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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.

While the problem of estimating a probability density function (pdf) from its observations is classical, the estimation under additional shape constraints is both important and challenging. We introduce an efficient, geometric approach for estimating pdfs given the number of its modes. This approach explores the space of constrained pdf's using an action of the diffeomorphism group that preserves their shapes. It starts with an initial template, with the desired number of modes and arbitrarily chosen heights at the critical points, and transforms it via: (1) composition by diffeomorphisms and (2) normalization to obtain the final density estimate. The search for optimal diffeomorphism is performed under the maximum-likelihood criterion and is accomplished by mapping diffeomorphisms to the tangent space of a Hilbert sphere, a vector space whose elements can be expressed using an orthogonal basis. This framework is first applied to shape-constrained univariate, unconditional pdf estimation and then extended to conditional pdf estimation. We derive asymptotic convergence rates of the estimator and demonstrate this approach using a synthetic dataset involving speed distribution for different traffic flow on Californian driveways., Comment: 31 pages, Initial version. Presented at IISA 2017, Shape Constrained Methods Workshp, BIRS,2018