Monitoring an Exponential Smoothing Forecasting System

The first part of this paper is concerned with the variance of the smoothed error when the forecasting system being used is exponential smoothing. The expression derived for this variance involves the variance of the noise, the smoothing constant and a sum of squared binomial coefficients. It is also shown that the variance of the sum of errors equals the variance of the smoothed error for one less degree of freedom divided by the square of the smoothing constant.The second part of the paper considers the practical application of the above result and also gives values for the tracking signal limits, obtained by simulation, which could be used in the automatic monitoring of a forecasting system.     

Exponential Smoothing with an Adaptive Response Rate

A modification is proposed to forecasting systems employing exponential smoothing whereby the response rate is varied and made to depend on the value of a tracking signal. In a simple system, this is equivalent to varying α the smoothing constant according to the extent to which biased forecasts are being obtained. Such a system is shown to react much faster to, for example, step changes whilst still retaining the facility to filter out random noise.     

CUSUM vs Smoothed-Error Forecast Monitoring Schemes: Some Simulation Results

This paper compares the performance of CUSUM and smoothed-error tracking signals for monitoring the adequacy of exponential smoothing forecasts. Previous research has favoured the CUSUM. However, there is some evidence that the performance of the smoothed-error signal can be improved by a simple modification in its application: the use of different smoothing parameters in the tracking signal and the forecasting model. The effects of this modification are tested using simulated time series. We conclude that the CUSUM is robust to the choice of forecasting parameter, while the smoothed-error signal is not. The CUSUM is also more responsive to small changes in the time series, regardless of the parameters used.     

Differential geometrical structures related to forecasting error variance ratios

Differential geometrical structures (Riemannian metrics, pairs of dual affine connections, divergences and yokes) related to multi-step forecasting error variance ratios are introduced to a manifold of stochastic linear systems. They are generalized to nonstationary cases. The problem of approximating a given time series by a specific model is discussed. As examples, we use the established scheme to discuss the AR (1) approximations and the exponential smoothing of ARMA series for multi-step forecasting purpose. In the process, some interesting results about spectral density functions are derived and applied.     

The Dynamic Behaviour of Linear Forecasting and Scheduling Rules

In the first section of this paper, some important results of Ward⁴ concerning trendcorrected exponential smoothing models are developed and extended. In the second section a linear production and stock control scheme, in which exponential smoothing is used for forecasting, is examined. Some results analogous to Ward's are obtained for the combined system which throws some interesting light on the interaction between the forecasting and decision-making aspects of such a system.     

An Examination of the Use of Adaptive Filtering in Forecasting

Adaptive filtering is a technique for preparing short- to medium-term forecasts based on the weighting of historical observations, in a similar way to moving average and exponential smoothing. However, adaptive filtering, as it has been developed in electrical engineering, attempts to distinguish a signal pattern from random noise, rather than simply smoothing the noise of past data. This paper reviews the technique of adaptive filtering and investigates its applications and limitations for the forecasting practitioner. This is done by looking at the performance of adaptive filtering in forecasting a number of time series and by comparing it with other forecasting techniques.     

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.     

SIOPRED: a prediction and optimisation integrated system for demand

This paper presents a forecasting support system based on the generalised Holt-Winters exponential smoothing scheme to forecast time series of levels of demand. It is conceived as an integrated tool which has been implemented in Visual Basic. For improving the accuracy of automatic forecasting it uses an optimisation-based scheme which unifies the stages of estimation of the parameters and model selection. Based on this scheme, suitable forecasts and prediction intervals are obtained. The performance of the proposed system is compared with a number of well-established automatic forecasting procedures with respect to the 3003 time series included in the M3-competition.      

Forecasting sales of slow and fast moving inventories

Traditional computerised inventory control systems usually rely on exponential smoothing to forecast the demand for fast moving inventories. Practices in relation to slow moving inventories are more varied, but the Croston method is often used. It is an adaptation of exponential smoothing that (1) incorporates a Bernoulli process to capture the sporadic nature of demand and (2) allows the average variability to change over time. The Croston approach is critically appraised in this paper. Corrections are made to underlying theory and modifications are proposed to overcome certain implementation difficulties. A parametric bootstrap approach is outlined that integrates demand forecasting with inventory control. The approach is illustrated on real demand data for car parts.     

Multi-item sales forecasting with total and split exponential smoothing

Efficient supply chain management relies on accurate demand forecasting. Typically, forecasts are required at frequent intervals for many items. Forecasting methods suitable for this application are those that can be relied upon to produce robust and accurate predictions when implemented within an automated procedure. Exponential smoothing methods are a common choice. In this empirical case study paper, we evaluate a recently proposed seasonal exponential smoothing method that has previously been considered only for forecasting daily supermarket sales. We term this method ‘total and split’ exponential smoothing, and apply it to monthly sales data from a publishing company. The resulting forecasts are compared against a variety of methods, including several available in the software currently used by the company. Our results show total and split exponential smoothing outperforming the other methods considered. The results were also impressive for a method that trims outliers and then applies simple exponential smoothing.     

Adaptive Exponential Smoothing Revisited

Adaptive exponential smoothing models are designed to improve performance by letting the smoothing parameter vary according to the most recent forecasting accuracy. This paper argues that the constant exponential smoothing results used in two comparative studies are inadequate as benchmarks. A reexamination does not indicate that adaptive exponential smoothing methods provide superior forecasts compared to those obtainable from constant exponential smoothing with a considerate choice of the smoothing constant. No support was found for the alleged advantages of the Dennis run based adaptive procedure.     

Adaptive robust synchronization of Rossler systems in the presence of unknown matched time-varying parameters

This paper deals with the problem of adaptive robust synchronization of chaotic systems based on the Lyapunov theory. A controller is designed for a feedback linearizable error system with matched uncertainties. The proposed method shows that the drive and response systems are synchronized and states of the response system track the states of the drive system as time tends to infinity. Since this approach does not require any information about the bound of uncertainties, this information is not needed in advance. In order to prevent the frequent switching phenomenon in the control signal, the method is modified such that the norm of tracking error is bounded. Numerical simulations on two uncertain Rossler systems with matched uncertainties show fast responses of tracking error, while the errors are Uniformly Ultimately Bounded, and the control signal is reasonably smooth.     

Adaptive Filtering: Once More with Gusto

Adaptive filtering, when used as a forecasting method, proposes to be able to distinguish a "signal pattern" of a time series instead of just smoothing out the random noise introduced by the data. Adaptive filtering is claimed by its creators to "...always do as well if not better than either moving averages, exponential smoothing,...". In order to see whether this claim could be substantiated, the author has taken the approach of a casual user of forecasting methods and has sought to determine whether adaptive filtering is useful, or not, as a forecasting method. The method was used to compute forecasts for ten sets of data on monthly insurance payments in a Finnish insurance company, and the experience gained from this work is compared with criticisms of the method expressed by a number of writers. It is shown that the method performs quite well for practical purposes, despite the fact that it has some major theoretical shortcomings.     

一种改进的基于指数平滑神经网络模型的时间序列预测方法

基于指数平滑模型与误差反传神经网络法提出了一个改进的时间序列预测方法.将神经网络模型移植入指数加权滑动平均模型中,充分考虑了时间序列的部分线性性和非线性性对预测结果的影响,是传统的混合模型的一个更合理的改进.最后通过对上证指数时间序列的实证分析,以预测均方误差为检验标准,对五种常用的时间序列预测模型进行了预测精度的比较,而且经验证所提出的改进的时间序列预测模型相对来说具有更小的预测均方误差.     

Lead time demand for simple exponential smoothing: an adjustment factor for the standard deviation

A new simple formula is found to correct the underestimation of the standard deviation for total lead time demand when using simple exponential smoothing. The traditional formula for the standard deviation of lead time demand is to multiply the standard deviation for the one-period-ahead forecast error (estimated by using the residuals) by the square root of the number of periods in the lead time. It has been shown by others that the traditional formula significantly underestimates variation in the lead time demand when the mean of the process is somewhat changing and simple exponential smoothing is appropriate. This new formula allows one to see readily the significant size of the underestimation of the traditional formula and can easily be implemented in practice. The formula is derived by using a state-space model for simple exponential smoothing.     

Exponential smoothing — I

Some new insights and results for exponential smoothing with application to forecasting and parameter estimation. Multidimensional exponential smoothing is described by a Kalman filter model with a limiting stationary gain matrix.     

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.     

基于GM-ES-GASVR组合模型的丽江国内旅游需求预测

构建适合于预测丽江国内旅游需求的预测模型,对推动丽江旅游业的发展具有重要意义.研究发现灰色GM(1,1)模型、三次指数平滑模型与GA-SVR模型都适用于预测丽江国内旅游需求,且GA-SVR模型为这三个单项模型中的最优模型.在此基础上,利用变权方法建立GM-ES-GASVR组合预测模型.通过对拟合与测试结果的对比分析,表明GM-ES-GASVR变权组合预测模型比单一模型的拟合与测试效果都有较大改善.     

Progressive Tuning of Simple Exponential Smoothing Forecasts

The paper outlines a finite sample version of exponential smoothing, and proposes a formula for estimating the smoothing parameter. The resulting method, which can be implemented on a recursive basis over time, is compared with alternative approaches, such as progressive numerical optimization of the smoothing parameter and adaptive forecasting on both synthetic and real data.     

组合预测模型在能源消费预测中的应用

能源的需求预测是一个复杂的非线形系统,其发展变化具有增长性和波动性,组合预测对于信息不完备的复杂经济系统具有一定的实用性.本文利用我国能源消费的历史数据,采用灰色预测的GM(1,1)模型、BP神经网络模型和三次指数平滑模型进行优化组合,建立了能源消费组合预测模型,实证分析结果表明预测值和实际结果有很好的一致性,可以作为能源消费预测的有效工具.