An application of copulas to improve PCA biplots for multivariate extremes

[摘要] ENGLISH SUMMARY : Principal Component Analysis (PCA) biplots is a valuable means of visualising high dimensional data. The application of PCA biplots over a wide variety of research areas containing multivariate data is well documented. However, the application of biplotsto financial data is limited. This is partly due to PCA being an inadequate means of dimension reduction for multivariate data that is subject to extremes. This implies that its application to financial data is greatly diminished since extreme observations are common in financial data. Hence, the purpose of this research is to develop a method to accommodate PCA biplots for multivariate data containing extreme observations.This is achieved by fitting an elliptical copula to the data and deriving a correlation matrix from the copula parameters. The copula parameters are estimated from only extreme observations and as such the derived correlationmatrices contain the dependencies of extreme observations. Finally, applying PCA to such an 'extremal correlation matrix more efficiently preserves the relationships underlying the extremes and a more refined PCA biplot can be constructed.

[发布机构] Stellenbosch University