Reconstruction of the state of a nonlinear dynamic system

The stock market behaves unpredictably and is an example of a non-linear dynamic system. In this study, researchers at TUS and Saitama University have made significant discoveries in the field of data science by developing new ways to predict how such nonlinear systems will change over time. Credit: Katrina.Tuliao

We often encounter non-linear dynamic systems with unpredictable behavior, such as the Earth’s climate and stock markets. To analyze them, use the measurements taken over time to reconstruct the state of the system. However, this depends on the quality of the data. Currently, Japanese researchers are proposing a whole new method for determining the necessary parameters that result in accurate reconstruction. Their new technology has a wide range of impacts in the field of data science.

Many of the frequently observed real-world phenomena are non-linear in nature. This means that their outputs do not change in a way that is proportional to their inputs. There is some unpredictability in these models, and it is unclear how the system responds to input changes. This is especially important when: Dynamic system, The output of the model changes over time. For such a system Data in chronological orderOr measurements from the system over time should be analyzed to determine how the system changes or evolves over time.

Due to the commonality of the problem, many solutions have been proposed to analyze the time series data to gain an understanding of the system. One way to reconstruct the state of a system based on time series data is state-space reconstruction. You can use it to rebuild your system to be stable or unchanged over time. This condition is known as an “attractor”. However, the accuracy of the reconstructed attractor depends on the parameters used for the reconstruction, and the finite nature of the data makes it difficult to identify such parameters and the reconstruction is inaccurate. Become.

Now, in a new study published on April 1, 2022, Nonlinear theory and its applications, IEICE, Professor Toru Ikeguchi, Tokyo University of Science, Ph.D. Kazuya Sawada of Tokyo University of Science and Professor Yutaka Shimada of Saitama University estimated the reconstruction parameters using the geometric structure of the attractor.

“To reconstruct the state space using the time-delayed coordinate system, two parameters, the dimension of the state space and the delay time, need to be set appropriately. This is currently being actively studied in this field. It’s an important issue. One way to solve this problem is to optimally set these parameters by focusing on the geometric structure of the attractor, “explains Professor Ikeguchi.

To obtain the optimum values ​​for the parameters, the researchers used a five-dimensional nonlinear dynamic system to maximize the similarity of the point-to-point distance distribution between the reconstructed attractor and the original attractor. As a result, the parameters were obtained in a way that produced a reconstructed attractor that was as geometrically as close as possible to the original attractor.

Although this method was able to generate the appropriate reconstruction parameters, the researchers did not consider the noise that would normally occur in actual data that could have a significant impact on the reconstruction. “Mathematically, this method has proven to be a good method, but there are many considerations to consider before applying it to actual data analysis. This is, in fact. This is because the data in the above contains noise. The length and accuracy of the observed data are finite, ”explains Professor Ikeguchi.

Nonetheless, this method solves one of the limitations associated with determining the state of nonlinear dynamic systems encountered in various disciplines of science, economics, and engineering. “This research has created an important analytical method in the field of data science today, and I think it is important for handling a wide variety of data in the real world,” concludes Professor Ikeguchi.

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For more information:
Kazuya Sawada et al., Similarity of point-to-point distance distribution between the original attractor and the reconstructed attractor, Nonlinear theory and its applications, IEICE (2022). DOI: 10.1587 / nolta.13.385

Quote: Https: // The state of the nonlinear dynamical system (April 7, 2022) obtained on April 7, 2022. Rebuilding

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Reconstruction of the state of a nonlinear dynamic system

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