Package {AIDA}

Type:	Package
Title:	Analysis of Interval DAta
Version:	0.1.5
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. (2026) <doi:10.48550/arXiv.2604.26769>.
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
RoxygenNote:	7.3.3
LazyData:	true
LazyDataCompression:	xz
VignetteBuilder:	knitr
Language:	en-US
Imports:	ggplot2, ggrepel, CerioliOutlierDetection, cellWise, geigen, kde1d, plotly, robustbase, MASS, assertthat, methods
Depends:	R (≥ 3.6)
Suggests:	knitr, rmarkdown, testthat (≥ 3.0.0), corrplot
NeedsCompilation:	no
Packaged:	2026-05-07 22:19:07 UTC; catar
Author:	Catarina P. Loureiro [aut, cre]
Maintainer:	Catarina P. Loureiro <catarinapadrela@tecnico.ulisboa.pt>
Repository:	CRAN
Date/Publication:	2026-05-12 19:30:02 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

credit_card_IMCD <- IMCD(credit_card_int, floor(0.75*credit_card_int@NObs), "farness", 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 is defined according to the LatentCase:

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

  d_{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:

  d_{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)\\ +\dfrac{\mathbb{E}(U)}{2}(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)+\dfrac{\mathbb{E}(U)}{2}(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C),

  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_{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)\\ +\dfrac{1}{2}(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}\boldsymbol{\Sigma}_{B}^{-1}\boldsymbol{\Psi}(\boldsymbol{r}-\boldsymbol{\mu}_R)+\dfrac{1}{2}(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\boldsymbol{\Psi}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C),

  where:

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

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

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{ii}=\mathbb{E}(U_i^2), i,j=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

z <- rep(1, nrow(credit_card_int))
credit_card_dist<-IMah_dist(credit_card_int,z)

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 is defined according to the LatentCase:

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

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

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

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

  d_{IMah}(\boldsymbol{x}_i,\boldsymbol{x}_j)^2=(\boldsymbol{c}_i-\boldsymbol{c}_j)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}_i-\boldsymbol{c}_j)+\delta(\boldsymbol{r}_i-\boldsymbol{r}_j)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}_i-\boldsymbol{r}_j)\\ +\dfrac{\mathbb{E}(U)}{2}(\boldsymbol{c}_i-\boldsymbol{c}_j)^\top\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}_i-\boldsymbol{r}_j)+\dfrac{\mathbb{E}(U)}{2}(\boldsymbol{r}_i-\boldsymbol{r}_j)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}_i-\boldsymbol{c}_j),

  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_{IMah}(\boldsymbol{x}_i,\boldsymbol{x}_j)^2=(\boldsymbol{c}_i-\boldsymbol{c}_j)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}_i-\boldsymbol{c}_j)+\dfrac{1}{4}(\boldsymbol{r}_i-\boldsymbol{r}_j)^{\top}\left(\boldsymbol{\mathfrak{E}}_{UU}\bullet\boldsymbol{\Sigma}_{B}^{-1}\right)(\boldsymbol{r}_i-\boldsymbol{r}_j)\\ +\dfrac{1}{2}(\boldsymbol{c}_i-\boldsymbol{c}_j)^{\top}\boldsymbol{\Sigma}_{B}^{-1}\boldsymbol{\Psi}(\boldsymbol{r}_i-\boldsymbol{r}_j)+\dfrac{1}{2}(\boldsymbol{r}_i-\boldsymbol{r}_j)^{\top}\boldsymbol{\Psi}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}_i-\boldsymbol{c}_j),

  where:

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

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

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{ii}=\mathbb{E}(U_i^2), i,j=1,\dots,p,

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

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)

Kullback-Leibler (KL) Divergence

Description

Computes the Kullback-Leibler (KL) divergence between an estimated covariance matrix and the ground truth. Assumes normal multivariate distributions.

Usage

KL_divergence(est_cov, ground_truth_cov)

Arguments

Details

The KL divergence between two p-dimensional Gaussians \mathcal{N}(\boldsymbol{\mu}, \hat{\boldsymbol{\Sigma}}) and \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma}) is given by:

\dfrac{1}{2}\left(\text{tr}(\boldsymbol{\Sigma}^{-1}\hat{\boldsymbol{\Sigma}}) + \log\left(\dfrac{\det(\boldsymbol{\Sigma})}{\det(\hat{\boldsymbol{\Sigma}})}\right) - p\right),

where \hat{\boldsymbol{\Sigma}} and \boldsymbol{\Sigma} are the estimated and ground truth covariance matrices, respectively.

Value

KL divergence between the two matrices.

References

Yufeng Zhang, Wanwei Liu, Zhenbang Chen, Ji Wang, and Kenli Li. On the properties of Kullback-Leibler divergence between multivariate gaussian distributions, 2023. https://arxiv.org/abs/2102.05485

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_{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_{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_{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.

Symbolic Biplot for Interval-valued Data

Description

Create a biplot 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

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

Arguments

Value

A biplot is drawn in the graphic window. The biplot 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

SYMB.biplot(credit_card_int[,c(3,5)])

# Highlight outliers in the biplot
credit_card_IMCD <- IMCD(credit_card_int, floor(0.75*credit_card_int@NObs), "farness", 0.9)
credit_card_outliers <- int_outliers(credit_card_IMCD$robust_dist, "farness", 0.9)
outliers_colors<-rep('gray50',credit_card_int@NObs)
names(outliers_colors)<-rownames(credit_card_int)
outliers_colors[credit_card_outliers$outliers_names] = 'red'
SYMB.biplot(credit_card_int[,c(3,5)], palette = outliers_colors, 
            is_outlier = credit_card_outliers$is_outlier)

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

SYMB.pairs.panels(
  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

credit_card_cov<-int_cov(credit_card_int)
credit_card_cor<-cov2cor(credit_card_cov)
SYMB.pairs.panels(credit_card_int,corr=credit_card_cor,labels=colnames(credit_card_int))

# Highlight outliers in the biplot
credit_card_IMCD <- IMCD(credit_card_int, floor(0.75*credit_card_int@NObs), "farness", 0.9)
credit_card_outliers <- int_outliers(credit_card_IMCD$robust_dist, "farness", 0.9)
outliers_colors<-rep('gray50',credit_card_int@NObs)
names(outliers_colors)<-rownames(credit_card_int)
outliers_colors[credit_card_outliers$outliers_names] = 'red'
SYMB.pairs.panels(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)

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.

Angle Error

Description

Computes the angle error between eigenvalues of the estimated covariance matrix and of the ground truth covariance matrix.

Usage

angle_error(est_cov, ground_truth_cov)

Arguments

Details

The angle error is given by:

1-\dfrac{\hat{\boldsymbol{a}}^\top\boldsymbol{a}}{\sqrt{\hat{\boldsymbol{a}}^\top\hat{\boldsymbol{a}}}\sqrt{\boldsymbol{a}^\top\boldsymbol{a}}},

where \hat{\boldsymbol{a}} and \boldsymbol{a} are the eigenvalues of the estimated and ground truth covariance matrices, respectively.

Value

Angle error between eigenvalues.

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}]_{ij}=\mathcal{E}(U_i,U_j) , i\neq j, with \mathcal{E}(U_i,U_j)=\int_0^1 F_{U_i}^{-1}(t) F_{U_j}^{-1}(t) \, dt

• [\boldsymbol{\mathfrak{E}}_{UU}]_{ii}=\mathbb{E}(U_i^2), i,j=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
z <- rep(1, nrow(credit_card_int))
credit_card_dist<-IMah_dist(credit_card_int,z)

credit_card_farness <- farness(credit_card_dist, 0.9)

Relative Frobenius Error

Description

Computes the relative Frobenius error between an estimated covariance matrix and the ground truth.

Usage

frobenius_error(est_cov, ground_truth_cov)

Arguments

Details

The relative Frobenius error is given by:

\dfrac{\|\boldsymbol{A} - \boldsymbol{B}\|_F}{\|\boldsymbol{B}\|_F}=\dfrac{\sqrt{\sum\limits_{i=1}^{p}\sum\limits_{j=1}^{p}|[\boldsymbol{A}]_{ij}-[\boldsymbol{B}]_{ij}|^2}}{\sqrt{\sum\limits_{i=1}^{p}\sum\limits_{j=1}^{p}|[\boldsymbol{B}]_{ij}|^2}},

where \boldsymbol{A} and \boldsymbol{B} are the estimated and ground truth covariance matrices, respectively.

Value

Frobenius error between the two matrices.

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}]_{ij}=\mathcal{E}(U_i,U_j), i\neq j, with \mathcal{E}(U_i,U_j)=\int_0^1 F_{U_i}^{-1}(t) F_{U_j}^{-1}(t) \, dt, and [\boldsymbol{\mathfrak{E}}_{UU}]_{ii}=\mathbb{E}(U_i^2), i,j=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
credit_agrby<-paste(CreditCard_microdata$Name,CreditCard_microdata$Month,sep = "_")
credit_card_U<-get_latent_var(CreditCard_microdata[,3:7], CreditCard_min_max, credit_agrby, 
                              agrlevels = row.names(CreditCard_min_max), Seq="LbUb_VarbyVar")
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_{ij}, 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_{ij}=C_j+U_{ij}\dfrac{R_j}{2},\quad j=1,\dots,p,\ i=1,\dots,m_j

the microdata with U_{ij} 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
credit_agrby<-paste(CreditCard_microdata$Name,CreditCard_microdata$Month,sep = "_")
credit_card_U<-get_latent_var(CreditCard_microdata[,3:7], CreditCard_min_max, 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:

min_max

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$min_max)
head(intCars$intData)

Interval Data Constructor

Description

Constructs an interval data object.

Usage

intData(
  Data,
  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(Data),
  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).

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:

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

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

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

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 indicating the number of micro units.

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).

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}]_{ij}=\mathcal{E}(U_i,U_j), i\neq j, with \mathcal{E}(U_i,U_j)=\int_0^1 F_{U_i}^{-1}(t) F_{U_j}^{-1}(t) \, dt,

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{ii}=\mathbb{E}(U_i^2), i,j=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

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:

  \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\\ +\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+\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,

  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{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{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]\\ +\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}+\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,

  where:

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

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

  • [\boldsymbol{\mathfrak{E}}_{UU}]_{ii}=\mathbb{E}(U_i^2), i,j=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

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

credit_card_IMCD <- IMCD(credit_card_int, floor(0.75*credit_card_int@NObs), "farness", 0.9)
credit_card_outliers <- int_outliers(credit_card_IMCD$robust_dist, "farness", 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(
  MicDtDF,
  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$microdata
credit_agrby<-factor(paste(CreditCard_microdata$Name,CreditCard_microdata$Month,sep = "_"))
credit_agr<-micro2intData(CreditCard_microdata[,3:7],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.

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 = TRUE,
  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

#Estimate the mean and covariance matrix
credit_card_IMCD<-IMCD(credit_card_int, floor(nrow(credit_card_int)*0.75), "farness", 0.9)
credit_card_outliers <- int_outliers(credit_card_IMCD$robust_dist, 
                                           p=credit_card_int@NIVar, cutoff_lvl = 0.9)

#Plot Distance-Distance plot
class_dist <- IMah_dist(credit_card_int, z=rep(1,credit_card_int@NObs))
class_outliers <- int_outliers(class_dist,cutoff = "adjbox",p=p,cutoff_lvl = 1.5)
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$is_outlier] <- "Inlier"
plot_dist_dist(class_dist, class_outliers$cutoff_value[2], "1.5 adjusted boxplot",
              credit_card_IMCD$robust_dist, credit_card_outliers$cutoff_value, "0.9 farness",
              color_class = credit_card_is_outliers, palette = c("grey50", "red"))

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

#Estimate the mean and covariance matrix
credit_card_IMCD<-IMCD(credit_card_int, floor(nrow(credit_card_int)*0.75), "farness", 0.9)
credit_card_outliers <- int_outliers(credit_card_IMCD$robust_dist, 
                                           p=credit_card_int@NIVar, cutoff_lvl = 0.9)
credit_card_is_outliers <- as.character(credit_card_outliers$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_IMCD$robust_dist,
                   cutoff = credit_card_outliers$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"))

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.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.

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, available at https://www.kaggle.com/datasets/maharshipandya/-spotify-tracks-dataset.

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.