Reconstruction of phase space for time series and detecting chaos: An application on exchange rates

In this study a broad area of chaos theory from mathematical roots to the experimental data analysis was mentioned, in the application part not the mathematical concepts but some practical issues were investigated. For this purpose the exchange rates are chosen to apply the methods. Firstly it is assumed that the only information is the observed time series and all the information required to evolve the analysis again gathered from the time series itself. To reconstruct the phase space auto mutual information was investigated. To determine the number of dimensions for the embedding space, the false nearest neighbors and the Cao approach with suitable delay time was used. The new space gathered from the embedding space was used in order to determine the correlation dimension and maximum Lyapunov exponent of the original attractor to determine whether the observed system chaotic or not. Surrogate data, Kaplan and BDS test approaches showed that the logarithmic Turkish Lira- Dolar exchange rate returns come from a nonlinear deterministic system. At the end of the study, to reduce the noise level the Singular Spectrum Analysis approach was used and the noisy space transformed in to a less noisy new space with lower dimension. The visual properties of embedding space have shown that the 1994 and 2001 economical crises both have similar trajectories. Keywords: Chaos, embedding space, fractals, Lyapunov exponent, exchange rates.