Principal component analysis (PCA) is a ubiquitous method of multivariate statistics that focuses on the eigenvalues $\lambda$ and eigenvectors of the sample covariance matrix of a data set. We consider p, N-dimensional data vectors $\mathit{\xi }$ drawn from a distribution with covariance matrix $\mathbf{C}.$ We use the replica method to evaluate the expected eigenvalue distribution $\rho \left(\lambda \right)$ as $\overrightarrow{N}\infty$ with $p=\alpha N$ for some fixed $\alpha .$ In contrast to existing studies we consider the case where $\mathbf{C}$ contains a number of symmetry-breaking directions, so that the sample data set contains some definite structure. Explicitly we set $\mathbf{C}={\sigma }^2\mathbf{I}+{\sigma }^2{\sum }_{m=1\mathrm{}}^S{A}_m{\mathbf{B}}_m{\mathbf{B}}_m^T,$ with $A_m>0\mathrm{}\forall m.\mathrm{}$ We find that the bulk of the eigenvalues are distributed as for the case when the elements of $\mathit{\xi }$ are independent and identically distributed. With increasing $\alpha$ a series of phase transitions are observed, at $\alpha {=A}_m^{-2},m=1,2,\dots ,S,$ each time a single $\delta$ function, $\delta \left(\lambda -{\lambda }_u{(A}_m\right)),$ separates from the upper edge of the bulk distribution, where ${\lambda }_u\mathrm{}\left(A\right)={\sigma }^2[1+A][1+(\alpha {A)}^{-1}].$ We confirm the results of the replica analysis by studying the Stieltjes transform of $\rho \left(\lambda \right).\mathrm{}$ This suggests that the results obtained from the replica analysis are universal, irrespective of the distribution from which $\mathit{\xi }$ is drawn, provided the fourth moment of each element of $\mathit{\xi }$ exists.

• Received 2 April 2003

DOI:https://doi.org/10.1103/PhysRevE.69.026124