evaluation {parsec}

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, comprising 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,
    inequality = FALSE
)
inequality(profiles = NULL, zeta = getzeta(profiles), ...)

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. Function inequality does not require its definition since its results do not depend on it.

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.

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 pairs of profiles. The matrix must be square, with dimensions equal to the number of profiles. Even if the poset is complete, the distance between two profiles is computed only if one profile covers the other.

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 positions.

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. Users can set a lower value to maxint in case of low RAM availability.

inequality

boolean parameter (by default FALSE) to make the evaluation function return also a measure of inequality (which can make computations quite lengthy). It is TRUE in function inequality and can not be modified.

...

further optional graphical parameters. See plot.default.

Value

profiles

an object of S3 class wprof reporting poset profiles and their associated frequencies (number of statistical units in each profile).

number_of_profiles

number of profiles.

number_of_variables

number of variables.

incidence

S3 class incidence, incidence matrix of the poset.

cover

S3 class cover, cover matrix of the poset.

threshold

boolean vector specifying whether a profile belongs to the threshold.

number_of_iterations

number of iterations performed by the Bubley-Dyer algorithm.

rank_dist

matrix reporting by rows the relative frequency distributions of the ranks of each profile, over the set of sampled linear extensions.

thr_dist

vector reporting the relative frequency a profile is used as threshold in the sampled linear extensions.

prof_w

vector of weights assigned to each profile.

edg_w

matrix of distances between profiles, used to evaluate the gap measures.

idn_f

vector reporting the identification function, computed as the fraction of sampled linear extensions where a profile is in the downset of the threshold.

svr_abs

vector 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_rel

equal to svr_abs divided by its maximum, that is svr_abs of the minimal element in the linear extension.

wea_abs

vector 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_rel

the 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_gap

Population mean of svr_rel

wealth_gap

Population mean of wea_rel

inequality

when the argument inequality is TRUE, the average value of the inequality index over the linear extensions (see Fattore and Arcagni, 2013). Function inequalty returns only this result.

References

Bubley R., Dyer M. (1999), Faster random generation of linear extensions, Discrete Math., 201, 81-88.

Fattore M., Arcagni A. (2013), 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

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

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

summary(res)
plot(res)