  Motivated by dimension reduction in regression analysis and signal detection,
we investigate the order determination for large dimension matrices including
spiked models of which the numbers of covariates are proportional to the sample
sizes for different models. Because the asymptotic behaviour of the estimated
eigenvalues of the corresponding matrices differ completely from those in fixed
dimension scenarios, we then discuss the largest possible number we can
identify and introduce a "valley-cliff" criterion. We propose two versions of
the criterion: one based on the original differences of eigenvalues and the
other based on the transformed differences, which reduces the effects of ridge
selection in the former one. This generic method is very easy to implement and
computationally inexpensive, and it can be applied to various matrices. As
examples, we focus on spiked population models, spiked Fisher matrices and
factor models with auto-covariance matrices. Numerical studies are conducted to
examine the method's finite sample performances and to compare it with existing
methods.
