Often, researchers will conduct a few high-resolution time course experiments (or densely sample points along a spatial axis), but then they must select a subset of points to sample in follow-up experiments due to financial constraints. NITPicker is a tool to select the best points to subsample. We present three different definitions of what constitutes a `good’ set of time points:
Please note that this is not a very fast value to compute– it may take a few hours to complete on a large dataset. The precise definition and motivation for these three criteria can be found at: https://doi.org/10.1101/301796
Please note that this package relies heavily on the fdasrvf package, which is used to generate probability distributions of curves, based on a set of example functions.
Please cite the following if you use this R package:
Ezer, D. and Keir J.C. Selection of time points for costly experiments: a comparison between human intuition and computer-aided experimental design. bioarxiv, :10.1101/301796 (2018).
It might also be advisable to cite this paper, which presents the fdasrvf package which is an important part of this project:
Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.
For more background on why it might be useful to minimise the L2-error for finding optimal time points to sample, please see:
Michael Kleyman, Emre Sefer, Teodora Nicola, Celia Espinoza, Divya Chhabra, James S Hagood, Naftali Kaminski, Namasivayam Ambalavanan, Ziv Bar-Joseph. Selecting the most appropriate time points to profile in high-throughput studies. eLife 2017;6:e18541 (2017).
In this example, we try to find 4 months to subsample that will let us estimate the shape of the curves as accurately as possible (minimise L2-error). In this case, we sample only 3 example curves from the distribution of curves when estimating the integral– this should be much higher (100 curves is usually suitable), but we make it 3 here to speed up loading the vignette.
Note that the tables that are printed as the evaluation are the optimisation tables that are used as part of the NITPicker algorithm’s dynamic programming algorithm.
library(fda)
#> Loading required package: splines
#> Loading required package: Matrix
#>
#> Attaching package: 'fda'
#> The following object is masked from 'package:graphics':
#>
#> matplot
library(NITPicker)
#> Loading required package: fdasrvf
#> Loading required package: fda.usc
#> Loading required package: MASS
#> Loading required package: mgcv
#> Loading required package: nlme
#> This is mgcv 1.8-23. For overview type 'help("mgcv-package")'.
#> Loading required package: rpart
mat=CanadianWeather$monthlyTemp #load data - a matrix with 12 rows, representing months (time); and 35 columns, representing cities (experiments)
a=findPathF1(c(1:12), mat, 4, numPerts=3) #find a set of points that help predict the shape of the curve
#> lambda = 0.0
#>
#> Initializing...
#> Computing Karcher mean of 35 functions in SRSF space...
#> updating step: r=1
#> updating step: r=2
#> updating step: r=3
#> updating step: r=4
#> updating step: r=5
#> updating step: r=6
#> updating step: r=7
#> updating step: r=8
#> updating step: r=9
#> updating step: r=10
#> updating step: r=11
#> updating step: r=12
#> updating step: r=13
#> updating step: r=14
#> updating step: r=15
#> updating step: r=16
#> updating step: r=17
#> updating step: r=18
#> [,1]
#> [1,] 0
#> [2,] Inf
#> [3,] Inf
#> [4,] Inf
#> [5,] Inf
#> [,1] [,2]
#> [1,] 3.621742 Inf
#> [2,] Inf 0
#> [3,] Inf Inf
#> [4,] Inf Inf
#> [5,] Inf Inf
#> [,1] [,2] [,3]
#> [1,] 133.3451 Inf Inf
#> [2,] Inf 5.668084 3.621742
#> [3,] Inf Inf 0.000000
#> [4,] Inf Inf Inf
#> [5,] Inf Inf Inf
#> [,1] [,2] [,3] [,4]
#> [1,] 748.3137 Inf Inf Inf
#> [2,] Inf 28.1367 4.5472787 133.345107
#> [3,] Inf Inf 0.9255366 3.621742
#> [4,] Inf Inf Inf 0.000000
#> [5,] Inf Inf Inf Inf
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 2263.264 Inf Inf Inf Inf
#> [2,] Inf 69.35294 8.477366 133.6449597 748.3137091
#> [3,] Inf Inf 4.855624 3.9215948 4.5472787
#> [4,] Inf Inf Inf 0.2998527 0.9255366
#> [5,] Inf Inf Inf Inf 0.0000000
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 4825.478 Inf Inf Inf Inf Inf
#> [2,] Inf 112.3561 13.219538 134.575208 748.4339064 2263.2644594
#> [3,] Inf Inf 9.597795 4.851843 4.6674760 8.4773663
#> [4,] Inf Inf Inf 1.230101 1.0457339 3.9215948
#> [5,] Inf Inf Inf Inf 0.1201973 0.2998527
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 7404.123 Inf Inf Inf Inf Inf
#> [2,] Inf 99.70077 13.75664 143.37729 755.372079 2265.390453
#> [3,] Inf Inf 10.13489 13.65392 11.605649 10.603359
#> [4,] Inf Inf Inf 10.03218 7.983907 6.047588
#> [5,] Inf Inf Inf Inf 7.058370 2.425846
#> [,7]
#> [1,] Inf
#> [2,] 4825.478228
#> [3,] 13.219538
#> [4,] 4.667476
#> [5,] 1.045734
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 7056.881 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 178.6286 223.3720 344.0645 893.9640 2330.78733 4839.28299
#> [3,] Inf Inf 219.7503 214.3412 150.1976 76.00024 27.02430
#> [4,] Inf Inf Inf 210.7194 146.5758 71.44446 18.47224
#> [5,] Inf Inf Inf Inf 145.6503 67.82272 14.85050
#> [,8]
#> [1,] Inf
#> [2,] 7404.123489
#> [3,] 13.756635
#> [4,] 10.134893
#> [5,] 4.667476
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 4318.885 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 1236.897 1384.188 1303.690 1553.1187 2684.8633 4959.6385
#> [3,] Inf Inf 1380.566 1173.967 809.3522 430.0762 147.3798
#> [4,] Inf Inf Inf 1170.345 805.7305 425.5204 138.8277
#> [5,] Inf Inf Inf Inf 804.8050 421.8987 135.2060
#> [,8] [,9]
#> [1,] Inf Inf
#> [2,] 7417.24509 7056.88056
#> [3,] 26.87823 178.62863
#> [4,] 23.25649 13.75664
#> [5,] 17.78907 10.13489
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 3082.905 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 3679.203 3727.011 3162.400 2821.955 3386.232 5226.4995
#> [3,] Inf Inf 3723.389 3032.677 2078.188 1131.445 414.2408
#> [4,] Inf Inf Inf 3029.055 2074.567 1126.889 405.6887
#> [5,] Inf Inf Inf Inf 2073.641 1123.267 402.0670
#> [,8] [,9] [,10]
#> [1,] Inf Inf Inf
#> [2,] 7463.72923 7058.45530 4318.88463
#> [3,] 73.36237 180.20337 1236.89744
#> [4,] 69.74063 15.33138 26.87823
#> [5,] 64.27322 11.70963 13.75664
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 4690.557 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 6935.959 6694.838 5451.350 4347.958 4211.230 5533.5129
#> [3,] Inf Inf 6691.216 5321.627 3604.192 1956.442 721.2543
#> [4,] Inf Inf Inf 5318.005 3600.570 1951.887 712.7022
#> [5,] Inf Inf Inf Inf 3599.644 1948.265 709.0805
#> [,8] [,9] [,10] [,11]
#> [1,] Inf Inf Inf Inf
#> [2,] 7516.2068 7060.16373 4318.97264 3082.90475
#> [3,] 125.8399 181.91180 1236.98545 2821.95469
#> [4,] 122.2182 17.03980 26.96625 73.36237
#> [5,] 116.7508 13.41806 13.84465 15.33138
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 9869.331 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 11175.46 10458.46 8314.885 6242.657 5240.497 5931.348
#> [3,] Inf Inf 10454.84 8185.161 5498.891 2985.710 1119.089
#> [4,] Inf Inf Inf 8181.539 5495.269 2981.154 1110.537
#> [5,] Inf Inf Inf Inf 5494.344 2977.533 1106.915
#> [,8] [,9] [,10] [,11] [,12]
#> [1,] Inf Inf Inf Inf Inf
#> [2,] 7595.4135 7066.60905 4321.06485 3084.02904 4690.5565
#> [3,] 205.0467 188.35712 1239.07766 2823.07898 3082.9047
#> [4,] 201.4249 23.48513 29.05846 74.48666 125.8399
#> [5,] 195.9575 19.86338 15.93686 16.45567 17.0398
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,] 0 Inf Inf Inf Inf Inf Inf Inf
#> [2,] Inf 13072.04 11191.11 6930.932 3972.131 3819.494 6708.756 10488.089
#> [3,] Inf Inf 11187.48 6801.208 3228.365 1564.707 1896.497 3097.723
#> [4,] Inf Inf Inf 6797.587 3224.743 1560.151 1887.945 3094.101
#> [5,] Inf Inf Inf Inf 3223.817 1556.529 1884.323 3088.633
#> [,9] [,10] [,11] [,12] [,13]
#> [1,] Inf Inf Inf Inf Inf
#> [2,] 9959.539 5774.459 3509.7067 4751.3009 9869.33147
#> [3,] 3081.287 2692.472 3248.7567 3143.6491 3084.02904
#> [4,] 2916.415 1482.453 500.1644 186.5843 188.35712
#> [5,] 2912.793 1469.331 442.1334 77.7842 23.48513
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 0 0 0 0
#> [2,] 0 1 0 0 0
#> [3,] 0 2 2 0 0
#> [4,] 0 2 2 3 0
#> [5,] 0 2 3 3 4
#> [6,] 0 2 4 4 4
#> [7,] 0 2 2 6 6
#> [8,] 0 1 7 7 7
#> [9,] 0 1 7 8 8
#> [10,] 0 4 7 8 8
#> [11,] 0 10 7 8 8
#> [12,] 0 10 8 8 9
#> [13,] 0 10 5 11 12
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.000000 Inf Inf Inf Inf
#> [2,] 3.621742 0.000000 Inf Inf Inf
#> [3,] 133.345107 3.621742 0.0000000 Inf Inf
#> [4,] 748.313709 4.547279 0.9255366 0.0000000 Inf
#> [5,] 2263.264459 8.477366 3.9215948 0.2998527 0.0000000
#> [6,] 4825.478228 13.219538 4.6674760 1.0457339 0.1201973
#> [7,] 7404.123489 13.756635 10.1348930 4.6674760 1.0457339
#> [8,] 7056.880563 178.628628 13.7566351 10.1348930 4.6674760
#> [9,] 4318.884625 1236.897436 26.8782331 13.7566351 10.1348930
#> [10,] 3082.904746 2821.954693 73.3623746 15.3313760 11.7096340
#> [11,] 4690.556549 3082.904746 125.8399416 17.0398025 13.4180604
#> [12,] 9869.331468 3084.029036 188.3571201 23.4851271 15.9368638
#> [13,] 0.000000 3509.706729 1564.7069898 186.5843393 23.4851271
print(a) #indices of months to select for follow-up experiments
#> [1] 2 7 8 12
print(rownames(CanadianWeather$monthlyTemp)[a]) #month names selected
#> [1] "Feb" "Jul" "Aug" "Dec"In this example, we consider Canadian cities to be different experimental conditions, and we consider Resolute, Canada to be the control condition. We want to find a set of points that will enable us to estimate the profile of the difference in temperature between Resolute and other cities in Canada. In this case, we sample only 3 example curves from the distribution of curves when estimating the integral– this should be much higher (100 curves is usually suitable), but we make it 3 here to speed up loading the vignette.
Note that the tables that are printed as the evaluation are the optimisation tables that are used as part of the NITPicker algorithm’s dynamic programming algorithm.
library(fda)
library(NITPicker)
mat=CanadianWeather$monthlyTemp #load data - a matrix with 12 rows, representing months (time); and 35 columns, representing cities (experiments)
y=CanadianWeather$monthlyTemp[,"Resolute"]
a=findPathF2(c(1:12), y, mat, 4, numPerts=3) #find a set of points that help predict the shape of the curve
#> lambda = 0.0
#>
#> Initializing...
#> Computing Karcher mean of 35 functions in SRSF space...
#> updating step: r=1
#> updating step: r=2
#> updating step: r=3
#> updating step: r=4
#> updating step: r=5
#> updating step: r=6
#> updating step: r=7
#> updating step: r=8
#> updating step: r=9
#> updating step: r=10
#> updating step: r=11
#> updating step: r=12
#> updating step: r=13
#> updating step: r=14
#> updating step: r=15
#> updating step: r=16
#> updating step: r=17
#> updating step: r=18
#> [,1]
#> [1,] 0
#> [2,] Inf
#> [3,] Inf
#> [4,] Inf
#> [5,] Inf
#> [,1] [,2]
#> [1,] 6.355158 Inf
#> [2,] Inf 0
#> [3,] Inf Inf
#> [4,] Inf Inf
#> [5,] Inf Inf
#> [,1] [,2] [,3]
#> [1,] 118.4347 Inf Inf
#> [2,] Inf 2.027377 6.355158
#> [3,] Inf Inf 0.000000
#> [4,] Inf Inf Inf
#> [5,] Inf Inf Inf
#> [,1] [,2] [,3] [,4]
#> [1,] 231.2051 Inf Inf Inf
#> [2,] Inf 9.411251 12.264336 118.434700
#> [3,] Inf Inf 5.909178 2.027377
#> [4,] Inf Inf Inf 0.000000
#> [5,] Inf Inf Inf Inf
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 423.7435 Inf Inf Inf Inf
#> [2,] Inf 196.328 123.5027 137.94313 231.205142
#> [3,] Inf Inf 117.1476 21.53580 9.411251
#> [4,] Inf Inf Inf 19.50843 2.027377
#> [5,] Inf Inf Inf Inf 0.000000
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1389.967 Inf Inf Inf Inf Inf
#> [2,] Inf 519.9946 296.2477 168.00575 231.3226847 423.743512
#> [3,] Inf Inf 289.8925 51.59843 9.5287936 123.502712
#> [4,] Inf Inf Inf 49.57105 2.1449193 9.411251
#> [5,] Inf Inf Inf Inf 0.1175425 2.027377
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 2114.2 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 645.3614 343.2179 166.08117 247.52608 428.780044 1389.967041
#> [3,] Inf Inf 336.8628 49.67385 25.73219 128.539243 168.005755
#> [4,] Inf Inf Inf 47.64647 18.34832 14.447782 9.528794
#> [5,] Inf Inf Inf Inf 16.32094 7.063908 2.144919
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 1789.303 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 555.4289 309.5598 229.3602 373.0257 480.75672 1396.430905
#> [3,] Inf Inf 303.2047 112.9529 151.2318 180.51592 174.469618
#> [4,] Inf Inf Inf 110.9255 143.8479 66.42446 15.992657
#> [5,] Inf Inf Inf Inf 141.8205 59.04059 8.608783
#> [,8]
#> [1,] Inf
#> [2,] 2114.199861
#> [3,] 166.081168
#> [4,] 25.732190
#> [5,] 9.528794
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 1144.156 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 610.4569 454.5879 499.8046 670.7499 613.9047 1423.62736
#> [3,] Inf Inf 448.2328 383.3972 448.9560 313.6639 201.66608
#> [4,] Inf Inf Inf 381.3699 441.5721 199.5724 43.18912
#> [5,] Inf Inf Inf Inf 439.5447 192.1886 35.80524
#> [,8] [,9]
#> [1,] Inf Inf
#> [2,] 2115.68638 1789.30335
#> [3,] 167.56769 229.36024
#> [4,] 27.21871 112.95292
#> [5,] 11.01532 15.99266
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 848.888 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 999.461 953.8747 1148.214 1284.524 917.3409 1513.6125
#> [3,] Inf Inf 947.5196 1031.807 1062.730 617.1001 291.6512
#> [4,] Inf Inf Inf 1029.779 1055.346 503.0087 133.1742
#> [5,] Inf Inf Inf Inf 1053.319 495.6248 125.7904
#> [,8] [,9] [,10]
#> [1,] Inf Inf Inf
#> [2,] 2131.31746 1791.46860 1144.15587
#> [3,] 183.19876 231.52549 454.58793
#> [4,] 42.84978 115.11817 167.56769
#> [5,] 26.64639 18.15791 27.21871
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 1176.465 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 1417.835 1386.389 1616.562 1682.371 1078.0452 1539.6483
#> [3,] Inf Inf 1380.033 1500.155 1460.577 777.8044 317.6871
#> [4,] Inf Inf Inf 1498.128 1453.193 663.7129 159.2101
#> [5,] Inf Inf Inf Inf 1451.166 656.3291 151.8262
#> [,8] [,9] [,10] [,11]
#> [1,] Inf Inf Inf Inf
#> [2,] 2130.68494 1792.20633 1147.7862 848.88796
#> [3,] 182.56625 232.26322 458.2182 917.34093
#> [4,] 42.21727 115.85590 171.1980 183.19876
#> [5,] 26.01387 18.89564 30.8490 42.84978
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 1515.19 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 1660.6 1532.703 1651.363 1651.141 1021.3012 1537.3541
#> [3,] Inf Inf 1526.348 1534.955 1429.347 721.0604 315.3928
#> [4,] Inf Inf Inf 1532.928 1421.963 606.9689 156.9159
#> [5,] Inf Inf Inf Inf 1419.936 599.5851 149.5320
#> [,8] [,9] [,10] [,11] [,12]
#> [1,] Inf Inf Inf Inf Inf
#> [2,] 2195.32323 1862.91421 1198.24912 856.41686 1176.46550
#> [3,] 247.20454 302.97110 508.68118 924.86982 848.88796
#> [4,] 106.85556 186.56378 221.66094 190.72766 182.56625
#> [5,] 90.65216 89.60352 81.31196 50.37868 42.21727
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 1881.577 1798.239 2477.160 2488.608 1031.2624 1835.1374
#> [3,] Inf Inf 1791.884 2360.753 2266.814 731.0216 613.1761
#> [4,] Inf Inf Inf 2358.725 2259.430 616.9301 454.6991
#> [5,] Inf Inf Inf Inf 2257.403 609.5463 447.3152
#> [,8] [,9] [,10] [,11] [,12] [,13]
#> [1,] Inf Inf Inf Inf Inf Inf
#> [2,] 2862.6417 2381.4535 1422.4112 890.20539 1187.14978 1515.1903
#> [3,] 914.5230 821.5104 732.8432 958.65835 859.57225 856.4169
#> [4,] 774.1741 705.1031 445.8230 224.51619 193.25053 247.2045
#> [5,] 757.9707 608.1428 305.4740 84.16721 52.90155 106.8556
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 0 0 0 0
#> [2,] 0 1 0 0 0
#> [3,] 0 1 2 0 0
#> [4,] 0 1 3 3 0
#> [5,] 0 2 4 4 4
#> [6,] 0 3 4 4 4
#> [7,] 0 3 4 6 6
#> [8,] 0 3 3 6 6
#> [9,] 0 2 7 7 7
#> [10,] 0 5 7 7 8
#> [11,] 0 10 7 7 8
#> [12,] 0 10 7 7 11
#> [13,] 0 10 6 11 11
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.000000 Inf Inf Inf Inf
#> [2,] 6.355158 0.000000 Inf Inf Inf
#> [3,] 118.434700 2.027377 0.000000 Inf Inf
#> [4,] 231.205142 9.411251 2.027377 0.000000 Inf
#> [5,] 423.743512 123.502712 9.411251 2.027377 0.0000000
#> [6,] 1389.967041 168.005755 9.528794 2.144919 0.1175425
#> [7,] 2114.199861 166.081168 25.732190 9.528794 2.1449193
#> [8,] 1789.303348 229.360240 112.952917 15.992657 8.6087826
#> [9,] 1144.155870 454.587928 167.567692 27.218713 11.0153172
#> [10,] 848.887963 917.340929 183.198763 42.849784 18.1579119
#> [11,] 1176.465499 848.887963 182.566250 42.217271 18.8956367
#> [12,] 1515.190253 856.416855 247.204539 106.855560 42.2172710
#> [13,] 0.000000 890.205386 613.176076 193.250533 52.9015540
print(a) #indices of months to select for follow-up experiments
#> [1] 1 4 7 11
print(rownames(CanadianWeather$monthlyTemp)[a])
#> [1] "Jan" "Apr" "Jul" "Nov"In this example, we try to identify points that can predict the shape of the curve of the difference between the temperatures of Canadian cities that are along the Atlantic with those that are Continental– however, we don’t care too much about sampling time points where there is lots of noise, so we normalise this by the variance at each point.
In this case, we sample only 3 example curves from the distribution of curves when estimating the integral– this should be much higher (100 curves is usually suitable), but we make it 3 here to speed up loading the vignette.
Note that the tables that are printed as the evaluation are the optimisation tables that are used as part of the NITPicker algorithm’s dynamic programming algorithm.
library(fda)
library(NITPicker)
#Set up data:
atlanticCities=which(CanadianWeather$region[as.character(colnames(CanadianWeather$monthlyTemp))]=="Atlantic")
matAtlantic=CanadianWeather$monthlyTemp[, names(atlanticCities)]
continentalCities=which(CanadianWeather$region[as.character(colnames(CanadianWeather$monthlyTemp))]=="Continental")
matContinental=CanadianWeather$monthlyTemp[, names(continentalCities)]
#find a set of points that helps capture the difference between Atlantic and Continental cities, normalised by the variance
a=findPathF3(c(1:12), matAtlantic, matContinental, 4, numPerts=3) #find a set of points that help predict the shape of the curve
#> lambda = 0.0
#>
#> Initializing...
#> Computing Karcher mean of 15 functions in SRSF space...
#> updating step: r=1
#> updating step: r=2
#> updating step: r=3
#> updating step: r=4
#> updating step: r=5
#> updating step: r=6
#> updating step: r=7
#> updating step: r=8
#> updating step: r=9
#> updating step: r=10
#> updating step: r=11
#> lambda = 0.0
#>
#> Initializing...
#> Computing Karcher mean of 12 functions in SRSF space...
#> updating step: r=1
#> updating step: r=2
#> updating step: r=3
#> updating step: r=4
#> updating step: r=5
#> updating step: r=6
#> lambda = 0.0
#>
#> Initializing...
#> Computing Karcher mean of 3 functions in SRSF space...
#> updating step: r=1
#> updating step: r=2
#> updating step: r=3
#> updating step: r=4
#> updating step: r=5
#> updating step: r=6
#> updating step: r=7
#> updating step: r=8
#> updating step: r=9
#> updating step: r=10
#> updating step: r=11
#> updating step: r=12
#> updating step: r=13
#> updating step: r=14
#> updating step: r=15
#> updating step: r=16
#> updating step: r=17
#> [,1]
#> [1,] 0
#> [2,] Inf
#> [3,] Inf
#> [4,] Inf
#> [5,] Inf
#> [,1] [,2]
#> [1,] 0.01085616 Inf
#> [2,] Inf 0
#> [3,] Inf Inf
#> [4,] Inf Inf
#> [5,] Inf Inf
#> [,1] [,2] [,3]
#> [1,] 0.0882301 Inf Inf
#> [2,] Inf 4.398408e-06 0.01085616
#> [3,] Inf Inf 0.00000000
#> [4,] Inf Inf Inf
#> [5,] Inf Inf Inf
#> [,1] [,2] [,3] [,4]
#> [1,] 0.2954144 Inf Inf Inf
#> [2,] Inf 1.226947e-05 1.085776e-02 8.823010e-02
#> [3,] Inf Inf 1.607228e-06 4.398408e-06
#> [4,] Inf Inf Inf 0.000000e+00
#> [5,] Inf Inf Inf Inf
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.6853104 Inf Inf Inf Inf
#> [2,] Inf 4.722347e-05 1.087729e-02 8.823336e-02 2.954144e-01
#> [3,] Inf Inf 2.112856e-05 7.663471e-06 1.226947e-05
#> [4,] Inf Inf Inf 3.265062e-06 1.607228e-06
#> [5,] Inf Inf Inf Inf 0.000000e+00
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1.294239 Inf Inf Inf Inf
#> [2,] Inf 0.0002712687 0.0110306359 8.828748e-02 2.954240e-01
#> [3,] Inf Inf 0.0001744783 6.178628e-05 2.187124e-05
#> [4,] Inf Inf Inf 5.738787e-05 1.120900e-05
#> [5,] Inf Inf Inf Inf 9.601768e-06
#> [,6]
#> [1,] Inf
#> [2,] 6.853104e-01
#> [3,] 4.722347e-05
#> [4,] 7.663471e-06
#> [5,] 1.607228e-06
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 2.15068 Inf Inf Inf Inf
#> [2,] Inf 0.001250506 0.0117628505 0.0886603751 0.2955652136
#> [3,] Inf Inf 0.0009066929 0.0004346769 0.0001630451
#> [4,] Inf Inf Inf 0.0004302785 0.0001523828
#> [5,] Inf Inf Inf Inf 0.0001507756
#> [,6] [,7]
#> [1,] Inf Inf
#> [2,] 6.853353e-01 1.294239e+00
#> [3,] 7.220059e-05 2.712687e-04
#> [4,] 3.264059e-05 2.187124e-05
#> [5,] 2.658435e-05 7.663471e-06
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 3.369037 Inf Inf Inf Inf
#> [2,] Inf 0.002506565 0.012575216 0.0890486652 0.2957251227
#> [3,] Inf Inf 0.001719058 0.0008229670 0.0003229542
#> [4,] Inf Inf Inf 0.0008185686 0.0003122919
#> [5,] Inf Inf Inf Inf 0.0003106847
#> [,6] [,7] [,8]
#> [1,] Inf Inf Inf
#> [2,] 6.853942e-01 1.294280e+00 2.150680e+00
#> [3,] 1.310418e-04 3.125521e-04 1.250506e-03
#> [4,] 9.148183e-05 6.315459e-05 7.220059e-05
#> [5,] 8.542558e-05 4.894682e-05 2.187124e-05
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 4.920554 Inf Inf Inf Inf Inf
#> [2,] Inf 0.006144414 0.015137421 0.090566118 0.296536769 0.6857530711
#> [3,] Inf Inf 0.004281264 0.002340420 0.001134601 0.0004899286
#> [4,] Inf Inf Inf 0.002336021 0.001123938 0.0004503686
#> [5,] Inf Inf Inf Inf 0.001122331 0.0004443124
#> [,7] [,8] [,9]
#> [1,] Inf Inf Inf
#> [2,] 1.2944146353 2.150698e+00 3.369037e+00
#> [3,] 0.0004470093 1.268247e-03 2.506565e-03
#> [4,] 0.0001976118 8.994168e-05 1.310418e-04
#> [5,] 0.0001834041 3.961232e-05 6.315459e-05
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 6.422067 Inf Inf Inf Inf Inf
#> [2,] Inf 0.02249463 0.02752082 0.09898234 0.301824258 0.688765118
#> [3,] Inf Inf 0.01666466 0.01075664 0.006422090 0.003501975
#> [4,] Inf Inf Inf 0.01075225 0.006411428 0.003462415
#> [5,] Inf Inf Inf Inf 0.006409820 0.003456359
#> [,7] [,8] [,9] [,10]
#> [1,] Inf Inf Inf Inf
#> [2,] 1.296066128 2.1515862802 3.3692717542 4.920554e+00
#> [3,] 0.002098502 0.0021567557 0.0027416657 6.144414e-03
#> [4,] 0.001849105 0.0009784506 0.0003661422 4.470093e-04
#> [5,] 0.001834897 0.0009281213 0.0002982550 8.994168e-05
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 7.029903 Inf Inf Inf Inf Inf
#> [2,] Inf 0.1141843 0.10234348 0.15672151 0.34456938 0.71903299
#> [3,] Inf Inf 0.09148732 0.06849581 0.04916722 0.03376984
#> [4,] Inf Inf Inf 0.06849141 0.04915655 0.03373028
#> [5,] Inf Inf Inf Inf 0.04915495 0.03372423
#> [,7] [,8] [,9] [,10] [,11]
#> [1,] Inf Inf Inf Inf Inf
#> [2,] 1.31667277 2.16475382 3.375170083 4.921719820 6.4220669647
#> [3,] 0.02270514 0.01532429 0.008639994 0.007310587 0.0224946338
#> [4,] 0.02245574 0.01414599 0.006264471 0.001613182 0.0020985021
#> [5,] 0.02244154 0.01409566 0.006196584 0.001256115 0.0003661422
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 5.329122 Inf Inf Inf Inf Inf Inf
#> [2,] Inf 0.6969646 0.5992045 0.5668602 0.6734905 0.9740162 1.5063195
#> [3,] Inf Inf 0.5883483 0.4786345 0.3780883 0.2887530 0.2123519
#> [4,] Inf Inf Inf 0.4786301 0.3780776 0.2887135 0.2121025
#> [5,] Inf Inf Inf Inf 0.3780760 0.2887074 0.2120883
#> [,8] [,9] [,10] [,11] [,12]
#> [1,] Inf Inf Inf Inf Inf
#> [2,] 2.2971194 3.45151528 4.95321273 6.428421120 7.029902896
#> [3,] 0.1476898 0.08498519 0.03880350 0.028848789 0.102343482
#> [4,] 0.1465115 0.08260967 0.03310610 0.008452658 0.007310587
#> [5,] 0.1464612 0.08254178 0.03274903 0.006720298 0.001613182
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,] 0 Inf Inf Inf Inf Inf Inf Inf
#> [2,] Inf 11.88921 8.603930 6.010222 4.166324 3.056789 2.629357 2.8110388
#> [3,] Inf Inf 8.593074 5.921996 3.870922 2.371525 1.335389 0.6616093
#> [4,] Inf Inf Inf 5.921992 3.870911 2.371486 1.335140 0.6604310
#> [5,] Inf Inf Inf Inf 3.870909 2.371480 1.335126 0.6603806
#> [,9] [,10] [,11] [,12] [,13]
#> [1,] Inf Inf Inf Inf Inf
#> [2,] 3.6089103 4.96819152 6.430026738 7.03848773 5.329122207
#> [3,] 0.2423802 0.05378229 0.030454407 0.11092832 0.566860224
#> [4,] 0.2400047 0.04808488 0.010058276 0.01589542 0.028848789
#> [5,] 0.2399368 0.04772782 0.008325916 0.01019802 0.007310587
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 0 0 0 0
#> [2,] 0 1 0 0 0
#> [3,] 0 1 2 0 0
#> [4,] 0 1 2 3 0
#> [5,] 0 1 3 4 4
#> [6,] 0 1 4 5 5
#> [7,] 0 1 5 6 6
#> [8,] 0 1 5 6 7
#> [9,] 0 1 6 7 7
#> [10,] 0 1 6 8 9
#> [11,] 0 2 9 9 10
#> [12,] 0 3 10 11 11
#> [13,] 0 6 10 10 12
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.00000000 Inf Inf Inf Inf
#> [2,] 0.01085616 0.000000e+00 Inf Inf Inf
#> [3,] 0.08823010 4.398408e-06 0.000000e+00 Inf Inf
#> [4,] 0.29541444 1.226947e-05 1.607228e-06 0.000000e+00 Inf
#> [5,] 0.68531037 4.722347e-05 7.663471e-06 1.607228e-06 0.000000e+00
#> [6,] 1.29423889 2.712687e-04 2.187124e-05 7.663471e-06 1.607228e-06
#> [7,] 2.15068003 1.250506e-03 7.220059e-05 2.187124e-05 7.663471e-06
#> [8,] 3.36903665 2.506565e-03 1.310418e-04 6.315459e-05 2.187124e-05
#> [9,] 4.92055365 6.144414e-03 4.470093e-04 8.994168e-05 3.961232e-05
#> [10,] 6.42206696 2.249463e-02 2.098502e-03 3.661422e-04 8.994168e-05
#> [11,] 7.02990290 1.023435e-01 7.310587e-03 1.613182e-03 3.661422e-04
#> [12,] 5.32912221 5.668602e-01 2.884879e-02 7.310587e-03 1.613182e-03
#> [13,] 0.00000000 2.629357e+00 3.045441e-02 1.005828e-02 7.310587e-03
print(a) #indices of months to select for follow-up experiments
#> [1] 1 9 11 12
print(rownames(CanadianWeather$monthlyTemp)[a]) #month names selected
#> [1] "Jan" "Sep" "Nov" "Dec"