# Shapley value examples

#### Catarina P. Loureiro

    library(AIDA)

This vignette reproduces the examples presented in Loureiro et al. (2026). It provides examples of how to compute Shapley values for interval-valued data and visualize the results. The datasets included here are the Cars and Spotify Tracks datasets, which are discussed in Sections 5.1 and 5.2 of the paper, respectively. The datasets are available in the package and can be loaded using `data("intCars")` and `data("spotify_tracks")`.

The `intData` class is used to create objects that represent interval-valued data. The `int_Shapley` function computes the Shapley value for each observation, based on the robust squared Interval-Mahalanobis distances. The `int_Shapley_interaction` function computes the Shapley interaction values for pairs of variables. The `int_Shapley_decomp` function decomposes the Shapley values into centers, ranges, and their interactions’ contributions. The functions `plot_bar_int_Shapley`, `plot_beeswarm_int_Shapley`, `plot_tile_int_Shapley`, and `plot_radar_int_Shapley` are used to visualize the Shapley values in different ways. The functions `plot_int_Shapley_inter` and `plot_bar_int_Shapley_decomp` are used to visualize the Shapley interaction values and the decomposition of the Shapley values, respectively. The examples provided here demonstrate how to apply these methods to real datasets, and the results can be compared to those presented in the paper for validation.

For more details and examples on the `intData` class, please see the vignette: Class `intData` examples.

For extra insight into the `IMCD` estimator and outlier detection methods’ application on the same two datasets, please see the vignette: IMCD estimator examples.

## Cars Dataset

This dataset contains interval data of car specifications (Duarte Silva and Brito (2025)), including min-max values. The aggregation of the microdata was done by car model, resulting in \(n=27\) observations. It is composed of \(5\) variables:

- Engine Capacity (EngCap)
- Top Speed
- Acceleration
- Price (lnPrice after log transformation)
- Class (values: Berlina, Luxury, Sportive, and Utilitarian)

As no microdata are available, and following the standard assumption in the literature, we adopt a continuous uniform distribution for the latent variables, corresponding to the symmetric and i.d. case with \(\delta = 1/12\). This is the default setting for the `intData` class.

    data(intCars)
    cars_microdata <- intCars$microdata
    cars_int <- intCars$intData

The `IMCD` function is used to compute the reweighted IMCD estimates and the robust squared Interval-Mahalanobis distances of each observation from the estimated barycenter. The `int_outliers` function identifies potential outliers based on the robust distances obtained from the IMCD estimates.

    cars_IMCD <- IMCD(cars_int, m=floor(0.75*cars_int@NObs), cutoff = "farness", cutoff_lvl = 0.9)
    cars_outliers <- int_outliers(cars_IMCD$robust_dist, p = cars_int@NIVar,
                                        cutoff = "farness", cutoff_lvl = 0.9)
    cars_outliers$outliers_names
    #> [1] "Ferrari"         "HondaNSK"        "MercedesSL"      "MercedesClasseS"
    #> [5] "Porsche"

Plot of the robust squared Interval-Mahalanobis distances, ordered from largest to smallest. The shape represents the car model class, with the horizontal line corresponding to the \(0.9\) farness cutoff value, and the outliers being marked in blue.

    cars_is_outliers <- as.character(cars_outliers$is_outlier)
    cars_is_outliers[cars_outliers$is_outlier] <- "Outlier"
    cars_is_outliers[!cars_outliers$is_outlier] <- "Inlier"

    plot_interval_dist(
      dist = cars_IMCD$robust_dist,
      cutoff = cars_outliers$cutoff_value,
      cutoff_label = "0.9 Farness",
      obs_names = rownames(cars_int),
      color_class = cars_is_outliers,
      palette =c("gray50","dodgerblue"),
      shape_class = cars_microdata$class,
      shape_label = "Class",
      sort.obs = TRUE
    )

The `int_Shapley` function is used to compute the Shapley value decomposition of the squared robust Interval-Mahalanobis distances, using the IMCD estimates as the reference values for the mean and covariance parameters. The `int_Shapley` function can also be used without specifying the reference values, in which case it will run the `IMCD` estimator internally to obtain the necessary parameters for the Shapley value computation.

    cars_shapley <- int_Shapley(cars_int, mean_c = cars_IMCD$mean_IMCD_c,
                                mean_r = cars_IMCD$mean_IMCD_r, cov = cars_IMCD$cov_IMCD)

    cars_shapley2 <- int_Shapley(cars_int)

Barplot of the Shapley value decomposition of the squared robust Interval-Mahalanobis distances. Only the six observations with the largest squared distances are shown, ordered from greatest to smallest squared distance represented by the black line, with the dotted line representing the \(0.9\) farness cutoff.

    plot_bar_int_Shapley(cars_shapley[c(cars_outliers$outliers_names,"Bmwserie7"),],
                        cutoff_value = cars_outliers$cutoff_value,
                        cutoff_label = "0.9 Farness Cutoff",
                        palette = scales::hue_pal()(4))

Beeswarm of the Shapley value decomposition of the squared robust Interval-Mahalanobis distances. The outlying observations are marked in blue and the shape represents the car model class.

    plot_beeswarm_int_Shapley(cars_shapley, cars_is_outliers, color_label = NULL,
                          shape_class = cars_microdata$class, shape_label = "Class",
                          palette = c("gray50","dodgerblue"), ggplotly = FALSE,
                          label_obs = c(cars_outliers$outliers_names), rotate_x = FALSE)

Tile plot of the Shapley value decomposition of the squared robust Interval-Mahalanobis distances. The observations are ordered from largest to smallest squared distance.

    plot_tile_int_Shapley(cars_shapley, abbrev.var = 15, sort.obs = TRUE)

Radar plot of the Shapley value decomposition of the squared robust Interval-Mahalanobis distances. The outlying observations are marked in blue.

    outliers_colors <- rep('gray50', cars_int@NObs)
    names(outliers_colors) <- rownames(cars_int)
    outliers_colors[cars_outliers$outliers_names] = 'dodgerblue'

    plot_radar_int_Shapley(cars_shapley,outliers_colors)

Tile plot of the Shapley interaction index for observation Ferrari.

    cars_shapley_inter <- int_Shapley_interaction(cars_int, mean_c = cars_IMCD$mean_IMCD_c,
                                                  mean_r = cars_IMCD$mean_IMCD_r,
                                                  cov = cars_IMCD$cov_IMCD)

    plot_int_Shapley_inter(cars_shapley_inter[["Ferrari"]], abbrev = 15, title = "Ferrari")

Barplot of the decomposition of the Shapley values by centers and ranges. Only the six observations with the largest squared distances are shown.

    cars_shapley_decomp <- int_Shapley_decomp(cars_int, mean_c = cars_IMCD$mean_IMCD_c,
                                              mean_r = cars_IMCD$mean_IMCD_r, cov = cars_IMCD$cov_IMCD)

    plot_bar_int_Shapley_decomp(cars_shapley_decomp[c(cars_outliers$outliers_names,"Bmwserie7")],
                              rotate_x = FALSE, palette = scales::hue_pal()(4))

## Spotify Tracks Dataset

This dataset contains interval data of Spotify tracks’ audio features (Pandya (2022)), including min-max values and trimmed intervals, as well as the microdata. The aggregation of the microdata was done by track genre, resulting in \(n=111\) observations. It is composed of 11 audio features:

- duration
- danceability
- energy
- loudness
- speechiness
- acousticness
- instrumentalness
- liveness
- valence
- tempo
- popularity

Prior to aggregation, logarithmic transformations were applied to loudness and tempo, duration_ms in milliseconds was converted to duration in minutes, and popularity was scaled to the range \([0,1]\). The latent variables’ parameters are estimated from the microdata, using Kernel Density Estimation (KDE) (package `kde1d`). Here, we will use the data aggregated into trimmed intervals, taking as lower and upper bounds the \(1\%\) and \(99\%\) quantiles, respectively.

    data(spotify_tracks)
    spotify_int <- spotify_tracks$intData_trimmed

The IMCD estimates and robust squared Interval-Mahalanobis distance are computed using the `IMCD` function, while the `int_outliers` function applies the outlier detection rule identifying strong (`cutoff_lvl=0.95`) and mild outliers (`cutoff_lvl=0.9`).

    spotify_IMCD <- IMCD(spotify_int, m=round(0.75*nrow(spotify_int)),
                                  cutoff="farness", cutoff_lvl = 0.95)

    # Strong outliers
    spotify_outliers <- int_outliers(spotify_IMCD$robust_dist,cutoff="farness", cutoff_lvl = 0.95)
    spotify_outliers$outliers_names
    #> [1] "classical" "comedy"    "grindcore" "sleep"

    # Mild outliers
    spotify_outliers_2 <- int_outliers(spotify_IMCD$robust_dist,cutoff="farness", cutoff_lvl = 0.9)
    spotify_outliers_2$outliers_names[!spotify_outliers_2$outliers_names%in%spotify_outliers$outliers_names]
    #> [1] "ambient"       "chicago-house" "death-metal"   "iranian"

Plot of the robust squared Interval-Mahalanobis distances for the Spotify dataset. The horizontal lines correspond to the \(0.9\) and \(0.95\) farness cutoff values, and the mild and strong outliers are marked in green and blue, respectively.

    spotify_is_outliers <- as.character(spotify_outliers$is_outlier)
    spotify_is_outliers[!spotify_outliers_2$is_outlier] <- "Regular"
    spotify_is_outliers[spotify_outliers_2$is_outlier] <- "Mild Outlier"
    spotify_is_outliers[spotify_outliers$is_outlier] <- "Extreme Outlier"
    palette_outliers <- c(
      "Regular"        = "gray50",
      "Mild Outlier"   = 'forestgreen', #"darkorange",
      "Extreme Outlier"= 'dodgerblue' #"red"
    )

    plot_interval_dist(
      spotify_IMCD$robust_dist,
      c(spotify_outliers$cutoff_value,spotify_outliers_2$cutoff_value),
      c("0.95 Farness", "0.90 Farness"),
      sort.obs = FALSE,
      color_class = spotify_is_outliers,
      color_label = "Outlier Status",
      palette = palette_outliers,
      label_obs = spotify_outliers_2$outliers_names)

Barplot of the Shapley value decomposition of the squared robust Interval-Mahalanobis distances for the \(12\) observations with the largest squared distances, ordered from greatest to smallest squared distance represented by the black line, with the dotted lines representing the \(0.9\) and \(0.95\) farness cutoff values.

    spotify_Shapley <- int_Shapley(spotify_int, mean_c = spotify_IMCD$mean_IMCD_c,
                                  mean_r = spotify_IMCD$mean_IMCD_r, cov = spotify_IMCD$cov_IMCD)

    high_dist_12 <- names(spotify_IMCD$robust_dist[order(spotify_IMCD$robust_dist, decreasing = TRUE)[1:12]])

    plot_bar_int_Shapley(spotify_Shapley[high_dist_12,],
                        cutoff_value = c(spotify_outliers$cutoff_value, spotify_outliers_2$cutoff_value),
                        cutoff_label = c("0.95 Farness Cutoff", "0.90 Farness Cutoff"),
                        sort.obs = TRUE, abbrev.obs = 20)

Beeswarm plot of the Shapley value decomposition of the squared robust Interval-Mahalanobis distances for the Spotify dataset. The extreme outliers are marked in blue and the mild ones in green.

    plot_beeswarm_int_Shapley(spotify_Shapley, spotify_is_outliers, "Outlier Status",
                          palette = palette_outliers, ggplotly = FALSE,
                          label_obs = c("sleep","classical"))

Tile plot of the Shapley value decomposition of the squared robust Interval-Mahalanobis distances for the \(12\) observations with largest squared distances. The observations are ordered from largest to smallest squared distance.

    plot_tile_int_Shapley(spotify_Shapley[high_dist_12,], sort.obs = TRUE, abbrev.var = 15)

Tile plot of the Shapley interaction index for observation grindcore.

    spotify_shapley_inter <- int_Shapley_interaction(spotify_int, mean_c = spotify_IMCD$mean_IMCD_c,
                                                      mean_r = spotify_IMCD$mean_IMCD_r,
                                                      cov = spotify_IMCD$cov_IMCD)

    plot_int_Shapley_inter(spotify_shapley_inter[["grindcore"]], abbrev = 20, title = "grindcore")

Barplot of the decomposition of the Shapley values by centers, ranges, and cross centers-ranges. The four extreme outliers and the four mild outliers are shown, ordered from greatest to smallest squared Interval-Mahalanobis distance.

    spotify_shapley_decomp <- int_Shapley_decomp(spotify_int, mean_c = spotify_IMCD$mean_IMCD_c,
                                                  mean_r = spotify_IMCD$mean_IMCD_r,
                                                  cov = spotify_IMCD$cov_IMCD)

    plot_bar_int_Shapley_decomp(spotify_shapley_decomp[spotify_outliers_2$outliers_names])

## References

Duarte Silva, Pedro, and Paula Brito. 2025. MAINT.Data: Model and Analyse Interval Data. https://doi.org/10.32614/CRAN.package.MAINT.Data.
Loureiro, Catarina P., M. Rosário Oliveira, Paula Brito, and Lina Oliveira. 2026. “Explainable Outlier Detection for Interval-valued Data.” https://arxiv.org/abs/2606.26307.
Pandya, Maharshi. 2022. “Spotify Tracks Dataset.” https://doi.org/10.34740/KAGGLE/DSV/4372070.