Climate models have become the primary tools for scientists to project climate-change into the future and to understand its potential impact. Continental-scale General Circulation Models (GCMs) oversimplify the regional climate processes and geophysical features such as topography and land cover. The consequences of local/regional climate change are particularly relevant to natural resource management and environmental-policy decisions, for which Regional Climate Models (RCMs) have been developed. RCMs simulate, for example, three-hourly "weather" over long time periods, from which long-run averages (e.g., over 30 years) are commonly computed to estimate a region's future climate. With anthropogenic forcings incorporated, RCMs provide a means to assess a combination of natural and anthropogenic influences on climate variability. The North American Regional Climate Change Assessment Program ran RCMs into the future, until 2070, for 11,760 contiguous regions, each of which is approximately (Formula presented.) in area. Using the 94,080 temperature changes projected to 2070 for all regions, for two RCMs, and for the four seasons, we present both an exploratory and a Bayesian inferential spatial analysis. Climate-model output is deterministic, but we capture its spatial variability using a hierarchy of conditional probability models. The exploratory Spatial Proportion Over Threshold (SPOT) function and the inferential PRedictive probability Over Threshold (PROT) function are defined and contrasted through videos available online in the Supplementary Materials, showing regions of North America that attain or exceed temperature change thresholds as a function of increasing threshold. The preponderance of our results throughout all regions of North America is one of warming by 2070, usually more (and sometimes much more) than (Formula presented.).