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Conformalized Signal Temporal Logic Inference under Covariate Shift

Yixuan Wang, Danyang Li, Matthew Cleaveland, Roberto Tron, Mingyu Cai

Abstract

Signal Temporal Logic (STL) inference learns interpretable logical rules for temporal behaviors in dynamical systems. To ensure the correctness of learned STL formulas, recent approaches have incorporated conformal prediction as a statistical tool for uncertainty quantification. However, most existing methods rely on the assumption that calibration and testing data are identically distributed and exchangeable, an assumption that is frequently violated in real-world settings. This paper proposes a conformalized STL inference framework that explicitly addresses covariate shift between training and deployment trajectories dataset. From a technical standpoint, the approach first employs a template-free, differentiable STL inference method to learn an initial model, and subsequently refines it using a limited deployment side dataset to promote distribution alignment. To provide validity guarantees under distribution shift, the framework estimates the likelihood ratio between training and deployment distributions and integrates it into an STL-robustness-based weighted conformal prediction scheme. Experimental results on trajectory datasets demonstrate that the proposed framework preserves the interpretability of STL formulas while significantly improving symbolic learning reliability at deployment time.

Conformalized Signal Temporal Logic Inference under Covariate Shift

Abstract

Signal Temporal Logic (STL) inference learns interpretable logical rules for temporal behaviors in dynamical systems. To ensure the correctness of learned STL formulas, recent approaches have incorporated conformal prediction as a statistical tool for uncertainty quantification. However, most existing methods rely on the assumption that calibration and testing data are identically distributed and exchangeable, an assumption that is frequently violated in real-world settings. This paper proposes a conformalized STL inference framework that explicitly addresses covariate shift between training and deployment trajectories dataset. From a technical standpoint, the approach first employs a template-free, differentiable STL inference method to learn an initial model, and subsequently refines it using a limited deployment side dataset to promote distribution alignment. To provide validity guarantees under distribution shift, the framework estimates the likelihood ratio between training and deployment distributions and integrates it into an STL-robustness-based weighted conformal prediction scheme. Experimental results on trajectory datasets demonstrate that the proposed framework preserves the interpretability of STL formulas while significantly improving symbolic learning reliability at deployment time.

Paper Structure

This paper contains 25 sections, 4 theorems, 40 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Signal Temporal Logic Inference
  4. Misclassification rate (MCR)
  5. Differentiable STL learning (TLINet)
  6. Conformal Prediction
  7. Problem Formulation
  8. Data distributions and covariate shift
  9. Data partitioning
  10. Problem Formulation
  11. Conformalized Active STL Inference under Covariate Shift
  12. STL Inference Adaptation
  13. Density-Ratio Weighting on the Alignment Set
  14. Regularization of Robustness-Value Distributions via Jensen--Rényi Divergence
  15. Weighted Conformalized STL Inference
  16. ...and 10 more sections

Key Result

Lemma 1

Under Assumption sec:assump1, assume that $P_{\mathrm{dep}}$ is absolutely continuous with respect to $P_{\mathrm{train}}$ over $X$. Then, with ideal density-ratio weight $\omega^\star(X)$, for any measurable function $h:\mathcal{X}\times\{-1,+1\}\to\mathbb{R}$ with finite expectation, we have

Figures (5)

  • Figure 1: Pipeline of proposed shift-aware conformal STL framework under covariate shift. The framework consists of four sequential stages integrating STL formula learning, distribution alignment, and weighted conformal calibration.
  • Figure 2: Schematic Diagram of the Dataset
  • Figure 3: Weighted CP with Covariate Shift
  • Figure 4: Comparison across datasets with weighted CP.
  • Figure 5: Comparison between weighted CP and Standard CP using our method in different dataset.

Theorems & Definitions (10)

  • Remark 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Definition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof