Kannan Rohit, Tangirala Arun K
Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600004, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Jun;89(6):062144. doi: 10.1103/PhysRevE.89.062144. Epub 2014 Jun 30.
Identification of directional influences in multivariate systems is of prime importance in several applications of engineering and sciences such as plant topology reconstruction, fault detection and diagnosis, and neurosciences. A spectrum of related directionality measures, ranging from linear measures such as partial directed coherence (PDC) to nonlinear measures such as transfer entropy, have emerged over the past two decades. The PDC-based technique is simple and effective, but being a linear directionality measure has limited applicability. On the other hand, transfer entropy, despite being a robust nonlinear measure, is computationally intensive and practically implementable only for bivariate processes. The objective of this work is to develop a nonlinear directionality measure, termed as KPDC, that possesses the simplicity of PDC but is still applicable to nonlinear processes. The technique is founded on a nonlinear measure called correntropy, a recently proposed generalized correlation measure. The proposed method is equivalent to constructing PDC in a kernel space where the PDC is estimated using a vector autoregressive model built on correntropy. A consistent estimator of the KPDC is developed and important theoretical results are established. A permutation scheme combined with the sequential Bonferroni procedure is proposed for testing hypothesis on absence of causality. It is demonstrated through several case studies that the proposed methodology effectively detects Granger causality in nonlinear processes.
在工程和科学的多个应用领域，如工厂拓扑结构重建、故障检测与诊断以及神经科学中，识别多变量系统中的方向性影响至关重要。在过去二十年中，出现了一系列相关的方向性度量方法，从诸如偏相干性（PDC）等线性度量到诸如转移熵等非线性度量。基于PDC的技术简单有效，但作为一种线性方向性度量，其适用性有限。另一方面，转移熵尽管是一种强大的非线性度量，但计算量很大，实际上仅适用于双变量过程。这项工作的目标是开发一种非线性方向性度量，称为核偏相干性（KPDC），它具有PDC的简单性，但仍适用于非线性过程。该技术基于一种称为核熵的非线性度量，这是最近提出的一种广义相关度量。所提出的方法相当于在一个核空间中构建PDC，在该核空间中使用基于核熵构建的向量自回归模型来估计PDC。开发了KPDC的一致估计量并建立了重要的理论结果。提出了一种结合顺序邦费罗尼程序的排列方案，用于检验因果关系不存在的假设。通过几个案例研究表明，所提出的方法能够有效地检测非线性过程中的格兰杰因果关系。
