Chapter 14
Patterns in Time: SSA
and MSSA
by Robert Vautard
14.1 Introduction
Singular Spectrum Analysis (SSA) is a particular application of the EOF
expansion. In classical EOF analysis, the random field X to be studied,
called also the state veetor, contains values measured or estimated at a given
time, that is, the coordinates of X represent different locations in space at the
same time. By diagonalising the covariance matrix of X, one tries therefore
to capture the dominant spatial patterns. The SSA expansion (Vautard et
al., 1992) is an EOF expansion, but the state vector X now contains values
at the same loeation but at different lags. The leading eigenelernents of the
corresponding covariance matrix represent thus the leading time patterns of
the random field. SSA is a time senes analysis, in the sense that a single
signal is analysed.
When both space and lag vary in the state vector X, the analysis is called
Multichannel Singular Spectrum Analysis (MSSA; Plaut and Vautard, 1994),
or Extended EOF analysis (EEOF; Weare and Nasstrom,1982). These latter
two techniques are mathematically equivalent, but differ in their practical
domain of application: In general, MSSA deals with more temporal degrees
Acknowledgements: The NMC data were kindly provided by Kingtse Mo at the
Climate Analysis Center. Most of the results shown here were carried out by Guy Plaut at
the Institut Non-Linaire de Nice. The long-range forecasting scheme has been developed
recently by Carlos Pires in his Ph.D. work.
Patterns in Time: SSA
and MSSA
by Robert Vautard
14.1 Introduction
Singular Spectrum Analysis (SSA) is a particular application of the EOF
expansion. In classical EOF analysis, the random field X to be studied,
called also the state veetor, contains values measured or estimated at a given
time, that is, the coordinates of X represent different locations in space at the
same time. By diagonalising the covariance matrix of X, one tries therefore
to capture the dominant spatial patterns. The SSA expansion (Vautard et
al., 1992) is an EOF expansion, but the state vector X now contains values
at the same loeation but at different lags. The leading eigenelernents of the
corresponding covariance matrix represent thus the leading time patterns of
the random field. SSA is a time senes analysis, in the sense that a single
signal is analysed.
When both space and lag vary in the state vector X, the analysis is called
Multichannel Singular Spectrum Analysis (MSSA; Plaut and Vautard, 1994),
or Extended EOF analysis (EEOF; Weare and Nasstrom,1982). These latter
two techniques are mathematically equivalent, but differ in their practical
domain of application: In general, MSSA deals with more temporal degrees
Acknowledgements: The NMC data were kindly provided by Kingtse Mo at the
Climate Analysis Center. Most of the results shown here were carried out by Guy Plaut at
the Institut Non-Linaire de Nice. The long-range forecasting scheme has been developed
recently by Carlos Pires in his Ph.D. work.