Uncertainty Intervals for Prediction Errors in Time Series Forecasting
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时间序列预测误差的不确定性区间
- 研究问题
- 在时间相关或非平稳数据中，如何分别对近期实际测试误差及其期望进行推断，避免普通前向交叉验证区间失效？
- 方法与思路
- 作者提出基于分位数的前向交叉验证，对随机测试误差构造预测区间，并结合自适应保形预测生成非平稳滚动区间。
- 材料陈述的贡献
- 摘要称遍历性条件下区间渐近覆盖有效且短于边际经验分位数区间；非平稳扩展具有时间平均覆盖保证。
- 阅读关注点
- 应核对两类误差目标的定义、遍历性及其他条件，并区分渐近覆盖、时间平均覆盖与特定时点覆盖的含义。
- 阅读边界
- 这是 arXiv 预印本，同行评审与接收未核实。摘要未给出完整条件和有限样本误差界，时间平均覆盖不保证每期覆盖。
原文摘要
Inference for prediction errors is critical in time series forecasting pipelines. However, providing statistically meaningful uncertainty intervals for prediction errors remains relatively under-explored. Practitioners often resort to forward cross-validation (FCV) for obtaining point estimators and constructing confidence intervals based on the Central Limit Theorem (CLT). The naive version assumes independence, a condition that is usually invalid due to time correlation. These approaches lack statistical interpretations and theoretical justifications even under stationarity. This paper systematically investigates uncertainty intervals for prediction errors in time series forecasting. We first distinguish two key inferential targets: the stochastic test error over near future data points, and the expected test error as the expectation of the former. The stochastic test error is often more relevant in applications needing to quantify uncertainty over individual time series instances. To construct prediction intervals for the stochastic test error, we propose the quantile-based forward cross-validation (QFCV) method. Under an ergodicity assumption, QFCV intervals have asymptotically valid coverage and are shorter than marginal empirical quantiles. In addition, we also illustrate why naive CLT-based FCV intervals fail to provide valid uncertainty intervals, even with certain corrections. For non-stationary time series, we further provide rolling intervals by combining QFCV with adaptive conformal prediction to give time-average coverage guarantees. Overall, we advocate the use of QFCV procedures and demonstrate their coverage and efficiency through simulations and real data examples.