Class `intData` examples"
In AIDA: Analysis of Interval DAta

knitr::opts_chunk$set(
    collapse = TRUE,
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library(AIDA)

This vignette provides examples of how to use the intData class and related functions for handling interval-valued data. The intData class is designed to represent interval-valued data. The examples included here demonstrate how to create intData objects, compute summary statistics, and visualize interval-valued data using the plot_pairs_int() function. The dataset used in these examples is the Credit Card dataset, which is available in the package and can be loaded using data("creditcard"). The examples illustrate the basic functionalities of the intData class and how to work with interval-valued data in R.

For more details on the interval-valued data framework implemented in the package, please refer to @oliveira2025.

Credit Card Dataset

This dataset contains interval data of credit card expenses (@billard_diday_2006), including min-max values, centers and ranges, centers and logranges, and microdata. The aggregation of the microdata was done by person-month, resulting in $n=36$ observations. It is composed of $5$ variables:

Food
Social
Travel
Gas
Clothes

The creditcard dataset includes 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 interval 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.

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

There are different ways to create an intData object from the dataset, depending on the assumptions about the latent distribution of the microdata. In this example, we will create an intData object using the min_max component of the dataset, assuming a continuous uniform distribution for the latent variables, which corresponds to the symmetric and i.d. case with $\delta = 1/12$. 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])

# Check the parameters of the latent distribution                          
credit_card_int_unif@LatentParam

Since the microdata are available, we can take a closer look at the distribution of the latent variables. The get_latent_var() function can be used to obtain the latent variables observed values, by normalizing the microdata into the $[-1,1]$ interval. In this example, we will use the min_max component of the dataset to normalize the microdata. The aggregation criterion is by month and name, so we will create a new variable that combines the name and month to use as the grouping variable. We can then visualize the distribution of the latent variables using histograms and density plots.

credit_agrby <- factor(paste(CreditCard_microdata$Name,CreditCard_microdata$Month, sep = "_"))
credit_card_U <- get_latent_var(CreditCard_microdata[,3:7], CreditCard_min_max, credit_agrby,
                                rownames(CreditCard_min_max), Seq = "LbUb_VarbyVar")

oldpar <- par(no.readonly = TRUE)
par(mfrow=c(1,5), mar=c(2, 2, 2, 1))
for (i in 1:5){
    hist(credit_card_U[,i], xlab = NULL, ylab = NULL, 
            main = colnames(credit_card_U)[i], probability = TRUE)
    lines(density(credit_card_U[,i], na.rm = TRUE), col = '#009de0', lwd = 2)
}
par(oldpar)

After examining the distribution of the latent variables, we can assume the distributions are approximately triangular and symmetric. Then, we can create an intData object using the min_max component of the dataset, specifying the latent distribution as "Triang".

credit_card_int_triang <- intData(CreditCard_min_max, Seq = "LbUb_VarbyVar", LatentDist = "Triang", 
                                    VarNames = colnames(CreditCard_microdata)[3:7])

# Check the parameters of the latent distribution
credit_card_int_triang@LatentParam

head(credit_card_int_triang)

The intData object contains the centers and ranges of the interval data, as well as the parameters of the latent distribution. The centers and ranges can be accessed using the @Centers and @Ranges slots, respectively, while the lower and upper bounds can be obtained using the LowerBounds() and UpperBounds() functions.

credit_card_int_triang@Centers[1:5,]
credit_card_int_triang@Ranges[1:5,]
LowerBounds(credit_card_int_triang)[1:5,]
UpperBounds(credit_card_int_triang)[1:5,]

Alternatively, we can create an intData object using the centers_ranges component of the dataset, which contains the centers and ranges of the interval data. As for the latent variables' parameters, since we have the microdata available, another option is to estimate the parameters directly based on the microdata by setting LatentCase = "General" and LatentDist = "KDE" to use a kernel density estimation for the latent distribution. In this case, it is necessary to specify the Umicro argument, which contains the normalized microdata values.

credit_card_int_KDE <- intData(CreditCard_CR, Seq = "AllCen_AllRng", 
                                VarNames = colnames(CreditCard_microdata)[3:7], 
                                LatentCase = "General", LatentDist = "KDE", Umicro = credit_card_U)

# Check the parameters of the latent distribution
credit_card_int_KDE@LatentParam

In the majority of the cases, the macrodata has to be constructed from the microdata. The micro2intData() function can be used to create an intData object from the microdata, by aggregating the microdata according to a specified criterion. In this example, we will aggregate the microdata by month and name, using the same grouping variable created earlier. We will also specify the latent distribution as "General" to estimate the parameters based on the microdata. If the LatentCase argument is omitted, it assumes the latent distribution is uniform (i.d. and symmetric), which is the default setting for the intData class. It is also possible to specify the latent distribution, as seen in the previous examples.

credit_card_int_agr <- micro2intData(CreditCard_microdata[,3:7], credit_agrby, LatentCase = "General")

# Check the parameters of the latent distribution
credit_card_int_agr@LatentParam

head(credit_card_int_agr)

Now that we have created the intData object, we can, for instance, compute the sample barycenter and the symbolic covariance and correlation matrices of the interval data using the int_mean(), int_cov(), and int_cor() functions, respectively.

credit_card_mean <- int_mean(credit_card_int_agr)
credit_card_cov <- int_cov(credit_card_int_agr)
credit_card_cor <- int_cor(credit_card_int_agr)

# Check the sample barycenter
credit_card_mean

# Check the covariance matrix
credit_card_cov

Finally, we can visualize the interval data using the plot_pairs_int() function, which creates a pairs plot for interval-valued data. The lower triangular shows scatter plots of the variables, while the upper triangular shows the interval correlation matrix.

plot_pairs_int(credit_card_int_agr, type = "rectangles", 
                    corr = credit_card_cor, labels = colnames(credit_card_int_agr))

Considering the Mallows distance is a special case of the Interval-Mahalanobis distance when the covariance matrix is the identity matrix, we can compute both distances and check that they are equal using the Mallows_dist() and IMah_dist() functions, respectively. We can also compute the pairwise Mallows distances using the IMah_dist_pairs() function, and applying the same logic.

creditcard_Mallows <- Mallows_dist(credit_card_int_agr)
creditcard_Mallows2 <- IMah_dist(credit_card_int_agr, 
                                 cov = diag(credit_card_int_agr@NIVar), 
                                 mean = credit_card_mean)

# Check that the Mallows distance is equal to the Interval-Mahalanobis distance 
# when the covariance matrix is the identity matrix
all.equal(creditcard_Mallows, creditcard_Mallows2)

creditcard_pairs <- IMah_dist_pairs(credit_card_int_agr, cov = diag(credit_card_int_agr@NIVar))