Hierarchical Forecasting Research Articles

Forecasting to support EMS tactical planning: what is important and what is not.

Forecasting emergency medical service (EMS) call volumes is critical for resource allocation and planning. The development of many commercial and free software packages has made a variety of forecasting methods accessible. Practitioners, however, are left with little guidance on selecting the most appropriate method for their needs. Using 5 years of data from 3 cities in Alberta, we compute exponential smoothing and benchmark forecasts for 8-hour periods for each ambulance station catchment area and with a forecast horizon of two weeks-a spatio-temporal resolution appropriate for tactical planning. The methods that we consider differ on three spectra: the number and type of time-series components, whether forecasts are computed individually or jointly, and the way in which forecasts at a specific resolution are converted to forecasts at the resolution of interest. We find that it is important to include a weekly seasonal component when forecasting EMS demand. Multiplicative seasonality, however, shows no benefit over additive seasonality. Adding other time-series components (e.g., trend, ARMA errors, Box-Cox transformation) does not improve performance. Spatial resolutions of station catchment area and lower, and temporal resolution of 4-24 hours perform similarly. We adapt an existing hierarchical forecasting framework to a two-dimensional spatio-temporal hierarchy, but find that hierarchical reconciliation of forecasts does not improve performance at the forecast resolution of interest for tactical planning. Neither does jointly forecasting time series. We show that added complexity does not materially improve forecasting performance. The simple methods that we find perform well are easy to implement and interpret, making implementation in practice more likely. In a simulation study we alter the empirical weekly patterns and demonstrate how extreme differences between the weekly seasonality patterns of different regions cause hierarchically-reconciled bottom-up approaches to outperform top-down approaches.

A Hybrid Approach for Hierarchical Forecasting of Industrial Electricity Consumption in Brazil

The Brazilian industrial sector is the largest electricity consumer in the power system. Energy planning in this sector is important mainly due to its economic, social, and environmental impact. In this context, electricity consumption analysis and projections are highly relevant for the decision-making of the industrial sectorand organizations operating in the energy system. The electricity consumption data from the Brazilian industrial sector can be organized into a hierarchical structure composed of each geographic region (South, Southeast, Midwest, Northeast, and North) and their respective states. This work proposes a hybrid approach that incorporates the projections obtained by the exponential smoothing and Box–Jenkins models to obtain the hierarchical forecasting of electricity consumption in the Brazilian industrial sector. The proposed approach was compared with the bottom-up, top-down, and optimal combination approaches, which are widely used for time series hierarchical forecasting. The performance of the models was evaluated using the mean absolute percentage error (MAPE) and root mean squared error (RMSE) precision measures. The results indicate that the proposed hybrid approach can contribute to the projection and analysis of industrial sector electricity consumption in Brazil.

GMP-AR: Granularity Message Passing and Adaptive Reconciliation for Temporal Hierarchy Forecasting

Time series forecasts of different temporal granularity are widely used in real-world applications, e.g., sales prediction in days and weeks for making different inventory plans. However, these tasks are usually solved separately without ensuring coherence, which is crucial for aligning downstream decisions. Previous works mainly focus on ensuring coherence with some straightforward methods, e.g., aggregation from the forecasts of fine granularity to the coarse ones, and allocation from the coarse granularity to the fine ones. These methods merely take the temporal hierarchical structure to maintain coherence without improving the forecasting accuracy. In this paper, we propose a novel granularity message-passing mechanism (GMP) that leverages temporal hierarchy information to improve forecasting performance and also utilizes an adaptive reconciliation (AR) strategy to maintain coherence without performance loss. Furthermore, we introduce an optimization module to achieve task-based targets while adhering to more real-world constraints. Experiments on real-world datasets demonstrate that our framework (GMP-AR) achieves superior performances on temporal hierarchical forecasting tasks compared to state-of-the-art methods. In addition, our framework has been successfully applied to a real-world task of payment traffic management in Alipay by integrating with the task-based optimization module.

The Impact of Age Group in Hierarchical Forecasting of Monthly Blood Donations in Zimbabwe.

To meet the blood requirements for transfusion therapy, blood banks need to ensure that blood inventories are maintained at desirable levels. There is a rising global need for optimal ways to manage blood supply and demand using statistical models in blood inventory planning and management. Thus, blood donation forecasting using donor-specific characteristics such as donor type and age is critical in managing the blood bank inventory. The monthly blood donation data covering the period 2007 to 2018, collected from the National Blood Service Zimbabwe (NBSZ) was used in this study. The data is first disaggregated based on donor age, and further disaggregation is performed for each age group based on donor type. The hierarchical forecasting approaches, namely the bottom-up, top-down and the optimal combination methods were used in the data analysis. The Error-Trend-Seasonality (ETS) and Autoregressive Integrated Moving Average (ARIMA) methods are used in the hierarchical forecasting approaches to generate the forecasts. New blood donors account for more than 55% of blood donations in Zimbabwe. The younger donors (16-29 years) dominate the blood donations, accounting for 89.2% of the donations. Young and new donors account for nearly 50% of the donations. The middle-aged and older donors have lower blood donations. The bottom-up approach under the ARIMA model outperformed all the other approaches. The future projections show that new and young donors will increase in blood donations, regular donations will decline slightly while the occasional donations are projected to remain constant. Hierarchical forecasting is a unique approach in that the different aggregation levels reveal important features of the blood donation data. The lower percentage of regular donations is worrisome to blood authorities as it points to new blood donors not returning for further donations. Blood authorities need to develop policies that will encourage new and young donor categories to become regular donors.

Properties of the reconciled distributions for Gaussian and count forecasts

Reconciliation enforces coherence between hierarchical forecasts, in order to satisfy a set of linear constraints. While most works focus on the reconciliation of point forecasts, we consider probabilistic reconciliation and we analyze the properties of distributions reconciled via conditioning. We provide a formal analysis of the variance of the reconciled distribution, treating the case of Gaussian and count forecasts separately. We also study the reconciled upper mean in the case of one-level hierarchies, again treating Gaussian and count forecasts separately. We then show experiments on the reconciliation of intermittent time series related to the count of extreme market events. The experiments confirm our theoretical results and show that reconciliation largely improves the performance of probabilistic forecasting.

Optimal reconciliation of hierarchical wind energy forecasts utilizing temporal correlation

Independent wind energy forecasts of a wind farm at different time horizons have limited accuracy, and they show disagreement despite relating to the same wind farm. The limited forecast accuracy is attributable to the insufficient information at a particular time horizon of the wind energy time series, whereas applying distinct forecasting methods to several time series of non-identical patterns at different time scales causes disagreement among forecasts. Mutual disagreement among less accurate forecasts negatively impacts the decision-making capabilities in associated power systems activities at distinct time scales. The configuration of time series expressing different time horizons at different levels of a non-overlapped hierarchically aggregated framework manifests a temporal hierarchy. Forecast combination through reconciliation of time series forecasts drawn at different hierarchical levels of temporal hierarchy using any state-of-the-art method facilitates the sharing of diverse information across the hierarchy; consequently, accuracy and mutual agreement of forecasts improve. Such benefits may be further enhanced by embedding intra- and inter-level forecast error correlations in the forecast reconciliation process. However, the forecast error covariance matrix of temporal hierarchy becomes a complex high-dimensional structure while accommodating intra- and inter-level correlations. Estimating such a matrix is challenging since the high-dimensional structure severely impedes the identifiability of model parameters. Besides, in the hierarchical forecast reconciliation process, the number of predictor variables is generally higher than the number of samples. This condition gives rise to a singular covariance matrix, making it non-invertible, and thus obstructs its parameter estimation. This work employs the MinT(shrinkage) covariance matrix estimator that considers all correlations and shrinks the non-diagonal components of the matrix toward zero to avert the complexity and, therefore, the non-identifiability. Additionally, the shrinkage parameter λ of MinT(shrinkage) conveniently obtains the invertible matrix. The case study validates that while incorporating the intra- and inter-level forecast error correlations, MinT(shrinkage) provides competitively accurate and mutually agreed forecasts over other reconciliation methods.

Using Quadratic Programming to Optimally Adjust Hierarchical Load Forecasting

Electric load is hierarchically organized based on geography, which requires hierarchical forecasts covering all levels to support decision makings of power system operations. A trivial way to implement hierarchical forecasts is to independently generate load forecasts at each level using state-of-the-art techniques. However, these independently-generated forecasts may not satisfy hierarchical structures, i.e., the sum of lower-level forecasts cannot add up exactly to upper-level forecasts. To deal with this problem, this letter presents a quadratic programming (QP) model to optimally adjust load forecasts independently generated at each level of a hierarchy. The proposed model obtains the “best” forecasts as close as possible to base forecasts but also satisfy the aggregate consistency defined by hierarchical structures. Numerical results using real-world data validate the effectiveness of the proposed approach in two application scenarios (i.e., bulk power systems and power distribution networks). The accuracy improvement of the proposed approach over either base forecasts or bottom-up forecasts can be observed at all levels of the hierarchy.

Least Squares-based Optimal Reconciliation Method for Hierarchical Forecasts of Wind Power Generation

Wind power generation is hierarchically organized and can be aggregated/disaggregated based on geography and electrical network structure. Forecasts are required at all levels of such a hierarchy for different operational requirements, which are also known as “hierarchical forecasts”. In this paper, we propose a new method to produce hierarchical wind power forecasts which performs better than the conventional top-down or bottom-up method. At first, wind power generation at all levels of the hierarchy is independently forecasted using state-of-the-art techniques. Then, a least squares regression model is used to optimally reconcile these base forecasts. Adjusted forecasts are as close as possible to base forecasts but also satisfy the requirement of the aggregate consistency (i.e., the lower-level forecasts add up exactly to the higher-level forecasts). Case studies validate the effectiveness of the proposed approach for very short-term wind power forecasts. The accuracy improvement of reconciled forecasts over benchmarks is confirmed at all levels of the hierarchy, regardless of forecast horizons and hierarchical structures.

Optimal reconciliation of hierarchical wind power forecasts of correlated wind farms

Wind power forecasts of widely distributed wind farms in a spatial hierarchy are required at different hierarchical levels of power system architecture for various operational decisions. Accurate hierarchical forecasts necessitate aggregation consistency across hierarchy i.e. lower-level forecasts summate precisely to higher-level forecasts. Aggregation consistency substantially improves through optimal reconciliation of forecasts. Reconciliation can further improve forecast accuracy by incorporating spatial correlation among wind farms in the hierarchy. However, covariance matrix formed in reconciliation observes matrix order to increase manifold while including correlation. Estimation of such high-dimensional matrix is computationally complex. Additionally, aggregation of estimation errors leads to poor estimation accuracy. Moreover, in this reconciliation process, covariance matrix based on the observed data is non-invertible as the number of predictors usually happens to be larger than sample size. This work proposes MinT(Shrink) estimator for covariance matrix estimation for such systems. This estimator captures correlation among wind farms while shrinks off-diagonal elements of covariance matrix toward zero to alleviate computational complexity of high-dimensional matrix. It also provides invertible covariance matrix by employing shrinkage parameter λ. Case study presents efficacy of considering spatial correlation in hierarchical wind power forecasts in terms of improved forecast accuracy at all hierarchy levels and time horizons.

A novel multi-phase hierarchical forecasting approach with machine learning in supply chain management

Hierarchical time series demands are often associated with products, time frames, or geographic aggregations. Traditionally, these hierarchies have been forecasted using “top-down,” “bottom-up,” or “middle-out” approaches. This study advocates using child-level forecasts in a hierarchical supply chain to improve parent-level forecasts. Improved forecasts can considerably reduce logistics costs, especially in e-commerce. We propose a novel multi-phase hierarchical approach for independently forecasting each series in a hierarchy using machine learning. We then combine all forecasts to allow a second-phase model estimation at the parent level. Sales data from a logistics solutions provider is used to compare our approach to “bottom-up” and “top-down” methods. Our results demonstrate an 82–90% improvement in forecast accuracy. Using the proposed method, supply chain planners can derive more accurate forecasting results by exploiting the benefit of multivariate data.

Exploring Hierarchical Forecasting of Data Popularity in High-Energy Physics Experiments

Exploring Hierarchical Forecasting of Data Popularity in High-Energy Physics Experiments

Enhancing forecast accuracy using combination methods for the hierarchical time series approach.

This study aims to investigate whether combining forecasts generated from different models can improve forecast accuracy rather than individual models using the hierarchical time series. Various approaches of hierarchical forecasting have been considered; a bottom-up, top-down, and an optimal combination approach. Autoregressive moving averages (ARIMA) and exponential smoothing (ETS) were used as forecasting models in creating forecasting for all levels in the hierarchy to show the effect of different forecasting methods for each hierarchical model. The results indicated that the Minimum Trace Sample estimator (MinT-Sample) and the bottom-up approaches with the ARIMA model have good predictive performance than other approaches. Moreover, the forecasts from the MinT-Sample and bottom-up approaches were combined using five different combining methods. The experimental results showed that the (AC) method is superior to all other combining methods and more accurate than other individual models at level zero (international total trade in Egypt) and level one (total exports, and total imports). So, combining forecasts generated from different models by hierarchical time series leads to more accurate forecasting of the value of imports and exports which will improve the overall international trade performance, and that is through using the forecasting values of imports and exports to plan for improving the trade balance and drawing up a more efficient production policy. Finally, the study recommends using hierarchical forecasting methods in the areas of international trade, and the Ministry of Commerce and Industry could adopt the results of this study to produce precise forecasts for international trade. Moreover, the results of this study are to be a guide for the researchers to apply these approaches in other fields to improve the performance of forecasting.

Hierarchical learning, forecasting coherent spatio-temporal individual and aggregated building loads

Optimal decision-making compels us to anticipate the future at different horizons. However, in many domains connecting together predictions from multiple time horizons and abstractions levels across their organization becomes all the more important, else decision-makers would be planning using separate and possibly conflicting views of the future. To this end, this work proposes a novel multi-dimensional hierarchical forecasting method built upon structurally-informed machine-learning regressors. A generic formulation of multi-dimensional hierarchies, reconciling spatial and temporal dimensions under a common frame is initially defined. Next, a coherency-informed hierarchical learner is developed built upon a custom loss function leveraging optimal reconciliation methods. The coherency of the produced hierarchical forecasts is then secured using similar reconciliation techniques, granting decision-makers a common view of the future serving aligned decision-making. The method is evaluated on two different case studies to predict building electrical loads across spatial, temporal, and spatio-temporal hierarchies. Although the regressor natively profits from computationally efficient learning, results displayed disparate performances, demonstrating the value of hierarchical-coherent learning in only one setting. Yet, existing obstacles were clearly delineated, presenting distinct pathways for future work. Overall, the paper expands and unites traditionally disjointed hierarchical forecasting methods providing a fertile route toward a novel generation of forecasting regressors.

A novel reconciliation approach for hierarchical electricity consumption forecasting based on resistant regression

Power systems have become increasingly complex throughout the years, with divisions that may vary across classes of consumption and geographic regions. Hence, accurate demand forecasts across sub-national levels have grown in importance for decision making. Official planning reports and most forecasting methods, however, have been tailored to deliver consumption forecasts only at the most aggregate level, i.e., total energy consumption. This work proposes a hierarchical forecast reconciliation approach aimed at delivering accurate forecasts of energy demand across all divisions in a power system. Its novelty relies on the use of resistant-based estimators to aid in the process of forecast reconciliation, so that outliers and influential points that may be present in the base forecasts have minimal or even null effects on reconciliation weights. Data from the Brazilian National Interlinked System are considered in multiple experiments that compare the performance of the proposed approach with those from traditional and state-of-the-art hierarchical forecasting methods. Overall, the proposed reconciliation approach shows promising forecasting results under multiple settings and through the lens of several evaluation metrics. The average gains brought forth by the proposed methodology, in terms of accumulated energy that can be ‘saved’ across all divisions of the power system given more accurate forecasts, is greater than 900,000 MWh for forecast horizons of three and four months ahead. This is particularly important for decision-making in the energy industry, since the more accurate the forecasts are, the easier it is to plan and dispatch, thus the method has potential for real savings in the sector. Furthermore, the developed methodology is flexible, as it can be readily applied to other sets of hierarchical time series. Findings and policy implications are further discussed.

Imbalance Reduction of P2P Energy Market by Closed-Loop Clustering and Forecasting

Energy forecasting at the individual level remains a technical barrier for the peer-to-peer (P2P) energy market. Forecasting errors will cause energy imbalances between real-time delivery and contracts. To reduce the economic loss, this paper proposes an imbalance reduction method based on hierarchical energy forecasting. Existing hierarchical forecasting usually cluster prosumers based on the similarities of their energy profiles. However, minimising the variance cannot guarantee reductions in energy imbalances. This paper explores a novel closed-loop clustering (CLC) algorithm, which aligns the mutual objectives of clustering and imbalance reduction through the introduction of a feedback mechanism. The proposed method is applied in a simulated P2P energy market to cluster photovoltaics (PVs) into virtual groups (VG). The output of each VG is predicted and traded with flexible demands as an entity. The results show that CLC could reduce up to 25.70% energy imbalance compared with individual trading. Network constraints have shown minor impact on the trading results.

Optimal reconciliation with immutable forecasts

The practical importance of coherent forecasts in hierarchical forecasting has inspired many studies on forecast reconciliation. Under this approach, base forecasts are produced for every series in the hierarchy and are subsequently adjusted to be coherent in a second reconciliation step. Reconciliation methods have been shown to improve forecast accuracy but will generally adjust the base forecast of every series. However, in an operational context, it is sometimes necessary or beneficial to keep forecasts of some variables unchanged after forecast reconciliation. In this paper, we formulate a reconciliation methodology that keeps forecasts of a pre-specified subset of variables unchanged or “immutable”. In contrast to existing approaches, these immutable forecasts need not all come from the same level of a hierarchy, and our method can also be applied to grouped hierarchies. We prove that our approach preserves unbiasedness in base forecasts. Our method can also account for correlations between base forecasting errors and ensure the non-negativity of forecasts. We also perform empirical experiments, including an application to a large-scale online retailer’s sales, to assess our proposed methodology’s impacts.

A novel forecast framework for unsteady flows based on a convolutional neural network

Fluid mechanics is an important area where deep learning produces excellent results and can bring about scientific innovation because of its high dimensionality, significant nonlinearity, and ability to process an enormous amount of data. Deep learning technology is currently being used to study fluid mechanics, and its application potential is gradually being demonstrated. We propose a novel multi-resolution convolutional interaction network (MCIN), a hierarchical forecast framework based on a convolutional neural network. This structure can capture temporal dependencies at multiple temporal resolutions to enhance the forecasting performance of the original time series. The high-dimensional data of the flow around a cylinder are projected into a low-dimensional subspace using a variational autoencoder (VAE) as a nonlinear order-reduction technique. Then, the data of the subspace are used as the input to MCIN to forecast future velocity fields. The proposed MCIN is compared to non-intrusive reduced-order models based on dynamic mode decomposition and long short-term memory, combined with a VAE. The results demonstrate that MCIN has superior stability to other models in forecasting the evolution of complicated fluid flows and has the potential to forecast a greater number of future outcomes.

On the evaluation of hierarchical forecasts

The aim of this paper is to provide a thinking road-map and a practical guide to researchers and practitioners working on hierarchical forecasting problems. Evaluating the performance of hierarchical forecasts comes with new challenges stemming from both the structure of the hierarchy and the application context. We discuss several relevant dimensions for researchers and analysts: the scale and units of the time series, the issue of intermittency, the forecast horizon, the importance of multiple evaluation windows and the multiple objective decision context. We conclude with a series of practical recommendations.

An Unsupervised 3D Image Registration Network for Brain MRI Deformable Registration.

In recent years, deep learning has made successful applications and remarkable achievements in the field of medical image registration, and the method of medical image registration based on deep learning has become the current research hotspot. However, the performance of convolutional neural networks may not be fully exploited due to neglect of spatial relationships between distant locations in the image and incomplete updates of network parameters. To avoid this phenomenon, MHNet, a multiscale hierarchical deformable registration network for 3D brain MR images, was proposed in this paper. This network was an unsupervised end-to-end convolutional neural network. After training, the dense displacement vector field can be predicted almost in real-time for the unseen input image pairs, which saves a lot of time compared with the traditional algorithms of independent iterative optimization for each pair of images. On the basis of the encoder-decoder structure, this network introduced the improved Inception module for multiscale feature extraction and expanding the receptive field and the hierarchical forecast structure to promote the update of the parameters of the middle layers, which achieved the best performance on the augmented public dataset compared with the existing four excellent registration methods.

Machine learning applications in hierarchical time series forecasting: Investigating the impact of promotions

Hierarchical forecasting is needed in many situations in the supply chain to support decision making. Top-down, bottom-up, and optimal linear combination methods are common in hierarchical forecasting. There is no universally optimal solution for hierarchical forecasting, and each method has some advantages and disadvantages. While top-down and bottom-up methods use only the information at the top and bottom levels, respectively, linear combinations use the individual sales forecasts from all series and levels and combine them linearly, often outperforming the conventional top-down and bottom-up methods. These methods do not directly utilise the explanatory information such as price and promotion status that may be available across different levels in the hierarchy, and their performance may be impacted by these external factors. We propose to use a multi-output regression model that utilises the explanatory variables from across hierarchical levels to simultaneously generate forecasts for all the series at the bottom level. We perform an in-depth analysis of 55 sets of fast-moving consumer goods time series and 3049 products of the M5 forecasting competition data. Our results show that our proposed algorithm effectively utilises explanatory variables from across the hierarchy to generate reliable forecasts for different hierarchical levels, especially in the presence of deep promotional discounts.