Robust exponential smoothing of multivariate time series

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Abstract

Multivariate time series may contain outliers of different types. In the presence of such outliers, applying standard multivariate time series techniques becomes unreliable. A robust version of multivariate exponential smoothing is proposed. The method is affine equivariant, and involves the selection of a smoothing parameter matrix by minimizing a robust loss function. It is shown that the robust method results in much better forecasts than the classic approach in the presence of outliers, and performs similarly when the data contain no outliers. Moreover, the robust procedure yields an estimator of the smoothing parameter less subject to downward bias. As a byproduct, a cleaned version of the time series is obtained, as is illustrated by means of a real data example.

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 y1,,yT, the smoothed values are given by yˆt=Λyt+(IΛ)yˆt1, for t=2,,T, where Λ is the smoothing matrix. The forecast that we can make at moment T for the next value yT+1 is then given by yˆT+1|T=yˆT=Λk=0T1(IΛ)kyTk. The forecast in (2) is a weighted linear combination of the past values of the series. Assuming the matrix sequence (IΛ)k converges to zero, the weights decay exponentially fast and sum to the identity matrix I. 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 yt 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 t is said to be outlying if its corresponding one-step-ahead prediction error ytyˆt|t1 is large, say larger than twice the robust scale estimate of the prediction errors. A large prediction error means that the value of yt 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 Λ 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 t we observe a p-dimensional vector yt, for t=1,,T. Exponential smoothing is defined in a recursive way. Assume that we already computed the smoothed values of y1,,yt1. To obtain a robust version of the update Eq. (1), we simply replace yt in (1) by a “cleaned” version yt for any t. We now detail how this cleaned value can be computed. Define the one-step-ahead forecast errorrt=ytyˆt|t1, being a vector of length p, for t=2,,T. 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 yt 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

References (22)

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