find_markers: Find marker genes for each cell type.
In dtangle: Cell Type Deconvolution from Gene Expressions
Description
Usage
Arguments
Value
Examples
View source: R/find_markers_fn.R
Description
Find marker genes for each cell type.
Usage
1
2
Arguments
Y
Expression matrix.
(Required) Two-dimensional numeric. Must implement as.matrix.
Each row contains expression measurements for a particular sample. Each columm contains the measurements of the same gene over all individuals. Can either contain just the mixture samples to be deconvolved or both the mixture samples and the reference samples. See pure_samples and references for more details.
references
Cell-type reference expression matrix.
(Optional) Two-dimensional numeric. Must implement as.matrix. Must have same number of columns as Y. Columns must correspond to columns of Y.
Each row contains expression measurements for a reference profile of a particular cell type. Columns contain measurements of reference profiles of a gene. Optionally may merge this matrix with Y and use pure_samples to indicate which rows of Y are pure samples. If pure_samples is not specified references must be specified. In this case each row of references is assumed to be a distinct cell-type. If both pure_samples and references are specified then multiple rows of references may refer be the same cell type, and pure_samples specifies to which cell-type each row of references corresponds.
pure_samples
The pure sample indicies.
(Optional) List of one-dimensional integer. Must implement as.list.
The i-th element of the top-level list is a vector of indicies (rows of Y or references) that are pure samples of type i. If references is not specified then this argument identifies which rows of Y correspond to pure reference samples of which cell-types. If references is specified then this makes same idenficiation but for the references matrix instead.
data_type
Type of expression measurements.
(Optional) One-dimensional string.
An optional string indicating the type of the expression measurements. This is used to set gamma to a pre-determined value based upon the data type. Valid values are for probe-level microarray as “microarray-probe”, gene-level microarray as “microarray-gene” or rna-seq as “rna-seq”. Alternatively can set gamma directly.
gamma
Expression adjustment term.
(Optional) One-dimensional positive numeric.
If provided as a single positive number then that value will be used for gamma and over-ride the value of gamma chosen by the data_type argument. If neither gamma nor data_type are specified then gamma will be set to one.
marker_method
Method used to rank marker genes.
(Optional) One-dimensional string.
The method used to rank genes as markers. If not supplied defaults to “ratio”. Only used if markers are not provided to argument “markers”. Options are
'ratio' selects and ranks markers by the ratio of the mean expression of each gene in each cell type to the mean of that gene in all other cell types.
'regression ' selects and ranks markers by estimated regression coefficients in a series of regressions with single covariate that is indicator of each type.
'diff' selects and ranks markers based upon the difference, for each cell type, between the median expression of a gene by each cell type and the median expression of that gene by the second most highly expressed cell type.
'p.value' selects and ranks markers based upon the p-value of a t-test between the median expression of a gene by each cell type and the median expression of that gene by the second most highly expressed cell type.
Value
List with four elements. “L” is respective ranked markers for each cell type and “V” is the corresponding values of the ranking method (higher are better) used to determine markers and sort them, “M” is the matrix used to create the other two arguments after sorting and subsetting, and “sM” is a sorted version of M.
Examples
1
2
3
4
5
6
7
truth = shen_orr_ex$annotation$mixture
pure_samples <- lapply(1:3, function(i) {
which(truth[, i] == 1)
})
Y <- shen_orr_ex$data$log
find_markers(Y=Y,pure_samples=pure_samples,
data_type='microarray-gene',marker_method='ratio')
Example output
$L
$L[[1]]
[1] 120 253 479 446 529 90 61 66 174 77 316 162 215 551 488 473 574 91
[19] 185 41 439 558 187 89 54 365 475 16 441 81 323 392 130 210 121 207
[37] 24 135 428 86 486 44 461 133 332 459 403 217 442 537 575 431 358 105
[55] 107 170 416 9 119 112 515 360 34 189 541 450 161 425 110 140 118 153
[73] 599 82 58 134 72 116 221 43 423 26 195 589 103 132 146 173 549 94
[91] 468 499 201 445 115 67 398 467 443 254 192 219 595 455 568 193 212 409
[109] 211 65 93 92 51 31 59 222 183 440 477 565 137 79 88 460 550 68
[127] 52 591 220 175 39 123 528 470 197 114 576 596 512 87 138 415 334 330
[145] 144 167 527 6 364 218 402 385 216 235 317 191 42 71 11 457 449 414
[163] 590 331 117 136 85 139 326 466 295 518 453 80 33 530 311 401 209 463
[181] 124 377 421 177 49 53 190 545 20 213 464 15 342 281 390 579 319 351
[199] 437 539
$L[[2]]
[1] 180 95 429 14 451 430 56 396 598 349 165 313 259 23 593 12 21 30
[19] 151 84 296 434 152 184 264 18 419 143 577 341 588 186 570 388 199 597
[37] 196 22 278 36 567 480 310 363 306 277 352 424 373 375 247 344 246 395
[55] 525 444 62 267 472 276 100 493 366 542 357 148 294 320 83 413 244 485
[73] 456 181 348 29 534 378 581 172 125 101 393 46 301 129 131 288 370 32
[91] 237 356 359 353 580 374 28 329 19 399 258 99 502 45 478 166 522 145
[109] 263 452 501 408 292 496 507 435 157 291 327 582 289 338 379 520 384 532
[127] 243 232 492 410 233 285 141 498 300 367 307 154 372 369 321 509 600 108
[145] 257 337 572 160 387 57 283 17 422 411 76 482 113 571 241 521 381 312
[163] 505 569 274 149 256 122 128 98 78 293 547 97 510 412 74 438 245 427
[181] 242 386 584 309 299 188 484 426 371 182 287 333 325 304 594 514 583 406
[199] 540 328
$L[[3]]
[1] 1 168 481 109 454 206 368 487 179 476 227 405 262 504 586 339 552 362
[19] 447 7 35 290 249 64 230 345 503 382 433 490 265 169 73 250 592 255
[37] 553 150 269 205 239 279 25 546 60 555 252 361 407 354 75 560 55 27
[55] 50 271 158 5 305 69 336 554 282 420 394 266 376 231 489 8 268 391
[73] 13 223 270 102 111 465 260 156 491 383 495 389 563 272 208 346 557 526
[91] 448 397 37 214 497 543 533 404 564 204 297 556 531 38 63 298 417 523
[109] 350 284 355 513 155 251 228 587 418 432 163 538 380 2 400 469 315 275
[127] 500 48 142 194 516 261 176 474 483 585 343 198 286 458 280 104 226 318
[145] 508 535 562 506 225 248 566 517 347 147 324 224 240 200 3 471 106 340
[163] 573 164 10 47 462 494 302 96 308 335 127 561 4 559 322 544 178 40
[181] 519 202 126 436 511 524 234 314 171 203 548 236 159 303 70 238 229 578
[199] 536 273
$V
$V[[1]]
[1] 242.46044 222.91446 215.28060 212.62594 190.36244 179.93137 177.40565
[8] 176.45311 175.58377 169.06212 168.91950 166.69446 165.22035 161.95655
[15] 161.55278 159.79985 159.74249 158.35153 156.87935 151.98703 150.50922
[22] 150.46284 149.69603 148.85105 148.51559 142.83828 142.65469 142.27505
[29] 137.95910 136.94153 131.70865 131.49109 131.41156 131.35372 131.27233
[36] 130.53878 128.57801 128.50012 127.76585 127.18317 127.07961 126.72713
[43] 126.46012 126.01171 124.84977 123.89228 122.63971 120.93857 120.44310
[50] 120.12139 120.11153 116.22404 115.67489 114.57137 114.24835 113.60556
[57] 113.57707 109.90386 109.13497 106.42688 105.87362 104.44035 104.16316
[64] 103.82184 103.28702 102.45259 101.60793 101.06642 99.15316 98.02395
[71] 98.01168 97.86853 97.37893 97.34471 96.94657 96.60282 95.25480
[78] 94.64394 94.16090 92.69600 92.25281 91.77234 91.51184 91.32743
[85] 88.80062 88.76333 85.70033 84.93646 83.81182 83.47830 82.66748
[92] 81.23226 81.20376 81.12640 80.83079 79.15479 78.43056 77.96049
[99] 77.92796 77.85051 76.94363 74.07114 73.22549 70.77145 69.80654
[106] 69.75112 68.99806 67.32110 65.78814 65.48942 65.46591 64.74478
[113] 63.49607 63.30770 63.27366 63.12980 62.32523 62.27060 62.13281
[120] 61.42215 61.32473 61.21393 60.90930 60.45943 60.16125 59.34351
[127] 59.08206 58.97573 58.67167 58.25329 58.22823 56.89037 56.47105
[134] 56.29660 55.92282 55.79668 55.22149 54.92905 52.60543 52.56832
[141] 52.18818 52.00880 51.79277 51.73631 50.82337 50.51428 50.24619
[148] 49.51434 49.49994 49.44598 49.29493 48.74805 48.74533 48.67492
[155] 48.20965 48.00628 46.91765 46.15000 45.53533 45.07823 44.16316
[162] 43.21879 42.58285 42.38112 41.32710 41.07108 40.18016 40.11522
[169] 39.73634 39.35721 38.98250 38.95818 37.63731 37.55797 37.50083
[176] 37.39889 37.16827 37.06380 36.73642 36.51131 36.32393 35.76912
[183] 35.76039 35.72624 35.05150 35.00339 34.37153 34.22095 33.57015
[190] 33.47629 33.34217 33.16317 32.95812 32.91718 32.37610 31.81997
[197] 31.80661 31.52158 31.23896 31.02910
$V[[2]]
[1] 390.19030 294.52534 289.42511 211.60346 179.41701 158.19397 131.49242
[8] 123.63070 119.23271 111.99647 111.35206 109.30619 109.04861 105.44800
[15] 100.71713 99.10636 98.68013 98.49159 96.09401 94.32064 92.92460
[22] 92.54878 90.55789 88.78434 87.42231 86.97392 85.76897 85.12225
[29] 81.30495 81.01972 80.85329 78.24764 77.78914 75.57237 75.45814
[36] 72.68299 72.66110 71.04771 69.17088 68.96705 68.31585 67.93476
[43] 67.85902 67.84151 65.80222 65.45358 64.98173 64.86867 64.78556
[50] 64.50785 63.00920 62.85190 61.92110 60.68372 60.62235 60.40268
[57] 59.65553 59.50531 59.37996 58.41836 57.67876 57.65701 57.41476
[64] 56.02591 55.84920 55.82340 55.56612 55.45506 54.43731 53.89955
[71] 53.46451 52.85443 51.56035 51.42869 49.33334 48.64347 47.17041
[78] 47.03579 46.53588 46.48954 45.67293 45.48738 45.45811 45.26787
[85] 44.81681 44.53747 44.16689 43.94625 43.92715 43.60422 43.40342
[92] 43.19170 42.33798 41.64620 41.60630 41.56925 41.22958 41.13779
[99] 40.68209 40.54566 40.51217 40.44051 40.15617 39.48073 38.89161
[106] 38.58163 38.32676 37.39784 37.32332 36.96166 36.59479 35.66014
[113] 35.33126 34.89791 34.63511 34.58097 34.57378 34.02238 34.02216
[120] 34.01025 33.75146 33.56285 33.34646 33.33405 33.03530 32.89263
[127] 32.80499 32.68100 32.52682 32.39797 32.37715 32.05847 32.02188
[134] 31.60134 31.44846 31.37748 31.25118 31.21922 31.20006 30.96671
[141] 30.95432 30.92634 30.85068 30.62921 30.56571 30.50997 30.47105
[148] 30.30428 30.12850 29.89497 29.75437 29.74617 29.67407 29.52082
[155] 29.41821 29.33677 29.22136 29.15955 28.76715 28.70126 28.68984
[162] 28.61602 28.60645 28.58729 28.41796 28.38620 28.23036 28.18920
[169] 28.15200 28.14781 27.99149 27.84600 27.83905 27.76898 27.57398
[176] 27.10134 27.08080 26.82859 26.78222 26.48656 26.44805 26.01144
[183] 25.96563 25.76010 25.53078 25.41195 25.32042 25.17598 25.13946
[190] 24.83849 24.81233 24.74204 24.70858 24.38719 24.23100 24.20882
[197] 24.16933 24.11786 24.09025 24.02758
$V[[3]]
[1] 599.359668 415.052034 399.963851 381.248319 374.727418 256.423912
[7] 230.604336 163.415396 158.182651 157.721323 112.184000 88.804000
[13] 87.452901 84.910460 80.330621 73.057847 64.330416 58.340685
[19] 57.158386 54.716876 52.784128 52.479113 49.642932 47.850118
[25] 44.680645 42.936093 42.353402 39.640173 39.573834 38.086928
[31] 37.324679 37.124587 37.107287 32.926376 31.751032 31.552985
[37] 31.325350 31.297013 30.838452 30.602287 30.497611 30.165257
[43] 28.105785 27.279288 26.515057 26.014952 25.987005 24.230101
[49] 23.804567 23.474905 23.208677 23.136930 22.705552 22.169292
[55] 21.925015 21.663595 21.545603 21.369208 21.203289 21.028854
[61] 20.267297 20.182648 19.917142 19.661400 19.556162 19.532939
[67] 19.191427 19.061141 18.933729 18.415950 18.229073 18.180313
[73] 17.926005 17.786169 17.767557 17.721657 17.482702 17.466888
[79] 17.383790 17.308960 17.161262 16.848290 16.699105 16.501361
[85] 16.434573 16.064486 16.020370 15.984595 15.806086 15.793794
[91] 15.632803 15.504091 15.403655 15.390841 15.073799 15.051252
[97] 15.021739 14.915415 14.827576 14.791638 14.783764 14.577726
[103] 14.535886 14.502059 14.156666 14.145969 13.936527 13.935458
[109] 13.841901 13.835335 13.833905 13.745383 13.488255 13.485368
[115] 13.418174 13.393770 13.329310 13.143679 13.046603 12.997175
[121] 12.992419 12.688339 12.507877 12.468493 12.429145 12.225565
[127] 12.170224 12.137673 12.107105 11.983704 11.899821 11.887084
[133] 11.814632 11.691487 11.605199 11.524067 11.471270 11.348010
[139] 11.308793 11.294299 11.224036 11.210236 11.168526 11.131958
[145] 11.129044 11.105642 11.010161 10.890382 10.861988 10.853521
[151] 10.824968 10.777357 10.678714 10.436097 10.368669 10.357307
[157] 10.323396 10.279038 10.158425 10.141870 10.118077 10.016287
[163] 10.015201 10.007869 9.982499 9.961879 9.901653 9.896991
[169] 9.891591 9.888104 9.875304 9.833395 9.820412 9.813056
[175] 9.788083 9.775021 9.700780 9.691315 9.652485 9.623129
[181] 9.606010 9.580741 9.578879 9.574901 9.528133 9.474883
[187] 9.462522 9.354159 9.337003 9.325040 9.299281 9.289342
[193] 9.282107 9.277748 9.177035 9.110043 9.075984 9.055262
[199] 9.055119 9.044122
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Description Usage Arguments Value Examples
View source: R/find_markers_fn.R
Description
Find marker genes for each cell type.
Usage
1 2 |
Arguments
Y |
Expression matrix. (Required) Two-dimensional numeric. Must implement Each row contains expression measurements for a particular sample. Each columm contains the measurements of the same gene over all individuals. Can either contain just the mixture samples to be deconvolved or both the mixture samples and the reference samples. See |
references |
Cell-type reference expression matrix. (Optional) Two-dimensional numeric. Must implement Each row contains expression measurements for a reference profile of a particular cell type. Columns contain measurements of reference profiles of a gene. Optionally may merge this matrix with |
pure_samples |
The pure sample indicies. (Optional) List of one-dimensional integer. Must implement The i-th element of the top-level list is a vector of indicies (rows of |
data_type |
Type of expression measurements. (Optional) One-dimensional string. An optional string indicating the type of the expression measurements. This is used to set gamma to a pre-determined value based upon the data type. Valid values are for probe-level microarray as “microarray-probe”, gene-level microarray as “microarray-gene” or rna-seq as “rna-seq”. Alternatively can set |
gamma |
Expression adjustment term. (Optional) One-dimensional positive numeric. If provided as a single positive number then that value will be used for |
marker_method |
Method used to rank marker genes. (Optional) One-dimensional string. The method used to rank genes as markers. If not supplied defaults to “ratio”. Only used if markers are not provided to argument “markers”. Options are
|
Value
List with four elements. “L” is respective ranked markers for each cell type and “V” is the corresponding values of the ranking method (higher are better) used to determine markers and sort them, “M” is the matrix used to create the other two arguments after sorting and subsetting, and “sM” is a sorted version of M.
Examples
1 2 3 4 5 6 7 | truth = shen_orr_ex$annotation$mixture
pure_samples <- lapply(1:3, function(i) {
which(truth[, i] == 1)
})
Y <- shen_orr_ex$data$log
find_markers(Y=Y,pure_samples=pure_samples,
data_type='microarray-gene',marker_method='ratio')
|
Example output
$L
$L[[1]]
[1] 120 253 479 446 529 90 61 66 174 77 316 162 215 551 488 473 574 91
[19] 185 41 439 558 187 89 54 365 475 16 441 81 323 392 130 210 121 207
[37] 24 135 428 86 486 44 461 133 332 459 403 217 442 537 575 431 358 105
[55] 107 170 416 9 119 112 515 360 34 189 541 450 161 425 110 140 118 153
[73] 599 82 58 134 72 116 221 43 423 26 195 589 103 132 146 173 549 94
[91] 468 499 201 445 115 67 398 467 443 254 192 219 595 455 568 193 212 409
[109] 211 65 93 92 51 31 59 222 183 440 477 565 137 79 88 460 550 68
[127] 52 591 220 175 39 123 528 470 197 114 576 596 512 87 138 415 334 330
[145] 144 167 527 6 364 218 402 385 216 235 317 191 42 71 11 457 449 414
[163] 590 331 117 136 85 139 326 466 295 518 453 80 33 530 311 401 209 463
[181] 124 377 421 177 49 53 190 545 20 213 464 15 342 281 390 579 319 351
[199] 437 539
$L[[2]]
[1] 180 95 429 14 451 430 56 396 598 349 165 313 259 23 593 12 21 30
[19] 151 84 296 434 152 184 264 18 419 143 577 341 588 186 570 388 199 597
[37] 196 22 278 36 567 480 310 363 306 277 352 424 373 375 247 344 246 395
[55] 525 444 62 267 472 276 100 493 366 542 357 148 294 320 83 413 244 485
[73] 456 181 348 29 534 378 581 172 125 101 393 46 301 129 131 288 370 32
[91] 237 356 359 353 580 374 28 329 19 399 258 99 502 45 478 166 522 145
[109] 263 452 501 408 292 496 507 435 157 291 327 582 289 338 379 520 384 532
[127] 243 232 492 410 233 285 141 498 300 367 307 154 372 369 321 509 600 108
[145] 257 337 572 160 387 57 283 17 422 411 76 482 113 571 241 521 381 312
[163] 505 569 274 149 256 122 128 98 78 293 547 97 510 412 74 438 245 427
[181] 242 386 584 309 299 188 484 426 371 182 287 333 325 304 594 514 583 406
[199] 540 328
$L[[3]]
[1] 1 168 481 109 454 206 368 487 179 476 227 405 262 504 586 339 552 362
[19] 447 7 35 290 249 64 230 345 503 382 433 490 265 169 73 250 592 255
[37] 553 150 269 205 239 279 25 546 60 555 252 361 407 354 75 560 55 27
[55] 50 271 158 5 305 69 336 554 282 420 394 266 376 231 489 8 268 391
[73] 13 223 270 102 111 465 260 156 491 383 495 389 563 272 208 346 557 526
[91] 448 397 37 214 497 543 533 404 564 204 297 556 531 38 63 298 417 523
[109] 350 284 355 513 155 251 228 587 418 432 163 538 380 2 400 469 315 275
[127] 500 48 142 194 516 261 176 474 483 585 343 198 286 458 280 104 226 318
[145] 508 535 562 506 225 248 566 517 347 147 324 224 240 200 3 471 106 340
[163] 573 164 10 47 462 494 302 96 308 335 127 561 4 559 322 544 178 40
[181] 519 202 126 436 511 524 234 314 171 203 548 236 159 303 70 238 229 578
[199] 536 273
$V
$V[[1]]
[1] 242.46044 222.91446 215.28060 212.62594 190.36244 179.93137 177.40565
[8] 176.45311 175.58377 169.06212 168.91950 166.69446 165.22035 161.95655
[15] 161.55278 159.79985 159.74249 158.35153 156.87935 151.98703 150.50922
[22] 150.46284 149.69603 148.85105 148.51559 142.83828 142.65469 142.27505
[29] 137.95910 136.94153 131.70865 131.49109 131.41156 131.35372 131.27233
[36] 130.53878 128.57801 128.50012 127.76585 127.18317 127.07961 126.72713
[43] 126.46012 126.01171 124.84977 123.89228 122.63971 120.93857 120.44310
[50] 120.12139 120.11153 116.22404 115.67489 114.57137 114.24835 113.60556
[57] 113.57707 109.90386 109.13497 106.42688 105.87362 104.44035 104.16316
[64] 103.82184 103.28702 102.45259 101.60793 101.06642 99.15316 98.02395
[71] 98.01168 97.86853 97.37893 97.34471 96.94657 96.60282 95.25480
[78] 94.64394 94.16090 92.69600 92.25281 91.77234 91.51184 91.32743
[85] 88.80062 88.76333 85.70033 84.93646 83.81182 83.47830 82.66748
[92] 81.23226 81.20376 81.12640 80.83079 79.15479 78.43056 77.96049
[99] 77.92796 77.85051 76.94363 74.07114 73.22549 70.77145 69.80654
[106] 69.75112 68.99806 67.32110 65.78814 65.48942 65.46591 64.74478
[113] 63.49607 63.30770 63.27366 63.12980 62.32523 62.27060 62.13281
[120] 61.42215 61.32473 61.21393 60.90930 60.45943 60.16125 59.34351
[127] 59.08206 58.97573 58.67167 58.25329 58.22823 56.89037 56.47105
[134] 56.29660 55.92282 55.79668 55.22149 54.92905 52.60543 52.56832
[141] 52.18818 52.00880 51.79277 51.73631 50.82337 50.51428 50.24619
[148] 49.51434 49.49994 49.44598 49.29493 48.74805 48.74533 48.67492
[155] 48.20965 48.00628 46.91765 46.15000 45.53533 45.07823 44.16316
[162] 43.21879 42.58285 42.38112 41.32710 41.07108 40.18016 40.11522
[169] 39.73634 39.35721 38.98250 38.95818 37.63731 37.55797 37.50083
[176] 37.39889 37.16827 37.06380 36.73642 36.51131 36.32393 35.76912
[183] 35.76039 35.72624 35.05150 35.00339 34.37153 34.22095 33.57015
[190] 33.47629 33.34217 33.16317 32.95812 32.91718 32.37610 31.81997
[197] 31.80661 31.52158 31.23896 31.02910
$V[[2]]
[1] 390.19030 294.52534 289.42511 211.60346 179.41701 158.19397 131.49242
[8] 123.63070 119.23271 111.99647 111.35206 109.30619 109.04861 105.44800
[15] 100.71713 99.10636 98.68013 98.49159 96.09401 94.32064 92.92460
[22] 92.54878 90.55789 88.78434 87.42231 86.97392 85.76897 85.12225
[29] 81.30495 81.01972 80.85329 78.24764 77.78914 75.57237 75.45814
[36] 72.68299 72.66110 71.04771 69.17088 68.96705 68.31585 67.93476
[43] 67.85902 67.84151 65.80222 65.45358 64.98173 64.86867 64.78556
[50] 64.50785 63.00920 62.85190 61.92110 60.68372 60.62235 60.40268
[57] 59.65553 59.50531 59.37996 58.41836 57.67876 57.65701 57.41476
[64] 56.02591 55.84920 55.82340 55.56612 55.45506 54.43731 53.89955
[71] 53.46451 52.85443 51.56035 51.42869 49.33334 48.64347 47.17041
[78] 47.03579 46.53588 46.48954 45.67293 45.48738 45.45811 45.26787
[85] 44.81681 44.53747 44.16689 43.94625 43.92715 43.60422 43.40342
[92] 43.19170 42.33798 41.64620 41.60630 41.56925 41.22958 41.13779
[99] 40.68209 40.54566 40.51217 40.44051 40.15617 39.48073 38.89161
[106] 38.58163 38.32676 37.39784 37.32332 36.96166 36.59479 35.66014
[113] 35.33126 34.89791 34.63511 34.58097 34.57378 34.02238 34.02216
[120] 34.01025 33.75146 33.56285 33.34646 33.33405 33.03530 32.89263
[127] 32.80499 32.68100 32.52682 32.39797 32.37715 32.05847 32.02188
[134] 31.60134 31.44846 31.37748 31.25118 31.21922 31.20006 30.96671
[141] 30.95432 30.92634 30.85068 30.62921 30.56571 30.50997 30.47105
[148] 30.30428 30.12850 29.89497 29.75437 29.74617 29.67407 29.52082
[155] 29.41821 29.33677 29.22136 29.15955 28.76715 28.70126 28.68984
[162] 28.61602 28.60645 28.58729 28.41796 28.38620 28.23036 28.18920
[169] 28.15200 28.14781 27.99149 27.84600 27.83905 27.76898 27.57398
[176] 27.10134 27.08080 26.82859 26.78222 26.48656 26.44805 26.01144
[183] 25.96563 25.76010 25.53078 25.41195 25.32042 25.17598 25.13946
[190] 24.83849 24.81233 24.74204 24.70858 24.38719 24.23100 24.20882
[197] 24.16933 24.11786 24.09025 24.02758
$V[[3]]
[1] 599.359668 415.052034 399.963851 381.248319 374.727418 256.423912
[7] 230.604336 163.415396 158.182651 157.721323 112.184000 88.804000
[13] 87.452901 84.910460 80.330621 73.057847 64.330416 58.340685
[19] 57.158386 54.716876 52.784128 52.479113 49.642932 47.850118
[25] 44.680645 42.936093 42.353402 39.640173 39.573834 38.086928
[31] 37.324679 37.124587 37.107287 32.926376 31.751032 31.552985
[37] 31.325350 31.297013 30.838452 30.602287 30.497611 30.165257
[43] 28.105785 27.279288 26.515057 26.014952 25.987005 24.230101
[49] 23.804567 23.474905 23.208677 23.136930 22.705552 22.169292
[55] 21.925015 21.663595 21.545603 21.369208 21.203289 21.028854
[61] 20.267297 20.182648 19.917142 19.661400 19.556162 19.532939
[67] 19.191427 19.061141 18.933729 18.415950 18.229073 18.180313
[73] 17.926005 17.786169 17.767557 17.721657 17.482702 17.466888
[79] 17.383790 17.308960 17.161262 16.848290 16.699105 16.501361
[85] 16.434573 16.064486 16.020370 15.984595 15.806086 15.793794
[91] 15.632803 15.504091 15.403655 15.390841 15.073799 15.051252
[97] 15.021739 14.915415 14.827576 14.791638 14.783764 14.577726
[103] 14.535886 14.502059 14.156666 14.145969 13.936527 13.935458
[109] 13.841901 13.835335 13.833905 13.745383 13.488255 13.485368
[115] 13.418174 13.393770 13.329310 13.143679 13.046603 12.997175
[121] 12.992419 12.688339 12.507877 12.468493 12.429145 12.225565
[127] 12.170224 12.137673 12.107105 11.983704 11.899821 11.887084
[133] 11.814632 11.691487 11.605199 11.524067 11.471270 11.348010
[139] 11.308793 11.294299 11.224036 11.210236 11.168526 11.131958
[145] 11.129044 11.105642 11.010161 10.890382 10.861988 10.853521
[151] 10.824968 10.777357 10.678714 10.436097 10.368669 10.357307
[157] 10.323396 10.279038 10.158425 10.141870 10.118077 10.016287
[163] 10.015201 10.007869 9.982499 9.961879 9.901653 9.896991
[169] 9.891591 9.888104 9.875304 9.833395 9.820412 9.813056
[175] 9.788083 9.775021 9.700780 9.691315 9.652485 9.623129
[181] 9.606010 9.580741 9.578879 9.574901 9.528133 9.474883
[187] 9.462522 9.354159 9.337003 9.325040 9.299281 9.289342
[193] 9.282107 9.277748 9.177035 9.110043 9.075984 9.055262
[199] 9.055119 9.044122
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