Full text of DOI:10.1007/BF02900584

NOTES
be proved useful for modeling in the presence of noise
[
8,13]
Detecting deterministic dy-
namics of cardiac rhythm
and full detection of deterministic dynamics
.
To obtain our primary goal for modeling heart rate
variability, the cluster-weighted filtering (CWF) method is
proposed to model the deterministic component of the
approximately dynamical system from complex and noisy
time series. In order to approach the complexity of origi-
nal attractor, the normalized Gaussian network is used as
1
PEI Wenjiang , HE Zhenya , YANG Luxi , S. S. Hull
1
1
2
3
&
J. Y. Cheung
1
2
3
. Department of Radio Engineering, Southeast University, Nanjing
10096, China;
. University of Oklahoma Health-Sciences Center, Oklahoma City, OK
3190, USA;
. School of Electrical & Computer Engineering, University of Okla-
homa, Norman, OK 73019, USA
[
9]
2
function approximator . By putting the problem of
[14,15]
nonlinear noise reduction into a Bayesian framework
,
7
we introduce an appropriate way of combining model
estimation with noise reduction procedures. It enables us
to determine the correct amount of noise reduction from
the data itself and thereby avoiding over-fitting and over-
Correspondence should be addressed to Pei Wenjiang (e-mail:
wjpei@jlonline.com)
[
16,17]
filtering
.
With the applications of CWF to the analysis of
Abstract Under the acceptable hypothesis that cardiac
rhythm is an approximately deterministic process with a
small scale noise component, an available way is provided to
construct a model that can reflect its prominent dynamics of
the deterministic component. When applied to the analysis of
[
18]
heartbeat intervals , we investigate the free-run dynam-
ics of the models to see if they exhibit linear and nonlin-
ear features consistent with those of original time series.
This involved comparing the time delay plots of the
free-run behavior derived from the models with that of
originals. We estimates the dynamical and measurement
noise levels to confirm that CWF neither changes the un-
derlying deterministic structure nor introduces spurious
deterministic structure into the data. In order to test
whether the deterministic component of cardiac dynamics
is sensible to initial conditions, we estimate the largest
Lyapunov exponents from the filtered time series and
free-run behavior of the models. We also use surrogate
data analysis to confirm the hypothesis that the heart rate
variability is not consistent with a linear Gaussian proc-
ess.
1
9 heart rate data sets, three main findings are stated. The
obtained model can reflect prominent dynamics of the de-
terministic component of cardiac rhythm; Cardiac chaos is
stated in a reliable way; Dynamical noise plays an important
role to the generation of complex cardiac rhythm.
Keywords: cardiac rhythm, nonlinear dynamics, Bayesian estima-
tion.
It has been proposed that the heart rate variability
may display complex nonlinear dynamics, including de-
[
1ü5]
terministic chaos
. Furthermore, it may be more rea-
sonable that cardiac dynamics is neither chaotic nor sto-
chastic, but rather both. In such a system, the unpredict-
ability of the output will be due to the uncertainty gener-
ated by the deterministic component, and to the stochastic
1
Method of cluster-weighted filtering
[
6,7]
inputs
.
N
i 1
Let {xi }
be the observed time series of an ap-
The primary goal of this study is the desire to obtain
a model that can reflect the prominent dynamics of the
deterministic component for heart rate variability under
such an assumption. That is, the model should not only fit
data and make a good short-term prediction, but should
also have dynamical behavior similar to that of the ob-
proximately dynamical system.
y ꢀH ,
xi
i
i
(1)
(2)
yi f ( y ) ꢀe ,
i
i
where yi (y , y ,ꢀ, y ). f denotes system dy-
iꢁ1
iꢁ2
iꢁd
[
8ü12]
namics with unknown parameters that need to be esti-
mated, e a stochastic element in the dynamics and H acts
as measurement noise.
Our goal is thus to estimate y and f from x. In order
to find the dependencies of y on y and y on x based on the
joint density p(x, y, y) (for simplicity, y and x are defined
as input and output of an approximately dynamical system
under consideration respectively), we expend the joint
served variable
. If we can obtain such a model, the
dynamics of the model will then present a reliable detec-
tion of deterministic dynamics for the heart rate variability.
To satisfy this stringent criterion, some available methods
[
8ü11]
with partial solutions have been provided
.
Since dynamical noise is a major source of variabil-
ity in the feedback system and the reliability of the result-
ing model as a long-term predictor is highly dependent on
the amount of measurement and dynamical noises present,
some questions remain concerning the significance of the
models obtained by such methods directly in the analysis
…
, M,
each of
density p(x, y, y) into M clusters c
m
, m = 1,
[
9]
which contains the product of four terms .
M
[
4,8]
p(x, y, y) =
m m m m
p (x| y, y, c )gp(y|y, c )gp(y|c )gp(c ),
of cardiac dynamics . It has been also stated that the
new directions for such methods, including elimination of
the influences of measurement and dynamical noises, may
¦
m 1
(
3)
1568
Chinese Science Bulletin Vol. 46 No. 1 January 2001

where p(c
main of influence in the input space of the cluster. p(y|y,
) denotes the dependence of y on y and c , the remain-
ing item expresses the output distribution of the cluster.
Generally, p(y|c ) is taken to be Gaussian with a
mean P and covariance matrix C . The dynamical noise
m
) is the weight of a cluster, and p(y|c
m
) the do-
new
chose D(yi , y ) as the criterion to accept or reject
new
i
c
m
m
yi
i
as a new sample of y :
new
yi , yi
D
ꢂ
ꢃ
m
M
­
§
¨
©
new
2
mİ
2
me
·
¸
¹
½
m
m
p y
^ j
y
`jzi , x,
^
E ,V ,V
`
i
m
m 1
(9)
°
°
°
°
e and measurement noise Hare also taken to be Gaussian
min
, 1 ,
®
¾
M
2
me
2
mH
§
©
2
2
me
·
¹
with variances V
and V
associated with cluster
° p y ^ j
y
`jzi , x,
^
E , V ,V
`
°
¨
i
m
mH
¸
m 1
°
°
¯
¿
m
C respectively. Here the mean value of y is replaced by
the output of local model f
f ( y, E )
m
(y, E
m
),
I
§
©
2
mH
2
M ·
p y ^ j
y
`jzi, x,
^
Em, V , V
`
exp
¨
i
me m 1 ¸
Emi fmi ( y).
(4)
¹
m
m
¦
i 1
§
¨
§
2
2 ··
§
¨
¨
©
·
ꢁ
(y ꢁx ) ꢁ>yi ꢁ f ( y , E )
@
¨
i
i
m
i
m
¸
¸¸
An iterative algorithm, the expectation-maximization
EM) algorithm search of the maximization of the model
likelihood given a data set and initial conditions, is used
p
ꢂ
c x , y , y
m
ꢃ
ꢀ
i
i
i
¨
¨
¨
¨
M ¨
2
mH
2
me
¸¸¸
V
V
(
¹
¨
¸¸
,
¦
iꢀd
2
¨
¸
¸
ꢁ
>yj ꢁ f ( y , E )
@
[
9,19]
m 1
m
j
m
¨
ꢀ
p(c | x , y , y )
m j j j
¸¸
for modeling and filtering
.
¦
2
me
¨
©
¨
©
V
¸¸
¹¹
In the E-step, we assume the current cluster parame-
ters to be correct and evaluate the posterior probabilities
that relate each cluster to each data point. These posteriors
can be interpreted as the probability that particular data
are generated by a particular cluster.
j
iꢀ1
(
10)
new
new
iꢁ1 iꢁd
where yi
(y ,ꢀ, yi ,ꢀ, y ). For the fast
convergence of the algorithm, the candidate sample y in ew
can be obtained based on taking the derivative of the total
log-likelihood function with respect to the data point
(terms associated with measurement noise are ignored so
p(x, y, y | c ) p(c )
m
m
p(c | x, y, y)
m
.
(5)
M
p(x, y, y | c ) p(c )
m
¦
m
m 1
[
14]
In the M-step, we assume the current data distribu-
tion to be correct and find the cluster parameters that
maximize the likelihood of the data. The new estimate for
the unconditioned cluster probabilities is
that the algorithm will proceed ).
Finally, the model estimation and the noise reduction
process by cluster-weighted filtering can be summarized:
(ν) Initialization (choose 1/M as the initial cluster prob-
abilities. Pick randomly M points from data set as the
cluster input means); (ii) evaluation of the probability of
N
1
p(c ) |
p(c | x , y , y ).
i
(6)
m
¦
m
i
i
N
i 1
the data p (x, y, y|c
probability of the clusters p (c
cluster weights p(c ), the cluster-weighted exceptions for
m
); (iii) calculation of the posterior
Given p(c
m
), we define a cluster-weighted expectation of
m
|x, y, y); (iv) update of the
function T(x, y, y)
m
N
input means P
m
and covariance matrix C
m
, the maximum
T(x , y , y ) p(c | x , y , y )
¦
i
i
i
m
i
i
i
log-likelihood model parameters E
m
, dynamical noise
2
i 1
2
me
T(x, y, y) m
|
.
(7)
variance
and measurement noise variance
; (v)
V
V
mH
N
N
i 1
p(cm | x , y , y )
i
¦
The cluster-weighted expectation is used to update
i
i
update of
^
y
`
based on M-H algorithm; (vi) return
i 1
to (ii) until the total data likelihood does not increase any
more.
2
me
2
mH
[14]
P
m
, C
m
, V and V discussed above . An expression
2
Experimental materials
to update E
m
= [Em1, Em2, …, EmI] for each cluster c
be obtained by maximizing the log-likelihood function.
N
m
can
1
9 mixed breed male dogs (15 ü 24 kg in weight, 2
ü 4 years old by dentition) free of heartworms were con-
ditioned for chronic study. Each dog was prepared surgi-
cally with an anteroseptal myocardial infarction or was
shamed operated. Heart rate variability was studied before
infarction. After the last heart rate variability test, each
dog was identified as being at either high or low risk for
the development of ventricular fibrillation during a sub-
maximal exercise and transient myocardial ischemia test.
log
p(x , y , y ) .
i i i
3
i 1
ꢁ
1
m
Em
fmi ( y) fmi ( y) m , [Am ]i
In this note, Metropolis-Hastings algorithm is used
B
A ,
(8)
m
where [Bm ]ij
yfmi (y) m .
N
[14]
for updating time series ^ i
y
`
. For simplicity we
i 1
Chinese Science Bulletin Vol. 46 No. 1 January 2001
1569

NOTES
The procedure describing the surgically created
[
3
Results and discussions
18]
.
myocardial infarction was presented in detail elsewhere
From the results in ref. [7], when applying parsimo-
During surgical plane anesthesia, the heart was exposed
and the fatty tissue was dissected from the vessel ap-
proximately 2 cm from the origin surrounding the circum-
flex branch of the left coronary artery. A loose fitting
nious Volterra series model method directly to the analysis
of heart beat intervals, we find that the largest Lyapunov
exponents are negative for most (89%) subjects as shown
in table. 1 (Lyap_kor). These results suggest that cardiac
chaos is not prevalent in heart. In spite of the fact that this
method had been successfully tested by artificial data
(
diameter 3.0 ü 3.5 mm) Doppler flow probe (20 MHz,
Hartley) and, immediately distally, a pneumatic vascular
occluder were implanted. To produce the myocardial in-
farction, the anterior intraventricular branch of the left
coronary artery immediately proximal to the first major
diagonal artery perforator was critically stenosed for 20
min and then permanently ligated. A catheter was planted
in the descending aorta for the later direct measurement of
arterial pressures.
[
7]
sets , it does not eliminate the influences of noise essen-
tially. So the conclusion that the deterministic component
of cardiac dynamics is chaotic or not obtained by such
methods is suspicious. In order to approach the complex-
ity of original attractor and eliminate the influences of
noise in modeling the deterministic component of cardiac
dynamics, we introduce the method of CWF. The parsi-
monious Volterra series models, which are used as local
models for cluster-weighted filtering, are fitted to the time
series using a search space that included linear, bilinear,
and trilinear terms of up to 3th order. Despite the large
search space (1 constant, 4 linear, 10 bilinear, 20 trilinear
terms), only six significant terms are included in the local
models. By searching the space of the candidate terms, the
one that gives the greatest reduction in the MSE is se-
lected. By repeating this greatest reduction in MSE at
each step, a parsimonious model can be constructed. Em-
pirically, six to eight embedding data points, which are
chosen randomly from the training set, are used as the
cluster input means initially. We discuss the power of
CWF as follows.
30 d after myocardial infarction, the dogs were stud-
ied consecutively and characterized for developing ven-
tricular fibrillation during an exercise and myocardial
ischemia test on a motor-driven treadmill. Briefly, each
dog was exercised submaximally for 12ü15 min while
the work load was increased progressively every 3 min.
(
4.8 km/h at 0% grade and 6.4 km/h at grades of 0%, 4%,
8%, 12%) until heart rate reached a target range of 215ü
2
25 beats/min. At that time, the left circumflex artery was
pneumatically occluded for 2 min; the treadmill was
stopped after the 1st min of occlusion, while ischemia was
maintained for an additional minute. The 2 min period
myocardial ischemia was verified by a zero flow trace
signal from circumflex artery Doppler flow probe. During
the 2-minute of exercises a myocardial ischemia, 8 dogs
(S_MI group) are susceptible to a sudden death because of
developing ventricular fibrillation. 11 dogs that did not
develop ventricular fibrillation were considered to be at
low risk and were defined to have resistance to a sudden
death (R_MI group).
A few days after myocardial infarction, 30 min ECG
samples at rest were obtained. All recordings were col-
lected in the late morning or early afternoon without the
use of sedation or physical restraint and before feeding.
After a 10ü20 min daily transition period in the labora-
tory, ECG data were obtained while the dog was quiet and
lying down but not sleeping on a padded examination
table. Specific care was taken to eliminate extraneous
noise, unfamiliar personnel and other environmental dis-
tractions. Data were not recorded when rectal temperature
was >39ć or the dog was judged to be behaviorally up-
set. A transthoracic modified lead I surface ECG was ob-
tained with the use of self-adhesive pads, amplified and
filtered at a low frequency (5 Hz) and digitized at 400 Hz.
All digitally encoded files were analyzed with a commer-
cially available program (Corazonix). Aberrant ECG
complexes such as premature ventricular beats, electrical
noise, or other aberrant ECG signals and their adjacent
RR intervals were rejected by software.
Fig. 1. 1200 samples of dynamical noise variance (a) and measurement
noise variance (b).
1570
Chinese Science Bulletin Vol. 46 No. 1 January 2001

First, cluster-weighted filtering yields a property of
automatic noise level estimation. For example, we apply
CWF to 1000 points observed from a stochastically driven
Henon map (the variances of dynamical and measurement
noises are 0.004 and 0.01 in delay coordinates respective-
ly ). Fig.1 shows 1200 samples of dynamical and meas-
urement noise variances. One can find that both dynami-
cal and measurement noise variances do converge ap-
proximately to the correct values after 400 iterations. To
some extent, CWF not only eliminates influences of
noises but also avoids the tendency to over-clean the data
in the process of filtering.
Second, fig. 2 shows delay plots of original (a, d)
and filtered (b, e) time series of heart beat intervals of two
subjects respectively. One can see that the attractors be-
tween the original and the filtered are very similar. Addi-
tionally, table 1 shows values of estimated relative dy-
namical (Amp_dyn) and measurement noise amplitudes
(Amp_mea) for all time series of heart beat intervals, each
of which is not larger than 3.5% of the original signal am-
plitudes. It implies that CWF does neither change the un-
derlying dynamics nor introduce spurious dynamics for
[20]
[
14]
the heart beat intervals
.
Third, fig. 2 (c), (f) also shows delay plots of the free
runs of the model obtained by CWF. That is, the model is
iterated 500 times without any corrections from the data.
Attractors similar to that of originals are also obtained.
For both subjects, free runs show complex deterministic
structures. Results can tell us that the deterministic com-
ponents of cardiac dynamics can be primarily described
by only a few of local nonlinear dynamical models.
Comparison between the originals and the model dynam-
ics also discovers that dynamical noise plays an important
role in the generation of complex cardiac rhythm.
Fig. 2. Delay plots of the time series of RR intervals from two subjects (No. 732v2, No. 683v2). (a) and (d) Original heartbeat intervals; (b) and (e)
filtered time series by CMF; (c) and (f) free runs from models.
[
7]
It is of considerable interest to study whether the
heartbeat series are chaotic. However, presence of dy-
namical and measurement noise can often lead to false
positive or negative identifications of chaos when tradi-
tional methods of nonlinear dynamics analysis are used.
Discussions listed above suggest that the models obtained
by CWF can reflect prominent dynamics of the determi-
nistic component of an approximately dynamical system,
and CWF does neither change the underlying dynamics
nor introduce spurious dynamics. So the characteristic
exponents of the deterministic components of cardiac dy-
namics can be estimated reliably from filtered time series.
By evaluating the Jacobians over all the filtered time se-
ries of heartbeat intervals, we estimate the largest
Lyapunov exponent by the algorithm based on QR de-
composition . The numerical results are shown in table 1.
Let us first consider the largest Lyapunov exponents de-
rived from filtered time series of RR intervals (Lyap_fil).
17 of 19 time series show positive values. When consid-
ering that for free run behavior of the models, again we
find 16 of 19 time series show positive values. The re-
mainders show negative values but very near to zero. It
implies that the deterministic components of cardiac dy-
namics for most subjects under studying are sensible to
initial conditions. What should be noted is that there are
not apparent differences between R_MI group and S_MI
group in the values of the largest Lyapunov exponents.
Finally, the widespread surrogate data method is
[
21,22]
used to test the nonlinearity for heartbeat intervals . In
this study, the amplitude-adjusted Fourier transform
Chinese Science Bulletin Vol. 46 No. 1 January 2001
1571

NOTES
[
21]
[23,24]
(
and the short-term predictability calculated by Sugi-
AAFT) algorithm is used to generate surrogate data
,
hara-May method acts as a statistic test
cance can be measured by S:
. The signifi-
Table 1 Calculated characters of heart rate variability for each subject
Lyap fil Lyap pre Amp dyn Amp mea
1.26 1.43 0.0232 0.0126
Subject
Lyap
ꢁ
kor
ꢁ
ꢁ
ꢁ
ꢁ
S
ꢁ
SM
ꢁ
S CWF
7
6
6
6
7
7
6
6
6
6
6
7
7
6
7
6
7
7
6
32v2
79v2
65v2
87v2
17v2
01v2
82v2
73v2
71v2
69v2
83v2
29v4
24v4
75v4
25v4
70v4
06v4
13v4
67v4
0.21
2.96
1.88
1.73
4.34
8.62
4.24
19.73
16.92
18.26
12.78
8.25
ꢁ0.34
ꢁ0.11
ꢁ0.07
ꢁ0.02
ꢁ0.26
ꢁ0.45
ꢁ0.05
ꢁ0.62
ꢁ0.14
0.59
0.49
0.21
0.07
0.14
-0.07
0.11
0.02
0.07
0.11
0.08
0.14
0.07
0.07
1.43
0.08
0.37
0.13
0.60
0.21
0.36
0.06
0.0241
0.0104
0.0208
0.0064
0.0125
0.0107
0.0034
0.0302
0.0066
0.0157
0.0195
0.0144
0.0028
0.0186
0.0350
0.0144
0.0346
0.0077
0.0172
0.0106
0.0024
0.0074
0.0047
0.0277
0.0042
0.0197
0.0077
0.0093
0.0121
0.0077
0.0041
0.0057
0.0078
0.0062
0.0188
0.0022
ꢁ0.08
0.10
9.52
ꢁ3.35
3.52
8.82
ꢁ0.02
0.10
0.10
0.11
0.16
0.07
0.11
1.41
0.16
0.45
0.21
5.75
ꢁ2.97
3.18
1.66
3.34
1.32
2.13
2.24
1.18
0.31
10.43
5.93
7.83
ꢁ0.23
ꢁ0.19
ꢁ0.12
ꢁ0.08
ꢁ0.19
ꢁ0.25
ꢁ0.16
ꢁ0.11
15.07
12.39
7.42
28.26
2.96
6.48
ꢁ0.67
1.71
21.34
5.92
ꢁ0.04
ꢁ0.01
cardiac rhythm behaves as a nonlinear deterministic sys-
tem with finite degrees of freedom. At the smaller scales
the system is likely to be dominated by noise (determinis-
ꢂ
M surr ꢁM org
ꢃ
S
,
(11)
V
[
4,6,25,26]
tic high dimensional motion or true noise source)
.
where M surr and V are the mean value and standard
This is in agreement with the physiology of the system.
The heart rate is determined by the activity in the auto-
nomic nerves that supplies the sinus. The activity in these
fibres is not only determined by feedback from the
baroreceptors in the cardiovascular system, but also in-
fluenced by inputs from many other systems, including
hormonal systems, and higher centers such as the cerebral
deviation of the predicative error associated with 20 sur-
rogate data sets, Morg is the predicative error of original
time series.
Results are shown in table 1 (S_SM). Only for 47%
of the subjects (S˚1.96) can we reject the null hypothesis
(
P˘0.05) that the data arise from a linear Gaussian proc-
ess. By such a method, no obvious evidences of nonlin-
earity in cardiac rhythms are found. Similar results were
[
4,6]
cortex . Because of its intrinsic complex mechanism,
although many researchers have studied cardiac rhythm in
a variety of ways in an attempt to determine whether it is
deterministic chaos or stochastic noise, the conclusions
[
24]
also obtained by Lefebvre . He concluded that the evi-
dence of deterministic chaos in cardiac rhythms is not
strong or persistent, and this relatively small effect was
believed to be a consequence of many rapidly changing
physiological inputs (can be viewed as dynamics noises)
to the sinus node. Fortunately, CWF shows power in
modeling deterministic components of such systems from
complex and noisy time series. When the one step predi-
cative errors of filtered time series of RR intervals and 20
surrogate data sets are obtained by CWF, we test nonlin-
earity reliably for all the subjects under studying. The
values of significance are obtained in table 1 (S_CWF).
[
4,6,24ü27]
have been mixed, sometimes contradictory
. In this
note, a powerful method has been presented for modeling
deterministic component of such an approximately dy-
namical system from complex and noisy heartbeat inter-
vals. After the reliability of the proposed method is both
supplied from the point view of analysis and tested by
artificial data, we applied it to the analysis of 19 heart rate
data sets. The main findings are listed. The obtained mod-
els can reflect prominent dynamics of the deterministic
component of cardiac rhythms; cardiac chaos is stated in a
reliable way; Dynamical noise plays an important role to
the generation of complex cardiac rhythm.
For each subject, we can reject the null hypothesis (P˘
0
.05).
Conclusions
An acceptable hypothesis is that cardiac rhythm is a
complex deterministic process with a small scale noise
high dimensional) components. On the largest scales,
4
Acknowledgements This work was supported by the National Natural
Science Foundation of China (Grant Nos. 69735101 and 69872009).
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(
Received March 14, 2001)
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