Noise-level estimation of time series using coarse-grained entropy

We present a method of noise-level estimation that is valid even for high noise levels. The method makes use of the functional dependence of coarse-grained correlation entropy $K_2\left(\varepsilon \right)$ on the threshold parameter $\varepsilon .$ We show that the function $K_2\left(\varepsilon \right)$ depends, in a characteristic way, on the noise standard deviation $\sigma .$ It follows that observing $K_2\left(\varepsilon \right)$ one can estimate the noise level $\sigma .$ Although the theory has been developed for the Gaussian noise added to the observed variable we have checked numerically that the method is also valid for the uniform noise distribution and for the case of Langevin equation corresponding to the dynamical noise. We have verified the validity of our method by applying it to estimate the noise level in several chaotic systems and in the Chua electronic circuit contaminated by noise.