Estimating a smooth function from noisy observations is a core problem in mathematical statistics and supports the areas of nonparametric regression and spatial data analysis. However, there are still gaps in our knowledge of the statistical properties of methods such as smoothing splines and geostatistical estimators (Kriging), and there is also limited understanding of estimators that adapt to heterogeneous structure in the function. In this proposal, the investigators address some of the issues in nonparametric function estimation through a spatial statistics framework. The approach is based on a new model for nonstationary covariance functions that combines a multiresolution (or wavelet) basis with a multivariate lattice model. This multiresolution lattice (MRL) model builds off previous work on lattice models for spatial fields and the use of multiresolution bases for representing nonstationary covariance functions. The key innovation is that the lattice model describes dependence on the coefficients of the basis, not the spatial field. The multivariate extension allows for connections of basis coefficients between different scales, and the localization of the basis functions in space facilitates modeling nonstationary covariance. An important component is the extension of large sample statistical theory to analyze spatial estimators applied to irregular locations and with nonstationary covariance. Ultimately, in addition to methodological and practical advances, this research seeks to break new ground in the theoretical understanding of how nonparametric and spatial smoothers behave, and, in effect, unifying a broad area of mathematical statistics.
The interpretation of spatial observations or fields is a fundamental data analysis problem that is ubiquitous in the geosciences. A specific example is the study of regional climate change where complex numerical models are coupled to simulate climate at local scales. These numerical models are a primary tool to quantify specific impacts of climate change at a scale that can be understood by the general public. The fields produced by these simulations have a great deal of large-scale structure, are noisy, and often exhibit heteroscedastic and nonstationary behavior. Drawing inferences about these fields to provide, for example, a probabilistic assessment of the projected climate change, requires a deliberate statistical approach. The research outlined in this proposal seeks to expand the tools available to analyze geophysical data, in particular the complex outputs of regional climate models.
