Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding

Abstract

The identification of partially observed continuous nonlinear systems from noisy and incomplete data series is an actual problem in many branches of science, for example, biology, chemistry, physics, and others. Two stages are needed to reconstruct a partially observed dynamical system. First, one should reconstruct the entire phase space to restore unobserved state variables. For this purpose, the integration or differentiation of the observed data series can be performed. Then, a fast-algebraic method can be used to obtain a nonlinear system in the form of a polynomial dynamical system. In this paper, we extend the algebraic method proposed by Kera and Hasegawa to Laurent polynomials which contain negative powers of variables, unlike ordinary polynomials. We provide a theoretical basis and experimental evidence that the integration of a data series can give more accurate results than the widely used differentiation. With this technique, we reconstruct Lorenz attractor from a one-dimensional data series and B. Muthuswamy’s circuit equations from a three-dimensional data series.

Deterministic Chaos Detection and Simplicial Local Predictions Applied to Strawberry Production Time Series

“…For example, if we apply the SINDy method [11], we must consider many non-linear functions within the matrix of possible functions, which makes it computationally difficult. On the other hand, we have obtained good results without having to rebuild the time series, such as in [16], making the method lighter, although certainly less successful. However, we leave these objectives as some very interesting future lines of research that can potentially improve upon the typical linear research carried out on the modeling of primary sector activities.…”

“…In this article, we discuss an alternative non-linear method for the cases in which the resulting series is not stationary. Although we find many emerging non-linear techniques that can be used to make both short-term and long-term predictions on non-stationary chaotic data, such as the sparse identification of nonlinear dynamics (SINDy) algorithm [11] widely used to model non-linear dynamic systems and make predictions on them [12][13][14][15], or non-linear systems reconstruction techniques that allow the regeneration of time series subjected to white noise, which would allow a new study of stationarity and eliminate the disturbances associated with the observed variable [16,17], in this work, we focus on maximal Lyapunov exponents.…”

Border Basis Computation with Gradient-Weighted Normalization

“…The approximate computation of bases of vanishing ideals has been extensively studied [1, 3, 5, 10-12, 14, 16-18, 23] in the last decade, where a basis comprises approximately vanishing polynomials, i.e., 𝑔(x) ≈ 0, (∀x ∈ 𝑋 ). Approximate basis computation and approximately vanishing polynomials are exploited in various fields such as dynamics reconstruction, signal processing, and machine learning [2,6,7,9,13,[20][21][22]. A wide variety of applications is possible because the approximate basis computation takes a set of noisy points as its input-suitable for the recent data-driven applications-and efficiently computes a set of multivariate polynomials that characterize the given data.…”

Border basis computation with gradient-weighted normalization

“…In the last decade, approximate computation of bases of vanishing ideals has been extensively studied [1,4,6,11,12,13,15,17,18,19], where a basis comprises approximately vanishing polynomials, i.e., g(x) ≈ 0, (∀x ∈ X). Such approximate basis computation and approximately vanishing polynomials have been exploited in various fields such as dynamics reconstruction, signal processing, and machine learning [2,7,8,10,14,22,21,23]. The wide variety of applications is based on the fact that the approximate basis computation takes a set of noisy points as its input-which is suitable for the recent data-driven science-and efficiently computes a set of multivariate polynomials that characterize the given data.…”