# Approximating high-dimensional dynamics by barycentric coordinates with linear programming

## Abstract

The increasing development of novel methods and techniques facilitates the measurement of high-dimensional time series but challenges our ability for accurate modeling and predictions. The use of a general mathematical model requires the inclusion of many parameters, which are difficult to be fitted for relatively short high-dimensional time series observed. Here, we propose a novel method to accurately model a high-dimensional time series. Our method extends the barycentric coordinates to high-dimensional phase space by employing linear programming, and allowing the approximation errors explicitly. The extension helps to produce free-running time-series predictions that preserve typical topological, dynamical, and/or geometric characteristics of the underlying attractors more accurately than the radial basis function model that is widely used. The method can be broadly applied, from helping to improve weather forecasting, to creating electronic instruments that sound more natural, and to comprehensively understanding complex biological data.

- Authors:

- Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505 (Japan)
- (Japan)
- Department of Mathematical Informatics, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656 (Japan)
- Center for Research in Agricultural Genomics (CRAG), Consorci CSIC-IRTA-UAB-UB, Barcelona 08193 (Spain)

- Publication Date:

- OSTI Identifier:
- 22403372

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 25; Journal Issue: 1; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; ATTRACTORS; COORDINATES; DYNAMICS; ERRORS; FORECASTING; LINEAR PROGRAMMING; MANY-DIMENSIONAL CALCULATIONS; MATHEMATICAL MODELS; PHASE SPACE; SIMULATION; TIME-SERIES ANALYSIS; TOPOLOGY; WEATHER

### Citation Formats

```
Hirata, Yoshito, E-mail: yoshito@sat.t.u-tokyo.ac.jp, Aihara, Kazuyuki, Suzuki, Hideyuki, Department of Mathematical Informatics, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, CREST, JST, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Shiro, Masanori, Mathematical Neuroinformatics Group, Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8568, Takahashi, Nozomu, and Mas, Paloma.
```*Approximating high-dimensional dynamics by barycentric coordinates with linear programming*. United States: N. p., 2015.
Web. doi:10.1063/1.4906746.

```
Hirata, Yoshito, E-mail: yoshito@sat.t.u-tokyo.ac.jp, Aihara, Kazuyuki, Suzuki, Hideyuki, Department of Mathematical Informatics, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, CREST, JST, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Shiro, Masanori, Mathematical Neuroinformatics Group, Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8568, Takahashi, Nozomu, & Mas, Paloma.
```*Approximating high-dimensional dynamics by barycentric coordinates with linear programming*. United States. doi:10.1063/1.4906746.

```
Hirata, Yoshito, E-mail: yoshito@sat.t.u-tokyo.ac.jp, Aihara, Kazuyuki, Suzuki, Hideyuki, Department of Mathematical Informatics, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, CREST, JST, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Shiro, Masanori, Mathematical Neuroinformatics Group, Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8568, Takahashi, Nozomu, and Mas, Paloma. Thu .
"Approximating high-dimensional dynamics by barycentric coordinates with linear programming". United States.
doi:10.1063/1.4906746.
```

```
@article{osti_22403372,
```

title = {Approximating high-dimensional dynamics by barycentric coordinates with linear programming},

author = {Hirata, Yoshito, E-mail: yoshito@sat.t.u-tokyo.ac.jp and Aihara, Kazuyuki and Suzuki, Hideyuki and Department of Mathematical Informatics, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656 and CREST, JST, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012 and Shiro, Masanori and Mathematical Neuroinformatics Group, Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8568 and Takahashi, Nozomu and Mas, Paloma},

abstractNote = {The increasing development of novel methods and techniques facilitates the measurement of high-dimensional time series but challenges our ability for accurate modeling and predictions. The use of a general mathematical model requires the inclusion of many parameters, which are difficult to be fitted for relatively short high-dimensional time series observed. Here, we propose a novel method to accurately model a high-dimensional time series. Our method extends the barycentric coordinates to high-dimensional phase space by employing linear programming, and allowing the approximation errors explicitly. The extension helps to produce free-running time-series predictions that preserve typical topological, dynamical, and/or geometric characteristics of the underlying attractors more accurately than the radial basis function model that is widely used. The method can be broadly applied, from helping to improve weather forecasting, to creating electronic instruments that sound more natural, and to comprehensively understanding complex biological data.},

doi = {10.1063/1.4906746},

journal = {Chaos (Woodbury, N. Y.)},

number = 1,

volume = 25,

place = {United States},

year = {Thu Jan 15 00:00:00 EST 2015},

month = {Thu Jan 15 00:00:00 EST 2015}

}