{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Zipf's Law and US Metro Population Growth\n", "*Jeremy A. Seibert*\n", "\n", "

\n", "\n", "George Zipf (Pictured) was a lingustic from Harvard in the early 20th century who postulated and found that within languages certian words are used in a higher frequency, while the rest are hardly ever used. Though initially intended only for use in analyzing word frequencies, the generalized form known as the Zipf-Madelbrot Law and its associated have been found throughout many unrelated diciplines. \n", "\n", "As it turns out Zipf's Law explains and interesting question in Urban Economics, City growth. In this notebook, we will be showing the (Rank-Size) distributions of the United States Metropolitan Area's Population, and how Zipf's law explains the population distribution of Metro Areas.\n", "\n", "If Zipf's law holds then it would state that the correlation coeffecient of between Metro Rank and the Metro Population would be approiximately -1.0. \n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "#Gather the tools\n", "import pandas as pd\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import seaborn as sns\n", "import warnings\n", "warnings.filterwarnings(\"ignore\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1. Metro Rank-Size Distribution\n", "\n", "Using Population Data collected from the US Census Bureau we can begin to construct the Rank-Size Distribution. In this notebook we are using the 2017 Population estimates as our base year. The methodology included in the repo explains how the Census Bureau derives their estimates for the populations in the metro areas. \n", "\n", "As a quick overview, they use the most recent census year (2000, 2010, 2020, etc.) as their base year and in conjuction with other population-based information, and then derive the estimate.\n", "\n", "Population Base + Births - Deaths + Migration = Population Estimate\n", "\n", "I would be remiss if I did not point out that there is obviously room for error in this calculation. However for our use case in this notebook the error is really a non-issue. Also within this notebook, where ever there is a metion of \"city\" this can be thought of synonomously as an agglomeration entity charaterized by the Census Metropolitan Statistical Areas (Metros)." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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POPESTIMATE2017RankLogRankLogPop
NAME
New York-Newark-Jersey City, NY-NJ-PA2032087611.00000016.827159
Los Angeles-Long Beach-Anaheim, CA1335390721.69314716.407320
Chicago-Naperville-Elgin, IL-IN-WI953304032.09861216.070274
Dallas-Fort Worth-Arlington, TX739966242.38629415.816945
Houston-The Woodlands-Sugar Land, TX689242752.60943815.745934
Washington-Arlington-Alexandria, DC-VA-MD-WV621658962.79175915.642732
Miami-Fort Lauderdale-West Palm Beach, FL615882472.94591015.633396
Philadelphia-Camden-Wilmington, PA-NJ-DE-MD609612083.07944215.623163
Atlanta-Sandy Springs-Roswell, GA588473693.19722515.587872
Boston-Cambridge-Newton, MA-NH4836531103.30258515.391708
\n", "
" ], "text/plain": [ " POPESTIMATE2017 Rank LogRank \\\n", "NAME \n", "New York-Newark-Jersey City, NY-NJ-PA 20320876 1 1.000000 \n", "Los Angeles-Long Beach-Anaheim, CA 13353907 2 1.693147 \n", "Chicago-Naperville-Elgin, IL-IN-WI 9533040 3 2.098612 \n", "Dallas-Fort Worth-Arlington, TX 7399662 4 2.386294 \n", "Houston-The Woodlands-Sugar Land, TX 6892427 5 2.609438 \n", "Washington-Arlington-Alexandria, DC-VA-MD-WV 6216589 6 2.791759 \n", "Miami-Fort Lauderdale-West Palm Beach, FL 6158824 7 2.945910 \n", "Philadelphia-Camden-Wilmington, PA-NJ-DE-MD 6096120 8 3.079442 \n", "Atlanta-Sandy Springs-Roswell, GA 5884736 9 3.197225 \n", "Boston-Cambridge-Newton, MA-NH 4836531 10 3.302585 \n", "\n", " LogPop \n", "NAME \n", "New York-Newark-Jersey City, NY-NJ-PA 16.827159 \n", "Los Angeles-Long Beach-Anaheim, CA 16.407320 \n", "Chicago-Naperville-Elgin, IL-IN-WI 16.070274 \n", "Dallas-Fort Worth-Arlington, TX 15.816945 \n", "Houston-The Woodlands-Sugar Land, TX 15.745934 \n", "Washington-Arlington-Alexandria, DC-VA-MD-WV 15.642732 \n", "Miami-Fort Lauderdale-West Palm Beach, FL 15.633396 \n", "Philadelphia-Camden-Wilmington, PA-NJ-DE-MD 15.623163 \n", "Atlanta-Sandy Springs-Roswell, GA 15.587872 \n", "Boston-Cambridge-Newton, MA-NH 15.391708 " ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#Read in the Population Estimates\n", "file = './Dataset/US_Metro_Pop_Est.csv'\n", "pop = pd.read_csv(file,usecols=list(range(3,15)),encoding='latin-1')\n", "\n", "#Filter Metro-Only\n", "pop = pop.loc[pop['LSAD'] == 'Metropolitan Statistical Area'].set_index('NAME')\n", "\n", "#Filter for 2017\n", "pop_17 = pop[['POPESTIMATE2017']].sort_values(by='POPESTIMATE2017', ascending=False)\n", "\n", "#Rank the Metros\n", "pop_17['Rank'] = range(1,len(pop_17.index)+1)\n", "\n", "# Define Functions to Convert Values\n", "log_pop = lambda x: np.log(x)\n", "log_rank = lambda x: np.log(x) + 1\n", "\n", "# Convert to Log values\n", "pop_17['LogRank'] = pop_17['Rank'].apply(log_rank)\n", "pop_17['LogPop'] = pop_17['POPESTIMATE2017'].apply(log_pop)\n", "\n", "#Show the DataFrame\n", "pop_17.head(10)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. Find the Regression Coeffecient\n", "\n", "In works done by Paul Krugman in 1996 and Gabaix in 1999 they found that regressing the size of the metro's against the rank of metro's yeilded a coeffecient very close to -1.0. Let's see in the next steps if we can do the same. \n" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "R^2: 0.9741162743633364\n", "Coeffecient: [[-0.88246969]]\n" ] } ], "source": [ "# Import LinearRegression\n", "from sklearn.linear_model import LinearRegression\n", "\n", "# Create the regressor: reg\n", "reg = LinearRegression()\n", "\n", "# Create arrays for features and target variable\n", "y = pop_17['LogRank']\n", "X = pop_17['LogPop']\n", "\n", "# Reshape X and y\n", "y = y.reshape(-1,1)\n", "X_LogPop = X.reshape(-1,1)\n", "\n", "# Create the prediction space\n", "prediction_space = np.linspace(min(X_LogPop), max(X_LogPop)).reshape(-1,1)\n", "\n", "# Fit the model to the data\n", "reg.fit(X_LogPop, y)\n", "\n", "# Compute predictions over the prediction space: y_pred\n", "y_pred = reg.predict(prediction_space)\n", "\n", "#Visualize the Distribution\n", "sns.lmplot(x='LogPop',y='LogRank', data=pop_17)\n", "plt.show()\n", "\n", "# Print R^2 \n", "print('R^2: {}'.format(reg.score(X_LogPop, y)))\n", "print('Coeffecient: {}'.format(reg.coef_))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3. Not Quite -1.0 , What is different?\n", "\n", "In the regression above we obtained a coeffecient value close to -0.9, which is not quite the -1.0 that was reported by Krugman and Gabaix. We see that within the middle section of the distribution the fit is near perfect, this section includes metro's such as Ashville, NC and Lexington, KY, however towards the tails we find that the fit is not so nice. Gabaix in his 1999 paper provides an explaiantion for the smaller metros, in that they are more succeptible to industry-level shocks due to a lack of indusrty diversity (Gabaix, 1999).\n", "\n", "Back to the regression, What did we miss? The key here is that both Krugman and Gabaix both ran their regressions on the top-135 largest metros, rather than all 382 metros (Gabiax,1999) (Krugman,1996). \n", "\n", "Let's re-run our regression and check out the result." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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NNG7rcklqyNEArFI2vaKIpV86njNmjgTgpaparnx4DYUBD/99xjTKC/00h2IcOSKfkjwPbpdgjCEYjRNLGC4/ZSLjSvMIxZxGQHWtEQzGaZMpzh2zTn1WQ4mmIFSfhaJx7nn5He5dvgXbQHHAw7c/fhQfnDyCsgIfliX7GsLvaAwytlMVxPKq2n054G317c7ySDhr2RmclpgAM0YV6godaqDRHLDqH7GEzbMbdnHzMxtpCcdxWcJV8yfx6bljGVnsx+d2dbtvR3Be814jkbiNS94PvAbnu9iyhAc/O0/zwWog0QCs+o9tG9a818iNT2ygem87AGfPGsW1C6ZSUeKnyO/pcf/lVbVc/tBrRA5cqgNndY5pI/UuWA0o+hBO9R/LEuZOKOVXXziOU6aOAOC5N/bwtcfW8fbuFmpbwthdNPjpMH96OVfNn9Tl52wDb+1p1QdyalDTAKwOi4gwYXg+d140hy+edAQCvLm7ha8sXcOqrQ3sbAoRiXe9+gY4yyVNH1mw31JJ+44N+kBODWoagFValOR7uf6MaXzvk7PI87qob4vy1cfW8eyGXexqCtMa7nplZoCbzpqByMG/vxmc9el++HxVRseuVLZoAFZpk+d186m5Y7n/0rmMLQ0QSxh++Pzb3PXSZvY0h6ltDdPVM4f508uZUlZAV8kKATbVtuldsBqUNACrtPK5XZw4aTi/+Nw8jj9yGAB/WLOTG5/cwM6GEDubQkS7eOh201kz8LkP/nY0OLPo9C5YDUYagFXauV0W00YWcsdFs1l4/DgA1r7XxBVL1/DW7hZ2NYVoi8T320cfyKmhSAOwyggRoaIkwA1nTGfJOc7d7Z6WMNc8vJaXqmqpbQk7M+E6pST0gZwaarQOWGVcazjGyup6vvn0G9S0RABY9IHxfOGkI/B7XJQX+pO9hJ3a4C/9dhV2coZcB4/l3Ann+9wUBTzaY1jlIq0DVrmn0O/hQ1PKeOAzc5k9rhiApf95j28+/QYN7dH9UhIHPpCzBDyWYHAa/ASjiS57DCs1EGkAVv3C73Fx1OhifnLhbM6fMwaAldUNXLV0DVvr26ltCbO3zUlJdDyQ87gEr8vCEqfxO4BtDG/XtLKnOUwskeD+FdXZ/LKUOiwagFW/8bgsxg/P54Yzp3HDGdPwuITtjSGuWrqGldX1tIRi7GwKcdKUEVw1fxKWCHHb4HY5v9kZnDtil+W8v7c1yuba1ux+UUodBs0Bq35njGFvW5T/VNdz87KN1LdHEeBLJx/JwuPH4bIsRhT6WP1uw76OaruanRrizk1+ookEIFQU+zUnrHKBNuNRA0dTMMqmmla+vWwjb+527mTnTy3jhjOnEfC4KAo4C3uKCHNv/Sst4TgWggjEbUPcdpY/Omp0kS5tpHKBPoRTA0dJnpeZo4v5yUVzOGvWKACWb6pj8SNr2dMcpiUUY1dzmFjCZurIIobne3G7hIQx2MZgCQS8LkSEPK8bj0s0J6wGFA3AKqvyfW4mDM/jpjOnc81pk7EEttS185WHXnP6BccS7GwM8bkTJ+B1uxhV7GfayEKnd4Q4i4ZW7Wmhuq6NeMJmR2Mw21+SUinTAKyyzu9xMaY0wEXHjeNHFxxLccBDSzjOjU9s4Mk1O0jYNlNHFXL96VMpK/DRHIrhc7swNiCy76HczqYwBT53tr8cpVKmAVjlBLfLYnRxgA9OGsF9iyqZXFaAbeCel7dw21/eJhq3mTW2mB9fNJuXrp/P+NIAluWUp0ViNpG4Tdw2PXZdUyrXaABWOcOyxEkxjCrkzoWzOXVaGQB/2VjDVx9bR11rhEgswa6mEC2ROKV5bhK2eb88TWB3S0QnZ6gBQwOwyjnDC3yMHZbHt845ii+ffCQCVO1p5Yqla9i4q5mEbSgv8NMSiu+brAGQMNo5TQ0sGoBVTirye6goDnDpCRP43idnke9z0dAe5brH1/Pc67u5+LhxxGxDwjZEE/a+RT21f7AaSDQAq5wV8LoYXRLgQ1PKuPeSSsYPyyOWMPzor5tY+W49E0oD+wVejyW4LWu/crTlVbUsfGAlJ//wJRY+sFIDs8opGoBVTvO6LUaXBJg8spC7L5nDiROHA/D0ul1gOd++Hgs8LkEAG8PIQh87GoMsr6plybKN1LaGtYGPykkagFXOc1nC6GI/o4r83PqJmVx6wngA3t3bjssSjDhNelwuobzQh8sSxpbmcf+KajwuZ5KGTtZQuSjjAVhEXCKyVkSezfS51OAlIpQX+RmW7+OLJx3JzR8/Cr/HcqYkJ6A44GFsaQBLhFDM5vMfnMD2xiDxhE11XZtO1lA5qT/ugK8F3uqH86ghYFi+l7JCH/OnlXP3wjlUFPsBqG+PsaspzLA8L9eeNoUpIwsJeFzsbAoTt41O1lA5KaUALCJf6uK9H6Sw31jgY8Av+j40pbpW6PdQUexnyshC7l1USeX4EgDaowl8XhdHjS4CIJGwMcZgbJNc3dPZP5caUKmhLdU74E+JyKKOD0TkHqAshf1+CtwIHLwM7vvHukxEVovI6rq6uhSHo4Y6v8epkBhR4OOHnzqGT891mry/+m4DVz68hm317bTHEows8uFyCdGETcy2sW3Du/VBfRCnckLKARj4vIgsFJHfAHFjzEF3xZ2JyDlArTHmtZ62M8Y8YIyZZ4yZV1aWSkxXyuFxORUSBX43V86fzE1nOk3edzSGuOrhteR53Lgsi2H5XlwiuC3B7XLaWWo1hMoFPfYDFpFhnT4sBJ4GXgGWABhjGnrY9/vAZ4A44AeKgD8YYy7tbh/tB6wOhTGGurYIbeE4VXtaWPLMRva2RQEo8LmJxOLJLQWDYXRxgJht0x5J6AKfKp3S25BdRN7FyZxJpz87GGPMxJROIjIfuN4Yc05P22kAVoejKRilod153bxsIxt3tQDON60lTk1xaZ4XEahpDoMI00cVajN3lS7pbchujDnSGDPxgD87XikFX6X6S0mel/IiP8MLfPz4gmP52NEVQHItOUsYWeSnwOemoT0KAl6XaH2wyqqU63FE5IPAEZ33Mcb8NpV9jTHLgeV9G5pSfVfgc+O2hJqWMNedPoXJ5QXc9dJmYgnDew1BKor9ROI2ljgBO5awcVtCwOPaN3vu/hXVbG8MampCZVxKa8KJyO+AScA6IJF82xhjFqdzMJqCUOkSS9jJpett1u9o4ptPvUF71PnW9bqEQp+L4QVODbGIEEsk8LldtEcTeFxOQNbUhOqjzCzKKSJvAUeZDBdQagBW6WTbhprWMKFogpqWMEue2cjm2jYAAh6nOiLP6yIcc5q5F3hd2ECe9/1fDIPROOWFfh657IQsfRVqAMnYopxvAKP6enClssmyhFFFfgr9HkYW+bnj4tl8JHknG4rZNLTHaArGGJ7v49rTptASieO2ZL+JGh2pCaUyIdUc8AjgTRF5FYh0vGmMOTcjo1IqTUSEskIfXpdFfXuE/z17OpPLC/j5P6oJxRL4PRZfPPkIZo0ppmJVgPr2CHlewe0i2VciQYHPzcIHVmpeWKVdqimID3f1vjHm7+kcjKYgVCYFo3FqWyLYxrBqawO3PvsWbcm73sUfmUx5gZ87XtqM2xL8HhfxhE1bJI7BafajeWHVi8zkgPuLBmCVaZF4gprmCHHbZmdjiG8+8wbb6p0Uw8ePreADRwzjidd2sqclxKiiAG3hKDaQ7/PsO4bmhVU3MvYQ7gTgLmAG4AVcQLsxpqivJ+yJBmDVH+IJm5rkAp/BaJzv/7mKV96pB+DoMUXc/PGZDMv3ArDw5ysp9nvwui1EnJ8vYwx7WsJMKS/UtITqLGMP4e4GFgKbgQDwZeCevp5MqVzgdlmMLvaT73OT53XznXNn8rkTJwDw+s4WrnhoDZtqWgGoKAoQiiWIJmwSyfWP6tsjtIbjutKGOmwp9wM2xrwDuIwxCWPMr4AzMzcspTJLxJkZV5LnxRLhcx88glvOnUnA46KuLcLiR9fxt7dquPi4ccRtQyiaIJZI0BqOUd8WpTTPoyttqMOWagAOiogXWCcit4nI1/qwr1I5q6PBu4hw8pQR3H3JHEaX+InGbb73XBVrtjdyzamTGZ7vozUcpyTg1A4PT6YoOmi5mjoUqZahfQYn4F4NfA0Yh9OiUqkBr9DvweOyqGkJc+SIfO5bVMmtz77F6m2NPL56B9V17Xz73KMoCjgP4q57bD0NwQiFPg+W5aT9QrEEY0vzAHQ6s0rZIVdBiMhJxphX0jkYfQinsqnz9OWEbfjFP6p5bPUOAEaX+Ln1vFkcOSKfV6sb9pWr5ftcxBJmX2kaOL2GdTrzkJTeh3DJBTUXisj1IjIr+d45IvIvnAdzSg0aHQ3eA14XLku4/MOT+MbZ0/G6LXY1hbn64bX8Y/Nejp84jGtPm8LwfB9NwRjFAQ/fPuco5k8v15WYVZ/01g/41zjphleBDwC7gHnATcaYp9M9GL0DVrnAGMPetiit4RgAm2paWfLMRmpbnUmgnz1xAp89cQKWvH/D47Ysyot8LLj975QEPPtK1jqOp2VrQ0Lay9DmAacbY/4HOBs4BzgpE8FXqVzRMX25oxZ46shC7ru0kqPHFAPw239v4+ZnNhKMxvftE7dtdjWFGFXkJxRL7Hc8LVtT3ektAEeNMTaAMSYMVBtj6jM/LKWyryTPy8giPyJCaZ6XH11wDOceOxqAV7bUc9XDaw+qfPh05VhCsQTtkRjGGILROA3tMS1bU13qLQBPF5ENydfrnT5+XUQ29McAlcqmfJ+bimI/bsvC47L46oIpXHf6FNyWsK0+yJVL17Jq6/tLIx4/cRiLT51CccBLUzBKeaGfAp+LEQW+/Y6rZWsKes8BT+hpZ2PMtnQORnPAKlfFEzZ7WsJE4zYAb+xs5uZlG2kMxrAEvvyhiVw0b+x+uV8RYViel6889Bq1rWHtMzz4pX1NuG09vQ59nEoNLM705cC+IDprTDE/u3Qu00YWYht4YEU133uuinCn/K8xhvr2CBfOG0s0bhOMxmkJRdlc08rW+nYa2yOaBx7iUprNJiKtItJywGu7iDwlIro4pxoSLEsYVeynODkho6zQx08vOpbTjxoJwEtVtSx+dB01LeH99jt2XAmLT3PSFjuawiAwtiRAzDb6MG6IS3U68U+BG4AxwFjgeuBh4FHgl5kZmlK5aXiBj+EFzvRln8fFTWdO44r5k7AE3qlt44qH1rBhR9N++8w9opSA18240gBTygspCnj1YZxKOQCfa4y53xjTaoxpMcY8AJxhjHkMKM3g+JTKScUBDyOLfFjiLG1/wdyx/OD8oyn0u2kKxfj67zfwzLpd+y1vtLs5hMclxBL2vvf1YdzQ1pdmPBeKiJV8XQh0/J6VOx3dlepHeV43o0sCeFzOj9G8I4Zx76JKjhyRT8I23PG3zdz+wmZiCefBXUVRgKZglG317VTtaWFLbSv17ZF9PSTU0JNqAF6E05CnNvn6DHCpiARwGvQoNSR53c70ZZ/HBcCYkgB3L5zDh6aMAOBPr+/musfX09AeZc64YurbY8QSNiIQTdjUtEQ4ceKwbH4JKot0SSKl0sAYQ11rhLaIMzvONoalK9/jV//aCsCIAi/D8rw0tEdojyaIJWw8Lot8r4sJwwv4/VdO3NdZTQ1YmVkRQ0TGJiseapOvJ0VkbN/Hp9TgJCKUF/kpzXOmL1sifObECdx63kzyvC72tkXZVNuGyxLGleYxcUQB40rzKM33srMpyM6m0L4aYzV0pJqC+BWwDBidfP0x+Z5SqpPSTg3eAU6aPIJ7LpnD2NIAADWtUWpbI/sewoVjNqOKAsQSTi+JjgZAamhINQCXGWN+ZYyJJ1+/BsoyOC6lBqxCv4eKYj+uZEphwvB87r2kkqkjCwBoCsXYsredzbVt7GkJM2ec0+THTqYx6joFaDW4pRqA60Xk0mR/YJeIXApoUx6luuH3uParkCjwu7nnkkqOHu0sJJ5c35N8r4vn36zh1er3+0m0hmOakhgiUg3AXwQuBPYAu4FPA5/P0JiUGhQ8LosxyQbvAC5LcFkWw/M9CE79ZnM4TjRu8+iq7fvtG407KYmOh3pqcEopACd7P5xrjCkzxpQbYz6BrgmnVK8sSxhV5KfQ70xf3t0SYli+l3EC9icYAAAc9ElEQVSlAdyWYAzUt0fZXNuKfUDawTaG2pYwe9s0JTFYHc7KxtelbRRKDWIdDd6H5/uoKAoQjtn4PS7GD8sj4HF+BNujCb719Ebau7jjbQnF2JVcq04NLocTgLVoUak+KM7z8JUPTyRuG0KxBC4Lhuc7y9wD/LvaafL+XsPBU5MjsQQ7G0NdBmg1cKW6LH1X9HcipfrozKMrcFnCvcu3sLs5xKiiAFefOoW97RHu+Ntm3msIctXSNXzjYzM4YeLw/fZduaWeR1dtp6Y1zIRheXzlw5N0XbkBrreG7K10HWgFCBhjDieAH0Rnwqmh4sAG7wAbdzVz87I3aWiPIsCXTj6ShcePQ0R4tbqBO17ajNsS/B6LSNzGGLj1vFkahHNH2huyFxpjirp4FaY7+Co1lBzY4B1g5uhi7ltUyfRRhRjgF/98l+/+6S1CsQSPrtqO2xICHheC4He7EIF7l2/J3hehDtvh5ICVUofhwAbv0NHkfTZnzhwFwMtv17H4kbXsaAri9+z/4+pzW7zX0E5De7Rfx63SRwOwUlk2vMDHiE7Tl71uixvOmMrVpzpN3rfUtdMYjNEU3H+acsc05qZglF1NIeJaJTHgaABWKgcU+T2MKvJjJYOwiHB+5Vhu+/QxFPndJGxDXZvTR8I2NqFYgrhtuPi4cQCEYwl2NoUIRRM9nUblGA3ASuWIgHf/6csAleNLue/SSiaV5QNOH4n3GkLUt0UIReM8umr7vmnMCduwuzlEo6YkBgwNwErlkI4G7/5kg3eAiuIAdy6cw4enOv2voglD3DaU5nmob49wx0ub9+sl0RiMsrs5RMLWStFcpwFYqRzjsoSKYj8FvvcrJAIeF0vOmUFFkR9wgvB7jc6qYG5LDuolEYo6EzfCMU1J5DINwErloAMbvHe8Z2MYXezDEiflsKMxRCSeYE9L6KBjxG2noU9TUFMSuSpjAVhE/CLyqoisF5GNIvKdTJ1LqcGqNN9LeZF/X4VERVEAl2UxvjQPj0swQG1rlLhtuq2CaGiPsqc5rCmJHJTJO+AIcJox5lhgNnCmiJyQwfMpNSgV+Nz7GrxffNw44rYhYQzjSgP7aoP3tkW58ckN3d7tBqNxTUnkoIwFYONoS37oSb70v2ClDkFHg/eTpozg2tOmMDzfRzCaYFp5IadOcx7OrdvezBVL1/BObVuXx4jbNrubwzQHddmjXJHRHHBy9Yx1OEvZv2CM+U8X21wmIqtFZHVdXV0mh6PUgOZxORUSH55exsXHjWNUUYA9rWEa22MsOn48frdFTUuEax5Zy8tVtV0ewxhDfXuEmpYwtqYksi6jAdgYkzDGzAbGAseLyKwutnnAGDPPGDOvrEyXmVOqJy5LqNrVwl0vv0N9e4Qiv5v69ggvvV3LZadMZFSRn0jc5tY/vcUDK6q7zfu2R+LsbNKURLb1SxWEMaYJeBk4sz/Op9Rg9sA/3sXvsSjwuxGcBj1uS1ixaS/3LapkzvgSAB5dtZ1vPP0GbeGuewjHEsmUREhTEtmSySqIMhEpSf49AJwOVGXqfEoNFdsbg8mga+F2WSDg91jsaQlRnOfhtk8dw/mVYwB49d0Grnx4Ddvq27s8ljGG+rYItZqSyIpM3gFXAC+LyAZgFU4O+NkMnk+pIWFcaR6hZOrAZQkel9MfeFRRYN97V586mRvPmIbHJexoDHHVw2v515a93R6zLZmSiMQ1JdGfMlkFscEYM8cYc4wxZpYx5pZMnUupoeTyUyYSSxiC0TjGGMKxBMbAog+M32+7M2eN4qcXzWZ4gZdgcs25363c1u0Cn7GEza6mMC1hTUn0F50Jp9QAM396ObecO5PyQj/NoRjlhX5uPW8W588du18PCYAZFUX8bFElR1UUYYBfvbKV7/zxzW67phlj2NsaobY1rCsx94MelyTqb7okkVKHxxhDXVvkoAdv0bjNnS9t5rnX9wAwcUQ+t5w3k9ElgW6P5XFZjCzy43XrfVqK+rwkkQZgpQao5VW13L+imu2NQcaV5nH5KRP3rQ/X2B6l8YBZccYYlq3fxd0vbyFhG4r8bpaccxSVE0q7PYclwvACL4V+T7fbqH3SuyacUio3La+qZcmyjdS2hikJeKhtDbNk2UaWJydgHNhDApxmPufNHsOPLjiGkoCHlnCcG5/cwBOv7eg23WAbQ11rhLrWiKYkMkADsFID0P0rqvG4hDyvGxHnT49LuH9F9b5tOveQ6OzYsSXcd2klk8sLsI2zsOdtf3l7vxWaD9QajrGzKdTjNqrvNAArNQB11AJ3FvC42NEY3O+9jh4SnVfZABhZ5OfOi2dzWjJl8ZeNNVz72DrqWiPdnjMad9pbtkW6ntih+k4DsFIDUOda4A6hWIKxpXkHbetxWYwpCRDw7h+w/R4X3zh7OpedMhEB3t7TyhVL1/DGzuZuz2sbQ21LmL1tmpJIBw3ASg1AB9YCB6NxYgnD5adM7HJ7yxJGFfkpCuz/ME3EaXH5/fOPpsDnpqE9ynWPr+dPG3b3eP6WUIxdzWFiuhLzYdEArNQA1FUt8C3nztxXBdEVEWFEgY/h+b6DPnf8kcO4d9EcJgzLI24bfvzCJu54cXOPS91HYs6yR+2akjhkWoam1BAUjMapbYlgH/Dz3x6J8/0/V/GvLfUAHDO2mJs/ftR+SyN1pTjgYVi+d7+qiyFIy9CUUr3L87qpKPHjtvYPAfk+N7ecN5PPnjABgA07mrnioTVsqmnt8XjNyZRET3fM6mB6B6zUINTTJI3O4gmbmtYIkS76Aq/YXMcP/lxFOGbjc1tc/9FpfGRG9ykOcBoBlRX6yPO6e9xukNI7YKWGut4maXTmdlmMLvaT7zs4YJ4ypYx7Lqmkothp8v6953pu8g7OSs17msM0tOtKzKnQAKzUIJPKJI3ORISRRX5KusjzHjkin/sWVTI3OV350VXb+d+nXqe1l45pTcEou5pCmpLohQZgpQaZVCdpHGhYvpeyQt9BD9KKAh5+cP7RXDB3LACrtjZy5dK1vLu36ybvHcKxBDubQt12XlMagJUadPoySeNAhX5Pl9OXXZZwxfxJ/M9Z0/G4hJ1NIa5+eC3/3Nx9k3dwUhK7m0M0akqiSxqAlRpk+jpJ40DdTV8GOP2okdx58RzKCnyEYgmWLNvIb/619aBytgM1BqPsbg71mD8eijQAKzXIHMokjQN5XBaju5i+DDBtVCH3XVrJrNFFAPzm39u4edlGgtGeJ2SEos7EDV2J+X1ahqaU6pYxhr1t0S4fusUSNne/9A5/TE5bPmJ4Hrd+YhZjemjy3mFYvrfLh34DnJahKaXSR8Sp6+1q+rLHZfG106fytQVTcFnC1vogVy5dw+qtDb0et6E9yp7m8JBPSWgAVkr1qjjPw8gDGrx3+Pixo7n9gmMpzfPQGo5z0x9e5/HV23vtlhaMxod8SkIDsFIqJfk+N6O7mL4McPTYYu5bVMnUkU6T95/9vZrv/7mqyxl2ncVtm93NYZqDQ3MlZg3ASqmU+dwuRpd0vVBneZGfOy6azYLkdOUX36rl2sfWUdsS7vGYxhjq2yPUtISxh1hKQh/CKTVEpNofIhW27ay+3FUrSmMMT7y2g/tXVGMbKM3zcPPHj+KYsSW9Htfjsigr9OH3HFx9MQDoQzil1MH60h8iFZblTF8uDhy8WrKIcME8p8l7od9NYzDG13+/gWXrd/V63FgimZIIDY2UhAZgpYaAvvaHSNXwAh8jupi+DHDcEcO4d1ElRwzPI2EbfvriZm5/YVOvq2gYY6hvi1A7BFISGoCVGgIOtT9EKor8HkYV+bG6CMJjSgLcfckcTp48AoBnN+zm64+vT6lbWlskzs6mEJH44K2S0ACs1BBwOP0hUhHwdj99Oc/r5tvnHsXnP+g0eX9jVwtXPLSGt/f03OQdnJTErqYwLb10XxuoNAArNQQcbn+IVHjdzvRlXxcP0CwRPnviEdx63kwCHhd1bRGufWwdL7xZ0+txjTHsbY1Q2zr4UhIagJUaAtLRHyIVLksYXeynoIsG7wAnTR7BPYvmMKYkQDRu8/0/V3Hf8i0pzYhrCzspiWh88PQY1jI0pVRGNLRHaQp2nettDce49dm3WL2tEYC5E0r51sdmUNRFVcWBLBGGF3gp9Pe+bT/TMjSlVG4Ylu/ttkKi0O/h++cfzUXznCbvr21r5Iqla3pt8g5gG0Nda4S61kiv051znQZgpVTG9FQh4bKEyz88iW+cPR2v22J3c5irHl7Dis11KR27NRwb8CkJDcBKqYzqqUIC4CMzRnLnxbMpL/QRjtl8e9mb/OqVd3tt8g4QjdvsagrR1sWMvIFAA7BSKuN6qpAAmDrSafJ+zNhiAH638j2+9fTGLqc6H8g2htqWMHvbBl5KQgOwUqpf9FYhUZrn5UefPobzjh0NwL+r67nq4bVsb0htskhLKMau5nCvM+1yiQZgpVS/ERHKi/zdrobhdllcu2AKXz99Km5LeK8hyJUPr+HVd3tv8g4QiTnLHqVy55wLNAArpfrdsHwvZd1USAB87JgKbr/QafLeHknwP394nUdffS+lFINtDDUtYeoHQEpC64CVUilLZ0tLcBbqrG3tfmmiutYIS5Zt3Ddt+dRpZdxwxrSU21X6PC5GFvpwd/MAMM20DlgplRnpbmkJToVERXH3FRJlhT7uuGg2Z8wcCcDLb9ex+NF17OmlyXuHSCzBzqZQrys2Z4sGYKVUSjLV0rKjQqK7u1qv2+LGM6Zx1amTsATeqW3jiofWsH57U0rHT9iGPc3hlDqw9TcNwEqplGSypaXLEip6qJAQET5VOZbbPnUMRX43zaEY1z+xgafX7kw5z9sUjLKrKUQ8h6okNAArpVKS6ZaWHRUSpd1USABUTijlvksrmViWT8I23PnSO/z4r5tSng0XTqYkQtHc6DGcsQAsIuNE5GUReVNENorItZk6l1Iq8/qjpSVAaS8VEhXFAe66eA6nTHWavD/3xh6ue3w99W2RlI6fsA27m0M05kBKImNVECJSAVQYY9aISCHwGvAJY8yb3e2jVRBK5baOKogdjUHGpqEKoifhWIKalu4rJIwxLP3Pe/zqla0YYHiBl1vOncmMiqKUzxHwuigv9OOy+lzA0JU+H6TfytBE5BngbmPMC91towFYKdVZNG5T09Lz7LZ/bdnL/z1XRTCawOMSvrZgKmfOGpXyOdyWRXlRWlZizs0yNBE5ApgD/KeLz10mIqtFZHVdXWpdkJRSQ0NvFRIAH5w0gnsvqWRsaYBYwnDbX97m7pfeSflhW9x2Gvp017s4kzJ+BywiBcDfge8ZY/7Q07Z6B6yU6ooxhrq2CG3h7ut528JxvvfcW/wnOW159rgSbj7nKIrzUm/cnud1U1boO9SURG7dAYuIB3gSWNpb8FVKqe6ICOWFPVdIFPjdfPcTs7jk+HEArNvexBVL17Clti3l8wSjcXY2hgjH+qdKIpNVEAI8CLxljLk9U+dRSg0dvVVIuCzhyx+ayJJzZuB3W+xpCXPNI2tZ/nbq6c24bbO7OUxzMPMrMWfyDvgk4DPAaSKyLvk6O4PnU0oNAYV+DxXFPVcuzJ9Wzl0L5zCyyEc4bnPLs2/y4D/fTWnxT3BSHvXtEWpaMrsSszbjUUoNSLGEzZ5e+v82B2N859k3WZectnzCxGH879kzup1x1xWPy6KsMKUqidzKASulVKZ4XL1XSBTnebjtU0dz/pwxAKysbuCqpWt4L8Um7+AE+t3NYZpD6U9JaABWSg1Y+3pI+Lu/o3W7LK4+bTI3nDENj0vY3hjiqqVrWFldn/J5jDHUt0WoTXNKQgOwUmpAS6VCAuCsWaP46UWzGZ7vpT2a4BtPvcHS/2zrU9P2tkicnU0hIvH0VEloAFZKDQql+V7Ki/zdVkgAzKgo4r5LKzmqohADPPjPrdzy7FsHNRnqSSxhs6spTEv48FMSGoCVUoNGgc/da4XEiAIft184m7OS05X/vqmOax5Zy+7mUMrnMcawtzVCbevhpSQ0ACulBhW/x8Xoku5X2QBnivP1H53K1adOxhKormvniofWsOa9xj6dqy3spCRSbYd5IA3ASqlBp6NCIuDtvkJCRDi/cgw/uuBYigMeWsJxbnxiA0+u2dGnvLCTkkj97rkzDcBKqZyxvKqWhQ+s5OQfvsTCB1Ye1npzLksYVdRzhQQ4PSPuW1TJ5LICbAP3vLyF2/7ydp/uau1DnE+hAVgplRMysehnR4XEsPyeKyRGFfu5c+FsTp1WBsBfNtbw1cfWUdeaWpP3Q6UBWCmVEzK16CdASV7vFRJ+j4tvfmwG//WhIxGgak8rVyxdw8ZdzYd9/u5oAFZK5YRMLvoJqVVIiAgLjx/P/50/i3yfi4b2KNc9vp7nXt+dljEcSAOwUionZHrRT0itQgLgA0cO595LKhk/LI9YwvCjv27ijr9tTvuKyhqAlVI5ob8W/fS4LMb0UiEBMG5YHvdcMocTJw4H4Jl1u7jhiQ1pXTlDA7BSKifMn17OLefOpLzQT3MoRnmhn1vOnZmRRT+tZIVEob/n1TLyfW5u/cRMLj1hPADrdzTzlYfWsLmmNS3j0HaUSqkhrSkYpSGFJepXbKrjB89XEY7Z+NwWN5wxjdM6/ecwsaxA21EqpVRfpFIhAXDK1DLuXjiHimI/kbjNd//0Fg+sqE65yXtXNAArpYa8VCokACaWFXDvokoqx5cA8Oiq7XzjqddpPcTGPBqAlVKK9yskvO6ew2JxwMMPP3UMn57rNHl/dWsjVz289pDOqQFYKaWSPC6L0cUB8rw9T192WcKV8ydz05lOk/cdjdoLQimlDptlCaOK/RQFeq6QAPjozFHccfFsRhT0PNW523Md0l5KKTXIjSjwMTzf1+t200cV8bNL5x7SOTQAK6VUN4rzPIwq9mP1UiHRW7Of7mgAVkqpHuR53VSU+HFb6Q+XGoCVUqoXPreL0SX+Xisk+koDsFJKpcCdrJDI9/VcIdEXGoCVUipFliWMLPJTnEKFRErHS8tRlFJqCBle4GNEoa/X6cu90QCslFKHoMjvYVRR7xUSPdEArJRShyjgTa3Be3c0ACul1GHwui1GlwQOaV8NwEopdZh666LWHQ3ASimVJRqAlVIqSzQAK6VUlmgAVkqpLNEArJRSWaIBWCmlskQDsFJKZYkGYKWUyhINwEoplSUagJVSKks0ACulVJZoAFZKqSxJ39oaBxCRXwLnALXGmFmZOo9SSh2O5VW13L+imu2NQcaV5nH5KROZP728X86dyTvgXwNnZvD4Sil1WJZX1bJk2UZqW8OUBDzUtoZZsmwjy6tq++X8GQvAxpgVQEOmjq+UUofr/hXVeFxCnteNiPOnxyXcv6K6X86f9RywiFwmIqtFZHVdXV22h6OUGkK2NwYJeFz7vRfwuNjRGOyX82c9ABtjHjDGzDPGzCsrK8v2cJRSQ8i40jxCscR+74ViCcaW5vXL+bMegJVSKlsuP2UisYQhGI1jjPNnLGG4/JSJ/XJ+DcBKqSFr/vRybjl3JuWFfppDMcoL/dxy7sx+q4LIZBnaI8B8YISI7ABuNsY8mKnzKaXUoZg/vbzfAu6BMhaAjTELM3VspZQaDDQFoZRSWaIBWCmlskQDsFJKZYkGYKWUyhINwEoplSUagJVSKks0ACulVJZoAFZKqSwRY0y2x7CPiNQB2zJ0+BHA3gwdu69yZSy5Mg7InbHkyjggd8aSK+OA3B7LXmNMn3qg51QAziQRWW2MmZftcUDujCVXxgG5M5ZcGQfkzlhyZRww+MaiKQillMoSDcBKKZUlQykAP5DtAXSSK2PJlXFA7owlV8YBuTOWXBkHDLKxDJkcsFJK5ZqhdAeslFI5RQOwUkplyYAPwCLySxGpFZE3Or13q4hsEJF1IvJXERndzb6fE5HNydfnsjyWRHKbdSKyLN3j6PS5r4uIEZER3eyb8WvSh7Fk9JqIyLdFZGenc5zdzb5nisjbIvKOiNx0OONIw1i2isjryW1Wp3scyfevEZEqEdkoIrd1s2/Gr0kfxpK2a9LdWETksU7/NltFZF03+/btuhhjBvQLOAWoBN7o9F5Rp78vBn7WxX7DgOrkn6XJv5dmYyzJz7Vl8pok3x8H/AVnssuIbF2TVMbSH9cE+DZwfS/7uYAtwETAC6wHjsrGWJLbbe3ueqVpHKcCLwK+5MflWbwmvY4l3dekp+/ZTp//MbAkHddlwN8BG2NWAA0HvNfS6cN8oKsnjWcALxhjGowxjcALQJ9msaRxLGnV1TiSfgLc2MMY+uWapDiWtOphHL05HnjHGFNtjIkCjwLnZWksadXNOK4AfmCMiSS3qe1i1/66JqmMJe16+vcREQEuBB7p4tN9vi4DPgB3R0S+JyLbgUXAki42GQNs7/TxjuR72RgLgF9EVovIShH5RAbGcB6w0xizvofN+uWapDgWyPA1Sbo6mSL6pYiUdvH5fvs+SWEs4PyH9VcReU1ELsvAGKYCHxKR/4jI30XkuC626a9rkspYIPPXpLMPATXGmM1dfK7P12XQBmBjzDeMMeOApcDVA2AsE4wzrfES4KciMild5xeRPOB/6T7495s+jiVj1yTpPmASMBvYjfOrZbakOpaTjTGVwFnAVSJySprH4cZJQZ0A3AA8nrzry4ZUx5Lpa9LZQrq++z0kgzYAd7IU+FQX7+/EyUN2GJt8LxtjwRizM/lnNbAcmJPG804CjgTWi8hWnK91jYiMOmC7/rgmqY4l09cEY0yNMSZhjLGBn+P8Cnmgfvk+SXEsna9JLfBUd9sdhh3AH4zjVcDGaTrTWX/97KQylv64JgCIiBs4H3ism036fF0GZQAWkSmdPjwPqOpis78AHxWR0uSvex9NvtfvY0mOwZf8+wjgJODNdI3BGPO6MabcGHOEMeYInG/sSmPMngM2zfg1SXUsmb4myeNWdPrwk8BBlRrAKmCKiBwpIl7gYuCwKjIOdSwiki8ihR1/x/n36WrMh+NpnIdfiMhUnIdJB3Yf65drkspY+umadFgAVBljdnTz+b5fl3Q9OczWC+fXgd1ADOeH+UvAkzj/CBuAPwJjktvOA37Rad8vAu8kX1/I1liADwKv4zw1fR34UrrHccDnt5J8apyNa5LKWPrjmgC/Sx57Q/IHpSK57WjguU77ng1swnnC/Y0MfZ/0Ohacp+vrk6+NhzuWbsbhBR5Kfs+uAU7L4jXpdSzpviY9fc8Cvwa+csC2h3VddCqyUkplyaBMQSil1ECgAVgppbJEA7BSSmWJBmCllMoSDcBKKZUlGoDVgCAibWk4xhEiEkp2tHpTRH4mIvozoLJGv/nUULPFGDMbOAY4CshUjwmleqUBWA1YyTval5INbP4mIuOT709KNvB5XUS+29XdszEmDvwLmCyO/ycibyT3uSh5nPkiskJE/pTs8ap3zCqt9JtJDWR3Ab8xxhyD02fjzuT7dwB3GGOOxpnJdJBkU6CP4Mw+Ox+nCc6xONNN/1+nqcHHA9fg3C1PSm6rVFpoAFYD2YnAw8m//w44udP7v0/+/eED9pmUXM3gFeBPxpg/J/d7xDjNcGqAvwMdrQ9fNU5/1wTOFNWTUSpN3NkegFL9rCMHnKoD5+rr3H2VNnoHrAayf+F0nAKn2f0/kn9fyfttPy8+cKcu/AO4SERcIlKGsyTNq8nPHZ/sbmUBFwH/TMvIlUIDsBo48kRkR6fXdTi52S+IyAbgM8C1yW2/ClyXfH8y0NzLsZ/C6UK2HngJuNG83x5zFXA38BbwbnJbpdJCu6GpQSf5gC1kjDEicjGw0BjT5zXLRGQ+zkKZ56R7jEqB5oDV4DQXuDu5fE0TTo9jpXKO3gErpVSWaA5YKaWyRAOwUkpliQZgpZTKEg3ASimVJRqAlVIqS/4/63/hYGs9EDkAAAAASUVORK5CYII=\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "R^2: 0.9762999583583303\n", "Coeffecient: [[-1.07343662]]\n" ] } ], "source": [ "# Import LinearRegression\n", "from sklearn.linear_model import LinearRegression\n", "\n", "# Create the regressor: reg\n", "reg = LinearRegression()\n", "\n", "# Create arrays for features and target variable\n", "y = pop_17['LogRank'][:134]\n", "X = pop_17['LogPop'][:134]\n", "\n", "# Reshape X and y\n", "y = y.reshape(-1,1)\n", "X_LogPop = X.reshape(-1,1)\n", "\n", "# Create the prediction space\n", "prediction_space = np.linspace(min(X_LogPop), max(X_LogPop)).reshape(-1,1)\n", "\n", "# Fit the model to the data\n", "reg.fit(X_LogPop, y)\n", "\n", "# Compute predictions over the prediction space: y_pred\n", "y_pred = reg.predict(prediction_space)\n", "\n", "#Visualize the Distribution\n", "sns.lmplot(x='LogPop',y='LogRank', data=pop_17[:134])\n", "plt.show()\n", "\n", "# Print R^2 \n", "print('R^2: {}'.format(reg.score(X_LogPop, y)))\n", "print('Coeffecient: {}'.format(reg.coef_))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4. One more try!\n", "\n", "We didn't exactly get the -1.0 coeffecient with the Top 135 either, but it we are pretty close. In this run we got a correlation coeffcient of approximately 1.1%. This means that we can confirm that somewhere between using all of the metro areas at -0.9 and the top-135 at -1.1, there exists a line of best fit that closely appoximates -1.0. \n", "\n", "If we look at the Top-203 largest metros we find the arbitrialy best fit to rank-size regression coeffecient of -1.0 for the data from 2017. " ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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8fuV7mFRZBNDhaom932uJJdTUR/KOArBkzPiKIh763Ls59aiKHh1f0xTVTFjyilZBSMa1xRL8bvkmfvCX12iOJro9/ohBBXxixkieW7eLjbUtjC4v0GoJ6W+0DE1yRzyRZOn6Wq68/yW2N0Y6Pc6AYaVBdjXHGFUeJuz3srM5wq7mGEVBrwo5pL/QMjTJHT6vh5PGDeLBS05i2sjSTo8LeI36lhiJpKMg4KMpEmdnU4ykc7TFklQ3tnHto2uUppABQQFY+oyZMXZwEb/5zAl8+JhhB34fGFYaJpJI7l5PXNMYwQy8ZrTFk2yrb2NzXQuXP/CSgrD0e1oHLH2uvDDI9eccx8jyML965m3StRkUBb04l8RrRnEo9asZTSTxeox4wpFMOuLJVBVdcyTOwnuWKy0h/ZpywJI10XiSlzfXcdUDK9lQ2wpAScjHR6YN5+m1Owj6PGytayWWdMQTDq8H/F4vsUSSRNLh8xoBr4dhpSFiCcd1Z04FYNGSdXp4J9mmh3CS+xJJx6baFn76xBs8umoLACPLwnxyxiieeWsH7+xsprEtTjSeJOAzcJaeFaca/SScY/KwElqicQJeD83RBH6v6eGdZJsewknu83qMMYMLufYjU/jyB4/C7zU217Vy+5K3mHfMMB699BRuvWAGxSEfiST4vIZZ6qGec+yutgv7vazb0Yzfa3p4J/2GArDkhCFFQRa8awy3XDCDoSVB2uJJvvfn1/jpE68zdkgB1589jRFlYYaVhgj5PCSSjiSOIUVBAFpjqfXFYb8X2PPwDgct0QQbdrVQ3dCm3ZslpygAS84oCfl576QK/vdTs5g5phyA363YzFW/XcXg4iDXfGASFUVBCoI+PGYMLgxQHPLREo0TSzjGDS7YHYijiSTOOWLJVIotkXS0xZO8tq1R3dckZygHLDknnkiyua6VO5as494XNgAwuCjAtz8yhRljBjG0OMgza3ewaMk6NtW2MCr9sA3g2kfX4PcaW+taadmrQXH7hqEGBHweFi2YqXywZJIewkn/lUw6apoi/O2Vbfzo8Sqaowl8HuMLp07grONHUlEcojB44CrKxVXVLFqyjrXVjexoinZ6/aOHFfPXK2dn8iNIflMAlv6vtjnK6k11XPvoGtbvbAHgA1OGctXciVSWhBhUGMCs49/17nZwvnruRC6fOykj45a8p1UQ0v+VFwaYOXYQt82fwZxJqa5qT7y6ncvuX0nVtga21LcRSyQ7PPcrpx+9zw4dezPgF4vf0qoIySoFYMl5RUEfEyqLuO6jU7nkvePxGLxZ08Ql96zgX2/UsLm2leZI/IDz5kyu5AtzJnR4TUcq17xoyboMj16kcwrA0i8EfV5GlhfwqXeN5fpPTKMs7KexLc5Xf/8yv3nuHbbWt7KzKcL+KbXL505i8tCiDq+ZcLBmS30fjF6kYwrA0m94Pcbw0hCzJ1Wy6MKZTB5WjAN+/ew7fPMPa9hc29phSuIrpx/daUKuMRJXGkKyRgFY+hUzo6I4yOThJfz83Ol8ZNpwAJ5bt5PP3buCqq0NB6Qk5kyupDDY8a+6c6g4Q7JGAVj6pdKwnzGDC7nmg0fxpb1KmL9w7wqefG072xva2LFXSuLYkeUYBz6aNuCN6ibNgiUrFICl3woHvAwvDXPmcSO4+fzjqSzeU8J86+I32dUU2Z2SWDh7PAGf54DNQQ1wzrHwnuWcf8fzCsTSpxSApV8L+DyMLAszfXQ5ixbMZOYRZQA8vHwz1zy8mq11rWyubWXWuEH7rIjwWOqXP0kqCCedU8Me6XM9CsBm9tkO3vth7w9H5OB5PMaw0hBHDC7gh2dP44ITRwOwelM9C+9Zzsub66huaOOCk8ZwVGUhIb8Hr8fAwOdJ5ZUTSaeGPdLnejoDPtvM5rd/YWa/AHq217hIHxlcFGRYaYj/mj2B75w5lYKAl51NUa56cBV/XLmF+pYo/zV7AhVFQY4YVIAnnRCOJx0eS62yiCdTDXtmfvcJpSQk43ocgIGLzOx8M/sNEHfOHTArFsm24pCf4aUhTj2qklsvmMGYQQXEk44b/7GWH//tdaaOKOHSU49kcGEQr8dD0qVmwX6vl2QS4slUSkI9hKUvdNkLwswG7fVlMfAH4FngWgDn3K7eHIx6QUhviSeSbG+MUNsc4cd/e50lb+wA4MjKIr5z5hSGl4Z5eVM9V/12JT6P4TEjEk/u85CuMOClJOxj7OAi7r/4pOx8EOmveqUXxHJgWfrPfwJlwIf3el8kJ/m8HkaUhqgsDvGtM6Zw8ex0CXN1E5+7ZwVL39nFsaNKGTu4ADNI7DcR8VgqNbGjMcra6sYsfQoZ6NQNTQa8upYou5qjrNhQy3cfe4361hgGfPrksRxZUcTN/3yToM/D+p0t+8yA23sIFwS8vHrdvOwMXvqr3u2GZmbvNrMLzOxT7a9DH5tI3ykrCDC0JMSsMYNYtGAGR6VLmO989h3+/PJWFr5nPGXhwAHntQfj1mhCeWDJiJ4uQ7sb+AlwCnBC+jUrg+MS6VWFQR/Dy0KMLC/gxnOn8+FjUyXMz761k18++zaXve9IjhtVhq+DeYvfa+qaJhlx4JYCHZsFTHG5lK8QOUhBn5cRZWG2N7TxxQ9MYvKwYm56ai2balv5wn0r+Pj0kazaVHfAefGEUx5YMqKnKYhXgGGZHIhIX2jvqFYc8vPhacO56bx0CXMsyX1LN+5eGwypJJ7PwOs1ovGOm76LHI6ezoCHAK+a2YtApP1N59yZGRmVSAa1d1QL+DwcNQxuXzCD7/35NVZsqCPhUoHXa6kKO1yqTDngTUXm9j3nNta2MDq9Gag295RD1dMA/O1MDkIkG0rDfgLeVFnyj86exq+eeZsHlm5M7ZbhwJt0BHweSgI+xg0pYnFV9e5dl8vC/t2FGteBgrAckh4FYOfc04d6AzPzklozvNk5d8ahXkckE9o7qm1vaOPi2ePxmXHvixtwpHbMCPm9+Lwezp4xipufepNoPM7OpgTRRJKA10NJ2MeiJesUgOWQ9HQVxElmttTMmswsamYJM2vo4T2uAF479CGKZFZ7R7WCgI/PvGccV82duHszz9qWGCPLCjhudCnrdjSxozFKLJFM941QoYYcnp4+hLsFOB9YC4SB/wR+0d1JZjaKVOXcLw91gCJ9ob2jWmnYzxnHjeB3n3sXsycOAWD5hlouf2AlbbEEWGoXjUgsSTSeJJZ0tEQTWR699Fc9LsRwzr0JeJ1zCefcr4GelAb9HPgyqbarHTKzi81smZktq6mp6elwRDJicFGQiuIghUE/3/rIFC5+zzg8Bmurm2iLJUkkIZZ0OFSoIYevpwG4xcwCwEoz+7GZXdXduWZ2BlDtnFve1XHOuTucc7Occ7MqKtThUrKvvaOa3+vlvBOP4MdnT6M07N8n6EJ6tQQQ8KlQQw5Nj3pBmNkYYDsQAK4CSoFb07Pizs75H+BCIA6EgBLg9865BZ2do14QkkviiSTbGtqIxpNsb2jjSw+tZlNdK7BnqRoGxSEfzZEElSUhLU2Tdj3qBXHIzXjM7GTn3LM9PHYOcE13qyAUgCXXOOeoaYzQFIkTjSf55O3/pjGSyvkaUBLy0RSJ4/Ua4wYXEkskiSfhujOnKgjnt8NvxmNm3nQT9mvM7Jj0e2eY2b9JPZgTGdDMjMqSEOUFAQI+D1//0BRKw6nVmw6ob4vjHAwpDOJcqrG714NSEtIj3TVkvwsYDbwI/AewhVRfiK845/7Q24PRDFhyWVMkTk1jhBfe2smvnn2bt2qaSKb/+ZQX+BlSGMDMcDh2Nkc5amiJKuby1+GnIMzsFWCacy5pZiFgGzDBObezd8a4LwVgyXVtsQTVDRHiySR1LVE+dedSmiJxAMJ+L8NLgzS2xahrjTOiNERR0EdbPEks4ZSWyC+90g846pxLAjjn2oB1mQq+Iv1ByO9lRFmIgM9DWUGAr86bTGHAC0BrLMH6na3saolRGvIR9HmJJ1PnqKWldKS7GXAL0L7SwYAJ6a8NcM65ab05GM2Apb9wzlHdGKE5EufFdbu4/em3WL9rz44alcUBSkN+zAwMPGbsaIowsbJYaYn80CspiDFdneycW3+Qg+qSArD0N7uao9S1RAFYv7OZa/+4ho21qaVqJSEflcVBPGbUtkSob40zqjxV8twaSygtMbAdfgrCObe+q1fvjFOk/xpUGKCyJISZMWZwIbfOn8HU4SUANLTF2VjbujsnXBLy4fd60/vM+ZSWkB4XYjSybxEQQD2pLmdfdM71ym+RZsDSX+39cM45x/V/e4PH12wDUk3gQz5jRFkYS0+MvB6jJRpnW0OEiuKgUhIDT69uyvlz4EvASGAUcA1wH/AAcOehjE5kINn74ZyZ8eV5R3H9J6ZREvKRSDqao0mqGyO0T3ga2mJsqm0F3D69hdVTIr/0NACf6Zxb5JxrdM41OOfuAD7onHsQKM/g+ET6DZ831dayMJgq1Jg5ppzbL5zJpKFFANS3xtlU10o8maSmMbWxzOB0AYdSEvnpYJrxnGNmnvTrHKAt/T1t1CmSZmYMTVfOAQwrCXHTecdz+jGpLRVbY0nW72ol6RxDS4IUBX2p8uVEkrDfy6baFiC19dH5dzzPKT96ivPveF4z4wGqpwF4PqnGOtXp14XAAjMLA5dmaGwi/Vb5Xg/nAj4P13xgEle/fyJ+r5FIOpIOIvE9c5dE0tHYFmNkWXj31kfVjW1KTwxwh9yMJxP0EE4Gmkg8wfb61MM5gNe2NvDtR1+lpimVgigO+hhaEiASd8STjqveN5GHV2xmZ3OEgsCeHcNaonEqi0Pcf/FJWfkcctB67yGcmY0ys0fMrDr9+l16twsR6ULQl3o4F/SnquWOHl7C7RfOYProMgAaI3E27GqlNOTnitMmMmvcINbvasbvsd0P7BrbYmyta+XFd3YpHTHA9DQF8WvgUWBE+vWn9Hsi0g2f15PqCxFKzWjLCwJc/4lpnDMrNYeJJhzbGyMUp7usDS8J0xxNFWo0tEbZUtdGLOkI+TxKRwwwPQ3AFc65Xzvn4unXXYC2rxDpITOjsjjEoMLUwzmvx7jkvRO49owphPweapoiXPngSh5bvZXzThhNPOloicbZ3tBGMtWOhSFFQa2WGGB6GoB3mtmCdH9gr5ktANSUR+QglRUEGFoSwmOpFOGcoyq4df4MRpWHiSUcP/v7GyxZW8Pn3zuBwYVBogmH3+thaHGI4vQMeu/VEtK/9TQAfwY4h1Q7yq3AJ4CLMjQmkQGtMOhjeFkIvzf1z2/s4EJumz+DU45M7cL8l1e2cc8LG/jv049i2siy9MzXS11LlLeqG3ltWwP1rTGlIQaAw9mS6Ern3M97czBaBSH5JJF0bG9oS213T6rD2v0vbuTOZ98m6VLNfD45czR/eWUr8USC2ubY7mfrQ4qCBHxeNfPJXb1aityRqw/jXJG85/UYw0tDFIf8QCpPfMF/HMEPP34sJSEfDW1xfv3vtzluVBnNkQQO8Hs9VBaHKCsI4NPWR/3e4QTgHkV4EemcmVFRHGRwUXD3e7PGDtpdwpx08PiabSQcjBlcwOjyAoqCPpJJh8/jYeOu5iyOXg7X4QTg3KngEOnnSsN+hpeGdz+cay9hnjc1VcIciSdZv7OF9buaWbejiY21LdS2RBhSHNq9JZL0P93titxoZg0dvBpJrQcWkV4SDngZURbe/XAu4PPwpQ9O4qq5E/EYJNLlyw6IJZLsbI5x/KhSqhvaqNmr05r0H901ZC92zpV08Cp2zvm6OldEDl7Al+qoFk7vM2dmfOS4EUwYUoQnnfRLpJYFUx728dLGeiBVLbe5rpVoPJmNYcshOpwUhIhkgMdjDC8NUxr2736vIRJjzOACwumS5ngSWmJJttTtWQ8cjSfZUpfagUP6BwVgkRw1uCjIkOIgZsbwkjDxhGNUWYjydGBujSWpbY1Rta1h9zlJ56hpjFDd2KaURD+gACySw0pCfoaXhrjgxFR5cls8STjgwZtOR8QSjsvuX8mfV2/d57ymtjibapWSyHUKwCI5LuT38vGZo/ji+yfh93rY1hDB6zUqivz4PKn+wj/9+xv85InX9wm4sUSSzUpJ5DQFYJF+wOf18PEZoygvCDCyLMyF68t8AAAXvklEQVTYQYWUFwQZM6iAsD/1z/gvL2/jigdXUt3Qtvs8t1dKIplUSiLXKACL9BMej7G9sY3CoHf3e16PMbIsRFHQiwGvb2tk4T0rWLGhdp9zm9ribK5rJRJP9PGopSsKwCL9yOjyAmIJh8/r2V2LGok7jqwo5odnp0qY61tjfPnh1TywdOM+D+JiiSRb6tpoUEoiZ2hLIpF+pH2/OL/XCPo8NEXixBKOK06byInjB7Gtvo1vPbqGtdVNABw7ogQwqpvaGF4S5rwTRnPi+EEUBX0MKQri8aijQIb06H9YBWCRfmZxVTWLlqxjU20LI8vCfHLmKI4fU777+5FYgp//Yy1/W7MdAF86TZF0EE/uCdZ+r4fKkiBBn7ezW8mhUwAWyQfOOWqaIjS1xfd57//duZRNda0AeAyGloTweYzBhUF+du5xQKrSblBhYJ+iD+kVGW9HKSI5oH27o8GFwX3eiyWTjCpPBd2kg631bTRF4myt31M955xjZ1OE6gatksgGBWCRAaK0wM+w0j3bHQ0vCWMYRwwK716qVtsSozmapL5l3wdxTRGtksgGBWCRAaQg4NvdUa19c89YwlEW9u1u5tMUifPpu5by+rbGfc5tXyVR36pVEn1FAVhkgAn4PIwoC/PeyRVccdpE/F4P2xuj+LxGediHAXWtMS67/yX++vK+JcztKYntSkn0CQVgkQHI6zGGlYSYO3UoJSH/7uq5iuIQRwwK4/MY8aTj+ife4Gd/f+OAnhHN6ZRE+351khkKwCIDlJkxpChIdVMbIf+ef+pBn5cjBoUJ+lLvPbZ6K1fuV8IMqZTE1vq2A/LF0nsyFoDNLGRmL5rZKjNbY2bfydS9RKRzYwYVEk+6fRZGxRKOyUOL+czJYzGgalsjl9yzgpf2K2F2zrGzOZWSSCgl0esyOQOOAKc5544DpgPzzOykDN5PRDqwcPZ4EkmIp7fSaI0liCcd5594BAtOGrO7hLmuNcaXHl7Ng/uVMEM6JVGrlERvy1gAdilN6S/96Zf+EyrSx+ZMruS6M6cytCRMSzROZXFodzUcwAljB3HbghkcWZHahXnRknVc99hrtET33ewznkylJOpaotn4GANSRnPAZuY1s5VANfB359wLHRxzsZktM7NlNTU1mRyOSN6aM7mShbPHM3pQIdsb23h4xSZeXLdr9/eHl4a5+fzpfHDqUACefqOGL9z7Eht2texzHeccu5qjbKtXSqI39EkpspmVAY8AlznnXunsOJUii2TG3k18wn4vrbEEkXiSS089khPHDdp9nHOOR1dt5Rf/fJN40lEQ8PKVeZM5ZeKQA67p86R6SYT86iXRgdwpRXbO1QH/BOb1xf1EZF+LlqzD7zUKAj7MUn8GfR4eWbEZn2dPGDAzPjp9BDecexyDiwK0RBNc++gafvmvdQfMeOPJ1CagSkkcukyugqhIz3wxszDwfqAqU/cTkc5trG3ZvaNyu7Dfy5b6VkaUhQju972pI0pZtGAm00aVAnDfixv5yu9f7rBKbldzlK31rUpJHIJMzoCHA/80s9XAUlI54McyeD8R6cTo8gJa91vB0BpLMKq8AJ/Xw4jSEEUh3z7fH1QY4CefmMYnZo4EYPn6Wi65ZzlvbN+3hBmgNZpgc20rrVGtkjgYmVwFsdo5d7xzbppz7hjn3HWZupeIdG3h7PHEEo6WaBznUn/GEo6Fs8cDezqqDSoM7HOez+vh83OO5BsfPpqQz8P2hkiqhPmVbQfcI7VKopXaZqUkekqVcCJ5oH0pWmVxiPrWGJXFIa47cypzJlfuc1xZQWCfjmrtTptcyS0XHM/IsjCxhOP6v73ODR2UMAPUtqRSEu3rjqVzCsAieaJ9Kdqo8gI21rawaMk6FldVH3Dc3h3V9ja+oojb5s/gXeMHA/Cn1Vu56rcrqWmMHHCN1miCzXVKSXRHAVgkT7QvRatubKMs7Ke6sY1rH13TYRBu76gWDuz7cK4o5OO7H5vKp9MlzK9tbWTh3csPKGEGSCQdW+tb2aWURKcUgEXyREdL0fxeY9GSdR0e395RrWS/7Yo8Zlx40hh+8PFjKN6rhPm3yw4sYQaoa4mypU4piY4oAIvkic6Wom2qbenkjD0d1QYXBbH98sL/MW4wt83fU8J8+9Pr+O5jr3WYdmiLpVIS+5c35zsFYJE80dVStO6Uhv0MKwnh3W8b+xFlYW46fzrvn5IqYV78Rg2fv2/FASXMkEpJbKtvY1dztMOZcj5SABbJE90tRetOOODt8OFcyO/lK/OO4vLTjsTrMdbvbOHz967gmbU7OrxOXUuUrfVtSkmgACySN3q6FK0rfq+HkWVhCoP7Fm2YGR87fiQ3nHMcgwv3lDD/6pm3O6yQU0oipU+a8fSUmvGI9B+7mqMd9oHY1RzlO396lZc31wNwwthyvvahoynd72Feu9Kwn0GFgQNyzP1c7jTjEZGBZ1BhgIriAx/ODSoM8NNPTuPsGakS5qXv1PK5e1Z0WMIMUN8aY0t9G7E8TEloBiwiLK6qZtGSdWysbWF0eQELZ4/vcWqiLZaguiFCPHlgAP3Ha9X85InXicSTBHwernzfROYdM6zD63g9qRUX+6c3+inNgEWkewdToNGRkN/bYUc1gPcdXckvLjieEWUhovEkP/7b69zwZMclzImkY3tDGzubInmzSkIBWCTPHWyBRkd2d1TrYPY6vqKI2+fP5KT0Fkh/WrWVqzspYYb8SkkoAIvkuUMp0OiImVFZcmBHNUiVMH/vY8dw0bvHYMCrWxu55J7lrNpY1+G1IrFUe8umyMBeJaEALJLnDqdAoyNlBQGGlhzYUc1jxqfeNZbvn3UMRUEftS0xvvjQKh7qpIQ56RzVDW3sGMApCQVgkTx3uAUaHSkM+hheFjqgaAPgpPGDuW3BDMZXFJJ0cNvT6/jenzsuYQZoaI2xua51QKYkFIBF8lxvFGh0JOhLVc51tGnnyLIwt5x/PHOPTt3jn6/X8IX7VnSa9ojGkwMyJaFlaCKSUc45djRFaWw7cD855xx/WLmFWxe/RSLpKAx4+eqHJvPuCQfuwtyuOORnSFHOF25oGZqIZJ+ZUVHccUc1M+OsdAnzoMIAzdEE3/jDGu58tuMSZoDGtlRKoqOlbP2NArCI9InOOqoBHDOylEULZnDsyBIA7nl+A19/5GUaOtiFGVIpiS11rR3OqvsTpSBE5ACHUxnXnVgiybZO1vnGE0luf3odv39pMwDDSkJ858wpTBxa3On1cjQloRSEiBy8w62M6057R7WCwIFFGz6vh0tPO5KvfWgyQZ+HbQ1tXPbASp5Yc+AuzO0a22Jsqu2fKQkFYBHZR29UxnXH4zGGlYY67ZA29+ih3HLB8QwvTZUw//Dx17nxybWdLkWLJZJs7ocpCQVgEdlHb1XG9cTgomCHHdUAJlQUcfuCGfzHuFQJ8x9XbeGqB1d1WsLsnKOmMUJ1YxvJTh7g5RoFYBHZR29XxnWnOORneGkIn+fAcFQc8vP9s47hU+8aA8CrWxtSJcybOi5hBmhqi7O5rpVIvOPCjlyiACwi+8hEZVx3uuqo5jHjoneP5Qd7lzD/dhUPL9/UaYlyLJFkS10bDTmeklAAFpF9ZKoyrjtddVSDvUqYh6RKmG9d/FaqhDnW8UzXOceOxgjVDbmbktAyNBHJOXUtUXY1H7jdEaTSIT974g3+kV6VMW5IId85c0qXKRK/10NlSZCg78AZdoZoGZqI9E+ddVSD1APBr31oMpeeOgGvx3h7RzOfu3cFz721s9Pr5WpKQgFYRHJSVx3VzIyPzxjFzz6ZLmGOJPj6H17h112UMOdiSkIBWERyVlcd1QCOHZUqYT5mRKqE+e7nN/D1P7zSaQkzQFMkd1ZJKACLSE7zeozhpSGKQx0XbQwuCvLTc47jY9NHAPDi27v43L0reKu6qdNrtqck6rsI1H1BAVhEct7eHdU64vd6uPx9E/nq6akS5q31bVx6/0v8/dXtnV7TOcfOpgjbs5iSUAAWkX6jNOxneGm4w45qAO+fMpSbz0+VMEfiSf7nr1Xc9I/OS5gBmtMpibZOlrNlkpahiUivymQntXbReJLtDZ3vnNzQGuMHf63ixbd3ATB1RAnf+sgUhnQyg4bULHtQQYDSgo5THQdJy9BEpG9lupNau4Cv845qACVhPz/Yq4R5zZYGFt69nNVdlDA759jZnEpJdLaSorcpAItIr+mLTmrtuuuo1l7C/P2PHUNh0JvehXk1v1vReQkzpFMStX2TklAAFpFe05ed1Np11VEN4F0TBnP7/JmMH1JIIun4xT/f4gd/qeq0hBkgnkyytb6N+pbMrpLIWAA2s9Fm9k8ze9XM1pjZFZm6l4jkhr7upNauvaNaZw/nRpaHufmC43lfOhf9j6pqLrvvJTbXtXZ6zfaUxLb6zKUkMjkDjgNfdM5NAU4CvmBmUzJ4PxHJsmx0UmsX8nsZWRYm4Os4rLWXMH/h1Al4DNbtaOaSe5bz/LrOS5gBWqKZS0lkLAA757Y651ak/94IvAaMzNT9RCT7stVJrZ0vvd1RZx3VzIyzZ4zip+ccR3mBn+ZIgq898gp3/fsdkl3khdtTEnUtHTcIOlR9sgzNzMYCS4BjnHMNnR2nZWgi0ltqm6PUdhEwdzRF+Pajr/Lq1lRI+o9xg/jahyZ3WnHXriDgo6I42Gm6Iy03lqGZWRHwO+DKjoKvmV1sZsvMbFlNTU2mhyMieaK8MEBlSajTh3NDioLccO6eEuYX3t7FJfd0XcIMvZuSyOgM2Mz8wGPA35xzP+vueM2ARaS3ReIJttdHiCc7r4Z74tXt/OzvbxCNJwn6PFz9/km8f8rQbq9dXhCgvDDQ0beyOwO21H92fgW81pPgKyKSCUGfl5HlnXdUA/jAlKHcsl8J881PvdllCTNAbUuUrfWth7xKIpMpiJOBC4HTzGxl+vWhDN5PRKRD3XVUAziysojb5s/gxLHlADzy0ma++NtV7GzqeBfmdq3RBJtrW2mNHnxKQr0gRCSv1LfE2NnceVBNJB3/99w73P38BgAGFQb41hlTOHZUabfX3islkRsP4UREcklpgZ9hpR1vdwSp2fKnTx7H9z42lcKgl13NUa5+aBW/X7G5yxJm2JOS6CkFYBHJOwUBHyPKwh1ud9Tu3ROGcNv8GYxLlzDf8s83+Z+/VnW7+uFgUhEKwCKSl9o7qoUDnT+cG1VewC0XHM+pR1UA8ORr1Vx6f9clzAdDAVhE8pbHYwwvDXfaUQ1SJczf+PDRfH5OuoS5ppnP3bOi2xLmHt3/sK8gItLPDS4KMqSLjmpmxidm7ilhborE+fojr/CbbkqYu6MALCIClHTTUQ3guFFl3L5gJlOGF+OA3zy3nm/84RUa2w6tbaUCsIjkvMVV1Zx/x/Oc8qOnOP+O53t9h412Ib+XEV10VAOoKA5yw7nTOfO4VAnz8+vSuzDXdF3C3BEFYBHJaX21zVE7v9fDiNIwhZ10VGs/5sq5E/nveUcR8HnYUtfGpfe9xJOvdb4Lc0cUgEUkp/XlNkftPB5jaEmI8oIO+zzs9sGpw7j5vOkMK0mVMP/gL1Xc8tSbPb/P4Q5URCSTsrHNUbvuOqoBTBxazG0LZnBCuoT59y9t7vH1FYBFJKdla5ujdkVBHyPKQvg8nYfL0rCfH5x1LAtOOuKgrq0ALCI5LZvbHLUL+ryMKAsR7KKjmtdjfObkcXz3o1N7fF0FYBHJadne5qidz+thRGmIolDnD+cATj5ySM+vebiDEhHJtDmTK/s84HbEzKgsDhH0dt1Rrac0AxYROUjddVTrKQVgEZFD0JOOat1RABYROUQBn4cR3XRU64oCsIjIYfB6jGElIUq66KjWGQVgEZHDZGYM6aajWke0CkJEpJeUhPwEDiInrBmwiEgvCnVRrLE/BWARkSxRABYRyRIFYBGRLFEAFhHJEgVgEZEsUQAWEckSBWARkSxRABYRyRIFYBGRLFEAFhHJEgVgEZEsUQAWEckSBWARkSxRABYRyRIFYBGRLFEAFhHJEgVgEZEsUQAWEcmSjO0JZ2Z3AmcA1c65YzJ1HxGR3rC4qppFS9axsbaF0eUFLJw9njmTKzN6z0zOgO8C5mXw+iIivWJxVTXXPrqG6sY2ysJ+qhvbuPbRNSyuqs7ofTMWgJ1zS4Bdmbq+iEhvWbRkHX6vURDwYZb60+81Fi1Zl9H7Zj0HbGYXm9kyM1tWU1OT7eGISB7aWNtCeL/djMN+L5tqWzJ636wHYOfcHc65Wc65WRUVFdkejojkodHlBbTGEvu81xpLMKq8IKP3zXoAFhHJtoWzxxNLOFqicZxL/RlLOBbOHp/R+yoAi0jemzO5kuvOnEplcYj61hiVxSGuO3NqxldBZHIZ2v3AHGCImW0CvuWc+1Wm7icicjjmTK7MeMDdX8YCsHPu/ExdW0RkIFAKQkQkSxSARUSyRAFYRCRLFIBFRLJEAVhEJEsUgEVEskQBWEQkSxSARUSyxJxz2R7DbmZWA6zP9jj6yBBgR7YH0Yfy7fNC/n3mfPu80Pln3uGc67Yfek4F4HxiZsucc7OyPY6+km+fF/LvM+fb54XD/8xKQYiIZIkCsIhIligAZ88d2R5AH8u3zwv595nz7fPCYX5m5YBFRLJEM2ARkSxRABYRyRIF4D5gZneaWbWZvbLXe981s9VmttLMnjCzEdkcY2/q6PPu9b0vmpkzsyHZGFumdPIz/raZbU7/jFea2YeyOcbe1NnP2MwuM7MqM1tjZj/O1vh6Wyc/3wf3+tm+Y2YrD/a6CsB94y5g/0XZ1zvnpjnnpgOPAdf2+agy5y4O/LyY2WjgA8CGvh5QH7iLDj4zcINzbnr69Zc+HlMm3cV+n9fMTgU+ChznnJsK/CQL48qUu9jv8zrnzm3/2QK/A35/sBdVAO4DzrklwK793mvY68tCYMA8De3o86bdAHyZAfRZ23XxmQekTj7v54AfOuci6WOq+3xgGdLVz9fMDDgHuP9gr6sAnEVm9n0z2wjMZ2DNgA9gZh8FNjvnVmV7LH3s0nSq6U4zK8/2YDJsEvAeM3vBzJ42sxOyPaA+8h5gu3Nu7cGeqACcRc65rzvnRgP3ApdmezyZYmYFwNcY4P+R6cBtwARgOrAV+Gl2h5NxPmAQcBLwJeC36dnhQHc+hzD7BQXgXHEvcHa2B5FBE4BxwCozewcYBawws2FZHVWGOee2O+cSzrkk8L/AidkeU4ZtAn7vUl4EkqSa1QxYZuYDPg48eCjnKwBniZlN3OvLjwJV2RpLpjnnXnbOVTrnxjrnxpL6hzrDObcty0PLKDMbvteXZwEHrAoZYP4AnApgZpOAAAO/O9pcoMo5t+lQTvb18mCkA2Z2PzAHGGJmm4BvAR8ys6NIzRLWA5dkb4S9q6PP65z7VXZHlVmd/IznmNl0Ug8d3wEWZm2AvayTz3sncGd6qVYU+H9ugJTadvE7fR6HmH4AlSKLiGSNUhAiIlmiACwikiUKwCIiWaIALCKSJQrAIiJZogAs/YaZNfXCNcaaWWu6g9WrZna7menfgWSFfvEkH72V7mA1DZgCfCzL45E8pQAs/Vp6RvtUuuHNP8zsiPT7E8zseTN72cy+19Hs2TkXB/4NHGkp15vZK+lzzk1fZ46ZLTGzP5vZ65oxS2/SL5L0dzcDv3HOTSPVU+Om9Ps3Ajc6544lVfp8gHSToPcBL5Oq558OHEeqvPT6vUqJTwQuIzVbnpA+VuSwKQBLf/cu4L703+8GTtnr/YfSf79vv3MmpHcveBb4s3Pur+nz7k83z9kOPA20t1N80Tm3zjmXIFV2egoivUC9ICQfteeAe2r/en3V70uv0AxY+rt/k2qIAqnG9v9K//159rT4PG//kzrwL+BcM/OaWQUwG3gx/b0TzWxcOvd7LvBMr4xc8p4CsPQnBWa2aa/X1aRys582s9XAhcAV6WOvBK5Ov38kUN/NtR8BVgOrgKeAL+/VLnMpcAvwGvB2+liRw6ZuaDIgpR+wtTrnnJmdB5zvnPvoIVxnDnCNc+6M3h6jiHLAMlDNBG5Jb4lTB3wmy+MROYBmwCIiWaIcsIhIligAi4hkiQKwiEiWKACLiGSJArCISJb8fzGI/gK0PfCSAAAAAElFTkSuQmCC\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "R^2: 0.977705640850799\n", "Coeffecient: [[-1.00146052]]\n" ] } ], "source": [ "# Import LinearRegression\n", "from sklearn.linear_model import LinearRegression\n", "\n", "# Create the regressor: reg\n", "reg = LinearRegression()\n", "\n", "# Create arrays for features and target variable\n", "y = pop_17['LogRank'][:202]\n", "X = pop_17['LogPop'][:202]\n", "\n", "# Reshape X and y\n", "y = y.reshape(-1,1)\n", "X_LogPop = X.reshape(-1,1)\n", "\n", "# Create the prediction space\n", "prediction_space = np.linspace(min(X_LogPop), max(X_LogPop)).reshape(-1,1)\n", "\n", "# Fit the model to the data\n", "reg.fit(X_LogPop, y)\n", "\n", "# Compute predictions over the prediction space: y_pred\n", "y_pred = reg.predict(prediction_space)\n", "\n", "#Visualize the Distribution\n", "sns.lmplot(x='LogPop',y='LogRank', data=pop_17[:202])\n", "plt.show()\n", "\n", "# Print R^2 \n", "print('R^2: {}'.format(reg.score(X_LogPop, y)))\n", "print('Coeffecient: {}'.format(reg.coef_))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Eureka! \n", "\n", "There is our -1.0 regression coeffecient! \n", "\n", "Through all of this fun, and several beautiful R^2 and coeffecient values, we found that US Metro Areas do closely follow the distribution that George Zipfs found almost 90 years ago.\n", "\n", "This notebook could be re-run using (2016,2015,2014,etc.) which are contained in the csv file, just change the pandas population column and run!\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Citations\n", "\n", "* Gabiax, X., ZIPF’S LAW FOR CITIES: AN EXPLANATION (Cambridge, MA, 1999). [Actual Paper](http://www.casa.ucl.ac.uk/mike-michigan-april1/mike%27s%20stuff/attach/Gabaix.pdf)\n", "\n", "* Krugman, P., The Self-Organizing Economy (Cambridge, MA: Blackwell, 1996a)., ‘‘Confronting the Urban Mystery,’’ \n", "\n", "* [Census Population Estimation Methodology](./Metro_Est_Methodology.pdf)\n" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.5" } }, "nbformat": 4, "nbformat_minor": 2 }