In this paper we tackle the problem of point and probabilistic forecasting by describing a blending methodology of machine learning models that belong to gradient boosted trees and neural networks families. These principles were successfully applied in the recent M5 Competition on both Accuracy and Uncertainty tracks. The keypoints of our methodology are: a) transform the task to regression on sales for a single day b) information rich feature engineering c) create a diverse set of state-of-the-art machine learning models and d) carefully construct validation sets for model tuning. We argue that the diversity of the machine learning models along with the careful selection of validation examples, where the most important ingredients for the effectiveness of our approach. Although forecasting data had an inherent hierarchy structure (12 levels), none of our proposed solutions exploited that hierarchical scheme. Using the proposed methodology, our team was ranked within the gold medal range in both Accuracy and the Uncertainty track. Inference code along with already trained models are available at https://github.com/IoannisNasios/M5_Uncertainty_3rd_place

Multivariate time series forecasting with hierarchical structure is pervasive in real-world applications, demanding not only predicting each level of the hierarchy, but also reconciling all forecasts to ensure coherency, i.e., the forecasts should satisfy the hierarchical aggregation constraints. Moreover, the disparities of statistical characteristics between levels can be huge, worsened by non-Gaussian distributions and non-linear correlations. To this extent, we propose a novel end-to-end hierarchical time series forecasting model, based on conditioned normalizing flow-based autoregressive transformer reconciliation, to represent complex data distribution while simultaneously reconciling the forecasts to ensure coherency. Unlike other state-of-the-art methods, we achieve the forecasting and reconciliation simultaneously without requiring any explicit post-processing step. In addition, by harnessing the power of deep model, we do not rely on any assumption such as unbiased estimates or Gaussian distribution. Our evaluation experiments are conducted on four real-world hierarchical datasets from different industrial domains (three public ones and a dataset from the application servers of Alipay's data center) and the preliminary results demonstrate efficacy of our proposed method.

Vaccine supply chain optimization can benefit from hierarchical time series forecasting, when grouping the vaccines by type or location. However, forecasts of different hierarchy levels become incoherent when higher levels do not match the sum of the lower levels forecasts, which can be addressed by reconciliation methods. In this paper, we tackle the vaccine sale forecasting problem by modeling sales data from GSK between 2010 and 2021 as a hierarchical time series. After forecasting future values with several ARIMA models, we systematically compare the performance of various reconciliation methods, using statistical tests. We also compare the performance of the forecast before and after COVID. The results highlight Minimum Trace and Weighted Least Squares with Structural scaling as the best performing methods, which provided a coherent forecast while reducing the forecast error of the baseline ARIMA.

A central problem in multivariate forecasting is the need to forecast a large group of time series arranged in a natural hierarchical structure, such that time series at higher levels of the hierarchy are aggregates of time series at lower levels. For example, hierarchical time series are common in retail forecasting applications [Fildes et al., 2019], where the time series may capture retail sales of a company at different granularities such as item-level sales, category-level sales, and department-level sales. In electricity demand forecasting [Van Erven and Cugliari, 2015], the time series may correspond to electricity consumption at different granularities, starting with individual households, which could be progressively grouped into city-level, and then state-level consumption time-series. The hierarchical structure among the time series is usually represented as a tree, with leaf-level nodes corresponding to time series at the finest granularity, while higher-level nodes represent coarser-granularities and are obtained by aggregating the values from its children nodes. Since businesses usually require forecasts at various different granularities, the goal is to obtain accurate forecasts for time series at every level of the hierarchy. Furthermore, to ensure decisionmaking at different hierarchical levels are aligned, it is essential to generate predictions that are coherent [Hyndman et al., 2011] with respect to the hierarchy, that is, the forecasts of a parent time-series should be equal to the sum of forecasts of its children time-series.

This paper discusses the prediction of hierarchical time series, where each upper-level time series is calculated by summing appropriate lower-level time series. Forecasts for such hierarchical time series should be coherent, meaning that the forecast for an upper-level time series equals the sum of forecasts for corresponding lower-level time series. Previous methods for making coherent forecasts consist of two phases: first computing base (incoherent) forecasts and then reconciling those forecasts based on their inherent hierarchical structure. With the aim of improving time series predictions, we propose a structured regularization method for completing both phases simultaneously. The proposed method is based on a prediction model for bottom-level time series and uses a structured regularization term to incorporate upper-level forecasts into the prediction model. We also develop a backpropagation algorithm specialized for application of our method to artificial neural networks for time series prediction. Experimental results using synthetic and real-world datasets demonstrate the superiority of our method in terms of prediction accuracy and computational efficiency.

In this paper, we propose a machine learning approach for forecasting hierarchical time series. Rather than using historical or forecasted proportions, as in standard top-down approaches, we formulate the disaggregation problem as a non-linear regression problem. We propose a deep neural network that automatically learns how to distribute the top-level forecasts to the bottom level-series of the hierarchy, keeping into account the characteristics of the aggregate series and the information of the individual series. In order to evaluate the performance of the proposed method, we analyze hierarchical sales data and electricity demand data. Besides comparison with the top-down approaches, the model is compared with the bottom-up method and the optimal reconciliation method. Results demonstrate that our method does not only increase the average forecasting accuracy of the hierarchy but also addresses the need of building an automated procedure generating coherent forecasts for many time series at the same time.