Two generalizations of the common principal component model

Under the common principal component model the covariance matrices Ψi, of k populations are assumed to have identical eigenvectors, that is, the same orthogonal matrix diagonalizes all Ψi, simultaneously. This paper modifies the common principal component model by assuming that only q out of p eigenvectors are common to all Ψi, while the remaining p–q elgenvectors are specific in each group. This is called a partial common principal component model. A related modification assumes that q eigenvectors of each matrix span the same subspace, a problem that was first considered by Krzanowski (1979). For both modifications this paper derives the normal theory maximum likelihood estimators. It is shown that approximate maximum likelihood estimates can easily be computed if estimates of the ordinary common principal component model are available. The methods are illustrated by numerical examples.