Modelo estocastico croston

Stochastic models underlying Croston’s method for intermittent demand forecasting

2 February 2005

Lydia Shenstone

Department of Econometrics and Business Statistics Monash University, VIC 3800, Australia. Email: Lydia.Shenstone@buseco.monash.edu.au Phone: +61-3-9905 2383 Fax: +61-3-9905 5474

Rob J. Hyndman

Department of Econometrics and Business Statistics Monash University, VIC3800, Australia. Email: Rob.Hyndman@buseco.monash.edu.au

1

Authors’ Biographies:
Lydia Shenstone is a PhD student at the Department of Econometrics and Business Statistics, Monash University. She holds a Master’s degree in Statistics from the University of Auckland, New Zealand, and a Bachelor’s degree in Mathematical Statistics from Jilin University, China. She was awarded the 1997 AnnualPrize in Statistics by the University of Auckland. Rob Hyndman is Professor of Statistics and Director of the Business and Economic Forecasting Unit, Monash University. He holds a PhD in Statistics from the University of Melbourne. He is co-author of the textbook Forecasting: methods and applications (Wiley, 1998) with Makridakis and Wheelwright, and has had published papers in many journalsincluding the International Journal of Forecasting, the Journal of Forecasting, and the Journal of the Royal Statistical Society, Series B. He is Editor-in-Chief of the International Journal of Forecasting.

2

Stochastic models underlying Croston’s method for intermittent demand forecasting

2 February 2005

1

Stochastic models underlying Croston’s method for intermittent demandforecasting

Abstract:
Croston’s method is widely used to predict inventory demand when it is intermittent. However, it is an ad hoc method with no properly formulated underlying stochastic model. In this paper, we explore possible models underlying Croston’s method and three related methods, and we show that any underlying model will be inconsistent with the properties of intermittent demand data.However, we find that the point forecasts and prediction intervals based on such underlying models may still be useful.

Keywords:
Croston’s method, exponential smoothing, forecasting, intermittent demand, prediction intervals.

2 February 2005

2

Stochastic models underlying Croston’s method for intermittent demand forecasting

1. Introduction
Inventories with intermittent demands arequite widespread in practice. Data for such items consist of time series of non-negative integer values where some values are zero. We shall denote the historical demand series Y1 , Y2 , . . . , Yn and assume these take non-negative integer values. Croston’s (1972) method is the most widely used approach for intermittent demand forecasting (IDF), and involves separate simple exponential smoothing(SES) forecasts on the size of a demand and the time period between demands. Other authors, including Johnston & Boylan (1996) and Syntetos & Boylan (2001), have suggested a few modifications to Croston’s method that can provide improved forecast accuracy. One such modification is to apply Croston’s method to the logarithms of the demand data and to the logarithms of the inter-demand time. However, allof these methods provide only point forecasts and are not based on a stochastic model. In fact, no underlying model for Croston’s method has ever been properly formulated. Consequently, there are no forecast distributions and prediction intervals associated with forecasts obtained using these methods. This paper aims to identify stochastic models that underly Croston’s method and related methods,and hence obtain forecast distributions and other properties. However, we end up showing that the models that might be considered as underlying Croston’s and related methods are inconsistent with the properties of intermittent demand data. In particular, the possible models underlying Croston’s and related methods must be non-stationary and defined on a continuous sample space. For Croston’s...