Exploiting the information content of noise in complex systems: Bayesian inference of nonlinear stochastic models and applications to human blood flow

在科学和工程的许多分支中，一个经久不衰的问题是从非线性随机系统产生的信号中识别和描述非线性随机系统。通常，系统本身是不可接近的（例如，超流氦中的剩余量子涡旋环或半导体激光器中的反转居群），或者难以直接测量（例如，心血管系统，其测量通常必须是非侵入性的）。此外，除了系统本身的任何动态波动（动态噪声）外，结果总是会在一定程度上受到外部噪声（测量噪声）的破坏。这样的例子也出现在生态学和其他科学领域，如分子马达和天体物理学中的耦合物质辐射系统。主要的困难源于这样一个事实，即在许多重要的问题中，不可能从第一原理推导出一个合适的模型，因此人们面临着相当广泛的可能的参数模型来考虑。此外，实验数据往往高度偏斜，因此，由于噪声和非线性之间复杂的相互作用，模型的重要隐藏特征（例如耦合参数）可能很难提取。尽管许多科学家付出了巨大的努力，仍然没有可靠的分析方法。这一问题的解决方案是迫切需要的，不仅要获得对所考虑的系统复杂动力学的物理洞察力，而且要促进现实模型的发展，从而提高预测未来的可靠性和准确性。我们现在提议解决这个问题。虽然这一直是统计物理学中最具挑战性的问题之一，但我们现在可以通过开发随机非线性动力系统的贝叶斯推理的完整理论来解决它，并有很高的成功期望。它利用了动态噪声的信息量，在测量噪声面前具有较强的鲁棒性。提出的方法是基于我们与V.N.斯梅扬斯基博士合作开发的新想法。该技术涉及动态事件似然的路径积分计算，一般适用于复杂的随机非线性系统。在发展基本思想的同时，我们建议将它们应用于一个复杂系统的特定例子，这个系统在实践中是不可接近的：人类心血管系统（CVS）。在这里，测量的时间序列数据（时间间隔相等的测量序列）可归因于身体深处发生的生理过程。兰开斯特大学和美国宇航局/艾姆斯研究中心之间已经完成了足够的初步工作，以证明这项研究的可行性。利用包含在看似随机波动本身中的关于原始系统的大量信息，并经过一系列创新，我们已经成功地从时间序列数据的测量中重建了随机非线性模型。我们的CVS示例应用程序不仅提供了对基本技术进行迭代改进所需的实践经验，而且对企业本身也非常有用。合适的CVS数据目前正在被记录为与皇家兰开斯特医院合作的单独研究项目的一部分。如果一个好的随机非线性CVS模型可以重建，这对医学有潜在的重要意义，因为我们预计它将证明有可能将模型中的参数值与系统的状态联系起来。这对心血管疾病的早期诊断和治疗效果的定量评估都有潜力。我们强调，我们的新推理方法的适用性范围可能非常广泛，涵盖了上面提到的广泛问题和许多其他问题。

An enduring problem in many branches of science and engineering is that of identifying and characterising a nonlinear stochastic system from the signals it produces. Often, the system itself is inaccessible (e.g. remanent quantum vortex loops in superfluid helium or the inversion population in a semiconductor laser), or difficult to measure directly (e.g. the cardiovascular system, where measurements must usually be non-invasive). Furthermore, in addition to any dynamical fluctuations in the system itself (dynamical noise), the results will always be to some extent corrupted by external noise (measurement noise). Examples also arise in ecology, and in other scientific areas as diverse as molecular motors and coupled matter-radiation systems in astrophysics. The chief difficulty stems from the fact that, in a great number of important problems, it is not possible to derive a suitable model from first principles, and one is therefore faced with a rather broad range of possible parametric models to consider. Furthermore, experimental data can often be highly skewed, so that important hidden features of a model (e.g. coupling parameters) can be very difficult to extract due to the intricate interplay between noise and nonlinearity. There is still no reliable method of analysis, despite intensive effort by many scientists.A solution to this problem is sorely is needed, not only to gain physical insight into the complex dynamics of the system under consideration, but also to facilitate the development of realistic models and thus to improve the reliability and accuracy with which the future can be predicted.We now propose to solve the problem. Although it has been one of the most challenging in statistical physics, we can now tackle it - with a high expectation of success - by developing a full theory of Bayesian inference for stochastic nonlinear dynamical systems. It will exploit the information content of dynamical noise and will be robust in the face of measurement noise. The proposed approach is based on novel ideas developed in our collaboration with Dr. V.N. Smelyanskiy. The technique involves path-integral calculations of the likelihood of dynamical events, and it is applicable to complex stochastic nonlinear systems quite generally. As well as developing the fundamental ideas, we propose to apply them to a particular example of a complex system that is in practice inaccessible: the human cardiovascular system (CVS). Here, the measured time series data (a sequence of measurements equally spaced in time) are attributable to physiological processes occurring deep within the body.Enough initial work has already been completed between Lancaster and the NASA/Ames Research Center to demonstrate the feasibility of the research. Exploiting the huge amount of information about the originating system that is contained within the seemingly random fluctuations themselves, and following a number of innovations, we have succeeded in reconstructing stochastic nonlinear models from measurements of time series data.Our exemplary application to the CVS will not only provide the practical experience needed for iterative improvement in the basic technique, but the enterprise will also be intrinsically extremely useful. Suitable CVS data are currently being recorded as part of a separate research project in collaboration with the Royal Lancaster Infirmary. If a good stochastic nonlinear model of the CVS can be reconstructed, there are potentially important implications for medicine because we anticipate that it will prove possible to relate parameter values in the model to the state of the system. There potential for both early diagnosis of cardiovascular disease and for quantitative assessment of the effect of treatment. We emphasize that the range of applicability of our new inference method is potentially very broad, encompassing the wide range of problems mentioned above and many others.