Machine Learning algorithms are notorious for providing point predictions but not prediction intervals. There are many applications where one requires confidence in predictions and prediction intervals. Stringing together, these intervals give rise to joint prediction regions with the desired significance level. It is an easy task to compute Joint Prediction regions (JPR) when the data is IID. However, the task becomes overly difficult when JPR is needed for time series because of the dependence between the observations. This project aims to implement Wolf and Wunderli's method for constructing JPRs and compare it with other methods (e.g. NP heuristic, Joint Marginals). The method under study is based on bootstrapping and is applied to different datasets (Min Temp, Sunspots), using different predictors (e.g. ARIMA and LSTM). One challenge of applying the method under study is to derive prediction standard errors for models, it cannot be obtained analytically. A novel method to estimate prediction standard error for different predictors is also devised. Finally, the method is applied to a synthetic dataset to find empirical averages and empirical widths and the results from the Wolf and Wunderli paper are consolidated. The experimental results show a narrowing of width with strong predictors like neural nets, widening of width with increasing forecast horizon H and decreasing significance level alpha, controlling the width with parameter k in K-FWE, and loss of information using Joint Marginals.

翻译：机器学习算法因通常仅提供点预测而非预测区间而备受诟病。在许多应用场景中，人们需要对预测结果及其区间具有置信度。将这些区间串联起来，便形成了具有所需显著性水平的联合预测区域。当数据满足独立同分布条件时，计算联合预测区域相对简单。然而，由于时间序列观测值之间存在依赖性，为其构建联合预测区域的任务变得异常困难。本项目旨在实现Wolf和Wunderli提出的联合预测区域构建方法，并将其与其他方法（如NP启发式方法、联合边缘法）进行比较。所研究的方法基于自助法，并应用于不同数据集（最低温度、太阳黑子），同时采用多种预测器（如ARIMA和LSTM）。应用该方法的一个挑战在于推导模型的预测标准误，因其无法通过解析方法获得。为此，本文还设计了一种新颖的方法来估计不同预测器的预测标准误。最后，将该方法应用于合成数据集以获取经验平均值和经验宽度，并整合了Wolf和Wunderli论文中的结果。实验结果表明：使用神经网络等强预测器时宽度收窄；随着预测范围H的增加和显著性水平α的降低，宽度扩大；通过K-FWE中的参数k可控制宽度；而使用联合边缘法则会导致信息损失。