Stable and Privacy-Preserving Synthetic Educational Data with Empirical Marginals: A Copula-Based Approach

Gabriel Diaz Ramos; Lorenzo Luzi; Debshila Basu Mallick; Richard Baraniuk

cs.LGApr 5, 2026

Stable and Privacy-Preserving Synthetic Educational Data with Empirical Marginals: A Copula-Based Approach

Authors: Gabriel Diaz Ramos, Lorenzo Luzi, Debshila Basu Mallick, Richard Baraniuk

Organizations: Rice University · OpenStax · Rice University, OpenStax

Abstract

To advance Educational Data Mining (EDM) within strict privacy-protecting regulatory frameworks, researchers must develop methods that enable data-driven analysis while protecting sensitive student information. Synthetic data generation is one such approach, enabling the release of statistically generated samples instead of real student records; however, existing deep learning and parametric generators often distort marginal distributions and degrade under iterative regeneration, leading to distribution drift and progressive loss of distributional support that compromise reliability. In response, we introduce the Non-Parametric Gaussian Copula (NPGC), a plug-and-play synthesis method that replaces deep learning and parametric optimization with empirical statistical anchoring to preserve the observed marginal distributions while modeling dependencies through a copula framework. NPGC integrates Differential Privacy (DP) at both the marginal and correlation levels, supports heterogeneous variable types, and treats missing data as an explicit state to retain informative absence patterns. We evaluate NPGC against deep learning and parametric baselines on five benchmark datasets and demonstrate that it remains stable across multiple regeneration cycles and achieves competitive downstream performance at substantially lower computational cost. We further validate NPGC through deployment in a real-world online learning platform, demonstrating its practicality for privacy-preserving research.

Figures & tables

Figure 1: Marginal density of the Age variable in the Adult [ 2 ] dataset. The shaded region denotes the empirical density of the real data. Curves correspond to synthetic samples generated by a deep learning model, a Parametric Gaussian Copula (PGC), a Copula-GAN hybrid, and our proposed Non-Parametric Gaussian Copula (NPGC). Existing methods exhibit visible deviations from the empirical marginal, whereas NPGC aligns perfectly with the real distribution.

Figure 2: Marginal density of the Capital-gain variable in the Adult [ 2 ] dataset across three sequential regeneration iterations of a Tabular Variational Autoencoder (TVAE). The shaded region denotes the empirical density of the original data, and red curves (iterations 1–3) represent synthetic samples generated under repeated synthetic feedback. The progressive concentration of probability mass indicates variance collapse and loss of distributional support across iterations.

Dataset	Instances ( $n$ )	Features ( $p$ )
Adult [ 2 ]	48,842	14
Balance Scale [ 43 ]	625	4
Nursery [ 32 ]	12,960	8
Student Dropout [ 30 ]	4,424	36
Student Performance [ 6 ]	649	33

Table 1: Benchmark datasets used for evaluation, showing the number of instances and features for each dataset.

Figure 3: Overview of NPGC. FIT: Empirical marginal estimation via empirical Cumulative Distribution Functions (CDFs) with Gaussian projection, followed by correlation estimation

\widehat{R}

with positive semi-definite (PSD) correction; the collection of marginals

\{\widehat{F}_{j}\}_{j=1}^{p}

, where

p

denotes the number of features (shown as

\widehat{F}

in the figure for clarity), and

\widehat{R}

are stored. SAMPLE: Independent Gaussian samples are correlated using Cholesky factorization

LL^{\top}

\widehat{R}

and mapped back through inverse empirical CDF reconstruction.

Figure 4: Marginal density of the Admission grade variable in the Student Dropout Success [ 30 ] dataset. The shaded region denotes the empirical density of the real data. Curves correspond to TGAN, TVAE, PGC, Copula-GAN, and the proposed NPGC. Deep learning and parametric baselines distort the marginal shape, while NPGC preserves it through empirical anchoring.

Model	Overall Score	Column Shapes	Pair Trends
NPGC	0.962	0.985	0.939
CTGAN	0.895	0.918	0.872
Copula-GAN	0.888	0.914	0.863
PGC	0.897	0.913	0.881
TVAE	0.832	0.883	0.781

Table 2: Average fidelity metrics across datasets. Overall Score reflects univariate (Column Shapes) and bivariate (Pair Trends) agreement between synthetic and real data. NPGC attains the highest fidelity across all metrics.

Model	Discriminator AUC	DCR Share
NPGC	0.801	0.527
CTGAN	0.853	0.567
Copula-GAN	0.871	0.558
PGC	0.850	0.549
TVAE	0.910	0.545

Table 3: Average privacy metrics across datasets. Lower values indicate reduced real–synthetic separability and lower memorization risk. NPGC achieves the strongest privacy performance.

Figure 5: Marginal density of the Education-num variable in the Adult [ 2 ] dataset across three sequential regeneration iterations of NPGC. The shaded region denotes the empirical density of the original data, and purple curves (iterations 1–3) represent synthetic samples generated under repeated regeneration. The overlap across iterations demonstrates stability of the marginal distribution.

Model	Synthetic Score	Accuracy Drop (%)
NPGC	0.739	16.99
CTGAN	0.619	25.73
Copula-GAN	0.647	22.80
PGC	0.675	17.04
TVAE	0.797	8.66

Table 4: Average Train-on-Synthetic, Test-on-Real (TSTR) utility metrics across datasets using XGBoost. Synthetic Score denotes classification accuracy when training on synthetic data and testing on real data, and Accuracy Drop measures the relative decrease compared to training on real data. NPGC achieves the second-highest utility.

Model	Train Time (s)	Eval Time (s)
NPGC	0.650	1.19
CTGAN	821.01	0.68
Copula-GAN	791.90	0.75
PGC	11.860	1.03
TVAE	268.46	0.34

Table 5: Average computational runtimes across datasets. Train Time denotes model fitting time, and Eval Time denotes sampling and evaluation time. NPGC achieves the lowest training time.

Figure 6: Overall Score (%) across 10 synthetic regeneration iterations on the Adult [ 2 ] dataset. The Overall Score aggregates univariate and bivariate distributional agreement between synthetic and real data. Curves correspond to CTGAN, Copula-GAN, PGC, TVAE, and NPGC.

Figure 7: Proportion of activity types in the real and synthetic interaction dataset. The shaded bars represent empirical proportions from 7,506 real records. Synthetic proportions from 10,332 samples generated with NPGC closely match the empirical distribution, including low-frequency categories such as Hint and Extra. NPGC illustrates stable marginal preservation under deployment.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Value
Default numerical distribution	beta
Enforce min/max values	True
Enforce rounding	True

Appendix

Table 6: Parametric Gaussian Copula (PGC) Hyperparameters

Parameter	Value
Epochs	300
Batch size	500
Embedding dimension	128
Generator hidden dimensions	(256, 256)
Discriminator hidden dimensions	(256, 256)
Generator learning rate	$2\times 10^{-4}$

Appendix

Table 7: Copula-GAN Hyperparameters

Parameter	Value
Epochs	300
Batch size	500
Embedding dimension	128
Generator hidden dimensions	(256, 256)
Discriminator hidden dimensions	(256, 256)
Generator learning rate	$2\times 10^{-4}$

Appendix

Table 8: CTGAN Hyperparameters

Parameter	Value
Epochs	300
Embedding dimension	128
Compression layers	(128, 128)
Decompression layers	(128, 128)
L2 regularization scale	$1\times 10^{-5}$
Loss factor	2

Appendix

Table 9: TVAE Hyperparameters

Parameter	Value
Differential privacy parameter ( $\epsilon$ )	1.0
Neural optimization	None
Marginal modeling	Empirical
Dependency modeling	Gaussian copula relation

Appendix

Table 10: NPGC Hyperparameters

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Apr 22, 2026J/K move · Enter open · S save

Synthetic Data in Education: Empirical Insights from Traditional Resampling and Deep Generative Models

Tapiwa Amion Chinodakufa, Ashfaq Ali Shafin, Khandaker Mamun Ahmed

The Beacom College of Computer & Cyber Sciences, Dakota State University · Knight Foundation School of Computing & Information Sciences, Florida International University

Synthetic Data Generation Educational Technology Synthetic Tabular Data Generation Synthetic Data Evaluation

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Synthetic data generation offers promise for addressing data scarcity and privacy concerns in educational technology, yet practitioners lack empirical guidance for selecting between traditional resampling techniques and modern deep learning approaches. This study presents the first systematic benchmark comparing these paradigms using a 10,000-record student performance dataset. We evaluate three resampling methods (SMOTE, Bootstrap, Random Oversampling) against three deep learning models (Autoencoder, Variational Autoencoder, Copula-GAN) across multiple dimensions: distributional fidelity (Kolmogorov-Smirnov distance, Jensen-Shannon divergence), machine learning utility such as Train-on-Synthetic-Test-on-Real scores (TSTR), and privacy preservation (Distance to Closest Record). Our findings reveal a fundamental trade-off: resampling methods achieve near-perfect utility (TSTR: 0.997) but completely fail privacy protection (DCR ~ 0.00), while deep learning models provide strong privacy guarantees (DCR ~ 1.00) at significant utility cost. Variational Autoencoders emerge as the optimal compromise, maintaining 83.3% predictive performance while ensuring complete privacy protection. We also provide actionable recommendations: use traditional resampling for internal development where privacy is controlled, and VAEs for external data sharing where privacy is paramount. This work establishes a foundational benchmark and practical decision framework for synthetic data generation in learning analytics.

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