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Natural Hazards and Earth System Sciences (2004) 4: 615–631SRef-ID: 1684-9981/nhess/2004-4-615© European Geosciences Union 2004

Natural Hazardsand Earth

System Sciences

A unified approach of catastrophic events

S. Nikolopoulos1, P. Kapiris2, K. Karamanos3, and K. Eftaxias2

1National Technical Univ. Athens, Dept. of Electrical & Computer Engineering Zografou Campus, Zografou 15773, Greece2Dept. of Solid State Section, Faculty of Physics, Univ. of Athens Panepistimioupolis Zografou, 157 84 Athens, Greece3Centre for Nonlinear Phenomena and Complex Systems, Univ. Libre de Bruxelles, CP 231, Campus Plaine, B-1050Brussels, Belgium

Received: 20 July 2004 – Revised: 8 October 2004 – Accepted: 11 October 2004 – Published: 14 October 2004

Part of Special Issue “Precursory phenomena, seismic hazard evaluation and seismo-tectonic electromagnetic effects”

Abstract. Although there is an accumulated charge of theo-retical, computational, and numerical work, like catastrophetheory, bifurcation theory, stochastic and deterministic chaostheory, there is an important feeling that these matters donot completely cover the physics of real catastrophic events.Recent studies have suggested that a large variety of com-plex processes, including earthquakes, heartbeats, and neu-ronal dynamics, exhibits statistical similarities. Here we arestudying in terms of complexity and non linear techniqueswhether isomorphic signatures emerged indicating the tran-sition from the normal state to the both geological and bi-ological shocks. In the last 15 years, the study of Com-plex Systems has emerged as a recognized field in its ownright, although a good definition of what a complex systemis, actually is eluded. A basic reason for our interest in com-plexity is the striking similarity in behaviour close to irre-versible phase transitions among systems that are otherwisequite different in nature. It is by now recognized that thepre-seismic electromagnetic time-series contain valuable in-formation about the earthquake preparation process, whichcannot be extracted without the use of important computa-tional power, probably in connection with computer Algebratechniques. This paper presents an analysis, the aim of whichis to indicate the approach of the global instability in the pre-focal area. Non-linear characteristics are studied by apply-ing two techniques, namely the Correlation Dimension Esti-mation and the Approximate Entropy. These two non-lineartechniques present coherent conclusions, and could cooper-ate with an independent fractal spectral analysis to provide adetection concerning the emergence of the nucleation phaseof the impending catastrophic event. In the context of similarmathematical background, it would be interesting to augmentthis description of pre-seismic electromagnetic anomalies inorder to cover biological crises, namely, epileptic seizure andheart failure.

Correspondence to:K. A. Eftaxias([email protected])

1 Introduction

Prediction of natural phenomena has always been a well-pondered problem. In physics, the predictability degree ofa phenomenon is often measured by how well we understandit. Despite the large amount of experimental data and theconsiderable effort that has been undertaken by the materialscientists, many questions about the fracture remain stand-ing.

When a heterogeneous material is strained, its evolutiontoward breaking is characterized by the nucleation and co-alescence of micro-cracks before the final break-up. Bothacoustic as well as electromagnetic (EM) emission in awide frequency spectrum ranging from very low frequen-cies (VLF) to very high frequencies (VHF), is producedby micro-cracks, which can be considered as the so-calledprecursors of general fracture. These precursors are de-tectable both at a laboratory and a geological scale. Sev-eral experimental results, which illustrate the connectionbetween anomalous VLF-VHF electromagnetic phenomenaand acoustic phenomena with earthquake preparation, werepresented in a rather comprehensive collection of papersedited by (Hayakawa and Fujinawa, 1994; Hayakawa, 1999;Hayakawa and Molchanov, 2002).

Aiming at recording VLF-VHF electromagnetic precur-sors, since 1994 a station was installed at a mountainoussite of Zante island (37.76◦ N–20.76◦ E) in western Greece(Fig. 1).

An important earthquake (Ms=5.9) occurred on 7 Septem-ber 1999 at 11:56 GMT at a distance of about 20 km from thecenter of the city of Athens, the capital of Greece. Very clearelectromagnetic anomalies have been detected in the VLFband (Fig.2), i.e. at 3 kHz and 10 kHz, before the AthensEQ (Eftaxias et al., 2000, 2001). The whole EM precursorswere emerged from 31 August to 7 September 1999 (Fig.2).It is characterized by an accelerating emission rate (Fig.2),while, this radiation is embedded in a long duration quies-cence period concerning the detection of EM disturbancesat the VLF frequency band. These emissions have a rather

616 S. Nikolopoulos et al.: A unified approach of catastrophic events

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Fig. 1. The map demonstrates the location of the Zante RF station (�) and the epicentres of the Athens and Kozani-Grevena earthquakes(©).

long duration, (the data were sampled at 1 Hz), and thus itprovides sufficient data for statistical analysis.

Recently, in a series of papers (Eftaxias et al., 2001, 2002;Kapiris et al., 2003, 2004b,a; Eftaxias et al., 2004), we at-tempt to establish the hypothesis that the pre-seismic elec-tromagnetic emissions offer a potential window for a stepby step monitoring of the last stages of earthquake prepara-tion processes. However, it is difficult to prove associationbetween any two events (possible precursor and earthquake)separated in times. As a major result, the present study indi-cates that it seems useful to combine various computationalmethods to enhance the association of the pre-seismic EMphenomena with micro-fracturing in the pre-focal area. Theachievement of converging estimations would definitely im-prove the chances for an understanding of the physics behindthe generation of earthquakes.

2 Background information

In this section, we briefly describe the algorithms that wereused and compared in this study. Their main characteristicsas well as the reasons they were chosen are discussed.

2.1 The Delay Times Method

The Delay Times method is an important tool in non-linearanalysis and gives both a qualitative and quantitative mea-sure of the complexity of the time-series under examination.It was first established by (Grassberger and Procaccia, 1983)and is based on the Takens Theorem (Takens, 1981). A time-series is constructed from a set of successive and experimen-tally derived values. From the original time-series we thenconstruct a new series, which in this case is composed ofvectors. For the construction of each of the vectors the es-timation of two parameters, the embedding dimension,m,and the time lag,τ , is required. The time lag representsthe window that is used for the computation of the coordi-nates of these vectors. It is estimated from the decorrelationtime, which is the window beyond which the signal ceasesto present periodicities. The decorrelation time is calculatedeither from the first zero-value of the autocorrelation func-tion, or from the first value of the mutual information func-tion (Farmer and Swinney, 1986) that is close to zero. Themutual information function is a widely accepted method thatcomputes non-linear and linear correlation of a signal. The

S. Nikolopoulos et al.: A unified approach of catastrophic events 617

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parameterm is assigned increasing integer values, in a rangethat satisfies both the Takens criterion and the maximum ad-mitted window length, according to basic non-linear dynam-ics theory. Appendix A includes analytical information onthese parameters, as well as a detailed description of the en-tire Delay Times method.

Once the above is completed, the correlation integral,C(r), is computed for increasing values ofr. This integralbasically computes how many of the above vectors have adistance between them less thanr, wherer is a ray in thevector space. We are then able to plot ln(C) vs. ln(r), whereln is the natural logarithm function. From this plot, we se-lect a scaling region and compute the slope of the curve inthat region. This process is repeated for increasing values ofthe embedding dimension,m, and if the values of the slopesconverge, then we have found the Correlation DimensionD2of the time-seriesX(t). The convergence value of the slopeis an estimation of the Correlation Dimension. A time-seriesthat results from a complex non-linear dynamic system yieldsa larger value for the Correlation Dimension, as opposed to atime-series which results from a regular and linear dynamicsystem, lower Correlation Dimension values. Generally, the

Correlation Dimension,D2, represents the independent de-grees of freedom that are required for the proper descriptionof a system or for the construction of its model.

2.2 The Approximate Entropy

There are various definitions of entropy, most of which usu-ally arise from entropy computation such as the Shannonentropy or the Kolmogorov-Sinai entropy. From all knownmethods, Approximate Entropy (ApEn) is chosen, since ithas been introduced as a quantification of regularity in dataand as the natural information parameter for an approximat-ing Markov Chain to a process (Pincus, 1991).

Given the original time-seriesX(t), we construct a seriesof vectors, and then we find the heuristic estimation of an in-teger parameter,m, which in this case represents a windowsize. We then, one again, heuristically estimate a thresh-old, r, which arises from the product of the standard devi-ation of the time series and an arbitrary constant form 0 to1, which is kept the same for all time-series. We then ap-ply an iterative procedure which finally produces an approx-imation ofApEn(m, r). Generally, random time-series pro-duce increasing values ofApEn(m, r), compared to regular


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time-series, a property which we exploit here. More detailsas well as a more analytical description of the method areincluded in Appendix B.

2.3 The fractal spectral analysis

The concept of fractal is most often associated with irregulargeometric objects that display self-similarity. Fractal formsare composed of subunits (and sub-sub-units, etc.) that re-semble the structure of the overall object. The fractal analy-sis can be applied not just in irregular geometric forms thatlack a characteristic (single) scale of length, but also to cer-tain complex processes that generate irregular fluctuationsacross multiple time scales, analogous to scale-invariant ob-jects that have a branching or wrinkly structure across mul-tiple length scales. Earthquakes happen in self-organizingcomplex systems consisting of many non linear interactingunits, namely opening micro-cracks. Self-organized com-plexity manifests itself in linkages between space and timeproducing fractal processes and structures. Herein, we con-centrate on the question whether distinctive alterations in as-sociated scaling parameters emerge as earthquakes are ap-proaching.

We focus on the statistics of the detected electromagneticfluctuations with respect to their amplitude, let’s sayA(ti).We attempt to investigate autocorrelation structures in thesetime-series. Any time series may exhibit a variety of autocor-relation structures; successive terms may show strong (brownnoise), moderate (pink noise) or no (white noise) correlationwith previous terms. The strength of these correlations pro-vides useful information about the inherent “memory” of thesystem. The power spectrum,S(f ), which measures the rela-tive frequency content of a signal, is probably the most com-monly used technique to detect structure in time-series. Ifthe time-seriesA(ti) is a fractal time series that series cannothave a characteristic time scale. But a fractal time series can-not have any characteristic frequency either. The only pos-sibility is then that the power spectrumS(f ) has a scalingform:

S(f ) ∼ f −β (1)

where the power spectrumS(f ) quantifies the correlations atthe time scaleτ∼1/f andf is the frequency of the Fourier

transform. In a lnS(f )− ln(f ) representation the powerspectrum is a line with linear spectral slopeβ. The linearcorrelation coefficient,r, is a measure of the goodness of fitto the power law Eq. (1).

Our approach is to calculate the fractal parameterβ andthe linear correlation coefficientr of the power law fit di-viding the signal into successive segments of 1024 sampleseach, in order to study not only the presence of a power lawS(f )∼f −β but, mainly, the temporal evolution of the asso-ciated parametersβ andr. The Continuous Wavelet Trans-form (CWT), using Morlet wavelet, is applied to compute thepower spectrum, since being superior to the Fourier spectralanalysis providing excellent decompositions within the max-imum admitted window length (Kaiser, 1994).

3 Methods – Results

A convenient way to examine transient phenomena is to di-vide the measurements in time windows and analyze thesewindows. If this analysis yields different results for someprecursory time intervals (epochs), then a transient behaviourcan be extracted. We apply this technique for each of themethods used below.

We discriminate four epochs in the EM time series understudy (Fig. 3). The first epoch refers to the electromagnet-ically quiescent period preceding the emergence of the EManomaly. The second and third epochs include the precur-sory (possibly seismogenic) EM activity. We separate twotime intervals during the detection of this EM anomaly, be-cause we mainly search for the appearance of transient phe-nomena during the last preparation stage of the main shock.Finally, the fourth epoch refers to the period after the abrupttermination of the recorded EM anomaly.

3.1 Application of the Delay Times method

Through the use of the autocorrelation and mutual informa-tion functions, a value was determined for the time lag,τ ,that is most suitable for this study and that wasτ=7. The di-mensions chosen for the phase space reconstruction startedat m=3 and went tom=20. Both m and τ values werebased on the fact that after several trials these values yieldthe best reconstruction and thus lead to more accurate results


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Fig. 4. We first estimate the Correlation Dimension,D2, in consecutive segments of 3000 samples each. Then, we trace the distribution oftheseD2-values for four consecutive epochs. The four epochs are depicted in Fig.3. The epochs 1 and 4 correspond to the EM quiescencethat precedes and follows respectively the EM precursory activity. The allmost similar distributions in the epochs 1 and 4 characterizethe EM background (noise). In epoch 2, the little deformation of the distribution to the left side in respect to the distribution of the purenoise indicates that the initial part of the precursory emission is characterized by a little reduction of the complexity in respect to the highcomplexity of the pure noise. The right lobe that appears in the period 3 corresponds to the EM background, while the left lobe correspondsto the EM precursory activity. We observe a dramatic shift of the distribution of theD2-values in epoch 3. This evidence indicates a strongreduction of complexity during the emergence of the two strong EM bursts in the tail of the precursory emission.

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Fig. 5. We first estimate the Approximate Entropy,ApEn, in consecutive segments of 3000 samples each. Then, we trace the distribution oftheseApEn-values for four consecutive epochs that correspond exactly to the four epochs of Fig.4. In the epoch 2, we observe an importantshift of ApEn-values to lower values. This indicates that the emerged EM emission has a behavior far from this of the EM background. Theright lobe that appears in epoch 3 corresponds to the EM background, while the left lobe corresponds to the EM precursory activity. Weobserve a dramatic reduction of complexity during the emergence of the two strong EM bursts in the tail of the precursory emission.

and subject discrimination. The correlation integral was thencalculated for an extended range ofr (up to 108, experimen-tally determined).

We calculate the correlation dimension,D2, associatedwith successive segments of 3000 samples each and studythe distributions of correlation dimensionD2 in four con-secutive time intervals (Fig.3). We recall that the recordedVLF EM anomaly of gradual increasing activity has beenlaunched through a long duration kilohertz EM quiescence,while they ceased a few hours before the Athens earthquake.The first time interval corresponds to the quiescence EM pe-riod preceding the EM anomaly. The second and third timeintervals correspond to the period of the recorded precursoryanomaly; the third time interval includes the two strong im-pulsive bursts in the tail of the precursory emission. Thefourth time interval refers to the quiescence period after thecessation of the precursory emission.

We underline the similarity of the distributions of theD2-values in the first and fourth time intervals (Fig.4). Thisallmost common distribution characterizes the order of com-

plexity in the background noise of the EM time series. Theassociated predominanceD2-values, from 7 up to 10, indi-cate a strong complexity and non linearity. Notice, that arelevant lobe remains in the distributions ofD2-values in thesecond and third time interval, as it was expected.

Now, we focus on the second and third time intervals,namely during the emergence of the precursory emission.We observe a significant decrease of theD2-values as wemove from the second to the third time window. The ob-served significant decrease of theD2-values signals a strongloss of complexity in the underlying fracto-electromagneticmechanism during the launching of the two strong EM burstsin the tail of the precursory emission. This evidence mightbe indicated by the appearance of a new phase in the tail ofthe earthquake preparation process, which is characterized bya higher order of organization. Sufficient experimental evi-dence seems to support the association of the aforementionedtwo EM bursts with the nucleation phase of the impendingearthquake (Eftaxias et al., 2001; Kapiris et al., 2004b).


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Fig. 6. (a)We first estimate the exponentβ, in consecutive segments of 1024 samples each. Then, we trace the distribution of theseβ-valuesfor four consecutive epochs. The four epochs are depicted in Fig.3 and correspond exactly to the four epochs of Fig.4 and Fig.5. Insets showpercentage of segments withr>0.85. It is evident that the closer the final stage of seismic process, the larger the percentage of segmentswith r>0.85, and the larger shift ofβ to higher values. Notice that in epoch 3 the signal becomes persistent.(b) The propability distributionsof linear coef.r beyond 0.85.

These findings suggest that there is important informa-tion in terms of correlation dimension hidden in the hetero-geneities of the pre-seismic time series. The correlation di-mensionD2 in the sequence of the precursory EM pulsesseems to measure the distance from the global instability: thelarger theD2-values the larger the distance from the criticalpoint.

3.2 Application of the Approximate Entropy method

The Approximate Entropy was computed for a variety ofr-values proposed by previous researchers and it was foundthat the optimum value yielding clearest discrimination wasthe valuer=0.65ST D, whereST D is the standard deviationof the time-series.

We calculate the Approximate Entropy associated withsuccessive segments of 3000 samples each and study the dis-tributions of theApEn-values in four consecutive time in-tervals, as in the case of the study in terms of CorrelationDimension (Fig.5).

We observe again the similarity of the distributions of theApEn-values in the first and fourth time intervals: this all-most common distribution refers to the background noise ofthe EM time series. A relevant lobe remains in the distribu-tions of theApEn-values in the second and third time inter-val, as it was expected.

Now, we concentrate on the second and third time inter-vals, namely during the emergence of the precursory emis-sion. We observe a significant decrease of theApEn-valuesas we move from the second to the third time window. The

observed considerable decrease of theApEn-values in thethird time interval reveals a strong loss of complexity inthe underlying mechano-electromagnetic transduction dur-ing the launching of the two strong EM bursts in the tailof the precursory emission. In other words, the pre-focalarea seems to be less responsive to the external stimuli whenthe pre-seismic EM signals are characterized by lowApEn-values.

In summary, in the pre-seismic EM time-series the valuesof the Correlation Dimension and Approximate Entropy arereduced as the main event is approached. This evidenceindicates that the underlying fracto-electromagnetic mech-anism exhibits a strong complexity and non-linearity farfrom the global failure. A significant loss of complexity andnon-linearity is observed close to the global instability. Thisconsiderable alteration in bothD2-values andApEn-valuesmight be considered as candidate precursor of the impendingevent.

Remark

According to the appendixes, the method of Correla-tion Dimension (CorrDim) embeds the original time seriesinto a phase space of dimension 3 to 20, examining thusthe probability distribution of a norm defined in this phasespace, contrary to the Approximate Entropy method whichembeds the original time series in a 2-dimensional phasespace only. As a result, the CorrDim method yields a moredetailed description of a system’s complexity, comparing to


theApEn method, which focuses mainly to coarse grainedcharacteristics. When the examined time series is generatedby a low dimensional process, it is better to use theApEnmethod. On the other hand, when the complexity of theexamined time series increases, the CorrDim method ismore suitable as it is more sensitive to high complexity.In the cases of epochs 2 and 3 in Figs. 4 and 5, we areable to observe the above mentioned property. Focusingto epoch 2 in Fig. 5, the shift ofApEn-values to lowervalues witnesses the reduction of complexity, and thus theemergence of the precursor. In epoch 3 the complexityhas further been diminished. Thus theApEn method isthe proper one to describe the associated grouping activityof the structures. Indeed, we observe that the probabilitydistributions of EQ-D2 and Background-D2 values seem tobe similar (epoch 3 in Fig. 4), while in the case ofApEnmethod, the probability of EQ-ApEn values is larger thanthe BackgroundApEn-values.

3.3 Application of fractal-dynamics

The spectral fractal analysis reveals that the pre-seismic elec-tromagnetic fluctuations exhibit hidden scaling structure. Weobserve alterations in the associated dynamical parameters,which seem to uncover important features of the underlyingearthquake preparation process (Kapiris et al., 2002; Eftax-ias et al., 2003; Kapiris et al., 2003, 2004b,a; Eftaxias et al.,2004).

Figure6 exhibits the temporal evolution ofr as the mainevent is approached. We observe a gradual increase of thecorrelation coefficient with time: at the tail of the precursoryactivity the fit to the power law is excellent. The fact thatthe data are well fitted by the power-law (1) suggests that thepre-seismic EM activity could be ascribed to a multi-time-scale cooperative activity of numerous activated fundamentalunits, namely, emitting-cracks, in which an individual unit’sbehavior is dominated by its neighbours so that all units si-multaneously alter their behavior to a common large scalefractal pattern. On the other hand, the gradual increase ofr indicates that the clustering in more compact fractal struc-tures of activated cracks is strengthened with time.

Now we focus on the behavior ofβ-exponent. Two classesof signal have been widely used to model stochastic frac-tal time series (Heneghan and McDarby, 2000): fractionalGaussian noise (fGn) and fractional Brownian motion (fBm).These are, respectively, generalizations of white Gaussiannoise and Brownian motion. The nature of fractal behav-ior (i.e. fGn versus fBm) provides insight into the physicalmechanism that generates the correlations: the fBm repre-sents cumulative summation or integration of a fGn. A for-mal mathematical definition of continuous fBm was first of-fered by (Mandelbrot and Ness, 1968).

For the case of the fBm model the scaling exponentβ liesbetween 1 and 3, while, the range ofβ from−1 to 1 indicatesthe regime of fGn (Heneghan and McDarby, 2000). Fig. 6reveals that during the epochs 2 and 3 (Fig. 3) theβ-values

are distributed in the region from 1 to 3. This means that thepossible seismogenic EM activity follows the fBm model.

We concentrate on the quiescent EM period (first epoch inFig. 3), preceding the emergence of the EM anomaly (sec-ond and third epochs in Fig. 3). We observe that only a verysmall number of segments, approximately 5%, follows thepower law (1) (see inserts in Fig. 6). We can conclude thatduring the epoch 1 the associated time series do not behaveas a temporal fractal. Moreover, if we concentrate on the 5%of the segments, the associatedβ-values range from 0 to 1,namely, this minority of segments may follow the fGn model.We conclude that regime of the quiescent period is quite dif-ferent from those of the possible seismogenic emission. Thetransition to the fractal structure and fBm class further iden-tify the launch of the fracto-electromagnetic emission fromthe background (noise) of EM activity.

The distribution ofβ-exponents is also shifted to highervalues (Fig.6) during the precursory period.

The precursory shift of the distribution of bothβ-exponentandr-coefficient to higher values reveals important featuresof the underlying mechanism. The fractal-laws observed cor-roborate to the existence of memory; the system refers toits history in order to define its future. As theβ-exponentincreases the spatial correlation in the time-series also in-creases. This behaviour signals the gradual increase of thememory, and thus the gradual loss of complexity in the pro-cess.Maslov et al.(1994) have formally established the re-lationship between spatial fractal behaviour and long-rangetemporal correlations for a broad range of critical phenom-ena. By studying the time correlations in the local activ-ity, they show that the temporal and spatial activity can bedescribed as different cuts in the same underlying fractal.Laboratory results support this hypothesis:Ponomarev et al.(1997) have reported in phase changes of the temporal andspatial Hurst exponents during sample deformation in labo-ratory acoustic emission experiments. Consequently, the ob-served increase of the temporal correlation in the pre-seismictime-series may also reveal that the opening-cracks are cor-related at larger scale length with time.

The following feature goes to the heart of the prob-lem: first, single isolated micro-cracks emerge which, subse-quently, grow and multiply. This leads to cooperative effects.Finally, the main shock forms. The challenge is to determinethe “critical time-window” during which the “short-range”correlations evolve into “long-range” ones. Fig.6a indicatesthat the closer the global instability the larger the percent-ages of segments withr close to 1 and the larger the shiftof β-exponent to higher values; theβ-values are maximal atthe tail of the pre-seismic state (Fig6b). This behaviour mayreveal the “critical time-window”.

The exponentβ is related to the Hurst parameter,H , bythe formula (Turcotte, 1992)

β = 2H + 1 with 0 < H < 1 and 1< β < 3 (2)

for the fBm model (Mandelbrot and Ness, 1968; Heneghanand McDarby, 2000). Consequently, segments with Hurst-


exponents estimated by the previous formula out of the range0


β-values showed a tendency to gradually decrease during theprocess of the earthquake preparation. We think that thisdifference supports the hypothesis that the ULF signals onone hand and the VLF-VHF signals on the other hand mayhave originated on different mechanisms. Indeed, it has beensuggested that the ULF geo-electical signals could be ex-plained in these terms: (i) “Pressure Stimulated Currents”that are transient currents emitted from a solid containingelectric dipole upon a gradual variation of pressure (Varotsosand Alexopoulos, 1984a,b; Varotsos et al., 1996). (ii) Theelectro-kinetic effect e.g (Mizutani and Ishido, 1976; Dobro-volsky et al., 1989; Gershenzon and Bambakidis, 2001). Be-cause electro-kinetic effect is controlled by the diffusion ofwater with the diffusion time comparable to the period ofULF emissions, more energy is provided to the ULF range(Gershenzon and Bambakidis, 2001). We note that recently,(Surkov et al., 2002) have explained the logarithmic depen-dence of electric field amplitudeE on the earthquake mag-nitudeM that is indicated by experimental results (Varotsoset al., 1996).

4 From the normal state to the seismic shock or epilep-tic seizure in terms of complexity

The world is made of highly interconnected parts on manyscales, the interactions of which results in a complex behav-ior that requires separate interpretation of each level. Thelaws that describe the behavior of a complex system are qual-itatively different from those that govern its units. New fea-tures emerge as one moves from one scale to another, so itfollows that the science of complexity is about revealing theprinciples that govern the ways in which these new propertiesappear.

A basic reason for our interest in complexity is the strikingsimilarity in behaviour close to irreversible phase transitionsamong systems that are otherwise quite different in nature(Stanley, 1999, 2000; Sornette, 2002; Vicsek, 2001, 2002;Turcotte and Rudle, 2002). Recent studies have demon-strated that a large variety of complex processes, includingearthquakes (Bak and Tang, 1989; Bak, 1997), forest fires(Malamud et al., 1998), heartbeats (Peng et al., 1995), humancoordination (Gilden et al., 1995), neuronal dynamics (Wor-rell et al., 2002), financial markets (Mantegna and Stanley,1995) exhibits statistical similarities, most commonly power-law scaling behaviour of a particular observable.Stanley(2000) offer a brief and somewhat parochial overview ofsome “exotic” statistical physics puzzles of possible interestto biophysicists, medical physicists, and econophysics.

Interestingly, authors have suggested that earthquake’s dy-namics and neurodynamics could be analyzed within similarmathematical frameworks (Rundle et al., 2002). Character-istically, slider block models are simple examples of drivennon-equilibrium threshold systems on a lattice. It has beennoted that these models, in addition to simulating the as-pects of earthquakes and frictional sliding, may also repre-sent the dynamics of neurological networks (Rundle et al.,

1995, and references therein). A few years ago,Bak et al.(1987) coined the term self-organized criticality (SOC) todescribe the phenomenon observed in a particular automa-ton model, nowadays known as the sandpile-model. Thissystem is critical in analogy with classical equilibrium criti-cal phenomena, where neither characteristic time nor lengthscales exist. In general, the strong analogies between the dy-namics of the “self-organized-criticality” (SOC) model forearthquakes and that of neurobiology have been realized bynumerous of authors (Hopfield, 1994), (Herz and Hopfield,1995, and references there in), (Usher et al., 1995), (Zhaoand Chen, 2002, and references there in), (Beggs and Plenz,2003).

Complexity does not have a strict definition, but a lot ofwork on complexity centers around statistical power laws,which describe the scaling properties of fractal processes andstructures that are common among systems that at least qual-itative are considered complex. The big question is whetherthere is a unified theory for the ways in which elements ofa system organize themselves to produce a behavior that isfollowed by a large class of systems (Vicsek, 2002).

The aforementioned concepts motivated us to investigatewhether common precursory patterns are emerged during theprecursory stage of both epileptic seizure and earthquake (Liet al., 2004).

The brain possesses more than billions neurons and neu-ronal connections that generate complex patterns of be-haviour. Electroencephalogram (EEG) provides a window,through which the dynamics of epilepsy preparation can beinvestigated. Fig.7 exhibits rat epileptic seizure.

As in the case of the pre-seismic EM emission, we moni-tor the evolution of fractal characteristics of pre-epileptic ac-tivities toward criticality in consecutive time windows. Ouranalysis reveals that numerous distinguishing features wereemerged during the transition from normal states to epilep-tic seizures (Li et al., 2004): (i) appearance of long rangepower-law correlations, i.e. strong memory effects; (ii) in-crease of the spatial correlation in the time-series with time;(iii) gradual enhancement of lower frequency fluctuations,which indicates that the electric events interact and coalesceto form larger fractal structures; (iv) decrease of the fractaldimension of the time series; (v) decrease with time of theanti-persistent behavior in the precursory electric time series;(vi) appearance of persistent properties in the tail of the pre-epileptic period. Fig.7 shows the aforementioned precursorybehavior.

Notice that the aforementioned candidate precursors ofthe impending epileptic seizure or earthquake are launchedin a way striking similar to those occurring just beforethe “critical point” of phase transition in statistical physics.Based on this similarity, it might be argued that the earth-quake/epilepsy may be also viewed as “a generalized kind ofphase transition” (Kapiris et al., 2004b; Contoyiannis et al.,2004).

Our results indicate that an individual firing neuron or anopening crack is dominated by its neighbours so that all acti-vated biological or geological units simultaneously alter their


0 5 10 15 20 25 30−500

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Fig. 7. A rat epileptic seizure (red signal) in EEG time-series (upper part). Two electrodes were placed in epidural space to record the EEGsignals from temporal lobe. EEG signals were recorded using an amplifier with band-pass filter setting of 0.5–100 Hz. The sampling ratewas 200 Hz. Bicuculline i.p injection was used to induce the rat epileptic seizure. The injection time is at 7:49 (m-s) and the seizure at 12:55(m-s), respectively. This pre-seizure period is depicted by the yellow part of the EEG time series. We estimate the exponentβ, in consecutivesegments of 1024 samples each. Then, we trace the distribution of theseβ-values for four consecutive epochs (lower part). The four epochsare depicted in the upper part with numbered dashed frames. Insets show percentage of segments withr>0.97. Notice that at the last stageof the pre-ictal period (epoch 3) the signal emerges persistent behavior.

behavior to a common fractal pattern as the epileptic seizureor the earthquake is approaching. Interestingly, common al-terations in the associated parameters are emerged indicatingthe approach to the global instability in harmony with rele-vant theoretical suggestions (Hopfield, 1994; Herz and Hop-field, 1995; Usher et al., 1995; Zhao and Chen, 2002; Run-dle et al., 2002; Beggs and Plenz, 2003). Consequently, thepresent analysis seems to support the concept that, indeed,a unified theory may describe the ways in which elementsof a biological or geological system organize themselves toproduce a catastrophic event.

5 From the normal state to the seismic shock or heart-failure in terms of Correlation Dimension, Approxi-mate Entropy and Multifractality

Recently,Fukuda et al.(2003) have investigated similaritiesbetween communication dynamics in the Internet and the au-tonomic nervous system. They found quantitative similari-ties between the statistical properties of (i) healthy heart ratevariability and non-congested Internet traffic, and (ii) dis-eased heart rate variability and congested Internet traffic. Theauthors conclude that their finding suggest that the under-standing of the mechanisms underlying the “human-made”Internet could help to understand the “natural” network thatcontrols the heart. In the sense of this approach, we search

for similarities from the normal state to the seismic shock orheart-failure.

5.1 Similarities in terms of multifractality

Mathematical analysis of both long-term heart-rate fluctua-tions (Ivanov et al., 1999; Stanley et al., 1999; Goldbergeret al., 2002) and pre-seismic EM emissions (Kapiris et al.,2004b) show that they are members of a special class of com-plex processes, termed multi-fractals, which require a largenumber of exponents to characterize their scaling properties.In general, the detection of multi-fractal scaling may indicatethat the underlying nonlinear mechanism regulating the sys-tem might interact as part as a coupled cascade of feedbackloops in a system operating far from equilibrium (Meneveauand Sreenivasan, 1987).

Monofractal signals can be indexed by a single global ex-ponent, i.e. the Hurst exponentH (Hurst, 1951). Multifrac-tal signals, on the other hand, can be decomposed into manysubsets characterized by different local Hurst exponentsh,which quantify the local singular behavior and thus relateto the local scaling to the time series. Thus, multifractalsignals require many exponents to characterize their scal-ing properties fully (Vicsek, 1993). The statistical proper-ties of the different subsets characterized by these differentexponentsh can be quantified by the functionD(h) , whereD(h0) is the fractal dimension of the subset of the time series


15:20 15:52 16:24

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300

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Time (UT)0 0.1 0.2 0.3

0.6

0.7

0.8

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1

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D(h

)

Fig. 8. Two segments of the precursory 41MHz electromagnetic signal, recorded on 12 May 1995 (upper row) and 13 May 1995 (lower row)before the Kozani-Grevena earthquakeMs=6.6 on May 13, 1995 at 08:47:12.9 UTC. On the right part of the figure the corresponding fractaldimensionsD(h) are presented.

characterized by the local Hurst exponenth0. Ivanov et al.(1999) have uncovered a loss of multifractality, as well asa loss of the anti-persistent behaviour, for a life-threateningcondition, congestive heart failure. Following the method ofmultifractal analysis used byIvanov et al.(1999), we exam-ined multi-fractal properties in the VHF time series, namely,the spectrum of the fractal dimensionD(h), as a candidateprecursor of the Kozani-Grevena earthquake (Kapiris et al.,2004b). Figure8 shows that as the main event approaches,the EM time series manifest: a significant loss of multi-fractal complexity and reduction of non-linearities, display-ing a narrow (red) multifractal spectrum, and their fluctua-tions become less anti-correlated, as the dominant local Hurstexponents is shifted to higher values. These results reflectthat for both the heart and pre-focal area at high risk themulti-fractal organization allmost breaks down.

In summary, the multifractality of the heart-beat time se-ries and pre-seismic EM time series further enables us toquantify the greater complexity of the “healthy” dynamicscompared to those of “pathological” conditions in both heartand pre-focal area.

5.2 Similarities in terms of Correlation Dimension and Ap-proximate entropy

Recently, we have studied several methods which have beenused for the categorization of two subjects groups, one whichrepresents subjects with no prior occurrence of coronary dis-ease events and another group who have had a coronarydisease event (Nikolopoulos et al., 2003; Karamanos et al.,2004). It is worth mentioning that the Delay Times methodand the computation of the Approximate Entropy present co-herent results and succeed in clearly and accurately differ-entiating healthy subject ECGs from those of unhealthy sub-jects and coronary patients.

Heart Rate Variability (HRV) time series coming fromcoronary patients exhibit more regular and periodical be-

haviour compared to ones coming from healthy subjects. Thecorrelation dimensions of healthy time series are aboutD2 ≈9 when the respective ones for the patients are aboutD2≈6(Nikolopoulos et al., 2003). Similarly, the meanApEn valuefor the healthy time series was aboutApEn≈1.2 and for thepatientsApEn≈0.4 (Nikolopoulos et al., 2003). A simi-lar reduction of complexity for heart failures has been ob-served in terms of Block-Entropy by some of the present au-thors (Karamanos et al., 2004). It is important to note thattheD2-values andApEn-values associated with the secondtime interval of the pre-seismic EM time series are close tothe ones coming from healthy subjects, while, theD2-valuesandApEn-values associated with the third time interval areclose to the ones coming from patient subjects. Based on thisanalogy, we could say that the EM emissions in second andthird time interval implies a kind of “healthy” and “patient”pre-focal area correspondingly. We focus on this analogy.

We recall that the EM time series in the second time inter-val is characterized by strong anti-persistence and multifrac-tality. The multifractality indicates that the underlying non-linear mechanism regulating the system might interact as partas a coupled cascade of feedback loops in a system operat-ing far from equilibrium (Meneveau and Sreenivasan, 1987).The anti-persistent properties during this period imply a setof fluctuations tending to induce a greater stability in the sys-tem. Thus, by the term “healthy pre-focal area”, we meana candidate focal area, which is consistent with a non-linearnegative feedback system that “kicks” the cracking rate awayfrom extremes.

By the term “patient pre-focal area”, we mean a pre-focalarea in which the system has been starting to self-organizeby a non-linear positive feedback process, and thus, thisacquires to a great degree the property of irreversibility. Thisbehaviour may imply that the nucleation stage, the most in-teresting phase in the preparation process of the catastrophicfracture, has already been emerged.


The study of pre-failure EM signals seems to provide away for observing the Earth’s crust ability to respond tostresses. Hallmarks of the “patient pre-focal area” are: thepersistence behavior, the low multifractality, the low Corre-lation Dimension and the low Approximate Entropy. Thesehallmarks characterize a life-threatening condition for thehuman heart, too. Briefly, the “patient pre-focal area” andthe “patient human heart” are characterized by a low com-plexity. On the other hand, signatures indicating the “healthypre-focal area” are: the anti-persistence behavior, the highmultifractality, the high Correlation Dimension, and the highApproximate Entropy. These signatures also characterizehealthy human heartbeat. Briefly, the “healthy pre-focalarea” and the healthy human heart are characterized by highcomplexity.

In a geometrical sense, the dynamical parameterβ speci-fies the strength of the signal’s irregularity as well. Qualita-tively speaking, the irregularity of the signal decreases as thememory in the time-series increases. For the fBm model thefractal dimensiond is found from the relationd=(5−β)/2,which, after considering the aforementioned shift ofβ ex-ponent to higher values, leads to a decrease of fractal di-mension as the earthquake approaches. We recall the West-Goldberger hypothesis that a decrease in healthy variabilityof a physiological system is manifest in a decreasing fractaldimension (Goldberger et al., 2002, and references there in).Our results imply that this hypothesis could be extended togeological systems as well.

6 Conclusions

A method to asses the approach to the global instability hasbeen applied in EM pre-seismic anomalies. The study ofthese pre-failure signals seems to provide a way for observ-ing the Earth’s crust ability to respond to stresses. The DelayTimes method, the computation of the Approximate Entropy,and the monitoring of alteration of Fractal Spectral charac-teristics of pre-seismic EM activity toward global instabilityin consecutive time windows, present coherent results andsucceed in a potential differentiation of the nucleation phasefrom previous stages of the earthquake preparation process.More precisely, the emergence of long-range correlations,i.e. appearance of long memory effects, the increase of thespatial correlation in the time series with time, the predom-inance of large events with time, as well as the gradual de-crease of the anti-persistent behaviour may indicate the ap-proach to the nucleation phase of the impending catastrophicevent. The appearance of persistent properties in the tail ofthe precursory time series, the significant divergence of theenergy release, the sharp significant decrease of the Approx-imate Entropy, and the quick reduction of the Correlation Di-mension as well, all these, may hints that a new phase, prob-ably the nucleation phase of the earthquake, has been started.This analysis may provide a useful way to the understandingof the fracture in the disordered media. The agreement ofthe “diagnostic” information given by each one of the meth-

ods indicates the necessity of further investigation, combineduse, and complementary application of different approaches.

The performed analysis reveals that common precursorysigns emerge in terms of fractal dynamics as the epilepticseizure and earthquake are approaching: common distinc-tive alterations in associated scaling dynamical parametersemerge as biomedical or geophysical shock is approaching.The experimental results verify relevant theoretical sugges-tions that earthquake dynamics and neural seizure dynam-ics should have many similar features and should be an-alyzed within similar mathematical frameworks (Hopfield,1994; Herz and Hopfield, 1995; Usher et al., 1995; Zhao andChen, 2002; Rundle et al., 2002; Beggs and Plenz, 2003).

In principle, it is difficult to prove associations betweenevents separated in time, such as EQs and their precursors.The present state of research in this area requires a refineda definition of a possible pre-seismic anomaly in the recordof EM radiation, and also the development of more objec-tive methods of distinguishing seismogenic emissions fromnon-seismic EM events. A study in terms of complexitywould seem to be useful in this regard. EEG time-seriesprovide a window through which the dynamics of biologi-cal shock preparation can be investigated in the absence ofnon-biological events. We observe that both kinds of catas-trophic events under investigation follow common behaviorin their pre-catastrophic stage. This evidence may supportthe seismogenic origin of the detected EM anomaly.

We find also quantitative similarities between the prop-erties of (i) healthy heart rate variability and initial anti-persistence part of the pre-seismic EM time series, and (ii)diseased heart rate variability and terminal persistence partof the pre-seismic EM activity. These similarities have beenemerged in terms of Correlation Dimension, Approximateentropy, and multifractal dynamics.

Fukuda et al.(2003) recall that very simple models ofvery complex systems in many cases provide deep insights.For example the Ising model and its simple variants as theHeisenberg model are sufficient to quantitatively describe awealth of very complex systems in regions of their respectivephase diagrams where scale invariant is displayed. The prin-ciple of “universality” in chemistry and physics, whereby di-verse systems are described by the identical (simple) model,may have its counterpart in physiology (Stanley, 1999). Eventhe numerical values of the critical-point exponents describ-ing the quantitative nature of the singularities are identicalfor large groups of apparently diverse physical systems. Itwas found empirically that one could form an analog of theMendeleev table if one partitions all critical systems into“universality classes”. Two systems with the same valuesof critical-point exponents and scaling functions are said tobelong to the same universality class. In the frame of thisapproach we have shown that the pre-seismic VHF emis-sion belongs to the 3D-Ising-transition class (Contoyianniset al., 2004). Fukuda et al.(2003) argue that their findingsuggest that the understanding of the mechanism underly-ing the “human-made” internet could help to understand the“natural” network that controls the heart. In this sense, it


appears that the fracture in the disordered systems may pro-vide another useful “model system” to investigate the mecha-nism responsible for the dynamics of the autonomic nervoussystem (ANS), which controls involuntary the heart or theepilepsy generation. In terms of complexity, this possibilityis not implausible.

The science of complexity is in its infancy, and some re-search directions that today seem fruitful might eventuallyprove to be academic cul-de-sacs.Sethna et al.(2001) showthat the seemingly random, impulsive events by which manyphysical systems evolve exhibit universal, and , to some ex-tend, predictable behavior. Nevertheless, it is reasonable tobelieve that the results of the present study indicate that itis useful to transfer knowledge from the domain of biomed-ical shock preparation to the domain of earthquake genera-tion and vice versa. This work could serve as an invitationto other specialists in these areas to transfer knowledge fromthe one field of research to the other.

Appendix A Delay Times method

Given a timeseries,X(t), t is an integer,t ∈ (1, N) andNis the total number of timeseries points. The Delay Timesmethod was first established byGrassberger and Procaccia(1983) and based on the Takens Theorem (Takens, 1981).According to this method, the timeseriesx(t) is a measure ofa single coordinate of anm-dimensional system’s underlyingdynamics. Assuming m is the embedding dimension (the di-mension of space in which the assumed system’s trajectory isunfolded) andτ is the time lag, then phase space reconstruc-tion (described below) is performed with time delays and thefollowing m-dimensional vectors are constructed:

x(t) = [X(t), X(t−r), X(t−2τ ), . . . , X(t−(m−1)τ )](A1)

In this way, using the original timeseries,X(t), we are ableto construct a new vector timeseries,x(t), which representsthe trajectory fromx(0) up to and includingx(t) within thereconstructed phase space.

These vectors are defined in anm-dimensional phase spaceand are used in constructing the trajectory of the signal dy-namics to this space. If the original phase space of the dy-namics produce the attractorA, then the reconstruction ofthe phase space with the Delay Times method produces thereconstructed attractor,A′. If the reconstruction is accurate,thenA′ is the topological conjugate of the original attractor,A. Consequently, all dynamic properties ofA are projectedto A′. The criterion of the Takens Theorem (Takens, 1981)for a precise phase space reconstruction of an experimen-tal trajectory dictates that m must be greater then[2mc+1],wheremc is the estimated dimension of the attractor.

According to the Takens Theorem, this is efficient whenthe number of points of the timeseries,N , is infinite, mean-ing that for an infinite number of points,A andA′ have thesame properties. However, for most experimental methods,N is a finite number and in many cases is confined to 3000–4000 points. Therefore, onlyA′ is estimated in the recon-

structed space and retains only some of the properties ofA(not all). Essential to phase space reconstruction, especiallyfor the Delay Times method, is the estimation of the timelag, τ . There is a range of methods for estimatingτ , themost popular being the calculation of the decorrelation time.

The decorrelation time is calculated either from the firstzero-value of the Autocorrelation function or from the firstminimum value of the mutual information function. The Au-tocorrelation function has been described in the previous sec-tion. The mutual information method is widely accepted andit computes the nonlinear and linear correlation of the Auto-correlation function. Once the Autocorrelation function hasbeen normalized, the decorrelation time is found from thesmallest time lag for which the function tends to zero. Simi-larly, the decorrelation time can also be found from the small-est time lag for which the mutual information function tendsto zero.

According toBroomhead and King(1986); Albano et al.(1988); Kugiumtzis(1996) the results of a time-series anal-ysis depends on the window length(m−1)τ , which incorpo-rates both the embedding dimensionm and the time lagτ .Therefore, the constraint to the above methods is the limiton the size of the window,(m−1)τ . A proper value for thewindow size provides good phase space reconstruction andensures that all the points of the reconstructed phase spacecome from the same trajectory. As mentioned above, theTakens theorem dictates that proper phase space reconstruc-tion is achieved whenm is greater than[2mc+1]. This crite-rion is difficult to satisfy for increased values ofτ due to thesubsequently larger values of(m−1)τ . A consistent windowarises from the decorrelation time, seen as the time neededfor the first decay of the Autocorrelation function. A timelag,τ , is chosen and the reconstructed dynamics are embed-ded in them-dimensional phase space.

After the phase space reconstruction of the system’s as-sumed dynamics, non-linear dynamics algorithms are de-veloped for the experimental analysis of a timeseries. Themost popular algorithmic method is the Delay Times method,also known as the Algorithm ofGrassberger and Procaccia(1983), which estimates the Correlation Dimension from thecomputation of the correlation integral.

The Grassberger Procaccia Algorithm (Grassberger andProcaccia, 1983) assumes a time-series,X(i), which is ameasure over timei of a parameter of anm-dimensional dy-namic system, fori∈[1, N ]. The phase space reconstruc-tion of this system is done according to the Takens theo-rem. Once again, the vector coordinates are constructedas in Eq. (8) and it is assumed that this vector is the tra-jectory vector of thei-th time point of the reconstructedphase space of the dynamic system. The whole trajectoryis x(1), x(2), . . ., x(i), P . . ., x(ρ) where ρ=N−(m−1)τ .As mentioned and according to the Takens theorem,A′ isthe attractor to the reconstructed system dynamics and thetopological conjugate to the original attractorA. Propertiessuch as the Correlation Dimension are maintained after theprojection ofA to A′. The Correlation Dimension is defined


as

D2 = limN→∞

log(C(m, r, τ ))

log(r)(A2)

wherer is a distance radius in the reconstructed phase space.The index 2 inD2 is used because the Correlation Dimensionis a special case of the generalized dimensionDq whereqinteger. C(m, r, τ ) is the correlation integral and is definedas

C(m, r, τ ) =2

N − 1

N∑i=1

N∑j=i+1

2[r − ||xi − xj ||

], (A3)

wherexi andxj are as in Eq. (8).2 is the Heavyside func-tion:

2(i) =

{1, if i ≥ 00 if i ≤ 0

(A4)

The Euclidean norm used in the above equation states thatthe difference betweenxi andxj is the maximum differenceamong their coordinates:

||xi − xj || ={∣∣X(i) − X(j)∣∣2 + ∣∣X(i + τ) − X(j + τ)∣∣2+

. . . +∣∣X(i + (m − 1)τ ) − X(j + (m − 1)τ )∣∣2} 12 . (A5)

The formula Eq. (10) simply says: for specificm, r, τ findall pairs ofxi andxj in the reconstructed time-seriesx(t) forwhich the distance||xi−xj || is smaller thanr.

According to this algorithm theC(m, r, τ ) is computed forincreasing values ofm and for a steady range ofr. For eachlog(C) versus log(r) plot a scaling region is been selectedand the slope of the curve is calculated for this scaling regionwith a simple method (i.e. least squares). If the slope valuesestimated for eachm converge in a steady value, then thissteady value corresponds to the correlation dimension of thetimeseries.

Appendix B Approximate Entropy

Given N data points, X(1), X(2), X(3), . . ., X(N), theApEn(m, r, N) is estimated, wherer is a thresholdand m a window size. The vector sequences neces-sary for phase space reconstruction,x(i), are constructedwith x(N−m+1), defined byx(i)=[X(i), . . ., X(i+m−1)].These vectors representm consecutiveX values, using thei-th point as the starting point. The distance||x(i), x(j)||is defined between the vectorsx(i) andx(j) as the infinitynorm

||X(i) − X(j)|| = max{|X(i) − X(j)|,

|X(i + 1) − X(j + 1)|,

. . . , |X(i + m − 1) − X(j + m − 1)|}. (B1)

The probability that|X(i+m−1)−X(j+m−1)|≤r giventhat |X(i)−X(j)|≤r and |X(i+1)−X(j+1)|≤r and|X(i+2)−X(j+2)|≤r and . . . is true is termedCmr (i),

where, once again,r is the a threshold andm the win-dow size. For example, ifm=2, C2r (i) for i=1, . . ., Nis the probability that|X(i+1)−X(j+1)|≤r given that|X(i)−X(j)|≤r.

The sequence in Eq. (14) is used to construct theCmi (r)for eachi≤N−m+1 as in

Cmr (i) =

[no.ofj ≤ N−m + 1, suchthat||x(i)−x(j)|| ≤ r]

N − m + 1. (B2)

8m(r) is defined as

8m(r) =1

N − m + 1

N−m+1∑i=1

ln Cmi (r) , (B3)

where ln is the natural logarithm. Then Approximate Entropyis defined as

ApEn(m, r) = limN→∞

[8m(r) − 8m+1(r)]. (B4)

It is therefore found that−ApEn=8m+1(r) − 8m(r) andis equal to the average overi of the natural log of the con-ditional probability that|X(j+m)−X(i+m)|≤r, given that|X(j+k)−X(i+k)|≤r, for k=0, 1, 2, . . ., m−1.

Several trials of this algorithm were run on the HRV dataand it was adjusted accordingly in order to obtain a betterdistinction between the two subject groups. The first stepin computing the Approximate Entropy is finding the lengthvector form=2, which is[X(i), X(i+1)], denotedx(i). Allvectors that are close tox(i), x(j)=[X(j), X(j+1)], areidentified. As has already been stated, the vectorx(j) isclose tox(i) if ||x(i), x(j)||≤r. This, by definition, meansthat both|X(i)−X(j)|≤r and|X(i+1)−X(j+1)|≤r apply.A count of all the vectorsx(j) close tox(i) is found andcalledB. The next step is to compute the rest of thex(j) vec-tors for which|X(i+2)−X(j+2)|≤r, and call itA. The ratioof A/B represents the conditional probability thatX(j+2) isclose toX(i+2), given that the vectorx(j) is close tox(i).

The above process is repeated for each length 2 vectorx(i), calculating the conditional probability. TheApEn isfound by calculating the average of the logarithm of theseconditional probabilities and taking its negative (to make itpositive), as seen in Eq. (18)

−ApEn = 8m+1r − 8mr

=

[ 1N − m

N−m∑i=1

ln(Cm+1r (i))]−

[ 1N − m

N−m+1∑i=1

ln(Cmr (i))]

'1

N − m

N−m∑i=1

[ln(Cm+1r (i)) − ln(C

mr (i))

]=

1

N − m

N−m∑i=1

ln(Cm+1r (i))

ln(Cmr (i))(B5)


The calculation of the conditional probabilities will resultin values between 0 and 1. If the timeseries is regular,the valuesX(i), X(i+1), X(i+2) are expected to be closeto each other, as areX(j), X(j+1), X(j+2). There-fore, the differences|X(i)−X(j)|, |X(i+1)−X(j+1)| and|X(i+2)−X(j+2)| will be close to each other for many val-ues ofi, j . This means that the conditional probabilities areexpected to be closer to 1 for time-series coming from moreregular processes. The negative logarithm of such a valuewill be closer to 0.

Conversely, random processes will produce conditionalprobabilities closer to 0, the negative logarithms of whichwill be closer to 1. The comparison of subsequent vectorsin a random signal will result in different values in the suc-cessive vector distances. Thus, theApEn values for signalscoming from regular processes will be lower than theApEnvalues coming from random signals. In this application, thisimplies that lowApEn values are to be clinically associatedwith cardiac pathology, while high values indicate a healthyand robust heart.

The previous algorithm calculates an estimation of thevalue of the Approximate Entropy, which is equal to the the-oretical one, when theN tends to infinity. An examination ofthis algorithm reveals that it is analogous to the Grassberger& Procaccia Algorithm (the Delay Times method) for theCorrelation Dimension estimation. Theoretical calculationsby Wolf et al.(1965) indicate that reasonable estimations areachieved with anN value of at least 10m and preferably 30m.In the experimental analysis,N=2000 andm=2 were used,producing satisfactory statisticalApEn validity.

However, theApEn is a biased statistic. The expectedvalue ofApEn(m, r, N) increases asymptotically withN toApEn(m, r) for all processes. The choice of window foreach vectorx is also important for theApEn estimation.However, the interest in this method is not in the recon-structed space, but rather in having a sufficient number ofvectors in close proximity to each other, so that accurateconditional probabilities can be found. This work was partlysupported by the PYTHAGORAS fellowships.

Edited by: M. ContadakisReviewed by: two referees

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