Robust Variants of PASTIS

• The paper introduces robust extensions to the PASTIS criterion that correct estimation bias in SDEs and SPDEs under non-ideal sampling conditions.
• It employs a higher-order trapeze rule (PASTIS–Δt) and noise-cancelling techniques (PASTIS–σ) to rectify drift and likelihood estimation errors.
• Empirical tests show these methods outperform traditional criteria, ensuring reliable model recovery in noisy, coarsely sampled data.

Robust variants of PASTIS refer to statistically and algorithmically grounded extensions of the Parsimonious Stochastic Inference (PASTIS) model selection criterion, which are specifically developed to ensure reliable model identification for stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) under imperfect sampling and measurement conditions. These variants address and correct for estimation biases introduced by large temporal sampling intervals and measurement noise, thereby extending the applicability and reliability of PASTIS in practical scenarios characterized by noisy or coarsely sampled data (Gerardos, 5 Jul 2025).

1. Foundations of PASTIS and the Need for Robustness

PASTIS is an information criterion derived from extreme value theory and designed to select the simplest adequate model from a large set of candidates when learning stochastic dynamics. The central insight motivating the PASTIS penalty structure is that, as the size of the library of candidate basis functions ($n_0$) increases, the probability of spuriously improving the likelihood by chance alone also increases, but does so only logarithmically. The PASTIS criterion takes the form:

$$I_{\mathrm{PASTIS}}(\mathcal{B}) = \hat{\ell}(\mathcal{B}) - n_{\mathcal{B}} \ln (n_0/p)$$

where $\hat{\ell}(\mathcal{B})$ is the maximized log-likelihood for basis $\mathcal{B}$, $n_{\mathcal{B}}$ is the number of parameters in the candidate model, $n_0$ is the size of the initial library, and $p$ is a user-chosen significance threshold.

However, the derivation of this criterion assumes data are ideally sampled—i.e., with infinitesimal time steps and negligible observation noise. In practical experimental settings, large timestep intervals ($\Delta t$) and nontrivial measurement noise ($\sigma$) are unavoidable and can lead to biased or inconsistent parameter estimation. Robust variants of PASTIS are introduced to address these issues and maintain the fidelity of the model selection process under real-world data acquisition regimes.

2. PASTIS–$\Delta t$: Robustness to Finite Sampling Intervals

When observed time series data are sampled at finite (potentially large) time intervals, standard drift estimation for SDEs employs the Euler–Maruyama method, which suffers from first-order accuracy in $\Delta t$. This error can bias both the drift reconstruction and likelihood evaluation, leading to suboptimal model selection.

The robust variant PASTIS–$\Delta t$ corrects for this by employing a higher-order "trapeze" rule for the approximation of the stochastic integral. Instead of using

$$\int_{t}^{t+\Delta t} f(x(s))\,ds \approx f(x_t)\,\Delta t$$

the method approximates the integral as

$$\int_{t}^{t+\Delta t} f(x(s))\,ds \approx \frac{1}{2}\left[ f(x_t) + f(x_{t+\Delta t}) \right] \Delta t$$

This modification yields a reconstructed estimator whose bias decreases as $\sim (\Delta t)^4$ (from the leading-order Taylor error), which is a substantial improvement over the $\sim (\Delta t)^2$ bias of the basic method. As a result, PASTIS–$\Delta t$ maintains accurate and robust model selection even when time series data are sampled with moderate to large $\Delta t$ (Gerardos, 5 Jul 2025).

3. PASTIS–$\sigma$: Robustness to Measurement Noise

Measurement noise is pervasive in experimental data, often modeled as additive Gaussian noise: $y_t = x_t + \eta_t$ with $\eta_t \sim \mathcal{N}(0, \sigma^2)$. When finite differences are computed for drift estimation, the noise increment $\Delta \eta_t / \Delta t$ has variance scaling inversely with $\Delta t$, causing divergence as $\Delta t \rightarrow 0$ and introducing spurious correlations with the basis functions evaluated at noisy points.

PASTIS–$\sigma$ introduces two bias-canceling estimators to confront this issue:

• Shift Estimator: Evaluates the drift basis at a time point uncorrelated with the noise increment (e.g., $f(y_{t-\Delta t})$ for estimating the drift at time $t$), thereby breaking the spurious correlation.
• Stratonovich-like Averaging: Utilizes the average value of the process at the beginning and end of the interval, i.e., $f\big( \frac{y_t + y_{t+\Delta t}}{2} \big)$, along with a correction term akin to the Itô–Stratonovich conversion, thereby eliminating leading order bias due to noise.

These corrections are integrated into the computation of the log-likelihood used by PASTIS, preventing measurement noise from artificially boosting likelihoods and thereby making overfitting less probable. As a result, PASTIS–$\sigma$ reliably maintains model selection accuracy across a broad range of signal-to-noise ratios (Gerardos, 5 Jul 2025).

4. Unified Criterion and Penalty Structure

Robust variants of PASTIS retain the same penalty structure as the original criterion. The log-likelihood $\hat{\ell}$ employed in $I_{\mathrm{PASTIS}}$ is replaced by its robustly estimated counterpart, while the penalty term

$n_{\mathcal{B}} \ln(n_0/p)$

remains unchanged. This ensures that the fundamental property of PASTIS—that the expected spurious increase in log-likelihood from redundant parameters is properly penalized—holds even under imperfect data conditions. Extreme value theory continues to provide the asymptotic justification for the penalty, as the maximal gain from noise remains order $\ln n_0$ regardless of the particular estimator used for the drift or likelihood.

5. Empirical Performance and Practical Recommendations

Benchmarks conducted on canonical systems (including the Lorenz system, Ornstein-Uhlenbeck process, Lotka-Volterra equations for SDEs, and Gray-Scott system for SPDEs) demonstrate that both PASTIS–$\Delta t$ and PASTIS–$\sigma$ outperform traditional criteria (AIC, BIC) and methods such as cross-validation and SINDy in terms of exact model recovery and predictive accuracy (Gerardos, 5 Jul 2025). Specifically, these robust variants prevent the inclusion of spurious model terms that would otherwise be selected due to estimation bias under large $\Delta t$ or sizable measurement noise.

For practitioners, the adoption of PASTIS–$\Delta t$ and/or PASTIS–$\sigma$ is advised when:

• The available time series are sampled at intervals not negligible compared to the system's intrinsic timescales.
• There is significant measurement noise, or when the signal-to-noise ratio is uncertain.
• Model discovery is applied to experimental data outside tightly controlled settings.

In such contexts, the robust variants ensure parsimonious model selection with principled false discovery rate control, reflecting only genuine dynamical structure in the underlying process.

6. Significance for Model Discovery in Stochastic Dynamics

The robust extensions to PASTIS exemplify a methodological advance in the intersection of information criteria, extreme value theory, and practical model selection for stochastic dynamics. They address fundamental challenges that have historically hampered the application of sparse model discovery to experimental SDE and SPDE systems, namely: setting an appropriate penalty as the library of candidate models grows, and correcting for estimation biases induced by non-ideal data.

This approach offers a coherent and unified pipeline for:

• Sparse identification of differentiated drift terms and nonlinearities under realistic data limitations;
• Statistically valid model selection with user-controllable probabilities for including superfluous basis terms;
• Application to both temporal and spatial (SPDE) systems, given the generality of the underlying theory.

A plausible implication is that wider adoption of robust PASTIS variants will enable more reliable inference of governing equations from diverse, noisy, or irregularly sampled observations in physics, biology, quantitative finance, and beyond.

7. Summary Table: PASTIS and Its Robust Variants

Variant	Data Challenge Addressed	Key Adjustment
PASTIS	Ideally sampled, low-noise	Extreme value–motivated penalty; standard drift reconstruction
PASTIS–$\Delta t$	Large sampling intervals	Trapeze rule for drift integral; reduces bias
PASTIS–$\sigma$	Measurement noise	Shifted or averaged estimators to break noise-basis correlation

The robust variants of PASTIS thus provide a statistically principled and practically validated framework for model selection in stochastic dynamical systems, tailored for the complications inherent in real-world experimental data (Gerardos, 5 Jul 2025).