🤖 AI Summary
This study addresses the insufficient uncertainty quantification in standard ReLU deep neural networks for time series forecasting by proposing a forward bootstrap-based prediction interval (PPI) method. By constructing PPIs and establishing the consistency of DNN estimators alongside the mixing properties and stationary distribution characteristics of bootstrap sequences, this approach effectively captures future variability and enables nonparametric statistical inference. Both simulation studies and empirical analyses demonstrate that the proposed method maintains theoretical rigor while significantly outperforming existing standard nonparametric approaches in terms of predictive performance and uncertainty quantification accuracy.
📝 Abstract
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.