Robust optimal reconciliation for hierarchical time series forecasting with M-estimation

TL;DR

The robust reconciliation process for hierarchical time series (HTS) forecasting is explored, and M-estimation is incorporated to obtain the reconciled forecasts by minimizing a robust loss function of transforming a group of base forecasts subject to the aggregation constraints.

Abstract

Aggregation constraints, arising from geographical or sectoral division, frequently emerge in a large set of time series. Coherent forecasts of these constrained series are anticipated to conform to their hierarchical structure organized by the aggregation rules. To enhance its resilience against potential irregular series, we explore the robust reconciliation process for hierarchical time series (HTS) forecasting. We incorporate M-estimation to obtain the reconciled forecasts by minimizing a robust loss function of transforming a group of base forecasts subject to the aggregation constraints. The related minimization procedure is developed and implemented through a modified Newton-Raphson algorithm via local quadratic approximation. Extensive numerical experiments are carried out to evaluate the performance of the proposed method, and the results suggest its feasibility in handling numerous abnormal cases (for instance, series with non-normal errors). The proposed robust reconciliation also demonstrates excellent efficiency when no outliers exist in HTS. Finally, we showcase the practical application of the proposed method in a real-data study on Australian domestic tourism.

Figures (12)

• Figure 1: Paradigm of a two-level HTS depicted by the hierarchical tree. This hierarchy contains $n = 6$ series at the bottom level, where the first 2 (in green) and the other 4 (in blue) of them separately aggregate two series (in red) at the middle level, and all series finally constitute the most aggregated one (in black) at the top level. The summing matrix is $\bm{S} = (\mathbf{1}_6^{\top}; \mathbf{D}_{2, 4}; \mathbf{I}_6)$, where $\mathbf{1}_6$, $\mathbf{D}_{2, 4}$ and $\mathbf{I}_6$ correspond to the 1, 2 and 6 series at the top ("L2"), middle ("L1") and bottom ("L0") levels, respectively.
• Figure 2: Curves of LS, LAD, Huber's loss functions and their derivatives, where "Huber(0.5)", "Huber(1.345)" and "Huber(2)" correspond to the Huber's loss function with $k = 0.5$, 1.345 and 2, respectively.
• Figure 3: Hierarchical tree for the numerical experiments in Section \ref{['Section_NonGaussian']}. This hierarchy contain $n=13$ series, with 9 series ("L0-$i$", $i = 1, 2, \cdots, 9$) at the bottom level, 3 series ("L1-1", "L1-2" and "L1-3") at the middle level, and the most aggregated one ("L2-1") at the top level. The "Changeable" group (in red) indicates the series of which errors have non-normal distributions, the "Stable" one (in blue) indicates the other Gaussian series, and the "All" group indicates the whole HTS.
• Figure 4: Hierarchical trees for the numerical experiments in Section \ref{['Section_EfficiencyLoss']}. The summing matrix corresponding to the hierarchy of (a) in the left panel is $\bm{S} = (\mathbf{1}_6^{\top}; \text{diag} (\mathbf{1}_3^{\top}, \mathbf{1}_3^{\top}); \mathbf{I}_6)$, indicating the 1, 2 and 6 series ("L2-1"; "L1-1" and "L1-2"; and "L0-$i$" for $i = 1, 2, \cdots, 6$) at the top, middle and bottom levels, respectively. For (b) in the right panel, the family of the two-level hierarchies follow the aggregation rules that series (e.g., "L1-A1") at the middle level are aggregated in order by every 5 series (e.g., "L0-A$i$", $i = 1, 2, \cdots, 5$) at the bottom level.
• Figure 5: Hierarchical tree for the numerical experiments in Section \ref{['Section_Proportion']}. This hierarchy, following the similar aggregation rules of that in Figure \ref{['Figure_EfficiencyLoss']}(b) that series at the middle level are aggregated in order by every 6 series at the bottom level, contains $m = 36$ series, with 30 series ("L0-$i$", $i = 1, 2, \cdots, 30$) at the bottom level, 5 series ("L1-$j$", $j = 1, 2, \cdots, 5$) in the middle level, and the most aggregated one ("L2-1") in the top level.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3