Application of Non-Gaussian Statistics to Heart Rate Variability

Non-Gaussian characteristics of heart rate variability in health and diseaseKen KiyonoDivision of BioengineeringGraduate School of Engineering Science Osaka University

Multifractal Analysis: From Theory to Applications and Back1) about characterization of heart rate variability. 2) Basic idea is based on the relation between multifractality and non-Gaussian probability density function.3) papers reporting multifractality of heart rate variability.4) Purpose not to characterize multifractal time series. 5) To characterize HRV and real-world time series regardless of whether it is multifractal process or not. 6) As a possible approach, pay attention to non-Gaussian properties of the observed time series. 1

OutlineHeart rate variability (HRV)

Relation between multifractality and non-GaussiantyMultiplicative random cascade model

Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution

Non-Gaussian properties of HRVPrediction of mortality risk

Summary1) a brief outline of my talk. 2) (First of all) explain briefly about HRV; 3) (Next) describe the relation between multifractality and non-Gaussianty using a simple random cascade model;4) (After that) Characterization method of non-Gaussian distributions5) discuss the non-Gaussian properties of HRV. 2

OutlineHeart rate variability (HRV)

Relation between multifractality and non-GaussiantyMultiplicative random cascade model

Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution

Non-Gaussian properties of HRVPrediction of mortality risk

Summaryexplain about heart rate variability3

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)

1) HRV the temporal fluctuation of inter-beat intervals. 2) HRV can be derived from ECG (electronic signal originating from heart)3) Figure shows the normal and standard ECG.4) By detecting large spikes called R waves, we can obtain RR intervals.5) (In most cases) HRV means the RR interval time series. 6) (figure) 24-hour RRI during daily life.4

[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 ms2119.0%50-100 ms47213.8%< 50 ms12534.4%Lower HRV was associated with ahigher risk.Mortality rate of patients after myocardial infarctionImportance of HRV study is1) HRV has a potential to predict mortality risk.2) In 1987, Kleiger et al reported the relation between HRV and mortality risk.In their study, about 800 patients after myocardial infarction are divided into 3 subgroups based on the SD.3) This table shows the mortality rate in each subgroup.4) This group having small SD less than 50ms showed the Highest mortality risk. 5) This graph shows survival rate during the folloe-up period. By tracking study of the patient after the measurement, survival curves was estimated.6) The fastest decrease of mortality rate was observed the low HRV group.7) SD is not always good. The predictive power depends on the type of disease.

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Physiological cause of heart rate variabilityHRV is mainly controlled by autonomic nervous system (ANS).Parasympathetic blockade reduces heart rate fluctuation.

[Reprinted from ()2001, ]Pharmacological blockade experiment Intravenous administration of propranolol and atropine.

Physiological cause1) HRV is mainly controlled by ANS. ()2) The ANS consists of sympathetic and parasympathetic nervous systems.3) Sympathetic accelerateparasympathetic decelerate

4) () an experimental result using pharmacological blockade of ANS. 5) First, the sympathetic nervous system wasblockaded sufficientlywith propranolol.6) After that, the parasympathetic nervous system wasblockadedstep by step.7) As the blockade level increases, the RRI baseline becomes shorter. Fluctuation width changes much smaller.8) Through the HRV analysis, we can evaluate activity of cardiovascular control system including ANS. (ECG cardiac conduction)

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[Kleiger et al., Am. J. Cardiol., 59 (1987) 256]

Frequency characteristics of HRVThrough the Fourier transform, observed signals can be decomposed into the superposition of sinusoidal signal with different frequencies and amplitudes.

Periodic components

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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) bandMayer wave (~10 s) , Blood pressure oscillationssympathetic controlparasympathetic control Two characteristic frequencies HF: Respiration period LF: Mayer wave known the relation between these periodic component and ANS HF and LF powers have been used as index of ANS. 8

1/f fluctuation of HRVHealthy HRV power spectrum shows a 1/f-type power-law scaling

[Kobayashi, Musha, IEEE Trans. Biomed. Eng. 29, 456 (1982)]1/f scaling is related to fractal9

1/f noise and fractal (1)

A simple repeating rule can produce 1/f noise. Deterministic fractalityStochastic fractalityKoch curve1/f noise simple rule makes fractal

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1/f noise and fractal (2)

Random variables lie on a dyadic grid.

1) Prepare random numbers on a dyadic grid.2) we sum the random numbers across the branching.

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Multifractality of Heart Rate Variability

Since I read this paper,I have come to consider about Multifractality of HRV.12

Analogy between HRV and cascade model

As a model of HRV, cascade-type model was proposed.

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OutlineHeart rate variability (HRV)

Relation between multifractality and non-GaussiantyMultiplicative random cascade model

Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution

Non-Gaussian properties of HRVPrediction of mortality risk

Summary1) using a simple random cascade model, remind you the relation between the Multifractal and non-Gaussian properties. 14

Multiplicative random cascade (1)

(cf. Additive random cascade can generate 1/f noise) Multifractal time series can be generated by multiplicative cascade. As already mentioned, to generate 1/f noise, we used additive cascade. To generate multifractal time series (discrete time series), we assume multiplicative random cascadeIn our model, using multiplication of random weights, the local variances are modulated. (1) Gaussian noise (2) whole interval -> two equal subintervals; variance of each subinterval is modulated by random weight exponential of Y super 15) The same procedure is repeated. 15

Multiplicative random cascade (2)(Again) 16

Non-Gaussian PDF of cascade model

Our cascade model can be described by this equation. (bracket = floor function)PDF is given by this form. The summation of random variables correspond to m-fold convolution. Time series shows variance heterogeneity.PDF shows non-Gaussian shape with fat tails.17

Multiscaling property of structure functionThe structure function is defined as,

,

where

1) To characterize multifractal time series, we consider structure function.2) z is the increment of the integrated time series.18

Multiscaling property of cascade-type modelUsing aGaussian approximation of the partial sums, we obtain

where Y is the cumulant-generating function of Y(j)

(If Y(j) is a Gaussian variable, Y is a quadratic function.)

white Gaussian

In our model, the scaling exponent can be approximately described by this equation. (To obtain) PDF of local sum was approximated by a Gaussin.19

Multiscale PDF analysis

Fine resolutionCoarse resolutionPartial sum process {DsZi}Deformation of PDFs across scalesConvergence to a Gaussian

[Castaing et al., Physica D, 46, 177 (1990); Kiyono et al., IEEE TBME 53, 95-102 (2006)]To characterize the cascade process, here we consider the deformaton process of PDFs across scale.From the observed time series, we compute partial sum or local average. The corase grained time series can be obtained using partial sum or local average.Scale means number of variables for summation.

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Convergence process to a Gaussian

Log-normal cascade modeliid sequence

uiui

s = 1, 2, 4, 8, 16, 32from top to bottomUsing examples, explain the deformation process pf the PDFsA sample time series of Cascade model.IID sequence.Both distributions of U are the same.but the deformation process across scale are difference.To characterize this deformation process, we have to quantify the observed non-Gaussain distribution

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OutlineHeart rate variability (HRV)

Relation between multifractality and non-GaussiantyMultiplicative random cascade model

Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution

Non-Gaussian properties of HRVPrediction of mortality risk

SummaryCharacterization of non-Gussian distribution22

Parameter estimation problem of non-Gaussian processes Conventional models of non-Gaussain distributions Castaings 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 distributionf (b): fluctuations of intensive parameterPL(x): PDF at integral scale LG(s): fluctuations through energy cascade

Many models describing non-Gaussian distribution

By assuming a model, we can estimate its parameters.Nonparametric way23

Models for non-Gaussian fluctuationsConventional models of non-Gaussain distributions Castaings 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 distributionf(b): fluctuations of intensive parameterPL(x): PDF at integral scale LG(ln s): fluctuations through energy cascade

The Mellin convolution of f and g :

X ~ PL(x)sX s ~ G(ln s) Come from multiplication of two random variables

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Parameter estimation problem of non-Gaussian processes Conventional models of non-Gaussain distributions Castaings 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 distributionf (b): fluctuations of intensive parameterPL(x): PDF at integral scale LG(s): fluctuations through energy cascade

Ut = Xt exp Yt observed non-Gaussian noiseMultiplicative decomposition of non-Gaussian noiseutAssumethat U has a unimodal symmetric distribution.(cf. Multifractal random walk [Bacry et al., Phys. Rev. E 64, 026103 (2001)])

To describe the observed times series, we assume this form. Base on this relation, we decompose the observed time series U into X and Y.26

Ut = Xt exp Yt Gaussian noiselog-amplitude fluctuationobserved non-Gaussian noiseMultiplicative decomposition of non-Gaussian noiseutytxt

amplitude fluctuation

In this case, the envelop is modulated by log-amplitude.In our approach, using cumulants of Y, non-Gaussian process is characterized.27

Log-amplitude cumulants (1)cumulant of Yt = cumulant of ln|Ut| - cumulant of ln|Xt |U is observableX is a GaussianBy assuming Ut = Xt exp Yt , we can obtain the following relation,

Log-amplitude cumulants Ck (cumulants of Y) can be estimated from {Ut}.

through logarithmic transformC(Y) = log absolute value of U log absoute value of X28

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

Under this assumptions Define log-amplitude cumulants as the cumulants of Y29

Gaussian distribution on (C2 , C3 ) plane Third log-amplitude cumulant C3 vs. second log-amplitude cumulant C2 Using log-amplitude cumulants, we can characterize variou non-Gaussian distribution. 30

Castaings model on (C2 , C3 ) plane Third log-amplitude cumulant C3 vs. second log-amplitude cumulant C2 deviation from a Gaussian shapeX ~ Gaussianlog-normal

Castaings model using log-normal distribution

[Castaing, Gagne & Hopfinger, Physica D, 46 (1990) 177] log-normal x GaussianlGaussian (l 0):log10 P(x) ~ -x 2non-Gaussian PDFwith fat tails

Superstatistics on (C2 , C3 ) plane Third log-amplitude cumulant C3 vs. second log-amplitude cumulant C2

power-law tailsexponential tails(n = 2)33

Superstatistical distributions[Beck & Cohen, Physica A, 322, 267 (2003)]

Local equilibrium distributionIntensity parameter fluctuationMarginal distribution

Superstatistical distributions[Beck & Cohen, Physica A, 322, 267 (2003)]

bq-Gaussian distribution (t distribution)

b

Bessel function distribution of the second kind

Stable distributions on (C2 , C3 ) plane Third log-amplitude cumulant C3 vs. second log-amplitude cumulant C2 All of Cn have closed-form expressions.

(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 cascademodel(multifracrtal noise)iid variables

C2(s)Back to the cascade model.

PDFCC2

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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 cascademodel(multifracrtal noise)iid variables[A. Arneodo et al., Phys. Rev. Lett. 80, 708 (1998); K. Kiyono et al., Phys. Rev. Lett. 95, 058101 (2005)]39

OutlineHeart rate variability (HRV)

Relation between multifractality and non-GaussiantyMultiplicative random cascade model

Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution

Non-Gaussian properties of HRVPrediction of mortality risk

Summary a brief outline of my talk. (First of all) explain briefly about HRV; (Next) describe the relation between multifractality and non-Gaussianty using a simple random cascade model; (After that) discuss the non-Gaussian properties of HRV. 40

Detrending procedure of non-stationary time seriesLocal 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

To analyze HRV time series, detrending procedure is required. baseline drift

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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)survivornonsurvivornonsurvivor

Healthy subjects vs heart failure patientsHealthy subjects vs heart failure patients

CHF patients (n = 108)Healthy subjects (n = 123)

C290C3043

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-Gaussianity25 sec25 sec(n = 69)(n = 39)(n = 45)(n = 625)(follow-up of mean of 33 months)(follow-up for median of 25 months)(CHF) 39 of 108 patients died in the observation period,(MI) 45 of 670 patients dies.follow-up for a median of 25 months44

Non-Gaussian heart rate as a risk factor for mortality

scale = 40, 100, 400, 1000 beatsNon-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 = 2,when Y is a Gaussian

Non-Gaussian heart rate as a risk factor for cardiac mortalityIncreased non-Gaussianity of heart rate variability predicts cardiac mortality after an acute myocardial infarction. [J. Hayano et al, Frontiers in Physiology. 2, 65 (2011)].

survivornonsurvivornonsurvivor

Non-Gaussian index 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)]

W: Gaussian variableY: log-amplitudeIf Y is a Gaussian, lq = const,C2 = lq2.If X is a Gaussian,C2 = lq2 = 0

Prediction of sudden cardiac death

[Heart Rhythm, 5, 269-270 (2008)]editorial commentary

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OutlineHeart rate variability (HRV)

Relation between multifractality and non-GaussiantyMultiplicative random cascade model

Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution

Non-Gaussian properties of HRVPrediction of mortality risk

Summary a brief outline of my talk. (First of all) explain briefly about HRV; (Next) describe the relation between multifractality and non-Gaussianty using a simple random cascade model; (After that) discuss the non-Gaussian properties of HRV. 49

SummaryMultiscale 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)]

Collaborators

Junichiro 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)