36 research outputs found
Partial Coherence Estimation via Spectral Matrix Shrinkage under Quadratic Loss
Partial coherence is an important quantity derived from spectral or precision
matrices and is used in seismology, meteorology, oceanography, neuroscience and
elsewhere. If the number of complex degrees of freedom only slightly exceeds
the dimension of the multivariate stationary time series, spectral matrices are
poorly conditioned and shrinkage techniques suggest themselves. When true
partial coherencies are quite large then for shrinkage estimators of the
diagonal weighting kind it is shown empirically that the minimization of risk
using quadratic loss (QL) leads to oracle partial coherence estimators superior
to those derived by minimizing risk using Hilbert-Schmidt (HS) loss. When true
partial coherencies are small the methods behave similarly. We derive two new
QL estimators for spectral matrices, and new QL and HS estimators for precision
matrices. In addition for the full estimation (non-oracle) case where certain
trace expressions must also be estimated, we examine the behaviour of three
different QL estimators, the precision matrix one seeming particularly robust
and reliable. For the empirical study we carry out exact simulations derived
from real EEG data for two individuals, one having large, and the other small,
partial coherencies. This ensures our study covers cases of real-world
relevance
Multitaper estimation on arbitrary domains
Multitaper estimators have enjoyed significant success in estimating spectral
densities from finite samples using as tapers Slepian functions defined on the
acquisition domain. Unfortunately, the numerical calculation of these Slepian
tapers is only tractable for certain symmetric domains, such as rectangles or
disks. In addition, no performance bounds are currently available for the mean
squared error of the spectral density estimate. This situation is inadequate
for applications such as cryo-electron microscopy, where noise models must be
estimated from irregular domains with small sample sizes. We show that the
multitaper estimator only depends on the linear space spanned by the tapers. As
a result, Slepian tapers may be replaced by proxy tapers spanning the same
subspace (validating the common practice of using partially converged solutions
to the Slepian eigenproblem as tapers). These proxies may consequently be
calculated using standard numerical algorithms for block diagonalization. We
also prove a set of performance bounds for multitaper estimators on arbitrary
domains. The method is demonstrated on synthetic and experimental datasets from
cryo-electron microscopy, where it reduces mean squared error by a factor of
two or more compared to traditional methods.Comment: 28 pages, 11 figure