Kernel Smoothing for Bivariate Copula Densities

This package provides fast implementations of kernel estimators for the copula density. Due to its several plotting options it is particularly useful for the exploratory analysis of dependence structures. It can be further used for flexible nonparametric estimation of copula densities and resampling.

Details

The function kdecop can be used to estimate a copula density with a number of popular kernel estimators. The density estimate can be evaluated on arbitrary points with dkdecop; the cdf with pkdecop. Furthermore, synthetic data can be simulated with rkdecop, and several plot options are provided by plot.kdecopula.

References

Nagler, T. (2018) kdecopula: An R Package for the Kernel Estimation of Bivariate Copula Densities. Journal of Statistical Software 84(7), 1-22 Gijbels, I. and Mielniczuk, J. (1990). Estimating the density of a copula function. Communications in Statistics - Theory and Methods, 19(2):445-464. Charpentier, A., Fermanian, J.-D., and Scaillet, O. (2006). The estimation of copulas: Theory and practice. In Rank, J., editor, Copulas: From theory to application in finance. Risk Books. Geenens, G., Charpentier, A., and Paindaveine, D. (2017). Probit transformation for nonparametric kernel estimation of the copula density. Bernoulli, 23(3), 1848-1873. Nagler, T. (2014). Kernel Methods for Vine Copula Estimation. Master's Thesis, Technische Universitaet Muenchen, https://mediatum.ub.tum.de/node?id=1231221 Wen, K. and Wu, X. (2015). Transformation-Kernel Estimation of the Copula Density, Working paper, http://agecon2.tamu.edu/people/faculty/wu-ximing/agecon2/public/copula.pdf

Examples

## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x)/(length(x)+1))

## estimation of copula density of variables 5 and 6
dens.est <- kdecop(udat[, 5:6])
plot(dens.est)

## evaluate density estimate at (u1,u2)=(0.123,0.321)
dkdecop(c(0.123, 0.321), dens.est)
#> [1] 1.28916

## evaluate cdf estimate at (u1,u2)=(0.123,0.321)
pkdecop(c(0.123, 0.321), dens.est)
#> [1] 0.09779868

## simulate 500 samples from density estimate
rkdecop(500, dens.est)
#>               [,1]        [,2]
#>   [1,] 0.657731941 0.419708252
#>   [2,] 0.207837910 0.150848389
#>   [3,] 0.694639961 0.867218018
#>   [4,] 0.711432453 0.447479248
#>   [5,] 0.943706420 0.969635010
#>   [6,] 0.573340682 0.582244873
#>   [7,] 0.852291450 0.872283936
#>   [8,] 0.311149699 0.544342041
#>   [9,] 0.147333510 0.654937744
#>  [10,] 0.731730036 0.478424072
#>  [11,] 0.214588573 0.340667725
#>  [12,] 0.254973439 0.398284912
#>  [13,] 0.679856723 0.895294189
#>  [14,] 0.973047823 0.977996826
#>  [15,] 0.710151324 0.892364502
#>  [16,] 0.830522863 0.704559326
#>  [17,] 0.247942575 0.763092041
#>  [18,] 0.398540270 0.210906982
#>  [19,] 0.002595307 0.079437256
#>  [20,] 0.617420135 0.560638428
#>  [21,] 0.139052498 0.051849365
#>  [22,] 0.797792856 0.672760010
#>  [23,] 0.255559014 0.536346436
#>  [24,] 0.641577568 0.363189697
#>  [25,] 0.166200074 0.568023682
#>  [26,] 0.070029189 0.109893799
#>  [27,] 0.660109937 0.466461182
#>  [28,] 0.762222615 0.994110107
#>  [29,] 0.179988140 0.574554443
#>  [30,] 0.259728940 0.303680420
#>  [31,] 0.326962577 0.233123779
#>  [32,] 0.331927526 0.265960693
#>  [33,] 0.212289615 0.468292236
#>  [34,] 0.554588144 0.651947021
#>  [35,] 0.154508260 0.787139893
#>  [36,] 0.073787603 0.046539307
#>  [37,] 0.844944446 0.746002197
#>  [38,] 0.826472592 0.919586182
#>  [39,] 0.257359410 0.095001221
#>  [40,] 0.129501529 0.694915771
#>  [41,] 0.515242205 0.047637939
#>  [42,] 0.523661483 0.680389404
#>  [43,] 0.573024127 0.669464111
#>  [44,] 0.691912591 0.479644775
#>  [45,] 0.674265571 0.666595459
#>  [46,] 0.892217924 0.610076904
#>  [47,] 0.620913206 0.721099854
#>  [48,] 0.292761225 0.104522705
#>  [49,] 0.612123621 0.643951416
#>  [50,] 0.799682476 0.508514404
#>  [51,] 0.746831156 0.702056885
#>  [52,] 0.586956183 0.237762451
#>  [53,] 0.370331767 0.388214111
#>  [54,] 0.069886389 0.020050049
#>  [55,] 0.932097737 0.674652100
#>  [56,] 0.409898654 0.585174561
#>  [57,] 0.568316692 0.886688232
#>  [58,] 0.631424568 0.852935791
#>  [59,] 0.575799994 0.475921631
#>  [60,] 0.475059796 0.588165283
#>  [61,] 0.389886441 0.660430908
#>  [62,] 0.922457936 0.701141357
#>  [63,] 0.780835950 0.606781006
#>  [64,] 0.121897611 0.013458252
#>  [65,] 0.368493767 0.227996826
#>  [66,] 0.238006620 0.104400635
#>  [67,] 0.495799420 0.078826904
#>  [68,] 0.358797438 0.140838623
#>  [69,] 0.444671729 0.617218018
#>  [70,] 0.899668560 0.663848877
#>  [71,] 0.478680335 0.551910400
#>  [72,] 0.459139855 0.686614990
#>  [73,] 0.908157840 0.845428467
#>  [74,] 0.269798030 0.891632080
#>  [75,] 0.838856573 0.945098877
#>  [76,] 0.298786547 0.037750244
#>  [77,] 0.952000294 0.931671143
#>  [78,] 0.259967701 0.936553955
#>  [79,] 0.017007930 0.039398193
#>  [80,] 0.634379990 0.413421631
#>  [81,] 0.556254944 0.396270752
#>  [82,] 0.673443982 0.408477783
#>  [83,] 0.429437592 0.554962158
#>  [84,] 0.372892456 0.198333740
#>  [85,] 0.617104122 0.840911865
#>  [86,] 0.354520694 0.446746826
#>  [87,] 0.872738776 0.919219971
#>  [88,] 0.289147252 0.608673096
#>  [89,] 0.550425930 0.469329834
#>  [90,] 0.844691871 0.971221924
#>  [91,] 0.313501220 0.169952393
#>  [92,] 0.103925865 0.594879150
#>  [93,] 0.157729878 0.146026611
#>  [94,] 0.218890225 0.178802490
#>  [95,] 0.962804668 0.918670654
#>  [96,] 0.799536007 0.894622803
#>  [97,] 0.192931402 0.335418701
#>  [98,] 0.764232948 0.743072510
#>  [99,] 0.165880032 0.496673584
#> [100,] 0.245985393 0.228240967
#> [101,] 0.114512391 0.103118896
#> [102,] 0.818968804 0.529937744
#> [103,] 0.356692594 0.392852783
#> [104,] 0.555327459 0.606597900
#> [105,] 0.505494700 0.756011963
#> [106,] 0.630283260 0.462860107
#> [107,] 0.906597301 0.922760010
#> [108,] 0.696335648 0.727935791
#> [109,] 0.687676023 0.493316650
#> [110,] 0.237453076 0.157989502
#> [111,] 0.114608110 0.127227783
#> [112,] 0.856139370 0.985198975
#> [113,] 0.191179732 0.069427490
#> [114,] 0.054347266 0.569732666
#> [115,] 0.544570074 0.674835205
#> [116,] 0.083126266 0.043060303
#> [117,] 0.935016418 0.851776123
#> [118,] 0.166938091 0.140594482
#> [119,] 0.646749625 0.781951904
#> [120,] 0.967858623 0.945098877
#> [121,] 0.902732816 0.933380127
#> [122,] 0.591863465 0.557891846
#> [123,] 0.994801769 0.888214111
#> [124,] 0.901540259 0.903350830
#> [125,] 0.158165844 0.427337646
#> [126,] 0.035102877 0.150238037
#> [127,] 0.788337865 0.924896240
#> [128,] 0.695797757 0.758026123
#> [129,] 0.150103830 0.012298584
#> [130,] 0.174470032 0.293060303
#> [131,] 0.914011287 0.792144775
#> [132,] 0.743700803 0.841827393
#> [133,] 0.409348551 0.507781982
#> [134,] 0.960719446 0.997039795
#> [135,] 0.394421594 0.367889404
#> [136,] 0.442035154 0.435577393
#> [137,] 0.645682902 0.690887451
#> [138,] 0.209150874 0.444061279
#> [139,] 0.836589083 0.803985596
#> [140,] 0.337702142 0.042877197
#> [141,] 0.988734397 0.844757080
#> [142,] 0.218829713 0.194488525
#> [143,] 0.702749531 0.382293701
#> [144,] 0.032482995 0.023040771
#> [145,] 0.691050702 0.294769287
#> [146,] 0.197899779 0.236907959
#> [147,] 0.561234753 0.728668213
#> [148,] 0.436827192 0.720916748
#> [149,] 0.532984084 0.951629639
#> [150,] 0.421125719 0.847442627
#> [151,] 0.056372226 0.111358643
#> [152,] 0.735138231 0.795989990
#> [153,] 0.063063403 0.062591553
#> [154,] 0.937402864 0.983001709
#> [155,] 0.957660115 0.683990479
#> [156,] 0.588638176 0.453399658
#> [157,] 0.581045673 0.827850342
#> [158,] 0.907547265 0.751312256
#> [159,] 0.892106560 0.570770264
#> [160,] 0.166937509 0.320159912
#> [161,] 0.595983259 0.855499268
#> [162,] 0.194694750 0.226959229
#> [163,] 0.844599718 0.449249268
#> [164,] 0.751394781 0.390838623
#> [165,] 0.900164531 0.612396240
#> [166,] 0.498886962 0.856536865
#> [167,] 0.613789340 0.249908447
#> [168,] 0.647996982 0.766082764
#> [169,] 0.008115353 0.016510010
#> [170,] 0.498883670 0.202789307
#> [171,] 0.988353483 0.986053467
#> [172,] 0.157562933 0.341094971
#> [173,] 0.270586539 0.542388916
#> [174,] 0.655376394 0.149322510
#> [175,] 0.321510235 0.149993896
#> [176,] 0.512051571 0.476348877
#> [177,] 0.069208545 0.208221436
#> [178,] 0.664904679 0.798614502
#> [179,] 0.380510255 0.299041748
#> [180,] 0.660036862 0.425384521
#> [181,] 0.622960399 0.645782471
#> [182,] 0.488363212 0.664886475
#> [183,] 0.395497584 0.404205322
#> [184,] 0.820455792 0.807891846
#> [185,] 0.086861419 0.530059814
#> [186,] 0.457899384 0.863739014
#> [187,] 0.653359031 0.778900146
#> [188,] 0.460978405 0.115386963
#> [189,] 0.704958505 0.783111572
#> [190,] 0.662697790 0.476409912
#> [191,] 0.983712873 0.996978760
#> [192,] 0.955352355 0.947601318
#> [193,] 0.582111517 0.463714600
#> [194,] 0.361883245 0.600189209
#> [195,] 0.100864813 0.025543213
#> [196,] 0.153684766 0.401153564
#> [197,] 0.303042646 0.791290283
#> [198,] 0.767962630 0.908905029
#> [199,] 0.468557819 0.091033936
#> [200,] 0.239253261 0.437347412
#> [201,] 0.738376567 0.757171631
#> [202,] 0.357404279 0.325653076
#> [203,] 0.791869481 0.881988525
#> [204,] 0.216180131 0.354339600
#> [205,] 0.185879116 0.047515869
#> [206,] 0.281771073 0.619171143
#> [207,] 0.343990844 0.355438232
#> [208,] 0.412954396 0.789215088
#> [209,] 0.795066570 0.920379639
#> [210,] 0.599917300 0.426300049
#> [211,] 0.096114965 0.081268311
#> [212,] 0.337990663 0.232818604
#> [213,] 0.808740987 0.617279053
#> [214,] 0.635674817 0.858734131
#> [215,] 0.754810228 0.669097900
#> [216,] 0.313890051 0.084625244
#> [217,] 0.196720287 0.188140869
#> [218,] 0.932861990 0.830718994
#> [219,] 0.743497721 0.747161865
#> [220,] 0.634435491 0.500946045
#> [221,] 0.919244660 0.930755615
#> [222,] 0.650056635 0.365570068
#> [223,] 0.841932576 0.310150146
#> [224,] 0.069647716 0.292144775
#> [225,] 0.001174629 0.009613037
#> [226,] 0.018116738 0.001129150
#> [227,] 0.404283878 0.441009521
#> [228,] 0.877155792 0.749359131
#> [229,] 0.967973433 0.416046143
#> [230,] 0.450604922 0.391998291
#> [231,] 0.046792598 0.172393799
#> [232,] 0.298032616 0.396881104
#> [233,] 0.264996193 0.228240967
#> [234,] 0.868323128 0.608062744
#> [235,] 0.599994787 0.762664795
#> [236,] 0.708643445 0.960479736
#> [237,] 0.489952185 0.281707764
#> [238,] 0.887757181 0.608734131
#> [239,] 0.425979024 0.322601318
#> [240,] 0.366604714 0.647979736
#> [241,] 0.407733270 0.040069580
#> [242,] 0.734742536 0.941558838
#> [243,] 0.996777193 0.982452393
#> [244,] 0.539621820 0.956329346
#> [245,] 0.002614375 0.004364014
#> [246,] 0.023925265 0.376007080
#> [247,] 0.950673287 0.886383057
#> [248,] 0.479319974 0.721160889
#> [249,] 0.995466883 0.842742920
#> [250,] 0.829248670 0.835968018
#> [251,] 0.273172236 0.207672119
#> [252,] 0.417163700 0.706085205
#> [253,] 0.659255172 0.747222900
#> [254,] 0.049940002 0.125640869
#> [255,] 0.709480459 0.874664307
#> [256,] 0.857282013 0.834197998
#> [257,] 0.016912709 0.016998291
#> [258,] 0.460721354 0.246124268
#> [259,] 0.454658773 0.639984131
#> [260,] 0.344883882 0.190704346
#> [261,] 0.392168555 0.153533936
#> [262,] 0.240786120 0.222808838
#> [263,] 0.582532714 0.185394287
#> [264,] 0.386746131 0.210540771
#> [265,] 0.990981744 0.967926025
#> [266,] 0.983495619 0.939483643
#> [267,] 0.945815283 0.930816650
#> [268,] 0.729219155 0.546905518
#> [269,] 0.950924951 0.908966064
#> [270,] 0.270184667 0.755096436
#> [271,] 0.986594948 0.829925537
#> [272,] 0.687626511 0.821746826
#> [273,] 0.329563093 0.399261475
#> [274,] 0.799381141 0.600494385
#> [275,] 0.713398703 0.848968506
#> [276,] 0.587649311 0.915496826
#> [277,] 0.881017672 0.704986572
#> [278,] 0.976083268 0.439056396
#> [279,] 0.175210413 0.133819580
#> [280,] 0.949069972 0.554656982
#> [281,] 0.491221151 0.263641357
#> [282,] 0.477989193 0.485870361
#> [283,] 0.407784974 0.256561279
#> [284,] 0.939232257 0.943328857
#> [285,] 0.110448556 0.256744385
#> [286,] 0.199768789 0.153228760
#> [287,] 0.713729821 0.934661865
#> [288,] 0.002615845 0.007904053
#> [289,] 0.763658846 0.495513916
#> [290,] 0.737885153 0.563690186
#> [291,] 0.220059186 0.863861084
#> [292,] 0.065191952 0.093231201
#> [293,] 0.667940494 0.746734619
#> [294,] 0.660442005 0.362640381
#> [295,] 0.616446525 0.916595459
#> [296,] 0.867069320 0.607025146
#> [297,] 0.165612917 0.286590576
#> [298,] 0.893604258 0.888580322
#> [299,] 0.260793115 0.152557373
#> [300,] 0.486605743 0.154693604
#> [301,] 0.821976096 0.791412354
#> [302,] 0.361090764 0.331390381
#> [303,] 0.138674784 0.504425049
#> [304,] 0.665960320 0.238739014
#> [305,] 0.677274422 0.558074951
#> [306,] 0.004783282 0.184600830
#> [307,] 0.413656918 0.126373291
#> [308,] 0.435630046 0.506927490
#> [309,] 0.907084145 0.858612061
#> [310,] 0.858154554 0.712005615
#> [311,] 0.612293901 0.749420166
#> [312,] 0.428257959 0.499053955
#> [313,] 0.185534871 0.120697021
#> [314,] 0.944276621 0.783111572
#> [315,] 0.492915109 0.337066650
#> [316,] 0.483609826 0.129058838
#> [317,] 0.663598058 0.660308838
#> [318,] 0.314112334 0.613494873
#> [319,] 0.549093229 0.716705322
#> [320,] 0.197071938 0.531585693
#> [321,] 0.048610585 0.226470947
#> [322,] 0.931645285 0.761871338
#> [323,] 0.449127802 0.800384521
#> [324,] 0.884333966 0.984283447
#> [325,] 0.186577184 0.024627686
#> [326,] 0.720167688 0.330780029
#> [327,] 0.815136651 0.177032471
#> [328,] 0.375621317 0.374664307
#> [329,] 0.911640968 0.502166748
#> [330,] 0.742152855 0.572174072
#> [331,] 0.151023658 0.267486572
#> [332,] 0.217902921 0.505279541
#> [333,] 0.696253272 0.846771240
#> [334,] 0.438522304 0.372283936
#> [335,] 0.632962317 0.740447998
#> [336,] 0.772230370 0.801361084
#> [337,] 0.395848406 0.054718018
#> [338,] 0.180986919 0.275726318
#> [339,] 0.671877165 0.699432373
#> [340,] 0.511174022 0.692352295
#> [341,] 0.914346990 0.799285889
#> [342,] 0.273138616 0.272796631
#> [343,] 0.205677347 0.157318115
#> [344,] 0.918303049 0.625091553
#> [345,] 0.575862022 0.476226807
#> [346,] 0.006541255 0.054595947
#> [347,] 0.926026067 0.826629639
#> [348,] 0.401354799 0.542755127
#> [349,] 0.157827018 0.404571533
#> [350,] 0.257497411 0.371429443
#> [351,] 0.730845864 0.365081787
#> [352,] 0.187580498 0.145904541
#> [353,] 0.596611367 0.960418701
#> [354,] 0.110780826 0.045379639
#> [355,] 0.240295378 0.224090576
#> [356,] 0.309665438 0.379608154
#> [357,] 0.059236037 0.205718994
#> [358,] 0.727988177 0.413177490
#> [359,] 0.224311077 0.196380615
#> [360,] 0.340468831 0.106842041
#> [361,] 0.938475170 0.992401123
#> [362,] 0.306318262 0.726287842
#> [363,] 0.474218420 0.082305908
#> [364,] 0.218359346 0.676910400
#> [365,] 0.166877309 0.036895752
#> [366,] 0.127630359 0.720672607
#> [367,] 0.646492120 0.914642334
#> [368,] 0.269159423 0.445343018
#> [369,] 0.021649705 0.013275146
#> [370,] 0.861158334 0.700164795
#> [371,] 0.596691484 0.832183838
#> [372,] 0.635841775 0.179656982
#> [373,] 0.509716243 0.168914795
#> [374,] 0.966367833 0.940399170
#> [375,] 0.174273640 0.586456299
#> [376,] 0.160947754 0.522003174
#> [377,] 0.503939518 0.391815186
#> [378,] 0.093671985 0.394805908
#> [379,] 0.111433149 0.185943604
#> [380,] 0.199010950 0.185760498
#> [381,] 0.755100048 0.725982666
#> [382,] 0.730374678 0.847503662
#> [383,] 0.677603594 0.517791748
#> [384,] 0.750071903 0.197967529
#> [385,] 0.714257048 0.795806885
#> [386,] 0.330361486 0.402740479
#> [387,] 0.483104306 0.871246338
#> [388,] 0.949262396 0.852264404
#> [389,] 0.689520664 0.363616943
#> [390,] 0.860060370 0.800445557
#> [391,] 0.219707371 0.416412354
#> [392,] 0.107680341 0.378753662
#> [393,] 0.943045341 0.907012939
#> [394,] 0.800711072 0.565826416
#> [395,] 0.033988178 0.125091553
#> [396,] 0.483852512 0.514862061
#> [397,] 0.628118857 0.835845947
#> [398,] 0.924579103 0.845367432
#> [399,] 0.389474344 0.193817139
#> [400,] 0.116767381 0.133026123
#> [401,] 0.815433128 0.837554932
#> [402,] 0.666248807 0.563812256
#> [403,] 0.260590234 0.291534424
#> [404,] 0.129437695 0.095733643
#> [405,] 0.556144329 0.255035400
#> [406,] 0.315790283 0.292633057
#> [407,] 0.049011773 0.085906982
#> [408,] 0.499392647 0.459564209
#> [409,] 0.023587344 0.074249268
#> [410,] 0.602022762 0.670135498
#> [411,] 0.303512095 0.139678955
#> [412,] 0.305454134 0.082672119
#> [413,] 0.024806744 0.089080811
#> [414,] 0.555375675 0.576019287
#> [415,] 0.619119491 0.914398193
#> [416,] 0.963634601 0.976593018
#> [417,] 0.730314263 0.777679443
#> [418,] 0.238394985 0.638885498
#> [419,] 0.772974213 0.780303955
#> [420,] 0.714960999 0.815155029
#> [421,] 0.489733362 0.533355713
#> [422,] 0.162705973 0.083953857
#> [423,] 0.763296986 0.336151123
#> [424,] 0.818664324 0.982940674
#> [425,] 0.859732515 0.535430908
#> [426,] 0.972163037 0.847137451
#> [427,] 0.450814878 0.323944092
#> [428,] 0.884531691 0.989105225
#> [429,] 0.767194388 0.482025146
#> [430,] 0.316549950 0.486297607
#> [431,] 0.583144925 0.661895752
#> [432,] 0.258412898 0.278656006
#> [433,] 0.838255492 0.681243896
#> [434,] 0.206289321 0.571319580
#> [435,] 0.064704170 0.227691650
#> [436,] 0.352894880 0.558319092
#> [437,] 0.492675916 0.553558350
#> [438,] 0.229140382 0.108612061
#> [439,] 0.277803772 0.560882568
#> [440,] 0.182672088 0.491668701
#> [441,] 0.581704484 0.265167236
#> [442,] 0.255053670 0.475799561
#> [443,] 0.674416862 0.353851318
#> [444,] 0.528603422 0.779937744
#> [445,] 0.747311271 0.919158936
#> [446,] 0.375851842 0.711944580
#> [447,] 0.299104743 0.407562256
#> [448,] 0.941396771 0.950775146
#> [449,] 0.510023732 0.173370361
#> [450,] 0.319198924 0.333709717
#> [451,] 0.709524267 0.477203369
#> [452,] 0.518141436 0.619903564
#> [453,] 0.660930820 0.627593994
#> [454,] 0.054996178 0.478790283
#> [455,] 0.638340546 0.925628662
#> [456,] 0.146986039 0.224334717
#> [457,] 0.222254897 0.911651611
#> [458,] 0.226401961 0.523468018
#> [459,] 0.305341359 0.358734131
#> [460,] 0.169927668 0.110992432
#> [461,] 0.036133090 0.123382568
#> [462,] 0.590899826 0.098602295
#> [463,] 0.382101703 0.240509033
#> [464,] 0.153191030 0.023284912
#> [465,] 0.326275888 0.848663330
#> [466,] 0.940808590 0.641021729
#> [467,] 0.822869403 0.828704834
#> [468,] 0.400723770 0.213470459
#> [469,] 0.080555990 0.052154541
#> [470,] 0.854218204 0.376312256
#> [471,] 0.897257377 0.390472412
#> [472,] 0.953241835 0.851531982
#> [473,] 0.406108873 0.579864502
#> [474,] 0.164540590 0.619415283
#> [475,] 0.180278668 0.046966553
#> [476,] 0.654383833 0.554901123
#> [477,] 0.925541070 0.547271729
#> [478,] 0.062046801 0.066497803
#> [479,] 0.494534026 0.162384033
#> [480,] 0.968916612 0.734344482
#> [481,] 0.297918237 0.242767334
#> [482,] 0.168882789 0.277862549
#> [483,] 0.790972360 0.966156006
#> [484,] 0.196490364 0.764495850
#> [485,] 0.368727054 0.266693115
#> [486,] 0.270207003 0.096466064
#> [487,] 0.075054670 0.412872314
#> [488,] 0.614200671 0.205841064
#> [489,] 0.968648615 0.818206787
#> [490,] 0.113356322 0.338287354
#> [491,] 0.521208054 0.776763916
#> [492,] 0.242325973 0.213958740
#> [493,] 0.519286813 0.250762939
#> [494,] 0.252147327 0.335540771
#> [495,] 0.097435059 0.262542725
#> [496,] 0.337602619 0.531097412
#> [497,] 0.141859992 0.418731689
#> [498,] 0.622819271 0.928802490
#> [499,] 0.491809397 0.851837158
#> [500,] 0.295275017 0.521148682