Stationary random fields play a fundamental role in both theoretical and applied spatial statistics. Unfortunately, stationarity is often violated when working with real data. This presents a challenge for the spatial statistician who is interested in estimating and modeling dependence structure in random fields. The investigator and his collaborator will study, and subsequently apply, a recently developed version of local likelihood estimation for estimating parameters that govern the local dependency in nonstationary random fields. The application is to use local likelihoods for the estimation of the local distortion of the cosmic microwave background (CMB) due to gravitational lensing. Detecting and estimating this lensing is important for two reasons. First, measuring the distortion gives an indirect measurement of matter in the Universe which can provide detailed maps of the distribution of Dark Matter. Secondly, one can use the estimates of gravitational lensing to obtain a more accurate measurement of the un-lensed CMB. This will provide scientists with a deeper probe of fundamental cosmological questions. Specific aims of this proposal include: 1) develop computational techniques which will make local likelihood estimation applicable to large data sets such as the CMB; 2) develop the theory of estimating equations for local weight construction with a particular focus on mitigating bias and automatically adjusting for boundaries and uneven observation locations; 3) construct estimates of uncertainty in the local likelihood estimates of the parameter function; 4) use the principle irregular term to develop flexible and universal local models for general nonstationary random fields.
Nonstationary random fields are becoming a ubiquitous feature in the recent data deluge and high resolution sensing. Unfortunately, the techniques for modeling, estimating and predicting nonstationary random fields have yet to be fully developed and analyzed. This proposal will make steps towards the goal of developing a complete set of methodological techniques for analyzing nonstationary random fields. Moreover, the tools thus constructed will be useful, not only for Astronomy and Cosmology, but other branches of science and technology as well. This makes the broader impact of the scientific consequences of this proposal two fold. On the one hand, accurately estimating gravitational lensing in the CMB has far reaching consequences for the understanding of cosmic structure and the beginnings of the Universe. On the other hand, the tools from this project are expected to have broad use in other areas of real world application. Recently developed technologies such as fMRI and diffusion tensor imaging are two examples of other scientific areas that will benefit from the methodologies developed for nonstationary random fields.
The fundamental focus of this grant is the research and development of statistical methodology for estimating local information contained in random field observations. It is often the case that the majority of the statistical information in a random field observation is local. Moreover, with the growing importance of non-stationary models and the ubiquity of large data sets, global statistical analyses are often impractical if not impossible. This grant develops two main lines of research, the statistical methodology for the cosmic microwave background in cosmology and the use of warpings or nonlinear distortions for general statistics procedures. These two lines of research probe both local and, comparatively, global statistical procedures. The cosmic microwave background (CMB) is a distant cosmological random field which contains information on the nature of dark matter, dark energy and the cosmological parameters that govern the universe. Much of this information is local in nature. A particularly cogent example is the subtle distortions due to the gravitational influence of intervening dark matter on the CMB. These distortions are local in nature and have a non-stationary effect on the isotropic CMB. In Anderes, Knox and van Engelen (2011) and Anderes, van Engelen (2012) we develop statistical estimates of the local curvature of the gravitational potential on sliding local neighborhoods of the observed CMB and polarization fields. The local analysis allows one to avoid using the typical first order Taylor expansion for the quadratic estimator and avoids the likelihood approximations used in global estimates. We push the range of applicability of local methods to the experimental conditions available today, where the local information is so weak that the accurate construction of gravitational lensing becomes extremely difficult. As a complement to the local analysis we also made contributions, in Anderes and Paul (2012) and Anderes (2013), to a refinement and extension of the state-of-the-art quadratic estimator of weak lensing. We gain further understanding of the statistical properties of the quadric estimator and develop a Bayesian adaptation. This Bayesian adaptation has wider statistical applicability as was shown in Anderes (2012). A second focus of this grant is to push further into the development of warpings as a general statistical procedure. One of the advantages of warping is that it can induce locally varying stretching and squeezing of Euclidean space. In Anderes, Huser et al (2013) we develop a technique to generate non-stationary covariance tapers such that the taper neighborhoods can depend on observation density: larger neighborhoods for sparsely observed areas; smaller neighborhood for densely observed areas. In Anderes and Coram (2012), we develop a computationally tractable way, available to general statistical practitioners, for estimating a warping or deformation of d dimensional Euclidean space which pushes forward an unknown sampling distribution to some known target probability measure. One of the broad impacts of this work is to foster interdisciplinary training of graduate students in both statistics and physics. These students worked on a broad range of projects such as: Bayesian techniques for photometric redshifts; development of a goodness-of-fit test for spectral estimation from the observed CMB; computational techniques for remote sensing detection of cloud height. One of goals of this effort was to energize and educate the both the faculty and the graduate students on a diverse range of topics including; high dimensional data challenges, weak lensing surveys, the CMB, machine learning, etc. The PI organized (with Paul Baines, UC Davis) a day long interdisciplinary workshop/conference focusing on problems at the intersection of statistics, astronomy and cosmology (Oct, 2011). The goal of the conference was to bring together experts from diverse fields to foster collaboration on current statistical challenges in astronomy and cosmology. Another contribution to training and development of practitioners in the field of statistics was the submission of an invited review article for the Encyclopedia of Enviornmetrics 2nd ed which is an educational document giving a overview of Kriging in Spatial Statistics. The tools which have resulted from this project will be useful, not only in statistics and cosmology, but also in other branches of science and technology. Indeed, smooth invertible transformations, or deformations, are fast becoming important tools in modern data analysis. They have been used with spectacular success in the field of computational anatomy where time varying vector field flows, which generate deformations, are used to statistically analyze medical fMRI images and quantify abnormal morphological structure. In cosmology, deformations are used to model gravitational distortions of the CMB from dark matter density fluctuations and have resulted in a deeper understanding of cosmic structure. Even though these important tools are becoming integrated in modern scientific methods, the statistical properties of estimated deformations have been largely unexplored. The research that has been conducted for this grant furthers the development and the theoretical understanding of deformations applied to science and statistics.
