Identification and analysis of directed influences in multivariate systems is an important problem in many scientific areas. Recent studies in neuroscience have provided measures to determine the network structure of the process and to quantify the total effect in terms of energy transfer. These measures are based on joint stationary representations of a multivariate process using vector auto-regressive (VAR) models. A few important issues remain unaddressed though. The primary outcomes of this study are (i) a theoretical proof that the total coupling strength consists of three components, namely, the direct, indirect, and the interference produced by the direct and indirect effects, (ii) expressions to estimate/calculate these effects, and (iii) a result which shows that the well-known directed measure for linear systems, partial directed coherence (PDC) only aids in structure determination but does not provide a normalized measure of the direct energy transfer. Simulation case studies are shown to illustrate the theoretical results.
In this paper, we propose the use of non-negative matrix factorization (NMF) of multivariate spectra for
plantwide oscillation detection. One of the key features of NMF is that it provides a parts-based representation
that allows us to retain the causal basis spectral shapes or parts that constitute the spectra of measurements,
unlike the popular principal component analysis (PCA)-based methods. The contributions of this paper are as
follows: (i) a novel measure known as the pseudo-singular value (PSV) to assess the order of the basis space
(the PSV is also useful in determining the most dominant features of a data set); (ii) a power decomposition
plot that contains the total power (defined in this work) and its decomposition by NMF (the power plot is a
useful and compact visual tool that provides overall spectral characteristics of the plant and shows the
decomposition of these characteristics into well-localized frequency components); and (iii) a novel measure
defined as the strength factor (SF) to assess the strength of the localized features in the variables (it can be
also used in isolating the root cause). Finally, it is shown that the proposed implementation of NMF is powerful
and sensitive enough to capture small oscillations in the measurements. As a result, it largely eliminates the
need to filter the data. Industrial case studies are presented to illustrate the applications of NMF and to
demonstrate the utility and practicality of the proposed measures.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.