The COVID-19 pandemic has had a devastating effect on many industries around the world including tourism and policy makers are interested in mapping out what the recovery path will look like. We propose a novel statistical methodology for generating scenario-based probabilistic forecasts based on a large survey of 443 tourism experts and stakeholders. The scenarios map out pessimistic, most-likely and optimistic paths to recovery. Taking advantage of the natural aggregation structure of tourism data due to geographic locations and purposes of travel, we propose combining forecast reconciliation and forecast combinations implemented to historical data to generate robust COVID-free counterfactual forecasts, to contrast against. Our empirical application focuses on Australia, analyzing international arrivals and domestic flows. Both sectors have been severely affected by travel restrictions in the form of international and interstate border closures and regional lockdowns. The two sets of forecasts, allow policy makers to map out the road to recovery and also estimate the expected effect of the pandemic.
Collections of time series formed via aggregation are prevalent in many fields. These are commonly referred to as hierarchical time series and may be constructed cross-sectionally across different variables, temporally by aggregating a single series at different frequencies, or even generalised beyond aggregation as time series that respect linear constraints. When forecasting such time series, a desirable condition is for forecasts to be coherent: to respect the constraints. The past decades have seen substantial growth in this field with the development of reconciliation methods that ensure coherent forecasts and improve forecast accuracy. This paper serves as a comprehensive review of forecast reconciliation and an entry point for researchers and practitioners dealing with hierarchical time series. The scope of the article includes perspectives on forecast reconciliation from machine learning, Bayesian statistics and probabilistic forecasting, as well as applications in economics, energy, tourism, retail demand and demography.
This paper introduces the concept of Temporal Hierarchies for time series forecasting. A temporal hierarchy can be constructed for any time series by means of non-overlapping temporal aggregation. Predictions constructed at all aggregation levels are combined with the proposed framework to result in temporally reconciled, accurate and robust forecasts. The implied combination mitigates modelling uncertainty, while the reconciled nature of the forecasts results in a unified prediction that supports aligned decisions at different planning horizons: from short-term operational up to long-term strategic planning. The proposed methodology is independent of forecasting models. It can embed high level managerial forecasts that incorporate complex and unstructured information with lower level statistical forecasts. Our results show that forecasting with temporal hierarchies increases accuracy over conventional forecasting, particularly under increased modelling uncertainty. We discuss organisational implications of the temporally reconciled forecasts using a case study of Accident & Emergency departments.
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.
Inventory control systems rely on accurate and robust forecasts of future demand to support decisions such as setting of safety stocks. The combination of multiple forecasts is shown to be effective not only in reducing forecast errors, but also in being less sensitive to limitations of a single model. Research on forecast combination has primarily focused on improving accuracy, largely ignoring the overall shape and distribution of forecast errors. Nonetheless, these are essential for managing the level of aversion to risk and uncertainty for companies. This study examines the forecast error distributions of base and combination forecasts and their implications for inventory performance. It explores whether forecast combinations transform the forecast error distribution towards desired properties for safety stock calculations, typically based on the assumption of normally distributed errors and unbiased forecasts. In addition, it considers the similarity between in- and out-of-sample characteristics of such errors and the impact of different lead times. The effects of established combination methods are explored empirically using a representative set of forecasting methods and a dataset of 229 weekly demand series from a leading household and personal care UK manufacturer. Findings suggest that forecast combinations make the in- and out-of-sample behaviour more consistent, requiring less safety stock on average than base forecasts. Furthermore we find that using in-sample empirical error distributions of combined forecasts approximates well the out-of-sample ones, in contrast to base forecasts.
A key challenge for call centres remains the forecasting of high frequency call arrivals collected in hourly or shorter time buckets. In addition to the complex intraday, intraweek and intrayear seasonal cycles, call arrival data typically contain a large number of anomalous days, driven by the occurrence of holidays, special events, promotional activities and system failures. This study evaluates the use of a variety of univariate time series forecasting methods for forecasting intraday call arrivals in the presence of such outliers. Apart from established, statistical methods, we consider artificial neural networks (ANNs). Based on the modelling flexibility of the latter, we introduce and evaluate different methods to encode the outlying periods. Using intraday arrival series from a call centre operated by one of Europe's leading entertainment companies, we provide new insights on the impact of outliers on the performance of established forecasting methods. Results show that ANNs forecast call centre data accurately, and are capable of modelling complex outliers using relatively simple outlier modelling approaches. We argue that the relative complexity of ANNs over standard statistical models is offset by the simplicity of coding multiple and unknown effects during outlying periods.
A major challenge in automating the production of a large number of forecasts, as often required in many business applications, is the need for robust and reliable predictions. Increased noise, outliers and structural changes in the series, all too common in practice, can severely affect the quality of forecasting. We investigate ways to increase the reliability of exponential smoothing forecasts, the most widely used family of forecasting models in business forecasting. We consider two alternative sets of approaches, one stemming from statistics and one from machine learning. To this end, we adapt M-estimators, boosting and inverse boosting to parameter estimation for exponential smoothing. We propose appropriate modifications that are necessary for time series forecasting while aiming to obtain scalable algorithms. We evaluate the various estimation methods using multiple real datasets and find that several approaches outperform the widely used maximum likelihood estimation. The novelty of this work lies in (1) demonstrating the usefulness of M-estimators, (2) and of inverse boosting, which outperforms standard boosting approaches, and (3) a comparative look at statistics versus machine learning inspired approaches.
In forecasting research, the focus has largely been on decision support systems for enhancing performance, with fewer studies in learning support systems. As a remedy, Intelligent Tutoring Systems (ITSs) offer an innovative solution in that they provide one-on-one online computer-based learning support affording student modelling, adaptive pedagogical response, and performance tracking. This study provides a detailed description of the design and development of the first Forecasting Intelligent Tutoring System, aptly coined FITS, designed to assist students in developing an understanding of time series forecasting using classical time series decomposition. The system’s impact on learning is assessed through a pilot evaluation study, and its usefulness in understanding how students learn is illustrated through the exploration and statistical analysis of a small sample of student models. Practical reflections on the system’s development are also provided to better understand how such systems can facilitate and improve forecasting performance through training.
This chapter elucidates liquidity forecasting within the context of technical assistance. The audience for this chapter is central bank staff with a strong quantitative background. Liquidity forecasting entails a process of estimating the near-term path of a bank’s reserves using a centralized framework. Short-term liquidity forecasts are used to calibrate the volume of central bank monetary operations to align liquidity with the announced stance of monetary policy, whether expressed as an interest rate or as a quantity. The best practice would be for the central bank to receive accurate information for counterparties that have accounts in its books, including its monetary counterparties (banks) or non-monetary counterparties, such as the government. However, the central bank may not have direct access to some counterparties (e.g., the public which demands banknotes) or the information could include significant errors. This chapter presents the statistical methods that have been used in technical assistance to forecast liquidity factors and the demand for liquidity. It also proposes solutions to select the best models, measure forecast accuracy, and reconcile forecasts. Some liquidity factors are relatively easy to forecast due to regular patterns (currency in circulation) while others require more sophisticated models, such as the government account. There is a tradeoff between the cost of implementing complex models and the accuracy gains.
Modelling artificial neural networks for accurate time series prediction poses multiple challenges, in particular specifying the network architecture in accordance with the underlying structure of the time series. The data generating processes may exhibit a variety of stochastic or deterministic time series patterns of single or multiple seasonality, trends and cycles, overlaid with pulses, level shifts and structural breaks, all depending on the discrete time frequency in which it is observed. For heterogeneous datasets of time series, such as the 2008 ESTSP competition, a universal methodology is required for automatic network specification across varying data patterns and time frequencies. We propose a fully data driven forecasting methodology that combines filter and wrapper approaches for feature selection, including automatic feature evaluation, construction and transformation. The methodology identifies time series patterns, creates and transforms explanatory variables and specifies multilayer perceptrons for heterogeneous sets of time series without expert intervention. Examples of the valid and reliable performance in comparison to established benchmark methods are shown for a set of synthetic time series and for the ESTSP'08 competition dataset, where the proposed methodology obtained second place.
Forecasting researchers, with few exceptions, have ignored the current major forecasting controversy: global warming and the role of climate modelling in resolving this challenging topic. In this paper, we take a forecaster's perspective in reviewing established principles for validating the atmospheric-ocean general circulation models (AOGCMs) used in most climate forecasting, and in particular by the Intergovernmental Panel on Climate Change (IPCC). Such models should reproduce the behaviours characterising key model outputs, such as global and regional temperature changes. We develop various time series models and compare them with forecasts based on one well-established AOGCM from the UK Hadley Centre. Time series models perform strongly, and structural deficiencies in the AOGCM forecasts are identified using encompassing tests. Regional forecasts from various GCMs had even more deficiencies. We conclude that combining standard time series methods with the structure of AOGCMs may result in a higher forecasting accuracy. The methodology described here has implications for improving AOGCMs and for the effectiveness of environmental control policies which are focussed on carbon dioxide emissions alone. Critically, the forecast accuracy in decadal prediction has important consequences for environmental planning, so its improvement through this multiple modelling approach should be a priority.
Intermittent demand appears when demand events occur only sporadically. Typically such time series have few observations making intermittent demand forecasting challenging. Forecast errors can be costly in terms of unmet demand or obsolescent stock. Intermittent demand forecasting has been addressed using established forecasting methods, including simple moving averages, exponential smoothing and Croston's method with its variants. This study proposes a neural network (NN) methodology to forecast intermittent time series. These NNs are used to provide dynamic demand rate forecasts, which do not assume constant demand rate in the future and can capture interactions between the non-zero demand and the inter-arrival rate of demand events. This overcomes the limitations of Croston's method. In order to mitigate the issue of limited fitting sample, which is common in intermittent demand, the proposed models use regularised training and median ensembles over multiple training initialisations to produce robust forecasts. The NNs are evaluated against established benchmarks using both forecasting accuracy and inventory metrics. The findings of forecasting and inventory metrics are conflicting. While NNs achieved poor forecasting accuracy and bias, all NN variants achieved higher service levels than the best performing Croston's method variant, without requiring analogous increases in stock holding volume. Therefore, NNs are found to be effective for intermittent demand applications. This study provides further arguments and evidence against the use of conventional forecasting accuracy metrics to evaluate forecasting methods for intermittent demand, concluding that attention to inventory metrics is desirable.
Intermittent demand time series involve items that are requested infrequently, resulting in sporadic demand. Crostons method and its variants have been proposed in the literature to address this forecasting problem. Recently other novel methods have appeared. Although the literature provides guidance on the suggested range for model parameters, a consistent and valid optimisation methodology is lacking. Growing evidence in the literature points against the use of conventional accuracy error metrics for model evaluation for intermittent demand time series. Consequently these may be inappropriate for parameter or model selection. This paper contributes to the discussion by evaluating a series of conventional time series error metrics, along with two novel ones for parameter optimisation for intermittent demand methods. The proposed metrics are found to not only perform best, but also provide consistent parameters with the literature, in contrast to conventional metrics. Furthermore, this work validates that employing different parameters for smoothing the non-zero demand and the inter-demand intervals of Crostons method and its variants is beneficial. The evaluated error metrics are considered for automatic model selection for each time series. Although they are found to perform similar to theory driven model selection schemes, they fail to outperform single models substantially. These findings are validated using both out-of-sample forecast evaluation and inventory simulations.
Key to ensuring a successful tourism sector is timely policy making and detailed planning. National policy formulation and strategic planning requires long-term forecasts at an aggregate level, while regional operational decisions require short-term forecasts, relevant to local tourism operators. For aligned decisions at all levels, supporting forecasts must be ‘coherent’ that is they should add up appropriately, across relevant demarcations (e.g., geographical divisions or market segments) and also across time. We propose an approach for generating coherent forecasts across both cross-sections and planning horizons for Australia. This results in significant improvements in forecast accuracy with substantial decision making benefits. Coherent forecasts help break intra- and inter-organisational information and planning silos, in a data driven fashion, blending information from different sources. This article also launches the Annals of Tourism Research Curated Collection on Tourism Demand Forecast, a special selection of research in this field.
Intermittent demand forecasting has been widely researched in the context of spare parts management. However, it is becoming increasingly relevant to many other areas, such as retailing, where at the very disaggregate level time series may be highly intermittent, but at more aggregate levels are likely to exhibit trends and seasonal patterns. The vast majority of intermittent demand forecasting methods are inappropriate for producing forecasts with such features. We propose using temporal hierarchies to produce forecasts that demonstrate these traits at the various aggregation levels, effectively informing the resulting intermittent forecasts of these patterns that are identifiable only at higher levels. We conduct an empirical evaluation on real data and demonstrate statistically significant gains for both point and quantile forecasts.
The combination of forecasts resulting from an ensemble of neural networks has been shown to outperform the use of a single "best" network model. This is supported by an extensive body of literature, which shows that combining generally leads to improvements in forecasting accuracy and robustness, and that using the mean operator often outperforms more complex methods of combining forecasts. This paper proposes a mode ensemble operator based on kernel density estimation, which unlike the mean operator is insensitive to outliers and deviations from normality, and unlike the median operator does not require symmetric distributions. The three operators are compared empirically and the proposed mode ensemble operator is found to produce the most accurate forecasts, followed by the median, while the mean has relatively poor performance. The findings suggest that the mode operator should be considered as an alternative to the mean and median operators in forecasting applications. Experiments indicate that mode ensembles are useful in automating neural network models across a large number of time series, overcoming issues of uncertainty associated with data sampling, the stochasticity of neural network training, and the distribution of the forecasts.
Forecast selection and combination are regarded as two competing alternatives. In the literature there is substantial evidence that forecast combination is beneficial, in terms of reducing the forecast errors, as well as mitigating modelling uncertainty as we are not forced to choose a single model. However, whether all forecasts to be combined are appropriate, or not, is typically overlooked and various weighting schemes have been proposed to lessen the impact of inappropriate forecasts. We argue that selecting a reasonable pool of forecasts is fundamental in the modelling process and in this context both forecast selection and combination can be seen as two extreme pools of forecasts. We evaluate forecast pooling approaches and find them beneficial in terms of forecast accuracy. We propose a heuristic to automatically identify forecast pools, irrespective of their source or the performance criteria, and demonstrate that in various conditions it performs at least as good as alternative pools that require additional modelling decisions and better than selection or combination.
Sales data often only represent a part of the demand for a service product owing to constraints such as capacity or booking limits. Unconstraining methods are concerned with estimating the true demand from such constrained sales data. This paper addresses the frequently encountered situation of observing only a few sales events at the individual product level and proposes variants of small demand forecasting methods to be used for unconstraining. The usual procedure is to aggregate data; however, in that case we lose information on when restrictions were imposed or lifted within a given booking profile. Our proposed methods exploit this information and are able to approximate convex, concave or homogeneous booking curves. Furthermore, they are numerically robust due to our proposed group-based parameter optimization. Empirical results on accuracy and revenue performance based on data from a major car rental company indicate revenue improvements over a best practice benchmark by statistically significant 0.5–1.4% in typical scenarios.
Demand forecasting is central to decision making and operations in organisations. As the volume of forecasts increases, for example due to an increased product customisation that leads to more SKUs being traded, or a reduction in the length of the forecasting cycle, there is a pressing need for reliable automated forecasting. Conventionally, companies rely on a statistical baseline forecast that captures only past demand patterns, which is subsequently adjusted by human experts to incorporate additional information such as promotions. Although there is evidence that such process adds value to forecasting, it is questionable how much it can scale up, due to the human element. Instead, in the literature it has been proposed to enhance the baseline forecasts with external well-structured information, such as the promotional plan of the company, and let experts focus on the less structured information, thus reducing their workload and allowing them to focus where they can add most value. This change in forecasting support systems requires reliable multivariate forecasting models that can be automated, accurate and robust. This paper proposes an extension of the recently proposed Multiple Aggregation Prediction Algorithm (MAPA), which uses temporal aggregation to improve upon the established exponential smoothing family of methods. MAPA is attractive as it has been found to increase both the accuracy and robustness of exponential smoothing. The extended multivariate MAPA is evaluated against established benchmarks in modelling a number of heavily promoted products and is found to perform well in terms of forecast bias and accuracy. Furthermore, we demonstrate that modelling time series using multiple temporal aggregation levels makes the final forecast robust to model mis-specification.
Identifying the most appropriate time series model to achieve a good forecasting accuracy is a challenging task. We propose a novel algorithm that aims to mitigate the importance of model selection, while increasing the accuracy. Multiple time series are constructed from the original time series, using temporal aggregation. These derivative series highlight different aspects of the original data, as temporal aggregation helps in strengthening or attenuating the signals of different time series components. In each series, the appropriate exponential smoothing method is fitted and its respective time series components are forecast. Subsequently, the time series components from each aggregation level are combined, then used to construct the final forecast. This approach achieves a better estimation of the different time series components, through temporal aggregation, and reduces the importance of model selection through forecast combination. An empirical evaluation of the proposed framework demonstrates significant improvements in forecasting accuracy, especially for long-term forecasts.
Recent advances have demonstrated the benefits of temporal aggregation for demand forecasting, including increased accuracy, improved stock control and reduced modelling uncertainty. With temporal aggregation a series is transformed, strengthening or attenuating different elements and thereby enabling better identification of the time series structure. Two different schools of thought have emerged. The first focuses on identifying a single optimal temporal aggregation level at which a forecasting model maximises its accuracy. In contrast, the second approach fits multiple models at multiple levels, each capable of capturing different features of the data. Both approaches have their merits, but so far they have been investigated in isolation. We compare and contrast them from a theoretical and an empirical perspective, discussing the merits of each, comparing the realised accuracy gains under different experimental setups, as well as the implications for business practice. We provide suggestions when to use each for maximising demand forecasting gains.
COVID-19 disrupted international tourism worldwide, subsequently presenting forecasters with a challenging conundrum. In this competition, we predict international arrivals for 20 destinations in two phases: (i) Ex post forecasts pre-COVID; (ii) Ex ante forecasts during and after the pandemic up to end 2021. Our results show that univariate combined with cross-sectional hierarchical forecasting techniques (THieF-ETS) outperform multivariate models pre-COVID. Scenarios were developed based on judgemental adjustment of the THieF-ETS baseline forecasts. Analysts provided a regional view on the most likely path to normal, based on country-specific regulations, macroeconomic conditions, seasonal factors and vaccine development. Results show an average recovery of 58% compared to 2019 tourist arrivals in the 20 destinations under the medium scenario; severe, it is 34% and mild, 80%.
Using leading indicators for business forecasting-in contrast to macroeconomic forecasting-has been relatively rare, partly because our traditional time-series methods do not readily allow incorporation of external variables. Nowadays, however, we have an abundance of potentially useful indicators, and there is evidence that utilizing relevant ones in a forecasting model can significantly improve forecast accuracy and transparency. In this article, Nikolaos and Yves show how to find appropriate leading indicators and make good use of them for sales forecasting.
Inaccurate forecasts can be costly for company operations, in terms of stock-outs and lost sales, or over-stocking, while not meeting service level targets. The forecasting literature, often disjoint from the needs of the forecast users, has focused on providing optimal models in terms of likelihood and various accuracy metrics. However, there is evidence that this does not always lead to better inventory performance, as often the translation between forecast errors and inventory results is not linear. In this study, we consider an approach to parametrising forecasting models by directly considering appropriate inventory metrics and the current inventory policy. We propose a way to combine the competing multiple inventory objectives, i.e. meeting demand, while eliminating excessive stock, and use the resulting cost function to identify inventory optimal parameters for forecasting models. We evaluate the proposed parametrisation against established alternatives and demonstrate its performance on real data. Furthermore, we explore the connection between forecast accuracy and inventory performance and discuss the extent to which the former is an appropriate proxy of the latter.
Intermittent demand is characterised by infrequent demand arrivals, where many periods have zero demand, coupled with varied demand sizes. The dual source of variation renders forecasting for intermittent demand a very challenging task. Many researchers have focused on the development of specialised methods for intermittent demand. However, apart from a case study on hierarchical forecasting, the effects of combining, which is a standard practice for regular demand, have not been investigated. This paper empirically explores the efficiency of forecast combinations in the intermittent demand context. We examine both method and temporal combinations of forecasts. The first are based on combinations of different methods on the same time series, while the latter use combinations of forecasts produced on different views of the time series, based on temporal aggregation. Temporal combinations of single or multiple methods are investigated, leading to a new time-series classification, which leads to model selection and combination. Results suggest that appropriate combinations lead to improved forecasting performance over single methods, as well as simplifying the forecasting process by limiting the need for manual selection of methods or hyper-parameters of good performing benchmarks. This has direct implications for intermittent demand forecasting in practice.
In most business forecasting applications, the decision-making need we have directs the frequency of the data we collect (monthly, weekly, etc.) and use for forecasting. In this article, Fotios and Nikolaos introduce an approach that combines forecasts generated by modeling the different frequencies (levels of temporal aggregation). Their technique augments our information about the data used for forecasting and, as such, can result in more accurate forecasts. It also automatically reconciles the forecasts at different levels.
In this paper we focus on forecasting for intermittent demand data. We propose a new aggregation framework for intermittent demand forecasting that performs aggregation over the demand volumes, in contrast to the standard framework that employs temporal (over time) aggregation. To achieve this we construct a transformed time series, the inverse intermittent demand series. The new algorithm is expected to work best on erratic and lumpy demand, as a result of the variance reduction of the non-zero demands. The improvement in forecasting performance is empirically demonstrated through an extensive evaluation in more than 8000 time series of two well-researched spare parts data sets from the automotive and defence sectors. Furthermore, a simulation is performed so as to provide a stock-control evaluation. The proposed framework could find popularity among practitioners given its suitability when dealing with clump sizes. As such it could be used in conjunction with existing popular forecasting methods for intermittent demand as an exception handling mechanism when certain types of demand are observed.
In this paper, we explored how judgment can be used to improve the selection of a forecasting model. We compared the performance of judgmental model selection against a standard algorithm based on information criteria. We also examined the efficacy of a judgmental model-build approach, in which experts were asked to decide on the existence of the structural components (trend and seasonality) of the time series instead of directly selecting a model from a choice set. Our behavioral study used data from almost 700 participants, including forecasting practitioners. The results from our experiment suggest that selecting models judgmentally results in performance that is on par, if not better, to that of algorithmic selection. Further, judgmental model selection helps to avoid the worst models more frequently compared to algorithmic selection. Finally, a simple combination of the statistical and judgmental selections and judgmental aggregation significantly outperform both statistical and judgmental selections.
Forecasting has always been at the forefront of decision making and planning. The uncertainty that surrounds the future is both exciting and challenging, with individuals and organisations seeking to minimise risks and maximise utilities. The large number of forecasting applications calls for a diverse set of forecasting methods to tackle real-life challenges. This article provides a non-systematic review of the theory and the practice of forecasting. We provide an overview of a wide range of theoretical, state-of-the-art models, methods, principles, and approaches to prepare, produce, organise, and evaluate forecasts. We then demonstrate how such theoretical concepts are applied in a variety of real-life contexts. We do not claim that this review is an exhaustive list of methods and applications. However, we wish that our encyclopedic presentation will offer a point of reference for the rich work that has been undertaken over the last decades, with some key insights for the future of forecasting theory and practice. Given its encyclopedic nature, the intended mode of reading is non-linear. We offer cross-references to allow the readers to navigate through the various topics. We complement the theoretical concepts and applications covered by large lists of free or open-source software implementations and publicly-available databases.
Exponential smoothing is widely used in practice and has shown its efficacy and reliability in many business applications. Yet there are cases, for example when the estimation sample is limited, where the estimated smoothing parameters can be erroneous, often unnecessarily large. This can lead to over-reactive forecasts and high forecast errors. Motivated by these challenges, we investigate the use of shrinkage estimators for exponential smoothing. This can help with parameter estimation and mitigating parameter uncertainty. Building on the shrinkage literature, we explore ℓ1 and ℓ2 shrinkage for different time series and exponential smoothing model specifications. From a simulation and an empirical study, we find that using shrinkage in exponential smoothing results in forecast accuracy improvements and better prediction intervals. In addition, using bias–variance decomposition, we show the interdependence between smoothing parameters and initial values, and the importance of the initial value estimation on point forecasts and prediction intervals.
Hierarchical forecasting has been receiving increasing attention in the literature. The notion of coherency is central to this, which implies that the hierarchical time series follows some linear aggregation constraints. This notion, however, does not take several modelling uncertainties into account. We propose to redefine coherency as stochastic. This allows to accommodate overlooked uncertainties in forecast reconciliation. We show analytically that there are two potential sources of uncertainty in forecast reconciliation. We use simulated data to demonstrate how these uncertainties propagate to the covariance matrix estimation, introducing uncertainty in the reconciliation weights matrix. This then increases the uncertainty of the reconciled forecasts. We apply our understanding to modelling accident and emergency admissions in a UK hospital. Our analysis confirms the insights from stochastic coherency in forecast reconciliation. It shows that we gain accuracy improvement from forecast reconciliation, on average, at the cost of the variability of the forecast error distribution. Users may opt to prefer less volatile error distributions to assist decision making.
Retailers depend on accurate forecasts of product sales at the Store × SKU level to efficiently manage their inventory. Consequently, there has been increasing interest in identifying more advanced statistical techniques that lead to accuracy improvements. However, the inclusion of multiple drivers affecting demand into commonly used ARIMA and ETS models is not straightforward, particularly when many explanatory variables are available. Moreover, regularization regression models that shrink the model’s parameters allow for the inclusion of a lot of relevant information but do not intrinsically handle the dynamics of the demand. These problems have not been addressed by previous studies. Nevertheless, multiple simultaneous effects interacting are common in retailing. To be successful, any approach needs to be automatic, robust and efficiently scaleable. In this study, we design novel approaches to forecast retailer product sales taking into account the main drivers which affect SKU demand at store level. To address the variable selection challenge, the use of dimensionality reduction via principal components analysis (PCA) and shrinkage estimators was investigated. The empirical results, using a case study of supermarket sales in Portugal, show that both PCA and shrinkage are useful and result in gains in forecast accuracy in the order of 10% over benchmarks while offering insights on the impact of promotions. Focusing on the promotional periods, PCA-based models perform strongly, while shrinkage estimators over-shrink. For the non-promotional periods, shrinkage estimators significantly outperform the alternatives.
Tactical forecasting in supply chain management supports planning for inventory, scheduling production, and raw material purchase, amongst other functions. It typically refers to forecasts up to 12 months ahead. Traditional forecasting models take into account univariate information extrapolating from the past, but cannot anticipate macroeconomic events, such as steep increases or declines in national economic activity. In practice this is countered by using managerial expert judgement, which is well known to suffer from various biases, is expensive and not scalable. This paper evaluates multiple approaches to improve tactical sales forecasting using macro-economic leading indicators. The proposed statistical forecast selects automatically both the type of leading indicators, as well as the order of the lead for each of the selected indicators. However as the future values of the leading indicators are unknown an additional uncertainty is introduced. This uncertainty is controlled in our methodology by restricting inputs to an unconditional forecasting setup. We compare this with the conditional setup, where future indicator values are assumed to be known and assess the theoretical loss of forecast accuracy. We also evaluate purely statistical model building against judgement aided models, where potential leading indicators are pre-filtered by experts, quantifying the accuracy-cost trade-off. The proposed framework improves on forecasting accuracy over established time series benchmarks, while providing useful insights about the key leading indicators. We evaluate the proposed approach on a real case study and find 18.8% accuracy gains over the current forecasting process.
We propose a forecasting method to improve the accuracy of tactical sales predictions for a major supplier to the tire industry. This level of forecasting, which serves as direct input to the demand-planning process and steers the global supply chain, is typically done up to a year in advance. The product portfolio of the company for which we did our research is sensitive to external events. Univariate statistical methods, which are commonly used in practice, cannot be used to anticipate and forecast changes in the market; and forecasts by human experts are known to be biased and inconsistent. The method we propose allows us to automate the identification of key leading indicators, which drive sales, from a massive set of macroeconomic indicators, across di erent regions and markets; thus, we can generate accurate forecasts. Our method also allows us to handle the additional complexity that results from short-term and long-term dynamics of product sales and external indicators. For the company we study, accuracy improved by 16.1 percent over its current practice. Furthermore, our method makes the market dynamics transparent to company managers, thus allowing them to better understand the events and economic variables that a ect the sales of their products.
Tactical capacity planning relies on future estimates of demand for the mid- to long-term. On these forecast horizons there is increased uncertainty that the analysts face. To this purpose, we incorporate macroeconomic variables into microeconomic demand forecasting. Forecast accuracy metrics, which are typically used to assess improvements in predictions, are proxies of the real decision associated costs. However, measuring the direct impact on decisions is preferable. In this paper, we examine the capacity planning decision at plant level of a manufacturer. Through an inventory simulation setup, we evaluate the gains of incorporating external macroeconomic information in the forecasts, directly, in terms of achieving target service levels and inventory performance. Furthermore, we provide an approach to indicate capacity alerts, which can serve as input for global capacity pooling decisions. Our work has two main contributions. First, we demonstrate the added value of leading indicator information in forecasting models, when evaluated directly on capacity planning. Second, we provide additional evidence that traditional metrics of forecast accuracy exhibit weak connection with the real decision costs, in particular for capacity planning. We propose a more realistic assessment of the forecast quality by evaluating both the first and second moment of the forecast distribution. We discuss implications for practice, in particular given the typical over-reliance on forecast accuracy metrics for choosing the appropriate forecasting model.
Safety stock is necessary for firms in order to manage the uncertainty of demand. A key component in its determination is the estimation of the variance of the forecast error over lead time. Given the multitude of demand processes that lack analytical expressions of the variance of forecast error, an approximation is needed. It is common to resort to finding the one-step ahead forecast errors variance and scaling it by the lead time. However, this approximation is flawed for many processes as it overlooks the autocorrelations that arise between forecasts made at different lead times. This research addresses the issue of these correlations first by demonstrating their existence for some fundamental demand processes, and second by showing through an inventory simulation the inadequacy of the approximation. We propose to monitor the empirical variance of the lead time errors, instead of estimating the point forecast error variance and extending it over the lead time interval. The simulation findings indicate that this approach provides superior results to other approximations in terms of cycle-service level. Given its lack of assumptions and computational simplicity, it can be easily implemented in any software, making it appealing to both practitioners and academics.
Recently, there has been substantial research on the augmentation of aggregate forecasts with individual consumer data from internet platforms, such as search traffic or social network shares. Although the majority of studies have reported increases in accuracy, many exhibit design weaknesses, including a lack of adequate benchmarks or rigorous evaluation. Furthermore, their usefulness over the product life-cycle has not been investigated, even though this may change, as consumers may search initially for pre-purchase information, but later for after-sales support. This study begins by reviewing the relevant literature, then attempts to support the key findings using two forecasting case studies. Our findings are in stark contrast to those in the previous literature, as we find that established univariate forecasting benchmarks, such as exponential smoothing, consistently perform better those that include online information. Our research underlines the need for a thorough forecast evaluation and argues that the usefulness of online platform data for supporting operational decisions may be limited.
In today's competitive market environment, it is vital for companies to gain insight about competitors' new product launches. Past studies have demonstrated the predictive value of prerelease online search traffic (PROST) for new product forecasting. Relying on these findings and the public availability of PROST, we investigate its usefulness for estimating sales of competing products. We propose a model for predicting the success of competitors' product launches, based on own past product sales data and competitor's prerelease Google Trends. We find that PROST increases predictive accuracy by more than 18% compared to models that only use internally available sales data and product characteristics of video game sales. We conclude that this inexpensive source of competitive intelligence can be helpful when managing the marketing mix and planning new product releases.
Achieving high accuracy in energy consumption forecasting is critical for improving energy management and planning. However, this requires the selection of appropriate forecasting models, able to capture the individual characteristics of the series to be predicted, which is a task that involves a lot of system and region level, not only the model selection problem is expanded to multiple time series, but we also require aggregation consistency of the forecasts across levels. Although hierarchical forecasting, such as the bottom-up, the top-down, and the optimal reconciliation methods, can address the aggregation consistency concerns, it does not resolve the model selection uncertainty. To address this issue, we rely on Multiple Temporal Aggregation (MTA), which has been shown to mitigate the model selection problem for low-frequency time series. We propose a modification of the Multiple Aggregation Prediction Algorithm, a special implementation of MTA, for high-frequency time series to better handle the undesirable effect of seasonality shrinkage that MTA implies and combine it with conventional cross-sectional hierarchical forecasting. The impact of incorporating temporal aggregation in hierarchical forecasting is empirically assessed using a real data set from five bank branches. We show that the proposed MTA approach, combined with the optimal reconciliation method, demonstrates superior accuracy, aggregation consistency, and reliable automatic forecasting.
Despite improvements in statistical forecasting, human judgment remains fundamental to business forecasting and demand planning. Typically, forecasters do not rely solely on statistical forecasts; they also adjust forecasts according to their knowledge, experience, and information that is not available to statistical models. However, we have limited understanding of the adjustment mechanisms employed, particularly how people use additional information (e.g., special events and promotions, weather, holidays) and under which conditions this is beneficial. Using a multi-method approach, we first analyse a UK retailer case study exploring its operations and the forecasting process. The case study provides a contextual setting for the laboratory experiments that simulate a typical supply chain forecasting process. In the experimental study, we provide past sales, statistical forecasts (using baseline and promotional models) and qualitative information about past and future promotional periods. We include contextual information, with and without predictive value, that allows us to investigate whether forecasters can filter such information correctly. We find that when adjusting, forecasters tend to focus on model-based anchors, such as the last promotional uplift and the current statistical forecast, ignoring past baseline promotional values and additional information about previous promotions. The impact of contextual statements for the forecasting period depends on the type of statistical predictions provided: when a promotional forecasting model is presented, people tend to misinterpret the provided information and over-adjust, harming accuracy.
Many modern statistical models are used for both insight and prediction when applied to data. When models are used for prediction one should optimise parameters through a prediction error loss function. Estimation methods based on multiple steps ahead forecast errors have been shown to lead to more robust and less biased estimates of parameters. However, a plausible explanation of why this is the case is lacking. In this paper, we provide this explanation, showing that the main benefit of these estimators is in a shrinkage effect, happening in univariate models naturally. However, this can introduce a series of limitations, due to overly aggressive shrinkage. We discuss the predictive likelihoods related to the multistep estimators and demonstrate what their usage implies to time series models. To overcome the limitations of the existing multiple steps estimators, we propose the Geometric Trace Mean Squared Error, demonstrating its advantages. We conduct a simulation experiment showing how the estimators behave with different sample sizes and forecast horizons. Finally, we carry out an empirical evaluation on real data, demonstrating the performance and advantages of the estimators. Given that the underlying process to be modelled is often unknown, we conclude that the shrinkage achieved by the GTMSE is a competitive alternative to conventional ones.
Exponential smoothing has been one of the most popular forecasting methods usedto support various decisions in organizations, in activities such as inventory man-agement, scheduling, revenue management, and other areas. Although its relativesimplicity and transparency have made it very attractive for research and practice,identifying the underlying trend remains challenging with significant impact on theresulting accuracy. This has resulted in the development of various modifications oftrend models, introducing a model selection problem. With the aim of addressingthis problem, we propose the complex exponential smoothing (CES), based on thetheory of functions of complex variables. The basic CES approach involves only twoparameters and does not require a model selection procedure. Despite these simpli-fications, CES proves to be competitive with, or even superior to existing methods.We show that CES has several advantages over conventional exponential smooth-ing models: it can model and forecast both stationary and non-stationary processes,and CES can capture both level and trend cases, as defined in the conventionalexponential smoothing classification. CES is evaluated on several forecasting com-petition datasets, demonstrating better performance than established benchmarks.We conclude that CES has desirable features for time series modeling and opens newpromising avenues for research.
Supply chain risk management has drawn the attention of practitioners and academics alike. One source of risk is demand uncertainty. Demand forecasting and safety stock levels are employed to address this risk. Most previous work has focused on point demand forecasting, given that the forecast errors satisfy the typical normal i.i.d. assumption. However, the real demand for products is difficult to forecast accurately, which means that—at minimum—the i.i.d. assumption should be questioned. This work analyzes the effects of possible deviations from the i.i.d. assumption and proposes empirical methods based on kernel density estimation (non-parametric) and GARCH(1,1) models (parametric), among others, for computing the safety stock levels. The results suggest that for shorter lead times, the normality deviation is more important, and kernel density estimation is most suitable. By contrast, for longer lead times, GARCH models are more appropriate because the autocorrelation of the variance of the forecast errors is the most important deviation. In fact, even when no autocorrelation is present in the original demand, such autocorrelation can be present as a consequence of the overlapping process used to compute the lead time forecasts and the uncertainties arising in the estimation of the parameters of the forecasting model. Improvements are shown in terms of cycle service level, inventory investment and backorder volume. Simulations and real demand data from a manufacturer are used to illustrate our methodology.
The safety stock calculation requires a measure of the forecast error uncertainty. Such errors are usually assumed to be Gaussian iid (independently and identically distributed). However, deviations from iid lead to a deterioration in the performance of the supply chain. Recent research has shown that, contrary to theoretical approaches, empirical techniques that do not rely on the aforementioned assumptions can enhance the calculation of safety stocks. In particular, GARCH models cope with time-varying heterocedastic forecast error, and kernel density estimation does not need to rely on a determined distribution. However, if the forecast errors are time-varying heterocedastic and do not follow a determined distribution, the previous approaches are inadequate. We overcome this by proposing an optimal combination of the empirical methods that minimizes the asymmetric piecewise linear loss function, also known as the tick loss. The results show that combining quantile forecasts yields safety stocks with a lower cost. The methodology is illustrated with simulations and real data experiments for different lead times.
The bullwhip effect (BE) consists of the demand variability amplification that exists in a supply chain when moving upwards. This undesirable effect produces excess inventory and poor customer service. Recently, several research papers from either a theoretical or empirical point of view have indicated the nature of the demand process as a key aspect to defining the BE. Nonetheless, they reached different conclusions. On the one hand, theoretical research quantified the BE depending on the lead time and ARIMA parameters, where ARIMA functions were employed to model the demand generator process. In turn, empirical research related nonlinearly the demand variability extent with the BE size. Although, it seems that both results are contradictory, this paper explores how those conclusions complement each other. Essentially, it is shown that the theoretical developments are precise to determine the presence of the BE based on its ARIMA parameter estimates. Nonetheless, to quantify the size of the BE, the demand coefficient of variation should be incorporated. The analysis explores a two-staged serially linked supply chain, where weekly data at SKU level from a manufacturer specialized in household products and a major UK grocery retailer have been collected.
Forecasts of demand are crucial to drive supply chains and enterprise resource planning systems. Usually, well-known univariate methods that work automatically such as exponential smoothing are employed to accomplish such forecasts. The traditional Supply Chain relies on a decentralized system where each member feeds its own Forecasting Support System (FSS) with incoming orders from direct customers. Nevertheless, other collaboration schemes are also possible, for instance, the Information Exchange framework allows demand information to be shared between the supplier and the retailer. Current theoretical models have shown the limited circumstances where retailer information is valuable to the supplier. However, there has been very little empirical work carried out. Considering a serially linked two-level supply chain, this work assesses the role of sharing market sales information obtained by the retailer on the supplier forecasting accuracy. Weekly data from a manufacturer and a major UK grocery retailer have been analyzed to show the circumstances where information sharing leads to improved forecasting accuracy. Without resorting to unrealistic assumptions, we find significant evidence of benefits through information sharing with substantial improvements in forecast accuracy.