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γ
= 0.050
γ
= 0.129
5
4
phase
median
binary
phase
median
binary
4
3
3
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
−4
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
Time (d)
Time (d)
(a)
LZHP
complexity (
cos
)
(b)
LZHP
complexity (
sfa4
)
Fig. 1.
Examples of instantaneous phases and
LZHP
complexity
shifts (Figure 1(b)). In this case, the binary version for the
LZ
algorithm is a
rectangular signal with varying pulse widths. The
LZ
complexity index measures
the degree of disorder on this quasi-rectangular signal. The
ApEn
complexity
index measures the degree of short-term unpredictability in phase evolution by
comparing small fractions of the irregular sawtooth shaped instantaneous phase.
4 Results and Discussion
Table 1 shows the results achieved for instantaneous phase complexity measures
LZHP
,
ApEnHP
and
ApEnWP
. They have been applied to the
sfa4
and
sflp
signals, in order to obtain some insight about the dependence on the signal
bandwidth. Results for PPS surrogate means (
PPS sfa4
and
PPS sflp
)arealso
shown. Surrogates have been preprocessed in the same way that the original sig-
nal. The parameter
β
1
is the slope of the linear regression,
r
is the corresponding
correlation coe
cient and
ρ
is the Spearman correlation coe
cient. Figures 2(a)
and 2(b) show that
LZHP
and
ApEnWP
complexity indexes decrease with age.
The values
m
=2and
r
=0
.
2
× std
have been selected for the
ApEn
(
m, r
)
statistic (see Section 3.1). In the case of LZ complexity, the median is used as the
threshold for binary conversion because of its robustness to outliers and norma-
lization properties [9]. In the case of
ApEn
complexity of CWT instantaneous
phase (
ApEnWP
), complex gaussian wavelets have been selected because of its
symmetry and smoothness. The particular combination of order and resolution
(1st order cgau1 at scale 43) has been selected because it corresponds to the
frequency 1.01
c/d
, quite close to the circadian normal frequency. CWT allows
the use of a continuous scale, while in DWT only discrete 2
l
scales are allowed.
In general, results show that phase complexity significantly decreases with
age. Our results suggest that circadian phase irregularities and phase shifts are
less frequent when ageing.
LZHP
and
ApEnHP
appear as the best behaved
statistics for phase complexity. In fact, both indexes, when rounded, provide
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