Confidence Regions for Spatial Excursion Sets From Repeated Random Field Observations, With an Application to Climate

Abstract

The goal of this paper is to give confidence regions for the excursion set of a spatial function above a given threshold from repeated noisy observations on a fine grid of fixed locations. Given an asymptotically Gaussian estimator of the target function, a pair of data-dependent nested excursion sets are constructed that are sub- and super-sets of the true excursion set, respectively, with a desired confidence. Asymptotic coverage probabilities are determined via a multiplier bootstrap method, not requiring Gaussianity of the original data nor stationarity or smoothness of the limiting Gaussian field. The method is used to determine regions in North America where the mean summer and winter temperatures are expected to increase by mid 21st century by more than 2 degrees Celsius.

Assessing NARCCAP climate model effects using spatial confidence regions

“…We will estimate these effects using ANOVA regression models fit across the spatial domain, then use the CoPE method (Sommerfeld et al, 2015) to make valid simultaneous inference for the effects over the spatial domain. We describe these topics in more details below.…”

“…The functional nature of these effects implies that they vary spatially with location. To account for this, and because it is easily facilitated by the gridded nature of the data, we estimate the effects in each grid cell independently of other grid cells (as previously done by Sommerfeld et al (2015)). Doing so has the important consequence of allowing us, as in Sommerfeld et al (2015)), to avoid making a stationarity assumption about the spatial noise process.…”

“…To account for this, and because it is easily facilitated by the gridded nature of the data, we estimate the effects in each grid cell independently of other grid cells (as previously done by Sommerfeld et al (2015)). Doing so has the important consequence of allowing us, as in Sommerfeld et al (2015)), to avoid making a stationarity assumption about the spatial noise process. Spatial pooling, which would require assuming stationarity, is not actually necessary given that the estimated effects vary fairly smoothly across the spatial domain.…”

“…These approaches vary in complexity and computational efficiency, but all require the random field under consideration to be Gaussian. In contrast, Sommerfeld et al (2015) recently proposed a new approach for addressing this problem based on constructing Co verage P robability E xcursion (CoPE) sets when the data are repeated noisy observations on a fine grid of fixed locations. The confidence sets are constructed using the multiplier bootstrap method, which requires only mild assumptions about the data structure while also making the method very fast to apply.…”

“…Utilizing the recently developed CoPE methodology of Sommerfeld et al (2015), one can objectively assess whether the difference in average temperature effects for different climate models varies across large spatial domains over long periods time. In contrast to assessing these differences pointwise at each location, the CoPE method properly adjusts for the simultaneous inference problem caused by performing many tests across the spatial domain.…”