Empirical copula-based data augmentation for mixed-type datasets: a robust approach for synthetic data generation

Treatment of missing data

Real-world datasets often contain missing values, necessitating preprocessing to enable subsequent modeling. The ECG offers two strategies: imputation, which fills missing entries with statistically derived substitutes, and exclusion, which removes incomplete observations.

• Imputation: For numerical variables (continuous or integer), missing values are replaced with the column mean or any convenient function), leveraging central tendency to maintain distributional coherence. For categorical variables, the mode, the most frequent category is used, preserving discrete structure. This maximizes data retention, suitable for moderate deficiency.

• Exclusion: Rows with any missing values are discarded, ensuring a complete dataset at the cost of reduced sample size, this is ideal when data incompleteness is minimal.

The algorithm takes as input a dataset D ∈ ${\mathbb{R}}^{n\ast m}$ and a strategy S ∈ {impute, drop}, and outputs a processed dataset D′. Initially, D′ is set equal to D. If the strategy S is “impute”, the algorithm iterates over each column j = 1,…,m. For each numerical columns D[:,j], it computes the mean ${\mu }_j$ of the non-missing values and replaces missing entries in D′[:,j] with ${\mu }_j$. For categorical columns, it computes the mode $m_j$ and fills in missing values with $m_j$. If the strategy is “drop”, the algorithm removes all rows D[i,:] that contain missing values in any column j, resulting in a cleaned dataset D′. Finally, it returns D′ as result.

Handling mixed data types

The heterogeneity of data types—continuous, integer, and categorical—needs a nuanced approach to modeling and transformation. The ECG adeptly categorizes and processes each variable according to its intrinsic properties, a foundational step that enables the separation of marginal probability distributions from their dependence structure, a hallmark of copula theory.

• Detection type: The method begins by classifying each variable. Continuous variables are identified as those capable of assuming any real value within a range, typically represented numerically with floating-point precision. Integer variables, while numeric, are discrete and confined to whole numbers, often requiring special handling to approximate continuity. Categorical variables, encompassing nominal or ordinal data such as labels or categories, resist numerical conversion and are treated as discrete variables. This classification is pivotal, as it dictates the subsequent transformation strategy, ensuring that the probability distribution characteristics of each variable are appropriately captured.

• Transformation to uniform margins: Copula theory states that any multivariate probability distribution can be decomposed into its marginal probability distributions and a copula that encapsulates their dependence. To isolate this dependence, each variable is transformed to a uniform distribution over [0,1], a process that standardizes the margins while preserving inter-variable relationships. For continuous variables, this transformation employs the empirical cumulative probability distribution function (ECPDF), where the rank of each observation within the sorted data is normalized by the sample size. This rank-based approach yields a pseudo-uniform probability distribution, reflecting the shape of the ECPDF without parametric assumptions. Integer variables, inherently discrete, undergo a preliminary jittering process, where small random perturbations are added to break ties and simulate a continuous probability distribution, followed by the same rank-based transformation. Categorical variables, lacking a natural ordering, are mapped to the unit interval based on their cumulative frequencies: each category is assigned a uniform value proportional to its position in the cumulative probability distribution, effectively discretizing the [0,1] range according to category prevalence.

The algorithm takes as input a dataset D ∈ ${\mathbb{R}}^{n\ast m}$ and outputs a list T = [t₁,…, t_m], where each t_j indicates the data type of column j. For each column j = 1,…,m, it attempts to convert the values D[:,j] to float type, stored in F[:,j]. If the conversion is successful, it further tries to convert them to integers I[:,j]. If all values in F[:,j] equal those in I[:,j], the column is classified as ‘integer’; otherwise, it is ‘continuous’. If the initial float conversion fails, the column is labeled as ‘categorical’. The algorithm returns the list T, indicating the detected type for each column.

The algorithm transforms a column C ∈ ${\mathbb{R}}^n$into a uniform distribution on [0, 1], based on its type ∈ {continuous, integer, categorical}, with optional noise scale $\mathrm{\epsilon }$ . If t = “continuous”, it computes the rank R[i] of each value as the proportion of entries less than or equal to C[i], then sets U[i] = (R[i]−1)/n. If t = “integer” it adds small random noise ${\eta }_i\sim \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\left(-\mathrm{\epsilon },\mathrm{\epsilon }\right)$ to each value (jittering), forming C′[i] = C[i] + ${\eta }_i$, then computes ranks R′[i] and uniform values as before. If t = “categorical”, it first determines the set of unique values V = {v₁,…,v_k}, then estimates their relative frequencies N[v_l], and computes the cumulative probability P(v_l). Each value C[i] is then mapped to its corresponding cumulative probability U[i] = P(v_l). The output is the transformed column U ∈ ${\left[0,1\right]}^n$, uniformly distributed.

Perturbing continuous data

To ensure that generated data introduces novelty rather than merely replicating the original observations, the method incorporates a controlled perturbation mechanism for continuous variables. This step is theoretically motivated by the need to balance fidelity to the original probability distribution with the generation of plausible variations, a critical aspect of data augmentation.

• Perturbation mechanism: For each uniform value derived from a continuous variable, the method identifies its position within the sorted original data. Adjacent values—termed neighbors—define the local context, establishing bounds within which perturbation can occur without disrupting the order of the data. The maximum allowable noise is calculated as the minimum distance to these neighbors, ensuring that the perturbed value remains consistent with its rank. A small random noise, scaled by a predefined factor, is then added within this range, and the perturbed uniform value is mapped back to the original scale via an inverse transformation. This process introduces subtle variations, mimicking natural variability while preserving the distributional properties and dependence structure.

The algorithm perturbs a continuous column X ∈ ${\mathbb{R}}^n$, using a reference sorted column C, its corresponding uniform transformation U ∈ ${\left[0,1\right]}^n$, and a noise scale ε, to generate perturbed data X′ ∈ ${\mathbb{R}}^n$. It begins by sorting C to obtain the ordered list S = [s₁,…,s_n]. For each value v = X[i], it finds the interval [s_k, s_k+1] that contains v, and sets the bounding values v₁ = s_max(1,k-1), and v₂ = s_min(k+1,n). The maximum noise magnitude η is the smaller of the distance from v to either bound, or 0.01 if both bounds are equal. Then, a uniform random noise z∼Unif (−1, 1) is scaled by η⋅ε and added to v to produce U′[i], which is clipped to [0,1]. Finally, X′[i] is obtained by applying the inverse CDF of C to U′[i], and the perturbed vector X′ is returned.

Computing the empirical copula

The empirical copula serves as the linchpin of this methodology, providing a non-parametric estimate of the dependence structure among variables. Grounded in copula theory, this construct captures the joint behavior of the transformed uniform margins, enabling the generation of new data that respects observed interdependencies.

• Construction process: After transforming all variables to uniform margins, the empirical copula is implicitly defined by the joint probability distribution of these uniform variables. Rather than fitting a parametric copula model, the method retains the empirical joint probability distribution as observed in the data, leveraging the ranks and their multivariate configuration. This non-parametric approach eschews assumptions about the functional form of the copula, relying instead on the inherent structure of the data.

• Sampling mechanism: To generate new samples, the method employs resampling with replacement from the uniform dataset. Each sampled row represents a realization of the empirical copula, preserving the observed dependence through the co-occurrence of uniform values across variables. This resampling mirrors the bootstrap technique, adapted here to synthesize new multivariate observations rather than estimate statistical properties.

Algorithm 5 takes a dataset D ∈ ${\mathbb{R}}^{n\ast m}$ with a noise scale ε, and transforms it into a matrix U ∈ ${\left[0,1\right]}^{n\ast m}$ with uniform margins, suitable for copula modeling. First, it applies the Handling Missing Value procedure to produce a clean dataset D′. Then, it uses Detecting Column Names to infer the type T = [t₁,…,t_m] for each column in D′. For every column j = 1,…, it applies Transforming to Uniform Margins to the column D′[:,j], using its type T[j] and noise scale ε, resulting in a uniform column U[:,j]. The final output U is a uniformly transformed version of the data, preserving the dependence structure across features.

Algorithm 6 generates new samples U_s ∈ ${\left[0,1\right]}^{n\ast m}$ from a previously fitted copula represented by U ∈ ${\left[0,1\right]}^{n\ast m}$, using a specified sample size n_s. It begins by drawing n_s random indices I = [i₁,…,i_ns], each sampled independently with replacement from the set {1,…,n}, following a uniform distribution. For each k = 1,…,n_s, it sets the new sample row U_s[k,:] = U[i_k,:]. The resulting matrix u_spreserves the empirical dependency structure from U, making it a valid synthetic dataset on the uniform scale.

Inverse transformation via the empirical copula

Generating synthetic data in the original space requires reversing the uniform transformation, a process that reconstructs each variable’s marginal probability distribution from the sampled copula values.

• Continuous variables: The inverse transformation for continuous variables approximates the quantile function (inverse CPDF) using the original data’s sorted values. Uniform samples are mapped to this empirical quantile function, often via interpolation to handle values between observed points. Post-mapping, the perturbation step described earlier introduces controlled noise, ensuring that the resulting values reflect both the original probability distribution and added variability.

• Integer variables: For integers, the uniform samples are mapped to the nearest corresponding quantile in the original data, effectively rounding to the closest integer value. This preserves the discrete nature of the variable, aligning the synthetic data with its empirical probability distribution.

• Categorical variables: Categorical variables are reconstructed by partitioning the [0,1] interval according to the original categories’ cumulative frequencies. Each uniform sample is assigned to the category whose cumulative probability range it falls within, replicating the discrete probability distribution observed in the input data.

This algorithm reverses the copula-based uniform transformation, mapping uniform samples U_s ∈ ${\left[0,1\right]}^{n_s}$ back to their original data space using a reference column C ∈ ${\mathbb{R}}^n$and its type t ∈ {“continuous”, “integer”, “categorical”}. If t= “continuous”, it first sorts C to obtain S, defines a function f(u) that interpolates u ∈ [0,1] over S, then computes X = f(U_s). The result is refined using the Perturbing Continuous Data algorithm to add realistic noise, producing X_s . If t = "integer", each X_s[i] is computed as the empirical quantile of C at U_s[i]. For categorical data, it retrieves the unique values V and their cumulative probabilities P (as in Algorithm 3), and assigns X_s[i] = v_l such that P(v_l−1) < U_s[i] ≤ P(v_l). The final output is X_s, a column of data samples in the original scale.

Generating augmented data

The culmination of the ECG is the production of augmented data, achieved through an integrated workflow that synthesizes the preceding components. The process commences with preprocessing to handle missing data, followed by type detection and transformation to uniform margins, effectively fitting the empirical copula. New samples are then drawn from this copula via resampling, and each uniform sample is transformed back to the original space using the appropriate inverse method. The result is a synthetic dataset that mirrors the original’s marginal probability distributions and dependence structure, augmented with controlled variations for continuous variables.

This algorithm creates a synthetic dataset D_s ∈ ${\mathbb{R}}^{n_s\ast m}$ from an original dataset D ∈ ${\mathbb{R}}^{n\ast m}$, using a sample size n_s. First, it applies fitting the Copula to D to obtain the uniform representation U ∈ ${\left[0,1\right]}^{n\ast m}$, which captures the dependence structure between features. Then, it uses sampling from the Copula to draw n_snew samples U_s ∈ ${\left[0,1\right]}^{n_s\ast m}$from U. For each column j = 1,…,m, the algorithm applies Inverse Transformation to U_s[:,j], using the original column D[:,j] and its type T[j], to produce the synthetic column D_s[:,j]. The result is a fully synthetic dataset D_s, with the same feature structure and dependency patterns as the original data.

Advantages and theoretical considerations

The ECG offers several theoretical advantages:

• Flexibility with mixed types: By tailoring transformations to each data type, it seamlessly accommodates heterogeneity, a common feature of real-world datasets.

• Non-parametric nature: Its reliance on empirical probability distributions avoids restrictive parametric assumptions, enhancing applicability across diverse domains.

• Dependence preservation: The resampling strategy ensures that multivariate relationships are retained, a critical factor in multivariate analysis.

• Controlled variation: Perturbation introduces novelty without compromising statistical fidelity, enriching the synthetic output.

The ECG emerges as a robust theoretical framework for synthetic data generation, harmonizing copula theory with practical adaptations for mixed-type data. Its meticulous handling of missing values, type-specific transformations, perturbation mechanisms, and empirical copula construction culminates in a method that balances fidelity and innovation. This approach holds significant promise for advancing research in data science, offering a versatile tool for augmentation, privacy preservation, and beyond.

Assessment methodology

To rigorously evaluate the synthetic data produced by the ECG, we developed a multifaceted assessment protocol that examines both univariate and multivariate properties. This approach ensures that the augmented data not only mirrors the individual feature distributions of the original dataset but also maintains the intricate interdependencies among variables. Below, we describe the theoretical constructs and statistical methodologies underpinning our evaluation, drawing from established techniques in probability distribution comparison and dependence analysis (Wang, Wang & Liu, 2025).

Empirical results

Experiments on synthetic datasets

Synthetic datasets provide a controlled setting to assess the generator’s ability to replicate structured patterns, making them an ideal starting point for evaluation. We begin with experiments on the Star Shape dataset and the Multiple Forms dataset, conducted in sequence to test the method’s precision and robustness under varying noise conditions.

Star Shape dataset experiment

The first experiment focuses on the Star Shape dataset, comprising 100 points arranged in a five-armed star pattern with an initial noise perturbation of 0.05. This dataset tests the generator’s capacity to preserve a simple yet distinct geometric structure. The experiment progresses through three stages, each visualized in Figs. 2–4. The experiment begins with the original dataset, depicted in Fig. 2 as a scatter plot in blue. The five arms of the star are clearly defined, radiating symmetrically from the center despite the slight noise. This figure serves as the baseline, illustrating the target structure that the generator must replicate. The clarity of the star’s shape, even with minor perturbations, establishes a straightforward yet effective reference for evaluating synthetic outputs.

Next, we generate synthetic data with a minimal noise level of 0.1, overlaid in red on the original dataset (blue) in Fig. 3.

The experiment concludes with synthetic data generated at a high noise level of 5.0, shown in red alongside the original (blue) in Fig. 4. Here, the synthetic points exhibit greater dispersion, spreading outward from the star’s arms in a cloud-like pattern. Despite this variability, the five-armed structure remains discernible, with each arm still traceable amid the noise. This outcome highlights the generator’s robustness: it introduces significant diversity while retaining the core geometric essence of the original dataset. The trade-off between variability and fidelity is evident, as the increased noise broadens the synthetic distribution but does not obliterate the underlying pattern. This adaptability makes the method suitable for scenarios where controlled diversity is beneficial, such as data augmentation for machine learning, while still anchoring the synthetic output to the original structure.

Multiple Forms dataset experiment

The second synthetic experiment targets the Multiple Forms dataset, a more complex collection of 200 points distributed across four distinct shapes: a crescent, a circle, an asterisk, and a Gaussian cloud. This experiment, visualized in Fig. 5, tests the generator’s ability to handle multi-modal probability distributions with overlapping and diverse patterns.

The experiment starts with Fig. 5 (right-hand), a scatter plot of the original dataset. The crescent’s smooth curvature, the circle’s closed boundary, the asterisk’s radial arms, and the Gaussian cloud’s dense cluster are all distinctly visible. This diversity of forms poses a significant challenge, requiring the generator to capture multiple structural characteristics simultaneously. The clarity of each shape in the original data sets a high bar for the synthetic outputs, making this dataset an excellent test of the method’s versatility in replicating complex patterns.

In the next stage, synthetic data generated with a noise level of 0.01 is overlaid in red on the original (blue) in Fig. 5 (left-hand). The synthetic points replicate each shape with remarkable accuracy: the crescent’s curve remains intact, the circle’s outline is precise, the asterisk’s arms are sharply defined, and the Gaussian cloud’s density is consistent. This close correspondence underscores the generator’s ability to preserve intricate, multi-modal probability distributions under low noise. The synthetic data mirrors the original so closely that distinguishing between them visually is challenging, highlighting the method’s precision in capturing both the marginal probability distributions and spatial relationships of diverse forms. This level of detail is particularly valuable for applications requiring faithful reproduction of complex datasets, such as simulation studies or generative modeling.

In Fig. 5 (right-hand) synthetic data at a noise level of 5.0 (red) is plotted against the original (blue). The increased noise level introduces noticeable dispersion, with synthetic points spreading outward from each shape. Nevertheless, the core features persist: the crescent’s arc, the circle’s boundary, the asterisk’s radial pattern, and the Gaussian cloud’s concentration remain recognizable. This resilience under high noise demonstrates the generator’s capacity to maintain structural integrity despite significant perturbation. The synthetic data introduces variability that enriches the dataset without erasing its defining characteristics, a balance that enhances its utility for tasks like robustness testing or diversity-driven analysis.

Experiments on real-world datasets

Following the synthetic experiments, we shift to real-world datasets, which introduce practical complexities such as mixed data types, varying sample sizes, and high-dimensional feature spaces. The experiments proceed in sequence across the Adult, Ecoli, Forest Fires, and Wisconsin Diagnostic Breast Cancer (WDBC) datasets, with analyses tied to Figs. 6–13.

Adult dataset experiment

The Adult dataset is a widely used benchmark dataset in the field of machine learning and data science. Based on 1,000 records from this dataset and using 6 numeric and eight categorical features, we will generate 5,000 new records mimicking the original dataset using a level noise of 0.01. The experiment unfolds across Figs. 6, 7 for some features. The evaluation leverages the Kolmogorov-Smirnov (KS) test for numeric features, Jensen-Shannon (JS) divergence for categorical features and overall joint probability distributions, and correlation heatmaps for dependency structures

The experiment begins with Fig. 6 (left-hand), comparing histograms of the “age” feature from the original (blue) and synthetic (red) datasets. The probability distributions align closely, with the synthetic histogram replicating the original’s shape, peaks, and spread. A KS statistic of 0.0084 and p-value of 1.0000 indicate no significant difference, confirming the generator’s accuracy in preserving this numeric feature’s marginal probability distribution. Figure 6 (right-hand), quantifies the similarity of the categorical “workclass” feature using a JS divergence of 0.0107. This low value—close to 0, the ideal for identical probability distributions reflects the synthetic data’s ability to mirror the original frequency distribution across workclass categories (e.g., private, self-employed).

The experiment concludes with Fig. 7, presenting correlation heatmaps for the numeric features. The original (left) and synthetic (right) heatmaps are nearly indistinguishable, with correlation coefficients differing by less than 0.05 on average. Features like age and hours-per-week retain their interrelationships, demonstrating the generator’s success in capturing joint dependencies. This preservation is essential for applications where feature interactions inform outcomes, such as income prediction. The average JS divergence for the two datasets is insignificant (0.0117).

The Adult experiment (Figs. 6, 7) highlights the generator’s prowess with mixed-type data. Figure 5 confirms marginal fidelity for both examples: Numeric feature (age) and categorical feature (work class). Figure 6 validates joint fidelity for numeric features through the correlation heatmap. The low KS and JS metrics, paired with consistent correlations, highlight the method’s robustness in demographic contexts, where both individual and relational properties must be accurately replicated.

Ecoli dataset experiment

The Ecoli dataset involves 336 records with seven continuous predictor features. The proposed method synthesizes 5,000 new records that replicate the statistical properties of the original dataset, incorporating a noise parameter of 0.01 to introduce controlled variability. The experiment unfolds across Figs. 8, 9 for some features. The evaluation leverages the Kolmogorov-Smirnov (KS) test for numeric features, Jensen-Shannon (JS) divergence for categorical features and overall joint distributions, and correlation heatmaps for dependency structures.

The experiment starts with Fig. 8, comparing histograms of the “mcg” and “gvh” features. The original (blue) and synthetic (red) distributions overlap tightly, with a KS statistic of 0.0174 and p-value of 1.0000 for the mcg feature and a KS statistic of 0.0073 and p-value of 1.0000 for gvh feature, indicating statistical equivalence. This precision in replicating a continuous feature’s probability distribution is vital for biological datasets, where small sample sizes amplify the importance of fidelity.

In Fig. 9, we show the correlation heatmaps for all numeric features. The original (left) and synthetic (right) heatmaps align closely, with minimal deviations in correlations among features like mcg and gvh. This consistency underscores the generator’s ability to preserve joint dependencies, critical for scientific analyses relying on inter-feature relationships.

The average JS divergence for the two datasets is insignificant (0.0134). The Ecoli experiment (Figs. 8, 9) demonstrates the generator’s effectiveness in small, continuous datasets. Figure 8 affirms the marginal fidelity, while Fig. 9 confirms joint fidelity. The low KS statistics and consistent heatmaps highlight the method’s utility in data-scarce scientific domains, where precision is paramount.

Forest Fires dataset experiment

The Forest Fires dataset experiment with 517 records, 10 numeric features, and two categorical features, assesses the generator on environmental data with temporal elements. The proposed method synthesizes 5,000 new records that replicate the statistical properties of the original dataset, incorporating a noise parameter of 0.01 to introduce controlled variability. The experiment begins with Fig. 10 (left-hand), comparing histograms of the Duff Moisture Code (DMC) feature. The original (blue) and synthetic (red) distributions are nearly identical, with a KS statistic of 0.012 and p-value of 1.0000. This fidelity ensures that key environmental indicators are preserved, supporting applications like fire risk modeling.

Figure 10 (right-hand), quantifies the similarity of the categorical “month” feature using a JS divergence of 0.0134 indicating high similarity between the original and synthetic distributions of this categorical variable. The low divergence preserves temporal patterns, essential for analyzing seasonal trends in environmental data. The experiment concludes with Fig. 11, presenting correlation heatmaps for numeric features. The original (left) and synthetic (right) heatmaps align closely, with features like DMC, temp, and wind showing consistent correlations. This preservation of dependencies enhances the synthetic data’s utility for environmental studies.

The experiment concludes with Fig. 11, presenting correlation heatmaps for numeric features. The original (left) and synthetic (right) heatmaps align closely, with features like DMC, temp, and wind showing consistent correlations. This preservation of dependencies enhances the synthetic data’s utility for environmental studies. The low KS and JS metrics, paired with consistent correlations, affirm the method’s robustness in practical settings.

Wisconsin Diagnostic Breast Cancer dataset experiment

The final experiment targets the Wisconsin Diagnostic Breast Cancer (WDBC) dataset, It contains 569 records and 29 continuous features, testing the generator in a high-dimensional medical context. The experiment unfolds across Figs. 12, 13 for some features.

The proposed method synthesizes 5,000 new records that replicate the statistical properties of the original dataset, incorporating a noise parameter of 0.001 to introduce controlled variability.

The experiment starts with Fig. 12, comparing histograms of the “texture1” feature. The original (blue) and synthetic (red) istributions align closely, with a KS statistic of 0.0118 and p-value of 1.0000, confirming marginal fidelity in a high-dimensional setting, it continues with the “perimeter1” feature, where the synthetic distribution matches the original (KS statistic = 0.0126, p-value = 1.0000). This consistency across features highlights the generator’s scalability to high-dimensional data.

In the WDBC dataset, the ECG achieved an average JS divergence of 0.0292 between the original and synthetic datasets. This exceptionally low value highlights the method’s proficiency in replicating the intricate statistical structure of a 29-feature dataset, with minimal deviations that affirm its robustness for high-dimensional applications. Correlation heatmaps show that features align closely (Fig. 13).

Dataset complexity effect on augmentation data quality

The quality of augmented data hinges on factors like the noise level, controlled by the noise level epsilon, and the Q ratio, calculated as the number of records divided by the number of features, which reflects the dataset’s informational density. Across four real-world datasets: Adult, Wisconsin Diagnostic Breast Cancer (WDBC), Ecoli, and Forest Fires, we see clear trends in how these elements shape the preservation of feature distributions. In the Adult dataset, boasting a high Q ratio of 71, the augmentation shines in Table 1, representing lower and higher noise levels, respectively. For continuous features like “age,” the KS statistic barely budges (0.0112 for ε = 0.01, 0.0120 for ε = 5), with p-values holding steady at 0.999, signaling robust similarity despite rising epsilon. Categorical features, such as “workclass,” also fare well, with JSD shifting only slightly from 0.0065 to 0.0093, hinting that a high Q ratio acts as a shield against noise, preserving fidelity across conditions. In contrast, the WDBC dataset, with a leaner Q ratio of 19.62, struggles as noise escalates. Table 2 shows “perimeter1” closely aligned (KS = 0.0151, p = 0.9997) for ε = 0.001, but reveals a stark drop-off (KS = 0.1088, p = 0) for ε = 0.01, exposing how low Q ratios leave datasets vulnerable to distributional drift under higher noise. The Ecoli dataset, sitting at a moderate Q ratio of 48, strikes a middle ground—Table 3 reports “mcg” with a KS of 0.0125 and p-value of 1.0000 for ε = 0.01, it rises to 0.0441 for ε = 5, yet retains a p-value of 0.5577, suggesting resilience that does not quite match the consistency of the Adult dataset. Likewise, the Forest Fires dataset, with a Q ratio of 43.08, displays varied outcomes: “DMC” in Table 4 holds tight (KS = 0.0150, p = 0.9999) for ε = 0.01, but “FFMC” diverges sharply (KS = 0.4717, p = 0) for ε = 1, though categorical “month” stays stable (JSD = 0.0141 and 0.0117), unsurprising since noise targets only continuous features. Together, these patterns reveal that higher Q ratios bolster robustness against noise, while lower ratios amplify sensitivity, particularly for continuous features under larger epsilon values, offering practical guidance for applying the augmentation method effectively.

Comprehensive synthesis and implications

Across all experiments, the ECG exhibits the following:

• Marginal fidelity: Low KS statistics (0.0073–0.0174) and JS divergences (0.0107–0.0134) across datasets.

• Joint fidelity: Consistent heatmaps, with minor challenges in high-dimensional cases (Wisconsin).

• Noise flexibility: Synthetic experiments (Figs. 2–5) show adaptability from precision to variability. Bounded noise perturbation procedure ensures stable outputs across varying levels (0.01 to 5.0) except for leaner ratio.

• Versatility: Success across synthetic and real-world contexts.

Limitations include computational scalability for extremely large datasets, an aspect that does not compromise its statistical precision but suggests potential for optimization, suggesting areas for future enhancement. In high-dimensional settings with a low sample-to-feature ratio, the ECG faces significant challenges. However, this difficulty is not unique to the generator; it reflects a broader issue in data analysis where the limited number of data points fails to fully capture the underlying probability distribution of the data. Nonetheless, this extensive evaluation confirms the generator’s broad applicability for synthetic data generation.

Computational complexity and empirical runtime evaluation

To investigate the scalability and efficiency of our proposed framework, we analyze the computational complexity of each stage in the pipeline. The full process consists of three main components: transforming the marginals to a uniform domain, generating synthetic samples from the copula-defined dependence structure, and inverting the synthetic samples back to the original data space. Each of these steps incurs specific computational costs, which depend on the number of features, original samples, and synthetic points. We detail these complexities below to highlight the algorithm’s performance characteristics and identify the dominant operations in practical use cases. In fact the proposed framework operates in three sequential stages: The first stage transforms each marginal feature into the uniform domain via empirical cumulative distribution functions (ECDFs). This requires sorting each feature column, which has a worst-case complexity of O(n. log (n)) per feature. Hence, for d features, the total complexity of this transformation is O(d.n .log (n)).

In the second stage, we generate m synthetic samples from the copula model defined over the uniform marginals. This process involves sampling from a dependence structure (e.g., Gaussian or empirical copula) and has linear complexity O(d⋅m), assuming constant-time sampling per dimension.

The third and final stage inverts the synthetic uniform samples back into the original data space using interpolation over the inverse ECDFs. Since each synthetic point is transformed dimension-wise, this stage also exhibits a linear cost of O(d⋅m). Aggregating all stages, the total runtime complexity of the full pipeline becomes:

$$\mathrm{O}(\mathrm{d}.\mathrm{n}.\mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{n}))+\mathrm{O}(\mathrm{d}\cdot \mathrm{m}).$$which simplifies to O(d⋅m) in high-throughput settings where synthetic data generation dominates (i.e., m ≫ n).

Runtime assessment on synthetic and real-world mixed-type datasets

Empirical runtimes obtained from a random (n,d) sample with the three mixed types are visualized in Fig. 14. The plot clearly demonstrates the expected linear scaling of computation time with respect to sample size n, across all three-feature dimensions. Additionally, increasing the number of features d results in proportional increases in runtime, validating the multiplicative role of dimensionality in the algorithm. Importantly, no super-linear behavior was observed, confirming the framework’s suitability for efficient augmentation in large-scale, mixed-type datasets.

The benchmark experiment conducted here consists of using three mixed-type synthetic datasets. Each dataset composed of a combination of (i) continuous features drawn from Gaussian or uniform distributions, (ii) ordinal features mimicking ordered categories, and (iii) nominal categorical features representing non-ordered classes type. All categorical and ordinal variables were preprocessed using appropriate encoding schemes to allow unified treatment. We systematically varied the number of features d ∈ {2, 5, 10} while incrementing the sample size n ∈ [100, 10,000] uniformly to assess the computational impact of both data dimensionality and original sample size.

Furthermore, the runtime performance of our empirical copula-based data generation approach was evaluated across multiple datasets and varying feature dimensions. We measured the average generation time for producing 10,000 synthetic samples (averaged over 100 runs) using four benchmark datasets. For the Adult dataset (1,000 samples, 10 features), the mean execution time was 0.192 s. The Ecoli dataset (336 samples, eight features) required 0.137 s, while the Forest Fires dataset (517 samples, 12 features) took 1.2036 s. The most computationally intensive case was the WDBC dataset (569 samples, 30 features), reaching 5.1162 s. These results confirm that the method is efficient and scalable for low and high dimensional data (refer to Endres, Mannarapotta Venugopal & Tran (2022) to compare with other methods).