贝叶斯小波去噪的分辨率自适应紧支撑先验

中文摘要 AI 辅助

我们针对含噪一维信号提出了一种分辨率自适应的贝叶斯小波去噪方法。该模型采用尖峰-板条先验，其连续板条部分是紧支撑Wendland型多项式核与半圆密度的混合，具有数据自适应的分辨率特定截断尺度。Wendland分量将质量集中在零点附近并在支撑边界处平滑消失，而半圆分量则更为分散，使得收缩规则能够在不同分辨率水平上调整其行为。在平方误差损失下，我们推导了后验均值估计量，建立了关键的对称性、有界性、连续性和极限性质，定义了逐点固定超参数的偏差、方差和风险，并开发了经验贝叶斯估计程序。在拉普拉斯工作似然下，Wendland贡献具有有限和表达式，而半圆贡献则通过稳定的一维积分进行评估。使用Bumps、Blocks、Doppler和HeaviSine信号进行的模拟将提出的高斯似然版本和拉普拉斯似然版本与通用阈值法、错误发现率（FDR）阈值法、交叉验证（CV）、斯坦无偏风险估计（SURE）、贝叶斯自适应多分辨率收缩器（BAMS）以及基于非局部先验（NLP）的方法进行了比较。在主要的高斯误差模拟研究中，高斯似然版本在36个设计单元中的26个单元（包括所有低信噪比（SNR）单元）中是性能最优的非NLP方法，且其计算特性比拉普拉斯似然版本显著更优。对2008年奇诺山地震的地震加速度迹线的分析表明，在所选诊断方法下，该方法能够衰减快速波动并保留主导加速度事件。

英文摘要

We propose a resolution-adaptive Bayesian wavelet denoising method for noisy one-dimensional signals. Its central innovation is a spike-and-slab prior whose continuous slab mixes a compactly supported Wendland-type polynomial density with the more dispersed semicircle density. A low-dimensional empirical-Bayes trend produces data-adaptive mixture weights by resolution, while a data-adaptive support scale controls the common bounded interval. Thus, the method combines sparsity, explicit support control, and interpretable resolution-dependent shrinkage. Under squared-error loss, we derive the posterior-mean estimator and establish symmetry, boundedness, continuity, and limiting properties. We define fixed-hyperparameter bias, variance, and risk and develop an empirical-Bayes fitting procedure. Under a Laplace working likelihood, the Wendland contribution has finite-sum expressions, while the semicircle contribution is evaluated by stable one-dimensional integration. Simulations with the Bumps, Blocks, Doppler, and HeaviSine signals compare Gaussian- and Laplace-likelihood versions with universal thresholding, false-discovery-rate (FDR) thresholding, cross-validation (CV), Stein's unbiased risk estimate (SURE), the Bayesian adaptive multiresolution shrinker (BAMS), and a nonlocal-prior (NLP) method. In the primary Gaussian-error study, WS--Gaussian was the best non-NLP method in 24 of 36 cells, including 11 of 12 low-SNR cells, with a much more favorable computational profile than WS--Laplace. A real seismic acceleration record from the 2008 Chino Hills earthquake illustrates attenuation of rapid fluctuations and preservation of the dominant event. A semi-synthetic study using the processed trace as surrogate truth showed improvement over the noisy observation at lower and moderate SNRs, but not at the highest SNR.

发表机构

• The University of Texas at El Paso（德克萨斯大学埃尔帕索分校）

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