Privacy-Preserving Synthetic Mobility Data

• Privacy-preserving synthetic mobility datasets are artificially generated mobility traces that retain key spatio-temporal patterns while safeguarding individual privacy.
• Advanced methods like phase-type dwell time modeling and quantile deep learning ensure accurate uncertainty estimation and high-fidelity data replication.
• These datasets support research in traffic prediction and system optimization by providing calibrated uncertainty measures and robust privacy guarantees.

A privacy-preserving synthetic mobility dataset is an artificially generated collection of mobility traces, typically constructed for the purpose of sharing spatial-temporal movement data (e.g., public transit, ride-sharing, individual trajectories) while safeguarding the privacy of the entities present in the original data. Such datasets are essential for research in traffic prediction, dwell/travel time quantile modeling, and system optimization, where access to granular real-world data would otherwise pose significant privacy risks. The development and validation of these datasets require precise quantification of uncertainty, explicit statistical modeling of delay and dwell time distributions, and well-calibrated mechanisms for estimating and verifying quantile estimates across time and space.

1. Statistical Foundations for Mobility Data Synthesis

Synthetic mobility datasets are constructed to replicate the essential statistical properties of observed mobility traces, such as spatio-temporal correlation, dwell time distributions, and traffic flow variability. Phase-type (PH) dwell time distributions provide a rigorous mathematical mechanism for modeling these features: a PH distribution is defined by a subgenerator matrix $T$ (encoding transitions between $k$ transient states in a Markov chain) and an initial distribution $\alpha$, with survival function $S(t) = \alpha e^{T t} \mathbf{1}$, density $$f(t) = -\frac{d}{dt} S(t) = \alpha e^{T t} (-T \mathbf{1})$$, and CDF $F(t) = 1 - S(t)$ (Hurtado et al., 2020). Embedding PH dwell times into ODE-based system models via the Generalized Linear Chain Trick (GLCT) enables the generation of synthetic traces that preserve realistic temporal dependencies and sojourn time characteristics.

The generation process typically involves:

• Specification of initial distributions and Markovian subgenerator matrices to calibrate sojourn times.
• Simulation of agent and vehicle dynamics under these stochastic rules.
• Extraction of quantiles and higher moments for uncertainty calibration.

2. Uncertainty and Quantile Modeling in Synthetic Datasets

Accurate modeling of uncertainty in synthetic datasets is critical, both to reflect natural stochastic variability and to prevent statistical artifacts that may risk re-identification. Deep learning models such as the Quantile Graph WaveNet are directly designed to estimate conditional $\tau$-quantiles of the output variable (e.g., dwell or travel times) as a function of recent trajectories and covariates (Maas et al., 2020). The pinball loss (quantile loss) is used in training to ensure that the output at a specific quantile index $\tau$ corresponds to the empirical $\tau$-quantile of the target:

$\rho_\tau(u) = (\tau - 1_{u<0}) u, \quad u = y - q$

Comprehensive uncertainty modeling includes:

• Design of asymmetric losses to capture distribution skewness in mobility traces.
• Direct parameterization of models to regress quantile functions $f_\theta(X,\tau) \to \widehat{y}_\tau$.
• Calibration and assessment via empirical coverage, pinball loss, and interval coverage metrics.

Well-calibrated quantile outputs are essential in synthetic data to avoid both underrepresentation of high-variance behavior and the introduction of anomalous features that might compromise privacy.

3. Construction and Evaluation of Synthetic Mobility Datasets

The construction of a privacy-preserving synthetic mobility dataset typically adopts the following workflow:

1. Graph Representation: Mobility networks are represented as weighted directed graphs $G=(V,E,W)$, with nodes for locations (stops, stations) and edges for transitions. Edge and node features include historical average flows, time-of-day, and event indicators.
2. Dynamic Simulation: Agent-based simulation or stochastic differential equation (SDE) models utilize calibrated dwell/travel time distributions to sample trajectories, dwell times, and interaction patterns.
3. Quantile Estimation: At each node (or edge), the network estimates a suite of quantiles for dwell or travel times, leveraging methods such as Quantile Graph WaveNet or ODE-embedded PH distributions (Hurtado et al., 2020, Maas et al., 2020).
4. Empirical Verification: For large $n$, empirical central limit theorems (CLTs) for quantiles provide the asymptotic joint distribution of empirical quantile curves over $(t, \alpha)$, enabling the formation of pointwise and simultaneous confidence bands (Kuelbs et al., 2011):

$$\sqrt{n}\bigl(\tau_n(\alpha, t) - \tau(\alpha, t)\bigr) \longrightarrow H(\alpha, t)$$

with $H$ a mean-zero Gaussian process depending on the underlying density at the quantile.

1. Privacy Assessment: Measures are taken to obscure re-identifiable patterns in the synthetic trajectories, typically via downsampling, perturbation, or synthetic-to-real mapping constraints, although such protocol details may not be explicit in every modeling study.

4. Model Calibration, Confidence Bands, and Empirical CLTs

Empirical CLTs for quantile processes ensure the statistical fidelity of synthetic mobility datasets by quantifying the sampling variability of quantile curves $\tau_n(\alpha, t)$ across both quantile level $\alpha$ and time or location $t$ (Kuelbs et al., 2011). Provided the marginal distribution at each $t$ is strictly increasing and smooth, and the empirical process satisfies suitable uniform entropy and bracketing conditions, the quantile process converges in distribution to a Gaussian process with explicit covariance structure.

Confidence intervals and bands for quantile estimates are constructed as:

$$\tau_n(\alpha, t) \pm \frac{z_{1-\gamma/2} \sqrt{\alpha(1-\alpha)}}{\sqrt{n}\, \hat{f}_t(\tau_n(\alpha, t))}$$

Uniform bands over $(t, \alpha)$ are derived from the supremum distribution of the Gaussian limit process, ensuring simultaneous statistical validity of all quantile estimates in the synthetic dataset. This is particularly relevant for mobility datasets where researchers require robust uncertainty quantification for downstream tasks.

5. Practical Applications in Traffic and Mobility Research

Privacy-preserving synthetic mobility datasets serve as vital resources for algorithm development and benchmarking in predictive modeling, real-time uncertainty quantification, and system optimization. Applications include:

• Construction and validation of spatio-temporal traffic prediction models—e.g., direct estimation of $\tau$-quantile travel or dwell time at graph nodes over time using pretrained deep neural architectures (Maas et al., 2020).
• Calibrated uncertainty intervals for graph-based traffic prediction, essential for robust transit system operation, informed scheduling, and passenger information systems.
• Simulation and validation of mechanistic transport processes (e.g., queueing, vehicle boarding, or spatial flow) with explicit PH-type sojourn times embedded in mean-field ODEs (Hurtado et al., 2020).
• Empirical analysis of coverage rates, pinball losses, and calibration curves on benchmark datasets, enabling standardized comparison between algorithms.

The utility of these datasets is contingent on rigorous adherence to privacy guarantees and validated statistical calibration, as achieved by the combination of distributional modeling, empirical quantile CLTs, and uncertainty estimation methods.

6. Challenges and Limitations

While synthetic mobility datasets address key privacy concerns, challenges persist in achieving both high-fidelity statistical realism and strong privacy protection. The dependence structures (spatio-temporal autocorrelation), marginal and joint quantile distributions, and intrinsic stochasticity of real-world data are nontrivial to capture in synthetic form without risking leakage or introducing artifacts.

A plausible implication is that quantitative privacy assessment (e.g., membership inference risk) and the preservation of distributional properties (e.g., calibration of uncertainty bands, alignment of dwell/travel time quantiles across the mobility network), as facilitated by statistically robust ODE, PH-distributions, and quantile deep learning methods, must be iteratively harmonized. This aligns with the observed approach of calibrating synthetic data with empirical CLTs and coverage-based metrics (Kuelbs et al., 2011, Maas et al., 2020).