Modeling the correlations among errors is closely associated with how accurately the model can quantify predictive uncertainty in probabilistic time series forecasting. Recent multivariate models have made significant progress in accounting for contemporaneous correlations among errors, while a common assumption on these errors is that they are temporally independent for the sake of statistical simplicity. However, real-world observations often deviate from this assumption, since errors usually exhibit substantial autocorrelation due to various factors such as the exclusion of temporally correlated covariates. In this work, we propose an efficient method, based on a low-rank-plus-diagonal parameterization of the covariance matrix, which can effectively characterize the autocorrelation of errors. The proposed method possesses several desirable properties: the complexity does not scale with the number of time series, the resulting covariance can be used for calibrating predictions, and it can seamlessly integrate with any model with Gaussian-distributed errors. We empirically demonstrate these properties using two distinct neural forecasting models -- GPVar and Transformer. Our experimental results confirm the effectiveness of our method in enhancing predictive accuracy and the quality of uncertainty quantification on multiple real-world datasets.

翻译：在概率时间序列预测中，建模误差之间的相关性直接关系到模型量化预测不确定性的准确性。近期多变量模型在考虑误差的同期相关性方面取得了显著进展，但为了统计简便性，这些误差通常被假设为时间独立。然而，由于排除时间相关协变量等因素，实际观测数据中的误差往往呈现出显著的自相关特性，这与该假设存在偏差。本文提出一种基于协方差矩阵低秩加对角参数化的高效方法，能够有效刻画误差的自相关结构。该方法具备多项理想特性：计算复杂度不随时间序列数量增加而增长、所得协方差矩阵可用于校准预测结果、并能无缝集成任意采用高斯误差分布的模型。我们通过两种不同的神经预测模型——GPVar和Transformer——对该方法进行了实证验证。实验结果表明，该方法在多个真实数据集上能够有效提升预测精度和不确定性量化质量。