An application of the stationary phase method for estimating probability densities of function derivatives

TL;DR

This paper introduces a novel method using the stationary phase approximation to estimate the probability density functions of derivatives of functions, linking the density to the power spectrum of a complex exponential function.

Contribution

It presents a new theoretical result connecting the density of function derivatives to the power spectrum of a phase-modulated exponential, with proof and experimental validation.

Findings

Density of derivatives approximated by power spectrum as tau approaches zero

Theoretical proof using stationary phase approximation

Experimental evidence supports the approximation

Abstract

We prove a novel result wherein the density function of the gradients---corresponding to density function of the derivatives in one dimension---of a thrice differentiable function S (obtained via a random variable transformation of a uniformly distributed random variable) defined on a closed, bounded interval \Omega \subset R is accurately approximated by the normalized power spectrum of \phi=exp(iS/\tau) as the free parameter \tau-->0. The result is shown using the well known stationary phase approximation and standard integration techniques and requires proper ordering of limits. Experimental results provide anecdotal visual evidence corroborating the result.