Gravity Changes over Russian River Basins from GRACE - Planetary Exploration and Science: Recent Results and Advances - page 49

covariance matrix). SSA algorithm has four stages: (a) formation of trajectory
matrix, (b) its singular value decomposition (SVD), (c) singular numbers grouping,
and (d) principal components recovering through Hankelization. SSA algorithm is
described in details in Golyandina et al. ( 2001 ) and Golyandina ( 2004 ). MSSA
includes similar iterations.
Firstly (a), we select lag parameter L. For every Stokes coefficient, we have time
series of length N. The trajectory matrix can be built for one time series component
(channel); let's say for the Stokes coefficient C ij .t k /, k D 0;:::;N 1,as
follows:

0 1

C ij .t 0 /C ij .t 1 /:::C ij .t K1 /
C ij .t 1 /C ij .t 2 /::: C ij .t K /
: : : : : : : : : : : :
C ij .t L1 /C ij .t L /:::C ij .t N1 /

@ A

X C ij D ; (3.3)

here K D N L C 1.
The trajectory matrices X C ij ; X S ij for every Stokes coefficient C ij and S ij should
be incorporated into large block matrix X as follows:

X D ΠX C 2;0 ; X S 2;0 :::;; X C ij ; X S ij ;:::; X C 60;60 X S 60;60 T :

(3.4)

Thus, in our realization, we put blocks one behind another. This multichannel
trajectory matrix, composed of blocks for every channel, can be used to calculate
the lagged covariance matrix A D X T X .
At the second stage (b), SVD should be applied to X .

X D USV T :

As a result, a sequence of singular numbers (SNs) s i standing along the diagonal
of matrix S in order of decreasing values (see Fig. 3.2 ) and the corresponding
eigenvectors v i (left) and u i (right) are obtained. If to solve the eigenvalue problem
for A D VS T SV T , then the eigenvalues will be squared singular numbers i D s i
and left eigenvectors v i , included as columns in V , form empirical orthogonal
functions (EOFs).
The ith component corresponds to the matrix

X i D s i u i v i :

In MSSA we reconstruct the vectorial PCs from this matrix, knowing its structure,
similar to the structure of X . It is done through Hankelization (d), allowing to
reconstruct every channel of ith component from the corresponding blocks of matrix
X i , organized as in ( 3.4 ). Suppose we need to reconstruct the C lm channel. Then
each kth count can be obtained from the averaging along the side diagonals of
the corresponding matrix block Y

X i C lm . The first and the last L elements are

D