用普通ReLU网络预测时间序列并量化不确定性，兼顾未来波动与模型误差。
Prediction Inference of Time Series with Standard ReLU Deep Neural Networks
- 基于标准ReLU DNN构建预测区间，融合未来变异性与训练数据估计误差。
- 证明DNN估计器在β-混合数据下具有一致性，且前向自助序列保持相同平稳分布。
- 只需最小化对预测根极限分布的假设，适合需要可靠置信区间的实证研究者。
我们提出一种基于标准ReLU深度神经网络（DNN）的方法，用于时间序列的预测及其不确定性量化。传统方法多依赖线性、非线性或非参数核方法进行拟合与预测。随着DNN被证明具有通用逼近能力，其在各科学领域预测任务中应用日益广泛，但相应的不确定性量化研究仍不充分。预测不确定性由两部分构成：(1) 未来波动性；(2) 训练数据内的估计变异性。为捕捉这两部分，我们构建了相关的置信预测区间（PPI），基于DNN模型估计器。首先，研究了在β-混合依赖数据下DNN估计器的一致性；随后，证明推导出的前向自助序列仍为β-混合，并以概率收敛于原时间序列的相同平稳分布，这是实现PPI的关键条件；最后，在对预测根极限分布施加最小条件后，构建出所需的预测区间。通过模拟实验和真实数据分析，验证了该方法相较于标准非参数方法的有效性。
原文摘要 · Abstract (English)
We propose a methodology based on the standard ReLU Deep Neural Networks (DNN) to make predictions and quantify their uncertainty. Classically, people rely on linear, non-linear, or non-parametric kernel methods to fit and then predict the time series. As the universal approximation ability was revealed for DNN, its application has become more and more popular for prediction tasks in various scientific areas. However, the corresponding uncertainty quantification has not been studied thoroughly. Particularly, the uncertainty in prediction will consist of two parts: (1) the future variability; (2) the estimation variability within training data. To capture both variabilities, we build the so-called pertinent prediction interval (PPI) with the DNN model estimator. We first explore the consistency property of the DNN estimator with beta-mixing dependent data. Subsequently, we show that the implied forward bootstrap series is still beta-mixing and possesses the same stationary distribution as the original time series in probability, which is a key condition to enable the PPI. Lastly, the desired PPI is built after imposing minimal conditions on the limiting distribution of predictive roots. Simulations and real-data analysis are deployed to challenge our approach with standard non-parametric methods.
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