GPS-X Technical Reference

Optimizer 406
The Portmanteau statistic for response variable j is calculated as the number of
observations times the sum of the squared autocorrelations between the weighted
residuals up to a certain number of time lags (Brockwell and Davis, 1996):
Equation 14.43
where n j is the number of data points for response j, τ is the number of time lags, and ρ j(k)
is the sample autocorrelation among the weighted residuals up to lag k for response j.
The sample autocorrelation is defined as:
Equation 14.44
where Ω j(k) is the sample auto covariance function between the weighted residuals up to
lag k for response j and is defined as:
Equation 14.45
The sample autocorrelations are approximately independent, normally distributed random
variables with mean zero and a variance of 1/n j if the weighted residuals are independent
and identically distributed (Brockwell and Davis, 1996). This approximation gets better
as the number of measurements increases. Although the residuals themselves are not
necessarily identically distributed (they can have different variances), the weighted
residuals should be identically distributed because they are scaled.
GPS-X Technical Reference