Krylov Subspace Method for Nonlinear Dynamical Systems with Random Noise
Authors: Yuka Hashimoto, Isao Ishikawa, Masahiro Ikeda, Yoichi Matsuo, Yoshinobu Kawahara
JMLR 2020 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | The empirical performance of our methods is investigated using synthetic and real-world healthcare data. Keywords: Nonlinear dynamical system, Transfer operator, Krylov subspace methods, Operator theory, Time-series data ... 7. Numerical Results We empirically evaluate the behavior of the proposed Krylov subspace methods in Subsection 7.1 then describe their application to anomaly detection using real-world time-series data in Subsection 7.2. |
| Researcher Affiliation | Collaboration | Yuka Hashimoto EMAIL NTT Network Technology Laboratories, NTT Corporation 3-9-11, Midori-cho, Musashinoshi, Tokyo, 180-8585, Japan / Graduate School of Science and Technology, Keio University 3-14-1, Hiyoshi, Kohoku, Yokohama, Kanagawa, 223-8522, Japan Isao Ishikawa EMAIL Faculty of Science, Ehime University 2-5, Bunkyo-cho, Matsuyama, Ehime, 790-8577, Japan / Center for Advanced Intelligence Project, RIKEN 1-4-1, Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan Masahiro Ikeda EMAIL Center for Advanced Intelligence Project, RIKEN 1-4-1, Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan / Faculty of Science and Technology, Keio University 3-14-1, Hiyoshi, Kohoku, Yokohama, Kanagawa, 223-8522, Japan Yoichi Matsuo EMAIL NTT Network Technology Laboratories, NTT Corporation 3-9-11, Midori-cho, Musashinoshi, Tokyo, 180-8585, Japan Yoshinobu Kawahara EMAIL Institute of Mathematics for Industry, Kyushu University 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan / Center for Advanced Intelligence Project, RIKEN 1-4-1, Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan |
| Pseudocode | Yes | Appendix C. Pseudo-codes of Arnoldi and shift-invert Arnoldi methods Let RS:T be the matrix composed of ri,t (S t T, 0 i S 1). The pseudo-codes for computing KS with the Arnoldi method and shift-invert Arnoldi method are shown in Algorithms 1 and 2, respectively. Algorithm 1 Arnoldi method for Perron-Frobenius operator K in an RKHS ... Algorithm 2 Shift-invert Arnoldi method for Perron-Frobenius operator K in an RKHS |
| Open Source Code | No | The paper does not provide explicit statements about open-source code availability or links to a code repository for the described methodology. |
| Open Datasets | Yes | We used electrocardiogram (ECG) data (Keogh et al., 2005).1 1. Available at http://www.cs.ucr.edu/~eamonn/discords/. We used chfdb chf01 275.txt , chfdb chf13 45590.txt and mitdbx mitdbx 108.txt in the experiment. |
| Dataset Splits | Yes | Using the synthetic data, K was first estimated, then the empirical abnormalities at,S,N were computed using all time-series data with t = 1601, N = 50, 75, 100 and S = 1, . . . , 12. We chose time points t = 1601 for evaluation because the estimation of K requires { x0, . . . , x N (S+1)} and 1601 > N (S + 1) for all N = 50, 75, 100 and S = 1, . . . , 12. ... We first computed KS,N with S = 10, N = 40 then computed the empirical abnormality ˆat,S,N for each t with the shift-invert Arnoldi method. ... KS,N was computed using the data { x0, . . . , x399}, and the empirical abnormalities ˆat,S,N at t = 430, 431, . . . were computed. |
| Hardware Specification | No | The paper does not provide specific details about the hardware used for running the experiments. It only discusses the empirical evaluation of methods using synthetic and real-world data without mentioning computational resources. |
| Software Dependencies | No | The paper does not provide specific software dependencies with version numbers (e.g., library names like PyTorch or scikit-learn with their versions) that were used to implement the methods or run experiments. |
| Experiment Setup | Yes | We used 100 synthetic time-series datasets { x0, . . . , x T } randomly generated by the following three dynamical systems: x0 = 2, xt+1 = 0.9995xt + 0.1ξt, (18) x0 = 0.5, xt+1 = 0.99xt cos(0.1xt) + ξt, (19) x0 = 0.1, x1 = x0 + 0.5x3 0 + ξ1, xt+1 = xt + 0.5(xt xt 1)3 + ξt, (20) where {ξt} is i.i.d with ξt N(0, 0.01). ... Using the synthetic data, K was first estimated, then the empirical abnormalities at,S,N were computed using all time-series data with t = 1601, N = 50, 75, 100 and S = 1, . . . , 12. ... The Gaussian kernel was used, and γ = 1+1i, where i denotes the imaginary unit, was set for the shift-invert Arnoldi method. ... We first computed KS,N with S = 10, N = 40 then computed the empirical abnormality ˆat,S,N for each t with the shift-invert Arnoldi method. The Laplacian kernel and γ = 1.25 were used. ... In this example, p was set as p = 15, 30, KS,N was computed using the data { x0, . . . , x399}, and the empirical abnormalities ˆat,S,N at t = 430, 431, . . . were computed. Also, the results obtained using long short-term memory (LSTM) (Malhotra et al., 2015) and autoregressive (AR) model (Takeuchi and Yamanishi, 2006) were evaluated for comparison. LSTM with 15, 30 time-series and 10 neurons, with the tanh activation function and the AR model xt+1 = Pp 1 i=0 cixt i + ξt with p = 15, 30 were used. |