Modular and Adaptive Conformal Prediction for Sequential Models via Residual Decomposition

TL;DR

This work extends conformal prediction to modular two-stage sequential models by decomposing end-to-end residuals into upstream and downstream components, enabling explicit attribution of uncertainty to each stage. It develops a risk-controlled calibration framework that selects scaling parameters via FWER-based hypothesis testing on a calibration set, ensuring valid coverage and interpretable diagnostics, and it introduces an adaptive variant that updates parameters in response to nonstationary shifts while preserving long-run coverage. The approach is validated on synthetic distribution shifts and real-world data (supply chains, finance, auto indicators), showing robust coverage under stage-specific disturbances where black-box conformal methods falter, along with actionable insights for targeted retraining. The framework thus provides both reliable predictive intervals and transparent, stage-wise uncertainty diagnostics that support diagnostic intervention and modular robustness in practical pipelines.

Abstract

Conformal prediction offers finite-sample coverage guarantees under minimal assumptions. However, existing methods treat the entire modeling process as a black box, overlooking opportunities to exploit modular structure. We introduce a conformal prediction framework for two-stage sequential models, where an upstream predictor generates intermediate representations for a downstream model. By decomposing the overall prediction residual into stage-specific components, our method enables practitioners to attribute uncertainty to specific pipeline stages. We develop a risk-controlled parameter selection procedure using family-wise error rate (FWER) control to calibrate stage-wise scaling parameters, and propose an adaptive extension for non-stationary settings that preserves long-run coverage guarantees. Experiments on synthetic distribution shifts, as well as real-world supply chain and stock market data, demonstrate that our approach maintains coverage under conditions that degrade standard conformal methods, while providing interpretable stage-wise uncertainty attribution. This framework offers diagnostic advantages and robust coverage that standard conformal methods lack.

Figures (15)

• Figure 1: Visualization of a sequential two-stage model with residual components $R,\Delta R_1, R_2$.
• Figure 2: Comparison of interval width and coverage under synthetic distribution shifts
• Figure 3: Coverage of prediction intervals for $\alpha = 0.2$, $\delta = 0.3$, $\gamma = 0.01$, $\eta = 0.01$, $k=40$
• Figure 4: Coverage of prediction intervals for $\alpha = 0.1$, $\delta = 0.3$, $\gamma = 0.01$, $\eta = 0.01$, $k=24$
• Figure 5: Performance of various intervals on the stocks dataset for $\alpha = 0.1$, $\delta = 0.3$, $\gamma = 0.01$, $\eta = 0.01$, $k=24$.

Theorems & Definitions (37)

• Definition 1: Second-stage residual
• Definition 2: First-stage delta
• Proposition 1
• Definition 3: Separate component quantiles
• Theorem 1: Coverage of separate component quantiles
• Definition 4: Scaled component quantiles
• Corollary 1: Coverage with $a = b = 1$
• proof
• Proposition 2: Existence of ideal scaling parameters
• Theorem 2: Risk control via FWER calibration