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arXiv 1803.08784cs.AIcs.LGstat.MEstat.ML
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动力系统的因果建模

Causal Modeling of Dynamical Systems

Informatics Institute, University of Amsterdam(阿姆斯特丹大学信息学研究所);Korteweg-de Vries Institute, University of Amsterdam(阿姆斯特丹大学科尔特韦格-德弗里斯研究所);Pattern Recognition Laboratory, Delft University of Technology(代尔夫特理工大学模式识别实验室)

Stephan Bongers, Tineke Blom, Joris M. Mooij

摘要:

动力系统广泛应用于科学和工程中,用于对由多个相互作用的组件组成的系统进行建模。通常,它们可以被给予因果解释,因为它们不仅模拟了系统组件的状态随时间的演变,而且还描述了它们的演变如何受到干扰动态系统的外部干预的影响。我们介绍了结构动态因果模型(SDCMs)的正式框架,阐明了因果语义系统的组件作为模型的一部分。SDCM代表了一个动态系统作为一个随机过程的集合,并指定了基本的因果机制,管理动态的每个组件作为一个结构化的系统的随机微分方程的任意顺序。SDCM扩展了结构因果模型(SCM)的通用因果建模框架,也称为结构方程模型(SEM),通过明确允许时间依赖性。SDCM可以被认为是SCM的随机过程版本,其中SCM的静态随机变量被动态随机过程及其衍生物取代。我们提供了SDCM理论的基础,(i)正式定义SDCM,他们的解决方案,随机干预,和一个图形表示;(ii)研究的存在性和唯一性的解决方案,给定的初始条件;(iii)提供马尔可夫性质的SDCM与初始条件;(iv)讨论在什么条件下SDCM的平衡SCM的时间趋于无穷大;(v)将SDCM的性质与平衡SCM的性质联系起来。这种对应关系使人们能够利用丰富的统计工具和发现方法,可用于SCM研究的因果语义的一大类随机动力系统时。该理论是从不同的科学领域的例子说明。

英文摘要:

Dynamical systems are widely used in science and engineering to model systems consisting of several interacting components. Often, they can be given a causal interpretation in the sense that they not only model the evolution of the states of the system's components over time, but also describe how their evolution is affected by external interventions on the system that perturb the dynamics. We introduce the formal framework of structural dynamical causal models (SDCMs) that explicates the causal semantics of the system's components as part of the model. SDCMs represent a dynamical system as a collection of stochastic processes and specify the basic causal mechanisms that govern the dynamics of each component as a structured system of random differential equations of arbitrary order. SDCMs extend the versatile causal modeling framework of structural causal models (SCMs), also known as structural equation models (SEMs), by explicitly allowing for time-dependence. An SDCM can be thought of as the stochastic-process version of an SCM, where the static random variables of the SCM are replaced by dynamic stochastic processes and their derivatives. We provide the foundations for a theory of SDCMs, by (i) formally defining SDCMs, their solutions, stochastic interventions, and a graphical representation; (ii) studying existence and uniqueness of the solutions for given initial conditions; (iii) providing Markov properties for SDCMs with initial conditions; (iv) discussing under which conditions SDCMs equilibrate to SCMs as time tends to infinity; (v) relating the properties of the SDCM to those of the equilibrium SCM. This correspondence enables one to leverage the wealth of statistical tools and discovery methods available for SCMs when studying the causal semantics of a large class of stochastic dynamical systems. The theory is illustrated with examples from different scientific domains.

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