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.
Flexible signal denoising via flexible empirical Bayes shrinkage
信号去噪-也称为非参数回归-通常通过变换后的收缩估计来执行（例如，小波）域;变换域中的收缩对应于原始域中的平滑。这类应用中的一个关键问题是要收缩多少，或者说要平滑多少。经验贝叶斯收缩方法为这个问题提供了一个有吸引力的解决方案;它们使用数据来估计潜在“效应”的分布，从而自动选择适当的收缩量。然而，经验贝叶斯收缩的大多数现有实现都不如它们所能做到的灵活-无论是在它们对效应的潜在分布的假设上，还是在它们处理异方差的能力上-这限制了它们的信号去噪应用。在这里，我们通过采取一种特别灵活，稳定和计算方便的经验贝叶斯收缩方法来解决这个问题，并将其应用于几个信号去噪问题。这些应用包括泊松数据和异方差高斯数据的平滑。我们通过实证比较表明，与其他方法，包括简单的阈值规则和专门建立的经验贝叶斯程序的结果是有竞争力的。我们的方法是在R包smashr中实现的，“Smoothing by Adaptive Shrinkage in R”，可在https://www.github.com/stephenslab/smashr获得
