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The TIMESERIES Procedure |
Singular Spectrum Analysis |
Given a time series, , for
, and a window length,
, singular spectrum analysis Golyandina, Nekrutkin, and Zhigljavsky (2001) decompose the time series into spectral groupings using the following steps:
Using the time series, form a trajectory matrix,
, with elements
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such that for
and
and where
. By definition
, because
.
Using the trajectory matrix, , apply singular value decomposition to the trajectory matrix
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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,
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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.
For each group index, , define a group of window indices
. Let
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represent the grouped trajectory matrix for group . If groupings represent a spectral partition,
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then according to the singular value decomposition theory,
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For each group index, , compute the diagonal average of
,
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where
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If the groupings represent a spectral partition, then by definition
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Hence, singular spectrum analysis additively decomposes the original time series, , into
component series
for
.
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
.
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
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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,
.
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