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，一个长期存在的问题是如何从产生的信号中识别和描述一个非线性随机系统。通常，系统本身是不可访问的(例如，超流氦中的剩余量子涡圈或半导体激光器中的反转布居)，或者难以直接测量(例如，心血管系统，在那里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，结果总是在一定程度上受到外部噪声(测量噪声)的破坏。在生态学和其他科学领域也出现了这样的例子，如天体物理学中的分子马达和物质-辐射耦合系统。主要的困难来自这样一个事实：在许多重要问题中，不可能从第一原理推导出合适的模型，因此人们面临着相当广泛的可能的参数模型来考虑。此外，由于噪声和非线性之间错综复杂的相互作用，实验数据往往会高度倾斜，因此模型的重要隐藏特征(例如耦合参数)可能很难提取。尽管许多科学家进行了大量的努力，但仍然没有可靠的分析方法。这个问题迫切需要一个解决方案，不仅要对所考虑的系统的复杂动力学有物理上的了解，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。我们现在提出解决这个问题。尽管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发展随机非线性动力系统的贝叶斯推理的完整理论来解决它--人们对它抱有很高的成功期望。该算法充分利用了动态噪声的信息含量，对测量噪声具有较强的鲁棒性。拟议的方法是基于我们与V.N.Smelyanskiy博士合作开发的新想法。该技术涉及动态事件的可能性的路径积分计算，它相当普遍地适用于复杂的随机非线性系统。除了发展基本思想外，我们还建议将它们应用于一个实际上无法访问的复杂系统的特定例子：人类心血管系统(CVS)。在这里，测量的时间序列数据(在时间上等间隔的测量序列)归因于发生在身体深处的生理过程。兰开斯特和NASA/Ames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。利用看似随机波动本身包含的关于原始系统的大量信息，并遵循一些创新，我们成功地从时间序列数据的测量重建随机非线性模型。我们对CVS的示范应用不仅将提供迭代改进基本技术所需的实践经验，而且本质上也将是非常有用的。目前正在记录合适的CVS数据，作为与皇家兰开斯特医院合作的一个单独研究项目的一部分。如果CVS的一个良好的随机非线性模型能够被重建，这对医学具有潜在的重要意义，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相关联。这对心血管疾病的早期诊断和治疗效果的量化评估都是有潜力的。我们强调，我们的新推理方法的适用范围可能非常广泛，包括上述广泛的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.