Adapting the Singular Spectrum Analysis Trajectory Matrix Technique to Identify Multiple Additive Time-Series Outliers

Abstract

Singular spectrum analysis is a powerful non-parametric time series method that unfolds an observed time series into a special structured matrix, called a trajectory matrix. The trajectory matrix has a Hankel structure, whereby off-diagonal elements are non-unique. Singular value decomposition is then applied to the trajectory matrix to extract time series structures. The method can handle complex time series structures that include combinations of polynomials, sinusoids and exponentials. Time series structures that SSA can handle are very typical of seasonal series researched by economists and econometricians alike. The possible presence of multiple additive time series outliers can adversely affect time series forecasting and construction of bootstrap results. An additive time series outlier typically involves a single time series observation which either resulted due to some recording error or temporal shock (upwards/downwards) to the time series at a specific time. When more than a single additive outlier is present in a time series, we are faced with multiple additive outliers. An algorithm is proposed in this paper whereby Robust Principal Component Analysis (ROBPCA) methods are applied to the trajectory matrix, in order to identify multiple additive outliers and also estimate their size and appropriate imputations, based on the underlying time series structure. The procedure is iterative in the sense that, following identification of a single additive outlier, a signal processing procedure (Cadzow signal reconstruction) is used to impute identified outliers at each stage of the iterations. Monte Carlo simulations are used to illustrate the effectiveness of the iterative approach to handle identification of multiple additive time series outliers. The practical usefulness of the method is also illustrated by application to a real-life time series containing multiple additive outliers in a South African tourism time series.