Dhar Mainak, Dickinson Joseph A, Berg Mark A
Department of Chemistry and Biochemistry, University of South Carolina, Columbia, South Carolina 29208, USA.
J Chem Phys. 2023 Aug 7;159(5). doi: 10.1063/5.0158199.
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.
单分子及相关实验会产生一个可观测值随热运动波动的时间序列。在此类数据中，区分波动信号与波动噪声可能会很困难。我们提出了一种使用非线性相关函数从噪声中分离信号的方法。该方法完全是非参数的：不需要系统的先验模型，不需要知道系统是连续的还是离散的，状态数不固定，并且系统可以是马尔可夫的或非马尔可夫的。经噪声校正的非线性相关函数可以转换为系统的格林函数；经噪声校正的矩产生系统的平衡概率分布。作为演示，我们分析了来自一个三态系统的合成数据。将相关方法与另一种完全非参数的方法——时间分箱去噪和直方图法获取分布——进行了比较。相关方法在时间和状态空间上具有显著更好的分辨率。我们推导了从时间序列中恢复信号所需的数据质量极限公式，并在不同大小和信噪比的数据集上进行了测试。公式表明，在收敛以实际速率进行之前，信噪比需要达到或大于二分之一左右。利用实验基准数据，状态的位置、占据数及其交换率能够以与参数方法相似的精度被恢复。这里展示的方法是仅使用高阶相关函数对时间序列进行完整分析的重要组成部分。
