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Medical Engineering & Physics 36 (2014) 761–767

Contents lists available at ScienceDirect

Medical Engineering & Physics

jo ur nal ho me p ag e: www.elsev ier .com/ locate /medengphy

echnical note

erformance analysis of four nonlinearity analysis methods using aodel with variable complexity and application to uterine EMG

ignals

hmad Diaba,b,∗, Mahmoud Hassanc, Catherine Marquea, Brynjar Karlssonb

UMR CNRS 7338, Biomécanique et Bio-ingénierie, Université de Technologie de Compiègne, Compiègne, FranceSchool of Science and Engineering, Reykjavik University, Reykjavik, IcelandLaboratoire Traitement du Signal et de L’Image, INSERM, Université de Rennes 1, Campus de Beaulieu, Rennes, France

r t i c l e i n f o

rticle history:eceived 29 November 2012eceived in revised form 4 October 2013ccepted 27 January 2014

eywords:onlinear time series analysisterine electromyogramontraction discriminationurrogates

a b s t r a c t

Several measures have been proposed to detect nonlinear characteristics in time series. Results on timeseries, multiple surrogates and their z-score are used to statistically test for the presence or absence ofnon-linearity. The z-score itself has sometimes been used as a measure of nonlinearity. The sensitivity ofnonlinear methods to the nonlinearity level and their robustness to noise have rarely been evaluated inthe past. While surrogates are important tools to rigorously detect nonlinearity, their usefulness for eval-uating the level of nonlinearity is not clear. In this paper we investigate the performance of four methodsarising from three families that are widely used in non-linearity detection: statistics (time reversibility),predictability (sample entropy, delay vector variance) and chaos theory (Lyapunov exponents). We usedsensitivity to increasing complexity and the mean square error (MSE) of Monte Carlo instances for quan-titative comparison of their performances. These methods were applied to a Henon nonlinear syntheticmodel in which we can vary the complexity degree (CD). This was done first by applying the methodsdirectly to the signal and then using the z-score (surrogates) with and without added noise. The methodswere then applied to real uterine EMG signals and used to distinguish between pregnancy and labor

contraction bursts. The discrimination performances were compared to linear frequency based methodsclassically used for the same purpose such as mean power frequency (MPF), peak frequency (PF) andmedian frequency (MF). The results show noticeable difference between different methods, with a clearsuperiority of some of the nonlinear methods (time reversibility, Lyapunov exponents) over the linearmethods. Applying the methods directly to the signals gave better results than using the z-score, exceptfor sample entropy.

Crown Copyright © 2014 Published by Elsevier Ltd on behalf of IPEM. All rights reserved.

. Introduction

One of the most common ways of obtaining information on neu-ophysiologic systems is to study the features of the signal(s) usingime series analysis techniques. This traditionally rely on linear

ethods in both time and frequency domains [1]. Unfortunately,

hese methods cannot give information about purely nonlineareatures of the signal. Due to the intrinsic nonlinearity of mostiological systems, these nonlinear features may be present in

∗ Corresponding author at: UMR CNRS 7338, Biomécanique et Bio-ingénierie,niversité de Technologie de Compiègne, Compiègne, France. Tel.: +33 612295638.

E-mail addresses: ahmad.diab@utc.fr, ahmaddiab 87@hotmail.com (A. Diab),ahmoud.hassan@univ-rennes1.fr (M. Hassan), catherine.marque@utc.fr

C. Marque), brynjar@ru.is (B. Karlsson).

350-4533/$ – see front matter. Crown Copyright © 2014 Published by Elsevier Ltd on bettp://dx.doi.org/10.1016/j.medengphy.2014.01.009

physiological data and even be a characteristic of major interest.Recently, much attention has been paid to the use of nonlinear anal-ysis techniques for the characterization of a biological signal [2].Indeed, this type of analysis gives information about the nonlinearfeatures of these signals, which arise from the underlying physio-logical processes, many of which have complex behavior. There is agrowing literature reporting nonlinear analysis of various biosignaltypes (EEG [3], ECG [4], HRV [5] and EMG [6]).

The EHG or electrohysterogram (electrical uterine activityrecorded on woman’s abdomen) has been widely studied [7–11].Nonlinear characteristics have been observed in the EHG andsome success has been achieved by using these characteristics to

obtain information of potential clinical usefulness. Radomski et al.show that nonlinear analysis of EHG based on the sample entropystatistic could differentiate dynamic states of uterine contractions[12]. A comparison between linear and nonlinear analysis with

half of IPEM. All rights reserved.

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(r) as (N − m − 1)−1 times the number of

vectors Xm+1j

within r of Xm+1i

, where j ranges from 1 to N − m,where j /= i, and set

62 A. Diab et al. / Medical Engine

ifferent conditions was done in [13]. It was concluded thatedian frequency is the best method among linear methods and

hat sample entropy is the best method among nonlinear methodsor term/preterm EHG contractions classification. Sample entropys superior to median frequency, which indicates that nonlinearnalysis is more suitable than linear analysis for studying EHGignals. In [14] the progress of labor was evaluated using samplentropy. Our team has examined nonlinear EHG analysis methods.ur results confirm the presence of nonlinearity in EHG signals.his character of the signals is useful in discriminating betweenregnancy and labor contractions [15,2,16]. Practical disadvan-ages of the nonlinear analysis methods have been reported in [16].hey include excessive calculation time due to surrogates analysisnd promising but inconclusive results due to the small amount ofata that can practically be used due to heavy calculation times.

This paper presents work that extends previous work done inur group in comparing approximate entropy, correntropy andime reversibility [16]. In this work we implemented additionalonlinear analysis methods (delay vector variance, Lyapunov expo-ents) and new ways of testing them. We also used a larger databasef real signals than in the previous work and we investigated theensitivity of the methods to the varying complexity of signals andheir robustness. The kind of sensitivity and robustness analyses ofon-linearity measures presented in this paper, are rare or absent

n the literature.Four nonlinear methods: time reversibility [17], sample entropy

18], delay vector variance [19] and Lyapunov exponents [20] weresed in this work. Sensitivity of these methods to the complexityegree (CD) of a signal as well as robustness analysis was done onenon model synthetic signals where CD can be controlled. The

ensitivity to CD was first studied using the direct value providedy the method. It was then studied using surrogates and z-score, ashe measure permitting evaluation of the nonlinearity. One objec-ive of this study is to show which method(s) is most sensitive tohe change of signal complexity. A second objective is to determinehether the use of surrogates gives better overall results than theirect application of the methods. This is of major practical impor-ance for clinical application, as the generation of surrogates is veryomputationally expensive. The methods are also compared usinghe Mean square error (MSE) of the method results for 30 Montearlo instances of the signal. Finally, these non-linear methods areompared to three linear frequency based characteristics of the sig-al, MPF, PF and MF, when applied to real EHG signals, in order toiscriminate pregnancy and labor contractions.

. Materials and methods

.1. Data

.1.1. Synthetic signalsThe Henon map is a well-known two-dimensional discrete-time

ystem given by:

t+1 = c − Y2t + CD × Xt,

t+1 = Yt,

here Yt and Xt represent dynamical variables, CD is the complex-ty degree and c is the dissipation parameter. In this paper we use

= 1 as in [21] and CD ∈ [0,1] to change the model complexity [22]Fig. 1). The number of generated points is fixed to 1000. For the

obustness analysis, we add to the synthetic signals a white Gauss-an noise with the same duration, with a fixed 5db SNR with CDarying between 0 and 1 with a step 0.1. In the Monte Carlo analysis,e use 30 signals generated for each CD value.

Physics 36 (2014) 761–767

2.1.2. Real signalsEHG signals were recorded from 38 subjects using a 4 × 4 elec-

trode matrix located on the subject’s abdomen (Fig. 2), during 1 heither at rest (woman lying on a bed) or during labor. One signalchannel (bipolar vertical 7: BP7), located on the median vertical axisof the uterus was used for subsequent analysis (see [23] for details).After segmentation we obtained 115 labor bursts (recorded dur-ing delivery) and 174 pregnancy bursts (recorded more than 24 hbefore delivery).

2.2. Non-linear analysis methods

2.2.1. Statistics family2.2.1.1. Time reversibility. A time series is said to be reversible onlyif its probabilistic properties are invariant with respect to timereversal. Time irreversibility can be taken as a strong signatureof nonlinearity [17]. In this paper we used the simplest method,described in [24] to compute the time reversibility of a signal Sn:

Tr(�) =(

1N − �

) N∑n=�+1

(Sn − Sn−�)3

where N is the signal length and � is the time delay.

2.2.2. Chaos theory family2.2.2.1. Lyapunov exponents. Lyapunov exponent (LE) is a quantita-tive indicator of system dynamics, which characterizes the averageconvergence or divergence rate between adjacent tracks in phasespace [20]. We used the method described in [13] to compute LE:

� = limt→∞

lim||�y0||→0

(1t

)log

||�yt ||||�y0|| ,

where ||�y0|| and ||�yt|| represent the Euclidean distance betweentwo states of the system, respectively to an arbitrary time t0 and alater time t.

2.2.3. Predictability family2.2.3.1. Sample entropy. Sample entropy (SampEn) is the negativenatural logarithm of the conditional probability that a dataset oflength N, having repeated itself for m samples within a tolerancer, will also repeat itself for m + 1 samples. Thus, a lower value ofSampEn indicates more regularity in the time series [18]. We usedthe method described in [12] to compute SampEn:

For a time series of N points, x1, x2, . . ., xN, we define sub-sequences, also called template vectors, of length m, given by:yi(m) = (xi, xi+1, . . ., xi+m−1) where i = 1, 2, . . ., N − m + 1.

Then the following quantity is defined: Bmi

(r) as (N − m − 1)−1

times the number of vectors Xmj

within r of Xmi

, where j ranges from1 to N − m, and j /= i, to exclude self-matches, and then define:

Bm(r) = 1N − m

N−m∑Bm

i (r)

Am(r) = 1N − m

N−m∑i=1

Ami (r)

A. Diab et al. / Medical Engineering & Physics 36 (2014) 761–767 763

differe

N

S

wt

2d

ebmXtˇEvpmo

o|ˇl

Fig. 1. Simulated signal generated using Henon model with

The parameter SampEn(m, r) is then defined aslim→∞

{−ln[Am(r)/Bm(r)]}, which can be estimated by the statistic:

ampleEn(m, r, N) = −lnAm(r)Bm(r)

here N is the length of the time series, m is the length of sequenceso be compared, and r is the tolerance for accepting matches.

.2.3.2. Delay vector variance. We use the measure of unpre-ictability �∗2 described in [25]:

Time series can be represented in phase space using time delaymbedding. When time delay is embedded into a time series, it cane represented by a set of delay vectors (DVs) of a given dimension. The dimension of the delay vectors can then be expressed as(k) = [x(k−m�), . . ., x(k−�)], where � is the time lag. For every DV X(k),here is a corresponding target, namely the next sample xk. A setk(m, d) is generated by grouping those DVs that are within a certainuclidean distance d to DV X(k). This Euclidean distance will bearied in a standardized manner with respect to the distribution ofairwise distances between DVs. For a given embedding dimension, a measure of unpredictability �*2 (target variance) is computed

ver all sets of ˇk.The mean �d and the standard deviation �d are computed

ver all pair wise Euclidean distances between DVs given by|x(i) − x(j)||(i /= j). The sets ˇk(m, d) are generated such thatk = {x(i)||x(k) − x(j)|| ≤ d}, i.e., sets which consist of all DVs that

ie closer to X(k) than a certain distance d, taken from the interval

Fig. 2. Electrode placement (left), monopolar configuratio

nt complexity degrees (CD). Top: CD = 0.1, Bottom: CD = 0.9.

[�d − nd × �d; �d + nd × �d] where nd is a parameter controllingthe span over which to perform DVV analysis.

For every set ˇk(m, d) the variance of the corresponding tar-gets �k

2(m, d) is computed. The average over the N sets ˇk(m, d)is divided by the variance of the time series signal �2

x , �k gives theinverse measure of predictability, namely target variance �*2.

�∗2 = (1/N)∑N

k=1�2k

�2x

2.2.3.3. Surrogates and z-score. The most commonly used nullhypothesis considers that a given time series is generated by aGaussian linear stochastic process and collected through a non-linear measurement static function. Thus surrogates must havethe same linear properties (autocorrelation and amplitude distri-bution) as the original signal. However, any underlying nonlineardynamic structure within the original data is altered in the surro-gates by phase randomization [16].

The statistics of significance z-score is,

zscore = |q0 − 〈qs(i)〉|�q(i)

where q0 stands for the statistic on the original time series, 〈qs(i)〉for the mean and �q(i) for the standard deviation of the surrogate,for i = 1, 2, . . ., M (number of generated surrogate). The critical valueof z-score is 1.96 [26].

n and the corresponding bipolar signals BPi (right).

7 ering & Physics 36 (2014) 761–767

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Table 1Comparison of ROC curves for labor detection (direct method).

Parameter AUC Specificity Sensitivity

Time reversibility 0.842 0.721 0.860Sample entropy 0.478 0.382 0.643Lyapunov exponent 0.758 0.643 0.756Delay vector variance 0.615 0.582 0.600

In the case of added noise and direct application of the method,as expected, a decrease in the sensitivity of all methods occurredat a low signal to noise ratio (SNR = 5 db). Indeed, at this low SNR,none of the methods detected the varying complexity of the signal,

Table 2Comparison of ROC curves for labor detection (with surrogate use).

Parameter AUC Specificity Sensitivity

64 A. Diab et al. / Medical Engine

. Results

.1. Results on synthetic signals

In this section we study the evolution of the values generatedy the four methods with variable complexity degree (CD) of theenon synthetic model in four cases: (1) direct application of theethod with no added noise, (2) using surrogates with no added

oise, (3) direct application of the method with added noise and4) using surrogates with added noise. The added noise is a whiteaussian noise (SNR = 5 db) while CD varies between 0 and 1, for

he Henon model. Our first objective is to test the sensitivity of theethods to varying CD for signals with and without noise. The use

f surrogates is computationally very expensive and therefore ourecond objective is to test if the use of surrogates improves theethod sensitivity or not.We compare the methods using two criteria, the method’s sen-

itivity to the change of CD (slope of the curve “value of the method”s. “CD”) and the MSE of the method for different values of CD.

Fig. 3A1 presents the mean value for each method (directethod value) as a function of CD computed from the 30 Monte

arlo instances of the signal generated by the Henon model. Fig. 3A2resents the MSE of the methods for each CD. We see in Fig. 3A1hat in the direct case without noise, the four methods evolve wellut with differences in their sensitivity (slopes). Tr and LE are moreensitive than the other methods. In Fig. 3A2 we observe that Tras a much lower MSE than LE.

Fig. 3B1 presents the effect of adding noise (SNR = 5 db) on theethods. We notice no significant slope for the LE and SampEn.

he sensitivity of Tr and DVV also decreases with the addition ofoise. In the other hand we find, Fig. 3B2, that DVV and SampEnive the lowest MSE. However SampEn does not demonstrate anyensitivity to the variation of CD so this method is useless for theoisy signal. Tr gives an intermediate MSE and the highest sensi-ivity when compared to the other methods when applied to noisyignals.

We then applied the methods to the synthetic signals with sur-ogates using the z-score as measure, in order to test if the usef surrogates improves the results or not. Fig. 3C1 presents the-score for each method versus CD. We note that all the meth-ds reflect the non-linearity of the signal generated by the Henonodel as theirs z-score are always above 1.96. In terms of sensitiv-

ty to CD variation, SampEn is the best, but with the highest MSEFig. 3C2). Tr presents an acceptable evolution for lower CD. Buteyond CD = 0.4 an unexpected decrease occurs in the curve andhe Tr value remains constant after CD = 0.7. This method however,ives the lowest MSE (Fig. 3C2). The DVV method presents an inter-ediate slope, contrary to the LE that presents no change with CD.

oth DVV and LE have low MSE under these conditions.The methods were then applied to the signals using again surro-

ates and z-score but with added noise (SNR = 5 db). All the methodstill reveal the nonlinearity of the model. Indeed z-score is above.96 for all the methods, except for DVV where it gives a z-scorealue lower then 1.96 for CD between 0.4 and 0.6. We can clearlyotice an increase in the sensitivity of Tr, Fig. 3D1, compared to thease in Fig. 3C1. SampEn has a good evolution beyond CD = 0.4 but,n the other hand, it presents a rapid increase in MSE (Fig. 3D2).he LE and DVV do not evolve as a function of CD (Fig. 3D1) andive similar MSE as Tr (Fig. 3D2).

.2. Results on real signals

The different nonlinear methods were applied to real uterineMG signals (EHG), first direct application of the method, and thenith surrogates. We also computed three classical linear frequency

ased parameters from these real signals. The values were then

Mean power frequency 0.778 0.678 0.730Peak frequency 0.561 0.582 0.600Median frequency 0.654 0.556 0.704

used to discriminate the pregnancy and labor contractions. Weused ROC curves in order to test the discriminating power of eachcase.Our first objective was to test if the use of surrogates improvesthe discrimination of EHG bursts recorded during pregnancy orlabor. Our second objective was to compare the performances oflinear and nonlinear methods and to verify that the nonlinear meth-ods reveal the evolution of EHG characteristics better than thelinear ones. The ROC curves obtained with the different methodswithout and with use of surrogates are presented Fig. 4B and C,respectively. The characteristics of all the ROC curves without andwith use of surrogates are presented in Tables 1 and 2, respec-tively. The bold characters in the tables refer to the highest valueof AUC, specificity, and sensitivity for nonlinear methods in Tables1 and 2, and for linear methods in table 1. From these data, it isclear that nonlinear methods improve the discrimination of preg-nancy and labor signals. Indeed, the highest area under curve AUC(0.842), sensitivity (0.86) and specificity (0.72) are obtained for theTr method whatever the nonlinear or linear methods used. TheMPF and LE methods also give an acceptable performance (Fig. 4B)with AUC = 0.778 and AUC = 0.758, respectively. The performancesin correct discrimination of labor varies markedly from AUC = 0.478with SampEn to AUC = 0.842 with Tr. When surrogates are used,all ROC curves present approximately the same appearance withthe highest AUC = 0.650 obtained for SampEn. Using surrogates wenotice that the performance of SampEn improves while that of DVVremains approximately the same. On the other hand, the perfor-mance of Tr and LE seem to decrease with the use of surrogates.Finally, we can conclude from Fig. 4 and Table 1 that nonlinearmethods can provide better discrimination between pregnancy andlabor contractions compared to the linear methods. Furthermore,even if the use of surrogates improves the performance of somemethods, it does not generally improve the discrimination results.

4. Discussion and conclusion

We analyzed, quantitatively and as comprehensively as possi-ble, four different nonlinear analysis methods (Tr, SampEn, DVVand LE). These methods were applied on synthetic signals, in orderto test their sensitivity to the change in signal complexity, in nor-mal and noisy conditions, with or without using surrogates. All fourmethods were found to reflect correctly the increasing complexityof the signals in the noise free case, but with different sensitivities.

Time reversibility 0.560 0.513 0.626Sample entropy 0.650 0.593 0.643Lyapunov exponent 0.614 0.591 0.530DVV 0.642 0.573 0.669

A. Diab et al. / Medical Engineering & Physics 36 (2014) 761–767 765

Evolu�on Error

Fig. 3. Results obtained for Henon model using Monte-Carlo simulation. On the left: Evolution of the methods with variable complexity in different cases. On the right: MSEof the methods function of complexity degree in different cases. (A) Direct method with no added noise, (B) direct method with added noise, (C) with surrogate use and noadded noise, (D) with surrogate use and added noise.

766 A. Diab et al. / Medical Engineering & Physics 36 (2014) 761–767

F nt linm

eTisbirnata

ediunamdpo

sp(bmw

ig. 4. Example of ROC curves obtained for the detection of labor with the differeethod, (C) with surrogate use.

xcept for Tr, which clearly reflected the increasing non-linearity.he sensitivity of SampEn increased with the use of surrogates andt gave the highest sensitivity of all the methods, in the case ofurrogate use with no added noise. Indeed SampEn has previouslyeen shown to be sensitive to many aspects of the signal character-

stics, including the sampling rate of the signal [14,11]. Unexpectedesults were obtained in the case of surrogate use and with addedoise. Tr was more sensitive when compared to the previous case,nd SampEn still presented a good sensitivity. We noticed that inhe case of surrogate use, SampEn gave the highest sensitivity butlso had the highest MSE, making it unreliable.

In this paper we also presented results obtained using nonlin-ar and linear methods for discrimination of EHG bursts recordeduring pregnancy and labor. Comparison between the methods

ndicated that Tr, which is a nonlinear method, applied withoutsing surrogates is clearly better in discriminating correctly preg-ancy and labor contractions than the other methods. We can seelso that the use of surrogates improves the performance of someethods like SampEn. These results confirm the results obtained

uring the study on synthetic signal, since the sensitivity of Sam-En increases if surrogates are used, a posteriori justifying the usef the Henon model.

To sum up, the main findings of this study are the following: (i)ome of the studied methods are insensitive to varying signal com-lexity; (ii) SampEn performance depends on the use of surrogates;

iii) generally speaking, none of the studied methods performedest in all the studied situations; (iv) Tr is very sensitive to change ofodel complexity, giving average or good performances, associatedith the lowest MSE in most situations.

ear and nonlinear methods. (A) Real pregnancy and labor contractions, (B) direct

This leads to the conclusion that, of the four methods tested,Tr performed best for our application on real EHG. Indeed Tr dealsrobustly with real, usually noisy, signals and has a good sensitivityto complexity, one of the EHG characteristics that permits discrim-ination of uterine contraction efficiency. Using surrogates and thez-score, as a measure of nonlinearity, does not seem to bring anyimprovement to Tr. Therefore we will not use them for further workon EHG when using Tr.

There are some weaknesses in our study of which we are awareand aim to improve. Tr is dependent on the length of the signal andon the choice of the time delay (�) and we aim to find a methodto optimize these parameters. In further work we also aim to useall of the available bipolar channels (VA1, . . ., VA12) instead of onlyone channel, as in this work. This has been shown to dramaticallyincrease the discrimination rate as evidenced in prior work [27].

Sources of funding

French Ministry of Research, French ministry of foreign affairand the EraSysBio+ program.

Ethical approval

The measurements in Iceland were approved by the relevantethical committee (VSN 02-0006-V2), those in France approvedby the regional ethical committee (ID-RCB 2011-A00500-41) ofAmiens Hospital.

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A. Diab et al. / Medical Engine

cknowledgment

The authors wish to thank Dr. Jeremy Terrien for his helpfuldvice.

onflict of interest

No conflict of interest.

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