Correlation in the magnitude of heart.beat increments as a measure of nonlinearity

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Correlations in the magnitude of

heartbeat increments

as a measure of nonlinearity

Pedro A. Bernaola-Galv´

an

Dpt. of Applied Physics II

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Introduction Correlations in the magnitude Linear Gaussian Noise Heartbeat data

Colaborators

Dpt. of Applied Physics II. University of M´

alaga

Pedro J. Carpena

Manuel G´

omez-Extremera

Dpt. of Physical Education. University of M´

alaga

A. Ram´

on Romance-Garc´ıa

Javier Ben´ıtez-Porres

EADE University of Wales Trinity Saint David (M´

alaga)

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The talk

We propose a measure nonlinearity for time series based

on the analysis of the autocorrelation of the magnitude of

the series

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Why nonlinearity of heartbeat time series?

A periodic heart is just the

opposite to a healthy heart

Peng et al. (1993): heartbeat

series have 1

/

f

power-spectrum

In Physics 1

/

f

⇒

non equilibrium, complexity, fractals, etc.

Ivanov et al. (1999): heartrate is

multifractal

⇒

nonlinear

Lack of nonlinearity

⇒

PROBLEMS

C. K. Peng, et al.: Long-range anti-correlations and non-Gaussian behavior of the heartbeat. Phys. Rev. Lett. 70, 1343 (1993).

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Why nonlinearity of heartbeat time series?

A periodic heart is just the

opposite to a healthy heart

Peng et al. (1993): heartbeat

series have 1

/

f

power-spectrum

In Physics 1

/

f

⇒

non equilibrium, complexity, fractals, etc.

Lack of nonlinearity

⇒

PROBLEMS

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Why nonlinearity of heartbeat time series?

A periodic heart is just the

opposite to a healthy heart

Peng et al. (1993): heartbeat

series have 1

/

f

power-spectrum

In Physics 1

/

f

⇒

non equilibrium, complexity, fractals, etc.

Ivanov et al. (1999): heartrate is

multifractal

⇒

nonlinear

Lack of nonlinearity

⇒

PROBLEMS

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Why nonlinearity of heartbeat time series?

A periodic heart is just the

opposite to a healthy heart

Peng et al. (1993): heartbeat

series have 1

/

f

power-spectrum

In Physics 1

/

f

⇒

non equilibrium, complexity, fractals, etc.

Ivanov et al. (1999): heartrate is

multifractal

⇒

nonlinear

Lack of nonlinearity

⇒

PROBLEMS

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Nonlinear time series

Generated by nonlinear dynamical equations

There exist correlations beyond the autocorrelation

function (i.e. beyond lineal correlations)

C

x

(

`

) =

hx

i

· x

i+`

i − hx

i

ihx

i+`

i

σ

2

x

Multifractality

Non-random Fourier phases (Schreiber & Schmitz, 2000)

Correlations in the magnitude series

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Nonlinear time series

Generated by nonlinear dynamical equations

There exist correlations beyond the autocorrelation

function (i.e. beyond lineal correlations)

C

x

(

`

) =

hx

i

· x

i+`

i − hx

i

ihx

i+`

i

σ

2

x

Multifractality

Non-random Fourier phases (Schreiber & Schmitz, 2000)

Correlations in the magnitude series

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Correlations in the magnitude

Given a time series

{

y

i

}

,

i

= 1

, ...,

N

its magnitude series

(also called

volatility

) is given by:

|

x

i

|

=

|

y

i+1

−

y

i

|

Correlations in

|

x

i

| →

Related to nonlinearity

The decomposition into magnitude and sign has

Physiological meaning

(e.g. heartbeat fluctuations)

Such correlations are quantified using DFA (Detrended

Fluctuation An´

alysis)

Y. Ashkenazy, et al.: Magnitude and Sign Correlations in Heartbeat Fluctuations.

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Detrended Fluctuation Analysis

Indirect measure of correlations (actually it measures

fluctuations)

Smooths out the noise in the autocorrelation function.

Is it always good?

Only when autocorrelation function is a power law the

results can be properly interpreted

Even having power laws

there are problems with

correlations in the magnitude

(Carpena et al. 2017)

We propose here a direct study of

the autocorrelation function of the magnitude

P. Carpena et al.: Spurious Results of Fluctuation Analysis Techniques in

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Model for linearity

⇒

Linear Gaussian Noise

Let{xi}be a series ofN(0,1) random variables with only linear

correlations

If we denote byCx(`) its autocorrelation function at distance`, the

autocorrelation function of its magnitudes,C|x|(`), is given by:

C|x|=

2hCxarcsinCx −1 +

p

1−C2

x

i

π−2

C|x|(`)≥0 yC|x|(`) = 0⇔Cx(`) = 0

For small values ofCx, we have: C|x|= π−12C

2

x +O(Cx4)

M. G´omez-Extremera et al.: Correlations in magnitude series to assess

nonlinearities: Application to multifractal models and heartbeat fluctuations.

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Model for linearity

⇒

Linear Gaussian Noise

Let{xi}be a series ofN(0,1) random variables with only linear

correlations

If we denote byCx(`) its autocorrelation function at distance`, the

autocorrelation function of its magnitudes,C|x|(`), is given by:

C|x|=

2hCxarcsinCx −1 +

p

1−C2

x

i

π−2

C|x|(`)≥0 yC|x|(`) = 0⇔Cx(`) = 0

For small values ofCx, we have: C|x|= π−12C

2

x +O(Cx4)

M. G´omez-Extremera et al.: Correlations in magnitude series to assess

nonlinearities: Application to multifractal models and heartbeat fluctuations.

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Examples of

C

|x|

vs.

C

x

for Gaussian Noises

-1 -0.5 0 0.5 1

C

x

0 0.2 0.4 0.6 0.8 1

C|x|

Theoretical curve fGn with H = 0.05 fGn with H = 0.95

The magnitude of a linear noise

can be correlated

C

|x|

6 = 0

=

6 ⇒

Nonlinearity

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Examples of

C

|x|

vs.

C

x

for Gaussian Noises

-1 -0.5 0 0.5 1

C

x

0 0.2 0.4 0.6 0.8 1

C|x|

Theoretical curve fGn with H = 0.05 fGn with H = 0.95 Nonlinear noise with H = 0.95

The magnitude of a linear noise

can be correlated

C

|x|

6 = 0

=

6 ⇒

Nonlinearity

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Heartrate during exercise

-10 0 10

∆ tRR 0

0.05 0.1

Data Gaussian fit

0 1000 2000

Beat number -20 -10 0 10 20 400 440 480 Exercise

0 1000 2000 3000

Beat number 500

1000

tRR

(ms)

-300 -200 -100 0 100 200 300 ∆ tRR 0 0.002 0.004 0.006 0.008 p( ∆ tRR )

0 200 400 600

Beat number -400 -200 0 200 400 ∆ tRR (ms) 800 1000 1200 tRR (ms) Rest (a) (b) (c) (d) (e) (f) (g) Rest (10 min)

Exercise (20 min)

Heartrate increases and heartrate variability is reduced

Power spectrum is reduced, specially at low frequencies (respiration rate dominates)

Sample entropyis reduced

Short-range correlations are reduced (Not clear)

Multifractal spectrum disappears (?)

In general:

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Heartrate during exercise

-10 0 10

∆ tRR 0

0.05 0.1

0 1000 2000

0 1000 2000 3000

Beat number 500

1000

tRR

(ms)

-300 -200 -100 0 100 200 300 ∆ tRR 0 0.002 0.004 0.006 0.008 p( ∆ tRR )

0 200 400 600

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Heartrate during exercise

-10 0 10

∆ tRR 0

0.05 0.1

0 1000 2000

0 1000 2000 3000

Beat number 500

1000

tRR

(ms)

-300 -200 -100 0 100 200 300 ∆ tRR 0 0.002 0.004 0.006 0.008 p( ∆ tRR )

0 200 400 600

In general:

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-0.4 -0.2 0 0.2 0.4 C_x 0 0.1 0.2 C|x| Linear Gaussian Rest Exercise -0.4 -0.2 0 0.2 0.4 Cx 0 0.1 0.2 C|x|

0 5 10 15 20

l -0.05 0 0.05 0.1 0.15 δ C (a) (b) (c) (d)

¿What about nonlinearity?

More lineal⇒less complex

Measure of nonlinearity:

deviation from linear Gaussian expectation

∆ =

`max

X

`=1

δC(`)2

where:

δC(`) =C|x|(`)−C|x|,linear[Cx(`)]

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0 5 10 15 20

l -0.05 0 0.05 0.1 0.15 δ C (a) (b) (c) (d)

C_|x| >>

δ_{C <<}

∆ =

`max

X

`=1

δC(`)2

where:

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0 5 10 15 20

l -0.05 0 0.05 0.1 0.15 δ C (a) (b) (c) (d)

C_|x| >>

δ_{C <<}

∆ =

`max

X

`=1

δC(`)2

where:

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Rest vs. exercise for football players

0 0.05 0.1 0.15 0.2

Nonlinearity index

∆

Amateur football players

Rest vs. warming up running

Amateurs:

10 males 22.1 ± 3.4 y/o

Exercise: running for 20 min at warming up pace Resting for 10 min

on football court

We choose`max= 10 beats

Prior to the analysis data is converted into Gaussian

For all subjects ∆ is greater for rest, this is also true for group average.

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Rest vs. exercise for football players

0 0.05 0.1 0.15 0.2 Nonlinearity index ∆

Amateur football players

Rest vs. warming up running

Amateurs:

10 males 22.1 ± 3.4 y/o

Exercise: running for 20 min at warming up pace Resting for 10 min

on football court

0 0.1 0.2 0.3 0.4 0.5 Nonlinearity index ∆

Professional football players Rest vs. ’yo-yo’ test

Professionals: 12 males 23.8 ± 2.9 y/o Professional

players rest

Professional players ’yo-yo’ test

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Rest vs. exercise for football players

0 0.1 0.2 0.3 0.4 0.5

∆

Amateur football payers vs. professionals

Professionals: 12 males 23.8 ± 2.9 y/o

Amateur: 10 males 22.1 ± 3.4 y/o Professional

players rest

Amateur players rest

Professional

players ’yo-yo’ test

Amateur players run

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Permanent effects of exercise on HR nonlinearity

(cardio vs. strength training)

0 0.1 0.2 0.3 0.4 0.5

∆

Football players vs. bodybuilders 10 minutes rest

Bodybuilders

31 males 28.0 ± 6.1 y/o Soccer players

22 males 23.0 ± 4.1 y/o

Higher nonlinearity during rest for football players

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Permanent effects of exercise on HR nonlinearity

(cardio vs. strength training)

0 0.1 0.2 0.3 0.4 0.5

∆

Football players vs. bodybuilders 10 minutes rest

Bodybuilders

31 males 28.0 ± 6.1 y/o Soccer players

22 males 23.0 ± 4.1 y/o

Higher nonlinearity during rest for football players

(statistically significantp= 7.6×10−4₎

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