Hierarchical Time Series Research Articles

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.

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.

Learning hierarchical time series data augmentation invariances via contrastive supervision for human activity recognition

Human activity recognition (HAR) using wearable sensors is always a research hotspot in ubiquitous computing scenario, in which feature learning has played a crucial role. Recent years have witnessed outstanding success of contrastive learning in image data, which learns invariant representations by adding contrastive loss to the last layer of deep neural networks. However, the advantages of contrastive loss have been rarely leveraged in time series data for activity recognition. A fundamental obstacle to contrastive learning in HAR is that image-based augmentation could not fit well with sensor data, which raises a critical issue: the distortions induced by augmentation might be further enlarged by intermediate layers of a network and thus severely harm semantic structure of original activity instance. In this paper, taking an inspiration from deeply-supervised learning, we propose a novel approach called Contrastive Supervision by considering “where” to contrast, which aims to learn time series augmentation invariances by forcing positive pairs nearby and negative pairs far apart at different depths of neural network. Our approach can be seen as a generalization of contrastive learning in a deeply-supervised setting, where the contrastive loss is used to supervise the intermediate layers instead of only the last layer, allowing us to effectively leverage label information so as to better fuse the multi-level features. Experiments on popular benchmarks demonstrate that our approach can learn better representations and improve classification accuracy without additional inference cost for various HAR tasks in supervised and semi-supervised learning paradigms.

SLOTH: Structured Learning and Task-Based Optimization for Time Series Forecasting on Hierarchies

Multivariate time series forecasting with hierarchical structure is widely used in real-world applications, e.g., sales predictions for the geographical hierarchy formed by cities, states, and countries. The hierarchical time series (HTS) forecasting includes two sub-tasks, i.e., forecasting and reconciliation. In the previous works, hierarchical information is only integrated in the reconciliation step to maintain coherency, but not in forecasting step for accuracy improvement. In this paper, we propose two novel tree-based feature integration mechanisms, i.e., top-down convolution and bottom-up attention to leverage the information of the hierarchical structure to improve the forecasting performance. Moreover, unlike most previous reconciliation methods which either rely on strong assumptions or focus on coherent constraints only, we utilize deep neural optimization networks, which not only achieve coherency without any assumptions, but also allow more flexible and realistic constraints to achieve task-based targets, e.g., lower under-estimation penalty and meaningful decision-making loss to facilitate the subsequent downstream tasks. Experiments on real-world datasets demonstrate that our tree-based feature integration mechanism achieves superior performances on hierarchical forecasting tasks compared to the state-of-the-art methods, and our neural optimization networks can be applied to real-world tasks effectively without any additional effort under coherence and task-based constraints.

Long term 5G network traffic forecasting via modeling non-stationarity with deep learning

5G cellular networks have recently fostered a wide range of emerging applications, but their popularity has led to traffic growth that far outpaces network expansion. This mismatch may decrease network quality and cause severe performance problems. To reduce the risk, operators need long term traffic prediction to perform network expansion schemes months ahead. However, long term prediction horizon exposes the non-stationarity of series data, which deteriorates the performance of existing approaches. We deal with this problem by developing a deep learning model, Diviner, that incorporates stationary processes into a well-designed hierarchical structure and models non-stationary time series with multi-scale stable features. We demonstrate substantial performance improvement of Diviner over the current state of the art in 5G network traffic forecasting with detailed months-level forecasting for massive ports with complex flow patterns. Extensive experiments further present its applicability to various predictive scenarios without any modification, showing potential to address broader engineering problems.

What do consumers want? A methodological framework to identify determinant product attributes from consumers’ online questions

Determinant attributes play an important role in consumers’ purchase decisions. Firms rely on them to differentiate their products. Determinant attributes are typically identified using direct methods or indirect methods. However, the voice of the customer online can also provide key insights regarding attributes that are part of the consumers’ pre-purchase search process. The purpose of this study is to propose a framework to identify determinant attributes from online consumer questions. Our method uses semi-supervised Latent Dirichlet Allocation to identify product attributes initially. This is followed by the application of sequence pattern mining to identify temporal sequences of determinant attributes. Finally, hierarchical time series is used to forecast consumer interest in determinant attributes over time. The results show that our study can be used to identify determinant attributes of competing brands and can also be used to forecast consumer interest in these attributes. Brands can use this framework to have a pulse on what consumers look for before purchase. This will aid their decisions related to promotion as well as product development processes.

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.

Hierarchical Joint Graph Learning and Multivariate Time Series Forecasting

Hierarchical Joint Graph Learning and Multivariate Time Series Forecasting

Modelling and Forecasting Wheat Production in Punjab State of India using Hierarchical Time Series Models

Modelling and Forecasting Wheat Production in Punjab State of India using Hierarchical Time Series Models

Kinder, gentler – and crisis-proof? Consensus democracy, inclusive institutions and COVID-19 pandemic performance

Which patterns of democracy perform best? Although a decades-long research tradition has fairly robustly suggested that consensual democracies outperform their Westminster-style majoritarian counterparts, the scope of previous studies has been limited to ‘normal times’. In this article, the endogenous context of the COVID-19 pandemic is leveraged to study whether the alleged superiority of consensualism also holds during crises. It is hypothesised that, in addition to consensus democracy, inclusive institutions – i.e. cabinet size and interest-group corporatism – enhance crisis-related performance. Drawing on new and original data, cross-sectional and hierarchical time-series regression analyses show that horizontal power-sharing and the number of ministers substantively reduced excess mortality, while the structure of the interest-group system had no effect. Although established consensus democracies can draw on their built-in institutional assets even during crises, our findings indicate that majoritarian systems may, as a compensatory performance-enhancing tool, flexibly gear up for crisis-induced necessities by adding more ministers to the cabinet.

Levels and trends estimate of sex ratio at birth for seven provinces of Pakistan from 1980 to 2020 with scenario-based probabilistic projections of missing female birth to 2050: A Bayesian modeling approach

Most evidence on son preference in Pakistan is reflected in the higher child mortality among females than males. The sex discrimination before birth is rarely reported in Pakistan. This is the first study to quantify prenatal sex discrimination in Pakistan on a subnational level. We provide annual estimates of the sex ratio at birth (SRB) from 1980 to 2020 and scenario-based projections of the number of missing female births up to 2050 by Pakistan province. The results are based on a comprehensive database consisting of 832,091 birth records from all available surveys and censuses. We adopted a Bayesian hierarchical time series model to synthesize different data sources. We identified Balochistan with an existing imbalanced SRB since 1980. For the rest provinces without past or ongoing SRB inflation, we projected the largest female birth deficit to occur in Punjab in 2033 under the scenario that the SRB transition process starts in 2021. We demonstrated important disparities in the occurrence and quantification of missing female births up to 2050.

O‘ZBEKISTON RESPUBLIKASI TASHQI SAVDOSI TARKIBINING 2022–2026 YILLAR UCHUN PROGNOZ KUTILMALARI

Today, foreign trade is becoming one of the important factors in ensuring stable economic growth when the world is deepening the processes of globalization. The article considers the structure of foreign trade of the Republic of Uzbekistan and analyzes the relevant indicators on its basis. It is noted that in the foreign trade, as well as in all spheres in recent years there have been radical changes due to deep reforms in our country, as well as increasing foreign trade through measures aimed at eliminating existing problems in this area and strengthening the export competitiveness of our country. The study developed a forecast of the structure of foreign trade of the Republic of Uzbekistan for 2022–2026 using the method of hierarchical time series in the computer program R. At the same time, the structure of export and import of goods and services of our country was divided into two parts by geographical direction – forecasts of export and import structure with CIS and non-CIS countries were developed.

Exploring dynamic change in arterial base excess with patient outcome and lactate clearance in the intensive care unit by hierarchical time-series clustering.

Hyperlactatemia is common in the intensive care unit (ICU) and relevant to prognosis, while the process of lactate normalization requires a relatively long period. We hypothesized that the dynamic change in base excess (BE) would be associated with ICU mortality and lactate clearance. We performed a retrospective cohort study of adult patients with hyperlactatemia admitted to the ICU from 2016 to 2021. The patients were divided into two groups according to whether the peak BE in 12 h was reached in the first 6 h. We compared ICU mortality and lactate clearance at 6 and 12 h after ICU admission. During the study period, 1,608 patients were admitted to the ICU with a lactate concentration of >2.0 mmol/L and stayed in the ICU for >24 h. The mortality rate was 11.2%. The patients were divided into two groups according to whether the peak BE was reached in the first 6 h following ICU admission: Peak BE12h ≤ 6h and Peak BE12h > 6h. The patients were also recorded as whether bicarbonate treatment was received (bicarbonate group, CRRT included) or not (non-bicarbonate group). Furthermore, lactic acid clearance patterns were identified by time-series clustering (TSC) using various algorithms and distance measures. We compared ICU mortality and lactate clearance at 6 and 12 h after ICU admission with logistic regression. After adjustment for other confounding factors, we found that Peak BE12h > 6h was independently associated with ICU mortality with an odds ratio of 2.231 (p = 0.036) in the bicarbonate group and 2.359 (p < 0.005) in the non-bicarbonate group. In addition, based on the definition of >10% lactate clearance at 6 h or >30% at 12 h, we found that Peak BE12h ≤ 6h had 85.2% sensitivity and 38.1% specificity for effective lactate clearance. In time-series clustering analysis, four categories were discriminated, and pattern of lactic acid clearance reveals the early prognostic value of BE in clearance of lactic acid. A prolonged time to reaching the peak BE was independently associated with ICU mortality. In patients with hyperlactatemia, Peak BE12h ≤ 6h could be used as an indicator to predict effective lactate clearance.

Pooling information across levels in hierarchical time series forecasting via Kernel methods

In this paper, we present a novel method that extends the kernel-based support vector regression to hierarchical time series analysis. This predictive task consists of taking advantage of the hierarchical structure of a set of related time series. This is a common challenge in retail, for example, in which product sales are grouped according to categories with multiple levels. The proposed strategy constructs several predictors in a single optimization problem, pooling information across the different levels. In addition to the traditional two objectives included in support vector machines, model fit and Tikhonov regularization, data pooling is performed by including a third objective in the formulation. Originally presented as a linear method, a kernel machine is derived using duality theory. Experiments on benchmark datasets for hierarchical time series forecasting demonstrate the virtues of our all-together strategy over the well-known strategies for handling this task, namely, the bottom-up and top-down approaches.

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.

Assessing the associations between Aedes larval indices and dengue risk in Kalutara district, Sri Lanka: a hierarchical time series analysis from 2010 to 2019

BackgroundDengue is a major public health problem in Sri Lanka. Aedes vector surveillance and monitoring of larval indices are routine, long-established public health practices in the country. However, the association between Aedes larval indices and dengue incidence is poorly understood. It is crucial to evaluate lagged effects and threshold values of Aedes larval indices to set pragmatic targets for sustainable vector control interventions.MethodsMonthly Aedes larval indices and dengue cases in all 10 Medical Officer of Health (MOH) divisions in Kalutara district were obtained from 2010 to 2019. Using a novel statistical approach, a distributed lag non-linear model and a two-staged hierarchical meta-analysis, we estimated the overall non-linear and delayed effects of the Premise Index (PI), Breteau Index (BI) and Container Index (CI) on dengue incidence in Kalutara district. A set of MOH division-specific variables were evaluated within the same meta-analytical framework to determine their moderator effects on dengue risk. Using generalized additive models, we assessed the utility of Aedes larval indices in predicting dengue incidence.ResultsWe found that all three larval indices were associated with dengue risk at a lag of 1 to 2 months. The relationship between PI and dengue was homogeneous across MOH divisions, whereas that with BI and CI was heterogeneous. The threshold values of BI, PI and CI associated with dengue risk were 2, 15 and 45, respectively. All three indices showed a low to moderate accuracy in predicting dengue risk in Kalutara district.ConclusionsThis study showed the potential of vector surveillance information in Kalutara district in developing a threshold-based, location-specific early warning system with a lead time of 2 months. The estimated thresholds are nonetheless time-bound and may not be universally applicable. Whenever longitudinal vector surveillance data areavailable, the methodological framework we propose here can be used to estimate location-specific Aedes larval index thresholds in any other dengue-endemic setting.Graphical

A hybrid approach with step‐size aggregation to forecasting hierarchical time series

AbstractHierarchical time series arise in various fields such as manufacturing and services when the products or services can be hierarchically structured. “Top‐down” and “bottom‐up” forecasting approaches are often used for forecasting such hierarchical time series. In this paper, we develop a new hybrid approach (HA) with step‐size aggregation for hierarchical time series forecasting. The new approach is a weighted average of the two classical approaches with the weights being optimally chosen for all the series at each level of the hierarchy to minimize the variance of the forecast errors. The independent selection of weights for all the series at each level of the hierarchy makes the HA inconsistent while aggregating suitably across the hierarchy. To address this issue, we introduce a step‐size aggregate factor that represents the relationship between forecasts of the two consecutive levels of the hierarchy. The key advantage of the proposed HA is that it captures the structure of the hierarchy inherently due to the combination of the hierarchical approaches instead of independent forecasts of all the series at each level of the hierarchy. We demonstrate the performance of the new approach by applying it to the monthly data of ‘Industrial’ category of M3‐Competition as well as on Pakistan energy consumption data.

Probabilistic forecast reconciliation: Properties, evaluation and score optimisation

We develop a framework for forecasting multivariate data that follow known linear constraints. This is particularly common in forecasting where some variables are aggregates of others, commonly referred to as hierarchical time series, but also arises in other prediction settings. For point forecasting, an increasingly popular technique is reconciliation, whereby forecasts are made for all series (so-called base forecasts) and subsequently adjusted to cohere with the constraints. We extend reconciliation from point forecasting to probabilistic forecasting. A novel definition of reconciliation is developed and used to construct densities and draw samples from a reconciled probabilistic forecast. In the elliptical case, we prove that true predictive distributions can be recovered using reconciliation even when the location and scale of base predictions are chosen arbitrarily. Reconciliation weights are estimated to optimise energy or variogram score. The log score is not considered since it is improper when comparing unreconciled to reconciled forecasts, a result also proved in this paper. Due to randomness in the objective function, optimisation uses stochastic gradient descent. This method improves upon base forecasts in simulated and empirical data, particularly when the base forecasting models are severely misspecified. For milder misspecification, extending popular reconciliation methods for point forecasting results in similar performance to score optimisation.

Systematizing Macroframework Forecasting: High-Dimensional Conditional Forecasting with Accounting Identities

Forecasting a macroframework, which consists of many macroeconomic variables and accounting identities, is widely conducted in the policy arena to present an economic narrative and check its consistency. Such forecasting, however, is challenging because forecasters should extend limited information to the entire macroframework in an internally consistent manner. This paper proposes a method to systematically forecast macroframework by integrating (1) conditional forecasting with machine-learning techniques and (2) forecast reconciliation of hierarchical time series. We apply our method to an advanced economy and a tourism-dependent economy using France and Seychelles and show that it can improve the WEO forecast.

A novel robust structural quadratic forecasting model and applications

Recently, big data have been collected and stored for the purpose of forecasting in modern operations management. Nevertheless, a massive amount of data information does not provide additional advantages as expected when implementing forecasting or decision-making tasks but causes a waste of time and memory for collection, storage, and computing. Besides hierarchical time series forecasting, strong and weak hierarchies in variable space have also been studied for years. To this end, structural variable selection methods that extract vital information from data by distinguishing important variable from nuisance variable with strong or weak hierarchy are favored by scholars. However, the existing structural variable selection methods focus on establishing models with Gaussian or binomial noises without considering heavy-tailed distributions. Therefore, in this paper, we investigate a robust structured variable selection approach called structural quadratic quantile regression. The model hierarchies are achieved using a proper design of an optimization with check loss function for robustness, regularization term, and one constraint. In computation, a novel algorithm is derived based on Dykstra's algorithm and the proximal alternating direction method of multipliers (pADMM), and its convergence is theoretically guaranteed. Finally, the efficacy of the proposed approach is demonstrated using both simulation and empirical applications from numbers of scientific domains in forecasting.