DDAWASS

Discrete Distribution Data Analysis with Wasserstein distance

DDAWASS is an R package for the analysis of data tables whose cells are discrete distributions with a finite support: for example the distributions of the scores given on a Likert scale by groups of respondents, of grades, of ratings or of counts. Each row is a statistical unit (a group, a territory, a company, …), each column a distributional variable, and distributions are compared through the L2 Wasserstein distance between their quantile functions.

It is the discrete counterpart of HistDAWass (histogram data) and provides:

• S4 classes distributionD (a discrete distribution) and MatD (a matrix of discrete distributions), built from raw microdata, frequency tables, histograms or legacy nested tibbles;
• exact Wasserstein distances (computed in C++), with the decomposition into position, size and shape components;
• barycenters (Fréchet means), Wasserstein variance, covariance and correlation, and a barycenter projected on the original categories of the scale for presentation;
• hierarchical clustering and k-means;
• quantile-based principal component analysis and multiple factor analysis;
• two-component linear regression;
• a distributional Cronbach’s alpha.

Installation

The development version can be installed from GitHub:

# install.packages("remotes")
remotes::install_github("Airpino/DDAWASS", build_vignettes = TRUE)

The package contains C++ code: on Windows, Rtools is needed to install it from source.

A quick example

The univer data of the CUB package contain the 1-7 evaluations of the orientation services of the University of Naples Federico II given by 2179 students. We describe each faculty-by-gender group by the distributions of its scores.

library(DDAWASS)
data(univer, package = "CUB")
univer$Faculty <- factor(sprintf("F%02d", univer$faculty))
univer$Gender  <- factor(univer$gender, levels = 0:1, labels = c("M", "W"))
items <- c("informat", "willingn", "officeho", "compete", "global")

M <- data2MatD(univer, unit = c("Faculty", "Gender"), vars = items,
               support = 1:7, min.n = 10)
M[1:4, 1:3]
#> a matrix of discrete distributions
#> 4 rows x 3 columns
#>       informat             willingn             officeho
#> F01_M [m=5.35 s=1.73 k=7]  [m=6.19 s=1.23 k=7]  [m=5.77 s=1.41 k=7]
#> F01_W [m=5.16 s=1.72 k=7]  [m=6.42 s=0.674 k=7] [m=5.42 s=1.57 k=7]
#> F02_M [m=6.62 s=0.785 k=7] [m=6.65 s=0.748 k=7] [m=6.3 s=1.19 k=7]
#> F02_W [m=6.47 s=1.07 k=7]  [m=6.49 s=0.965 k=7] [m=6.23 s=1.34 k=7]

Distance between two groups, decomposed into the contributions of the means (POSITION), of the standard deviations (SIZE) and of the shapes:

WassSqDistD(get.cell.MatD(M, "F02_M", "global"),
            get.cell.MatD(M, "F06_M", "global"), details = TRUE)
#>      SQ.W  POSITION      SIZE     SHAPE       rQQ
#> 5.4861184 4.0232281 0.9578624 0.5050279 0.7773405

Barycenter of the global satisfaction, expressed on the 1-7 scale, and Wasserstein correlations between the items:

WD.vec.mean.simplified(M[, "global"], support = 1:7)
#> Discrete distribution (7 support values)
#>  x       p     cdf
#>  1 0.00000 0.00000
#>  2 0.00870 0.00870
#>  3 0.04008 0.04878
#>  4 0.10507 0.15385
#>  5 0.15385 0.30769
#>  6 0.45012 0.75781
#>  7 0.24219 1.00000
#> mean = 5.72317   std = 1.11335

round(WD.correlation(M), 2)
#>          informat willingn officeho compete global
#> informat     1.00     0.64     0.56    0.78   0.75
#> willingn     0.64     1.00     0.51    0.66   0.72
#> officeho     0.56     0.51     1.00    0.53   0.58
#> compete      0.78     0.66     0.53    1.00   0.83
#> global       0.75     0.72     0.58    0.83   1.00

Clustering and regression:

set.seed(123)
km <- WD_kmeans(M, k = 3, rep = 20)
km$solution$cardinality
#>  1  2  3
#>  4 12 10

pars <- WD.regression.two.components(M, Yvar = "global",
                                     Xvars = c("informat", "willingn", "officeho", "compete"))
round(pars, 3)
#> (AV_Intercept)    AV_informat    AV_willingn    AV_officeho     AV_compete
#>         -0.217          0.049          0.232          0.156          0.610
#>   CEN_informat   CEN_willingn   CEN_officeho    CEN_compete
#>          0.162          0.240          0.056          0.419

Vignettes

1. Building discrete distributional data: from microdata or frequency tables to a MatD.
2. Basic statistics: statistics of the cells, barycenters, variance decomposition, covariance, correlation and reliability.
3. Clustering: Wasserstein distances, hierarchical clustering and k-means.
4. Principal component analysis: quantile PCA, Spanish-fan plot and multiple factor analysis.
5. Linear regression: the two-component model, predictions, goodness of fit and bootstrap.

browseVignettes("DDAWASS")

Main functions

References

• Crocetta C., Irpino A., Antonucci L., Marin C. (2025). A performance indicator and its decomposition according to the impacts of different aspects based on distributional data. Quality & Quantity, 59(3), 2091–2110. https://doi.org/10.1007/s11135-024-01956-0
• Irpino A., Verde R. (2015). Basic statistics for distributional symbolic variables: a new metric-based approach. Advances in Data Analysis and Classification, 9(2), 143–175. https://doi.org/10.1007/s11634-014-0176-4
• Irpino A., Verde R. (2015). Linear regression for numeric symbolic variables: a least squares approach based on Wasserstein distance. Advances in Data Analysis and Classification, 9(1), 81–106. https://doi.org/10.1007/s11634-015-0197-7

License

GPL (>= 3)

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