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Robust Predictions with Ambiguous Time Delays: A Bootstrap Strategy

Jiajie Wang, Zhiyuan Jerry Lin, Wen Chen

TL;DR

The paper tackles predictive modeling for multivariate time series with non-deterministic time delays by introducing Time Series Model Bootstrap (TSMB). By treating delays as a random variable and drawing bootstrap samples to infer delay realizations $\hat{\bm{\delta}}^b$, TSMB trains an ensemble of predictors $f_{\hat{\bm{\delta}}^b}$ and aggregates their outputs to approximate $E[Y|X]$, providing a model-agnostic, uncertainty-aware framework. Across nine real-world datasets and with base models including GBDT and TFT, TSMB consistently outperforms traditional time delay estimation baselines (TDMI, GCC) in predictive accuracy and demonstrates meaningful insights into delay distributions and coverage. The work highlights both the practical benefits of accommodating delay uncertainty and the challenges of calibration for prediction intervals, offering a scalable, extensible approach for robust forecasting under delay variability.

Abstract

In contemporary data-driven environments, the generation and processing of multivariate time series data is an omnipresent challenge, often complicated by time delays between different time series. These delays, originating from a multitude of sources like varying data transmission dynamics, sensor interferences, and environmental changes, introduce significant complexities. Traditional Time Delay Estimation methods, which typically assume a fixed constant time delay, may not fully capture these variabilities, compromising the precision of predictive models in diverse settings. To address this issue, we introduce the Time Series Model Bootstrap (TSMB), a versatile framework designed to handle potentially varying or even nondeterministic time delays in time series modeling. Contrary to traditional approaches that hinge on the assumption of a single, consistent time delay, TSMB adopts a nonparametric stance, acknowledging and incorporating time delay uncertainties. TSMB significantly bolsters the performance of models that are trained and make predictions using this framework, making it highly suitable for a wide range of dynamic and interconnected data environments.

Robust Predictions with Ambiguous Time Delays: A Bootstrap Strategy

TL;DR

The paper tackles predictive modeling for multivariate time series with non-deterministic time delays by introducing Time Series Model Bootstrap (TSMB). By treating delays as a random variable and drawing bootstrap samples to infer delay realizations , TSMB trains an ensemble of predictors and aggregates their outputs to approximate , providing a model-agnostic, uncertainty-aware framework. Across nine real-world datasets and with base models including GBDT and TFT, TSMB consistently outperforms traditional time delay estimation baselines (TDMI, GCC) in predictive accuracy and demonstrates meaningful insights into delay distributions and coverage. The work highlights both the practical benefits of accommodating delay uncertainty and the challenges of calibration for prediction intervals, offering a scalable, extensible approach for robust forecasting under delay variability.

Abstract

In contemporary data-driven environments, the generation and processing of multivariate time series data is an omnipresent challenge, often complicated by time delays between different time series. These delays, originating from a multitude of sources like varying data transmission dynamics, sensor interferences, and environmental changes, introduce significant complexities. Traditional Time Delay Estimation methods, which typically assume a fixed constant time delay, may not fully capture these variabilities, compromising the precision of predictive models in diverse settings. To address this issue, we introduce the Time Series Model Bootstrap (TSMB), a versatile framework designed to handle potentially varying or even nondeterministic time delays in time series modeling. Contrary to traditional approaches that hinge on the assumption of a single, consistent time delay, TSMB adopts a nonparametric stance, acknowledging and incorporating time delay uncertainties. TSMB significantly bolsters the performance of models that are trained and make predictions using this framework, making it highly suitable for a wide range of dynamic and interconnected data environments.
Paper Structure (23 sections, 1 theorem, 5 equations, 7 figures, 5 tables, 1 algorithm)

This paper contains 23 sections, 1 theorem, 5 equations, 7 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Time Series Model Bootstrap
  4. Problem Definition
  5. Method
  6. Experiment
  7. Experiment Setup
  8. Datasets
  9. Baselines
  10. Experiment Setup
  11. Predictive Performance
  12. Choice of Bootstrap Sample Size
  13. Prediction Coverage
  14. Bootstrap Distribution Diagnostic Analysis
  15. TSMB Results with Temporal Fusion Transformer
  16. ...and 8 more sections

Key Result

Proposition 1

Assume the time delay $\bm{\delta}$ is a random variable and $f_{\bm{\delta}}(x) = \mathbb{E}[Y|X=x, \bm{\delta}]$ is the model prediction given a realized $\bm{\delta}$, the TSMB estimator is a finite sample approximation of $\mathbb{E}[Y|X=x]$.

Figures (7)

  • Figure 1: An illustration of the challenge of time delay estimation in multivariate time series. In this scenario, $\bm{\delta}=\{\delta_A, \delta_B\}$ are well-defined and have unique values, which isn't always the case in real-world applications involving unpredictable events and fluctuating noise.
  • Figure 2: Performance visualization of GBDT models applied across different datasets, showcasing the efficacy of various methods in handling time delays. Each point indicates the AUC (for classification tasks) or $R^2$ (for regression tasks). Across all datasets, TSMB methods consistently outperform traditional TDE techniques like TDMI and GCC. Error bars represent 95% CIs for TSMB-based methods and are generally small.
  • Figure 3: Ablation on bootstrap sample size $B$ for TSMB. The horizontal axis depicts the bootstrap sample size $B$. $B$ is minimally impacting the predictive performance of TSMB estimators.
  • Figure 4: Bootstrap percentile $1-\alpha$ confidence interval coverage under TSMB. For classification tasks where we only observe binary values, we examine TSMB coverage using the corresponding point estimates given by TDMI or GCC.
  • Figure 5: Bootstrap distribution of normalized estimated time delays using TDMI and GCC as the score function respectively. Vertical lines represent the point estimates given by optimizing TDMI and GCC as well as the ground truth (when applicable) time delays from each dataset. The fact that many of these distributions are not even close to point distirbution suggests that there exists a significant amount of uncertainty in the estimated time delays that is being ignored by traditional TDE methods.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Proposition 1