evaluation: Multidimensional evaluation on posets

Description

Given a partial order (arguments profiles and/or zeta) and a selected threshold, the function returns an object of S3 class parsec. In particular, the function returns the identification function and different severity measures, computed by uniform sampling of the linear extensions of the poset, through a C implementation of the Bubley - Dyer (1999) algorithm.

Usage

evaluation(
    profiles = NULL,
    threshold,
    error = 10^(-3),
    zeta = getzeta(profiles),
    weights = {
        if (!is.null(profiles)) 
            profiles$freq
        else rep(1, nrow(zeta))
    },
    distances = {
        n <- nrow(zeta)
        matrix(1, n, n) - diag(1, n)
    },
    linext = lingen(zeta),
    nit = floor({
        n <- nrow(zeta)
        n^5 * log(n) + n^4 * log(error^(-1))
    }),
    maxint = 2^31 - 1)

Arguments

profiles

an object of S3 class wprof.

threshold

a vector identifying the threshold. It can be a vector of indexes (numeric), a vector of profile names (character) or a boolean vector of length equal to the number of profiles.

error

the "distance"" from uniformity in the sampling distribution of linear extensions.

zeta

the incidence matrix of the poset. An object of S3 class incidence. By default, extracted from profiles and the order of variable modalities.

weights

weights assigned to profiles. If the argument profiles is not NULL, weights are by default set equal to profile frequencies, otherwise they are set equal to 1.

distances

matrix of distances between profiles. The matrix must be square, the dimensions must be equal to the number of profiles and complete. Even if the matrix is complete, the distance between two profiles profiles is considered only if one profile covers th

linext

the linear extension initializing the sampling algorithm. By default, it is generated by lingen(zeta). Alternatively, it can be provided by the user through a vector of profile ranks.

nit

Number Of Iterations in the Bubley-Dyer algorithm, by default evaluated from a formula of Karzanov and Khachiyan based on the number of profiles and the argument error (see Bubley and Dyer, 1999).

maxint

Maximum integer. By default the maximum integer obtainable in a 32bit system. This argument is used to group iterations and run the compiled C code more times, so as to avoid memory indexing problems. User can set a lower value to maxint i

Value

profilesan object of S3 class wprof reporting poset profiles and their associated frequencies (number of statistical units in each profile).
number_of_profilesnumber of profiles.
number_of_variablesnumber of variables.
incidenceS3 class incidence, incidence matrix of the poset.
coverS3 class cover, cover matrix of the poset.
thresholdboolean vector specifying whether a profile belongs to the threshold.
number_of_iterationsnumber of iterations performed by the Bubley-Dyer algorithm.
rank_distmatrix reporting by rows the relative frequency distribution of the ranks of each profile, over the set of sampled linear extensions.
thr_distvector reporting the relative frequency a profile is used as threshold in the sampled linear extensions. This result is useful for a posteriori valuation of the poset threshold.
prof_wvector of weights assigned to each profile.
edg_wmatrix of distances between profiles, used to evaluate the measures of gap.
idn_fvector reporting the identification function, computed as the fraction of sampled linear extensions where a profile is in the downset of the threshold.
svr_absvector reporting, for each profile, the average graph distance from the first profile above all threshold elements, over the sampled linear extensions. In each linear extension, the distance is set equal to 0 for profiles above the threshold.
svr_relequal to svr_abs divided by its maximum, that is svr_abs of the minimal element in the linear extension.
wea_absvector reporting, for each profile, the average graph distance from the maximum threshold element, over the sampled linear extensions. In each linear extension, the distance is set equal to 0 for profiles in the downset of threshold elements.
wea_relthe previous absolute distance is divided by its maximum possible value, that is the absolute distance of the threshold from the maximal element in the linear extension.
poverty_gapPopulation mean of svr_rel
wealth_gapPopulation mean of wea_rel

References

Bubley R., Dyer M. (1999), Faster random generation of linear extensions, Discrete Math., 201, 81-88. Fattore M., Arcagni A. (2013), http://meetings.sis-statistica.org/index.php/sis2013/ALV/paper/view/2633{Measuring multidimensional polarization with ordinal data}, SIS 2013 Statistical Conference, BES-M3.1 - The BES and the challenges of constructing composite indicators dealing with equity and sustainability

Examples

Run this code

profiles <- var2prof(varlen = c(3, 2, 4))
threshold <- c("311", "112")

res <- evaluation(profiles, threshold, maxint = 10^5)

summary(res)
plot(res)

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