Robust exponential smoothing

Abstract

The paper is devoted to robust modifications of exponential smoothing for time series with outliers or long-tailed distributions. Classical exponential smoothing applied to such time series is sensitive to the presence of outliers or long-tailed distributions and may give inadequate smoothing and forecasting results. First, simple and double exponential smoothing in the L1 norm (i.e. based on the least absolute deviations) are discussed in detail. Then, general exponential smoothing is made robust, replacing the least squares approach by M-estimation in such a way that the recursive character of the final formulas is preserved. The paper gives simple algorithmic procedures which preserve advantageous features of classical exponential smoothing and, in addition, which are less sensitive to outliers. Robust versions are compared numerically with classical ones. KEY WORDS Exponential smoothing Outliers Long-tailed distributions Robust methods L1 norm Least absolute deviations M-estimation Time seriesExponential smoothing is a popular recursive smoothing and prediction technique frequently used for routine treatment of time series. Despite its numerical simplicity, it gives good practical results in comparison with more complicated smoothing and forecasting methods (see e.g. Gardner, 1985). Although exponential smoothing has an adaptive character due to the exponential weighting of observations in the past, its results may be distorted by the presence of outliers in time series. For instance, Ledolter (1989) has investigated the effect of outliers on the forecasts in ARIMA models, the connection between these models and exponential smoothing being known (e.g. between the ARIMA(O,l, 1) model and simple exponential smoothing). Outliers are frequently explained as a contamination of time series by long-tailed distributions according to various models (e.g. so-called additive and innovation outliers), and there exist corresponding methods of robust time-series analysis suitable for particular methods (see e.g. Tsay, 1988, or the survey paper by Stockinger and Dutter, 1987). In the framework of exponential smoothing, indirect methods of outliers treatment consisting of identification of outliers and interpolation of missing values have been preferred (see e.g. Aldrin and Damsleth, 1989;Cipra, 1989; Jun, 1989). This paper suggests direct approaches to making the exponential smoothing more robust in order to become less sensitive to outliers.First, exponential smoothing in the L I norm is suggested, since this norm is known to be

Robust forecasting with exponential and Holt–Winters smoothing

“…The approach we recommend in this paper is different from that of Cipra (1992), and turns out to yield a better performance than the WR method, as will be shown in the third section. The idea is to replace the observed y t in equation (1) …”

“…To obtain better forecast accuracy in the presence of outliers, robust versions of this technique are needed. Cipra (1992) proposes a robust exponential and Holt-Winters smoothing method based on M-estimation. In the case of ordinary exponential smoothing, the value of the smoothed series at time t, ỹ t , is the solution of the following minimization problem:…”

“…After algebraic manipulations and making use of an iterated weighted least squares algorithm, recursive formulae are obtained for the smoothing methods proposed by Cipra (1992). In the sequel of the paper we refer to this approach based on M-estimation as the weighted regression (WR) method.…”

“…The classical Holt method, denoted by HW, is compared to the two robust schemes presented above. The fi rst is the weighted regression approach of Cipra (1992) (WR), and the second the robust Holt method (RHW) that we recommend. To illustrate the importance of the use of an appropriate robust scale estimator in the recursive equations, we also apply the proposed RHW method with the non-robust scale estimator (7) instead of (12).…”

“…8, for a review), surprisingly little attention has been given to robust alternatives to exponential and Holt-Winters smoothing and forecasting. A fi rst attempt was made by Cipra (1992), who presents the Holt-Winters method as a weighted regression problem, and then replaces the traditional least squares fi t by a robust regression fi t. This yields a reasonable method, but it can be improved upon. Another proposal was made by Taylor (2007), who uses quantile regression to obtain robust forecasts.…”