Frontiers in Applied Mathematics and Statistics (Oct 2022)
A unified framework for analyzing complex systems: Juxtaposing the (Kernel) PCA method and graph theory
Abstract
In this article, we present a unified framework for the analysis and characterization of a complex system and demonstrate its application in two diverse fields: neuroscience and astrophysics. The framework brings together techniques from graph theory, applied mathematics, and dimensionality reduction through principal component analysis (PCA), separating linear PCA and its extensions. The implementation of the framework maps an abstract multidimensional set of data into reduced representations, which enable the extraction of its most important properties (features) characterizing its complexity. These reduced representations can be sign-posted by known examples to provide meaningful descriptions of the results that can spur explanations of phenomena and support or negate proposed mechanisms in each application. In this work, we focus on the clustering aspects, highlighting relatively fixed stable properties of the system under study. We include examples where clustering leads to semantic maps and representations of dynamic processes within the same display. Although the framework is composed of existing theories and methods, its usefulness is exactly that it brings together seemingly different approaches, into a common framework, revealing their differences/commonalities, advantages/disadvantages, and suitability for a given application. The framework provides a number of different computational paths and techniques to choose from, based on the dimension reduction method to apply, the clustering approaches to be used, as well as the representations (embeddings) of the data in the reduced space. Although here it is applied to just two scientific domains, neuroscience and astrophysics, it can potentially be applied in several other branches of sciences, since it is not based on any specific domain knowledge.
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