Application of Singular Spectrum Analysis (SSA)
for the Reconstruction of Annual Phenological
Profiles of NDVI Time Series Data
1. METHODS
1. 1. Profile Extraction: The Singular Spectrum Analysis (SSA)
The method of SSA is a relatively new technique, and is sufficiently effective and comparative
with numerous smoothing techniques (Percival et al., 1993 , Theiler et al., 1992; Kaplan and
Glass, 1992). Moreover, in certain cases forecasting the system evolution on its basis gives
much more reliable results in comparison with the other known algorithms (Casdagli, 1989;
Danilov et al., 1997; Deppish et al., 1991; Murray, 1993; Cao et al., 1995; Keppenne and Ghil,
1992, 1993, 1994, 1995).
The SSA is usually regarded as a method of identifying and extracting oscillatory components
from the original time series (Yiou et al., 1996; Ghil and Tarrico, 1997; Fowler and Kember,
1998). It is also used in particular tasks, such as detection of “change-point” in time series
(Moskvina and Zhigljavsky, 2002a, 2002b); “extraction of signal from noise”, and smoothing
(Golyandina, 2001). This current study is a demonstration of the power of SSA in doing the
aforesaid tasks. Filtering “noisy” data using SSA have been performed by Allen et al. (1996,
1997) and Vautard et al. (1992) on earth’s global mean temperature; and on the El Niño-Southern
Oscillation (ENSO) (Mann and Lees, 1996).
Suppose that we have a time series x of length N. Suppose that we choose some aximum lag M
and compute the M X ..M lag autocovariance matrix,
If we do an eigenanalysis of this covariance matrix, we will obtain temporal structures that
explain the maximum possible amount of the temporal autocovariance on an interval of measure
M. The eigenvalues will be the amount of covariance in time explained y each eigenvector. You
can then reconstruct the time series at any point n, by expanding the data in the basis set of
eigenfunctions.
where n=i+j, ek ( j) is the jth element of the kth eigenvector, and ak (i) is an expansion coefficient
for the kth eigenvector beginning at the point i.
1.2. Application on NOAA-AVHRR Pathfinder data
NOAA-AVHRR Pathfinder NDVI 10-day time series data for the year 1984 are considered in
this study and were geometrically corrected. Sample data of 3X3 pixel size of different land
cover types namely, for Africa/Saharan (desert), Eastern Canada (deciduous/boreal), Philippines
(two-time cropping), and Central Brazil (tropical) were collected and annual NDVI profiles
were constructed. Agbu et al. (1994) describes the Pathfinder format. By default, we choose the
lag value M < [N/2+1] , which is equivalent to 17.
The typically noisy profile can be seen in the original data shown in the Figure 1. The SSA
algorithm assumes (1) that cloud and cloud shadow and changing atmospheric conditions (such
as the persistence of clouds in images) will usually depress NDVI values (Ariño et al,. 1991,
Simpson et al., 2000); (2) that there exists vestiges (relics) of various systematic and data
transmissions errors which tend to increase the NDVI values.
