Test for Determining the Degree of Nonlinearity of Time Series - Multimedia and Signal Processing - page 551

misjudge the nonlinearity of the tested time series. This phenomenon, we think, is
reasonable. The signal constituted by the tested time series and the added noise can be
considered as a whole, when , the tested time series

is dominant, and the property of the tested time series determines the property of the
whole signal, thus the property of the whole signal is nonlinear at this time.

lon < 1 , i.e., Power

n Power

<

tts

4.3 Influence of the Embedding Dimension of the Tested Time Series on the

Results of V -Test

σ

Figures 3 (a), (b) and (c) show the V values corresponding to the time series gener-

ated by Hénon map, Logistic map and AR model with the embedding dimension
values from 1 to 10, respectively. For the time series generated by Hénon map and
Logistic map, the largest

σ

V value appears when the embedding dimension is the

σ

optimal value = m (the optimal embedding dimensions for Hénon map and Logistic

map are both equal to 2), and the further the taken embedding dimension value is
away from the optimal embedding dimension value, the smaller the corresponding

2

V value is the smallest
one when the embedding dimension is taken as the optimal value

V value is; for the time series generated by AR model, the

σ σ

= m , and the
further the taken embedding dimension value is away from the optimal embedding
dimension value is, the larger the corresponding

4

V value is basically (except when

σ

V values are still larger than that obtained with the op-
timal embedding dimension value

m = 2 9 and 10 , but these

σ

= m ).
By observing Figs. 3 (a), (b) and (c), it can be found that for the time series gener-
ated by Hénon map, Logistic map and AR model,

4

V curves show extreme point
when the embedding dimension is taken as the optimal one, the only difference is that
the

σ

V curves corresponding to the time series generated by Hénon map and Logistic
map have maximum values, whereas the

σ

V curve corresponding to the time series
generated by AR model has minimum value. Therefore, we summarize that for a
nonlinear time series, the corresponding embedding dimension-

σ

V curve shows max-
imum value, whereas for a linear time series, the corresponding embedding dimen-
sion-

σ

V curve shows minimum value. According to the results above, we propose a
simple procedure based on

σ

V -test for determining whether a time series with known
information about embedding dimension is linear or nonlinear: if the range of embed-
ding dimension values of the tested time series is known, it is not needed to calculate
its optimal embedding dimension, whereas it is only needed to pick up an embedding
dimension within the known range randomly to compute the corresponding temporary
temp

σ

V . If the obtained V > 0 . 01 , the tested time series must be nonlinear,

σ − σ − temp

since for a nonlinear time series, the V value obtained with the optimal em-

σ − optimal

bedding dimension value is larger than other V values obtained with non-

σ − temp

optimal embedding dimension values, namely V

optimal V

> > 0 . 01 , and in

σ − σ − temp