A generalized quadratic estimate for random field nonstationarity

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.