Fitting a function to time-dependent ensemble averaged data

Area of Science:

• Physics
• Physical Chemistry
• Biophysics

Background:

• Time-dependent ensemble averages are crucial in scientific data analysis.
• Fitting these averages to functions requires accurate parameter and error estimation.
• Temporal correlations in data fluctuations are often overlooked, leading to inaccurate fitting.

Purpose of the Study:

• To address the limitations of existing methods for fitting time-dependent ensemble averages.
• To develop a robust method for parameter and error estimation that accounts for temporal correlations.
• To provide a reliable tool for analyzing complex dynamic processes.

Main Methods:

• Derivation of a closed-form error estimation formula for weighted least square fitting.
• Inclusion of the full covariance matrix to rigorously account for temporal correlations.
• Development of the weighted least squares including correlation in error estimation (WLS-ICE) method.

Main Results:

• Existing methods (correlated chi-square, standard weighted least squares) fail in robust parameter or accurate error estimation.
• The new WLS-ICE method provides accurate parameter and error estimation by including temporal correlations.
• Demonstrated accuracy across diverse models including Brownian motion and fractional Brownian motion.
• Successful application to real-world particle tracking data.

Conclusions:

• The WLS-ICE method offers a robust and accurate approach for analyzing time-dependent ensemble averages.
• This method overcomes limitations of previous techniques by rigorously incorporating temporal correlations.
• WLS-ICE is applicable to various fit functions and has been made publicly available.