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Abstract

Correlation analysis serves as an easy-to-implement estimation approach for the quantification of the interaction or connectivity between different units. Often, pairwise correlations estimated by sliding windows are time-varying (on different window segments) and window size-dependent (on different window sizes). Still, how to choose an appropriate window size remains unclear. This paper offers a framework for studying this fundamental question by observing a critical transition from a chaotic-like state to a nonchaotic state. Specifically, given two time series and a fixed window size, we create a correlation-based series based on nonlinear correlation measurement and sliding windows as an approximation of the time-varying correlations between the original time series. We find that the varying correlations yield a state transition from a chaotic-like state to a nonchaotic state with increasing window size. This window size-dependent transition is analyzed as a universal phenomenon in both model and real-world systems (e.g., climate, financial, and neural systems). More importantly, the transition point provides a quantitative rule for the selection of window sizes. That is, the nonchaotic correlation better allows for many regression-based predictions.
Complex connections between different units can be simply approximated by correlation analysis between corresponding time series. When the complete information (the entire time series) is considered for analysis, dynamic connections are aggregated into a single value, reflecting the overall macro linkage. When segmented information (a sliced time series) is combined with sliding windows, the underlying dynamic connections can be approximated by time-varying correlations. Intuitively, the longer the segments are, the more likely to capture cyclic behavior. A typical example is that in climate science, large-scale climate phenomena, such as seasonal changes induced by the annual cycle of solar radiation, are not observable on the timescale of diurnal cycles. Similarly, for correlation analysis, choosing a suitable window scale to capture the necessary patterns hidden in the time series is fundamental; yet, how to do so is unclear. We intend to address this issue in our work.