An application of copulas to improve PCA biplots for multivariate extremes

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Stellenbosch : Stellenbosch University
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 biplots to 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.
AFRIKAANSE OPSOMMING : Hoofkomponent Analise (HKA) bistippings is ’n nuttige metode ommeer dimensionele data te visualiseer. Die toepassing van HKA bistippings is al goed gedokumenteer oor ’n wye verskeidenheid van navorsingsareas waar meerveranderlike data voorkom, maar die toepassing van bistippings op finansiële data is beperk. Dit is deels te wyte aan HKA wat ‘n onvoldoende metode is van dimensie reduksie van meerveranderlike data wat ekstreme waarnemings bevat. Dit impliseer dat die toepassing daarvan op finansiële data aansienlik beperk is, gegee dat ekstreme waarnemings algemeen voorkom in finansiële data. Die doel van hierdie navorsing is om ’n metode te ontwikkel om HKA- bistippings toe te pas op meerveranderlike data wat ekstreme waarnemings bevat. Dit word gedoen deur ’n elliptiese copula op die data te pas en ‘n korrelasiematriks uit die copula parameters af te lei. Die copula parameters word beraam deur slegs die ekstreme waarnemings te gebruik en dus dui die afgeleide korrelasiematrikse die afhanklikhede van slegs ekstreme waarnemings aan. Laastens, deur HKA op so ’n “ekstreme” korrelasie matriks toe te pas, word die verwantskappe onderliggend aan die ekstreme waardes meer doeltreffend behou en kan ’n meer onderskeidende HKA bistipping gekonstrueer word.
Thesis (MCom)--Stellenbosch University, 2018.
Principal components analysis, Biplots, Copulas (Mathematical statistics), Extreme value theory, Financial data -- Statistical methods, UCTD