Estimating signals underlying noisy data is a significant problem in statistics and engineering. Numerous estimators are available in the literature, depending on the observation model and estimation criterion. This paper introduces a framework that estimates the shape of the unknown signal and the signal itself. The approach utilizes a peak-persistence diagram (PPD), a novel tool that explores the dominant peaks in the potential solutions and estimates the function's shape, which includes the number of internal peaks and valleys. It then imposes this shape constraint on the search space and estimates the signal from partially-aligned data. This approach balances two previous solutions: averaging without alignment and averaging with complete elastic alignment. From a statistical viewpoint, it achieves an optimal estimator under a model with both additive noise and phase or warping noise. We also present a computationally-efficient procedure for implementing this solution and demonstrate its effectiveness on several simulated and real examples. Notably, this geometric approach outperforms the current state-of-the-art in the field.

翻译：估计隐藏在噪声数据中的信号是统计学和工程领域的重要问题。针对不同的观测模型和估计准则，文献中存在多种估计器。本文提出了一种框架，用于同时估计未知信号的形状及其本身。该方法利用峰值-持久性图（PPD）这一新型工具，探索潜在解中的主导峰，并估计函数形状（包括内部峰谷数量）。随后，该约束被施加于搜索空间，并从部分对齐的数据中估计信号。该方法平衡了两种既有方案：无对齐平均和完全弹性对齐平均。从统计角度而言，它能在同时包含加性噪声和相位扭曲噪声的模型下实现最优估计。我们还提出了一种高效计算方案来实现该求解过程，并通过模拟和真实数据示例验证其有效性。值得注意的是，这种几何方法优于当前领域内最先进技术。