Further Development of SSA Was Carried out Independently by Several Groups of Applied

Appendix

A British marine biologist Colebrook (1978), analysing long-term observations on zooplankton in North-Eastern Atlantic, was one of the first to use PCA (Principle Component Analysis) technique for analysis of time series. Regrettably, this fruitful approach did not achieve wide circulation in hydrobiology. Eight years after Colebrook’s study, English physicists Broomhead and King (1986) suggested the SSA (Singular Spectrum Analysis). SSA may be considered as a modification of the multidimensional PCA adapted for processing one-dimensional data—thetimeseries (BroomheadandKing, 1986; Golyandinaetal, 2001). The key stage of calculations during SSA is, similarly to PCA, Singular Value Decomposition (SVD) (for SSA see 1.1.2.). The tasks solved by SVDin PCA and SSA are also the same: presentation of the initial data as a sum of independent summands and determination of the contribution of each summand into the variation of the initial data. However, the stagespreceding SVD and the treatment of SVD results differ in these two techniques, which is due to the differences in the initial data: a matrix consisting of heterogeneous variables in case of PCA and a one-dimensional time series in case of SSA.

Further development of SSA was carried out independently by several groups of applied mathematicians in the UK, the USA and Russia. At present, the papers dealing with methodological aspects and applications of SSA number several hundreds. This method is broadly applied to analysis of climatic, meteorological, geophysical and econometric time series (Golyandina et al, 2001).

1.SSAexecution sequence

According to Golyandina et al. (2001), SSA is executed in 2 stages (decomposition and reconstruction) following the scheme:

1.1. First stage: Decomposition

1.1.1. Embedding

Time series consisting ofNobservations (N>2) may be represented as vector x (1).

(1)

The researcher chooses the lag (window length)L (1LN).Embedding procedure transforms the time series x (1) into the matrix (2),.The greatest number of components of the series xmay be extracted atL≈N/2, since the rank of the matrix Ycannot exceed eitherLor K — r≤min(L,K) (see the next step below).

(2)

The matrix consists of Kvectors (yj,1≤j≤K).Each vector yjis a fragment of the initial series x(1) with the length L, starting with the element (xj) whose number(j) corresponds to that of the vector (yj)of the matrix Y (2). In this way, each element of the matrix Yis determined as yi,j=xi+j–1 (2). As is easy to see, elements Y, situated on the diagonals perpendicular to the main one, are equal (yi,j=const if i+j=const — matrix Y is a Hankel matrix).

1.1.2. Singular Value Decomposition

The matrix Yis represented as a product of three matrices (3):

,(3)

where — matrix consisting of rleft singular vectors uj (1≤j≤r) of the matrix Y (the vectors are orthogonal — if i≠j);Σr×r — diagonal matrix containing singular values of the matrix Yin the order of decreasing magnitude (); — matrix consisting of rright singular vectors of the matrix Y (the vectors are orthogonal — if i≠j);r — rank of the matrix Y, andT – the symbol of transposition. To be specific, we claim that the norms of all singular vectors be equal to 1 (||um|| = ||vm|| = 1, 1≤m≤r).

Many iterative algorithms of singular value decomposition are presently known. Most of them amount to finding eigen vectors and eigen values (λm) of the matrix YYT(Golub and van Loan, 1996), since the matrix of eigen vectors YYTis equivalent to the matrix of left singular vectors Y, squares of singular values of the matrix Yare equal to the eigen values of the matrix YYT () and the right singular vectors of the matrix Ycan be easily found with the use of the formula (4).

(4)

Singular decomposition allows one to represent Yas a sum (5) ofrelementary matrices Ym(6):

(5)

(6)

The collection (, um, vm) is referred to as the eigentriple of the SVD.

1.2. Second stage: Reconstruction

1.2.1. Grouping

The obtained r eigentriples may be grouped into several (dr) sets of eigentriples, each of which corresponds to a separate component of the time series x.This step is difficult to formalise (see for details: Golyandina et al., 2001), and we skipped it, since the objective of our study was to reveal only the trend and not all the possible components of the series.

1.2.2. Diagonal averaging (Hankelization)

The obtained matrices Ymserve for construction of the corresponding modes xm. Each element of the mode m, for allyi,j if i+j=const (7). This operation is, in fact, opposite to the first step of the analysis, the embedding.

(7)

2. Determination of thetrend contribution tothe analyzed time series

For the trend (the first mode), the ratio of the first eigen value in the sum of eigen values (normalised eigen value ) would correspond to the explained variance of the initial time series only if the average has been previously subtracted from the initial series. Since each eigen valueλmis equal to the square of Frobenius norm of the elementary matrix Ym () corresponding to it, it is obvious thatλmwould correspond to the dispersion of the matrix Ym (,where — the average value of the matrix Ym) only if the average of this matrix is equal to 0 (only if, here SSm is the sum of the squares of deviations of Ym elements from the average). Therefore, in order to assess the contribution of the mode xm to the analyzed time series, SSm of the matrix of Ym (corresponding to xm) was calculated ().Thus, the explained variance was calculated as the normalised sums of the squares ().In particular, the variance of the initial series explained by the trend was calculated as .

References

Broomhead, D. S. & G. P. King, 1986. Extracting qualitative dynamics from experimental data. Physica D, 20: 217–236.

Colebrook, J. M., 1978. Continuous plankton records: zooplankton and environment, North-East Atlantic and North Sea, 1948-1975. Oceanologica Acta 1: 9–23.

Golub, G. H. & C.F van Loan, 1996. Matrix Computations (Third edition). The JohnsHopkinsUniversity Press, Baltimore.

Golyandina, N., Nekrutkin, V. & A. Zhigljavsky, 2001. Analysis of Time Series Structure: SSA and Related Techniques. Chapman & Hall/CRC, Boca Raton.