We introduce functional adaptive shrinkage (FASH), an empirical Bayes method for joint analysis of observation units in which each unit estimates an effect function at several values of a continuous condition variable. The ideas in this paper are motivated by dynamic expression quantitative trait locus (eQTL) studies, which aim to characterize how genetic effects on gene expression vary with time or another continuous condition. FASH integrates a broad family of Gaussian processes defined through linear differential operators into an empirical Bayes shrinkage framework, enabling adaptive smoothing and borrowing of information across units. This provides improved estimation of effect functions and principled hypothesis testing, allowing straightforward computation of significance measures such as local false discovery and false sign rates. To encourage conservative inferences, we propose a simple prior- adjustment method that has theoretical guarantees and can be more broadly used with other empirical Bayes methods. We illustrate the benefits of FASH by reanalyzing dynamic eQTL data on cardiomyocyte differentiation from induced pluripotent stem cells. FASH identified novel dynamic eQTLs, revealed diverse temporal effect patterns, and provided improved power compared with the original analysis. More broadly, FASH offers a flexible statistical framework for joint analysis of functional data, with applications extending beyond genomics. To facilitate use of FASH in dynamic eQTL studies and other settings, we provide an accompanying R package at https: //github.com/stephenslab/fashr.

翻译：本文提出功能自适应收缩（FASH）方法，这是一种用于联合分析观测单元的经验贝叶斯方法，其中每个单元在连续条件变量的多个取值处估计效应函数。本研究的动机源于动态表达数量性状基因座（eQTL）研究，该研究旨在表征遗传效应对基因表达的影响如何随时间或其他连续条件变化。FASH将通过线性微分算子定义的广泛高斯过程族整合到经验贝叶斯收缩框架中，实现跨单元的自适应平滑和信息共享。这提供了改进的效应函数估计和原则性假设检验，能够直接计算局部错误发现率和错误符号率等显著性指标。为促进保守推断，我们提出一种具有理论保证的简单先验调整方法，该方法可更广泛地应用于其他经验贝叶斯方法。通过重新分析诱导多能干细胞分化为心肌细胞的动态eQTL数据，我们阐明了FASH的优势。相较于原始分析，FASH识别出新的动态eQTL，揭示了多样化的时间效应模式，并提供了更高的统计功效。更广泛而言，FASH为功能数据的联合分析提供了灵活的统计框架，其应用可延伸至基因组学之外的领域。为促进FASH在动态eQTL研究及其他场景中的应用，我们在https://github.com/stephenslab/fashr提供了配套的R软件包。