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

Abstract

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 (ε) on the threshold parameter ε. We show that the function K 2 (ε) depends in a characteristic way on the noise standard deviation σ. It follows that observing K 2 (ε) one can estimate the noise level σ.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 Langevine 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.

How random is your heart beat?

“…If noise is added to the trajectory of a deterministic dynamical system (measurement noise) or if noise is present in the equation of motion (dynamical noise affecting the dynamics of the system) then the complexity measure called coarse-grained correlation entropy K 2 [21] increases. Knowing the analytical dependence of this entropy on the standard deviation of the uncorrelated noise σ [20], we can estimate the noise level from the calculation of the entropy K 2 . The method also allows to estimate the error of the standard deviation of the random noise σ.…”

“…To estimate the level N of random noise in the analyzed data, we applied a method developed by Urbanowicz and Hołyst [20]. All data sets were analyzed by means of a 1000 data point sliding window shifted by 200 RR intervals.…”

“…We express it and all similar quantities in percent. 6 The noise level estimation method [20] was developed for uncorrelated noise. In the case of a highly correlated stochastic process, our analysis may underestimate the stochastic component of the variability.…”

“…[17,18,19]). In this paper, using a general method [20], we measure the content of random, uncorrelated noise in heart rate data. We analyzed 70 24-hour recordings of heart rate variability measured in patients of the Institute of Cardiology at Warszawa.…”

Anti-deterministic behaviour of discrete systems that are less predictable than noise

“…For this reason one can alternatively use terms embedding dimension and length of a line (the embedding dimension used in RP is one for all the calculations). Let us define the number of lines in RP of length n or longer as DET n (ε) [16]. Then the coarse-grained block correlation entropy can be defined as [17] …”