Ralph D. Snyder

A state space framework for automatic forecasting using exponential smoothing methods

We provide a new approach to automatic business forecasting based on an extended range of exponential smoothing methods. Each method in our taxonomy of exponential smoothing methods can be shown to be equivalent to the forecasts obtained from a state space model. This allows (1) the easy calculation of the likelihood, the AIC and other model selection criteria; (2) the 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.

Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models

Abstract A class of nonlinear state-space models, characterized by a single source of randomness, is introduced. A special case, the model underpinning the multiplicative Holt-Winters method of forecasting, is identified. Maximum likelihood estimation based on exponential smoothing instead of a Kalman filter, and with the potential to be applied in contexts involving non-Gaussian disturbances, is considered. A method for computing prediction intervals is proposed and evaluated on both simulated and real data.

Forecasting Time Series With Complex Seasonal Patterns Using Exponential Smoothing

An innovations state space modeling framework is introduced for forecasting complex seasonal time series such as those with multiple seasonal periods, high-frequency seasonality, non-integer seasonality, and dual-calendar effects. The new framework incorporates Box–Cox transformations, Fourier representations with time varying coefficients, and ARMA error correction. Likelihood evaluation and analytical expressions for point forecasts and interval predictions under the assumption of Gaussian errors are derived, leading to a simple, comprehensive approach to forecasting complex seasonal time series. A key feature of the framework is that it relies on a new method that greatly reduces the computational burden in the maximum likelihood estimation. The modeling framework is useful for a broad range of applications, its versatility being illustrated in three empirical studies. In addition, the proposed trigonometric formulation is presented as a means of decomposing complex seasonal time series, and it is shown that this decomposition leads to the identification and extraction of seasonal components which are otherwise not apparent in the time series plot itself.

A new look at models for exponential smoothing

Exponential smoothing (ES) forecasting methods are widely used but are often dis-cussed 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 forecasters 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 appli-cations where strategies that allow for risk and uncertainty are needed.

Forecasting models and prediction intervals for the multiplicative Holt–Winters method

A new class of models for data showing trend and multiplicative seasonality is presented. The models allow the forecast error variance to depend on the trend and/ or the seasonality. It can be shown that each of these models has the same updating equations and forecast functions as the multiplicative Holt-Winters method, regardless of whether the error variation in the model is constant or not. While the point forecasts from the different models are identical, the prediction intervals will, of course, depend on the structure of the error variance and so it is essential to be able to choose the most appropriate form of model. Two methods for making this choice are presented and examined by simulation.

Control of inventories with intermittent demand

Abstract The problem of controlling inventories with intermittent demands is considered. A method for determining re-order levels consistent with a specified customer service level is proposed. The distinguishing feature is the use of a probability distribution with a spike at zero to represent the relative frequency of periods with no transactions.

Forecasting for Inventory Control with Exponential Smoothing

Exponential smoothing, often used for sales forecasting in 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 the appropriate approach under more general conditions where the variances are allowed to grow and contract with corresponding movements in the underlying level. The implications for estimation and prediction are explored. In particular the problem of finding the prediction distribution of aggregate lead- time demand for use in inventory control calculations is considered. It is found that unless a drift term is added to simple exponential smoothing, the prediction distribution is largely unaffected by the variance assumption. A method for establishing order-up-to levels and reorder levels directly from the simulated prediction distributions is also proposed.

Structural time series models in inventory control

Abstract Exponential smoothing methods are often used to forecast demand in computerized inventory control systems. These methods, by themselves, are rather ad hoc, but they can be given a proper statistical foundation by setting up a class of structural time series models. The purpose of the paper is to highlight the potential role of these models in inventory control. In particular they are used as the basis for deriving formulae for estimating the mean and variance of the lead time demand distribution under both constant and stochastic lead time assumptions.

Inventory control with the gamma probability distribution

Abstract Application of inventory theory often rely on the normal and negative exponential distributions to represent the lead time demand of fast and slow moving items respectively. Yet it is now accepted that both distributions, when taken together, are incapable of adequately describing the demand characteristics of all items found in the typical inventory. Instead there has been a growing interest in the use of the gamma probability distribution because it not only encompasses both former distributions as special cases but also covers the gaps left by them. In the process a number of methods for calculating control parameters have appeared in the literature for items with gamma distributed lead time demand. As knowledge about the problem has increased there has been a general tendency towards greater simplification. This paper continues the trend by introducing an approach that depends only on concepts from basic statistics. The aim is to eliminate unnecessary complexity and make the associated theory easier to understand.

Prediction Intervals for Arima Models

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.