I. Introduction: Basic concepts.- Getting started. II. Essentials: Linear innovations state space models.- Non-linear and heteroscedastic innovations state space models.- Estimation of innovations state space models.- Prediction distributions and intervals.- Selection of models. III. Further topics: Normalizing seasonal components.- Models with regressor variables.- Some properties of linear models.- Reduced forms and relationships with ARIMA models.- Linear innovations state space models with random seed states.- Conventional state space models.- Time series with multiple seasonal patterns.- Non-linear models for positive data.- Models for count data.- Vector exponential smoothing. IV. Applications: Inventory control application.- Conditional heteroscedasticity and finance applications.- Economic applications: the Beveridge-Nelson decomposition.

Applications of exponential smoothing to forecasting time series usually rely on three basic methods: simple exponential smoothing, trend corrected exponential smoothing and a seasonal variation thereof. A common approach to selecting the method appropriate to a particular time series is based on prediction validation on a withheld part of the sample using criteria such as the mean absolute percentage error. A second approach is to rely on the most appropriate general case of the three methods. For annual series this is trend corrected exponential smoothing: for sub-annual series it is the seasonal adaptation of trend corrected exponential smoothing. The rationale for this approach is that a general method automatically collapses to its nested counterparts when the pertinent conditions pertain in the data. A third approach may be based on an information criterion when maximum likelihood methods are used in conjunction with exponential smoothing to estimate the smoothing parameters. In this paper, such approaches for selecting the appropriate forecasting method are compared in a simulation study. They are also compared on real time series from the M3 forecasting competition. The results indicate that the information criterion approaches provide the best basis for automated method selection, the Akaike information criteria having a slight edge over its information criteria counterparts.

In this article, we propose a new empirical information criterion (EIC) for model selection which penalizes the likelihood of the data by a non-linear function of the number of parameters in the model. It is designed to be used where there are a large number of time series to be forecast. However, a bootstrap version of the EIC can be used where there is a single time series to be forecast. The EIC provides a data-driven model selection tool that can be tuned to the particular forecasting task. We compare the EIC with other model selection criteria including Akaike’s information criterion (AIC) and Schwarz’s Bayesian information criterion (BIC). The comparisons show that for the M3 forecasting competition data, the EIC outperforms both the AIC and BIC, particularly for longer forecast horizons. We also compare the criteria on simulated data and find that the EIC does better than existing criteria in that case also.

Three general classes of state space models are presented, using the single source of error formulation. The first class is the standard linear model with homoscedastic errors, the second retains the linear structure but incorporates a dynamic form of heteroscedasticity, and the third allows for non-linear structure in the observation equation as well as heteroscedasticity. These three classes provide stochastic models for a wide variety of exponential smoothing methods. We use these classes to provide exact analytic (matrix) expressions for forecast error variances that can be used to construct prediction intervals one or multiple steps ahead. These formulas are reduced to non-matrix expressions for 15 state space models that underlie the most common exponential smoothing methods. We discuss relationships between our expressions and previous suggestions for finding forecast error variances and prediction intervals for exponential smoothing methods. Simpler approximations are developed for the more complex schemes and their validity examined. The paper concludes with a numerical example using a non-linear model. Copyright © 2005 John Wiley & Sons, Ltd.

The state space approach to modelling univariate time series is now widely used both in theory and in applications. However, the very richness of the framework means that quite different model formulations are possible, even when they purport to describe the same phenomena. In this paper, we examine the single source of error [SSOE] scheme, which has perfectly correlated error components. We then proceed to compare SSOE to the more common version of the state space models, for which all the error terms are independent; we refer to this as the multiple source of error [MSOE] scheme. As expected, there are many similarities between the MSOE and SSOE schemes, but also some important differences. Both have ARIMA models as their reduced forms, although the mapping is more transparent for SSOE. Further, SSOE does not require a canonical form to complete its specification. An appealing feature of SSOE is that the estimates of the state variables converge in probability to their true values, thereby leading to a formal inferential structure for the ad-hoc exponential smoothing methods for forecasting. The parameter space for SSOE models may be specified to match that of the corresponding ARIMA scheme, or it may be restricted to meaningful sub-spaces, as for MSOE but with somewhat different outcomes. The SSOE formulation enables straightforward extensions to certain classes of non-linear models, including a linear trend with multiplicative seasonals version that underlies the Holt-Winters forecasting method. Conditionally heteroscedastic models may be developed in a similar manner. Finally we note that smoothing and decomposition, two crucial practical issues, may be performed within the SSOE framework.

Irregular demand in retail is characterized by stock keeping units (SKUs) that present high intermittency, and simultaneously, high erraticness (classified as lumpy demand) or low erraticness (classified as intermittent demand), following the classification of Syntetos et al. (2005). These SKUs are basically defined by periods of zero sales interleaved with positive sales producing series with some degree of variability. Many SKUs at the store/daily level can be characterized as presenting such a type of demand. Therefore methods for adequately forecasting irregular time series are necessary for proper inventory management. This article derives models for intermittent and lumpy time series using the framework of score-driven models as developed by Creal et al. (2013) and Harvey (2013). More precisely we derive Poisson, negative binomial, hurdle Poisson, and hurdle negative binomial models and apply them to real data obtained from a large Brazilian retail chain, comparing the performance of the proposed models to adequate competing methods from the 'slow'/intermittent demand forecasting literature. Forecasting accuracy is evaluated based on point forecasts and the entire predictive distribution. Our results show that the score-driven models perform well compared to intermittent traditional forecasting methods, providing competitive forecasting models for irregular demand in retailing.

Historically, seasonal factors have been concerned with estimates of historical data, not with forecasting. The Miller-Williams procedures enable organizations to adjust seasonal factors so as to make more accurate forecasts. This has an immediate payoff to forecasters - they can reduce forecast errors by about four percent. In addition, their papers have paved the way for improvements in the estimation of seasonal factors for forecasting, such as through the use of domain knowledge and analogous series.

Exponential smoothing is often used to forecast lead-time demand (LTD) for inventory control. In this paper, formulae are provided for calculating means and variances of LTD for a wide variety of exponential smoothing methods. A feature of many of the formulae is that variances, as well as the means, depend on trends and seasonal effects. Thus, these formulae provide the opportunity to implement methods that ensure that safety stocks adjust to changes in trend or changes in season. An example using weekly sales shows how safety stocks can be seriously underestimated during peak sales periods.

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.

We provide a new approach to automatic forecasting based on an extended range of exponential smoothing methods. Each method in our taxonomy of exponential smoothing methods provides forecasts that are equivalent to forecasts from a state space model. This equivalence allows: (1) easy calculation of the likelihood, the AIC and other model selection criteria; (2) computation of prediction intervals for each method; and (3) random simulation from the underlying state space model. We demonstrate the methods by applying them to the data from the M-competition and the M3-competition. The method provides forecast accuracy comparable to the best methods in the competitions; it is particularly good for short forecast horizons with seasonal data.

Exponential smoothing, often used in sales forecasting for inventory control, has always been rationalized in terms of statistical models that possess errors with constant variances. It is shown in this paper that exponential smoothing remains appropriate under more general conditions, where the variance is allowed to grow or contract with corresponding movements in the underlying level. The implications for estimation and prediction are explored. In particular, the problem of finding the predictive distribution of aggregate lead-time demand, for use in inventory control calculations, is considered using a bootstrap approach. A method for establishing order-up-to levels directly from the simulated predictive distribution is also explored.

Many processes must be monitored by using observations that are correlated. An approach called algorithmic statistical process control can be employed in such situations. This involves fitting an autoregressive/moving average time series model to the data. Forecasts obtained from the model are used for active control, while the forecast errors are monitored by using a control chart. In this paper we consider using an exponentially weighted moving average (EWMA) chart for monitoring the residuals from an autoregressive model. We present a computational method for finding the out-of-control average run length (ARL) for such a control chart when the process mean shifts. As an application, we suggest a procedure and provide an example for finding the control limits of an EWMA chart for monitoring residuals from an autoregressive model that will provide an acceptable out-of-control ARL. A computer program for the needed calculations is provided via the World Wide Web.

The present study focuses on the application of stochastic modeling technique in analyzing the future trends of aerosol optical properties. For this, the Box–Jenkins ARIMA (Autoregressive Integrated Moving Average) model has been used for simulating the monthly average Aerosol Optical Depth (AOD550 nm) retrieved from Terra MODIS (Moderate Resolution Imaging Spectroradiometer) over New Delhi, the urban capital of India. The satellite dataset has been collected for a period of ten years from 2004 to 2014. The analysis of autocorrelation function indicates existence of seasonality in the AOD time series. Several seasonal ARIMA models have been generated and their validation has been verified by assessing various estimation parameters, using the Statistical Package for the Social Sciences (SPSS, version 20). After rigorous evaluation of the selected models, the ARIMA (1,0,0)x(0,1,2)12 is identified as the best fit model w.r.t. measures of goodness-of-fit like Stationary R-square (0.530), R-square (0.674), Root Mean Squared Error (0.128); Mean Absolute Error (0.095); Mean Absolute Percentage Error (16.942); and normalized Bayesian Information Criteria (−3.941). The selected models have been further used to forecast AOD values for the year 2014 at 95% level of confidence. However, the ARIMA (1,0,0)x(2,1,1)12 model is found to have minimum forecasting error, calculated as Mean Percentage Error (0.220). As the difference in BIC of both the models is minimal (0.046), so both the models have been considered as best fit models and utilized for prediction of AOD. Satisfactory results have been obtained using the selected ARIMA models, suggesting that a simplistic modeling technique for determining the future values of AOD is feasible.

Exponential smoothing (ES) forecasting methods are widely used but are often discussed without recourse to a formal statistical framework. This paper reviews and compares a variety of potential models for ES. As well as autoregressive integrated moving average and structural models, a promising class of dynamic non-linear state space models is described that allows for a changing variance. The richness of possible models helps to explain why ES methods seem to be robust in practice. A modelling approach can enhance the forecaster's ability to identify pertinent components of time series variation, and to obtain more reliable estimates of prediction error variances. The paper should be of particular interest to those engaged in forecasting applications where strategies that allow for risk and uncertainty are needed.

The problem of constructing prediction intervals for linear time series (ARIMA) models is examined. The aim is to find prediction intervals that incorporate an allowance for sampling error associated with parameter estimates. The effect of constraints on parameters arising from stationarity and invertibility conditions is also incorporated. Two new methods, based on varying degrees of first-order Taylor approximations, are proposed. These are compared in a simulation study to two existing methods, a heuristic approach and the "plug-in" method whereby parameter values are set equal to their maximum likelihood estimates. A comparison of the four methods is also made for quarterly retail sales for 10 Organization for Economic Cooperation and Development countries. The new approaches provide a systematic improvement over existing methods.