Bootstrap nonlinear prediction

Estimating the Jacobian matrix of a nonlinear dynamical system through observed time-series data is one of the important steps in predicting future states of the time series. The Jacobian matrix is estimated using local information about divergences of nearby trajectories. Although the basic algorithm for estimating the Jacobian matrix generally works well, it often fails for short or noisy data series. In this paper, we proposed a scheme to effectively use near-neighbor information for more accurate estimation of the Jacobian matrix using the bootstrap resampling method. Then, to confirm the validity of the proposed method, we applied it to a mathematical model and several real time series. As a result, we confirmed that the proposed method greatly improves nonlinear predictability, not only for noise-corrupted mathematical models but also for real time series.

Figure 1

(Color online) (a) Time-series data of the Japanese vowel /a/. The original sampling rate is $48\mathrm{kHz}$. The mean value was subtracted from the original time series. (b) Reconstructed attractor of the Japanese vowel /a/ with $d=3$ and $\tau =5$.

Figure 2

(Color online) Time-series data of the difference of the number of (a) measles and (b) chickenpox patients [6].

Figure 3

(Color online) Improvement ratio $R$ of the proposed prediction based on the bootstrap resampling prediction to several conventional prediction methods: the Jacobian-matrix estimate prediction (solid lines with circles), the Lorenz method of analogs (dotted lines with crosses), the weighted average prediction (dash-dotted lines with triangles), and the negative exponent prediction (dashed lines with rectangles). Predicted data are $\mathbf{(}x\left(t\right),y\left(t\right)\mathbf{)}$ of the Ikeda map. The near-neighbor radius is set to (a) 1%, (b) 3%, (c) 5%, and (d) 10%, and the prediction step $s$ is 1.

Figure 4

(Color online) Properties of the improvement ratio $R$ to the variables $r$ and $s$. Improvements by the proposed method are indicated by closed circles. As the conventional method, we use the original Jacobian-matrix estimate prediction to the Ikeda map. $N$ is the data length. $N=$ (a) 100, (b) 200, (c) 500, and (d) 1000.

Figure 5

(Color online) Properties of the improvement ratio $R$ to the variables of $r$ and $s$ for an Ikeda map that is corrupted by observational noise with different noise levels: (a) noiseless, and $R_{\mathrm{sn}}=$ (b) 40, (c) 30, and (d) $20\mathrm{dB}$. Improvements by the proposed method are indicated by closed circles. The data length $N$ is fixed at 200.

Figure 6

(Color online) Properties of the improvement ratio $R$ to the variables of $r$ and $s$ for the Ikeda map that is corrupted by observational noise with different noise levels: (a) noiseless and $R_{\mathrm{sn}}$= (b) 40, (c) 30, and (d) $=20\mathrm{dB}$. Improvements by the proposed method are indicated by closed circles. The data length $N$ is fixed at 500.

Figure 7

(Color online) Properties of the improvement ratio $R$ to the variables of $r$ and $s$ for the Japanese vowel /a/ embedded in a $d$-dimensional state space: $d=$ (a) 2, (b) 3, (c) 4, and (d) 5. Improvements by the proposed method are indicated by closed circles. The data length $N$ is 1000.

Figure 8

(Color online) Properties of the improvement ratio $R$ to the variables of $r$ and $s$ for the difference of the number of measles patients embedded in a $d$-dimensional state space: $d=$ (a) 2, (b) 3, (c) 4, and (d) 5. Improvements by the proposed method are indicated by closed circles.

Figure 9

(Color online) Properties of the improvement ratio $R$ to the variables of $r$ and $s$ for the difference of the number of chickenpox patients embedded in a $d$-dimensional state space: $d=$ (a) 2, (b) 3, (c) 4, and (d) 5. Improvements by the proposed method are indicated by closed circles.