Robust exponential smoothing of multivariate time series
Introduction
Exponential smoothing is a popular technique used to forecast time series. Thanks to its very simple recursive computing scheme, it is easy to implement. It has been shown to be competitive with respect to more complicated forecasting methods. A multivariate version of exponential smoothing was introduced by Jones (1966) and further developed by Pfefferman and Allon (1989). For a given multivariate time series , the smoothed values are given by for , where is the smoothing matrix. The forecast that we can make at moment for the next value is then given by The forecast in (2) is a weighted linear combination of the past values of the series. Assuming the matrix sequence converges to zero, the weights decay exponentially fast and sum to the identity matrix . The forecast given in (2) is optimal when the series follows a vector IMA(1, 1) model; see Reinsel (2003, page 51). The advantage of a multivariate approach is that for forecasting one component of the multivariate series, information from all components is used. Hence the covariance structure can be exploited to get more accurate forecasts. In this paper, we propose a robust version of the multivariate exponential smoothing scheme.
Classic exponential smoothing is sensitive to outliers in the data, since they affect both the update Eq. (1) for obtaining the smoothed values and Eq. (2) for computing the forecast. To alleviate this problem, Gelper et al. (in press) proposed a robust approach for univariate exponential smoothing. In the multivariate case the robustness problem becomes even more relevant, since an outlier in one component of the multivariate series will affect the smoothed values of all series. Generalizing the approach of Gelper et al. (in press) to the multivariate case raises several new issues.
In the univariate case, the observation at time is said to be outlying if its corresponding one-step-ahead prediction error is large, say larger than twice the robust scale estimate of the prediction errors. A large prediction error means that the value of is very different from what one expects, and hence indicates a possible outlier. In a multivariate setting the prediction errors are vectors. We declare then an observation as outlying if the robust Mahalanobis distance between the corresponding one-step-ahead prediction error and zero becomes too large. Computing this Mahalanobis distance requires a local estimate of multivariate scale.
Another issue is the selection of the smoothing matrix used in Eq. (1). The smoothing matrix needs to be chosen such that a certain loss function computed from the one-step-ahead prediction errors is minimized. As loss function we propose the determinant of a robust estimator of the multivariate scale of the prediction errors.
In Section 2 of this paper we describe the robust multivariate exponential smoothing procedure. Its recursive scheme allows us both to detect outliers and to “clean” the time series. It then applies classic multivariate exponential smoothing to the cleaned series. The method is affine equivariant, making it different from the approach of Lanius and Gather (in press). In Section 3 we show by means of simulation experiments the improved performance of the robust version of exponential smoothing, both for forecasting and for selecting the optimal smoothing matrix. Section 4 elaborates on the use of the cleaned time series, an important byproduct of applying robust multivariate exponential smoothing. This cleaned time series can be used as an input for more complicated time series methods. We illustrate this in a real data example, where the parameters of a Vector AutoRegressive (VAR) model are estimated from the cleaned time series. Finally, Section 5 contains some conclusions and ideas for further research.
Section snippets
Robust multivariate exponential smoothing
At each time point we observe a -dimensional vector , for . Exponential smoothing is defined in a recursive way. Assume that we already computed the smoothed values of . To obtain a robust version of the update Eq. (1), we simply replace in (1) by a “cleaned” version for any . We now detail how this cleaned value can be computed. Define the one-step-ahead forecast error being a vector of length , for . The multivariate cleaned series is given by
Simulation study
In this section we study the effect of additive outliers and correlation outliers on both the classic and the robust multivariate exponential smoothing method. We compare the one-step-ahead forecast accuracy, and the selection of the smoothing parameter matrix by both methods. Forecast accuracy is measured by the determinant of the MCD estimator on the scatter of the one-step-ahead forecast errors. We prefer to use a robust measure of forecast accuracy, since we want to avoid the forecasts made
Real data example
The robust multivariate exponential smoothing scheme provides a cleaned version of the time series. As a result, an affine equivariant data cleaning method for multivariate time series is obtained. In this example, we illustrate how a cleaned series can be used as input for further time series analysis.
Consider the housing data set from the book of Diebold (2001) and used in Croux and Joossens (2008). It concerns a bivariate time series of monthly data. The first component contains housing
Conclusion
For univariate time series analysis, robust estimation procedures are well developed; see Maronna et al. (2006, Chapter 8) for an overview. To avoid the propagation effect of outliers, a cleaning step is advised, that goes along with the robust estimation procedure (e.g. Muler et al. (2009)). For resistant analysis of multivariate time series much less work has been done. Estimation of robust VAR models is proposed in Ben et al. (1999) and Croux and Joossens (2008), and a projection–pursuit
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