Reconstructing the states of a nonlinear dynamical system
We often encounter nonlinear dynamical systems that behave unpredictably, such as the Earth's climate and the stock market. To analyze them, measurements taken over time are used to reconstruct the state of the system. However, this depends on the quality of the data. Now, researchers from Japan have proposed an all-new method for determining the necessary parameters that results in an accurate reconstruction. Their new technique has far-reaching implications for the field of data science.
Many frequently observed real-world phenomena are nonlinear in nature. This means that their output does not change in a manner that is proportional to their input. These models have a degree of unpredictability, where it is unclear how the system will respond to any changes in its input. This is especially important in the case of dynamical systems, where the output of the model changes with time. For such systems, the time series data, or the measurements from the system over time, have to be analyzed to determine how the system changes or evolves with time.
Due to the commonality of the problem, many solutions have been proposed to analyze time-series data to gain an understanding of the system. One method of reconstructing the state of a system based on time series data is state space reconstruction, which can be used to reconstruct those states where the system remains stable or unchanged with time. Such states are known as "attractors." However, the accuracy of the reconstructed attractors depends on the parameters used for reconstruction, and due to the finite nature of the data, such parameters are difficult to ascertain, resulting in inaccurate reconstructions.
Now, in a new study to be published on April 1, 2022, in Nonlinear Theory and Its Applications, IEICE, Professor Tohru Ikeguchi from Tokyo University of Science, his Ph.D. student Mr. Kazuya Sawada from Tokyo University of Science, and Prof. Yutaka Shimada from Saitama University, Japan, have used the geometric structure of the attractor to estimate the reconstruction parameters.
"To reconstruct the state space using time-delay coordinate systems, two parameters, the dimension of the state space and the delay time, must be set appropriately, which is an important issue that is still being actively studied in this field. We discuss how to set these parameters optimally by focusing on the geometric structure of the attractor as one way to solve this problem," explains Prof. Ikeguchi.
To obtain the optimal values of the parameters, the researchers used five three-dimensional nonlinear dynamical systems and maximized the similarity of the inter-point distance distributions between the reconstructed attractor and the original attractor. As a result, the parameters were obtained in a way that produced a reconstructed attractor which was geometrically as close as possible to the original.
While the method was able to generate the appropriate reconstruction parameters, the researchers did not factor in the noise that is normally encountered in real-world data, which can significantly affect the reconstruction. "Mathematically, this method has been proven to be a good one, but there are many considerations that need to be made before applying this method to real-world data analysis. This is because real-world data contains noise, and the length and accuracy of the observed data is finite," explains Prof. Ikeguchi.
Despite this, the method resolves one of the limitations involved in determining the state of nonlinear dynamical systems that are encountered in various fields of science, economics, and engineering. "This research has yielded an important analysis technique in the current data science field, and we believe that it is important for handling a wide variety of data in the real world," concludes Prof. Ikeguchi.