Influence functions in multivariate analysis

In this thesis we derive and apply influence functions for the detection of observations in multivariate analysis which when omitted from, or added to, the data lead to substantial changes in some aspect of our analysis. Emphasis is placed on the influence functions for the eigenvalues and eigenvectors in principal component analysis, from both the covariance and correlation matrices, and correspondence analysis. Also considered are the influence functions for the bivariate, multiple and partial correlation coefficients and the eigenvalues and eigenvectors in canonical correlation analysis. We derive algebraic expressions, in terms of the original analysis, for the theoretical influence function in all cases and it is compared with the sample influence function when this has a 'simple' algebraic form. Only limited sample expressions can be derived for the changes in the eigenvalues and eigenvectors in principal component analysis and correspondence analysis, but the functions are compared numerically when applied to datasets. Problems in assessing the influence on eigenvectors when we have close eigenvalues, due to rotation within a relatively unchanged subspace, are highlighted in both principal component analysis and correspondence analysis and are discussed.