Probabilistic forecast reconciliation under the Gaussian framework

Forecast reconciliation of multivariate time series maps a set of incoherent forecasts into coherent forecasts to satisfy a given set of linear constraints. Available methods in the literature either follow a projection matrix-based approach or an empirical copula-based reordering approach to revise the incoherent future sample paths to obtain reconciled probabilistic forecasts. The projection matrices are estimated either by optimizing a scoring rule such as energy or variogram score or simply using a projection matrix derived for point forecast reconciliation. This paper proves that (a) if the incoherent predictive distribution is jointly Gaussian, then MinT (minimum trace) minimizes the logarithmic scoring rule for the hierarchy; and (b) the logarithmic score of MinT for each marginal predictive density is smaller than that of OLS (ordinary least squares). We illustrate these theoretical results using a set of simulation studies and the Australian domestic tourism data set. The estimation of MinT needs to estimate the covariance matrix of the base forecast errors. We have evaluated the performance using the sample covariance matrix and shrinkage estimator. It was observed that the theoretical properties noted above are greatly impacted by the covariance matrix used and highlighted the importance of estimating it reliably, especially with high dimensional data.

多变量时间序列预测协调（Forecast reconciliation of multivariate time series）旨在将一组不协调的预测转换为协调的预测，以满足给定的线性约束集。现有文献中的方法大致分为两类：一类基于投影矩阵（projection matrix），另一类基于经验Copula（copula）的重排序方法，用于修正不协调的未来样本路径以得到协调后的概率预测。投影矩阵的估计方式主要有两种：一是通过优化评分规则（如能量评分或变异函数评分），二是直接采用针对点预测协调所推导得到的投影矩阵。本文证明了两项结论：(a) 若不协调的预测分布为联合高斯分布，则MinT（最小迹，minimum trace）可使层级结构下的对数评分规则达到最小；(b) 针对每个边缘预测密度，MinT的对数评分均小于OLS（普通最小二乘，ordinary least squares）的对数评分。本文通过一组仿真实验与澳大利亚国内旅游数据集对上述理论结果进行了验证。MinT的估计过程需要对基础预测误差的协方差矩阵进行估算，本文分别采用样本协方差矩阵与收缩估计量（shrinkage estimator）对模型性能进行了评估。实验结果表明，前述理论特性会受到所采用协方差矩阵的显著影响，凸显了可靠估计协方差矩阵的重要性，在高维数据场景下尤为如此。