TABLE OF CONTENTS In this chapter, we first consider various methods for dealing with a matrix of stressed correlations. We start with scenario analysis to define target stressed correlations, motivated by data (see also Ch. 37). We introduce the concept of the average correlation stress. Naturally, these are target correlation stresses for which the correlation matrix will not be positive definite. The technique of finding an optimal positive-definite approximation to a non-positive-definite target correlation matrix is treated in the next chapter.
We then show how to generate random correlation matrices using two techniques. The first method is a direct application of historical data. Historical correlation matrices are in principle positive definite, and in practice are close to positive definite. The second method uses historical data to construct a model for stochastic random correlation matrices. The model contains Gaussian multivariate techniques on the space of angles of the 
-sphere that is equivalent to the correlation matrices. This second method always yields a positive definite correlation matrix.
In this section, we look at data for correlations to get an idea of the variability of the correlations ( ? ??) in practice over many variables. We need this in order to perform the stressed correlation matrix analysis. We need to get a target stressed matrix ( ? ??) (Target) that has the property that individual matrix elements are stressed using information from historical data. We will then find the best-fit positive-definite correlation matrix ? ?? (BestFit,PosDef)
