Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Temporal Conformal Prediction (TCP): A Distribution-Free Statistical and Machine Learning Framework for Adaptive Risk Forecasting

Agnideep Aich, Ashit Baran Aich, Dipak C. Jain

TL;DR

Overall, TCP provides a practical, theoretically grounded solution to calibrated uncertainty quantification under distribution shift, bridging statistical inference and machine learning for risk forecasting.

Abstract

We propose \textbf{Temporal Conformal Prediction (TCP)}, a distribution-free framework for constructing well-calibrated prediction intervals in nonstationary time series. TCP couples a modern quantile forecaster with a split-conformal calibration layer on a rolling window and, in its \textbf{TCP-RM} variant, augments the conformal threshold with a single online Robbins-Monro (RM) offset to steer coverage toward a target level in real time. We benchmark TCP against GARCH, Historical Simulation, a rolling tree-based Quantile Regression (QR) model, a classical linear quantile regression baseline (QR-Linear), and an adaptive conformal method (ACI) across equities (S\&P 500), cryptocurrency (Bitcoin), and commodities (Gold). Three results are consistent across assets. First, both QR and QR-Linear yield the sharpest intervals but are materially under-calibrated, and even ACI remains below the nominal 95\% target in our full-sample backtests. Second, TCP (and TCP-RM) achieves near-nominal coverage across assets, with intervals that are wider than Historical Simulation in this evaluation (e.g., S\&P 500: 5.21 vs.\ 5.06). Third, the RM update changes calibration and width only marginally at our default hyperparameters. Crisis-window visualizations around March 2020 show TCP/TCP-RM expanding and then contracting their interval bands promptly as volatility spikes and recedes, with \textbf{red dots} marking days where realized returns fall outside the reported 95\% interval (miscoverage). A sensitivity study confirms robustness to window size and step-size choices. Overall, TCP provides a practical, theoretically grounded solution to calibrated uncertainty quantification under distribution shift, bridging statistical inference and machine learning for risk forecasting.

Temporal Conformal Prediction (TCP): A Distribution-Free Statistical and Machine Learning Framework for Adaptive Risk Forecasting

TL;DR

Overall, TCP provides a practical, theoretically grounded solution to calibrated uncertainty quantification under distribution shift, bridging statistical inference and machine learning for risk forecasting.

Abstract

We propose \textbf{Temporal Conformal Prediction (TCP)}, a distribution-free framework for constructing well-calibrated prediction intervals in nonstationary time series. TCP couples a modern quantile forecaster with a split-conformal calibration layer on a rolling window and, in its \textbf{TCP-RM} variant, augments the conformal threshold with a single online Robbins-Monro (RM) offset to steer coverage toward a target level in real time. We benchmark TCP against GARCH, Historical Simulation, a rolling tree-based Quantile Regression (QR) model, a classical linear quantile regression baseline (QR-Linear), and an adaptive conformal method (ACI) across equities (S\&P 500), cryptocurrency (Bitcoin), and commodities (Gold). Three results are consistent across assets. First, both QR and QR-Linear yield the sharpest intervals but are materially under-calibrated, and even ACI remains below the nominal 95\% target in our full-sample backtests. Second, TCP (and TCP-RM) achieves near-nominal coverage across assets, with intervals that are wider than Historical Simulation in this evaluation (e.g., S\&P 500: 5.21 vs.\ 5.06). Third, the RM update changes calibration and width only marginally at our default hyperparameters. Crisis-window visualizations around March 2020 show TCP/TCP-RM expanding and then contracting their interval bands promptly as volatility spikes and recedes, with \textbf{red dots} marking days where realized returns fall outside the reported 95\% interval (miscoverage). A sensitivity study confirms robustness to window size and step-size choices. Overall, TCP provides a practical, theoretically grounded solution to calibrated uncertainty quantification under distribution shift, bridging statistical inference and machine learning for risk forecasting.

Paper Structure

This paper contains 33 sections, 2 theorems, 15 equations, 9 figures, 12 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. Adaptive and Time-Series Conformal Prediction.
  4. NOTATION
  5. MATHEMATICAL BACKGROUND
  6. Problem Formulation
  7. Classical Conformal Prediction
  8. Temporal Conformal Prediction (TCP)
  9. Adaptive Calibration (TCP vs. TCP-RM)
  10. Clarifications on Assumptions (A1--A4).
  11. Proof of Asymptotic Calibration
  12. METHODOLOGY
  13. Data and Feature Construction
  14. Forecasting Models
  15. Temporal Conformal Prediction (TCP)
  16. ...and 18 more sections

Key Result

Theorem 4.1

If the sequence of pairs $\{(X_i,Y_i)\}_{i=1}^{n+1}$ is exchangeable, then the conformal prediction set $\Gamma_{1-\alpha}$ satisfies:

Figures (9)

  • Figure 1: A comparison of 95% prediction intervals from six models (TCP, TCP-RM, QR, GARCH, Hist, ACI) for S&P 500 daily returns during the COVID-19 market crash (Feb–Apr 2020). Shaded bands show the interval; the red dots mark days where the realized return falls outside the 95% interval (miscoverage). TCP/TCP-RM bands widen rapidly into the March spike and contract in April, while ACI maintains relatively narrow bands and under-covers more frequently. Analogous crisis-window panels for BTC-USD and Gold appear in Appendix \ref{['app:Appendix B']}.
  • Figure 2: S&P 500: TCP vs TCP–RM 95% prediction intervals during the COVID crisis window (Feb–Apr 2020).
  • Figure 3: BTC-USD: TCP vs TCP-RM 95% prediction intervals during the COVID crisis window (Feb–Apr 2020).
  • Figure 4: Gold: TCP vs TCP-RM 95% prediction intervals during the COVID crisis window (Feb–Apr 2020).
  • Figure 5: A comparison of 95% prediction intervals from five models (TCP, TCP-RM, QR, GARCH, Hist, ACI) for BTC-USD daily returns during the COVID-19 market crash (February-April 2020).
  • ...and 4 more figures

Theorems & Definitions (4)

  • Definition 4.1: Non-Conformity Score
  • Theorem 4.1: Finite-Sample Validity
  • Theorem 4.2
  • Remark 4.1: Practical stability