  In this paper, we attempt to shed light on a new class of nonstationary
random fields which exhibit, what we call, local invariant nonstationarity. We
argue that the local invariant property has a special interaction with a new
generalized quadratic estimate---also derived here---which extends an estimate
originally developed for gravitational lensing of the Cosmic Microwave
Background in Cosmology \cite{hu2001mapping, hu2002mass}. The nature of this
interaction not only encourages low estimation bias but also enables accurate
(and fast) quantification of Frequentist mean square error quantification of
the estimated nonstationarity. These quadratic estimates are interesting, in
their own right, as they detect and estimate nonstationarity by probing
correlation among Fourier frequencies, the absence of which is the
characterizing feature of weak stationarity (by Bochner's Theorem). Moreover,
this generalized quadratic estimate can be computed with a Fourier
characterization that runs in $\mathcal O(n\log n)$ time when observing the
field on a uniform grid of size $n$ in $\Bbb R^d $. Finally, the work presented
here partially addresses two other problems associated with the statistical
theory of nonstationarity: 1) estimating the phase of a spatially varying
modulated stationary random field and 2) identifying a larger class of
nonstationary random fields which admit an extension of the quadratic estimator
of gravitational lensing that extends the same attractive statistical
properties.
