Exponentially smoothed regression analysis for demand forecasting

This article proposes a new technique for estimating trend and multiplicative seasonality in time series data. The technique is computationally quite straightforward and gives better forecasts (in a sense described below) than other commonly used methods. Like many other methods, the one presented here is basically a decomposition technique, that is, it attempts to isolate and estimate the several subcomponents in the time series. It draws primarily on regression analysis for its power and has some of the computational advantages of exponential smoothing. In particular, old estimates of base, trend, and seasonality may be smoothed with new data as they occur. The basic technique was developed originally as a way to generate initial parameter values for a Winters exponential smoothing model [4], but it proved to be a useful forecasting method in itself.The objective in all decomposition methods is to separate somehow the effects of trend and seasonality in the data, so that the two may be estimated independently. When seasonality is modeled with an additive form (Datum = Base + Trend + Seasonal Factor), techniques such as regression analysis with dummy variables or ratio-to-moving-average techniques accomplish this task well. It is more common, however, to model seasonality as a multiplicative form (as in the Winters model, for example, where Datum = [Base + Trend] * Seasonal Factor). In this case, it can be shown that neither of the techniques above achieves a proper separation of the trend and seasonal effects, and in some instances may give highly misleading results. The technique described in this article attempts to deal properly with multiplicative seasonality, while remaining computationally tractable.The technique is built on a set of simple regression models, one for each period in the seasonal cycle. These models are used to estimate individual seasonal effects and then pooled to estimate the base and trend. As new data occur, they are smoothed into the least-squares formulas with computations that are quite similar to those used in ordinary exponential smoothing. Thus, the full least-squares computations are done only once, when the forecasting process is first initiated. Although the technique is demonstrated here under the assumption that trend is linear, the trend may, in fact, assume any form for which the curve-fitting tools are available (exponential, polynomial, etc.).The method has proved to be easy to program and execute, and computational experience has been quite favorable. It is faster than the RTMA method or regression with dummy variables (which requires a multiple regression routine), and it is competitive with, although a bit slower than, ordinary triple exponential smoothing.     

Normalization of seasonal factors in Winters’ methods

In Winters’ seasonal exponential smoothing methods, a time series is decomposed into: level, trend and seasonal components, that change over time. The seasonal factors are initialized so that their average is 0 in the additive version or 1 in the multiplicative version. Usually, only one seasonal factor is updated each period, and the average of the seasonal factors is no longer 0 or 1; the ‘seasonal factors’ no longer meet the usual meaning of seasonal factors. We provide an equivalent reformulation of previous equations for renormalizing the components in the additive version. This form of the renormalization equations is then adapted to new renormalization formulas for the multiplicative Winters’ method. For both the standard and renormalized equations we make a minor change to the seasonal equation. Predictions from our renormalized smoothing values are the same as for the original smoothed values. The formulas can be applied every period, or when required. However, we recommend renormalization every time period. We show in the multiplicative version that the level and trend should be adjusted along with the seasonal component.     

Short-term forecasting of crime

The major question investigated is whether it is possible to accurately forecast selected crimes 1 month ahead in small areas, such as police precincts. In a case study of Pittsburgh, PA, we contrast the forecast accuracy of univariate time series models with naïve methods commonly used by police. A major result, expected for the small-scale data of this problem, is that average crime count by precinct is the major determinant of forecast accuracy. A fixed-effects regression model of absolute percent forecast error shows that such counts need to be on the order of 30 or more to achieve accuracy of 20% absolute forecast error or less. A second major result is that practically any model-based forecasting approach is vastly more accurate than current police practices. Holt exponential smoothing with monthly seasonality estimated using city-wide data is the most accurate forecast model for precinct-level crime series.     

Incorporating a tracking signal into a state space model

It is a common practice to complement a forecasting method such as simple exponential smoothing with a monitoring scheme to detect those situations where forecasts have failed to adapt to structural change. It will be suggested in this paper that the equations for simple exponential smoothing can be augmented by a common monitoring statistic to provide a method that automatically adapts to structural change without human intervention. The resulting method, which turns out to be a restricted form of damped trend corrected exponential smoothing, is compared with related methods on the annual data from the M3 competition. It is shown to be better than simple exponential smoothing and more consistent than traditional damped trend exponential smoothing.     

The inventory performance of forecasting methods: Evidence from the M3 competition data

Forecasting competitions have been a major driver not only of improvements in forecasting methods’ performances, but also of the development of new forecasting approaches. However, despite the tremendous value and impact of these competitions, they do suffer from the limitation that performances are measured only in terms of the forecast accuracy and bias, ignoring utility metrics. Using the monthly industry series of the M3 competition, we empirically explore the inventory performances of various widely used forecasting techniques, including exponential smoothing, ARIMA models, the Theta method, and approaches based on multiple temporal aggregation. We employ a rolling simulation approach and analyse the results for the order-up-to policy under various lead times. We find that the methods that are based on combinations result in superior inventory performances, while the Naïve, Holt, and Holt-Winters methods perform poorly.     

Holt’s exponential smoothing and neural network models for forecasting interval-valued time series

Interval-valued time series are interval-valued data that are collected in a chronological sequence over time. This paper introduces three approaches to forecasting interval-valued time series. The first two approaches are based on multilayer perceptron (MLP) neural networks and Holt’s exponential smoothing methods, respectively. In Holt’s method for interval-valued time series, the smoothing parameters are estimated by using techniques for non-linear optimization problems with bound constraints. The third approach is based on a hybrid methodology that combines the MLP and Holt models. The practicality of the methods is demonstrated through simulation studies and applications using real interval-valued stock market time series.     

Seasonal exponential smoothing with damped trends : An application for production planning

This paper reviews a spreadsheet-based forecasting approach which a process industry manufacturer developed and implemented to link annual corporate forecasts with its manufacturing/distribution operations. First, we consider how this forecasting system supports overall production planning and why it must be compatible with corporate forecasts. We then review the results of substantial testing of variations on the Winters three-parameter exponential smoothing model on 28 actual product family time series. In particular, we evaluate whether the use of damping parameters improves forecast accuracy. The paper concludes that a Winters four-parameter model (i.e. the standard Winters three-parameter model augmented by a fourth parameter to damp the trend) provides the most accurate forecasts of the models evaluated. Our application confirms the fact that there are situations where the use of damped trend parameters in short-run exponential smoothing based forecasting models is beneficial.     

Improving time series forecasting: An approach combining bootstrap aggregation,clusters and exponential smoothing

Some recent papers have demonstrated that combining bagging (bootstrap aggregating) with exponential smoothing methods can produce highly accurate forecasts and improve the forecast accuracy relative to traditional methods. We therefore propose a new approach that combines the bagging, exponential smoothing and clustering methods. The existing methods use bagging to generate and aggregate groups of forecasts in order to reduce the variance. However, none of them consider the effect of covariance among the group of forecasts, even though it could have a dramatic impact on the variance of the group, and therefore on the forecast accuracy. The proposed approach, referred to here as Bagged.Cluster.ETS, aims to reduce the covariance effect by using partitioning around medoids (PAM) to produce clusters of similar forecasts, then selecting several forecasts from each cluster to create a group with a reduced variance. This approach was tested on various different time series sets from the M3 and CIF 2016 competitions. The empirical results have shown a substantial reduction in the forecast error, considering sMAPE and MASE.     

Improving forecasting by estimating time series structural components across multiple frequencies

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.     

Focus forecasting reconsidered

Focus Forecasting is a popular heuristic methodology for production and inventory control although there has never been a rigorous test of accuracy using real time series. We compare Focus Forecasting to damped-trend, seasonal exponential smoothing using five time series of cookware demand in a production planning application. We also make comparisons using 91 time series from the M-Competition study of forecast accuracy. Exponential smoothing was more accurate in both cases.     

Enhancing capacity planning through forecasting: An integrated tool for maintenance of complex product systems

Despite the extensive amount of data generated and stored during the maintenance capacity planning process, Maintenance, Repair, and Overhaul (MRO) organizations have yet to explore their full potential in forecasting the required capacity to face future and unprecedented maintenance interventions. This paper explores the integration of time series forecasting capabilities in a tool for maintenance capacity planning of complex product systems (CoPS), intended to value data that is routinely generated and stored, but often disregarded by MROs. State space formulations with multiplicative errors for the simple exponential smoothing (SES), Holt’s linear method (HLM), additive Holt-Winters (AHW), and multiplicative Holt-Winters (MHW) are assessed using real data, comprised of 171 maintenance projects collected from a major Portuguese aircraft MRO. A state space formulation of the MHW, selected using the bias-corrected Akaike information criterion (AICc), is integrated in a Decision Support System (DSS) for capacity planning with probabilistic inference capabilities and used to forecast the workload probability distribution of a future and unprecedent maintenance intervention. The developed tool is validated by comparing forecasted values with workloads of a particular maintenance intervention and with a model simulating current forecasting practices employed by MROs.     

Restarting a forecasting system when demand suddenly changes

Exponential smoothing is commonly used in automatic forecasting systems. However, when only a small amount of historical data is relevant to future demands, the ad hoc startup methods used in exponential smoothing produce unexpected results. With large data sets, an exponentially smoothed average implicitly weights the data in a declining manner, similar to discounting. This pattern is important in that it minimizes a measure of forecast error. However, restarting with limited data distorts the weighting pattern. A new technique, termed the declining alpha method, is presented and shown to preserve the exponential weight pattern. The key is a formula that changes the smoothing constant each period. Examples are given to illustrate the method and contrast it to other startup techniques.     

GROEC: Combination method via Generalized Rolling Origin Evaluation

Combination methods have performed well in time series forecast competitions. This study proposes a simple but general methodology for combining time series forecast methods. Weights are calculated using a cross-validation scheme that assigns greater weights to methods with more accurate in-sample predictions. The methodology was used to combine forecasts from the Theta, exponential smoothing, and ARIMA models, and placed fifth in the M4 Competition for both point and interval forecasting.     

指数趋势模型在工程成本预测中的应用

在实施"低成本战略"市场环境下,各种成本预测方法层出不穷。针对工程项目的实际情况,把工程项目进行从粗到细的层次划分;然后提出了一种新的预测算法——指数趋势模型算法,对施工企业的工程成本进行预测;最后,以一个算例对该模型的运用进行分析。     

Forecasting correlated time series with exponential smoothing models

This paper presents the Bayesian analysis of a general multivariate exponential smoothing model that allows us to forecast time series jointly, subject to correlated random disturbances. The general multivariate model, which can be formulated as a seemingly unrelated regression model, includes the previously studied homogeneous multivariate Holt-Winters’ model as a special case when all of the univariate series share a common structure. MCMC simulation techniques are required in order to approach the non-analytically tractable posterior distribution of the model parameters. The predictive distribution is then estimated using Monte Carlo integration. A Bayesian model selection criterion is introduced into the forecasting scheme for selecting the most adequate multivariate model for describing the behaviour of the time series under study. The forecasting performance of this procedure is tested using some real examples.     

Treating and Pruning: New approaches to forecasting model selection and combination using prediction intervals

We propose a new way of selecting among model forms in automated exponential smoothing routines, consequently enhancing their predictive power. The procedure, here addressed as treating, operates by selectively subsetting the ensemble of competing models based on information from their prediction intervals. By the same token, we set forth a pruning strategy to improve the accuracy of both point forecasts and prediction intervals in forecast combination methods. The proposed approaches are respectively applied to automated exponential smoothing routines and Bagging algorithms, to demonstrate their potential. An empirical experiment is conducted on a wide range of series from the M-Competitions. The results attest that the proposed approaches are simple, without requiring much additional computational cost, but capable of substantially improving forecasting accuracy for both point forecasts and prediction intervals, outperforming important benchmarks and recently developed forecast combination methods.     

动态数列直线趋势预测方法和参数研究

本文通过采取步长和初始时间序列不同的两种情况,根据运用公式1计算的结果初步推断出动态数列直线趋势预测方法和参数的取值的规律,并对此规律进行数学证明和实例验证,并由此提出公式2和3。通过本文的论述可以得出,按照直线趋势预测法进行预测,预测值与取时间序列的第一个取值无关,也与时间序列间的步长大小无关,只要时间序列间的步长相等即可,预测值都是一样的,且预测值呈等差数列。     

A two–stage approach to additive time series models

For nonlinear additive time series models, an appealing approach used in the literature to estimate the nonparametric additive components is the projection method. In this paper, it is demonstrated that the projection method might not be efficient in an asymptotic sense. To estimate additive components efficiently, a two–stage approach is proposed together with a local linear fitting and a new bandwidth selector based on the nonparametric version of the Akaike information criterion. It is shown that the two–stage method not only achieves efficiency but also makes bandwidth selection relatively easier. Also, the asymptotic normality of the resulting estimator is established. A small simulation study is carried out to illustrate the proposed methodology and the two–stage approach is applied to a real example from econometrics.     

河南农村物流需求的组合预测及策略研究

河南作为农业大省大力发展农村物流对其率先实现中部崛起有着很重要的现实意义,预测河南农村物流需求对于制定发展战略显得尤为重要。文中以河南农村消费品零售总额为河南农村物流需求预测指标,综合一元线性回归、时间序列双指数平滑法、移动平均法,建立组合预测模型,追求预测误差平方和最小,预测出河南农村物流需求呈良性发展趋势,并就进一步发展河南农村物流提出建议。     

基于农村物流需求量的组合预测分析

农村物流需求量的预测对于农村物流的发展有重要意义。文章把农村消费品零售总额作为农村物流需求量预测指标．通过分析各影响因素,建立了多元回归、双指数平滑及移动平均单预测模型。根据得出的单项预测误差数据,采用折扣系数法建立组合预测模型,使得组合预测模型预测误差平方和最小。预测能力明显优于单项预测模型。