ECDF and Mahalanobis Distance for Niche Modeling

Introduction

This vignette shows how to use the ECDFniche package to reproduce the simulations from the original ECDF_MahalDist.R script, comparing Mahalanobis distance-based suitability transformations using the chi-squared distribution and the empirical cumulative distribution function (ECDF).

library(ECDFniche)

Core simulation: ecdf_niche()

The function ecdf_niche() simulates a multivariate normal “environmental space”, computes Mahalanobis distances for a sample of points, and then maps those distances to suitability using:

• \(1 - F_{\chi^2}(D^2)\) (theoretical chi-squared transformation)
• \(1 - \text{ECDF}(D^2)\) (empirical transformation)

set.seed(3)
res1 <- ecdf_niche(n = 2)
res1
#> $corplot

#> 
#> $sample_data
#>                  V1           V2
#>   [1,]  0.563046563  0.129469420
#>   [2,] -1.310246067 -0.387356875
#>   [3,] -2.255161603  0.181921742
#>   [4,]  1.452975451  0.415048996
#>   [5,] -0.826324836 -0.509311770
#>   [6,]  0.227709132  0.162699868
#>   [7,]  0.918197262 -1.589179021
#>   [8,] -0.162917623 -0.754037841
#>   [9,]  0.220791294 -0.404621501
#>  [10,]  0.460137977 -0.572527396
#>  [11,] -1.070056523  0.211483078
#>  [12,] -0.441982312 -1.265239660
#>  [13,] -0.295798680  0.710874974
#>  [14,]  0.819383860  0.204655190
#>  [15,] -0.063584099  1.165171436
#>  [16,] -0.506972675  0.782557360
#>  [17,]  1.371248097  0.501120756
#>  [18,]  1.386229238  0.661738456
#>  [19,]  0.213417289  2.378020395
#>  [20,]  0.232300765 -0.589192973
#>  [21,]  1.341067101  0.172173745
#>  [22,] -0.125781517  0.949390909
#>  [23,]  0.538106919 -0.104203286
#>  [24,]  0.808075415  1.122216135
#>  [25,]  0.579053748 -1.314052033
#>  [26,]  0.975449335 -0.099073115
#>  [27,]  0.562934899  0.453402251
#>  [28,]  1.029468071 -0.009220813
#>  [29,]  0.931980655  0.184101408
#>  [30,]  0.267009777  1.474313101
#>  [31,]  0.914112067 -1.304359692
#>  [32,]  0.534954782 -0.028397116
#>  [33,] -0.311014427 -0.192121157
#>  [34,] -1.637973164  1.446213362
#>  [35,]  0.207461870 -0.751071250
#>  [36,] -0.290427784 -0.406185105
#>  [37,]  0.832593527 -1.143355239
#>  [38,] -1.350377002  0.539858886
#>  [39,] -1.696418932 -0.269469689
#>  [40,] -1.097708789  1.128206855
#>  [41,] -0.240048209  0.845113799
#>  [42,]  0.671470904 -0.899290292
#>  [43,] -1.325980229 -1.329929851
#>  [44,] -1.073433767  1.180647609
#>  [45,] -1.402318948  1.649628491
#>  [46,] -1.741160685 -0.714009262
#>  [47,] -0.958590217  0.717673176
#>  [48,] -1.820055959  0.093212446
#>  [49,] -1.934225700  0.054737852
#>  [50,]  0.449985398  1.022034601
#>  [51,] -1.354960693 -0.458508634
#>  [52,] -0.097122854 -0.372202712
#>  [53,]  0.411552252 -1.528705255
#>  [54,] -1.287156802  0.989052457
#>  [55,] -0.008507477 -1.179655970
#>  [56,]  0.473089651 -0.936402700
#>  [57,] -0.453678495 -1.489742101
#>  [58,] -0.491495643  0.609977314
#>  [59,]  0.766086723 -0.465992167
#>  [60,]  1.708705839  1.437579484
#>  [61,]  0.549405156 -0.559816010
#>  [62,] -0.885118121 -1.377569263
#>  [63,]  0.879766738 -0.746131667
#>  [64,] -1.459550313 -0.283260528
#>  [65,] -1.521176493 -0.939934610
#>  [66,] -0.315647053  0.828035656
#>  [67,]  0.573753998 -0.832022006
#>  [68,] -0.485939619  0.048414330
#>  [69,]  0.674875169  0.791327672
#>  [70,] -1.516705292 -1.586931273
#>  [71,]  0.728181225 -0.412467186
#>  [72,]  1.961939255 -0.810888856
#>  [73,] -0.487717204 -0.258649236
#>  [74,]  1.627365687  0.022513344
#>  [75,] -0.913696176 -1.175647537
#>  [76,] -1.177135987  2.604774247
#>  [77,]  0.417482877 -0.162328952
#>  [78,]  1.478917756 -1.490084433
#>  [79,] -0.585383589 -1.028637068
#>  [80,]  0.405755074 -0.125635414
#>  [81,]  0.929052937 -0.729507119
#>  [82,]  1.200377588  2.102048385
#>  [83,]  1.097302297 -0.709609287
#>  [84,] -0.527642505  0.459091055
#>  [85,] -0.746039554  0.527928757
#>  [86,] -0.422520834  0.429485389
#>  [87,] -2.989971911 -1.705856647
#>  [88,]  0.970218904  1.035983526
#>  [89,] -0.675771676  1.599537495
#>  [90,] -1.213337948 -0.124223974
#>  [91,]  0.498548094 -0.433690751
#>  [92,] -0.620364312 -1.192707045
#>  [93,]  0.687470173  1.159054048
#>  [94,]  0.458912664  0.052183531
#>  [95,] -0.977989017  0.071687827
#>  [96,] -0.286462545  1.115875997
#>  [97,]  0.662340129 -1.676065241
#>  [98,]  1.463966257  0.881734077
#>  [99,]  0.031284822 -0.081901748
#> [100,]  0.603479049 -0.002261627
#> [101,]  1.462397352 -0.470960443
#> [102,]  2.910907118 -1.490249355
#> [103,] -1.168345675  0.551475709
#> [104,]  0.070297860 -0.633887176
#> [105,]  1.379724769 -1.549922268
#> [106,]  1.658359975  0.038850942
#> [107,]  1.714739347 -0.696489141
#> [108,] -0.518229132 -2.105714792
#> [109,]  0.130940476 -0.285846108
#> [110,]  2.737543124  0.130741995
#> [111,]  0.308796501 -2.081828611
#> [112,]  0.113608513 -1.540035457
#> [113,]  1.329762766  1.487418238
#> [114,] -1.050244885 -1.129080237
#> [115,] -2.074630153  0.835247606
#> [116,]  0.147890472 -0.936670276
#> [117,] -0.082297041 -0.597370183
#> [118,] -0.369796463  0.712930652
#> [119,] -0.094800662 -0.019349777
#> [120,] -1.403821849 -0.432038161
#> [121,]  0.188036085  0.142951921
#> [122,] -0.480672703 -2.581106516
#> [123,] -0.012122427  1.772917388
#> [124,]  0.239121031  0.428447143
#> [125,]  0.561661627  0.967706640
#> [126,]  2.069702717  0.411321249
#> [127,] -0.562404085 -0.610429320
#> [128,]  1.169220953  2.414414167
#> [129,]  1.133007505  0.971399454
#> [130,] -0.271108219  0.276187608
#> [131,] -0.437070833 -0.207808769
#> [132,]  0.180358408  1.294558849
#> [133,] -0.590252287 -0.218730717
#> [134,] -0.509031169  0.293323441
#> [135,]  1.157296285 -0.506192138
#> [136,]  0.315998342  0.964786800
#> [137,] -3.240603750 -0.592741783
#> [138,] -0.913477103  1.861363572
#> [139,] -2.483079083 -1.158950774
#> [140,]  0.233809239 -0.538666049
#> [141,]  0.211137193  0.193568011
#> [142,] -0.030227774 -2.006746950
#> [143,]  0.107445189 -0.960844540
#> [144,]  0.604560580 -0.065928374
#> [145,]  0.848470909 -1.032035289
#> [146,] -0.322217054  2.959177875
#> [147,]  0.953110399  1.022095176
#> [148,] -0.866157228 -0.229131114
#> [149,]  2.307508012 -0.501532454
#> [150,]  2.467275125  0.028019244
#> [151,] -1.166856559  2.062692755
#> [152,] -0.723301122  0.080197063
#> [153,]  0.786869809  0.769107756
#> [154,]  0.610291431 -0.315790505
#> [155,]  2.467171643  0.930089716
#> [156,]  0.200925633  1.363316150
#> [157,]  0.159852509  1.654788822
#> [158,]  1.138871363 -0.193611794
#> [159,] -1.128498611 -0.738851391
#> [160,]  0.487927244 -0.251156500
#> [161,] -1.322755153  0.394487221
#> [162,] -1.463272431  0.474989755
#> [163,]  1.221661970  0.456832963
#> [164,]  0.158747878 -1.853806545
#> [165,] -0.482520123 -1.747920575
#> [166,]  0.201701374  0.450112059
#> [167,] -1.045456306  1.376006517
#> [168,] -0.714118435  0.752360977
#> [169,]  0.246175539 -1.065371344
#> [170,] -0.778778557  1.301239482
#> [171,] -0.964752872  0.941600302
#> [172,]  0.690774755 -0.896131559
#> [173,]  0.478838237 -0.480822662
#> [174,]  1.051052453 -1.526867269
#> [175,]  0.280164403  0.875830393
#> [176,] -0.068716320  0.091675692
#> [177,]  0.491642085  0.387112348
#> [178,]  1.197284839 -0.429318145
#> [179,]  1.926684420  0.803702899
#> [180,] -1.928349850 -0.111363461
#> [181,]  1.074442242 -0.372176797
#> [182,] -0.312159369 -0.312574693
#> [183,] -0.349993688  1.145491014
#> [184,]  0.041336055  2.703200525
#> [185,]  0.424374698  0.667288319
#> [186,]  0.076073579  0.551156540
#> [187,] -0.051339679  0.674891694
#> [188,]  0.653071563  0.377420689
#> [189,]  1.717464536 -0.099586245
#> [190,]  0.427948152  0.659378101
#> [191,] -0.060621256 -0.132006255
#> [192,]  1.408201483 -0.027211629
#> [193,]  0.108016403  0.480790844
#> [194,]  0.117652033  1.171020165
#> [195,] -1.810504760  1.513727404
#> [196,]  1.061547793  1.346438434
#> [197,] -3.078024881  0.848616017
#> [198,] -1.290503049 -0.968069681
#> [199,] -1.511402441 -0.469266839
#> [200,] -0.551603254 -0.336314709
#> [201,] -0.951989679 -0.187589652
#> [202,]  0.898162027 -0.615713565
#> [203,]  1.840215513 -0.490984448
#> [204,] -0.603528466 -0.696644709
#> [205,] -0.202227934  1.051448289
#> [206,]  1.296430256 -0.563830479
#> [207,] -1.518270143  0.468475749
#> [208,] -0.942618020 -1.048475963
#> [209,] -1.640193256 -1.879545891
#> [210,]  1.726813175 -0.151598378
#> [211,]  0.040206318 -0.157654529
#> [212,] -2.606919529  0.023479846
#> [213,]  0.298744912  0.156001465
#> [214,]  0.592994415 -0.746456711
#> [215,] -0.935377681  1.804657889
#> [216,] -0.244752248  0.300069778
#> [217,] -0.415332589 -0.023334460
#> [218,] -0.244908437  0.120557859
#> [219,]  2.146205944 -1.836984929
#> [220,]  1.135258532  1.759169281
#> [221,]  1.118139761  0.391790668
#> [222,]  1.714686272  0.955940233
#> [223,] -0.971304244 -0.900120979
#> [224,] -0.688032910 -1.358720701
#> [225,] -0.309988382 -0.004359969
#> [226,]  0.819227777  1.676228810
#> [227,]  0.286623588  1.540877826
#> [228,] -1.302545151 -0.560877014
#> [229,] -2.803989276  0.090556002
#> [230,] -0.142833819 -1.628232931
#> [231,]  1.226436053  0.408363832
#> [232,]  0.496217421  0.265521595
#> [233,]  0.248486557 -1.519463606
#> [234,]  0.072663346  0.426867638
#> [235,] -0.391589546 -0.870593641
#> [236,]  0.597549098  0.019124002
#> [237,] -0.791586249 -1.455320420
#> [238,]  1.307554716 -0.656059904
#> [239,]  0.169251810  1.094531642
#> [240,]  0.466500113  1.032649544
#> [241,] -0.677607341 -1.041700276
#> [242,] -0.818146329 -1.294667689
#> [243,] -1.971986702  1.479931111
#> [244,]  2.014359406 -0.997563946
#> [245,]  1.465237739 -2.609888083
#> [246,]  1.306671927  1.091729179
#> [247,] -1.391158564  1.615724452
#> [248,] -2.044858747 -0.025474797
#> [249,]  0.063555657  1.074360382
#> [250,] -0.177768430 -0.599288352
#> [251,] -1.164337414 -0.523475879
#> [252,]  0.472081599  0.004750261
#> [253,] -3.041326886 -0.061224205
#> [254,] -0.110879815 -1.048207277
#> [255,]  0.701030597  0.242942251
#> [256,]  0.487741496  0.067075904
#> [257,]  0.805403407  1.001941146
#> [258,] -0.169331282  0.743432876
#> [259,] -0.250335821  0.049992498
#> [260,]  0.311778585  1.284925417
#> [261,]  0.051617445 -0.058694503
#> [262,]  1.418247564 -0.377776146
#> [263,]  0.544398007 -1.913014897
#> [264,]  0.297822771 -0.625265366
#> [265,]  3.176497665 -0.376321012
#> [266,] -0.053829120 -0.046157738
#> [267,]  1.545656718  0.175914192
#> [268,]  1.434028364 -1.602566277
#> [269,] -0.328002153 -0.829160734
#> [270,]  0.278737737 -0.848253533
#> [271,]  0.714102966  0.123533262
#> [272,] -1.027329062 -0.850032598
#> [273,] -0.914140678 -0.362404904
#> [274,] -0.783440751  1.457166160
#> [275,] -1.420056044  0.901376762
#> [276,] -0.048447452 -1.165142066
#> [277,]  2.540405420  0.780376915
#> [278,]  0.983550806 -0.336222799
#> [279,]  0.847041978  0.435364852
#> [280,] -0.729357219  0.124518929
#> [281,] -0.326949603 -1.252251112
#> [282,] -0.997621560 -1.766828744
#> [283,]  0.506524409  2.304047743
#> [284,]  0.009021070  1.791410184
#> [285,] -0.675834964 -1.581102543
#> [286,]  0.149847840  1.865138556
#> [287,]  1.427395596 -0.039090101
#> [288,]  0.236379049  0.553603212
#> [289,] -0.224093616  0.864685983
#> [290,]  0.068540155 -1.319259574
#> [291,] -0.007469834 -1.197258237
#> [292,]  1.380920084 -0.673244927
#> [293,] -0.020854603 -1.691225335
#> [294,] -1.084409006  0.706557385
#> [295,]  0.848334149  0.520013030
#> [296,]  1.503586239 -0.691393757
#> [297,]  0.771799373 -0.400085154
#> [298,] -0.055427571 -0.312524679
#> [299,]  0.858477145  0.483591618
#> [300,] -0.380693467  0.620695497
#> [301,]  0.327786881  0.665631409
#> [302,]  0.032230912 -1.632541720
#> [303,]  0.380670164  0.870182320
#> [304,]  1.668907567  1.131032189
#> [305,]  0.244512556  0.443871565
#> [306,]  1.034812746  0.470860985
#> [307,] -0.047143447  0.178143292
#> [308,]  0.923935817 -0.426623681
#> [309,] -0.648924439  0.779568527
#> [310,] -0.888622124 -2.077137807
#> [311,]  0.073787184 -0.717243375
#> [312,]  1.150010822 -1.843447672
#> [313,]  0.684874426  1.187641309
#> [314,] -0.083830824  0.837383766
#> [315,]  0.365969189 -0.154849088
#> [316,] -0.457422325  0.446112912
#> [317,] -1.271930396 -0.029582844
#> [318,]  0.729897880 -0.205872580
#> [319,] -0.446588083 -0.064694105
#> [320,] -0.135004631 -1.514920457
#> [321,]  0.686366975 -0.333717388
#> [322,]  0.155454802  0.345304573
#> [323,]  0.051748906  1.321183226
#> [324,]  0.796539854  0.422305762
#> [325,] -1.204576041  1.945129571
#> [326,] -1.470038912 -1.064997378
#> [327,]  1.084783853 -0.236406611
#> [328,]  0.498621582  0.831905655
#> [329,]  2.052026963 -0.213485408
#> [330,] -0.553327829 -0.859779183
#> [331,] -0.322477546 -0.007822616
#> [332,]  0.414339595  1.236887993
#> [333,]  0.274734763  1.262285365
#> [334,] -0.187917796  0.484477446
#> [335,] -0.286924040  0.319856469
#> [336,] -0.373053925  1.790214738
#> [337,] -1.824166108  0.390740070
#> [338,] -0.708776165 -0.683525308
#> [339,]  1.378006705  1.290622792
#> [340,] -0.558234623 -0.682949755
#> [341,] -0.870347777  0.265978816
#> [342,] -0.282948867  1.357148569
#> [343,]  0.402587029  0.602616909
#> [344,]  0.155676899  0.266179731
#> [345,]  1.440685570  0.899089566
#> [346,]  0.503125244  0.904763385
#> [347,]  0.581395890 -1.439205751
#> [348,] -0.301211503 -0.267178788
#> [349,] -1.365679285 -0.588424518
#> [350,] -0.398812707 -1.592061613
#> [351,]  0.181871211 -1.858063545
#> [352,] -0.448931757 -0.896138721
#> [353,]  0.494961257  0.587361281
#> [354,] -0.765686860 -0.842418308
#> [355,] -0.065824567 -1.201096493
#> [356,] -0.734511300 -1.440196753
#> [357,] -0.300533945 -2.309278116
#> [358,]  0.575481988  0.538658003
#> [359,] -1.535082572 -0.958244382
#> [360,]  0.532540163  0.393450427
#> [361,] -0.951548744 -0.158674317
#> [362,] -1.053935650  0.954274945
#> [363,]  0.037210578  0.362285290
#> [364,] -0.227445966 -0.740161332
#> [365,]  0.576152568  1.565148643
#> [366,] -0.097900706  0.111162984
#> [367,]  1.323195674 -0.851769618
#> [368,]  0.090502627 -0.593601818
#> [369,]  0.364404233 -0.300881456
#> [370,]  0.203665216  0.902305445
#> [371,]  0.176658268  0.721386639
#> [372,]  0.945202339 -0.496680575
#> [373,] -1.208284907 -1.463527445
#> [374,] -1.272128764 -1.548459136
#> [375,] -0.844135366 -0.932146742
#> [376,]  1.289969520 -0.378370538
#> [377,] -0.061371151 -0.743933357
#> [378,]  1.534410907  1.019960395
#> [379,] -1.187215716 -0.093743526
#> [380,]  0.426463659 -0.280748365
#> [381,]  1.092457590  0.623648069
#> [382,] -1.338657348  0.124534857
#> [383,] -0.860442946  0.250217073
#> [384,]  1.062360229  0.574095079
#> [385,]  1.440309922  0.862041620
#> [386,]  0.376034404  1.871051689
#> [387,]  0.016856071  0.402629560
#> [388,] -0.271838635  0.897479184
#> [389,] -0.631437505 -0.656672699
#> [390,] -1.091954937 -0.847672514
#> [391,]  2.531299431 -0.150242832
#> [392,] -1.066324189 -0.973921320
#> [393,]  1.166172931  0.388551517
#> [394,]  1.041879730 -0.241488108
#> [395,] -0.570748510  0.477356823
#> [396,]  0.685308453 -0.711524003
#> [397,]  0.811791557 -0.122594164
#> [398,] -0.372718363  1.028273523
#> [399,]  1.139527627 -0.389091800
#> [400,]  0.265615705 -1.870535628
#> [401,]  0.741670831  0.978584408
#> [402,]  0.390670661  0.064998698
#> [403,] -2.330920906  0.050810046
#> [404,]  1.178959926 -0.900153724
#> [405,]  0.332808831 -0.427324983
#> [406,]  1.147994103 -0.783726601
#> [407,] -0.953642770 -0.110747362
#> [408,] -2.071571546 -0.260042108
#> [409,] -0.538808591 -1.075275258
#> [410,]  0.382139376  0.823832767
#> [411,] -0.305176141  0.297145814
#> [412,]  1.010784030  1.683157431
#> [413,] -0.969646485  0.459818568
#> [414,] -2.204733343 -0.418649506
#> [415,]  0.236947339 -1.054718457
#> [416,] -0.675286597 -0.303764777
#> [417,] -0.195585637 -0.480003295
#> [418,] -1.432097272  1.686749219
#> [419,] -0.148904992  0.752637559
#> [420,] -0.696450806  2.050875935
#> [421,] -1.741395673 -0.191815302
#> [422,]  1.716999847 -1.350046781
#> [423,] -1.260500024 -0.882749996
#> [424,] -1.140266185  0.653539519
#> [425,] -0.938453019  0.873429851
#> [426,]  1.387515324 -0.561724678
#> [427,]  0.283356696 -0.008632493
#> [428,] -1.342070164 -0.441329017
#> [429,]  0.448706184 -1.815878785
#> [430,]  0.074979047  0.706694190
#> [431,] -0.339360552  0.679897401
#> [432,] -0.585732088  0.711178217
#> [433,]  0.959290695  0.548006138
#> [434,]  1.007800500  1.139779123
#> [435,]  0.698224500 -0.813853720
#> [436,] -0.981617684  0.442337696
#> [437,] -0.939916822 -0.873774825
#> [438,] -0.513400978 -1.944705410
#> [439,] -0.037288310 -0.312027733
#> [440,]  0.605404380  0.541900212
#> [441,]  2.461406622 -0.525134098
#> [442,]  0.306296433 -0.173616515
#> [443,] -0.162803591  0.819505472
#> [444,] -1.471069827 -0.844461030
#> [445,] -1.037456462  2.367159905
#> [446,] -1.262527368  1.137082739
#> [447,] -0.425105873 -0.442344153
#> [448,]  0.255216633 -1.102307576
#> [449,]  0.330240028 -0.452451704
#> [450,] -1.029210665  0.117725876
#> [451,] -2.208984112  1.659268033
#> [452,] -0.155648816 -0.570393176
#> [453,]  1.404372296  0.047806945
#> [454,]  2.668892487  1.004115973
#> [455,]  0.372794176 -0.335085555
#> [456,]  0.393691092 -1.508556409
#> [457,]  0.286338768 -0.883494158
#> [458,] -1.722847891  0.334690258
#> [459,] -1.612446887  1.783595608
#> [460,] -0.502205969  0.963510939
#> [461,]  0.251246075  0.501804997
#> [462,] -1.296884509 -0.700194707
#> [463,] -1.618299173 -1.331882794
#> [464,]  0.316153535  1.512697472
#> [465,]  0.260239469 -0.526843248
#> [466,]  2.018478198  1.645361146
#> [467,] -0.792458627  0.252166612
#> [468,] -0.240530734  0.523646490
#> [469,] -0.780912930 -0.441322973
#> [470,] -1.687208922 -0.861379026
#> [471,] -1.278828406 -0.254345117
#> [472,]  0.498274273  1.107054035
#> [473,] -0.694018101  1.360261409
#> [474,]  1.618229678 -0.078220444
#> [475,] -1.098264498  0.627888090
#> [476,]  0.935300420  0.240872527
#> [477,] -0.783517518  2.474340516
#> [478,]  0.669878380 -0.691533017
#> [479,] -1.559328147 -0.086479164
#> [480,] -0.728314214 -0.558270836
#> [481,]  0.334340378 -1.104922207
#> [482,] -1.993990338  0.152427161
#> [483,] -0.533536804  0.114963770
#> [484,] -0.439082998  0.643526729
#> [485,] -0.494843913 -2.530015811
#> [486,]  0.666843480  1.868431816
#> [487,] -0.514701043  0.057658850
#> [488,]  0.095907481 -1.015931305
#> [489,]  0.940763489 -0.833160685
#> [490,] -0.636162782  0.409559194
#> [491,] -1.472622489  0.075430483
#> [492,]  0.289646939  0.340303107
#> [493,]  0.236494593  0.217454412
#> [494,]  0.046199548  0.498360435
#> [495,] -0.239356267 -2.052588539
#> [496,] -0.965181284  1.008670110
#> [497,]  0.796915333 -0.485746248
#> [498,] -0.032685110  0.160379828
#> [499,] -0.209461860 -1.977267023
#> [500,]  0.337311233 -0.090143151
#> 
#> $sample_niche
#>   [1] 0.841059754 0.400824786 0.079795638 0.314438293 0.634999450 0.957103615
#>   [7] 0.198563273 0.758361709 0.906876997 0.774098445 0.556847003 0.424882502
#>  [13] 0.741356042 0.693811763 0.504343495 0.647902477 0.338809585 0.301543966
#>  [19] 0.059041600 0.830191744 0.397050720 0.629337230 0.859803659 0.378038461
#>  [25] 0.374227378 0.617158358 0.761469270 0.585974713 0.631167880 0.323783916
#>  [31] 0.295697399 0.864015221 0.940863171 0.095434136 0.753403955 0.892305676
#>  [37] 0.382986450 0.352393023 0.233947948 0.294475772 0.677597572 0.547562825
#>  [43] 0.179632493 0.284705273 0.099650072 0.175493071 0.492255006 0.194303890
#>  [49] 0.157610760 0.529425465 0.367103944 0.936890696 0.303058803 0.272722509
#>  [55] 0.518320823 0.593472310 0.313652130 0.735739542 0.675649500 0.080602116
#>  [61] 0.744995451 0.274424962 0.525208307 0.337615610 0.209185846 0.673805310
#>  [67] 0.614756783 0.890594523 0.573634704 0.095274551 0.710239726 0.108469028
#>  [73] 0.866090822 0.264527838 0.343154455 0.018100251 0.905516854 0.118119000
#>  [79] 0.512641314 0.913913734 0.508231971 0.052927121 0.434678814 0.783510625
#>  [85] 0.661111609 0.833431034 0.002875169 0.358519731 0.225273468 0.482068714
#>  [91] 0.811025905 0.421176718 0.397340483 0.895154516 0.623885670 0.514822027
#>  [97] 0.211555781 0.227141131 0.997340179 0.830560868 0.310930623 0.005221495
#> [103] 0.438903052 0.829349966 0.124620494 0.251155976 0.184738861 0.104127787
#> [109] 0.956970164 0.023286847 0.120056123 0.321663072 0.133837382 0.316218736
#> [115] 0.084894430 0.655953905 0.846705601 0.723068210 0.996195562 0.347274114
#> [121] 0.968541442 0.036056590 0.209049987 0.879344894 0.527333400 0.106238742
#> [127] 0.721386293 0.027379635 0.321884435 0.926443392 0.895891359 0.422954097
#> [133] 0.827432453 0.842478712 0.455877093 0.590851471 0.004638425 0.120252084
#> [139] 0.024619175 0.852410706 0.954979640 0.145480710 0.645041446 0.829484791
#> [145] 0.424128443 0.012703775 0.369780999 0.677180330 0.062752336 0.047532631
#> [151] 0.063191407 0.771729900 0.537279228 0.793624057 0.030137177 0.384720684
#> [157] 0.251130784 0.513564665 0.412959326 0.863020731 0.390911009 0.311271913
#> [163] 0.420697817 0.191431942 0.206617904 0.878371202 0.229287663 0.586277609
#> [169] 0.568971742 0.320675756 0.407339382 0.541905086 0.802701624 0.191476276
#> [175] 0.648329003 0.992260802 0.814082074 0.449526505 0.110567057 0.158736405
#> [181] 0.527530438 0.914807069 0.488647938 0.026919618 0.722889931 0.850267990
#> [187] 0.790314985 0.744220965 0.227415036 0.725591066 0.992309926 0.369343247
#> [193] 0.879197594 0.496992384 0.064600164 0.225719726 0.006515504 0.281400673
#> [199] 0.292502818 0.820458240 0.631956913 0.561592601 0.165525686 0.667573911
#> [205] 0.562321371 0.373762189 0.287951659 0.382977914 0.047902810 0.222742018
#> [211] 0.989579813 0.034733703 0.940002751 0.648026859 0.130572232 0.925892827
#> [217] 0.920515370 0.963408916 0.020526506 0.110067824 0.488909582 0.142143898
#> [223] 0.428396221 0.328478036 0.955392642 0.173469466 0.291506048 0.374138811
#> [229] 0.020401370 0.279707769 0.427462842 0.846555731 0.323816225 0.904577865
#> [235] 0.650569563 0.832984113 0.266807651 0.349669941 0.537189677 0.519642146
#> [241] 0.477498624 0.323334813 0.050255208 0.083482260 0.013286468 0.229610374
#> [247] 0.106821372 0.126902495 0.556689726 0.835612297 0.451774590 0.891732063
#> [253] 0.010267840 0.592765639 0.752542040 0.881835910 0.430412737 0.744290307
#> [259] 0.968944142 0.413677680 0.997521152 0.343349047 0.150691641 0.799193822
#> [265] 0.006141657 0.998545646 0.295271616 0.106722970 0.688327908 0.687796974
#> [271] 0.763835886 0.422683887 0.625625770 0.258603082 0.247902953 0.526215721
#> [277] 0.028589000 0.585857220 0.627178307 0.764584573 0.451175188 0.136474190
#> [283] 0.062557822 0.202282981 0.241501232 0.174614272 0.359423094 0.826657005
#> [289] 0.668721782 0.437463657 0.508022506 0.313607421 0.255586789 0.437325946
#> [295] 0.600980961 0.259943723 0.690557943 0.957575362 0.607032419 0.765835907
#> [301] 0.751234984 0.280791049 0.629456934 0.127842251 0.872138708 0.516397818
#> [307] 0.980560939 0.600999801 0.599693339 0.084855580 0.786070406 0.103073380
#> [313] 0.384988944 0.697828272 0.925080138 0.815105007 0.451417952 0.751172909
#> [319] 0.907413660 0.332480566 0.751409524 0.924721296 0.415688374 0.657754793
#> [325] 0.076199997 0.199903128 0.541136625 0.616600717 0.119601528 0.608437825
#> [331] 0.951767344 0.422570123 0.430578809 0.870250277 0.910255201 0.190747567
#> [337] 0.179597423 0.628809567 0.164533748 0.691425845 0.665034947 0.383731572
#> [343] 0.760467708 0.948231208 0.231351501 0.577429929 0.316835574 0.929000601
#> [349] 0.338938269 0.275775187 0.189269385 0.621586496 0.735666438 0.536959424
#> [355] 0.504639969 0.284646280 0.073798958 0.724013401 0.201391797 0.794928066
#> [361] 0.635117125 0.368569904 0.930612258 0.756422813 0.246613951 0.987854359
#> [367] 0.298355638 0.847529097 0.898848768 0.645754481 0.751998224 0.571935967
#> [373] 0.173951694 0.141983833 0.467114379 0.408225638 0.772439866 0.178952551
#> [379] 0.498732351 0.881546937 0.445670327 0.410741116 0.673451276 0.474598709
#> [385] 0.239225289 0.161980825 0.916723763 0.642632546 0.673447409 0.395636159
#> [391] 0.040370624 0.363972596 0.463331727 0.565788953 0.759097310 0.625868029
#> [397] 0.713130256 0.549987125 0.488074524 0.181772898 0.463067691 0.920893547
#> [403] 0.068184648 0.343025050 0.871297951 0.389947813 0.637544611 0.116357711
#> [409] 0.501929388 0.654294228 0.912078979 0.143478597 0.566402012 0.083250972
#> [415] 0.576417866 0.768724470 0.885080444 0.089987844 0.741383315 0.098940470
#> [421] 0.220432122 0.097898354 0.315495963 0.426355194 0.443725384 0.331263369
#> [427] 0.958811661 0.376264095 0.187953414 0.770703336 0.747538403 0.655232714
#> [433] 0.534994461 0.308453132 0.576269853 0.564361549 0.451409432 0.143143655
#> [439] 0.958428697 0.709888965 0.043072277 0.941594190 0.702153411 0.244747447
#> [445] 0.037494211 0.240958587 0.838845224 0.546372561 0.863151019 0.590248388
#> [451] 0.023439886 0.852149467 0.370231979 0.016691454 0.887245551 0.314314930
#> [457] 0.666689573 0.218853511 0.058327080 0.555249833 0.846678672 0.346318333
#> [463] 0.116687829 0.301320478 0.851839016 0.032788802 0.711336626 0.844098768
#> [469] 0.678890741 0.171909526 0.434581428 0.472533864 0.315334545 0.268589130
#> [475] 0.453846553 0.620935531 0.036345156 0.640829096 0.300966095 0.667931156
#> [481] 0.532440238 0.138904091 0.864280176 0.737657521 0.040634653 0.139074366
#> [487] 0.877602804 0.613041278 0.465792903 0.752820802 0.342730975 0.898036774
#> [493] 0.944399880 0.876315820 0.129048174 0.381745048 0.653883230 0.984195678
#> [499] 0.150487744 0.940529346
#> 
#> $chisq_suits
#>   [1] 0.841059754 0.400824786 0.079795638 0.314438293 0.634999450 0.957103615
#>   [7] 0.198563273 0.758361709 0.906876997 0.774098445 0.556847003 0.424882502
#>  [13] 0.741356042 0.693811763 0.504343495 0.647902477 0.338809585 0.301543966
#>  [19] 0.059041600 0.830191744 0.397050720 0.629337230 0.859803659 0.378038461
#>  [25] 0.374227378 0.617158358 0.761469270 0.585974713 0.631167880 0.323783916
#>  [31] 0.295697399 0.864015221 0.940863171 0.095434136 0.753403955 0.892305676
#>  [37] 0.382986450 0.352393023 0.233947948 0.294475772 0.677597572 0.547562825
#>  [43] 0.179632493 0.284705273 0.099650072 0.175493071 0.492255006 0.194303890
#>  [49] 0.157610760 0.529425465 0.367103944 0.936890696 0.303058803 0.272722509
#>  [55] 0.518320823 0.593472310 0.313652130 0.735739542 0.675649500 0.080602116
#>  [61] 0.744995451 0.274424962 0.525208307 0.337615610 0.209185846 0.673805310
#>  [67] 0.614756783 0.890594523 0.573634704 0.095274551 0.710239726 0.108469028
#>  [73] 0.866090822 0.264527838 0.343154455 0.018100251 0.905516854 0.118119000
#>  [79] 0.512641314 0.913913734 0.508231971 0.052927121 0.434678814 0.783510625
#>  [85] 0.661111609 0.833431034 0.002875169 0.358519731 0.225273468 0.482068714
#>  [91] 0.811025905 0.421176718 0.397340483 0.895154516 0.623885670 0.514822027
#>  [97] 0.211555781 0.227141131 0.997340179 0.830560868 0.310930623 0.005221495
#> [103] 0.438903052 0.829349966 0.124620494 0.251155976 0.184738861 0.104127787
#> [109] 0.956970164 0.023286847 0.120056123 0.321663072 0.133837382 0.316218736
#> [115] 0.084894430 0.655953905 0.846705601 0.723068210 0.996195562 0.347274114
#> [121] 0.968541442 0.036056590 0.209049987 0.879344894 0.527333400 0.106238742
#> [127] 0.721386293 0.027379635 0.321884435 0.926443392 0.895891359 0.422954097
#> [133] 0.827432453 0.842478712 0.455877093 0.590851471 0.004638425 0.120252084
#> [139] 0.024619175 0.852410706 0.954979640 0.145480710 0.645041446 0.829484791
#> [145] 0.424128443 0.012703775 0.369780999 0.677180330 0.062752336 0.047532631
#> [151] 0.063191407 0.771729900 0.537279228 0.793624057 0.030137177 0.384720684
#> [157] 0.251130784 0.513564665 0.412959326 0.863020731 0.390911009 0.311271913
#> [163] 0.420697817 0.191431942 0.206617904 0.878371202 0.229287663 0.586277609
#> [169] 0.568971742 0.320675756 0.407339382 0.541905086 0.802701624 0.191476276
#> [175] 0.648329003 0.992260802 0.814082074 0.449526505 0.110567057 0.158736405
#> [181] 0.527530438 0.914807069 0.488647938 0.026919618 0.722889931 0.850267990
#> [187] 0.790314985 0.744220965 0.227415036 0.725591066 0.992309926 0.369343247
#> [193] 0.879197594 0.496992384 0.064600164 0.225719726 0.006515504 0.281400673
#> [199] 0.292502818 0.820458240 0.631956913 0.561592601 0.165525686 0.667573911
#> [205] 0.562321371 0.373762189 0.287951659 0.382977914 0.047902810 0.222742018
#> [211] 0.989579813 0.034733703 0.940002751 0.648026859 0.130572232 0.925892827
#> [217] 0.920515370 0.963408916 0.020526506 0.110067824 0.488909582 0.142143898
#> [223] 0.428396221 0.328478036 0.955392642 0.173469466 0.291506048 0.374138811
#> [229] 0.020401370 0.279707769 0.427462842 0.846555731 0.323816225 0.904577865
#> [235] 0.650569563 0.832984113 0.266807651 0.349669941 0.537189677 0.519642146
#> [241] 0.477498624 0.323334813 0.050255208 0.083482260 0.013286468 0.229610374
#> [247] 0.106821372 0.126902495 0.556689726 0.835612297 0.451774590 0.891732063
#> [253] 0.010267840 0.592765639 0.752542040 0.881835910 0.430412737 0.744290307
#> [259] 0.968944142 0.413677680 0.997521152 0.343349047 0.150691641 0.799193822
#> [265] 0.006141657 0.998545646 0.295271616 0.106722970 0.688327908 0.687796974
#> [271] 0.763835886 0.422683887 0.625625770 0.258603082 0.247902953 0.526215721
#> [277] 0.028589000 0.585857220 0.627178307 0.764584573 0.451175188 0.136474190
#> [283] 0.062557822 0.202282981 0.241501232 0.174614272 0.359423094 0.826657005
#> [289] 0.668721782 0.437463657 0.508022506 0.313607421 0.255586789 0.437325946
#> [295] 0.600980961 0.259943723 0.690557943 0.957575362 0.607032419 0.765835907
#> [301] 0.751234984 0.280791049 0.629456934 0.127842251 0.872138708 0.516397818
#> [307] 0.980560939 0.600999801 0.599693339 0.084855580 0.786070406 0.103073380
#> [313] 0.384988944 0.697828272 0.925080138 0.815105007 0.451417952 0.751172909
#> [319] 0.907413660 0.332480566 0.751409524 0.924721296 0.415688374 0.657754793
#> [325] 0.076199997 0.199903128 0.541136625 0.616600717 0.119601528 0.608437825
#> [331] 0.951767344 0.422570123 0.430578809 0.870250277 0.910255201 0.190747567
#> [337] 0.179597423 0.628809567 0.164533748 0.691425845 0.665034947 0.383731572
#> [343] 0.760467708 0.948231208 0.231351501 0.577429929 0.316835574 0.929000601
#> [349] 0.338938269 0.275775187 0.189269385 0.621586496 0.735666438 0.536959424
#> [355] 0.504639969 0.284646280 0.073798958 0.724013401 0.201391797 0.794928066
#> [361] 0.635117125 0.368569904 0.930612258 0.756422813 0.246613951 0.987854359
#> [367] 0.298355638 0.847529097 0.898848768 0.645754481 0.751998224 0.571935967
#> [373] 0.173951694 0.141983833 0.467114379 0.408225638 0.772439866 0.178952551
#> [379] 0.498732351 0.881546937 0.445670327 0.410741116 0.673451276 0.474598709
#> [385] 0.239225289 0.161980825 0.916723763 0.642632546 0.673447409 0.395636159
#> [391] 0.040370624 0.363972596 0.463331727 0.565788953 0.759097310 0.625868029
#> [397] 0.713130256 0.549987125 0.488074524 0.181772898 0.463067691 0.920893547
#> [403] 0.068184648 0.343025050 0.871297951 0.389947813 0.637544611 0.116357711
#> [409] 0.501929388 0.654294228 0.912078979 0.143478597 0.566402012 0.083250972
#> [415] 0.576417866 0.768724470 0.885080444 0.089987844 0.741383315 0.098940470
#> [421] 0.220432122 0.097898354 0.315495963 0.426355194 0.443725384 0.331263369
#> [427] 0.958811661 0.376264095 0.187953414 0.770703336 0.747538403 0.655232714
#> [433] 0.534994461 0.308453132 0.576269853 0.564361549 0.451409432 0.143143655
#> [439] 0.958428697 0.709888965 0.043072277 0.941594190 0.702153411 0.244747447
#> [445] 0.037494211 0.240958587 0.838845224 0.546372561 0.863151019 0.590248388
#> [451] 0.023439886 0.852149467 0.370231979 0.016691454 0.887245551 0.314314930
#> [457] 0.666689573 0.218853511 0.058327080 0.555249833 0.846678672 0.346318333
#> [463] 0.116687829 0.301320478 0.851839016 0.032788802 0.711336626 0.844098768
#> [469] 0.678890741 0.171909526 0.434581428 0.472533864 0.315334545 0.268589130
#> [475] 0.453846553 0.620935531 0.036345156 0.640829096 0.300966095 0.667931156
#> [481] 0.532440238 0.138904091 0.864280176 0.737657521 0.040634653 0.139074366
#> [487] 0.877602804 0.613041278 0.465792903 0.752820802 0.342730975 0.898036774
#> [493] 0.944399880 0.876315820 0.129048174 0.381745048 0.653883230 0.984195678
#> [499] 0.150487744 0.940529346
#> 
#> $ecdf_suits
#>   [1] 0.838 0.420 0.080 0.322 0.648 0.966 0.206 0.772 0.914 0.794 0.570 0.446
#>  [13] 0.744 0.714 0.516 0.662 0.352 0.306 0.064 0.826 0.416 0.640 0.860 0.396
#>  [25] 0.392 0.624 0.778 0.596 0.644 0.340 0.298 0.866 0.950 0.096 0.768 0.900
#>  [37] 0.402 0.370 0.242 0.294 0.702 0.562 0.188 0.286 0.102 0.182 0.508 0.204
#>  [49] 0.164 0.544 0.378 0.944 0.308 0.272 0.532 0.606 0.318 0.740 0.698 0.082
#>  [61] 0.752 0.274 0.536 0.350 0.218 0.696 0.620 0.896 0.586 0.094 0.722 0.114
#>  [73] 0.870 0.266 0.360 0.018 0.912 0.124 0.524 0.922 0.522 0.060 0.460 0.796
#>  [85] 0.680 0.832 0.000 0.372 0.228 0.500 0.810 0.436 0.418 0.902 0.630 0.528
#>  [97] 0.220 0.232 0.994 0.828 0.312 0.004 0.466 0.822 0.132 0.258 0.192 0.106
#> [109] 0.964 0.024 0.128 0.334 0.142 0.328 0.090 0.676 0.848 0.732 0.992 0.366
#> [121] 0.976 0.042 0.216 0.886 0.540 0.108 0.728 0.032 0.336 0.938 0.904 0.442
#> [133] 0.820 0.840 0.484 0.602 0.002 0.130 0.028 0.858 0.960 0.158 0.658 0.824
#> [145] 0.444 0.012 0.384 0.700 0.068 0.054 0.070 0.790 0.554 0.802 0.036 0.406
#> [157] 0.256 0.526 0.428 0.862 0.412 0.314 0.434 0.200 0.214 0.882 0.236 0.598
#> [169] 0.582 0.332 0.422 0.558 0.808 0.202 0.666 0.988 0.812 0.472 0.118 0.166
#> [181] 0.542 0.924 0.504 0.030 0.730 0.852 0.800 0.748 0.234 0.736 0.990 0.382
#> [193] 0.884 0.510 0.072 0.230 0.008 0.282 0.292 0.816 0.646 0.572 0.172 0.686
#> [205] 0.574 0.388 0.288 0.400 0.056 0.226 0.986 0.040 0.946 0.664 0.140 0.936
#> [217] 0.928 0.974 0.022 0.116 0.506 0.152 0.452 0.344 0.962 0.176 0.290 0.390
#> [229] 0.020 0.278 0.450 0.844 0.342 0.910 0.668 0.830 0.268 0.368 0.552 0.534
#> [241] 0.498 0.338 0.058 0.086 0.014 0.238 0.112 0.134 0.568 0.834 0.480 0.898
#> [253] 0.010 0.604 0.764 0.890 0.454 0.750 0.978 0.430 0.996 0.362 0.162 0.806
#> [265] 0.006 0.998 0.296 0.110 0.708 0.706 0.780 0.440 0.632 0.262 0.254 0.538
#> [277] 0.034 0.594 0.636 0.782 0.474 0.144 0.066 0.212 0.248 0.180 0.374 0.818
#> [289] 0.690 0.464 0.520 0.316 0.260 0.462 0.610 0.264 0.710 0.968 0.614 0.784
#> [301] 0.758 0.280 0.642 0.136 0.876 0.530 0.980 0.612 0.608 0.088 0.798 0.104
#> [313] 0.408 0.716 0.934 0.814 0.478 0.756 0.916 0.348 0.760 0.932 0.432 0.678
#> [325] 0.078 0.208 0.556 0.622 0.126 0.616 0.958 0.438 0.456 0.872 0.918 0.198
#> [337] 0.186 0.638 0.170 0.712 0.682 0.404 0.776 0.956 0.240 0.592 0.330 0.940
#> [349] 0.354 0.276 0.196 0.628 0.738 0.550 0.518 0.284 0.076 0.734 0.210 0.804
#> [361] 0.650 0.380 0.942 0.770 0.252 0.984 0.300 0.850 0.908 0.660 0.762 0.584
#> [373] 0.178 0.150 0.492 0.424 0.792 0.184 0.512 0.888 0.470 0.426 0.694 0.496
#> [385] 0.244 0.168 0.926 0.656 0.692 0.414 0.048 0.376 0.488 0.578 0.774 0.634
#> [397] 0.726 0.564 0.502 0.190 0.486 0.930 0.074 0.358 0.874 0.410 0.652 0.120
#> [409] 0.514 0.672 0.920 0.156 0.580 0.084 0.590 0.786 0.892 0.092 0.746 0.100
#> [421] 0.224 0.098 0.326 0.448 0.468 0.346 0.972 0.394 0.194 0.788 0.754 0.674
#> [433] 0.548 0.310 0.588 0.576 0.476 0.154 0.970 0.720 0.052 0.952 0.718 0.250
#> [445] 0.046 0.246 0.836 0.560 0.864 0.600 0.026 0.856 0.386 0.016 0.894 0.320
#> [457] 0.684 0.222 0.062 0.566 0.846 0.364 0.122 0.304 0.854 0.038 0.724 0.842
#> [469] 0.704 0.174 0.458 0.494 0.324 0.270 0.482 0.626 0.044 0.654 0.302 0.688
#> [481] 0.546 0.146 0.868 0.742 0.050 0.148 0.880 0.618 0.490 0.766 0.356 0.906
#> [493] 0.954 0.878 0.138 0.398 0.670 0.982 0.160 0.948
#> 
#> $mahal_dists
#>   [1]  0.346185141  1.828461778  5.056572873  2.313934860  0.908262293
#>   [6]  0.087687246  3.233294951  0.553189636  0.195496906  0.512112447
#>  [11]  1.170929513  1.711885229  0.598548562  0.731109181  1.368995410
#>  [16]  0.868030186  2.164634052  2.397678899  5.659026013  0.372197175
#>  [21]  1.847382498  0.926176059  0.302102438  1.945518680  1.965783410
#>  [26]  0.965259262  0.545010924  1.068957285  0.920366795  2.255357826
#>  [31]  2.436837294  0.292329788  0.121915116  4.698637893  0.566307469
#>  [36]  0.227893038  1.919511340  2.086016369  2.905313262  2.445117094
#>  [41]  0.778403437  1.204556150  3.433684439  2.512601530  4.612181021
#>  [46]  3.480311442  1.417516783  3.276663809  3.695253664  1.271925776
#>  [51]  2.004220491  0.130377313  2.387656843  2.598600900  1.314321757
#>  [56]  1.043529444  2.318941544  0.613758213  0.784161657  5.036460743
#>  [61]  0.588754333  2.586154835  1.287920641  2.171694558  3.129064410
#>  [66]  0.789628136  0.973057130  0.231732071  1.111524979  4.701985097
#>  [71]  0.684305448  4.442581203  0.287531000  2.659617569  2.139149251
#>  [76]  8.023658949  0.198498779  4.272125385  1.336357742  0.180038191
#>  [81]  1.353634602  5.877678793  1.666295757  0.487941312  0.827665209
#>  [86]  0.364408647 11.703287864  2.051543167  2.980880406  1.459337232
#>  [91]  0.418910567  1.729405552  1.845923452  0.221517864  0.943576296
#>  [96]  1.327868034  3.106533146  2.964367466  0.005326730  0.371308124
#> [101]  2.336370939 10.509942979  1.646953456  0.374226118  4.164964418
#> [106]  2.763362230  3.377624027  4.524272833  0.087966129  7.519733180
#> [111]  4.239591902  2.268501282  4.022259573  2.302642205  4.932693597
#> [116]  0.843329518  0.332804447  0.648503436  0.007623386  2.115281714
#> [121]  0.063928014  6.645331256  3.130363764  0.257156175  1.279844585
#> [126]  4.484132862  0.653161021  7.195911608  2.267125388  0.152804667
#> [131]  0.219872248  1.720983246  0.378855607  0.342813770  1.571064079
#> [136]  1.052381221 10.746761000  4.236330087  7.408459347  0.319373639
#> [141]  0.092130517  3.855423551  0.876881414  0.373901010  1.715437878
#> [146]  8.731712231  1.989688687  0.779635349  5.537118943  6.092677691
#> [151]  5.523173913  0.518241320  1.242474685  0.462290820  7.003991510
#> [156]  1.910475406  2.763562845  1.332758654  1.768812351  0.294633132
#> [161]  1.878550688  2.334176864  1.731680956  3.306445858  3.153768133
#> [166]  0.259371987  2.945555785  1.067923732  1.127849017  2.274649547
#> [171]  1.796217157  1.225328821  0.439544421  3.305982730  0.866713982
#> [176]  0.015538603  0.411388181  1.599120923  4.404266178  3.681020560
#> [181]  1.279097426  0.178084178  1.432226026  7.229799911  0.648996614
#> [186]  0.324407395  0.470647395  0.590834585  2.961957158  0.641537385
#> [191]  0.015439590  1.992057718  0.257491226  1.398361155  5.479076667
#> [196]  2.976922400 10.067141307  2.535951488  2.458561957  0.395784530
#> [201]  0.917868127  1.153957201  3.597257792  0.808210332  1.151363517
#> [206]  1.968271084  2.489925326  1.919555917  6.077162231  3.003482090
#> [211]  0.020949714  6.720089586  0.123744954  0.867646268  4.071657402
#> [216]  0.153993578  0.165643163  0.074554660  7.772076486  4.413317049
#> [221]  1.431155420  3.901830731  1.695413523  2.226570611  0.091265760
#> [226]  3.503507362  2.465389049  1.966256794  7.784306433  2.548019805
#> [231]  1.699775826  0.333158484  2.255158259  0.200573784  0.859814099
#> [236]  0.365481419  2.642454574  2.101531193  1.242808063  1.309229771
#> [241]  1.478388000  2.258133837  5.981282183  4.966242261  8.642018409
#> [246]  2.942742863  4.473194514  4.128672488  1.171494480  0.359181066
#> [251]  1.589143835  0.229179138  9.157477172  1.045912340  0.568596833
#> [256]  0.251498567  1.686021355  0.590648247  0.063096627  1.765336321
#> [261]  0.004963852  2.138015441  3.785039283  0.448303563 10.185321470
#> [266]  0.002910825  2.439719228  4.475037730  0.746979887  0.748523161
#> [271]  0.538804643  1.722261385  0.938005796  2.704921786  2.789435853
#> [276]  1.284088070  7.109466487  1.069358341  0.933048796  0.536845270
#> [281]  1.591799141  3.983239533  5.543328009  3.196175329  2.841761407
#> [286]  3.490351801  2.046510097  0.380730833  0.804774352  1.653523295
#> [291]  1.354459060  2.319226652  2.728386487  1.654152982  1.018384047
#> [296]  2.694580241  0.740510789  0.086701708  0.998346162  0.533574707
#> [301]  0.572073564  2.540288969  0.925795681  4.113916379  0.273613598
#> [306]  1.321755692  0.039260968  1.018321352  1.022673712  4.933609056
#> [311]  0.481417831  4.544628236  1.909081326  0.719564472  0.155749820
#> [316]  0.408876663  1.590723294  0.572238830  0.194313715  2.202347736
#> [321]  0.571608943  0.156525777  1.755638803  0.837846145  5.148787713
#> [326]  3.219844782  1.228166983  0.967067202  4.247179321  0.993721099
#> [331]  0.098869321  1.722799749  1.685249816  0.277948868  0.188060556
#> [336]  3.313608724  3.434074944  0.927853648  3.609279152  0.737998744
#> [341]  0.815831375  1.915624006  0.547643259  0.106313832  2.927634150
#> [346]  1.098336357  2.298744666  0.147291786  2.163874572  2.576338570
#> [351]  3.329167920  0.950960409  0.613956946  1.243665494  1.367820071
#> [356]  2.513015986  5.212821339  0.645890753  3.205006063  0.459007302
#> [361]  0.907891697  1.996249772  0.143825134  0.558309564  2.799862236
#> [366]  0.024440003  2.418938173  0.330860215  0.213280962  0.874671816
#> [371]  0.570042634  1.117456478  3.497955279  3.904084155  1.522362258
#> [376]  1.791870447  0.516402234  3.441269169  1.391371397  0.252154064
#> [381]  1.616351555  1.779584300  0.790679260  1.490571311  2.860699082
#> [386]  3.640554629  0.173898185  0.884364371  0.790690746  1.854520561
#> [391]  6.419305777  2.021353402  1.538624016  1.139068291  0.551250603
#> [396]  0.937231494  0.676182377  1.195720819  1.434574344  3.409994369
#> [401]  1.539764068  0.164821668  5.371071682  2.139903606  0.275542563
#> [406]  1.883484725  0.900262052  4.302172239  1.378591660  0.848396276
#> [411]  0.184057386  3.883138807  1.136902368  4.971790961  1.101844838
#> [416]  0.526045340  0.244153482  4.816161369  0.598474987  4.626473851
#> [421]  3.024330938  4.647651082  2.307218788  1.704964983  1.625098827
#> [426]  2.209683086  0.084121229  1.954928004  3.343122289  0.520903516
#> [431]  0.581939201  0.845529637  1.250997771  2.352370737  1.102358466
#> [436]  1.144120378  1.590761039  3.887813145  0.084920218  0.685293417
#> [441]  6.289751445  0.120361787  0.707206730  2.815056854  6.567137464
#> [446]  2.846260396  0.351458132  1.208908379  0.294331221  1.054423668
#> [451]  7.506632323  0.319986673  1.987250997  8.185716807  0.239267003
#> [456]  2.314719665  0.810861498  3.038705343  5.683377602  1.176674235
#> [461]  0.332868057  2.120793776  4.296506070  2.399161742  0.320715436
#> [466]  6.835336454  0.681219014  0.338971536  0.774590151  3.521573910
#> [471]  1.666743891  1.499291728  2.308242314  2.629144931  1.579992255
#> [476]  0.953056033  6.629388715  0.889984959  2.401515321  0.807140341
#> [481]  1.260569235  3.947943155  0.291716570  0.608551252  6.406268090
#> [486]  3.945492960  0.261122350  0.978646015  1.528028314  0.567856116
#> [491]  2.141618938  0.215088520  0.114411202  0.264057457  4.095138998
#> [496]  1.926004613  0.849652981  0.031861084  3.787747261  0.122624857

The returned list contains:

• corplot: correlation vs sample size between the “true” niche and both suitability transformations
• sample_data: matrix with the last sample of environmental predictors
• sample_niche, chisq_suits, ecdf_suits: suitability values
• mahal_dists: Mahalanobis distances for the last sample

You can directly plot the correlation object:

res1$corplot

Reproducing the full analysis

The convenience function run_ecdf_mahal_analysis() wraps the original workflow: it runs ecdf_niche() for several dimensions (by default 1 to 5) and produces three figures analogous to those in the script.

set.seed(3)
full_res <- run_ecdf_mahal_analysis(dims = 1:5)

full_res$figure1 |> plot()

full_res$figure2 |> plot()

full_res$figure3 |> plot()

Customizing simulations

You can customize key aspects of the simulation by passing arguments to ecdf_niche():

res_custom <- ecdf_niche(
n = 3,
n_population = 20000,
sample_sizes = seq(50, 1000, 50),
seed = 123
)

res_custom$corplot

These arguments control the dimensionality, the size of the environmental “background”, and the grid of sample sizes used to compute correlations.