Package {AIDA}

Type:	Package
Title:	Analysis of Interval DAta
Version:	0.2.0
Description:	Tools for the analysis of interval-valued data, including construction, visualization, and statistical modeling. The package provides the 'intData' class for representing interval-valued data, along with functions to aggregate microdata and to estimate parameters of latent distributions. Barycenter and covariance matrix estimation is implemented based on the Mallows distance (Oliveira et al. (2025) <doi:10.48550/arXiv.2407.05105>). Robust estimation of the symbolic covariance matrix is implemented via the Interval Minimum Covariance Determinant (IMCD) estimator, enabling outlier detection based on the robust squared Interval-Mahalanobis distance, as proposed by Loureiro et al. (2026b) <doi:10.48550/arXiv.2604.26769>. Explainable outlier detection is supported through Shapley value based decomposition of the squared robust Interval-Mahalanobis distance, allowing assessment of variable contributions to outlyingness (Loureiro et al. (2026a) <doi:10.48550/arXiv.2606.26307>). Shapley interaction indices are also implemented, along with visualization tools to support interpretation of the results.
License:	MIT + file LICENSE
Encoding:	UTF-8
URL:	https://github.com/catarinaploureiro/AIDA, https://catarinaploureiro.github.io/AIDA/
BugReports:	https://github.com/catarinaploureiro/AIDA/issues
LazyData:	true
LazyDataCompression:	xz
VignetteBuilder:	knitr
Language:	en-US
Imports:	cellWise, cowplot, fmsb, ggbeeswarm, ggplot2, kde1d, MASS, methods
Depends:	R (≥ 3.6)
Suggests:	CerioliOutlierDetection, corrplot, ggrepel, knitr, plotly, RColorBrewer, rmarkdown, robustbase, scales, testthat (≥ 3.0.0)
Config/roxygen2/version:	8.0.0
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-06-30 10:42:32 UTC; catar
Author:	Catarina P. Loureiro [aut, cre]
Maintainer:	Catarina P. Loureiro <catarinapadrela@tecnico.ulisboa.pt>
Repository:	CRAN
Date/Publication:	2026-06-30 11:42:13 UTC

Equality Comparison for intData Objects

Description

Compare two intData objects for equality.

Compare two intData objects for inequality.

Usage

## S4 method for signature 'intData,intData'
e1 == e2

## S4 method for signature 'intData,intData'
e1 != e2

Arguments

Value

A logical matrix indicating which elements are equal between the two intData objects.

A logical matrix indicating element-wise inequality of the two intData objects.

Computes [\boldsymbol{\mathfrak{E}}_{UU}]_{ij}=\mathcal{E}(U_i,U_j) for the latent variables inherent to the macrodata, where they follow a Beta distribution.

Description

Computes [\boldsymbol{\mathfrak{E}}_{UU}]_{ij}=\mathcal{E}(U_i,U_j) for the latent variables inherent to the macrodata, where they follow a Beta distribution.

Usage

CalE.beta.beta(a1, b1, a2, b2)

Arguments

Value

Value

Computes [\boldsymbol{\mathfrak{E}}_{UU}]_{ij}=\mathcal{E}(U_i,U_j) for the latent variables inherent to the macrodata, where U_1 follows a Beta(a_1,b_1) and the PDF of U_2 is estimated by a KDE.

Description

Computes [\boldsymbol{\mathfrak{E}}_{UU}]_{ij}=\mathcal{E}(U_i,U_j) for the latent variables inherent to the macrodata, where U_1 follows a Beta(a_1,b_1) and the PDF of U_2 is estimated by a KDE.

Usage

CalE.beta.kde(micro, a1, b1)

Arguments

Value

Value

Computes [\boldsymbol{\mathfrak{E}}_{UU}]_{ij}=\mathcal{E}(U_i,U_j) for the latent variables inherent to the macrodata, where the PDF is estimated by a KDE.

Description

Computes [\boldsymbol{\mathfrak{E}}_{UU}]_{ij}=\mathcal{E}(U_i,U_j) for the latent variables inherent to the macrodata, where the PDF is estimated by a KDE.

Usage

CalE.kde.kde(micro1, micro2)

Arguments

Value

Value

Computes [\boldsymbol{\mathfrak{E}}_{UU}]_{ij}=\mathcal{E}(U_i,U_j) for the latent variables inherent to the macrodata, where they follow a Triangular distribution.

Description

Computes [\boldsymbol{\mathfrak{E}}_{UU}]_{ij}=\mathcal{E}(U_i,U_j) for the latent variables inherent to the macrodata, where they follow a Triangular distribution.

Usage

CalE.triang.triang(mo1 = 0, mo2 = 0)

Arguments

Value

Value

Centers Method for intData

Description

Centers Method for intData

Usage

Centers(Sdt)

## S4 method for signature 'intData'
Centers(Sdt)

Arguments

Value

A data.frame containing the centers of the intervals.

Interval-valued data Minimum Covariance Determinant (IMCD) estimation

Description

Applies an adaptation of the FAST-MCD algorithm to estimate location and scatter for interval-valued data.

Usage

IMCD(
  data,
  m = 0,
  cutoff = c("farness", "adjbox", "chi-squared", "F-dist", "raw"),
  cutoff_lvl = NULL
)

Arguments

Value

A list containing the robustly estimated parameters:

References

Loureiro, C. P., Oliveira, M. R., Brito, P., & Oliveira, L. (2026). Minimum Covariance Determinant Estimator and Outlier Detection for Interval-valued Data. arXiv preprint arXiv:2604.26769. https://arxiv.org/abs/2604.26769

Adapted from https://github.com/frankp-0/fastMCD.

The case cutoff=="F-dist" is adapted from package CerioliOutlierDetection (https://cran.r-project.org/package=CerioliOutlierDetection).

Examples

# Example using creditcard dataset
data(creditcard)
credit_card_int <- creditcard$intData

# Obtain reweighted IMCD estimates using farness cutoff
credit_card_IMCD <- IMCD(credit_card_int, 
                         m = floor(nrow(credit_card_int)*0.75), 
                         cutoff = "farness", 
                         cutoff_lvl = 0.9)

Interval-Mahalanobis Distance

Description

Calculate the squared Interval-Mahalanobis distance of all rows in the data and the barycenter.

Usage

IMah_dist(data, z = NULL, mean_c = NULL, mean_r = NULL, cov = NULL)

Arguments

Details

The squared Interval-Mahalanobis distance between \boldsymbol{x}=(\boldsymbol{c}^\top,\boldsymbol{r}^\top)^\top and the barycenter \boldsymbol{\mu}_B=(\boldsymbol{\mu}_C^\top,\boldsymbol{\mu}_R^\top)^\top of a population with symbolic covariance matrix \boldsymbol{\Sigma}_B (see int_cov) is defined according to the LatentCase:

• "U_id_symmetric": The latent variables are identically distributed and symmetric:

  d_\mathrm{IMah}(\boldsymbol{x})^2=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)+\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R),

  where \delta=\mathbb{E}(U^2)/4 is the parameter of the latent variables.

• "U_id": The latent variables are identically distributed:

  \begin{aligned} d_\mathrm{IMah}(\boldsymbol{x})^2&=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)+\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)\\ &\quad+\mathbb{E}(U)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R), \end{aligned}

  where \delta=\mathbb{E}(U^2)/4 and \mathbb{E}(U) are the parameter of the latent variables.

• "General": The latent variables do not have any nice properties:

  \begin{aligned} d_\mathrm{IMah}(\boldsymbol{x})^2&=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)+\dfrac{1}{4}(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\left(\boldsymbol{\mathfrak{E}}_{UU}\bullet\boldsymbol{\Sigma}_{B}^{-1}\right)(\boldsymbol{r}-\boldsymbol{\mu}_R)\\ &\quad+(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}\boldsymbol{\Sigma}_{B}^{-1}\boldsymbol{\Psi}(\boldsymbol{r}-\boldsymbol{\mu}_R), \end{aligned}

  where:

  • \boldsymbol{\Psi}=\text{diag}(\mathbb{E}(U_1),\dots,\mathbb{E}(U_p)),

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{j\ell}=\mathcal{E}(U_j,U_\ell), j\neq \ell, with \mathcal{E}(U_j,U_\ell)=\int_0^1 F_{U_j}^{-1}(t) F_{U_\ell}^{-1}(t) \, dt,

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{jj}=\mathbb{E}(U_j^2), j,\ell=1,\dots,p,

  • \bullet denotes the Schur (or entrywise) product of matrices.

Value

A vector with the squared Interval-Mahalanobis distance of each observation.

References

Loureiro, C. P., Oliveira, M. R., Brito, P., & Oliveira, L. (2026). Minimum Covariance Determinant Estimator and Outlier Detection for Interval-valued Data. arXiv preprint arXiv:2604.26769. https://arxiv.org/abs/2604.26769

Examples

data(creditcard)
credit_card_int <- creditcard$intData

# Compute squared Interval-Mahalanobis distance using IMCD estimates of mean and covariance
credit_card_dist <- IMah_dist(credit_card_int)

Interval-Mahalanobis distance for all pairs

Description

Calculate the squared Interval-Mahalanobis distance of all pairs of observations in the data.

Usage

IMah_dist_pairs(data, cov = NULL)

Arguments

Details

The squared Interval-Mahalanobis distance between \boldsymbol{x}_1=(\boldsymbol{c}_1^\top,\boldsymbol{r}_1^\top)^\top and \boldsymbol{x}_2=(\boldsymbol{c}_2^\top,\boldsymbol{r}_2^\top)^\top of a population with symbolic covariance matrix \boldsymbol{\Sigma}_B (see int_cov) is defined according to the LatentCase:

• "U_id_symmetric": The latent variables are identically distributed and symmetric:

  d_\mathrm{IMah}(\boldsymbol{x}_1,\boldsymbol{x}_2)^2=(\boldsymbol{c}_1-\boldsymbol{c}_2)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}_1-\boldsymbol{c}_2)+\delta(\boldsymbol{r}_1-\boldsymbol{r}_2)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}_1-\boldsymbol{r}_2),

  where \delta=\mathbb{E}(U^2)/4 is the parameter of the latent variables.

• "U_id": The latent variables are identically distributed:

  \begin{aligned} d_\mathrm{IMah}(\boldsymbol{x}_1,\boldsymbol{x}_2)^2&=(\boldsymbol{c}_1-\boldsymbol{c}_2)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}_1-\boldsymbol{c}_2)+\delta(\boldsymbol{r}_1-\boldsymbol{r}_2)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}_1-\boldsymbol{r}_2)\\ &\quad+\mathbb{E}(U)(\boldsymbol{c}_1-\boldsymbol{c}_2)^\top\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}_1-\boldsymbol{r}_2), \end{aligned}

  where \delta=\mathbb{E}(U^2)/4 and \mathbb{E}(U) are the parameter of the latent variables.

• "General": The latent variables do not have any nice properties:

  \begin{aligned} d_\mathrm{IMah}(\boldsymbol{x}_1,\boldsymbol{x}_2)^2&=(\boldsymbol{c}_1-\boldsymbol{c}_2)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}_1-\boldsymbol{c}_2)+\dfrac{1}{4}(\boldsymbol{r}_1-\boldsymbol{r}_2)^{\top}\left(\boldsymbol{\mathfrak{E}}_{UU}\bullet\boldsymbol{\Sigma}_{B}^{-1}\right)(\boldsymbol{r}_1-\boldsymbol{r}_2)\\ &\quad+(\boldsymbol{c}_1-\boldsymbol{c}_2)^{\top}\boldsymbol{\Sigma}_{B}^{-1}\boldsymbol{\Psi}(\boldsymbol{r}_1-\boldsymbol{r}_2), \end{aligned}

  where:

  • \boldsymbol{\Psi}=\text{diag}(\mathbb{E}(U_1),\dots,\mathbb{E}(U_p)),

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{j\ell}=\mathcal{E}(U_j,U_\ell), j\neq \ell, with \mathcal{E}(U_j,U_\ell)=\int_0^1 F_{U_j}^{-1}(t) F_{U_\ell}^{-1}(t) \, dt,

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{jj}=\mathbb{E}(U_j^2), j,\ell=1,\dots,p,

  • \bullet denotes the Schur (or entrywise) product of matrices.

If cov is not provided, it will be computed using the IMCD function. Additionally, if cov is set as the identity matrix, the computed distance is the Mallows distance between pairs of observations.

Value

A matrix with the squared Interval-Mahalanobis distance of each pair of observations.

References

Loureiro, C. P., Oliveira, M. R., Brito, P., & Oliveira, L. (2026). Minimum Covariance Determinant Estimator and Outlier Detection for Interval-valued Data. arXiv preprint arXiv:2604.26769. https://arxiv.org/abs/2604.26769

Examples

data(creditcard)
credit_card_int <- creditcard$intData

credit_card_dist <- IMah_dist_pairs(credit_card_int)

Latent Case Method for intData

Description

Latent Case Method for intData

Usage

LatentCase(Sdt)

## S4 method for signature 'intData'
LatentCase(Sdt)

Arguments

Value

A character with the latent case.

Latent Distribution Method for intData

Description

Latent Distribution Method for intData

Usage

LatentDist(Sdt)

## S4 method for signature 'intData'
LatentDist(Sdt)

Arguments

Value

A character with the latent distribution(s).

Latent Parameters Method for intData

Description

Latent Parameters Method for intData

Usage

LatentParam(Sdt)

## S4 method for signature 'intData'
LatentParam(Sdt)

Arguments

Value

A list with the latent parameters.

LogRanges Method for intData

Description

LogRanges Method for intData

Usage

LogRanges(Sdt)

## S4 method for signature 'intData'
LogRanges(Sdt)

Arguments

Value

A data.frame containing the logarithms of the ranges.

Lower Bounds Method for intData

Description

Lower Bounds Method for intData

Usage

LowerBounds(Sdt)

## S4 method for signature 'intData'
LowerBounds(Sdt)

Arguments

Value

A data.frame containing the lower bounds of the intervals.

Mallows Distance

Description

Calculate the squared Mallows distance between all rows in data and the barycenter.

Usage

Mallows_dist(data, mean_c = NULL, mean_r = NULL)

Arguments

Details

The squared Mallows distance is defined according to the LatentCase:

• "U_id_symmetric": The latent variables are identically distributed and symmetric:

  d_\mathrm{M}(\boldsymbol{x})^2=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}(\boldsymbol{c}-\boldsymbol{\mu}_C)+\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}(\boldsymbol{r}-\boldsymbol{\mu}_R),

  where \delta=\mathbb{E}(U^2)/4 is the parameter of the latent variables.

• "U_id": The latent variables are identically distributed:

  d_\mathrm{M}(\boldsymbol{x})^2=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}(\boldsymbol{c}-\boldsymbol{\mu}_C)+\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}(\boldsymbol{r}-\boldsymbol{\mu}_R) +\mathbb{E}(U)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top(\boldsymbol{r}-\boldsymbol{\mu}_R),

  where \delta=\mathbb{E}(U^2)/4 and \mathbb{E}(U) are the parameter of the latent variables.

• "General": The latent variables do not have any nice properties:

  d_\mathrm{M}(\boldsymbol{x})^2=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}(\boldsymbol{c}-\boldsymbol{\mu}_C)+(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\boldsymbol{\Delta}(\boldsymbol{r}-\boldsymbol{\mu}_R) +(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}\boldsymbol{\Psi}(\boldsymbol{r}-\boldsymbol{\mu}_R),

  where:

  • \boldsymbol{\Psi}=\text{diag}(\mathbb{E}(U_1),\dots,\mathbb{E}(U_p)),

  • \boldsymbol{\Delta}=\text{diag}(\mathbb{E}(U^2_1),\dots,\mathbb{E}(U^2_p))/4.

Value

A vector with the squared Mallows distance of each observation.

References

Oliveira, M. R., Pinheiro, D., & Oliveira, L. (2025). Location and association measures for interval-valued data based on Mallows' distance. arXiv preprint arXiv:2407.05105. https://arxiv.org/abs/2407.05105

Examples

data(creditcard)
credit_card_int <- creditcard$intData

credit_card_dist <- Mallows_dist(credit_card_int)

Number of Micro Units Method for intData

Description

Number of Micro Units Method for intData

Usage

NbMicroUnits(x)

## S4 method for signature 'intData'
NbMicroUnits(x)

Arguments

Value

An integer specifying the number of micro units.

Ranges Method for intData

Description

Ranges Method for intData

Usage

Ranges(Sdt)

## S4 method for signature 'intData'
Ranges(Sdt)

Arguments

Value

A data.frame containing the ranges of the intervals.

Upper Bounds Method for intData

Description

Upper Bounds Method for intData

Usage

UpperBounds(Sdt)

## S4 method for signature 'intData'
UpperBounds(Sdt)

Arguments

Value

A data.frame containing the upper bounds of the intervals.

Subset an intData Object

Description

Extract a subset of rows and columns from an intData object.

Usage

## S4 method for signature 'intData'
x[i, j, ..., drop = TRUE]

Arguments

Value

An intData object containing the specified subset of rows and columns.

Obtain unweighted estimates for data with > 600 observations

Description

Obtain unweighted estimates for data with > 600 observations

Usage

bigIMCD(m, p, n, data)

Arguments

Value

A list of estimated location and scatter

Perform single iteration of C-step

Description

Perform single iteration of C-step

Usage

c_step(z, m, data)

Arguments

Value

A list of z, covariance, barycenter and robust distances

Compute Cal.E Latent Variables

Description

Computes \boldsymbol{\mathfrak{E}}_{UU} for the latent variables inherent to the macrodata.

Usage

cal.E.UU(
  LatentDist = c("Unif", "Triang", "TNorm", "InvTri", "Beta", "KDE", "Degenerated"),
  TriangParam = 0,
  BetaParam.a = 1,
  BetaParam.b = 1,
  Umicro = NULL,
  p = NULL
)

Arguments

Details

The matrix \boldsymbol{\mathfrak{E}}_{UU} is defined as follows:

• [\boldsymbol{\mathfrak{E}}_{UU}]_{j\ell}=\mathcal{E}(U_j,U_\ell), j\neq \ell, with \mathcal{E}(U_j,U_\ell)=\int_0^1 F_{U_j}^{-1}(t) F_{U_\ell}^{-1}(t) \, dt

• [\boldsymbol{\mathfrak{E}}_{UU}]_{jj}=\mathbb{E}(U_j^2), j,\ell=1,\dots,p.

Value

A p\times p matrix.

Column Names Method for intData

Description

Column Names Method for intData

Usage

## S4 method for signature 'intData'
colnames(x)

Arguments

Value

A character vector of column names.

Credit Card Dataset

Description

This dataset contains interval data of credit card expenses, including min-max values, centers and ranges, microdata, and an intData object. It is composed of 5 variables: Food, Social, Travel, Gas, and Clothes. It was aggregated by person-month.

Usage

data(creditcard)

Format

A list with the following components:

microdata

A data frame with 1000 rows and 7 columns. It contains the microdata, with individual measurements of each variable for all observations.

min_max

A data frame with 36 rows and 10 columns. Each row corresponds to a different observation, and each column gives the minimum and maximum values for each variable.

centers_ranges

A data frame with 36 rows and 10 columns. Each row corresponds to the centers and ranges of the interval data.

intData

An intData object with 36 interval-valued observations and 5 variables, constructed assuming the microdata follow symmetric triangular distributions.

References

This data was retrieved from Billard, L. and Diday, E. (2006). Symbolic Data Analysis: Conceptual Statistics and Data Mining. John Wiley & Sons. doi:10.1002/9780470090183.

Examples

data(creditcard)
head(creditcard$min_max)
head(creditcard$microdata)
head(creditcard$intData)

Dimensions Method for intData

Description

Dimensions Method for intData

Usage

## S4 method for signature 'intData'
dim(x)

Arguments

Value

A vector of the number of rows and columns.

Randomly draw a subset of observations

Description

Randomly draw a subset of observations

Usage

draw_z(m, data)

Arguments

Value

A vector representing an m-length subset of X

Entrecampos Air Quality Dataset

Description

This dataset contains interval data of air pollutants' concentrations, including min-max values and microdata. This air quality dataset was obtained from a monitoring station in Entrecampos, Lisbon. It is composed of 9 pollutants' concentration measures in µg/m3 during the years 2019, 2020, and 2021: sulphur dioxide (SO2), particles < 10µm, ozone (O3), nitrogen dioxide (NO2), carbon monoxide (CO), benzene (C6H6), particles < 2.5µm, nitrogen oxides (NOx), and nitrogen monoxide (NO). For the microdata_transformed, min_max, and intData, the pollutant "benzene" was removed due to a high number of missing values. The aggregation of the microdata was done by day.

Usage

data(entrecampos_air_quality)

Format

A list with the following components:

microdata_raw

A data frame with 26304 rows and 11 columns. It contains the raw microdata, with individual measurements of each variable for all observations.

microdata_transformed

A data frame with 26304 rows and 10 columns. It contains the microdata, with individual measurements of each variable for all observations. Logarithmic transformations were applied to all variables and interpolation to deal with missing values.

min_max

A data frame with 1096 rows and 17 columns. Each row corresponds to a different observation, and each column gives the minimum and maximum values for each variable. The first column corresponds to the day, the next 8 to the minimum and the last 8 to the maximum.

intData

An intData object, constructed using KDE for estimating the parameters of the latent distributions.

References

This data was retrieved from the Portuguese Environment Agency database available at https://qualar.apambiente.pt/.

Examples

data(entrecampos_air_quality)
head(entrecampos_air_quality$microdata_raw)
head(entrecampos_air_quality$microdata_transformed)
head(entrecampos_air_quality$min_max)
head(entrecampos_air_quality$intData)

Farness Estimation

Description

Estimate farness from a distance vector in order to identify outlier observations.

Usage

farness(dist, cutoff_value = NULL)

Arguments

Value

Farness of each observation. Values between 0 and 1. If cutoff_value is provided, a list with the farness probabilities and the cutoff distance value in the original distance scale is returned.

References

J. Raymaekers and P.J. Rousseeuw (2021). Transforming variables to central normality. Machine Learning. doi:10.1007/s10994-021-05960-5

Based on the cellWise package: Raymaekers J, Rousseeuw P (2023). cellWise: Analyzing Data with Cellwise Outliers. R package version 2.5.3, https://CRAN.R-project.org/package=cellWise.

Examples

data(creditcard)
credit_card_int <- creditcard$intData

# Compute squared Interval-Mahalanobis distance
credit_card_dist <- IMah_dist(credit_card_int)

credit_card_farness <- farness(credit_card_dist, cutoff_value = 0.9)

Compute Latent Variables Parameters

Description

Obtain the parameters of the latent variables inherent to the macrodata.

Usage

get_latent_param(
  LatentCase = c("U_id_symmetric", "U_id", "General"),
  LatentDist = c("Unif", "Triang", "TNorm", "InvTri", "Beta", "KDE", "Degenerated"),
  TriangParam = 0,
  BetaParam.a = 1,
  BetaParam.b = 1,
  Umicro = NULL,
  p = NULL,
  estimate.DistParam = FALSE
)

Arguments

Details

The parameters of the latent variables inherent to the macrodata are defined according to the LatentCase:

• "U_id_symmetric": The latent variables are identically distributed and symmetric, so its parameters are:

  • \delta=\mathbb{E}(U^2)/4

• "U_id": The latent variables are identically distributed, so its parameters are:

  • \delta=\mathbb{E}(U^2)/4

  • \mathbb{E}(U)

• "General": The latent variables do not have any nice properties, so its parameters are:

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{j\ell}=\mathcal{E}(U_j,U_\ell), j\neq \ell, with \mathcal{E}(U_j,U_\ell)=\int_0^1 F_{U_j}^{-1}(t) F_{U_\ell}^{-1}(t) \, dt, and [\boldsymbol{\mathfrak{E}}_{UU}]_{jj}=\mathbb{E}(U_j^2), j,\ell=1,\dots,p

  • \boldsymbol{\Psi}=\text{Diag}(\mathbb{E}(U_1),\dots,\mathbb{E}(U_p))

Value

A list with the parameters of the latent variables.

References

Oliveira, M. R., Pinheiro, D., & Oliveira, L. (2025). Location and association measures for interval-valued data based on Mallows' distance. arXiv preprint arXiv:2407.05105. https://arxiv.org/abs/2407.05105

Examples

data(creditcard)
CreditCard_min_max <- creditcard$min_max
CreditCard_microdata <- creditcard$microdata

# Define grouping variable for microdata aggregation
credit_agrby <- paste(CreditCard_microdata$Name, CreditCard_microdata$Month, sep = "_")

# Obtain latent variables inherent to the macrodata (standardized to [-1,1])
credit_card_U <- get_latent_var(microdata = CreditCard_microdata[,3:7], 
                                macrodata = CreditCard_min_max, 
                                agrby = credit_agrby, 
                                agrlevels = row.names(CreditCard_min_max), 
                                Seq = "LbUb_VarbyVar")

# Obtain parameters of the latent variables
credit_card_param <- get_latent_param(LatentCase = "General",
                                      LatentDist = "KDE",
                                      Umicro = credit_card_U)

Compute Latent Variables

Description

Obtain the latent variables inherent to the macrodata.

Usage

get_latent_var(
  microdata,
  macrodata,
  agrby,
  agrlevels,
  Seq = c("AllLb_AllUb", "AllCen_AllRng", "LbUb_VarbyVar", "CenRng_VarbyVar")
)

Arguments

Details

The latent variables, U_{j}, are defined according to the following model:

Let X_j=(C_j,R_j)^\top=\left[C_j-\dfrac{R_j}{2}, C_j+\dfrac{R_j}{2}\right] represent the macrodata and

V_{j}=C_j+U_{j}\dfrac{R_j}{2},\quad j=1,\dots,p,

the microdata with U_{j} being random variables with support on [-1,1], uncorrelated with (C_j,R_j).

Value

A matrix with the same size as the microdata.

References

Oliveira, M.R., Azeitona, M., Pacheco, A., Valadas, R.. Association measures for interval variables. Advances in Data Analysis and Classification 16, 491–520 (2022). doi:10.1007/s11634-021-00445-8

Examples

data(creditcard)
CreditCard_min_max <- creditcard$min_max
CreditCard_microdata <- creditcard$microdata

# Define grouping variable for microdata aggregation
credit_agrby <- paste(CreditCard_microdata$Name, CreditCard_microdata$Month, sep = "_")

# Obtain latent variables inherent to the macrodata (standardized to [-1,1])
credit_card_U <- get_latent_var(microdata = CreditCard_microdata[,3:7], 
                                macrodata = CreditCard_min_max, 
                                agrby = credit_agrby, 
                                agrlevels = row.names(CreditCard_min_max), 
                                Seq = "LbUb_VarbyVar")

Head Method for intData

Description

Returns the first n rows of an intData object.

Usage

## S4 method for signature 'intData'
head(x, n = min(nrow(x), 6L))

Arguments

Value

A subset of the intData object.

Cars Dataset

Description

This dataset contains interval data of car specifications, including min-max values. It is composed of 5 variables: Engine Capacity, Top Speed, Acceleration, Price and Class. The aggregation of the microdata was done by car model.

Usage

data(intCars)

Format

A list with the following components:

microdata

A data frame with 27 rows and 9 columns. It contains the lower and upper bounds for each variable.

intData

An intData object with 27 interval-valued observations and 4 variables. The variable "Price" was log-transformed into "lnPrice". The microdata are not available, thus the default parameters of the latent distributions were used assuming a uniform distribution.

References

This data was retrieved from the MAINT.Data package, available at https://cran.r-project.org/package=MAINT.Data.

Examples

data(intCars)
head(intCars$microdata)
head(intCars$intData)

Interval Data Constructor

Description

Constructs an interval data object.

Usage

intData(
  macrodata,
  Seq = c("AllLb_AllUb", "AllCen_AllRng", "LbUb_VarbyVar", "CenRng_VarbyVar"),
  LatentParam = NULL,
  LatentCase = c("U_id_symmetric", "U_id", "General"),
  LatentDist = c("Unif", "Triang", "TNorm", "InvTri", "Beta", "KDE", "Degenerated"),
  TriangParam = 0,
  BetaParam.a = 1,
  BetaParam.b = 1,
  Umicro = NULL,
  estimate.DistParam = FALSE,
  VarNames = NULL,
  ObsNames = row.names(macrodata),
  NbMicroUnits = integer(0)
)

Arguments

Value

An object of class intData.

References

Oliveira, M. R., Pinheiro, D., & Oliveira, L. (2025). Location and association measures for interval-valued data based on Mallows' distance. arXiv preprint arXiv:2407.05105. https://arxiv.org/abs/2407.05105

Adapted from package MAINT.Data (https://cran.r-project.org/package=MAINT.Data).

Examples

# Load microdat and macrodata
data(creditcard)
CreditCard_microdata <- creditcard$microdata
CreditCard_min_max <- creditcard$min_max

# Create an intData object using the min_max component of the dataset 
# Assume a continuous uniform distribution for the latent variables 
# This corresponds to LatentCase="U_id_symmetric"
# This is the default setting for the intData class
credit_card_int_unif <- intData(CreditCard_min_max, 
                                Seq = "LbUb_VarbyVar", 
                                VarNames = colnames(CreditCard_microdata)[3:7])

Interval Data Class

Description

A class to represent interval data.

Slots

Centers

A data frame of centers of the intervals.

Ranges

A data frame of ranges of the intervals.

LatentParam

A list with the parameters of the latent variables.

LatentCase

A string specifying which of the three scenarios applies to the latent variables:

• "U_id_symmetric": The case where the latent variables are identically distributed and symmetric.

• "U_id": The case where the latent variables are identically distributed.

• "General": The case where the latent variables do not have any nice properties.

Defaults to "U_id_symmetric".

LatentDist

A string or vector of strings specifying the distribution(s) of the latent variables. If the variables are identically distributed it can be one of ("Unif", "Triang", "TNorm", "InvTri", "Beta", "KDE", "Degenerated"), if not, it is a vector with the distribution for each variable.

ObsNames

A character vector of observation names.

VarNames

A character vector of variable names.

NObs

A numeric value indicating the number of observations.

NIVar

A numeric value indicating the number of interval variables.

NbMicroUnits

An integer vector indicating the number of individual observations (microdata) aggregated by interval (macrodata).

References

Oliveira, M. R., Pinheiro, D., & Oliveira, L. (2025). Location and association measures for interval-valued data based on Mallows' distance. arXiv preprint arXiv:2407.05105. https://arxiv.org/abs/2407.05105

Adapted from package MAINT.Data (https://cran.r-project.org/package=MAINT.Data).

Compute Shapley Values for Interval-valued Data

Description

Outlier explanation based on Shapley values for interval-valued data. Decomposes the squared interval-valued Mahalanobis distance into additive outlyingness contributions of the variables.

Usage

int_Shapley(data, mean_c = NULL, mean_r = NULL, cov = NULL)

Arguments

Details

The Shapley value decomposes the squared Interval-Mahalanobis distance (see IMah_dist) into additive outlyingness contributions of the variables. Let \boldsymbol{\mu}_B=(\boldsymbol{\mu}_C^\top,\boldsymbol{\mu}_R^\top)^\top be the barycenter and \boldsymbol{\Sigma}_B the symbolic covariance matrix (see int_cov). The Shapley value of an interval-valued observation \boldsymbol{x}=(\boldsymbol{c}^\top,\boldsymbol{r}^\top)^\top, for the Interval-Mahalanobis distance, is defined according to the LatentCase:

• "U_id_symmetric": The latent variables are identically distributed and symmetric:

  \boldsymbol{\phi}(\boldsymbol{x})=(\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right]+\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)\right],

  where \delta=\mathbb{E}(U^2)/4 is the parameter of the latent variables.

• "U_id": The latent variables are identically distributed:

  \begin{aligned} \boldsymbol{\phi}(\boldsymbol{x})&=(\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right]+\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)\right]\\ &\quad+\dfrac{\mathbb{E}(U)}{2}(\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)\right]+\dfrac{\mathbb{E}(U)}{2}(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right], \end{aligned}

  where \delta=\mathbb{E}(U^2)/4 and \mathbb{E}(U) are the parameter of the latent variables.

• "General": The latent variables do not have any nice properties:

  \begin{aligned} \boldsymbol{\phi}(\boldsymbol{x})&=(\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right] +\dfrac{1}{4}(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\left(\boldsymbol{\mathfrak{E}}_{UU}\bullet\boldsymbol{\Sigma}_B^{-1}\right)(\boldsymbol{r}-\boldsymbol{\mu}_R)\right]\\ &\quad+\dfrac{1}{2}(\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}\boldsymbol{\Psi}(\boldsymbol{r}-\boldsymbol{\mu}_R)\right] +\dfrac{1}{2}(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\boldsymbol{\Psi}\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right], \end{aligned}

  where:

  • \boldsymbol{\Psi}=\text{Diag}(\mathbb{E}(U_1),\dots,\mathbb{E}(U_p)),

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{j\ell}=\mathcal{E}(U_j,U_\ell), j\neq \ell, with \mathcal{E}(U_j,U_\ell)=\int_0^1 F_{U_j}^{-1}(t) F_{U_\ell}^{-1}(t) \, dt,

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{jj}=\mathbb{E}(U_j^2), j,\ell=1,\dots,p,

  • \bullet denotes the Schur (or entrywise) product of matrices.

Value

A matrix of Shapley values with row and column names corresponding to the rows and columns of the input data.

References

Loureiro, C. P., Oliveira, M. R., Brito, P., & Oliveira, L. (2026). Explainable Outlier Detection for Interval-valued Data. arXiv preprint arXiv:2606.26307. https://arxiv.org/abs/2606.26307

Examples

# Create intData object
data(creditcard)
credit_card_int <- creditcard$intData

# Compute Shapley values based on IMCD estimates of mean and covariance
credit_card_shapley <- int_Shapley(credit_card_int)

Compute Shapley Decomposition into contributions of (Centers, Ranges, and CrossCentersRanges) for Interval-valued Data

Description

Decomposes the squared interval-valued Mahalanobis distance of each observation into outlyingness contributions of (Centers, Ranges, and CrossCentersRanges) per variable for interval-valued data.

Usage

int_Shapley_decomp(data, mean_c = NULL, mean_r = NULL, cov = NULL)

Arguments

Details

Let \boldsymbol{\mu}_B=(\boldsymbol{\mu}_C^\top,\boldsymbol{\mu}_R^\top)^\top be the barycenter and \boldsymbol{\Sigma}_B the symbolic covariance matrix (see int_cov). Based on the Shapley value (see int_Shapley), we can further decompose the Interval-Mahalanobis distance of an interval-valued observation \boldsymbol{x}=(\boldsymbol{c}^\top,\boldsymbol{r}^\top)^\top into contributions of the centers, ranges and cross-centers-ranges of the variables. The decomposition is defined according to the LatentCase:

• "U_id_symmetric": The latent variables are identically distributed and symmetric:

  • Centers contribution:

    (\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right],

  • Ranges contribution:

    \delta(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)\right],

  where \delta=\mathbb{E}(U^2)/4 is the parameter of the latent variables.

• "U_id": The latent variables are identically distributed:

  • Centers contribution:

    (\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right],

  • Ranges contribution:

    \delta(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)\right],

  • CrossCentersRanges contribution:

    \dfrac{\mathbb{E}(U)}{2}(\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)\right]+\dfrac{\mathbb{E}(U)}{2}(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right],

  where \delta=\mathbb{E}(U^2)/4 and \mathbb{E}(U) are the parameter of the latent variables.

• "General": The latent variables do not have any nice properties:

  • Centers contribution:

    (\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right],

  • Ranges contribution:

    \dfrac{1}{4}(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\left(\boldsymbol{\mathfrak{E}}_{UU}\bullet\boldsymbol{\Sigma}_B^{-1}\right)(\boldsymbol{r}-\boldsymbol{\mu}_R)\right],

  • CrossCentersRanges contribution:

    \dfrac{1}{2}(\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}\boldsymbol{\Psi}(\boldsymbol{r}-\boldsymbol{\mu}_R)\right]+\dfrac{1}{2}(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\boldsymbol{\Psi}\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right],

  where:

  • \boldsymbol{\Psi}=\text{Diag}(\mathbb{E}(U_1),\dots,\mathbb{E}(U_p)),

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{j\ell}=\mathcal{E}(U_j,U_\ell), j\neq \ell, with \mathcal{E}(U_j,U_\ell)=\int_0^1 F_{U_j}^{-1}(t) F_{U_\ell}^{-1}(t) \, dt,

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{jj}=\mathbb{E}(U_j^2), j,\ell=1,\dots,p,

  • \bullet denotes the Schur (or entrywise) product of matrices.

Value

A list containing the matrix of Shapley value decomposition into contributions of (Centers, Ranges, and CrossCentersRanges) per variable for each observation.

References

Loureiro, C. P., Oliveira, M. R., Brito, P., & Oliveira, L. (2026). Explainable Outlier Detection for Interval-valued Data. arXiv preprint arXiv:2606.26307. https://arxiv.org/abs/2606.26307

Examples

# Create intData object
data(creditcard)
credit_card_int <- creditcard$intData

# Compute Shapley decomposition into contributions of (Centers, Ranges, and CrossCentersRanges) 
# based on IMCD estimates of mean and covariance
credit_card_shap_decomp <- int_Shapley_decomp(credit_card_int)

Compute Shapley interaction indices for Interval-valued Data

Description

Obtains a p \times p matrix containing pairwise outlyingness scores based on Shapley interaction indices for each observation. Decomposes the squared interval-valued Mahalanobis distance of each observation into outlyingness contributions of pairs of variables.

Usage

int_Shapley_interaction(data, mean_c = NULL, mean_r = NULL, cov = NULL)

Arguments

Details

Let \boldsymbol{\mu}_B=(\boldsymbol{\mu}_C^\top,\boldsymbol{\mu}_R^\top)^\top be the barycenter and \boldsymbol{\Sigma}_B the symbolic covariance matrix (see int_cov). Let also \boldsymbol{\phi}(\boldsymbol{x}) be the Shapley value of \boldsymbol{x} (see int_Shapley) and \mathrm{diag}(\boldsymbol{v}) be the diagonal matrix whose main diagonal is the vector \boldsymbol{v}. The Shapley interaction index of an interval-valued observation \boldsymbol{x}=(\boldsymbol{c}^\top,\boldsymbol{r}^\top)^\top, for the Interval-Mahalanobis distance, is defined according to the LatentCase:

• "U_id_symmetric": The latent variables are identically distributed and symmetric:

  \boldsymbol{\Phi}(\boldsymbol{x})=2(\boldsymbol{c}-\boldsymbol{\mu}_C)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\bullet\boldsymbol{\Sigma}_B^{-1} + 2\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)(\boldsymbol{r}-\boldsymbol{\mu}_R)^\top\bullet\boldsymbol{\Sigma}_B^{-1}-\mathrm{diag}\left(\boldsymbol{\phi}(\boldsymbol{x})\right),

  where \delta=\mathbb{E}(U^2)/4 is the parameter of the latent variables.

• "U_id": The latent variables are identically distributed:

  \begin{aligned} \boldsymbol{\Phi}(\boldsymbol{x})&=2(\boldsymbol{c}-\boldsymbol{\mu}_C)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\bullet\boldsymbol{\Sigma}_B^{-1} + 2\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)(\boldsymbol{r}-\boldsymbol{\mu}_R)^\top\bullet\boldsymbol{\Sigma}_B^{-1}\\ &\quad+\mathbb{E}(U)(\boldsymbol{c}-\boldsymbol{\mu}_C)(\boldsymbol{r}-\boldsymbol{\mu}_R)^\top\bullet\boldsymbol{\Psi} + \mathbb{E}(U)(\boldsymbol{r}-\boldsymbol{\mu}_R)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\bullet\boldsymbol{\Sigma}_B^{-1}-\mathrm{diag}\left(\boldsymbol{\phi}(\boldsymbol{x})\right), \end{aligned}

  where \delta=\mathbb{E}(U^2)/4 and \mathbb{E}(U) are the parameter of the latent variables.

• "General": The latent variables do not have any nice properties:

  \begin{aligned} \boldsymbol{\Phi}(\boldsymbol{x})&=2(\boldsymbol{c}-\boldsymbol{\mu}_C)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\bullet\boldsymbol{\Sigma}_B^{-1} + \dfrac{1}{2}(\boldsymbol{r}-\boldsymbol{\mu}_R)(\boldsymbol{r}-\boldsymbol{\mu}_R)^\top\bullet\boldsymbol{\mathfrak{E}}_{UU}\bullet\boldsymbol{\Sigma}_B^{-1}\\ &\quad+(\boldsymbol{c}-\boldsymbol{\mu}_C)(\boldsymbol{r}-\boldsymbol{\mu}_R)^\top\bullet\boldsymbol{\Sigma}_B^{-1}\boldsymbol{\Psi} + (\boldsymbol{r}-\boldsymbol{\mu}_R)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\bullet\boldsymbol{\Psi}\boldsymbol{\Sigma}_B^{-1}-\mathrm{diag}\left(\boldsymbol{\phi}(\boldsymbol{x})\right), \end{aligned}

  where:

• \boldsymbol{\Psi}=\text{Diag}(\mathbb{E}(U_1),\dots,\mathbb{E}(U_p)),

• [\boldsymbol{\mathfrak{E}}_{UU}]_{j\ell}=\mathcal{E}(U_j,U_\ell), j\neq \ell, with \mathcal{E}(U_j,U_\ell)=\int_0^1 F_{U_j}^{-1}(t) F_{U_\ell}^{-1}(t) \, dt,

• [\boldsymbol{\mathfrak{E}}_{UU}]_{jj}=\mathbb{E}(U_j^2), j,\ell=1,\dots,p,

• \bullet denotes the Schur (or entrywise) product of matrices.

Value

A list containing the matrix of Shapley interaction indices for each observation.

References

Loureiro, C. P., Oliveira, M. R., Brito, P., & Oliveira, L. (2026). Explainable Outlier Detection for Interval-valued Data. arXiv preprint arXiv:2606.26307. https://arxiv.org/abs/2606.26307

Examples

# Create intData object
data(creditcard)
credit_card_int <- creditcard$intData

# Compute Shapley interaction indices based on the mean and covariance matrix estimated by IMCD
credit_card_shap_inter <- int_Shapley_interaction(credit_card_int)

Interval-valued Covariance

Description

Calculate the interval-valued covariance matrix based on the covariance matrices of the centers and ranges or data.

Usage

int_cov(
  data = NULL,
  sigma_cc = NULL,
  sigma_rr = NULL,
  sigma_cr = NULL,
  LatentParam = NULL,
  LatentCase = c("U_id_symmetric", "U_id", "General")
)

Arguments

Details

This function calculates the interval-valued covariance matrix, \boldsymbol{\Sigma}_B, based on the covariance matrices of the centers, \boldsymbol{\Sigma}_{CC}, ranges, \boldsymbol{\Sigma}_{RR}, and the covariance matrix between the centers and ranges, \boldsymbol{\Sigma}_{CR}=\boldsymbol{\Sigma}_{RC}^\top. The covariance matrix is defined according to the LatentCase:

• "U_id_symmetric": The latent variables are identically distributed and symmetric:

  \boldsymbol{\Sigma}_B=\boldsymbol{\Sigma}_{CC}+\delta\boldsymbol{\Sigma}_{RR},

  where \delta=\mathbb{E}(U^2)/4 is the parameter of the latent variables.

• "U_id": The latent variables are identically distributed:

  \boldsymbol{\Sigma}_B=\boldsymbol{\Sigma}_{CC}+\delta\boldsymbol{\Sigma}_{RR}+\dfrac{\mathbb{E}(U)}{2}\left(\boldsymbol{\Sigma}_{CR}+\boldsymbol{\Sigma}_{RC}\right),

  where \delta=\mathbb{E}(U^2)/4 and \mathbb{E}(U) are the parameters of the latent variables.

• "General": The latent variables do not have any nice properties:

  \boldsymbol{\Sigma}_B=\boldsymbol{\Sigma}_{CC}+\dfrac{1}{4}\boldsymbol{\mathfrak{E}}_{UU}\bullet\boldsymbol{\Sigma}_{RR}+\dfrac{1}{2}\boldsymbol{\Sigma}_{CR}\boldsymbol{\Psi}+\dfrac{1}{2}\boldsymbol{\Psi}\boldsymbol{\Sigma}_{RC}

  where:

  • \boldsymbol{\Psi}=\text{diag}(\mathbb{E}(U_1),\dots,\mathbb{E}(U_p)),

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{j\ell}=\mathcal{E}(U_j,U_\ell), j\neq \ell, with \mathcal{E}(U_j,U_\ell)=\int_0^1 F_{U_j}^{-1}(t) F_{U_\ell}^{-1}(t) \, dt,

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{jj}=\mathbb{E}(U_j^2), j,\ell=1,\dots,p,

  • \bullet denotes the Schur (or entrywise) product of matrices.

The covariance matrix can be calculated either based on the covariance matrices of the centers and ranges or based on the data. If the data is provided, the covariance matrices are calculated using the sample covariance of the centers and ranges and the sample covariance between centers and ranges. For the robust estimation of the covariance matrix, see IMCD.

Value

The symbolic covariance matrix.

References

Oliveira, M. R., Pinheiro, D., & Oliveira, L. (2025). Location and association measures for interval-valued data based on Mallows' distance. arXiv preprint arXiv:2407.05105. https://arxiv.org/abs/2407.05105

Examples

data(creditcard)
credit_card_int <- creditcard$intData

credit_card_cov <- int_cov(credit_card_int)

Sample Interval-valued Covariance

Description

Calculate the interval-valued covariance matrix in function of z

Usage

int_cov_z(z, data)

Arguments

Details

Let \boldsymbol{z}\in\{0,1\}^n be a vector indicating which m observations are “active”. This function calculates the sample interval-valued covariance matrix in function of \boldsymbol{z}: \boldsymbol{S}_B(\boldsymbol{z}). Let \boldsymbol{C}, \boldsymbol{R} be the matrices of centers and ranges, respectively. Additionally, set:

\overline{\boldsymbol{c}}_B(\boldsymbol{z})=\dfrac{1}{m}\boldsymbol{C}^{\top}\boldsymbol{z}, \qquad \overline{\boldsymbol{r}}_B(\boldsymbol{z})=\dfrac{1}{m}\boldsymbol{R}^{\top}\boldsymbol{z}.

The sample interval-valued covariance matrix is obtained according to the LatentCase:

• "U_id_symmetric": The latent variables are identically distributed and symmetric:

  \boldsymbol{S}_B(\boldsymbol{z})=\left(\dfrac{1}{m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{c}_{h}\boldsymbol{c}_{h}^{\top}\right)-\overline{\boldsymbol{c}}_B(\boldsymbol{z})\overline{\boldsymbol{c}}_B(\boldsymbol{z})^\top+\left(\dfrac{\delta}{m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{r}_{h}\boldsymbol{r}_{h}^{\top}\right)-\delta\overline{\boldsymbol{r}}_B(\boldsymbol{z})\overline{\boldsymbol{r}}_B(\boldsymbol{z})^\top,

  where \delta=\mathbb{E}(U^2)/4 is the parameter of the latent variables.

• "U_id": The latent variables are identically distributed:

  \begin{aligned} \boldsymbol{S}_B(\boldsymbol{z})&=\left(\dfrac{1}{m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{c}_{h}\boldsymbol{c}_{h}^{\top}\right)-\overline{\boldsymbol{c}}_B(\boldsymbol{z})\overline{\boldsymbol{c}}_B(\boldsymbol{z})^\top+\left(\dfrac{\delta}{m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{r}_{h}\boldsymbol{r}_{h}^{\top}\right)-\delta\overline{\boldsymbol{r}}_B(\boldsymbol{z})\overline{\boldsymbol{r}}_B(\boldsymbol{z})^\top\\ &\quad+\left(\dfrac{\mathbb{E}(U)}{2m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{c}_{h}\boldsymbol{r}_{h}^{\top}\right)-\dfrac{\mathbb{E}(U)}{2}\overline{\boldsymbol{c}}_B(\boldsymbol{z})\overline{\boldsymbol{r}}_B(\boldsymbol{z})^\top\\ &\quad+\left(\dfrac{\mathbb{E}(U)}{2m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{r}_{h}\boldsymbol{c}_{h}^{\top}\right)-\dfrac{\mathbb{E}(U)}{2}\overline{\boldsymbol{r}}_B(\boldsymbol{z})\overline{\boldsymbol{c}}_B(\boldsymbol{z})^\top, \end{aligned}

  where \delta=\mathbb{E}(U^2)/4 and \mathbb{E}(U) are the parameters of the latent variables.

• "General": The latent variables do not have any nice properties:

  \begin{aligned} \boldsymbol{S}_B(\boldsymbol{z})&=\left(\dfrac{1}{m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{c}_{h}\boldsymbol{c}_{h}^{\top}\right)-\overline{\boldsymbol{c}}_B(\boldsymbol{z})\overline{\boldsymbol{c}}_B(\boldsymbol{z})^\top\\ &\quad+\left(\dfrac{1}{4m}\boldsymbol{\mathfrak{E}}_{UU}\bullet\sum\limits_{h=1}^{n}z_{h}\boldsymbol{r}_{h}\boldsymbol{r}_{h}^{\top}\right)-\dfrac{1}{4}\boldsymbol{\mathfrak{E}}_{UU}\bullet\left[\overline{\boldsymbol{r}}_B(\boldsymbol{z})\overline{\boldsymbol{r}}_B(\boldsymbol{z})^\top\right]\\ &\quad+\left(\dfrac{1}{2m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{c}_{h}\boldsymbol{r}_{h}^{\top}\right)\boldsymbol{\Psi}-\dfrac{1}{2}\overline{\boldsymbol{c}}_B(\boldsymbol{z})\overline{\boldsymbol{r}}_B(\boldsymbol{z})^\top\boldsymbol{\Psi}\\ &\quad+\boldsymbol{\Psi}\left(\dfrac{1}{2m}\sum\limits_{h=1}^{n}z_{h}\boldsymbol{r}_{h}\boldsymbol{c}_{h}^{\top}\right)-\dfrac{1}{2}\boldsymbol{\Psi}\overline{\boldsymbol{r}}_B(\boldsymbol{z})\overline{\boldsymbol{c}}_B(\boldsymbol{z})^\top, \end{aligned}

  where:

  • \boldsymbol{\Psi}=\text{diag}(\mathbb{E}(U_1),\dots,\mathbb{E}(U_p)),

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{j\ell}=\mathcal{E}(U_j,U_\ell), j\neq \ell, with \mathcal{E}(U_j,U_\ell)=\int_0^1 F_{U_j}^{-1}(t) F_{U_\ell}^{-1}(t) \, dt,

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{jj}=\mathbb{E}(U_j^2), j,\ell=1,\dots,p,

  • \bullet denotes the Schur (or entrywise) product of matrices.

Value

The symbolic covariance matrix

References

Oliveira, M. R., Pinheiro, D., & Oliveira, L. (2025). Location and association measures for interval-valued data based on Mallows' distance. arXiv preprint arXiv:2407.05105. https://arxiv.org/abs/2407.05105

Loureiro, C. P., Oliveira, M. R., Brito, P., & Oliveira, L. (2026). Minimum Covariance Determinant Estimator and Outlier Detection for Interval-valued Data. arXiv preprint arXiv:2604.26769. https://arxiv.org/abs/2604.26769

Examples

data(creditcard)
credit_card_int <- creditcard$intData

# Compute sample interval-valued covariance matrix using the all the observations
z <- rep(1, nrow(credit_card_int))
credit_card_cov <- int_cov_z(z, credit_card_int)

Sample Mean

Description

Calculate the mean of X in function of z

Usage

int_mean_z(z, X)

Arguments

Details

This function calculates the mean of \boldsymbol{X} in function of \boldsymbol{z}. If \boldsymbol{z} is a vector of 0 and 1, the mean is calculated for the m observations that are equal to 1:

\bar{\boldsymbol{x}}(\boldsymbol{z}) = \dfrac{1}{m} \boldsymbol{X}^\top \boldsymbol{z}.

Value

A vector where each element is the mean for each variable

Examples

n <- 100
p <- 4
X <- matrix(rnorm(n * p), ncol = p)

# if we consider all the observations the result obtained is the same as colMeans()
z <- c(rep(1, n))
int_mean_z(z, X)
colMeans(X)

Outlier Detection for Interval-Valued Data Based on Robust Distances

Description

Identifies potential outliers in interval-valued data using robust distance-based methods with customizable cutoff criteria.

Usage

int_outliers(
  robust_dist,
  cutoff = c("farness", "adjbox", "chi-squared", "F-dist"),
  cutoff_lvl = NULL,
  p = NULL,
  z = NULL
)

Arguments

Details

This function classifies observations as outliers based on robust distances and user-defined cutoff methods. It supports various approaches, including Chi-Squared quantiles, adjusted boxplots, F distribution quantiles, and farness probabilities.

Value

A list with the following components:

References

Loureiro, C. P., Oliveira, M. R., Brito, P., & Oliveira, L. (2026). Minimum Covariance Determinant Estimator and Outlier Detection for Interval-valued Data. arXiv preprint arXiv:2604.26769. https://arxiv.org/abs/2604.26769

Case cutoff=="F-dist" is adapted from package CerioliOutlierDetection (https://cran.r-project.org/package=CerioliOutlierDetection).

Examples

# Example of detecting outliers using robust distances
set.seed(42)
robust_dist <- abs(rnorm(100))
result <- int_outliers(robust_dist, cutoff = "chi-squared", p = 5)

# Example using creditcard dataset
data(creditcard)
credit_card_int <- creditcard$intData

# Compute robust distances using IMCD estimates of mean and covariance
credit_card_dist <- IMah_dist(credit_card_int)

# Detect outliers using farness cutoff
credit_card_outliers <- int_outliers(credit_card_dist, 
                                     cutoff = "farness", 
                                     cutoff_lvl = 0.9)

Compute Mean Latent Variables

Description

Obtain the mean of the latent variables inherent to the macrodata.

Usage

meanU(
  LatentDist = c("Unif", "Triang", "TNorm", "InvTri", "Beta", "KDE", "Degenerated"),
  TriangParam = 0,
  BetaParam.a = 1,
  BetaParam.b = 1,
  Umicro = NULL,
  p = NULL
)

Arguments

Value

Either a diagonal matrix with the mean of each variable or a value if the variables are identically distributed.

Compute Mean Square Latent Variables

Description

Obtain the mean of the square of the latent variables inherent to the macrodata.

Usage

meanU2(
  LatentDist = c("Unif", "Triang", "TNorm", "InvTri", "Beta", "KDE", "Degenerated"),
  TriangParam = 0,
  BetaParam.a = 1,
  BetaParam.b = 1,
  Umicro = NULL,
  p = NULL
)

Arguments

Value

Either a diagonal matrix with the mean of the square of each variable or a value if the variables are identically distributed.

Aggregate Microdata into Interval-Valued Data

Description

Aggregates microdata from a data frame into interval-valued data using various criteria and latent distribution settings.

Usage

micro2intData(
  microdata,
  agrby,
  agrcrt = "minmax",
  LatentParam = NULL,
  LatentCase = c("U_id_symmetric", "U_id", "General"),
  LatentDist = c("Unif", "Triang", "TNorm", "InvTri", "Beta", "KDE", "Degenerated"),
  TriangParam = 0,
  BetaParam.a = 1,
  BetaParam.b = 1,
  estimate.DistParam = FALSE
)

Arguments

Details

This function processes a data frame of microdata and aggregates it into interval-valued data according to the specified grouping factor and aggregation criteria. It can handle different latent distribution cases and parameter settings.

If some rows contain invalid (non-finite or missing) values, those rows are removed before aggregation. If all rows in the resulting interval-valued data are degenerate (i.e., the lower bound equals the upper bound), the function will return NULL.

Value

An intData object containing the aggregated interval-valued data, or NULL if all units lead to degenerate intervals.

References

Adapted from package MAINT.Data (https://cran.r-project.org/package=MAINT.Data).

Examples

data(creditcard)
CreditCard_microdata <- creditcard$microdata

# Define grouping variable for microdata aggregation
credit_agrby <- factor(paste(CreditCard_microdata$Name, CreditCard_microdata$Month, sep = "_"))

# Create intData object by aggregating microdata using the default minmax criterion 
# and using KDE for estimation of the latent distribution in the general case
credit_agr <- micro2intData(CreditCard_microdata[,3:7],
                            agrby = credit_agrby,
                            LatentCase = "General")

Variable Names Method for intData

Description

Variable Names Method for intData

Usage

## S4 method for signature 'intData'
names(x)

Arguments

Value

A character vector of variable names.

Number of Columns Method for intData

Description

Number of Columns Method for intData

Usage

## S4 method for signature 'intData'
ncol(x)

Arguments

Value

The number of columns.

Number of Rows Method for intData

Description

Number of Rows Method for intData

Usage

## S4 method for signature 'intData'
nrow(x)

Arguments

Value

The number of rows.

Choose the 10 best estimates after iterating twice through initial sets

Description

Choose the 10 best estimates after iterating twice through initial sets

Usage

pick10(z_all, m, data)

Arguments

Value

A list of z, covariance, barycenter and robust distances

Plot Method for Two intData Objects

Description

Plots one intData object against another, with options to visualize the intervals as crosses or rectangles.

Plots a single intData object, either in a vertical or horizontal layout.

Usage

## S4 method for signature 'intData,intData'
plot(
  x,
  y,
  type = c("crosses", "rectangles", "crosses2"),
  append = FALSE,
  palette = rainbow(x@NObs),
  ...
)

## S4 method for signature 'intData,missing'
plot(
  x,
  casen = NULL,
  layout = c("vertical", "horizontal"),
  append = FALSE,
  ...
)

Arguments

Value

A plot showing the relationship between the two intData objects.

A plot showing the intervals of the intData object.

Barplot of Shapley values for Interval-valued Data

Description

Barplot of Shapley values for Interval-valued Data

Usage

plot_bar_int_Shapley(
  x,
  cutoff_value = NULL,
  cutoff_label = NULL,
  palette = NULL,
  abbrev.var = 20,
  abbrev.obs = 20,
  sort.obs = TRUE,
  plot_IMah = TRUE,
  IMah_label = expression(Robust ~ d[IMah]^2 * (bold(x))),
  rotate_x = TRUE
)

Arguments

Value

Returns a barplot that displays the Shapley values (int_Shapley) for each observation and optionally (plot_IMah = TRUE) includes the squared (robust) Interval-Mahalanobis distance (IMah_dist) (black bar) and the corresponding outlier detection cut-off value (dotted line).

References

Adapted from package ShapleyOutlier (https://CRAN.R-project.org/package=ShapleyOutlier).

Examples

# Create intData object
data(creditcard)
credit_card_int <- creditcard$intData

# Estimate the mean and covariance matrix
credit_card_IMCD <- IMCD(credit_card_int, 
                         m = floor(nrow(credit_card_int)*0.75), 
                         cutoff = "farness", 
                         cutoff_lvl = 0.9)

# Detect outliers using farness cutoff
credit_card_outliers <- int_outliers(credit_card_IMCD$robust_dist, 
                                     cutoff = "farness", 
                                     cutoff_lvl = 0.9)

# Compute Shapley values
credit_card_shapley <- int_Shapley(credit_card_int, 
                                   mean_c = credit_card_IMCD$mean_IMCD_c, 
                                   mean_r = credit_card_IMCD$mean_IMCD_r, 
                                   cov = credit_card_IMCD$cov_IMCD)

# Plot Shapley values with cutoff line and Interval-Mahalanobis distance
plot_bar_int_Shapley(credit_card_shapley, 
                    cutoff_value = credit_card_outliers$cutoff_value,
                    cutoff_label = "Farness 0.9",
                    palette = rainbow(credit_card_int@NIVar))

Barplot of Shapley value decomposition into contributions of (Centers, Ranges, and CrossCentersRanges) for interval-valued data.

Description

Barplot of Shapley value decomposition into contributions of (Centers, Ranges, and CrossCentersRanges) for interval-valued data.

Usage

plot_bar_int_Shapley_decomp(
  shapley_decomp,
  palette = NULL,
  rotate_x = TRUE,
  abbrev.obs = 20,
  sort.obs = TRUE,
  plot_IMah = FALSE
)

Arguments

Value

Returns a barplot that displays the Shapley value decomposition into contributions of (Centers, Ranges, and CrossCentersRanges) for each observation.

Examples

# Create intData object
data(creditcard)
credit_card_int <- creditcard$intData

# Compute Shapley decomposition into contributions of Centers, Ranges, and CrossCentersRanges
# based on IMCD estimates of mean and covariance matrix
credit_card_shap_decomp <- int_Shapley_decomp(credit_card_int)

# Plot Shapley decomposition with contributions of Centers, Ranges, and CrossCentersRanges
plot_bar_int_Shapley_decomp(credit_card_shap_decomp, palette = rainbow(credit_card_int@NIVar))

Beeswarm plot of Shapley values for interval-valued data.

Description

Beeswarm plot of Shapley values for interval-valued data.

Usage

plot_beeswarm_int_Shapley(
  shapley,
  color_class,
  color_label = NULL,
  palette = NULL,
  rotate_x = TRUE,
  shape_class = NULL,
  shape_label = NULL,
  ggplotly = FALSE,
  label_obs = NULL
)

Arguments

Value

Returns a beeswarm plot that displays the Shapley values (int_Shapley) for each observation and feature.

Examples

# Create intData object
data(creditcard)
credit_card_int <- creditcard$intData

# Estimate the mean and covariance matrix
credit_card_IMCD <- IMCD(credit_card_int, 
                         m = floor(nrow(credit_card_int)*0.75), 
                         cutoff = "farness", 
                         cutoff_lvl = 0.9)

# Detect outliers using farness cutoff
credit_card_outliers <- int_outliers(credit_card_IMCD$robust_dist, 
                                     cutoff = "farness", 
                                     cutoff_lvl = 0.9)

# Compute Shapley values
credit_card_shapley <- int_Shapley(credit_card_int, 
                                   mean_c = credit_card_IMCD$mean_IMCD_c,
                                   mean_r = credit_card_IMCD$mean_IMCD_r, 
                                   cov = credit_card_IMCD$cov_IMCD)

# Beeswarm plot of Shapley values colored by outlier status
plot_beeswarm_int_Shapley(credit_card_shapley, 
                     color_class = credit_card_outliers$is_outlier, 
                     palette = c("gray50", "darkred"), 
                     color_label = "Outlier Status")

Distance-Distance plot for interval-valued data.

Description

Distance-Distance plot for interval-valued data.

Usage

plot_dist_dist(
  class_dist,
  class_cutoff = NULL,
  class_cutoff_label = NULL,
  rob_dist,
  rob_cutoff = NULL,
  rob_cutoff_label = NULL,
  obs_names = NULL,
  ggplotly = FALSE,
  color_class = NULL,
  color_label = NULL,
  palette = NULL,
  shape_class = NULL,
  shape_label = NULL,
  label_obs = NULL
)

Arguments

Value

Returns a Distance-Distance plot that displays the classical distances against the robust distances for each observation, highlighting outliers.

Examples

# Create intData object
data(creditcard)
credit_card_int <- creditcard$intData

# Compute robust distances using IMCD estimates of mean and covariance
credit_card_dist <- IMah_dist(credit_card_int)

# Detect outliers using farness cutoff
credit_card_outliers <- int_outliers(credit_card_dist, 
                                     cutoff = "farness", 
                                     cutoff_lvl = 0.9)

# Compute classical distances and outliers
class_dist <- IMah_dist(credit_card_int, z = rep(1,credit_card_int@NObs))
class_outliers <- int_outliers(class_dist, 
                               cutoff = "chi-squared", 
                               p = credit_card_int@NIVar)

# Create a vector indicating if the observations are outliers or inliers 
# based on the robust distance outlier detection
credit_card_is_outliers <- as.character(credit_card_outliers$is_outlier)
credit_card_is_outliers[credit_card_outliers$is_outlier] <- "Outlier"
credit_card_is_outliers[!credit_card_outliers$is_outlier] <- "Inlier"

# Plot Distance-Distance plot 
plot_dist_dist(class_dist, 
               class_cutoff = class_outliers$cutoff_value, 
               class_cutoff_label = "0.975 chi-squared",
               rob_dist = credit_card_dist, 
               rob_cutoff = credit_card_outliers$cutoff_value, 
               rob_cutoff_label = "0.9 farness",
               color_class = credit_card_is_outliers, 
               palette = c("grey50", "red"))

Plot Shapley interaction indices

Description

Plot Shapley interaction indices

Usage

plot_int_Shapley_inter(
  x,
  abbrev = 10,
  title = NULL,
  legend = TRUE,
  text_size = 22
)

Arguments

Value

Returns a figure consisting of two panels. The right panel shows the Shapley values, and the left panel the Shapley interaction indices.

References

Adapted from package ShapleyOutlier (https://CRAN.R-project.org/package=ShapleyOutlier).

Examples

# Create intData object
data(creditcard)
credit_card_int <- creditcard$intData

# Estimate the mean and covariance matrix
credit_card_IMCD <- IMCD(credit_card_int, 
                         m = floor(nrow(credit_card_int)*0.75), 
                         cutoff = "farness", 
                         cutoff_lvl = 0.9)

# Compute Shapley interaction indices
credit_card_shap_inter <- int_Shapley_interaction(credit_card_int, 
                                                  mean_c = credit_card_IMCD$mean_IMCD_c, 
                                                  mean_r = credit_card_IMCD$mean_IMCD_r, 
                                                  cov = credit_card_IMCD$cov_IMCD)

# Plot Shapley interaction for 1st observation
plot_int_Shapley_inter(credit_card_shap_inter[[1]])

Interval-Mahalanobis distance plot for interval-valued data.

Description

Interval-Mahalanobis distance plot for interval-valued data.

Usage

plot_interval_dist(
  dist,
  cutoff = NULL,
  cutoff_label = NULL,
  obs_names = NULL,
  sort.obs = TRUE,
  color_class = NULL,
  color_label = NULL,
  palette = NULL,
  shape_class = NULL,
  shape_label = NULL,
  label_obs = NULL
)

Arguments

Value

Returns a plot that displays the Interval-Mahalanobis distances for each observation, highlighting outliers based on specified cutoffs.

Examples

# Create intData object
data(creditcard)
credit_card_int <- creditcard$intData

# Compute robust distances using IMCD estimates of mean and covariance
credit_card_dist <- IMah_dist(credit_card_int)

# Detect outliers using farness cutoff
credit_card_outliers <- int_outliers(credit_card_dist, 
                                     cutoff = "farness", 
                                     cutoff_lvl = 0.9)

# Create a vector indicating if the observations are outliers or inliers 
# based on the robust distance outlier detection
credit_card_is_outliers <- as.character(credit_card_outliers$is_outlier)
credit_card_is_outliers[credit_card_outliers$is_outlier] <- "Outlier"
credit_card_is_outliers[!credit_card_outliers$is_outlier] <- "Inlier"

# Plot Interval-Mahalanobis distance plot
plot_interval_dist(credit_card_dist,
                   cutoff = credit_card_outliers$cutoff_value,
                   cutoff_label = c("0.9 farness"),
                   obs_names = rownames(credit_card_int),
                   sort.obs = FALSE,
                   color_class = credit_card_is_outliers,
                   palette = c("grey50", "red"))

Pairs-plot for Interval-valued Symbolic data.

Description

Adapted from pairs.panels (R package "psych") shows a scatter plot of matrices, with bivariate symbolic scatter plots below the diagonal, variables' names on the diagonal, and all the symbolic correlations above the diagonal. Useful for descriptive statistics of symbolic objects described by interval variables.

Usage

plot_pairs_int(
  data,
  type = c("rectangles", "crosses", "crosses2"),
  cex.cor = 2,
  corr = NULL,
  palette = rainbow(nrow(data)),
  fill_col = "gray50",
  is_outlier = NULL,
  ...
)

Arguments

Value

A scatter plot matrix is drawn in the graphic window. The lower off diagonal draws scatter plots, the diagonal variables' names, the upper off diagonal reports all the symbolic correlations.

Examples

data(creditcard)
credit_card_int <- creditcard$intData

# Compute covariance and correlation matrices
credit_card_cov <- int_cov(credit_card_int)
credit_card_cor <- cov2cor(credit_card_cov)
plot_pairs_int(credit_card_int,
                  corr = credit_card_cor,
                  labels = colnames(credit_card_int))

# Alternatively, highlight outliers in the scatter plot and use the robust correlation matrix
# Obtain reweighted IMCD estimates using farness cutoff
credit_card_IMCD <- IMCD(credit_card_int, 
                         m = floor(nrow(credit_card_int)*0.75), 
                         cutoff = "farness", 
                         cutoff_lvl = 0.9)

# Detect outliers using farness cutoff
credit_card_outliers <- int_outliers(credit_card_IMCD$robust_dist, 
                                     cutoff = "farness", 
                                     cutoff_lvl = 0.9)

outliers_colors <- rep('gray50',credit_card_int@NObs)
names(outliers_colors) <- rownames(credit_card_int)
outliers_colors[credit_card_outliers$outliers_names] = 'red'

plot_pairs_int(credit_card_int, 
                  corr = cov2cor(credit_card_IMCD$cov_IMCD), 
                  palette = outliers_colors,
                  labels = colnames(credit_card_int),
                  type = "rectangles",
                  is_outlier = credit_card_outliers$is_outlier)

Radar plot of Shapley values for interval-valued data.

Description

Radar plot of Shapley values for interval-valued data.

Usage

plot_radar_int_Shapley(shapley, palette = NULL, sort.obs = FALSE)

Arguments

Value

Returns a radar plot that displays the Shapley values (int_Shapley) for each observation.

Examples

# Create intData object
data(creditcard)
credit_card_int <- creditcard$intData

# Estimate the mean and covariance matrix
credit_card_IMCD <- IMCD(credit_card_int, 
                         m = floor(nrow(credit_card_int)*0.75), 
                         cutoff = "farness", 
                         cutoff_lvl = 0.9)

# Detect outliers using farness cutoff
credit_card_outliers <- int_outliers(credit_card_IMCD$robust_dist, 
                                     cutoff = "farness", 
                                     cutoff_lvl = 0.9)

# Compute Shapley values
credit_card_shapley <- int_Shapley(credit_card_int, 
                                   mean_c = credit_card_IMCD$mean_IMCD_c,
                                   mean_r = credit_card_IMCD$mean_IMCD_r, 
                                   cov = credit_card_IMCD$cov_IMCD)

# colors
outliers_colors <- rep('black',credit_card_int@NObs)
names(outliers_colors) <- rownames(credit_card_int)
outliers_colors[credit_card_outliers$outliers_names] = '#009de0'

plot_radar_int_Shapley(credit_card_shapley, palette = outliers_colors)

Scatter Plot for Interval-valued Data

Description

Create a scatter plot for interval-valued symbolic data, visualizing the symbolic data as rectangles or crosses, with the first two variables on the x and y axes. The function allows customization of colors, fill colors, and outlier representation.

Usage

plot_scatter_int(
  data,
  type = c("rectangles", "crosses", "crosses2"),
  palette = rainbow(nrow(data)),
  fill_col = "gray50",
  is_outlier = NULL,
  ...
)

Arguments

Value

A scatter plot is drawn in the graphic window. The scatter plot shows the symbolic data as rectangles or crosses, with the first two variables on the x and y axes.

Examples

data(creditcard)
credit_card_int <- creditcard$intData

plot_scatter_int(credit_card_int[, c(3, 5)])

# Alternatively, highlight outliers in the scatter plot
# Compute robust distances using IMCD estimates of mean and covariance
credit_card_dist <- IMah_dist(credit_card_int)

# Detect outliers using farness cutoff
credit_card_outliers <- int_outliers(credit_card_dist, "farness", 0.9)

outliers_colors <- rep('gray50', credit_card_int@NObs)
names(outliers_colors) <- rownames(credit_card_int)
outliers_colors[credit_card_outliers$outliers_names] = 'red'

plot_scatter_int(credit_card_int[, c(3, 5)], 
            palette = outliers_colors, 
            is_outlier = credit_card_outliers$is_outlier)

Tileplot of Shapley values for interval-valued data.

Description

Tileplot of Shapley values for interval-valued data.

Usage

plot_tile_int_Shapley(
  shapley,
  outliers = NULL,
  rotate_x = TRUE,
  abbrev.var = FALSE,
  abbrev.obs = FALSE,
  sort.var = FALSE,
  sort.obs = FALSE,
  show_values = FALSE
)

Arguments

Value

Returns a tileplot that displays the Shapley values (int_Shapley) for each observation and variable. Optionally, only the outliers are highlighted in the plot.

References

Adapted from package ShapleyOutlier (https://CRAN.R-project.org/package=ShapleyOutlier).

Examples

# Create intData object
data(creditcard)
credit_card_int <- creditcard$intData

# Estimate the mean and covariance matrix
credit_card_IMCD <- IMCD(credit_card_int, 
                         m = floor(nrow(credit_card_int)*0.75), 
                         cutoff = "farness", 
                         cutoff_lvl = 0.9)

# Detect outliers using farness cutoff
credit_card_outliers <- int_outliers(credit_card_IMCD$robust_dist, 
                                     cutoff = "farness", 
                                     cutoff_lvl = 0.9)

# Compute Shapley values
credit_card_shapley <- int_Shapley(credit_card_int, 
                                   mean_c = credit_card_IMCD$mean_IMCD_c, 
                                   mean_r = credit_card_IMCD$mean_IMCD_r, 
                                   cov = credit_card_IMCD$cov_IMCD)

plot_tile_int_Shapley(credit_card_shapley, 
                     outliers = credit_card_outliers, 
                     sort.var = TRUE, 
                     sort.obs = TRUE)

Print Method for Summary intData

Description

Print Method for Summary intData

Usage

## S4 method for signature 'summaryintData'
print(x, ...)

Arguments

Value

The object itself, returned invisibly. Called for its side effects (printing).

Row Bind for intData

Description

Combine multiple intData objects by rows.

Usage

rbind(..., deparse.level = 1)

## S4 method for signature 'intData'
rbind(..., deparse.level = 1)

Arguments

Value

An intData object with rows combined from the input intData objects.

Row.Names Method for intData

Description

Row.Names Method for intData

Usage

## S4 method for signature 'intData'
row.names(x)

Arguments

Value

A character vector of row names.

Row Names Method for intData

Description

Row Names Method for intData

Usage

## S4 method for signature 'intData'
rownames(x)

Arguments

Value

A character vector of row names.

Safely invert a covariance matrix with Moore-Penrose generalized inverse fallback

Description

Computes a numerically stable inverse of a covariance matrix. The function:

1. Attempts standard inversion via solve().

2. If the matrix is ill-conditioned, falls back to a Moore-Penrose generalized inverse.

Usage

safe_solve_cov(cov, verbose = TRUE)

Arguments

Details

When the covariance matrix is singular or nearly singular, direct inversion may fail or produce unstable results. This function ensures robustness by using Moore-Penrose generalized inverse (via MASS::ginv()).

The pseudo-inverse effectively ignores directions with negligible variance, which may slightly affect interpretations (e.g., Mahalanobis distances or Shapley values).

Value

A matrix representing:

• The inverse of cov if well-conditioned

• A Moore–Penrose generalized inverse if inversion fails

Examples

set.seed(1)

# Example where inversion fails
X <- matrix(rnorm(20), ncol = 5)
cov_X <- cov(X)

#solve(cov_X)  # Standard inversion fails
safe_solve_cov(cov_X) # Returns a generalized inverse

# Example where inversion does not fail
Y <- cbind(rnorm(20), rnorm(20, mean=1, sd=2))
cov_Y <- cov(Y)
solve(cov_Y)  # Standard inversion succeeds
safe_solve_cov(cov_Y)  # Returns same result

Show Method for intData

Description

Show Method for intData

Show Method for Summary intData

Usage

## S4 method for signature 'intData'
show(object)

## S4 method for signature 'summaryintData'
show(object)

Arguments

Value

The object itself, returned invisibly. Called for its side effects (printing).

Obtain unweighted estimates for data with <= 600 observations

Description

Obtain unweighted estimates for data with <= 600 observations

Usage

smallIMCD(m, data)

Arguments

Value

A list of estimated barycenter and symbolic covariance matrix

Spotify Tracks Dataset

Description

This dataset contains interval data of Spotify tracks' audio features, including min-max values and trimmed intervals, as well as the microdata. It is composed of 11 audio features: duration, danceability, energy, loudness, speechiness, acousticness, instrumentalness, liveness, valence, tempo, and popularity. The aggregation of the microdata was done by track genre.

Usage

data(spotify_tracks)

Format

A list with the following components:

microdata

A data frame with 81033 rows and 20 columns. It contains the microdata, with individual measurements of each variable for all observations.

microdata_transformed

A data frame with 81033 rows and 20 columns. It contains the transformed microdata, with individual measurements of each variable for all observations. Logarithmic transformations were applied to "loudness" and "tempo". "duration_ms" in milliseconds was converted to "duration" in minutes. "popularity" was scaled to the range ⁠[0,1]⁠.

intData_minmax

An intData object with 111 interval-valued observations and 11 variables, constructed using min-max aggregation based on the transformed microdata.

intData_trimmed

An intData object with 111 interval-valued observations and 11 variables, constructed using trimmed aggregation (⁠1\%⁠ trimming) based on the transformed microdata.

References

This data was retrieved from Kaggle (DOI:10.34740/KAGGLE/DSV/4372070; Spotify Tracks Dataset by Maharshi Pandya).

Examples

data(spotify_tracks)
head(spotify_tracks$intData_minmax)
head(spotify_tracks$intData_trimmed)
head(spotify_tracks$microdata)
head(spotify_tracks$microdata_transformed)

Iterate through C-step

Description

Iterate through C-step

Usage

step_it(z, m, data, it = 0)

Arguments

Value

A list of z, covariance, barycenter and robust distances

Summary Method for intData

Description

Summary Method for intData

Usage

## S4 method for signature 'intData'
summary(object)

Arguments

Value

An object of class summaryintData.

Summary Interval Data Class

Description

A class to represent the summary of interval data.

Slots

Centersumar

A table summarizing the centers.

Rngsumar

A table summarizing the ranges.

Tail Method for intData

Description

Returns the last n rows of an intData object.

Usage

## S4 method for signature 'intData'
tail(x, n = min(nrow(x), 6L))

Arguments

Value

A subset of the intData object.