non-gaussian characteristics of heart rate variability in health and disease ken kiyono division of...
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Non-Gaussian characteristics of heart rate variability in health and disease
Ken KiyonoDivision of Bioengineering
Graduate School of Engineering Science Osaka University
Multifractal Analysis: From Theory to Applications and Back
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Outline
1. Heart rate variability (HRV)
2. Relation between multifractality and non-GaussiantyMultiplicative random cascade model
3. Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution
4. Non-Gaussian properties of HRVPrediction of mortality risk
5. Summary
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Outline
1. Heart rate variability (HRV)
2. Relation between multifractality and non-GaussiantyMultiplicative random cascade model
3. Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution
4. Non-Gaussian properties of HRVPrediction of mortality risk
5. Summary
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Heart rate variability (HRV)
HRV is the temporal fluctuation of heart rhythm. The times series is derived from the QRS to QRS (RR) interval sequence of
the ECG, by extracting only normal-to-normal interbeat intervals. The normal ECG is composed of a P wave, a QRS complex and a T wave.
RR intervals fluctuate beat by beat.
Electrocardiogram (ECG)
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[Kleiger et al., Am. J. Cardiol., 59 (1987) 256]
HRV as a predictor of mortality
Reduced heart-rate variability has been shown to be a risk factor for increased mortality after myocardial infarction
[Reprinted from Kleiger et al., Am. J. Cardiol., 59 (1987) 256]
SD of RRINo. of
PatientsMortality rate
> 100 ms 211 9.0%
50-100 ms 472 13.8%
< 50 ms 125 34.4%
Lower HRV was associated with a higher risk.
Mortality rate of patients after myocardial infarction
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Physiological cause of heart rate variability
HRV is mainly controlled by autonomic nervous system
(ANS). Parasympathetic blockade reduces heart rate fluctuation.
[Reprinted from 井上博編,循環器疾患と自律神経機能 ( 第2版 ) , 2001, 医学書院 ]
Pharmacological blockade experiment Intravenous administration of propranolol and atropine.
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[Kleiger et al., Am. J. Cardiol., 59 (1987) 256]
Frequency characteristics of HRV
Through the Fourier transform, observed signals can be decomposed into the
superposition of sinusoidal signal with different frequencies and amplitudes.
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HF and LF components of HRV
short-term HRV ( ~ 5 min)
High frequency (HF; 0.15-0.4 Hz) band : Synchronization between respiration (~ 4 s) and HRV
Low frequency (LF; 0.04-0.15 Hz) band :Mayer wave (~10 s) , Blood pressure oscillations
sympathetic control
parasympathetic control
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1/f fluctuation of HRV
Healthy HRV power spectrum shows a 1/f-type power-law scaling
[Kobayashi, Musha, IEEE Trans. Biomed. Eng. 29, 456 (1982)]
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1/f noise and fractal (1)
A simple repeating rule can produce 1/f noise.
Deterministic fractality Stochastic fractality
Koch curve
1/f noise
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1/f noise and fractal (2)
Random variables lie on a dyadic grid.
function.floor theis where
,1
)(
12/)1(
・
m
j
ji jmi
WX
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Multifractality of Heart Rate Variability
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Analogy between HRV and cascade model
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Outline
1. Heart rate variability (HRV)
2. Relation between multifractality and non-GaussiantyMultiplicative random cascade model
3. Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution
4. Non-Gaussian properties of HRVPrediction of mortality risk
5. Summary
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Multiplicative random cascade (1)
(cf. Additive random cascade can generate 1/f noise)
Multifractal time series can be generated by multiplicative cascade.
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Multiplicative random cascade (2)
where is the floor function.
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Non-Gaussian PDF of cascade model
Let us consider a multiplicative cascade-type model,
where is the floor function. When and , the probability density function of is given by
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Multiscaling property of structure function
The structure function is defined as,
,
where
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Multiscaling property of cascade-type model
Using a Gaussian approximation of the partial sums, we obtain
where ΨY is the cumulant-generating function of Y(j)
(If Y(j) is a Gaussian variable, ΨY is a quadratic function.)
white Gaussian
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Multiscale PDF analysis
i
kki
s
jjiisiis
UZ
UZZZ
1
1
. where
,
Fine resolution
Coarse resolution
Partial sum process {DsZi}
Deformation of PDFs across scales
Convergence to a Gaussian
[Castaing et al., Physica D, 46, 177 (1990); Kiyono et al., IEEE TBME 53, 95-102 (2006)]
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Convergence process to a Gaussian
Log-normal cascade model iid sequence
u iu i
s = 1, 2, 4, 8, 16, 32from top to bottom
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Outline
1. Heart rate variability (HRV)
2. Relation between multifractality and non-GaussiantyMultiplicative random cascade model
3. Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution
4. Non-Gaussian properties of HRVPrediction of mortality risk
5. Summary
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Parameter estimation problem of non-Gaussian processes
Conventional models of non-Gaussain distributions ■ Castaing’s model [Castaing, Gagne & Hopfinger, Physica D, 46, 177 (1990)]
This model is involved with turbulent cascade picture.
■ Superstatistics [Beck & Cohen, Physica A, 322, 267 (2003)]
Superstatistics considers a driven nonequilibrium system that consists of many subsystems with different values of some intensive parameter b (the inverse effective temperature).
■ Heavy tailed distributions (independently and identically distributed process)
◇ symmetric Levy stable distribution, P(x) ~ |x|-(a+1) (0 < a < 2) for large |x|
◇ stretched exponential distribution, P(x) exp(-∝ g|x|a)
PL(x): local equilibrium distribution
f (b): fluctuations of intensive parameter
PL(x): PDF at integral scale L
G(s): fluctuations through energy cascade
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Models for non-Gaussian fluctuations
Conventional models of non-Gaussain distributions ■ Castaing’s model [Castaing, Gagne & Hopfinger, Physica D, 46, 177 (1990)]
This model is involved with turbulent cascade picture.
■ Superstatistics [Beck & Cohen, Physica A, 322, 267 (2003)]
Superstatistics considers a driven nonequilibrium system that consists of many subsystems with different values of some intensive parameter b (the inverse effective temperature).
■ Heavy tailed distributions
◇ symmetric Levy stable distribution, P(x) ~ |x|-(a+1) (0 < a < 2) for large |x|
◇ stretched exponential distribution, P(x) exp(-∝ g|x|a)
PL(x): local equilibrium distribution
f(b): fluctuations of intensive parameter
PL(x): PDF at integral scale L
G(ln s): fluctuations through energy cascade
The Mellin convolution of f and g :
X ~ PL(x) sX s ~ G(ln s)
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Parameter estimation problem of non-Gaussian processes
Conventional models of non-Gaussain distributions ■ Castaing’s model [Castaing, Gagne & Hopfinger, Physica D, 46, 177 (1990)]
This model is involved with turbulent cascade picture.
■ Superstatistics [Beck & Cohen, Physica A, 322, 267 (2003)]
Superstatistics considers a driven nonequilibrium system that consists of many subsystems with different values of some intensive parameter b (the inverse effective temperature).
■ Heavy tailed distributions (independently and identically distributed process)
◇ symmetric Levy stable distribution, P(x) ~ |x|-(a+1) (0 < a < 2) for large |x|
◇ stretched exponential distribution, P(x) exp(-∝ g|x|a)
PL(x): local equilibrium distribution
f (b): fluctuations of intensive parameter
PL(x): PDF at integral scale L
G(s): fluctuations through energy cascade
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Ut = Xt exp Yt
observed non-Gaussian noise
Multiplicative decomposition of non-Gaussian noise
u t
Assume that U has a unimodal symmetric distribution.
(cf. Multifractal random walk [Bacry et al., Phys. Rev. E 64, 026103 (2001)])
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Ut = Xt exp Yt
Gaussian noise
log-amplitude fluctuationobserved non-Gaussian noise
Multiplicative decomposition of non-Gaussian noise
u t
yt
x t
amplitude fluctuation
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Log-amplitude cumulants (1)
cumulant of Yt = cumulant of ln|Ut| - cumulant of ln|Xt | U is observable X is a Gaussian
By assuming Ut = Xt exp Yt , we can obtain the following relation,
Log-amplitude cumulants Ck (cumulants of Y) can be estimated from {Ut}.
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Log-amplitude cumulants (2)
■ Definition of log-amplitude cumulants Consider a process {Ut} described by a multiplication of random variables,
where Xt and Yt are random variables independent of each other and Xt is a standard
Gaussian random variable with zero mean and unit variance.
In this process, log-amplitude cumulants are defined as cumulants of Yt .
Ut = Xt exp Yt
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Gaussian distribution on (C2 , C3 ) plane
Third log-amplitude cumulant C3 vs. second log-amplitude cumulant C2
X ~ N(0, ),(Y = const.)
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Castaing’s model on (C2 , C3 ) plane
Third log-amplitude cumulant C3 vs. second log-amplitude cumulant C2
→ deviation from a Gaussian shape
X ~ Gaussian×log-normal
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Castaing’s model using log-normal distribution
[Castaing, Gagne & Hopfinger, Physica D, 46 (1990) 177]
log-normal x Gaussian
lGaussian (l → 0):
log10 P(x) ~ -x 2
non-Gaussian PDFwith fat tails
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Superstatistics on (C2 , C3 ) plane
Third log-amplitude cumulant C3 vs. second log-amplitude cumulant C2
power-law tails
exponential tails(n = 2)
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Superstatistical distributions
[Beck & Cohen, Physica A, 322, 267 (2003)]
Local equilibrium distribution Intensity parameter fluctuation
Marginal distribution
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Superstatistical distributions
[Beck & Cohen, Physica A, 322, 267 (2003)]
b
q-Gaussian distribution (t distribution)
b
Bessel function distribution of the second kind
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Stable distributions on (C2 , C3 ) plane
Third log-amplitude cumulant C3 vs. second log-amplitude cumulant C2
All of Cn have closed-form expressions.
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(C2 , C3 ) plane
Third log-amplitude cumulant C3 vs. second log-amplitude cumulant C2
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Scale dependence of log-amplitude cumulants
Log-normal cascade model(multifracrtal noise) iid variables
ssmsC log~)log()( 2022
C2(s
)
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If {Xi} is a white noise process, the autocovariance of {Yi} can be estimated by
where t > 0.
Autocovariance of the log-amplitude(magnitude correlation)
Log-normal cascade model(multifracrtal noise) iid variables
[A. Arneodo et al., Phys. Rev. Lett. 80, 708 (1998); K. Kiyono et al., Phys. Rev. Lett. 95, 058101 (2005)]
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Outline
1. Heart rate variability (HRV)
2. Relation between multifractality and non-GaussiantyMultiplicative random cascade model
3. Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution
4. Non-Gaussian properties of HRVPrediction of mortality risk
5. Summary
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Detrending procedure of non-stationary time series
Local detrending by fitting and subtracting a polynomial(e.g. detrended fluctuation analysis [C.-K. Peng et al., Chaos 5, 82-87 (1995)])
Log-returns,
Using a wavelet with vanishing moments
High-pass filtering
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Detrended time series of HRV
[J. Hayano et al, Frontiers in Physiology. 2, 65 (2011)]
heart rate variability
← Heart rate variabilityafter acute myocardial infarction
← Detrended serieswith zero mean and unit variance
← Probability distribution of the detrended series (on log-linear plot)
survivor nonsurvivor nonsurvivor
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Healthy subjects vs heart failure patients
Healthy subjects vs heart failure patients
CHF patients (n = 108)Healthy subjects (n = 123)
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Non-Gaussianity at the time scale of 25 sec is important for risk stratification [J. Hayano et al, Frontiers in Physiology. 2, 65 (2011); K. Kiyono et al., Heart Rhythm, 5, 261-268, (2008)]
Scale dependence of non-Gaussianity
25 sec 25 sec
(n = 69)
(n = 39) (n = 45)
(n = 625)
(follow-up of mean of 33 months) (follow-up for median of 25 months)
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Non-Gaussian heart rate as a risk factor for mortality
scale = 40, 100, 400, 1000 beats
Non-Gaussian index of HRV was a significant and independent mortality predictor in patients with congestive heart failure (CHF) [Kiyono et al., Heart Rhythm, 5, 261-268, (2008)].
C2 = λ2,when Y is a Gaussian
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Non-Gaussian heart rate as a risk factor for cardiac mortality
Increased non-Gaussianity of heart rate variability predicts cardiac mortality after an acute myocardial infarction. [J. Hayano et al, Frontiers in Physiology. 2, 65 (2011)].
survivor nonsurvivor nonsurvivor
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Non-Gaussian index λ
■ Multiplicative Stochastic Process
■ q th non-Gaussian index [Kiyono et al., Phys. Rev. E, 76, 041113 (2007)]
Assumption: Y is a Gaussian (multiplicative log-normal process)
■ Log-amplitude cumulants
[Kiyono, Konno, Phys. Rev. E., 87, 052104 (2013)]
ttt YWX expW: Gaussian variable Y: log-amplitude
If Y is a Gaussian,
lq = const,
C2 = lq2.
If X is a Gaussian,
C2 = lq2 = 0
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Prediction of sudden cardiac death
[Heart Rhythm, 5, 269-270 (2008)]
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Outline
1. Heart rate variability (HRV)
2. Relation between multifractality and non-GaussiantyMultiplicative random cascade model
3. Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution
4. Non-Gaussian properties of HRVPrediction of mortality risk
5. Summary
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Summary
Multiscale PDF analysis is applicable to a variety of real-world time series.[K. Kiyono et al. Phys. Rev. Lett. 96, 068701 (2006); Phys. Rev. Lett. 95, 058101; Phys. Rev. Lett. 93, 178103 (2004)]
Non-Gauusain PDF can be characterized by log-amplitude cumulants.[K. Kiyono, Phys. Rev. E 79, 031129 (2009); K. Kiyono, H. Konno, Phys. Rev. E 87, 052104 (2013)]
Increased non-Gaussianity of heart rate variability predicts mortality.[K. Kiyono et al., Heart Rhythm 5, 261-268, (2008); J. Hayano et al. Front Physiol. 2, 65 (2011)]
CollaboratorsJunichiro Hayano (Nagoya City University)
Eiichi Watanabe (Fujita Health University)
Hidetoshi Konno (Tsukuba University)
Yosiharu Yamamoto (University of Tokyo)
Taishin Nomura (Osaka University)
Akihiro Azuma (Osaka University)Tetsutaro Endo (Osaka University)Koichi Takeuchi (Osaka University)Syota Fujii (Osaka University)Ysutaka Moriwaki (Osaka University)Yauyuki Suzuki (Osaka University)