Digital Signal Processing Reference
In-Depth Information
respectively, and then the
V
values for the time series with different number of
sampling points are computed. As shown in Figs. 1 (a), (b) and (c), when
σ
n
<
1000
,
p
V
values increase rapidly as
n
increases for the time series generated by Hénon
map and Logistic map, whereas
σ
V
value descends rapidly as
n
increases for the
σ
time series generated by AR model; when
V
values tend to be stable
and fluctuate in a small interval for all three kinds of time series, the fluctuation inter-
vals for Hénon map, Logistic map and AR model are 0.25~0.28, 0.25~0.28 and
0.002~0.007, respectively. Apparently,
n
≥
1000
,
p
σ
V
-test has no high requirement on the num-
ber of sampling points of the tested time series, and stable results can be obtained
only if the number of sampling points of the tested time series is more than 1000.
Moreover, for the time series generated by AR model,
σ
V
values are smaller than
σ
n
values, therefore, it is considered to set 0.01 as a threshold value, i.e.,
0.01 for all
if
V
<
0
.
01
, the tested time series is linear ; conversely, the tested time series is
σ
nonlinear if
V
>
0
.
01
.
σ
4.2
Influence of Noise on the Results of
V
-Test
σ
Different levels of Gaussian white noise are added to the tested time series, and then
Power
n
V
values are computed. The level of noise is measured by
lon
=
, where
σ
Power
tts
Power
is the power of the added noise, and
Power
is the power of the tested time
n
tts
series. Figure 2 shows the results of
V
values of the tested time series with different
level of noise. With the power of the added noise increasing, the
σ
V
values descend
for both kinds of tested time series. When
lon
is equal to 1, i.e., the power of the
added noise is equal to that of the tested time series, the corresponding
σ
V
values for
σ
both Hénon map and Logistic map are close to 0.01, at this time
V
-test may
σ
0.3
Logistic map
Henon map
0.25
0.2
0.15
0.1
0.05
0.01
0
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
lon
V
values of tested time series with different level of noise. (
lon
stands
for the level of noise)
Fig. 2.
Results of
σ
