3. Unfolding Nonlinear Characteristics of Noise-Contaminated Real-World Data

The success of dynamic characterization of a system mostly depends on how far the nonlinear structure of the data is identified. In most of the practical systems, the dynamic behaviour is to some extent dominated by stochastic processes. In such a scenario, unfolding the hidden determinism and nonlinearity from the real world data which is acquired from a complex physical phenomenon is a challenging task. The tools from nonlinear time series analysis facilitate a systematic investigation of complex dynamics mostly observed in real life phenomena. The present chapter proposes a survey on different such tools which are used for the dynamic characterization of experimental time series. To take the first step in this course, detecting noise contamination in the time series data is discussed, highlighting the tests for determinism such as local flow test (Kaplan-Glass test), translation error, correlation dimension, and correlation entropy. Once the determinism of a time series is identified, figuring out the nonlinear nature is the next concern. There are a few direct tests such as Lyapunov exponent, correlation dimension to confirm chaos, however, they have their inherent limitations while analysing experimental data contaminated with noise. In such a situation, a statistical test popularly known as surrogate test is adopted. The test is based on different (null) hypotheses and examining their validity through any discriminating statistics such as translation error, permutation entropy. Permutation spectrum can further be used to characterize the dynamic nature of the time series. Other aspects which help further understanding of the data sets in hand are the fractal features and the predictability. The fractal features of a time series are identified by using singularity spectrum. Finally, the role of local predictor such as Sugihara-May algorithm for forecasting the dynamics of a deterministic system is discussed.