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Single-molecule and related experiments yield time series of an observable as it fluctuates due to thermal motion. In such data, it can be difficult to distinguish fluctuating signal from fluctuating noise. We present a method of separating signal from noise using nonlinear-correlation functions. The method is fully nonparametric: No a priori model for the system is required, no knowledge of whether the system is continuous or discrete is needed, the number of states is not fixed, and the system can be Markovian or not. The noise-corrected, nonlinear-correlation functions can be converted to the system’s Green’s function; the noise-corrected moments yield the system’s equilibrium-probability distribution. As a demonstration, we analyze synthetic data from a three-state system. The correlation method is compared to another fully nonparametric approach—time binning to remove noise, and histogramming to obtain the distribution. The correlation method has substantially better resolution in time and in state space. We develop formulas for the limits on data quality needed for signal recovery from time series and test them on datasets of varying size and signal-to-noise ratio. The formulas show that the signal-to-noise ratio needs to be on the order of or greater than one-half before convergence scales at a practical rate. With experimental benchmark data, the positions and populations of the states and their exchange rates are recovered with an accuracy similar to parametric methods. The methods demonstrated here are essential components in building a complete analysis of time series using only high-order correlation functions.
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This content will become publicly available on November 14, 2026
Nonparametric analysis of noisy, multivariable time series using high-order correlation functions: Single-molecule FRET as an example
High-order correlation functions offer a model-free (nonparametric) method of analyzing single-molecule data with high resolution in both time and state space. However, they have only been demonstrated for single-channel experiments, whereas many single-molecule experiments measure multiple data channels. This paper identifies the central problem with multichannel datasets and presents a roadmap for its general solution. The process is demonstrated using the specific example of fluorescence resonance energy transfer (FRET), one of the most common single-molecule experiments. The method’s practicality is demonstrated on FRET data published as a data-analysis benchmark. The paper emphasizes the need to work at high noise levels to optimize single-molecule experiments and the importance of effective noise removal in their analysis. Overall, an additional step is taken toward making correlation analysis a general, model-free method of treating experimental time series with optimum performance.
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- Award ID(s):
- 2003619
- PAR ID:
- 10653230
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- The Journal of Chemical Physics
- Volume:
- 163
- Issue:
- 18
- ISSN:
- 0021-9606
- Page Range / eLocation ID:
- 184113
- Subject(s) / Keyword(s):
- single-molecule kinetics time-series analysis nonparametric statistics single-molecule spectroscopy correlation spectroscopy signal processing
- Format(s):
- Medium: X Size: 7MB Other: pdf
- Size(s):
- 7MB
- Sponsoring Org:
- National Science Foundation
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