基于标准ReLU深度神经网络的时间序列预测推断

中文摘要 AI 辅助

我们提出一种基于标准ReLU深度神经网络（DNN）的方法，用于进行预测并量化其不确定性。传统上，人们依赖线性、非线性或非参数核方法来拟合时间序列并进行预测。随着DNN通用逼近能力的揭示，其在各科学领域的预测任务中应用愈发广泛，但对应的不确定性量化尚未得到充分研究。预测中的不确定性包含两部分：（1）未来的变异性；（2）训练数据内的估计变异性。为捕获这两种变异性，我们用DNN模型估计器构建了所谓的相关预测区间（PPI）。我们首先探讨了beta混合相依数据下DNN估计器的一致性性质；随后证明，隐含的向前自助序列仍为beta混合，且在概率上具有与原始时间序列相同的平稳分布，这是实现PPI的关键条件；最后，在对预测根的极限分布施加 minimal 条件后，构建出所需的PPI。我们通过模拟和真实数据分析，将所提方法与标准非参数方法进行对比验证。

英文摘要

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.

发表机构

• Loyola University Chicago（芝加哥洛约拉大学）

机构由 AI 辅助整理，请以论文原文为准。