Gravity Changes over Russian River Basins from GRACE - Planetary Exploration and Science: Recent Results and Advances - page 50

1E-008

1E-008

singular numbers,
MSSA , L=36

8E-009

6E-009

4E-009

2E-009

12 3 4 5 6 7 8 910

number

1E-009

0 10 20 30 40 50 60 70 80 90

number

Fig. 3.2 Distribution of singular numbers, defining MSSA components energy

calculated from the fewer number of Y values, so the first and the last points of PCs
will be less consistent. It is supposed that elements on the side diagonals of matrix
Y are approximately equal and it is almost Hankel. In case when it doesn't hold
strictly, some kind of edge effect appears.
Grouping (c) of components is required, when some of SNs are related to one
and the same PC and have similar behavior that could be detected with the use
of ! correlations and other techniques (Golyandina et al. 2001 ). Then, SNs (s i )
should be grouped together and reconstructed as one PC before stage (d). It can
be done after Hankelization (d) by simple item-by-item summation of components.
Details on grouping and theorems about separability of components can be found in
(Golyandina 2004 ).
As a result, the set of PCs with decreasing amplitudes representing different
modes of time series variability are obtained.
The main parameter of the algorithm - the time lag L, which determines the
dimensionality of the time series embedding space, should be chosen heuristically,
using recommendations given in Golyandina ( 2004 ). It should not be larger than
N=2, and it is better to choose it as a multiplier of periods, expected in the time
series. In earlier works, we used L D 24 (Zotov and Shum 2009 ;Zotov 2012 ). But
with extend of period of observations, we have chosen L D 36 months (3 years) that
allows to better separate the components. No other filters like Gaussian smoothing
were used, though it is possible; see Eq. ( 3.2 ).
In Zotov and Shum ( 2009 )andZotov( 2012 ), we found MSSA more flexible than
simple EOF for recognition of trend, modulated oscillations of different periods, and
denoising of multidimensional time series. Different channels “help” each other to
capture spatiotemporal correlation patterns. Lagged matrix X allows to find them
in L-dimensional space. The obtained PCs extract correlations, which present in all
the channels simultaneously.