Estimation of Dynamical Noise Power in Unknown Systems
Andrea Scarciglia, Fulvio Gini, Vincenzo Catrambone, Claudio Bonanno, Gaetano Valenza
Abstract
Noise can be modeled as a sequence of random variables defined on a probability space that may be added to a given dynamical system <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> , which is a map on a phase space. In the non-trivial case of dynamical noise <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lbrace \varepsilon _{n}\rbrace _{n}$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon _{n}$</tex-math></inline-formula> follows a Gaussian distribution <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {N}(0,\sigma ^{2})$</tex-math></inline-formula> and the system output is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x_{n} = T(x_{n-1};x_{0})+\varepsilon _{n}$</tex-math></inline-formula> , without any specific knowledge or assumption about <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> , the quantitative estimation of the noise power <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma ^{2}$</tex-math></inline-formula> is a challenge. Here, we introduce a formal method based on the nonlinear entropy profile to estimate the dynamical noise power <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma ^{2}$</tex-math></inline-formula> without requiring knowledge of the specific <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> 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.