Abstract
In this paper, we apply multidimensional scaling (MDS) and parametric similarity indices (PSI) in the analysis of complex systems (CS). Each CS is viewed as a dynamical system, exhibiting an output time-series to be interpreted as a manifestation of its behavior. We start by adopting a sliding window to sample the original data into several consecutive time periods. Second, we define a given PSI for tracking pieces of data. We then compare the windows for different values of the parameter, and we generate the corresponding MDS maps of ‘points’. Third, we use Procrustes analysis to linearly transform the MDS charts for maximum superposition and to build a globalMDS map of “shapes”. This final plot captures the time evolution of the phenomena and is sensitive to the PSI adopted. The generalized correlation, theMinkowski distance and four entropy-based indices are tested. The proposed approach is applied to the Dow Jones Industrial Average stock market index and the Europe Brent Spot Price FOB time-series.
Highlights
IntroductionComplex systems (CS) are frequent in many natural (e.g., geophysics, cosmology, ecology, biology, genetics) and man-made (e.g., economy, computer science, chemical and physical apparatus) systems [1,2,3,4,5,6,7,8,9,10]
Complex systems (CS) are frequent in many natural and man-made systems [1,2,3,4,5,6,7,8,9,10]
Since we can explore indices with several parameters, do they lead to useful multidimensional “shapes” in the Multidimensional scaling (MDS) maps? How can we analyze more assertively the sensitivity of different time windows? While the standard visualization of the flux of time as a constant velocity process, is MDS pointing to a variable speed time arrow? We plan to apply the new methodology to several data series in the future, aiming to clarify some of these issues
Summary
Complex systems (CS) are frequent in many natural (e.g., geophysics, cosmology, ecology, biology, genetics) and man-made (e.g., economy, computer science, chemical and physical apparatus) systems [1,2,3,4,5,6,7,8,9,10]. Machado et al [18] adopted MDS to study fifteen stock markets and to unveil time-varying correlations between them. Standard MDS representations capture the system dynamics by means of a single similarity index. Such an index depends of the researcher’s choice, and we can define distinct criteria. We propose subdividing each time-series into several smaller slices, to capture time dynamics, and to adopt PSI in each MDS map, providing distinct comparisons for the same windows. The interpretation of the MDS map is based on “objects” consisting of “shapes” (a collection of points) instead of “points”, capturing the time evolution of the phenomena and being sensitive to the parametric index adopted.
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