Practical Principal Components Analysis

Nigel Da Costa Lewis

Principal components analysis (PCA) is a method of transforming a given set of risk factor variables into a new set of composite variables. These new variables are uncorrelated to each other and account for the entire variance in the original data. In risk modelling practice, it is often found much fewer principal components than the number of original variables are sufficient to summarise the variability in the data. The objective of PCA is to take p random variables, X₁, X_{2, …,} X_p, on which we have N measurements, and find linear combinations of these to produce a new set of variables, Z₁, Z_2, …, Z_p, that are uncorrelated with each other – ie, correlation (Z_i, Z_j) = 0∀i ≠ j. The lack of correlation is convenient for market risk modelling because it implies each of the are measuring different aspects of the data. The Z_i s are known as the principal components. The principal components are ordered so that variance (Z₁) > variance (Z₂) > … > variance (Z_p). Therefore, Z₁ explains the largest amout of variation, followed by Z₂and so on. It is hoped that, by carrying out PCA on financial risk factors, the variances on most of the principal components are negligible. If so, the