Probabilistic Forecast Reconciliation under the Gaussian Framework
Probabilistic Forecast Reconciliation under the Gaussian Framework
复制标题
高斯框架下的概率预测调节
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发表时间：
2021
期刊：
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通讯作者：
Shanika L. Wickramasuriya
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文献类型：
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作者：
Shanika L. Wickramasuriya
多变量时间序列预测调和是将一组不连贯的预测映射为满足一组给定线性约束的连贯预测。文献中可用的方法要么采用基于投影矩阵的方法，要么采用基于经验公式的重排序方法来修正不一致的未来样本路径，以获得协调的概率预测。投影矩阵要么通过优化评分规则（如能量或方差分数）来估计，要么简单地使用为点预测调和而派生的投影矩阵。本文证明了(a)如果非相干预测分布是联合高斯分布，则MinT （minimum trace）对层次的对数评分规则最小化；(b) MinT对每个边缘预测密度的对数得分小于OLS（普通最小二乘）。我们使用一组模拟研究和澳大利亚国内旅游数据集来说明这些理论结果。MinT的估计需要估计基本预测误差的协方差矩阵。我们使用样本协方差矩阵和收缩估计器评估了性能。观察到，上述理论性质很大程度上受到所使用的协方差矩阵的影响，并强调了可靠估计协方差矩阵的重要性，特别是对于高维数据。
Abstract 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 article 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 dataset. 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.