The Combination of Forecasts

Two separate sets of forecasts of airline passenger data have been combined to form a composite set of forecasts. The main conclusion is that the composite set of forecasts can yield lower mean-square error than either of the original forecasts. Past errors of each of the original forecasts are used to determine the weights to attach to these two original forecasts in forming the combined forecasts, and different methods of deriving these weights are examined.     

预测组合权重更新的过程控制及分析

源于与决策分析的相关性,预测组合已经逐渐形成了一个重要的研究领域。为此,本文引进EWMA技术对预测组合权重更新的过程进行控制,从而提出一种能够应用于实际且简单有效的EWMA赋权方法。这种赋权方法能够确定预测组合权重应该何时更新,而不是机械地更新预测组合权重。本文额外针对各种赋权方法在旅游预测组合模型中的预测性能(全面预测性能和总均方根误差)和预测效率(权重更新频率)进行了经验评估。结果显示:EWMA赋权方法的预测性能优于传统的赋权方法,并与CUSUM赋权方法相似,同时该赋权方法获得了最小的权重更新频率。综合考虑预测性能和预测效率,EWMA赋权方法相比于其他赋权方法在旅游实际应用过程中更具优势。     

An application of vector GARCH model in semiconductor demand planning

Demand planning has been the key to supply chain management in semiconductor industry. With an appropriate weight assignment scheme, the planning accuracy resulting from combinational forecasts can be improved by merging several individual candidate methods. In this paper we discuss the applicability of vector generalized autoregressive conditional heteroskedasticity (GARCH) model to determine the optimal combinational weights of component forecasts, where the conditional variances and correlations of forecast errors from candidate methods are represented and estimated by a maximum-likelihood procedure. The asymptotical properties of parameter estimators for GARCH model are investigated by simulation experiments. An example of the proposed method to real time series of electronic products demonstrates that this weight-varying combinational method produces less prediction errors, compared to other commonly used forecasting approaches that are based on single model selection criteria or fixed weights.     

Combining ranked mean value forecasts

A methodology is developed for combining mean value forecasts using not only all the important statistics related to the past performance and the dependence of the individual forecasts, but also a rank ordering of the individual forecasts representing the belief of a decision maker about the future performance of the forecasts. The maximum likelihood combination of the forecasts turns out to be weighted linear combination of the individual forecasts, where the weights are a function of the rank order of the forecasts, correlation coefficients between the forecasts, and relative entropy information measures between the individual forecasts and the actual values. These weights are assessed once in the most general case and once in a special case where the forecasts are normally distributed. The sensitivity of the weights is also investigated. A sample application of this method for predicting U.S. hog prices is also presented.     

A Method for Pooling Forecasts

A group of experts is to produce a joint forecast of a set of unknowns. Each expert is asked to distribute subjectively a given sum of confidence weights over his own forecasts. A joint forecast is computed as the product sum of the individual forecasts and weights deduced from the individual's weights. A probabilistic interpretation of this procedure is provided and a measure of the reliability of the joint forecasts is suggested. A Bayesian variant can be constructed by introducing sample information.     

Using combined forecasts with changing weights for electricity demand profiling

Day-ahead half-hourly demand forecasts are required for scheduling and for calculating the daily electricity pool price. One approach predicts turning points on the demand curve and then produces half-hourly forecasts by a heuristic procedure, called profiling, which is based on a past demand curve. This paper investigates possible profiling improvements. Using a cubic smoothing spline in the heuristic leads to a slight improvement. Often, several past curves could reasonably be used in the profiling method. Consequently, there are often several demand curve forecasts available. Switching and smooth transition forecast combination models are considered. These models enable the combining weights to vary across the 48 half-hours, which is appealing as different forecasts may be more suitable for different periods. Several criteria are used to control the changing weights, including weather, and the methodology is extended to the case of more than two forecasts. Empirical analysis gives encouraging results.     

Predicting ambulance demand using singular spectrum analysis

This paper demonstrates techniques to generate accurate predictions of demand exerted upon the Emergency Medical Services (EMS) using data provided by the Welsh Ambulance Service Trust (WAST). The aim is to explore new methods to produce accurate forecasts that can be subsequently embedded into current OR methodologies to optimise resource allocation of vehicles and staff, and allow rapid response to potentially life-threatening emergencies. Our analysis explores a relatively new non-parametric technique for time series analysis known as Singular Spectrum Analysis (SSA). We explain the theory of SSA and evaluate the performance of this approach by comparing the results with those produced by conventional time series methods. We show that in addition to being more flexible in approach, SSA produces superior longer-term forecasts (which are especially helpful for EMS planning), and comparable shorter-term forecasts to well established methods.     

Forecasting corporate failure using ensemble of self-organizing neural networks

For more than a decade, the number of research works that deal with ensemble methods applied to bankruptcy prediction has been increasing. Ensemble techniques present some characteristics that, in most situations, allow them to achieve better forecasts than those estimated with single models. However, the difference between the performance of an ensemble and that of its base classifier but also between that of ensembles themselves, is often low. This is the reason why we studied a way to design an ensemble method that might achieve better forecasts than those calculated with traditional ensembles. It relies on a quantification process of data that characterize the financial situation of a sample of companies using a set of self-organizing neural networks, where each network has two main characteristics: its size is randomly chosen and the variables used to estimate its weights are selected based on a criterion that ensures the fit between the structure of the network and the data used over the learning process. The results of our study show that this technique makes it possible to significantly reduce both the type I and type II errors that can be obtained with conventional methods.     

Using investment portfolio return to combine forecasts: A multiobjective approach

This study investigates the usefulness and efficacy of a multiobjective decision method for financial trading guided by a set of seemingly diverse analysts' forecasts. The paper proposes a goal programming (GP) approach which combines various forecasts based on the performance of their previous investment returns. In our experiment, several series of financial analysts' forecasts are generated by different forecasting techniques. Investment returns on each series of forecasts are measured and then evaluated by three performance criteria, namely, mean, variance, and skewness. Subsequently, these distributional properties of the returns are used to construct a GP model. Results of the GP model provide a set of weights to compose an investment portfolio using various forecasts. To examine its practicality, the approach is tested on several major stock market indices. The performance of the proposed GP approach is compared with those of individual forecasting techniques and a number of forecast combination models suggested by previous studies. This comparison is conducted with respect to different levels of investor preference over return, variance, and skewness. Statistical significance of the results are accessed by bootstrap re-sampling. Empirical results indicate that, for all examined investor preference functions and market indices, the GP approach is significantly better than all other models tested in this study.     

A Simplification of the Matrix Beta Distribution for Combining Estimators

The problem of estimating a simplex of weights with a direct probabilistic interpretation for combining a set of estimators, either forecasts or expert opinions on an uncertain quantity, is approached via the outperformance construct. Thus, each weight represents the probability that its estimator will perform best. These outperformance probability weights are derived from a generalization of the beta distribution, i.e. the Matrix Beta. A simplified procedure for estimating the outperformance probability weights from this distribution is presented.     

Experts' adjustment to model-based SKU-level forecasts: does the forecast horizon matter?

Experts (managers) may have domain-specific knowledge that is not included in a statistical model and that can improve short-run and long-run forecasts of SKU-level sales data. While one-step-ahead forecasts address the conditional mean of the variable, model-based forecasts for longer horizons have a tendency to convert to the unconditional mean of a time series variable. Analysing a large database concerning pharmaceutical sales forecasts for various products and adjusted by a range of experts, we examine whether the forecast horizon has an impact on what experts do and on how good they are once they adjust model-based forecasts. For this, we use regression-based methods and we obtain five innovative results. First, all horizons experience managerial intervention of forecasts. Second, the horizon that is most relevant to the managers shows greater overweighting of the expert adjustment. Third, for all horizons the expert adjusted forecasts have less accuracy than pure model-based forecasts, with distant horizons having the least deterioration. Fourth, when expert-adjusted forecasts are significantly better, they are best at those distant horizons. Fifth, when expert adjustment is down-weighted, expert forecast accuracy increases.     

Electricity price forecasting accounting for renewable energies: optimal combined forecasts

Electricity price forecasting is an interesting problem for all the agents involved in electricity market operation. For instance, every profit maximisation strategy is based on the computation of accurate one-day-ahead forecasts, which is why electricity price forecasting has been a growing field of research in recent years. In addition, the increasing concern about environmental issues has led to a high penetration of renewable energies, particularly wind. In some European countries such as Spain, Germany and Denmark, renewable energy is having a deep impact on the local power markets. In this paper, we propose an optimal model from the perspective of forecasting accuracy, and it consists of a combination of several univariate and multivariate time series methods that account for the amount of energy produced with clean energies, particularly wind and hydro, which are the most relevant renewable energy sources in the Iberian Market. This market is used to illustrate the proposed methodology, as it is one of those markets in which wind power production is more relevant in terms of its percentage of the total demand, but of course our method can be applied to any other liberalised power market. As far as our contribution is concerned, first, the methodology proposed by García-Martos et al (2007 and 2012) is generalised twofold: we allow the incorporation of wind power production and hydro reservoirs, and we do not impose the restriction of using the same model for 24?h. A computational experiment and a Design of Experiments (DOE) are performed for this purpose. Then, for those hours in which there are two or more models without statistically significant differences in terms of their forecasting accuracy, a combination of forecasts is proposed by weighting the best models (according to the DOE) and minimising the Mean Absolute Percentage Error (MAPE). The MAPE is the most popular accuracy metric for comparing electricity price forecasting models. We construct the combination of forecasts by solving several nonlinear optimisation problems that allow computation of the optimal weights for building the combination of forecasts. The results are obtained by a large computational experiment that entails calculating out-of-sample forecasts for every hour in every day in the period from January 2007 to December 2009. In addition, to reinforce the value of our methodology, we compare our results with those that appear in recent published works in the field. This comparison shows the superiority of our methodology in terms of forecasting accuracy.     

The value of combining forecasts in inventory management – a case study in banking

Managing inventories in the face of uncertain stochastic demand requires an investment in safety stocks. These are related to the accuracy in forecasting future demands and the noise in the demand generation process. Reducing the demand forecasting error can free up capital and space, and reduce the operating costs of managing the inventories. A leading bank in Hong Kong consumes more than three hundred kinds of printed forms for its daily operations. A major problem of its inventory control system for the forms management is to forecast the monthly demand of these forms. In this study the idea of combining forecasts is introduced and its practical application is addressed. The individual forecasts come from well established time series models and the weights for combination are estimated with Quadratic Programming. The combined forecast is found to perform better than any of the individual forecasts.     

Time series forecasting with neural network ensembles: an application for exchange rate prediction

This paper investigates the use of neural network combining methods to improve time series forecasting performance of the traditional single keep-the-best (KTB) model. The ensemble methods are applied to the difficult problem of exchange rate forecasting. Two general approaches to combining neural networks are proposed and examined in predicting the exchange rate between the British pound and US dollar. Specifically, we propose to use systematic and serial partitioning methods to build neural network ensembles for time series forecasting. It is found that the basic ensemble approach created with non-varying network architectures trained using different initial random weights is not effective in improving the accuracy of prediction while ensemble models consisting of different neural network structures can consistently outperform predictions of the single ‘best’ network. Results also show that neural ensembles based on different partitions of the data are more effective than those developed with the full training data in out-of-sample forecasting. Moreover, reducing correlation among forecasts made by the ensemble members by utilizing data partitioning techniques is the key to success for the neural ensemble models. Although our ensemble methods show considerable advantages over the traditional KTB approach, they do not have significant improvement compared to the widely used random walk model in exchange rate forecasting.     

Coherent mortality forecasting with generalized linear models: A modified time-transformation approach

In this paper, we propose an alternative approach for forecasting mortality for multiple populations jointly. Our contribution is developed upon the generalized linear models introduced by Renshaw et al., (1996) and Sithole et al., (2000), in which mortality forecasts are generated within the model structure, without the need of additional stochastic processes. To ensure that the resulting forecasts are coherent, a modified time-transformation is developed to stipulate the expected mortality differential between two populations to remain constant when the long-run equilibrium is attained. The model is then further extended to incorporate a structural change, an important property that is observed in the historical mortality data of many national populations. The proposed modeling methods are illustrated with data from two different pairs of populations: (1) Swedish and Danish males; (2) English and Welsh males and U.K. male insured lives.     

On the convergence of multiattribute weighting methods

The convergent validity of five multiattribute weighting methods is studied in an Internet experiment. This is the first experiment where the subjects created the alternatives and attributes themselves. Each subject used five methods to assess attribute weights – one version of the analytic hierarchy process (AHP), direct point allocation, simple multiattribute rating technique (SMART), swing weighting, and tradeoff weighting. They can all be used following the principles of multiattribute value theory. Furthermore, SMART, swing, and AHP ask the decision makers to give directly the numerical estimates of weight ratios although the elicitation questions are different. In earlier studies these methods have yielded different weights. Our results suggest that the resulting weights are different because the methods explicitly or implicitly lead the decision makers to choose their responses from a limited set of numbers. The other consequences from this are that the spread of weights and the inconsistency between the preference statements depend on the number of attributes that a decision maker considers simultaneously.     

Temporal hierarchies with autocorrelation for load forecasting

We propose four different estimators that take into account the autocorrelation structure when reconciling forecasts in a temporal hierarchy. Combining forecasts from multiple temporal aggregation levels exploits information differences and mitigates model uncertainty, while reconciliation ensures a unified prediction that supports aligned decisions at different horizons. In previous studies, weights assigned to the forecasts were given by the structure of the hierarchy or the forecast error variances without considering potential autocorrelation in the forecast errors. Our first estimator considers the autocovariance matrix within each aggregation level. Since this can be difficult to estimate, we propose a second estimator that blends autocorrelation and variance information, but only requires estimation of the first-order autocorrelation coefficient at each aggregation level. Our third and fourth estimators facilitate information sharing between aggregation levels using robust estimates of the cross-correlation matrix and its inverse. We compare the proposed estimators in a simulation study and demonstrate their usefulness through an application to short-term electricity load forecasting in four price areas in Sweden. We find that by taking account of auto- and cross-covariances when reconciling forecasts, accuracy can be significantly improved uniformly across all frequencies and areas.     

Determination of an Optimal Forecast Model for Ambulance Demand Using Goal Programming

A multistep approach to determining the optimal parameters of an exponential smoothing model was used to forecast emergency medical service (E.M.S.) demand for four counties of South Carolina. Daily emergency and routine (non-emergency) demand data were obtained and forecast statistics generated for each county sampled, using Winters' exponential smoothing model. A goal programme was formulated to combine forecast results for emergency calls with routine call forecasts. The goal programme gave a higher priority to accurate forecasting of emergency demand. The forecast model generated implicitly weights demand by severity and provides a reliable estimate of demand overall. The optimal parameter values for the smoothing model were obtained by minimizing the objective function value of the goal programming problem. The parameter values obtained were used to forecast demand for E.M.S. in the selected counties. The results of the model were compared to those using a multiple linear regression model and a single-objective-based exponential smoothing model for 2 months of data. When compared with two single-objective forecast models, the multiple-objective approach yielded more accurate forecasts and, therefore, was more cost-effective for the planner. The model presents and demonstrates a theoretical approach to improving the accuracy of ambulance demand forecasts. The possible impact of this approach on planning efficiency is discussed.     

The admissible parameter space for exponential smoothing models

We discuss the admissible parameter space for some state space models, including the models that underly exponential smoothing methods. We find that the usual parameter restrictions (requiring all smoothing parameters to lie between 0 and 1) do not always lead to stable models. We also find that all seasonal exponential smoothing methods are unstable as the underlying state space models are neither reachable nor observable. This instability does not affect the forecasts, but does corrupt the state estimates. The problem can be overcome with a simple normalizing procedure. Finally we show that the admissible parameter space of a seasonal exponential smoothing model is much larger than that for a basic structural model, leading to better forecasts from the exponential smoothing model when there is a rapidly changing seasonal pattern.     

Forecasting daily supermarket sales using exponentially weighted quantile regression

Inventory control systems typically require the frequent updating of forecasts for many different products. In addition to point predictions, interval forecasts are needed to set appropriate levels of safety stock. The series considered in this paper are characterised by high volatility and skewness, which are both time-varying. These features motivate the consideration of forecasting methods that are robust with regard to distributional assumptions. The widespread use of exponential smoothing for point forecasting in inventory control motivates the development of the approach for interval forecasting. In this paper, we construct interval forecasts from quantile predictions generated using exponentially weighted quantile regression. The approach amounts to exponential smoothing of the cumulative distribution function, and can be viewed as an extension of generalised exponential smoothing to quantile forecasting. Empirical results are encouraging, with improvements over traditional methods being particularly apparent when the approach is used as the basis for robust point forecasting.