Evaluation of deterministic property of time series by the method of surrogate data and the trajectory parallel measure method

It is now known that a seemingly random irregular time series can be determ inistic chaos thereafter, chaos). However: there can be various kind of noi se superimposed into signals from real systems. Other factors affecting a s ignal include sampling intervals and finite length of observation. Perhaps, there may be cases in which a chaotic time series is considered as noise. J. Theiler proposed a method of surrogating data to address these problems. The proposed method is one of a number of approaches for testing a statist ical hypothesis. The method can identify the deterministic characteristics of a time series. In this approach, a surrogate data is formed to have stoc hastic characteristics with the statistic value associated with the origina l data. When the characteristics of the original data differs from that of a surrogate data, the null hypothesis is no longer valid. In other words, t he original data is deterministic. In comparing the characteristics of an o riginal time series data and that of a surrogate data, the maximum Lyapunov exponents, correlation dimensions and prediction accuracy are utilized. Th ese techniques, however, can not calculate the structure in local subspaces on the attractor and the Row of trajectories, In deal with these issues; w e propose the trajectory parallel measure (TPM) method to determine whether the null hypothesis should be rejected. In this paper, we apply the TPM me thod and the method of surrogate data to test a chaotic time series and a r andom time series. We also examine whether a practical time series has a de terministic property or not. The results demonstrate that the TPM method is useful for judging whether the original and the surrogate data sets are di fferent, For illustration, the TPM method is applied to a practical time se ries, tap water demand data.