Evaluation of nonlinearity and validity of nonlinear modeling for complex time series

Even if an original time series exhibits nonlinearity, it is not always effective to approximate the time series by a nonlinear model because such nonlinear models have high complexity from the viewpoint of information criteria. Therefore, we propose two measures to evaluate both the nonlinearity of a time series and validity of nonlinear modeling applied to it by nonlinear predictability and information criteria. Through numerical simulations, we confirm that the proposed measures effectively detect the nonlinearity of an observed time series and evaluate the validity of the nonlinear model. The measures are also robust against observational noises. We also analyze some real time series: the difference of the number of chickenpox and measles patients, the number of sunspots, five Japanese vowels, and the chaotic laser. We can confirm that the nonlinear model is effective for the Japanese vowel /a/, the difference of the number of measles patients, and the chaotic laser.

Figure 1

(Color online) Estimation of (a) nonlinearity with Eq. (3) and (b) efficiency of nonlinear modeling with Eq. (5) for the Rössler model with and without observational noise and its FT surrogate.

Figure 2

(Color online) The same as Fig. 1, but for the real data described in the text.

Figure 3

(Color online) Reconstructions of the five Japanese vowels $(N=8000)$: (a) /a/ $(\tau =4)$, (b) /i/ $(\tau =15)$, (c) /u/ $(\tau =13)$, (d) /e/ $(\tau =10)$, and (e) /g/ $(\tau =8)$ [13].

Figure 4

(Color online) The same as Fig. 1, but for the five Japanese vowels.