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Statistical modeling and interpolation of space–time processes has gained increasing relevance over the last few years. However, real world data often exhibit characteristics that challenge conventional methods such as nonstationarity and temporal misalignment. For example, high frequency solar irradiance data are typically observed at fine temporal scales, but at sparse spatial sampling, so space–time interpolation is necessary to support solar energy studies. The nonstationarity and phase misalignment of such data challenges extant approaches. We propose random elastic space–time (REST) prediction, a novel method that addresses temporally-varying phase misalignment by combining elastic alignment and conventional kriging techniques. Moreover, uncertainty in both amplitude and phase alignment can be readily quantified in a conditional simulation framework, whereas conventional space–time methods only address am- plitude uncertainty. We illustrate our approach on a challenging solar irradiance dataset, where our method demonstrates superior predictive distributions compared to existing geostatistical and functional data analytic techniques.