Civil Engineering Reference
In-Depth Information
For two random variables‚
and
given by
The covariance of
and
can be computed as
In other words‚ the number of multiplications is linear in the dimension of
the space‚ since orthogonality of the principal components implies that the
products of terms and for need not be considered.
The work in [CS03a] uses these properties to perform SSTA under the gen-
eral spatial correlation model of Figure 6.3. The method assumes that the
fundamental process parameters are in the form of correlated Gaussians‚ so
that the delay‚ given by Equation (6.7) is a weighted sum of Gaussians‚ which
is therefore Gaussian.
As in the work of Berkelaar‚ this method maintains the invariant that‚ all ar-
rival times are approximated as Gaussians‚ although in this case the Gaussians
are correlated and are represented in terms of their principal components. Since
the delays are considered as correlated Gaussians‚ the sum and max operations
that underlie this block-based CPM-like traversal must yield Gaussians in the
form of principal components.
The computation of the distribution of the sum function‚ is
simple. Since this function is a linear combination of normally distributed ran-
dom variables‚
is a normal distribution whose mean‚
and variance‚
are given by
Strictly speaking‚ the max function of normally distributed random vari-
ables‚ is not Gaussian; however‚ like [Ber97]‚ it is
approximated as one. The approximation here is in the form of a correlated
Gaussian‚ and the procedure in Appendix B is employed. The result is char-
acterized in terms of its principal components‚ so that it is enough to find
the mean of the max function and the coeficients associated with the principal
components.
The utility of using principal components is twofold:
As described earlier‚ it implies that covariance calculations between paths
are of linear complexity in the number of variables‚ obviating the need for
the expensive pair-wise delay computation methods used in other methods.
