  In this paper, we develop a method for the simultaneous estimation of
spectral density functions (SDFs) for a collection of stationary time series
that share some common features. Due to the similarities among the SDFs, the
log-SDF can be represented using a common set of basis functions. The basis
shared by the collection of the log-SDFs is estimated as a low-dimensional
manifold of a large space spanned by a pre-specified rich basis. A collective
estimation approach pools information and borrows strength across the SDFs to
achieve better estimation efficiency. Also, each estimated spectral density has
a concise representation using the coefficients of the basis expansion, and
these coefficients can be used for visualization, clustering, and
classification purposes. The Whittle pseudo-maximum likelihood approach is used
to fit the model and an alternating blockwise Newton-type algorithm is
developed for the computation. A web-based shiny App found at
"https://ncsde.shinyapps.io/NCSDE" is developed for visualization, training and
learning the SDFs collectively using the proposed technique. Finally, we apply
our method to cluster similar brain signals recorded by the
electroencephalogram for identifying synchronized brain regions according to
their spectral densities.
