poseticDataAnalysis (version 1.0.0)

BubleyDyerMRP: Approximating MRP matrix computation, using the Bubley-Dyer procedure.

Description

Computes an approximated MRP matrix, starting from a set of linear extensions sampled according to the Bubley-Dyer procedure.

Usage

BubleyDyerMRP(generator, n = NULL, error = NULL, output_every_sec = NULL)

Value

A list of two elements: 1) the matrix of approximated MRP and 2) the number of linear extensions generated to compute it.

Arguments

generator

The approximated MRP matrix generator created by function BubleyDyerMRPGenerator().

n

Number of linear extensions generated to compute the approximated MRP matrix. See Details for further information on this argument.

error

A real number in \((0,1)\), representing the "distance" from uniformity of the sampling distribution of the linear extensions. This parameter is used to determine the number of linear extensions to be sampled, in order to achieve the desired "distance". According to Bubley and Dyer (1999), if error\(=\epsilon\) and \(E\) is the number of elements in the poset, then the number \(n_\epsilon\) of sampled linear extensions is given by

\(n_\epsilon=E^4(\ln(E))^2+E^3\ln(E)\ln(\epsilon^{-1})\).

If both arguments n and error are specified by the user, the number of linear extensions actually generated is n.

output_every_sec

Integer specifying a time interval (in seconds). By specifying this argument, during the execution of BubleyDyerMRP, a message reporting the number of linear extensions progressively generated is printed on the R-Console, every output_every_sec seconds.

Details

The function BubleyDyerMRP allows the user to update a previously computed approximated MRP matrix, so as to improve estimation accuracy. Specifically, the argument generator internally stores the approximated MRP matrix computed at each execution of BubleyDyerMRP. At the subsequent call (with the same generator argument), the previously computed MRP are updated, based on the newly sampled linear extensions. In this case, the number of additional linear extensions is determined either directly, by parameter n, or indirectly, by specifying parameter error, which sets the desired "distance" from uniformity of the sampling distribution of linear extensions, in the Bubley-Dyer procedure. In the latter case, the number of additional linear extensions is computed as \(n_\epsilon-n_a\), where \(n_\epsilon\) is the number of linear extensions necessary to achive the desired "distance" and \(n_a\) is the total number of linear extensions generated in the previous calls of BubleyDyerEvaluation. If \(n_\epsilon-n_a\leq 0\), no further linear extensions are generated and a warning message is displayed.

In case a newly computed approximated MRP matrix is desired, run BubleyDyerMRP with a generator argument newly generated by function BubleyDyerMRPGenerator().

References

Bubley, R., Dyer, M. (1999). Faster random generation of linear extensions. Discrete Mathematics, 201, 81-88. https://doi.org/10.1016/S0012-365X(98)00333-1

Examples

# \donttest{
el1 <- c("a", "b", "c")
el2 <- c("x", "y", "z")
el3 <- c("h", "k")
dom <- matrix(c(
  "a", "b",
  "c", "b"
), ncol = 2, byrow = TRUE)

pos1 <- POSet(elements = el1, dom = dom)

pos2 <- LinearPOSet(elements = el2)

pos3 <- LinearPOSet(elements = el3)

pos <- ProductPOSet(pos1, pos2, pos3)

BDgen <- BubleyDyerMRPGenerator(pos)
#MRP computation with parameter n
res <- BubleyDyerMRP(BDgen, n=700000, output_every_sec=1)
#MRP refinement with parameter n
res <- BubleyDyerMRP(BDgen, n=100000, output_every_sec=1)
#MRP refinement with parameter error
res <- BubleyDyerMRP(BDgen, error=0.05, output_every_sec=1)
#Attempt to further refine MRP with parameter `error=0.05` does not modify previous result
res <- BubleyDyerMRP(BDgen, error=0.05, output_every_sec=1)

#new MRP computation with parameter error
BDgen <- BubleyDyerMRPGenerator(pos)
res <- BubleyDyerMRP(BDgen, error=0.05, output_every_sec=1)
# }