Independent statistical observables for ultrametric disordered populations

It is not exceptional that a sample of N random data X_i,i=1,…,N contains ultrametric covariations, namely the matrix C with matrix elements 〈X_iX_j〉-〈X_i〉〈X_j〉 is ultrametric. We define independent (decorrelated) “collective” observables by diagonalizing this matrix. Symmetry properties of such eigenvectors are discussed. Often also, however, while the existence of an ultrametric tree is known, the degrees of parentage of the data are unknown, because a random perturbation confuses the labeling of the leaves of the tree. We sort out those observables which are more robust with respect to such labeling mistakes.