A biplot method for multivariate normal populations with unequal covariance matrices

Some previous ideas about non-linear biplots to achieve a joint representation of multivariate normal populations and any parametric function without assumptions about the covariance matrices are extended. Usual restrictions on the covariance matrices (such as homogeneity) are avoided. Variables are represented as curves corresponding to the directions of maximum means variation. To demonstrate the versatility of the method, the representation of variances and covariances as an example of further possible interesting parametric functions have been developed. This method is illustrated with two different data sets, and these results are compared with those obtained using two other distances for the normal multivariate case: the Mahalanobis distance (assuming a common covariance matrix for all populations) and Rao’s distance, assuming a common eigenvector structure for all the covariance matrices.