4. Einsum operation by DelayedTensor (original) (raw)
By using multiple arrays or DelayedArray objects as input and writing “,” on the right side of ->, multiplication will be performed.
Hadamard Product
Hadamard Product can also be implemented in einsum, multiplying by the product of each element.
einsum::einsum('i,i->i', arrA, arrA)## [1] 0.07704605 0.10918153 0.29595504DelayedTensor::einsum('i,i->i', darrA, darrA)## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 0.07704605 0.10918153 0.29595504einsum::einsum('ij,ij->ij', arrC, arrC)## [,1] [,2] [,3] [,4]
## [1,] 0.06649487 0.03829802 0.2090746 0.62292755
## [2,] 0.64033113 0.36491696 0.1547516 0.04184106
## [3,] 0.23794107 0.45191493 0.3038045 0.14081255DelayedTensor::einsum('ij,ij->ij', darrC, darrC)## <3 x 4> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4]
## [1,] 0.06649487 0.03829802 0.20907465 0.62292755
## [2,] 0.64033113 0.36491696 0.15475155 0.04184106
## [3,] 0.23794107 0.45191493 0.30380447 0.14081255einsum::einsum('ijk,ijk->ijk', arrE, arrE)## , , 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.02416239 0.4044326 0.7688279 0.13952609
## [2,] 0.02270761 0.6329686 0.6150065 0.02478527
## [3,] 0.13877490 0.5582126 0.7203108 0.07069093
##
## , , 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.32564197 0.1726612 0.03170875 0.1373284
## [2,] 0.19084119 0.1154033 0.05974271 0.2418514
## [3,] 0.04953964 0.1133054 0.37894355 0.7443683
##
## , , 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.8187318719 2.987619e-01 0.581433727 0.0004421427
## [2,] 0.0003065689 6.750267e-01 0.003512518 0.2762386296
## [3,] 0.7063421303 1.774016e-06 0.713009992 0.0503153214
##
## , , 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.6204730 0.1657464 0.4955939 0.087357927
## [2,] 0.1970712 0.3857979 0.5118365 0.202830289
## [3,] 0.2005848 0.2303923 0.7997802 0.005921693
##
## , , 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.02239586 0.17689301 0.0005421553 0.4600547
## [2,] 0.86040209 0.11445206 0.9309201924 0.1202748
## [3,] 0.06658372 0.01083404 0.2671739765 0.3070143DelayedTensor::einsum('ijk,ijk->ijk', darrE, darrE)## <3 x 4 x 5> HDF5Array object of type "double":
## ,,1
## [,1] [,2] [,3] [,4]
## [1,] 0.02416239 0.40443263 0.76882785 0.13952609
## [2,] 0.02270761 0.63296865 0.61500646 0.02478527
## [3,] 0.13877490 0.55821258 0.72031076 0.07069093
##
## ,,2
## [,1] [,2] [,3] [,4]
## [1,] 0.32564197 0.17266116 0.03170875 0.13732838
## [2,] 0.19084119 0.11540329 0.05974271 0.24185143
## [3,] 0.04953964 0.11330543 0.37894355 0.74436832
##
## ,,3
## [,1] [,2] [,3] [,4]
## [1,] 8.187319e-01 2.987619e-01 5.814337e-01 4.421427e-04
## [2,] 3.065689e-04 6.750267e-01 3.512518e-03 2.762386e-01
## [3,] 7.063421e-01 1.774016e-06 7.130100e-01 5.031532e-02
##
## ,,4
## [,1] [,2] [,3] [,4]
## [1,] 0.620473042 0.165746368 0.495593926 0.087357927
## [2,] 0.197071237 0.385797867 0.511836482 0.202830289
## [3,] 0.200584809 0.230392344 0.799780170 0.005921693
##
## ,,5
## [,1] [,2] [,3] [,4]
## [1,] 0.0223958620 0.1768930086 0.0005421553 0.4600547083
## [2,] 0.8604020896 0.1144520582 0.9309201924 0.1202748459
## [3,] 0.0665837178 0.0108340374 0.2671739765 0.3070142605Outer Product
The outer product can also be implemented in einsum, in which the subscripts in the input array are all different, and all of them are kept.
einsum::einsum('i,j->ij', arrA, arrA)## [,1] [,2] [,3]
## [1,] 0.07704605 0.09171698 0.1510039
## [2,] 0.09171698 0.10918153 0.1797577
## [3,] 0.15100386 0.17975768 0.2959550DelayedTensor::einsum('i,j->ij', darrA, darrA)## <3 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 0.07704605 0.09171698 0.15100386
## [2,] 0.09171698 0.10918153 0.17975768
## [3,] 0.15100386 0.17975768 0.29595504einsum::einsum('ij,klm->ijklm', arrC, arrE)## , , 1, 1, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.04008335 0.03041992 0.07107561 0.12268421
## [2,] 0.12438620 0.09390029 0.06114873 0.03179591
## [3,] 0.07582364 0.10449566 0.08567754 0.05832981
##
## , , 2, 1, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.03885794 0.02948994 0.06890272 0.11893357
## [2,] 0.12058353 0.09102962 0.05927932 0.03082386
## [3,] 0.07350559 0.10130107 0.08305825 0.05654658
##
## , , 3, 1, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.09606154 0.0729027 0.1703359 0.29401821
## [2,] 0.29809711 0.2250363 0.1465457 0.07620032
## [3,] 0.18171475 0.2504285 0.2053301 0.13979001
##
## , , 1, 2, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1639899 0.1244547 0.2907862 0.5019285
## [2,] 0.5088917 0.3841671 0.2501731 0.1300842
## [3,] 0.3102114 0.4275151 0.3505260 0.2386403
##
## , , 2, 2, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2051565 0.1556966 0.3637825 0.6279280
## [2,] 0.6366392 0.4806048 0.3129742 0.1627393
## [3,] 0.3880841 0.5348345 0.4385188 0.2985464
##
## , , 3, 2, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1926610 0.1462137 0.3416257 0.5896830
## [2,] 0.5978636 0.4513327 0.2939120 0.1528274
## [3,] 0.3644471 0.5022595 0.4118100 0.2803629
##
## , , 1, 3, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2261042 0.1715942 0.4009269 0.6920434
## [2,] 0.7016441 0.5296776 0.3449309 0.1793560
## [3,] 0.4277099 0.5894445 0.4832943 0.3290298
##
## , , 2, 3, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2022246 0.1534716 0.3585837 0.6189543
## [2,] 0.6275411 0.4737365 0.3085015 0.1604136
## [3,] 0.3825380 0.5271912 0.4322519 0.2942798
##
## , , 3, 3, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2188538 0.1660918 0.3880705 0.6698518
## [2,] 0.6791446 0.5126925 0.3338700 0.1736046
## [3,] 0.4139946 0.5705429 0.4677966 0.3184789
##
## , , 1, 4, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.09632118 0.07309975 0.1707963 0.29481290
## [2,] 0.29890283 0.22564450 0.1469418 0.07640628
## [3,] 0.18220589 0.25110540 0.2058850 0.14016784
##
## , , 2, 4, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.04059672 0.03080953 0.07198592 0.12425550
## [2,] 0.12597929 0.09510292 0.06193189 0.03220314
## [3,] 0.07679475 0.10583399 0.08677486 0.05907688
##
## , , 3, 4, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.06856081 0.05203194 0.1215717 0.20984597
## [2,] 0.21275715 0.16061233 0.1045922 0.05438551
## [3,] 0.12969301 0.17873525 0.1465477 0.09977059
##
## , , 1, 1, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1471514 0.1116756 0.2609281 0.4503902
## [2,] 0.4566385 0.3447206 0.2244852 0.1167271
## [3,] 0.2783588 0.3836176 0.3145338 0.2141366
##
## , , 2, 1, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1126497 0.08549175 0.1997500 0.34479013
## [2,] 0.3495734 0.26389617 0.1718516 0.08935882
## [3,] 0.2130938 0.29367326 0.2407871 0.16392936
##
## , , 3, 1, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.05739453 0.04355767 0.10177172 0.17566903
## [2,] 0.17810608 0.13445392 0.08755762 0.04552792
## [3,] 0.10857033 0.14962521 0.12267993 0.08352128
##
## , , 1, 2, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1071498 0.08131778 0.1899976 0.32795639
## [2,] 0.3325061 0.25101193 0.1634613 0.08499604
## [3,] 0.2026899 0.27933520 0.2290311 0.15592581
##
## , , 2, 2, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.08759981 0.06648096 0.1553316 0.2681192
## [2,] 0.27183877 0.20521359 0.1336370 0.0694881
## [3,] 0.16570812 0.22836915 0.1872433 0.1274764
##
## , , 3, 2, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.08679994 0.06587392 0.1539133 0.26567099
## [2,] 0.26935663 0.20333980 0.1324167 0.06885361
## [3,] 0.16419506 0.22628393 0.1855335 0.12631242
##
## , , 1, 3, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.04591807 0.03484799 0.08142172 0.14054272
## [2,] 0.14249246 0.10756887 0.07004983 0.03642428
## [3,] 0.08686089 0.11970655 0.09814918 0.06682058
##
## , , 2, 3, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.06302844 0.04783333 0.11176173 0.19291288
## [2,] 0.19558915 0.14765205 0.09615236 0.04999698
## [3,] 0.11922770 0.16431257 0.13472231 0.09171980
##
## , , 3, 3, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1587382 0.1204690 0.2814738 0.4858543
## [2,] 0.4925945 0.3718641 0.2421613 0.1259182
## [3,] 0.3002769 0.4138239 0.3393004 0.2309978
##
## , , 1, 4, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.09555958 0.07252176 0.1694458 0.29248185
## [2,] 0.29653943 0.22386035 0.1457799 0.07580215
## [3,] 0.18076521 0.24911994 0.2042571 0.13905955
##
## , , 2, 4, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1268144 0.09624152 0.2248666 0.3881442
## [2,] 0.3935289 0.29707859 0.1934603 0.1005948
## [3,] 0.2398883 0.33059986 0.2710637 0.1845419
##
## , , 3, 4, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2224785 0.1688426 0.3944978 0.6809461
## [2,] 0.6903928 0.5211839 0.3393997 0.1764799
## [3,] 0.4208513 0.5799924 0.4755443 0.3237536
##
## , , 1, 1, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2333270 0.1770757 0.4137343 0.7141503
## [2,] 0.7240577 0.5465978 0.3559495 0.1850854
## [3,] 0.4413728 0.6082739 0.4987328 0.3395405
##
## , , 2, 1, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.004515004 0.003426512 0.008005984 0.013819197
## [2,] 0.014010910 0.010576965 0.006887816 0.003581504
## [3,] 0.008540804 0.011770431 0.009650751 0.006570293
##
## , , 3, 1, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2167213 0.1644734 0.3842893 0.6633249
## [2,] 0.6725272 0.5076970 0.3306169 0.1719131
## [3,] 0.4099607 0.5649837 0.4632385 0.3153757
##
## , , 1, 2, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1409473 0.1069672 0.2499271 0.4314013
## [2,] 0.4373861 0.3301868 0.2150206 0.1118057
## [3,] 0.2666228 0.3674438 0.3012727 0.2051083
##
## , , 2, 2, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2118627 0.1607861 0.3756740 0.6484541
## [2,] 0.6574501 0.4963151 0.3232049 0.1680590
## [3,] 0.4007700 0.5523175 0.4528533 0.3083054
##
## , , 3, 2, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.0003434574 0.0002606555 0.0006090170 0.0010512295
## [2,] 0.0010658131 0.0008045921 0.0005239577 0.0002724458
## [3,] 0.0006497009 0.0008953794 0.0007341349 0.0004998037
##
## , , 1, 3, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1966275 0.1492239 0.3486589 0.6018231
## [2,] 0.6101722 0.4606246 0.2999630 0.1559737
## [3,] 0.3719502 0.5125998 0.4202882 0.2861349
##
## , , 2, 3, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.01528281 0.01159838 0.02709942 0.04677653
## [2,] 0.04742546 0.03580192 0.02331454 0.01212301
## [3,] 0.02890973 0.03984168 0.03266678 0.02223975
##
## , , 3, 3, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2177418 0.1652479 0.3860988 0.6664485
## [2,] 0.6756941 0.5100877 0.3321738 0.1727226
## [3,] 0.4118912 0.5676441 0.4654198 0.3168608
##
## , , 1, 4, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.005422197 0.004114996 0.009614615 0.016595868
## [2,] 0.016826102 0.012702179 0.008271775 0.004301130
## [3,] 0.010256895 0.014135447 0.011589863 0.007890452
##
## , , 2, 4, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1355303 0.1028562 0.2403216 0.4148212
## [2,] 0.4205760 0.3174967 0.2067568 0.1075087
## [3,] 0.2563757 0.3533219 0.2896939 0.1972254
##
## , , 3, 4, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.05784212 0.04389735 0.10256538 0.17703898
## [2,] 0.17949503 0.13550245 0.08824043 0.04588297
## [3,] 0.10941701 0.15079206 0.12363664 0.08417261
##
## , , 1, 1, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2031213 0.1541522 0.3601738 0.6216991
## [2,] 0.6303239 0.4758373 0.3098696 0.1611250
## [3,] 0.3842343 0.5295291 0.4341687 0.2955848
##
## , , 2, 1, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1144737 0.08687599 0.2029842 0.35037281
## [2,] 0.3552335 0.26816905 0.1746341 0.09080567
## [3,] 0.2165441 0.29842827 0.2446858 0.16658362
##
## , , 3, 1, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1154897 0.08764703 0.2047857 0.35348240
## [2,] 0.3583862 0.27054907 0.1761840 0.09161158
## [3,] 0.2184659 0.30107685 0.2468574 0.16806206
##
## , , 1, 2, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1049823 0.07967282 0.1861541 0.32132224
## [2,] 0.3257799 0.24593426 0.1601546 0.08327667
## [3,] 0.1985897 0.27368460 0.2243981 0.15277162
##
## , , 2, 2, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1601673 0.1215537 0.2840080 0.4902286
## [2,] 0.4970296 0.3752122 0.2443416 0.1270519
## [3,] 0.3029805 0.4175498 0.3423553 0.2330776
##
## , , 3, 2, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1237736 0.09393386 0.2194748 0.37883735
## [2,] 0.3840929 0.28995530 0.1888215 0.09818279
## [3,] 0.2341363 0.32267281 0.2645642 0.18011700
##
## , , 1, 3, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1815336 0.1377689 0.3218946 0.5556250
## [2,] 0.5633331 0.4252654 0.2769367 0.1440006
## [3,] 0.3433979 0.4732508 0.3880253 0.2641701
##
## , , 2, 3, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1844844 0.1400083 0.3271269 0.5646566
## [2,] 0.5724900 0.4321780 0.2814383 0.1463413
## [3,] 0.3489798 0.4809434 0.3943326 0.2684641
##
## , , 3, 3, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2306107 0.1750143 0.4089178 0.7058365
## [2,] 0.7156285 0.5402345 0.3518057 0.1829307
## [3,] 0.4362345 0.6011926 0.4929268 0.3355877
##
## , , 1, 4, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.07621584 0.05784147 0.1351456 0.23327593
## [2,] 0.23651216 0.17854520 0.1162703 0.06045782
## [3,] 0.14417364 0.19869160 0.1629102 0.11091029
##
## , , 2, 4, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1161343 0.08813625 0.2059288 0.35545545
## [2,] 0.3603867 0.27205921 0.1771674 0.09212293
## [3,] 0.2196854 0.30275739 0.2482353 0.16900015
##
## , , 3, 4, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.01984344 0.01505952 0.03518630 0.06073537
## [2,] 0.06157795 0.04648576 0.03027195 0.01574071
## [3,] 0.03753683 0.05173105 0.04241505 0.02887644
##
## , , 1, 1, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.03859028 0.02928681 0.06842812 0.11811435
## [2,] 0.11975294 0.09040260 0.05887100 0.03061154
## [3,] 0.07299928 0.10060330 0.08248614 0.05615709
##
## , , 2, 1, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2391910 0.1815260 0.4241324 0.7320985
## [2,] 0.7422548 0.5603350 0.3648953 0.1897370
## [3,] 0.4524655 0.6235612 0.5112671 0.3480739
##
## , , 3, 1, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.06653928 0.05049777 0.1179871 0.20365862
## [2,] 0.20648396 0.15587664 0.1015083 0.05278194
## [3,] 0.12586898 0.17346520 0.1422267 0.09682883
##
## , , 1, 2, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1084550 0.08230827 0.1923118 0.33195109
## [2,] 0.3365562 0.25406940 0.1654523 0.08603134
## [3,] 0.2051587 0.28273767 0.2318208 0.15782508
##
## , , 2, 2, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.08723804 0.0662064 0.1546901 0.26701187
## [2,] 0.27071612 0.2043661 0.1330851 0.06920113
## [3,] 0.16502377 0.2274260 0.1864700 0.12694993
##
## , , 3, 2, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.02684042 0.02036964 0.04759330 0.08215120
## [2,] 0.08329088 0.06287705 0.04094611 0.02129102
## [3,] 0.05077265 0.06997187 0.05737098 0.03905853
##
## , , 1, 3, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.006004211 0.004556696 0.01064664 0.018377255
## [2,] 0.018632201 0.014065620 0.00915966 0.004762810
## [3,] 0.011357861 0.015652734 0.01283391 0.008737406
##
## , , 2, 3, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2488000 0.1888184 0.4411710 0.7615089
## [2,] 0.7720733 0.5828452 0.3795541 0.1973593
## [3,] 0.4706423 0.6486114 0.5318061 0.3620570
##
## , , 3, 3, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1332880 0.1011545 0.2363457 0.4079584
## [2,] 0.4136180 0.3122440 0.2033361 0.1057300
## [3,] 0.2521342 0.3474765 0.2849011 0.1939625
##
## , , 1, 4, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1749036 0.1327373 0.3101383 0.5353324
## [2,] 0.5427590 0.4097338 0.2668224 0.1387414
## [3,] 0.3308563 0.4559667 0.3738538 0.2545221
##
## , , 2, 4, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.08942964 0.06786964 0.1585762 0.2737198
## [2,] 0.27751708 0.20950019 0.1364284 0.0709396
## [3,] 0.16916952 0.23313944 0.1911545 0.1301392
##
## , , 3, 4, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1428806 0.1084345 0.2533553 0.4373187
## [2,] 0.4433856 0.3347159 0.2179700 0.1133393
## [3,] 0.2702800 0.3724840 0.3054052 0.2079218DelayedTensor::einsum('ij,klm->ijklm', darrC, darrE)## <3 x 4 x 3 x 4 x 5> HDF5Array object of type "double":
## ,,1,1,1
## [,1] [,2] [,3] [,4]
## [1,] 0.04008335 0.03041992 0.07107561 0.12268421
## [2,] 0.12438620 0.09390029 0.06114873 0.03179591
## [3,] 0.07582364 0.10449566 0.08567754 0.05832981
##
## ,,2,1,1
## [,1] [,2] [,3] [,4]
## [1,] 0.03885794 0.02948994 0.06890272 0.11893357
## [2,] 0.12058353 0.09102962 0.05927932 0.03082386
## [3,] 0.07350559 0.10130107 0.08305825 0.05654658
##
## ,,3,1,1
## [,1] [,2] [,3] [,4]
## [1,] 0.09606154 0.07290270 0.17033588 0.29401821
## [2,] 0.29809711 0.22503625 0.14654566 0.07620032
## [3,] 0.18171475 0.25042853 0.20533007 0.13979001
##
## ...
##
## ,,1,4,5
## [,1] [,2] [,3] [,4]
## [1,] 0.1749036 0.1327373 0.3101383 0.5353324
## [2,] 0.5427590 0.4097338 0.2668224 0.1387414
## [3,] 0.3308563 0.4559667 0.3738538 0.2545221
##
## ,,2,4,5
## [,1] [,2] [,3] [,4]
## [1,] 0.08942964 0.06786964 0.15857623 0.27371977
## [2,] 0.27751708 0.20950019 0.13642844 0.07093960
## [3,] 0.16916952 0.23313944 0.19115448 0.13013918
##
## ,,3,4,5
## [,1] [,2] [,3] [,4]
## [1,] 0.1428806 0.1084345 0.2533553 0.4373187
## [2,] 0.4433856 0.3347159 0.2179700 0.1133393
## [3,] 0.2702800 0.3724840 0.3054052 0.2079218