Testing for high frequency features in a noisy signal

TL;DR

This paper introduces a statistical method to detect high frequency features in noisy, nonstationary signals, with applications demonstrated on protein fibril measurements and prion disease data.

Contribution

It proposes a new empirical definition and a Fourier-based estimator for high frequency features, along with a statistical test for their detection in noisy signals.

Findings

The test detects HF features even when noise exceeds signal amplitude by five times.

Application to experimental data confirms the presence of HF features with high confidence.

The method effectively distinguishes transient oscillations from noise in nonstationary signals.

Abstract

Given nonstationary data, one generally wants to extract the trend from the noise by smoothing or filtering. However, it is often important to delineate a third intermediate category, that we call high frequency (HF) features: this is the case in our motivating example, which consists in experimental measurements of the time-dynamics of depolymerising protein fibrils average size. One may intuitively visualise HF features as the presence of fast, possibly nonstationary and transient oscillations, distinct from a slowly-varying trend envelope. The aim of this article is to propose an empirical definition of HF features and construct estimators and statistical tests for their presence accordingly, when the data consists of a noisy nonstationary 1-dimensional signal. We propose a parametric characterization in the Fourier domain of the HF features by defining a maximal amplitude and distance to low frequencies of significant energy. We introduce a data-driven procedure to estimate these parameters, and compute a p-value proxy based on a statistical test for the presence of HF features. The test is first conducted on simulated signals where the ratio amplitude of the HF features to the level of the noise is controlled. The test detects HF features even when the level of noise is five times larger than the amplitude of the oscillations. In a second part, the test is conducted on experimental data from Prion disease experiments and it confirms the presence of HF features in these signals with significant confidence.