Embedding Step

Using the time series, form a trajectory matrix, , with elements

such that for and and where . By definition , because .

Decomposition Step

Using the trajectory matrix, , apply singular value decomposition to the trajectory matrix

where represents the matrix that contains the left-hand-side (LHS) eigenvectors, where represents the diagonal matrix that contains the singular values, and where represents the matrix that conatins the right-hand-side (RHS) eigenvectors.

Therefore,

where represents the principal component matrix, represents the left-hand-side (LHS) eigenvector, represents the singular value, and represents the right-hand-side (RHS) eigenvector associated with the th window index.

Grouping Step

For each group index, , define a group of window indices . Let

represent the grouped trajectory matrix for group . If groupings represent a spectral partition,

then according to the singular value decomposition theory,

Averaging Step

For each group index, , compute the diagonal average of ,

where

If the groupings represent a spectral partition, then by definition

Hence, singular spectrum analysis additively decomposes the original time series, , into component series for .

Specifying the Window Length

You can explicitly specify the maximum window length, , using the LENGTH= option or implicitly specify the window length using the INTERVAL= option in the ID statement or the SEASONALITY= option in the PROC TIMESERIES statement.

Either way the window length is reduced based on the accumulated time series length, , to enforce the requirement that .

Specifying the Groups

You can use the GROUPS= option to explicitly specify the composition and number of groups, or use the THRESHOLDPCT= option in the SSA statement to implicitly specify the grouping. The THRESHOLDPCT= option is useful for removing noise or less dominant patterns from the accumulated time series.

Let be the cumulative percent singular value THRESHOLDPCT=. Then the last group, , is determined by the smallest value such that

Using this rule, the last group, , describes the least dominant patterns in the time series and the size of the last group is at least one and is less than the window length, .