplify()\n", "print (\"logfx = \", logfx)\n", "\n", "## STEP 3 - find second derivate of expression from STEP 2 w.r.t. parameter\n", "d2logfx = logfx.diff(lam,2).full_simplify()\n", "print (\"d2logfx = \", d2logfx)\n", "\n", "## STEP 4 - to get Fisher Information of one sample\n", "## integrate d2logfx * f(x) over x in [0,Infinity), f(x) id PDF lam*exp(-lam*x)\n", "assume(lam>0) # usually you need make such assume's for integrate to work - see suggestions in error messages\n", "FisherInformation1 = -integrate(d2logfx*lam*exp(-lam*x),x,0,Infinity)\n", "print (\"FisherInformation1 = \",FisherInformation1)\n", "\n", "## STEP 5 - get Standard Error from FisherInformation1\n", "StdErr = 1/sqrt(n*FisherInformation1)\n", "print (\"StdErr = \",StdErr)\n", "\n", "## STEP 6 - get Standard Error from Standard Error and MLE or lamHat\n", "# lamHat = 1/xBar = 1/sampleMean; know from before\n", "lamHat,sampMean = var('lamHat','sampMean')\n", "lamHat = 1/sampMean\n", "EstStdErr = StdErr.subs(lam=lamHat)\n", "print (\"EstStdErr = \",EstStdErr)\n", "\n", "## STEP 7 - Get lower and upper 95% CI\n", "(lamHat-2*EstStdErr, lamHat+2*EstStdErr)" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The 95% CI for lambda in the Orbiter Waiting time experiment = \n", "0.09100312972775282 0.12936414907024382\n" ] } ], "source": [ "# Sample Exam Problem 5 Solution\n", "# Only replace the XXX below, do not change the function naemes or parameters\n", "import numpy as np\n", "sampleWaitingTimes = np.array([8,3,7,18,18,3,7,9,9,25,0,0,25,6,10,0,10,8,16,9,1,5,16,6,4,1,3,21,0,28,3,8,6,6,11,\\\n", " 8,10,15,0,8,7,11,10,9,12,13,8,10,11,8,7,11,5,9,11,14,13,5,8,9,12,10,13,6,11,13,0,\\\n", " 0,11,1,9,5,14,16,2,10,21,1,14,2,10,24,6,1,14,14,0,14,4,11,15,0,10,2,13,2,22,10,5,\\\n", " 6,13,1,13,10,11,4,7,9,12,8,16,15,14,5,10,12,9,8,0,5,13,13,6,8,4,13,15,7,11,6,23,1])\n", "\n", "def SampleExamProblem5(exponentialSamples):\n", " '''return the 95% confidence interval as a 2-tuple for the unknown rate parameter lambda* \n", " from n IID Exponential(lambda*) trials in the input numpy array called exponentialSamples'''\n", " sampleMean = exponentialSamples.mean()\n", " n=len(exponentialSamples)\n", " correction=RR(2/(sqrt(n)*sampleMean)) # you can also replace RR by float here or you get expressions\n", " lower95CI=1.0/sampleMean - correction\n", " upper95CI=1.0/sampleMean + correction\n", " return (lower95CI,upper95CI)\n", "\n", "# do NOT change anything below\n", "lowerCISampleExamProblem5,upperCISampleExamProblem5 = SampleExamProblem5(sampleWaitingTimes)\n", "print (\"The 95% CI for lambda in the Orbiter Waiting time experiment = \")\n", "print (lowerCISampleExamProblem5,upperCISampleExamProblem5)" ] }, { "cell_type": "markdown", "metadata": { "lx_assignment_number": "3", "lx_problem_cell_type": "PROBLEM" }, "source": [ "---\n", "## Assignment 3, PROBLEM 5\n", "Maximum Points = 3" ] }, { "cell_type": "markdown", "metadata": { "deletable": false, "lx_assignment_number": "3", "lx_assignment_type": "ASSIGNMENT", "lx_assignment_type2print": "Assignment", "lx_problem_cell_type": "PROBLEM", "lx_problem_number": "5", "lx_problem_points": "3" }, "source": [ "\n", "Obtain the 95% CI based on the asymptotic normality of the MLE for the mean paramater $\\lambda$ based on $n$ IID $Poisson(\\lambda^*)$ trials.\n", "\n", "Recall that a random variable $X \\sim Poisson(\\lambda)$ if its probability mass function is:\n", "\n", "$$\n", "f(x; \\lambda) = \\exp{(-\\lambda)} \\frac{\\lambda^x}{x!}, \\quad \\lambda > 0, \\quad x \\in \\{0,1,2,\\ldots\\}\n", "$$\n", "\n", "The MLe $\\widehat{\\lambda}_n = \\overline{x}_n$, the sample mean.\n", "\n", "Work out your answer and express it in the next cell by replacing `XXX`s." ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "deletable": false, "lx_assignment_number": "3", "lx_assignment_type": "ASSIGNMENT", "lx_assignment_type2print": "Assignment", "lx_problem_cell_type": "PROBLEM", "lx_problem_number": "5", "lx_problem_points": "3" }, "outputs": [ { "ename": "NameError", "evalue": "name 'XXX' is not defined", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 15\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 16\u001b[0m \u001b[0;31m# do NOT change anything below\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 17\u001b[0;31m \u001b[0mlowerCISampleExamProblem5\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mupperCISampleExamProblem5\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mAssignment3Problem5\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msamplePoissonCounts\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 18\u001b[0m \u001b[0mprint\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m\"The 95% CI for lambda based on IID Poisson(lambda) data in samplePoissonCounts = \"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 19\u001b[0m \u001b[0mprint\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mlowerCISampleExamProblem5\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mupperCISampleExamProblem5\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m\u001b[0m in \u001b[0;36mAssignment3Problem5\u001b[0;34m(poissonSamples)\u001b[0m\n\u001b[1;32m 7\u001b[0m '''return the 95% confidence interval as a 2-tuple for the unknown parameter lambda* \n\u001b[1;32m 8\u001b[0m from n IID Poisson(lambda*) trials in the input numpy array called samplePoissonCounts'''\n\u001b[0;32m----> 9\u001b[0;31m \u001b[0mXXX\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 10\u001b[0m \u001b[0mXXX\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0mXXX\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mNameError\u001b[0m: name 'XXX' is not defined" ] } ], "source": [ "# Only replace the XXX below, do not change the function naemes or parameters\n", "import numpy as np\n", "samplePoissonCounts = np.array([0,5,11,5,6,8,9,0,1,14,2,4,4,11,2,12,10,5,6,1,7,9,8,0,5,7,11,6,0,1])\n", "\n", "def Assignment3Problem5(poissonSamples):\n", " '''return the 95% confidence interval as a 2-tuple for the unknown parameter lambda* \n", " from n IID Poisson(lambda*) trials in the input numpy array called samplePoissonCounts'''\n", " XXX\n", " XXX\n", " XXX\n", " lower95CI=XXX\n", " upper95CI=XXX\n", " return (lower95CI,upper95CI)\n", "\n", "# do NOT change anything below\n", "lowerCISampleExamProblem5,upperCISampleExamProblem5 = Assignment3Problem5(samplePoissonCounts)\n", "print (\"The 95% CI for lambda based on IID Poisson(lambda) data in samplePoissonCounts = \")\n", "print (lowerCISampleExamProblem5,upperCISampleExamProblem5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Hypothesis Testing\n", "\n", "The subset of *all posable hypotheses* that have the property of *[falsifiability](https://en.wikipedia.org/wiki/Falsifiability)* constitute the space of *scientific hypotheses*. \n", "Roughly, a falsifiable statistical hypothesis is one for which a statistical experiment can be designed to produce data or empirical observations that an experimenter can use to falsify or reject it. \n", "In the *statistical decision problem of hypothesis testing*, we are interested in empirically falsifying a scientific hypothesis, i.e. we attempt to reject a hypothesis on the basis of empirical observations or data. \n", "Thus, hypothesis testing has its roots in the *philosophy of science* and is based on *Karl Popper's falsifiability criterion for demarcating scientific hypotheses from the set of all posable hypotheses*.\n", "\n", "## Introduction\n", "Usually, the hypothesis we **attempt to reject or falsify** is called the **null hypothesis** or $H_0$ and its complement is called the **alternative hypothesis** or $H_1$. \n", "For example, consider the following two hypotheses:\n", "\n", "- $H_0$: The average waiting time at an Orbiter bus stop *is less than or equal to* $10$ minutes.\n", "- $H_1$: The average waiting time at an Orbiter bus stop *is more than* $10$ minutes.\n", "\n", "If the sample mean $\\overline{x}_n$ is much larger than $10$ minutes then we may be inclined to reject the null hypothesis that the average waiting time is less than or equal to $10$ minutes. \n", "\n", "Suppose we are interested in the following slightly different hypothesis test for the Orbiter bus stop problem:\n", "\n", "- $H_0$: The average waiting time at an Orbiter bus stop *is equal to* $10$ minutes.\n", "- $H_1$: The average waiting time at an Orbiter bus stop *is not* $10$ minutes.\n", "\n", "Once again we can use the sample mean as the test statistic, but this time we may be inclined to reject the null hypothesis if the sample mean $\\overline{x}_n$ is much larger than *or* much smaller than $10$ minutes. \n", "The procedure for rejecting such a null hypothesis is called the **Wald test** we are about to see.\n", "\n", "More generally, suppose we have the following parametric experiment based on $n$ IID trials:\n", "$$\n", "X_1,X_2,\\ldots,X_n \\overset{IID}{\\sim} F(x_1;\\theta^*), \\quad \\text{ with an unknown (and fixed) } \\theta^* \\in \\mathbf{\\Theta} \\ .\n", "$$\n", "\n", "Let us partition the parameter space $\\mathbf{\\Theta}$ into $\\mathbf{\\Theta}_0$, the null parameter space, and $\\mathbf{\\Theta}_1$, the alternative parameter space, i.e.,\n", "$$\\mathbf{\\Theta}_0 \\cup \\mathbf{\\Theta}_1 = \\mathbf{\\Theta}, \\qquad \\text{and} \\qquad \\mathbf{\\Theta}_0 \\cap \\mathbf{\\Theta}_1 = \\emptyset \\ .$$\n", "\n", "Then, we can formalise testing the null hypothesis versus the alternative as follows:\n", "$$\n", "H_0 : \\theta^* \\in \\mathbf{\\Theta}_0 \\qquad \\text{versus} \\qquad H_1 : \\theta^* \\subset \\mathbf{\\Theta}_1 \\ .\n", "$$\n", "\n", "The basic idea involves finding an appropriate **rejection region** $\\mathbb{X}_R$ within the **data space** $\\mathbb{X}$ and rejecting $H_0$ if the observed data $x:=(x_1,x_2,\\ldots,x_n)$ falls inside the rejection region $\\mathbb{X}_R$,\n", "$$\n", "\\text{If $x:=(x_1,x_2,\\ldots,x_n) \\in \\mathbb{X}_R \\subset \\mathbb{X}$, then reject $H_0$, else do not reject $H_0$.}\n", "$$\n", "Typically, the rejection region $\\mathbb{X}_R$ is of the form:\n", "$$\n", "\\mathbb{X}_R := \\{ x:=(x_1,x_2,\\ldots,x_n) : T(x) > c \\}\n", "$$\n", "where, $T$ is the **test statistic** and $c$ is the **critical value**. Thus, the problem of finding $\\mathbb{X}_R$ boils down to that of finding $T$ and $c$ that are appropriate. Once the rejection region is defined, the possible outcomes of a hypothesis test are summarised in the following table.\n", "\n", "\n", "The outcomes of a hypothesis test, in general, are:\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "
'true state of nature'Do not reject $H_0$
Reject $H_0$
\n", "

$H_0$ is true

\n", "

 

\n", "		\n", "

OK 

\n", "		\n", "

Type I error

\n", "
\n", "

$H_0$ is false

\n", "		Type II errorOK
\n", "\n", "So, intuitively speaking, we want a small probability that we reject $H_0$ when $H_0$ is true (minimise Type I error). Similarly, we want to minimise the probability that we fail to reject $H_0$ when $H_0$ is false (type II error). Let us formally see how to achieve these goals.\n", "\n", "## Power, Size and Level of a Test\n", "\n", "### Power Function \n", "\n", "The **power function** of a test with rejection region $\\mathbb{X}_R$ is\n", "$$\n", "\\boxed{\n", "\\beta(\\theta) := P_{\\theta}(x \\in \\mathbb{X}_R)\n", "}\n", "$$\n", "So $\\beta(\\theta)$ is the power of the test if the data were generated under the parameter value $\\theta$, i.e. the probability that the observed data $x$, sampled from the distribution specified by $\\theta$, falls in the rejection region $\\mathbb{X}_R$ and thereby leads to a rejection of the null hypothesis.\n", "\n", "### Size of a test\n", "The $\\mathsf{size}$ of a test with rejection region $\\mathbb{X}_R$ is the supreme power under the null hypothesis, i.e.~the supreme probability of rejecting the null hypothesis when the null hypothesis is true:\n", "$$\n", "\\boxed{\n", "\\mathsf{size} := \\sup_{\\theta \\in \\mathbf{\\Theta}_0} \\beta(\\theta) := \\sup_{\\theta \\in \\mathbf{\\Theta}_0} P{\\theta}(x \\in \\mathbb{X}_R) \\ .\n", "}\n", "$$\n", "The $\\mathsf{size}$ of a test is often denoted by $\\alpha$. A test is said to have $\\mathsf{level}$ $\\alpha$ if its $\\mathsf{size}$ is less than or equal to $\\alpha$.\n", "\n", "\n", "## Wald test\n", "\n", "The Wald test is based on a direct relationship between the $1-\\alpha$ confidence interval and a $\\mathsf{size}$ $\\alpha$ test. It can be used for testing simple hypotheses involving a scalar parameter.\n", "\n", "### Definition\n", "\n", "Let $\\widehat{\\Theta}_n$ be an asymptotically normal estimator of the fixed and possibly unknown parameter $\\theta^* \\in \\mathbf{\\Theta} \\subset \\mathbb{X}$ in the parametric IID experiment:\n", "\n", "$$\n", "X_1,X_2,\\ldots,X_n \\overset{IID}{\\sim} F(x_1;\\theta^*) \\enspace .\n", "$$\n", "\n", "Consider testing:\n", "\n", "$$\n", "\\boxed{H_0: \\theta^* = \\theta_0 \\qquad \\text{versus} \\qquad H_1: \\theta^* \\neq \\theta_0 \\enspace .}\n", "$$\n", "\n", "Suppose that the null hypothesis is true and the estimator $\\widehat{\\Theta}_n$ of $\\theta^*=\\theta_0$ is asymptotically normal:\n", "\n", "$$\n", "\\boxed{\n", "\\theta^*=\\theta_0, \\qquad \\frac{\\widehat{\\Theta}_n - \\theta_0}{\\widehat{\\mathsf{se}}_n} \\overset{d}{\\to} Normal(0,1) \\enspace .}\n", "$$\n", "\n", "Then, **the Wald test based on the test statistic $W$** is:\n", "$$\n", "\\boxed{\n", "\\text{Reject $H_0$ when $|W|>z_{\\alpha/2}$, where $W:=W((X_1,\\ldots,X_n))=\\frac{\\widehat{\\Theta}_n ((X_1,\\ldots,X_n)) - \\theta_0}{\\widehat{\\mathsf{se}}_n}$.}\n", "}\n", "$$\n", "The rejection region for the Wald test is:\n", "$$\n", "\\boxed{\n", "\\mathbb{X}_R = \\{ x:=(x_1,\\ldots,x_n) : |W (x_1,\\ldots,x_n) | > z_{\\alpha/2} \\} \\enspace .\n", "}\n", "$$\n", "\n", "### Asymptotic $\\mathsf{size}$ of a Wald test\n", "\n", "As the sample size $n$ approaches infinity, the $\\mathsf{size}$ of the Wald test approaches $\\alpha$ :\n", "\n", "$$\n", "\\boxed{\n", "\\mathsf{size} = P_{\\theta_0} \\left( |W| > z_{\\alpha/2} \\right) \\to \\alpha \\enspace .}\n", "$$\n", "\n", "**Proof:** Let $Z \\sim Normal(0,1)$. The $\\mathsf{size}$ of the Wald test, i.e.~the supreme power under $H_0$ is:\n", "\n", "$$\n", "\\begin{align}\n", "\\mathsf{size}\n", "& := \\sup_{\\theta \\in \\mathbf{\\Theta}_0} \\beta(\\theta) := \\sup_{\\theta \\in \\{\\theta_0\\}} P_{\\theta}(x \\in \\mathbb{X}_R) = P_{\\theta_0}(x \\in \\mathbb{X}_R) \\\\\n", "& = P_{\\theta_0} \\left( |W| > z_{\\alpha/2} \\right) = P_{\\theta_0} \\left( \\frac{|\\widehat{\\theta}_n - \\theta_0|}{\\widehat{\\mathsf{se}}_n} > z_{\\alpha/2} \\right) \\\\\n", "& \\to P \\left( |Z| > z_{\\alpha/2} \\right)\\\\\n", "& = \\alpha \\enspace .\n", "\\end{align}\n", "$$\n", "\n", "Next, let us look at the power of the Wald test when the null hypothesis is false.\n", "\n", "### Asymptotic power of a Wald test\n", "\n", "Suppose $\\theta^* \\neq \\theta_0$. The power $\\beta(\\theta^*)$, which is the probability of correctly rejecting the null hypothesis, is approximately equal to:\n", "\n", "$$\n", "\\boxed{\n", "\\Phi \\left( \\frac{\\theta_0-\\theta^*}{\\widehat{\\mathsf{se}}_n} - z_{\\alpha/2} \\right) +\n", "\\left( 1- \\Phi \\left( \\frac{\\theta_0-\\theta^*}{\\widehat{\\mathsf{se}}_n} + z_{\\alpha/2} \\right) \\right) \\enspace ,\n", "}\n", "$$\n", "where, $\\Phi$ is the DF of $Normal(0,1)$ RV. Since ${\\widehat{\\mathsf{se}}_n} \\to 0$ as $n \\to 0$ the power increase with sample $\\mathsf{size}$ $n$. Also, the power increases when $|\\theta_0-\\theta^*|$ is large.\n", "\n", "Now, let us make the connection between the $\\mathsf{size}$ $\\alpha$ Wald test and the $1-\\alpha$ confidence interval explicit.\n", "\n", "### The $\\mathsf{size}$ Wald test\n", "\n", "The $\\mathsf{size}$ $\\alpha$ Wald test rejects:\n", "\n", "$$\n", "\\boxed{\n", "\\text{ $H_0: \\theta^*=\\theta_0$ versus $H_1: \\theta^* \\neq \\theta_0$ if and only if $\\theta_0 \\notin C_n := (\\widehat{\\theta}_n-{\\widehat{\\mathsf{se}}_n} z_{\\alpha/2}, \\widehat{\\theta}_n+{\\widehat{\\mathsf{se}}_n} z_{\\alpha/2})$.\n", "}}\n", "$$\n", "\n", "$$\\boxed{\\text{Therefore, testing the hypothesis is equivalent to verifying whether the null value $\\theta_0$ is in the confidence interval.}}$$\n", "\n", "\n", "### Example: Wald test for the mean waiting times at our Orbiter bus-stop\n", "\n", "Let us use the Wald test to attempt to reject the null hypothesis that the mean waiting time at our Orbiter bus-stop is $10$ minutes under an IID $Exponential(\\lambda^*)$ model. Let $\\alpha=0.05$ for this test. We can formulate this test as follows:\n", "$$\n", "H_0: \\lambda^* = \\lambda_0= \\frac{1}{10} \\quad \\text{versus} \\quad H_1: \\lambda^* \\neq \\frac{1}{10}, \\quad \\text{where, } \\quad X_1\\ldots,X_{132} \\overset{IID}{\\sim} Exponential(\\lambda^*) \\enspace .\n", "$$\n", "We already obtained the $95\\%$ confidence interval based on its MLE's asymptotic normality property to be $[0.0914, 0.1290]$. \n", "\n", "$$\\boxed{\\text{Since our null value $\\lambda_0=0.1$ belongs to this confidence interval, we fail to reject the null hypothesis from a $\\mathsf{size}$ $\\alpha=0.05$ Wald test.}}$$\n", "\n", "We will revisit this example in a more computationally explicit fasion soon below." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### A Live Example: Simulating Bernoulli Trials to understand Wald Tests\n", "\n", "Let's revisit the MLE for the $Bernoulli(\\theta^*)$ model with $n$ IID trails, we have already seen, and test the null hypothesis that the unknown $\\theta^* = \\theta_0 = 0.5$.\n", "\n", "Thus, we are interested in the null hypothesis $H_0$ versus the alternative hypothesis $H_1$:\n", "\n", "$$\\displaystyle{H_0: \\theta^*=\\theta_0 \\quad \\text{ versus } \\quad H_1: \\theta^* \\neq \\theta_0, \\qquad \\text{ with }\\theta_0=0.5}$$\n", "\n", "We can test this hypothesis with Type I error at $\\alpha$ using the **size-$\\alpha$ Wald Test** that builds on the asymptotic normality of the MLE, i.e., \n", "$$\\displaystyle{ \\frac{\\widehat{\\theta}_n - \\theta_0}{\\widehat{se}_n} \\overset{d}{\\to} Normal(0,1)}$$\n", "\n", "The size-$\\alpha$ Wald test is:\n", "\n", "$$\n", "\\boxed{\n", "\\text{Reject } \\ H_0 \\quad \\text{ when } |W| > z_{\\alpha/2}, \\quad \\text{ where, } \\quad W = \\frac{\\widehat{\\theta}_n - \\theta_0}{\\widehat{se}_n}\n", "}\n", "$$" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1.4064216928154865\n" ] }, { "data": { "text/plain": [ "False" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import numpy as np\n", "# do a live simulation ... to implement this test...\n", "# simulate from Bernoulli(theta0) n samples\n", "# make mle\n", "# construct Wald test\n", "# make a decision - i.e., decide if you will reject or fail to reject the H0: theta0=0.5\n", "trueTheta=0.45\n", "n=20\n", "myBernSamples=np.array([floor(random()+trueTheta) for i in range(0,n)])\n", "#myBernSamples\n", "mle=myBernSamples.mean() # 1/mean\n", "mle\n", "NullTheta=0.5\n", "se=sqrt(mle*(1.0-mle)/n)\n", "W=(mle-NullTheta)/se\n", "print (abs(W))\n", "alpha = 0.05\n", "abs(W) > 2 # alpha=0.05, so z_{alpha/2} =1.96 approx=2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Sample Exam Problem 6 \n", "\n", "Consider the following model for the parity (odd=1, even=0) of the first Lotto ball to pop out of the NZ Lotto machine. We had $n=1114$ IID trials:\n", "\n", "$$\\displaystyle{X_1,X_2,\\ldots,X_{1114} \\overset{IID}{\\sim} Bernoulli(\\theta^*)}$$\n", "\n", "and know from this dataset that the number of odd balls is $546=\\sum_{i=1}^{1114} x_i$.\n", "\n", "Your task is to perform a Wald Test of size $\\alpha=0.05$ to try to reject the null hypothesis that the chance of seeing an odd ball out of the NZ Lotto machine is exactly $1/2$, i.e.,\n", "\n", "$$\\displaystyle{H_0: \\theta^*=\\theta_0 \\quad \\text{ versus } \\quad H_1: \\theta^* \\neq \\theta_0, \\qquad \\text{ with }\\theta_0=0.5}$$\n", "\n", "Show you work by replacing `XXX`s with the right expressions in the next cell." ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "ename": "NameError", "evalue": "name 'XXX' is not defined", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0;31m## STEP 1: get the MLE thetaHat\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mthetaHat\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mXXX\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 5\u001b[0m \u001b[0mprint\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m\"mle thetaHat = \"\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mthetaHat\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 6\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mNameError\u001b[0m: name 'XXX' is not defined" ] } ], "source": [ "# Sample Exam Problem 6 Problem\n", "\n", "## STEP 1: get the MLE thetaHat\n", "thetaHat=XXX \n", "print (\"mle thetaHat = \",thetaHat)\n", "\n", "## STEP 2: get the NullTheta or theta0\n", "NullTheta=XXX\n", "print (\"Null value of theta under H0 = \", NullTheta)\n", "\n", "## STEP 3: get estimated standard error\n", "seTheta=XXX # for Bernoulli trials from earleir in 10.ipynb\n", "print (\"estimated standard error\",seTheta)\n", "\n", "# STEP 4: get Wald Statistic\n", "W=XXX\n", "print (\"Wald staatistic = \",W)\n", "\n", "# STEP 5: conduct the size alpha=0.05 Wald test\n", "# do NOT change anything below\n", "rejectNullSampleExamProblem6 = abs(W) > 2.0 # alpha=0.05, so z_{alpha/2} =1.96 approx=2.0\n", "if (rejectNullSampleExamProblem6):\n", " print (\"we reject the null hypothesis that theta_0=0.5\")\n", "else:\n", " print (\"we fail to reject the null hypothesis that theta_0=0.5\")" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "mle thetaHat = 273/557\n", "Null value of theta under H0 = 0.500000000000000\n", "estimated standard error 0.0149776163832414\n", "Wald staatistic = -0.659272243178650\n", "we fail to reject the null hypothesis that theta_0=0.5\n" ] } ], "source": [ "# Sample Exam Problem 6 Solution\n", "\n", "## STEP 1: get the MLE thetaHat\n", "n=1114 # sample size\n", "thetaHat=546/n # MLE is sample mean for IID Bernoulli trials\n", "print (\"mle thetaHat = \",thetaHat)\n", "\n", "## STEP 2: get the NullTheta or theta0\n", "NullTheta=0.5\n", "print (\"Null value of theta under H0 = \", NullTheta)\n", "\n", "## STEP 3: get estimated standard error\n", "seTheta=sqrt(thetaHat*(1.0-thetaHat)/n) # for Bernoulli trials from earleir in 10.ipynb\n", "print (\"estimated standard error\",seTheta)\n", "\n", "# STEP 4: get Wald Statistic\n", "W=(thetaHat-NullTheta)/seTheta\n", "print (\"Wald staatistic = \",W)\n", "\n", "# STEP 5: conduct the size alpha=0.05 Wald test\n", "rejectNullSampleExamProblem6 = abs(W) > 2.0 # alpha=0.05, so z_{alpha/2} =1.96 approx=2.0\n", "if (rejectNullSampleExamProblem6):\n", " print (\"we reject the null hypothesis that theta_0=0.5\")\n", "else:\n", " print (\"we fail to reject the null hypothesis that theta_0=0.5\")" ] }, { "cell_type": "markdown", "metadata": { "lx_assignment_number": "3", "lx_problem_cell_type": "PROBLEM" }, "source": [ "---\n", "## Assignment 3, PROBLEM 6\n", "Maximum Points = 3" ] }, { "cell_type": "markdown", "metadata": { "deletable": false, "lx_assignment_number": "3", "lx_assignment_type": "ASSIGNMENT", "lx_assignment_type2print": "Assignment", "lx_problem_cell_type": "PROBLEM", "lx_problem_number": "6", "lx_problem_points": "3" }, "source": [ "\n", "For the Orbiter waiting time problem, assuming IID trials as follows: \n", "\n", "$$\\displaystyle{X_1,X_2,\\ldots,X_{n} \\overset{IID}{\\sim} Exponential(\\lambda^*)}$$\n", "\n", "Your task is to perform a Wald Test of size $\\alpha=0.05$ to try to reject the null hypothesis that the waiting time at the Orbiter bus-stop, i.e., the inter-arrival time between buses, is exactly $10$ minutes:\n", "\n", "$$\\displaystyle{H_0: \\lambda^*=\\lambda_0 \\quad \\text{ versus } \\quad H_1: \\lambda^* \\neq \\lambda_0, \\qquad \\text{ with }\\lambda_0=0.1}$$\n", "\n", "Show you work by replacing `XXX`s with the right expressions in the next cell." ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "deletable": false, "lx_assignment_number": "3", "lx_assignment_type": "ASSIGNMENT", "lx_assignment_type2print": "Assignment", "lx_problem_cell_type": "PROBLEM", "lx_problem_number": "6", "lx_problem_points": "3" }, "outputs": [ { "ename": "NameError", "evalue": "name 'XXX' is not defined", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;31m#test H0: lambda=0.1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 6\u001b[0m \u001b[0;31m## STEP 1: get the MLE thetaHat\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 7\u001b[0;31m \u001b[0mlambdaHat\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mXXX\u001b[0m \u001b[0;31m# you need to use sampleWaitingTimes here!\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 8\u001b[0m \u001b[0mprint\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m\"mle lambdaHat = \"\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mlambdaHat\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 9\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mNameError\u001b[0m: name 'XXX' is not defined" ] } ], "source": [ "import numpy as np\n", "sampleWaitingTimes = np.array([8,3,7,18,18,3,7,9,9,25,0,0,25,6,10,0,10,8,16,9,1,5,16,6,4,1,3,21,0,28,3,8,6,6,11,\\\n", " 8,10,15,0,8,7,11,10,9,12,13,8,10,11,8,7,11,5,9,11,14,13,5,8,9,12,10,13,6,11,13,0,\\\n", " 0,11,1,9,5,14,16,2,10,21,1,14,2,10,24,6,1,14,14,0,14,4,11,15,0,10,2,13,2,22,10,5,\\\n", " 6,13,1,13,10,11,4,7,9,12,8,16,15,14,5,10,12,9,8,0,5,13,13,6,8,4,13,15,7,11,6,23,1])\n", "\n", "#test H0: lambda=0.1\n", "## STEP 1: get the MLE thetaHat\n", "lambdaHat=XXX # you need to use sampleWaitingTimes here!\n", "print (\"mle lambdaHat = \",lambdaHat)\n", "\n", "## STEP 2: get the NullLambda or lambda0\n", "NullLambda=XXX\n", "print (\"Null value of lambda under H0 = \", NullLambda)\n", "\n", "## STEP 3: get estimated standard error\n", "seLambda=XXX # see Sample Exam Problem 5 in 10.ipynb\n", "print (\"estimated standard error\",seLambda)\n", "\n", "# STEP 4: get Wald Statistic\n", "W=XXX\n", "print (\"Wald statistic = \",W)\n", "\n", "# STEP 5: conduct the size alpha=0.05 Wald test\n", "# do NOT change anything below\n", "rejectNullAssignment3Problem6 = abs(W) > 2.0 # alpha=0.05, so z_{alpha/2} =1.96 approx=2.0\n", "if (rejectNullAssignment3Problem6):\n", " print (\"we reject the null hypothesis that lambda0=0.1\")\n", "else:\n", " print (\"we fail to reject the null hypothesis that lambda0=0.1\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## P-value\n", "\n", "It is desirable to have a more informative decision than simply reporting \"reject $H_0$\" or \"fail to reject $H_0$.\"\n", "\n", "For instance, we could ask whether the test rejects $H_0$ for each $\\mathsf{size}=\\alpha$. \n", "Typically, if the test rejects at $\\mathsf{size}$ $\\alpha$ it will also reject at a larger $\\mathsf{size}$ $\\alpha' > \\alpha$. \n", "Therefore, there is a smallest $\\mathsf{size}$ $\\alpha$ at which the test rejects $H_0$ and we call this $\\alpha$ the $\\text{p-value}$ of the test.\n", "\n", "$$\\boxed{\\text{The smallest $\\alpha$ at which a $\\mathsf{size}$ $\\alpha$ test rejects the null hypothesis $H_0$ is the $\\text{p-value}$.}}$$\n" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 11 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "p=text('Reject $H_0$?',(12,12)); p+=text('No',(30,10)); p+=text('Yes',(30,15)); p+=text('p-value',(70,10))\n", "p+=text('size',(65,4)); p+=text('$0$',(40,4)); p+=text('$1$',(90,4)); p+=points((59,5),rgbcolor='red',size=50)\n", "p+=line([(40,17),(40,5),(95,5)]); p+=line([(40,10),(59,10),(59,15),(90,15)]);\n", "p+=line([(68,9.5),(59.5,5.5)],rgbcolor='red'); p.show(axes=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Definition of p-value\n", "Suppose that for every $\\alpha \\in (0,1)$ we have a $\\mathsf{size}$ $\\alpha$ test with rejection region $\\mathbb{X}_{R,\\alpha}$ and test statistic $T$. Then,\n", "$$\n", "\\text{p-value} := \\inf \\{ \\alpha: T(X) \\in \\mathbb{X}_{R,\\alpha} \\} \\enspace .\n", "$$\n", "That is, the p-value is the smallest $\\alpha$ at which a $\\mathsf{size}$ $\\alpha$ test rejects the null hypothesis.\n", "\n", "### Understanding p-value\n", "If the evidence against $H_0$ is strong then the p-value will be small. However, a large p-value is not strong evidence in favour of $H_0$. This is because a large p-value can occur for two reasons:\n", "\n", "- $H_0$ is true.\n", "- $H_0$ is false but the test has low power (i.e., high Type II error).\n", "\n", "Finally, it is important to realise that *p-value is not the probability that the null hypothesis is true*, i.e. $\\text{p-value} \\, \\neq P(H_0|x)$, where $x$ is the data. The following itemisation of implications for the evidence scale is useful.\n", "\n", "The scale of the evidence against the null hypothesis $H_0$ in terms of the range of the p-values has the following interpretation that is commonly used:\n", "\n", "- P-value $\\in (0.00, 0.01]$ $\\implies$ Very strong evidence against $H_0$\n", "- P-value $\\in (0.01, 0.05]$ $\\implies$ Strong evidence against $H_0$\n", "- P-value $\\in (0.05, 0.10]$ $\\implies$ Weak evidence against $H_0$\n", "- P-value $\\in (0.10, 1.00]$ $\\implies$ Little or no evidence against $H_0$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next we will see a convenient expression for the p-value for certain tests.\n", "\n", "### The p-value of a hypothesis test\n", "\n", "Suppose that the $\\mathsf{size}$ $\\alpha$ test based on the test statistic $T$ and critical value $c_{\\alpha}$ is of the form:\n", "\n", "$$\n", "\\text{Reject $H_0$ if and only if $T:=T((X_1,\\ldots,X_n))> c_{\\alpha}$,}\n", "$$\n", "\n", "then\n", "\n", "$$\n", "\\boxed{\n", "\\text{p-value} \\, = \\sup_{\\theta \\in \\mathbf{\\Theta}_0} P_{\\theta}(T((X_1,\\ldots,X_n)) \\geq t:=T((x_1,\\ldots,x_n))) \\enspace ,}\n", "$$\n", "\n", "where, $(x_1,\\ldots,x_n)$ is the observed data and $t$ is the observed value of the test statistic $T$. \n", "\n", "In words, **the p-value is the supreme probability under $H_0$ of observing a value of the test statistic the same as or more extreme than what was actually observed.**\n", "\n", "\n", "Let us revisit the Orbiter waiting times example from the p-value perspective.\n", "\n", "### Example: p-value for the parametric Orbiter bus waiting times experiment\n", "\n", "Let the waiting times at our bus-stop be $X_1,X_2,\\ldots,X_{132} \\overset{IID}{\\sim} Exponential(\\lambda^*)$. Consider the following testing problem:\n", "\n", "$$\n", "H_0: \\lambda^*=\\lambda_0=\\frac{1}{10} \\quad \\text{versus} \\quad H_1: \\lambda^* \\neq \\lambda_0 \\enspace .\n", "$$\n", "\n", "We already saw that the Wald test statistic is:\n", "\n", "$$\n", "W:=W(X_1,\\ldots,X_n)= \\frac{\\widehat{\\Lambda}_n-\\lambda_0}{\\widehat{\\mathsf{se}}_n(\\widehat{\\Lambda}_n)} = \\frac{\\frac{1}{\\overline{X}_n}-\\lambda_0}{\\frac{1}{\\sqrt{n}\\overline{X}_n}} \\enspace .\n", "$$\n", "\n", "The observed test statistic is:\n", "\n", "$$\n", "w=W(x_1,\\ldots,x_{132})=\n", "\\frac{\\frac{1}{\\overline{X}_{132}}-\\lambda_0}{\\frac{1}{\\sqrt{132}\\overline{X}_{132}}}\n", "= \\frac{\\frac{1}{9.0758}-\\frac{1}{10}}{\\frac{1}{\\sqrt{132} \\times 9.0758}} = 1.0618 \\enspace .\n", "$$\n", "Since, $W \\overset{d}{\\to} Z \\sim Normal(0,1)$, the p-value for this Wald test is:\n", "\n", "$$\n", "\\begin{align}\n", "\\text{p-value} \\, \n", "&= \\sup_{\\lambda \\in \\mathbf{\\Lambda}_0} P_{\\lambda} (|W|>|w|)= \\sup_{\\lambda \\in \\{\\lambda_0\\}} P_{\\lambda} (|W|>|w|) = P_{\\lambda_0} (|W|>|w|) \\\\\n", "& \\to P (|Z|>|w|)=2 \\Phi(-|w|)=2 \\Phi(-|1.0618|)=2 \\times 0.1442=0.2884 \\enspace .\n", "\\end{align}\n", "$$\n", "\n", "Therefore, there is little or no evidence against $H_0$ that the mean waiting time under an IID $Exponential$ model of inter-arrival times is exactly ten minutes.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Preparation for Nonparametric Estimation and Testing\n", "### YouTry Later\n", "\n", "Python's `random` for sampling and sequence manipulation\n", "\n", "The Python `random` module, available in SageMath, provides a useful way of taking samples if you have already generated a 'population' to sample from, or otherwise playing around with the elements in a sequence. See http://docs.python.org/library/random.html for more details. Here we will try a few of them.\n", "\n", "The aptly-named sample function allows us to take a sample of a specified size from a sequence. We will use a list as our sequence:" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[16, 95, 54, 24, 19, 51, 5, 50, 74, 70]" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "popltn = range(1, 101, 1) # make a population\n", "sample(popltn, 10) # sample 10 elements from it at random" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Each call to sample will select unique elements in the list (note that 'unique' here means that it will not select the element at any particular position in the list more than once, but if there are duplicate elements in the list, such as with a list [1,2,4,2,5,3,1,3], then you may well get any of the repeated elements in your sample more than once). sample samples with replacement, which means that repeated calls to sample may give you samples with the same elements in." ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\n" ] } ], "source": [ "popltnWithDuplicates = list(range(1, 11, 1))*4 # make a population with repeated elements\n", "print(popltnWithDuplicates)" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[7, 6, 3, 5, 8, 7, 4, 7, 1, 4]\n", "[1, 2, 8, 3, 5, 2, 4, 9, 1, 4]\n", "[8, 5, 9, 10, 6, 1, 7, 3, 2, 3]\n", "[9, 1, 10, 7, 8, 7, 5, 5, 3, 4]\n", "[7, 4, 5, 10, 1, 1, 10, 2, 2, 8]\n" ] } ], "source": [ "for i in range (5):\n", " print( sample(popltnWithDuplicates, 10))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Try experimenting with choice, which allows you to select one element at random from a sequence, and shuffle, which shuffles the sequence in place (i.e, the ordering of the sequence itself is changed rather than you being given a re-ordered copy of the list). It is probably easiest to use lists for your sequences. See how `shuffle` is creating permutations of the list. You could use `sample` and `shuffle` to emulate *permuations of k objects out of n* ...\n", "\n", "You may need to check the documentation to see how use these functions." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "#?sample" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "#?shuffle" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "#?choice" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.1", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.8" }, "lx_course_instance": "2020", "lx_course_name": "Introduction to Data Science: A Comp-Math-Stat Approach", "lx_course_number": "1MS041" }, "nbformat": 4, "nbformat_minor": 2 }