On The Evaluation Of Noise Levels For Quantized Data

In recent years, quantitative methods for evaluating chaotic properties have been developed in the field of nonlinear time-series analysis. The embedding theorem, which is a mathematical background for the methods, assumes an ideal situation in which noiseless time series are observed with infinite resolution and an infinite amount of data points. However, under real situations we cannot ignore two classes of noises which are included in really observed data. The first one is observational and dynamical noises which depend on internal nonlinear systems and performance of observational instruments. The second one is quantization error included in the time series since we usually use digital computers for applying the methods. In the present paper, we derive formulae for evaluating the levels of observational and quantization noises in the case of embedding observed time series in reconstructed state spaces. By measuring a distance between noiseless and noisy attractors, we also confirm that the derived formulae are appropriate for quantifying the noise included in the reconstructed attractor.