Ethan Anderes

Estimating deformations of isotropic Gaussian random fields on the plane

This paper presents a new approach to the estimation of the deformation of an isotropic Gaussian random field on R 2 based on dense observations of a single realization of the deformed random field. Under this framework we investigate the identification and estimation of deformations. We then present a complete methodological package-from model assumptions to algorithmic recovery of the deformation-for the class of nonstationary processes obtained by deforming isotropic Gaussian random fields.

On the consistent separation of scale and variance for Gaussian random fields

We present fixed domain asymptotic results that establish consistent estimates of the variance and scale parameters for a Gaussian random field with a geometric anisotropic Matern autocovariance in dimension d > 4. When d 4, we show that one can estimate the coefficient on the principle irregular term accurately enough to get a consistent estimate of the coefficient on the second irregular term. These two coefficients can then be used to separate the scale and variance. We extend our results to the general problem of estimating a variance and geometric anisotropy for more general autocovariance functions. Our results illustrate the interaction between the accuracy of estimation, the smoothness of the random field, the dimension of the observation space and the number of increments used for estimation. As a corollary, our results establish the orthogonality of Matern Gaussian random fields with different parameters when d > 4. The case d = 4 is still open.

Local likelihood estimation for nonstationary random fields

We develop a weighted local likelihood estimate for the parameters that govern the local spatial dependency of a locally stationary random field. The advantage of this local likelihood estimate is that it smoothly downweights the influence of faraway observations, works for irregular sampling locations, and when designed appropriately, can trade bias and variance for reducing estimation error. This paper starts with an exposition of our technique on the problem of estimating an unknown positive function when multiplied by a stationary random field. This example gives concrete evidence of the benefits of our local likelihood as compared to unweighted local likelihoods. We then discuss the difficult problem of estimating a bandwidth parameter that controls the amount of influence from distant observations. Finally we present a simulation experiment for estimating the local smoothness of a local Matern random field when observing the field at random sampling locations in [0,1]^2. The local Matern is a fully nonstationary random field, has a closed form covariance, can attain any degree of differentiability or Holder smoothness and behaves locally like a stationary Matern. We include an appendix that proves the positive definiteness of this covariance function.

Consistent estimates of deformed isotropic Gaussian random fields on the plane

This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation f: R 2 -R 2 when observing the deformed random field Z o f on a dense grid in a bounded, simply connected domain Ω, where Z is assumed to be an isotropic Gaussian random field on R 2 is constructed on a simply connected domain U, such that Ū Ω and is defined using kernel smoothed quadratic variations, Bergman projections and results from quasiconformal theory. We show, under mild assumptions on the random field Z and the deformation f, that R θ f + c uniformly on compact subsets of U with probability one as the grid spacing goes to zero, where R θ is an unidentifiable rotation and c is an unidentifiable translation.

Discrete Wasserstein barycenters: optimal transport for discrete data

Wasserstein barycenters correspond to optimal solutions of transportation problems for several marginals, and as such have a wide range of applications ranging from economics to statistics and computer science. When the marginal probability measures are absolutely continuous (or vanish on small sets) the theory of Wasserstein barycenters is well-developed [see the seminal paper (Agueh and Carlier in SIAM J Math Anal 43(2):904–924, 2011)]. However, exact continuous computation of Wasserstein barycenters in this setting is tractable in only a small number of specialized cases. Moreover, in many applications data is given as a set of probability measures with finite support. In this paper, we develop theoretical results for Wasserstein barycenters in this discrete setting. Our results rely heavily on polyhedral theory which is possible due to the discrete structure of the marginals. The results closely mirror those in the continuous case with a few exceptions. In this discrete setting we establish that Wasserstein barycenters must also be discrete measures and there is always a barycenter which is provably sparse. Moreover, for each Wasserstein barycenter there exists a non-mass-splitting optimal transport to each of the discrete marginals. Such non-mass-splitting transports do not generally exist between two discrete measures unless special mass balance conditions hold. This makes Wasserstein barycenters in this discrete setting special in this regard. We illustrate the results of our discrete barycenter theory with a proof-of-concept computation for a hypothetical transportation problem with multiple marginals: distributing a fixed set of goods when the demand can take on different distributional shapes characterized by the discrete marginal distributions. A Wasserstein barycenter, in this case, represents an optimal distribution of inventory facilities which minimize the squared distance/transportation cost totaled over all demands.

Decomposing CMB lensing power with simulation

The reconstruction of the CMB lensing potential is based on a Taylor expansion of lensing effects which is known to have poor convergence properties. For lensing of temperature fluctuations, an understanding of the higher order terms in this expansion which is accurate enough for current experimental sensitivity levels has been developed in Hanson et. al. (2010), as well as a slightly modified Okamoto and Hu quadratic estimator which incorporates lensed rather than unlensed spectra into the estimator weights to mitigate the effect of higher order terms. We extend these results in several ways: (1) We generalize this analysis to the full set of quadratic temperature/polarization lensing estimators, (2) We study the effect of higher order terms for more futuristic experimental noise levels, (3) We show that the ability of the modified quadratic estimator to mitigate the effect of higher order terms relies on a delicate cancellation which occurs only when the true lensed spectra are known. We investigate the sensitivity of this cancellation to uncertainties in or knowledge of these spectra. We find that higher order terms in the Taylor expansion can impact projected error bars at experimental sensitivities similar to those found in future ACTpol/SPTpol experiments.

Nonstationary positive definite tapering on the plane

A common problem in spatial statistics is to predict a random field f at some spatial location t 0 using observations f(t 1), …, f(tn ) at . Recent work by Kaufman et al. and Furrer et al. studies the use of tapering for reducing the computational burden associated with likelihood-based estimation and prediction in large spatial datasets. Unfortunately, highly irregular observation locations can present problems for stationary tapers. In particular, there can exist local neighborhoods with too few observations for sufficient accuracy, while others have too many for computational tractability. In this article, we show how to generate nonstationary covariance tapers T(s, t) such that the number of observations in {t: T(s, t) > 0} is approximately a constant function of s. This ensures that tapering neighborhoods do not have too many points to cause computational problems but simultaneously have enough local points for accurate prediction. We focus specifically on tapering in two dimensions where quasi-conformal theory can be used. Supplementary materials for the article are available online.

Mapping gravitational lensing of the CMB using local likelihoods

We present a new estimation method for mapping the gravitational lensing potential from observed CMB intensity and polarization fields. Our method uses Bayesian techniques to estimate the average curvature of the potential over small local regions. These local curvatures are then used to construct an estimate of a low pass filter of the gravitational potential. By utilizing Bayesian/likelihood methods one can easily overcome problems with missing and/or nonuniform pixels and problems with partial sky observations (E- and B-mode mixing, for example). Moreover, our methods are local in nature, which allow us to easily model spatially varying beams, and are highly parallelizable. We note that our estimates do not rely on the typical Taylor approximation which is used to construct estimates of the gravitational potential by Fourier coupling. We present our methodology with a flat sky simulation under nearly ideal experimental conditions with a noise level of 1 {mu}K-arcmin for the temperature field, {radical}(2) {mu}K-arcmin for the polarization fields, with an instrumental beam full width at half maximum (FWHM) of 0.25 arcmin.

Robust adaptive Wiener filtering

A recent paper by Anderes and Paul [1] analyze a regression characterization of a new estimator of lensing from cosmic microwave observations, developed by Hu and Okamoto [2, 3, 4]. A key tool used in that paper is the application of the robust generalized shrinkage priors developed 30 years ago in [5, 6, 7] to the problem of adaptive Wiener filtering. The technique requires the user to propose a fiducial model for the spectral density of the unknown signal but the resulting estimator is developed to be robust to misspecification of this model. The role of the fiducial spectral density is to give the estimator superior statistical performance in a “neighborhood of the fiducial model” while controlling the statistical errors when the fiducial spectral density is drastically wrong. One of the main advantages of this adaptive Wiener filter is that one can easily obtain posterior samples of the true signal given the unknown data. These posterior samples are particularly advantageous when studying non-linear functions of the signal, cross correlating with other independent measurements of the same signal and can be used to propagate uncertainty when the filtering is done in a scientific pipeline. In this paper we explore these advantages with simulations and examine the possibility of widespread application in more general image and signal processing problems.