Variance and eigenbasis
A gentle introduction to PCA
by Arturo Frisina
Starting dataset
Let’s start generating 3000 random datapoints normally distributed on
only two dimensions (corresponding to two variables X1 and
X2)
X1 = rnorm(1000, mean = 10, sd= 4)
X2 = rnorm(1000, mean = 0, sd=2)+(X1)
X3 = rnorm(1000, mean=10, sd=1)
df = data.frame(X1,X2,X3,row.names = c(1:1000))
kable(df) %>% kable_styling('striped',fixed_thead = T, full_width = FALSE) %>%
scroll_box( height = "400px")| X1 | X2 | X3 |
|---|---|---|
| 1.4909588 | -0.6389581 | 10.709623 |
| 8.8661404 | 8.7073640 | 11.471004 |
| 12.8810614 | 13.9365537 | 10.230943 |
| 10.8436140 | 10.4047321 | 9.134547 |
| 9.3043068 | 7.1836192 | 9.685626 |
| 7.7831814 | 6.9816489 | 10.534997 |
| 5.9432539 | 5.6614064 | 9.548322 |
| 4.9467533 | 6.9337704 | 10.902752 |
| 6.9220214 | 6.2107450 | 9.832676 |
| 10.0966817 | 9.3955005 | 9.228428 |
| 12.4121890 | 14.1294338 | 10.459287 |
| 11.2689822 | 5.0779426 | 10.395555 |
| 9.4643349 | 11.7299677 | 9.100904 |
| 7.0386780 | 9.6461749 | 11.586456 |
| 11.5538911 | 12.3370426 | 9.795076 |
| 7.3112210 | 10.5857706 | 10.961347 |
| 5.1261357 | 6.9426609 | 9.593046 |
| 6.8497627 | 8.8087887 | 11.433069 |
| 12.4774752 | 9.7037982 | 11.447245 |
| 10.9943912 | 10.4998876 | 7.901226 |
| 4.4965188 | 6.2414803 | 7.924243 |
| 8.9918887 | 8.3513003 | 10.872180 |
| 3.3979956 | 5.2446425 | 10.448392 |
| 7.1527506 | 7.6706340 | 9.897177 |
| 14.1351298 | 12.3260921 | 9.753251 |
| 4.9257138 | 4.9702961 | 10.902371 |
| 9.8736833 | 10.6587654 | 9.534753 |
| 6.0530397 | 4.8881948 | 9.313907 |
| 8.4598991 | 5.1527676 | 10.094032 |
| 10.9090450 | 12.2330325 | 10.046435 |
| 6.2581585 | 8.7584469 | 8.264087 |
| 6.4994857 | 6.4202630 | 8.492995 |
| 10.0170744 | 11.8054749 | 10.059009 |
| 11.4093763 | 14.2107900 | 8.638742 |
| 11.6994672 | 10.8767603 | 10.323034 |
| 10.4783430 | 13.0698156 | 8.991778 |
| 7.0820853 | 8.1625625 | 10.324515 |
| 16.5773473 | 13.4558469 | 10.618150 |
| 5.9651965 | 3.8706449 | 12.286403 |
| 10.8336376 | 12.1097501 | 11.698716 |
| 3.5466281 | 4.3594412 | 9.074896 |
| 10.9875047 | 12.7873971 | 8.557177 |
| 4.8694575 | 5.5827304 | 9.494495 |
| 12.1088881 | 12.4950181 | 10.430490 |
| 5.4460932 | 4.4460587 | 11.124682 |
| 12.6399700 | 8.6359556 | 10.931346 |
| 10.6266983 | 10.1012932 | 8.969173 |
| 14.0036475 | 11.8254686 | 9.443371 |
| 10.3046093 | 9.5407052 | 8.283380 |
| 3.1448992 | 1.6492712 | 9.966247 |
| 13.9960147 | 15.0436850 | 9.558890 |
| 14.0059720 | 14.2783639 | 10.294165 |
| 7.5234001 | 7.1477796 | 11.178862 |
| 8.8869599 | 11.5652081 | 11.188987 |
| 5.3467236 | 3.5777234 | 10.997024 |
| 8.7482258 | 6.8120298 | 9.574035 |
| 20.0308595 | 20.7585669 | 9.649527 |
| 7.6233408 | 8.9479791 | 11.181825 |
| 16.5969767 | 15.2837727 | 9.325711 |
| 6.5026075 | 2.8191043 | 11.673504 |
| 6.3223045 | 7.8028827 | 10.024004 |
| 7.2293570 | 5.1479968 | 8.798702 |
| 6.7188751 | 7.9843427 | 8.550795 |
| 15.9219684 | 17.0831428 | 8.891647 |
| 13.0538399 | 14.2979386 | 11.247331 |
| 2.4050958 | -0.4557150 | 9.142496 |
| 21.0122571 | 21.9707826 | 10.191486 |
| 15.3259107 | 14.9541510 | 10.780581 |
| 12.0080748 | 9.3653300 | 9.400608 |
| 9.4182007 | 8.4712807 | 7.802521 |
| 8.8539972 | 6.0909222 | 10.498696 |
| 9.2389550 | 9.9140137 | 10.637811 |
| 8.3020347 | 6.8674215 | 9.464821 |
| 11.5014216 | 12.8381791 | 10.894521 |
| 4.5452927 | 3.5506048 | 10.616071 |
| 5.2134867 | 4.6052357 | 11.259886 |
| 10.0111845 | 10.4340597 | 10.201728 |
| 11.8338853 | 9.3582959 | 9.901110 |
| 8.0586383 | 7.3173203 | 10.400552 |
| 11.3506444 | 11.6774648 | 9.489989 |
| 14.1084819 | 11.7230324 | 11.683733 |
| 11.2328742 | 8.9207590 | 9.780919 |
| 4.7061705 | 5.9708429 | 11.263618 |
| 2.6301488 | 0.7224523 | 9.862005 |
| 16.5962359 | 14.6412097 | 10.160043 |
| 6.3489102 | 6.6656994 | 11.173606 |
| 11.7250973 | 16.1332336 | 10.329215 |
| 12.7088167 | 13.1810042 | 9.688160 |
| 3.2865816 | 4.4935483 | 9.063980 |
| 9.9365339 | 10.4496779 | 11.439412 |
| 9.8318150 | 8.8957332 | 12.059693 |
| 5.3576888 | 3.7747028 | 9.979358 |
| 10.2844123 | 10.5753169 | 11.202658 |
| 17.6358096 | 17.9537155 | 9.327340 |
| 4.4576912 | 6.3941086 | 10.551267 |
| 6.5038644 | 2.1277925 | 10.366587 |
| 12.4544011 | 10.3089511 | 11.384161 |
| 2.4820515 | 1.5670534 | 10.338174 |
| 12.5702740 | 10.4874465 | 9.735121 |
| 13.7963063 | 11.3761341 | 10.372152 |
| 2.5378127 | 1.8939610 | 10.896983 |
| 10.4261673 | 10.5488772 | 9.149140 |
| 15.0067867 | 14.7077756 | 8.944410 |
| 12.9340796 | 14.9261367 | 10.486089 |
| 13.6311816 | 10.3414719 | 11.948300 |
| 10.7158885 | 9.6252148 | 10.378235 |
| 5.9417729 | 7.8296579 | 10.618738 |
| 9.8726225 | 11.0361835 | 10.291287 |
| 12.7819617 | 14.7604046 | 8.124759 |
| 16.2812170 | 15.1102968 | 10.529285 |
| 13.1698850 | 16.9927275 | 9.173697 |
| 12.6480701 | 11.1893071 | 11.426168 |
| 7.6525087 | 7.8726254 | 9.413732 |
| 10.5413378 | 12.2544557 | 9.464386 |
| 11.7023837 | 10.8324938 | 9.322382 |
| 7.3165911 | 5.2229696 | 10.944692 |
| 8.9879119 | 7.4639354 | 9.523481 |
| 6.4819762 | 7.9459722 | 10.001118 |
| 10.7661187 | 13.0074908 | 9.895431 |
| 9.3386948 | 8.9099625 | 9.804283 |
| 11.7414185 | 11.0650301 | 10.456464 |
| 11.0225344 | 12.4816370 | 10.056040 |
| 13.5548612 | 12.8439916 | 12.886254 |
| 12.5593890 | 14.9199990 | 9.499501 |
| 10.4420280 | 10.0675517 | 8.602741 |
| 10.4072273 | 8.4694782 | 10.267263 |
| 8.2949404 | 8.4800364 | 10.527556 |
| 6.8263458 | 8.7768736 | 11.935713 |
| 7.0673012 | 4.6355924 | 9.642320 |
| 14.5597353 | 15.7830026 | 9.035205 |
| 13.3678756 | 13.1185655 | 8.500535 |
| 13.5968520 | 13.4477081 | 9.859218 |
| 13.5791810 | 15.1602622 | 8.183574 |
| 15.1131045 | 12.5315531 | 10.327759 |
| 8.0699674 | 9.5698615 | 11.263011 |
| 15.5654468 | 12.6361911 | 9.668805 |
| 10.6585288 | 9.5913091 | 10.321656 |
| -1.7623321 | -2.6885333 | 10.644788 |
| 6.6256828 | 6.6346638 | 8.943923 |
| 16.4577011 | 18.5111446 | 10.228546 |
| 7.2615853 | 6.3607463 | 11.360006 |
| 11.8344016 | 10.2697098 | 10.343649 |
| 9.2685793 | 12.5521327 | 10.101639 |
| 12.9913444 | 14.0470024 | 8.420625 |
| 13.6544885 | 13.9611619 | 9.722917 |
| 10.1000842 | 12.8595163 | 8.758705 |
| 6.7970089 | 5.5964919 | 9.762397 |
| 12.1246336 | 15.1714444 | 9.012775 |
| 11.1275522 | 11.1767525 | 11.208538 |
| 9.7055173 | 9.3426581 | 9.824741 |
| 8.8605399 | 9.8232996 | 10.949301 |
| 12.2457325 | 12.5387897 | 8.690040 |
| 11.1844895 | 10.6940730 | 10.694741 |
| 6.5973983 | 4.6187643 | 10.218819 |
| 8.2842140 | 10.1601260 | 9.297192 |
| 14.2366549 | 14.9039233 | 10.223940 |
| 7.3547605 | 10.9435669 | 9.421317 |
| 14.2569717 | 15.2011335 | 10.501986 |
| 6.8963229 | 5.0632032 | 11.619684 |
| 5.8955844 | 4.7472728 | 8.202959 |
| 11.2954381 | 12.4799114 | 9.851451 |
| 9.8559129 | 9.0513027 | 10.857762 |
| 9.5701946 | 8.5290857 | 10.458161 |
| 4.9963424 | 4.9703352 | 9.215633 |
| 11.3236917 | 15.4031462 | 8.320276 |
| 10.6123034 | 12.9368071 | 10.377056 |
| 0.1854947 | -2.4343085 | 9.990988 |
| 16.4187796 | 16.8815238 | 10.192109 |
| 11.0892414 | 8.5276548 | 10.215945 |
| 7.7017689 | 5.4540864 | 8.276279 |
| 13.1817135 | 16.9491096 | 10.454233 |
| 12.5966882 | 11.8824754 | 9.995958 |
| 4.9556644 | 3.4791608 | 9.915157 |
| 10.7866145 | 13.2826106 | 9.488736 |
| 9.1729844 | 10.7706718 | 10.042710 |
| 10.9721223 | 10.6800096 | 9.690614 |
| 9.7222145 | 10.4013347 | 9.245404 |
| 9.0730359 | 9.5853178 | 9.207158 |
| 11.2511586 | 13.5697558 | 8.395999 |
| 11.4092026 | 11.9132083 | 11.339951 |
| 8.4244610 | 7.2027173 | 10.330524 |
| 18.2339749 | 20.8640154 | 10.899649 |
| 12.2595600 | 8.8948648 | 11.900167 |
| 12.6258066 | 18.7156319 | 9.782942 |
| 10.0295910 | 11.5950886 | 9.476076 |
| 15.4093993 | 14.5973649 | 11.824482 |
| 8.0870112 | 11.3395974 | 9.950527 |
| 8.4601872 | 8.8085177 | 9.919723 |
| 2.6416853 | 3.2850831 | 10.801697 |
| 3.2831628 | 2.8528549 | 11.590493 |
| 7.9189650 | 10.3065768 | 12.294019 |
| 7.8938442 | 9.6345388 | 9.576934 |
| 11.4693911 | 13.8729497 | 10.105551 |
| 0.8085023 | 3.9114011 | 10.200695 |
| 9.9768068 | 12.4765841 | 9.705118 |
| 0.3085731 | 0.2969684 | 9.287101 |
| 12.9178805 | 14.5372039 | 8.192500 |
| 13.2069053 | 9.6456828 | 8.572788 |
| 9.7399495 | 12.8862305 | 12.466873 |
| 13.4628285 | 11.1574667 | 9.692570 |
| 5.2828637 | 4.6840917 | 9.236272 |
| 14.5609259 | 11.3769924 | 9.898125 |
| 11.8260500 | 12.1553196 | 10.445208 |
| 5.3535655 | 6.1804934 | 9.737553 |
| 8.1293321 | 9.8190953 | 11.072710 |
| 13.1230567 | 11.2567747 | 8.874631 |
| 11.9256471 | 9.9474865 | 11.134224 |
| 4.3935609 | 2.8045371 | 9.551009 |
| 13.9456545 | 14.6782531 | 8.241109 |
| 12.0952942 | 12.8867354 | 8.971018 |
| 8.7810405 | 10.1403737 | 11.460629 |
| 7.7005010 | 8.7188233 | 9.835289 |
| 6.5979039 | 6.2458473 | 9.708072 |
| 8.1711484 | 10.6385544 | 10.383178 |
| 14.7762449 | 17.1101180 | 10.767952 |
| 7.5643191 | 3.9865081 | 12.000453 |
| 9.6689115 | 9.8836355 | 9.512696 |
| 7.7844008 | 7.3445180 | 10.726642 |
| 10.2130084 | 11.0846823 | 8.503927 |
| 13.7941021 | 14.0264297 | 9.444221 |
| 6.8384716 | 7.2292029 | 11.865453 |
| 12.7325645 | 12.5406209 | 8.530679 |
| 17.4025693 | 15.1903116 | 10.397487 |
| 1.2946229 | 2.8099226 | 10.326512 |
| 10.0009617 | 14.0555087 | 10.531557 |
| 19.9286933 | 19.8734435 | 10.947919 |
| 11.9545532 | 9.2160180 | 9.719627 |
| 6.3394771 | 2.1131538 | 8.914133 |
| 12.8073326 | 12.4685143 | 9.687574 |
| 11.5282558 | 13.1362970 | 10.550027 |
| 9.8675326 | 8.1083441 | 7.747620 |
| 10.2766291 | 11.3762311 | 9.059866 |
| 12.5562877 | 12.6791600 | 10.695300 |
| 15.1719131 | 16.4641763 | 7.789139 |
| 14.9580789 | 16.2278999 | 10.135527 |
| 12.3720171 | 14.4301733 | 11.723879 |
| 14.6403840 | 14.5581490 | 7.574319 |
| 5.1895017 | 2.2783536 | 8.687378 |
| 11.5970394 | 10.4846804 | 9.129014 |
| 9.4807550 | 10.9758987 | 10.368493 |
| 10.0818765 | 12.0423007 | 9.965605 |
| 5.7537586 | 8.0543660 | 10.410221 |
| 8.4866732 | 8.2523409 | 10.615868 |
| 6.3709784 | 6.9684358 | 10.982509 |
| 10.7889274 | 12.4358750 | 8.713812 |
| 5.0630675 | 6.2063873 | 10.251944 |
| 10.7137423 | 13.6868886 | 9.997985 |
| 15.9410372 | 17.7501818 | 9.986724 |
| 7.8161134 | 7.5564794 | 10.162106 |
| 8.8381324 | 7.2981224 | 11.580652 |
| 8.4990172 | 6.8662854 | 9.756463 |
| 8.5823784 | 13.3774871 | 11.372020 |
| 8.4878185 | 9.5060824 | 10.779898 |
| 8.5626441 | 8.7062717 | 9.038320 |
| 8.0450959 | 9.1041841 | 10.298579 |
| 11.7780150 | 12.7569069 | 8.191127 |
| 2.2494483 | 5.8489281 | 11.776350 |
| 9.9003875 | 11.3421366 | 10.030533 |
| 8.5025373 | 5.5620334 | 10.411976 |
| 7.2024426 | 7.8172858 | 10.432225 |
| 14.7852983 | 15.2080015 | 8.246940 |
| 9.1955790 | 5.6602506 | 9.649601 |
| 14.7178051 | 14.1299545 | 9.194136 |
| 14.4455234 | 14.9265762 | 9.146901 |
| 9.1829883 | 7.4320508 | 10.435381 |
| 6.4781968 | 6.6775380 | 9.357885 |
| 8.9800930 | 5.6895690 | 9.661573 |
| 6.3216736 | 4.4188718 | 9.206619 |
| 1.3976719 | 2.7535171 | 9.012864 |
| 8.9680626 | 9.5962201 | 10.875255 |
| 11.1935545 | 12.4938880 | 10.257062 |
| 14.4127318 | 16.1755010 | 9.815529 |
| 12.6654139 | 12.4544808 | 9.620384 |
| 9.9720800 | 8.7054804 | 9.591905 |
| 12.7510585 | 9.1212718 | 12.247961 |
| 14.0991803 | 16.8484847 | 9.974698 |
| 9.9028728 | 12.2206389 | 10.403971 |
| 10.1996181 | 8.4917620 | 10.357818 |
| 9.9203603 | 10.3338728 | 12.303684 |
| 13.5586467 | 14.7056227 | 9.128771 |
| 14.7231170 | 14.7502141 | 9.111765 |
| 5.7739550 | 9.1119584 | 10.157638 |
| 4.7065563 | 4.3814860 | 10.808025 |
| 7.3163130 | 8.6416065 | 10.144435 |
| 14.0462965 | 13.0337968 | 8.665644 |
| 14.8913041 | 17.2163456 | 9.869654 |
| 5.7815287 | 5.6640317 | 11.484240 |
| 8.5076194 | 5.0386328 | 10.110891 |
| 17.8249134 | 15.5924434 | 8.865232 |
| 11.2355553 | 12.3202705 | 10.607450 |
| 2.3296314 | 1.7359762 | 12.110265 |
| 15.0895733 | 15.1502108 | 11.644574 |
| 10.3769009 | 12.3285001 | 9.075187 |
| 17.6077824 | 18.7252516 | 9.540314 |
| 10.8046739 | 10.7162424 | 9.675367 |
| 3.9969171 | 3.2915792 | 10.177407 |
| 12.4805690 | 12.2953485 | 9.185658 |
| 8.7937238 | 9.6133852 | 10.779702 |
| 3.1725327 | 1.2038426 | 10.920082 |
| 7.0122328 | 8.0376016 | 10.225442 |
| 11.6452210 | 9.6016604 | 10.384878 |
| 8.7959630 | 11.0551107 | 9.348979 |
| 6.1664505 | 5.8040434 | 11.839659 |
| 12.6169968 | 10.8821370 | 10.854780 |
| 6.1469767 | 5.9204724 | 10.307662 |
| 16.8747266 | 17.9829690 | 10.406625 |
| 13.0564072 | 12.1983541 | 9.993719 |
| 11.7102068 | 9.1151820 | 8.582378 |
| 12.5902543 | 12.3257706 | 10.039596 |
| 11.1780205 | 8.1459869 | 9.682916 |
| 8.2872343 | 7.3171051 | 8.501763 |
| 11.1240363 | 12.2193122 | 10.306861 |
| 6.8775600 | 6.9591520 | 9.578339 |
| 13.9332123 | 11.8323760 | 9.120768 |
| 3.3314235 | 1.6711157 | 10.011087 |
| 17.5834818 | 16.9725293 | 10.011633 |
| 10.0115978 | 8.8147448 | 9.598892 |
| 9.2917059 | 6.6469303 | 9.909386 |
| 4.6115207 | 1.8090884 | 10.694811 |
| -2.9264863 | 0.6728550 | 10.858056 |
| 12.7580745 | 12.0626759 | 9.428239 |
| 12.5729447 | 12.5655701 | 9.595496 |
| 5.4047812 | 8.8741360 | 11.067021 |
| 8.4826386 | 11.3767256 | 8.776779 |
| 7.8409668 | 6.1243952 | 11.500371 |
| 7.2323805 | 7.7031171 | 9.563275 |
| 6.3539028 | 7.9391895 | 8.595427 |
| 11.8727890 | 10.6035395 | 10.201974 |
| 7.1845153 | 7.0112698 | 9.562410 |
| 7.9285714 | 9.7785169 | 10.504202 |
| 11.2493335 | 11.3372270 | 9.561984 |
| 9.9874005 | 9.5507564 | 9.428083 |
| 13.1646745 | 12.5752892 | 10.549372 |
| 12.5282703 | 11.2988593 | 8.851055 |
| 7.5209958 | 9.1446738 | 9.914931 |
| 16.5325636 | 17.9159901 | 9.272659 |
| 9.9230198 | 9.6732827 | 10.086483 |
| 15.5167564 | 15.9806234 | 12.098173 |
| 10.8454231 | 10.7539537 | 9.910651 |
| 13.3609349 | 14.4750878 | 10.106438 |
| 4.8686300 | 7.1695349 | 9.183227 |
| 5.1981263 | 5.0528831 | 9.499130 |
| 17.0072423 | 18.7211063 | 8.887308 |
| 10.0173185 | 7.2743077 | 10.613763 |
| 6.0731280 | 5.3571618 | 11.845741 |
| 1.3724907 | 2.9522721 | 9.455200 |
| 12.7995730 | 13.8660168 | 9.119985 |
| 7.4944906 | 8.5176280 | 9.181150 |
| 5.2620616 | 5.8083663 | 8.979508 |
| 11.7879006 | 11.2503164 | 10.144148 |
| 11.5641651 | 16.3782257 | 9.255743 |
| 14.1828897 | 15.2902096 | 11.729578 |
| 6.3705197 | 7.3245531 | 10.375376 |
| 11.4873093 | 9.6368661 | 11.422506 |
| 14.0494779 | 12.8203135 | 9.289360 |
| 10.6037045 | 9.6027391 | 8.834974 |
| 7.2335496 | 2.9340784 | 6.857445 |
| 3.5354565 | 0.2442785 | 10.237863 |
| 8.6074760 | 9.5347559 | 10.967982 |
| 8.8138879 | 8.0032375 | 10.123572 |
| 3.8530878 | 3.7341892 | 11.769426 |
| 13.9268255 | 13.8206090 | 7.456638 |
| 12.9287883 | 12.4149589 | 10.959235 |
| 15.3888167 | 15.8623315 | 10.750744 |
| 8.0470718 | 9.6597781 | 9.507318 |
| 9.6016789 | 13.3277378 | 8.314076 |
| 6.5547550 | 7.1065502 | 11.547911 |
| 16.3369591 | 19.2800507 | 9.973123 |
| 14.3676842 | 16.3082762 | 9.886447 |
| 8.0391921 | 6.2309962 | 9.305704 |
| 10.4061073 | 10.6205888 | 9.318768 |
| 9.3509098 | 8.6716802 | 12.455845 |
| 4.2677384 | 5.6855050 | 9.427536 |
| 5.7867074 | 8.0771220 | 10.207116 |
| 16.3404414 | 14.0945225 | 9.344262 |
| 18.1259525 | 16.8037281 | 10.915700 |
| 9.7308076 | 5.9339257 | 12.156376 |
| 3.2771086 | 3.7023046 | 11.447219 |
| 14.8630298 | 16.6878353 | 9.642906 |
| 19.5118591 | 17.7183275 | 11.969430 |
| 1.5286825 | 0.4102832 | 9.067825 |
| 10.5911938 | 9.9156709 | 8.255581 |
| 12.6183008 | 14.7537696 | 9.163429 |
| 11.4206239 | 9.7835261 | 9.201572 |
| 15.9564136 | 12.7564407 | 9.440954 |
| 8.4072554 | 4.8940080 | 8.862370 |
| 16.6060400 | 16.5010767 | 10.569273 |
| 11.2428629 | 11.0746208 | 11.153761 |
| 10.4838057 | 11.6114824 | 11.140848 |
| 5.9520057 | 5.6115120 | 10.057621 |
| 12.4604024 | 11.6136474 | 10.016933 |
| 5.4602091 | 3.1574651 | 10.228950 |
| 9.8216121 | 9.5085669 | 8.957423 |
| 8.2091080 | 10.2567281 | 9.608897 |
| 11.3436768 | 8.8590613 | 11.125772 |
| 7.9214462 | 5.4767545 | 10.701387 |
| 11.9909710 | 10.7464616 | 8.966577 |
| 13.9933130 | 12.0850625 | 8.878817 |
| 7.3213722 | 8.0925016 | 9.771880 |
| 9.3614268 | 9.0959317 | 10.929376 |
| 5.0647628 | 7.9310951 | 9.703585 |
| 8.3063392 | 7.3119631 | 9.611049 |
| 10.7084072 | 9.7870336 | 9.777873 |
| 10.3932704 | 7.5709582 | 9.106032 |
| 5.5136333 | 5.9192315 | 10.003073 |
| 5.2181726 | 3.6879819 | 11.527813 |
| 11.3855968 | 13.2069677 | 10.484561 |
| 12.8818658 | 11.5969697 | 8.810835 |
| 10.3928456 | 11.4484758 | 10.963827 |
| 14.1361725 | 14.5115058 | 10.334878 |
| 7.0293344 | 6.2957795 | 11.527066 |
| 8.3738456 | 9.7900231 | 9.210700 |
| 3.0794275 | 3.8455750 | 11.698528 |
| 8.9037923 | 9.9428110 | 10.129973 |
| 9.4216478 | 8.7615355 | 8.535801 |
| 4.5298293 | 3.1194673 | 9.684449 |
| 6.6845007 | 7.3301144 | 10.504980 |
| 9.8543423 | 9.7871248 | 9.508397 |
| 13.7405319 | 11.1827358 | 9.896803 |
| 13.6151072 | 14.1411871 | 10.564578 |
| 7.5773697 | 9.7551868 | 9.523077 |
| 14.9378410 | 16.9406503 | 8.828520 |
| 12.4805825 | 12.8016492 | 10.603405 |
| 9.1121528 | 9.7643436 | 9.612765 |
| 7.0660231 | 9.6610732 | 9.908298 |
| 8.7006530 | 9.3053378 | 9.322445 |
| 11.7138662 | 11.7088805 | 9.011800 |
| 11.4330197 | 11.7370351 | 7.938000 |
| 10.1183403 | 10.5528153 | 10.069685 |
| 15.2275900 | 14.5448767 | 11.029003 |
| 13.5160217 | 10.1701531 | 10.317861 |
| 7.6224997 | 6.8102443 | 9.021301 |
| 11.6281142 | 12.9435895 | 9.772408 |
| 11.0856057 | 10.1234794 | 8.573417 |
| 11.7365220 | 12.5720986 | 9.952327 |
| 4.8120286 | 5.7664922 | 10.396878 |
| 14.8295934 | 15.2347417 | 9.632205 |
| 9.6942207 | 12.4584050 | 10.221941 |
| 10.3910310 | 10.5765957 | 9.865824 |
| 5.4375446 | 4.9474808 | 12.465346 |
| 15.0056124 | 12.8817388 | 9.858408 |
| 14.6460361 | 15.5888525 | 11.890997 |
| 9.2422320 | 7.3157614 | 9.374597 |
| 9.5746904 | 9.2969703 | 10.503386 |
| 10.0286504 | 7.0699152 | 9.722644 |
| 13.0126865 | 14.6897440 | 9.148162 |
| 13.3431004 | 14.3644744 | 10.183553 |
| 18.1583183 | 16.7296558 | 9.132059 |
| 7.5289871 | 8.2693828 | 9.820781 |
| 11.1632339 | 14.6705248 | 8.036066 |
| 12.7452491 | 12.9101322 | 10.834834 |
| 3.8656109 | 4.9183552 | 10.895645 |
| 7.9031401 | 6.1436692 | 11.949807 |
| 16.5979389 | 17.1271733 | 10.279290 |
| 9.2465557 | 9.3228939 | 9.741906 |
| 11.3162361 | 12.7048530 | 9.007874 |
| 10.3471846 | 13.4874662 | 10.606642 |
| 10.5750818 | 12.3599574 | 9.767329 |
| 7.8908811 | 8.9362046 | 11.420670 |
| 8.2103465 | 10.9157146 | 10.251212 |
| 4.3558573 | 3.1723915 | 9.083797 |
| 17.0165125 | 16.2326402 | 9.220187 |
| 17.0618217 | 17.0673764 | 9.008228 |
| 8.1459735 | 5.9045689 | 8.690499 |
| 19.3256000 | 20.6624167 | 9.176115 |
| 16.6835403 | 18.9005619 | 9.895711 |
| 0.9675794 | -1.0407440 | 9.669160 |
| 11.3328730 | 12.3771510 | 8.312394 |
| 14.8547359 | 13.5173788 | 8.818014 |
| 8.6098282 | 11.0828863 | 8.617932 |
| 9.6566643 | 8.6524688 | 9.235712 |
| 13.8626599 | 13.6336936 | 9.632229 |
| 7.2644851 | 7.9764259 | 11.106981 |
| 9.6695350 | 13.5818482 | 9.308055 |
| 12.1243997 | 13.8106055 | 9.411216 |
| 11.5985183 | 14.6180481 | 9.977975 |
| 10.8276696 | 11.9828803 | 11.324249 |
| 5.1595715 | 4.4817402 | 8.988899 |
| 6.4878922 | 5.9848498 | 9.552347 |
| 9.2660367 | 10.9425968 | 9.591490 |
| 15.2157591 | 19.0791948 | 12.067307 |
| 16.3372630 | 16.1744515 | 9.218720 |
| 10.1191227 | 12.7070120 | 10.040299 |
| 11.1225011 | 11.4060465 | 10.256264 |
| 13.3577367 | 17.5107993 | 11.077966 |
| 9.9188290 | 13.4980326 | 10.371727 |
| 7.4433242 | 9.0308196 | 12.926746 |
| 7.5176980 | 7.8761269 | 10.111553 |
| 12.4357043 | 11.8310221 | 10.897562 |
| 10.4780077 | 12.0744181 | 10.299336 |
| 12.5756855 | 10.6684254 | 10.143678 |
| 7.8139426 | 2.8123347 | 10.209686 |
| 12.7435749 | 13.3532957 | 7.970736 |
| 8.4687074 | 7.8122913 | 10.557721 |
| 11.0359040 | 12.0825764 | 9.485476 |
| 9.8420464 | 5.0416252 | 10.220693 |
| 3.6930708 | -0.2394072 | 7.967416 |
| 7.1297894 | 5.6024828 | 10.088557 |
| 6.3592121 | 8.7754980 | 10.120866 |
| 9.7443131 | 10.8380537 | 10.274751 |
| 13.6376131 | 11.0875668 | 9.867968 |
| 4.9652816 | 4.2333561 | 10.399828 |
| 9.6234801 | 8.2173189 | 8.860615 |
| 13.5670567 | 12.1855711 | 10.828229 |
| 8.1140084 | 7.9030517 | 8.391306 |
| 8.6195600 | 11.4507255 | 10.110521 |
| 4.6169007 | 5.5256782 | 10.724655 |
| 11.2981838 | 12.0376167 | 9.935078 |
| 12.3573563 | 15.4806057 | 12.001539 |
| 7.6811959 | 3.1856118 | 9.641675 |
| 9.7288846 | 11.0848948 | 9.508001 |
| 13.8299590 | 12.3025901 | 11.138382 |
| 11.2780660 | 8.1641107 | 9.041701 |
| 4.5111512 | -0.7847685 | 9.055219 |
| 8.2324550 | 8.5790011 | 8.886452 |
| 4.2420511 | 1.6554450 | 11.143887 |
| 18.4541131 | 20.3616825 | 8.755100 |
| 5.7814657 | 6.9178607 | 9.993405 |
| 6.5054877 | 9.3641638 | 10.551417 |
| 7.4768443 | 9.0843409 | 10.223067 |
| 9.5686605 | 8.2505906 | 9.389559 |
| 16.7730836 | 15.2345359 | 10.302773 |
| 10.1384849 | 8.3364911 | 11.234580 |
| 9.5374157 | 7.2120194 | 10.031883 |
| 8.6098738 | 8.6478803 | 9.920670 |
| 15.0878710 | 17.0282135 | 9.364089 |
| 3.1045440 | 1.6999714 | 9.600704 |
| 13.1614721 | 14.3319027 | 8.449530 |
| 11.9866391 | 16.0230871 | 9.874506 |
| 11.9618980 | 14.4151721 | 10.124788 |
| 7.3653389 | 3.7294576 | 9.205371 |
| 11.6746380 | 6.4771500 | 10.385229 |
| 7.6596400 | 7.7632604 | 10.531323 |
| 4.3143030 | 9.2668677 | 10.994500 |
| 14.5035136 | 15.1020301 | 10.104057 |
| 11.4146424 | 7.4102002 | 11.952200 |
| 9.1056038 | 14.0466906 | 10.268225 |
| 8.7966803 | 5.1207106 | 7.749341 |
| 10.2094281 | 9.9170912 | 11.270072 |
| 6.5996552 | 5.0637256 | 10.493248 |
| 7.9520110 | 8.6691477 | 10.575724 |
| 9.9180294 | 8.2558096 | 8.482893 |
| 12.4423239 | 13.0565601 | 9.023594 |
| 4.4353559 | 5.6819182 | 12.184934 |
| 12.9704169 | 14.3380035 | 10.121481 |
| 5.1989221 | 2.5038696 | 9.567028 |
| 11.7316123 | 13.1059153 | 12.708795 |
| 2.1894377 | 1.7044993 | 9.961818 |
| 7.7460702 | 9.7590827 | 11.066108 |
| 9.0658520 | 6.8798681 | 10.060122 |
| 4.3277263 | 7.5336307 | 9.968665 |
| 6.4414175 | 5.7191351 | 11.725462 |
| 12.1039115 | 11.0420339 | 11.887432 |
| 11.5304417 | 8.3334863 | 10.473566 |
| 5.1607329 | 1.3459986 | 9.372957 |
| 9.6308419 | 10.1949427 | 7.425060 |
| 3.3539796 | 6.5803509 | 11.320369 |
| 10.5014729 | 10.1064373 | 11.213255 |
| 4.8043179 | 8.1117374 | 9.959665 |
| 17.1132544 | 19.2102572 | 10.973733 |
| 10.9164667 | 11.9347756 | 9.441087 |
| 3.8951958 | 3.2095531 | 9.125298 |
| 3.5854702 | 5.8869141 | 9.335212 |
| 9.8897899 | 7.3637152 | 9.071449 |
| 12.2670422 | 10.1436538 | 9.127694 |
| 4.5330434 | 5.4253580 | 10.600080 |
| 11.5479375 | 12.8004855 | 11.002012 |
| 5.3446865 | 9.1038377 | 10.275961 |
| 12.2588794 | 15.2296313 | 10.926321 |
| 11.1275413 | 12.7954605 | 9.714038 |
| 7.3754766 | 5.8871715 | 10.045565 |
| 6.5239573 | 7.2309785 | 10.445669 |
| 11.5154368 | 8.3078254 | 11.083604 |
| 3.4702926 | 5.8031580 | 10.121888 |
| 6.1354118 | 6.4932912 | 9.216973 |
| 7.6786847 | 7.5743666 | 10.052774 |
| 4.2036705 | 1.9051615 | 9.941319 |
| 14.2873511 | 16.7241232 | 9.649482 |
| 17.0100947 | 19.3657354 | 11.252327 |
| 8.6097820 | 5.0246231 | 9.850792 |
| 11.1089319 | 7.9312779 | 8.814936 |
| 7.2059466 | 4.3217963 | 11.205422 |
| 14.0353425 | 13.1900274 | 10.355386 |
| 6.8164037 | 7.0582551 | 9.250544 |
| 10.1618763 | 11.7449499 | 8.783730 |
| 10.3007655 | 10.8451264 | 10.542169 |
| 8.2629836 | 8.5066332 | 9.051784 |
| 7.0030472 | 6.8063018 | 9.794901 |
| 9.7956461 | 9.9417811 | 10.997299 |
| 8.0929015 | 7.5742351 | 10.500101 |
| 12.7788482 | 10.6433560 | 8.703264 |
| 10.2137039 | 5.3516904 | 8.230764 |
| 12.6220764 | 11.7913166 | 10.904279 |
| 5.0885480 | 5.3147042 | 10.073516 |
| 4.6732955 | 3.4984139 | 9.948080 |
| 9.4321981 | 11.3222967 | 10.480282 |
| 3.9961838 | 4.3386145 | 8.312383 |
| 4.8049074 | 8.4565815 | 8.612397 |
| 9.2266976 | 9.1488874 | 10.648434 |
| 6.7924111 | 7.1044197 | 9.006450 |
| 11.7316288 | 11.4433062 | 9.672605 |
| 9.8444344 | 8.7198378 | 10.025905 |
| 3.2323016 | 6.7714864 | 10.240023 |
| 10.6401025 | 9.9029990 | 11.738003 |
| 9.5967526 | 9.2977024 | 8.964505 |
| 6.7368918 | 8.5376816 | 9.756878 |
| 11.8040948 | 11.5864658 | 9.117885 |
| 11.8168634 | 11.5347929 | 11.416472 |
| 8.3210233 | 11.6714745 | 8.479842 |
| 11.7415225 | 13.2851357 | 9.054593 |
| 11.4138394 | 9.8058497 | 10.680251 |
| 17.1121498 | 17.6876925 | 11.255000 |
| 7.9189420 | 10.1231582 | 10.105469 |
| 13.0552520 | 16.5330757 | 10.058565 |
| 17.3340001 | 18.6747548 | 10.969364 |
| 10.9140311 | 9.9817215 | 8.560603 |
| 12.8474609 | 12.3763373 | 12.734923 |
| 10.8916176 | 9.0092545 | 10.615180 |
| 12.6701387 | 14.4322463 | 9.535456 |
| 5.4790989 | 4.3162912 | 9.023393 |
| 1.4049228 | 0.4902850 | 9.398555 |
| 8.7408541 | 9.5948150 | 8.663794 |
| 9.5945619 | 11.0273236 | 11.268860 |
| 5.0046959 | 3.8004722 | 7.829608 |
| 14.9270184 | 16.5325471 | 9.915528 |
| 2.8559839 | 5.7969312 | 9.592244 |
| 5.9081697 | 6.9451012 | 9.991188 |
| 11.8505353 | 10.2841017 | 11.084313 |
| 11.2312889 | 13.0656798 | 8.918682 |
| 16.6689218 | 14.8452671 | 9.154539 |
| 8.4392202 | 9.6063355 | 9.647034 |
| 11.6647908 | 12.1135341 | 9.711263 |
| 4.3408198 | 4.0496701 | 9.062157 |
| 7.3747881 | 9.3607032 | 9.425535 |
| 13.4663185 | 15.9180471 | 8.429809 |
| 9.9492265 | 6.0482718 | 10.984152 |
| 8.2997543 | 9.6307292 | 10.673925 |
| 10.8329326 | 13.3701328 | 11.697390 |
| 12.8660207 | 10.8972150 | 10.004004 |
| 4.2091295 | 0.7290111 | 10.169965 |
| 6.6111700 | 6.0561203 | 10.971188 |
| 14.7718704 | 15.1982753 | 11.224714 |
| 8.8769067 | 8.3449899 | 9.560958 |
| 14.3759753 | 11.4609267 | 9.728628 |
| 6.8145558 | 5.6844716 | 9.521164 |
| 12.7763252 | 14.6146221 | 9.138430 |
| 6.1453207 | 5.8428398 | 9.786125 |
| 14.2069199 | 12.5911988 | 10.192608 |
| 10.9279435 | 7.6478393 | 9.547566 |
| 6.9159711 | 8.4603940 | 9.435019 |
| 11.8436555 | 10.9711603 | 9.436410 |
| 4.2019170 | 4.3086145 | 12.090244 |
| 6.7325938 | 8.4596299 | 10.376541 |
| 8.1645698 | 7.2000672 | 10.977947 |
| 5.4486245 | 6.1037508 | 10.171266 |
| 11.3276771 | 11.1747945 | 9.843888 |
| 5.8006836 | 5.9591146 | 9.066661 |
| 11.3637500 | 11.4259796 | 10.633184 |
| 17.6601941 | 15.3635777 | 10.180700 |
| 8.2018543 | 7.3416218 | 9.762866 |
| 3.3101996 | 5.9402239 | 10.187970 |
| 4.3261274 | 1.9884129 | 11.702478 |
| 9.4191561 | 13.1283597 | 10.217256 |
| 4.5992096 | 5.7393613 | 11.279295 |
| 15.2392579 | 12.5241417 | 9.858362 |
| 5.6287172 | 5.9063315 | 9.161968 |
| 12.0691608 | 13.7417119 | 9.358653 |
| 12.8974583 | 12.6794679 | 9.550518 |
| 7.7799651 | 8.3559576 | 8.944121 |
| 6.6137793 | 5.0277868 | 9.376746 |
| 13.2723762 | 10.6030964 | 10.333618 |
| 8.5143267 | 12.4853671 | 10.219670 |
| 9.7561054 | 11.2323643 | 10.240687 |
| 7.6169428 | 6.1086858 | 10.234927 |
| 8.2849728 | 3.2281663 | 10.541214 |
| 5.7300272 | 5.3630818 | 10.984624 |
| 14.1629038 | 12.1587589 | 10.774774 |
| 9.8684576 | 14.3668296 | 9.271445 |
| 14.7246704 | 13.1612347 | 8.731097 |
| 10.4487100 | 10.0547411 | 9.249580 |
| 7.0066906 | 6.4987856 | 8.513232 |
| 9.2946079 | 12.7861762 | 9.727494 |
| 6.4700044 | 5.1446470 | 9.815604 |
| 14.3836790 | 13.4168687 | 10.010211 |
| 12.6841642 | 16.4961496 | 8.489889 |
| 10.6081242 | 10.1422229 | 8.769782 |
| 10.2880649 | 10.9534520 | 10.161100 |
| 3.7086017 | 4.7136970 | 10.328058 |
| 12.3770268 | 9.6261189 | 9.852677 |
| 17.3283522 | 15.7857120 | 9.114161 |
| 15.8971113 | 17.8584495 | 12.093745 |
| 10.2186963 | 10.3058758 | 11.382159 |
| 13.9219296 | 17.3904823 | 10.981800 |
| 2.1663558 | -0.0220269 | 8.702988 |
| 12.4822718 | 7.6534713 | 10.635799 |
| 9.2338407 | 7.8256429 | 10.685476 |
| 9.6965597 | 11.3497646 | 9.788082 |
| 11.0010153 | 14.1409985 | 8.555241 |
| 13.2426161 | 12.7212937 | 10.775006 |
| 10.2125270 | 11.8299179 | 8.960548 |
| 8.4732473 | 7.8000750 | 12.671392 |
| 6.3622898 | 8.8580374 | 11.947259 |
| 2.6410986 | -0.2614394 | 10.727650 |
| 10.6684070 | 10.6376439 | 10.018764 |
| 9.0120382 | 7.3228603 | 10.885425 |
| 4.8487167 | 8.0021196 | 11.879087 |
| 12.3775940 | 11.5305087 | 9.994118 |
| 17.6566014 | 17.5090203 | 9.885914 |
| 13.6133712 | 16.5275103 | 11.516482 |
| 12.1140771 | 11.9379428 | 10.443304 |
| 10.3941885 | 9.5671256 | 10.995979 |
| 11.7067120 | 9.7149302 | 9.841203 |
| 12.1576766 | 13.2922459 | 8.690045 |
| 9.6160220 | 5.2193938 | 12.219909 |
| 11.4653024 | 11.3176389 | 11.109099 |
| 4.5088080 | 5.6467272 | 10.763316 |
| 15.4593663 | 17.6695763 | 10.102850 |
| 14.5985750 | 15.9929515 | 9.380967 |
| 9.9224646 | 9.8787057 | 7.941949 |
| 9.9614322 | 7.9852858 | 9.677915 |
| 1.9858482 | 8.1655869 | 9.080084 |
| 8.7724658 | 9.2270203 | 10.092662 |
| 8.3699524 | 12.7649385 | 9.315528 |
| 14.3087232 | 13.1923942 | 8.095500 |
| 5.0497386 | 1.8792572 | 9.492756 |
| 9.3501552 | 6.7701566 | 9.598250 |
| 0.5350346 | 2.6356243 | 11.940637 |
| 4.1408982 | 3.1505799 | 10.540124 |
| 5.0821985 | 1.8880561 | 10.185638 |
| 17.8246593 | 15.7639332 | 11.844484 |
| 15.3736131 | 14.2519300 | 9.964136 |
| 10.8683398 | 12.3493342 | 11.508134 |
| 14.6856886 | 14.0438756 | 9.888895 |
| 6.4039347 | 6.0402886 | 10.668469 |
| 5.4270315 | 6.4760771 | 9.329402 |
| 6.9146644 | 7.9245542 | 9.473423 |
| 10.7273996 | 11.0367056 | 9.480907 |
| 7.0849809 | 5.9075391 | 11.154513 |
| 16.7983666 | 18.1625220 | 11.767593 |
| 14.2132940 | 12.7096017 | 9.706971 |
| 9.5572452 | 8.2813540 | 7.833005 |
| 11.6815455 | 8.2300557 | 10.086836 |
| 10.9261160 | 9.6607448 | 10.951147 |
| 6.4991126 | 8.1545395 | 10.368222 |
| 11.3453595 | 11.7448003 | 8.394049 |
| 8.4112638 | 8.7748591 | 9.657944 |
| 15.0269575 | 14.6200994 | 9.146785 |
| 6.3007057 | 5.5939181 | 10.689866 |
| 14.0253816 | 16.2506887 | 8.545091 |
| 16.6845201 | 11.7358598 | 10.197442 |
| 14.2910239 | 16.5973213 | 12.502343 |
| 13.2470910 | 15.9447160 | 9.897173 |
| 6.9891315 | 6.0496441 | 9.892985 |
| 8.6370833 | 12.9164714 | 10.772656 |
| 14.4557295 | 11.1870959 | 10.002415 |
| 7.7985805 | 5.6557922 | 9.032455 |
| 9.1017769 | 9.4556679 | 10.363475 |
| 9.8721506 | 11.6723860 | 9.282190 |
| 6.6780436 | 7.7099112 | 10.909337 |
| 6.8848939 | 5.4972960 | 8.280911 |
| 10.7466844 | 9.8262996 | 9.401369 |
| 18.8888509 | 21.2348658 | 9.391650 |
| 13.6796499 | 15.1680118 | 10.130928 |
| 7.7470560 | 8.9938982 | 11.186358 |
| 12.2085150 | 9.2364034 | 9.583692 |
| 14.9630400 | 12.6177238 | 7.115939 |
| 5.7928029 | 7.8867035 | 8.445230 |
| 12.0492380 | 10.4208627 | 10.825331 |
| 4.7898928 | 7.1706555 | 8.756444 |
| 13.3868668 | 14.3514512 | 10.575718 |
| 10.0516605 | 9.8486351 | 11.315914 |
| 6.9699959 | 6.9977244 | 10.772135 |
| 6.2914127 | 6.4686920 | 9.488776 |
| 9.6930972 | 9.4667924 | 10.215380 |
| 6.9751259 | 9.3509871 | 10.843787 |
| 11.3229729 | 10.7183807 | 9.721535 |
| 4.0176599 | 5.7130204 | 10.108986 |
| 7.0186268 | 3.6743038 | 10.199689 |
| 12.6238793 | 12.9666218 | 11.050416 |
| 15.6323921 | 17.3146996 | 10.805215 |
| 9.1364970 | 11.3506371 | 9.397455 |
| 3.9413670 | 2.2046862 | 10.005543 |
| 17.2191190 | 17.8472551 | 11.505797 |
| 20.3978717 | 18.7548019 | 10.593946 |
| 14.4221993 | 14.6912111 | 10.172908 |
| 5.6518306 | 2.9693986 | 9.761665 |
| 2.0094935 | 3.5187913 | 10.375002 |
| 10.0991824 | 10.3933702 | 10.278051 |
| 15.0263500 | 11.6512575 | 9.829766 |
| 7.6243036 | 7.9010932 | 10.371275 |
| 6.6458931 | 5.6943695 | 8.080281 |
| 13.5853365 | 12.1442941 | 10.689056 |
| 9.3518068 | 13.4419573 | 8.703402 |
| 5.0450231 | 7.1061948 | 9.331137 |
| 12.1339700 | 11.0707032 | 11.026477 |
| 7.6022791 | 7.1570958 | 10.794852 |
| 11.5544200 | 13.5178517 | 10.208971 |
| 9.0974730 | 6.2826785 | 11.793988 |
| 7.3821108 | 6.6169198 | 10.422196 |
| 1.4170253 | 1.2336912 | 10.746884 |
| 18.3644814 | 17.0202375 | 10.576442 |
| 15.5477742 | 17.0343885 | 9.689350 |
| 1.5698086 | -1.5347856 | 8.953845 |
| 6.5729543 | 4.9161911 | 9.457498 |
| 11.3733720 | 13.2376450 | 9.756458 |
| 15.5208952 | 15.8739227 | 10.038682 |
| 8.0009616 | 9.9552670 | 8.750561 |
| 8.9411047 | 7.4909172 | 10.536542 |
| 7.3004604 | 4.7723257 | 10.114183 |
| 4.9387344 | 4.9036235 | 8.626765 |
| 12.7048744 | 11.0034054 | 10.687102 |
| 7.3588915 | 5.5611632 | 9.211927 |
| 15.7344196 | 10.0971091 | 10.288107 |
| 8.4218641 | 10.3514515 | 9.015701 |
| 12.0718345 | 12.5924408 | 10.411492 |
| 5.9196390 | 1.9271646 | 9.859441 |
| 8.7691837 | 12.5698462 | 7.784129 |
| 10.2343046 | 8.8974741 | 10.533850 |
| 8.0360949 | 10.7956057 | 11.352095 |
| 12.7183437 | 10.8810200 | 11.030304 |
| 9.1593065 | 8.7555054 | 10.421857 |
| 10.0775911 | 6.3959064 | 10.901516 |
| 11.7098247 | 10.8627988 | 9.824191 |
| 5.8158801 | 8.1385278 | 8.604800 |
| 1.9549774 | 2.3698850 | 10.054570 |
| 12.7606835 | 11.0077761 | 9.846111 |
| 5.4983952 | 5.4723696 | 9.874036 |
| 7.3275652 | 6.7767945 | 10.492721 |
| 12.3810422 | 12.7074250 | 10.012970 |
| 11.9871982 | 13.0982813 | 9.266071 |
| 7.8020130 | 5.4738611 | 10.143984 |
| 5.3249288 | 5.7057416 | 11.962245 |
| 1.7196803 | 1.7033602 | 9.024309 |
| 19.6614238 | 17.5724520 | 8.237204 |
| 11.6491358 | 14.3463863 | 11.613147 |
| 5.6752094 | 5.5165495 | 8.609122 |
| 10.8686325 | 8.7561970 | 10.022731 |
| 12.6765791 | 11.7271709 | 9.926925 |
| 14.1393928 | 16.1482069 | 10.032523 |
| 8.8757410 | 8.5907711 | 8.833402 |
| 1.1303913 | 4.6908344 | 10.321996 |
| 12.1842307 | 10.6789674 | 9.909078 |
| 7.5787223 | 5.2684228 | 9.589380 |
| 4.6565746 | 3.4912547 | 10.614561 |
| 17.0445525 | 14.4776787 | 10.943802 |
| 14.7219230 | 15.0223685 | 9.983696 |
| 17.5982859 | 16.9428604 | 8.717461 |
| 8.0116691 | 10.3760178 | 10.672209 |
| 15.4636080 | 17.3871700 | 9.225940 |
| 10.5611233 | 8.2635703 | 8.568831 |
| 12.1221658 | 11.5798554 | 9.690936 |
| 13.7538879 | 16.3918252 | 8.814427 |
| 10.3207457 | 9.4083205 | 11.822636 |
| 9.1263326 | 8.6572432 | 11.403034 |
| 11.8016119 | 13.1408181 | 9.775305 |
| 13.4433660 | 12.1713322 | 8.254470 |
| 4.0472676 | 7.8468515 | 11.000664 |
| 12.9829909 | 11.8756227 | 10.113189 |
| 8.9665867 | 8.5724369 | 10.398141 |
| 5.7077018 | 5.8194759 | 10.134961 |
| 11.7180619 | 12.0868329 | 9.608587 |
| 12.8895268 | 11.8327099 | 9.202267 |
| 3.1135552 | 0.6113114 | 9.431592 |
| 3.0256686 | -2.1038017 | 9.813040 |
| 15.1034479 | 12.0062436 | 10.567009 |
| 16.6948250 | 16.6987174 | 10.613143 |
| 9.8892482 | 6.9220014 | 8.973916 |
| 18.5494373 | 19.9337068 | 9.053328 |
| 7.2229705 | 7.6700820 | 10.556468 |
| 2.8104295 | 3.0214804 | 8.246579 |
| 7.2872443 | 3.6619946 | 9.149129 |
| 9.4306904 | 6.7342818 | 8.861815 |
| 11.3442568 | 10.6294914 | 11.980732 |
| 6.5849500 | 4.7622979 | 10.444377 |
| 16.3609667 | 16.0678894 | 9.476815 |
| 5.5168893 | 5.3785773 | 11.386265 |
| 10.2639241 | 9.8891438 | 9.868259 |
| 9.6294210 | 10.5454956 | 8.582970 |
| 7.0309316 | 9.3164396 | 12.091511 |
| 10.4581266 | 10.8594627 | 10.519711 |
| 7.9429293 | 10.2603688 | 9.932533 |
| 4.4507937 | 5.2821483 | 9.812973 |
| 8.3041104 | 5.5666432 | 10.094316 |
| 12.8483956 | 14.0678728 | 10.496607 |
| 11.1588761 | 12.3459495 | 10.404245 |
| 10.6085807 | 12.6888866 | 8.984637 |
| 4.4588103 | 5.1615986 | 9.498537 |
| 6.4027510 | 7.8689446 | 7.977405 |
| 15.9461913 | 17.7897972 | 9.945959 |
| 16.8967308 | 12.9319482 | 9.699929 |
| 16.9058757 | 15.3691318 | 8.363673 |
| 12.2319230 | 14.2620913 | 9.853102 |
| 11.5495996 | 11.6939306 | 10.317926 |
| 17.1357742 | 16.3572742 | 10.544338 |
| 9.1524803 | 8.7122919 | 10.825238 |
| 2.5360347 | 4.0272128 | 10.615152 |
| 13.6994963 | 14.6682211 | 8.438240 |
| 9.8212832 | 9.3444553 | 11.483048 |
| 15.7556161 | 16.4445036 | 9.079870 |
| 5.9305872 | 7.2479710 | 9.283617 |
| 11.0808536 | 13.2868783 | 11.075598 |
| 6.8183193 | 7.0346528 | 10.950995 |
| 8.5718263 | 6.3714109 | 10.234464 |
| 12.0748376 | 7.4195380 | 10.979132 |
| 6.9204449 | 6.6774351 | 10.064719 |
| 4.6539317 | 4.5402173 | 11.846766 |
| 11.5815985 | 10.0136867 | 10.294231 |
| 6.8334930 | 7.0178975 | 10.012803 |
| 14.4205921 | 13.0352951 | 10.614363 |
| 11.8281129 | 13.1967944 | 10.104034 |
| 14.1764885 | 13.3826245 | 9.239959 |
| 11.4915329 | 13.3390902 | 10.890396 |
| 14.2801822 | 11.7329706 | 9.711738 |
| 5.2155991 | 4.3230762 | 9.924641 |
| 9.5425587 | 9.1459009 | 9.721926 |
| 9.5767530 | 8.8290638 | 9.983013 |
| 21.9731524 | 23.9302035 | 9.904114 |
| 3.4136170 | 6.8984582 | 10.741152 |
| 5.9487429 | 7.6488919 | 8.767769 |
| 9.6713090 | 14.0602366 | 9.389404 |
| -0.9099282 | -1.0048141 | 8.851437 |
| 9.7125421 | 13.2802435 | 9.294742 |
| 11.2281587 | 9.4557477 | 10.968568 |
| 13.8290823 | 11.6237675 | 10.003799 |
| 9.5344285 | 14.7243159 | 10.911425 |
| 3.4091018 | 2.2723223 | 8.147142 |
| 14.9818974 | 15.8558091 | 10.317356 |
| 14.4628626 | 14.2377061 | 8.674906 |
| 6.3218615 | 8.3580478 | 9.879770 |
| 6.7645510 | 6.3172950 | 9.879965 |
| 13.5518213 | 14.6546471 | 10.575422 |
| 7.1674847 | 6.6456323 | 9.654808 |
| 5.4950759 | 3.4214260 | 8.998009 |
| 13.8436434 | 13.0187867 | 10.574394 |
| 11.5120616 | 12.6242927 | 9.942859 |
| 9.5946828 | 9.0643648 | 8.037029 |
| 9.0777359 | 7.5876134 | 10.831029 |
| 13.0180532 | 14.7268090 | 11.158064 |
| 10.2668677 | 11.6735254 | 9.981412 |
| 13.6887330 | 13.5260161 | 9.992270 |
| 3.2751260 | 2.1170206 | 9.500418 |
| 11.5854491 | 14.7531942 | 8.831352 |
| 8.1633616 | 9.0876237 | 8.893885 |
| 7.9765102 | 6.8111679 | 11.331588 |
| 12.3233157 | 13.6444916 | 12.381135 |
| 7.1685275 | 6.9039198 | 10.772948 |
| 9.5387475 | 13.2576412 | 9.974691 |
| 14.5607344 | 11.2906663 | 11.202767 |
| 12.1050876 | 13.9647819 | 10.235030 |
| 4.4555067 | 1.8832556 | 11.554996 |
| 7.6169662 | 9.0448644 | 9.308438 |
| 3.3070220 | 4.4078185 | 10.698471 |
| 5.0992079 | 4.9488223 | 9.963070 |
| 8.2363816 | 4.4070045 | 9.269677 |
| 11.3236512 | 9.9518041 | 10.055856 |
| 10.1562227 | 9.4292973 | 11.073055 |
| 11.3476432 | 11.7371411 | 10.742966 |
| 9.7826278 | 13.7361096 | 8.983404 |
| 1.1824546 | 1.5082695 | 8.844103 |
| 16.5319305 | 13.5130863 | 10.443175 |
| 13.8889147 | 16.0058005 | 10.515182 |
| 10.0656734 | 10.0131498 | 10.138251 |
| 13.8562246 | 11.9989727 | 12.259664 |
| 13.9715235 | 15.0007625 | 9.940886 |
| 13.8168620 | 12.5668466 | 10.409207 |
| 12.0683070 | 12.0397939 | 10.578609 |
| 10.2997871 | 9.0990525 | 9.040938 |
| 9.2197060 | 9.0040769 | 9.978597 |
| 7.7158831 | 10.9173142 | 10.378647 |
| 11.6970183 | 11.4844030 | 9.931428 |
| 12.4648866 | 11.6527273 | 9.847117 |
| 11.3256069 | 11.0615573 | 10.852062 |
| 10.9224254 | 6.8806038 | 8.039382 |
| 7.7482346 | 7.2277663 | 10.726829 |
| 15.4013534 | 15.8341037 | 11.528444 |
| 6.2891499 | 7.8143445 | 10.370561 |
| 8.6503558 | 9.5204039 | 10.729181 |
| 19.5732695 | 19.1780049 | 11.197212 |
| 11.5489007 | 10.7795017 | 10.326631 |
| 6.5335934 | 5.4788297 | 10.123073 |
| 4.8379679 | 5.1858628 | 9.082168 |
| 11.2348729 | 12.0335386 | 10.298124 |
| 14.6721145 | 12.4994207 | 10.835795 |
| 14.4117302 | 16.9542621 | 10.160365 |
| 8.2865719 | 7.3930510 | 9.410131 |
| 13.5035226 | 10.9778873 | 8.772734 |
| 8.5170672 | 6.5208795 | 10.288619 |
| 13.2128347 | 13.5958243 | 11.808395 |
| 15.9817276 | 14.9986840 | 9.426795 |
| 11.9778315 | 10.4199589 | 9.688980 |
| 10.7339553 | 10.5070032 | 9.979141 |
| 9.3394071 | 11.7833203 | 10.890564 |
| 1.1944117 | 0.8155077 | 9.988236 |
| 6.3040093 | 5.2484415 | 11.495680 |
| 10.2648521 | 10.1210595 | 9.279155 |
| 2.7413010 | 2.3315670 | 8.291667 |
| 12.0354059 | 11.3022123 | 9.059469 |
| 6.8502923 | 4.4531242 | 8.984131 |
| 16.1211721 | 16.2178521 | 10.590746 |
| 12.5740054 | 10.3193883 | 9.912629 |
And here’s how they look like
plot_ly(df, x = ~X1, y = ~X2, z = ~X3) %>%
add_markers(size= 0.5) %>%
layout(scene = list(xaxis = list(title = 'X1'),
yaxis = list(title = 'X2'),
zaxis = list(title = 'X3')))
We can see there’s a some sort of correlation between \(X1\) and \(X2\) that are mainly spread along one
direction. We can think of this direction as the place where there is
the most of the total variability.
If we look instead to the \(X3\)
axis we may see there’s another source of variation, smaller than the
previous one.
We might say there are three principal components of variation, one bigger than the others.
To correctly perform PCA main components must pass through the origin
. The so-called mean centering is carried out,
which consist in subtracting for all values within each column their
mean.
This procedure is mandatory in most of the cases when you carry out a
PCA because we need major component to go through the centroid of the
data.
\[ X^*_{i,j} = X_{i,j} - \overline{X_i} \]
df_scaled = scale(df, scale = F) %>% data.frame()
plot_ly(df_scaled, x = ~X1, y = ~X2, z = ~X3) %>%
add_markers(size= 0.5) %>%
layout(scene = list(xaxis = list(title = 'X1'),
yaxis = list(title = 'X2'),
zaxis = list(title = 'X3')))
Now we can generate the covariance matrix.
cor_df = cov(df_scaled) %>% data.frame()
kable(cor_df) %>% kable_styling(full_width = FALSE)| X1 | X2 | X3 | |
|---|---|---|---|
| X1 | 15.3790481 | 15.0952951 | -0.1034882 |
| X2 | 15.0952951 | 18.6198889 | -0.1043137 |
| X3 | -0.1034882 | -0.1043137 | 0.9753565 |
pc = prcomp(df_scaled)
pca = summary(pc)ggcorrplot(cor_df,
type = 'full',
method = 'square',
lab = TRUE,
lab_size = 5,
colors = c("tomato2", "white", "springgreen3"),
title="Correlogram",
ggtheme=theme_bw)
General formula for covariance between two variables \(X\) and \(Y\) is
\[
cov(X,Y) = \frac{1}{n}\sum_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})
\]
We call this matrix \({\bf C}\) and,
since is a square simmetric matrix, exist in \(\mathbb{R}\) an eigenvalue
\(\lambda\) and an
eigenvector \({\bf
v}\) such that
\[
{\bf Cv} = {\bf \lambda v}
\]
In other words, vector \({\bf v}\)
is scaled by the matrix of a factor \(\lambda\), which is basically the amount of
the total variance.
In our case there are three eigenvalues (one for each variable) and
three unit eigenvector.
eig = eigen(cor_df)
eig## eigen() decomposition
## $values
## [1] 32.1821776 1.8175124 0.9746036
##
## $vectors
## [,1] [,2] [,3]
## [1,] 0.668299939 0.743830506 0.009558751
## [2,] 0.743877057 -0.668312644 -0.002265958
## [3,] -0.004702745 -0.008624876 0.999951747
A nice property of eigenvalues is that their sum is always
equal to the trace of the starting matrix.
The total variance in our case is then divided into three components
according to the size of each eigenvalues, which are in fact the measure
of the variance for each component.
When you have a lot of variables may be convenient to express them in
percentage
eig_prc = (eig$values/sum(eig$values))*100
eig_df = data.frame('PC' = c(1:3), 'Var' = round(eig_prc, digits = 2))
eig_df1 = eig_df
eig_df1[2] = data.frame(lapply(eig_df[2], paste0, '%'))
kable(eig_df1) %>% kable_styling('striped', full_width = FALSE)| PC | Var |
|---|---|
| 1 | 92.02% |
| 2 | 5.2% |
| 3 | 2.79% |
And put those values onto the so-called
Scree plot
ggplot(eig_df, aes(PC, Var))+geom_col(width = 0.8, fill = 'cornflowerblue')+
geom_line()+
theme_bw()+
labs(y='Explained variance', x='Principal components')+
geom_text(label = eig_df1$Var, nudge_y = 5, nudge_x = 0.1, size = 3.5)
Eigenvectors however tell us the direction of the
variance, and there will be one unit eigenvector for
each source of variance, i.e. for each variable. They are also
necessarily orthogonal to each other.
Let’s see what happens if we plot them onto a graph.
eigenvectors = data.frame('v1'= eig$vectors[,1], 'v2'= eig$vectors[,2], 'v3' = eig$vectors[,3]) %>%
t() %>% data.frame()
plot_ly(eigenvectors)%>%
add_trace(x=~X1, y=~X2,z=~X3,
type = 'scatter3d',
mode= 'text',
text= c('V1','V2','V3'),
color = c('red', 'green', 'blue'),
showlegend= FALSE,
textfont = list(size=16) ) %>%
layout(scene = list(
xaxis = list(title = 'X1', range=c(-1.5,1.5)),
yaxis = list(title = 'X2', range=c(-1.5,1.5)),
zaxis = list(title = 'X3', range=c(-1.5,1.5)))) %>%
add_trace(x=c(0,~X1[1]), y= c(0,~X2[1]), z = c(0,~X3[1]) , mode= 'lines', type='scatter3d', name='V1', text= c('V1','V1')) %>%
add_trace(x=c(0,~X1[2]), y= c(0,~X2[2]), z = c(0,~X3[2]) , mode= 'lines', type='scatter3d', name='V2', text= c('V2','V2')) %>%
add_trace(x=c(0,~X1[3]), y= c(0,~X2[3]), z = c(0,~X3[3]) , mode= 'lines', type='scatter3d', name='V3', text= c('V3','V3')) %>% layout(scene = list(aspectmode = 'cube'))
Actually there are infinite eigenvectors but computer software
give us back only the ones with length 1.
Since we said eigenvalues express the amount of variance and
eigenvectors are his direction, we just combine these two pieces of
information to get vectors with length proportional to the
variance.
eigenvectorsM = eigenvectors %>% as.matrix()
# --- multiply each eigenvector by corresponding eigenvalue ---
loading_vectors = t(eigenvectorsM) %*% (sqrt(eig$values) * diag(3)) %>% data.frame()
colnames(loading_vectors) = c('P1','P2','P3')
loading_vectors## P1 P2 P3
## X1 3.79122129 1.00279618 0.009436592
## X2 4.21996528 -0.90098667 -0.002237000
## X3 -0.02667836 -0.01162764 0.987172494loading_vectors = t(loading_vectors) %>% data.frame()
plot_ly(loading_vectors)%>%
add_trace(x=~X1, y=~X2,z=~X3,
type = 'scatter3d',
mode= 'text',
text= c('P1','P2','P3'),
color = c('red', 'green', 'blue'),
showlegend= FALSE,
textfont = list(size=16) ) %>%
layout(scene = list(
xaxis = list(title = 'X1', range=c(-10,10)),
yaxis = list(title = 'X2', range=c(-10,10)),
zaxis = list(title = 'X3', range=c(-4,4)))) %>%
add_trace(x=c(0,~X1[1]), y= c(0,~X2[1]), z = c(0,~X3[1]) , mode= 'lines', type='scatter3d', name='P1', text= c('V1','V1')) %>%
add_trace(x=c(0,~X1[2]), y= c(0,~X2[2]), z = c(0,~X3[2]) , mode= 'lines', type='scatter3d', name='V2', text= c('V2','V2')) %>%
add_trace(x=c(0,~X1[3]), y= c(0,~X2[3]), z = c(0,~X3[3]) , mode= 'lines', type='scatter3d', name='V3', text= c('V3','V3'))IN PCA these linear combination are called loading
vectors and they’re generally indicated with the letter \({\bf p}\)
Now, let’s take a step forward and take a look what happens when we
project the original points on these new vectors we just found.
We want to know, in other words, how much each object weighs in
relation to principal components.
In mathematical terms this is a linear combination and it’s something
like this
\[ PC_{i,m} = p_{i1}x_1+p_{i2}x_2+...+p_{im}x_m \]
Where every \(p_{i,m}\) is a
loading vectors component and \(m\) is
for the number of total variables in the data matrix.
Or in vector notation
\[
{\bf PC_1} = {\bf X \cdot }{ \bf p}_1^T
\]
What we’re doing is a change of base. We want our
principal components vectors to be the axes of a new reference
system.
In math terms we say they are an eigenbase of the
original matrix, namely a set of vectors all linearly indipendent.
score_df = as.matrix(df_scaled) %*% t(eigenvectorsM) %>% data.frame()
colnames(score_df) = c('PC1', 'PC2', 'PC3')
kable(score_df) %>% kable_styling('striped', full_width = FALSE) %>% scroll_box(height = '500px')| PC1 | PC2 | PC3 |
|---|---|---|
| -13.4171015 | 0.7879534 | 0.6369618 |
| -1.5393342 | 0.0210064 | 1.4476258 |
| 5.0395433 | -0.4766111 | 0.2341523 |
| 1.0558323 | 0.3776907 | -0.8736625 |
| -2.3715902 | 1.3806645 | -0.3300249 |
| -3.5423936 | 0.3768586 | 0.5052218 |
| -5.7494751 | -0.1008909 | -0.4960008 |
| -5.4753236 | -1.7041371 | 0.8459553 |
| -4.6880618 | 0.2575638 | -0.2035494 |
| -0.1945283 | 0.4957722 | -0.7846391 |
| 4.8686011 | -0.9562464 | 0.4575672 |
| -2.6283009 | 4.2431773 | 0.4034207 |
| 1.1200307 | -1.5336407 | -0.9234914 |
| -2.0628102 | -1.9667308 | 1.5434761 |
| 2.9648056 | -0.3910680 | -0.2107552 |
| -1.1787864 | -2.3865572 | 0.9188736 |
| -5.3426696 | -1.5653526 | -0.4619930 |
| -2.8112533 | -1.5462930 | 1.3901882 |
| 1.6154570 | 2.0415027 | 1.4561298 |
| 1.2331807 | 0.4368870 | -2.1056991 |
| -6.2771868 | -1.5506798 | -2.1351446 |
| -1.7173480 | 0.3576685 | 0.8508387 |
| -7.7647250 | -1.7233660 | 0.3806404 |
| -3.4481908 | -0.5470313 | -0.1401546 |
| 4.6818981 | 1.5366151 | -0.2278794 |
| -6.9499660 | -0.4075689 | 0.8498222 |
| 0.5947152 | -0.5169987 | -0.4833229 |
| -6.2501772 | 0.4995401 | -0.7276035 |
| -4.4485323 | 2.1062897 | 0.0748915 |
| 2.4553021 | -0.8033809 | 0.0346638 |
| -3.2291676 | -1.9253701 | -1.7841818 |
| -4.8082865 | -0.1852000 | -1.5476798 |
| 1.5410889 | -1.1812222 | 0.0396794 |
| 4.2675020 | -1.7408384 | -1.3726609 |
| 1.9733408 | 0.6885874 | 0.3218777 |
| 2.7948875 | -1.6738866 | -1.0259552 |
| -3.1314918 | -0.9320421 | 0.2853726 |
| 7.1503632 | 2.5907220 | 0.6577619 |
| -7.0797938 | 1.0886036 | 2.2462150 |
| 2.3054302 | -0.7913308 | 1.6864231 |
| -8.3174156 | -1.0093712 | -0.9893626 |
| 2.9271195 | -1.1026644 | -1.4550289 |
| -6.5253653 | -0.8465690 | -0.5599108 |
| 3.4502362 | -0.0893018 | 0.4295759 |
| -6.9932101 | 0.3279421 | 1.0782849 |
| 0.9321347 | 2.8804736 | 0.9442281 |
| 0.6859240 | 0.4205606 | -1.0404146 |
| 4.2230834 | 1.7760603 | -0.5378670 |
| 0.0568885 | 0.5615439 | -1.7279829 |
| -10.6061161 | 0.4953637 | -0.0957531 |
| 6.6113966 | -0.3813882 | -0.4297194 |
| 6.0452884 | 0.1311504 | 0.3073498 |
| -3.5954526 | 0.0670449 | 1.1461960 |
| 0.6017905 | -1.8710084 | 1.1593452 |
| -7.7049532 | 0.8354485 | 0.9516502 |
| -3.0191111 | 1.2163348 | -0.4460842 |
| 14.8952262 | 0.2874039 | -0.2943506 |
| -2.1895491 | -1.0617378 | 1.1460359 |
| 8.5293116 | 1.3948441 | -0.6385691 |
| -7.4999767 | 2.1963904 | 1.6408658 |
| -3.9053975 | -1.2542198 | -0.0215715 |
| -5.2683609 | 1.2053354 | -1.2321280 |
| -3.4984573 | -1.0678043 | -1.4913291 |
| 9.4187550 | -0.3060457 | -1.0831411 |
| 5.4190573 | -0.5983775 | 1.2513247 |
| -12.6625037 | 1.3589690 | -0.9217672 |
| 16.4502850 | 0.2025839 | 0.2542167 |
| 7.4278183 | 0.6571287 | 0.8048283 |
| 1.0596028 | 1.9362029 | -0.5941284 |
| -1.3287573 | 0.6210634 | -2.2148686 |
| -3.4891880 | 1.7689611 | 0.4811766 |
| -0.3886649 | -0.5009157 | 0.6153022 |
| -3.2755824 | 0.8483673 | -0.5596840 |
| 3.2973537 | -0.7744949 | 0.8870002 |
| -8.2589307 | 0.2607292 | 0.5631166 |
| -7.0308887 | 0.0473763 | 1.2108982 |
| 0.5163172 | -0.2703000 | 0.1854430 |
| 0.9356057 | 1.8070197 | -0.0953003 |
| -3.1079753 | 0.3585780 | 0.3726556 |
| 2.3397658 | -0.0988136 | -0.5162754 |
| 4.2064085 | 1.9031759 | 1.7036206 |
| 0.2090415 | 1.6534174 | -0.2202389 |
| -6.3541018 | -1.2426658 | 1.2066865 |
| -11.6390728 | 0.7327805 | -0.2028109 |
| 8.0469050 | 1.8165300 | 0.1971718 |
| -4.7389478 | -0.4843509 | 1.1308066 |
| 5.9006101 | -2.8053690 | 0.3163928 |
| 4.3649488 | -0.0951072 | -0.3085381 |
| -8.3913941 | -1.2923330 | -1.0030678 |
| 0.4722257 | -0.3469402 | 1.4223190 |
| -0.7566188 | 0.6083378 | 2.0450901 |
| -7.5463108 | 0.7207382 | -0.0663079 |
| 0.7992862 | -0.1701018 | 1.1886162 |
| 11.2096651 | 0.3831892 | -0.6330605 |
| -6.2019528 | -1.7042221 | 0.4910349 |
| -8.0072415 | 2.6706098 | 0.3355901 |
| 2.0504926 | 1.6204523 | 1.3914569 |
| -11.1120061 | 0.0540567 | 0.2700058 |
| 2.2684641 | 1.6015741 | -0.2568001 |
| 3.7458999 | 1.9141189 | 0.3899050 |
| -10.8341899 | -0.1277626 | 0.8285801 |
| 0.8840102 | -0.0292787 | -0.8633877 |
| 7.0399099 | 0.6002471 | -1.0337461 |
| 5.8099035 | -1.1007259 | 0.4875509 |
| 2.8584734 | 2.4691779 | 1.9667436 |
| 0.3847595 | 0.7929192 | 0.3705107 |
| -4.1425862 | -1.5602946 | 0.5694364 |
| 0.8712011 | -0.7765461 | 0.2723086 |
| 5.5960635 | -1.0827488 | -1.8747434 |
| 8.1835846 | 1.2655279 | 0.5623219 |
| 7.5109536 | -2.2951363 | -0.8272072 |
| 2.8346006 | 1.1757939 | 1.4333180 |
| -2.9616722 | -0.3061198 | -0.6192568 |
| 2.2282370 | -1.0861900 | -0.5509208 |
| 1.9470668 | 0.7289712 | -0.6785977 |
| -5.1643838 | 1.2016086 | 0.9144222 |
| -2.3737536 | 0.9593800 | -0.4958221 |
| -3.6921403 | -1.2308822 | -0.0432541 |
| 2.9365964 | -1.4259718 | -0.1194540 |
| -1.0649795 | 0.2514830 | -0.2149569 |
| 2.1407988 | 0.5928182 | 0.4552758 |
| 2.7160331 | -0.8851924 | 0.0447897 |
| 4.6646245 | 0.7318531 | 2.8982521 |
| 5.5595718 | -1.3668212 | -0.5025575 |
| 0.5391326 | 0.3089075 | -1.4085179 |
| -0.6807228 | 1.3366780 | 0.2592128 |
| -2.0857340 | -0.2438065 | 0.4992782 |
| -2.8530076 | -1.5467172 | 1.8926563 |
| -5.7617959 | 1.4199636 | -0.3889384 |
| 7.5405551 | -0.4516543 | -0.9496652 |
| 4.7645362 | 0.4470925 | -1.4896645 |
| 5.1560132 | 0.3857235 | -0.1296043 |
| 6.4260134 | -0.7574900 | -1.8092168 |
| 5.4856145 | 2.1217953 | 0.3554834 |
| -1.4288463 | -1.1458354 | 1.2300776 |
| 5.8688516 | 2.3940138 | -0.2993527 |
| 0.3214705 | 0.7734009 | 0.3134629 |
| -17.1156027 | -0.2616274 | 0.5456769 |
| -4.5665818 | -0.2385068 | -1.0960529 |
| 10.8327559 | -0.8734336 | 0.2555782 |
| -4.3567313 | 0.3967210 | 1.3266129 |
| 1.6118495 | 1.1944775 | 0.3451578 |
| 1.5960908 | -2.2373442 | 0.0734613 |
| 5.2039191 | -0.4527797 | -1.5752736 |
| 5.5771191 | 0.0866233 | -0.2665111 |
| 2.3867565 | -1.8126912 | -1.2621568 |
| -5.2282059 | 0.5756950 | -0.2736282 |
| 5.4583582 | -1.8540516 | -0.9939857 |
| 1.8101229 | 0.0550537 | 1.2011928 |
| -0.4980559 | 0.2349843 | -0.1919741 |
| -0.7105047 | -0.7244536 | 0.9233657 |
| 3.5824348 | -0.0017546 | -1.3095819 |
| 1.4915362 | 0.4244177 | 0.6890583 |
| -6.0910612 | 1.0767096 | 0.1830791 |
| -0.8373363 | -1.3639985 | -0.7349359 |
| 6.6651232 | -0.1147242 | 0.2379158 |
| -0.8762900 | -2.5800084 | -0.6214770 |
| 6.8984813 | -0.3006394 | 0.5154695 |
| -5.5672699 | 0.9899525 | 1.5857272 |
| -6.4550087 | 0.4861817 | -1.8396837 |
| 2.8980931 | -0.6792805 | -0.1571772 |
| -0.6191372 | 0.5326599 | 0.8430943 |
| -1.1966688 | 0.6725847 | 0.4419648 |
| -6.8948035 | -0.3405114 | -0.8361589 |
| 5.0987031 | -2.5986932 | -1.6946319 |
| 2.7789567 | -1.4972994 | 0.3608372 |
| -15.6216836 | 1.0229628 | -0.0900485 |
| 9.5946785 | 0.1870258 | 0.2224639 |
| -0.1814152 | 1.8055434 | 0.2142840 |
| -4.7224981 | 1.3566721 | -1.7507036 |
| 7.4803902 | -2.2682319 | 0.4534795 |
| 3.3226200 | 0.6866567 | 0.0011150 |
| -8.0345287 | 0.6197683 | -0.1336785 |
| 3.1568617 | -1.5910847 | -0.5265575 |
| 0.2072939 | -1.1173694 | 0.0176582 |
| 1.3438720 | 0.2845117 | -0.3170181 |
| 0.3033525 | -0.4551261 | -0.7735222 |
| -0.7373300 | -0.3923206 | -0.8161232 |
| 3.6860560 | -1.4280206 | -1.6154515 |
| 2.5455645 | -0.2287623 | 1.3336232 |
| -2.9484172 | 0.7078828 | 0.3063875 |
| 13.7669301 | -1.1304284 | 0.9382965 |
| 0.8659473 | 2.4161248 | 1.9087794 |
| 8.4261099 | -3.8565318 | -0.2270955 |
| 1.3956935 | -1.0262804 | -0.5426285 |
| 7.2132996 | 0.9486712 | 1.8502852 |
| -0.0948177 | -2.3045746 | -0.0861902 |
| -1.7280914 | -0.3351766 | -0.1076903 |
| -9.7294999 | -0.9793816 | 0.7311397 |
| -9.6260346 | -0.2201708 | 1.5270085 |
| -0.9865841 | -1.7594040 | 2.2579230 |
| -1.4905083 | -1.3055236 | -0.4577482 |
| 4.0494002 | -1.4830657 | 0.0954168 |
| -10.4858860 | -2.7563516 | 0.1112242 |
| 2.0150650 | -1.6566330 | -0.3160998 |
| -13.5043858 | -0.7047636 | -0.7989145 |
| 5.5205457 | -0.8330647 | -1.8052011 |
| 2.0732221 | 2.6477062 | -1.4110844 |
| 2.1485120 | -2.1304063 | 2.4423295 |
| 3.3635710 | 1.8180675 | -0.2923363 |
| -6.9163484 | 0.0637339 | -0.8121343 |
| 4.2597628 | 2.4863810 | -0.0767921 |
| 3.0084524 | -0.0727874 | 0.4423587 |
| -5.7583168 | -0.8880635 | -0.3135923 |
| -1.2028786 | -1.2666029 | 1.0397887 |
| 3.2142210 | 1.5060207 | -1.1137080 |
| 1.4294164 | 1.4708760 | 1.1372970 |
| -8.9103070 | 0.6556590 | -0.5016535 |
| 6.3121016 | -0.1632592 | -1.7470896 |
| 3.7394044 | -0.3486151 | -1.0308437 |
| -0.5301745 | -0.9999025 | 1.4331903 |
| -2.3021142 | -0.8395823 | -0.1991792 |
| -4.8779716 | -0.0059093 | -0.3313253 |
| -0.5621133 | -1.7772064 | 0.3488330 |
| 8.6643105 | -1.1924806 | 0.7820598 |
| -5.9235675 | 2.2031133 | 1.9753021 |
| -0.1186315 | -0.1510949 | -0.5055804 |
| -3.2725500 | 0.1336028 | 0.6960474 |
| 1.1431636 | -0.5403533 | -1.5118209 |
| 5.7202847 | 0.1492566 | -0.5440075 |
| -3.9958502 | -0.5027638 | 1.8260232 |
| 3.9098962 | 0.3605165 | -1.4642857 |
| 8.9931252 | 2.0472858 | 0.4410675 |
| -10.9809679 | -1.6597135 | 0.2441775 |
| 3.2018470 | -2.7010090 | 0.5069528 |
| 14.1624196 | 0.7917502 | 1.0050075 |
| 0.9112642 | 1.9934276 | -0.2752984 |
| -8.1211605 | 2.5706439 | -1.1183322 |
| 3.9007847 | 0.4543430 | -0.3065684 |
| 3.5386701 | -0.9507995 | 0.5421039 |
| -1.2981908 | 1.1983188 | -2.2646499 |
| 1.3999434 | -0.6926709 | -0.9559613 |
| 3.8849669 | 0.1181395 | 0.6982321 |
| 8.4622429 | -0.4407876 | -2.1913635 |
| 8.1325426 | -0.4621748 | 0.1534030 |
| 5.0595204 | -1.1980223 | 1.7210322 |
| 6.6901823 | 0.4395198 | -2.4069346 |
| -8.7657342 | 1.6068078 | -1.3564422 |
| 1.6188442 | 0.8847288 | -0.8721751 |
| 0.5641085 | -1.0284059 | 0.3459024 |
| 1.7610046 | -1.2904884 | -0.0536365 |
| -4.1001004 | -1.8485222 | 0.3586231 |
| -2.1273918 | 0.0502203 | 0.5899346 |
| -4.4981023 | -0.6686103 | 0.9392437 |
| 2.5321845 | -1.0167966 | -1.2995027 |
| -5.9356138 | -1.1258867 | 0.1979384 |
| 3.4064994 | -1.9198656 | -0.0189448 |
| 9.9225438 | -0.7470972 | 0.0105530 |
| -3.0910284 | 0.0204041 | 0.1313612 |
| -2.6068700 | 0.9410415 | 1.5601939 |
| -3.1461556 | 0.9931329 | -0.2661703 |
| 1.7454907 | -3.3103130 | 1.3353513 |
| -1.1947682 | -0.7882338 | 0.7511263 |
| -1.7315329 | -0.1830317 | -0.9878394 |
| -1.7873392 | -0.8447994 | 0.2665092 |
| 3.4344579 | -0.4911246 | -1.8134353 |
| -8.0890299 | -2.9929957 | 1.6961864 |
| 1.1185741 | -0.9581170 | 0.0111400 |
| -4.1170890 | 1.8617456 | 0.3923003 |
| -3.3084069 | -0.6126929 | 0.3950104 |
| 7.2672757 | 0.1072056 | -1.7344339 |
| -3.5772825 | 2.3181867 | -0.3636362 |
| 6.4157812 | 0.7693051 | -0.7854858 |
| 6.8266262 | 0.0347887 | -0.8371262 |
| -2.2713907 | 1.1179276 | 0.4179706 |
| -4.6352003 | -0.3804351 | -0.6836175 |
| -3.6995388 | 2.1382045 | -0.3537914 |
| -6.4192633 | 1.0139380 | -0.8312551 |
| -10.9478813 | -1.5340360 | -1.0682939 |
| -0.8072182 | -0.4920762 | 0.8508653 |
| 2.8384938 | -0.7679038 | 0.2474091 |
| 7.7306136 | -0.8300418 | -0.1716735 |
| 3.7958173 | 0.3587378 | -0.3750794 |
| -0.7927991 | 0.8611039 | -0.4208075 |
| 1.3611991 | 2.6274060 | 2.2607420 |
| 8.0209358 | -1.5144073 | -0.0170342 |
| 1.7719767 | -1.5466034 | 0.3825924 |
| -0.8033174 | 1.1665785 | 0.3477276 |
| 0.3712076 | -0.2890308 | 2.2866570 |
| 6.0696495 | -0.4770749 | -0.8632319 |
| 6.8811154 | 0.3594393 | -0.8692073 |
| -3.2986966 | -2.5381234 | 0.1038488 |
| -7.5339875 | -0.1762620 | 0.7547207 |
| -2.6177607 | -1.0764144 | 0.1064546 |
| 5.1540908 | 1.0069507 | -1.3178865 |
| 8.8244493 | -1.1701415 | -0.1153352 |
| -5.8647073 | -0.2396385 | 1.4382719 |
| -4.5016224 | 2.2179178 | 0.0924638 |
| 9.5817201 | 2.1059039 | -1.0879871 |
| 2.7357650 | -0.6236535 | 0.5985752 |
| -11.0965445 | -0.1874953 | 2.0401713 |
| 7.4116554 | 0.3428527 | 1.6660760 |
| 2.1752540 | -1.2546313 | -0.9418405 |
| 11.7638610 | -0.1551225 | -0.4221122 |
| 1.2589906 | 0.1358751 | -0.3339468 |
| -8.8160304 | 0.0297242 | 0.1198186 |
| 3.5559550 | 0.3313441 | -0.8111912 |
| -0.9105108 | -0.6324023 | 0.7536115 |
| -10.9234784 | 0.8053773 | 0.8593085 |
| -3.2706638 | -0.8996331 | 0.1859203 |
| 0.9882796 | 1.4998695 | 0.3860895 |
| 0.1701805 | -1.5819203 | -0.6802877 |
| -5.5049840 | -0.0499590 | 1.7970354 |
| 2.5880247 | 1.3628944 | 0.8623565 |
| -5.4241849 | -0.1290419 | 0.2646632 |
| 10.7177187 | -0.2117868 | 0.4388317 |
| 3.8648356 | 0.8175232 | 0.0025547 |
| 0.6783062 | 1.8888739 | -1.4145997 |
| 3.6478722 | 0.3852348 | 0.0436856 |
| -0.4034914 | 2.1312509 | -0.3170061 |
| -2.9464351 | 0.5451353 | -1.5238556 |
| 2.5875499 | -0.6365405 | 0.2971632 |
| -4.1598564 | -0.2734841 | -0.4599952 |
| 4.1826670 | 1.7218345 | -0.8611433 |
| -10.4654232 | 0.6191204 | -0.0491820 |
| 10.4415944 | 0.9941033 | 0.0529230 |
| -0.6851429 | 0.8174154 | -0.4136908 |
| -2.7802942 | 1.7280376 | -0.1051805 |
| -9.5105149 | 1.4731899 | 0.6464321 |
| -15.3941502 | -3.3758585 | 0.7401900 |
| 3.5671913 | 0.6911670 | -0.5654420 |
| 3.8167740 | 0.2159287 | -0.4011021 |
| -3.7266025 | -2.6616297 | 1.0101977 |
| 0.2027187 | -2.0249847 | -1.2561844 |
| -4.1460069 | 0.9844285 | 1.4730442 |
| -3.3692404 | -0.5066292 | -0.4733522 |
| -3.7761667 | -1.3094903 | -1.4500857 |
| 1.8864983 | 1.0011506 | 0.2031000 |
| -3.9158740 | -0.0798551 | -0.4731075 |
| -1.3645587 | -1.3839125 | 0.4694806 |
| 2.0186260 | 0.0525926 | -0.4444815 |
| -0.1530085 | 0.3090040 | -0.5863901 |
| 4.2149710 | 0.6413529 | 0.5583620 |
| 2.8481419 | 1.0356780 | -1.1430639 |
| -2.1056717 | -1.2583919 | -0.1222218 |
| 10.4445600 | -0.4117549 | -0.6981981 |
| -0.1079859 | 0.1735512 | 0.0710848 |
| 8.3127335 | 0.1017170 | 2.1218543 |
| 1.3131694 | 0.1389534 | -0.0983702 |
| 5.7614314 | -0.4785019 | 0.1130207 |
| -5.3440671 | -1.9049814 | -0.8747673 |
| -6.6998790 | -0.2480314 | -0.5509347 |
| 11.3625073 | -0.5934202 | -1.0808178 |
| -1.8319882 | 1.8424110 | 0.6046768 |
| -5.8998050 | 0.1792291 | 1.8032376 |
| -10.8189408 | -1.6894121 | -0.6266701 |
| 4.9378383 | -0.4805021 | -0.8773710 |
| -2.5863793 | -0.8527158 | -0.8547998 |
| -6.0927208 | -0.7008917 | -1.0716322 |
| 2.3111618 | 0.5062575 | 0.1429993 |
| 5.9803515 | -3.0795480 | -0.7591213 |
| 6.9094609 | -0.4258624 | 1.7420917 |
| -4.2306462 | -0.9017127 | 0.3313290 |
| 0.9040562 | 1.3499321 | 1.4220786 |
| 4.9944785 | 1.1466112 | -0.6936865 |
| 0.3003255 | 0.7378046 | -1.1736979 |
| -6.9033128 | 2.7047868 | -3.1682349 |
| -11.3915259 | 1.7225039 | 0.1827659 |
| -1.0943561 | -0.7200140 | 0.9402799 |
| -2.0917014 | 0.4643375 | 0.1013545 |
| -8.5903909 | -0.3867933 | 1.7093838 |
| 5.6652256 | 0.4026755 | -2.5297599 |
| 3.9361347 | 0.5695093 | 0.9663134 |
| 8.1455734 | 0.0972289 | 0.7735364 |
| -1.3690039 | -1.2078157 | -0.5259536 |
| 2.4040625 | -2.4925037 | -1.7125888 |
| -4.2752032 | -0.6290917 | 1.5060621 |
| 11.3252368 | -1.4749119 | -0.0027289 |
| 7.7989432 | -0.9528966 | -0.1014908 |
| -3.9239140 | 1.0795604 | -0.7198633 |
| 0.9231511 | -0.0935888 | -0.6941215 |
| -1.2465386 | 0.3969465 | 2.4371334 |
| -6.8507276 | -1.3622540 | -0.6328517 |
| -4.0601979 | -1.8374702 | 0.1557911 |
| 7.4731259 | 1.9986563 | -0.6197756 |
| 10.6743086 | 1.5026241 | 0.9625148 |
| -3.0277973 | 2.5117849 | 2.1475136 |
| -8.9975207 | -0.7911363 | 1.3817585 |
| 8.4134732 | -0.8360070 | -0.3411453 |
| 12.2759040 | 1.9131771 | 2.0273686 |
| -12.6036633 | 0.1289526 | -1.0067742 |
| 0.5274720 | 0.5243596 | -1.7538916 |
| 5.4768688 | -1.2090090 | -0.8376728 |
| 0.9790318 | 1.2214700 | -0.7997176 |
| 6.2206571 | 2.6064276 | -0.5237275 |
| -4.6704073 | 2.2506870 | -1.1566278 |
| 9.4350450 | 0.5773204 | 0.6022614 |
| 1.8114693 | 0.2095537 | 1.1477521 |
| 1.7036111 | -0.7137363 | 1.1263676 |
| -5.7831367 | -0.0654286 | 0.0134706 |
| 3.0314666 | 0.7647634 | 0.0213952 |
| -7.9381193 | 1.2073510 | 0.1856511 |
| -0.2929753 | 0.2179408 | -1.0585167 |
| -0.8171354 | -1.4871135 | -0.4241831 |
| 0.2308709 | 1.7654699 | 1.1257485 |
| -4.5702302 | 1.4840091 | 0.6763360 |
| 2.0776055 | 1.0041964 | -1.0314314 |
| 4.4119378 | 1.5997525 | -1.1030805 |
| -3.0210941 | -0.7024642 | -0.2647897 |
| -0.9167405 | 0.1344024 | 0.9098768 |
| -4.6489315 | -2.2725401 | -0.3542855 |
| -2.9427091 | 0.5532151 | -0.4144294 |
| 0.5029565 | 0.6843869 | -0.2302608 |
| -1.3529776 | 1.9368043 | -0.9000604 |
| -5.8469389 | -0.5966857 | -0.0459633 |
| -7.7113412 | 0.6615632 | 1.4809352 |
| 3.4962094 | -1.1035791 | 0.4751173 |
| 3.3063963 | 1.0998091 | -1.1805782 |
| 1.5223982 | -0.6709290 | 0.9488546 |
| 6.3055388 | 0.0718347 | 0.3487772 |
| -4.5610575 | 0.2659430 | 1.4915916 |
| -1.0523398 | -1.0492374 | -0.8197282 |
| -9.0242373 | -1.0360945 | 1.6308416 |
| -0.5888441 | -0.7650855 | 0.1042194 |
| -1.1139881 | 0.4233221 | -1.4822484 |
| -8.5855970 | 0.5453967 | -0.3676316 |
| -4.0172852 | -0.6729986 | 0.4639152 |
| -0.0664800 | 0.0513706 | -0.5078873 |
| 3.5669967 | 2.0059825 | -0.0855155 |
| 5.6807591 | -0.0702420 | 0.5743243 |
| -1.6120076 | -1.6210931 | -0.5149013 |
| 8.6553627 | -0.9422957 | -1.1553494 |
| 3.9259222 | -0.0192409 | 0.6053401 |
| -0.5799224 | -0.4863678 | -0.4105674 |
| -2.0255610 | -1.9418734 | -0.1343732 |
| -1.1950062 | -0.4831905 | -0.7037665 |
| 2.6081251 | 0.1544906 | -0.9910406 |
| 2.4464288 | -0.0639662 | -2.0675375 |
| 0.6768899 | -0.2688213 | 0.0541613 |
| 7.0564926 | 0.8553754 | 1.0532254 |
| 2.6617394 | 2.5120753 | 0.3356702 |
| -3.7701627 | 0.3849460 | -1.0095483 |
| 3.4657118 | -0.7410261 | -0.2340863 |
| 1.0109767 | 0.7504959 | -1.4318155 |
| 3.2609710 | -0.4136688 | -0.0522983 |
| -6.4312925 | -1.0198796 | 0.3414631 |
| 7.3102551 | 0.1103351 | -0.3488728 |
| 1.8102592 | -1.8591372 | 0.1980386 |
| 0.8777775 | -0.0801202 | -0.1471367 |
| -6.6322315 | -0.0250863 | 2.4176658 |
| 5.6764799 | 1.8118540 | -0.1156664 |
| 7.4403761 | -0.2823390 | 1.9072532 |
| -2.3133145 | 1.2488617 | -0.6419327 |
| -0.6226652 | 0.1623518 | 0.4854912 |
| -1.9722673 | 1.9951240 | -0.2858282 |
| 5.6928813 | -0.8727320 | -0.8490247 |
| 5.6668672 | -0.4185084 | 0.1902116 |
| 10.6492260 | 1.5915859 | -0.8205634 |
| -2.7509972 | -0.6666676 | -0.2143068 |
| 4.4478255 | -2.2259752 | -1.9787014 |
| 4.1824086 | 0.1031296 | 0.8390421 |
| -7.6970387 | -1.1613351 | 0.8330815 |
| -4.0922326 | 1.0139175 | 1.9230094 |
| 9.8967336 | 0.1553674 | 0.3107965 |
| -0.8190926 | -0.0924823 | -0.2791473 |
| 3.0832885 | -0.8068659 | -1.0010241 |
| 3.0103208 | -2.0644954 | 0.5866305 |
| 2.3278436 | -1.1342111 | -0.2479085 |
| -2.0206339 | -0.8569242 | 1.3874526 |
| -0.3291234 | -1.9321412 | 0.2166195 |
| -8.6596687 | 0.3858018 | -0.9700372 |
| 9.5160243 | 1.0736776 | -0.7422277 |
| 10.1682424 | 0.5513433 | -0.9556347 |
| -4.0924806 | 1.3824492 | -1.3332778 |
| 14.3546038 | -0.1688482 | -0.7742627 |
| 11.2749280 | -0.9628294 | -0.0759644 |
| -14.0608623 | 0.6761403 | -0.4075437 |
| 2.8539076 | -0.5694850 | -1.6955692 |
| 6.0533799 | 1.2837944 | -1.1588929 |
| 0.0698863 | -1.7326306 | -1.4131420 |
| -1.0413503 | 0.6649885 | -0.7798778 |
| 5.4730704 | 0.4611009 | -0.3544632 |
| -3.1517365 | -0.6787187 | 1.0699665 |
| 2.6337632 | -2.6204285 | -0.7185852 |
| 4.4440312 | -0.9481963 | -0.5924823 |
| 4.6905575 | -1.8838752 | -0.0326067 |
| 2.2088272 | -0.7077515 | 1.3122048 |
| -7.1481058 | 0.1093931 | -1.0602146 |
| -5.1449101 | 0.0880317 | -0.4875028 |
| 0.3994938 | -1.1591622 | -0.4330399 |
| 10.4166783 | -2.1927221 | 2.0810919 |
| 9.0188036 | 0.6073320 | -0.7500554 |
| 2.2800085 | -1.7076628 | 0.0199029 |
| 1.9817922 | -0.0937303 | 0.2483965 |
| 8.0129213 | -2.5180644 | 1.0775922 |
| 2.7330157 | -2.3881550 | 0.3476087 |
| -2.2564369 | -1.2660527 | 2.8889643 |
| -3.0524431 | -0.4147548 | 0.0772343 |
| 3.1725196 | 0.5935226 | 0.9012530 |
| 2.0480610 | -1.0201768 | 0.2837914 |
| 2.4047853 | 1.4811252 | 0.1513779 |
| -6.6217633 | 3.1889509 | 0.1896688 |
| 4.5244179 | -0.1695850 | -2.0259375 |
| -2.4664677 | 0.3314490 | 0.5326158 |
| 2.4307992 | -0.6036295 | -0.5247150 |
| -3.6081154 | 3.2075584 | 0.2150100 |
| -11.6353177 | 2.1825778 | -2.0849682 |
| -5.0028860 | 0.8164104 | 0.0556837 |
| -3.1576815 | -1.8776134 | 0.0734346 |
| 0.6381455 | -0.7394312 | 0.2549965 |
| 3.4275577 | 1.9932798 | -0.1151178 |
| -7.4693522 | 0.1187034 | 0.3493514 |
| -1.3854613 | 0.9343564 | -1.1542878 |
| 4.1926692 | 1.1987055 | 0.8419351 |
| -2.6258104 | 0.0256418 | -1.6372906 |
| 0.3429979 | -1.9840968 | 0.0786348 |
| -6.7423740 | -1.0069096 | 0.6679041 |
| 2.5705220 | -0.3823683 | -0.0725255 |
| 5.8298094 | -1.9133395 | 1.9961583 |
| -6.4301344 | 2.8456431 | -0.3804299 |
| 0.8150599 | -0.9092613 | -0.5124232 |
| 4.4539559 | 1.3133799 | 1.1543210 |
| -0.3201336 | 2.1990858 | -0.9572742 |
| -11.4993919 | 3.1461806 | -0.9881627 |
| -2.0461578 | -0.3422701 | -1.1425680 |
| -9.8738352 | 1.2971758 | 1.0923032 |
| 13.5504598 | -0.6124711 | -1.2029064 |
| -4.9250437 | -1.0647769 | -0.0553332 |
| -2.6240553 | -2.1659354 | 0.5040299 |
| -2.1815074 | -1.2535695 | 0.1856149 |
| -1.3998347 | 0.8667820 | -0.6259688 |
| 8.6057828 | 1.5503163 | 0.3402412 |
| -0.9637984 | 1.2173132 | 1.2242154 |
| -2.1963056 | 1.5320914 | 0.0183785 |
| -1.7475548 | -0.1164874 | -0.1049482 |
| 8.8182454 | -0.8938376 | -0.6185704 |
| -10.5936517 | 0.4346154 | -0.4617796 |
| 5.5294103 | -0.5168853 | -1.5453898 |
| 5.9956015 | -2.5332921 | -0.1355444 |
| 4.7817988 | -1.4792640 | 0.1181324 |
| -6.2346153 | 2.2510031 | -0.8209640 |
| -1.3163143 | 3.6098975 | 0.3938024 |
| -3.0435162 | -0.2373644 | 0.4985961 |
| -4.1628840 | -3.7346026 | 0.9263671 |
| 6.9913958 | -0.0475899 | 0.1201408 |
| -0.8033637 | 2.7794206 | 1.9560983 |
| 2.5981581 | -3.3588390 | 0.2350946 |
| -4.2362815 | 2.3984446 | -2.2663947 |
| 0.2592182 | 0.2134418 | 1.2568017 |
| -5.7598470 | 0.7786483 | 0.4565084 |
| -2.1744647 | -0.6256888 | 0.5437377 |
| -1.1582054 | 1.1309852 | -1.5292632 |
| 4.0974058 | -0.2044333 | -0.9753380 |
| -6.7543443 | -1.2589601 | 2.1260237 |
| 5.3984036 | -0.6774956 | 0.1246402 |
| -8.5958191 | 1.4555129 | -0.4772564 |
| 3.6418209 | -0.7978514 | 2.7027799 |
| -11.2035472 | -0.2522092 | -0.1094401 |
| -1.5036234 | -1.5115207 | 1.0296592 |
| -2.7586642 | 1.4030653 | 0.0428611 |
| -5.4384042 | -2.5574261 | -0.0953628 |
| -5.3838479 | 0.2123001 | 1.6856651 |
| 2.3592171 | 0.8654783 | 1.8896920 |
| -0.0322101 | 2.2612650 | 0.4765500 |
| -9.4817420 | 2.2026002 | -0.6690589 |
| 0.0973185 | -0.3694571 | -2.5941841 |
| -6.8046387 | -2.6562980 | 1.2491275 |
| 0.5955091 | 0.3046213 | 1.2023504 |
| -4.6898154 | -2.5892012 | -0.1011167 |
| 11.7874115 | -0.8594660 | 1.0054117 |
| 2.2412424 | -0.5933105 | -0.5699080 |
| -8.9400803 | 0.0179542 | -0.9330257 |
| -7.1564296 | -2.0035539 | -0.7321489 |
| -1.8434543 | 1.7011015 | -0.9389838 |
| 1.8129313 | 1.6110210 | -0.8663181 |
| -6.8724558 | -1.0011656 | 0.5427613 |
| 3.2998954 | -0.7156309 | 0.9950155 |
| -3.5921739 | -2.8530197 | 0.2180805 |
| 5.5823596 | -1.8095867 | 0.9206199 |
| 3.0212637 | -1.0138676 | -0.2969028 |
| -4.6267177 | 0.8092698 | 0.0143971 |
| -4.1980424 | -0.7256504 | 0.4032971 |
| -0.0641953 | 2.2619919 | 1.0864732 |
| -7.2994065 | -2.0400362 | 0.0535776 |
| -5.0006777 | -0.5110592 | -0.8273827 |
| -3.1690519 | -0.0928308 | 0.0206810 |
| -9.7080711 | 1.1121104 | -0.1111400 |
| 8.0557100 | -1.2885228 | -0.3401546 |
| 11.8328164 | -1.0425102 | 1.2826540 |
| -4.4425455 | 2.3055157 | -0.1666142 |
| -0.6052985 | 2.2308396 | -1.1851175 |
| -5.9099158 | 1.7193246 | 1.1761248 |
| 5.2550402 | 0.8798181 | 0.3713147 |
| -4.1254651 | -0.3823787 | -0.7885834 |
| 1.5988341 | -1.0220653 | -1.2340161 |
| 1.0140262 | -0.3325582 | 0.5277046 |
| -2.0803658 | -0.2726236 | -0.9767884 |
| -4.1907135 | -0.0798590 | -0.2418980 |
| 0.0023367 | -0.1084898 | 0.9800310 |
| -2.8944323 | 0.2115060 | 0.4719462 |
| 2.5286842 | 1.6614213 | -1.2869675 |
| -3.1197282 | 3.2939510 | -1.7719733 |
| 3.2675044 | 0.7586296 | 0.9098414 |
| -6.5810486 | -0.5094714 | 0.0217832 |
| -8.2090687 | 0.3965828 | -0.1035003 |
| 0.7888097 | -1.2969903 | 0.4564361 |
| -8.0288841 | -0.6544826 | -1.7474941 |
| -4.4265639 | -2.8076063 | -1.4490956 |
| -0.9660663 | 0.0012188 | 0.6275407 |
| -4.1060106 | -0.4289723 | -1.0329992 |
| 2.4193336 | 0.3394903 | -0.3294962 |
| -0.8694654 | 0.7528187 | 0.0119195 |
| -6.7386942 | -2.8652262 | 0.1672382 |
| 0.5343544 | 0.5391727 | 1.7288593 |
| -0.6001394 | 0.1915457 | -1.0531064 |
| -3.0804726 | -1.4346086 | -0.2863859 |
| 2.5768645 | 0.3025018 | -0.8838206 |
| 2.5361497 | 0.3267081 | 1.4148947 |
| 0.3153646 | -2.3396224 | -1.5553198 |
| 3.7989466 | -0.8787381 | -0.9515567 |
| 0.9841500 | 1.1887510 | 0.6787740 |
| 10.6527495 | 0.1548357 | 1.2901042 |
| -1.1127482 | -1.6179643 | 0.0698945 |
| 7.0882586 | -2.0808446 | 0.0575641 |
| 11.5366084 | -0.3373479 | 1.0043655 |
| 0.7909233 | 0.7177226 | -1.4459474 |
| 3.8447033 | 0.5195117 | 2.7412262 |
| 0.0428863 | 1.3332423 | 0.6105195 |
| 5.2705887 | -0.9587809 | -0.4644405 |
| -7.0578016 | 0.4573415 | -1.0222932 |
| -12.6284158 | -0.0194224 | -0.6774237 |
| -0.9497070 | -0.6410682 | -1.3626573 |
| 0.6741853 | -0.9858863 | 1.2471974 |
| -7.7529369 | 0.4594907 | -2.2193869 |
| 8.3394394 | -0.6869806 | -0.0675731 |
| -7.7120902 | -2.4882482 | -0.4818988 |
| -4.8200934 | -0.9887166 | -0.0564008 |
| 1.6298543 | 1.1904718 | 1.0859072 |
| 3.2953484 | -1.1104281 | -1.0918417 |
| 8.2520032 | 1.7428943 | -0.8080516 |
| -1.1473430 | -0.8816123 | -0.3823746 |
| 2.8730512 | -0.1584810 | -0.2929969 |
| -8.0170291 | -0.2114933 | -0.9938082 |
| -2.0403814 | -1.5072998 | -0.6134804 |
| 6.9131283 | -1.3500015 | -1.5657899 |
| -2.7912559 | 2.6079429 | 0.9771757 |
| -1.2272313 | -1.0105107 | 0.6430788 |
| 3.2425351 | -1.6341734 | 1.6822346 |
| 2.7696644 | 1.5453870 | 0.0139683 |
| -10.5804095 | 1.9002351 | 0.1202124 |
| -5.0161796 | 0.1198612 | 0.9322863 |
| 7.2370630 | 0.0780347 | 1.2430905 |
| -1.7927184 | 0.2876679 | -0.4614044 |
| 4.1993943 | 2.2941767 | -0.2482387 |
| -5.1498987 | 0.5320295 | -0.5148811 |
| 5.4790854 | -0.9982559 | -0.8608454 |
| -5.4805880 | -0.0738927 | -0.2566889 |
| 4.9250160 | 1.4090512 | 0.2115417 |
| -0.9405419 | 2.2793217 | -0.4536110 |
| -3.0167728 | -1.2469757 | -0.6063425 |
| 2.1440933 | 0.7403975 | -0.5635388 |
| -7.9314754 | -0.5139862 | 2.0322189 |
| -3.1443200 | -1.3909872 | 0.3333828 |
| -3.1271186 | 0.5107549 | 0.9513017 |
| -5.7539155 | -0.7698084 | 0.1211829 |
| 1.9488275 | 0.2169912 | -0.1614737 |
| -5.6210314 | -0.4017468 | -0.9796763 |
| 2.1560740 | 0.0691456 | 0.6275593 |
| 9.2952040 | 2.1249888 | 0.2263609 |
| -2.9911878 | 0.4543654 | -0.2636852 |
| -7.3047473 | -2.2512911 | 0.1178157 |
| -9.5725867 | 1.1323698 | 1.6509161 |
| 2.1248195 | -2.5114375 | 0.1892058 |
| -6.5978512 | -1.1676597 | 1.2218642 |
| 5.5666171 | 2.2246336 | -0.1126683 |
| -5.7756688 | -0.4952071 | -0.8858981 |
| 4.3561139 | -0.9427889 | -0.6454144 |
| 4.1185839 | 0.3815803 | -0.4432339 |
| -2.5147449 | -0.5302805 | -1.0887219 |
| -5.7718912 | 0.8228022 | -0.6597236 |
| 2.8208937 | 2.0413669 | 0.3481162 |
| 1.0418033 | -2.7547779 | 0.1844275 |
| 0.9395051 | -0.9938887 | 0.2201536 |
| -4.3014570 | 0.8392057 | 0.2055559 |
| -5.9992054 | 3.2585527 | 0.5247409 |
| -6.1206459 | -0.0725091 | 0.9388697 |
| 4.5711802 | 1.6602947 | 0.7942388 |
| 3.3508049 | -2.9967610 | -0.7550703 |
| 5.7019384 | 1.4258131 | -1.2462413 |
| 0.5310267 | 0.3168603 | -0.7616169 |
| -4.4110055 | 0.1394222 | -1.5227734 |
| 1.7893448 | -2.3711705 | -0.3009469 |
| -5.7831102 | 0.6339738 | -0.2225258 |
| 5.6581988 | 0.9902976 | 0.0289717 |
| 6.8201692 | -2.3186632 | -1.5144989 |
| 0.7048953 | 0.3811105 | -1.2400670 |
| 1.0879114 | -0.4111140 | 0.1462872 |
| -7.9515392 | -1.1364522 | 0.2644843 |
| 1.4980423 | 2.0324532 | -0.1391460 |
| 9.3924657 | 1.6052357 | -0.8442545 |
| 9.9638173 | -0.8703001 | 2.1168076 |
| 0.5540930 | -0.0404606 | 1.3680916 |
| 8.3009226 | -2.0171617 | 0.9870967 |
| -12.4973761 | 0.8953383 | -1.3645184 |
| 0.0972874 | 3.4223286 | 0.6494148 |
| -1.9457980 | 0.8905536 | 0.6676477 |
| 0.9891706 | -1.1127371 | -0.2332655 |
| 3.9430707 | -1.9972270 | -1.4599022 |
| 4.3746076 | 0.5998056 | 0.7843994 |
| 1.6950582 | -1.0427000 | -1.0569153 |
| -2.4824611 | 0.3247600 | 2.6462555 |
| -3.1028146 | -1.9462386 | 1.8995821 |
| -12.3677205 | 1.3910065 | 0.6651268 |
| 1.1078410 | 0.0840822 | 0.0083094 |
| -2.4689774 | 1.0598615 | 0.8666068 |
| -4.7507124 | -2.4994718 | 1.8188853 |
| 2.9143881 | 0.7589273 | -0.0020210 |
| 10.8901348 | 0.6910325 | -0.0733066 |
| 7.4502535 | -1.6745534 | 1.5207584 |
| 3.0392482 | 0.2867480 | 0.4437011 |
| 0.1236512 | 0.5871227 | 0.9852812 |
| 1.1161896 | 1.4745978 | -0.1572281 |
| 4.0840659 | -0.5707977 | -1.3121261 |
| -3.6363312 | 2.9033866 | 2.2115656 |
| 2.1411112 | 0.2129841 | 1.1046681 |
| -6.7247486 | -1.1685444 | 0.7052560 |
| 9.5401365 | -1.0525107 | 0.1222523 |
| 7.7210618 | -0.5660578 | -0.6040249 |
| 0.0545377 | 0.0543478 | -2.0738161 |
| -1.3360557 | 1.3337570 | -0.3332716 |
| -6.5292047 | -4.7140670 | -1.0077187 |
| -1.2088945 | -0.3840765 | 0.0672767 |
| 1.1575366 | -3.0412110 | -0.7216841 |
| 5.4501288 | 1.1010765 | -1.8858546 |
| -9.1598040 | 1.7626226 | -0.5515348 |
| -2.6481041 | 1.6918439 | -0.4160223 |
| -11.6258481 | -2.1221543 | 1.8513593 |
| -8.8263997 | 0.2279249 | 0.4842140 |
| -9.1348242 | 1.7749109 | 0.1416037 |
| 9.6951070 | 1.9654105 | 1.8907298 |
| 6.9411713 | 1.1689562 | -0.0095299 |
| 2.5077391 | -0.9239916 | 1.4956395 |
| 6.3270184 | 0.7969512 | -0.0908723 |
| -5.1650281 | -0.0210953 | 0.6276367 |
| -5.4874222 | -1.0274393 | -0.7216906 |
| -3.4164256 | -0.8901704 | -0.5667387 |
| 1.4466480 | -0.1340963 | -0.5298624 |
| -4.8109201 | 0.5700137 | 1.1204687 |
| 10.7938525 | -0.4003214 | 1.7985971 |
| 5.0196369 | 1.3388509 | -0.2742792 |
| -1.3772594 | 0.8511564 | -2.1826263 |
| -0.0063484 | 2.4461201 | 0.0915179 |
| 0.5489903 | 0.9206064 | 0.9453235 |
| -3.5272661 | -1.3606901 | 0.3235235 |
| 2.3914772 | -0.1382935 | -1.6123655 |
| -1.7845937 | -0.3468150 | -0.3698477 |
| 6.9872179 | 0.6721005 | -0.8309894 |
| -5.5661615 | 0.2002503 | 0.6490581 |
| 7.5336524 | -1.1574562 | -1.4459233 |
| 5.9445063 | 3.8235580 | 0.2419958 |
| 7.9504233 | -1.2256532 | 2.5128913 |
| 6.7795563 | -1.5435486 | -0.1006525 |
| -4.7633349 | 0.4146280 | -0.1422369 |
| 1.4419295 | -2.9563498 | 0.7375836 |
| 4.0477100 | 2.5341435 | 0.0269175 |
| -4.5113107 | 1.2873590 | -0.9940955 |
| -0.8200037 | -0.2942686 | 0.3407070 |
| 1.3488877 | -1.1933759 | -0.7381858 |
| -3.7409799 | -0.9351121 | 0.8673300 |
| -5.2362948 | 0.7201380 | -1.7539780 |
| 0.5595167 | 0.6898640 | -0.6064697 |
| 14.4875425 | -0.8781494 | -0.5642109 |
| 6.4897637 | -0.7047330 | 0.1389851 |
| -2.0727333 | -1.0004420 | 1.1516469 |
| 1.0967904 | 2.1698808 | -0.4088451 |
| 5.4645312 | 1.9802854 | -2.8578111 |
| -4.1894865 | -1.6904811 | -1.6055205 |
| 1.8655986 | 1.2491076 | 0.8285276 |
| -5.3938465 | -1.9606164 | -1.3022850 |
| 5.6845843 | -0.3806324 | 0.5828057 |
| 0.1026435 | 0.1414442 | 1.3012888 |
| -4.0750025 | -0.2408023 | 0.7345394 |
| -4.9159993 | -0.3809253 | -0.5540456 |
| -0.4158528 | 0.1394161 | 0.1982454 |
| -2.3213729 | -1.8103196 | 0.8009044 |
| 1.6067434 | 0.5195744 | -0.2828318 |
| -7.0005917 | -1.5725363 | 0.0461124 |
| -6.5120267 | 2.0213922 | 0.1701163 |
| 4.1423046 | -0.0267610 | 1.0533258 |
| 9.3884820 | -0.6926981 | 0.8270415 |
| 0.6173668 | -1.5265428 | -0.6292293 |
| -9.6608610 | 0.7162710 | -0.0501053 |
| 10.8417526 | 0.1256019 | 1.5415506 |
| 13.6455043 | 1.8913947 | 0.6580715 |
| 6.6311309 | 0.1658878 | 0.1891414 |
| -7.9477594 | 1.4796024 | -0.2793544 |
| -9.9761368 | -1.6021350 | 0.2978928 |
| 0.5444993 | -0.1783095 | 0.2626955 |
| 4.7751467 | 2.6498725 | -0.1413199 |
| -2.9638482 | -0.3543837 | 0.3379057 |
| -5.2484770 | 0.4123857 | -1.9573298 |
| 4.1748351 | 1.2410888 | 0.7030369 |
| 2.3202074 | -2.7580584 | -1.3259287 |
| -5.2739964 | -1.7327195 | -0.7250353 |
| 2.4046805 | 0.8761023 | 1.0290006 |
| -3.5340017 | 0.1228034 | 0.7629376 |
| 3.8415894 | -1.1833940 | 0.2004492 |
| -3.1899214 | 1.8107410 | 1.7782990 |
| -4.0812123 | 0.3232561 | 0.3894195 |
| -12.0736657 | -0.5188772 | 0.6692713 |
| 10.9963694 | 1.5382792 | 0.6250619 |
| 9.1286625 | -0.5586797 | -0.2889436 |
| -14.0225348 | 1.4604405 | -1.1159476 |
| -5.8825678 | 0.8663197 | -0.5791133 |
| 3.5142838 | -1.1268946 | -0.2531370 |
| 8.2458125 | 0.1938679 | 0.0627443 |
| -1.1764531 | -1.4330659 | -1.2837846 |
| -2.3897278 | 0.8977936 | 0.5166809 |
| -5.5064819 | 1.4979442 | 0.0848200 |
| -6.9801589 | -0.3336988 | -1.4253984 |
| 2.7377506 | 1.3486615 | 0.6952522 |
| -4.8763912 | 1.0219988 | -0.8186205 |
| 4.0900988 | 4.2112602 | 0.3272883 |
| -0.6016983 | -1.3870476 | -1.0155317 |
| 3.4980331 | -0.1818083 | 0.4100040 |
| -8.5445369 | 2.3745013 | -0.1766611 |
| 1.2864200 | -2.6006597 | -2.2487511 |
| -0.4791643 | 0.9197185 | 0.5231637 |
| -0.5400992 | -1.9909795 | 1.3160565 |
| 2.6540984 | 1.4375120 | 1.0388430 |
| -1.3026661 | 0.2159475 | 0.4012225 |
| -2.4464838 | 2.4718086 | 0.8949823 |
| 1.9722229 | 0.7099248 | -0.1768105 |
| -3.9874881 | -1.8429891 | -1.4463085 |
| -10.8657080 | -0.8720935 | -0.0204422 |
| 2.7822540 | 1.3945064 | -0.1451745 |
| -6.1889260 | -0.3082639 | -0.1741271 |
| -3.9990695 | 0.1752288 | 0.4590572 |
| 3.7920848 | -0.0252184 | 0.0141958 |
| 3.8231405 | -0.5729438 | -0.7373176 |
| -4.6495784 | 1.4019123 | 0.1178246 |
| -6.1410739 | -0.6112695 | 1.9117948 |
| -11.5139246 | -0.5927819 | -1.0513924 |
| 12.2848964 | 2.1541083 | -1.7029174 |
| 4.5146124 | -1.6787726 | 1.6035855 |
| -6.0319482 | -0.1953603 | -1.4373893 |
| -0.1579322 | 1.4903767 | 0.0184529 |
| 3.2608083 | 0.8504693 | -0.0667993 |
| 7.5266172 | -1.0169902 | 0.0427589 |
| -1.6072449 | 0.1288171 | -1.1894934 |
| -9.6915355 | -3.0388722 | 0.2338303 |
| 2.1521213 | 1.1849271 | -0.0869757 |
| -4.9490163 | 1.3779022 | -0.4384216 |
| -8.2287031 | 0.3831815 | 0.5628046 |
| 8.2211822 | 2.2525316 | 0.9855492 |
| 7.0786664 | 0.1691467 | 0.0020536 |
| 10.4355042 | 1.0361053 | -1.2409777 |
| -0.8653474 | -1.7228683 | 0.6369202 |
| 9.3370196 | -0.8530566 | -0.7539352 |
| -0.7230570 | 1.6034102 | -1.4371997 |
| 2.7818191 | 0.5385680 | -0.3077421 |
| 7.4559348 | -1.4560478 | -1.1795155 |
| -0.0474497 | 0.6314953 | 1.8115563 |
| -1.4024118 | 0.2486279 | 1.3822598 |
| 3.7283605 | -0.7438085 | -0.2299785 |
| 4.1115185 | 1.1384149 | -1.7328503 |
| -5.3976902 | -2.9842702 | 0.9331951 |
| 3.5751372 | 0.9775690 | 0.1220483 |
| -1.5675297 | 0.1951483 | 0.3760802 |
| -5.7920691 | -0.3868012 | 0.0880003 |
| 2.8892727 | -0.1001260 | -0.3950990 |
| 3.4850372 | 0.9445834 | -0.7896257 |
| -11.3966634 | 1.1703421 | -0.6283300 |
| -13.4769022 | 2.9162238 | -0.2415889 |
| 5.0872702 | 2.4636200 | 0.5958202 |
| 9.6411940 | 0.5108973 | 0.6465302 |
| -2.1719384 | 1.9967426 | -1.0355173 |
| 13.2944012 | -0.2581166 | -0.9028117 |
| -3.4047740 | -0.5001170 | 0.5197776 |
| -9.8008001 | -0.6556580 | -1.8216446 |
| -6.3367257 | 2.2384854 | -0.8777973 |
| -2.6175056 | 1.7820757 | -1.1515706 |
| 1.5442203 | 0.5753266 | 1.9766608 |
| -5.9936697 | 0.9695795 | 0.4081822 |
| 8.9541620 | 0.6939543 | -0.4915050 |
| -6.2534480 | -0.2448676 | 1.3384188 |
| 0.2814408 | 0.2847457 | -0.1443592 |
| 0.3516918 | -0.6147799 | -1.4371389 |
| -2.3156448 | -1.7564826 | 2.0491803 |
| 1.1299607 | -0.2248956 | 0.5067188 |
| -0.9938364 | -1.6903298 | -0.1031155 |
| -7.0302521 | -0.9598478 | -0.2447693 |
| -4.2447747 | 1.7138086 | 0.0727486 |
| 5.1141486 | -0.5909624 | 0.4991938 |
| 2.7045779 | -0.6960988 | 0.3945884 |
| 2.5985947 | -1.3223707 | -1.0309881 |
| -7.1130901 | -0.8706080 | -0.5588401 |
| -3.7928685 | -1.2208796 | -2.0674523 |
| 9.9556491 | -0.7693874 | -0.0302499 |
| 6.9784091 | 3.1863368 | -0.2561742 |
| 8.8037697 | 1.5758635 | -1.5978014 |
| 4.8496609 | -1.1737621 | -0.1506123 |
| 2.4810824 | 0.0310302 | 0.3134863 |
| 9.6822122 | 1.0676730 | 0.5827167 |
| -1.3412705 | 0.2362711 | 0.8046160 |
| -9.2471757 | -1.5523334 | 0.5419119 |
| 6.1392045 | -0.3413549 | -1.5522980 |
| -0.4271513 | 0.3055907 | 1.4673554 |
| 8.8316276 | -0.0045963 | -0.8950702 |
| -4.5764864 | -1.1683509 | -0.7644098 |
| 3.3492137 | -1.3887593 | 1.0630318 |
| -4.1497389 | -0.3798463 | 0.9118572 |
| -3.4678710 | 1.3738986 | 0.2136249 |
| -0.3506330 | 3.2726460 | 0.9893660 |
| -4.3430465 | -0.0575051 | 0.0274094 |
| -7.4559649 | -0.3304470 | 1.7925481 |
| 1.2526839 | 1.1779646 | 0.2939054 |
| -4.1476501 | -0.3492701 | -0.0261065 |
| 5.3961828 | 1.2675544 | 0.6343125 |
| 3.7861647 | -0.7643412 | 0.0988616 |
| 5.4978823 | 0.8657121 | -0.7431460 |
| 3.6633808 | -1.1165800 | 0.8816451 |
| 4.3378223 | 2.0412582 | -0.2666598 |
| -7.2330896 | 0.2490346 | -0.1236235 |
| -0.7528408 | 0.2461529 | -0.2958959 |
| -0.9669044 | 0.4810820 | -0.0337772 |
| 18.5513709 | -0.3897001 | -0.0283965 |
| -6.5254264 | -2.8195373 | 0.6697877 |
| -4.2636912 | -1.4183375 | -1.2809679 |
| 2.9904283 | -2.9395235 | -0.6383072 |
| -15.2850275 | -0.7373667 | -1.2432554 |
| 2.4382106 | -2.3867573 | -0.7308026 |
| 0.5982708 | 1.2821269 | 0.9660959 |
| 3.9537452 | 1.7761792 | 0.0213218 |
| 3.3857868 | -3.4982792 | 0.8808273 |
| -9.9575213 | 0.2911823 | -1.9136573 |
| 7.8708156 | -0.1973531 | 0.3362941 |
| 6.3279989 | 0.5121376 | -1.3073720 |
| -3.4920407 | -1.6243292 | -0.1670603 |
| -4.7142614 | 0.0688159 | -0.1580095 |
| 6.0203653 | -0.4605614 | 0.5833991 |
| -4.1996794 | 0.1510403 | -0.3800481 |
| -7.7126745 | 1.0674943 | -1.0454957 |
| 4.9985158 | 0.8497798 | 0.5888681 |
| 3.1498347 | -0.6154296 | -0.0640294 |
| -0.7707355 | 0.3539480 | -1.9800288 |
| -2.2278720 | 0.9322608 | 0.8122415 |
| 5.7145877 | -0.9108463 | 1.1607479 |
| 1.6102364 | -0.9065654 | -0.0352262 |
| 5.2750431 | 0.4005857 | 0.0041416 |
| -10.1689468 | 0.2836452 | -0.5613754 |
| 4.7877476 | -1.9740269 | -1.1796054 |
| -1.7140151 | -0.7336470 | -1.1369485 |
| -3.5437551 | 0.6277266 | 1.3040092 |
| 4.4394317 | -0.7148357 | 2.3795716 |
| -4.0121067 | -0.0304447 | 0.7374627 |
| 2.3020527 | -2.5067901 | -0.0524970 |
| 4.1892834 | 2.5326791 | 1.2279811 |
| 4.5419391 | -1.0727045 | 0.2307581 |
| -9.5636530 | 1.3001560 | 1.5049160 |
| -2.1129284 | -1.1150714 | -0.7275411 |
| -8.4491929 | -1.2339318 | 0.6317343 |
| -6.8455764 | -0.2560662 | -0.0877264 |
| -5.1487883 | 2.4455434 | -0.7498705 |
| 1.0353857 | 1.0295083 | 0.0532165 |
| -0.1382711 | 0.5015641 | 1.0603914 |
| 2.3762595 | -0.1517352 | 0.7364774 |
| 2.8256214 | -2.6366012 | -1.0424894 |
| -12.0172285 | -0.8604509 | -1.2362828 |
| 7.1634130 | 2.5201949 | 0.4822315 |
| 7.2510201 | -1.1122944 | 0.5233232 |
| 0.2399254 | 0.0520773 | 0.1234435 |
| 4.2403822 | 1.5261575 | 2.2764881 |
| 6.5613036 | -0.3742145 | -0.0478778 |
| 4.6452067 | 1.1333211 | 0.4244570 |
| 3.0837884 | 0.1834674 | 0.5783309 |
| -0.2784320 | 0.8465852 | -0.9695076 |
| -1.0753100 | 0.0985742 | -0.0420025 |
| -0.6589827 | -2.3021062 | 0.3393187 |
| 2.4255572 | 0.2840482 | -0.0711093 |
| 3.0643326 | 0.7434459 | -0.1484578 |
| 1.8584683 | 0.2824338 | 0.8468885 |
| -1.5078660 | 2.8009782 | -1.9600358 |
| -3.3835697 | 0.1847263 | 0.6961531 |
| 8.1292963 | 0.1187115 | 1.5513818 |
| -3.9206583 | -1.2895303 | 0.3246259 |
| -1.0752526 | -0.6764675 | 0.7019336 |
| 13.4063967 | 0.9899954 | 1.2524669 |
| 1.8003518 | 0.6415596 | 0.3242558 |
| -5.4934688 | 0.4552829 | 0.0847786 |
| -6.8396914 | -0.6012039 | -0.9716206 |
| 2.5234704 | -0.4298667 | 0.2899072 |
| 5.1646092 | 1.8108663 | 0.8593528 |
| 8.3076251 | -1.3542169 | 0.1713719 |
| -2.8946553 | 0.4860524 | -0.6157100 |
| 3.2615076 | 1.9762856 | -1.2113319 |
| -3.3935349 | 1.2328083 | 0.2669153 |
| 5.0003882 | -0.0157195 | 1.8154718 |
| 7.9055943 | 1.1268597 | -0.5427247 |
| 1.8225493 | 1.2063981 | -0.3084495 |
| 1.0546526 | 0.2204896 | -0.0303897 |
| 1.0678129 | -1.6776576 | 0.8647669 |
| -12.5299477 | -0.3984434 | -0.0905205 |
| -5.8247353 | 0.4266440 | 1.4556468 |
| 0.4573482 | 0.1355247 | -0.7339516 |
| -10.3604215 | -0.2463889 | -1.7756567 |
| 2.5202747 | 0.6650321 | -0.9393787 |
| -6.0394616 | 1.3861684 | -1.0487564 |
| 8.9002225 | 0.4057583 | 0.6197410 |
| 2.1451084 | 1.7151341 | -0.0788849 |
Let’s say we call all these new points scores
and here’s how they look when are plotted onto a graph.
plot_ly(score_df, x = ~PC1, y = ~PC2, z = ~PC3) %>%
add_markers(size= 0.5) %>%
layout(scene = list(xaxis = list(title = 'PC1'),
yaxis = list(title = 'PC2', range= c(-20,20)),
zaxis = list(title = 'PC3', range= c(-20,20))))
As we saw on the scree plot however, 97.22% of the total
variance is given by only two components, then maybe we don’t need to
keep the third one.
In fact the contribution of PC3 in this case is
minimal.
Score plot
ggplot(score_df, aes(PC1,PC2))+geom_point(color='cornflowerblue', size = 1, shape= 1)+
ylim(-10,10)+
theme_classic()+
xlab(paste('PC1 - ', eig_df1$Var[1]))+
ylab(paste('PC2 - ', eig_df1$Var[2]))
Loading plot
loading_vectors = t(loading_vectors) %>% data.frame()
ggplot(loading_vectors, aes(P1,P2))+geom_point(color='cornflowerblue', size = 1, shape= 1)+
ylim(-2,2)+
xlim(-5,5)+
theme_bw()+
geom_segment(aes(xend=0,yend=0), alpha= 1, color= 'darkred')+
xlab(paste('PC1 - ', eig_df1$Var[1]))+
ylab(paste('PC2 - ', eig_df1$Var[2]))+
geom_label(label= colnames(df), color='black', size=3, nudge_y = 0.3, fontface='bold')+
geom_vline(xintercept= 0, linetype="dashed", alpha = 0.3)+
geom_hline(yintercept= 0, linetype="dashed", alpha = 0.3)
Biplot
ggplot(loading_vectors, aes(P1,P2))+
geom_point(data=score_df, aes(PC1,PC2),color='cornflowerblue', size = 1, shape=1)+
geom_point(color='white')+
ylim(-10,10)+
geom_segment(aes(xend=0,yend=0), alpha= 1, color='darkred')+
xlab(paste('PC1 - ', eig_df1$Var[1]))+
ylab(paste('PC2 - ', eig_df1$Var[2]))+
geom_label(label= colnames(df), color='black', size=3, nudge_y = 0.3, fontface='bold')+
theme_bw()+
geom_vline(xintercept= 0, linetype="dashed", alpha = 0.3)+
geom_hline(yintercept= 0, linetype="dashed", alpha = 0.3)
As we see, most of the variation is carried by \(X_1\) and \(X_2\) and they’re clearly correlated one to
each other.