A unified framework for analyzing complex systems: Juxtaposing the (Kernel) PCA method and graph theory
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. Copyright
Classification of local ultraluminous infrared galaxies and quasars with kernel principal component analysis
We present a new diagnostic diagram for local ultraluminous infrared galaxies (ULIRGs) and quasars, analysing particularly the Spitzer Space Telescope's infrared spectrograph spectra of 102 local ULIRGs and 37 Palomar Green quasars. Our diagram is based on a special non-linear mapping of these data, employing the kernel principal component analysis method. The novelty of this map lies in the fact that it distributes the galaxies under study on the surface of a well-defined ellipsoid, which, in turn, links basic concepts from geometry to physical properties of the galaxies. Particularly, we have found that the equatorial direction of the ellipsoid corresponds to the evolution of the power source of ULIRGs, starting from the pre-merger phase, moving through the starburst-dominated coalescing stage towards the active galactic nucleus-dominated phase, and finally terminating with the post-merger quasar phase. On the other hand, the meridian directions distinguish deeply obscured power sources of the galaxies from unobscured ones. These observations have also been verified by comparison with simulated ULIRGs and quasars using radiative transfer models. The diagram correctly identifies unique galaxies with extreme features that lie distinctly away from the main distribution of the galaxies. Furthermore, special two-dimensional projections of the ellipsoid recover almost monotonic variations of the two main physical properties of the galaxies, the silicate and polycyclic aromatic hydrocarbon features. This suggests that our diagram naturally extends the well-known Spoon diagram and it can serve as a diagnostic tool for existing and future infrared spectroscopic data, such as those provided by the James Webb Space Telescope.
Graph Theoretical Analysis of local ultraluminous infrared galaxies and quasars
We present a methodological framework for studying galaxy evolution by utilizing Graph Theory and network analysis tools. We study the evolutionary processes of local ultraluminous infrared galaxies (ULIRGs) and quasars and the underlying physical processes, such as star formation and active galactic nucleus (AGN) activity, through the application of Graph Theoretical analysis tools. We extract, process and analyze mid-infrared spectra of local (z ¡ 0.4) ULIRGs and quasars between 5-38μm through internally developed Python routines, in order to generate similarity graphs, with the nodes representing ULIRGs being grouped together based on the similarity of their spectra. Additionally, we extract and compare physical features from the mid-IR spectra, such as the polycyclic aromatic hydrocarbons (PAHs) emission and silicate depth absorption features, as indicators of the presence of star-forming regions and obscuring dust, in order to understand the underlying physical mechanisms of each evolutionary stage of ULIRGs. Our analysis identifies five groups of local ULIRGs based on their mid-IR spectra, which is quite consistent with the well established fork classification diagram by providing a higher level classification. We demonstrate how graph clustering algorithms and network analysis tools can be utilized as unsupervised learning techniques for revealing direct or indirect relations between various galaxy properties and evolutionary stages, which provides an alternative methodology to previous works for classification in galaxy evolution. Additionally, our methodology compares the output of several graph clustering algorithms in order to demonstrate the best-performing Graph Theoretical tools for studying galaxy evolution.
A unified framework for analyzing complex systems: Juxtaposing the (Kernel) PCA method and graph theory
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.