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1 """Chapter 5.3 exercise 5 in Allen Downey's Think Complexity book.
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2
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3 "The distribution of populations for cities and towns has been proposed as an
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4 example of a real-world phenomenon that can be described with a Pareto
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5 distribution.
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6
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7 The U.S. Census Bureau publishes data on the population of every incorporated
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8 city and town in the United States. I wrote a small program that downloads this
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9 data and converts it into a convenient form. You can download it from
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10 thinkcomplex.com/populations.py.
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11
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12 Read over the program to make sure you know what it does and then write
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13 a program that computes and plots the distribution of populations for the 14593
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14 cities and towns in the dataset.
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15
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16 Plot the CDF on linear and log-x scales so you can get a sense of the shape of
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17 the distribution. Then plot the CCDF on a log-log scale to see if it has the
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18 characteristic shape of a Pareto distribution.
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19
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20 What conclusion do you draw about the distribution of sizes for cities and
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21 towns?"
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22
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23 """
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24 import sys
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25
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26 import matplotlib.pyplot as pyplot
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27
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28
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29 def plot_ccdf_log_log(x_vals, y_vals, title=''):
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30 """Given a set of x-values and y-values from a continuous distribution, plot
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31 the complementary distribution (CCDF) on a log-log scale.
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32
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33 """
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34 if len(x_vals) != len(y_vals):
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35 raise ValueError
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36
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37 ys = [1.0 - y for y in y_vals]
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38
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39 pyplot.clf()
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40 pyplot.xscale('log')
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41 pyplot.yscale('log')
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42 pyplot.title(title)
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43 pyplot.xlabel('x')
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44 pyplot.ylabel('1-y')
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45 pyplot.plot(x_vals, ys, label='1-y', color='green', linewidth=3)
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46 pyplot.legend(loc='upper right')
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47 pyplot.show()
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48
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49
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50 def main(script, filename):
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51
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52 with open(filename, 'r') as fp:
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53 y_vals = [int(y) for y in fp]
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54
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55 print 'Read {} populations'.format(len(y_vals))
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56 y_vals.sort(reverse=True)
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57
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58 x_vals = range(len(y_vals))
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59
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60 pyplot.clf()
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61 pyplot.xscale('linear')
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62 pyplot.yscale('linear')
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63 pyplot.title('Populations')
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64 pyplot.xlabel('x')
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65 pyplot.ylabel('y')
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66 pyplot.plot(x_vals, y_vals, label='Population Linear', color='green', linewidth=3)
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67 pyplot.legend(loc='upper right')
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68 pyplot.show()
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69
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70 pyplot.clf()
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71 pyplot.xscale('log')
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72 pyplot.yscale('linear')
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73 pyplot.title('Populations')
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74 pyplot.xlabel('x')
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75 pyplot.ylabel('y')
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76 pyplot.plot(x_vals, y_vals, label='Population Log-x', color='green', linewidth=3)
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77 pyplot.legend(loc='upper right')
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78 pyplot.show()
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79
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80 # normalize to 0-1
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81 max_p = y_vals[0]
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82 ys = [y / float(max_p) for y in y_vals]
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83
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84 plot_ccdf_log_log(x_vals, ys, 'Population CCDF log-log')
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85
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86 if __name__ == '__main__':
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87 main(*sys.argv)
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