Tsallis q distribution project

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Warka late Uruk site sizes

u=c(24,20,20,20,20,20,20,19,19,18,18,17,17,17,17,17,16,15,15,14,12,11,11, 10,10,10,10,10,10,10,10,9,9,9,9,9,9,8,8,8,8,6,6,6,6,6,6,6,6,5,5,5,5,5,5,5,5,4,4,4,3,3,3,2,2,2,2,2,2,1,1) in hectares times 10 (multiplying makes no difference to the scaling)


WarkaLateUruk71sites.jpg WarkaLateUruk71Excel.jpg


Laurent's method converges in results on q with Cosma's MLE, and qc=0.43 -> q=2-(1/0.43)= -0.33. But Laurent's kappa=14.83, while Cosma's 1567.631 seems way off. Since we had trouble before with Cosma's kappa, we need him to check his MLE program.

Tasks

LogLikelihood Comparisons

Tallis Matlab fit

Vuong LR test Log-Likelihood test - comparison of two models

Introduction

Using the estimating Tsallis q equations, the Tsallis q distribution project (tentative paper title Estimating Tsallis q and Power-Law Tails in Empirical Data) Tambayong, Clauset, Shalizi, and White compare two distributions to empirical data

1. q-Exponential (Tsallis) using Cosma Shalizi, 2007 Maximum Likelihood Estimation for Distributions. http://www.cscs.umich.edu/~crshalizi/research/tsallis-MLE These are classically known as Pareto II distributions. (Examples). For a standard book on Pareto II see Extreme Value Distributions: Theory and Applications 2001 Samuel Kotz and Saralees Nadarajah. The bibliography on Tsallis Entropy at http://tsallis.cat.cbpf.br/TEMUCO3.pdf lists entry 2329 as C. Tsallis, Entropy, Springer Encyclopedia of Complexity and Systems Science (2007), in press. For the probability distribution of q-entropy see Are citations of scientific papers a case of nonextensivity? Constantino Tsallis, Marcio P. de Albuquerque, 2000. European Physics Journal B 13:777. Where x is a continuous variable, Tsallis entropy has the form(s)
P_q(x) = [1-(1-q)\kappa x]^{1/(1-q)} \left(1\right) escort cdf.

Substituting q=1/Θ and κ=Θ/σ, this gives the Pareto II

P_q(x) = [1-x/\sigma]^{-\theta} \left(1\right). The probability density function is....
2. Pareto distributions using Aaron Clauset, Cosma Rohilla Shalizi, and M. E. J. Newman. 2007. Power-law distributions in empirical data. http://arxiv.org/abs/0706.1062. Submitted on 7 Jun 2007. R and Matlab software and documentation and R Code for The Estimation of Power Laws and Their Comparison to Heavy-Tailed Alternatives. Their results are summarized at http://en.wikipedia.org/wiki/Power_law. Where x is a continuous variable, the power law has the form
p(x) = \frac{\alpha-1}{x_{\mathrm{min}}} \left(\frac{x}{x_{\mathrm{min}}}\right)^{-\alpha} \left(2\right),

where the constant is necessary to guarantee that the distribution is properly normalized. Comments from Cosma's blog. Where the fitted distributions for the q-exponential give 2 < q ≤ 3, then the tail of the q distribution asymptotes to a power-law tail with exponent {\alpha} = 1/(1-q).

Here we are trying (courtesy of Cosma Shalizi's R software and plot routines) to plot a Pareto fit and a Pareto II fit (q-exponential) against the ranked frequency data for eight of the Clauset et al. (2007) datasets, those found to have a moderate power-law tendency or a power-law tendency with a secondary upper-tail cutoff (i.e. two power-law slopes). For the proposed article, our decision is to apply the same tests used to determine the cutoff for Pareto I to the the Pareto II distributions: i.e., to apply the Kolmogorov-Smirnov test to each fit of the Pareto II, varying the minxq cutoff. In most cases from the Clauset et al. 2007 study the K-S values forms a U-shaped distribution of fit with a single minimum or minimum region, varying the minxq cutoff for the Pareto II. Shalizi's Likilihood ratio tests can then be used for comparison of models. The challenge here will be to compare fit between Pareto I with one cutoff, minxp and Pareto II with another lower cutoff, minxq.

Birds

Birds.q This distribution is for the numbers of sightings of bird species in the North American Breeding Bird Survey for 2003.

Cosma proposes that the following provides a comparison of fit of the Pareto and Pareto II to the birds data. The Pareto I fit with minxp=6679 is good, but the Pareto II fit with minxq=1 is not, so in the next graph we reset minxq=40 to reflect species clustering.

minx=1 sightings of bird species in the U.S. from Clauset, Shalizi, Newman, fitted by Pareto I minxp=6679 and Pareto II minxq=1 (Tsallis q-exponential)

birds =c(138705,98611,83906,77656,69928,69745,59698,57377,42162,40134,35594,32482,30790,29974,28293,27548,27344,26177,21714,21168,20120,19133,18944,17831,15604,15192,14029,13987,13965,13862,12988,12920,12253,11825,11358,11313,11163,10975,10968,10851,10127,10094,9946,9860,9826,9774,9746,9270,9236,9134,9093,8671,8623,8436,8205,8046,7914,7869,7791,7266,7202,7071,6876,6793,6733,6679,6213,6101,6043,5964,5963,5860,5734,5530,5485,5464,5341,5229,5093,5061,5041,4956,4896,4859,4814,4811,4792,4783,4665,4501,4501,4411,4384,4206,4137,3955,3911,3883,3875,3782,3744,3728,3721,3522,3520,3480,3474,3471,3462,3461,3435,3425,3398,3344,3341,3312,3287,3271,3243,3194,3139,3111,3059,3039,3037,2899,2893,2797,2727,2705,2600,2595,2558,2557,2514,2501,2467,2442,2386,2363,2356,2346,2278,2262,2229,2229,2183,2158,2109,2047,2037,2035,2000,1973,1952,1923,1908,1886,1873,1868,1853,1845,1845,1770,1760,1754,1737,1684,1670,1666,1655,1654,1619,1617,1614,1539,1530,1526,1479,1458,1438,1428,1408,1380,1372,1361,1346,1344,1336,

+1282,1220,1217,1198,1161,1133,1128,1116,1069,1068,1044,1028,1012,1007,1004,1000,971,945,938,937,925,924,915,898,882,879,879,877,870,869,863,857,855,854,850,849,814,813,812,782,760,752,750,750,746,728,722,715,710,696,690,685,684,677,677,656,651,632,632,628,627,
+622,608,603,596,587,573,561,557,552,549,548,542,539,538,538,535,535,524,519,516,507,506,500,500,476,464,455,454,442,436,433,431,429,428,414,404,402,400,396,388,376,376,372,366,365,364,358,357,351,350,350,345,341,323,320,319,312,311,309,304,303,301,301,291,291,288,285,281,281,271,269,265,265,260,253,252,247,241,236,234,232,229,228,227,225,224,222,222,218,217,211,211,210,206,204,196,195,193,191,191,190,188,185,184,176,175,174,173,173,170,169,169,165,164,164,162,161,160,160,155,152,144,142,140,139,138,136,136,135,133,132,132,130,128,127,125,123,121,121,119,119,116,114,110,110,109,108,108,105,105,105,103,102,102,101,99,96,95,94,94,89,87,87,85,84,84,84,81,79,79,78,78,75,74,72,72,72,72,70,70,69,65,65,64,63,63,63,61,61,61,59,58,57,56,55,51,50,47,46,46,43,43,41,40,40,40,39,39,39,39,39,38,38,37,35,35,34,33,33,33,33,32,31,31,31,30,30,30,30,28,28,28,27,27,26,26,25,24,24,23,23,23,22,21,19,19,18,18,17,17,
+17,16,16,16,14,14,14,13,13,13,13,12,12,12,11,11,11,11,10,10,9,9,9,9,9,8,8,8,8,8,7,6,6,6,6,
+5,5,5,5,5,5,5,5,4,4,4,4,4,4,4,4,4,4,4,3,3,3,3,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
plot.survival.loglog(birds,ylab="Cumulative probability",main="Survival function of birds data", sub="zagged=empirical, solid=Tsallis, dashed=Pareto, ")
birds.tsal <- tsal.fit(birds,xmin=1)
birds.min <- 6679
birds.tailprob <- sum(birds>=birds.min)/length(birds)
birds.pareto <- pareto.fit(birds,birds.min)
curve(ptsal(x,birds.tsal$shape,birds.tsal$scale,lower.tail=FALSE),add=TRUE,col="blue")
curve(birds.tailprob*ppareto(x,birds.min,birds.pareto$exponent,lower.tail=FALSE),from=birds.min,col="red",lty="dashed",add=TRUE)

Now we truncate the data distribution deleting sightings under 40 (to reflect e.g., species clustering), changing the lower threshold on which Pareto II is fitted, and see a better fit.

minx=40 sightings of bird species in the U.S. from Clauset, Shalizi, Newman, fitted by Pareto I minxp=6679 and Pareto II minxq=40 (Tsallis q-exponential fit improves)

birds =c(138705,98611,83906,77656,69928,69745,59698,57377,42162,40134,35594,32482,30790,29974,28293,27548,27344,26177,21714,21168,20120,19133,18944,17831,15604,15192,14029,13987,13965,13862,12988,12920,12253,11825,11358,11313,11163,10975,10968,10851,10127,10094,9946,9860,9826,9774,9746,9270,9236,9134,9093,8671,8623,8436,8205,8046,7914,7869,7791,7266,7202,7071,6876,6793,6733,6679,6213,6101,6043,5964,5963,5860,5734,5530,5485,5464,5341,5229,5093,5061,5041,4956,4896,4859,4814,4811,4792,4783,4665,4501,4501,4411,4384,4206,4137,3955,3911,3883,3875,3782,3744,3728,3721,3522,3520,3480,3474,3471,3462,3461,3435,3425,3398,3344,3341,3312,3287,3271,3243,3194,3139,3111,3059,3039,3037,2899,2893,2797,2727,2705,2600,2595,2558,2557,2514,2501,2467,2442,2386,2363,2356,2346,2278,2262,2229,2229,2183,2158,2109,2047,2037,2035,2000,1973,1952,1923,1908,1886,1873,1868,1853,1845,1845,1770,1760,1754,1737,1684,1670,1666,1655,1654,1619,1617,1614,1539,1530,1526,1479,1458,1438,1428,1408,1380,1372,1361,1346,1344,1336,

+1282,1220,1217,1198,1161,1133,1128,1116,1069,1068,1044,1028,1012,1007,1004,1000,971,945,938,937,925,924,915,898,882,879,879,877,870,869,863,857,855,854,850,849,814,813,812,782,760,752,750,750,746,728,722,715,710,696,690,685,684,677,677,656,651,632,632,628,627,
+622,608,603,596,587,573,561,557,552,549,548,542,539,538,538,535,535,524,519,516,507,506,500,500,476,464,455,454,442,436,433,431,429,428,414,404,402,400,396,388,376,376,372,366,365,364,358,357,351,350,350,345,341,323,320,319,312,311,309,304,303,301,301,291,291,288,285,281,281,271,269,265,265,260,253,252,247,241,236,234,232,229,228,227,225,224,222,222,218,217,211,211,210,206,204,196,195,193,191,191,190,188,185,184,176,175,174,173,173,170,169,169,165,164,164,162,161,160,160,155,152,144,142,140,139,138,136,136,135,133,132,132,130,128,127,125,123,121,121,119,119,116,114,110,110,109,108,108,105,105,105,103,102,102,101,99,96,95,94,94,89,87,87,85,84,84,84,81,79,79,78,78,75,74,72,72,72,72,70,70,69,65,65,64,63,63,63,61,61,61,59,58,57,56,55,51,50,47,46,46,43,43,41,40,40,40)
plot.survival.loglog(birds,ylab="Cumulative probability",main="Survival function of birds data", sub="black=empirical, blue=Tsallis, dashed red=Pareto")
birds.tsal <- tsal.fit(birds,xmin=40)
birds.min <- 6679
birds.tailprob <- sum(birds>=birds.min)/length(birds)
birds.pareto <- pareto.fit(birds,birds.min)
curve(ptsal(x,birds.tsal$shape,birds.tsal$scale,lower.tail=FALSE),add=TRUE,col="blue")
curve(birds.tailprob*ppareto(x,birds.min,birds.pareto$exponent,lower.tail=FALSE),from=birds.min,col="red",lty="dashed",add=TRUE)

It turns out, however, that a better cutoff for Pareto II is in the range 300-400 (see discussion of cutoffs above).

Blackouts

Blackouts.q This distribution is for the number of customers affected in electrical blackouts in the United States between 1984 and 2002 (Newman, 2005).

We tested the Pareto II cutoff idea with these data.

blackouts <- c(7500000,2100000,2085000,1660000,1600000,1500000,1500000,1500000,1300000,1140000,899000,877000,875000,870000,725000,660000,650000,600000,600000,598000,570000,557354,500000,500000,500000,490000,464000,460000,404000,400000,400000,385000,375000,363476,360000,350000,350000,320831,315000,312000,300000,300000,300000,300000,290000,284000,272000,272000,258000,257718,250000,246000,246000,242910,240000,238000,235000,234000,230000,219000,210882,207200,206000,203000,200000,191000,190000,173000,173000,166000,164500,163000,160000,160000,160000,158000,148000,147000,146000,145000,145000,142000,133000,130000,130000,128000,126000,124000,122000,120000,120000,115000,114500,114000,113200,112000,106850,100000,100000,100000,100000,95630,95000,94285,92000,92000,91000,90000,88000,82500,81000,80000,80000,75000,75000,74000,71000,71000,71000,70000,70000,70000,70000,66005,65000,65000,63500,62000,60000,60000,60000,60000,60000,59000,58000,56000,56000,55000,55000,53000,51000,51000,50462,50000,50000,50000,

+ 50000,50000,50000,50000,50000,50000,48000,46000,45000,43696,43000,40911,40000,40000,40000,39500,38500,37000,36073,35000,33000,32000,32000,30500,30001,30000,29900,29000,29000,26334,25000,25000,25000,25000,25000,25000,24506,24000,20000,20000,19000,18819,18351,18000,18000,17000,15000,15000,14273,12000,11529,11000,10300,10000,10000,9000,8000,7500,5300,4150,2900,2000,1800,1646,1000)
plot.survival.loglog(blackouts,ylab="Cumulative probability",main="Survival function of blackouts data",sub="black=empirical, blue=Tsallis, red=Pareto")
blackouts.tsal <- tsal.fit(blackouts,xmin=1000)
blackouts.min <- 230000
blackouts.tailprob <- sum(blackouts>=blackouts.min)/length(blackouts)
blackouts.pareto <- pareto.fit(blackouts,blackouts.min)
curve(ptsal(x,blackouts.tsal$shape,blackouts.tsal$scale,blackouts.tsal$q,blackouts.tsal$kappa,blackouts.tsal$xmin,lower.tail=FALSE),add=TRUE,col="blue")
curve(blackouts.tailprob*ppareto(x,blackouts.min,blackouts.pareto$exponent,lower.tail=FALSE),from=blackouts.min,col="red",lty="dashed",add=TRUE)

Cities

Cities.q New fitting to Tsallis-q, ntail circa 150-200,000. KS method xqmin=75,000 p=.78 (good fit to KS, ntail=100) but at xmin=1000 p=.28 by KS (not good) (bigger ntail? and less accurate)

This distribution is for the populations of US cities in the 2000 US Census.

The raw city size data are expressed from 1 to 8 million. Cities of size 1 are unknown: Archaeologically, the smallest cities with a complex division of labor and intercity trade based on those productive specializations have a minimum size of 5,000. We take this to be the minx=5000 for the size distribution.

cities <- c(8008654,3694742,2896047,1953633,1517550,1321190,1223429,1188589,1151305,951270,895193,781864,776733,735617,711265,656562,651154,650100,596974,589141,572059,563657,563376,557834,553693,545535,541099,529184,506129,487341,484674,479639,477472,461522,448948,441545,428873,425257,416441,407018,399484,397776,393120,391019,382747,371657,362437,360988,351150,348189,337977,334563,332969,331285,328071,313782,303463,292648,286840,282956,277496,275923,272537,260512,260283,256207,255175,248408,243771,243082,242790,240055,234403,228520,226419,225638,224047,222008,220483,219773,218831,217070,215794,208903,203413,202596,201607,200172,199572,199191,199184,198768,197800,197790,196019,195629,195182,194973,193556,191615,190914,189627,189453,188864,187316,186291,185975,185787,185237,184256,183135,181767,180900,180150,177322,176643,175406,174501,173627,173618,173553,172648,170359,166196,165196,161029,158625,158518,158011,155681,154198,152082,151722,151404,151131,150349,149473,149431,149222,147854,

+ 146866,146437,146435,144137,143593,143148,143072,142685,142379,141674,140031,139715,139530,139529,138319,138247,137946,137415,136991,133936,133747,131844,131784,128937,128821,128409,128283,127743,127427,126003,125251,125056,124943,124523,124506,124471,124121,123626,122882,121872,121582,121271,120568,119371,118718,118716,117549,117083,117005,116829,116760,116273,115965,115930,115488,114117,113880,113866,113288,113019,112580,112495,111532,111473,111373,111365,109949,109578,108902,108896,108756,108685,107919,107323,107271,107006,106656,106632,106431,105441,105167,105080,104323,104197,103717,103625,103478,102767,102746,102743,102673,102361,102286,102238,102129,101355,100998,100920,100565,100545,100316,100266,99216,98359,97775,97255,96650,96375,96178,95694,94996,94911,94892,94666,94624,94580,94492,94304,94301,93768,93493,93102,92998,92779,92472,92335,91957,91938,91776,91628,90943,90607,90532,90405,90205,90032,89730,89692,89621,89050,88967,88774,88642,88443,88207,88116,88025,87933,87097,86911,
+ 86605,86602,86319,85979,85932,85808,85794,85779,85655,85616,85403,85172,85071,85025,84662,84362,84329,84112,84110,84084,83829,83629,83259,83048,82951,82934,82549,82376,82111,82026,81991,81855,81207,81111,80959,80874,80734,80537,80431,80268,80142,80114,80056,79928,79904,79554,79452,79426,79345,79269,79062,78721,78306,78296,77962,77753,77737,77685,77478,77423,77248,77145,77021,76947,76760,76415,76042,76021,75837,75720,75702,75402,75402,74848,74764,74239,73990,73539,73402,73344,72958,72878,72739,72684,72670,72664,72259,72182,72069,72043,71538,71393,71329,71283,71127,71118,71021,70705,70702,70517,70296,70126,69845,69824,69567,69473,69368,68991,68904,68825,68747,68652,68556,68393,68381,68315,68263,68015,67861,67624,67561,67504,67388,67249,67173,67088,67083,66869,66787,66715,66455,65936,65920,65894,65868,65660,65273,65269,64903,64583,64296,64249,64196,64112,64029,63900,63870,63828,63677,63603,63557,63428,63325,63317,63296,63285,63261,63096,63011,62749,62698,62590,62406,62226,62150,61968,61842,
+ 61821,61792,61714,61607,61348,61254,61182,60603,60578,60552,60525,60523,60448,60432,60389,60308,60220,60213,60062,60062,60051,60032,59880,59740,59702,59686,59684,59577,59226,58974,58969,58912,58898,58812,58757,58596,58590,58271,58266,58264,58244,58062,57955,57923,57870,57755,57746,57697,57695,57585,57247,57221,57104,56933,56929,56769,56694,56646,56521,56456,56372,56352,56340,56287,56265,56259,56255,56063,56052,55981,55976,55919,55901,55825,55765,55635,55593,55588,55566,55513,55502,55419,55266,55256,55245,55147,54978,54977,54901,54749,54653,54593,54552,54514,54370,54260,54239,54088,54059,54016,53948,53909,53505,53421,53364,53296,53205,53127,53109,53077,53071,53054,52975,52946,52913,52793,52717,52715,52648,52613,52524,52466,52457,52360,52029,52018,51988,51884,51765,51659,51599,51548,51508,51488,51460,51443,51104,51102,50868,50866,50790,50769,50767,50719,50673,50644,50624,50594,50365,50284,50211,49936,49816,49739,49693,49672,49553,49523,49435,49415,49399,49374,49366,49325,49286,49286,49170,
+ 49155,49121,48971,48950,48806,48702,48688,48501,48472,48411,48187,48151,48129,48129,48089,48048,48028,47996,47829,47821,47782,47662,47656,47637,47425,47388,47386,47380,47303,47283,47271,47214,47152,47045,46837,46822,46783,46707,46699,46614,46606,46549,46509,46506,46488,46463,46449,46332,46277,46258,46005,45946,45866,45793,45768,45697,45563,45527,45504,45444,45430,45409,45293,45287,45255,45175,45054,45027,45014,44925,44897,44870,44795,44752,44712,44681,44536,44503,44382,44292,44193,44102,44054,44003,43957,43858,43789,43783,43768,43638,43576,43567,43407,43370,43364,43250,43224,43128,43123,43081,43062,42928,42805,42776,42687,42677,42647,42584,42514,42471,42412,42358,42249,42236,42223,42097,42068,42059,41938,41872,41862,41773,41721,41715,41633,41633,41581,41550,41531,41464,41375,41303,41303,41207,41155,41145,41138,41103,41063,40866,40862,40853,40786,40687,40687,40580,40517,40513,40507,40453,40424,40407,40377,40340,40238,40236,40165,40105,40099,40073,40072,40018,40008,39968,39862,39838,39824,
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+ 27819,27819,27812,27802,27772,27765,27735,27718,27696,27674,27656,27651,27619,27599,27588,27508,27449,27433,27433,27407,27400,27387,27368,27362,27345,27324,27217,27215,27195,27152,27144,27136,27134,27109,27069,27044,26997,26992,26987,26984,26955,26940,26884,26865,26847,26839,26809,26782,26705,26653,26622,26614,26588,26588,26549,26530,26518,26502,26500,26475,26463,26451,26411,26409,26408,26407,26386,26365,26359,26325,26320,26309,26309,26263,26247,26237,26232,26225,26206,26203,26200,26186,26185,26156,26128,26118,26107,26052,26038,26009,26001,25974,25946,25945,25927,25924,25919,25912,25897,25894,25869,25866,25849,25756,25754,25737,25720,25709,25645,25642,25639,25619,25614,25605,25600,25573,25496,25478,25470,25464,25421,25410,25405,25363,25325,25267,25233,25216,25205,25178,25171,25144,25104,25099,25018,25011,24998,24948,24936,24927,24918,24881,24863,24848,24747,24731,24668,24662,24604,24598,24568,24555,24516,24508,24498,24477,24426,24331,24325,24312,24303,24302,24244,24243,24208,24183,24157,
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+ 10387,10380,10377,10375,10375,10374,10366,10360,10355,10350,10344,10343,10340,10331,10311,10307,10302,10300,10299,10297,10296,10295,10290,10278,10266,10260,10259,10255,10255,10255,10254,10242,10238,10232,10228,10207,10192,10182,10175,10174,10167,10163,10157,10155,10146,10122,10117,10116,10105,10103,10098,10087,10085,10082,10076,10074,10070,10063,10056,10050,10049,10042,10036,10034,10029,10011,10010,10001,9995,9992,9991,9991,9989,9986,9983,9975,9974,9952,9952,9929,9925,9923,9920,9917,9916,9896,9892,9888,9887,9885,9880,9873,9862,9861,9860,9860,9859,9850,9850,9848,9847,9844,9843,9839,9835,9826,9821,9813,9812,9810,9808,9800,9798,9798,9796,9792,9788,9774,9771,9764,9763,9761,9745,9743,9737,9736,9735,9733,9730,9728,9727,9722,9710,9708,9705,9703,9686,9684,9682,9679,9676,9670,9667,9664,9657,9653,9652,9652,9652,9640,9637,9628,9628,9622,9621,9621,9618,9614,9609,9602,9602,9600,9596,9592,9591,9577,9558,9557,9552,9548,9544,9541,9538,9532,9523,9523,9520,9519,9513,9511,9501,9501,9488,9488,9481,9476,9471,
+ 9470,9460,9451,9449,9445,9443,9441,9435,9426,9424,9424,9420,9417,9415,9413,9409,9405,9400,9400,9389,9388,9377,9371,9365,9363,9352,9338,9333,9331,9323,9321,9320,9311,9311,9305,9293,9291,9282,9280,9279,9279,9279,9278,9275,9275,9268,9266,9264,9257,9254,9247,9245,9242,9239,9239,9237,9237,9231,9228,9224,9222,9215,9214,9214,9212,9207,9206,9198,9189,9189,9188,9182,9175,9168,9157,9156,9149,9149,9146,9146,9144,9139,9131,9105,9100,9099,9099,9098,9097,9094,9090,9089,9086,9083,9081,9081,9056,9052,9042,9038,9029,9027,9022,9019,9017,9015,9012,9008,9008,9006,9006,9000,8998,8994,8993,8989,8987,8984,8979,8973,8969,8968,8960,8959,8958,8934,8931,8923,8914,8912,8910,8901,8895,8883,8881,8877,8876,8870,8870,8869,8860,8842,8838,8836,8835,8833,8829,8828,8811,8811,8806,8803,8803,8798,8795,8793,8789,8771,8770,8762,8758,8757,8751,8749,8748,8747,8737,8737,8733,8733,8731,8721,8716,8716,8714,8711,8708,8705,8704,8702,8695,8694,8693,8683,8674,8673,8670,8669,8667,8664,8662,8662,8658,8656,8654,8647,8647,8629,8626,8625,
+ 8620,8618,8610,8610,8609,8604,8600,8596,8595,8592,8591,8586,8586,8583,8578,8576,8571,8570,8562,8558,8556,8551,8551,8542,8540,8533,8531,8524,8516,8506,8498,8496,8495,8491,8489,8489,8488,8488,8487,8485,8463,8460,8442,8442,8438,8437,8432,8428,8426,8419,8413,8410,8409,8408,8405,8402,8399,8397,8389,8385,8385,8383,8382,8382,8378,8377,8376,8368,8364,8363,8357,8356,8354,8348,8346,8342,8342,8339,8334,8329,8314,8313,8312,8312,8303,8297,8297,8288,8286,8276,8276,8267,8264,8261,8258,8252,8250,8249,8244,8243,8242,8237,8227,8209,8206,8195,8193,8193,8192,8192,8188,8186,8183,8180,8178,8177,8175,8174,8170,8166,8165,8155,8154,8154,8153,8152,8144,8139,8137,8133,8132,8130,8129,8127,8123,8123,8104,8103,8098,8098,8089,8082,8080,8053,8052,8047,8047,8044,8042,8040,8038,8037,8035,8035,8029,8024,8023,8016,8014,8012,8009,8006,8000,7996,7996,7991,7989,7988,7985,7978,7975,7967,7962,7957,7957,7951,7948,7943,7943,7939,7936,7923,7920,7916,7916,7915,7913,7912,7901,7897,7896,7894,7888,7880,7870,7862,7861,7859,7857,7856,
+ 7853,7846,7845,7845,7844,7843,7837,7835,7829,7829,7825,7824,7823,7820,7817,7816,7812,7807,7804,7804,7799,7796,7793,7783,7782,7780,7780,7776,7774,7774,7772,7769,7765,7765,7761,7761,7759,7746,7744,7741,7741,7734,7734,7731,7728,7725,7723,7723,7723,7722,7715,7714,7712,7709,7698,7696,7696,7695,7694,7693,7678,7676,7676,7676,7675,7668,7666,7661,7661,7657,7655,7644,7643,7640,7638,7636,7633,7618,7614,7610,7610,7609,7609,7608,7606,7606,7605,7605,7603,7601,7597,7596,7595,7591,7589,7589,7587,7584,7579,7578,7576,7572,7570,7568,7563,7561,7554,7553,7547,7544,7538,7527,7524,7523,7516,7513,7511,7508,7506,7505,7502,7498,7495,7494,7490,7487,7476,7467,7465,7459,7459,7456,7452,7450,7450,7442,7438,7437,7436,7435,7435,7433,7430,7424,7422,7420,7407,7405,7403,7401,7400,7396,7393,7389,7388,7381,7375,7372,7371,7370,7356,7349,7347,7345,7341,7340,7338,7337,7336,7334,7332,7330,7325,7320,7319,7319,7315,7307,7305,7290,7289,7289,7288,7286,7281,7280,7277,7274,7271,7271,7270,7270,7266,7263,7261,7259,7257,7247,7246,7245,
+ 7244,7239,7231,7228,7228,7226,7224,7212,7210,7204,7197,7196,7194,7190,7174,7174,7174,7173,7172,7172,7170,7160,7159,7149,7149,7148,7145,7145,7141,7139,7139,7137,7135,7135,7132,7129,7127,7126,7125,7115,7111,7109,7106,7096,7095,7093,7093,7091,7091,7090,7090,7084,7080,7068,7064,7063,7057,7047,7042,7039,7036,7034,7034,7032,7027,7025,7022,7021,7015,7012,7009,7006,7000,6997,6994,6991,6989,6985,6982,6980,6978,6975,6973,6970,6964,6959,6956,6951,6950,6947,6944,6942,6928,6927,6924,6920,6918,6913,6911,6905,6903,6901,6895,6894,6890,6887,6884,6882,6880,6880,6871,6868,6867,6866,6863,6862,6862,6862,6861,6849,6847,6847,6846,6845,6842,6837,6835,6834,6834,6834,6831,6831,6830,6830,6828,6826,6825,6823,6822,6821,6820,6819,6818,6818,6815,6814,6812,6809,6808,6807,6806,6802,6802,6796,6796,6792,6789,6789,6789,6781,6781,6779,6773,6772,6765,6764,6762,6760,6757,6753,6749,6744,6744,6741,6737,6733,6733,6730,6729,6720,6716,6713,6712,6711,6710,6710,6709,6706,6704,6702,6699,6698,6696,6695,6692,6688,6686,6682,6681,6681,
+ 6676,6674,6673,6670,6668,6664,6662,6660,6658,6656,6653,6650,6650,6649,6645,6644,6644,6641,6635,6635,6632,6631,6631,6628,6626,6626,6622,6619,6619,6607,6602,6595,6595,6594,6592,6589,6588,6586,6585,6579,6576,6571,6571,6571,6569,6568,6563,6562,6561,6558,6554,6551,6543,6540,6537,6537,6537,6536,6535,6534,6530,6522,6518,6518,6508,6507,6502,6492,6486,6485,6485,6482,6480,6480,6480,6478,6476,6475,6473,6472,6471,6470,6465,6463,6460,6459,6457,6456,6452,6451,6448,6446,6440,6438,6437,6433,6433,6433,6433,6432,6426,6426,6419,6415,6414,6412,6410,6408,6407,6400,6396,6396,6395,6395,6392,6391,6390,6388,6386,6385,6383,6381,6380,6380,6372,6369,6364,6363,6361,6355,6355,6350,6349,6348,6347,6345,6338,6334,6331,6324,6323,6319,6318,6314,6314,6313,6311,6310,6310,6310,6310,6306,6303,6302,6299,6299,6298,6293,6288,6286,6284,6281,6272,6270,6266,6260,6260,6256,6256,6252,6249,6249,6239,6235,6231,6229,6224,6223,6220,6220,6217,6216,6216,6216,6212,6211,6208,6208,6205,6204,6198,6198,6189,6185,6183,6180,6179,6179,6170,6170,
+ 6168,6164,6164,6163,6160,6151,6149,6149,6148,6148,6145,6143,6138,6137,6131,6125,6120,6116,6116,6115,6115,6110,6106,6099,6098,6098,6098,6097,6097,6093,6080,6078,6078,6077,6076,6076,6075,6070,6065,6064,6062,6056,6049,6048,6045,6040,6038,6035,6034,6033,6033,6025,6023,6022,6020,6018,6017,6016,6015,6014,6013,6005,6004,6002,6002,5997,5996,5995,5993,5992,5991,5990,5989,5989,5987,5986,5985,5984,5982,5981,5980,5969,5966,5965,5963,5958,5958,5958,5955,5952,5952,5951,5950,5947,5942,5939,5937,5936,5929,5925,5924,5917,5917,5916,5915,5910,5909,5907,5906,5905,5904,5900,5900,5899,5897,5887,5877,5873,5869,5868,5866,5864,5863,5863,5860,5858,5858,5857,5856,5853,5852,5852,5851,5850,5847,5844,5843,5836,5836,5822,5817,5813,5813,5812,5809,5807,5806,5805,5802,5800,5795,5788,5786,5784,5780,5779,5778,5776,5774,5767,5766,5765,5764,5759,5751,5745,5741,5740,5740,5739,5735,5730,5728,5727,5723,5720,5717,5717,5716,5714,5714,5712,5711,5700,5699,5696,5693,5691,5690,5688,5687,5686,5682,5681,5676,5676,5676,5676,5675,5674,
+ 5670,5669,5667,5666,5665,5665,5659,5659,5655,5654,5653,5649,5645,5641,5638,5636,5634,5634,5634,5633,5630,5628,5626,5626,5624,5624,5623,5621,5620,5614,5613,5612,5607,5606,5605,5603,5600,5599,5596,5590,5589,5589,5588,5586,5586,5585,5584,5583,5582,5580,5576,5576,5575,5575,5574,5573,5572,5567,5562,5561,5558,5556,5549,5548,5546,5539,5538,5537,5537,5537,5536,5533,5533,5528,5527,5523,5523,5514,5502,5502,5500,5500,5498,5498,5496,5490,5489,5488,5488,5486,5485,5483,5482,5482,5478,5471,5470,5470,5470,5469,5468,5468,5466,5464,5463,5462,5460,5459,5458,5458,5456,5456,5455,5450,5448,5445,5441,5439,5438,5436,5436,5436,5434,5433,5430,5429,5424,5422,5419,5414,5413,5409,5409,5408,5408,5406,5395,5391,5389,5384,5383,5379,5376,5376,5376,5376,5373,5371,5369,5368,5359,5356,5352,5344,5344,5341,5336,5333,5332,5332,5332,5331,5330,5322,5319,5318,5314,5314,5312,5308,5307,5307,5305,5303,5302,5301,5298,5297,5295,5295,5294,5294,5292,5291,5289,5288,5287,5286,5285,5282,5280,5278,5276,5274,5273,5264,5262,5257,5255,5250,
+ 5248,5247,5245,5236,5236,5234,5233,5231,5228,5227,5221,5219,5218,5218,5216,5216,5213,5213,5204,5203,5203,5203,5197,5193,5192,5192,5190,5189,5189,5188,5187,5184,5182,5181,5180,5178,5177,5174,5173,5166,5160,5157,5156,5156,5156,5152,5151,5148,5146,5144,5143,5138,5137,5136,5136,5135,5135,5133,5132,5132,5131,5130,5128,5127,5127,5126,5126,5124,5121,5119,5119,5117,5117,5114,5111,5109,5107,5107,5105,5105,5105,5104,5102,5102,5099,5099,5098,5097,5092,5092,5088,5088,5088,5087,5083,5083,5080,5080,5078,5076,5073,5072,5070,5067,5066,5065,5062,5061,5059,5058,5057,5057,5056,5056,5054,5054,5049,5043,5039,5035,5034,5032,5032,5032,5029,5028,5021,5019,5019,5017,5005,5004)
plot.survival.loglog(cities,ylab="Cumulative probability",main="Survival function of cities data",sub="zagged=empirical, solid=Tsallis, dotted=Pareto, xq-min=5000, xp-min=52640")
cities.tsal <- tsal.fit(cities,xmin=5000)
cities.min <- 52640
cities.tailprob <- sum(cities>=cities.min)/length(cities)
cities.pareto <- pareto.fit(cities,cities.min)
curve(ptsal(x,cities.tsal$shape,cities.tsal$scale,cities.tsal$q,cities.tsal$kappa,cities.tsal$xmin,lower.tail=FALSE),add=TRUE,col="blue")
curve(cities.tailprob*ppareto(x,cities.min,cities.pareto$exponent,lower.tail=FALSE),from=cities.min,col="red",lty="dotted",add=TRUE)

Books

Books.q And with the books data

This distribution is for the numbers of copies sold in the United States of bestselling books for the period 1895 to 1965 (Hackett, 1967).

books <-c(19076822,11325299,11000000,9919785,8065398,8061812,7000000,6978211,6578314,6326470,6000000,5899000,5563841,5490651,5473710,5416857,5390105,5363909,5089472,5000000,5000000,4988225,4916074,4873563,4835966,4828044,4810418,4712588,4637734,4466200,4400000,4375620,4186935,4171838,4007158,4000000,3913341,3910155,3858948,3821608,3800000,3768144,3662089,3646004,3642411,3633467,3593268,3581210,3577729,3549276,3501866,3501281,3500000,3499948,3475947,3445000,3430505,3426000,3389940,3316791,3283000,3262193,3220246,3216600,3208361,3190334,3171512,3170056,3157592,3156000,3142184,3130306,3100000,3087532,3079790,3075123,3007000,3000000,3000000,3000000,2974030,2950807,2940211,2925268,2911111,2833993,2830061,2825368,2820313,2816772,2816662,2816028,2814394,2786223,2778577,2773841,2763486,2760000,2750000,2739427,2719500,2716816,2702597,2700261,2680597,2672065,2667977,2655208,2635250,2634472,2632358,2625000,2621842,2611000,2609236,2600000,2595125,2585397,2560806,2530964,2527756,2527459,2524649, + 2521250,2505000,2499660,2472710,2471140,2469306,2461874,2437336,2435928,2433154,2414460,2409163,2403403,2403269,2400000,2400000,2370720,2358945,2341531,2340000,2336004,2330000,2318230,2316989,2312378,2311904,2310000,2299778,2286000,2282322,2266268,2262024,2253982,2253453,2251960,2246396,2244552,2237449,2230000,2229000,2219645,2214669,2213167,2192133,2184145,2183109,2160481,2153000,2152210,2149000,2147000,2146812,2136177,2134289,2133810,2121708,2111873,2106804,2105951,2102522,2100908,2089523,2087837,2087173,2080985,2075000,2073434,2068000,2063775,2054928,2053892,2051488,2050000,2033679,2029248,2024107,2011251,2007000,2007000,2006808,2005000,2000000,2000000,2000000,2000000,1983000,1968799,1968297,1956000,1941769,1933949,1932000,1930000,1929391,1928513,1926110,1920000,1919214,1918865,1900000,1894000,1888785,1886465,1874961,1863625,1863260,1862387,1853917,1843756,1842000,1836815,1834704,1830060,1829364,1827493,1826348,1820308,1819000,1818649,1816000,1805277,1805000,1801097,1794020,1793366, + 1783923,1778831,1777027,1758760,1755011,1755000,1754975,1751498,1750000,1748000,1747908,1740930,1738597,1736299,1730000,1728151,1725939,1724658,1720328,1715993,1710000,1706019,1703816,1702500,1700000,1696589,1692921,1691878,1690126,1686367,1682891,1676588,1671968,1661578,1660000,1653742,1652235,1651713,1642905,1640314,1640000,1631630,1626135,1625000,1620464,1617462,1616783,1612339,1611700,1611007,1609011,1601079,1600000,1594000,1591489,1589451,1588972,1586529,1586260,1584793,1584135,1583612,1583561,1582738,1581685,1578234,1578169,1575396,1575304,1573479,1571570,1568601,1563117,1560980,1558878,1558175,1550223,1550000,1550000,1548471,1547000,1546000,1544166,1543000,1540572,1539524,1536793,1536764,1534371,1531786,1529095,1525000,1525000,1520000,1516156,1515880,1515825,1507502,1507000,1500000,1500000,1500000,1500000,1496118,1493167,1488097,1480130,1480000,1479047,1473496,1469432,1469428,1468250,1467635,1464357,1462045,1457000,1455452,1455110,1450000,1436000,1436000,1434203,1433477,1432000, + 1430000,1425000,1422000,1421157,1421000,1415572,1415551,1412000,1409790,1406700,1405936,1400000,1398445,1396015,1396000,1390895,1388320,1388220,1383440,1381956,1378500,1377373,1375000,1373288,1369660,1366600,1363909,1362139,1357000,1356863,1354347,1354000,1351993,1351254,1350620,1349000,1338950,1336425,1333417,1327895,1325000,1322402,1320000,1319610,1318000,1317137,1316228,1314077,1312700,1311484,1310000,1308883,1308014,1303028,1300201,1300000,1300000,1296140,1291961,1290453,1290291,1286463,1285000,1282414,1280000,1279518,1279000,1279000,1278951,1275953,1272563,1269007,1266957,1266206,1265336,1265000,1262676,1261000,1260725,1259390,1253793,1250000,1250000,1250000,1247000,1246693,1244400,1243307,1241743,1241550,1239916,1237767,1237060,1235000,1235000,1229000,1226720,1225498,1221986,1220874,1217322,1216938,1212426,1211394,1210994,1210900,1209029,1203889,1200190,1200000,1200000,1200000,1200000,1200000,1200000,1200000,1200000,1194440,1194000,1193650,1193492,1193330,1193230,1191374,1190086, + 1188395,1183560,1183063,1177618,1177060,1175323,1173500,1170042,1168181,1165499,1163900,1153749,1152000,1151029,1150000,1148669,1148260,1146449,1143675,1142000,1139572,1138891,1137563,1136917,1132856,1131270,1131018,1128840,1126365,1123136,1121000,1120000,1119134,1118800,1117418,1115657,1111520,1110000,1108000,1107754,1107064,1104600,1100000,1100000,1100000,1100000,1100000,1100000,1100000,1099270,1099000,1098001,1097716,1095155,1095000,1094941,1090383,1090000,1089767,1088744,1087000,1083623,1081906,1081810,1079655,1079000,1077453,1076380,1070392,1068363,1068139,1068004,1068000,1063000,1059000,1058890,1057837,1056992,1056196,1055923,1055000,1054910,1054500,1053700,1050000,1049006,1048247,1047991,1045000,1040000,1039000,1034716,1033525,1032681,1031180,1028109,1028000,1027505,1026322,1021233,1018942,1018000,1017351,1016784,1011000,1010437,1010000,1010000,1010000,1009606,1008937,1007997,1007136,1005381,1005374,1005000,1005000,1002754,1001000,1000054,1000000,1000000,1000000,1000000,1000000, + 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000)

plot.survival.loglog(books,ylab="Cumulative probability",main="Survival function of books data",sub="black=empirical, blue=Tsallis, red=Pareto")
books.tsal <- tsal.fit(books,xmin=1000000)
books.min <- 2400000
books.tailprob <- sum(books>=books.min)/length(books)
books.pareto <- pareto.fit(books,books.min)
curve(ptsal(x,books.tsal$shape,books.tsal$scale,books.tsal$q,books.tsal$kappa,books.tsal$xmin,lower.tail=FALSE),add=TRUE,col="blue")
curve(books.tailprob*ppareto(x,books.min,books.pareto$exponent,lower.tail=FALSE),from=books.min,col="red",add=TRUE)

Blackouts -- STOPPED EDITING here

blackouts <- c(7500000,2100000,2085000,1660000,1600000,1500000,1500000,1500000,1300000,1140000,899000,877000,875000,870000,725000,660000,650000,600000,600000,598000,570000,557354,500000,500000,500000,490000,464000,460000,404000,400000,400000,385000,375000,363476,360000,350000,350000,320831,315000,312000,300000,300000,300000,300000,290000,284000,272000,272000,258000,257718,250000,246000,246000,242910,240000,238000,235000,234000,230000,219000,210882,207200,206000,203000,200000,191000,190000,173000,173000,166000,164500,163000,160000,160000,160000,158000,148000,147000,146000,145000,145000,142000,133000,130000,130000,128000,126000,124000,122000,120000,120000,115000,114500,114000,113200,112000,106850,100000,100000,100000,100000,95630,95000,94285,92000,92000,91000,90000,88000,82500,81000,80000,80000,75000,75000,74000,71000,71000,71000,70000,70000,70000,70000,66005,65000,65000,63500,62000,60000,60000,60000,60000,60000,59000,58000,56000,56000,55000,55000,53000,51000,51000,50462,50000,50000,50000, + 50000,50000,50000,50000,50000,50000,48000,46000,45000,43696,43000,40911,40000,40000,40000,39500,38500,37000,36073,35000,33000,32000,32000,30500,30001,30000,29900,29000,29000,26334,25000,25000,25000,25000,25000,25000,24506,24000,20000,20000,19000,18819,18351,18000,18000,17000,15000,15000,14273,12000,11529,11000,10300,10000,10000,9000,8000,7500,5300,4150,2900,2000,1800,1646,1000) plot.survival.loglog(blackouts,ylab="Cumulative probability",main="Survival function of blackouts data",sub="black=empirical, blue=Tsallis, red=Pareto") blackouts.tsal <- tsal.fit(blackouts,xmin=1000) blackouts.min <- 230000 blackouts.tailprob <- sum(blackouts>=blackouts.min)/length(blackouts) blackouts.pareto <- pareto.fit(blackouts,blackouts.min) curve(ptsal(x,blackouts.tsal$shape,blackouts.tsal$scale,blackouts.tsal$q,blackouts.tsal$kappa,blackouts.tsal$xmin,lower.tail=FALSE),add=TRUE,col="blue") curve(blackouts.tailprob*ppareto(x,blackouts.min,blackouts.pareto$exponent,lower.tail=FALSE),from=blackouts.min,col="red",add=TRUE)



Emails

This distribution is for the sizes of email address books of computer users at a large university (Newman et al., 2002). The way Aaron et.al. fitted the emails data might be incorrect. They use xmin=57, but the fitting generated is way off. A more proper xmin for power law fitting is xmin=120. This also works better in comparing this fit to the q-exp fitting.

emails <- c(333,303,264,231,221,189,186,184,180,177,177,175,168,168,148,147,146,146,146,145, +143,142,141,139,135,132,129,127,126,125,122,121,120,116,115,114,111,111,110,110, +109,109,108,108,107,106,105,105,104,103,103,101,101,101,100,99,99,98,97,97, +96,96,94,93,92,92,92,92,91,90,90,90,88,88,87,87,86,85,85,83, +83,82,82,82,81,81,80,80,80,79,78,78,77,77,76,76,76,76,76,76, +76,76,75,74,73,73,73,72,72,72,72,72,71,71,71,71,71,70,70,70, +70,70,70,69,69,69,69,69,68,68,68,67,67,67,67,66,66,66,66,66, +66,65,65,65,65,65,65,65,64,64,64,64,64,63,63,63,63,63,63,62, +62,62,61,60,60,60,60,60,60,59,59,59,59,59,59,59,58,58,58,58, +58,58,58,58,58,57,57,57,57,57,57,57,57,57,57,57,56,55,55,55, +55,55,55,55,55,55,54,54,53,53,52,52,52,52,52,52,52,52,51,51, +51,51,51,51,51,51,51,50,50,50,50,50,50,50,49,49,49,49,49,49, +49,49,49,49,49,49,48,48,48,48,48,48,48,48,48,47,47,47,47,47, +47,47,47,47,46,46,46,46,46,45,45,45,45,45,45,45,44,44,44,44, +44,44,44,44,44,43,43,43,43,43,43,43,43,43,43,43,43,42,42,42, +42,42,42,42,42,41,41,41,41,41,41,40,40,40,40,40,40,40,40,40, +40,40,40,39,39,39,39,39,39,39,39,39,39,39,38,38,38,38,37,37, +37,37,37,37,37,37,37,36,36,36,36,36,36,36,36,36,36,36,36,35, +35,35,35,35,35,35,35,35,35,35,35,35,34,34,34,34,34,34,34,34, +34,34,34,34,34,34,34,34,33,33,33,33,33,33,33,33,33,33,33,32, +32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32, +31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31,31, +31,31,31,31,30,30,30,30,30,30,30,30,30,30,30,30,29,29,29,29, +29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,28,28,28, +28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,27,27, +27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,26,26,26,26, +26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26, +26,26,26,26,26,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25, +25,25,25,25,25,25,25,25,25,25,25,25,24,24,24,24,24,24,24,24, +24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24, +24,24,24,24,24,24,24,24,23,23,23,23,23,23,23,23,23,23,23,23, +23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23, +23,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22, +22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22, +22,22,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21, +21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,20, +20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20, +20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20, +20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,19,19,19,19, +19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19, +19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19, +19,19,19,19,19,19,19,18,18,18,18,18,18,18,18,18,18,18,18,18, +18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18, +18,18,18,18,18,18,17,17,17,17,17,17,17,17,17,17,17,17,17,17, +17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17, +17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17, +17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,16,16,16,16,16, +16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16, +16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16, +16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16, +16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,15,15, +15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15, +15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15, +15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,14,14,14, +14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14, +14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14, +14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14, +14,14,14,14,14,14,14,14,14,14,14,14,14,13,13,13,13,13,13,13, +13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13, +13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13, +13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13, +13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13, +13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13, +13,13,13,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12, +12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12, +12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12, +12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12, +12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12, +12,12,12,12,12,12,12,12,12,12,12,12,11,11,11,11,11,11,11,11, +11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11, +11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11, +11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11, +11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11, +11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11, +11,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10, +10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10, +10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10, +10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10, +10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10, +10,10,10,10,10,10,10,10,10,10,10,10,10,10,9,9,9,9,9,9, +9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9, +9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9, +9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9, +9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9, +9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9, +9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9, +9,9,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8, +8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8, +8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8, +8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8, +8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8, +8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8, +8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,7,7,7, +7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7, +7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7, +7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7, +7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7, +7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7, +7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7, +7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7, +7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,6,6, +6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6, +6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6, +6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6, +6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6, +6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6, +6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6, +6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6, +6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6, +6,6,6,6,6,6,6,6,6,6,6,6,6,6,5,5,5,5,5,5, +5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, +5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, +5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, +5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, +5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, +5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, +5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, +5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, +5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, +5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, +5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, +5,5,5,5,5,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, +4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, +4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, +4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 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plot.survival.loglog(emails,ylab="Cumulative probability",main="Survival function of emails data",sub="black=empirical, blue=Tsallis, red=Pareto")
emails.tsal <- tsal.fit(emails,xmin=1)
emails.min <- 57
emails.tailprob <- sum(emails>=emails.min)/length(emails)
emails.pareto <- pareto.fit(emails,emails.min)
curve(ptsal(x,emails.tsal$shape,emails.tsal$scale,emails.tsal$q,emails.tsal$kappa,emails.tsal$xmin,lower.tail=FALSE),add=TRUE,col="blue")
curve(emails.tailprob*ppareto(x,emails.min,emails.pareto$exponent,lower.tail=FALSE),from=emails.min,col="red",add=TRUE)

comparison of pareto to fitted R only (not actual data)

birds =c(138705,98611,83906,77656,69928,69745,59698,57377,42162,40134,35594,32482,30790,29974,28293,27548,27344,26177,21714,21168,20120,19133,18944,17831,15604,15192,14029,13987,13965,13862,12988,12920,12253,11825,11358,11313,11163,10975,10968,10851,10127,10094,9946,9860,9826,9774,9746,9270,9236,9134,9093,8671,8623,8436,8205,8046,7914,7869,7791,7266,7202,7071,6876,6793,6733,6679,6213,6101,6043,5964,5963,5860,5734,5530,5485,5464,5341,5229,5093,5061,5041,4956,4896,4859,4814,4811,4792,4783,4665,4501,4501,4411,4384,4206,4137,3955,3911,3883,3875,3782,3744,3728,3721,3522,3520,3480,3474,3471,3462,3461,3435,3425,3398,3344,3341,3312,3287,3271,3243,3194,3139,3111,3059,3039,3037,2899,2893,2797,2727,2705,2600,2595,2558,2557,2514,2501,2467,2442,2386,2363,2356,2346,2278,2262,2229,2229,2183,2158,2109,2047,2037,2035,2000,1973,1952,1923,1908,1886,1873,1868,1853,1845,1845,1770,1760,1754,1737,1684,1670,1666,1655,1654,1619,1617,1614,1539,1530,1526,1479,1458,1438,1428,1408,1380,1372,1361,1346,1344,1336,

+1282,1220,1217,1198,1161,1133,1128,1116,1069,1068,1044,1028,1012,1007,1004,1000,971,945,938,937,925,924,915,898,882,879,879,877,870,869,863,857,855,854,850,849,814,813,812,782,760,752,750,750,746,728,722,715,710,696,690,685,684,677,677,656,651,632,632,628,627,
+622,608,603,596,587,573,561,557,552,549,548,542,539,538,538,535,535,524,519,516,507,506,500,500,476,464,455,454,442,436,433,431,429,428,414,404,402,400,396,388,376,376,372,366,365,364,358,357,351,350,350,345,341,323,320,319,312,311,309,304,303,301,301,291,291,288,285,281,281,271,269,265,265,260,253,252,247,241,236,234,232,229,228,227,225,224,222,222,218,217,211,211,210,206,204,196,195,193,191,191,190,188,185,184,176,175,174,173,173,170,169,169,165,164,164,162,161,160,160,155,152,144,142,140,139,138,136,136,135,133,132,132,130,128,127,125,123,121,121,119,119,116,114,110,110,109,108,108,105,105,105,103,102,102,101,99,96,95,94,94,89,87,87,85,84,84,84,81,79,79,78,78,75,74,72,72,72,72,70,70,69,65,65,64,63,63,63,61,61,61,59,58,57,56,55,51,50,47,46,46,43,43,41,40,40,40,39,39,39,39,39,38,38,37,35,35,34,33,33,33,33,32,31,31,31,30,30,30,30,28,28,28,27,27,26,26,25,24,24,23,23,23,22,21,19,19,18,18,17,17,
+17,16,16,16,14,14,14,13,13,13,13,12,12,12,11,11,11,11,10,10,9,9,9,9,9,8,8,8,8,8,7,6,6,6,6,
+5,5,5,5,5,5,5,5,4,4,4,4,4,4,4,4,4,4,4,3,3,3,3,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
birds.tsal <- tsal.fit(birds,xmin=1)
tsal.sample <- rtsal(length(birds),birds.tsal$shape,birds.tsal$scale)
plot.survival.loglog(tsal.sample,ylab="Cumulative probability",main="Survival function of birds data",sub="black=empirical, blue=Tsallis, red=Pareto")
curve(ptsal(x,birds.tsal$shape,birds.tsal$scale,lower.tail=FALSE),add=TRUE,col="blue")
tsal.sample.min <- 6679
birds.tailprob <- sum(tsal.sample>=tsal.sample.min)/length(tsal.sample)
birds.pareto <- pareto.fit(tsal.sample,tsal.sample.min)
curve(birds.tailprob*ppareto(x, tsal.sample.min,birds.pareto$exponent,lower.tail=FALSE),from= tsal.sample.min,col="red",lty="dashed",add=TRUE)

A related project: Tsallis q historical cities and city-sizes

A Tsallis q historical cities and city-sizes is being undertaken by Doug Whte and Laurent Tambayong.