Correlation structure of a complex system from its subsystems’ perspectives

Detecting the correlation between records is a basic task in data science. Today, researchers try to estimate it from an objective perspective; i.e., they try to remain outside the investigated system and adopt a set of measures representing all the agents precisely and completely. This enables them to answer the questions, ‘how often’, ‘how many’, or ‘what is the relationship between variables’. In reality, the researcher is usually an agent in the investigated system; they themselves design the metrics for the system’s state to answer the questions of ‘how’ and ‘why’. In other words, the researcher conducts the investigation from a subjective perspective. Inspired by qualitative research in social science, in this paper we propose a scheme for investigating complex systems from a subjective perspective. Technically, one decomposes the researcher’s output time series into intrinsic mode functions that represent their specific features. Projecting all the agents’ time series onto the phase space spanned by this set of specific features, one obtains the coordinates of the agents, from which a similarity network is constructed. The properties of this network are then used to produce a portrait of the system’s correlation from the researcher’s perspective. As typical examples, we investigate three theoretical models, including the Lorenz system, the Rössler system, and their coupled systems. Although Takens’s theorem implies that different variables should be equivalent in reconstructing the dynamical behaviors, we find significant differences between them. We also investigate a financial system composed of 22 stock markets, one crude oil market, and one gold market. The similarity networks for markets distributed across the USA, Europe, and China turn out to be largely homogeneous and dominated by star patterns, while those corresponding to markets distributed across Russia, Brazil, India, and South Africa behave inhomogeneously and are dominated by cluster patterns.