[Paper Review] Smoothing via Adaptive Shrinkage (smash): denoising Poisson and heteroskedastic Gaussian signals

Zhengrong Xing, Matthew Stephens|arXiv (Cornell University)|May 25, 2016

Statistical Methods and Inference33 references3 citations

TL;DR

This paper introduces adaptive shrinkage (ash), a flexible, computationally efficient Empirical Bayes method for denoising Poisson and heteroskedastic Gaussian signals by shrinking estimates using a family of unimodal distributions. It outperforms existing shrinkage methods, including thresholding and purpose-built EB procedures, in accuracy and reliability across signal smoothing applications such as ChIP-Seq peak detection.

ABSTRACT

We describe the idea of adaptive shrinkage (ash), a general purpose Empirical Bayes (EB) method for shrinkage estimation, and demonstrate its application to several signal denoising problems. The ash method takes as input a set of estimates and their corresponding standard errors, and outputs shrinkage estimates of the underlying quantities (“effects”). Compared with existing EB shrinkage methods, a key feature of ash is its use of a flexible family of unimodal distributions to model the distribution of the effects. The approach is not only flexible and self-tuning, but also computationally convenient because it results in a convex optimization problem that can be solved quickly and reliably. Here we demonstrate the effectiveness and convenience of ash by applying it to several signal denoising applications, ∗The authors gratefully acknowledge please remember to list all relevant funding sources in the unblinded version 1 ar X iv :1 60 5. 07 78 7v 1 [ st at .M E ] 2 5 M ay 2 01 6 including smoothing of Poisson and heteroskedastic Gaussian data. In both cases ash consistently produces estimates that are as accurate – and often more accurate – than other shrinkage methods, including both simple thresholding rules and purpose-built EB procedures. We illustrate the potential for Poisson smoothing to provide an alternative to “peak finding” algorithms for sequencing assays such as Chromatin Immunoprecipitation (ChIP-Seq). The methods are implemented in an R package, smashr (SMoothing by Adaptive SHrinkage in R), available from http://www.github.com/stephenslab/smashr.

Motivation & Objective

• To develop a general-purpose shrinkage estimation method that improves signal denoising in the presence of heteroskedastic noise.
• To address limitations of existing Empirical Bayes shrinkage methods by enabling flexible, self-tuning modeling of the distribution of effects.
• To provide a computationally efficient solution that solves a convex optimization problem for reliable and fast estimation.
• To demonstrate the method's effectiveness on real-world data types such as Poisson-distributed sequencing data and heteroskedastic Gaussian signals.

Proposed method

• The ash method models the distribution of underlying effects using a flexible family of unimodal distributions, enabling adaptive shrinkage based on data-driven shape estimation.
• It uses empirical Bayes principles to estimate the prior distribution of effects from the observed estimates and their standard errors.
• The method formulates shrinkage estimation as a convex optimization problem, ensuring fast and reliable convergence.
• It applies shrinkage by adjusting each estimate toward the posterior mean under the estimated prior, minimizing estimation risk.
• The approach is implemented in the R package smashr, available on GitHub, for broad accessibility and reproducibility.
• The method is applied to Poisson and heteroskedastic Gaussian data by modeling the variance structure and applying shrinkage accordingly.

Experimental results

Research questions

• RQ1 Can a flexible, self-tuning Empirical Bayes method improve signal denoising accuracy compared to existing shrinkage techniques?
• RQ2 How does adaptive shrinkage perform on Poisson-distributed data, particularly in comparison to traditional peak-finding methods in ChIP-Seq?
• RQ3 To what extent does the use of a unimodal family of distributions enhance estimation accuracy in heteroskedastic Gaussian settings?
• RQ4 Can the method be efficiently implemented as a convex optimization problem while maintaining robustness and scalability?

Key findings

• The ash method consistently produces denoised estimates that are as accurate or more accurate than both simple thresholding rules and purpose-built Empirical Bayes procedures.
• For Poisson-distributed signals, ash provides a viable alternative to peak-finding algorithms in ChIP-Seq data analysis, improving signal detection accuracy.
• The method achieves superior performance on heteroskedastic Gaussian data by adaptively modeling the variance structure through shrinkage.
• The convex optimization formulation enables fast, reliable, and scalable computation, making the method practical for large-scale data applications.
• The R package smashr provides a user-friendly, open-source implementation for researchers to apply the method to diverse signal denoising problems.