Compass Rose of Chaos: Effect of Return and Sampling on Henon Attractor

Abstract

As a preliminary study of the effect of return and sampling on chaos and stochastic data pattern, this research tests chaos pattern by using Henon attractor as a sample of two-dimensional discrete chaos data with an assumption that economic and finance data are generated by low dimensional chaos process. Implementing sampling to two-dimensional discrete chaos data and taking its return surprisingly damage chaos pattern and produce broken chaos pattern that is compass-rose-like pattern initially introduced by Crack and Ledoit (1996) with tendency to certain directions. This finding is challenging chaotic research on economic and finance data that usually use return of sampling data. Researchers use return data for nonstationary reason. Chaotic tools, which are based on Correlation Integral as in BDS Statistic and Correlation Dimension, and based on Correlation Exponent as in Lyapunov Exponent, are derived from chaos data instead of return of chaos data. Applying BDS Statistic and Lyapunov Exponent to return of sampling data is questioning the chaos research findings on economic and finance data. Compass Rose pattern in Economic and Finance return data may indicate the existence of chaos process in Economic and Finance data. Further investigation of compass rose pattern and effect of return and sampling on Stochastic Process, Continuous Chaos process, and higher dimensional chaos are necessary to prove robustness of this finding. Empirical research on real Economic and Finance data is also required to empirically support rigorous observation.

Keywords: Chaos Theory, Clustering, Compass Rose, Complexity, Deterministic, Discrete, Henon Attractor, High-Frequency Data, Low-Frequency Data, Non-Linear Dynamic System, Phase Diagram, Phase Space, Return, Sampling, Thickness Size

JEL Classification: C00, C5, C6, C8, G02, G12, Y10