A robust approach for outlier imputation: Singular spectrum decomposition

Abstract

Singular spectrum analysis (SSA) is a nonparametric method for separating time series data into a sum of small numbers of interpretable components (signal + noise). One of the steps of the SSA method, which is referenced to Embedding, is extremely sensitive to contamination of outliers which are often founded in time series analysis. To reduce the effect of outliers, SSA based on Singular Spectrum Decomposition (SSD) method is proposed. In this article, the ability of SSA based on SSD and basic SSA are compared in time series reconstruction in the presence of outliers. It is noteworthy that the matrix norm used in Basic SSA is the Frobenius norm or L₂-norm. There is a newer version of SSA that is based on L₁-norm and called L₁-SSA. It was confirmed that L₁-SSA is robust against outliers. In this regard, this research is also introduced a new version of SSD based on L₁-norm which is called L₁-SSD. A wide empirical study on both simulated and real data verifies the efficiency of basic SSA based on SSD and L₁-norm in reconstructing the time series where polluted by outliers.

Keywords:

• Outlier
• trajectory matrix
• signal extraction
• singular spectrum analysis and singular spectrum decomposition

Acknowledgments

We would like to thank three anonymous referees for many helpful comments. However, any remaining errors are solely ours.