Discrete Distribution Data Analysis with Wasserstein Distance

S4 classes and methods for the analysis of data described by discrete distributions with finite support (e.g. frequency distributions of ratings, scores or counts observed on groups of individuals), using the L2 Wasserstein metric between quantile functions. It is the discrete counterpart of 'HistDAWass' and provides exact Wasserstein distances (in C++), barycenters, Wasserstein variance, covariance and correlation, k-means and hierarchical clustering, quantile-based principal component and multiple factor analysis, two-component linear regression and reliability (Cronbach's alpha) indices. See Crocetta, Irpino, Antonucci and Marin (2025) .

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)