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Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series

Zitong Yang, Emmanuel Candès, Lihua Lei

TL;DR

Bellman Conformal Inference (BCI) tackles calibrated uncertainty quantification for time series by wrapping any multi-step forecast into a stochastic control problem. It uses Model Predictive Control to explicitly optimize the trade-off between average interval length and short-term calibration, solving a dynamic programming problem to produce a sequence of nominal miscoverage levels while maintaining long-run calibration at $\overline{\alpha}$. The approach guarantees calibrated prediction intervals under distribution shifts without strong model assumptions, and empirically reduces interval lengths relative to Adaptive Conformal Inference (ACI), avoiding infinite or uninformative intervals when priors are poorly calibrated. Overall, BCI provides a practical, theoretically grounded framework for reliable online uncertainty quantification in nonstationary time series with broad applicability to finance and non-financial domains alike.

Abstract

We introduce Bellman Conformal Inference (BCI), a framework that wraps around any time series forecasting models and provides approximately calibrated prediction intervals. Unlike existing methods, BCI is able to leverage multi-step ahead forecasts and explicitly optimize the average interval lengths by solving a one-dimensional stochastic control problem (SCP) at each time step. In particular, we use the dynamic programming algorithm to find the optimal policy for the SCP. We prove that BCI achieves long-term coverage under arbitrary distribution shifts and temporal dependence, even with poor multi-step ahead forecasts. We find empirically that BCI avoids uninformative intervals that have infinite lengths and generates substantially shorter prediction intervals in multiple applications when compared with existing methods.

Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series

TL;DR

Bellman Conformal Inference (BCI) tackles calibrated uncertainty quantification for time series by wrapping any multi-step forecast into a stochastic control problem. It uses Model Predictive Control to explicitly optimize the trade-off between average interval length and short-term calibration, solving a dynamic programming problem to produce a sequence of nominal miscoverage levels while maintaining long-run calibration at . The approach guarantees calibrated prediction intervals under distribution shifts without strong model assumptions, and empirically reduces interval lengths relative to Adaptive Conformal Inference (ACI), avoiding infinite or uninformative intervals when priors are poorly calibrated. Overall, BCI provides a practical, theoretically grounded framework for reliable online uncertainty quantification in nonstationary time series with broad applicability to finance and non-financial domains alike.

Abstract

We introduce Bellman Conformal Inference (BCI), a framework that wraps around any time series forecasting models and provides approximately calibrated prediction intervals. Unlike existing methods, BCI is able to leverage multi-step ahead forecasts and explicitly optimize the average interval lengths by solving a one-dimensional stochastic control problem (SCP) at each time step. In particular, we use the dynamic programming algorithm to find the optimal policy for the SCP. We prove that BCI achieves long-term coverage under arbitrary distribution shifts and temporal dependence, even with poor multi-step ahead forecasts. We find empirically that BCI avoids uninformative intervals that have infinite lengths and generates substantially shorter prediction intervals in multiple applications when compared with existing methods.
Paper Structure (29 sections, 2 theorems, 42 equations, 6 figures, 2 tables)

This paper contains 29 sections, 2 theorems, 42 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. Multi-step ahead prediction intervals
  4. Calibrating prediction intervals
  5. Bellman Conformal Inference
  6. BCI as Model Predictive Control
  7. BCI update rule for Lg and Lg
  8. Solving Lg via dynamic programming
  9. Empirical results
  10. Dataset and model fitting
  11. Return forecasting.
  12. Volatility forecasting.
  13. Google trend popularity.
  14. Performance evaluation
  15. Evaluation metrics.
  16. ...and 14 more sections

Key Result

Theorem 1

Let $\lambda_t$ and $\alpha_t$ be defined by eqn:lambda-update and eqn:alpha-t-truncate, respectively. Assume $\alpha_{t|t}^{*}$ is obtained by an algorithm satisfying eqn:trivial_property. Under Assumption ass:pred-sets, when $\gamma = c \lambda_{\max}$ and $\gamma\in (0, \lambda_{\max})$, for any In particular, eqn:global-guarantee holds by letting $m=0$ and $T\rightarrow \infty$.

Figures (6)

  • Figure 1: Online time series forecasting for three different tasks: return forecasting on AMD stock data, volatility forecasting on Amazon stock data, and Google search popularity data for keyword "deep leanring". Top panel: moving averages of miscoverage rates over 500 data points. Bottom panel: moving averages of prediction interval lengths. In all figures, the red curves correspond to our proposed BCI algorithm; the blue curves correspond to the ACI algorithm with stepsize $0.1$; the gray curves correspond to setting $\alpha_t=\overline{\alpha}$ for all $t$.
  • Figure 2: Schematic illustrations of standard ACI and BCI.
  • Figure 3: Same as Figure \ref{['fig:exp-main-tight']}, except that the stepsize $\gamma$ for ACI is $0.08$ for looser control.
  • Figure 4: Expected calibration curve of nominal prediction intervals for Google trend forecasting and volatility forecasting.
  • Figure 5: Additional return forecasting problems.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Proposition 3.1
  • Definition 1: GARCH(1, 1)
  • proof
  • proof