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 subset size is set to
\(\lfloor 0.75\times 27\rfloor=20\),
and the reweighting cutoff to “farness” with a cutoff level of \(0.9\). The int_outliers
function identifies potential outliers based on the robust distances
obtained from the IMCD estimates, using the same cutoff criteria.
cars_IMCD <- IMCD(cars_int, m = floor(0.75*cars_int@NObs), cutoff = "farness", cutoff_lvl = 0.9)
cars_outliers <- int_outliers(cars_IMCD$robust_dist, cutoff = "farness", cutoff_lvl = 0.9)
cars_outliers$outliers_names
#> [1] "Ferrari" "HondaNSK" "MercedesSL" "MercedesClasseS"
#> [5] "Porsche"
Pairs plot, the lower triangular shows scatter plots of the four
variables, with the outlying observations highlighted in red, while the
upper triangular shows the interval correlation matrix.
cars_outliers_colors <- rep('gray50', cars_int@NObs)
names(cars_outliers_colors) <- rownames(cars_int)
cars_outliers_colors[cars_outliers$outliers_names] <- 'red'
SYMB.pairs.panels(cars_int, palette = cars_outliers_colors, type = "rectangles",
corr = cov2cor(cars_IMCD$cov_IMCD), labels = colnames(cars_int),
is_outlier = cars_outliers$is_outlier, gap = 0)
Distance-distance plot of the classical versus robust squared
Interval-Mahalanobis distances. The points’ shape represents the car
model class, the horizontal and vertical lines correspond to the 0.9
farness cutoff value and the 1.5 adjusted boxplot cutoff value,
respectively, with the outliers marked in red.
# Classical distances and outliers
cars_class_dist <- IMah_dist(cars_int, z = rep(1,cars_int@NObs))
cars_class_outliers <- int_outliers(cars_class_dist, cutoff = "adjbox", cutoff_lvl = 1.5)
cars_is_outliers <- as.character(cars_outliers$is_outlier)
cars_is_outliers[cars_outliers$is_outlier] <- "Outlier"
cars_is_outliers[!cars_outliers$is_outlier] <- "Inlier"
plot_dist_dist(cars_class_dist, cars_class_outliers$cutoff_value[[2]], class_cutoff_label = "1.5 Adjbox",
cars_IMCD$robust_dist, cars_outliers$cutoff_value, rob_cutoff_label = "0.9 Farness",
color_class = cars_is_outliers, ggplotly = FALSE, shape_class = cars_microdata$class,
shape_label = "Class", palette = c("gray50","red"),
label_obs = c(cars_outliers$outliers_names, "Bmwserie7"))
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 are computed using the IMCD function
with a subset size of \(\lfloor 0.75\times
111\rfloor=83\), and a reweighting cutoff based on “farness” with
a cutoff level of \(0.95\). The
int_outliers function applies the outlier detection rule
using a “farness” cutoff of \(0.95\) to
identify strong outliers and \(0.9\)
for mild outliers.
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"
Interval correlation matrix plot based on the IMCD estimates.
# Compute correlation matrix from the robust covariance matrix
spotify_corr <- cov2cor(spotify_IMCD$cov_IMCD)
colfunc <- colorRampPalette(c("deepskyblue4", "white", "red4"))
corrplot::corrplot(
spotify_corr,
method = "color",
type = "upper",
diag = FALSE,
col = colfunc(200),
tl.col = "black",
tl.srt = 45,
tl.offset = 1,
tl.cex = 1.2,
outline = TRUE,
addCoef.col = "black",
number.cex = 1
)
Distance-distance plot of the classical versus robust squared
Interval-Mahalanobis distances for the Spotify dataset. The horizontal
lines correspond to the \(0.9\) and
\(0.95\) farness cutoff values, with
the extreme outliers marked in red and the mild outliers in orange,
while the vertical line represents the \(1.5\) adjusted boxplot cutoff.
# Classical distances and outliers
spotify_class_dist <- IMah_dist(spotify_int, z = rep(1,spotify_int@NObs))
spotify_class_outliers <- int_outliers(spotify_class_dist, cutoff = "adjbox")
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" = "darkorange",
"Extreme Outlier" = "red"
)
plot_dist_dist(spotify_class_dist, spotify_class_outliers$cutoff_value[[2]],
"1.5 Adjbox", spotify_IMCD$robust_dist,
c(spotify_outliers_2$cutoff_value, spotify_outliers$cutoff_value),
c("0.90 Farness", "0.95 Farness"), color_class = spotify_is_outliers,
color_label = "Outlier Status", palette = palette_outliers, ggplotly = FALSE,
label_obs = spotify_outliers_2$outliers_names)
