Refining low-resolution (LR) spatial fields with high-resolution (HR) information, often known as statistical downscaling, is challenging as the diversity of spatial datasets often prevents direct matching of observations. Yet, when LR samples are modeled as aggregate conditional means of HR samples with respect to a mediating variable that is globally observed, the recovery of the underlying fine-grained field can be framed as taking an "inverse" of the conditional expectation, namely a deconditioning problem. In this work, we propose a Bayesian formulation of deconditioning which naturally recovers the initial reproducing kernel Hilbert space formulation from Hsu and Ramos (2019). We extend deconditioning to a downscaling setup and devise efficient conditional mean embedding estimator for multiresolution data. By treating conditional expectations as inter-domain features of the underlying field, a posterior for the latent field can be established as a solution to the deconditioning problem. Furthermore, we show that this solution can be viewed as a two-staged vector-valued kernel ridge regressor and show that it has a minimax optimal convergence rate under mild assumptions. Lastly, we demonstrate its proficiency in a synthetic and a real-world atmospheric field downscaling problem, showing substantial improvements over existing methods.
翻译:以高分辨率(HR)信息重新界定低分辨率(LR)空间域,通常称为统计缩放,这具有挑战性,因为空间数据集的多样性往往阻碍直接匹配观测。然而,当LR样本以全球观测的介质变量为模型,作为总有条件的HR样本样本,模拟全球观测的介质变量时,可将潜在的细微微粒场的恢复归类为“反”有条件期望,即调制问题。在这项工作中,我们提议一种巴耶斯式的调制配方,自然恢复Hsu和Ramos(2019年)的最初再生产内核Hilbert空间配方(Hilbert)的初产(Hilbert)空间配方。我们将调制扩大到降尺度的设置,并设计出高效的有条件嵌入多分辨率数据的手段。通过将有条件的预期作为基础字段的内在特征,可将潜在场的外表设定为“反”有条件的预期,即调制问题。此外,我们表明,这一解决方案可以被视为一种分层矢量级的矢量内脊向后退器,并表明,在温化的场中展示了一种真实的高度的合成方法。