lattice-bases: Compute a basis with 4ti2

Description

4ti2 provides several executables that can be used to generate bases for a configuration matrix A. See the references for details.

Usage

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, ...)

Arguments

exec

don't use this parameter

memoise

don't use this parameter

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 vec2tab() if format = "tab" is used; a vector of the number of levels of each variable in order

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. p = "arb" specifies the flag -parb; not setting this issues a common warning

Value

a matrix containing the Markov basis as its columns (for easy addition to tables)

References

Drton, M., B. Sturmfels, and S. Sullivant (2009). Lectures on Algebraic Statistics, Basel: Birkhauser Verlag AG.

Examples

Run this code

# NOT RUN {

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))



}


# }

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