Signal induced Symmetry Breaking in Noise Statistical Properties of Data Analysis

TL;DR

This paper introduces a novel operator-based method to detect signals in noisy time series by analyzing spectral properties, demonstrating improved sensitivity over traditional techniques.

Contribution

The authors develop a new spectral analysis approach using an Hilbert space operator to distinguish signal from noise regardless of noise type.

Findings

Signal presence causes a detectable discontinuity in the essential spectrum.

Method is effective with various noise types, including Levy noise.

Sensitivity surpasses standard noise detection techniques.

Abstract

From a time series whose data are embedded in heavy noise, we construct an Hilbert space operator (J-operator) whose discrete spectrum represents the signal while the essential spectrum located on the unit circle, is associated with the noise. Furthermore the essential spectrum, in the absence of signal, is built from roots of unity ("clock" distribution). These results are independent of the statistical properties of the noise that can be Gaussian, non-Gaussian, pink or even without second moment (Levy). The presence of the signal has for effect to break the clock angular distribution of the essential spectrum on the unit circle. A discontinuity, proportional to the intensity of the signal, appears in the angular distribution. The sensitivity of this method is definitely better than standard techniques. We build an example that supports our claims.