{ "cells": [ { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt, seaborn as sn, numpy as np, pandas as pd\n", "sn.set_context('notebook')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Bayesian Inference and Monte Carlo methods\n", "\n", "This notebook introduces a Bayesian approach to model calibration and also presents the basics of how to use Monte Carlo (MC) methods to sample from difficult distributions.\n", "\n", "## 1.1. A Bayesian model calibration framework\n", "\n", "At the end of the first notebook we introduced Bayes' Theorem:\n", "\n", "$$P(y \\mid x) = \\frac{P(x \\mid y)P(y)}{\\int_y P(x \\mid y)P(y) dy}$$\n", "\n", "and in the second notebook we discussed model auto-calibration, which led us to the concept of the likelihood function. \n", "\n", "In a typical calibration situation we have an environmental model that includes some poorly defined parameters, and we also have observed data for the variable we're trying to simulate, together with contemporary observations for all the input datasets we need to run the model. We often also have some idea of roughly what values the parameters should take (e.g. from ranges in the literature), but we don't know enough about them to be able to use the model without calibrating it first. \n", "\n", "There are lots of different ways of calibrating models, but here we're going to focus on the Bayesian approach. In this case, the aim of model calibration is to start off with whatever **prior information** we have about the parameters (e.g. literature values) and then **refine** these estimates using the observed data. If our model has $\\theta = \\{\\theta_1, .., \\theta_n\\}$ poorly constrained parameters, we can capture our **prior** beliefs by defining distributions for each of them. For example, in the simplest case we might define **uniform distributions** for each parameter with the maximum and minimum values for each distribution based upon literature ranges. The **joint distribution** over all the $\\theta_i$ is the **prior distribution**, $P(\\theta)$, of our parameters. The individual distributions for each parameter are known as **marginal priors**.\n", "\n", "We want to take the prior distribution and *update* it based on the observed data. In other words, we want to calculate a new distribution for the parameters **given** the observations, $P(\\theta \\mid D)$, where $D$ is the observed data. This is called the **posterior distribution**, because it represents what we know about the parameters *after* we've incorporated the information from the observations. Once we have the posterior distribution, we can calculate **marginal posterior probability densities** (\"marginal posteriors\") for each parameter, which we can compare to our prior distributions to see how much we've learned from the calibration process (see notebook 1 for a quick overview of marginalisation).\n", "\n", "We therefore want to calculate $P(\\theta \\mid D)$, but it's not immediately obvious how to do this. This is where Bayes' Theorem comes in. From the equation above, we can write:\n", "\n", "$$P(\\theta \\mid D) = \\frac{P(D \\mid \\theta)P(\\theta)}{\\int_\\theta P(D \\mid \\theta)P(\\theta) d\\theta}$$\n", "\n", "How does this help? Well, $P(\\theta)$ is the prior distribution, which we already know. $P(D \\mid \\theta)$ is the probability of the observations, *given* a particular set of parameters. In other words, if we run the model with a particular set of parameters, what is the probabiltiy that it produces the observed data? From notebook 2, you should be able to see that this is the **likelihood function**. This gives us everything we need to work out the right-hand side of the equation above: the numerator in the fraction is the prior multiplied by the likelihood, and the denominator is the integral of the numerator over the entire parameter space. This integral is actually just a constant: it's a scaling factor which \"normalises\" the numerator so that the total volume under the posterior surface is equal to one. It is properly referred to as **the probability of the data** or, sometimes, as the **evidence**, but it's also often just called the **normalising constant**.\n", "\n", "This means we can re-write Bayes' Theorem as:\n", "\n", "$$posterior = \\frac{likelihood \\times prior}{normalising\\;constant}$$\n", "\n", "In principle, at least, Bayes' Theorem gives us everything we need to calculate $P(\\theta \\mid D)$. In practice, the integral in the denominator tends to be tricky to evaluate because it frequently has no analytical solution. Also, because it involves integrating over the *entire* parameter space, numerical approximations of the integral are often *expensive* to calculate (i.e. they take a long time to run, even on a powerful computer). Fortunately, in many applications we don't need to know the exact posterior distribution - anything *proportional* to it will do. Because the denominator in Bayes' equation is ultimately just a constant, we can write instead:\n", "\n", "$$P(\\theta \\mid D) \\propto P(D \\mid \\theta)P(\\theta)$$\n", "

\n", "$$posterior \\propto likelihood \\times prior$$\n", "\n", "## 1.2. Conjugate priors\n", "\n", "The difficulty of evaluating the posterior depends on the complexity of the prior distribution and likelihood function. If the likelihood is something simple, like a Gaussian, we can often make life easier for ourselves by choosing a **[conjugate prior](https://en.wikipedia.org/wiki/Conjugate_prior)** to ensure that the posterior will also be a friendly function. \n", "\n", "Choosing your prior to make the calculations easier might seem a bit arbitrary, and in fact this is one area where the Bayesian approach often comes in for criticism. However it's worth emphasising that, regardless of whether you adopt a Bayesian or a Frequentist paradigm, **[\"you can't do inference without making assumptions\"](http://www.inference.phy.cam.ac.uk/itila/ \"Quote from page 51 of David MacKay's book\")**. Clearly you should only choose a conjugate prior if it also makes sense based on what you know about your parameters, but if you have a choice between two reasonable alternatives, one of which is conjugate, you might as well makes things as straightforward as possible.\n", "\n", "Once we have the posterior distribution, we can calculate marginal distributions for each parameter by integrating. Recall from notebook 1 that for a two-dimensional joint distribution, $P(x,y)$, the marginal distribution for $x$ was obtained by \"integrating out\" $y$:\n", "\n", "$$P(x) = \\int_y P(x,y) dy$$\n", "\n", "If our model has $n$ parameters, $\\{\\theta_1, .., \\theta_n\\}$, the marginal distribution for each parameter is found by \"integrating out\" all the others.\n", "\n", "## 1.3. Practical difficulties\n", "\n", "We're not going to delve into the details of conjugate priors any further because for the purposes of calibrating complex models we won't find them very useful. This is because in most cases the likelihood function will not be a simple distribution for which an obvious conjugate prior exists. More often than not, the likelihood function will be something awkward that makes the posterior density impossible to integrate analytically.\n", "\n", "In general, when working with conceptual environmental models, we will often find ourselves in a situation where we can evaluate the posterior density for any point in the parameter space, but we ***can't write down an equation to represent it***. If this sounds odd, consider the example of the \"black box\" model from notebook 2. This model has just two parameters, $\\alpha$ and $\\beta$, and we assumed a simple, independent and identically distributed (iid) Gaussian error model to construct the likelihood function. However, it is important to realise that the likelihood function itself is **not** a simple Gaussian: the likelihood function is the combination of the deterministic (\"black box\") environmental model, *plus* the (stochastic) Gaussian error structure. We don't know what calculations are performed inside the black box, so we can't write down an algebraic equation for our likelihood function. Even if we could look at the equations inside the box, we'd probably find a set of ordinary differential equations, and we still wouldn't be able to write down a neat equation for our likelihood. What we *can* do is evaluate the posterior density for any point in the parameter space, because we can draw a set of parameters from our prior distribution and then run the model with those parameters. Once we've run the model, we can evaluate the likelihood, then we can multiply the probability of the parameters by the likelihood to get something that is *proportional to* the posterior density.\n", "\n", "This distinction between what we can and can't do is very important. We're trying to map out the posterior density, and ultimately we want to integrate it to calculate marginal distributions for our model parameters. Unfortunately, we **can't** write down the posterior density in a form that we can visualise or integrate analytically, but we **can** evaluate the posterior density at any point within the model's parameter space.\n", "\n", "Going from being able to sample the posterior density at a discrete number of points to being able to describe it in its entirety will turn out to be pretty challenging. The first step towards achieving this is to introduce some basic **numerical methods**.\n", "\n", "# 2. Numerical methods\n", "\n", "This section will begin by taking a detour away from model calibration to look at the more general problem of how we can integrate functions that have no analytical solution. Once we've introduced some of the basic methods, we'll return to Bayiesin model calibration to see how they can be applied.\n", "\n", "In one dimension, the general problem (as noted above) is that we have *something*, $P(x)$, that is proportional to a posterior density and that we can evaluate at any point $x$, but which we cannot integrate **analytically**. Our aims are to:\n", "\n", "1. Come up with a method for estimating the shape of the distribution $P(x)$ (because even though we can evaluate it, we don't know what it looks like), and

\n", "\n", "2. Find **numerical approximations** to integrals over $P(x)$ that are accurate enough for our purposes and which can be calculated within reasonable time frames.\n", "\n", "## 2.1. Riemann sum\n", "\n", "The [Riemann sum](https://en.wikipedia.org/wiki/Riemann_sum) is one of the simplest ways of estimating definite integrals numerically. Suppose we want to integrate a one-dimensional function, $P(x)$, between the limits $a$ and $b$. First, we divide the x-axis between these limits using $n$ equally spaced points, $x_i = \\{x_1, .., x_n\\}$ with a separation of $\\Delta x$. The function is then evaluated at each of these points and the area beneath the curve is approximated by $n$ tall, thin rectangles, each with width $\\Delta x$ and height $P(x_i)$. The area, $I$, beneath the curve can then be approximated by summing the areas of these rectangles\n", "\n", "$$I \\approx \\sum_{i=1}^{n} P(x_i) \\Delta x$$\n", "\n", "As $\\Delta x$ becomes small and the number of rectangles increases, this sum tends towards the value of the true integral\n", "\n", "$$I = \\int_{a}^{b} P(x) dx$$\n", "\n", "To get a feel for how this works in practice, lets consider the following integral as an example\n", "\n", "$$I = \\int_{-4}^{4} (\\sin^2 x + 0.3)e^{-0.5x^2} dx$$\n", "\n", "This equation actually has an analytical solution, which you can either calculate by hand or by cheating with e.g. [Wolfram Alpha](http://www.wolframalpha.com/widgets/view.jsp?id=8ab70731b1553f17c11a3bbc87e0b605). Either way, the exact result is **$I = 1.836$** (to three decimal places). Let's have a go at approximating this using a Riemann sum." ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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+uDIRiduNTUUJBRvoaff+gi23F7rHDPk6eaCKcRhjcMYjtLU0sr3b+3fFAytG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\n2z27qxyJiHc1WTnexjQLBHjV11/tcMQ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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Estimated integral: 1.84 (-0.03% error)\n" ] } ], "source": [ "def f(x):\n", " \"\"\" This is the function we want to integrate.\n", " \"\"\"\n", " return (np.sin(x)**2 + 0.3)*np.exp(-0.5*x**2)\n", "\n", "def riemann_sum(func, a, b, n):\n", " \"\"\" Approximates an integral using a Riemann sum.\n", " \n", " func is the function to be integrated\n", " a and b are the lower and upper integration limits respectively\n", " n is the number of rectangles to use in the approximation\n", " \"\"\"\n", " # Values for plotting \"exact\" function\n", " x = np.linspace(a, b, 1000)\n", " f_x = func(x)\n", " \n", " # Calculate delta x\n", " Dx = (float(b) - float(a))/float(n)\n", " \n", " # Get a list of x_i\n", " x_i = np.arange(a, b, Dx)\n", " \n", " # Evaluate f(x) at x_i\n", " f_x_i = func(x_i)\n", " \n", " # Evaluate integral\n", " i_est = (f_x_i*Dx).sum()\n", " \n", " # Plot\n", " plt.plot(x, f_x)\n", " plt.bar(x_i, f_x_i, width=Dx, alpha=0.5, facecolor='red')\n", " plt.xlabel('$x$', fontsize=20)\n", " plt.ylabel('$f(x)$', fontsize=20)\n", " plt.show()\n", " \n", " return i_est\n", "\n", "i_true = 1.836 # Analytical solution\n", "i_est = riemann_sum(f, -4, 4, 50)\n", "pct_error = 100*(i_est - i_true)/i_true\n", "\n", "print 'Estimated integral: %.2f (%.2f%% error)' % (i_est, pct_error)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For a first attempt, the Riemann sum does a pretty good job of addressing the two aims listed above: \n", "\n", "1. The rectangles make a kind of \"histogram\", which gives us a pretty good feel for the shape of the function, and

\n", "\n", "2. By summing the rectangle areas we can approximate the integral fairly accurately.\n", "\n", "Unfortunately this approach does not generalise well for multi-dimensional problems. The one-diminsional example above uses a simple function, which we've integrated over a narrow range (between -4 and 4). In practice, our posterior densities may be much more complex, and it is not uncommon for parameter ranges in environmental models to span several orders of magnitude (i.e. the integration interval is much wider). Even if we could get away with approximating the integral using just 50 slices *per dimension*, for a model with ten dimensions the number of function evaluations required increases from 50 to $50^{10} \\approx 9.8.10^{16}$. How long might this take to run? In the cell below I've streamlined the `riemann_sum` function by removing the code used for plotting. Let's use `%timeit` to see how long the code takes to execute." ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "100000 loops, best of 3: 17.8 µs per loop\n" ] } ], "source": [ "def riemann_sum(func, a, b, n):\n", " \"\"\" Approximates an integral using a Riemann sum.\n", " \n", " func is the function to be integrated\n", " a and b are the lower and upper integration limits respectively\n", " n is the number of rectangles to use in the approximation\n", " \"\"\" \n", " # Calculate delta x\n", " Dx = (float(b) - float(a))/float(n)\n", " \n", " # Get a list of x_i\n", " x_i = np.arange(a, b, Dx)\n", " \n", " # Evaluate f(x) at x_i\n", " f_x_i = func(x_i)\n", " \n", " # Evaluate integral\n", " i_est = (f_x_i*Dx).sum()\n", " \n", " return i_est\n", "\n", "%timeit riemann_sum(f, -4, 4, 50)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Without worrying too much about the details of exactly what's being timed, or about the possibilities for optimisation etc., on my machine it takes about $20 \\; \\mu s$ to estimate the one-dimensional integral. If we scale this up to 10 dimensions we might therefore expect a runtime somewhere in the region of ***1200 years***! Hopefully this is enough to convince you that the Riemann sum approach is not very useful in practice for complex environmental models.\n", "\n", "## 2.2. Monte Carlo (MC)\n", "\n", "Monte Carlo (MC) is an umbrella term for a whole range of different methods, all of which involve random sampling. One big advantage of MC methods is that their accuracy is **independent of the dimensionality of the space to be sampled**, which represents a big improvement over the Riemann sum approach. MC methods are a huge subject in themselves and we're barely going to scratch the surface here.\n", "\n", "### 2.2.1. Integration and expected values\n", "\n", "MC methods make use of a handy link between sampling theory and integration. If we draw $n$ independent random samples, $x_i$, from a continuous random variable with distribution $P(x)$, then the **[Law of Large Numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers)** says that the average of the sample values converges to the **[expected value](https://en.wikipedia.org/wiki/Expected_value)** of the distribution as $n$ tends towards infinity.\n", "\n", "For continuous random variables, the expected value, $E[x]$ can also be expressed as an integral\n", "\n", "$$E[x] = \\int_{-\\infty}^{\\infty} x P(x) dx$$\n", "\n", "Furthermore, the expected value of any function, $g(x)$, given that $x$ has probability density function $P(x)$, can be written as\n", "\n", "$$E[g(x)] = \\int_{-\\infty}^{\\infty} g(x) P(x) dx$$\n", "\n", "This means there is a useful link between averages of sampled values and integrals. Suppose we have some function $f(x)$ which we want to integrate\n", "\n", "$$I = \\int_{-\\infty}^{\\infty} f(x) dx$$\n", "\n", "we can re-write this by multiplying the integrand by some other function, $\\frac{P(x)}{P(x)}$, where $P(x)$ is any probability density distribution. This is one of those funny mathematical tricks where you multiply something by one, but in a way that turns out to be useful\n", "\n", "$$I = \\int_{-\\infty}^{\\infty} \\frac{f(x)}{P(x)} P(x) dx$$\n", "\n", "By comparing this equation to the one above, you should be able to see that, in this form, $I$ is the expected value of $\\frac{f(x)}{P(x)}$ with respect to a random variable distributed according to $P(x)$\n", "\n", "$$I = E\\left[\\frac{f(x)}{P(x)}\\right]$$\n", "\n", "We can estimate $E\\left[\\frac{f(x)}{P(x)}\\right]$ by drawing samples, $x_i$, from $P(x)$, then calculating $\\frac{f(x_i)}{P(x_i)}$ and averaging the results. This means that as long as we can evaluate $f(x)$ for any $x$, and as long as we choose $P(x)$ to be something we can draw samples from easily, we can estimate the integral using a simple random sampling strategy.\n", "\n", "### 2.2.2. Importance sampling\n", "\n", "The first sampling strategy we will consider is called **importance sampling**. Note that although this approach provides a method for evaluating integrals (which is one of the aims listed at the start of section 2), it does not allow us to say much about the shape of the posterior distribution (which was our other main motivation for adopting numerical methods). **Rejection sampling** (see below) is a related technique that can do both.\n", "\n", "Importance sampling is a direct extension of the discussion above concerning expected values - all we need to do is choose a probability density function, $P(x)$, that we can sample from. But what should we choose? Let's consider the same example integral we used above to illustrate the Riemann sum\n", "\n", "$$I = \\int_{-4}^{4} (\\sin^2 x + 0.3)e^{-0.5x^2} dx$$\n", "\n", "where we already know the true value is **$I = 1.836$**. We'll try two different sampling distributions for $P(x)$: a **uniform distribution** between -4 and 4, and a **Gaussian distribution** with mean 0 and standard deviation 1.\n", "\n", "These three functions ($f(x)$ and the two possibilities for $P(x)$) are plotted below." ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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gVY5nWPRmqHf9668/WeWYzmLZsj9JTU2lXbsOFlvkdrtu3Qyfi1Wr/pIRFGE3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\nAY4PZ5q24pvD4aBr164ALFgw15QMwnhSyIVhCofVu3btbtq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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import scipy.stats as stats\n", "\n", "# Range of x values for plotting\n", "x = np.arange(-4, 4, 0.1)\n", "\n", "# We'll try two differing \"sampler densities\": a uniform distribution and a Gaussian\n", "norm = stats.norm(loc=0, scale=1) # mu=0 and sd=1\n", "unif = stats.uniform(loc=-4, scale=8) # min=-4, max=4\n", "\n", "# Get densities to plot\n", "sampler_norm = norm.pdf(x)\n", "sampler_unif = unif.pdf(x)\n", "f_x = f(x) # f is defined above in Riemann sum example\n", "\n", "# Plot\n", "plt.plot(x, f_x, 'k-', lw=2, label='Target distribution')\n", "plt.plot(x, sampler_norm, 'r-', label='Gaussian sampler density')\n", "plt.plot(x, sampler_unif, 'b-', label='Uniform sampler density')\n", "plt.xlabel('$x$')\n", "plt.ylabel('Probability density')\n", "plt.legend(loc='best')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Looking at the image above, it is clear that when we draw samples from the uniform distribution, values towards the margins of the plot will be over-represented compared to the target distribution (because $P(x) > f(x)$ in these regions), whereas towards the middle of the range values will be under-represented. To account for this we calculate weights, $\\frac{f(x)}{P(x)}$, which we use to adjust the **importance** of each point in our sample.\n", "\n", "If our sampling distribution is very different from the target distribution (as is the case for the uniform distribution above), then the algorithm will draw many points from *unimportant* regions of the parameter space, and for this reason it will take longer to achieve an accurate estimate of the integral. Based on this, we might expect the Gaussian sampler density to perform better than the uniform one, because the Gaussian assigns greater *importance* to roughly the right parts of the parameter space. Let's see if this is the case." ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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zarwYLCZAFfQ25UDMNHqtUI4Q9AKBQCDoKsTNdC/LcjWA3W5PQRX6dwds7gE4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\noub5V4gOHIxn7rs8+eftrJxxCXfdFf8+jhnj54MXNjLqpPH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NvyOWl4v5D9kWZtOnR5g/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\n0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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Importance sampling\n", "# Try a variety of different sample sizes\n", "samp_sizes = range(10, 2000, 20)\n", "\n", "# Lists to store results\n", "unif_est = []\n", "norm_est = []\n", "\n", "# Loop over sample sizes\n", "for item in samp_sizes:\n", " # Draw the desired number of samples, x_i, from the sampler densities\n", " uni_samp = unif.rvs(item)\n", " norm_samp = norm.rvs(item)\n", "\n", " # Evaluate P(x_i) and f(x_i) for uniform sampler density \n", " p_i_uni = unif.pdf(uni_samp)\n", " f_i_uni = f(uni_samp)\n", " \n", " # Estimate expected value from uniform density by averaging\n", " exptd_uni = (f_i_uni/p_i_uni).mean()\n", " unif_est.append(exptd_uni)\n", "\n", " # Repeat for Gaussian sampler density\n", " p_i_norm = norm.pdf(norm_samp)\n", " f_i_norm = f(norm_samp)\n", " exptd_norm = (f_i_norm/p_i_norm).mean()\n", " norm_est.append(exptd_norm)\n", "\n", "# Plot results based on different sample sizes and sampler densities\n", "plt.plot(samp_sizes, norm_est, 'r-', lw=1, label='Gaussian sampler density')\n", "plt.plot(samp_sizes, unif_est, 'b-', lw=1, label='Uniform sampler density')\n", "plt.axhline(i_true, lw=2, color='k', label='True (analytical) solution')\n", "plt.xlabel('Sample size')\n", "plt.ylabel('Estimated integral')\n", "plt.legend(loc='best')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First of all, it's nice to see that both sampler densities give results that are pretty close to the true value, and we can also see that as the number of samples increases the errors generally become smaller. Secondly note that, as expected, the Gaussian sampler density (red line) converges towards the true value more quickly than the uniform sampler density (blue line). Although the difference might not seem very pronounced in this one-dimensional example, the effect becomes much more exaggerated in higher dimensional spaces. This is because high-dimensional posteriors typically have a large proportion of their probability density located within a small region of the total parameter space. As the number of parameters increases, the amount of \"empty space\" with low probability density increases exponentially, so uniform sampling strategies spend ever-increasing fractions of their time sampling from regions with *low importance*. As a result, these strategies become highly inefficient - so much so that the only way to sample effectively is to choose a sampler density that is already very similar to your posterior density. Of course, we often can't do this in practice, because we don't know what the posterior distribution looks like. This is one of the reasons why, like the Riemann sum, importance sampling is generally not very helpful for complex modelling applications.\n", "\n", "### 2.2.3. Rejection sampling\n", "\n", "Rejection sampling is closely related to importance sampling, but with the advantage that it allows us to draw samples from the target distribution (and therefore approximate it's shape) as well as calculating integrals. \n", "\n", "For illustration purposes we'll continue to work with the same example integral as above\n", "\n", "$$I = \\int_{-4}^{4} (\\sin^2 x + 0.3)e^{-0.5x^2} dx$$\n", "\n", "#### Uniform rejection sampling\n", "\n", "To perform uniform rejection sampling, we first of all need to define a point that we know is **above** the maximum value of the function we want to integrate. We will call this value $F_{max}$. Working out $F_{max}$ can sometimes be tricky, but if necessary we could use numerical optimisation routines (like the maximum likelihood example in notebook 2), or we could make an educated guess. For the examples here, we'll assume we know that $f(x)$ never gets above 1 (in fact, based on looking at the plot of the function shown above, we know that it only gets a little above 0.6, although of course in a real application we wouldn't know this).\n", "\n", "As with importance sampling, we start off by choosing a **sampler distribution**, $P(x)$, which can be anything as long as we can draw from it easily. For uniform rejection sampling we choose a **uniform distribution** over the integration interval, $a$ to $b$, and draw $n$ independent samples, $x_i = \\{x_1, .., x_n\\}$, from it\n", "\n", "$$x_i \\sim U(a, b)$$\n", "\n", "We also draw $n$ independent samples, $y_i = \\{y_1, .., y_n\\}$, from a second uniform distribution which ranges from 0 to $F_{max}$\n", "\n", "$$y_i \\sim U(0, F_{max})$$\n", "\n", "Hopefully you can see that points with co-ordinates $(x_i, y_i)$ represent uniform random samples drawn from a rectangle with lower-left and upper-right corner co-ordinates of $(a, 0)$ and $(b, F_{max})$ respectively. For our example integral, we have\n", "\n", "$$x_i \\sim U(-4, 4) \\qquad and \\qquad y_i \\sim U(0, 1)$$\n", "\n", "Next, we evaluate $f(x_i)$. For each of the $x_i$, if $y_i \\leq f(x_i)$, the value $x_i$ is **accepted** as a sample from $f(x)$, otherwise it is **rejected**. This is much easier to illustrate with code than it is to describe in words." ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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6MYUZWbM0cfIPTCA+JpyVWfFU1nVSWd+Jki4L0gsRLKRVEQBGo5HkHTdzorcH\nvV6PkpxCb2uL27lw05FduJ7qrduorq4CID5vBXmf+dyEQ8Zcy5aXmsZEg0KMRiO5GzeTVbieWpf9\n+XOe0mT7CnTyDddhfttNkRw0m9kWFeW2vM7Cw8O54o7Pj/8/9667efuZpyhsbSHeiwQy3pbJdR+B\nqDPNbqfD6d9Wq3U8YNM0jTd/+XO2LVuO0RhCeWUFbPB+yQXX85nO0Et/z7GbLDB3952crfUzPZVX\nktkIMb8dr+9Es9spUJIIMU7/QezqnARONvZw4nQXWUnJfiihEMIf5Bc+QLIL11N7tIzV7e10lpfS\nClzd0EClH+fCGY1G8r58z3ijLs8l6PPUqHSULb+9nVPNTdTGxLAlf/Wkx/LUc+iPeUoT7SvYesAA\njHo9yTtuptpguKi8k37WaCRj+QriGhtnrCfU2zpzt+zFVDWUl472oI0th2Lr6WH/s89w7VjA09va\nwic7Oqg5f47c9Ayfgh9352NYs27KZfXl3vUmcPQmsPM1CdJMBWzB+H0SQswus8XKycZeok0hLEnz\nz9ItRoOe/Kx4itU2apr6uUZyWAkRFIJ3zN0lztHYfCszk+7UNFYWFBE2A3PhpjJPyGg0knXnXbxj\nNpMAXG+KpH7vninNWfLnPCVP+5psfpe3866mMz/L3Xy17ML1Uz53d/vzdu6cN/vwZj6nY9mLT/zP\nn/jE//yJQ1+9C4tl4nWlAsXd+eiA4thY1MazqI1nKY6N9akOvb13necPVl97nd8Cp4munyNgy9v/\nhsd5ilO9n6c611cIcemoPtOFZrezckk8+ikkjfFkWeYCwkMNnGweYMASnPPYhZhv5FFvADl6gJY2\nNk46t2y2NVdWsC0qCkNMDPBxT4a3axDOhKn0hHjby+HYbnV7Oz2tLRza9zxFD+waXwB+suP6O1uj\nN/ubaM0+r/ehadS3tqBpGlaXjLauy158sqOD15595oIhrb5w14O2fuftlI8NE41KTuEv3V1sS04Z\nD36yxhLAOD7vS53aGU1ckDD2/14/NmhcTdarN5VhpxNdP2+GADvue6um8ea+51i04xayprkGqBDi\n0jc8YqPmbDfhoQZyMmL8um+jQU9+djzFJ9p4u/wc/3BdwuQfEkLMKGkVBFig1//yJcCy+bAG4UyU\n09OxJ6pDb+ddNZSXjgaCR8tIHRzkerudl3c/zJYHHvGYhMRd3flzPtdE+3NXH0nf/7bX+8hcW0Dx\nkY8IP3goDCmHAAAgAElEQVSAtYODtEZE0FtWgnUGgwVPwY3za5u+9T1qKysA3xLAuLsHjEBBd/f4\nsNS47u5Zm3PnaqoPCxz3d0N5KQ1jD2O8+ZzjvrcDNUfL2GE209bbe8EaoJ4E+m+SECKwas52M2LV\nWLUsEYPB/w+qly1awLFTHbx77Dw7rsglKiLE78cQQnhPgsEgYFyzjv21Kim5Crmz+OTe1wDLAAFL\naDFRUOdtQ1vTNLo8LFoPH6+XqNPp0AG5vb0ccZrT5nzcqa6h6C+O+nBeD7Lm0KHx7KuTMRqNhBcU\nkVFdRadeT1JqGokuwdL6nbfzl/1v8MmO0ZQvf0lIYNPO26dVbk9zS51fc/y7rviw1/ebpwQswWQq\nDws8fUe9DdjqW1tYMziIXqcbH/JZWXIEw9hcVnffA1mTUIj5y2bTqDrdRYhRz7JFC2bkGEaDHmVh\nFBX1vRwsa2LHliUzchwhhHfkFz6AnBt6+UC52QyF6/2yX296+5x7D+pbW4hqbqKu5AjLNm6eE41r\nZxP1gJWXFk+aqCdzbQGH9j3P9XY7OuBoRATLUtM47eF4gcz06KBp2nhPps1up+K554j73kqvG+4G\ng4G49Izxc7BpF6795LzshWazkZyr0FxZ4fdkJf4adux8D1itVqw2Gwf6+9lqMqHX633q4QqWbJoT\n3WcTBWyO+z6quQm73U6ryUR8aho2TePsy/u4IWo0IYSnhxiOupxsKLIQ4tLScL4fy7CNFUviCA0x\nzNhxslJNqI39HCxt5IaNmRhnoAdSCOEd+fYF0EwkavAmscQF22sa1WUlKHWnWF5fx/mXX/S4OLe3\nSU1mYqHzqSZU8TZRj9FopOiBXby8Zi3VS7JYtmYdlYmJo3PappnIxcGf9ZK5toCDZjNJZjMaUGEy\ncV1kpE/3jzd1Gh4ezubb7yB5aIh1777t10XVJ7tXvV1E3tN+Vx08wFUmE6+bB6jcus3rnltfv0OB\nMlGCG8d9P/iZO3l9zTpi16zDDhw0m9k+tv6l698c1/tzrtSDEMJ/Tjb2AJC7cGZ6BR1CjHo2LE+g\nu3+Y0pq2GT2WEGJi0jN4ifGlxypzbQGvv/C/XHfuHEM6Hd1JSWw1mVB9GIrnKRGLv4dPTmfomreJ\nesLDw7ly12M0lJdSi/s5bVNdQ9Hf9eJuPUijj0mIvK3TiXqQveWup22ye9WR1Xb/2NIW627Z6dUc\nQteHLDdERVNtMHhd18HQ6+swnfl7RqORZRs3k124HnWs7pNtNowHD1y0raelOVzroa64mLiclf44\nNSFEkOk3j9DSYSY5LoLYqNAZP97l+cn87VgbB0oa2ZCXMuPHE0K4J8FgAE3W0PPnGm+e2O0wCBgY\nzb7oia9DT901pKc79G46CVq8bVR7M6fN8ZovwalzvVg1jajKY3z4zFNsvv0Oj8loJuNYD9JxThUJ\nCWSuLfBpX97WqVXTqDlaxprBQex2O6+//CLZXs5v9RQIe/O5+r17uHbscwf+ZTdXmUwYxo4ZyCBt\ntkzlIchEiY2sVivlTveM43vg7nu7v1Zl1UydmBAi6JxsGu0VXJoROyvHS4kLJz8rnsr6ThrO9ZGZ\n4p/1DIUQvpFgMICcG3o2mw0DUF9yBDug2Ww0PflLPtXVBcBf9r/Bpif3TBoQpuev5tV9z5Hb28uS\n1DQqExM99iQ0lJfyiahoalJSRhv5g4O8bjazyWX7iZZd8FagF7KeiaQYvgSnVpuNk81NWDU7zbUq\nRUNDxNnt1JrNZN15l9dZMz2dk81mY8ECE3UlRxgpK6Ggu9unfV1UXqeAIj1/NW/ue44dZjN6nY5W\nk2nCHmRXnh4QTHavun6uoLeH0709KGPZQT2ZbjbM2cqm6W3Q7tN9Nsn3zJfvQWquQrnZfEE9bCkq\noqtr0PuTFELMCZrdzsmmHkIMehanzl5Qtq1wIZX1newvaeSuG/Nm7bhCiI9JMBhgjiGHjmDLkeQk\n2mBgSW0tpKdj1Om8WuPN0ZNyvSmSnt5e3jGbKbrzrol7hvR68tYVoo6tNZe842a3w+7cLbtw5a7H\nLtrWU0M6GIbe+XvpB29ZrVZGykpIaWlm8EQ16YNmwhbEYT3Xyuq0dN7ykLHUm7I63z/K0ABHa+tI\na21BV1CEfmxOmK/17C6gSL9xB229vRj0euJT0ybsRfbqGDabz/dqbGoah8wDLB1LdDNR7+50Av/Z\nyKY51YcjkwWQ3nzP3H0P3H5vC9dD4XrJKirEPNDSbsZssZK7MJYQ4+ylk1iVk0DygggOHT/H31+d\nQ7Rp5oenCiEuJL/sQcDRgOs9f44Mi4U04F1N4yrrCCf7+1gW7d2ir+MNQaORxIWL2KZpVFdWeAwE\nnBuA2alpHzcA3XC37EKDmyBD0tJfrKG8lILubnrSMzA2nCHLZuNkVDTGtjbOVFZgy8jweZ/OQYHV\nZhu97tER6PV6UgcH6WxtIS7d9/06yusIKDRNY3HlMQ6HhWFbsZKC7m7s+NZb5mkNwMnuVdfPVSQm\nsuXO71E9tg7hRPfWdAP/mX5wMJWHIzPZuz7R9/ZSHoYrhBg1PkR04ewMEXXQ63RcvS6DPx88yaHj\n59i+fuKRH0II/5vfrfRZ5sjWB56HhVntdk719zOUkkpVeDgGzY5V03gpLo6M7KXUFR/2W6p7bwM3\nx7IL2zWNnoEBSsPC2JicQv0E+/Wq52Gepaq3AyERJuoGBjjb0c52gwFCQ7H1D1AcG0tRz+iPsa/J\naA7092M1mQDISk2jvLmJDKcMnFOtZ03T6CwrIclsJgfozltB5dZtGAwGnxP4TGWZEk/353wNTrwJ\nIKebcGa+1q0Q89mwVePsuX4WRIWSGOv/3AST2ZKfynPvnOJvFc1cW7QQnU4362UQYj6TYHCWWK1W\njv3qV+Q1NAMXJtGw2Wy82t/HVfEJfPDRIXJHhrkmMpK/5CrYt15DjcFAcv8Aa959+4LPejtEcyLe\nNACNRiNrf/AAez+zk8vsdq4yGnn9ZA2b8n/g9fnP5x7DzLUFlBYfZlHJEUz9fdQAuXY7tSmpZF61\nlcy+Xiq3bqN6bCFwX5LRAGwdWz5hZ3QEOsCydRtNBUW0jgVtwKQPIVzLW15azOLKYySZzVSYTOSl\nZ2Dr6uLtWpWM5Ssm/Ly7oYyu99l0Evr4IljWC3Q1Uw9H5vP3TAgxNS0dFjS7nay0mIAEYjGRoaxd\nmkhJTRunW/vISvNuNNRco2ka/f19U/58aKhGb+/UPh8VFY3ex4zjYv6QVsIsaSgvpaijA4vTU/3y\nw4fo+OsrFPT2sDw5hZfOnObygkL0RiPdOh3XJ6egrl4LQN7+NyYdUjaTDcHzJ6r4x41b6D9/jl5g\ne3IK6gRDUN2Zrz0PRqORkHWF9FQdpzctncT2NhY2N2NenofRaMSmaRgMhinXjV6vZ9GOWziVGENX\n1wB5Ttd9KkMLHffRh888xRIgb2y46YnyUgpbW4hrbPS4H9fjFR/5iPCCIgwGwwXB2FTvVV+Cu0An\nLZrIVM5/tgJoIcT8crZ9NCnUbCaOcXX56jRKatp4r6Llkg0G+/v7ePOjk0SYIqf0+ajITvoHhnz+\n3KB5gO0blxITM7tDgMXcEfhW0TylaRrH/vN3fKG1FYNOR+u5Vq5OTqFTp0PJWAiAbSxRhi9msiGo\n1+vH56FNpWzBaCZ6jtz2jBkMLE3PGF1aIj2D41YrGXb7+FDOrLHF1L0ph6dkH2lpcbS1XfjUcKqJ\ne4xGI5tvv4NasxldZycnm5tIBeLTMyZMTOO6hEb4wQNkVFcRl54x7WDM1+AuGJIWTcTX76r0+gkh\n/M08ZOVc1xBx0WHERAYueUt+djwLokI5VHWOT1+zlNAQQ8DKMpMiTJGYIqcWdEdGhaNh8XOJhJBg\ncNZkri2goqaSnLFhogfNZjaERYwmZNHpSB0cpBmojYlxmy0x0PPtHAFIfns7p1tbqI2JYUv+6lkt\ng7/NRM+Rp306B3CuQzmz8lf7tLTEbAUFzsdpOlHFNQ0NPg0zqW9tYe3gIJ1ji7+7rjnpa90He3A3\nG6TXTwjhT5X1PdjtsCSAvYIABr2ey1al8ZcPz1Ba08amlakBLY8Q84kMIJ4lRqORVffeS/W111F9\n7XUk77iZnPR0jkZEYLPbsdntlMcuYMsDj4xv42gcjzfKXV6f7fJn3XkX75jNJADXmyKp37sHq9Xq\n92M5Eu3UHDo0I/t3cA4uHMGKN4lNprJP12uY9+V7yN24meyiDTRXVowHib2tLSyuPEZ9yZEJj+MI\nCrKLNkzeixgfj80poUymmwcJjjqvKz58QZ07jrP59juoSEycdD/Ox9M0jdaICGJT07yuJ29ZNY2T\nzU00najyeI94e+5CCDFflZ8cfSAXyCGiDpevHv2t+FtFS4BLIsT8Ij2Ds8j5qb7VaqXyaBn5a9ZR\n7ehpG1vI3d2T/2DoEWiurGBbVBSGmNHx/DPRM2OxWCjevYuC3h7Sc5bwYcQ7sx78Wq1W6kuO0Fqr\nkpKrkF243uv135pOVKFo2njvlbOJrqGmaePrONrsdt5/8XmAi+ba+cqbXkRveul86Y00rlnH/lqV\nhCuvpreinMTubq8zm040bNe5d/pEeSmpwDUNDVTs+a3be0SGVQohhGcDlhHUxj4WRIYEdIioQ0qc\nCWXRAqrPdNHeM0hibESgiyTEvCAtowBxNFRrxxqqV14CDdXpzr+zWq18sPthdhwtx6DT0d7Vzuq8\nVagzNBTQ3dy7rPzVqL9/ksy39rN2cJDyiAiqt24j78v3eDwf52AqV9N4s+YE25ctR6/XexUAZa4t\n4OC+57nebEbT6SgNDyeztoaMp/cSm5rGwX3PE3/jTYQYDD4Fh95eD2+HX072QMK5HvKBcrOZ7M9/\nye26gJ7qfqKg1PGdefuZpyhsbZl0/qI3ZQ60YM12KoS49JXVtGPT7CxMmv3lJDzZkp+KerabQ8fP\ncdOWJYEujhDzgrQ8AijYG6quJspm6I/5dw3lpeT29o7Po0wZHORsaws2m82npRG85Wn9u6yq46Ra\nLOj0etZZLFRXV9EwQUDqOuRx27LlvJWZScbyFV71RhmNRpJ33MyJ3h70ej0mTWN5fR3tOh09R8vY\nPjDAa0c+oiA2lvi1BVR4UbfeXg+r1UrTiSpSm5vGg6upchtUesg466nuJwtKjUYjGctXENfYOOfT\nZAdztlMhxKXvyInzACxMDJ4euEIlmaferOHD4618cvNiWXNQiFkwt1tTYlZNNHexvuQIiyuP0dva\ngg6mPP9uSWraBfMoi6OisZQWk7f/DfL2v0Htnt/6dR6ht3PvfNqnXk/G8hU+7TO7cD39+avIHptf\n1xoxumZg6uAgPQMDLBseJsNiof/8Oa/q1ps5eY5g5OqGBppaW2grLWbEag36uXXBOhfQ07xLT6Y6\nb9LX4wghhKvBISvVZzpJT4ggKiJ4HkCZwo2sXZpIS4eZ061TX5NPCOG94PkLILwS6GFl7nozrVYr\nZ1/ex9r6utFlMlqaiV2zzud9Z64toLK0eHweZVNqEklXX8fad99221s0E3WRubYAtfgwttYWUseG\niVryVpA3QbAx2fpv3pTTuafMarPRW1ZCVtVxbHY7pWFhXB0SMmG5rVYrNYcO0dU14HVgNB6MGI2s\nLCjiVHMT5ZmZbL79jgvK6G09+7KQurtesaw77/J6HT1/zgX0x300W7180psohPCHqtOdWG128rMW\nBLooF9mcn8qRE+f5sLL1kl1zUIhgIi2IAJhq4zNYG4IN5aVsN0VSYTKxZnCQJLOZV/v7SR0b3unt\nObrOo7xh+1WUvPmO222nWheT1b3RaET50lepX1dI+VgCmbzC9QAeh6pOFJz4Us4LEgwVrqe+5AiH\nXt7H1vAIjlWUj67zl5ziNtis3fNbLhsaYGBgyKfAavzYej1L0zMYWb7iokDQl/J7G6R5GlLqdZIa\nPw2x9nR+jjKCd9/RqSx74Sl4nuge9XSczLFhtt6W11eOMnXFRRK9WAn43xwhxPSUn2wHIH9xLA3n\newNcmgvlZ8UTbQrho+pz/J9rlmI0yCA2IWaS/KLPsukEdMG8zppRrydvXSFqawsjVht2YNXBA4Bv\n5+jcyDcajR4bzO7qorLkCAbD6EK17hrE3ta90Wgkd+Nmcjdu9vpznoKT6Sz6nrtxM1mF66kvL8W+\n/XpagTaD4aJg88NnnmJJ5TFGcpZQ19pCVHMTDUfLJg2svOnJ87X80w3SZnsebV1xsdv7yHq0bFYe\nuhjGMq+m5irkjj108PXvg81m8+tDItdg1LlMkZFhvB82+xl+hRD+o2l2Kk51EBsZysJkU9AFg0aD\nng15KRwoaaTqdCercxIDXaSA6DMPc6a1j7Pn+7EMj7arNM1OYmw4GUmRZCRGYQqXv8Ni+uQummXB\nHNBNVXr+ag7se56C3h6WpKbxunmA602RfjlHb3ubNE3j7Mv7uCFqdK0kdw3iqdZ9IK/ZRMGRI0gt\nrDxGXN0p3igrZtuCeHTA6y+/SHbh+gnLGMilF3wZUupvzsFOdHTYRe+31qpc66devonK4AiwVjGa\neZXC9ZPea+6OYxjbzh/3p7sHH8Y16y6a2zjX/2YJMZ/VtfTSZx7hyjVp6IM0QcvmlakcKGnkg8rW\neRcMdvUNcbjqHOe6BgHQ6SA81IDRYAC7nYZz/TSc60fHOZZlLmBdbiKhIYYAl1rMZRIMziHTaUD7\nOjTV2+2tViv1e/dwlcnE6d4eDpkHSL9xB/p33/byrCbnLiByrYuDZjPb/RSA+tNMBj2OwEGXnkFp\nzQk+YTbTYQxlODmZrSaTV0tyTNYTN1Pln+lA1NNaka7BjpqRQumCBRR0dwOj55eSq0Bj44yej6eg\nbyrHmUqiJl/Ktb9WJd9vRxBCBNrRsSGia4I4yMpKiyYl3kRZbTuDQ1Yiwi795qrNplFR10llXQd2\nO6TGm8hKjyYzOZqwUAPRUeH09VvoHRimsa2fmrM9qA3dnGnto2h5MtnpMr9STM2l/+0KMtNpXE+1\nAe3r0FRftndOQqIsXMRSTeOYwUB5fDyr29vpaW2hNCaWovzVXp2jt1zrItlmwzg2LNWTqSZ6yVxb\nQGnxYbKqjgNQv2IlygxfM1/o9Xpili3HYrPSnZLGwvzV2Ke5T+e6yLrzLrdrBU51f4669WbdwqnO\nrfW0VmRdyRGiKo9Rp9eTlZpGQXc3hzZcTvXY8GLH/VDuPEzUy++oP4a4evP3wfU4M93LmpKrUG42\ns7azczx762z14goh/K/8ZDtGg54VS+IZsvQHujhu6XQ6tqxM4YW/1VOsnueK1emBLtKMMlusHChp\npKtvCFO4kU0rUliYHOV225jIUFZExqNkxlFV30nFqQ7eq2ihq89CwbIkWY5D+CwogkFFUfTAr4HV\nwBDwJVVVTzm9/xngW4AN2KOq6pMBKagfTDc4mEqD09dhjtMdFmk0GMi88y5e3/0wucBVJhOVe/f4\nfZ7RBQlXrNZJG/DTSfSi2e10jO1Hs/sWas3UPDjnIDVG0zi4ZAlXZeViZ3oBgb8TFU1lf9OdW+tu\nrci6kiOcf/lFrq+vQ6fTcbSlmXVbNmE0GC66PjMdwHsK4Hz9++AImA1r1lEJGFzmlPqlXIXroXA9\n1eWlxMVFkisJZGaMF7+F/wB8A7ACx4CvqapqVxSlFOgZ26xOVdUvzm7JxVzR3j1IU9sAq3MSCAs1\nMGQJdIk827QylRf+Vs+h4+cu6WCwzzzMm0ca6R8cYenCWNYvTybEOHnSHINex6qcBJakRXOguJHj\n9V0MjWhsWpkStMN/RXAKll/0W4BQVVW3KIqyEfjp2GsOPwFWAANAlaIof1JVtcfNfuaEubbY/EQm\nSvByQ1Q0hphYYPrDNr3JAupNI3oqiV4ayksp6unBsHARALaenqAYggpOQapeT/Jll1GVt2baAcF0\nM1a6XqupPFyYiXma52pVrjGZaDOZSB0cZLXZzBsDAxR5WL5iJq+v436tdBrK6uuxLwqY4+PJcgqY\nHdfBarOhYzRQnKyHdaLvUXbRBpKSomlrk7W/ZpDH30JFUSKAR4F8VVUtiqL8EbhJUZQ3AVRV3Rqo\nQou54+ip0ceaa5YG7xBRh6QFEeQujOXEmS46ey3Ex4QHukh+19lrYX9xI5ZhG2uWJrA6J8Hnnr1o\nUyjXb8zkQEkjJxt7GLFqXBHE80FF8AmWYPAy4DUAVVU/UhSlyOX9CmABoAE6mPYouHnF12Fkvmw/\nG8MgfckC6s8GvHVsaYymE1Xkatp4YOJatkCt++gIUnXpGfS0thBaXc3JvDUzssyA1cuMlZ4SkMwm\nT2tFpuYq6BsbiV9XSFtrCzZNI+e227wOamfi2lqPlnFtZyc0NlI+lgHW2+O4Dnl1XYOzds9vyW9v\n50R56eiyJGsLqPCih/VSelg1B030W2gBNquq6ujLMQKDwBrApCjK62Ov/ZOqqh/NYpnFHHL0lGO+\nYEKAS+Kdzfmp1Db2cKjqHDduWhzo4vhV/+DIeCC4IS+Z5YvjpryviDAj121YxFslTZxp7SMqIoRC\nJcmPpRWXsmBZvCUGcM5tbBsbLuNwHCgBKoGXVVUNrjzIfmS1WqkrPkxd8WGsVqtf9jkesF17HdXX\nXudVY9DX7bOLNpBdtOGCeXbl8fHYNG18npG3i6G7cu4lcmQz9GfSDHflLY6NZaSshLz9b3BNQwMH\nak4wZLVecC6OBnfe/jfI2/8GtXt+O+k18/f11TSNzrISkupOEXfyJKdffB7190/6VCZX7q6dDry6\nBu6ulWPYqi/3wnTuH8daka3f/j6vffofsH/7++R9+R6yCtdTHh+PHYhJTeNM/iqWbdrkdh9Tuba+\nms59bbVaOf/yiyyvr0OpO0V1WQlWTbto3w3nz7HOYiHDYqH//LkZ+e4Iv/L4W6iqql1V1TYARVHu\nByJVVd3P6IiZn6iqej3wVeBpl99PIQAYHrGhNnSzMClyzvSyrV+ejNGg48PKVuw+TtEIZsMjNt4q\n8U8g6BBqNLC1IINoUwjH6zupb7lkm8rCz4KlZ7AXiHb6v15VVQ1AUZTVwI3AYsAMPKUoyk5VVZ/1\ntLO4OBNGY3Cm2U1Kivb4ntVq5divfkVRx+gwjoqaSlbde6/feiTS0rbN6Paukr7/beqKiwHYUlQ0\npfNISoqmKy6SyMiw8Z45m6YRHR1G16nRhC7ZU9z3ROVNtVpR3ngDQ3QEALesW8M7ixeTuWrV+LnU\nHDrEZUMD49tcNjTAqTPqhAGGP69v3ParOPDGK1xjHUYfaqQiIoJVIxZS6muJycz0qkze1MWWoiLq\niosvugZxcZEX3c/urlVSYgzZU7gXvL1/RgPs0e2c74W0m64Drpt0n+6+k67XduNgH3975VkyV62a\n1v3mXNbo6DCv6tSdmkOHuCkpjvYFMaQMDlJoHeY1rNyw/SqMRuP4dTCFhxAaakQHRESEEhkZ5vUx\nPJnOZ8WkPP4Wwvicwv8LLAVuG3u5BjgJoKpqraIoHUAa0DTRgebydZzLZYfAlb9UPc+IVaNoRep4\nGUJDNaIiO4mM8i44jPZyu8noGSYxMZrY2InrIgnYsDKVDypa6BvWyFm4YFrHDeS946jrcFMYB9+r\no7t/mNVLE1m/Ms3rfUxW/9HATZdn8+xbtXxY2UpaUjRRkWFe1fVMk+9t8AqWYPB9YAfwv4qibGJ0\nWKhDD6NDYYZUVdUURTnP6JBRj7q6zDNW0OlwN9/GeSiazWYjv6EZy1jjMKehmZI335nTQ7biclYC\n0DW2Xo4vHPUVvVjh/bB3xoceFsfGoj/4t/HlAD444L9FsB3lrSs+zMDA0IUB6MJs4nJWjp9LV9fA\nRdt0dQ14nFNVV3yYPD9f36htN3CstQ29Xs+qnCVU1NYxCAwMDHlVpok4XzvXa1AeH0/uYuWi/Xra\nrqtrcEr3gqfPOM+HGykr8elecN6n83fS+btotdlIHbu2Vk3jeGkxuadOE3fi5JTvN4vFwge7Hya3\nt5clqWlUxMfTagynqGd0+rNznU42RLWra4DUwRGiVq6heWzIa8K2G8bryXEdVkQt4O2+flLQkR4Z\ny/thkW6vm7eCdc7gJfQjPdFvIcBvGB0u+neqqjq6Se4CVgH3KoqSzmjvYstkBwrG6+iNYL0HvRXI\n8r9fNrpkTk7qx2Xo7e2jf2AIjckzyTiWNvAH88AQ7e19DA9P3olduDSRDypa+Ot7ddy+LXfKxwz0\nveOo64+quzl7vp+FSZGszon3uk69rf8QPVy2KpW3y5r56/v1XLsuweu6nimBrvvpmsvl9+b3MViC\nwReA7YqivD/2/y+MZU2LUlX1d4qi/AZ4T1GUYUafgP4hQOX0K9f5Va/297HCaa08Mcp1XmK4zUb+\nwQOTJheZzpwvb+ZNzubC6Z7OJbtwPbVjWVR1QHfeCnp1OuKc1s2b7XUBDWvWsb9WJTVXIXdsfT9/\ncv7enGxuIq21BV1BEfppLIjuWI7CsXzIKSWP4thYinp6ONXcNDrnLj1jysewWq0U797FjqPlo5lM\nz7Wyes06TmzbfsGyFu7WQnQ3P9Nx7+W3t3Ne06iNiWGL09xMo9FI1p13cWD3w2TlKgyi403LIFvu\nvMvtPM9AzXsVF/H4WwgUMxr4vQu8pSgKwM+B/wD+oCjK3xidT/8F595EIRyOn+4k1Khn2aLYQBfF\nJ6tyEogMN/JR1Tn+fmvOnG4jtXRaqD7TRWxkKFesSZ+xJC+ZKdGszIrneH0nlaf7uHrtjBxGXCKC\n4ld/7AnnPS4v1zi9/xtGn4heUlwzJm43RXLQbGZb1OjaMqULFhAylsRkvjfSnJNa1BUfnnT76S6P\n4E3wM1FGSHemGjxaLBaKd++ioLeH2NS0C5KAOJczLi6SvMWjZZiJhD7erAvoqPNVQLnZDIXrfT7O\nZMGJ8/dGr9eTOjhIZ2sLcekZU95vfckRMt/aT6pl9KmrrbWFpm9+h+rQUJpOVHFNQwP6aTRAGspL\nKYn2Vd8AACAASURBVOjtwaDTodPpWDM4SHVrCwY3y1p4k0nVEey9s3sXBUC2KZIKl+VbmisrLsjo\nq2ga1ZUVF+zH38uIiOmZ7LcQ8DT/4TMzUyJxqejqG6KpbYD87HhCgnQajSdGg54NeSkcLGui+nQX\n+dlzI/mNq56BEY6o3ej1Oq5Yk+bV8hHTsWZpAmfP9XGyeYD61n7WxMythwBi9szdxyuXIKNeT/KO\nm6m+9joqt25jxGol4um9hOz9T6p/9+9+T2AxV3mTXMQfSWfcJcZxx3q0jGsbG1l18MCEiUZ8TcwD\no431D3Y/zPVHy0g9XU/P0TJWt7dfcC6Oci7btOmCxdwnK7en4001wY0/6tzXxC1ZqWmUR0RMmmhm\nsv221qqkDg6iGwvWUgcHaas7SXbRBjbffgcViYnTToYUm5pGY1gYNb291PT2ciIqaspJlWA02NsW\nFUXiwkWEGI1Tqu/ZSM4khAi8yvrRuer5WXMzkNqcnwrAB8dbA1ySqdHsdp4+UM+wVaNISZqVBD5G\ng57Nq0br7ZmDZxixyoAB4Z4EgwHkHNQMWa282t+Hbux1OxD57tvkna4n73Q94QcPUFdyxKf9z0Rm\n0pnkTXkvWGR76zavgyr4eKkIf9aHr41pXwO1hvJScnt7LwhSelonnQ40Jf7MoGnVNE42N9F0osqn\nfbjWZ357Ox8+89QF18z5e6MDLFu30fSZOye8Fya7Tim5ymhQabdjs9spj4gY7+mdShDvKnNtAaVx\ncbQCaXY7hrAwUjMWedzWH5l4/ZnRVwgxtx2vH+39X5kVH+CSTE1OegzJCyIorWnDMhz87RlXB0oa\nqWnsIy0+DCVzeklwfJESZyInzcS5LguvfHB61o4r5hYZCxRAjkbmsZIjnH/5Ra43RaI/eIDyo2Wc\nCwvjk4ODHw8VGxzktVqVZRs3e7XvuTb8y115k77/7Ym3cVlk25nrkMzi2Fj0ZSWscsylC/L6cLYk\nNY2y1hYWt7Uxomm8NjDAapsNq9XqU/l9GX4Jvi/07jyPzbG23dWnT/POrgdI3nEz2T7OH7RqGifK\nSylsbSGusfGCa+YYGmu12QjHuwXVJ5JduJ7qrduorq4CwJK3gjynIa7TXXvPaDQSXlBERnUVgxkL\nWZyaRmZfr9v69XZ+5mTDjj3tx2KxcOTZZwBYd8vOWZv3KoQIDE2zc7y+k7joMNITTIEuzpTodDo2\nrUzhpfdPU1rTxpZ87zNwBlpX3xAvvFuHKcxAYe4CnxeVn678JTF09I7w6kcNXLkmnYTYubGsiJg9\nhl27dgW6DH5nNg/vCnQZ3ImMDMNsHr7gNb1eT09rCxsaz2I0GNDrdKSYzdRFRxNdX0f0WG/I2bAw\nWnOXoVmtRCenTDp/6XRpMauOV47Oqxrb5ymTadJ5VdNhtVo5XVpMV3OTV2X0VF7sdsJOnaS0p4vE\npcvH9+PLOen1euLWrOOUyUR7dg6G1DTWnKj2e31EJ6dQdaKKFLMZu91OeXw82TfcNK35Za77r6yq\nhLpTDA8McKy3h8tMJpZYLFSr1cStWTd+LHf3l4MjkF51vJKkulNUnai64LMAXc1NJNWdQq/ToWka\nXc1NqGFhZKzI9+p8HHX+3tkzKD09xC9dRm9FOasbz9JztoHWpsaLjunufCuOH2Og7hTH6utYaB2h\nLzqG7oEBcnV66iMjiRtL5BKdnEL7a39hdXXVRefkei/Gpqa5vU7R0RGYzcPo9XoS1hXSlZKKtmYd\nSz/5Kb8/KOhpbWFRWxummFh0Oh12u5327BzP9296xvi5uuN6j2ffcBPA+HlHxCfQWFEOfBz8WywW\nDn31Lm48cpilxyt55+0D5H3/Ac7ExIzvY6LznugeC6TIyLBHAl2GOWZXMF5HbwTrPeitQJS/vqWP\nAyWNrF+ezLplFy5EPvT/s/fm4W2U597/RyN53+JdthMntmMrThzHW8jCmpU1QIG2KVDeNqe0UOjb\nU9rTltOF9Ad9y2lpTxdaOJSTtimFtFAIpEAgO1s227Edx7bsxE68yo73RfIijX5/WBKKLMkjWbYc\nZz7X1avBo5m5n2dmpOee772MjNDYMUhAYNCkxwkKVDHqI1VubGyU1MQIgoKkOybRkUEcKGlmZNTk\nlTPor3vnz+/W0Ng+yF3XLkApIGmuneHt/IumMZakRlNe34N+eIyCrJlvRi8/t/5Dyu/j7JdFrlCS\nlyyl0WBAWXUGoyhSNtDP5pYWhLa2GVW1pFYa9FaJtB6/paaKTEvYX/epEuL1evRHA6nr6PZ6rJ4W\nnfEGT6psent8q6LUGBTMlv5IlKOjXOxoJ08QJCt3UlQ/q9KU29lJd1kpOuCGxkYqd7zg9ho43iMp\nS5YS3dxMv64NtcGAqFBMWoXTvlUEoplYoEcUaWxp4ZqRURQKBadaWzBv2DTpmFLzCpzei1IKAk1n\nG5epVp919iza22z/DBpFkQPP/ppNWeMvU6xzcPK1Xdza1YXKMme3dnWxd/drXHv/l3w7WBkZmVnD\nGWu+4GVaeMVKYnQoGSmRVF3ooWdghOgI75yqmaSyvouTNR1kpESyKjuOTyr9k/NYlBXDB6c7+aRS\nx6aVC0hNnDPteGR8gJwzOAtwltuTVrgSzVceQvfAlylfvYbNmmwCVCrJRR58kS/kSQ6Z1Nw5+7zA\n4eFh2/HXNzZyoLaGjuYm4vV6KkJDyZw//5LjTGVM05k/NZWCLVJQKpVEJ6cQER9vCy8xm830eJGT\n5w6rY3swNZVedRLLCooImqQwibN7JDkn99O5NpspDwkhTe36La79MUL+tpNFRw4Sk5xCdEIiWWNj\njA4OIgBqQEpwjat7cbqv02RMJffQcZ61Lz5P7fGjl+RS2o+7saOdW7u6GOxolwvDyMhc4VSd70EB\nZC+M9rcpU2btMjVmMxyvave3KZMyOmbipfdrERQKHrhxybS1kZCCICj43LoMzMBrh8/5zQ6Z2Yms\nDM4C3KlL1rf+QnOzz44planmkDniqB4e2P0614eGorQ4uZuylvB30cRKQSA7OcWmXvhiTNOt4E0n\ntlw8UeRUawuJZjPm1hY6FIpLlDspx3BUpYxGI/UlJ2m39AVMK1xJypKlLG5ultTLyek9UllB5rav\n0lBykmN7drMpNAwFrpUwV60iBEEgTp1E07x5RMTFE5OQyEXlpyXRU/MKKC0+YesN2LB0GZq8glnt\n9HirPtrPkSiKpB7cT2/VGRYnp9hUPymsvGcrb+9/n1u7xpWCt2NjWX3PVo/tkZGRuTwYGTNxrrWP\n1MQIwkMC/G3OlFmZncgrB+r46HQbN161YMbz7zzhnWMX6Og1sHnlAhYkhNPf3+dXe5alxbB0UTSV\nDd2cOd/NskWXZzEhGd9zeayG5xiThXs54m142VTD3owmE2dbWxAEgTR1kltVRoqNjo5DQX8f5/v7\n0Mwfr6ooCALLN2xisPwUiu5um4LnWBTD2zH5KgxQSuisJ+G1k33O6sjWlZVi3rCJ0poqMkqKyVYn\nMdTRzkJdGw0lJ0m6bbNLm6196fZbCoestDgA1X98juBDB7jJYEAXEoJ2/UYyvvSVKRcVUalUZK5a\nQ1rhSuo8cMDT1EmUtbaQIoqkqpN4t7fn01BHOwe2sawUk8nEmNFIl2Vf0Wy2zeN0FkXxd5P2Pkv4\nbb+d8mkNj7WOOzUhkbd7e9iUkHjJc6RSqVj9/A72Wu6D1fdsJThYLiYgIzNXOdvch9FknhOqIEB4\nSAAFWfGcqO7gXEs/i+fPzt55um497xy7QHREEHdck+Zvc4DxIjyfvWExP/nzSXZ/UM/ShdGz2pmW\nmTlkZ3CG8Sa3zh+qltFoZOxUCUmWhWdZawvD6zaQmZNry7+zXwh7Y2OUOolj+iEWi+O9b8piYsgs\nXAmFK21N1DMXamaVgifl+km9xp7cC/aOrEqpJL2lhb7yU6gNBkxmM8f27MZ403q3djfs3MFG67l2\n7kC5Ip951VVkDw+jFARShodRVp2h1aLsSbmWUipaTuaA2x/D1iqioAilUsnqx76LtrLCZgdgm7Oe\n1hZadG1kFBShEgRMfX025Xq6nhd/Vem1nyOTKFIWEsIyh9Bbx2fQce6sNgYHB8s5gjIyVwhVF8a/\nq7IXzQ1nEOC6FcmcqO7gg/LWWekMms1mXnpfi9Fk5gsbMgkJmj1rmIXqCPIWx1F2thNtYy9L5shL\nApmpMXvu0CsEb0Mvp7u4hSONZaUU9PaiKCiiW9dGiihyYUU+DTt3uFwIT2ajo+NQERfH2ge+S7WT\nBWt60VXEx0dw8eLAdA7TY6RcP6nX2Nt7ITWvgEO7X+dGvR5RoaAiNJRNoWHUFxcTnbFMst3767Qs\nsmwXzWZ6BgYY6OrEZDJJvt988aLCeozKkpPo6rQkWsJVHUOlYbwQkHUcZiCso4MPTldw3fLcS5Rr\nlUpFqiVktNGimvnCYfN16LRU7OfZZDIxXFqMoq9vgnrueN2k2OVvpVNGRmb6qD7fg1JQkDV/5nrb\nTTdLFkYTFxXMiZp2vrBxdjlbACeqO6g638Py9FgKNTNfuXMybl2zkLKznbx97ILsDMoAsjM465ht\nCzNriXuTKFJSf5aNU1gIu3IcZtLJnc0YRZGWmvE+d+6uvUqlImHLHdT09yEIAtmThPC6Qp2poXdg\ngJK2VuY31BMO9I3GMVJajNGDnoA+C78tPzWuXDY3U1Z+yn0FU1GkVddG1uAAsS3NnBkbZXjdBrIt\nTtFs6rPpq2f6ksqhFvUcpqZ8zqZ5kpGR8S1Dw2NcaB8gMyWKoEDl5DtcJggKBdetSOb1D+o5XtXO\nDfnT1zLLU/TDRnYdqCNAJXDf5qxZGYaZkRJF9sJozjR009DWT1pSpL9NkvEzcjXRGcZdVUtPqnf6\nw051pkby/vZVQ+3H4MuKjq7OMZ1IqUrq6jOO9tp/bmh0lNePf0LB8WNo3t9L3Y4XGB4edjm+9MKV\nDOYsJ93iCJbFxJBeVDSp3SNGI9rmJt4dHGDBinyyH3yYtttu50KWhqa115BTuJKivr5pL8LiOBdS\nq9Fax3GutYU8g4HB9AzEFXmkqJMILiiy3VOTHc96/tpjxzy6dzytSjtdz7SvniOp8y4jI3P5oW3s\nxWyG7DlYKOTq5UkoFPBBeau/TbmENz6sp29olNvWLCRhXoi/zXHJLWsWAvDO0Qt+tkRmNiC//p1h\n3IXVNZaVktvZSX/HeMnkXFFE62EImi9VCEc7AcrKT01amGMm1AZ/KRpSwiJdzZ29vcUnj487Lyvy\nOWUyof3Ti9xnElFdOI+uo52c5SvY99QT3Bwe4XR8noZnWgvIHHlqOwVAemgYFTt3kLntq6QuW052\nW5tN8TWJIkaTyWluqC9wdu1UK/Il7Wsd99FdLxELxFqasptEEZ1S2ptv+/OHhQXxcdARyfeOp/Pu\nr7BSGRkZmerzPcDcaCnhSHREECsyxnPfGtsHZkXfvPO6fg6WNqOOCeWmVQv9bY5bli6MJi0pgpLa\ni7R0DpESF+Zvk2T8iKwM+gFXb/VNJhPdZaXE158jvv4c3Zb8IKn4WoVwtFNqn7SZUBscz5Hb2cnR\nXS/NiEooRZVx/Iy9vWYg+NABUv62k5xDB+h+51/kBQUhCAIKhQK1wUBr9Rky+/vdzqGn6lBrZQUb\nwsOJm7+AALv+gY5qV3FUFGOnSqZNoXZ2f5hBsuKmUqlYs/V+LuQsxwxOP+9OwZvq/envfoW+JDkn\nl3cHB9A2NzFiNPq0/6aMjIx/qbrQTWCAQHry3AwDvG5FMgAHS1v8bAmIopmde7WYzfDFzVkEqGb3\n8lqhUHDL6kUA7DvZ6F9jZPzO5b2SmWOYAR1grRGos/xNKjOhQniaHyaKoq05+nTlQIqiSHdZKYt0\nbSxubr6k79psyr+EcVsrTlewtKMDfVIySkGgoL+PDhSUh4SwwmDAbDZTHBrG7W4atfsSR7Ur2GQi\n59CBGVWzVEol6W4UN2eKtzuFzp99Je1tTc7JdVtt1Z85wtYKszeGhtHX388RvZ6iB7bNiudERkZm\navQOjtDWpScnPQaVcnY7Jt6SmxFL/Lxgjp7Rcff16USEBvrNlsNlLZzXDbB6aeJlE5abnxlHXFQw\nx86089l1iwkLvvz7UMp4h/yrP4tQKZVk5hWgtYSJLklIpE5i6Ntswlo1NLezk+6yUnRwSXN0Xyw2\n7SuT9rS2oAOWJafYlJ7KkpMYLSGtoihyaPfrJGy5g3QPCqP4EmuD9NSD+wlrb6e1uwtqqolMVNta\nbGxKTqZa10ZdZCTXf/9HVL6806e98ty1gbB38q3hoVLwxJmx7w9YHBVFUV/fJXa4etHgLiTYnYPq\navuENg0+7EPozNa0B7Y5rZg7naHOUq6L7eWRSkXc/AVsEEWqKyvkEFYZmTlA9YXxENGlCy8Px8Qb\nBEHBxsIFvHKgjsNlrWxZu8gvdvQNjvDPI/WEBKn4/PrFfrHBHaIoMjDQ73Tb2qWxvHW0hf0nGliX\nl+jzc4eHRyAIc/NlxFxCdgZnEal5BVSWFpNneXA8XaROd7PtybBfgKY9sI2Dr+1ika6NZckpqOya\nY3u72HRc4FqVn5aaKm5obERl94Wjq9Oy0dK3rq/8FDfq9dT091E3SZVKb+yQmmsWkF9IZ+VpWvv6\nUAcFk9eu4+yRQ3R+5m7WPvZd6iwOw3USlC9vkHpMqfeRJ86M42dL583j9LoNthcgM5l3Zz8Pvu5l\n6dRWFw7WdCn53jiZM6Hgy8jIzBzaxl4ANKlzp6WEM67JTeKND+s5WNrMzatS/aKC/v3QWQwjRu7b\nlEVUeNCMn38yhg16jpT2MC8mduJGs4ggwP7SNlSCyafVTw36ITatWkxk5OzrBSlzKfIv/ixiqg6A\nv0PjHBeg6hX5LG5uti12fX38zG1fJb3oqnEn2n5bTAyJmRpobqZP14baYEBUKBB85JB6q+aolEpU\ngsCNQUGYk5M5OzBAV2wsAfmFBAcHT7Bpqi0bHJ1WkBY2K/U+kuLMWG1oqalifWcnSstxCnp7qVYq\n/aZCWed2unpZiqJIn65tPGfRg7xfXyDVyZwJBV9GRsY/aJt6CQ5UkpoY7m9TppWQIBXX5iazr7iJ\nk9UdrMlRz+j5q893c+xMO4vUEaybRS0uHAkOCSU0bGKRnVAgPdnA2eY+evQC8xPm9v0i4xxZu51l\nSClO4a6lgr+KW0y1KIg3x7c6Ns4K26QXrrS1Uqjp7+ed0REWJLgOgZDapmIqxUdS8wqoi4zEbDaj\nBCISEliRk4vKi1Dgyex1LCakffF5qv/4nOSiMJ7eR0ZR5KxFWbIe196GRSeO011WiiiKHo/V03YO\n/iQ5J5d9/f1cOHSA6LN1tOjaGC4tdjrXvhqX1HvX8XPW5+Zgaiq96iSWFRQRZFdYSEZG5vKkd3CE\n9m49mfPn+eRl7GxnY9F8FAp4/2QTZrMnlRamxphR5K/v16IAvnijBkGYfT0FpbDEoh7XWNRkmSsP\n+dXvLEFq6OFsbBJtNBppqalC3dpCjKXUP7gvCuLrwhnOVLS0B7ZxdPsPCR0bI91s5kxpMcYNm2yN\nye3tn4k5ValUrP3hT3jvqe0U9PcRpU6iIi7O41BeV/ba46gOpVWdoQtQzl8A+CYk0aos5XR2UlNW\nihpY39hIhUVZsrchIzmFM7o2lK0tRCeneBTCPBOKty/uR2tBlgX9fZhGRihWQG5eAcq+Pqdz7Ytx\nucpRdAzzTcvJdXmPpyxZ6jMFX0ZGxv/UNl0ZIaJW4ueFkJ8ZT2ntRWqbetGkzkwrjb0nGtF161lf\nkHJZN26PiQwmITqE1s4h+odGiQzzXyEeGf8gO4OzAE+ckdnWt8xq+w2dndTo2jDp2ojJK7A5Oc6c\nNG+cL2/yIRvLTzG//ix5MTGMDA5SOtBPwIr8CefxZE69zcu0dzaKfrid1soKdHjnALiyNylpg0fH\nkWqvK+fI6swc3vUShbo224sAqz22Y4kiDbo2VAmJnCgsInXZcq9CoD29x2fiBYv9OUwmE3nd3dSr\nVGRFRiIAFzvaiXRTFXaqocCuchQdnUx397i/c41lZGR8i9biDGYtuDKcQYCbVqVSWnuRNz5s4Hv3\nzvNp7psz2rqG2PPxeSLDArnruvRpPddMoEmdR0ePgbrmPgo18f42R2aGkZ3BWcBsc/A8wb4i4bKC\nIs61tlCWmsqarfe7XEx7M15PVBTrAr1y317u1+vHc/UiI1kjiuytPwtrr/F6vN6oOf5Qcx0X+A1L\nlyGazZgcKng6c5g8sdeqLEU3N9sUYaMojivFmRqOh4cT9sFh8gwGdCEh6A0Gr5Q3T5U7o9GI9sXn\nSas6A4C2+ASarzwk6QVLbmcnB3e9RMqSpZM6kdV/fI551VUAnAsPZ2l4BGnqJMrbWsnV66elWqlj\nHqgzPHEy/ZlrLCMj43tqG3sJDBBYpPZ/I/aZYnFKFLkZsVSc6+LM+W5y0pwUS2E8n3twcGp54qLZ\nzIt7tBhNIndfMx/jqJ7+0cn3GxjoxyzOXBirJ6QmhBMYIHCupY/8zLjLNuRVxjvkX/wZxJqzA96H\nos2Wt/jDw8OcfG0XF883kIaCsMBAVILA4uQUxpYsnZbFpJQFrr0jI+jaOdfVSVZcPAqFAl1ICOpM\nzYR93M2pMyfEUzWnoeQkCytP0y8IRKmTpuzsS7kHHBf4Gst2+wU/4NTp89RZt7fHKIocqK1hEyA0\nN7Ovv5/0hES6VSri1UnE9fZ6PHZvnOmGkpOkHtyPengYAJOujYb8QjJXrXF7LvuelYsaG9m3+58s\n2HInaU5aktSXnCT40AGyLefQBwWxP1PD5shIslbks0c/xIItd5Lpo3YmUkNCnX0fTHbPWO9pf/Y9\nlJGRmToD+lFaOodYuih6zvYXdMVd16VTca6L14/Us2xRjFN1cHBwgH3HzxISGub1ec62DNKgGyIl\nNpgB/TAfnW6TtF93ZzuhYZGERcy+kFKlUiA9KZKaxl5aOodYIBeSuaKQf+lnCKPRyOnf/57sxlbg\n0gWtJw7eZG/xZ2IxNzw8zLGHtnFrVxdms5m/D/Rz26abCFapJDmn0+nQ2jsyWfPnc7q9jbOBgUTE\nxtG4dBmawpUT9nE1p+6cEE9CEJv27CavoR6lQoGurZWoFflTGqO7e8DRLkeny/6/64tPOHX6pmJP\nS00Vm4AAiz1FgwN0CQKa5PEqayYvCsg4c04rS06itBTecTb/ujoteQYDCss+aoOBsjqtU2fQWc9K\njTqJ2vJTbNHrudjf77QlSXudlpsMBptdBcPDvK3RUL1sOfBpixBfITUk1F1Ir7vPTWj/UXyCgPxC\nVEql7BjKyFwm1DaNR39cSSGiVlITIyhakkBxTQen6jopyHIe7hgSGua0sqYUBvVjVF5oIzBAYG1u\nCiFB0r8X9UODXp1zplg8P4qaxl7ONvfJzuAVhvzrPkM0lpVS1NXFsBO1xdMwLW+ac1u3S3Vg3H3u\n5Gu7uLWry9bX7/MRkbyEmbyNmyWFmM1UWJpKEFiaX8jh1FRSlixFM0k7Bcc5daWQpeYVeJTjuSk0\njIrQUFYYDMTr9byn11OUkzslldjbXEz7a2t00fLAG2fd3h6hudn29yh1Esf0Qyy2OIG+cPxFUaRp\nz25uDh//MXc2zsRMDaXBwaR1dgLQEBc33m7Ehe2OPSsbO9rJ1esZHRxE391FbmcnWsu1t85ffPpi\ndCEhpFiUQV1ICClLls54eLdUpXqyz9nf76IoknpwP71VZ1icnOK0SJGMjMzsQ9s03mxecwU6gwCf\nuTaNEm0Hb3xYT95i34Y7imYzH1e2YTSZuXppokeO4OVATGQwMZFBNF8cxDBinHPjk3GNfKVnCVMt\nJAHuc/GkhtoNDw/zyVNPkNnfzyJ1EpUSQvIUCgXxi9I8st8X43WGvSMjiiJH9HoSMzU+VTY8DaNU\nCQLZ+YVodW2IokjMLbfRsHOHz3MI64uL3drlrPF7cVQURQ55hFNx1h0dyYq4ONY+8F2qKys8Ppar\nYx7S69kUGuZ2/lNX5PNxSAhhosigwUBVRDibLYqdM6z3o7VnZVhrK/q2VgwKBYv6+mgrK2X0hvWX\nzN/x8HCOpS8mpfE882NiacpZ7lR59hUzGSJu7c/Zb9dCxddFimRkZHxPbWMvKqVAevLsC0WcCZJi\nw1ibo+bj0zoOlDazqWiBz45dcbaL9m4DCxLC5+z8Lk6J4kR1B+da+8lJi/G3OTIzhOwMzhCpeQVU\n1FaSYQ0T9XAh506ts2/snSmKTkvESwm1MxqNvPuNh1hVV0t6RARn2nXkrMinzmGhvfKerby9/31u\n7eoC4O3YWFbfs9XDGZkarubD6shUlpykac9uNoWGoTp0gDInYX6T4Wrx7UkPNvtjpKuTKIuJIUCp\nJMeLgkFTDQF2vAcKenupXLeBass9kOkwj944664cyak4/o7HTDCZUB064Haf1soKNmctob2jgzRB\noEA086+nn+S67T8FsM1j9KbrnZ7ro7/9hea6GjYGBoFCgY7x0NMbLfNnFEXCPjjM4kQ1Qlw8pZFR\nFH3pK9MaSjndirr9vWoSRcpCQljmphKqjIzM7EI/PEZTxyBZC+YRoPK8f+1MIooiAwP903LsmwoT\nKKu7yGuHz5KeGEh8VDAAgYGi10Vc2rqGqDjXRViwirXL1dNerdRfpCVHUqy9yNnmPpYtip6z45S5\nFNkZnCFUKhXLH3mEkn1HANcLOXfVHXM6Ozmva+OD3f9k7Q9/QnBw8CVqj0YU2Vdbw4asJagEwa3D\n6RhqV3zyOD1NTdxdV0t4bw/l+iGWJaqp001MjA4ODmb18zvY+9ouAFbfs5Xg4GCfzJMUJlM5VSoV\nSqWSm8MjplSh1dXie6o5nt409Jai7KYXFfHJgSMeKUdKpdLnCu10qL72xzQajZSVn5p0nAMd7aQH\nBqIICsJkNpPZ3099yUlM9vvWVDCSueySvDiVSjXe/uL69dR1tAOwJCERnfLTxVWDro08g4FuUK4s\nIwAAIABJREFUpZLo5BQ2iCLVlRXTHiI6XYq69djWe9VkMjFcWoyir8/nFVFlZGSmh9rmPsxcHvmC\nwwY9R0p7mBfjvOrnVMlZGMlxbQ//s6eO63NjUSgUhId103ih0eMiLoYRIx+Wt6FQwHV5yQQFzG5H\neyoEBShJTQznfNsAnb3DxEeH+NskmRlAdgZnkMkWcq4W/I1lpeR0dlJbfooVBgNLzGbee2o7q7c/\ndYnaoxQENmUt4aAlR87e4Zws1G5edRUJXZ2ERURg0A+xYnSUmoEB6iIjuc7JIjA4OJhr7/+SbydI\nIlNtxeGJwuaqyqKnbS7sz+VVz0QJY55MOfJFmOFsqDYpRSFLzSvg2O7XudFsRgGUh4SQpU7icJ2W\njXZ5ceq9e+k4UTIhL85oMnFEr2ddQiKC5cXKynu2UmYJ7xVFEV1ICPFzTDm7xOkuXCm3m5CRuYyo\nbby8ms0Hh4R6XchlMrIWhdPWO0Zj+yBNXSaWLIwmLDyY4BDPqoiOGUUOlrYwPGqiSBNP/Ly57xxl\nJEdxvm2A+rZ+2Rm8QpB/3f2Es0W1qwU/wHldGysMBpQKBWagoL/PqcIkCILTIhZSQu3CY2JpNxpJ\nTFQzODDA8cwsbv7hTy7LRaArx8ebNgWu9vGkzYXjuaYr3M/dC4epnnc6+iV661xO9mJFpVJR9MPt\n7LHkv2apk6iMixtvLWIpcNOnayPRYKDTLi+usuQkRotyaAwN5T2H9hDW+TOaTPSfKiGut3fKypnR\naKS+5CTtdVrUmRqnbSz8wXSqkDIyMr5H29SLUlCQkRLlb1P8jkKhYNXSRHTdekq0F4mODCIi3LMI\nJpNJ5FBpC119w2QkR5K9KHqarJ1dJMWGEhyo5HzbACuXJMg9B68A/L/iuAJxVcK9vU6LxknOX2pe\nAYdef5WE/n6UCgXD8fFEq5PQ4Zna4y7Urjd7Kf0KBXlJybTr2iiNjOLmH26f9vBPb5wBb/rsWR2f\n2uNHCa88Tb0gkCax55+3SqSz/U6XnESlVGI0mVCALWfTF2OWgjf95KzOSuW+vdzY3o4iZT6CXVER\nZ1VNpVat9bVzaU9wcDDXbf8pjWWl1PFpb0XrfW8SRcpDQsi0U/d0dsqhUhC4OTyCaqXSaS6lL5Qz\na+P64EMHuMlgQBcSgnb9RjRfeWhWOIQyMjKXB4YRIxd0A6QlR8zpMEZPCAlScU1uEodKWzhY0kKk\nB86gKJr5oLwNXbee1MRw1uTM3TxBRwRBwSJ1BDWNvbR2DTE/Xm4zMdeRVxt+wFUJ9xvUSeyrrWFT\n1hJbaJp1AZuYMh9tUCB5I6PozWbKYmJsrRK8UXsm7JeTS1P5KQ7WaUncsInVXqgTkzkBjtvBedNz\nX7WmcFQ2jEYjHXve5MaGehQKBeVtrWRNseefJxhFkY49b3J9aCg1ZaWogZi8AiokjNuXaqInTpi9\ns7K6o4PwwQG60jOILSia8rGnGu4rBWfqln1eXEBtJYqWdpu6l2inHHpzbE9pLCtlXnUV2cPDKAWB\nlOFhlFVnaPTxPMjIyMxtzrX0IZrNaBZcGeqVVObHh3NNbhIflrfx1of1XJURTFSYe2d5yDDGhxVt\ndPQYUMeGcm1u0hWnjqUljzegb2jtl53BK4BZ4QxqNBoB+AOQC4wAX9Fqtefstq8EfgkoAB1wv1ar\nHfGHrb7GvoR7kErFBic5f/XFJ1g5MIB53UYaLO0JAvILfVL50bHtRA5QpteDhyXypfQ4dNyuXJHv\ntTPgzZgby0pZFxrKxdBQ1AYDuXo9e/RDTnMi7bFX5bRdXfx3/VkWBgaSfLaO5OQUNJpsEhMT3e4H\nsHdwAE3/AB831HODwUCgQsHFjnbyBEHSuH0VtueJE2bvrJgjI6kcGiSjo4Pu1hYu5CyfoE42lpWS\ncOECH/T20DLQT3N/Py+Wn+LR7zxOWlq6ZBunMzfRfh6jb1p/SVEnYNLiNLMhb3K6mMtjuxyQ8Fv4\nBeCbgBE4DXyd8d9Fl/vIXBlomy6vfMGZJC0pEqPJzNFKHUe1QyxOEimMmRiFZTabaeoY5JNKHaNj\nIgsTw1m7PAmlcmKF9rlOXFQwEaEBNHUMMmYUCVBdeXNwJTFbfunvBAK1Wu1ajUazinHH704AjUaj\nAF4A7tZqtfUajebfgIVArd+snSLuSrirXOT8WbdlJqdgEkVbOwBf4I1C47honOwYzrbvr9PiuvPb\n9CAIAjH5hVzUtWESRRZsuVOSErn4yw/ys/96iv/e9RIjIyPw4RHb9sDAQJ577kW2bLlzwn72OWZB\nb75B9oUG1IODVA0OkHOZFR9RKRQsUydRHBFB11WrWLP1/glz986B9/npi89jMl9auvuNd9/h5z//\nFffc83nb3xyd5dJ58wgwmag9fpSxUyUU9I4vbrwJH5Xq1LhTDmGiAuvr0NbUvAKqTx7nlKU6qS4k\nhMaly9D4oXLndIftykjC3W9hCPAkkKPVaoc1Gs3LwG1AABDkbB+ZKwdtUy8KxXifOJmJZM6PIiwk\nkCOlTdS0GGjpOc/ilCjCQlQEBajo6NFzXjfAgH4MpaBg9bJEMudHXTGhoY4oFArSkiKpONdFU8fg\nnO2rKDPObPmVvxrYC6DVao9rNBr7GLQsoAt4TKPR5ABva7Xay9YRBM9LuHuTLzZdb/iNRiMN9j38\nBIGy0mJUXoRbqjM1lOn1M9JEGy6dx0hLz79MCQpob28P3/72N9mzZzdRUfN45pnfEBcXR1tbG83N\njfzP/zzHgw9+iV/96nfce+8XL9nX6mzUF59gQ3g4F0NDSQDGBvo5PjpKQULitI5bSjXT4qgogk0m\n6otPTLhXHJ2V9pAQzNfd4NQR/OMfn+P/++XPiQgJ4ZEV+aRERDKcnAw5uWzf/kO+/vUHOXz4IE8/\n/Qzh4RETngOxtJjlhw5wtrWFJF0bioIit7mJrsYI3oUfu5ove3zx4sT+mCqViuwHH6a+oIi9lgIy\nmhkqIOPpCx2ZGcHdb+EwsEar1Q5b/ltl+dsNwLsu9pG5AhgZM9HQ2s/CxAhCgmbLsm72oVkYjWKk\nh7O6Uc5fHOFUXecl21VKBQvVEazIiGVeRJCfrJw9pCePO4MNrf2yMzjHmS3fGpGAffdRk0ajEbRa\nrQjEAWuBR4BzwL80Gk2xVqs95Ac7fYbUQhTWBZtyRT6VjBccmSxfzNM3/FKdA+txF1aeJq+hnorQ\nULLzC8cLozDuzLly7Jw6tIUrQWIRDl84t97k3bW0NHP77TfR1NTIqlVreO65F5k/f8Eln7npplvZ\nuvUu/v3fH6Gvr4+HH350wnFMJhN9ujYEdRIXAfOiNLqLVqJdtnzayvZLqWZqNJkQTpWQY6ks66xn\n42TOitls5plnnuYXv/gZiYlqXnnlNUKHx9er1mt17bXX89BD2/jHP16hpOQkb731HvHx8Zc4y0V9\nfSgFAUEQUBsMdOvaiE5O8XiM3oYfT1e11MmOqVKpyFq1hqxVa7w+jy/scnyhI4oiLTVVgBwyOoO4\n/C3UarVm4CKARqP5BhCm1Wr3aTSaz7naZ+bMlvEn9S19mESzHCIqgQCVgmWpoeRnp9A7MMLQsBHD\niJHoiCBS4sNQXYEhoa6IDAskNjKY1q4hhkeNBAfKvwFzldlyZfsB+2Yz9j9kXcBZrVarBdBoNHuB\nIsClMxgdHYpKNTuraUVHh1BfXAyMNwm3LrCSkjZM+KzRaOT0739PUVcXABWxsSx/5JFJF2W1x45x\n9cgQyojx/jBXjwxx7oKWrNWrnZ6jvriYuOuvxlo2I/LkSfJPfDR+ztpK2zmtx+0NCyY4KICVpjHO\n9XaSkZJCfFwk6d/7tm1sa+3GZiXexXZnY7ftEx8xcR7sbPIGd+ezx2w2c99936SpqZHvf//7PPnk\nk07PuWnT9Xz44Yds3ryZJ574TxQKIz/60Y+A8fmtPXaM3n1vE9DVQd7wMO0hIXTcdBN3fuNRny+w\n4+M/fYwmuw+SkjZQe+wYGWOGSe+VpNs2A5udnvM//uM/eOaZZ0hOns9vf/tnrr121cRrH5/H8ePH\n+M53vsPvfvc7nnjie7z66qu27T3RYYSFBaEUBHIyFlHW2U5qkIrgkAAqYmMp3HS907lyNsZDbRds\nxwIwiSLR0WGXzI2zOZPy3ERvup7TtZXk2j2TrmwDqProI+LPadEpFKSnpLh9FmcSZ2PVzgvlXGoy\nuV1diKLIgXNabg0LQnX0ou2ZA1zOo4xPcPdbaM0p/DmwGLhbyj6uuJyv4+VsO/je/n2lLQBclZPs\n8bEDA0XCw7oJk1hp09P2DK4wDAUiCAE+O55UwsLGzxsXH0HSDN9HvhizN/tO5byaRdF8UtFGR+8I\ny9I9KyQjMEpcXARRUePzLD+3s5fZ4gx+DGwBXtVoNKuBCrtt9UC4RqPJsCTFXwu86O5gPT36aTN0\nKkRHh/DJf/3S9jb+kwNH3CoP9cUnyG5sZdiyqM1obKVk35FJFY6eniGGhkZQCgJGUeRcawvnj54k\nYqHGff5TTAyqFfnkNOucntN63MCoWJqU54nX6xnUj/BxUBiZCzX09BiIzlhmscHgfA4m2W5PfHwE\nFy8OeDUPvlAS//rXP7Nv3z42bNjEt771uFub4+MX8Oabe7nrri088cQTrFx5NStW5FO34wXCK09z\nQ0M9bcHBVKakYgZGMpdJmgNPxmOdLyv29wGMO0U9PUMef8YdH3xwmGeeeYb4+EXcdtt+PvkkiZqa\nE2zblu3Uxh/84ElOnCjmtdde48UX/8Idd9wFQMRCDR8HHbHdi/1XX8/ZgiIalEpS8wpczpUz+0OT\nFvJxR/eluYid/fS8e2DC3NnPmdS5UH/uAYrtroUr24xGI9q/vsKNVTUoFAqKGy6QtSLfo/mdLpyN\ndWBghFTL2Fpqqrhh4QjGERNGTLZnbtXNG/xuuzPm0I+0u99CgP9hPDT0MxalUMo+TpmN11EKjt9z\nlxuu7BdFkcFB78Z1srIFBRDCMOfOSauGbGVgoJ+BgWFEAif9bER4MAODw5N+TgpDQ6MIgomgEN8c\nTwoR4cF+Oa+VqZ7b2/mfynnVlqbz2gvdpCaEebSvfmiEzs4BRkeFOfvcXg5I+X2cLc7gG8AmjUbz\nseW/v2ypmhau1Wr/aCka87KlmMzHWq32Xb9ZOgXqi4s9Dl8TRZF+XRsA4QkTK1Y6wxqSmdPZaWth\nsL6xkYodL1zifLoq6pIzyXHzuruJWpHPe3o9CVvusDXkni14Gu7nzNFqbm7iiSd+QEREJL/85W8l\nJZGnpi7kd797jjvuuJnvf/87/OGpp8nr7qZeEFAoFMwfGeGiIBCpTvKoAJDRaET74vOkVZ0BQFt8\nQlIfOim5plPpXzg6Osrjj38HhULB5s0vERk5H4Du7jzKyrQUFS2esI8gCPzmN79n3bqr+f73v83a\ntdfawkXtQ3izp9Jz0i782D4XEdzfC1LnQmpVV2+r1/oKdy8QXI3VOjaTycTAiePoBYGoy6zQ0WWO\ny99CoBjYBnwAHNRoNAC/drbPzJos4wsGBwfYd/wsIaGeLbhNopl63SCRYSpK6y56fN7uznZCwyIJ\ni5BzwmQmEh4SQFxUMLpuvRwqOoeZFVfV8obzYYc/19ptPwSsmlGjZgHJObnse/bX3GoJSXu7t4fV\nj3130v2sC+vDu16iUNdGTHKKpEIcAIluiro4LthXe6C4TUWp89RhaSwrJbezk/6OdgByRRGti3E7\ncxwXf/lBHnvsGwwODvDrX/+e5Eny1uxZs+Zq7r77c/zzn//grffeJScwkDR1EuVtreTq9S6LBLmj\noeQkqQf3o7bk4Zl0bTTkF5LpJsdMaq7pVPoXvvDCc9TV1XLrrZ8lMVF6AaH09MX853/+mB/96HEe\nf/w7vPjiX2y2eFqsxJ39jrmI4P4FjC97OVrxpnqtL5jshYi7sRqNRoZLi2mxVjhtbaFx/Ua/VDi9\n0pjstxBw9RbJcR+Zy5CQ0DBCwzxTudu79YgiJMWFe7wvgH5o0ON9ZK4sFqoj6OwbprF9kKwFcl7q\nXGRWOINXCulFRXxy4Ihkp6a1soINWUuotTg1GxISqauskNyHL2XJUqKbmxEE5wnR3hR18WbBPtXC\nHJ4u0k0mE91lpaRYnKeW1hZMGzY5/awzdfTp//ophw8fZP36jXzhC/dLHqeV7duf4r333uWFv/2F\ntQ89ynUGA/HxCbw0MkL21vvIvGq1Rw6Brk5LnsGAwmKj2mCgrE7r0hl0Fv6b5ma+7ftNSnXYW1tb\neOaZp4mNjeXnP3+aN94oo7s7D4CYmDLy8rLdjunLX36QXbv+zltvvcEbb2zhM5+5x/0kuMFX/Rcn\nO5anLzS8rV7rC6RUBnU11sayUor6+jAXFFHrpK+pjIzM7KHdEqaeGB3qZ0tk5ioL1RGUaC9yQTcg\nO4NzFPnXfQbxRnmw9haE8bweT5hMUbPac7rkJO2WSpHWv/uynLwvStZ7YpMZ0AHW4Dad5W9S6DYY\n+O1f/peIiEh+9avfedVjKDFRzXe/+zg//vF/suv8eQwR4WQKAvekpVNZWQFXeVY8JDFTQ1lICPkW\n57YsJIREy7VyhrftDzxx2Ldv/wF6/RA//el/ER8fz7Zt0ZSVacfPl+c8X9D+XH/+s5arrvoLWu1q\nHnvsMTZs2ERkpO/7Y00lDNYeb15oTIfSOJNMV19TGRkZ39HePV4jITEmxM+WyMxV5FDRuY9cQ3eG\nsTo16UVXSVMWYmIwiaItvDDVg4WsbTG6cTPVGze7XLyayk+xsbmZnEMHqNvxAkaj0eNxeYrJ0rqi\nvviEz8+nUipZkleANj0DbXoGS/IKUDksZserqJ7AZDJRHBVlm+OnamsY0uv59re/51F4qCP/9m9f\nY8mSbHbteomEgUE08xcQpFKR191tU5ekkl64kuF1G6helEb1ojSG120g3ccKk9WBVAD9ujYWVp6m\noeSk089++OERdu9+ncLCIptyqlKpKCpaTFHR4knv67Ky83R35xEbq2Hlym8xNNTLL37xW9t267Xx\nxb0h9RmYDHsHW2kJuZZyHT153n3JVL47pvq9IyMjMzOIopmLvQaiwgPlBbrMtLJQHYHZDI3tcljx\nXET+9pjFOCoLaTm5HufdTaaoeaIieZv356jOlM6bh1haTFFfH+Cbfm6O56ssLSbPMiZHNchR5Smd\nN4/T6zZgNBrZ/ZcdREXN44EHvuT02EajkbKy8wDk5S1yaXNAQABPP/1L7rzzFn784WH2br3P6/FY\ne/1Z536y4ireqmGiKNJXfgq1wYDJbObYnt2kOekp+JOf/AiFQsHTT//SZQiyVPLzv87Jk//N66//\nlR//+PsoFAq3Cpw396Cvle7LASmqpKu5vNwVTRmZK4Wu/mGMJjOJ0bIqKDO9yKGicxv5F36WY5/P\n5euG2J4wlfM7Li4DTCaWHzogOYzRUwfA2WIWxlt1wLgqae8AF/T2Uq1U8tGZSrq6Ovm///cxamra\ngfZLHD6j0ciOHdW23LjS0jKXLRQA1q69hhtuWM/hwwcpbmslP1HN8fBw+s6cpqWmipX3bCU4WFrf\nH08cGm8W86l5BRza/To36vWICgUVoaFsCg2jzuG6fPzxh1RUlLFly52sWCG9aIw9OTnz2b17P/39\nBajV8ygquoVjx/7J7t3/pDAt3eXLCX8+A74KN51JJst/nKzAzJXmQMvIXG58GiIq5wvKTC/2oaIj\noyaCAuXUgbmEHCbqRzwJh/M2TG0yknNyOTA4SGdzE2NGo8uQsKme/5JwOQ/bKtTteIHs/e+Tvf99\nyWGsKpXKNo6GkpNU//E52zGa9uxGdMi/NJpM/OEPvyMoKIjQ0HXs369h/34NO3ZU285nDW8UBCWC\noLS0UDjv1o6HH/4GAP+vpZmTV19L3fvvkvmn/2Xjrpc59tA2hoenp9eRp+GJKpWKhC13UJOWjjY9\ng+z8QlROVL/nn38WgIcfftQru4xGIzt31hEaegPQhV6/l1/+8nGUSiXPPvsbzGbX2Z3T9QxIwRfh\npr4Mf50q9nNpEkXMhw/y1tNPTtv9KCMj43vk4jEyM8mCxHDMZmi+KIeKzjVkZdBPOL6ZLz55nOCC\nIpSWJtszVX6+YecOrg8N5Xx/H8f0Q6x94Ls+Obc3Pc6c4W3xGfv57WltoUXXhrmgCJUgsCk0jEN6\nPRvCw23nr2puorHxPDfffDejo+sRhHGH1V3PPGfndAwhveGG9WRnL+XgRx+wOTCQRxobERQKyvRD\n3Ajsf20X197/pUmPPRUmU1at2xVAb/ZSivr6EEWRg3o9CSYTRqMRlUrF2bN1vP/+XlauXEWRl6qR\n1aFWqZTMn78EUcykr0/LHXd8htdff41zPd2MxsR4rcBNpYXJZMeZilrmb2XfFaNGI6X/epOr9Xoy\nWls5eraOomdfoKOmCpjaHMrIyEwfotlMR4+BiNAAQoPlZ1Rm+klNCOdUbSdNHYNkpPi+4JuM/5C/\nQfyEvZNjFEWCDx0gpbqK6OQUpwtFX1ZFtC5yjdZwSZUKzfwFLBZFql20rvDk/N72OHNcgE+FCSqS\nwUCtro3M5BRUgkDCljtsFRIXr8jn0ZvWIwgCd9/9RaqqnB8zL28RpaXOWyi4CyF96KFH+eY3v86Z\n2hqUCgWCQkHe2BjVg9P/dm2ya+Esf7Lsuhtoe2cPm0LDUB06QFn5KTK3fZXnn/89AA895J0qaEUU\nRXS68XzRhITxvliPPPLvvP76a/zhD7/jtdfechriOtk96CuHazoct8leavjKiZWKdS7Nhw9ytV5P\nZ2AgcZGR3NzZycvfepStGeMvP2aL0yojI3MpPf0jjBlFFqo97y0oI+MNUeFBRIYF0to5hNEkolLK\nwYVzBY9/4TUajQAkAANarXbI9ybNXaxhYjDuiFlpsDR37rYLf3PWE2yqRR0cF7kHBgcxhobaFqju\n8OT83vQ4c7YAj//ety9xAIyiyD79EAvs1CopRKmT0LW2INpVR8y0K4xy+PBBTp8u5/bbP8Mtt1yH\nTufc4VOpVGzblu20hYJ9CClcqijedddn+cn2H/DP1jYenj+fvJERRLOZj6Mi2XLPVkljsJ8nT5yG\nya6F4/aC3l7215/l5vCIS/Y5euQQ//jHy6SmLuKWW26TZKezQjs5OfN59tn36Oq6FYDe3rd57LHr\nCA4OZt26DRw6dIDy8lMUuuh76O4e9EULE18eRyr+UA2tc/lWRzsZra3ERUYiKBT0Dg6iGRqcsbHL\nyMh4R3uPJV9QLh4jM4MsSAjnTEM3bV16FiSE+9scGR8habWh0WgWAF8BYoExYBCI0Gg0KmAA+JNW\nq62dNivnAEajkdO//z3Zja3AuAJTHBVlC8nThYQQr05ye4ypFnVwXOSuCw3lPf0QN4ePv1mcTG20\n5uE1lpXSWFbqUwXD2QK8vriY6Ixltl6IHXve5MbQMAQ7tcrV+R1VpMb1GwnIL6RaqZzgRPz+978B\n4NFHv+nW4bPOgZSQUXuCgoL4yoMP8/Of/z/2z19ASFAQxaFh3Pzfz0ouIAP+DTV8/Z09DA8P87Wv\nPYxykpxPe5VUFEV27z7Ili1JFBYuprKymaysTXR0nAUgIWETlZXnKCpazKOP/juHDh3g2Wd/wx//\n+OdLFcviE+ONz2cwjNqXWO/HnM5OzuvaqIuMZG1OLuC58+lMQfdGVVSpVNzyncc5eraOW7u6EM1m\n9kZFcufSnCmNVUZGZvpp77bkC8rFY2RmkNTEcWewqX1QdgbnEJOuGjQazY2M9+/+mVarnVBdQKPR\nKIG7NBpNjlarfX0abJwTNJaVUtTVxbCdAlO5bgPVSiVGk4n+UyXE9fZ+qlzNQKVCQRBYsOVOW7jk\nZGqjVGfE15UXVSoVKqWSDeHhkhfMjiqSxsXYLlw4z5Ejh1i9ei15Fhu9cfjchZACfPnLD/Lb3/6K\nZ2truP6Pf+FGh5YNUnDnNBiNRmqPHaOnZ+gSh2Cya+Fs+8p7tlK2c4ftb8cjInjjr38mMjLK1lfQ\nHVaVFBSUl/eg12+mv19LeXk1K1YEIAgqkpMzARDFTxXya665jmXLlrN379ucOnSAayxjFUWR1IP7\n6a06w2IXYdRSxioVKeGo3rS3SHtgG0ee2k4BkB4aRsXOHWRu+6pHtjnLNRYUCgp6e8dt9fAFQXBw\nMKuf38He13YBsOHOe6h6eedlVTVVRuZKw2w2096jJyxYRXhIgL/NkbmCiIsKJiRISVPHIKLZjKBQ\n+NskGR8gZcVwVqvVvudqo1arNQGvajQatUajCXbmMMo4R6lUfpozVLhy2vt6OV3keuCUSFUwvG1t\n4Gjb2qIieizV0rxBipL697+/DMC9935R0jFdhT9OpijGxsbyuc/dy86dO9BevEimD6+v1UG4emSI\noaGRSxyCya6Fq+32f6usqebixQ4eeeSbhIdLz0/R6XoxGNQoFKKt+ipUMW/eKaqqFgKwdOkF8vKW\nAqBQKLj33vv5wQ++x3uHDnBNZCQAfbo21AYD/W7CqN2NxVPcHcfVCxGYXJ1rrawYf6FhGVdOZyeH\nd72EOlNjixIAzwoqzauuIhZQzl8AeJeLGBwcbCtiZDQaUa7IZ3+dFnWmxqPvBxkZmZmhd3CU0TGR\n+fGyMiMzsygUChYkhFPb1MfFHoOsTM8RJv2V12q156z/1mg0vwB2a7XajzUazbXAca1WO2r5nG76\nzLz8Sc0roKK2kgxLmKjjgm8m+nrNZDNpT8czmW3T0edNFEX+8Y9XCA0N47bb7pj085P1GZxMUfzq\nVx9m584d/OUv/8ttt93usb2u5sDmIESEOHWWJrsWzrbb/+3rP/wegiDwb/8mTcWyqqStramYzSZC\nQytQq5fZtpvNJqDL7t+fctddn2P79h/y/kdHuOu+B8jv6RlXy0NCWDZJGLWUsUrF1XGcvRA5XXIS\nU/kpj8J3jaJITVkphbo2opubKZ03j9PrNqByEsbsLZ6GFQ8PD/PJU0+Q2d/PDeokKvU5V28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yQsTM3atUVUVARz4MBr/MM/fNvldenrdWctut519C1ytV0IcjmNy5ZPKDA/nULLWfIXezEa6PvG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\n7JwJVrWZYiqL1tHhY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JFD3IRgm4P9IllQKNgYGcnLF84jmkw8umw57dVVAY+ddOaS6Fic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\nb97Jz372X7zzzk4eeOAhsrPVcYnJbajPvfTyCTE+HE4nbxZWY1QMXLpE1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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Uniform rejection sampling\n", "\n", "# Number of samples to draw\n", "n = 1000\n", "\n", "# Plotting \"true\" function for comparison\n", "x = np.arange(-4, 4, 0.1)\n", "f_x = f(x) # f is defined above\n", "\n", "# Define uniform distributions\n", "x_dist = stats.uniform(loc=-4, scale=8) # min=a, max=b\n", "y_dist = stats.uniform(loc=0, scale=1) # min=0, max=F_max\n", "\n", "# Draw x_i and y_i\n", "x_i = x_dist.rvs(n)\n", "y_i = y_dist.rvs(n)\n", "\n", "# Evaluate f(x_i)\n", "f_x_i = f(x_i)\n", "\n", "# Build into dataframe\n", "df = pd.DataFrame({'x_i':x_i,\n", " 'y_i':y_i,\n", " 'f_x_i':f_x_i})\n", "\n", "# Add column based on whether point is accepted or not\n", "df['accept'] = df['y_i'] <= df['f_x_i']\n", "\n", "# Extract accepted values to new df\n", "accepted_df = df.loc[df['accept'] == 1]\n", "\n", "# Plot\n", "fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(15, 5))\n", "\n", "# Plot (x_i, y_i), coloured according to acceptance\n", "accepted = df[df['accept']==1].dropna()\n", "rejected = df[df['accept']==0].dropna()\n", "axes[0].scatter(accepted['x_i'], accepted['y_i'], c='b', label='Accepted', alpha=0.5)\n", "axes[0].scatter(rejected['x_i'], rejected['y_i'], c='r', label='Rejected', alpha=0.5) \n", "\n", "# Plot \"true\" function for comparison\n", "axes[0].plot(x, f_x, 'k-', label='Target function')\n", "axes[0].legend(loc='best', ncol=3)\n", "axes[0].set_xlabel('$x$')\n", "axes[0].set_ylabel('$f(x)$')\n", "\n", "# Plot normalised histogram of accepted values\n", "sn.distplot(accepted_df['x_i'], ax=axes[1])\n", "axes[1].set_xlabel('$x_i$')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The left-hand plot above gives an impression of how rejection sampling works. Because the sampling is **uniform** in the $f(x)$ (\"vertical\") direction, the number of \"accepted\" points in any vertical slice is proportional to the value of $f(x)$ over that slice. The right-hand plot shows a histogram of the accepted values, together with a smoothed version estimated via [Kernal Density Estimation](https://en.wikipedia.org/wiki/Kernel_density_estimation). As the number of samples increases, the histogram becomes a better representation of the target function. We can therefore use this method to get an idea of the shape of $f(x)$, which was one of our main aims at the start of this section.\n", "\n", "Furthermore, we know the points $(x_i, y_i)$ represent a uniform sample from the rectangle bounded by $(a, 0)$ and $(b, F_{max})$. This rectangle has area $A = (b - a) F_{max}$, which equals 8 units in the example considered here. The area beneath the curve (i.e. the integral) can therefore be approximated as\n", "\n", "$$I = A \\frac{n_{accept}}{n}$$\n", "\n", "where $n$ is the total number of samples and $n_{accept}$ is the number accepted. The ratio $\\frac{n_{accept}}{n}$ is called the **acceptance rate**." ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Acceptance rate: 23.1%\n", "Estimated integral: 1.85 (0.65% error)\n" ] } ], "source": [ "# Estimate integral based on results above\n", "\n", "# Calculate acceptance rate\n", "acc_rate = float(len(accepted_df))/float(n)\n", "\n", "# Estimate integral\n", "i_est = 8*acc_rate\n", "pct_error = 100*(i_est - i_true)/i_true # i_true is defined above\n", "\n", "print 'Acceptance rate: %.1f%%' % (acc_rate*100)\n", "print 'Estimated integral: %.2f (%.2f%% error)' % (i_est, pct_error)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Non-unifrom rejection sampling\n", "\n", "The example above illustrates why the unifrom sampling strategy is inefficient: towards the margins of the target function, the algorithm is generating lots of samples that are rejected, because the target function has low density here compared to the uniform density of our sampling distribution. \n", "\n", "As with importance sampling, we can make the algorithm more efficient by changing our sampling strategy. Suppose we can identify a distribution, $Q(x)$, that we can sample from easily, and we can also identify a constant, $m$, such that\n", "\n", "$$mQ(x) > f(x)$$\n", "\n", "everywhere in the integration interval. If $mQ(x)$ is broadly similar in shape to $f(x)$, then samples drawn from it are more likely to be accepted, and the algorithm will converge more rapidly. One possible choice for $mQ(x)$ for the example given above might be\n", "\n", "$$mQ(x) = 1.5e^{-0.5x^2}$$\n", "\n", "This is $1.5 \\sqrt{2 \\pi}$ times a Gaussian distribution with mean 0 and standard devation 1 (see notebook 1 for a reminder about the form of a Gaussian distribution). In other words, if we choose\n", "\n", "$$m = 1.5 \\sqrt{2 \\pi} \\qquad and \\qquad Q(x) \\sim \\mathcal{N}(0, 1)$$\n", "\n", "then we have a function, $mQ(x)$ that is greater than $f(x)$ everywhere within the integration interval." ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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2GM//g/nXbXqnEcJrpBgLkYnUhSdsbWSIWg+2NqkLgMisahG4pBgLkRFNw7p4\nAVpoqNwuUSeOBo1wFyiANWYxuN16xxHCK6QYC5EB04E/CDr6J7ZmLSA0VO84eZPZjK1VG0xnzxAU\nu0PvNEJ4RZaKsaIotRVFuW3FdkVR2iqK8quiKFsURXnJ8/GE0Jcs9OEbUr//qbPahQg0mRZjRVEG\nAl8A1lu2m4HJwBPAY8AriqLc442QQujFGrMILTgYW9PmekfJ0+wNG+POH5GyGpem6R1HCI/LSs/4\nCBAFGG7ZXhE4oqrqVVVVHcAmoJGH8wmhG5N6kCD1IPYmzSAsTO84eZvVir15S0x/nSJozy690wjh\ncZkWY1VV5wPprUWXH7h6w9fXgAgP5RJCd9a02yXKELUv+HcBEBmqFoEnKAfPvQqE3/B1OHA5sycV\nLRqe2S5+Tdrn325q38qlYDaT/9lOEBEY7fbr969zB3gjjNAVMYR+MhkMNw/W+XXbskDaF9hyUowP\nAuUVRSkIJJAyRD0hsyedP38tB4f0bUWLhkv7/NiN7TMeP0bhPXuwNWtOnN0IAdDuQHj/wps2J3jR\nfC5t2I7roUpp2wOhbRmR9vm3rPyhcTeXNmkAiqI8oyjKy9fPE/cFVgJbgK9UVT2bnaBC+BrrshgA\n7K3b6ZxE3Mjeui0A1qWLdU4ihGdlqWesqupxoN71/8+6YXsMEOOVZELoyBqzCM1oxNbiSb2jiBvY\nmzVHs1iwLl1C4oDBescRwmNk0Q8hbmE8dxbzzl9x1K2PVqSI3nHEDbSwcOyNHyfoj98wHv1T7zhC\neIwUYyFuYbk+RG27PiQqfIvt+qkD69IlOicRwnOkGAtxi9Rf8vZWbXROItJjb9EKzWTCukzOG4vA\nIcVYiBsYLl3EvGUjjho1cZcoqXcckQ6tUGEc9Rpijt2J8cxpveMI4RFSjIW4geXnFRhcLmytZaEP\nX5Z6CsGyXOaPisAgxViIG6SuumVvLUPUvsz+ZMr7I+eNRaCQYixEqmvXsKxbg7NiJVxlyumdRmTA\nfV8xHLUexbxlE4aLF/WOI0SOSTEWItWyZRjsdplF7SdsrdthcLuxrlymdxQhckyKsRCp5s8H/r10\nRvg22/WhaousxiUCgBRjIQCSk2HpUlwPPHjTmsfCd7kfLIOzUhUs69dCXJzecYTIESnGQkDKL/SE\nhJReseHWW3cLX2Vr3RaD3Q7LZKha+DcpxkLw740H5Hyxf0k7pXD9FIMQ/kqKsRAOB5YVS6FECZw1\naumdRtw2jHChAAAgAElEQVQFV4WKOMuWS+kZJyXpHUeIbJNiLPI885ZNGK9cgQ4dwCg/En7FYEi5\nzWVCApa1v+idRohsk988Is9Luzdux476BhHZknpqwbpMFgAR/kuKscjb3G4sy2JwFyoEDRvqnUZk\ng/PhGlCyJJaVy8Hh0DuOENkixVjkaUE7d2D6529sLVtDUJDecUR2GAwQFYXx6hXMmzfqnUaIbJFi\nLPK01CFqu8yi9m9RUYCsVS38lxRjkXdpGtalS3CHhWNv1ETvNCInGjTAXaRIynljl0vvNELcNSnG\nIs8y/bYf08nj2Ju3AKtV7zgiJ0wmbK3aYDz/D0E7ftU7jRB3TYqxyLP+XehD1qIOBGmzqmWtauGH\npBiLPMu6dDFacDD2Js30jiI8wNHgMdzh+VOGqjVN7zhC3BUpxiJPMh0+RJB6EHvjphAWpncc4QkW\nC/bmLTGdOknQ/r16pxHirkgxFnmS5foCEbIWdWBJPeUgt1UU/kaKsciTrEsXowUFYW/eUu8owoPs\nTZqihYRgjZFiLPyLFGOR5xj/OoV5z24c9RuiFSykdxzhSfnyYW/SjKDDhzAdUvVOI0SWSTEWeU7q\nGsa2Nu11TiK8wdYmZahaZlULfyLFWOQ5lqVL0AyGlCUwRcCxP9ECzWzGIqtxCT8ixVjkKYZ//sG8\nbQvOR+ug3Xuv3nGEF2gRBXA0fAzzvj0YT57QO44QWSLFWOQp1pXLMGiazKIOcKmzquW2isJfSDEW\neYo1ZhEAtielGAcyW8vWaEYj1iWL9I4iRJZkeM84RVGMwDSgKmADXlJV9c8bHu8G9AVcwAxVVT/z\nYlYhcsRw5TLmjetxVKuOu1RpveMIL9KKFsVRpx6WLZswnjuL+75iekcSIkOZ9Yw7ABZVVesBg4BJ\ntzw+AWgK1Af6KYoS4fmIQniGZcUyDE4ntrYyizovSH2fZSKX8AeZFeP6wAoAVVW3A7VueXwfUAAI\nAQyALAgrfFbqELW9jdwYIi+wp543jpGhauH7MhymBvIDcTd87VIUxaiqqvv6178DsUACME9V1bhb\nX+BWRYuGZyuov5D2+ai4OFi3BqpWpVDt6nfczW/bl0WB3L7b2lY0HOrVw7J1M0W1JLjnHn2CeUgg\nv3cQ+O3LTGbFOA648TuUVogVRakKPAmUBhKB7xRFeVpV1bkZveD589dyENe3FS0aLu3zUdZ5P5Hf\nbiehVVsS79AGf25fVgRy++7UtpCWbQjbsoVr384m+bkXdEjmGYH83kHeaF9mMhum3kxKwUVRlDqk\nDEunugokAbbrBfofUoashfA5qbNqbW076JxE5KbUVdasSxbqnESIjGXWM14APKEoyubrX7+gKMoz\nQJiqql8oijId2KQoih04AnztvahCZFN8PJY1q3BGKrgiFb3TiFzkLnk/juo1MG/agOHSRbRChfWO\nJES6MizGqqpqwGu3bD50w+PTgeleyCWEx1jWrMKQnCxrUedRtjYdMO/ehWXlcmzP/EfvOEKkSxb9\nEAFPhqjztrQbR8hQtfBhUoxFYEtKwrpqJc4Hy+B6qJLeaYQO3A+WwVG5Kpb1azFcvaJ3HCHSJcVY\nBDTL2l8wJCZgb9sBDAa94wid2Nu2x+BwYPl5hd5RhEiXFGMR0FKHJmXVrbwt9RSFrFUtfJUUYxG4\nbDYsK5fjKlUaZ9WH9U4jdOQqVx5nhYpY1q7GEB+417MK/yXFWAQsy/o1GOOvpdxOT4ao8zxbm/YY\nbDYsq1bqHUWI20gxFgHLGrMYkCFqkSJtqPr650IIXyLFWAQmux3L8qW4ipfAWePW+5uIvMhVoSLO\ncuWx/PIzJCToHUeIm0gxFgHJvGkDxqtXsLVuC0b5mAvAYEgZqk5MxLJmld5phLiJ/JYSAcm6aD4A\ntrZP6ZxE+JK0oepFC3ROIsTNpBiLwGO3Y10Wg6tYcZyP1tY7jfAhrspVcJYth3XVChmqFj5FirEI\nOJb1a1KGqNs9JUPU4mYGA7b2URiSklIKshA+Qn5TiYBjXXh9iLq9DFGL29naRwH/fk6E8AVSjEVg\nSU5OmUV9fymcNR/RO43wQa6KD+FUKmD55WcM1+L0jiMEIMVYBBjLmtUpC320e0oW+hB3ZGsflbIA\nyIplekcRApBiLAKMdfH1IeoOUTonEb7M1qEjANbFMqta+AYpxiJwJCZiXbEc1wMPylrUIkOucuVx\nVqqCZc1qua2i8AlSjEXAsPzyM4bEhJQJOjJELTJha/9Uym0Vly/VO4oQUoxF4Ai+Pjs2ub0MUYvM\npX5OghfO0zmJEFKMRaCIj8eyeiXOcuVxVaqsdxrhB9wPlsFRrTrmDeswXLqodxyRx0kxFgHBumoF\nhqQkGaIWd8XWPgqD04l1WYzeUUQeJ8VYBIR/F/qQIWqRdbZ219eqlgVAhM6kGAu/Z7gWh2XNKpwV\nKuKqUFHvOMKPuEuVxlGzFuZN6zGcP693HJGHSTEWfs+yfCkGm016xSJbbO2jMLjdWGMW6R1F5GFS\njIXfS124QYqxyA5bu5Q1zFNvuymEHqQYC79muHQRy5rVOCpXxVWuvN5xhB9yFy+Bo3ZdzFs3Yzx7\nRu84Io+SYiz8mnXxQgxOJ7aOnfWOIvxYcsfOGDQN6/y5ekcReZQUY+HXguf+iGYwYIt6Wu8owo/Z\n2nVAM5sJnvuj3lFEHiXFWPgt44njmH/dhqNBI9zFiusdR/gxrVBh7E2fIOj3/ZgO/KF3HJEHSTEW\nfit4/k8AJD/dReckIhCkfo6C583ROYnIi4IyelBRFCMwDagK2ICXVFX984bHHwEmAQbgHPAfVVVt\n3osrxHWahnXuj2hWK/bWbXP0Ui6Xi/Xr1xISYsJoDKFgwYIUKFCQggULYjabPRRYeIrD4eDKlStc\nuXKZy5cvc/nyJQoVCuPhh+vk6P2yP9ESd1g41vk/kfDucDBKX0XkngyLMdABsKiqWk9RlNqkFN4O\nAIqiGIDPgY6qqh5VFKUnUBo45M3AQgAE7d9L0OFDJLd7Ci1/RLZew2az8dNPs5ky5UOOHTua7j7F\ni5cgOnoIXbt2wyDLbOpG0zQWLJjLmDEjOXnyRLr73H9/Kd54422efbY7wcHBd3+QkBBsbdsTMus7\nzNu34qhbP4ephci6zIpxfWAFgKqq2xVFqXXDY5HARaCvoiiVgaWqqkohFrnCOjdlKDE7s6gTEhL4\n/vuZTJ36CWfPnsFisdC9ew+qVq3EX3+dS+ttXblymdjYnbz99uv88MO3jB//IRUrPuTppohMHDly\nmOjofmzcuI7g4GDq129IwYKFrv9LGcW4cOEsM2bMYNCgfkyaNI7XXutNjx4vEhYWflfHsnXsTMis\n77DOnSPFWOQqg6Zpd3xQUZQvgHmqqq64/vUJ4EFVVd2KotQHVgHVgT+BGGCcqqprMziedv78NY+F\n9zVFi4Yj7csFLheFHq6IwW7j4v7DYLFk+amzZ3/P++8P5eLFi4SGhvLccy/y2mtvUqxY8XTbd/r0\nXwwZEs2yZUsICgqiV6836NcvmrCwME+3yut85v3LoqSkJD7+eCKffvoxdrudZs2aM2bMBB544MHb\n9i1aNJzff/+Tzz+fxowZXxAff40CBQowaNAwXnzx5awf1OWiUPWHMCQlcfG3w2C1erBF2edv793d\nygPty3xYTdO0O/6LjIycFBkZ2emGr0/d8P8KkZGR+274uk9kZOSAjF5PE8ITVq3SNNC0Xr3u6mkf\nf/yxBmgRERHa8OHDtfPnz2f5uTExMdoDDzygAdr999+v/fzzz3ebWtyFDRs2aGXKlNEArWTJktq8\nefM0t9udpedevnxZGzVqlFaoUCEN0MaMGXN3B+/XL+XztWBBNpILka4Ma62maZn2jKOAtqqqvqAo\nSh1gmKqqra8/ZgEOAk+oqvqnoijzgC9VVV2eUe0P8L9+Av2vO59oX3jvVwn+8QcuL16Js07dLD3n\ns88+Zfjwd7n33vuYPz+G8uUjb9sns/YlJiby0UcTmTr1Y4xGIz/+uIB69Rpkux25zVfev8zExu6g\nY8e22Gw2XnnldQYMGJzpSER6bTt+/BhRUW34669TREcPoV+/6Cwd37R/H4WaNsDWpj1xM77Ndjs8\nyV/eu+zKA+3LtGecWTE28O9saoAXgJpAmKqqXyiK0gQYS8ps6s2qqr6TyfGkGPsxn2hfYiKFK5VD\nK1SISzv2ZWnG66effszIkcO4775iLFgQQ9my6S+bmdX2rVu3hm7dOhEcHMLChcuoUqVqps/xBT7x\n/mXi0CGVtm2bExcXx//+9z0tWz6ZpefdqW0nT54gKqoNJ0+eoH//QQwYMDjziXiaRsFGtTEdO8rF\n34+gRRTITlM8yh/eu5zIA+3LtBhnOIFLVVUNeO2WzYdueHwtUDtb6YTIBuvPyzEmxJPw8qtZKsQf\nfzyJ0aPfp3jxEsyfH0OZMmVznKFx48eZOvVzevV6ka5do4iJ+ZkHHyyT49fN606f/osuXZ7i8uXL\nfPzxtCwX4oyUKlWaBQuWEhXVhokTx+J2u4iOHppxQTYYSH66C2Gj38cas5jkbs/lOIcQmZEL6YRf\nsV5frjArs6gnTx7P6NHvU7Lk/SxcuMwjhThVhw4dGTNmPOfP/0Pnzh34+++/PfbaedGlSxfp0uUp\nTp/+i2HDRvLMM//x2Gvff38pFi5cxoMPlmHy5AmMHv0+GY0IAtiiOgH/ft6E8DYpxsJvGC5ev0NT\nlWq4lAoZ7jt79veMHTuKUqVKs3DhsnRn4OZUz5696NcvmhMnjvPMMx2Ji7vq8WPkBQkJCXTr1olD\nh1ReffVN3nzzbY8fo0SJkixcuIyyZcvxySeT+eqr6Rnu776/FPY69TBv2YTx9F8ezyPEraQYC79h\nXTQ/S3doOnbsKIMHDyA8PD9z5y6mVKnSXss0cOC7PP98T377bR/PPfcMycnJXjtWIHI4HPTs2Z3Y\n2J106tSVESNGeW1xlWLFijN/fgyFCxfm/feHcfDggQz3t6XeyWneT17JI8SNpBgLvxE8+zs0kwlb\nx0533MfpdPL66y+TkBDPuHGTvNIjvpHBYGDs2Im0bduBLVs2MWbMSK8eL9BMnjyeNWtW06xZcz76\naCpGLy9BWaxYcSZP/hSbzcarr/bEZrvz6r229k+hWa0E//g9ZDKsLUROSTEWfsH0+2+Y9+zG3qw5\n7nvvu+N+kyePJzZ2B1FRT/N0Lt1AwmQyMWXKZ5QtW47p06eyZcumXDmuv9u9O5aPPppIyZL3M336\njFxbB7xVq9Z0796DP/74LcM/nrQCBbG1bkvQ4UME7fw1V7KJvEuKsfALwbNSrvdMfqb7HffZsWM7\nkyePp2TJ+xk3bnJuRQMgNDSUKVM+w2Aw8NZbrxMfH7iXaXhCUlISvXu/isvl4uOPpxEenj9Xjz9y\n5P9RpkxZ/vvfKWzYsO6O+6V+3oJ/8I3rjUXgkmIsfJ/NRvDcH3EXKYr9iRbp7hIff43XX38ZTdP4\n9NPpROhwbWitWo/Su/c7nDx5nBEjhuX68f3J2LGjOHRI5aWXetGw4WO5fvx8+fLx3/9+SVBQEL17\nv8rly5fS3c/R8DFcJe/HunA+xMfnckqRl0gxFj7P8vNyjJcukdypK9xhKHPIkJRZzb17v6Prqlj9\n+w+iYsVKfPPNDNasWa1bDl+2bdsWPvvsU8qUKcvQoe/rlqN69ZoMGDCYs2fP0L9/n/QvdzIaSe7a\nDWNCPNaYRbkfUuQZUoyFz0sdIky+w7WnS5YsYtas76ha9WEGDnw3N6Pdxmq18umn0zGbzbzzzptc\nuXJZ1zy+Jj4+nt69X8VgMDBlymeEhobqmuett/pSu3ZdlixZyI8//pDuPslduwEyVC28S4qx8GnG\nM6exrP0FR81auCpUvO3xq1evMHBgH0JCQvjvf7/Echd3cPKWKlWq0r//IM6ePcO77w7UO45Pef/9\nYZw4cZw33+zDI4/ov3ifyWRi6tTPCQ/Pz5Ah0Zw/f/62fdylSmNv2BjLti2Y/jysQ0qRF0gxFj4t\neM4sDG73HSduTZw4josXL9KvX3S6N3/QS+/e71C9eg3mzv2RmJjFesfxCWvWrGbmzK+oWPEhBgwY\nrHecNKVKlebdd4dz7VocY8d+kO4+yc+mjMoEz06/9yxETmV4owgvkBtF+LFcb5+mUaj2wxj/PsfF\n/YfQ8kfc9PCRI4dp1Kg2JUqUZNOmHVhzeO9ZT7fv8OFDNG3agLCwcLZujdVlUtmN9Px8JiQk0KDB\nI/z99zlWrlxLlSrVPPr6OW2b0+nk8cfro6oHWb164+03/0hKonCVSLTQUC7t+h2CMlzW3+Pkd4t/\ny8qNIqRnLHyWedsWTMePYWvT/rZCDPDee+/idDoZMWJ0jguxN5QvH0nfvgO5cOE8EyeO1TuOrqZM\nmczp03/x+utvebwQe0JQUBAjR/4fmqYxbNig2ydzhYRgi3oa07mzWNb9oktGEdikGAuflTZx69nb\nh6jXrFnNqlUradCgEU8+2Sa3o2XZa6/15oEHHuTLL6dnuvxioDp+/BhTp35CsWLF6dOnv95x7qhx\n48dp0aIVW7ZsSvfUQurnMPiH73I7msgDpBgLn2S4Fod1yUJcpR/AUbf+TY85HA6GDx+M0Wjkgw/G\nem0tY0+wWq2MGjUWl8vFkCEDM71bUCAaPnwwNpuNESNGERYWpnecDL3//mjMZjPvvz/0tnXGndWq\n46xYCcvKZRguXNApoQhUUoyFT7IuWoAhMTHlcqZb1iueOfMrDh1S+c9/elCpUmWdEmZd8+ataNas\nORs3rmfJkoV6x8lVv/zyMytWLKNevQZ06NBR7ziZKlOmHC+99ConT55g+vSpNz9oMJD87H8wOBwE\nz5NbKwrPkglcHpQHJiHkWvsKPNmMoNgdXNr1O+4SJdO2X7p0kTp1quN2a2zbtpsiRYp47JjebN/R\no0do1KgORYvew6ZNO8iXL59XjpOR3P582mw2HnusDidOHGf16o1e/cPJk22Li7tKnTrVSUxMYtu2\nXdx3X7G0xwwXLlC4moKrXCSX122BXBqVkd8t/k0mcAm/ZDrwB+adv+Jo/PhNhRhg/PgxXLlyhX79\noj1aiL2tTJlyvPrqm5w+/RdTpuTuutl6mT59GkeP/skLL7zkFyMYqfLnj2DQoGEkJiYwevTNK4Rp\nRYpgb/EkQQd+Jyh2h04JRSCSYix8Tsj/vgAgqcdLN20/ePAAM2fOoGzZcvTs+Yoe0XKkT5/+FCtW\nnKlTP+H48WN6x/Gqs2fPMHnyeAoXLqz7qmjZ0a3bc1SqVIUff/yB3btjb3osqUdPAEL+96Ue0USA\nkmIsfIrhWhzWn37EVaLkbTeFGDNmJC6XixEjRvvESlt3KywsjPfe+wCbzcbw4b6z6IU3vP9+Ss9y\nyJARFChQUO84d81kMjFqVMrlaKNG3dw7djR8DGe58lgXzZeJXMJjpBgLn2KdMwtjQjzJz79408IK\ne/bsYsWKpTzySG2aN2+pY8Kceeqpp6lbtz4rVixjzZpVesfxim3btjB//k88/HB1nk3nsjR/Ub9+\nQxo3fpyNG9exefPGfx8wGEh+4SUMdrusVy08Roqx8B2aRsj/vkQzm0l69rmbHho7dhQAgwcP8+lL\nmTJjMBgYM2YCRqORYcMG43A49I7kUSmXcEUDpLXTnw0aNBRI+fzdONk1ufMzaKGhhMz8ClwuveKJ\nAOLfPykioJg3byTokIqtbQe0e+5J2759+zbWrFlNgwaNaNCgkY4JPaNSpcp07/4Chw8f4n/Xz48H\nitmzv2f//r08/XQXatV6VO84OVajRi1atGjF9u1bWbduTdp2LaIAyR27YDp1EssvP+uYUAQKKcbC\nZ6ROiEl64eWbto8bl9Irjo4emuuZvCU6egj580cwYcJYLl68qHccj7h2LY7Ro98nNDSUYcP0u0+x\npw0cOASAsWM/uKl3nPRCygTDkBmB9QeV0IcUY+ETjGfPYFm2BGelKjgf/ffWehs3rmfTpg08/ngz\nateuo2NCzypSpAj9+0dz9eoVxo8frXccj/jww4lcuHCe3r3foVix4nrH8ZgqVarStm0Hdu/exc8/\nr0jb7qpcBcejdbCsWY3x2FEdE4pAIMVY+ITgb7/G4HKl9DaunxPWNC3tXHHqubtA8uKLr1C2bDlm\nzpzBgQN/6B0nR44dO8rnn0+jZMn7ef31t/SO43EDBgzGYDAwduwo3G532va03vHMGXpFEwFCirHQ\nn8NB8Ldf4w7PT3LHzmmb165dzY4d22nZsjUPP1xDx4DeYbFYGDlyDG63m2HDBvv1utUjRgzFbrcz\nfPhIQkJC9I7jcRUqVCQqqhO//76fpUv/vYmErU173EWKEjzrW0hK0jGh8HdSjIXuLMtjMP19juSu\nz8L1ZSJv7BVHRw/RM55XNWvWgiZNmrJhw1pWrlyud5xs2bhxPcuXx1C7dl3at4/SO47XDBgwCJPJ\nxLhxo3GlzqC2Wkn6z/MYL1/Gumi+vgGFX5NiLHSXOgEm+YaJWytWLGPPnt20bx/lV0sp3i2DwcDI\nkf+HyWTivffexWaz6R3prjidToYOHYTBYGDUKN++g1ZOlSlTji5dnuXQIZUFC+ambU9+7gU0o5GQ\nGZ/rmE74OynGQlemgwewbNmEvWFjXOXKA+B2uxk3bjRGo5EBAwJ7pSoARanACy+8xLFjR/nyy+l6\nx7kr3303kwMHfqdr125Uq1Zd7zhe17fvQMxmMxMm/B9OpxMAd8n7sTdvhXnPboJuWTpTiKzKsBgr\nimJUFOUzRVG2KIqyVlGUsnfY73NFUf7POxFFIEvtTSS9+G+veNmyGP744zeiojoRGanoFS1XDRgw\nmIIFCzJ58nj+/vuc3nGy5PLlS4wbN4p8+cJ499339I6TK0qVKs2zzz7HsWNHmTdvTtr21M9viJ/9\nMSV8R2Y94w6ARVXVesAgYNKtOyiK0guoDPjv7BOhC8OFCwTP/h7X/aWwt2gFpJwrnjx5PEajkb59\nB+qcMPcULFiIQYOGce1aHMOGDdI7TpaMHDmcixcv0q9fNPfee6/ecXLN22/3xWw289FHE9POHTsa\nNcYZqWBdMBfjmdM6JxT+KLNiXB9YAaCq6nag1o0PKopSD3gUmA4E7ski4RUhMz7HkJxM0qtvpK1D\n/fPPK/jtt3106BBFuevD1nnF88+/SM2atVi4cL7Pr1u9bdsWvv/+Gx56qDK9er2ud5xcVbLk/XTt\n2o0//zzCotRJW0YjSa+/hcHpJOTz/+obUPilzIpxfiDuhq9diqIYARRFKQYMB95ECrG4W4mJhMz4\nHHfBgmnrUGuaxqRJKZOA+vQZoHPA3Gc0Gpk48RNMJhMDB/YjMTFR70jpstvt9O//NgaDgYkTP8Js\nNusdKdf17v0OJpOJyZPHp113nNyxM6577yP4m/9hiLuqc0LhdzRNu+O/yMjISZGRkZ1u+PrUDf/v\nHRkZuTMyMnJtZGTkgcjIyBORkZHPZfR6mhCpPv1U00DThg5N27Rs2TIN0Dp16qRjMP0NGDBAA7RB\ngwbpHSVdo0aN0gDttdde0zuKrnr06KEB2pw5c/7dOHZsyud63Dj9gglflGGt1TQNg5bBQgOKokQB\nbVVVfUFRlDrAMFVVW6ez3/NABVVVM5v6qp0/fy3bfzj4uqJFw5H2ZYHLRaE61TGeO8vFXX+gFS2K\npmk8+WRTYmN3snbtFl0uZ/KV9y8hIYFGjWpz9uwZfvllExUrPuSR1/VE+44e/ZPGjeuSP38Emzfv\nICKigEey5ZQe793Ro0eoV68WilKRtWs3YzQaMVy9QqHqldDCwri0Yx9YrR45lq98Nr0lD7Qv09Hj\nzIapFwDJiqJsJmXy1juKojyjKMrL6ewrE7hElliWLsZ04jjJXbqhFS0KwPr1a4mN3UmrVm0C+rri\nrMiXLx9jx07E6XTSv//bNy2/qCdN04iO7ktycjKjRo31mUKslzJlyhEV1YkDB35n+fKlwPW7OXXv\ngencWazzf9I5ofAnGfaMvUB6xn7MI+3TNAq0aEzQ3j1c3hqLq0w5NE2jXbuWbN++ldWrN1C16sOe\nCXyXfO3969nzOZYsWcjEiR/z3HMv5Pj1ctq+efPm8NprL/H4482YNWueTy3wodd7d/jwIRo0eITK\nlauyevUGDAYDxtN/UeiRqrjKluPy+m3ggXs6+9pn09PyQPty3DMWwqPMWzZh3rMb+5NtcZUpB8CW\nLZvYvn0rzZu31K0Q+6LRo8cRFhbOBx+8xz///KNrlitXLjNs2GBCQkIYN26yTxViPZUvH0n79k+x\nf/9eVq1KuaOTu0RJbFGdCFIPYlm9UueEwl9IMRa5KmTqxwAkvvHvnX0mTRoHkKeuK86K++4rxpAh\nw7l69QoDBvTR7UYSmqbx7rsDuXDhPP36RVO69AO65PBV77yT8rmdNGlc2nuUeP3OVSFTP9Etl/Av\nUoxFrjEd+APr6p9x1K6Ls9ajAGzbtpVNmzbQpElTatSolckr5D09erxE/foNWb48hmnTpuiSYebM\nGcyd+yM1atTk1Vff1CWDL6tY8SHatGnP7t27WLt2NQCuhyphf7wZlq2bCYrdoXNC4Q+kGItcEzot\npZeQ+GaftG2TJo0FoG/faF0y+TqTycT06f/j3nvvY9So99iyZVOuHn/Xrp0MHRpNoUKF+Oqrb7FY\nLLl6fH+ROqozceINvePrn/NQ6R2LLJBiLHKF8cxprPN/wlk+EvsTLQDYsWM769evpWHDxtSuXUfn\nhL7rnnvu4csvv8FgMPDyyz04d+5srhz34sWL9Oz5HA6Hg88+m0GJEiVz5bj+qHLlKrRs2ZqdO39l\n/fq1ADjqN8RRrXrK1QN/HtY5ofB1UoxFrgj9cCIGh4PE3u+kzS6dODGlVzxggH+sxayn2rXr8N57\nH3D+/D+89NLzOBwOrx7P5XLx2ms9OX36L6Kjh9C48eNePV4g6N8/ZXRn4sSxKb1jg4HEt/pi0DRC\nJ3lL/CcAACAASURBVI3XOZ3wdVKMhdcZTxwn+PuZOMuWw/Z0FwBiY3ewdu0vNGjQiDp16umc0D+8\n8srrtG8fxa+/bmPkyOFePdaECf/HunVreOKJFvTp09+rxwoUVas+TIsWrfj1121s3LgeAHvrtjgr\nVcE6bw4m9aDOCYUvk2IsvC508ngMTieJ/Qel3RAitVfcv7/0irPKYDDw4YdTKF8+kunTp7J48QKv\nHGfVqhVMnjyeUqUeYOrUzzF64DrZvCL185zWOzYaSYgektI7niB3mRV3Jj9lwqtMR48QPGcWTqUC\ntg4dgZRJQb/8sop69RpQr14DnRP6l7CwcGbM+I7Q0Hy8/fYbbN262aOvHxu7gzfeeAWr1cr//vct\nBQoU9OjrB7pq1arzxBMt2LZtC5s3bwTA3qIVjuo1CF68ANNv+3VOKHyVFGPhVaETxmJwuUgY+C6Y\nTMC/1xVLrzh7FKUCU6Z8hs2WzNNPt+OHH771yOvOn/8THTo8SVxcHJMnT6FKlWoeed285sbeMQAG\nAwnRQwDIN36MXrGEj5NiLLzGpB7EOv8nHJWrYm/dDoA9e3axatVK6tSpR/36DXVO6L/atm3PTz8t\nIiwsjD593mDEiKFpN7q/W263m7FjR/Hqqz2xWKz88MNPdOrU1cOJ847q1WvStOkTbNmyKe1SNEeT\nZjgeqY11xVKC9uzSOaHwRVKMhdfkGz8Gg6aRGD0kbQb1jb1iWVIxZ1IWA1lD+fKRTJv2Cc8//wzx\n8Xe3vm9iYiIvv9yDyZPHU7r0AyxbtprHH3/CS4nzjnR7x4OGApBv7Ci9YgkfJsVYeIVp/z6sSxbi\nqFETe/OWAOzbt4eVK5fz6KN1aNjwMZ0TBoYyZcqybNlqGjd+nJ9/XkHr1k/wZxavaT1x4jjt27di\nyZKF1KlTjxUr1qIoFbycOG+oWfMRmjRpyqZNG9i2bQsAjoaPYa/fEMua1QT9ul3nhMLXSDEWXpFv\nQsq5sYSBQ+B6D3jiROkVe0NERAF++GEuPXu+woEDf1C3bk0ee6wuH3zwHlu3bk67JtnpdLJ9+zbG\njBnJ44834JFHqrJ3726eeeY/zJ27mMKFC+vcksCS2jueMGFs2raE6Ou943HSOxY3k1soelAeuA1Y\nltoXtDuWgi2a4Hi0DleWrASDgf3799K0aUNq1XqUpUtX+WQxDoT3b8GCucyZM4vNmzeSnJwMQP78\nETz8cA3279/D5cuXAbBYLNSr14CoqE506fKsT74fd8NX37tOndqzfv1aFi9ekXY9fUSXp7Cs/YUr\n82NwNGiUpdfx1fZ5Sh5oX6Y/YFKMPSgPfKAyb5+mEdG5A5b1a7myYCmO65O0nnmmI7/8soo5cxb6\n7GpOgfT+JSYmsmXLxv9v777DoyrWAA7/tqcXSKgiSHG80q5SRLpIUzqoVxAp5l4URFDQCEiRHqmK\nNBFQFCvKpYnSrxoERBSIgkMo0oVASM9udrN7/9glJJCCEnKSzbzPc55kz8wm3+Rk9zs7Z84MW7Zs\nYuvWzZw+fYo77riDNm3a07Zte5o3b0lAQIDWYRaa4nrs9u7dQ6dO7WjSpClr136NTqfD+PNPhHZs\ng71hYxK+2pLVc5Sf4tq+wlIK2lfgQTYWRSBK6WHe/A3mb3eQ0bpNViLevfsHtm3bQvPmLWnV6iGN\nIywd/Pz8aNu2A23bdsDlchEfH48QVbl0KUXr0EqVRo0eoEOHR9i06Wt27HAPjnPc3xDbo12wbFyP\n5b9fYOv5uNZhKsWAumasFB6bjYBxo3AZDKRMdl8nc7lcTJ06EYDRo8eV+O7Qkkin01G2bFn1t9fI\nqFHjAJg6dRJOpxOAlNen4LJY8J84DlJTtQxPKSZUMlYKje+SRRj+OEF6xCAyPaNyd+zYyp49u+jQ\n4REaNXpA4wgVpejVrl2Hnj0fIybmAF99tQ4AZ7W7SBv8Aobz5/B7e67GESrFgUrGSqHQX/gTvzkz\ncJYp456DGvdkElOnTgKufTpQlNIoMnIMBoOBqKgpOBwOANKGjSCzQkX8Fs5Df+qkxhEqWlPJWCkU\n/lMnok9NIXX0eFye+Yw3bFhLTMwBevZ8jNq162gcoaJop3r1mvTp8zSxsUf44ovP3DsDAkgdNxGd\n1UrARHWyWtqpZKzcMuPPP+Hz6Uc47q2DtW9/wH1Pa1TUFAwGA5GRr2kcoaJob+TIV7FYLMycOR2b\nzQaArdcT2Bs0wrJ+Dabo7zSOUNGSSsbKrXE6CXgtEoCUaTOyFoNYtepTjh6NpU+fflSvXkPLCBWl\nWKhUqTIDB/6H06dPsXLl++6der37dQMEjB0Fni5spfRRyVi5JZYvPsO07yesXXtg9yyHaLPZmDlz\nOhaLhZEjIzWOUFGKj2HDRuDvH8CcOTNJ9YyidtzXgPTefTEe+hWfD9/XNkBFMyoZK39fSgr+kyfg\n8vEhdcLkrN0ffLCcM2dOM3Dgf6hUqbKGASpK8RIWFsZzzz1PXNxFli5dnLU/dcwEnAGB+L8xBd2V\neA0jVLSikrHyt/lHTcZw4U/ShgzDWeVOABITE5gzZwb+/gEMGzZC4wgVpfgZPHgooaGhzJs3l7i4\nOABc5cuTNiISfXy8+95jpdRRyVj5W0y7f8D33cU4atYibfjIrP2zZr3B5cuXeemllwkLC9MwQkUp\nnoKCgomMHENychJR2ZZTTB80GEftuvh+/CGm7Vs0jFDRgkrGyl+XlkbgsMEAJL+1EHx9AYiNPcKy\nZe9QtWo1Bg0aomWEilKs9e8fwT33/IOVK98nJuage6fZTNK8RbiMRgJHDEOXlKhtkEqRUslY+cv8\np09yz7T13FAc2WbVGj9+NA6Hg4kTp+Hj46NhhIpSvBmNRiZPjsLlcjF27KtcXbAns2490l58GcO5\ns/i/PlbjKJWipJKx8peYdv+A75JFOGrUJHXUtTeLrVs3sW3bFlq0aM0jj3TSMEJFKRlatXqIjh07\nsWvXTtavX5O1P+3Fl93d1StXYNq+VcMIlaKkkrFy89LSCBju7n5OfmtRVve03W5n/Pgx6PV6Jk+e\nrhYkUJSb9PrrUzCZTEycOI709HT3zhzd1S+o7upSIt9kLITQCyEWCyF+EELsEELUuK68txBitxAi\nWgixSAih3oW92dixGE8cd3dPN77WPb18+RKOHo2lf/9nuPfe2hoGqCglS/XqNXj22ec5ffoUixa9\nnbU/R3e1Gl1dKhT0ybg7YJZSNgVGAbOvFgghfIHJQGspZXMgGOh8uwJVtGXcvQvefPOG7ulLly4x\nc2YUISEhvPqqmvZSUf6ql156mfDwcsybN4fz589l7c/qrv7wfdi8WbsAlSJRUDJuBnwDIKXcAzTM\nVmYFHpRSWj2PjUB6oUeoaE6XnETg8Kujp691TwNERU0hKSmRyMgxlClTVqsQFaXECgwMYuzY10lL\nS2Py5AnXCrJ1VxMRgS7+snZBKred7uoovtwIId4FvpRSfuN5fBK4S0rpvK7eC0BHKWVBI3fy/mVK\n8eRywWOPwerV8OqrEBWVVbR//34aNGiAEIIDBw5gMpk0DFRRSi6n00njxo3Zt28f0dHRNGvW7Frh\n1Kkwdix06ABffZU1/7tSohR4CddYQHkSEJjtsT57IhZC6IEZQE2g181EFBeXfDPVSqTw8ECva5/v\nwrcJWL2ajKbNMU+ZktU+u91Ov34DcDqdTJw4nYQEK+7OkpLLG49fdt7cPm9o28SJUXTp0p6BA59h\n27boa7cH/nso4Tt3wtdfkzp6HGmvjNY20NvAG45ffsLDAwusU1A39U7gUQAhRBPg4HXl7wAWoEe2\n7mrFS5h27cR/8ngyy1cg6Z33wHjt3G3BgreIiTlA7959ad26jYZRKop3aNz4ASIiBhEbe4TZs9+4\nVqDXw8qVZFa5E79ZUWp2Li9VUDe1DlgI1PPsGgg0AAKAnzxb9kU435JSriFvLm8/+/GW9ukv/EnI\nwy3QX75Ewn834mjyYFb7fv/9MG3btiA0tAzff7+HkJBQrcMtFN50/HLjze3zlralpKTQunVTzp49\nzTffbKd+/fsAd/uubP2OkE7tcPn7c2Xr91nzwXsDbzl+eQkPD7y1bmoppQsYfN3uI9m+VxcvvJHd\nTuB/BmC4eIGUSdNwNHkwq8jhcPDii0PIyMhg1qy3vCYRK0pxEBAQwNy5b9OrVxeGDRvCli3fYjab\nAXDUv4+U6bMIHDmMoIinSVi/GSwWjSNWCoua9EO5gf/UiZh3/4CtS3fSn30+R9nixQv4+ed99Or1\nBB06PKJRhIrivVq0aEW/fs9w+PBvzJ07M0eZtW9/rE8+hWn/LwSMHaVRhMrtkG839W2guqmLOcva\n1QT9ZwCOmrVI2LQDV2BQVll8/Dnq169PYGAQ0dE/et2tTN5w/PLjze3ztrYlJyfRsmUTLlz4k82b\nv+Whh5pea19aGqGd2mH8LYaktxZi691X22ALgbcdv+vdTDe1+mSsZDFFf0fg84Nw+geQtHxljkSc\nmZlJREQENpuNN96Y43WJWFGKk8DAIGbPnofD4WD48CHY7fZrhX5+JC77AGdICIEjXsC8dZN2gSqF\nRiVjBQBjzAGC+vUGIGnFx2Te848c5cuWvcPOnTvp2rUHXbp00yJERSlV2rRpS+/efYmJOcCMGTNy\nlDmr1yBx5SowmwmK6Idx7x6NolQKi0rGCvoTxwl+she61BSSFr6LvWXrHOUxMQeZMuV1wsLCmD59\nlhYhKkqpNGnSNCpUqMjEiRPZe13CdTR+gKSlKyAjg+CnHscgf9coSqUwqGRcyukuXCDkie7o4y6S\nMn0WGV175ChPSLjCwIF9sVqtvPfee4SHh2sUqaKUPsHBISxYsITMzEz+/e/+xMXF5SjPaNeR5Lnz\n0SckEPyvHujPntEoUuVWqWRciumSkwju3QvDyT9IHRGJ9Zn/5Ch3Op08//wgTp36gxEjXqFzZ7UO\niKIUtRYtWjFt2jTOnz/Hs88OxOFw5Ci3PfkUKeMmYTh3luB/9VBzWJdQKhmXVqmpBPXvg+nXg6Q/\nPZC0XFZcmjt3Jlu2bKJ16za88soYDYJUFAUgMjKSRx/tQnT0d0ybNumG8vShw0l7bijGI5Lgp55A\nl5ykQZTKrVDJuBTSJVwh5InumKO/w9apKykz5oAu58j77du3MGPGNKpUuZPFi5dhUJPTK4pmdDod\n8+YtpEaNmsyf/ybr16+9vgKpr0/B+viTmPbtJbhHZ3TXdWkrxZtKxqWM/sKfhHR7BNPePVh7Pk7S\nkvduWAXm5Mk/eO65CEwmE8uWfaBuY1KUYiAoKJj33vsIPz8/hg8fQmzskZwV9HqS5y0ivW9/TAf3\nE9K1A/rTp7QJVvnLVDIuRfQnjhPSqT3Gw4dIjxhE8sJ34bplD9PT04mI6EdCQgJRUbP55z/v1yha\nRVGud889/2DOnLdJSUlm4MCnSElJyVnBYCBl9jzSXngJ47GjhHRur0ZZlxAqGZcShl9jCO3cHsOp\nP0iNHEPKtJnu1WCycTgcDB36LAcP7uepp/rRt29/jaJVFCUvPXs+zqBBgzlyRDJo0AAyMjJyVtDp\nSB03kZTxkzGcP0dI1w4Yf/5Jm2CVm6aScSlg2rWTkO6Poo+7SPL0maS9POqGa8ROp5Phw4ewfv0a\nHnywmbqfWFGKsQkTptCmTVu2bt3Mc89F3DDCGtyDupLfXIAuMZGQnl0wbd+qQaTKzVLJ2Ju5XPgu\nmk9wz87o0lJJWrQUa8SzuVRz8eqrI1m16lMaNGjIRx99fm1hc0VRih2TycTy5Stp2rQ5GzasZfjw\nITidzhvqWfs8TdKyD8FhJ7jPY/jNnQm51FO0p5Kxl9IlJRL0zNMETBiDq0xZEletxdbriRvquVwu\nJkx4jRUrllG7dl0++eRLAgICNYhYUZS/ws/Pj5UrP6NBg4asWvUpkZEjyG3hn4xOXUhYsxFnhYr4\nT59MUN8n0F2J1yBiJT8qGXshw68xhLRrheWrdWQ82Iwr26OxN2uRa90ZM6axePF8atW6m88/X6PW\nJ1aUEiQgIJBPPvmSOnXq8cEHy5kw4bVcE7KjYWOubIsmo3UbLFs3E9q2JcZf9mkQsZIXlYy9icuF\n5dOPCH30YYwnjpP2wkskfrkeZ/kKuVZ/++03mT37DapWrcYXX6xTU10qSgkUEhLK55+v4e67BYsX\nz+eNN6bkmpBdZcuS+MmXpL4yGv2Z04R06YDP8nehaJfRVfKgkrGX0J89Q1C/JwkaNhiXxYfEDz8j\nddxEMBpvqOtwOBg79lUmTx5PpUqV+fLL9VSsWEmDqBVFKQxhYWGsWrWWqlWrMWfOTF5+efiNo6wB\nDAbSXhlN4mf/xRUYSOCoke45rf84UfRBKzmoZFzSZWbi+84CyjRrhGXT12Q0a8GVLd+S0eGRXKsn\nJibQp89jLFmyCCHuYc2ajdx5Z9UiDlpRlMJWsWIl1qzZSJ069fjww/d5/PFuXL6c+zzV9tZt3N3W\nDz2M+X/bKdOqCb7z5kL2dZOVIqWScQlmjDlAyCNtCBg3GpfFTNK8RSSu3oCz2l251j92LJaOHdvw\nv/9tp337jmzcuJVqedRVFKXkqVz5Dtav30Tnzt3YtWsnHTq05tCh33Kt66xUmcRPV5O0eBku/wAC\npkwgtF0rjPv2FnHUCqhkXCLpLl7E/7VIQtq3xrT/F6yPP0n8zn3YnnzqhvuHr9qxYxsdOz7MsWNH\nGTr0RVas+ITAwKAijlxRlNvN39+fpUtX8PLLozh16iSdOrXjm2825l5Zp8PW83Hid+4lvW9/jId+\nJeTRtgS8/CL68+eKNvBSTiXjEkR3+TL+E8dRtlFd/N5djLPKnSSsWkvygiW4wsJyfY7dbufNN2fR\nu3cv0tPTmD//HcaPn6QWflAUL6bX64mMHMPSpStwOjPp3783UVGTsdlsudZ3hZYhZc7bJKz7hsxa\nd+P7wXLKNK6P/9hX0V24UMTRl04qGZcAuivx+E2bRJmGdfFb8BbOkFCSZ8wlPnov9lYP5fm8ffv2\n0q5dK6ZNm0RYWDhr1mzkiSd6F2HkiqJoqWvXHmzYsJnKle9gzpyZtGnTjN27f8izvr1JU678bxfJ\nby7AWa48fksWUbZxPfxfH4vu0qUijLz0Ucm4GDPEHsF/3CjKNKyH/5uzcPn7kzxtBvF79mMdEAFm\nc67PS05OYtSokTz6aFsOHfqVp58eQHT0jzRs2LiIW6Aoitbq1q3Pt9/uIiJiEEePxtK1a0dGjHiB\nhIQruT/BaMTa52nid/1M8sw3cYaWwW/hPMo2rEvAqJEYDh8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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Plot f(x) and Q(x)\n", "x = np.arange(-4, 4, 0.1)\n", "f_x = f(x) # f is defined above\n", "mQ_x = 1.5*np.exp(-0.5*x**2)\n", "\n", "plt.plot(x, f_x, 'k-', label='$f(x)$')\n", "plt.plot(x, mQ_x, 'r-', label='$mQ(x)$')\n", "plt.legend(loc='best', fontsize=20)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From the plot above, you can see that by drawing our samples, $x_i$, from $Q(x)$ and then multiplying by $m$, we should generate more samples from regions where $f(x)$ is large. This gives higher acceptance rates compared to the uniform sampling alternative.\n", "\n", "To perform rejection sampling using a non-unifrom distribution, we first draw a value $x_i$ from our sampling distribution, $Q(x)$ (which for this example is a Gaussian distribution with mean 0 and standard deviation 1). We then evaluate $Q(x_i)$ and multiply by $m$.\n", "\n", "Next, we draw a value $y_i$ **uniformly** from $U(0, mQ(x_i))$ and we accept $x_i$ as a sample from $f(x)$ if $y_i \\leq f(x_i)$. It is important to note that $y_i$ is **always** drawn from a **uniform** distribution: we need to sample uniformly in the \"vertical\" direction so that the number of accepted samples is proportional to the density below $f(x)$. It is the sampling distribution for the $x$ direction that we can change to improve the efficiency of the algorithm." ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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tjWA/f/L1dvrnxWO67j+Oqz46tqfhcdfT60mKiQXAKdNEhRBCCCEuaLJmUIhp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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Non-uniform rejection sampling\n", "\n", "# Number of samples to draw\n", "n = 1000\n", "\n", "# Define Gaussian Q(x), with sampling limited to integration interval (-4 to 4)\n", "Q_x = stats.truncnorm(-4, 4, loc=0, scale=1) # mean=0, sd=1\n", "\n", "# Draw from Q(x)\n", "x_i = Q_x.rvs(n)\n", "\n", "# Evaluate mQ(x_i)\n", "mQ_x_i = 1.5*np.exp(-0.5*x_i**2)\n", " \n", "# Generate n uniform distributions between 0 and mQ_x_i\n", "y_dist = stats.uniform(loc=0, scale=mQ_x_i) # min=0, max=mQ_x_i\n", "\n", "# Sample from these uniform distributions\n", "y_i = y_dist.rvs(n)\n", "\n", "# Evaluate f(x_i)\n", "f_x_i = f(x_i)\n", "\n", "# Build into dataframe\n", "df = pd.DataFrame({'x_i':x_i,\n", " 'y_i':y_i,\n", " 'f_x_i':f_x_i})\n", "\n", "# Add column based on whether point is accepted or not\n", "df['accept'] = df['y_i'] <= df['f_x_i']\n", "\n", "# Extract accepted values to new df\n", "accepted_df = df.loc[df['accept'] == 1]\n", "\n", "# Plot\n", "fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(15, 5))\n", "\n", "# Plot (x_i, y_i), coloured according to acceptance\n", "accepted = df[df['accept']==1].dropna()\n", "rejected = df[df['accept']==0].dropna()\n", "axes[0].scatter(accepted['x_i'], accepted['y_i'], c='b', label='Accepted', alpha=0.5)\n", "axes[0].scatter(rejected['x_i'], rejected['y_i'], c='r', label='Rejected', alpha=0.5) \n", "\n", "# Plot \"true\" function for comparison\n", "axes[0].plot(x, f_x, 'k-', label='Target function')\n", "axes[0].legend(loc='best', ncol=3)\n", "axes[0].set_xlabel('$x$')\n", "axes[0].set_ylabel('$f(x)$')\n", "\n", "# Plot normalised histogram of accepted values\n", "sn.distplot(accepted_df['x_i'], ax=axes[1])\n", "axes[1].set_xlabel('$x_i$')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As before, we can estimate the integral as the acceptance rate, $\\frac{n_{accept}}{n}$, multiplied by the area under the sampling function, $mQ(x)$. This area, $A$, is given by\n", "\n", "$$A = \\int_{-4}^{4} 1.5e^{-0.5x^2} dx = 3.760$$" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Acceptance rate: 48.2%\n", "Estimated integral: 1.81 (-1.29% error)\n" ] } ], "source": [ "# Estimate integral based on non-uniform results\n", "\n", "# Calculate acceptance rate\n", "acc_rate = float(len(accepted_df))/float(n)\n", "\n", "# Estimate integral\n", "i_est = 3.760*acc_rate\n", "pct_error = 100*(i_est - i_true)/i_true # i_true is defined above\n", "\n", "print 'Acceptance rate: %.1f%%' % (acc_rate*100)\n", "print 'Estimated integral: %.2f (%.2f%% error)' % (i_est, pct_error)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that this approach is associated with higher acceptance rates than the uniform sampling example.\n", "\n", "Unfortunately, like the other methods discussed so far, rejection sampling is challenging to use in practice when dealing with complex models. The main issues are similar to those described above for importance sampling: as the dimensionality of the problem increases, the volume of \"low density\" space expands exponentially. This means that to achieve reasonable acceptance rates you need a sampling function, $mQ(x)$ that is already very similar to your target function. Rejection sampling also has the additional constraint that $mQ(x)$ must be greater than $f(x)$ over the entire region of interest. One way to achieve this is simply to make $m$ very large, but this usually prevents $mQ(x)$ from being a good match for $f(x)$, so acceptance rates go down. \n", "\n", "Ultimately, rejection sampling in this form will not help us much in calibrating complex models, but it does introduce some basic principles that we can extend to create techniques that *can* cope with high dimensional parameter spaces. We'll get to some of these in the next notebook, which considers some basic **Markov chain Monte Carlo (MCMC)** methods.\n", "\n", "# 3. Summary\n", "\n", "1. Within a Bayesian framework, what we want to find is the **posterior probability density**, $P(\\theta \\mid D)$, which is proportional to the **prior**, $P(\\theta)$, multiplied by the **likelihood**, $P(D \\mid \\theta)$.

\n", "\n", "2. Because the likelihood function is often complex, we need to rely on **numerical methods** to investigate the form of the posterior distribution.

\n", "\n", "3. Techniques such as **importance** and **rejection** sampling can be used to estimate the shape of the posterior, as well as for calculating integrals related to it.

\n", "\n", "4. Most straightforward sampling techniques struggle to deal with high-dimensional parameter spaces in reasonable computational time frames. Improving performance typically involves identifying functions that are almost as complicated as the posterior we are trying to investigate.

\n", "\n", "5. **Markov chain Monte Carlo (MCMC)** methods (notebook 4) are able to address some (but not all) of these difficulties." ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 0 }