R-TDA package tutorial
Mathieu Carrière and Steve Oudot
(adapted from the following tutorial by F. Chazal and B. Michel)
July 2016
For this practical lesson about Persistence Homology Inference, we need the TDA R-package.
install.packages("TDA", dependencies=TRUE) library(TDA) Persistence diagrams for Rips filtration
The ripsDiag() function computes the persistence diagram of the Rips filtration built on top of a point cloud.
The following code generates 60 points from two circles:
Circle1 <- circleUnif(60)
Circle2 <- circleUnif(60, r = 2) + 3
Circles <- rbind(Circle1, Circle2)
plot(Circles, pch = 16, xlab = "",ylab = "")
We specify the limit of the Rips filtration and the max dimension of the homological features we are interested in (0 for components, 1 for loops, 2 for voids, etc.).
maxdimension <- 1 # components and loops
maxscale <- 5 # limit of the filtrationand we generate the persistence diagram using either the C++ library GUDHI (or Dionysus):
Diag <- ripsDiag(X = Circles,
maxdimension,
maxscale,
library = "GUDHI",
printProgress = FALSE)
print(Diag[["diagram"]])## dimension Birth Death
## [1,] 0 0.0000000 5.000000000
## [2,] 0 0.0000000 1.248655167
## [3,] 0 0.0000000 0.681882543
## [4,] 0 0.0000000 0.626945491
## [5,] 0 0.0000000 0.605573483
## [6,] 0 0.0000000 0.558305002
## [7,] 0 0.0000000 0.503336136
## [8,] 0 0.0000000 0.454175829
## [9,] 0 0.0000000 0.420407057
## [10,] 0 0.0000000 0.419658105
## [11,] 0 0.0000000 0.370848708
## [12,] 0 0.0000000 0.364224499
## [13,] 0 0.0000000 0.351474017
## [14,] 0 0.0000000 0.345539414
## [15,] 0 0.0000000 0.335991622
## [16,] 0 0.0000000 0.334634177
## [17,] 0 0.0000000 0.326365159
## [18,] 0 0.0000000 0.323037978
## [19,] 0 0.0000000 0.315383629
## [20,] 0 0.0000000 0.310536467
## [21,] 0 0.0000000 0.310262665
## [22,] 0 0.0000000 0.299689810
## [23,] 0 0.0000000 0.299347715
## [24,] 0 0.0000000 0.281637584
## [25,] 0 0.0000000 0.280511260
## [26,] 0 0.0000000 0.267956666
## [27,] 0 0.0000000 0.262500366
## [28,] 0 0.0000000 0.252993717
## [29,] 0 0.0000000 0.234222336
## [30,] 0 0.0000000 0.232796821
## [31,] 0 0.0000000 0.231153192
## [32,] 0 0.0000000 0.220700648
## [33,] 0 0.0000000 0.206815991
## [34,] 0 0.0000000 0.205821198
## [35,] 0 0.0000000 0.193307118
## [36,] 0 0.0000000 0.186722979
## [37,] 0 0.0000000 0.185064513
## [38,] 0 0.0000000 0.182864067
## [39,] 0 0.0000000 0.174142457
## [40,] 0 0.0000000 0.173349233
## [41,] 0 0.0000000 0.165982732
## [42,] 0 0.0000000 0.165499845
## [43,] 0 0.0000000 0.162600742
## [44,] 0 0.0000000 0.158470316
## [45,] 0 0.0000000 0.156601404
## [46,] 0 0.0000000 0.154588033
## [47,] 0 0.0000000 0.151872929
## [48,] 0 0.0000000 0.143354811
## [49,] 0 0.0000000 0.142104999
## [50,] 0 0.0000000 0.136520367
## [51,] 0 0.0000000 0.130379191
## [52,] 0 0.0000000 0.129089764
## [53,] 0 0.0000000 0.127437449
## [54,] 0 0.0000000 0.126321858
## [55,] 0 0.0000000 0.123633581
## [56,] 0 0.0000000 0.115843530
## [57,] 0 0.0000000 0.111836441
## [58,] 0 0.0000000 0.109250181
## [59,] 0 0.0000000 0.105035804
## [60,] 0 0.0000000 0.103326846
## [61,] 0 0.0000000 0.097845406
## [62,] 0 0.0000000 0.096205648
## [63,] 0 0.0000000 0.093575941
## [64,] 0 0.0000000 0.090009959
## [65,] 0 0.0000000 0.089763367
## [66,] 0 0.0000000 0.087647752
## [67,] 0 0.0000000 0.085860423
## [68,] 0 0.0000000 0.080253095
## [69,] 0 0.0000000 0.078960989
## [70,] 0 0.0000000 0.070546775
## [71,] 0 0.0000000 0.068519479
## [72,] 0 0.0000000 0.066857388
## [73,] 0 0.0000000 0.066597194
## [74,] 0 0.0000000 0.066324402
## [75,] 0 0.0000000 0.065312852
## [76,] 0 0.0000000 0.058724720
## [77,] 0 0.0000000 0.058459910
## [78,] 0 0.0000000 0.058093939
## [79,] 0 0.0000000 0.057353007
## [80,] 0 0.0000000 0.056634495
## [81,] 0 0.0000000 0.055391002
## [82,] 0 0.0000000 0.054714317
## [83,] 0 0.0000000 0.053954024
## [84,] 0 0.0000000 0.053309793
## [85,] 0 0.0000000 0.051547687
## [86,] 0 0.0000000 0.046986251
## [87,] 0 0.0000000 0.046791021
## [88,] 0 0.0000000 0.044507936
## [89,] 0 0.0000000 0.042748403
## [90,] 0 0.0000000 0.041996189
## [91,] 0 0.0000000 0.035464877
## [92,] 0 0.0000000 0.034858018
## [93,] 0 0.0000000 0.033600787
## [94,] 0 0.0000000 0.032788605
## [95,] 0 0.0000000 0.032157741
## [96,] 0 0.0000000 0.032049281
## [97,] 0 0.0000000 0.026104314
## [98,] 0 0.0000000 0.025962140
## [99,] 0 0.0000000 0.023294849
## [100,] 0 0.0000000 0.022391369
## [101,] 0 0.0000000 0.021238881
## [102,] 0 0.0000000 0.019214665
## [103,] 0 0.0000000 0.018691308
## [104,] 0 0.0000000 0.018131942
## [105,] 0 0.0000000 0.018115428
## [106,] 0 0.0000000 0.015066935
## [107,] 0 0.0000000 0.013243244
## [108,] 0 0.0000000 0.012367233
## [109,] 0 0.0000000 0.011430027
## [110,] 0 0.0000000 0.011348145
## [111,] 0 0.0000000 0.011169319
## [112,] 0 0.0000000 0.009955018
## [113,] 0 0.0000000 0.008876962
## [114,] 0 0.0000000 0.007695819
## [115,] 0 0.0000000 0.006985402
## [116,] 0 0.0000000 0.005454312
## [117,] 0 0.0000000 0.004970368
## [118,] 0 0.0000000 0.004721412
## [119,] 0 0.0000000 0.003096041
## [120,] 0 0.0000000 0.001325557
## [121,] 1 0.7037153 3.470094911
## [122,] 1 0.5479865 1.733566397
## [123,] 1 1.2554868 1.257751432print(summary.diagram(Diag[["diagram"]]))## Call:
## ripsDiag(X = Circles, maxdimension = maxdimension, maxscale = maxscale,
## library = "GUDHI", printProgress = FALSE)
##
## Number of features:
## [1] 123
##
## Max dimension:
## [1] 1
##
## Scale:
## [1] 0 5Finally we plot the diagram:
plot(Diag[["diagram"]])
Persistent homology of sublevel and superlevel sets of functions
Download the crater dataset on your laptop and import the data into R:
crater = read.table("./Data/crater.xy")
plot(crater, cex = 0.1,main = "Crater Dataset",xlab = "x",ylab="y")