research article methodological framework for estimating...

Research ArticleMethodological Framework for Estimating the CorrelationDimension in HRV Signals

Juan Bolea12 Pablo Laguna12 Joseacute Mariacutea Remartiacutenez34 Eva Rovira34

Augusto Navarro345 and Raquel Bailoacuten12

1 Communications Technology Group (GTC) Aragon Institute for Engineering Research (I3A) IIS AragonUniversity of Zaragoza 50018 Zaragoza Spain

2 CIBER de Bioingenierıa Biomateriales y Nanomedicina (CIBER-BBN) 50018 Zaragoza Spain3 Anaesthesiology Service Miguel Servet University Hospital 50009 Zaragoza Spain4Medicine School University of Zaragoza 50009 Zaragoza Spain5 Aragon Health Sciences Institute (IACS) 50009 Zaragoza Spain

Correspondence should be addressed to Juan Bolea jboleaunizares

Received 28 August 2013 Revised 17 December 2013 Accepted 20 December 2013 Published 30 January 2014

Academic Editor Mika P Tarvainen

Copyright copy 2014 Juan Bolea et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a methodological framework for robust estimation of the correlation dimension in HRV signals It includes (i)a fast algorithm for on-line computation of correlation sums (ii) log-log curves fitting to a sigmoidal function for robust maximumslope estimation discarding the estimation according to fitting requirements (iii) three different approaches for linear region slopeestimation based on latter point and (iv) exponential fitting for robust estimation of saturation level of slope series with increasingembedded dimension to finally obtain the correlation dimension estimate Each approach for slope estimation leads to a correlationdimension estimate called 119863

2 1198632(perp)

and 1198632(max) 1198632 and 1198632(max) estimate the theoretical value of correlation dimension for the

Lorenz attractor with relative error of 4 and 1198632(perp)

with 1The three approaches are applied to HRV signals of pregnant womenbefore spinal anesthesia for cesarean delivery in order to identify patients at risk for hypotension 119863

2keeps the 81 of accuracy

previously described in the literature while 1198632(perp)

and 1198632(max) approaches reach 91 of accuracy in the same database

1 Introduction

Heart rate variability (HRV) has beenwidely used as amarkerof the autonomic nervous system (ANS) regulation of theheart Classical HRV indices include global descriptive statis-tics which characterize HRV distribution in the time domain(mean heart rate and standard deviation of the normal-to-normal beat interval among others) and in the frequencydomain (power in the very low frequency and low frequency(LF) and high frequency (HF) bands) The activity of the twomain branches of the ANS sympathetic and parasympatheticsystems has been related with the power in the LF and HFbands respectively [1]

HRV data often present nonlinear characteristics possi-bly reflecting intrinsic physiological nonlinearities such aschanges in the gain of baroreflex feedback loops or delays

in conduction time which are not properly described byclassical HRV indices

The most widespread methods used to characterize non-linear system dynamics are based on chaos theory The ques-tion of whether HRV arises from a low-dimensional attractorassociated with a deterministic nonlinear dynamical systemor whether it has a stochastic origin is still under debate

Of great interest is the concept of system complexitywhich refers to the richness of process dynamics Complexitymeasures are based on the theory of nonlinear systems butmay be applied to both linear and nonlinear systems Severaltechniques attempting to assess complexity have been devel-oped such as detrended fluctuation analysis [2] Lempel-Zivcomplexity [3] Lyapunov exponents [4] the correlationdimension (119863

2) [5] and approximate and sample entropies

[6]

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2014 Article ID 129248 11 pageshttpdxdoiorg1011552014129248

2 Computational and Mathematical Methods in Medicine

The reduction of HRV complexity has been associatedwith age disease and unbalanced cardiovascular regulation[7] Complexity measures have been proven to characterizeHRV signals more successfully than linear approaches in cer-tain applications [8] In [9ndash11] point correlation dimensionof HRV signals predicted hypotension events in pregnantwomen during spinal anesthesia for cesarean section some-thing which time and frequency domain indices were unableto do

While these measurements are of considerable interesttheir application to HRV has some pitfalls that could misleadtheir interpretation One of such limitations arises from theirapplication to limited time series Correlation dimensionestimation is highly dependent on the length of the time series[12] Several studies have reported the effect of data length on119863

2estimation as well as proposals to alleviate this effect [13

14] Stationarity is another requirement that a time serieshas to fulfil to obtain reliable results However satisfying theconstraint of finite series and stationarity at the same time isusually difficult [15] Yet another limitation of these measure-ments is the long computational time required Data lengthexponentially increases the computational time cost in aclassical sequential approach In the case of 119863

2 several

attempts to try to reduce this factor have been reported byWidman et al [16] and Zurek et al [17] the latter proposingparallel computing using MPI (Message Passing Interface)

Themain goal of this study is to propose amethodologicalframework for robust and fast estimation of119863

2and its appli-

cation inHRV signals Section 2 starts with a definition of thecorrelation dimension and its classical estimation An algo-rithm for its fast computation is proposed Robustness isaddressed by fitting the log-log curve to a sigmoid functionafter which three alternative approaches for119863

2estimation are

presented Section 3 introduces synthetic and real (HRVsignals) data where the proposed estimates are evaluated andinterpreted Section 4 presents the results while Section 5 setsout the discussion and conclusions of the study

2 Methods

21 Correlation Dimension Let 119909(119899) 119899 = 1 119873 be thetime series of interest which in HRV analysis will be the 119877119877interval series normalized to unit amplitude with119873being thetotal number of beats A set of119898-dimensional vectors y

119898(119894)

called reconstructed vectors are generated [18]

y119898119894= [119909(119894) 119909(119894 + 120591) 119909(119894 + 2120591) 119909(119894 + (119898 minus 1)120591)]

119879 (1)

where 120591 represents the delay between consecutive samples inthe reconstructed space Then the amount of reconstructedvectors is 119873

119898= 119873 minus 120591(119898 minus 1) for each 119898-dimension The

distance between each pair of reconstructed vectors y119898119894 y119898119895

is denoted as

119889

119898

119894119895= 119889 (y119898

119894 y119898119895) (2)

and this can be computed as the norm of the difference vectorΔy119898119894119895= y119898119894minus y119898119895 (In Appendix A different norms and their

effect on correlation dimension estimates from finite time

series are discussed) The correlation sum which representsthe probability of the reconstructed vector pair distance beingsmaller than a certain threshold 119903 is computed as

119862

119898(119903) =

1

119873

119898(119873

119898minus 1)

119873119898

sum

119894119895=1

119867(119903 minus 119889

119898

119894119895)

=

1

119873

119898(119873

119898minus 1)

119873119898

sum

119894=1

119888

119898

119894(119903)

(3)

where119867(sdot) is the Heaviside function defined as

119867(119909) =

1 119909 ge 0

0 119909 lt 0

(4)

and 119888119898119894(119903) = sum

119873119898

119895=1119867(119903 minus 119889

119898

119894119895)

For deterministic systems 119862119898(119903) decreases monotoni-

cally to 0 as 119903 approaches 0 and it is expected that 119862119898(119903) is

well approximated by 119862119898(119903) asymp 119903

119863119898

2 Thus 1198631198982can be defined

as

119863

119898

2= lim119903rarr0

log119862119898(119903)

log (119903) (5)

For increasing1198981198631198982values tend to saturate to a value119863

2

which constitutes the final correlation dimension estimate

22 Fast Computation of Correlation Sums One importantlimitation of 119863

2estimation is the long computational time

required mainly due to the sequential estimation of corre-lation sums This section describes an algorithm for the fastcomputation of correlation sums based on matrix operations(MO) A matrix S which contains the differences between allpairs of samples of 119909(119899) is computed as

S = X minus X119879 (6)

where X is the119873 times119873matrix

X =(

119909(1) 119909 (2) sdot sdot sdot 119909 (119873)

119909 (1) 119909 (2) sdot sdot sdot 119909 (119873)

119909 (1) 119909 (2) sdot sdot sdot 119909 (119873)

d

119909 (1) 119909 (2) sdot sdot sdot 119909 (119873)

) (7)

=

x xi1 xi+ xi+mminus xN1

x1j xij xi+1j xi+mminus1j xNj

x1j+1 xij+1 xi+1j+1 xi+mminus1j+1 xNj+1

x1j+mminus1 xij+mminus1 xi+1j+mminus1 xi+mminus1j+mminus1 xNj+mminus1

x1N xiN xi+1N xi+mminus1N xNN

(8)

Computational and Mathematical Methods in Medicine 3

where 119909119894119895symbolizes 119909(119894)minus119909(119895) For instance the dashed box

contains the elements of the difference vector Δy119898119894119895

for 120591 =1 For each embedded dimension 119898 and the reconstructedvector 119894 the difference vectors Δy119898

119894119895generates a S119898

119894matrix

mi =

xi1 xi+ xi+mminus1m

xij xi+1j+1 xi+mminus1j+mminus1

xiN xi+1N +1

xi+mminus1N

=

Δ miN

T

Δ mi1

T

Δ mij

T (9)

The selected norm is applied to the matrix S119898119894 generating

the norm vector d119898119894 whose elements are distances d119898

119894119895 To

compute the limit in (5) distances should be compared witha set of thresholds which implies the repetition of the wholeprocess as many times as the number of thresholds Thisrepetition is avoided since distances in d119898

119894are compared with

a whole set of thresholds r = [1199031 119903

2 119903

119873119903

]

Γ

119898

119894= (

119867(1199031 minus 119889119898

1198941) 119867(1199032 minus 119889

119898

1198941) sdot sdot sdot 119867(119903119873

119903

minus 119889119898

1198941)

119867(1199031 minus 119889119898

1198942) 119867(1199032 minus 119889

119898

1198942) sdot sdot sdot 119867(119903119873

119903

minus 119889119898

1198942)

119867(1199031 minus 119889119898

119894119873119898

) 119867(1199032 minus 119889119898

119894119873119898

) sdot sdot sdot 119867(119903119873119903

minus 119889119898

119894119873119898

)

)

(10)

where Γ119898119894

is a 119873119898times 119873

119903matrix which contains ones and

zeros The accumulative addition of each column representsthe partial correlation sum of the 119894th reconstructed vector fora set of thresholds 119903

c119898119894= Γ

119898

119894

1198791 = (

119888

119898

119894(119903

1)

119888

119898

119894(119903

2)

119888

119898

119894(119903

119873119903

)

) (11)

where 1 is a119873119898times 1 vector whose elements are equal to one

Finally the procedure has to be repeated varying the 119894index119873

119898times to compute119862

119898(119903)This technique saves com-

putational time due to the usage of a set of thresholds in onestep

23 New Approaches for1198632Assessment

231 Sigmoid Curves as Surrogates of Log-Log Curves 1198632has

to be estimated from (5) whose numerator and denominatorboth tend to minusinfin as 119903 tends to 0 Therefore applyingLrsquoHopitalrsquos rule the equation can be rewritten as [19]

119863

119898

2= lim119903rarr0

120597 log119862119898(119903)

120597 log (119903) (12)

Since the size of the time series is finite choosing smallvalues of 119903 to evaluate this limit is problematic For values of119903 close to 0 very few distances contribute to the correlationsummaking the estimation unreliableThe evaluation of this

0

minus5

minus10

minus15minus5 minus4 minus3 minus2 minus1 0

log(r)

m = 1

m = 2

m = 3

m = 10

log(Cm(r))

1st linear region2nd linear region

Figure 1 Log-log curves for a dynamic system Data correspond toan RR interval series extracted from 30mins of ECG recording

expression is usually done in a linear region in the log(119862119898(119903))

versus log(119903) representation called the log-log curve Theslope of this linear region is considered an estimate of 119863119898

2

There are different approaches for estimating this slopeMaximum slope searching can be done by directly computingthe increments in the log-log curve Another approach is toestimate numerically the maximum of the first derivative ofthe log-log curve Nevertheless these approaches encountersome limitations due to the usual nonequidistant sampling of119903 values in the logarithmic scale Yet another limitation arisesin the presence of dynamic systems whose log-log curves dis-play several linear regions as can be seen in Figure 1where the data corresponds to an RR interval series extractedfrom a 30 minute ECG recording In order to estimate theslope of the linear region of the log-log curve an attempt toartificially extend the linear region is made by excluding theself-comparisons (119889

119894119894) from the correlation sums

However the basis of the approach proposed in thiswork to improve 119863

2estimation lies in considering self-com-

parisons Figure 2 illustrates how log-log curves behave inboth situations considering or not considering self-compar-isons As it is shown both share part of the linear region Ourproposal is to use sigmoidal curve fitting (SCF) over the log-log curves to obtain an analytic function whose maximumslope in the linear region is well defined These log-logcurves are reminiscent of the biasymptotic fractals studied byRigaut [20] and Dollinger et al in [21] in which exponentialfittings were proposed The sigmoidal fitting is applied to theinterpolated log-log curves computed with evenlyspaced 119903values

A modified Boltzmann sigmoid curve was used byNavarro-Verdugo et al [22] as a model for the phase transi-tion of smart gels

119891 (119909) = 119860

2minus

(119860

2minus 119860

1)

119861 + 119890

(minus119909+119909119900)120572

(13)


0 50 100 150 200 250 300Beats

Nor

mal

ized

RR

inte

rval

0

04

08

minus04

minus08

(a)

Discarding self-comparisonsAdmitting self-comparisonsShared region

minus4

minus3 minus2

minus10

minus5

minus12

minus1

minus8

minus6

minus4

minus2

0

log(r)

m = 1

m = 2

m = 3 m = 4m = 5

log(Cm(r))

(b)

Figure 2 Log-log curves discarding and accepting self-comparisons Arrows show the slope of the scaling range Data correspond to an RRinterval series of 300 beats

where11986011198602 119909119900 and 119861 are the parameter designedThe first

derivative of 119891(119909) is

119889119891 (119909)

119889119909

= minus

(119860

2minus 119860

1) 119890

(minus119909+119909119900)120572

[120572(119861 + 119890

(minus119909+119909119900)120572)

2

]

(14)

In our study the sigmoid curve 119891(119909) is fitted to log-logcurveThe first derivative (14) is determined analytically andits maximum constitutes the slope of the linear range that is

119863

119898

2 Note that hat notation refers the use of SCFIn order to achieve a goodfitting the thresholds r have to

guarantee that both asymptotes are reached In this work r isin[001 3] with a step of 001 The upper asymptote is reachedwhen all comparisons are above the threshold 119862119898(119903) asymp 1 and the lower when only the self-comparison is below119862

119898(119903) =1119873

119898(119873

119898minus 1)

The SCF approach is robust in the presence of dynamicsystems which exhibit log-log curves with more than onelinear region since when the fitting is not good enough noestimate is given In this work the requirement for a good fit-ting is to achieve a regression factor greater than 08

As the embedding dimension 119898 increases the linearregions of the log-log curves tend to be parallel to each otherThus 119863119898

2estimates tend to saturate to a certain value which

is considered the correlation dimension 1198632 The correlation

dimension is estimated fitting the 1198631198982versus119898 curve follow-

ing a modified version of that used by Carvajal et al [23]

119863

119898

2= 119863

2(1 minus 119860119890

minus119896119898) (15)

where 119860 introduced in this study in order to reach the satu-ration level more quickly than previously proposed and 119896 areexponential growth factors

232 New Approaches for 1198632Estimation As mentioned

previously we chose 1198631198982as themaximum slope on each fitted

sigmoid curve Nevertheless the linear range is composed ofmore than one point Instead of considering only one pointper curve in this study we propose a new approach for 119863

2

estimation considering a set of points extracted from theselinear ranges

The proposed strategy is based on selecting one point ofthe linear range in the SCF log-log curve of the lowest embed-ded dimension119898 and moving forward to the next embeddeddimension 119898 + 1 selecting the point of the correspondingSCF log-log curve with minimum distance to the formercurve (ie where the perpendicular to the 119898th log-log curveintersects the (119898 + 1)th log-log curve as in the gradientdescent technique) see Figure 3 The procedure is repeatedup to the maximum embedded dimension analyzed Thenseveral sets of slopes are computed (one for each point in thelinear region around the maximum slope of the SCF log-logcurve of the lowest embedded dimension) providing a set ofcorrelation dimension estimates per embedded dimension(1198631198982(perp)119903

) The dependence on 119903 in the notation indicates thateach set of correlation dimension estimates is linked to an 119903value corresponding to the first value of each set

Finally (15) is used to estimate the final correlationdimension (119863

2(perp)119903) for each set of pointsThese (119863

2(perp)119903) esti-

mates are linked to the log(119903) value of the lowest embeddeddimension Finally the maximum of the 119863

2(perp)119903is selected as

the new1198632estimate called 119863

2(perp)

Another new approach for 1198632estimation based on sam-

ple entropy (SampEn) is now presented SampEn was definedby Zurek et al [17] as

SampEn (119898 119903119873) = log (119862119898(119903)) minus log (119862

119898+1(119903)) (16)


0 10 201

15

2

m

Gradient descent

Cor

relat

ion

dim

ensio

n

minus8

minus6

minus4

minus2

0

minus5 minus4 minus3 minus2 minus1 0 1 2

m = 1

m = 21

log(r)

log(Cm(r))

Dm

2(perp)r119895

Dm

2(perp)r119894

Figure 3Maximum slope points aremarkedwith crosses over fittedsigmoid curves Points calculated using gradient descent criteriafrom two starting points are shown in dots and circles 119903

119895is the point

which corresponds to the maximum slope in the lowest embeddeddimension The inset illustrates the correlation dimension estima-tion of the three sets of points

where in this case 119862119898(119903) is computed as in (3) but without

considering self-comparisons Let us define SampEn119894=119895(119898 119903)

as the sample entropy considering self-pairs which is easilycomputed for all embedded dimensions 119898 and a huge set ofthresholds r using the fast algorithm described in Section 22We can generate a SampEn

119894=119895(119898 119903) surface from the fitted

sigmoid curves as can be seen in Figure 4 an example of a300-beat RR interval series extracted from one recording ofthe database used in [10] For each embedded dimension thevalue of 119903 which maximizes SampEn

119894=119895(119898 119903) is used to eval-

uate the slope of the linear region of the SCF log-log curves

119863

119898

2(max) yielding another 1198632estimate called in this paper

119863

2(max)

3 Materials

The selected time series chosen to validate the approachesproposed in this paper to estimate119863

2are the Lorenz attractor

the MIX(119875) process and real HRV signals respectively

Lorenz Attractor The Lorenz system is described by threecoupled first order differential equations whose solutionexhibits a chaotic behaviour system for certain parameter val-ues and initial conditions This is called the Lorenz attractor

119889119909

119889119905

= 120590 (119910 minus 119909)

119889119910

119889119905

= 120588119909 minus 119910 minus 119909119911

119889119911

119889119905

= minus120573119911 + 119909119910

(17)

05

15

0

1

0 2

m

1510

5

0

minus8 minus6 minus4 minus2

minus05

log(r)

SampE

n (i=j)(SCF)

Dm

2(max)

Dm

2(perp)

05

15

0

1

0 2

m

1510

5


minus05

log(r)

Dm

2(max)

Dm

2(perp)

Figure 4 SampEn119894=119895(119898 119903) surface for a 300-beat RR interval series

For each embedded dimensionmaximumpoint ismarkedwith solidtriangle Circles correspond to the 119903 values which define 119863

2(perp)

For parameter values 120590 = 10 120588 = 28 and 120573 = 83the theoretical119863

2value is 202 [24] In this study the system

equations are discretized with a time step of 001

119872119868119883(119875) Signals MIX(119875) is a family of stochastic processesthat samples a sine for 119875 = 0 and becomes more randomas 119875 increases (119875 = 1 completely random) [5] following theexpression

MIX(119875)119895= (1 minus 119885

119895)119883

119895+ 119885

119895119884

119895 (18)

where 119883119895=radic2 sin(212058711989512) 119884

119895equiv 119894119894119889 uniform random

variables on [minusradic3radic3] and 119885119895equiv 119894119894119889 random variables

with 119885119895= 1 with probability 119875 and 119885

119895= 0 with probability

1minus119875 MIX indicates amixture of deterministic and stochasticcomponents

HRV Signals The HRV representation used in this study isthe time difference between the occurrences of consecutivenormal heart beats the so-called RR interval Ectopic beatsas well as missed and false detections introduce some extravariability in theRR interval series which is not representativeof the ANS activity Thus they were detected and corrected[25]The RR interval series analysed in this study belongs to adatabase recorded at theMiguel Servet University Hospital inZaragoza (Spain)That database was used to predict hypoten-sion events during spinal anesthesia in elective cesareandelivery by HRV analysis [10] It consists of ECG signals from11 women with programmed cesarean section recorded at a1000Hz sampling frequency immediately before the cesareansurgery Five of them suffered a hypotension event during thesurgery (Hyp) and 6 did not (NoHyp) The series analysedcorrespond to 5 minutes in a lateral decubitus position See[10] for further database details

4 Results

All the results presented in this section are computed usingthe ℓinfin-norm The effect of different norms in 119863

2estimation

is discussed in Appendix A


Table 1 Computational time of correlation sums estimated forLorenz attractor series of different sample lengths 119878

119901is the speed-

up achieved and defined as 119878119901= 119879Seq119879MO where 119879Seq is the time

demand for a sequential algorithm and 119879MO the time demand forthe proposed technique based on matrix operations

N (samples) 119879Seq (s) 119879MO (s) 119878

119901

300 1086 09 asymp12005000 8691198905 50 asymp1600010000 3631198906 300 asymp12000

The computational time cost of the correlation sumsdepends on the length of the data the maximum embeddeddimension considered and the amount of thresholds usedThe results shown in Table 1 correspond to computationaltime required for different data lengths of Lorenz attractorseries a maximum embedded dimension of 16 and a set of300 threshold values 119903 evenlyspaced from 001 to 3The com-putational time required for a sequential approach is denotedby119879Seq whereas the time required for the proposed techniquebased on matrix operations is denoted by 119879MO This allowsdefining the speed up achieved by the novel approach as theratio between both measurements 119878

119901= 119879Sep119879MO As it is

shown in Table 1 119878119901increases with data length For a 300-

sample data (usual length for a 5minute RR interval series)correlation sums are estimated in approximately 1 s

The Lorenz attractor series was used to validate the newproposed methodologies Figure 5(a) displays the SCF log-log curves for embedded dimensions119898 from 1 to 10 The setsof points where the slope is evaluated according to (15) aredisplayed for different starting points For each starting pointthe corresponding set of points is selected following a gradi-ent descent technique Figure 5(b) shows the slope estimate(1198631198982(perp)119903

) versus119898 for each starting point Figure 5(c) displaysthe correlation dimension estimate (119863

2(perp)119903) versus log(119903) for

each starting point The maximum (1198632(perp) 119903

) constitutes thenovel119863

2estimate (119863

2(perp))

Table 2 displays correlation dimension estimates usingthe different approaches presented in this study Note thatalthough the three approaches give results close to the theo-retical value of Lorenz attractor correlation dimensionthe 119863

2(perp)approach is the closest one Relative errors for

approaches 1198632and 119863

2(max) are above 4 while for 1198632(perp)

itis just 1 119863

2estimated as described in [10] is also included

for comparison purposes

119863

2was applied to a set of MIX series with different 119875

values (01 04 and 08)These estimates can be considered asmeasures of the randomness of the signals when these signalsare finite stochastic processes see Figure 6

The same database for HRV analysis as in [10] was usedThe results shown in Table 2 are divided into hypotensionand nonhypotension groups The approaches proposed inthis paper were applied as well as the classical correlationdimension estimate described in [10] included for compari-son purposes The distribution of the data was found to benot normally distributed by the Kolmogorov-Smirnov test

Table 2 1198632estimated by different approaches for Lorenz attrac-

tor series (5000 samples) and HRV signals (300 samples) Dataexpressed as median | interquartile range

Lorenz HRVHyp NoHyp P value

119863

2[10] 193 592|057 439|126 0028

119863

2193 594|060 481|079 0028

119863

2(perp)201 641|065 511|077 0028

119863

2(max) 193 588|049 483|070 0010

Table 3 ROC area for the analysis of all studied correlationdimension estimates for the database used Accuracy sensibility andspecificity estimatedwith the correspondent cut points are expressedin percentage

ROC area Acc Sen Spe119863

2[10] 0900 818 714 100

119863

20900 818 714 100

119863

2(perp)0900 909 100 857

119863

2(max) 0967 909 833 100

and therefore the Mann-Whitney 119880 test was applied to eval-uate their statistical differences in medians The differencesbetween both groups for all estimates were found to bestatistically significantwith a119875 value lower than 003 In orderto evaluate the discriminant power of the proposedmeasuresROC analysis was performed Area of the ROC curveaccuracy sensitivity and specificity for all the proposedapproaches and the classical 119863

2estimate used in [10] were

displayed in Table 3The proposed 1198632estimates maintain the

accuracy achieved in [10] while the techniques based on theSampEn surface and the gradient descent actually increase it

5 Discussion and Conclusion

In this paper amethodological framework has been proposedto compute the correlation dimension (119863

2) of a limited time

series such as HRV signals which includes fast computationof the correlation sums sigmoidal curve fitting of log-logcurves three approaches for estimating the slope of the linearregion and exponential fitting of the 119863119898

2versus119898 curves

One important limitation for the application of 1198632to

HRV analysis is the long computational time required forthe correlation sums In an attempt to solve this probleman algorithm has been proposed based on matrix operationsIn [17] another approach was described based on parallelcomputing which decreased the time demand Neverthelessthe computational times achieved in the present work wereobtained with a regular computer (Windows 7 based PCIntel Core i7 35 GHz 16GbRAM with Matlab R2011a) Asan example for a signal of 300 sample length (a usual lengthin typical 5min HRV analysis asymp300 beats) the time demandwas reduced with respect to the sequential approach from 18minutes to 1 second which allows the online computation of119863

2in clinical practice Computational time required for the

proposed approaches is discussed in Appendix B


minus8

minus6

minus4

minus2

0

2minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

log(r)

log(Cm(r))

(a)

Embedded dimension m

22

2

18

16

14

12

11 2 3 4 5 6 7 8 9 10

Dm 2(perp)

(b)

18

185

19

195

2

205

minus5 minus45 minus4 minus35 minus3

log(r)

Dm 2(perp)r

D2(perp)

(c)

Figure 5 (a) Set of points where slope is estimated from the fitted sigmoid curves in the approach proposed in Section 23 (b) Set of 1198631198982(perp)119903

estimates for different starting points versus embedded dimensions are fitted by the exponential equation (15) (c) Correlation dimensionestimate for each set corresponding to different starting points Data extracted from Lorenz attractor of 5000-sample length

1

0

minus1

MIX

(08

)M

IX(04

)M

IX(01

)

1

0

minus1

1

0

minus1

20 60 100 140 180

i

(a)

6

5

4

3

2

10 20 40 60 80 100

D2 = 543

D2 = 370

D2 = 183

Embedding dimension m

Cor

relat

ion

dim

ensio

n

Mix(08)

Mix(04)

Mix(01)

(b)

Figure 6 MIX signals with different degrees of randomness and the effects on the estimation of the 1198632


HypNoHyp

2 25 3 35 4 45 5 55

Time (min)

RR in

terv

al (m

s)

150

02

04

06

08

1

(a)


3

4

5

6

7

log(r)

HypNoHyp

X minus2139Y 6653

X minus1893Y 5257

D2(perp)r

(b)

Figure 7 The left panel shows two RR intervals one corresponding to a patient who developed a hypotension event (Hyp) and the other toone who did not (NoHyp) the right panel shows the 119863

2(perp)estimation using the perpendicular points in the log-log curves

Another limitation of the1198632estimate is its reliability One

of the system characteristics that can lead to an unreliablemeasurement of119863

2is the nonstationarity of the data Several

techniques attempting to characterize these dynamic systemshave been reportedmainly focused on changing the 120591param-eter or even taking into account the time between the vectors[13 26 27] Searching the linear region of the log-log curvesbecomes a difficult challenge when the system is nonstation-ary sincemore than one linear region can appear and classical119863

2estimate is unreliable in those cases The SCF approach is

more robust since it does not give any estimate if the fitting isnot good enough

The novel approaches proposed in this study for the esti-mation of 119863

2use the SCF approach 119863

2(perp)exploits the fact

that the linear region of the log-log curves is almost parallelfor high embedded dimensions This allows a set of pointssurrounding the maximum slope point to be consideredand therefore several correlation dimension estimates areobtained for these starting points 119863

2(max) is based on the dif-ferences between two consecutive log-log curves that definethe SampEn

119894=119895surfaceThis surface showedmaximum values

for each embedded dimension119898 and a specific threshold 119903providing another estimation of the correlation dimension

119863

2(perp)was found to be the closest to the theoretical correlation

dimension value for the Lorenz attractor series with 5000 sizepoints and for ℎ = 001 with a relative error of 1 while 119863

2

and 1198632(max) obtained a relative error of 4with the same data

The correlation dimension is known to be a surrogate ofthe fractal dimension of a chaotic attractor [12] Howeverwhen applied to limited time series nonzero finite correlationdimension values do not imply the existence of an under-lying chaotic attractor For example when applied to MIXprocesses nonzero finite119863

2values were obtained higher for

more random processes Thus although 1198632cannot be inter-

preted as the fractal dimension of an underlying chaoticattractor it still gives a measure of the complexity of the pro-cess at least regarding its unpredictability

Thus the 1198632estimate in HRV signals may shed light on

the degree of complexity of the ANS or how many degreesof freedom it has The group of women (Hyp) sufferinghypotension events occurring during the surgery of a pro-grammed cesarean section under spinal anesthesia showedhigher 119863

2values than the group who did not (NoHyp) in

the lateral decubitus position As an example Figure 7 showsone patient of each group and the 119863

2(perp)estimate All the

proposed correlation dimension estimates not only maintainthe accuracy obtained in [10] they also increase it Predictinghypotension is a challenge since it occurs in the 60 of thecases producing fetal stress [28] If the goal is to predict thosewho are going to suffer hypotension then the estimates thatperformed 100 of specificity will be selected being119863

2[10]

119863

2 and 119863

2(max) Otherwise if the goal is to use prophylaxis inthe less number of patients to prevent hypotension then theestimates that performed 100 of sensitivity will be chosenand in this case it is 119863

2(perp) The effect of prophyilaxis on

patients who finally are not going to suffer a hypotensionevent and the relation with fetal stress needs further studies

The contribution of this paper to the field is the proposalof a methodological framework for a reliable estimation ofthe correlation dimension from a limited time series such asHRV signals avoiding or at least alleviating the misleadinginterpretations that can be made from classical correlationdimension estimates The computational speed-up achievedmay allow this framework to be considered for monitoringin clinical practice Nevertheless the main limitation for theapplication of these methodologies to HRV analysis lies inits relation with the underlying physiology which is stillunclear and needs further studies In spite of the fact that theframework proposed in this paper is focused on the charac-terization of HRV signals its applicability could be extendedto a wide range of fields However an evaluation would beneeded to ascertain whether the proposed approaches areappropriate in each particular case


1

06

02

minus02

minus06

minus1minus1

minus06 minus02 02 06 1

Δ 2

x(i) minus x(j)

xx

(i+120591)minus

(j+120591) y ij

middot 2 = 1

middot 1 = 1

(a)

0

minus1

minus2

minus3

minus4

minus5

minus6minus5 minus4 minus3 minus2 minus1 0 1

2-norminfin-norm

log(r)

log(Cm(r))

1-norm985747

985747

985747

(b)

Figure 8 In the left panel vector differences of any two reconstructed vectors are shown (ie for119898 = 2Δy2119894119895) where solid circles and dashed

lines represent the points whose ℓ2-norm and ℓ

1-norm are equal to 1 respectively The dots are the differences below ℓ

2-norm unity and the

dots with circles are below ℓ1-norm unity In the right panel log-log curves of one HRV signal (300 samples) used in the study are shown for

a119898 = 10 and ℓ1- ℓ2- and ℓ

infin-norms

Appendices

A Use of Norms and Thresholds inCorrelation Dimension Estimates

The correlation dimension is considered norm invariant [14]However the effect of selecting the norm in correlationdimension estimates deserves further attention when appliedto a finite data set The norm of the difference vector Δy119898

119894119895

defines the distance 119889119898119894119895in (2) Norms can be defined from ℓ

1

( sdot 1) to ℓinfin( sdot infin) Left panel in Figure 8 shows normunity

for ℓ1and for ℓ

2 Moreover it is illustrated how a distance 1198892

119894119895

can be lower than the norm unity or not depending on whichnorm is used The norm unity is chosen as an example ofany threshold 119903 used in the correlation dimension algorithmTherefore by fixing the set of thresholds the appearance ofthe linear region of the log-log curve can be compromised

In Figure 8 the right panel shows how the applicationof different norms shifts the log-log curves losing the entirelinear region in some cases due to the fixed range of thresh-olds The range of these thresholds should be long enough toensure that the linear regions are contained therein thus theelection of the norm compromises the set of thresholds used

In the SCF approach it is particularly important that thetwo asymptotic regions should be represented in the log-logcurve Therefore the correct selection of the norm and therange of the set of thresholds are critical to assure the good-ness of the SCF approach Table 4 shows the correlationdimension estimates for 5000 data length of Lorenz attractor

Table 4 Correlation dimension estimates for the different proposedapproaches using different norms for Lorenz attractor series (5000samples) using different norms

Lorenz attractorℓ

1ℓ

2ℓ

infin

119863

2[10] 195 194 193

119863

2169 170 193

119863

2(perp)184 174 201

119863

2(max) 199 171 193

series The effect of different norms is reflected in the esti-mates since the set of thresholds was fixed As it is shown theapplication of ℓ

infin-norm combined with the fixed set of

thresholds used achieves closest values with respect to thetheoretical correlation dimension value for Lorenz attractor201

B Computational Time Demandof Novel Approaches to CorrelationDimension Estimates

In Section 22 a new technique based on matrix operations(MO) was introduced in order to compute correlation sumswhich represent the core of the correlation dimension algo-rithm Nevertheless in the paper no computational time costwas considered for the new proposed approaches for corre-lation dimension estimates Table 5 shows the time required


Table 5 Computational time cost for correlation dimension esti-mates by all proposed approaches considering Lorenz attractorseries and HRV signals in which ℓ

infin-norm was applied Data

expressed as mean plusmn standard deviation

Lorenz HRV(5000 samples) (300 samples)

119879

1198632

(s) [10] (886 plusmn 035)e5 3314 plusmn 180119879

2

(s) 194 plusmn 29 466 plusmn 058119879

2(perp)

(s) 260 plusmn 38 214 plusmn 30119879

2(max)

(s) 194 plusmn 29 430 plusmn 056

for the correlation dimension estimates including that usedin [10]

10 realizations of Lorenz attractor series were generatedwhose initial conditions were randomly chosen It is notice-able that the time cost of 119863

2(perp)is higher compared to the

others in both cases the Lorenz series and the HRV signals(11 subjects) since it uses several sets of slope estimatedto compute correlation dimension Furthermore the ratiobetween them is higher for the HRV signals than for theLorenz series Each of the different sets of thresholds is asso-ciated with an 119903 value in an interval centred on the maximumslope for119898 = 1 This interval is defined as a decrease in 50of the amplitude of the maximum in the SCF first derivativeThemore abrupt the transition zone in the sigmoid the lowerthe amount of starting points Thus each realization is donewith a different number of points varying the computationaltime

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported by the Ministerio de Cienciae Innovacion Spain and FEDER under Project TEC2010-21703-C03-02 and by ISCIII Spain through Project PI1002851 (FIS)

References

[1] Task Force of the European Society of Cardiology and TheNorth American Society of Pacing and ElectrophysiologyldquoHeart rate variability Standards ofmeasurement physiologicalinterpretation and clinical userdquo European Heart Journal vol 17no 3 pp 354ndash381 1996

[2] J W Kantelhardt S A Zschiegner E Koscielny-Bunde SHavlin A Bunde and H E Stanley ldquoMultifractal detrendedfluctuation analysis of nonstationary time seriesrdquo Physica A vol316 no 1ndash4 pp 87ndash114 2002

[3] A Lempel and J Ziv ldquoOn the complexity of finite sequencesrdquoIEEE Transactions on Information Theory vol 22 no 1 pp 75ndash81 1976

[4] AWolf J B Swift H L Swinney and J A Vastano ldquoDetermin-ing Lyapunov exponents from a time seriesrdquo Physica D vol 16no 3 pp 285ndash317 1985

[5] S M Pincus and A L Goldberger ldquoPhysiological time-seriesanalysis what does regularity quantifyrdquo American Journal ofPhysiologymdashHeart and Circulatory Physiology vol 266 no 4pp H1643ndashH1656 1994

[6] S Pincus and B H Singer ldquoRandomness and degrees of irreg-ularityrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 93 no 5 pp 2083ndash2088 1996

[7] A Porta S Guzzetti R Furlan T Gnecchi-Ruscone N Mon-tano and A Malliani ldquoComplexity and nonlinearity in short-term heart period variability comparison of methods based onlocal nonlinear predictionrdquo IEEE Transactions on BiomedicalEngineering vol 54 no 1 pp 94ndash106 2007

[8] S Cerutti G Carrault P J M Cluitmans et al ldquoNon-linearalgorithms for processing biological signalsrdquoComputerMethodsand Programs in Biomedicine vol 51 no 1-2 pp 51ndash73 1996

[9] D Chamchad V A Arkoosh J C Horrow et al ldquoUsing heartrate variability to stratify risk of obstetric patients undergoingspinal anesthesiardquo Anesthesia amp Analgesia vol 99 no 6 pp1818ndash1821 2004

[10] L Canga A Navarro J Bolea J M Remartınez P Laguna andR Bailon ldquoNon-linear analysis of heart rate variability and itsapplication to predict hypotension during spinal anesthesia forcesareandeliveryrdquo inProceedings of theComputing inCardiology(CinC rsquo12) pp 413ndash416 Krakow Poland September 2012

[11] J Bolea R Bailon E Rovira JM Remartınez P Laguna andANavarro ldquoHeart rate variability in pregnant women before pro-grammed cesarean interventionrdquo inXIIIMediterranean Confer-ence onMedical and Biological Engineering and Computing 2013M L R Romero Ed vol 41 of IFMBE Proceedings pp 710ndash713Springer International 2014

[12] P Grassberger and I Procaccia ldquoCharacterization of strangeattractorsrdquo Physical Review Letters vol 50 no 5 pp 346ndash3491983

[13] J Theiler ldquoSpurious dimension from correlation algorithmsapplied to limited time-series datardquo Physical Review A vol 34no 3 pp 2427ndash2432 1986

[14] J Theiler ldquoEstimating fractal dimensionrdquo Journal of the OpticalSociety of America A vol 7 no 6 pp 1055ndash1073 1990

[15] H Kantz and T SchreiberNonlinear Time Series Analysis Cam-bridge University Press Cambridge UK 2004

[16] GWidman K Lehnertz P JansenWMeyerW Burr andC EElger ldquoA fast general purpose algorithm for the computation ofauto- and cross-correlation integrals from single channel datardquoPhysica D vol 121 no 1-2 pp 65ndash74 1998

[17] S Zurek P Guzik S Pawlak M Kosmider and J PiskorskildquoOn the relation between correlation dimension approximateentropy and sample entropy parameters and a fast algorithm fortheir calculationrdquo Physica A vol 391 no 24 pp 6601ndash66102012

[18] F Takens ldquoDetecting strange attractors in turbulencerdquo inDynamical Systems and TurbulenceWarwick 1980 D Rand andL-S Young Eds vol 898 of Lecture Notes in Mathematics pp366ndash381 Springer Berlin Germany 1981

[19] J A Lee andMVerleysenNonlinear Dimensionality ReductionSpringer Berlin Germany 2007

[20] J P Rigaut ldquoAn empirical formulation relating boundarylengths to resolution in specimens showing ldquonon-ideally frac-talrdquo dimensionsrdquo Journal ofMicroscopy vol 133 no 1 pp 41ndash541984


[21] JWDollinger RMetzler andT F Nonnenmacher ldquoBi-asymp-totic fractals fractals between lower and upper boundsrdquo Journalof Physics A vol 31 no 16 pp 3839ndash3847 1998

[22] A L Navarro-Verdugo F M Goycoolea G Romero-MelendezI Higuera-Ciapara and W Arguelles-Monal ldquoA modifiedBoltzmann sigmoidal model for the phase transition of smartgelsrdquo Soft Matter vol 7 no 12 pp 5847ndash5853 2011

[23] R Carvajal M Vallverdu R Baranowski E Orlowska-Bara-nowska J J Zebrowski and P Caminal ldquoDynamical non-linearanalysis of heart rate variability in patients with aortic stenosisrdquoin Proceedings of the Computers in Cardiology pp 449ndash452September 2002

[24] E N Lorenz ldquoDeterministic non-periodic flowrdquo Journal ofAtmospheric Science vol 20 no 2 pp 130ndash141 1963

[25] J Mateo and P Laguna ldquoAnalysis of heart rate variability in thepresence of ectopic beats using the heart timing signalrdquo IEEETransactions on Biomedical Engineering vol 50 no 3 pp 334ndash343 2003

[26] L AAguirre ldquoAnonlinear correlation function for selecting thedelay time in dynamical reconstructionsrdquo Physics Letters A vol203 no 2-3 pp 88ndash94 1995

[27] H S Kim R Eykholt and J D Salas ldquoNonlinear dynamicsdelay times and embedding windowsrdquo Physica D vol 127 no1-2 pp 48ndash60 1999

[28] A M Cyna M Andrew R S Emmett P Middleton and SW Simmons ldquoTechniques for preventing hypotension duringspinal anaesthesia for caesarean sectionrdquo Cochrane Database ofSystematic Reviews vol 18 no 4 Article ID CD002251 2006

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


The reduction of HRV complexity has been associatedwith age disease and unbalanced cardiovascular regulation[7] Complexity measures have been proven to characterizeHRV signals more successfully than linear approaches in cer-tain applications [8] In [9ndash11] point correlation dimensionof HRV signals predicted hypotension events in pregnantwomen during spinal anesthesia for cesarean section some-thing which time and frequency domain indices were unableto do

While these measurements are of considerable interesttheir application to HRV has some pitfalls that could misleadtheir interpretation One of such limitations arises from theirapplication to limited time series Correlation dimensionestimation is highly dependent on the length of the time series[12] Several studies have reported the effect of data length on119863

2estimation as well as proposals to alleviate this effect [13

14] Stationarity is another requirement that a time serieshas to fulfil to obtain reliable results However satisfying theconstraint of finite series and stationarity at the same time isusually difficult [15] Yet another limitation of these measure-ments is the long computational time required Data lengthexponentially increases the computational time cost in aclassical sequential approach In the case of 119863

2 several

attempts to try to reduce this factor have been reported byWidman et al [16] and Zurek et al [17] the latter proposingparallel computing using MPI (Message Passing Interface)

Themain goal of this study is to propose amethodologicalframework for robust and fast estimation of119863

2and its appli-

cation inHRV signals Section 2 starts with a definition of thecorrelation dimension and its classical estimation An algo-rithm for its fast computation is proposed Robustness isaddressed by fitting the log-log curve to a sigmoid functionafter which three alternative approaches for119863

2estimation are

presented Section 3 introduces synthetic and real (HRVsignals) data where the proposed estimates are evaluated andinterpreted Section 4 presents the results while Section 5 setsout the discussion and conclusions of the study

2 Methods

21 Correlation Dimension Let 119909(119899) 119899 = 1 119873 be thetime series of interest which in HRV analysis will be the 119877119877interval series normalized to unit amplitude with119873being thetotal number of beats A set of119898-dimensional vectors y

119898(119894)

called reconstructed vectors are generated [18]

y119898119894= [119909(119894) 119909(119894 + 120591) 119909(119894 + 2120591) 119909(119894 + (119898 minus 1)120591)]

119879 (1)

where 120591 represents the delay between consecutive samples inthe reconstructed space Then the amount of reconstructedvectors is 119873

119898= 119873 minus 120591(119898 minus 1) for each 119898-dimension The

distance between each pair of reconstructed vectors y119898119894 y119898119895

is denoted as

119889

119898

119894119895= 119889 (y119898

119894 y119898119895) (2)

and this can be computed as the norm of the difference vectorΔy119898119894119895= y119898119894minus y119898119895 (In Appendix A different norms and their

effect on correlation dimension estimates from finite time

series are discussed) The correlation sum which representsthe probability of the reconstructed vector pair distance beingsmaller than a certain threshold 119903 is computed as

119862

119898(119903) =

1

119873

119898(119873

119898minus 1)

119873119898

sum

119894119895=1

119867(119903 minus 119889

119898

119894119895)

=

1

119873

119898(119873

119898minus 1)

119873119898

sum

119894=1

119888

119898

119894(119903)

(3)

where119867(sdot) is the Heaviside function defined as

119867(119909) =

1 119909 ge 0

0 119909 lt 0

(4)

and 119888119898119894(119903) = sum

119873119898

119895=1119867(119903 minus 119889

119898

119894119895)

For deterministic systems 119862119898(119903) decreases monotoni-

cally to 0 as 119903 approaches 0 and it is expected that 119862119898(119903) is

well approximated by 119862119898(119903) asymp 119903

119863119898

2 Thus 1198631198982can be defined

as

119863

119898

2= lim119903rarr0

log119862119898(119903)

log (119903) (5)

For increasing1198981198631198982values tend to saturate to a value119863

2

which constitutes the final correlation dimension estimate

22 Fast Computation of Correlation Sums One importantlimitation of 119863

2estimation is the long computational time

required mainly due to the sequential estimation of corre-lation sums This section describes an algorithm for the fastcomputation of correlation sums based on matrix operations(MO) A matrix S which contains the differences between allpairs of samples of 119909(119899) is computed as

S = X minus X119879 (6)

where X is the119873 times119873matrix

X =(

119909(1) 119909 (2) sdot sdot sdot 119909 (119873)

119909 (1) 119909 (2) sdot sdot sdot 119909 (119873)

119909 (1) 119909 (2) sdot sdot sdot 119909 (119873)

d

119909 (1) 119909 (2) sdot sdot sdot 119909 (119873)

) (7)

=

x xi1 xi+ xi+mminus xN1

x1j xij xi+1j xi+mminus1j xNj

x1j+1 xij+1 xi+1j+1 xi+mminus1j+1 xNj+1

x1j+mminus1 xij+mminus1 xi+1j+mminus1 xi+mminus1j+mminus1 xNj+mminus1

x1N xiN xi+1N xi+mminus1N xNN

(8)





119894119895generates a S119898

119894matrix

mi =

xi1 xi+ xi+mminus1m


xiN xi+1N +1

xi+mminus1N

=

Δ miN

T

Δ mi1

T

Δ mij

T (9)



119894119895 To




2 119903

119873119903

]

Γ

119898

119894= (

119867(1199031 minus 119889119898

1198941) 119867(1199032 minus 119889

119898

1198941) sdot sdot sdot 119867(119903119873

119903

minus 119889119898

1198941)

119867(1199031 minus 119889119898

1198942) 119867(1199032 minus 119889

119898

1198942) sdot sdot sdot 119867(119903119873

119903

minus 119889119898

1198942)

119867(1199031 minus 119889119898

119894119873119898

) 119867(1199032 minus 119889119898

119894119873119898

) sdot sdot sdot 119867(119903119873119903

minus 119889119898

119894119873119898

)

)

(10)

where Γ119898119894

is a 119873119898times 119873



c119898119894= Γ

119898

119894

1198791 = (

119888

119898

119894(119903

1)

119888

119898

119894(119903

2)

119888

119898

119894(119903

119873119903

)

) (11)









119863

119898

2= lim119903rarr0

120597 log119862119898(119903)

120597 log (119903) (12)


0

minus5

minus10


log(r)

m = 1

m = 2

m = 3

m = 10

log(Cm(r))





2







119891 (119909) = 119860

2minus

(119860

2minus 119860

1)

119861 + 119890

(minus119909+119909119900)120572

(13)


0 50 100 150 200 250 300Beats

Nor

mal

ized

RR

inte

rval

0

04

08

minus04

minus08

(a)


minus4

minus3 minus2

minus10

minus5

minus12

minus1

minus8

minus6

minus4

minus2

0

log(r)

m = 1

m = 2

m = 3 m = 4m = 5

log(Cm(r))

(b)




119889119891 (119909)

119889119909

= minus

(119860

2minus 119860

1) 119890

(minus119909+119909119900)120572

[120572(119861 + 119890

(minus119909+119909119900)120572)

2

]

(14)


119863

119898



119898(119903) =1119873

119898(119873

119898minus 1)







119863

119898

2= 119863

2(1 minus 119860119890

minus119896119898) (15)





2










2(perp)




119898+1(119903)) (16)


0 10 201

15

2

m

Gradient descent

Cor

relat

ion

dim

ensio

n

minus8

minus6

minus4

minus2

0


m = 1

m = 21

log(r)

log(Cm(r))

Dm

2(perp)r119895

Dm

2(perp)r119894


119895is the point







119894=119895(119898 119903) is used to eval-


119863

119898


119863

2(max)

3 Materials





119889119909

119889119905

= 120590 (119910 minus 119909)

119889119910

119889119905

= 120588119909 minus 119910 minus 119909119911

119889119911

119889119905

= minus120573119911 + 119909119910

(17)

05

15

0

1

0 2

m

1510

5

0


minus05

log(r)

SampE

n (i=j)(SCF)

Dm

2(max)

Dm

2(perp)

05

15

0

1

0 2

m

1510

5


minus05

log(r)

Dm

2(max)

Dm

2(perp)



2(perp)





MIX(119875)119895= (1 minus 119885

119895)119883

119895+ 119885

119895119884

119895 (18)

where 119883119895=radic2 sin(212058711989512) 119884







4 Results


2estimation




119901is the speed-




119901



119901= 119879Sep119879MO As it is








2estimate (119863

2(perp))





itis just 1 119863



119863







119863

2[10] 193 592|057 439|126 0028

119863

2193 594|060 481|079 0028

119863

2(perp)201 641|065 511|077 0028

119863

2(max) 193 588|049 483|070 0010



2[10] 0900 818 714 100

119863

20900 818 714 100

119863

2(perp)0900 909 100 857

119863

2(max) 0967 909 833 100















minus8

minus6

minus4

minus2

0

2minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

log(r)

log(Cm(r))

(a)


22

2

18

16

14

12

11 2 3 4 5 6 7 8 9 10

Dm 2(perp)

(b)

18

185

19

195

2

205


log(r)

Dm 2(perp)r

D2(perp)

(c)



1

0

minus1

MIX

(08

)M

IX(04

)M

IX(01

)

1

0

minus1

1

0

minus1

20 60 100 140 180

i

(a)

6

5

4

3

2

10 20 40 60 80 100

D2 = 543

D2 = 370

D2 = 183


Cor

relat

ion

dim

ensio

n

Mix(08)

Mix(04)

Mix(01)

(b)



HypNoHyp

2 25 3 35 4 45 5 55

Time (min)

RR in

terv

al (m

s)

150

02

04

06

08

1

(a)


3

4

5

6

7

log(r)

HypNoHyp

X minus2139Y 6653

X minus1893Y 5257

D2(perp)r

(b)
















119863



2












2[10]

119863

2 and 119863






1

06

02

minus02

minus06

minus1minus1


Δ 2

x(i) minus x(j)

xx

(i+120591)minus

(j+120591) y ij

middot 2 = 1

middot 1 = 1

(a)

0

minus1

minus2

minus3

minus4

minus5


2-norminfin-norm

log(r)

log(Cm(r))

1-norm985747

985747

985747

(b)






a119898 = 10 and ℓ1- ℓ2- and ℓ

infin-norms

Appendices



119894119895


1


for ℓ1and for ℓ


119894119895





Lorenz attractorℓ

1ℓ

2ℓ

infin

119863

2[10] 195 194 193

119863

2169 170 193

119863

2(perp)184 174 201

119863

2(max) 199 171 193











119879

1198632

(s) [10] (886 plusmn 035)e5 3314 plusmn 180119879

2

(s) 194 plusmn 29 466 plusmn 058119879

2(perp)

(s) 260 plusmn 38 214 plusmn 30119879

2(max)








Acknowledgments


References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




















119894119895generates a S119898

119894matrix

mi =

xi1 xi+ xi+mminus1m


xiN xi+1N +1

xi+mminus1N

=

Δ miN

T

Δ mi1

T

Δ mij

T (9)



119894119895 To




2 119903

119873119903

]

Γ

119898

119894= (

119867(1199031 minus 119889119898

1198941) 119867(1199032 minus 119889

119898

1198941) sdot sdot sdot 119867(119903119873

119903

minus 119889119898

1198941)

119867(1199031 minus 119889119898

1198942) 119867(1199032 minus 119889

119898

1198942) sdot sdot sdot 119867(119903119873

119903

minus 119889119898

1198942)

119867(1199031 minus 119889119898

119894119873119898

) 119867(1199032 minus 119889119898

119894119873119898

) sdot sdot sdot 119867(119903119873119903

minus 119889119898

119894119873119898

)

)

(10)

where Γ119898119894

is a 119873119898times 119873



c119898119894= Γ

119898

119894

1198791 = (

119888

119898

119894(119903

1)

119888

119898

119894(119903

2)

119888

119898

119894(119903

119873119903

)

) (11)









119863

119898

2= lim119903rarr0

120597 log119862119898(119903)

120597 log (119903) (12)


0

minus5

minus10


log(r)

m = 1

m = 2

m = 3

m = 10

log(Cm(r))





2







119891 (119909) = 119860

2minus

(119860

2minus 119860

1)

119861 + 119890

(minus119909+119909119900)120572

(13)


0 50 100 150 200 250 300Beats

Nor

mal

ized

RR

inte

rval

0

04

08

minus04

minus08

(a)


minus4

minus3 minus2

minus10

minus5

minus12

minus1

minus8

minus6

minus4

minus2

0

log(r)

m = 1

m = 2

m = 3 m = 4m = 5

log(Cm(r))

(b)




119889119891 (119909)

119889119909

= minus

(119860

2minus 119860

1) 119890

(minus119909+119909119900)120572

[120572(119861 + 119890

(minus119909+119909119900)120572)

2

]

(14)


119863

119898



119898(119903) =1119873

119898(119873

119898minus 1)







119863

119898

2= 119863

2(1 minus 119860119890

minus119896119898) (15)





2










2(perp)




119898+1(119903)) (16)


0 10 201

15

2

m

Gradient descent

Cor

relat

ion

dim

ensio

n

minus8

minus6

minus4

minus2

0


m = 1

m = 21

log(r)

log(Cm(r))

Dm

2(perp)r119895

Dm

2(perp)r119894


119895is the point







119894=119895(119898 119903) is used to eval-


119863

119898


119863

2(max)

3 Materials





119889119909

119889119905

= 120590 (119910 minus 119909)

119889119910

119889119905

= 120588119909 minus 119910 minus 119909119911

119889119911

119889119905

= minus120573119911 + 119909119910

(17)

05

15

0

1

0 2

m

1510

5

0


minus05

log(r)

SampE

n (i=j)(SCF)

Dm

2(max)

Dm

2(perp)

05

15

0

1

0 2

m

1510

5


minus05

log(r)

Dm

2(max)

Dm

2(perp)



2(perp)





MIX(119875)119895= (1 minus 119885

119895)119883

119895+ 119885

119895119884

119895 (18)

where 119883119895=radic2 sin(212058711989512) 119884







4 Results


2estimation




119901is the speed-




119901



119901= 119879Sep119879MO As it is








2estimate (119863

2(perp))





itis just 1 119863



119863







119863

2[10] 193 592|057 439|126 0028

119863

2193 594|060 481|079 0028

119863

2(perp)201 641|065 511|077 0028

119863

2(max) 193 588|049 483|070 0010



2[10] 0900 818 714 100

119863

20900 818 714 100

119863

2(perp)0900 909 100 857

119863

2(max) 0967 909 833 100















minus8

minus6

minus4

minus2

0

2minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

log(r)

log(Cm(r))

(a)


22

2

18

16

14

12

11 2 3 4 5 6 7 8 9 10

Dm 2(perp)

(b)

18

185

19

195

2

205


log(r)

Dm 2(perp)r

D2(perp)

(c)



1

0

minus1

MIX

(08

)M

IX(04

)M

IX(01

)

1

0

minus1

1

0

minus1

20 60 100 140 180

i

(a)

6

5

4

3

2

10 20 40 60 80 100

D2 = 543

D2 = 370

D2 = 183


Cor

relat

ion

dim

ensio

n

Mix(08)

Mix(04)

Mix(01)

(b)



HypNoHyp

2 25 3 35 4 45 5 55

Time (min)

RR in

terv

al (m

s)

150

02

04

06

08

1

(a)


3

4

5

6

7

log(r)

HypNoHyp

X minus2139Y 6653

X minus1893Y 5257

D2(perp)r

(b)
















119863



2












2[10]

119863

2 and 119863






1

06

02

minus02

minus06

minus1minus1


Δ 2

x(i) minus x(j)

xx

(i+120591)minus

(j+120591) y ij

middot 2 = 1

middot 1 = 1

(a)

0

minus1

minus2

minus3

minus4

minus5


2-norminfin-norm

log(r)

log(Cm(r))

1-norm985747

985747

985747

(b)






a119898 = 10 and ℓ1- ℓ2- and ℓ

infin-norms

Appendices



119894119895


1


for ℓ1and for ℓ


119894119895





Lorenz attractorℓ

1ℓ

2ℓ

infin

119863

2[10] 195 194 193

119863

2169 170 193

119863

2(perp)184 174 201

119863

2(max) 199 171 193











119879

1198632

(s) [10] (886 plusmn 035)e5 3314 plusmn 180119879

2

(s) 194 plusmn 29 466 plusmn 058119879

2(perp)

(s) 260 plusmn 38 214 plusmn 30119879

2(max)








Acknowledgments


References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 50 100 150 200 250 300Beats

Nor

mal

ized

RR

inte

rval

0

04

08

minus04

minus08

(a)


minus4

minus3 minus2

minus10

minus5

minus12

minus1

minus8

minus6

minus4

minus2

0

log(r)

m = 1

m = 2

m = 3 m = 4m = 5

log(Cm(r))

(b)




119889119891 (119909)

119889119909

= minus

(119860

2minus 119860

1) 119890

(minus119909+119909119900)120572

[120572(119861 + 119890

(minus119909+119909119900)120572)

2

]

(14)


119863

119898



119898(119903) =1119873

119898(119873

119898minus 1)







119863

119898

2= 119863

2(1 minus 119860119890

minus119896119898) (15)





2










2(perp)




119898+1(119903)) (16)


0 10 201

15

2

m

Gradient descent

Cor

relat

ion

dim

ensio

n

minus8

minus6

minus4

minus2

0


m = 1

m = 21

log(r)

log(Cm(r))

Dm

2(perp)r119895

Dm

2(perp)r119894


119895is the point







119894=119895(119898 119903) is used to eval-


119863

119898


119863

2(max)

3 Materials





119889119909

119889119905

= 120590 (119910 minus 119909)

119889119910

119889119905

= 120588119909 minus 119910 minus 119909119911

119889119911

119889119905

= minus120573119911 + 119909119910

(17)

05

15

0

1

0 2

m

1510

5

0


minus05

log(r)

SampE

n (i=j)(SCF)

Dm

2(max)

Dm

2(perp)

05

15

0

1

0 2

m

1510

5


minus05

log(r)

Dm

2(max)

Dm

2(perp)



2(perp)





MIX(119875)119895= (1 minus 119885

119895)119883

119895+ 119885

119895119884

119895 (18)

where 119883119895=radic2 sin(212058711989512) 119884







4 Results


2estimation




119901is the speed-




119901



119901= 119879Sep119879MO As it is








2estimate (119863

2(perp))





itis just 1 119863



119863







119863

2[10] 193 592|057 439|126 0028

119863

2193 594|060 481|079 0028

119863

2(perp)201 641|065 511|077 0028

119863

2(max) 193 588|049 483|070 0010



2[10] 0900 818 714 100

119863

20900 818 714 100

119863

2(perp)0900 909 100 857

119863

2(max) 0967 909 833 100















minus8

minus6

minus4

minus2

0

2minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

log(r)

log(Cm(r))

(a)


22

2

18

16

14

12

11 2 3 4 5 6 7 8 9 10

Dm 2(perp)

(b)

18

185

19

195

2

205


log(r)

Dm 2(perp)r

D2(perp)

(c)



1

0

minus1

MIX

(08

)M

IX(04

)M

IX(01

)

1

0

minus1

1

0

minus1

20 60 100 140 180

i

(a)

6

5

4

3

2

10 20 40 60 80 100

D2 = 543

D2 = 370

D2 = 183


Cor

relat

ion

dim

ensio

n

Mix(08)

Mix(04)

Mix(01)

(b)



HypNoHyp

2 25 3 35 4 45 5 55

Time (min)

RR in

terv

al (m

s)

150

02

04

06

08

1

(a)


3

4

5

6

7

log(r)

HypNoHyp

X minus2139Y 6653

X minus1893Y 5257

D2(perp)r

(b)
















119863



2












2[10]

119863

2 and 119863






1

06

02

minus02

minus06

minus1minus1


Δ 2

x(i) minus x(j)

xx

(i+120591)minus

(j+120591) y ij

middot 2 = 1

middot 1 = 1

(a)

0

minus1

minus2

minus3

minus4

minus5


2-norminfin-norm

log(r)

log(Cm(r))

1-norm985747

985747

985747

(b)






a119898 = 10 and ℓ1- ℓ2- and ℓ

infin-norms

Appendices



119894119895


1


for ℓ1and for ℓ


119894119895





Lorenz attractorℓ

1ℓ

2ℓ

infin

119863

2[10] 195 194 193

119863

2169 170 193

119863

2(perp)184 174 201

119863

2(max) 199 171 193











119879

1198632

(s) [10] (886 plusmn 035)e5 3314 plusmn 180119879

2

(s) 194 plusmn 29 466 plusmn 058119879

2(perp)

(s) 260 plusmn 38 214 plusmn 30119879

2(max)








Acknowledgments


References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 10 201

15

2

m

Gradient descent

Cor

relat

ion

dim

ensio

n

minus8

minus6

minus4

minus2

0


m = 1

m = 21

log(r)

log(Cm(r))

Dm

2(perp)r119895

Dm

2(perp)r119894


119895is the point







119894=119895(119898 119903) is used to eval-


119863

119898


119863

2(max)

3 Materials





119889119909

119889119905

= 120590 (119910 minus 119909)

119889119910

119889119905

= 120588119909 minus 119910 minus 119909119911

119889119911

119889119905

= minus120573119911 + 119909119910

(17)

05

15

0

1

0 2

m

1510

5

0


minus05

log(r)

SampE

n (i=j)(SCF)

Dm

2(max)

Dm

2(perp)

05

15

0

1

0 2

m

1510

5


minus05

log(r)

Dm

2(max)

Dm

2(perp)



2(perp)





MIX(119875)119895= (1 minus 119885

119895)119883

119895+ 119885

119895119884

119895 (18)

where 119883119895=radic2 sin(212058711989512) 119884







4 Results


2estimation




119901is the speed-




119901



119901= 119879Sep119879MO As it is








2estimate (119863

2(perp))





itis just 1 119863



119863







119863

2[10] 193 592|057 439|126 0028

119863

2193 594|060 481|079 0028

119863

2(perp)201 641|065 511|077 0028

119863

2(max) 193 588|049 483|070 0010



2[10] 0900 818 714 100

119863

20900 818 714 100

119863

2(perp)0900 909 100 857

119863

2(max) 0967 909 833 100















minus8

minus6

minus4

minus2

0

2minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

log(r)

log(Cm(r))

(a)


22

2

18

16

14

12

11 2 3 4 5 6 7 8 9 10

Dm 2(perp)

(b)

18

185

19

195

2

205


log(r)

Dm 2(perp)r

D2(perp)

(c)



1

0

minus1

MIX

(08

)M

IX(04

)M

IX(01

)

1

0

minus1

1

0

minus1

20 60 100 140 180

i

(a)

6

5

4

3

2

10 20 40 60 80 100

D2 = 543

D2 = 370

D2 = 183


Cor

relat

ion

dim

ensio

n

Mix(08)

Mix(04)

Mix(01)

(b)



HypNoHyp

2 25 3 35 4 45 5 55

Time (min)

RR in

terv

al (m

s)

150

02

04

06

08

1

(a)


3

4

5

6

7

log(r)

HypNoHyp

X minus2139Y 6653

X minus1893Y 5257

D2(perp)r

(b)
















119863



2












2[10]

119863

2 and 119863






1

06

02

minus02

minus06

minus1minus1


Δ 2

x(i) minus x(j)

xx

(i+120591)minus

(j+120591) y ij

middot 2 = 1

middot 1 = 1

(a)

0

minus1

minus2

minus3

minus4

minus5


2-norminfin-norm

log(r)

log(Cm(r))

1-norm985747

985747

985747

(b)






a119898 = 10 and ℓ1- ℓ2- and ℓ

infin-norms

Appendices



119894119895


1


for ℓ1and for ℓ


119894119895





Lorenz attractorℓ

1ℓ

2ℓ

infin

119863

2[10] 195 194 193

119863

2169 170 193

119863

2(perp)184 174 201

119863

2(max) 199 171 193











119879

1198632

(s) [10] (886 plusmn 035)e5 3314 plusmn 180119879

2

(s) 194 plusmn 29 466 plusmn 058119879

2(perp)

(s) 260 plusmn 38 214 plusmn 30119879

2(max)








Acknowledgments


References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















119901is the speed-




119901



119901= 119879Sep119879MO As it is








2estimate (119863

2(perp))





itis just 1 119863



119863







119863

2[10] 193 592|057 439|126 0028

119863

2193 594|060 481|079 0028

119863

2(perp)201 641|065 511|077 0028

119863

2(max) 193 588|049 483|070 0010



2[10] 0900 818 714 100

119863

20900 818 714 100

119863

2(perp)0900 909 100 857

119863

2(max) 0967 909 833 100















minus8

minus6

minus4

minus2

0

2minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

log(r)

log(Cm(r))

(a)


22

2

18

16

14

12

11 2 3 4 5 6 7 8 9 10

Dm 2(perp)

(b)

18

185

19

195

2

205


log(r)

Dm 2(perp)r

D2(perp)

(c)



1

0

minus1

MIX

(08

)M

IX(04

)M

IX(01

)

1

0

minus1

1

0

minus1

20 60 100 140 180

i

(a)

6

5

4

3

2

10 20 40 60 80 100

D2 = 543

D2 = 370

D2 = 183


Cor

relat

ion

dim

ensio

n

Mix(08)

Mix(04)

Mix(01)

(b)



HypNoHyp

2 25 3 35 4 45 5 55

Time (min)

RR in

terv

al (m

s)

150

02

04

06

08

1

(a)


3

4

5

6

7

log(r)

HypNoHyp

X minus2139Y 6653

X minus1893Y 5257

D2(perp)r

(b)
















119863



2












2[10]

119863

2 and 119863






1

06

02

minus02

minus06

minus1minus1


Δ 2

x(i) minus x(j)

xx

(i+120591)minus

(j+120591) y ij

middot 2 = 1

middot 1 = 1

(a)

0

minus1

minus2

minus3

minus4

minus5


2-norminfin-norm

log(r)

log(Cm(r))

1-norm985747

985747

985747

(b)






a119898 = 10 and ℓ1- ℓ2- and ℓ

infin-norms

Appendices



119894119895


1


for ℓ1and for ℓ


119894119895





Lorenz attractorℓ

1ℓ

2ℓ

infin

119863

2[10] 195 194 193

119863

2169 170 193

119863

2(perp)184 174 201

119863

2(max) 199 171 193











119879

1198632

(s) [10] (886 plusmn 035)e5 3314 plusmn 180119879

2

(s) 194 plusmn 29 466 plusmn 058119879

2(perp)

(s) 260 plusmn 38 214 plusmn 30119879

2(max)








Acknowledgments


References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















minus8

minus6

minus4

minus2

0

2minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

1

log(r)

log(Cm(r))

(a)


22

2

18

16

14

12

11 2 3 4 5 6 7 8 9 10

Dm 2(perp)

(b)

18

185

19

195

2

205


log(r)

Dm 2(perp)r

D2(perp)

(c)



1

0

minus1

MIX

(08

)M

IX(04

)M

IX(01

)

1

0

minus1

1

0

minus1

20 60 100 140 180

i

(a)

6

5

4

3

2

10 20 40 60 80 100

D2 = 543

D2 = 370

D2 = 183


Cor

relat

ion

dim

ensio

n

Mix(08)

Mix(04)

Mix(01)

(b)



HypNoHyp

2 25 3 35 4 45 5 55

Time (min)

RR in

terv

al (m

s)

150

02

04

06

08

1

(a)


3

4

5

6

7

log(r)

HypNoHyp

X minus2139Y 6653

X minus1893Y 5257

D2(perp)r

(b)
















119863



2












2[10]

119863

2 and 119863






1

06

02

minus02

minus06

minus1minus1


Δ 2

x(i) minus x(j)

xx

(i+120591)minus

(j+120591) y ij

middot 2 = 1

middot 1 = 1

(a)

0

minus1

minus2

minus3

minus4

minus5


2-norminfin-norm

log(r)

log(Cm(r))

1-norm985747

985747

985747

(b)






a119898 = 10 and ℓ1- ℓ2- and ℓ

infin-norms

Appendices



119894119895


1


for ℓ1and for ℓ


119894119895





Lorenz attractorℓ

1ℓ

2ℓ

infin

119863

2[10] 195 194 193

119863

2169 170 193

119863

2(perp)184 174 201

119863

2(max) 199 171 193











119879

1198632

(s) [10] (886 plusmn 035)e5 3314 plusmn 180119879

2

(s) 194 plusmn 29 466 plusmn 058119879

2(perp)

(s) 260 plusmn 38 214 plusmn 30119879

2(max)








Acknowledgments


References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















HypNoHyp

2 25 3 35 4 45 5 55

Time (min)

RR in

terv

al (m

s)

150

02

04

06

08

1

(a)


3

4

5

6

7

log(r)

HypNoHyp

X minus2139Y 6653

X minus1893Y 5257

D2(perp)r

(b)
















119863



2












2[10]

119863

2 and 119863






1

06

02

minus02

minus06

minus1minus1


Δ 2

x(i) minus x(j)

xx

(i+120591)minus

(j+120591) y ij

middot 2 = 1

middot 1 = 1

(a)

0

minus1

minus2

minus3

minus4

minus5


2-norminfin-norm

log(r)

log(Cm(r))

1-norm985747

985747

985747

(b)






a119898 = 10 and ℓ1- ℓ2- and ℓ

infin-norms

Appendices



119894119895


1


for ℓ1and for ℓ


119894119895





Lorenz attractorℓ

1ℓ

2ℓ

infin

119863

2[10] 195 194 193

119863

2169 170 193

119863

2(perp)184 174 201

119863

2(max) 199 171 193











119879

1198632

(s) [10] (886 plusmn 035)e5 3314 plusmn 180119879

2

(s) 194 plusmn 29 466 plusmn 058119879

2(perp)

(s) 260 plusmn 38 214 plusmn 30119879

2(max)








Acknowledgments


References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















1

06

02

minus02

minus06

minus1minus1


Δ 2

x(i) minus x(j)

xx

(i+120591)minus

(j+120591) y ij

middot 2 = 1

middot 1 = 1

(a)

0

minus1

minus2

minus3

minus4

minus5


2-norminfin-norm

log(r)

log(Cm(r))

1-norm985747

985747

985747

(b)






a119898 = 10 and ℓ1- ℓ2- and ℓ

infin-norms

Appendices



119894119895


1


for ℓ1and for ℓ


119894119895





Lorenz attractorℓ

1ℓ

2ℓ

infin

119863

2[10] 195 194 193

119863

2169 170 193

119863

2(perp)184 174 201

119863

2(max) 199 171 193











119879

1198632

(s) [10] (886 plusmn 035)e5 3314 plusmn 180119879

2

(s) 194 plusmn 29 466 plusmn 058119879

2(perp)

(s) 260 plusmn 38 214 plusmn 30119879

2(max)








Acknowledgments


References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















119879

1198632

(s) [10] (886 plusmn 035)e5 3314 plusmn 180119879

2

(s) 194 plusmn 29 466 plusmn 058119879

2(perp)

(s) 260 plusmn 38 214 plusmn 30119879

2(max)








Acknowledgments


References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of






























of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















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