Anne B. Koehler

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.

A Comparison of the Akaike and Schwarz Criteria for Selecting Model Order

SUMMARY The object of this paper is to compare the Akaike information criterion (AIC) and the Schwarz information criterion (SIC) when they are applied to the crucial and difficult task of choosing an order for a model in time series analysis. These order selection criteria are used to fit state space models. Models are fitted to a set of monthly time series randomly selected from the series used in the Makridakis competition (1982). All series are composed of real data. The AIC and SIC indicate different model orders in 27% of the cases. The forecasting accuracy is compared for these cases. The results of this comparison show that it is preferable to apply the SIC, which leads to lower order models for forecasting.

Forecasting time series with multiple seasonal patterns

A new approach is proposed for forecasting a time series with multiple seasonal patterns. A state space model is developed for the series using the innovations approach which enables us to develop explicit models for both additive and multiplicative seasonality. Parameter estimates may be obtained using methods from exponential smoothing. The proposed model is used to examine hourly and daily patterns in hourly data for both utility loads and traffic flows. Our formulation provides a model for several existing seasonal methods and also provides new options, which result in superior forecasting performance over a range of prediction horizons. In particular, seasonal components can be updated more frequently than once during a seasonal cycle. The approach is likely to be useful in a wide range of applications involving both high and low frequency data, and it handles missing values in a straightforward manner.

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.

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.

Exponential smoothing models: Means and variances for lead-time demand

Abstract 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.

A comparison of results from state space forecasting with forecasts from the Makridakis Competition

Abstract State space models are fit to a subset of the time series modelled by the Box-Jenkins method in the Makridakis Competition. The state space approach to time series analysis is outlined, as is the use of a modelling package on a personal computer. Forecasts from one to 18 months beyond the fit set of each series are computed from the fitted models. Box-Jenkins forecasts and deseasonalized single exponential smoothing (final forecasts are reasonalized) forecasts for the same series are extracted from the M-Competition data tape. All three sets of forecasts are compared to the actual series values withheld in the check sets; forecasting accuracy is calculated on the basis of mean absolute percentage error and median absolute percentage error. The automatic procedure of single exponential smoothing and the semi-automatic procedure of the state space package produce forecasts which, in most cases, are as accurate or more accurate than those developed by an expert using the Box-Jenkins method.

A model selection strategy for time series with increasing seasonal variation

We propose a model selection strategy for time series with increasing seasonal variation. This strategy amounts to a selection of the most appropriate differencing filter to obtain a stationary time series without using a Box-Cox transformation. Hence, it is based on a sequence of tests for nonseasonal and seasonal unit roots. Through Monte Carlo replications, we provide new tables of critical values for the various test statistics. We apply our methods, which can be automated, to six example series and find that the results compare favorably to those of an expert.