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Abstract Stationary points embedded in the derivatives are often critical for a model to be interpretable and may be considered as key features of interest in many applications. We propose a semiparametric Bayesian model to efficiently infer the locations of stationary points of a nonparametric function, which also produces an estimate of the function. We use Gaussian processes as a flexible prior for the underlying function and impose derivative constraints to control the function's shape via conditioning. We develop an inferential strategy that intentionally restricts estimation to the case of at least one stationary point, bypassing possible mis-specifications in the number of stationary points and avoiding the varying dimension problem that often brings in computational complexity. We illustrate the proposed methods using simulations and then apply the method to the estimation of event-related potentials derived from electroencephalography (EEG) signals. We show how the proposed method automatically identifies characteristic components and their latencies at the individual level, which avoids the excessive averaging across subjects that is routinely done in the field to obtain smooth curves. By applying this approach to EEG data collected from younger and older adults during a speech perception task, we are able to demonstrate how the time course of speech perception processes changes with age.
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Peak Persistence Diagrams for Signal Estimation under Additive and Warping Noise
Addressing the fundamental challenge of signal estimation from noisy data is a crucial aspect of signal processing and data analysis. Existing literature offers various estimators based on distinct observation models and criteria for estimation. This paper introduces an innovative framework that leverages topological and geometric features of the data for signal estimation. The proposed approach introduces a topological tool -- {\it peak-persistence diagram} (PPD) -- to analyze prominent peaks within potential solutions. Initially, the PPD estimates the unknown shape, incorporating details such as the number of internal peaks and valleys. Subsequently, a shape-constrained optimization strategy is employed to estimate the signal. This approach strikes a balance between two prior approaches: signal averaging without alignment and signal averaging with complete elastic alignment. Importantly, the proposed method provides an estimator within a statistical model where the signal is affected by both additive and warping noise. A computationally efficient procedure for implementing this solution is presented, and its effectiveness is demonstrated through simulations and real-world examples, including applications to COVID rate curves and household electricity consumption curves. The results showcase superior performance of the proposed approach compared to several current state-of-the-art techniques.
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- Award ID(s):
- 1953087
- PAR ID:
- 10557061
- Publisher / Repository:
- arXiv
- Date Published:
- Format(s):
- Medium: X
- Institution:
- Florida State University
- Sponsoring Org:
- National Science Foundation
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