Synthetic Long-Horizon Data Construction

• Synthetic long-horizon data construction is the process of creating datasets that accurately replicate extended temporal transitions and structural properties for robust scientific inference.
• Methodologies such as complete and sequential conditional synthesis use generative models (e.g., GANs, VAEs, CART) to preserve both cross-sectional and longitudinal relationships.
• Advanced techniques including variance estimation, multi-resolution modeling, and rigorous validation protocols ensure synthetic datasets remain analytically useful and privacy-preserving.

Synthetic long-horizon data construction refers to the principled creation of synthetic datasets that capture, over extended time periods, the statistical, temporal, and structural properties essential for valid scientific inference, exploratory analysis, simulation, or privacy-preserving data release. These datasets are constructed so that they not only reproduce cross-sectional characteristics at each time point but also accurately encode the temporal transitions and dependencies that occur in real-world longitudinal or time-evolving phenomena.

1. Principles and Foundations

The foundational methodologies of synthetic long-horizon data construction are rooted in generative modeling frameworks that seek to emulate the joint distribution of the observed variables across time. Central to this paradigm is the synthesising distribution assumption (SDA), which stipulates that if the synthetic data are drawn from a correctly specified model of the joint distribution, then statistical relationships—including those governing temporal transitions—are preserved for inferential purposes (Raab et al., 2014, Drechsler et al., 2023).

There are two principal strategies:

• Complete synthesis: All observed values for the variables of primary interest ($Y_{obs}$) are replaced with new, model-generated synthetic values, while potentially retaining auxiliary or design variables ($X$) to anchor certain aspects of the data design (e.g., sampling weights, stratification).
• Sequential conditional synthesis: The joint distribution $P(Y_{1:T}|X)$ is modeled via a series of conditional distributions, synthesizing each time period's values conditional on previous values and $X$. This is especially suited to longitudinal datasets and is exemplified by sequential regression or nonparametric techniques such as Classification and Regression Trees (CART) (Raab et al., 2014).

Ensuring that synthetic data accurately mimics longitudinal transitions is essential. For instance, in the Scottish Longitudinal Study, variables like age progression, marital status, and health transitions are synthesized to closely follow their empirical evolution (Raab et al., 2014).

2. Methodologies for Synthetic Long-Horizon Data

The construction pipeline typically involves model building, simulation, and integration steps:

• Model specification: Parametric models (such as sequential logistic regression) or non-parametric models (such as CART) are fit to observed data at each time point, enabling the conditional generation of future responses. Machine learning-based generative models, such as GANs or VAEs, are also used for high-dimensional or non-linear time series (Pinceti et al., 2021, Drechsler et al., 2023).
• Posterior Predictive Distribution (PPD) sampling: Traditionally, synthetic data are drawn from the full PPD, accounting for parameter uncertainty. For computational efficiency, especially at scale, plug-in approaches where draws are based on fixed parameter estimates are also utilized, leading to simplified, yet valid, uncertainty quantification (Raab et al., 2014).
• Hierarchical or multi-resolution modeling: In settings with natural time-scale stratification (e.g., power load profiles), data are modeled at several resolutions (e.g., sub-second, hourly, weekly, annual), and synthetic long-horizon series are constructed by stacking or blending generated outputs across these levels (Pinceti et al., 2021).
• Stitching and smoothing: For time series assembled from many separately modeled segments (e.g., concatenated weekly or hourly profiles), smoothing filters or scaling operations are applied to correct discontinuities and ensure realistic transitions between adjacent profile segments (Pinceti et al., 2021).
• Domain-specific augmentations: In astrophysics, forward simulation via deterministic physical models (e.g., GRMHD for black hole accretion) combined with injection of realistic measurement corruptions (via, e.g., the Radio Interferometry Measurement Equation) yields synthetic time series that reproduce instrumental, atmospheric, and calibration errors (Natarajan et al., 2022, Janssen et al., 16 Jun 2025).

3. Statistical Inference and Variance Estimation

Accurate statistical inference from synthetic long-horizon data depends on robust variance estimators that account for both model-based uncertainty and additional variability induced by the synthesis process. Key developments include:

• Variance estimators: For fully synthesized data, with $M$ synthetic replications and per-replication estimates $q^{(l)}$ (variance $v^{(l)}$),
  • Average point estimate: $\bar{q}_M = \frac{1}{M} \sum_{l=1}^M q^{(l)}$
  • Within-synthesis variance: $\bar{v}_M = \frac{1}{M} \sum_{l=1}^M v^{(l)}$
  • Between-synthesis variance: $$b_M = \frac{1}{M-1} \sum_{l=1}^M (q^{(l)} - \bar{q}_M)^2$$
  • Simplified variance estimators (Raab et al., 2014, Drechsler et al., 2023):
  • When sampling from PPD: $T_s(\text{PPD}) = \bar{v}_M (1 + 2/M)$
  • Without PPD: $T_s = \bar{v}_M (1 + 1/M)$
  • If sample sizes differ between observed ($n$) and synthetic ($k$): $T_s = \bar{v}_M (k/n + 1/M)$
• Applicability: These estimators have been shown to yield nearly the same confidence interval coverage for regression coefficients and other estimands as those computed from the original confidential data, even when only a single synthetic dataset is generated (Raab et al., 2014).
• Practical benefit: The ability to compute valid uncertainty intervals from a single synthetic dataset without repeated PPD draws is especially valuable when synthesizing large, high-dimensional, or long-horizon data where computational cost is prohibitive.

4. Preservation of Temporal and Structural Relationships

A central concern in synthetic long-horizon data construction is the preservation, both at the macro (aggregate statistics, marginal distributions) and micro (conditional dynamics, state transitions, rare event trajectories) level, of relationships over time:

• Temporal coherence: Sequential conditional synthesis naturally maintains temporal trajectories. For example, synthetic trajectories of age, marital status, and health must reflect the actual transition dynamics as observed in longitudinal studies (Raab et al., 2014).
• Flexible interaction modeling: Non-parametric methods, notably CART, have been shown to better preserve complex interactions or "natural" patterns (e.g., deterministic increments, conditional effects) compared to purely parametric models in empirical studies (Raab et al., 2014).
• Multi-scale consistency: In multi-resolution generative frameworks, scaling and trend-recovery operations ensure that fine-grained series are consistent with slower (annual, seasonal) variations. For instance, week-long profiles may be rescaled to fit annual synthetic trends by matching weekly means (Pinceti et al., 2021).
• Synthetic data for physically governed processes: In domains with deterministic or partially deterministic generative mechanisms (e.g., astrophysics), the synthetic pipeline must incorporate both physical law (e.g., ray-tracing images) and realistic distortions (calibration errors, atmospheric turbulence) to produce credibly realistic long-horizon time series (Natarajan et al., 2022, Janssen et al., 16 Jun 2025).

5. Validation, Utility, and Disclosure Risk Assessment

Validation of synthetic long-horizon data encompasses statistical similarity to the real data and utility for intended analytical tasks:

• Global utility: Measures such as the Wasserstein distance, Kullback-Leibler divergence, and propensity score mean-squared error (pMSE) are applied to compare global distributions. For time series, spectral analysis (e.g., power spectral density) ensures that synthetic samples match the temporal dependence of the target process (Pinceti et al., 2021, Drechsler et al., 2023).
• Outcome-specific utility: The congruence of regression coefficients or other model-based estimators between synthetic and observed data is an expected threshold. Simulations confirm that synthetic data constructed through SDA-based methods can recover key relationships necessary for analysis (Raab et al., 2014, Drechsler et al., 2023).
• Graphical and task-based validation: For time-dependent processes, visual checks (e.g., trajectory overlays, trend plots) and out-of-sample forecasting experiments (e.g., using LSTMs on synthetic power load profiles) provide additional assurance (Pinceti et al., 2021).
• Disclosure risk: For fully synthetic data, risk is assessed via the probability that synthetic records can be matched or attributed to the original; metrics such as Within Equivalence Class Attribution Probability (WEAP) or match risk rates are used. Preserving uniqueness in time-evolving sequences (e.g., non-repetitive transitions) presents additional disclosure control challenges for long-horizon data (Drechsler et al., 2023).

6. Practical Applications and Implications

Synthetic long-horizon datasets are increasingly used across domains:

• Statistical agencies: For longitudinal national surveys and administrative databases (e.g., the Scottish Longitudinal Study), synthetic datasets with proper disclosure control enable public release and facilitate reproducible research while safeguarding confidentiality (Raab et al., 2014, Drechsler et al., 2023).
• Engineering and industrial systems: In power systems, synthetic load profiles spanning seconds to years, constructed from multi-resolution generative models, support tasks in planning, simulation, and forecasting where access to real world data is often restricted (Pinceti et al., 2021).
• Scientific modeling and instrumental calibration: In radio astronomy, synthetic VLBI datasets incorporating physical model variability and real-world calibration artifacts enable robust testing of imaging and parameter inference pipelines across long observational baselines (Natarajan et al., 2022, Janssen et al., 16 Jun 2025).
• Software tools: The release of open-source applications (such as LoadGAN for synthetic load generation) democratizes access and streamlines the creation of customized synthetic long-horizon datasets for non-expert users (Pinceti et al., 2021).
• Privacy preserving analysis: Strong theoretical guarantees on variance estimation and risk metrics position synthetic long-horizon data as crucial for balancing analytical utility and privacy protection in sensitive data domains (Raab et al., 2014, Drechsler et al., 2023).

In summary, synthetic long-horizon data construction leverages sequential conditional modeling, advanced variance estimation, and rigorous validation protocols to generate datasets that simultaneously preserve key statistical relationships and safeguard individual privacy over extended time frames. Practical methodologies—including both parametric and flexible machine learning-based techniques—yield synthetic data products suitable for complex statistical analysis, simulation, and secure open data dissemination.