4ti2 provides several executables that can be used to generate bases for a configuration matrix A. See the references for details.
basis(exec, memoise = TRUE) zbasis( A, format = c("mat", "vec", "tab"), dim = NULL, all = FALSE, dir = tempdir(), quiet = TRUE, shell = FALSE, dbName = NULL, ... ) markov( A, format = c("mat", "vec", "tab"), dim = NULL, all = FALSE, dir = tempdir(), quiet = TRUE, shell = FALSE, dbName = NULL, ... ) groebner( A, format = c("mat", "vec", "tab"), dim = NULL, all = FALSE, dir = tempdir(), quiet = TRUE, shell = FALSE, dbName = NULL, ... ) hilbert( A, format = c("mat", "vec", "tab"), dim = NULL, all = FALSE, dir = tempdir(), quiet = TRUE, shell = FALSE, dbName = NULL, ... ) graver( A, format = c("mat", "vec", "tab"), dim = NULL, all = FALSE, dir = tempdir(), quiet = TRUE, shell = FALSE, dbName = NULL, ... ) fzbasis( A, format = c("mat", "vec", "tab"), dim = NULL, all = FALSE, dir = tempdir(), quiet = TRUE, shell = FALSE, dbName = NULL, ... ) fmarkov( A, format = c("mat", "vec", "tab"), dim = NULL, all = FALSE, dir = tempdir(), quiet = TRUE, shell = FALSE, dbName = NULL, ... ) fgroebner( A, format = c("mat", "vec", "tab"), dim = NULL, all = FALSE, dir = tempdir(), quiet = TRUE, shell = FALSE, dbName = NULL, ... ) fhilbert( A, format = c("mat", "vec", "tab"), dim = NULL, all = FALSE, dir = tempdir(), quiet = TRUE, shell = FALSE, dbName = NULL, ... ) fgraver( A, format = c("mat", "vec", "tab"), dim = NULL, all = FALSE, dir = tempdir(), quiet = TRUE, shell = FALSE, dbName = NULL, ... )
| exec | don't use this parameter |
|---|---|
| memoise | don't use this parameter |
| A | The configuration matrix |
| format | How the basis (moves) should be returned. if "mat", the moves are returned as the columns of a matrix. |
| dim | The dimension to be passed to |
| all | If TRUE, all moves (+ and -) are given. if FALSE, only the + moves are given as returned by the executable. |
| dir | Directory to place the files in, without an ending / |
| quiet | If FALSE, messages the 4ti2 output |
| shell | Messages the shell code used to do the computation |
| dbName | The name of the model in the markov bases database, http://markov-bases.de, see examples |
| ... | Additional arguments to pass to the function, e.g. |
a matrix containing the Markov basis as its columns (for easy addition to tables)
Drton, M., B. Sturmfels, and S. Sullivant (2009). Lectures on Algebraic Statistics, Basel: Birkhauser Verlag AG.
if (has_4ti2()) { # basic input and output for the 3x3 independence example (A <- rbind( kprod(diag(3), ones(1,3)), kprod(ones(1,3), diag(3)) )) markov(A, p = "arb") # you can get the output formatted in different ways: markov(A, p = "arb", all = TRUE) markov(A, p = "arb", "vec") markov(A, p = "arb", "tab", c(3, 3)) tableau(markov(A, p = "arb"), dim = c(3, 3)) # tableau notation # you can add options by listing them off # to see the options available to you by function, # go to http://www.4ti2.de markov(A, p = "arb") # the basis functions are automatically cached for future use. # (note that it doesn't persist across sessions.) A <- rbind( kprod( diag(4), ones(1,4), ones(1,4)), kprod(ones(1,4), diag(4), ones(1,4)), kprod(ones(1,4), ones(1,4), diag(4)) ) system.time(markov(A, p = "arb")) system.time(markov(A, p = "arb")) # the un-cashed versions begin with an "f" # (think: "forgetful" markov) system.time(fmarkov(A, p = "arb")) system.time(fmarkov(A, p = "arb")) # you can see the command line code by typing shell = TRUE # and the standard output wiht quiet = FALSE # we illustrate these with fmarkov because otherwise it's cached (A <- rbind( kprod(diag(2), ones(1,4)), kprod(ones(1,4), diag(2)) )) fmarkov(A, p = "arb", shell = TRUE) fmarkov(A, p = "arb", quiet = FALSE) # compare the bases for the 3x3x3 no-three-way interaction model A <- rbind( kprod( diag(3), diag(3), ones(1,3)), kprod( diag(3), ones(1,3), diag(3)), kprod(ones(1,3), diag(3), diag(3)) ) str( zbasis(A, p = "arb")) # 8 elements = ncol(A) - qr(A)$rank str( markov(A, p = "arb")) # 81 elements str(groebner(A, p = "arb")) # 110 elements str( graver(A)) # 795 elements # the other bases are also cached A <- rbind( kprod( diag(3), ones(1,3), ones(1,2)), kprod(ones(1,3), diag(3), ones(1,2)), kprod(ones(1,3), ones(1,3), diag(2)) ) system.time( graver(A)) system.time( graver(A)) system.time(fgraver(A)) system.time(fgraver(A)) # LAS ex 1.2.1, p.12 : 2x3 independence (A <- rbind( kprod(diag(2), ones(1,3)), kprod(ones(1,2), diag(3)) )) markov(A, p = "arb", "tab", c(3, 3)) # Prop 1.2.2 says that there should be 2*choose(2, 2)*choose(3,2) # = 6 # moves (up to +-1) markov(A, p = "arb", "tab", c(3, 3), TRUE) # LAS example 1.2.12, p.17 (no 3-way interaction) (A <- rbind( kprod( diag(2), diag(2), ones(1,2)), kprod( diag(2), ones(1,2), diag(2)), kprod(ones(1,2), diag(2), diag(2)) )) plot_matrix(A) markov(A, p = "arb") groebner(A, p = "arb") graver(A) tableau(markov(A, p = "arb"), dim = c(2,2,2)) # using the markov bases database, must be connected to internet # commented out for predictable and fast cran checks time # A <- markov(dbName = "ind3-3") # B <- markov(rbind( # kprod(diag(3), ones(1,3)), # kprod(ones(1,3), diag(3)) # ), p = "arb") # all(A == B) # possible issues # markov(diag(1, 10)) # zbasis(diag(1, 10), "vec") # groebner(diag(1, 10), "vec", all = TRUE) # graver(diag(1, 10), "vec", all = TRUE) # graver(diag(1, 4), "tab", all = TRUE, dim = c(2,2)) }#>#> ------------------------------------------------- #> 4ti2 version 1.6.7 #> Copyright 1998, 2002, 2006, 2015 4ti2 team. #> 4ti2 comes with ABSOLUTELY NO WARRANTY. #> This is free software, and you are welcome #> to redistribute it under certain conditions. #> For details, see the file COPYING. #> ------------------------------------------------- #> Using arbitrary precision integers. #> Computing generating set (Hybrid) ... #> Phase 1: #> Computing generating set (Saturation) ... #> Saturating 8 variable(s). #> #> Sat 8: Col: 2 (U) Size: 5, ToDo: 5 #> Sat 8: Col: 2 (U) Size: 5, ToDo: 0 #> Sat 8: Col: 2 (U) Size: 5, Time: 0.00 / 0.00 secs. #> Saturated already on 1 variable(s). #> #> Sat 6: Col: 3 (F) Size: 5, Index: 0 #> Sat 6: Col: 3 (F) Size: 5, Time: 0.00 / 0.00 secs. #> Saturated already on 1 variable(s). #> #> Sat 4: Col: 6 (F) Size: 5, Index: 0 #> Sat 4: Col: 6 (F) Size: 5, Time: 0.00 / 0.00 secs. #> Saturated already on 3 variable(s). #> Done. Size: 5, Time: 0.00 / 0.00 secs #> Phase 2: #> Lifting 0 variable(s). #> Done. Size: 5, Time: 0.00 / 0.00 secs #> Computing Miminal Generation Set ... #> #> Size: 5, Time: 0.00 / 0.00 secs. Done. #> 4ti2 Total Time: 0.00 secs. #> int [1:27, 1:8] 1 -1 0 -1 1 0 0 0 0 -1 ... #> int [1:27, 1:81] 0 0 0 0 0 0 0 0 0 0 ... #> int [1:27, 1:110] -1 0 1 0 0 0 1 0 -1 0 ... #> int [1:27, 1:795] 1 -1 0 -1 1 0 0 0 0 -1 ...#> 1 1 1 1 1 2 #> 1 2 2 - 1 2 1 #> 2 1 2 2 1 1 #> 2 2 1 2 2 2