Appendix I

Transforming Random Variables from Correlated to Uncorrelated

Assume that *μ*_{x}*, σ*_{x}, and *C*_{x} represent a vector of mean values, a vector of standard deviations, and covariance matrix respectively, of a set of dependent random variables *x*, denoted as *x*_{1}, *x*_{2}, …, *x*_{n}. It is desired to obtain *μ*_{v}*, σ*_{v}, and *C*_{v}, where the new *C*_{v} is a diagonal matrix, i.e. off-diagonal elements are all zero.

In terms of two variables, *μ*_{x}, *σ*_{x}, and *C*_{x} will be:

where *ρ* is the correlation coefficient, defined as , and .

It is required to get

To obtain the uncorrelated ...

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