A general statistical test for correlations in a finite-length time series

Area of Science:

• Statistics
• Time Series Analysis
• Signal Processing

Background:

• Autocorrelation functions (ACF) are crucial for analyzing time series data.
• Understanding the statistical properties and uncertainties of ACF calculations is essential.
• Existing methods for ACF calculation have varying uncertainty characteristics.

Purpose of the Study:

• To analyze the statistical properties of the autocorrelation function (ACF) for time series data.
• To derive analytical expressions for the variance of the ACF.
• To propose a statistically robust method for testing time series correlations.

Main Methods:

• Studied statistical properties of ACF from time series of independent and identically distributed stochastic variables.
• Derived analytical expressions for ACF variance.
• Compared uncertainty characteristics of moving-average and Fourier transform methods for ACF calculation.
• Developed a new statistical test for time series correlations.

Main Results:

• Analytical expressions for ACF variance were derived.
• Moving-average and Fourier transform methods exhibit different uncertainty profiles.
• The Fourier transform method provides smaller, uniform uncertainties for periodic time series.
• A statistically robust correlation test was developed and verified.

Conclusions:

• The choice of ACF calculation method impacts uncertainty, with Fourier transform preferred for periodic data.
• The proposed statistical test offers a reliable way to detect correlations in time series.
• The findings have implications for fields like single-molecule fluorescence spectroscopy.