The benefits of multivariate singular spectrum analysis over the univariate version

doi:10.1016/j.jfranklin.2017.09.008

Journal of the Franklin Institute

Volume 355, Issue 1, January 2018, Pages 544-564

https://doi.org/10.1016/j.jfranklin.2017.09.008 Get rights and content

Abstract

Singular Spectrum Analysis (SSA) is a relatively simple and powerful method in the area of time series analysis that is mainly based on matrix analysis. In this paper, we present a methodological comparison between the univariate and multivariate versions of SSA. Additionally, we explore the advantages of multivariate SSA in terms of theoretical results and with application to a real data set on currency exchange rates.

Introduction

The analysis of time series is one of the major research topics in statistics and widely used and relevant in many fields of application. Since in some cases the interest lies in the analysis and forecast of a single time series (univariate case) and in other cases the interest is to analyse and forecast simultaneously multiple time series (multivariate case), there are several methodologies adapted to analyse and forecast both univariate and multivariate time series. The non-parametric singular spectrum analysis (SSA) methodology is no different, as it can be applied to a single time series or jointly to several time series, being the latter referred as multivariate singular spectrum analysis (MSSA). Similarly to the case of parametric modelling and in many real-life applications, two or more time series may be related, which in the context of MSSA has a correspondence in terms of matched components of the several series

Golyandina et al.[4] give an in-depth overview of SSA, and many examples of successful application of SSA and MSSA can be found in different fields such as industry [8], [11], [14], [15], business [2], airline traffic [16], and mortality forecasting [10], [13], among others.

Contrary to SSA, there is little research on the theoretical aspects of MSSA. This might be a result of the idea that multivariate models are less parsimonious in parametric methods, having more parameters to be estimated than the univariate counterparts. In the context of SSA/MSSA, although we are dealing with a non-parametric method, the increase in model complexity does not apply. In fact, every time the univariate SSA is applied to an univariate time series, it requires the user to choose two “parameters”, the window length [20] and the number of eigentriples used for reconstruction [1], which will sum up to the double of the number of single time series in the multivariate data. This represents a higher number of parameters than for MSSA, which does modelling and forecasting by blocks.

In this paper, we compare SSA and MSSA, both methodologically and with an application to a real data set, and point out the advantages and limitations of MSSA in comparison with SSA.

The rest of this paper is organised as follow: in Section 2 we give a brief description of MSSA. Section 3 provides the methodological comparison between SSA and MSSA. Section 4 presents a comparison between SSA and MSSA with a real data set based on daily currency exchange rates in four of the BRICS emerging economies: Brazil, India, China and South Africa. We finish the paper with a summary conclusion in Section 5.

Section snippets

Multivariate singular spectrum analysis

Multi-channel SSA, Multivariate SSA, or MSSA, is a natural extension of SSA for analysing multivariate time series. MSSA, similarly to SSA, has many applications such as, among others, denoising, trend extraction, causality and forecasting. MSSA is especially popular to analyse and to forecast economic and financial time series with short and long series length [6], [7], [14]. Other multivariate extension of SSA is the 2D-SSA which can be applied to two-dimensional data such as digital images

Choices in SSA and MSSA

When using SSA and MSSA to analyse and forecast time series, two choices must be made by the user: the window length, L, and the number of eigentriples used for reconstruction, r. When using SSA for a multivariate time series (one univariate model for each of the M time series), a value for L and another for r must be chosen for each time series, whereas these choices are done only once when using MSSA. Table 2 shows the choices that must be considered in both univariate and multivariate SSA,

Description of data

In this section, we analyse a real data set to assess whether the methodological results are confirmed in practice, i.e., whether MSSA outperforms SSA in terms of forecasting ability. We intended to consider daily currency exchange rate data for the BRICS countries (Brazil–BRL, Russia–RUB, India–IND, China–CHN and South Africa–RAND). However, since the complete data from Russia could not be found, this country was discarded from our study, which does not interfere with the results as the recent

Conclusion

In this paper, we presented a comparison between the univariate and multivariate versions of SSA. The advantages of MSSA were explored in terms of methodological results and in terms of an application to real data on currency exchange rates. An overall superiority of MSSA based algorithms, specially HMSSA-R, over SSA based algorithms, for forecasting, was observed in the application. In practice, as the co-integration between time series is considered in MSSA and not in SSA, the performance of

Acknowledgment

The authors would like to thank the Associate Editor and two anonymous reviewers for providing helpful suggestions which contributed to the improvement of the paper. The authors acknowledge financial support by CAPES – Fundação Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Coordination for the Improvement of Higher Education Personnel), Brazil, grant number 88881.062137/2014-01.

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View full text

The benefits of multivariate singular spectrum analysis over the univariate version

Abstract

Introduction

Section snippets

Multivariate singular spectrum analysis

Choices in SSA and MSSA

Description of data

Conclusion

Acknowledgment

Econ. Lett.

C. R. Math.

Int. J. Forecast.

IEEE Geosci. Remote Sens. Lett.

A new approach for selecting the number of the eigenvalues in singular spectrum analysis

J. Frankl. Inst.

A principal axis transformation for non-Hermitian matrices

Bull. Amer. Math. Soc.

Analysis of Time Series Structure: SSA and Related Techniques

Singular Spectrum Analysis for Time Series

Multivariate singular spectrum analysis: a general view and new vector forecasting algorithm

Int. J. Energy Stat.

On the singular values of the Hankel matrix with application in singular spectrum analysis

Chil. J. Stat.