Extraction of unknown signals in arbitrary noise

Area of Science:

• Signal Processing
• Statistical Analysis
• Data Science

Background:

• Extracting weak signals from noisy data is a significant challenge in many scientific fields.
• Existing methods often rely on assumptions about noise distribution, limiting their applicability.

Purpose of the Study:

• To develop a general method for extracting weak signals of unknown form from noise with arbitrary distributions.
• To establish a robust technique for signal-noise decomposition applicable to single time series.

Main Methods:

• Signal-noise decomposition based on rank and time properties of data.
• Analysis of the joint rank-time probability distribution of stationary white noise.
• Derivation of a distribution-independent covariance matrix for cumulative distributions.
• Utilizing eigenfunctions of the covariance matrix for signal extraction.

Main Results:

• Demonstrated that stationary white noise exhibits a jointly uniform rank-time probability distribution.
• Established a simple relation showing that averaged rank tracks the underlying weak signal.
• Derived an exact analytic, distribution-independent form for the discrete covariance matrix.
• Successfully employed eigenfunctions to extract unknown signals from single time series.

Conclusions:

• The proposed method offers a general and robust approach for weak signal extraction.
• This technique is independent of noise distribution, enhancing its applicability across diverse datasets.
• The rank-time decomposition provides a powerful tool for analyzing complex time series data.