estimate.missing: Estimate locations of missing landmarks

Description Usage Arguments Details Value Author(s) References Examples

View source: R/estimate.missing.r

Description

A function for estimating the locations of missing landmarks

Usage

1
estimate.missing(A, method = c("TPS", "Reg"))

Arguments

A

An array (p x k x n) containing landmark coordinates for a set of specimens

method

Method for estimating missing landmark locations

Details

The function estimates the locations of missing landmarks for incomplete specimens in a set of landmark configurations, where missing landmarks in the incomplete specimens are designated by NA in place of the x,y,z coordinates. Two distinct approaches are implemented.

The first approach (method="TPS") uses the thin-plate spline to interpolate landmarks on a reference specimen to estimate the locations of missing landmarks on a target specimen. Here, a reference specimen is obtained from the set of specimens for which all landmarks are present, Next, each incomplete specimen is aligned to the reference using the set of landmarks common to both. Finally, the thin-plate spline is used to estimate the locations of the missing landmarks in the target specimen (Gunz et al. 2009).

The second approach (method="Reg") is multivariate regression. Here each landmark with missing values is regressed on all other landmarks for the set of complete specimens, and the missing landmark values are then predicted by this linear regression model. Because the number of variables can exceed the number of specimens, the regression is implemented on scores along the first set of PLS axes for the complete and incomplete blocks of landmarks (see Gunz et al. 2009).

One can also exploit bilateral symmetry to estimate the locations of missing landmarks. Several possibilities exist for implementing this approach (see Gunz et al. 2009). Example R code for one implementation is found in Claude (2008).

NOTE: Because all geometric morphometric analyses and plotting functions implemented in geomorph require a full complement of landmark coordinates, the alternative to estimating the missing landmark coordinates is to proceed with subsequent analyses EXCLUDING specimens with missing values. To do this, see functions complete.cases (use: mydata[complete.cases(mydata),]) or na.omit (use: newdata <- na.omit(mydata)) to make a dataset of only the complete specimens. These functions require the dataset to be a matrix in the form of a 2d array (see two.d.array).

Value

Function returns an array (p x k x n) of the same dimensions as input A, including coordinates for the target specimens (the original landmarks plus the estimated coordinates for the missing landmarks). These data need to be Procrustes Superimposed prior to analysis (seesee gpagen).

Author(s)

Dean Adams

References

Claude, J. 2008. Morphometrics with R. Springer, New York.

Bookstein, F. L., K. Schafer, H. Prossinger, H. Seidler, M. Fieder, G. Stringer, G. W. Weber, J.-L. Arsuaga, D. E. Slice, F. J. Rohlf, W. Recheis, A. J. Mariam, and L. F. Marcus. 1999. Comparing frontal cranial profiles in archaic and modern Homo by morphometric analysis. Anat. Rec. (New Anat.) 257:217-224.

Gunz, P., P. Mitteroecker, S. Neubauer, G. W. Weber, and F. L. Bookstein. 2009. Principles for the virtual reconstruction of hominin crania. J. Hum. Evol. 57:48-62.

Examples

1
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8
data(plethodon)
plethland<-plethodon$land
  plethland[3,,2]<-plethland[8,,2]<-NA  #create missing landmarks
  plethland[3,,5]<-plethland[8,,5]<-plethland[9,,5]<-NA  
  plethland[3,,10]<-NA  
  
estimate.missing(plethland,method="TPS")
estimate.missing(plethland,method="Reg")

Example output

Loading required package: rgl
Loading required package: ape
Warning messages:
1: In rgl.init(initValue, onlyNULL) : RGL: unable to open X11 display
2: 'rgl_init' failed, running with rgl.useNULL = TRUE 
3: .onUnload failed in unloadNamespace() for 'rgl', details:
  call: fun(...)
  error: object 'rgl_quit' not found 

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, , 1

          [,1]     [,2]
 [1,]  8.89372 53.77644
 [2,]  9.26840 52.77072
 [3,]  5.56104 54.21028
 [4,]  1.87340 52.75100
 [5,]  1.28180 53.18484
 [6,]  1.24236 53.32288
 [7,]  0.84796 54.70328
 [8,]  3.35240 55.76816
 [9,]  6.29068 55.70900
[10,]  8.87400 55.25544
[11,] 10.74740 55.43292
[12,] 14.39560 52.75100

, , 2

           [,1]     [,2]
 [1,]  8.679762 54.57819
 [2,]  8.935628 53.83027
 [3,]  5.710713 54.47421
 [4,]  1.987882 52.68871
 [5,]  1.515514 53.02331
 [6,]  1.338376 53.22013
 [7,]  1.239966 54.44041
 [8,]  2.971883 55.68053
 [9,]  6.219512 55.99529
[10,]  8.758490 56.01497
[11,] 10.903828 55.83783
[12,] 14.033266 53.86963

, , 3

           [,1]     [,2]
 [1,]  9.805328 56.06903
 [2,] 10.137712 55.27961
 [3,]  6.647680 55.73664
 [4,]  3.448484 53.86698
 [5,]  3.012230 54.34478
 [6,]  2.825264 54.57330
 [7,]  2.721394 55.73664
 [8,]  4.549506 56.79612
 [9,]  7.520188 57.35701
[10,]  9.846876 57.25314
[11,] 11.986598 57.23237
[12,] 14.749540 55.11342

, , 4

           [,1]     [,2]
 [1,]  9.637164 58.03294
 [2,]  9.952104 56.77318
 [3,]  6.109836 57.94896
 [4,]  2.645496 55.89135
 [5,]  2.015616 56.62621
 [6,]  1.721672 57.21410
 [7,]  1.868644 58.51585
 [8,]  3.632308 59.48167
 [9,]  6.865692 59.92258
[10,]  9.763140 59.56565
[11,] 11.883736 59.46067
[12,] 15.369072 56.56322

, , 5

           [,1]     [,2]
 [1,] 11.035692 58.75009
 [2,] 11.335110 57.85184
 [3,]  8.470849 58.86529
 [4,]  4.555431 57.46687
 [5,]  3.956595 58.08709
 [6,]  3.871047 58.40790
 [7,]  3.871047 59.58418
 [8,]  5.828430 60.47664
 [9,]  9.198300 60.35732
[10,] 11.420658 59.94776
[11,] 12.768039 59.88360
[12,] 15.847767 57.46687

, , 6

           [,1]     [,2]
 [1,]  7.946625 55.71114
 [2,]  8.476400 54.82112
 [3,]  4.979885 54.90588
 [4,]  2.479347 52.85035
 [5,]  2.076718 53.10465
 [6,]  1.737662 53.14703
 [7,]  1.335033 54.20658
 [8,]  2.712448 55.54161
 [9,]  5.382514 56.60116
[10,]  7.861861 56.96141
[11,]  9.663096 57.49118
[12,] 13.138420 55.75352

, , 7

           [,1]     [,2]
 [1,]  8.849841 58.66961
 [2,]  9.396387 57.82877
 [3,]  5.990985 58.31225
 [4,]  2.501499 56.84078
 [5,]  2.102100 57.28223
 [6,]  1.933932 57.47141
 [7,]  1.975974 58.43838
 [8,]  3.741738 59.34228
 [9,]  6.642636 59.72066
[10,]  8.849841 59.82577
[11,] 10.951941 59.48943
[12,] 13.579566 57.89183

, , 8

           [,1]     [,2]
 [1,]  9.331504 56.36904
 [2,] 10.154872 55.31344
 [3,]  6.671392 56.62238
 [4,]  2.195648 55.08121
 [5,]  1.688960 55.79902
 [6,]  1.520064 56.28459
 [7,]  1.583400 57.34019
 [8,]  3.715712 58.22690
 [9,]  7.241416 58.26912
[10,]  9.774856 57.88910
[11,] 11.949392 57.59354
[12,] 15.432872 54.70119

, , 9

           [,1]     [,2]
 [1,]  9.840795 58.49453
 [2,] 10.179403 57.37289
 [3,]  6.031455 58.49453
 [4,]  2.624212 57.05545
 [5,]  2.137463 57.35173
 [6,]  1.566062 57.71150
 [7,]  1.693040 58.98128
 [8,]  3.745851 60.06059
 [9,]  7.110768 60.14525
[10,] 10.073588 60.12408
[11,] 11.787791 59.89129
[12,] 15.131545 57.41522

, , 10

           [,1]     [,2]
 [1,]  9.563510 56.51740
 [2,]  9.900550 55.50628
 [3,]  6.128195 56.33379
 [4,]  1.811590 54.45303
 [5,]  1.369225 55.02178
 [6,]  1.221770 55.54841
 [7,]  1.200705 56.87550
 [8,]  3.265075 57.88662
 [9,]  6.551215 58.05514
[10,]  9.437120 57.97088
[11,] 11.627880 57.52852
[12,] 15.419580 55.08498

, , 11

           [,1]     [,2]
 [1,]  8.211782 57.04143
 [2,]  8.337794 55.52929
 [3,]  4.830460 59.47766
 [4,]  0.336032 60.90580
 [5,]  0.462044 61.68287
 [6,]  0.378036 62.18692
 [7,]  1.176112 63.19502
 [8,]  4.053386 62.58596
 [9,]  6.993666 60.56977
[10,]  9.765930 58.61658
[11,] 11.803124 57.29346
[12,] 15.016430 53.55510

, , 12

           [,1]     [,2]
 [1,]  8.691998 54.71960
 [2,]  9.660114 53.56207
 [3,]  5.240454 55.43516
 [4,]  1.641588 54.76169
 [5,]  0.926024 55.37203
 [6,]  0.610334 55.85608
 [7,]  0.926024 57.09780
 [8,]  3.220038 57.72918
 [9,]  6.503214 57.11884
[10,]  9.533838 56.46642
[11,] 11.449024 56.48746
[12,] 15.889730 54.13031

, , 13

           [,1]     [,2]
 [1,] 10.707338 61.44578
 [2,] 11.724882 60.26636
 [3,]  7.030304 61.28390
 [4,]  2.890750 59.13318
 [5,]  2.220096 59.78071
 [6,]  2.058214 60.10447
 [7,]  2.127592 61.49203
 [8,]  4.393940 62.85647
 [9,]  7.932218 63.11085
[10,] 11.216110 63.29586
[11,] 13.228072 63.85089
[12,] 18.500800 63.06460

, , 14

           [,1]     [,2]
 [1,]  8.470779 53.32874
 [2,]  9.155484 52.66360
 [3,]  6.397101 53.21136
 [4,]  4.460364 52.42884
 [5,]  4.166919 52.76141
 [6,]  3.990852 53.21136
 [7,]  4.264734 54.07213
 [8,]  5.164632 54.50252
 [9,]  7.453503 54.32645
[10,]  8.842476 54.34601
[11,]  9.820626 54.26776
[12,] 12.735513 53.48524

, , 15

           [,1]     [,2]
 [1,] 11.532672 74.42845
 [2,] 12.226768 72.61312
 [3,]  7.635056 75.04246
 [4,]  2.776384 73.92122
 [5,]  2.322552 74.32166
 [6,]  1.681848 74.82889
 [7,]  2.215768 76.40395
 [8,]  4.965456 77.36501
 [9,]  9.396992 76.99126
[10,] 12.280160 76.59082
[11,] 14.389144 76.27047
[12,] 19.488080 73.89453

, , 16

           [,1]     [,2]
 [1,]  8.919900 55.17745
 [2,]  9.675468 53.01569
 [3,]  5.561820 55.57622
 [4,]  0.797544 54.17003
 [5,]  0.482724 54.98856
 [6,]  0.272844 55.49227
 [7,]  0.524700 56.81452
 [8,]  3.022272 57.63305
 [9,]  6.737148 57.27625
[10,]  9.864360 56.98242
[11,] 11.984148 56.45772
[12,] 16.454592 54.75769

, , 17

           [,1]     [,2]
 [1,]  9.914553 55.94491
 [2,] 10.459539 55.16935
 [3,]  7.524999 55.94491
 [4,]  5.009679 55.10647
 [5,]  4.716225 55.48377
 [6,]  4.527576 55.86107
 [7,]  4.527576 56.55278
 [8,]  5.743314 57.43314
 [9,]  8.405361 57.34930
[10,] 10.124163 57.22353
[11,] 11.738160 57.22353
[12,] 14.484051 56.04971

, , 18

           [,1]     [,2]
 [1,] 10.661782 62.89986
 [2,] 11.616221 61.66607
 [3,]  6.867305 63.76118
 [4,]  2.677085 62.24805
 [5,]  2.025273 62.80674
 [6,]  1.489856 63.36544
 [7,]  1.839041 64.78546
 [8,]  4.283336 65.83301
 [9,]  8.566672 65.55366
[10,] 11.616221 64.92513
[11,] 13.874284 64.76218
[12,] 18.623200 62.87658

, , 19

           [,1]     [,2]
 [1,]  8.313908 51.34480
 [2,]  9.735764 50.02168
 [3,]  4.838260 51.38430
 [4,]  1.303368 49.98219
 [5,]  0.710928 50.57463
 [6,]  0.572692 51.04858
 [7,]  0.829416 52.23346
 [8,]  2.784468 53.28010
 [9,]  6.516840 53.35910
[10,]  9.419796 53.26036
[11,] 11.098376 53.14187
[12,] 15.818148 51.44354

, , 20

           [,1]     [,2]
 [1,]  8.506270 57.07500
 [2,]  9.813331 55.99615
 [3,]  5.912895 57.30321
 [4,]  1.514531 56.24512
 [5,]  1.161832 56.84678
 [6,]  0.871374 57.19948
 [7,]  1.286314 58.36131
 [8,]  3.651472 59.00447
 [9,]  6.929498 59.08746
[10,]  9.979307 58.65177
[11,] 11.888031 58.77625
[12,] 16.556106 56.95052

, , 21

           [,1]     [,2]
 [1,] 10.310018 55.68670
 [2,] 11.086944 54.67879
 [3,]  7.895248 55.77069
 [4,]  4.388582 54.51081
 [5,]  3.884630 54.80478
 [6,]  3.569660 55.09875
 [7,]  3.548662 56.10666
 [8,]  5.291496 57.15656
 [9,]  8.294210 57.40853
[10,] 10.519998 57.34554
[11,] 13.732692 57.51352
[12,] 15.958480 55.93867

, , 22

           [,1]     [,2]
 [1,]  8.168564 57.41153
 [2,]  8.189617 56.04309
 [3,]  5.579045 59.41157
 [4,]  0.694749 60.42211
 [5,]  0.421060 61.11686
 [6,]  0.336848 61.58002
 [7,]  1.305286 62.52741
 [8,]  3.957964 62.61162
 [9,]  7.494868 60.90633
[10,] 10.084387 59.05366
[11,] 13.810768 56.90626
[12,] 15.052895 53.24304

, , 23

           [,1]     [,2]
 [1,]  8.514847 55.94275
 [2,]  9.163398 54.83394
 [3,]  5.732354 55.92183
 [4,]  2.238547 54.66657
 [5,]  1.882890 55.00131
 [6,]  1.694601 55.41973
 [7,]  1.799206 56.25657
 [8,]  3.640254 57.34446
 [9,]  6.673799 57.59551
[10,]  9.016951 57.51183
[11,] 11.883128 57.34446
[12,] 14.100754 55.37789

, , 24

          [,1]     [,2]
 [1,]  7.52874 58.00074
 [2,]  7.90728 57.24366
 [3,]  5.23647 58.12692
 [4,]  1.95579 56.94924
 [5,]  1.55622 57.26469
 [6,]  1.36695 57.79044
 [7,]  1.53519 58.73679
 [8,]  3.13347 59.49387
 [9,]  5.59398 59.51490
[10,]  7.90728 59.22048
[11,] 10.53603 59.17842
[12,] 12.38667 57.43293

, , 25

           [,1]     [,2]
 [1,]  9.660255 58.00344
 [2,] 10.498455 57.16524
 [3,]  7.543800 58.17108
 [4,]  4.253865 56.80901
 [5,]  3.876675 57.27002
 [6,]  3.583305 57.60530
 [7,]  3.604260 58.50636
 [8,]  5.196840 59.55411
 [9,]  7.858125 59.70080
[10,] 10.142220 59.61698
[11,] 12.824460 59.38647
[12,] 15.108555 57.66816

, , 26

           [,1]     [,2]
 [1,]  8.374318 56.47404
 [2,]  8.837220 55.10638
 [3,]  5.554824 57.02111
 [4,]  1.346624 55.94802
 [5,]  0.967886 56.74758
 [6,]  0.799558 57.33673
 [7,]  0.904763 58.51502
 [8,]  3.008863 59.20937
 [9,]  6.312300 58.97792
[10,]  9.110753 58.28357
[11,] 12.498354 57.84171
[12,] 15.465135 55.56928

, , 27

           [,1]     [,2]
 [1,]  8.634288 57.98208
 [2,]  9.663680 56.13338
 [3,]  5.945264 58.54930
 [4,]  1.491568 56.53253
 [5,]  0.924352 57.41486
 [6,]  0.630240 58.27619
 [7,]  0.861328 59.83078
 [8,]  3.361280 60.90219
 [9,]  7.016672 60.86018
[10,]  9.936784 59.95683
[11,] 14.180400 59.43163
[12,] 17.016480 57.01571

, , 28

          [,1]     [,2]
 [1,]  9.99570 57.01410
 [2,] 10.63920 55.74855
 [3,]  6.73530 57.70050
 [4,]  1.26555 55.83435
 [5,]  0.64350 56.49930
 [6,]  0.49335 56.99265
 [7,]  0.66495 58.25820
 [8,]  3.81810 59.56665
 [9,]  7.61475 59.48085
[10,] 10.85370 58.92315
[11,] 14.43585 58.53705
[12,] 17.54610 56.11320

, , 29

           [,1]     [,2]
 [1,]  9.293050 56.80955
 [2,]  9.860725 55.37985
 [3,]  6.664925 57.29312
 [4,]  2.733250 55.90547
 [5,]  2.291725 56.15777
 [6,]  2.018400 56.62032
 [7,]  2.060450 57.83977
 [8,]  4.205000 59.08025
 [9,]  7.758225 59.01717
[10,] 10.449425 58.40745
[11,] 13.645225 57.96592
[12,] 15.831825 56.15777

, , 30

           [,1]     [,2]
 [1,]  8.395014 57.54170
 [2,]  9.238734 56.52924
 [3,]  6.243528 57.96356
 [4,]  2.425695 57.01438
 [5,]  1.792905 57.26750
 [6,]  1.455417 57.49952
 [7,]  1.413231 58.51198
 [8,]  3.417066 59.65100
 [9,]  6.686481 59.54554
[10,]  9.133269 58.80728
[11,] 11.790987 58.51198
[12,] 14.090124 56.74017

, , 31

           [,1]     [,2]
 [1,]  9.536103 56.41668
 [2,] 10.188684 55.53254
 [3,]  6.862626 56.83770
 [4,]  2.778732 55.72200
 [5,]  2.315610 56.16407
 [6,]  1.978794 56.45878
 [7,]  2.084049 57.49028
 [8,]  4.189149 58.39547
 [9,]  7.431003 58.62703
[10,] 10.314990 58.35337
[11,] 13.809456 57.99551
[12,] 15.935607 56.03776

, , 32

          [,1]     [,2]
 [1,]  9.28998 57.66999
 [2,]  9.43845 56.39739
 [3,]  6.15090 60.06672
 [4,]  0.91203 61.10601
 [5,]  0.67872 61.63626
 [6,]  0.57267 62.16651
 [7,]  1.16655 63.16338
 [8,]  4.43289 63.26943
 [9,]  7.86891 61.44537
[10,] 10.90194 59.30316
[11,] 14.23191 57.26700
[12,] 15.84387 53.95824

, , 33

          [,1]     [,2]
 [1,]  9.45411 57.01684
 [2,] 10.10108 56.05682
 [3,]  6.63666 57.28815
 [4,]  2.69223 55.86899
 [5,]  2.35831 56.36987
 [6,]  2.02439 56.72466
 [7,]  2.08700 57.72642
 [8,]  3.88182 58.85340
 [9,]  7.57581 58.89514
[10,] 10.20543 58.39426
[11,] 13.44028 58.20643
[12,] 16.13251 56.32813

, , 34

           [,1]     [,2]
 [1,]  7.130627 56.33819
 [2,]  7.546407 55.63136
 [3,]  5.238828 56.50450
 [4,]  1.683909 55.36111
 [5,]  1.476019 55.67294
 [6,]  1.247340 55.75610
 [7,]  1.164184 56.75397
 [8,]  2.577836 57.81421
 [9,]  5.488296 57.98052
[10,]  7.691930 57.54395
[11,] 10.165821 57.39843
[12,] 12.847602 55.79768

, , 35

           [,1]     [,2]
 [1,]  9.093168 58.07419
 [2,]  9.745687 56.79020
 [3,]  6.209455 58.36888
 [4,]  1.852312 57.00069
 [5,]  1.283989 57.48482
 [6,]  1.094548 57.90580
 [7,]  1.305038 59.00035
 [8,]  3.388889 59.96860
 [9,]  7.114562 60.32643
[10,]  9.787785 59.71601
[11,] 12.945135 59.16874
[12,] 15.239476 57.50587

, , 36

           [,1]     [,2]
 [1,]  8.076244 58.16164
 [2,]  8.837356 57.21025
 [3,]  5.412352 58.26735
 [4,]  1.945064 56.51257
 [5,]  1.458798 56.93541
 [6,]  1.205094 57.29482
 [7,]  1.374230 58.20393
 [8,]  3.065590 59.45130
 [9,]  5.835192 59.85300
[10,]  8.625936 59.64158
[11,] 11.691526 59.76843
[12,] 14.672548 57.88680

, , 37

          [,1]     [,2]
 [1,]  8.46210 56.89815
 [2,]  9.28305 55.97195
 [3,]  6.44130 56.77185
 [4,]  2.31550 55.29835
 [5,]  1.91555 55.57200
 [6,]  1.78925 56.05615
 [7,]  1.76820 57.08760
 [8,]  3.17855 57.99275
 [9,]  6.31500 58.45585
[10,]  8.77785 58.37165
[11,] 11.97745 58.35060
[12,] 14.46135 57.08760

, , 38

          [,1]     [,2]
 [1,]  8.58024 57.72735
 [2,]  9.65277 56.27628
 [3,]  6.07767 58.31619
 [4,]  1.21974 57.41190
 [5,]  0.79914 58.27413
 [6,]  0.65193 58.77885
 [7,]  1.07253 60.12477
 [8,]  3.36480 60.73464
 [9,]  6.98196 60.27198
[10,] 10.01028 59.28357
[11,] 13.83774 58.88400
[12,] 16.71885 56.08701

, , 39

           [,1]     [,2]
 [1,]  9.389055 56.73521
 [2,] 10.000926 55.84905
 [3,]  6.751680 56.94620
 [4,]  2.869464 55.57477
 [5,]  2.363088 56.06004
 [6,]  2.088801 56.62972
 [7,]  2.236494 57.62137
 [8,]  4.198701 58.50753
 [9,]  7.532343 58.38093
[10,]  9.916530 58.21214
[11,] 12.638301 58.29654
[12,] 15.338973 56.27103

, , 40

           [,1]     [,2]
 [1,]  9.494001 56.66929
 [2,] 10.125531 55.57464
 [3,]  6.694218 57.25872
 [4,]  1.494621 56.47983
 [5,]  0.968346 56.77455
 [6,]  0.842040 57.34292
 [7,]  0.989397 58.33232
 [8,]  3.410262 59.30067
 [9,]  7.473105 58.92175
[10,]  9.957123 58.26917
[11,] 13.514742 57.72184
[12,] 15.956658 55.59569


  |                                                                            
  |                                                                      |   0%
  |                                                                            
  |==============                                                        |  20%
  |                                                                            
  |============================                                          |  40%
  |                                                                            
  |======================================================================| 100%
, , 1

            [,1]        [,2]
 [1,]  2.8386551 -0.43429746
 [2,]  3.1894567 -1.45291118
 [3,] -0.4910170  0.08618844
 [4,] -4.2275017 -1.28164603
 [5,] -4.8091671 -0.83113895
 [6,] -4.8458254 -0.69081159
 [7,] -5.2040862  0.70286340
 [8,] -2.6665570  1.70591980
 [9,]  0.2797956  1.57098927
[10,]  2.8590233  1.04978389
[11,]  4.7447179  1.17826251
[12,]  8.3325058 -1.60320209

, , 2

             [,1]        [,2]
 [1,]  2.32902258 -0.05835821
 [2,]  2.49103330 -0.83204954
 [3,] -0.60961457 -0.03933386
 [4,] -4.54433830 -1.11110347
 [5,] -4.97200536 -0.72099380
 [6,] -5.12361224 -0.50389627
 [7,] -5.07130850  0.71923165
 [8,] -2.59545145  1.71309928
 [9,]  0.06157839  1.65035788
[10,]  2.58372872  1.35786102
[11,]  4.69103465  0.91841283
[12,]  7.55486675 -1.41946209

, , 3

            [,1]        [,2]
 [1,]  2.5577461 -0.05210613
 [2,]  2.7842117 -0.88224586
 [3,] -0.6251294  0.03966150
 [4,] -4.0569430 -1.39151233
 [5,] -4.4271360 -0.85834867
 [6,] -4.5828010 -0.60569570
 [7,] -4.5304628  0.56464340
 [8,] -2.5712317  1.37434410
 [9,]  0.4580334  1.53477411
[10,]  2.7584810  1.12092177
[11,]  4.8846033  0.81488458
[12,]  7.3506284 -1.65932076

, , 4

            [,1]        [,2]
 [1,]  2.8539941 -0.14958308
 [2,]  3.0908265 -1.43118589
 [3,] -0.6828265 -0.00795801
 [4,] -4.2861914 -1.84601565
 [5,] -4.8698347 -1.07067284
 [6,] -5.1267814 -0.46368668
 [7,] -4.8967763  0.83046900
 [8,] -3.0673462  1.68585878
 [9,]  0.2004056  1.92016463
[10,]  3.0795956  1.37819670
[11,]  5.1994160  1.13775774
[12,]  8.5055188 -1.98334468

, , 5

            [,1]        [,2]
 [1,]  2.2966056  0.12692417
 [2,]  2.5889601 -0.77365357
 [3,] -0.5693514 -0.03753572
 [4,] -4.1935331 -1.10536442
 [5,] -4.7874798 -0.48045767
 [6,] -4.8705058 -0.15899072
 [7,] -4.8612680  1.01725800
 [8,] -2.4368072  1.60656755
 [9,]  0.4444252  1.43068675
[10,]  2.6909655  1.32153596
[11,]  4.0378011  1.24679547
[12,]  7.0984546 -1.19404723

, , 6

            [,1]       [,2]
 [1,]  2.3827210 -0.1285192
 [2,]  2.6104691 -1.1424519
 [3,] -0.6952283  0.0321380
 [4,] -3.7228094 -1.1447993
 [5,] -4.0273309 -0.7762820
 [6,] -4.3369558 -0.6291708
 [7,] -4.3887774  0.5062030
 [8,] -2.6581602  1.3478608
 [9,]  0.2183107  1.5220499
[10,]  2.6937112  1.0902564
[11,]  4.5775727  1.0314529
[12,]  7.3464773 -1.7087377

, , 7

            [,1]        [,2]
 [1,]  2.4914266 -0.02785107
 [2,]  2.9451110 -0.92602076
 [3,] -0.3968818 -0.07162678
 [4,] -4.0358004 -1.15845098
 [5,] -4.3860344 -0.67493088
 [6,] -4.5332908 -0.46751556
 [7,] -4.3860336  0.49208538
 [8,] -2.5271933  1.20120446
 [9,]  0.4062627  1.26200618
[10,]  2.6186987  1.12559140
[11,]  4.6786267  0.56225052
[12,]  7.1251085 -1.31674191

, , 8

            [,1]         [,2]
 [1,]  2.5693126 -0.222863924
 [2,]  3.4081877 -1.270654827
 [3,] -0.1017097 -0.003387122
 [4,] -4.5675389 -1.606186068
 [5,] -5.0847613 -0.893709949
 [6,] -5.2606470 -0.409266136
 [7,] -5.2106804  0.650105103
 [8,] -3.0846117  1.567080190
 [9,]  0.4491781  1.655175155
[10,]  2.9944959  1.307144542
[11,]  5.1797696  1.039367411
[12,]  8.7090052 -1.812804375

, , 9

            [,1]        [,2]
 [1,]  3.0114651 -0.26993016
 [2,]  3.2878945 -1.41279688
 [3,] -0.8014205 -0.05195625
 [4,] -4.2982246 -1.29883134
 [5,] -4.7688103 -0.97364659
 [6,] -5.3196256 -0.58054536
 [7,] -5.1200417  0.68371548
 [8,] -3.0022600  1.64751589
 [9,]  0.3731211  1.54025419
[10,]  3.3387880  1.34904603
[11,]  5.0457311  1.01825329
[12,]  8.2533829 -1.65107831

, , 10

            [,1]        [,2]
 [1,]  3.0765252 -0.07998333
 [2,]  3.3495340 -1.11023854
 [3,] -0.5595231 -0.07767071
 [4,] -4.7895337 -1.65446909
 [5,] -5.1953842 -1.05910838
 [6,] -5.3095449 -0.52427736
 [7,] -5.2473972  0.80152901
 [8,] -3.1237167  1.68128375
 [9,]  0.1665248  1.64352445
[10,]  3.0414760  1.37856548
[11,]  5.2002056  0.79977141
[12,]  8.8313111 -1.87659740

, , 11

            [,1]       [,2]
 [1,]  3.1296907 -0.9453947
 [2,]  3.9794036 -2.2085699
 [3,] -1.0213781 -0.4653197
 [4,] -5.6568281 -1.4140158
 [5,] -5.9273210 -0.6721883
 [6,] -6.2473260 -0.2714596
 [7,] -6.0413338  1.0020944
 [8,] -3.2243606  1.8764791
 [9,]  0.3384314  1.5491766
[10,]  3.7212447  1.1936182
[11,]  6.1550041  1.0305574
[12,] 10.7947729 -0.6749777

, , 12

            [,1]       [,2]
 [1,]  2.5738927 -0.8331962
 [2,]  3.6228326 -1.9245067
 [3,] -0.9298177 -0.3589195
 [4,] -4.4873670 -1.2862813
 [5,] -5.2453994 -0.7255149
 [6,] -5.5949877 -0.2631391
 [7,] -5.3661078  1.0017028
 [8,] -3.1142085  1.7949657
 [9,]  0.2157150  1.4149018
[10,]  3.2941762  0.9739229
[11,]  5.2128449  1.1276472
[12,]  9.8184267 -0.9215827

, , 13

            [,1]       [,2]
 [1,]  2.7839721 -0.7660797
 [2,]  3.5480110 -2.1294015
 [3,] -0.8626566 -0.1888891
 [4,] -5.3637828 -1.4741492
 [5,] -5.8933188 -0.7037490
 [6,] -5.9882255 -0.3525543
 [7,] -5.6424592  0.9975872
 [8,] -3.1412207  1.8854320
 [9,]  0.3895381  1.4276474
[10,]  3.6545544  0.9513029
[11,]  5.7470892  1.0923172
[12,] 10.7684988 -0.7394636

, , 14

            [,1]       [,2]
 [1,]  1.3814519 -0.3292149
 [2,]  2.0157500 -1.0488306
 [3,] -0.7057077 -0.2865869
 [4,] -2.7108906 -0.9232847
 [5,] -2.9796900 -0.5674436
 [6,] -3.1217561 -0.1038503
 [7,] -2.7823317  0.7375772
 [8,] -1.8447213  1.1013155
 [9,]  0.4360802  0.7506341
[10,]  1.8321198  0.6624727
[11,]  2.8109721  0.5088488
[12,]  5.6687234 -0.5016372

, , 15

            [,1]       [,2]
 [1,]  3.1493679 -0.7233783
 [2,]  3.8876163 -2.5261781
 [3,] -0.7717364 -0.1984713
 [4,] -5.6171711 -1.4360979
 [5,] -6.0817274 -1.0444802
 [6,] -6.7348620 -0.5507348
 [7,] -6.2369456  1.0402587
 [8,] -3.5029284  2.0679323
 [9,]  0.9478626  1.7965250
[10,]  3.8489306  1.4613574
[11,]  5.9740804  1.1890156
[12,] 11.1375129 -1.0757484

, , 16

            [,1]       [,2]
 [1,]  2.7458633 -0.4649230
 [2,]  3.5463450 -2.6149295
 [3,] -0.6289167 -0.1317885
 [4,] -5.3771361 -1.6362351
 [5,] -5.7095893 -0.8224859
 [6,] -5.9302232 -0.3216925
 [7,] -5.7041468  1.0089511
 [8,] -3.2166490  1.8799004
 [9,]  0.5153633  1.5966870
[10,]  3.6563958  1.3636027
[11,]  5.7947031  0.8803064
[12,] 10.3079906 -0.7373932

, , 17

            [,1]       [,2]
 [1,]  1.7907029 -0.4637049
 [2,]  2.2793698 -1.2776828
 [3,] -0.5960306 -0.2917732
 [4,] -3.1695811 -0.9488812
 [5,] -3.4357051 -0.5509137
 [6,] -3.5970637 -0.1605694
 [7,] -3.5472242  0.5306553
 [8,] -2.2690624  1.3225336
 [9,]  0.3839162  1.0477207
[10,]  2.0923015  0.7983993
[11,]  3.7051520  0.6822573
[12,]  6.3632247 -0.6880410

, , 18

            [,1]        [,2]
 [1,]  2.8217989 -0.89419143
 [2,]  3.7767299 -2.13167578
 [3,] -0.9790916 -0.02847398
 [4,] -5.1805109 -1.54071153
 [5,] -5.8329068 -0.98015991
 [6,] -6.3684094 -0.41993200
 [7,] -6.0174995  1.00255833
 [8,] -3.5671269  2.05011940
 [9,]  0.7241933  1.76658451
[10,]  3.7797306  1.13413693
[11,]  6.0441572  0.96847453
[12,] 10.7989351 -0.92672905

, , 19

            [,1]       [,2]
 [1,]  2.2913850 -0.5799429
 [2,]  3.6160547 -2.0066533
 [3,] -1.1827626 -0.2875617
 [4,] -4.8241431 -1.4343257
 [5,] -5.3737199 -0.7987185
 [6,] -5.4781455 -0.3145027
 [7,] -5.1356928  0.8527334
 [8,] -3.1025570  1.7583439
 [9,]  0.6377114  1.5665106
[10,]  3.5354445  1.2562088
[11,]  5.2104972  1.0155778
[12,]  9.8059281 -1.0276696

, , 20

            [,1]       [,2]
 [1,]  1.9971442 -0.5933256
 [2,]  3.2837419 -1.7035362
 [3,] -0.6016438 -0.3095019
 [4,] -5.0358260 -1.2796049
 [5,] -5.3774762 -0.6688190
 [6,] -5.6613799 -0.3085680
 [7,] -5.2215685  0.8484767
 [8,] -2.8358382  1.4452963
 [9,]  0.4555077  1.4592374
[10,]  3.5053154  0.9580843
[11,]  5.4259504  1.0416097
[12,] 10.0660731 -0.8893488

, , 21

            [,1]       [,2]
 [1,]  2.0586755 -0.5464193
 [2,]  2.7201666 -1.6356211
 [3,] -0.3358432 -0.1952450
 [4,] -3.9656541 -1.0609205
 [5,] -4.4345707 -0.7123124
 [6,] -4.7153985 -0.3847347
 [7,] -4.6245087  0.6208049
 [8,] -2.7734492  1.4726051
 [9,]  0.2431346  1.3906230
[10,]  2.4518104  1.0811152
[11,]  5.6668591  0.8921898
[12,]  7.7087783 -0.9220849

, , 22

            [,1]       [,2]
 [1,]  2.5502056 -0.7942396
 [2,]  3.1895019 -2.0080884
 [3,] -0.6711997 -0.1805921
 [4,] -5.4944554 -1.4911577
 [5,] -6.0538678 -0.9940123
 [6,] -6.3390512 -0.6179489
 [7,] -5.9033274  0.6678226
 [8,] -3.5702965  1.9448710
 [9,]  0.3638383  2.0232117
[10,]  3.5179448  1.5407820
[11,]  7.8192501  1.3100541
[12,] 10.5914573 -1.4007023

, , 23

            [,1]       [,2]
 [1,]  2.1411649 -0.2946308
 [2,]  2.7191239 -1.4432716
 [3,] -0.6399865 -0.1396220
 [4,] -4.2099330 -1.1731577
 [5,] -4.5442042 -0.8161529
 [6,] -4.7059622 -0.3862761
 [7,] -4.5486105  0.4434138
 [8,] -2.6405592  1.4138991
 [9,]  0.4060724  1.4729892
[10,]  2.7419014  1.2412156
[11,]  5.5944084  0.8931097
[12,]  7.6865846 -1.2115162

, , 24

             [,1]       [,2]
 [1,]  1.96808811 -0.3133458
 [2,]  2.32120175 -1.0841775
 [3,] -0.32168967 -0.1083650
 [4,] -3.64507753 -1.1743267
 [5,] -4.03409631 -0.8448301
 [6,] -4.20551671 -0.3124539
 [7,] -4.00479120  0.6288090
 [8,] -2.37928044  1.3317157
 [9,]  0.08413067  1.2683894
[10,]  2.38888093  0.8944505
[11,]  5.01751196  0.7621225
[12,]  6.81063843 -1.0479882

, , 25

            [,1]        [,2]
 [1,]  1.7909177 -0.41569706
 [2,]  2.5667436 -1.31334419
 [3,] -0.3102298 -0.09490434
 [4,] -3.6934360 -1.21608395
 [5,] -4.0366071 -0.72854860
 [6,] -4.3051210 -0.37255603
 [7,] -4.2188174  0.52555266
 [8,] -2.5528314  1.45594626
 [9,]  0.1148245  1.40940617
[10,]  2.3890299  1.15999639
[11,]  5.0496875  0.73559840
[12,]  7.2058394 -1.14536570

, , 26

             [,1]       [,2]
 [1,]  2.32608292 -0.6270818
 [2,]  2.87940248 -1.9629069
 [3,] -0.52791226 -0.2670561
 [4,] -4.66208234 -1.6178491
 [5,] -5.09353304 -0.8442974
 [6,] -5.30081523 -0.2669218
 [7,] -5.27347894  0.9174982
 [8,] -3.21669689  1.7508455
 [9,]  0.09996064  1.7381483
[10,]  2.94229436  1.2296299
[11,]  6.35575958  1.0122810
[12,]  9.47101872 -1.0622899

, , 27

            [,1]       [,2]
 [1,]  2.0218370 -0.5426721
 [2,]  3.1014599 -2.3699489
 [3,] -0.6950645 -0.0428223
 [4,] -5.1149685 -2.1789014
 [5,] -5.7067661 -1.3089190
 [6,] -6.0240289 -0.4533710
 [7,] -5.8320309  1.1121458
 [8,] -3.3502736  2.2517215
 [9,]  0.3212578  2.3029463
[10,]  3.2765043  1.4725596
[11,]  7.5450206  1.0549756
[12,] 10.4570529 -1.2977143

, , 28

            [,1]       [,2]
 [1,]  2.9431292 -0.5277151
 [2,]  3.5939201 -1.7917769
 [3,] -0.3250804  0.1429988
 [4,] -5.7915102 -1.7521549
 [5,] -6.4175360 -1.0895370
 [6,] -6.5705091 -0.5963058
 [7,] -6.4049543  0.6718021
 [8,] -3.2552075  1.9973075
 [9,]  0.5468932  1.9302588
[10,]  3.7926180  1.3880017
[11,]  7.3806481  1.0193306
[12,] 10.5075890 -1.3922099

, , 29

            [,1]        [,2]
 [1,]  2.0657650 -0.40410606
 [2,]  2.6398747 -1.83458920
 [3,] -0.5709769  0.07137553
 [4,] -4.5074028 -1.33251636
 [5,] -4.9508530 -1.08050794
 [6,] -5.2263332 -0.61788903
 [7,] -5.1880319  0.60440751
 [8,] -3.0427595  1.85436328
 [9,]  0.5186458  1.80315626
[10,]  3.2182656  1.20152616
[11,]  6.4218528  0.77025038
[12,]  8.6219534 -1.03547052

, , 30

             [,1]         [,2]
 [1,]  2.06647622 -0.384506719
 [2,]  2.93071578 -1.382288074
 [3,] -0.09750685 -0.003820901
 [4,] -3.90266500 -1.028710392
 [5,] -4.54096223 -0.787073600
 [6,] -4.88327777 -0.560951113
 [7,] -4.94479459  0.452342122
 [8,] -2.96003996  1.631442147
 [9,]  0.31621070  1.589165673
[10,]  2.78114180  0.897715400
[11,]  5.44834956  0.653520358
[12,]  7.78635234 -1.076834902

, , 31

            [,1]       [,2]
 [1,]  2.2447005 -0.6261710
 [2,]  2.8829775 -1.5238180
 [3,] -0.4266578 -0.1576928
 [4,] -4.5371648 -1.2040135
 [5,] -4.9933364 -0.7529145
 [6,] -5.3254132 -0.4515325
 [7,] -5.2017832  0.5801616
 [8,] -3.0771362  1.4501177
 [9,]  0.1747570  1.6246794
[10,]  3.0586706  1.2996724
[11,]  6.5514441  0.8801345
[12,]  8.6489420 -1.1186232

, , 32

            [,1]        [,2]
 [1,]  3.1793958 -0.80451417
 [2,]  3.8730936 -1.88426089
 [3,] -0.6983571 -0.03051212
 [4,] -5.8674563 -1.40116185
 [5,] -6.3107817 -1.02655874
 [6,] -6.6394634 -0.59617363
 [7,] -6.5438110  0.56211510
 [8,] -3.6529903  2.09428035
 [9,]  0.2405150  1.96562053
[10,]  3.9107325  1.37349726
[11,]  7.8010373  1.00743030
[12,] 10.7080856 -1.25976214

, , 33

            [,1]         [,2]
 [1,]  2.2333643 -0.349498295
 [2,]  2.8573402 -1.326569532
 [3,] -0.5802042 -0.008494057
 [4,] -4.5633167 -1.331660711
 [5,] -4.8853660 -0.822083920
 [6,] -5.2107596 -0.458661211
 [7,] -5.1233107  0.542669035
 [8,] -3.2987171  1.626288428
 [9,]  0.3993497  1.576998837
[10,]  3.0191683  1.011006907
[11,]  6.2517083  0.743304821
[12,]  8.9007435 -1.203300301

, , 34

            [,1]        [,2]
 [1,]  1.7668588 -0.30046594
 [2,]  2.1466520 -1.02993762
 [3,] -0.1190545 -0.03572064
 [4,] -3.7356000 -0.99555096
 [5,] -3.9278586 -0.67245195
 [6,] -4.1523663 -0.57680373
 [7,] -4.1834061  0.42608212
 [8,] -2.7129817  1.41330974
 [9,]  0.2082112  1.42843249
[10,]  2.3910628  0.87747090
[11,]  4.8588471  0.60349450
[12,]  7.4596353 -1.13785892

, , 35

            [,1]       [,2]
 [1,]  2.5040096 -0.3974722
 [2,]  3.1428864 -1.6904740
 [3,] -0.3805026 -0.0695374
 [4,] -4.7587793 -1.3899445
 [5,] -5.3222296 -0.8986445
 [6,] -5.5072408 -0.4749214
 [7,] -5.2840916  0.6188438
 [8,] -3.1863157  1.5647366
 [9,]  0.5481317  1.8801112
[10,]  3.2181932  1.2387219
[11,]  6.3730307  0.6552514
[12,]  8.6529080 -1.0366710

, , 36

              [,1]        [,2]
 [1,]  2.038356315 -0.33439296
 [2,]  2.689169902 -1.36656488
 [3,] -0.601081597  0.06920366
 [4,] -4.248503843 -1.28831333
 [5,] -4.685032020 -0.81301115
 [6,] -4.897272627 -0.42688116
 [7,] -4.627251328  0.45918056
 [8,] -2.804042565  1.51077666
 [9,] -0.002199743  1.59995186
[10,]  2.751099155  1.07699010
[11,]  5.815333769  0.85968431
[12,]  8.571424583 -1.34662368

, , 37

             [,1]       [,2]
 [1,]  2.04603850 -0.3656546
 [2,]  2.74011345 -1.3934070
 [3,]  0.02049493 -0.2260056
 [4,] -4.27009971 -1.1497467
 [5,] -4.63169227 -0.8251916
 [6,] -4.69407596 -0.3278893
 [7,] -4.57977924  0.6996201
 [8,] -3.05925319  1.4143715
 [9,]  0.11719082  1.4632730
[10,]  2.55296894  1.0569628
[11,]  5.72738093  0.6172858
[12,]  8.03071280 -0.9636184

, , 38

            [,1]       [,2]
 [1,]  2.0814144 -0.5538173
 [2,]  3.3143287 -1.8749568
 [3,] -0.4781248 -0.2545334
 [4,] -5.2088169 -1.7113334
 [5,] -5.7262479 -0.9018615
 [6,] -5.9307576 -0.4164446
 [7,] -5.6666142  0.9711003
 [8,] -3.4549436  1.8411995
 [9,]  0.1982786  1.7953708
[10,]  3.3253015  1.1591226
[11,]  7.1787788  1.1996810
[12,] 10.3674031 -1.2535273

, , 39

            [,1]       [,2]
 [1,]  2.2649267 -0.4407178
 [2,]  2.8434181 -1.3497920
 [3,] -0.3641150 -0.1306634
 [4,] -4.2970605 -1.3559451
 [5,] -4.7850775 -0.8517438
 [6,] -5.0378996 -0.2719651
 [7,] -4.8529908  0.7140041
 [8,] -2.8578653  1.5263089
 [9,]  0.4703274  1.2745767
[10,]  2.8478150  1.0161549
[11,]  5.5721444  0.9981573
[12,]  8.1963770 -1.1283746

, , 40

             [,1]        [,2]
 [1,]  2.79618546 -0.49564034
 [2,]  3.50248596 -1.54707380
 [3,] -0.04372609 -0.09812394
 [4,] -5.18663179 -1.23310174
 [5,] -5.73279385 -0.97392731
 [6,] -5.89817525 -0.41459480
 [7,] -5.81840771  0.58503891
 [8,] -3.46483738  1.71854754
 [9,]  0.62170987  1.61792806
[10,]  3.15033776  1.13563961
[11,]  6.74281245  0.83211422
[12,]  9.33104058 -1.12680640

geomorph documentation built on Aug. 10, 2017, 1:11 a.m.