SIAM Journal on Imaging Sciences, Volume 13, Issue 1, Page 234-264, January 2020.
Tracking of objects in cellular environments has become a vital tool in molecular cell biology. A particularly important example is single molecule tracking, which enables the study of the motion of a molecule in cellular environments by locating the molecule over time and provides quantitative information on the behavior of individual molecules in cellular environments, which were not available before through bulk studies. Here, we consider a dynamical system where the motion of an object is modeled by stochastic differential equations (SDEs), and measurements are the detected photons, emitted by the moving fluorescently labeled object, that occur at discrete time points, corresponding to the arrival times of a Poisson process, in contrast to equidistant time points, which have been commonly used in the modeling of dynamical systems. The measurements are distributed according to the optical diffraction theory, and therefore, they would be modeled by different distributions, e.g., an Airy profile for an in-focus and a Born and Wolf profile for an out-of-focus molecule with respect to the detector. For some special circumstances, Gaussian image models have been proposed. In this paper, we introduce a stochastic framework in which we calculate the maximum likelihood estimates of the biophysical parameters of the molecular interactions, e.g., diffusion and drift coefficients. More importantly, we develop a general framework to calculate the Cramér--Rao lower bound (CRLB), given by the inverse of the Fisher information matrix, for the estimation of unknown parameters and use it as a benchmark in the evaluation of the standard deviation of the estimates. There exists no established method, even for Gaussian measurements, to systematically calculate the CRLB for the general motion model that we consider in this paper. We apply the developed methodology to simulated data of a molecule with linear trajectories and show that the standard deviation of the estimates matches well with the square root of the CRLB. We also show that equally sampled and Poisson distributed time points lead to significantly different Fisher information matrices.

中文翻译：

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具有随机轨迹的单分子Fisher信息矩阵

SIAM影像科学杂志，第13卷，第1期，第234-264页，2020年1月。
在细胞环境中跟踪对象已成为分子细胞生物学中的重要工具。一个特别重要的例子是单分子跟踪，它可以通过随时间定位分子来研究分子在细胞环境中的运动，并提供有关单个分子在细胞环境中行为的定量信息，而这在批量研究之前是无法获得的。 。在这里，我们考虑一个动力学系统，其中对象的运动由随机微分方程（SDE）建模，而测量值是运动的荧光标记对象发出的检测到的光子，它们在离散的时间点出现，对应于到达时间与等距时间点相反，后者通常用于动力学系统建模。根据光学衍射理论来分配测量值，因此，将通过不同的分布对它们进行建模，例如，相对于光学系统，焦距的艾里分布和焦外分子的伯恩和沃尔夫分布。探测器。对于某些特殊情况，已经提出了高斯图像模型。在本文中，我们引入一个随机框架，在该框架中，我们计算分子相互作用的生物物理参数（例如扩散系数和漂移系数）的最大似然估计。更重要的是，我们开发了一个通用框架来计算费舍尔信息矩阵逆给出的Cramér-Rao下界（CRLB），用于估算未知参数，并将其用作评估标准偏差的基准的估计。即使对于高斯测量，也没有确定的方法可以为我们在本文中考虑的一般运动模型系统地计算CRLB。我们将开发的方法应用于具有线性轨迹的分子的模拟数据，并表明估计的标准偏差与CRLB的平方根很好地匹配。我们还表明，同样采样的时间和泊松分布的时间点会导致显着不同的Fisher信息矩阵。