# Introduction to dataSDA

### Datasets and Basic Statistics for Symbolic Data Analysis

#### Po-Wei Chen, Chun-houh Chen and Han-Ming Wu*

#### March 14, 2026

# 1 Introduction

The `dataSDA` package (v0.3.0) gathers various symbolic data tailored to different research themes and provides a comprehensive set of functions for reading, writing, converting, and analyzing symbolic data. The package is available on CRAN at https://CRAN.R-project.org/package=dataSDA and on GitHub at https://github.com/hanmingwu1103/dataSDA.

The package includes 114 datasets spanning seven types of symbolic data. Each dataset name uses a suffix that indicates its type:

Type | Suffix | Datasets | Description
--- | --- | --- | ---
Interval | `.int`, `.iGAP`, `.int.mm` | 57 | Interval-valued data in RSDA (54), iGAP (2), and min-max (1) formats
Histogram | `.hist` | 25 | Histogram-valued distributional data
Mixed | `.mix` | 11 | Datasets combining interval and categorical variables
Interval Time Series | `.its` | 9 | Interval-valued time series data
Modal | `.modal` | 7 | Modal multi-valued symbolic data
Distributional | `.distr` | 3 | Distributional symbolic data
Other | ??? | 2 | Auxiliary datasets (`bank_rates`, `hierarchy`)
Total |  | 114 |

The package provides functions organized into the following categories:

Category | Functions | Count
--- | --- | ---
Format detection & conversion | `int_detect_format`, `int_list_conversions`, `int_convert_format`, `RSDA_to_MM`, `iGAP_to_MM`, `SODAS_to_MM`, `MM_to_iGAP`, `RSDA_to_iGAP`, `SODAS_to_iGAP`, `MM_to_RSDA`, `iGAP_to_RSDA` | 11
Core statistics | `int_mean`, `int_var`, `int_cov`, `int_cor` | 4
Geometric properties | `int_width`, `int_radius`, `int_center`, `int_midrange`, `int_overlap`, `int_containment` | 6
Position & scale | `int_median`, `int_quantile`, `int_range`, `int_iqr`, `int_mad`, `int_mode` | 6
Robust statistics | `int_trimmed_mean`, `int_winsorized_mean`, `int_trimmed_var`, `int_winsorized_var` | 4
Distribution shape | `int_skewness`, `int_kurtosis`, `int_symmetry`, `int_tailedness` | 4
Similarity measures | `int_jaccard`, `int_dice`, `int_cosine`, `int_overlap_coefficient`, `int_tanimoto`, `int_similarity_matrix` | 6
Uncertainty & variability | `int_entropy`, `int_cv`, `int_dispersion`, `int_imprecision`, `int_granularity`, `int_uniformity`, `int_information_content` | 7
Distance measures | `int_dist`, `int_dist_matrix`, `int_pairwise_dist`, `int_dist_all` | 4
Histogram statistics | `hist_mean`, `hist_var`, `hist_cov`, `hist_cor` | 4
Utilities | `clean_colnames`, `RSDA_format`, `set_variable_format`, `read_symbolic_csv`, `write_symbolic_csv` | 5

# 2 Data Formats and Conversion

## 2.1 Interval data formats overview

The `dataSDA` package works with three primary formats for interval-valued data:

- RSDA format: `symbolic_tbl` objects where intervals are encoded as complex numbers (`min + max*i`). Used by the `RSDA` package.
- MM format: Standard data frames with paired `_min` / `_max` columns for each variable.
- iGAP format: Data frames where each interval is a comma-separated string (e.g., `"2.5,4.0"`).

    data(mushroom.int)
    head(mushroom.int, 3)
    class(mushroom.int)

    data(abalone.int)
    head(abalone.int, 3)
    class(abalone.int)

    data(abalone.iGAP)
    head(abalone.iGAP, 3)
    class(abalone.iGAP)

The `int_detect_format()` function automatically identifies the format of a dataset:

    int_detect_format(mushroom.int)
    int_detect_format(abalone.int)
    int_detect_format(abalone.iGAP)

Use `int_list_conversions()` to see all available format conversion paths:

    int_list_conversions()

## 2.2 Unified format conversion

The `int_convert_format()` function provides a unified interface for converting between formats. It auto-detects the source format and applies the appropriate conversion:

    # RSDA to MM
    mushroom.MM <- int_convert_format(mushroom.int, to = "MM")
    head(mushroom.MM, 3)

    # iGAP to MM
    abalone.MM <- int_convert_format(abalone.iGAP, to = "MM")
    head(abalone.MM, 3)

    # iGAP to RSDA
    data(face.iGAP)
    face.RSDA <- int_convert_format(face.iGAP, to = "RSDA")
    head(face.RSDA, 3)

## 2.3 Direct conversion functions

For explicit control, direct conversion functions are available:

    # RSDA to MM
    mushroom.MM <- RSDA_to_MM(mushroom.int, RSDA = TRUE)
    head(mushroom.MM, 3)

    # MM to iGAP
    mushroom.iGAP <- MM_to_iGAP(mushroom.MM)
    head(mushroom.iGAP, 3)

    # iGAP to MM
    data(face.iGAP)
    face.MM <- iGAP_to_MM(face.iGAP, location = 1:6)
    head(face.MM, 3)

    # MM to RSDA
    face.RSDA <- MM_to_RSDA(face.MM)
    head(face.RSDA, 3)
    class(face.RSDA)

    # iGAP to RSDA (direct, one-step)
    abalone.RSDA <- iGAP_to_RSDA(abalone.iGAP, location = 1:7)
    head(abalone.RSDA, 3)
    class(abalone.RSDA)

    # RSDA to iGAP
    mushroom.iGAP2 <- RSDA_to_iGAP(mushroom.int)
    head(mushroom.iGAP2, 3)

The `SODAS_to_MM()` and `SODAS_to_iGAP()` functions convert SODAS XML files but require an XML file path and are not demonstrated here.

## 2.4 Legacy workflow: creating symbolic_tbl from raw data

The traditional workflow for converting a raw data frame into the `symbolic_tbl` class used by `RSDA` involves several steps. We illustrate with the `mushroom` dataset, which contains 23 species described by 3 interval-valued variables and 2 categorical variables.

    data(mushroom.int.mm)
    head(mushroom.int.mm, 3)

First, use `set_variable_format()` to create pseudo-variables for each category using one-hot encoding:

    mushroom_set <- set_variable_format(data = mushroom.int.mm, location = 8,
                                        var = "Species")
    head(mushroom_set, 3)

Next, apply `RSDA_format()` to prefix each variable with `$I` (interval) or `$S` (set) tags:

    mushroom_tmp <- RSDA_format(data = mushroom_set,
                                sym_type1 = c("I", "I", "I", "S"),
                                location = c(25, 27, 29, 31),
                                sym_type2 = c("S"),
                                var = c("Species"))
    head(mushroom_tmp, 3)

Clean up variable names with `clean_colnames()` and write to CSV with `write_symbolic_csv()`:

    mushroom_clean <- clean_colnames(data = mushroom_tmp)
    head(mushroom_clean, 3)

    write_symbolic_csv(mushroom_clean, file = "mushroom_interval.csv")
    mushroom_int <- read_symbolic_csv(file = "mushroom_interval.csv")
    head(mushroom_int, 3)
    class(mushroom_int)

Note: The `MM_to_RSDA()` function provides a simpler one-step alternative to this workflow.

## 2.5 Histogram data: the MatH class

Histogram-valued data uses the `MatH` class from the `HistDAWass` package. The built-in `BLOOD` dataset is a `MatH` object with 14 patient groups and 3 distributional variables:

    BLOOD[1:3, 1:2]

Below we illustrate constructing a `MatH` object from raw histogram data:

    A1 <- c(50, 60, 70, 80, 90, 100, 110, 120)
    B1 <- c(0.00, 0.02, 0.08, 0.32, 0.62, 0.86, 0.92, 1.00)
    A2 <- c(50, 60, 70, 80, 90, 100, 110, 120)
    B2 <- c(0.00, 0.05, 0.12, 0.42, 0.68, 0.88, 0.94, 1.00)
    A3 <- c(50, 60, 70, 80, 90, 100, 110, 120)
    B3 <- c(0.00, 0.03, 0.24, 0.36, 0.75, 0.85, 0.98, 1.00)

    ListOfWeight <- list(
      distributionH(A1, B1),
      distributionH(A2, B2),
      distributionH(A3, B3)
    )

    Weight <- methods::new("MatH",
                           nrows = 3, ncols = 1, ListOfDist = ListOfWeight,
                           names.rows = c("20s", "30s", "40s"),
                           names.cols = c("weight"), by.row = FALSE)
    Weight

# 3 The Eight Interval Methods

Many `dataSDA` functions accept a `method` parameter that determines how interval boundaries are used in computations. The eight available methods (Wu, Kao and Chen, 2020) are:

Method | Name | Description
--- | --- | ---
CM | Center Method | Uses the midpoint (center) of each interval
VM | Vertices Method | Uses both endpoints of the intervals
QM | Quantile Method | Uses a quantile-based representation
SE | Stacked Endpoints Method | Stacks the lower and upper values of an interval
FV | Fitted Values Method | Fits a linear regression model
EJD | Empirical Joint Density Method | Joint distribution of lower and upper bounds
GQ | Symbolic Covariance Method | Alternative expression of the symbolic sample variance
SPT | Total Sum of Products | Decomposition of the SPT

Quick demonstration:

    data(mushroom.int)
    var_name <- c("Stipe.Length", "Stipe.Thickness")
    int_mean(mushroom.int, var_name, method = c("CM", "FV", "EJD"))

# 4 Descriptive Statistics for Interval-Valued Data

The core statistical functions `int_mean`, `int_var`, `int_cov`, and `int_cor` compute descriptive statistics for interval-valued data across any combination of the eight methods.

## 4.1 Mean and variance

We compute the mean and variance of `Pileus.Cap.Width` and `Stipe.Length` in the `mushroom.int` dataset using all eight interval methods.

    data(mushroom.int)
    var_name <- c("Pileus.Cap.Width", "Stipe.Length")
    method <- c("CM", "VM", "QM", "SE", "FV", "EJD", "GQ", "SPT")

    mean_mat <- int_mean(mushroom.int, var_name, method)
    mean_mat

    var_mat <- int_var(mushroom.int, var_name, method)
    var_mat

The means are identical across most methods because methods other than FV operate on the same midpoint or boundary values; only FV (which regresses upper bounds on lower bounds) produces a different mean. In contrast, the variances differ substantially across methods, reflecting how each method weighs interval width and position.

    cols <- c("#4E79A7", "#F28E2B")
    par(mfrow = c(2, 1), mar = c(5, 4, 3, 6), las = 2, xpd = TRUE)

    # --- Mean across eight methods ---
    bp <- barplot(t(mean_mat), beside = TRUE, col = cols,
                  main = "Interval Mean by Method (mushroom.int)",
                  ylab = "Mean",
                  ylim = c(0, max(mean_mat) * 1.25))
    legend("topright", inset = c(-0.18, 0),
           legend = colnames(mean_mat), fill = cols, bty = "n", cex = 0.85)

    # --- Variance across eight methods ---
    bp <- barplot(t(var_mat), beside = TRUE, col = cols,
                  main = "Interval Variance by Method (mushroom.int)",
                  ylab = "Variance",
                  ylim = c(0, max(var_mat) * 1.25))
    legend("topright", inset = c(-0.18, 0),
           legend = colnames(var_mat), fill = cols, bty = "n", cex = 0.85)

## 4.2 Covariance and correlation

We compute the covariance and correlation between `Pileus.Cap.Width` and `Stipe.Length` across all eight methods. Note that EJD, GQ, and SPT methods require character variable names (not numeric indices).

    cov_list <- int_cov(mushroom.int, "Pileus.Cap.Width", "Stipe.Length", method)
    cor_list <- int_cor(mushroom.int, "Pileus.Cap.Width", "Stipe.Length", method)

    # Collect scalar values into named vectors for display and plotting
    cov_vec <- sapply(cov_list, function(x) x[1, 1])
    cor_vec <- sapply(cor_list, function(x) x[1, 1])

    data.frame(Method = names(cov_vec), Covariance = round(cov_vec, 4),
               Correlation = round(cor_vec, 4), row.names = NULL)

The SE method yields the largest covariance because it doubles the effective sample by stacking both endpoints, amplifying joint variation. VM produces the lowest correlation (0.36) because the vertex expansion introduces \(2^p\) combinations per observation, many of which are non-informative.

    par(mfrow = c(2, 1), mar = c(5, 4, 3, 1), las = 2)

    # --- Covariance across eight methods ---
    bar_cols <- c("#4E79A7", "#59A14F", "#F28E2B", "#E15759",
                  "#76B7B2", "#EDC948", "#B07AA1", "#FF9DA7")
    bp <- barplot(cov_vec, col = bar_cols, border = NA,
                  main = "Cov(Pileus.Cap.Width, Stipe.Length) by Method",
                  ylab = "Covariance",
                  ylim = c(0, max(cov_vec) * 1.25))
    text(bp, cov_vec, labels = round(cov_vec, 2), pos = 3, cex = 0.8)

    # --- Correlation across eight methods ---
    bp <- barplot(cor_vec, col = bar_cols, border = NA,
                  main = "Cor(Pileus.Cap.Width, Stipe.Length) by Method",
                  ylab = "Correlation",
                  ylim = c(0, 1.15))
    text(bp, cor_vec, labels = round(cor_vec, 2), pos = 3, cex = 0.8)
    abline(h = 1, lty = 2, col = "grey50")

# 5 Geometric Properties

Geometric functions characterize the shape and spatial properties of individual intervals and relationships between interval variables.

## 5.1 Width, radius, center, and midrange

    data(mushroom.int)

    # Width = upper - lower
    head(int_width(mushroom.int, "Stipe.Length"))

    # Radius = width / 2
    head(int_radius(mushroom.int, "Stipe.Length"))

    # Center = (lower + upper) / 2
    head(int_center(mushroom.int, "Stipe.Length"))

    # Midrange
    head(int_midrange(mushroom.int, "Stipe.Length"))

## 5.2 Overlap and containment

These functions measure the degree to which intervals from two variables overlap or contain each other, observation by observation:

    # Overlap between two interval variables
    head(int_overlap(mushroom.int, "Stipe.Length", "Stipe.Thickness"))

    # Containment: proportion of var_name2 contained within var_name1
    head(int_containment(mushroom.int, "Stipe.Length", "Stipe.Thickness"))

# 6 Position and Scale Measures

## 6.1 Median and quantiles

    data(mushroom.int)

    # Median (default method = "CM")
    int_median(mushroom.int, "Stipe.Length")

    # Quantiles
    int_quantile(mushroom.int, "Stipe.Length", probs = c(0.25, 0.5, 0.75))

    # Compare median across methods
    int_median(mushroom.int, "Stipe.Length", method = c("CM", "FV"))

## 6.2 Range, IQR, MAD, and mode

    # Range (max - min)
    int_range(mushroom.int, "Stipe.Length")

    # Interquartile range (Q3 - Q1)
    int_iqr(mushroom.int, "Stipe.Length")

    # Median absolute deviation
    int_mad(mushroom.int, "Stipe.Length")

    # Mode (histogram-based estimation)
    int_mode(mushroom.int, "Stipe.Length")

# 7 Robust Statistics

Robust statistics reduce the influence of outliers by trimming or winsorizing extreme values.

## 7.1 Trimmed and winsorized means

    data(mushroom.int)

    # Compare standard mean vs trimmed mean (10% trim)
    int_mean(mushroom.int, "Stipe.Length", method = "CM")
    int_trimmed_mean(mushroom.int, "Stipe.Length", trim = 0.1, method = "CM")

    # Winsorized mean: extreme values are replaced (not removed)
    int_winsorized_mean(mushroom.int, "Stipe.Length", trim = 0.1, method = "CM")

## 7.2 Trimmed and winsorized variances

    int_var(mushroom.int, "Stipe.Length", method = "CM")
    int_trimmed_var(mushroom.int, "Stipe.Length", trim = 0.1, method = "CM")
    int_winsorized_var(mushroom.int, "Stipe.Length", trim = 0.1, method = "CM")

# 8 Distribution Shape

Shape functions characterize the distribution of interval-valued data.

    data(mushroom.int)

    # Skewness: asymmetry of the distribution
    int_skewness(mushroom.int, "Stipe.Length", method = "CM")

    # Kurtosis: tail heaviness
    int_kurtosis(mushroom.int, "Stipe.Length", method = "CM")

    # Symmetry coefficient
    int_symmetry(mushroom.int, "Stipe.Length", method = "CM")

    # Tailedness (related to kurtosis)
    int_tailedness(mushroom.int, "Stipe.Length", method = "CM")

# 9 Similarity Measures

Similarity functions quantify how alike two interval variables are across all observations. Available measures include Jaccard, Dice, cosine, and overlap coefficient.

    data(mushroom.int)

    int_jaccard(mushroom.int, "Stipe.Length", "Stipe.Thickness")
    int_dice(mushroom.int, "Stipe.Length", "Stipe.Thickness")
    int_cosine(mushroom.int, "Stipe.Length", "Stipe.Thickness")
    int_overlap_coefficient(mushroom.int, "Stipe.Length", "Stipe.Thickness")

Note: `int_tanimoto()` is equivalent to `int_jaccard()` for interval-valued data:

    int_tanimoto(mushroom.int, "Stipe.Length", "Stipe.Thickness")

The `int_similarity_matrix()` function computes a pairwise similarity matrix across all interval variables:

    int_similarity_matrix(mushroom.int, method = "jaccard")

# 10 Uncertainty and Variability

These functions measure the uncertainty, variability, and information content of interval-valued data.

## 10.1 Entropy, CV, and dispersion

    data(mushroom.int)

    # Shannon entropy (higher = more uncertainty)
    int_entropy(mushroom.int, "Stipe.Length", method = "CM")

    # Coefficient of variation (SD / mean)
    int_cv(mushroom.int, "Stipe.Length", method = "CM")

    # Dispersion index
    int_dispersion(mushroom.int, "Stipe.Length", method = "CM")

## 10.2 Imprecision, granularity, uniformity, and information content

    # Imprecision: based on interval widths
    int_imprecision(mushroom.int, "Stipe.Length")

    # Granularity: variability in interval sizes
    int_granularity(mushroom.int, "Stipe.Length")

    # Uniformity: inverse of granularity (higher = more uniform)
    int_uniformity(mushroom.int, "Stipe.Length")

    # Normalized information content (between 0 and 1)
    int_information_content(mushroom.int, "Stipe.Length", method = "CM")

# 11 Distance Measures

Distance functions compute dissimilarity between observations in interval-valued datasets. Available methods include: euclidean, hausdorff, ichino, de_carvalho, and others.

We use the interval columns of `car.int` for distance examples (excluding the character `Car` column):

    data(car.int)
    car_num <- car.int[, 2:5]
    head(car_num, 3)

## 11.1 Single distance method

    # Euclidean distance between observations
    int_dist(car_num, method = "euclidean")

## 11.2 Distance matrix

    # Return as a full matrix
    dm <- int_dist_matrix(car_num, method = "hausdorff")
    dm[1:5, 1:5]

## 11.3 Pairwise distance between variables

    int_pairwise_dist(car_num, "Price", "Max_Velocity", method = "euclidean")

## 11.4 All distance methods at once

    all_dists <- int_dist_all(car_num)
    names(all_dists)

# 12 Descriptive Statistics for Histogram-Valued Data

The `hist_mean`, `hist_var`, `hist_cov`, and `hist_cor` functions compute descriptive statistics for histogram-valued data (`MatH` objects). All four functions support the same four methods: BG (Bertrand and Goupil, 2000), BD (Billard and Diday, 2006), B (Billard, 2008), and L2W (L2 Wasserstein).

## 12.1 Mean and variance

We compute the mean and variance of `Cholesterol` and `Hemoglobin` in the `BLOOD` dataset using all four methods.

    all_methods <- c("BG", "BD", "B", "L2W")
    var_names <- c("Cholesterol", "Hemoglobin")

    # Compute mean for each variable and method
    mean_mat <- sapply(all_methods, function(m) {
      sapply(var_names, function(v) hist_mean(BLOOD, v, method = m))
    })
    rownames(mean_mat) <- var_names
    mean_mat

    # Compute variance for each variable and method
    var_mat <- sapply(all_methods, function(m) {
      sapply(var_names, function(v) hist_var(BLOOD, v, method = m))
    })
    rownames(var_mat) <- var_names
    var_mat

The BG, BD, and B means are identical because they share the same first-order moment definition; only L2W (quantile-based) differs slightly. The variances, however, show large differences: BG is the largest because it includes within-histogram spread, while BD, B, and L2W progressively decrease.

    bar_cols <- c("#4E79A7", "#59A14F", "#F28E2B", "#E15759")
    par(mfrow = c(2, 2), mar = c(4, 5, 3, 1), las = 1)

    # --- Mean: Cholesterol ---
    bp <- barplot(mean_mat["Cholesterol", ], col = bar_cols, border = NA,
                  main = "Mean of Cholesterol", ylab = "Mean",
                  ylim = c(0, max(mean_mat["Cholesterol", ]) * 1.15))
    text(bp, mean_mat["Cholesterol", ],
         labels = round(mean_mat["Cholesterol", ], 2), pos = 3, cex = 0.8)

    # --- Mean: Hemoglobin ---
    bp <- barplot(mean_mat["Hemoglobin", ], col = bar_cols, border = NA,
                  main = "Mean of Hemoglobin", ylab = "Mean",
                  ylim = c(0, max(mean_mat["Hemoglobin", ]) * 1.15))
    text(bp, mean_mat["Hemoglobin", ],
         labels = round(mean_mat["Hemoglobin", ], 2), pos = 3, cex = 0.8)

    # --- Variance: Cholesterol ---
    bp <- barplot(var_mat["Cholesterol", ], col = bar_cols, border = NA,
                  main = "Variance of Cholesterol", ylab = "Variance",
                  ylim = c(0, max(var_mat["Cholesterol", ]) * 1.25))
    text(bp, var_mat["Cholesterol", ],
         labels = round(var_mat["Cholesterol", ], 1), pos = 3, cex = 0.8)

    # --- Variance: Hemoglobin ---
    bp <- barplot(var_mat["Hemoglobin", ], col = bar_cols, border = NA,
                  main = "Variance of Hemoglobin", ylab = "Variance",
                  ylim = c(0, max(var_mat["Hemoglobin", ]) * 1.25))
    text(bp, var_mat["Hemoglobin", ],
         labels = round(var_mat["Hemoglobin", ], 4), pos = 3, cex = 0.8)

## 12.2 Covariance and correlation

We compute the covariance and correlation between `Cholesterol` and `Hemoglobin` using all four methods.

    cov_vec <- sapply(all_methods, function(m)
      hist_cov(BLOOD, "Cholesterol", "Hemoglobin", method = m))
    cor_vec <- sapply(all_methods, function(m)
      hist_cor(BLOOD, "Cholesterol", "Hemoglobin", method = m))

    data.frame(Method = all_methods,
               Covariance = round(cov_vec, 4),
               Correlation = round(cor_vec, 4),
               row.names = NULL)

All four methods yield a negative association between Cholesterol and Hemoglobin. Following Irpino and Verde (2015, Eqs. 30–32), the BG, BD, and B correlations all use the Bertrand-Goupil standard deviation in the denominator, so their values are similar (around ???0.20 to ???0.22). Only L2W uses its own Wasserstein-based variance, which produces a different correlation.

    par(mfrow = c(1, 2), mar = c(4, 5, 3, 1), las = 1)

    # --- Covariance ---
    bp <- barplot(cov_vec, col = bar_cols, border = NA,
                  main = "Cov(Cholesterol, Hemoglobin)",
                  ylab = "Covariance",
                  ylim = c(min(cov_vec) * 1.35, 0))
    text(bp, cov_vec, labels = round(cov_vec, 2), pos = 1, cex = 0.8)

    # --- Correlation ---
    bp <- barplot(cor_vec, col = bar_cols, border = NA,
                  main = "Cor(Cholesterol, Hemoglobin)",
                  ylab = "Correlation",
                  ylim = c(min(cor_vec) * 1.4, 0))
    text(bp, cor_vec, labels = round(cor_vec, 2), pos = 1, cex = 0.8)
    abline(h = -1, lty = 2, col = "grey50")

# 13 Application: Benchmarking Symbolic Data Methods

This section demonstrates how `dataSDA` datasets can be used for benchmarking symbolic data analysis methods across four analytical tasks: clustering (interval and histogram), classification, and regression. Five representative datasets are selected for each task, with no overlap among the interval-data tasks.

## 13.1 Exploratory Data Analysis and Visualization

### 13.1.1 Aggregating classical data to symbolic data

The `aggregate_to_symbolic()` function converts a classical data frame into interval-valued or histogram-valued symbolic data via grouping (clustering, resampling, or a categorical variable).

    set.seed(42)
    iris_int <- aggregate_to_symbolic(
      iris,
      type        = "int",
      group_by    = "kmeans",
      stratify_var = "Species",
      K           = 10
    )
    iris_int

### 13.1.2 Visualization with ggInterval

The `ggInterval` package provides specialized plots for symbolic data including index image plots, PCA biplots, and radar plots. The following examples require `ggInterval` to be installed. Note: with 30 observations the index image and PCA plots may take several minutes to render.

    library(ggInterval)
    library(ggplot2)

    # Keep only interval columns (drop the 'sample' label column).
    # Fix zero-width intervals from singleton clusters.
    iris_int_num <- iris_int[, sapply(iris_int, inherits, "symbolic_interval")]
    for (v in colnames(iris_int_num)) {
      cv <- unclass(iris_int_num[[v]])
      w  <- Im(cv) - Re(cv)
      fix <- which(w == 0)
      if (length(fix) > 0) {
        cv[fix] <- complex(real = Re(cv[fix]) - 1e-6, imaginary = Im(cv[fix]) + 1e-6)
        class(cv) <- c("symbolic_interval", "vctrs_vctr")
        iris_int_num[[v]] <- cv
      }
    }

Index image plot – a heatmap of all interval variables:

    ggInterval_indexImage(iris_int_num, plotAll = TRUE)

PCA biplot – principal component analysis for interval data:

    ggInterval_PCA(iris_int_num)

Radar plot – multivariate comparison of interval columns from `environment.mix` (observations 4 and 6):

    data(environment.mix)
    env_int <- environment.mix[, 5:17]
    ggInterval_radarplot(env_int, plotPartial = c(4, 6),
                         showLegend = FALSE, addText = FALSE)

### 13.1.3 Interval time series visualization

We plot the first 12 months of the `irish_wind.its` dataset, showing each station’s wind speed interval as a bar with midpoint lines.

    library(ggplot2)
    data(irish_wind.its)
    wind_sub <- irish_wind.its[1:12, ]

    # Reshape to long format
    stations <- c("BIR", "DUB", "KIL", "SHA", "VAL")
    wind_long <- do.call(rbind, lapply(stations, function(st) {
      data.frame(
        month_num = seq_len(12),
        Station   = st,
        lower     = wind_sub[[paste0(st, "_l")]],
        upper     = wind_sub[[paste0(st, "_u")]],
        mid       = (wind_sub[[paste0(st, "_l")]] + wind_sub[[paste0(st, "_u")]]) / 2
      )
    }))
    wind_long$Station <- factor(wind_long$Station, levels = stations)

    # Dodge bars for each station within each month
    n_st  <- length(stations)
    bar_w <- 0.6 / n_st
    wind_long$st_idx <- as.numeric(wind_long$Station)
    wind_long$x <- wind_long$month_num +
      (wind_long$st_idx - (n_st + 1) / 2) * bar_w

    ggplot(wind_long) +
      geom_rect(aes(xmin = x - bar_w / 2, xmax = x + bar_w / 2,
                    ymin = lower, ymax = upper, fill = Station),
                alpha = 0.4, color = NA) +
      geom_line(aes(x = x, y = mid, color = Station, group = Station),
                linewidth = 0.5) +
      geom_point(aes(x = x, y = mid, color = Station), size = 1) +
      scale_x_continuous(breaks = 1:12, labels = month.abb) +
      labs(title = "Irish Wind Speed Intervals (1961)",
           x = "Month", y = "Wind Speed (knots)") +
      theme_grey(base_size = 12)

## 13.2 Clustering for Interval-Valued Data

We benchmark three clustering algorithms on five interval-valued datasets using the quality index \(1 - \text{WSS}/\text{TSS}\):

- `RSDA::sym.kmeans()` – K-means for symbolic data
- `symbolicDA::DClust()` – Distance-based symbolic clustering
- `symbolicDA::SClust()` – Symbolic clustering

Each method independently determines its own optimal number of clusters \(k\) via an n-adaptive elbow method. For each method, we sweep \(k\) from 2 to \(k_{\max} = \min(n-1,\, 10,\, \max(3,\, \lfloor n/5 \rfloor))\) and compute the quality index at each \(k\). The elbow is detected using an absolute gain threshold \(\tau = \Delta_{\max} / (1 + n/100)\), where \(\Delta_{\max}\) is the largest quality gain across all \(k\). A 2-step lookahead skips temporary dips. This yields a higher threshold (fewer clusters) for small datasets and a lower threshold (more clusters allowed) for large datasets.

    library(symbolicDA)

    set.seed(123)

    datasets_clust_int <- list(
      list(name = "face.iGAP",              data = "face.iGAP"),
      list(name = "prostate.int",           data = "prostate.int"),
      list(name = "nycflights.int",         data = "nycflights.int"),
      list(name = "china_temp.int",         data = "china_temp.int"),
      list(name = "lisbon_air_quality.int", data = "lisbon_air_quality.int")
    )

    clust_int_results <- do.call(rbind, lapply(datasets_clust_int, function(ds) {
      tryCatch({
        data(list = ds$data)
        x <- get(ds$data)
        if (!inherits(x, "symbolic_tbl")) {
          x <- tryCatch(int_convert_format(x, to = "RSDA"), error = function(e) x)
          for (i in seq_along(x)) {
            if (is.complex(x[[i]]) && !inherits(x[[i]], "symbolic_interval"))
              class(x[[i]]) <- c("symbolic_interval", "vctrs_vctr")
          }
          if (!inherits(x, "symbolic_tbl"))
            class(x) <- c("symbolic_tbl", class(x))
        }
        x_int <- .get_interval_cols(x)
        n <- nrow(x_int); p <- ncol(x_int)
        k_max <- min(n - 1, 10, max(3, floor(n / 5)))

        d <- int_dist_matrix(x_int, method = "hausdorff")
        so <- simple2SO(.to_3d_array(x_int))

        km_qs <- dc_qs <- sc_qs <- setNames(rep(NA_real_, k_max - 1),
                                             as.character(2:k_max))
        for (k in 2:k_max) {
          set.seed(123)
          km_qs[as.character(k)] <- tryCatch({
            res <- sym.kmeans(x_int, k = k)
            1 - res$tot.withinss / res$totss
          }, error = function(e) NA)

          set.seed(123)
          dc_qs[as.character(k)] <- tryCatch({
            cl <- DClust(d, cl = k, iter = 100)
            .clust_quality(d, cl)
          }, error = function(e) NA)

          set.seed(123)
          sc_qs[as.character(k)] <- tryCatch({
            cl <- SClust(so, cl = k, iter = 100)
            .clust_quality(d, cl)
          }, error = function(e) NA)
        }

        km_k <- .find_optimal_k(km_qs, n); km_q <- km_qs[as.character(km_k)]
        dc_k <- .find_optimal_k(dc_qs, n); dc_q <- dc_qs[as.character(dc_k)]
        sc_k <- .find_optimal_k(sc_qs, n); sc_q <- sc_qs[as.character(sc_k)]

        data.frame(Dataset = ds$name, n = n, p = p,
                   sym.kmeans = sprintf("%.4f (%d)", km_q, km_k),
                   DClust     = sprintf("%.4f (%d)", dc_q, dc_k),
                   SClust     = sprintf("%.4f (%d)", sc_q, sc_k))
      }, error = function(e) NULL)
    }))

    kable(clust_int_results, row.names = FALSE,
          caption = "Table 4: Interval clustering quality (1 - WSS/TSS) with optimal k in parentheses")

## 13.3 Clustering for Histogram-Valued Data

We benchmark three clustering algorithms on five histogram-valued datasets from `dataSDA`. Each dataset is converted from dataSDA’s histogram string format to `HistDAWass::MatH` objects for analysis:

- `WH_kmeans()` – K-means for histogram data
- `WH_fcmeans()` – Fuzzy C-means for histogram data
- `WH_hclust()` – Hierarchical clustering with Wasserstein distance

The same n-adaptive elbow method from Section 4.2 is used for each method to independently select its optimal \(k\).

    set.seed(123)

    datasets_clust_hist <- list(
      list(name = "age_pyramids.hist"),
      list(name = "ozone.hist"),
      list(name = "china_climate_season.hist"),
      list(name = "french_agriculture.hist"),
      list(name = "flights_detail.hist")
    )

    clust_hist_results <- do.call(rbind, lapply(datasets_clust_hist, function(ds) {
      tryCatch({
        data(list = ds$name, package = "dataSDA")
        raw <- get(ds$name)
        x <- .dataSDA_hist_to_MatH(raw)
        n <- nrow(x@M); p <- ncol(x@M)
        k_max <- min(n - 1, 10, max(3, floor(n / 5)))

        # Precompute Wasserstein distance matrix and hclust tree (shared across k)
        dm <- WH_MAT_DIST(x)
        set.seed(123)
        hc <- WH_hclust(x, simplify = TRUE)

        km_qs <- fc_qs <- hc_qs <- setNames(rep(NA_real_, k_max - 1),
                                             as.character(2:k_max))
        for (k in 2:k_max) {
          set.seed(123)
          km_qs[as.character(k)] <- tryCatch({
            res <- WH_kmeans(x, k = k)
            res$quality
          }, error = function(e) NA)

          set.seed(123)
          fc_qs[as.character(k)] <- tryCatch({
            res <- WH_fcmeans(x, k = k)
            res$quality
          }, error = function(e) NA)

          set.seed(123)
          hc_qs[as.character(k)] <- tryCatch({
            cl <- cutree(hc, k = k)
            .clust_quality(dm, cl)
          }, error = function(e) NA)
        }

        km_k <- .find_optimal_k(km_qs, n); km_q <- km_qs[as.character(km_k)]
        fc_k <- .find_optimal_k(fc_qs, n); fc_q <- fc_qs[as.character(fc_k)]
        hc_k <- .find_optimal_k(hc_qs, n); hc_q <- hc_qs[as.character(hc_k)]

        data.frame(Dataset = ds$name, n = n, p = p,
                   WH_kmeans  = sprintf("%.4f (%d)", km_q, km_k),
                   WH_fcmeans = sprintf("%.4f (%d)", fc_q, fc_k),
                   WH_hclust  = sprintf("%.4f (%d)", hc_q, hc_k))
      }, error = function(e) NULL)
    }))

    kable(clust_hist_results, row.names = FALSE,
          caption = "Table 5: Histogram clustering quality (1 - WSS/TSS) with optimal k in parentheses")

## 13.4 Classification for Interval-Valued Data

We benchmark three classifiers on five interval-valued datasets and report resubstitution accuracy:

- `MAINT.Data::lda()` – Linear discriminant analysis for interval data
- `MAINT.Data::qda()` – Quadratic discriminant analysis for interval data
- `e1071::svm()` – Support vector machine on lower/upper bound features

    library(MAINT.Data)
    library(e1071)

    datasets_class <- list(
      list(name = "cars.int",        data = "cars.int",
           class_col = "class",
           class_desc = "class: Utilitarian(7), Berlina(8), Sportive(8), Luxury(4)"),
      list(name = "china_temp.int",  data = "china_temp.int",
           class_col = "GeoReg",
           class_desc = "GeoReg: 6 regions"),
      list(name = "mushroom.int",    data = "mushroom.int",
           class_col = "Edibility",
           class_desc = "Edibility: T(4), U(2), Y(17)"),
      list(name = "ohtemp.int",      data = "ohtemp.int",
           class_col = "STATE",
           class_desc = "STATE: 10 groups"),
      list(name = "wine.int",        data = "wine.int",
           class_col = "class",
           class_desc = "class: 1(21), 2(12)")
    )

    class_results <- do.call(rbind, lapply(datasets_class, function(ds) {
      tryCatch({
        data(list = ds$data)
        x <- get(ds$data)
        grp <- .get_class_labels(x, ds$class_col)

        idata <- .build_IData(x)

        int_cols <- sapply(x, function(col) inherits(col, "symbolic_interval"))
        svm_df <- data.frame(row.names = seq_len(nrow(x)))
        for (v in names(x)[int_cols]) {
          cv <- unclass(x[[v]])
          svm_df[[paste0(v, "_l")]] <- Re(cv)
          svm_df[[paste0(v, "_u")]] <- Im(cv)
        }

        set.seed(123)
        lda_acc <- tryCatch({
          res <- MAINT.Data::lda(idata, grouping = grp)
          pred <- predict(res, idata)
          mean(pred$class == grp)
        }, error = function(e) NA)

        set.seed(123)
        qda_acc <- tryCatch({
          res <- MAINT.Data::qda(idata, grouping = grp)
          pred <- predict(res, idata)
          mean(pred$class == grp)
        }, error = function(e) NA)

        set.seed(123)
        svm_acc <- tryCatch({
          svm_df$class <- grp
          res <- svm(class ~ ., data = svm_df, kernel = "radial")
          pred <- predict(res, svm_df)
          mean(pred == grp)
        }, error = function(e) NA)

        data.frame(Dataset = ds$name, Response = ds$class_desc,
                   LDA = lda_acc, QDA = qda_acc, SVM = svm_acc)
      }, error = function(e) NULL)
    }))

    kable(class_results, digits = 4, row.names = FALSE,
          caption = "Table 6: Classification accuracy (resubstitution)")

## 13.5 Regression for Interval-Valued Data

We benchmark five regression methods on five interval-valued datasets and report \(R^2\):

- `RSDA::sym.lm()` – Symbolic linear regression (center method)
- `RSDA::sym.glm()` – LASSO regression via `glmnet` (center method)
- `RSDA::sym.rf()` – Symbolic random forest
- `RSDA::sym.rt()` – Symbolic regression tree
- `RSDA::sym.nnet()` – Symbolic neural network

    datasets_reg <- list(
      list(name = "abalone.iGAP",      data = "abalone.iGAP",
           response = "Length",           n_x = 6),
      list(name = "cardiological.int",  data = "cardiological.int",
           response = "pulse",            n_x = 4),
      list(name = "nycflights.int",     data = "nycflights.int",
           response = "distance",         n_x = 3),
      list(name = "oils.int",           data = "oils.int",
           response = "specific_gravity", n_x = 3),
      list(name = "prostate.int",       data = "prostate.int",
           response = "lpsa",             n_x = 8)
    )

    reg_results <- do.call(rbind, lapply(datasets_reg, function(ds) {
      tryCatch({
        data(list = ds$data)
        x <- get(ds$data)

        if (!inherits(x, "symbolic_tbl")) {
          x2 <- tryCatch(int_convert_format(x, to = "RSDA"), error = function(e) NULL)
          if (!is.null(x2)) {
            x <- x2
            for (i in seq_along(x)) {
              if (is.complex(x[[i]]) && !inherits(x[[i]], "symbolic_interval"))
                class(x[[i]]) <- c("symbolic_interval", "vctrs_vctr")
            }
            if (!inherits(x, "symbolic_tbl"))
              class(x) <- c("symbolic_tbl", class(x))
          } else {
            cn <- colnames(x)
            l_cols <- grep("_l$", cn, value = TRUE)
            vars <- sub("_l$", "", l_cols)
            out <- data.frame(row.names = seq_len(nrow(x)))
            for (v in vars) {
              lv <- x[[paste0(v, "_l")]]; uv <- x[[paste0(v, "_u")]]
              si <- complex(real = lv, imaginary = uv)
              class(si) <- c("symbolic_interval", "vctrs_vctr")
              out[[v]] <- si
            }
            class(out) <- c("symbolic_tbl", class(out))
            x <- out
          }
        }
        x_int <- .get_interval_cols(x)

        fml <- as.formula(paste(ds$response, "~ ."))
        nc <- data.frame(row.names = seq_len(nrow(x_int)))
        for (v in names(x_int)) {
          cv <- unclass(x_int[[v]])
          nc[[v]] <- (Re(cv) + Im(cv)) / 2
        }
        actual <- nc[[ds$response]]
        resp_idx <- which(names(x_int) == ds$response)
        .r2 <- function(a, p) 1 - sum((a - p)^2) / sum((a - mean(a))^2)

        set.seed(123)
        lm_r2 <- tryCatch({
          res <- sym.lm(fml, sym.data = x_int, method = "cm")
          summary(res)$r.squared
        }, error = function(e) NA)

        set.seed(123)
        glm_r2 <- tryCatch({
          res <- sym.glm(sym.data = x_int, response = resp_idx, method = "cm")
          pred <- as.numeric(predict(res, newx = as.matrix(nc[, -resp_idx]),
                                     s = "lambda.min"))
          .r2(actual, pred)
        }, error = function(e) NA)

        set.seed(123)
        rf_r2 <- tryCatch({
          res <- sym.rf(fml, sym.data = x_int, method = "cm")
          tail(res$rsq, 1)
        }, error = function(e) NA)

        set.seed(123)
        rt_r2 <- tryCatch({
          res <- sym.rt(fml, sym.data = x_int, method = "cm")
          .r2(actual, predict(res))
        }, error = function(e) NA)

        set.seed(123)
        nnet_r2 <- tryCatch({
          res <- sym.nnet(fml, sym.data = x_int, method = "cm")
          pred_sc <- as.numeric(res$net.result[[1]])
          pred <- pred_sc * res$data_c_sds[resp_idx] + res$data_c_means[resp_idx]
          .r2(actual, pred)
        }, error = function(e) NA)

        data.frame(Dataset = ds$name, Response = ds$response, p = ds$n_x,
                   sym.lm = lm_r2, sym.glm = glm_r2, sym.rf = rf_r2,
                   sym.rt = rt_r2, sym.nnet = nnet_r2)
      }, error = function(e) NULL)
    }))

    kable(reg_results, digits = 4, row.names = FALSE,
          caption = "Table 7: Regression R-squared")

# 14 Symbolic Dataset Donation/Submission Guidelines

We welcome contributions of high-quality datasets for symbolic data analysis. Submitted datasets will be made publicly available (or under specified constraints) to support research in machine learning, statistics, and related fields. You can submit the related files via email to wuhm@g.nccu.edu.tw or through the Google Form at Symbolic Dataset Submission Form. The submission requirements are as follows.

1. Dataset Format:

   - Preferred formats: `.csv`, `.xlsx`, or any symbolic format in plain text.
   - Compressed (`.zip` or `.gz`) if multiple files are included.

2. Required Metadata: Contributors must provide the following details:

   Field | Description | Example
   --- | --- | ---
   Dataset Name | A clear, descriptive title. | “face recognition data”
   Dataset Short Name | A clear,abbreviation title. | “face data”
   Authors | Full names of donator. | “First name, Last name”
   E-mail | Contact email. | “abc123@gmail.com”
   Institutes | Affiliated organizations. | “-”
   Country | Origin of the dataset. | “France”
   Dataset Descriptions | Data descriptive | See ‘README’
   Sample Size | Number of instances/rows. | 27
   Number of Variables | Total features/columns (categorical/numeric). | 6 (interval)
   Missing Values | Indicate if missing values exist and how they are handled. | “None” / “Yes, marked as NA”
   Variable Descriptions | Detailed description of each column (name, type, units, range). | See ‘README’
   Source | Original data source (if applicable). | “Leroy et al. (1996)”
   References | Citations for prior work using the dataset. | “Douzal-Chouakria, Billard, and Diday (2011)”
   Applied Areas | Relevant fields (e.g., biology, finance). | “Machine Learning”
   Usage Constraints | Licensing (CC-BY, MIT) or restrictions. | “Public domain”
   Data Link | URL to download the dataset (Google Drive, GitHub, etc.). | “(https)”

3. Quality Assurance:

   - Datasets should be clean (no sensitive/private data).

4. Optional (Recommended):

   - A companion `README` file with:
     - Dataset background.
     - Suggested use cases.
     - Known limitations.

# 15 Citation

Po-Wei Chen, Chun-houh Chen, Han-Ming Wu (2026), dataSDA: datasets and basic statistics for symbolic data analysis in R (v0.3.0). Technical report.