TIMESERIES Procedure

Time Series Transformation

There are four transformations available for strictly positive series only. Let y Subscript t Baseline greater-than 0 be the original time series, and let w Subscript t be the transformed series. The transformations are defined as follows:

Log

is the logarithmic transformation.

w Subscript t Baseline equals normal l normal n left-parenthesis y Subscript t Baseline right-parenthesis
Logistic

is the logistic transformation.

w Subscript t Baseline equals normal l normal n left-parenthesis c y Subscript t Baseline slash left-parenthesis 1 minus c y Subscript t Baseline right-parenthesis right-parenthesis

where the scaling factor c is

c equals left-parenthesis 1 minus 10 Superscript negative 6 Baseline right-parenthesis 10 Superscript minus normal c normal e normal i normal l left-parenthesis log Super Subscript 10 Superscript left-parenthesis normal m normal a normal x left-parenthesis y Super Subscript t Superscript right-parenthesis right-parenthesis right-parenthesis

and normal c normal e normal i normal l left-parenthesis x right-parenthesis is the smallest integer greater than or equal to x.

Square root

is the square root transformation.

w Subscript t Baseline equals StartRoot y Subscript t Baseline EndRoot
Box Cox

is the Box-Cox transformation.

w Subscript t Baseline equals StartLayout Enlarged left-brace 1st Row 1st Column StartFraction y Subscript t Superscript lamda Baseline minus 1 Over lamda EndFraction comma 2nd Column lamda not-equals 0 2nd Row 1st Column normal l normal n left-parenthesis y Subscript t Baseline right-parenthesis comma 2nd Column lamda equals 0 EndLayout

More complex time series transformations can be performed by using the EXPAND procedure of SAS/ETS.

Last updated: June 19, 2025