Stationary Point Constrained Inference via Diffeomorphisms

TL;DR

This paper introduces a new framework for joint inference of multiple stationary points in regression functions, ensuring valid uncertainty quantification and interpretability by constraining the number of extrema using a diffeomorphic transformation.

Contribution

It develops a novel diffeomorphic parameterization that guarantees a fixed number of stationary points, enabling coherent joint inference and interpretability in functions with multiple extrema.

Findings

Non-asymptotic confidence bounds derived for stationary points

Approximate normality established for maximum likelihood estimators

Method demonstrated effective in brain signal analysis

Abstract

Stationary points or derivative zero crossings of a regression function correspond to points where a trend reverses, making their estimation scientifically important. Existing approaches to uncertainty quantification for stationary points cannot deliver valid joint inference when multiple extrema are present, an essential capability in applications where the relative locations of peaks and troughs carry scientific significance. We develop a principled framework for functions with multiple regions of monotonicity by constraining the number of stationary points. We represent each function in the diffeomorphic formulation as the composition of a simple template and a smooth bijective transformation, and show that this parameterization enables coherent joint inference on the extrema. This construction guarantees a prespecified number of stationary points and provides a direct, interpretable parameterization of their locations. We derive non-asymptotic confidence bounds and establish approximate normality for the maximum likelihood estimators, with parallel results in the Bayesian setting. Simulations and an application to brain signal estimation demonstrate the method's accuracy and interpretability.