Patterns in Time: SSA and MSSA

Singular Spectrum Analysis (SSA) is a particular application of the EOF expansion. In classical EOF analysis, the random field <m:math display='block'> <m:mover accent='true'> <m:mi>X</m:mi> <m:mo>&#x2192;</m:mo> </m:mover> </m:math>$$ \vec X $$ to be studied, called also the state vector, contains values measured or estimated at a given time, that is, the coordinates of <m:math display='block'> <m:mover accent='true'> <m:mi>X</m:mi> <m:mo>&#x2192;</m:mo> </m:mover> </m:math>$$ \vec X $$ represent different locations in space at the same time. By diagonalising the covariance matrix of <m:math display='block'> <m:mover accent='true'> <m:mi>X</m:mi> <m:mo>&#x2192;</m:mo> </m:mover> </m:math>$$ \vec X $$, one tries therefore to capture the dominant spatial patterns. The SSA expansion (Vautard et al., 1992) is an EOF expansion, but the state vector <m:math display='block'> <m:mover accent='true'> <m:mi>X</m:mi> <m:mo>&#x2192;</m:mo> </m:mover> </m:math>$$ \vec X $$ now contains values at the same location but at different lags. The leading eigenelements of the corresponding covariance matrix represent thus the leading time patterns of the random field. SSA is a time series analysis, in the sense that a single signal is analysed.