# Estimating function averages on linear extensions, by the Bubley-Dyer procedure. `BubleyDyerEvaluation` computes the averages of the input functions (defined on linear orders) over a subset of linear extensions of the input poset, randomly generated by the Bubley-Dyer procedure. ```r BubleyDyerEvaluation( generator, n = NULL, error = NULL, output_every_sec = NULL ) ``` ## Arguments - `generator`: S4 object of class `BubleyDyerEvaluationGenerator` created by function `BuildBubleyDyerEvaluationGenerator()`, implicitly containing the poset and the list of functions, whose averages are to be estimated. - `n`: Number of linear extensions to be generated. See the `Details`, for further information on this parameter. - `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 `BubleyDyerEvaluation`, the number of linear extensions progressively generated is printed on the R-Console, every `output_every_sec`seconds. ## Returns List of the estimated averages, along with the number of linear extensions used to compute them. ## Details The function `BubleyDyerEvaluation` allows the user to update previously computed averages, so as to improve estimation accuracy. The `generator` internally stores the averages computed at each call of `BubleyDyerEvaluation`. At the subsequent call (with the same `generator` argument), the previously computed averages 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 new function averages are desired, run `BubleyDyerEvaluation` with a `generator` argument newly generated by function `BuildBubleyDyerEvaluationGenerator`. ## Examples ```r el1 <- c("a", "b", "c", "d") el2 <- c("x", "y") el3 <- c("h", "k") dom <- matrix(c( "a", "b", "c", "b", "b", "d" ), ncol = 2, byrow = TRUE) pos1 <- POSet(elements = el1, dom = dom) pos2 <- LinearPOSet(elements = el2) pos3 <- LinearPOSet(elements = el3) pos <- ProductPOSet(pos1, pos2, pos3) # median_distr computes the frequency distribution of median profile elements <- POSetElements(pos) median_distr <- function(le) { n <- length(elements) if (n %% 2 != 0) { res <- (elements == le[(n + 1) / 2]) } else { res <- (elements == le[n / 2]) } res <- as.matrix(res) rownames(res) <- elements colnames(res) <- "median_distr" return (as.matrix(res)) } # computation with parameter n BDgen <- BuildBubleyDyerEvaluationGenerator(poset = pos, seed = NULL, median_distr) res <- BubleyDyerEvaluation(BDgen, n=40000, output_every_sec=1) #refinement results with parameter n res <- BubleyDyerEvaluation(BDgen, n=10000, output_every_sec=1) #refinement results with parameter error res <- BubleyDyerEvaluation(BDgen, error=0.2, output_every_sec=1) #Attempt to further refine results with parameter `error=0.2` does not modify previous result res <- BubleyDyerEvaluation(BDgen, error=0.2, output_every_sec=1) # computation with parameter error BDgen <- BuildBubleyDyerEvaluationGenerator(poset = pos, seed = NULL, median_distr) res <- BubleyDyerEvaluation(BDgen, error=0.2, output_every_sec=1) ``` ## 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

BubleyDyerEvaluation function