Conformal prediction for multi-dimensional time series by ellipsoidal sets (2403.03850v2)

Abstract: Conformal prediction (CP) has been a popular method for uncertainty quantification because it is distribution-free, model-agnostic, and theoretically sound. For forecasting problems in supervised learning, most CP methods focus on building prediction intervals for univariate responses. In this work, we develop a sequential CP method called $\texttt{MultiDimSPCI}$ that builds prediction $\textit{regions}$ for a multivariate response, especially in the context of multivariate time series, which are not exchangeable. Theoretically, we estimate $\textit{finite-sample}$ high-probability bounds on the conditional coverage gap. Empirically, we demonstrate that $\texttt{MultiDimSPCI}$ maintains valid coverage on a wide range of multivariate time series while producing smaller prediction regions than CP and non-CP baselines.

• The paper pioneers multidimensional conformal prediction with ellipsoidal sets to create tighter, adaptive prediction regions for time series.
• It introduces the MultiDimSPCI framework that leverages underlying correlations and temporal dependencies to calibrate prediction set sizes.
• Empirical results demonstrate consistent high coverage across up to 20 dimensions while producing significantly smaller prediction regions.

Conformal Prediction for Multi-Dimensional Time Series Using Ellipsoidal Sets

This paper explores the enhancement of conformal prediction (CP) methods to cover the niche of multivariate time series forecasting, where traditional univariate CP methods fall short. The authors propose a novel method, MultiDimSPCI, to create prediction regions for multivariate responses by evolving beyond the straightforward approach of univariate prediction intervals.

Main Contributions

1. Multidimensional Conformal Prediction: The work pioneers a sequential CP approach that adapts to time series' multivariate nature by employing ellipsoidal prediction regions. This is a significant shift from the typical axis-aligned approaches that handle each dimension independently and are prone to overestimation.
2. Adaptive Prediction Sets: By calibrating the size of the ellipsoidal sets dynamically during test time, MultiDimSPCI ensures these sets are both flexible and precise in capturing the intrinsic correlations within multivariate responses.
3. Theoretical Guarantees: The paper provides robust theoretical backing by estimating finite-sample high-probability bounds on the conditional coverage gap, ensuring that the prediction sets are not only valid but optimally small across a variety of situations.
4. Empirical Validation: Through extensive experimentation across different datasets with up to 20 dimensions, MultiDimSPCI is shown to consistently yield smaller prediction sets compared to traditional CP and non-CP baselines, without sacrificing coverage. This suggests that the method efficiently uses historical dependencies to make tighter predictions.

Theoretical Underpinnings

The authors build on the conformal prediction framework's strength in providing distribution-free probability guarantees while modifying the methodology to better accommodate multivariate outputs and non-exchangeable observations. A key theoretical contribution is adapting the CP framework's coverage guarantees to the multivariate and temporally dependent setting, thereby addressing limitations faced by previous CP models when dealing with non-exchangeable time series data.

Experimental Insights

Through simulations as well as real-world data applications involving wind, solar, and traffic series, the proposed method demonstrates superior performance. Notably, the empirical results underline its practical applicability where MultiDimSPCI not only achieves the desired coverage but also maintains significantly smaller prediction regions compared to the leading methods leveraging CP in multivariate contexts, such as Copula-based approaches.

Implications and Future Directions

The introduction of ellipsoidal prediction regions highlights the opportunity to better capture the dependencies among components in multivariate datasets, a move that could be valuable across various domains such as finance, meteorology, and engineering where multivariate time series data is common.

Future research could extend these ideas further by investigating the robustness and adaptability of alternative shapes for prediction sets beyond ellipsoids. Additionally, exploring combinations with copula methods could provide insights into even more succinct and accurate uncertainty quantification. Finally, extending this approach into higher dimensions efficiently remains a promising avenue, particularly in light of computational challenges presented by massive datasets.

The fusion of traditional machine learning with statistical methodology, as demonstrated, delineates a pathway for the progressive development of predictive models that are not only theoretically sound but also practically efficient.