Estimation of Dynamical Noise Power in Unknown Systems

Noise can be modeled as a sequence of random variables defined on a probability space that may be added to a given dynamical system $T$, which is a map on a phase space. In the non-trivial case of dynamical noise $\lbrace \varepsilon _{n}\rbrace _{n}$, where $\varepsilon _{n}$ follows a Gaussian distribution $\mathcal {N}(0,\sigma ^{2})$ and the system output is $x_{n} = T(x_{n-1};x_{0})+\varepsilon _{n}$, without any specific knowledge or assumption about $T$, the quantitative estimation of the noise power $\sigma ^{2}$ is a challenge. Here, we introduce a formal method based on the nonlinear entropy profile to estimate the dynamical noise power $\sigma ^{2}$ without requiring knowledge of the specific $T$ function. We tested the correctness of the proposed method using time series generated from Logistic maps and Pomeau-Manneville systems under different conditions. Our results demonstrate that the proposed estimation algorithm can properly discern different noise levels without any a priori information.