Comparison of fractal dimension estimation algorithms for epileptic seizure onset detection

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J. Neural Eng. 7 (2010) 046007 (18pp) doi:10.1088/1741-2560/7/4/046007

Comparison of fractal dimensionestimation algorithms for epileptic seizureonset detectionG E Polychronaki1, P Y Ktonas2, S Gatzonis2,3, A Siatouni2,3,P A Asvestas4, H Tsekou2,3, D Sakas2,3 and K S Nikita1

1 School of Electrical and Computer Engineering, National Technical University of Athens, 9, HeroonPolytechniou Str., Zografou, Athens 157 80, Greece2 Greek Center for Neurosurgical Research ‘Prof. Petros Kokkalis’, 3, Ploutarxou Str., Athens 106 75,Greece3 Epilepsy Surgery Unit, Department of Neurosurgery, University of Athens, ‘Evangelismos’ Hospital,45-47, Ipsilantou Str., Athens 106 76, Greece4 Department of Medical Instruments Technology, Faculty of Technological Applications, TechnologicalEducational Institute of Athens, Ag. Spyridonos Str., Egaleo, Athens 122 10, Greece

E-mail: gpoly@biosim.ntua.gr, knikita@cc.ece.ntua.gr and Periklis.Ktonas@mail.uh.edu

Received 19 October 2009Accepted for publication 26 May 2010Published 23 June 2010Online at stacks.iop.org/JNE/7/046007

AbstractFractal dimension (FD) is a natural measure of the irregularity of a curve. In this study theperformances of three waveform FD estimation algorithms (i.e. Katz’s, Higuchi’s and thek-nearest neighbour (k-NN) algorithm) were compared in terms of their ability to detect theonset of epileptic seizures in scalp electroencephalogram (EEG). The selection of parametersinvolved in FD estimation, evaluation of the accuracy of the different algorithms andassessment of their robustness in the presence of noise were performed based on syntheticsignals of known FD. When applied to scalp EEG data, Katz’s and Higuchi’s algorithms werefound to be incapable of producing consistent changes of a single type (either a drop or anincrease) during seizures. On the other hand, the k-NN algorithm produced a drop, startingclose to the seizure onset, in most seizures of all patients. The k-NN algorithm outperformedboth Katz’s and Higuchi’s algorithms in terms of robustness in the presence of noise andseizure onset detection ability. The seizure detection methodology, based on the k-NNalgorithm, yielded in the training data set a sensitivity of 100% with 10.10 s mean detectiondelay and a false positive rate of 0.27 h−1, while the corresponding values in the testing dataset were 100%, 8.82 s and 0.42 h−1, respectively. The above detection results comparefavourably to those of other seizure onset detection methodologies applied to scalp EEG in theliterature. The methodology described, based on the k-NN algorithm, appears to be promisingfor the detection of the onset of epileptic seizures based on scalp EEG.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Epilepsy belongs amongst the most common neurologicaldiseases, second only to strokes (Annegers 1997). Althoughcomputer detection of epileptic seizures is a relatively oldfield of research (Gotman 1982), its significance becomes

greater as newly available technologies generate increasingdemands. Surgical resection is nowadays evolving into abroadly applied strategy for confronting some types of focalepilepsies (Tellez-Zenteno et al 2005). Long-term video/EEGmonitoring during pre-surgical evaluation for epilepsysurgery produces huge numbers of continuous multiple-day

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J. Neural Eng. 7 (2010) 046007 G E Polychronaki et al

and multiple-channel digital electroencephalographic (EEG)recordings, which need to be reviewed for the existenceof seizures. Similar needs arise from ambulatory EEGmonitoring (Waterhouse 2003). For that reason, variousautomated seizure detection techniques have been developedwhich aim to detect the presence of a seizure using both scalp(Sazonov et al 2009, Schad et al 2008, van Putten et al 2005,Wilson et al 2004, Liu et al 2002, McSharry et al 2002,Schindler et al 2001, Gabor 1998, Gabor et al 1996, Gotman1990, 1982) and intracranial EEG (Schad et al 2008, Bhavarajuet al 2006, Gardner et al 2006, Meng et al 2004, Khan andGotman 2003, Schindler et al 2001, Harding 1993, Murro et al1991).

Exhaustive pre-surgical evaluation should commonlyprovide information about the localization of an epilepticfocus, which could be ascertained by the interaction of medicalstaff with the patient during the first few seconds of theevolving seizure. This leads to the more challenging taskof online seizure onset detection, with moreover as few falsedetections (false positives (FP)) as possible. Systems capableof defining seizures’ onsets with high sensitivity and low FPrates could also be utilized for automatic seizure termination,e.g. via electrical stimulation (Morrell 2006) or drug delivery(Eder et al 1997). The patient-specific algorithm of Qu andGotman (1997) for the detection of seizure onset in long-termscalp EEG monitoring was the first attempt at introducing sucha seizure warning system. Other important works in the fieldof seizure onset detection include Meier et al (2008), Saaband Gotman (2005) and Shoeb et al (2004) which have beendeveloped using scalp EEG, while intracranial EEG has beenused in Haas et al (2007), Grewal and Gotman (2005), Osorioet al (2002, 1998).

The term ‘fractal’ can be used to characterize objectsin space or fluctuations in time which show a form of self-similarity. A fractal is an object made of parts similar tothe whole and has the property that more fine structure isrevealed as the object is magnified (Klonowski 2000). Thefractal dimension (FD) is a measure of how complicated aself-similar object is, taking greater values for increasingcomplexity. Strict self-similarity is a property that onlyartificially generated mathematical objects are characterizedby; the Sierpinski triangle is such an example (Kaplan andGlass 1995). Natural objects show statistical self-similarity(Klonowski 2000). In the statistical sense, an object is self-similar if its parts, on average, are similar to the whole (Kaplanand Glass 1995), but there exist no exact replicas of particularparts. In statistical self-similarity, a measure of complexity fora given magnification will have the same statistical momentsas at any other magnification, although the details will not beidentical (Arle and Simon 1990).

In time series analysis, FD can be used to quantify theirregularity or complexity of a waveform (Katz and George1985). In this study, we compared different FD estimationalgorithms using scalp EEG data in order to assess theirusefulness in automatically detecting the onset of epilepticseizures of patients with refractory mesial temporal lobe(MTL) epilepsy. MTL epilepsy is the most common formof human epilepsy and is typically medically refractory

(Engel 2001). Interictal (between seizures) EEG patternsare described by electroencephalographers as low to mediumvoltage, irregular and arrhythmic, while during the ictal period(seizure) the EEG activity changes to have more organized,rhythmic and self-sustained characteristics (Sackellares et al2000). Therefore, intuitively, it seems that the FD is anappropriate measure for the characterization of changes in suchEEG dynamics, as it could be expected to drop in value duringseizures as compared to interictal periods. In the current study,the selection of scalp over intracranial EEG was motivated bythe fact that seizure detection using this type of EEG recordingswould be more practical in real life and, moreover, it would besuitable for types of epilepsy that do not warrant intracranialelectrode implantation (Le Van Quyen et al 2001). It has tobe pointed out, however, that scalp EEG is subject to manydifferent types of artefacts (e.g. eye blinks, muscle activity,etc) which make the detection of seizures’ onsets with a lowFP rate a challenging task.

The applicability of fractal analysis to EEG time series hasbeen examined in various studies (Andrzejak et al 2006, 2001,Li et al 2005, Liu et al 2005, Klonowski 2002, Esteller et al2001b, Accardo et al 1997, Bullmore et al 1994, Cabukovskiet al 1993, Arle and Simon 1990, and references therein)using different FD estimators including, but not limited to,algorithms by Petrosian (1995), Maragos and Sun (1993), Katz(1988), Higuchi (1988), Richardson in Voss (1988), Pickoverand Khorasani (1986), Grassberger and Procaccia (1983). Fora review of applications of fractal time series analysis inphysiological research, not limited to EEG analysis, see Ekeet al (2002).

In the current study, two of the most widely used FDmethods in the literature, i.e. the algorithms proposed by Katz(1988) and Higuchi (1988), are compared to the algorithmintroduced by our group (Asvestas et al 1999), called thek-nearest neighbour (k-NN) algorithm. The former twoalgorithms, or variations thereof, have been broadly appliedin various studies (Gomez et al 2009, Acharya et al 2005,Berryman et al 2005, Accardo et al 1997, Esteller et al 2001b,Bullmore et al 1994). In contrast, to the best of our knowledge,this is the first time that the k-NN algorithm has been appliedin a time series analysis sense.

Few works in the literature have utilized FD methodsparticularly for the study of epileptic seizures (Estelleret al 2001b, 1999, Accardo et al 1997, Bullmore et al1994, 1992). The main objective of those works was toexamine whether FD could provide a sensitive criterion fordiscriminating the seizure from the non-seizure state byobserving whether distinct changes in the FD would ariseduring seizures. Nevertheless, none of those studies usedlong-term EEG data to assess the ability of different FDalgorithms to be not only sensitive but also specific in detectingchanges during seizures. In all studies, the results usingonly short EEG segments containing seizures were reportedand no interictal data were analysed to provide a furtherunderstanding of how FD values would change during thedifferent physiological states (e.g. sleep, chewing, etc) thatwould normally exist during an inpatient or ambulatory setting.Analysis of long interictal EEG records would be necessary

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Table 1. Presentation of patients’ characteristics and EEG data.

Patient Sex Age (years) Seizure type Outcome Length of recording (h) No of seizures Average seizure duration (s)

1 f 36 CP I 13 4 78.482 m 62 CP I 162.5 5 52.473 f 32 CP, sGTC – 16.9 7 58.854 m 38 CP I 68.94 21 72.34

Sum training 261.34 37Mean training 65.34 ± 69.62 65.53 ± 11.96

5 f 33 CP I 101 3 31.576 m 25 CP, sGTC Ib 67 8 55.157 f 30 CP I 62.91 4 54.148 f 32 CP, sGTC Ib 60.89 3 71.7

Sum testing 291.8 18Mean testing 72.95 ± 18.87 53.18 ± 16.40

Sum overall 553.14 55Mean overall 69.14 ± 47.39 59.36 ± 14.84

m, male; f, female. Seizure types: complex partial (CP), secondarily generalized tonic–clonic (sGTC). Values presented are mean ±standard deviation. The outcome is according to Engel’s classification.

in order to demonstrate that the FD changes during seizurescould be far more distinct as compared to those during otherphysiological states and that, therefore, FD measures could beutilized effectively for the purpose of seizure detection.

The main objective of the current work was to performa comparison of the three FD methods in terms of theirability to detect the onset of epileptic seizures and at thesame time produce an acceptable number of FPs, using long-term, multichannel, continuous scalp EEG data, includingartefacts. In order to ensure an optimal EEG analysis usingthe algorithms and to facilitate the interpretation of the results,all three algorithms were first applied on synthetic signalsof known FD. This is related to the second objective of thecurrent work, which was, based on the synthetic signals, theselection of appropriate values for the parameters involvedin FD estimation algorithms (i.e. in Higuchi’s and the k-NNalgorithms) and the comparison of the different algorithmsbased on the assessment of their estimation accuracy androbustness in the presence of noise.

In section 2, the EEG data set used in the currentstudy is presented and an outline of the FD estimationalgorithms utilized is provided. Moreover, the seizuredetection methodology developed based on FD is described.This is followed in section 3 by the evaluation of the FDalgorithms and the selection of the parameters involved inFD calculations, using synthetic signals of known FD. Theapplication of the FD algorithms to scalp EEG data and acomparison of the different algorithms’ seizure onset detectionability are presented in section 4. Discussion of the results andsuggestions for further work follow in section 5.

2. Materials and methods

2.1. EEG data description

Continuous long-term scalp EEG data from eight patientssuffering from refractory epilepsy with mesial temporalsclerosis were used. All the data were collected in

the Epilepsy Telemetry Unit, Department of Neurosurgery,University of Athens, ‘Evangelismos’ Hospital, during long-term video/EEG for pre-surgical evaluation. For the dataacquisition, a Beehive Millennium Digital Recording Systemwith Grass Telefactor Twin Recording and Analysis Softwarewas used. The low- and high-pass filters of the amplifiersallowed EEG frequencies between 0.1 and 70 Hz to berecorded. Data were acquired at a sampling rate of 400 Hz,with 12 bits A/D resolution. The data set comprised 553.14 hof EEG recording and included 55 seizures in total. Table 1summarizes patients’ characteristics as well as details about theEEG recordings. Making the rough assumption that between11.30 p.m. and 7.30 a.m. the patient was asleep, 183.5 h weresleep recordings, originating from seven of eight patients in thedata set. No exclusion of records with artefacts, such as thoseintroduced by movement of the patient or transient electrodefailure, was undertaken. Twenty-five gold disk electrodes wereplaced according to the 10–20 international system in additionto six temporal electrodes. Two of these were sphenoidal,and the rest consisted of T1, T2 (placed according to theinternational extended 10–20 system) and 27, 28, which wereplaced at mastoid hilus. A referential electrode montage wasused both for the recording and the analysis, with the referenceelectrode being placed between Cz and Pz.

In order to design and develop the automatic seizuredetection methodology, the EEG records were divided intoa training and a testing data set. After sorting the patients inthe chronological order of admission for long-term video/EEGmonitoring, we selected the first four patients as our trainingdata set (261.34 h, 37 seizures) and the remaining four as thetesting data set (291.8 h, 18 seizures). Seizure onset timeswere marked by a single expert as the time points at which thefirst EEG changes occurred which led to a clear EEG seizuredischarge. Seizure offset times were also marked.

2.2. Fractal dimension estimation algorithms

All the algorithms utilized in the current study, i.e. Katz’s,Higuchi’s and the k-NN FD estimation algorithms, do not

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require a state-space reconstruction prior to FD estimation.This is a great advantage when a detailed temporal evolutionof the signal is of interest, as in our case where we wishedto detect the presence of seizures as closely to their onset aspossible. Such ‘time-domain’ approaches (as opposed to the‘phase-space’ approaches which actually calculate FD in phasespace) require a significantly lower number of points in order toprovide an estimate of the FD (Bullmore et al 1992, Accardoet al 1997, Esteller et al 2001b). This has the additionaladvantage that stationarity of EEG segments analysed can bemore easily assumed for shorter time windows. Moreover, thetechniques are faster in terms of required computational time(Esteller et al 2001b) and, therefore, could be better candidatesfor a possible online implementation.

The k-NN algorithm was considered an appropriatecandidate for scalp EEG analysis for epileptic seizure detectionfor two reasons: first, the aforementioned low demands ininput data points, and second, the fact that in image analysis,where it was originally applied, it proved to be superiorin terms of accuracy, dynamic range and computationaltime (Asvestas et al 1999) as compared to other traditionalFD estimation methods (e.g. box-counting dimension andcorrelation algorithm).

2.2.1. Katz’s algorithm. Waveforms can be viewed ascollections of points �pi = (xi , yi) with xi < xi+1, i = 1, 2, . . . ,N (N: number of points), and they are special cases of planarcurves (moving only forward in the x direction). Katz (1988)based his method of FD estimation on the measurement of thelength of such curves. He incorporated Mandelbrot’s originalcontribution (Mandelbrot 1982), in combination with a unit ofmeasure or ‘yardstick’ definition, in order to take into accountthat, to calculate FD, space must be discretized. According toMandelbrot (1982), the FD of a planar curve is given by

FD = log(L)/log(d), (1)

where L is the total length of the curve and d is its diameter.For waveforms, the total length L is the sum of the distancesbetween successive points

L =N∑

i=1

‖�pi+1 − �pi‖,

where ‖.‖ is the Euclidean distance. The diameter (planarextent) d can be considered to be the farthest distance betweenthe starting point and any other point of the waveform:

d = maxi

‖�pi − �p1‖.According to Katz, (1) needs to be corrected by making useof the average step a of the waveform (which is the averagedistance between successive points) viewed as a ‘yardstick’.Using a, (1) becomes

FD = log(L/a)

log(d/a). (2)

Defining n as the number of steps in the curve (1 less thanthe number of points N), n = L/a. Substituting n in (2), FDaccording to Katz’s approach is expressed as

FD = log(n)

log(n) + log(d/L)

2.2.2. Higuchi’s algorithm. Higuchi’s (1988) algorithm forFD estimation is also based on curve length measurement.The algorithm estimates the mean length of the curve, byusing a segment of k samples as a unit of measure. In practice,Higuchi’s method is a modification of an older similar FDestimation approach proposed by Burlaga and Klein (1986).In more detail, Higuchi’s FD estimation technique consists ofthe following steps.

Step 1. Let us define the values of a finite set of time seriesobservations, which are taken in a regular interval asy1, y2, y3, . . . , yi, . . . , yN = y(1), y(2), y(3), . . . ,

y(i), . . . , y(N), where i = 1, 2, . . . , N (N: number of pointsin the time series). In our case, y would be the successiveEEG values. For a range of k values ranging from 1 to kmax,construct k new times series yk

m defined as follows:

ykm : y(m), y(m + k), y(m + 2k), . . . , y(m + ik), . . . ,

y

(m + int

(N − m

k

)· k

), where m = 1, 2, . . . , k.

Step 2. Calculate the length Lm(k) of each curve ykm as follows:

Lm(k) =⎡⎣

⎛⎝int( N−m

k )∑i=1

|y(m + i · k) − y(m + (i − 1) · k)|⎞⎠

· N − 1

int(

N−mk

) · k

⎤⎦ · k−1.

The term (N − 1) ·(int(

N−mk

) · k)−1

serves as a normalizationfactor for the curve length of yk

m. For an illustrative exampleof Lm(k) calculation, the interested reader may refer to Accardoet al (1997).

Step 3. Calculate the mean length of the curve for each k,〈L(k)〉, as the average value over k sets of Lm(k), for m = 1,2, . . . , k, as 〈L(k)〉 = 1

k

∑km=1 Lm(k). Repeat the calculation

for k ranging from 1 to kmax.

Step 4. If 〈L(k)〉 ∝ k−FD, then the curve is fractal withdimension FD. In that case, the plot of ln(〈L(k)〉) againstln(k) should fall on a straight line with slope equal to −FD.Therefore, FD can be calculated by means of a least-squareslinear best-fitting procedure.

One important point that Higuchi did not extensivelyelaborate on in his original work (Higuchi 1988) is the selectionof kmax. In this study, we attempted to provide a means forselecting an optimum kmax, as illustrated in section 3.2.1.

2.2.3. k-nearest neighbour algorithm. In contrast to Katz’sand Higuchi’s algorithms, which are based on measurementsof the length of the waveform under investigation, the k-NNalgorithm belongs to a class of algorithms, called fixed-mass,according to which FD estimation is based on the sizes of cubeswhich are scaled appropriately as to contain the same numberof points (fixed-mass) (Asvestas et al 1999, and referencestherein). The average distance, 〈rk〉, of a point from itskth nearest neighbour can be expressed as a function of kas (Grassberger 1985)⟨

rγ

k

⟩ = G(k, γ )(k/N)γ/D(γ ), (3)

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where γ = (1 − q)Dq , D(γ ) = Dq , Dq is the multifractaldimension of order q, N is the number of points and G(k, γ ) isa function of k and γ , which is near unity for large k. For q =0, the FD is obtained, that is FD = D0, which means that FDis the fixed point of the function D(γ ), namely FD = D(FD)(for more details see Asvestas et al (1999)).

The FD of a waveform is estimated iteratively, using (3),for k = kmin, . . . , kmax (k integer), as follows (Asvestas et al1999).

Step 1. An initial value of γ , i.e. γ 0, is chosen arbitrarily andG(k,γ ) is set to unity for every k (G(k,γ ) was set to unity inAsvestas et al (1999) and we follow the same assumption in thecurrent study). Since the FD of waveforms lies theoreticallybetween 1 and 2 (Klonowski 2000), it would be better to chooseγ0 in this range, i.e. γ 0 = 1.5.

Step 2. For every point �pi = (xi , yi), i = 1, 2, . . . , N,we calculate the Euclidian distances rki

from its k-nearestneighbours, k = kmin, . . . , kmax.

Step 3. For j = 1, 2, . . . , the following recursive relations areapplied:

D(γj ) = γj−1

sj−1, γj = D(γj ), (4)

where sj−1 is the slope of the best-fitting line at the points(ln(k/N), ln

⟨r

γj−1

k

⟩)(least-squares sense) and

⟨r

γj−1

k

⟩ = 1

N

N∑i=1

rγj−1

ki.

The calculation of (4) is repeated until the quantity∣∣∣∣∣ D(γj ) − γj−112 [D(γj ) + γj−1]

∣∣∣∣∣drops below a certain value or a maximum number of iterationsis reached. FD is calculated as D(γ j ) for the last j before theabove criterion is met.

Similarly to Higuchi’s algorithm, the k-NN algorithmis also parametric: kmin and kmax need to be defined. Insection 3.2.2 we propose a method of selecting an optimumpair of kmin and kmax.

2.3. Description of the seizure detection methodology basedon FD

The first step of our approach included band-pass filtering ofthe EEG data between 3 and 30 Hz. The reason for this wasthreefold: (a) seizure activity during seizure onset lies moreoften between 3 and 29 Hz (Gotman 1982); (b) occurrencesof 0–3 Hz activity can be frequent in non-ictal sleep EEG(Saab and Gotman 2005) and (c) this filtering removed high-amplitude slow post-ictal EEG activity, which in our analysisof the training data proved to be the cause of a great numberof FPs (which were eliminated after filtering). A zero-phaseButterworth filter of order 4 was utilized. After filtering, asliding-window approach was employed for the productionof FD time profiles: 2 s length windows (corresponding to800 points) with no overlap were used. This led to theproduction of FD time profiles (time series), individual points

Table 2. EEG channels for which the results are presented for eachFD algorithm. Td denotes the value of the threshold which was usedin order to select those channels. Patients 1–4 are the training and5–8 the testing data set.

Katz’s Higuchi’s k-NNalgorithm algorithm algorithm

Patient Td = 1.89 Td = 1.39 Td = 1.27

1 T1 T3 T32 F8 T2 T23 T1 T5 T34 T4 28 T45 F7 F7 T36 T2 28 T27 F7 T5 T18 T6 T2 28

of which corresponded to one FD estimation for each 2 sdata window. Five FD time profiles were generated using astandard set of five EEG traces for each patient, depending onthe lateralization of the seizure origin (EEG channels T1, T3,T5, F7, 27 were used for left temporal epilepsy, and T2, T4, T6,F8, 28 for right temporal epilepsy). The procedure describedabove was repeated using all three different FD algorithms.

The results presented for each FD algorithm and eachpatient are the ones produced by the EEG channels whichachieved the earliest seizure detections. Those channels wereselected on the basis of a threshold Td. A different value forTd was defined for each FD algorithm based on the recordingsof the training data set as follows: we used the seizures ofthe channels associated with the maximum FD drops duringseizures and Td was computed as the average of the medianFD values during those seizures (all seizures of all patientsin the training data set were used for the averaging). Tdwas then applied to every patient in the training data setin order to detect seizures and define associated detectiondelays (DD) (the exact procedure for calculation of DDs isexplained in the next paragraph). The channel selected for eachpatient in the training data set was the one which detected thelargest number of seizures and at the same time produced theminimum average DD calculated using all detected seizures ofthe patient. The channels in the testing data set were similarlyselected, based on application of Td on the first two seizures ofeach patient. Channels selected and used for analysis as wellas the Td value for each FD method are depicted in table 2.

In order to detect seizure onset in the selected channels,a generic threshold Tg (different for each FD algorithm)was applied to the FD time profiles of both the trainingand the testing data set (the procedure for Tg selection willbe explained in the next paragraph). In order to produce‘alarms’ related to automatically detected seizures, a two-stepprocedure was followed. In the first step, a parameter w

was defined as the number of FD values (i.e. number of 2 swindows) that should have remained under the threshold inorder for a ‘detection mark’ to be produced and placed atthe end of the last 2 s window. This parameter w, whichhas been used in a similar manner in the seizure detectionliterature (Osorio et al 1998), was incorporated in order toeliminate possible detections due to short bursts of EEG

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J. Neural Eng. 7 (2010) 046007 G E Polychronaki et al

(a)

(b)

Figure 1. Demonstration of the automatic seizure detection methodology using the k-NN algorithm. (a) FD time profile for 1.5 h ofrecording of patient 4, including one seizure starting at about 3370 s (the start and the end of the seizure are denoted by solid vertical lines).Note the clear drop of the k-NN FD time profile during the seizure, as compared to interictal activity. (b) Detail of (a), zoomed around theseizure (again the start and the end of the seizure are denoted by solid vertical lines, while the seizure start is aligned with t = 0). A starindicates the FD value which is estimated based on the EEG values of the 2 s data window which starts at the point in time that the star lies.The solid horizontal line denotes the threshold. Dotted vertical lines depict the times that a ‘detection mark’ is produced, before thegrouping of ‘detection marks’ is performed. The first of them, which in the figure is dotted in bold, is the ‘alarm’. Note that the ‘alarm’ isproduced only after 2 FD points are found to be below the threshold.

activity or short artefacts. Using w was inspired by whatelectroencephalographers employ for visual inspection of EEGdata records: according to their assessment, whether someepileptiform activity is actually a seizure or an interictaldischarge critically depends on the duration of this activity. Inthe present study, we used w = 2 which ensured a reduction inFPs without significantly affecting the DDs. During seizures,multiple successive 4 s windows (defined by w = 2) produced‘detection marks’, since many successive FD values weresmaller than the threshold Tg. Therefore, in the second steptowards producing ‘alarms’ related to automatically detectedseizures, ‘detection marks’ separated by less than 40 s weregrouped in a single ‘alarm’ (the same approach as in Saaband Gotman (2005)). This ‘alarm’ was located at the sameposition in time as the first ‘detection mark’ of the group, i.e.at the end of the first 4 s window with corresponding FD valuesbelow the threshold Tg. The value of 40 s was selected basedon the fact that seizures in the examined data set (table 1)had durations that mostly exceeded 40 s. This groupingidentified ‘detection marks’ that should be attributable to thesame event (the user should be notified about one seizure withonly one ‘alarm’). ‘Alarms’ were then categorized as truepositives if they appeared during the seizures’ duration or asFPs otherwise. In the case of a true positive detection, DD wasdefined as the time elapsed between the beginning of a seizure,as found by the EEG specialist, and the position in time of the‘alarm’. The FP rate was calculated for each patient as the totalnumber of FPs generated divided by the total duration of therecording after excluding the seizures’ duration. The FP rateprovided a measure of specificity: the lower the FP rate, thehigher the achieved specificity. An example of the automaticdetection procedure for a seizure is illustrated in figure 1.

In the detection framework described so far, the selectionof threshold Tg for each FD method is of crucial importance.Since both the FP rate and DD are of interest, ideally athreshold should provide 100% success in the detection of

the seizures, with zero FPs and DDs as small as possible. Ina more realistic scenario, some FPs can be allowed, and thethreshold can be defined based on a maximum allowed numberof FPs per hour. In the recent seizure detection literatureusing scalp EEG, FP rates in the range 0.02 h−1 (Qu andGotman 1997) to 0.86 h−1 (Saab and Gotman 2005) have beenreported, with corresponding sensitivities/DD of 100%/9.35 sand 77.9%/9.8 s. In the current work, 0.3 h−1 was set asthe maximum FP rate allowed, following Saab and Gotman(2005). To that end, Tg was defined for each FD method asthe maximum threshold whose application produced a meanFP rate <0.3 h−1 in the training data set (the mean FP rate wascalculated using the channels selected based on the thresholdTd, as explained earlier in this section). The value of Tg foreach FD method is provided in section 4 and in the caption offigure 13.

3. Evaluation on synthetic signals

3.1. Purpose of study with synthetic signals and signalgeneration

In this section, Katz’s, Higuchi’s and the k-NN FD estimationalgorithms were applied to synthetic signals of known FD. Thepurpose of this application was twofold: in the first step, thedefinition of an objective and systematic way for determiningthe parameters of each of the parametric algorithms studied,i.e. Higuchi’s and the k-NN algorithms, was pursued; in thesecond step, using the selected parameter values, the differentalgorithms were assessed in terms of accuracy and noisesensitivity.

For the synthetic signal generation, a deterministicWeierstrass cosine function (Tricot 1995), sampled at Nequidistant points, was used:

WH (x) =M∑i=0

λ−iH cos(2πλix), 0 < H < 1, (5)

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J. Neural Eng. 7 (2010) 046007 G E Polychronaki et al

(a) (b)

Figure 2. Weierstrass cosine function for two different theoretical FD values (FDth = 1.2 (a) and FDth = 1.5 (b)).

where λ > 1 and we fixed λ = 5, M = 26, followingEsteller et al (2001b), and x ∈ [0,1]. N = 800 was used.The above-defined function is Weierstrass’s example of acontinuous function that is nowhere differentiable and has aknown theoretical FD (Falconer 2003). More specifically,parameter H, and this parameter alone, is connected tothe theoretical FD (FDth) of the Weierstrass waveform byFDth = 2 − H. Using (5), Weierstrass sequences, each having adifferent theoretical FD value (i.e. 1.1, 1.2, 1.3, . . . , 1.9), weregenerated. In figure 2, two of those sequences are depicted.In the current study, N = 800 was used both for the syntheticsignals and the EEG data analysis. This data window lengthwas selected as an acceptable compromise between includingan adequate number of points for accurate FD estimation andbeing relatively short in time, as to enable us to reasonablyassume that the EEG signal is stationary during that datawindow.

3.2. Selection of parameters for FD estimation algorithms

3.2.1. Higuchi’s algorithm. Even though Higuchi did notelaborate extensively on the selection of kmax in his originalwork (Higuchi 1988), it seems that this parameter has adecisive role for the FD estimation when utilizing his method,as illustrated in the current work. A few studies in the past haveattempted to address the issue of kmax selection: the authorsin Accardo et al (1997) selected kmax = 6 as the optimumkmax value in the range kmax = 3–10. Other studies havesuggested that the selection of the kmax range should probablybe subjected to further consideration if a large N is to beused. In his paper Higuchi selected in illustrative examplesmuch greater values of kmax, i.e. kmax = 211, for N = 217.In another study (Paramanathan and Uthayakumar 2008), theauthors provided an algorithmic estimation of kmax, inspired bya divider method for FD estimation. In their approach, kmax ofHiguchi’s method was recalculated for every FD estimation.In that study too, the authors suggested increasing kmax forincreasing N.

In the current study, a wide range of kmax values wasconsidered, i.e. 2–80 (recall N = 800 is used). Using eachof those values, the FDs using Higuchi’s algorithm werecalculated for different Weierstrass sequences. In order to

Figure 3. MSE for Higuchi’s FD estimations for increasing kmax

values. MSE takes minimum values for kmax equal to 9, 25 and 50.

assess the estimation accuracy of the algorithm for differentkmax values, a mean square error (MSE) was estimatedaccording to (6)

MSE =∑n

i=1 (FDei − FDthi)2

n, (6)

where FDth is the theoretical FD value for the synthetic signals,FDe is the estimated one and n is the number of Weierstrasssequences (of different theoretical FD values) used for theMSE estimation. Figure 3 shows the MSE against kmax. Valuesof kmax lower than 5 led to a clearly poor FD estimation whichresulted in very high MSEs (one order of magnitude higher, upto 0.085, not displayed in the figure). According to figure 3,there are three troughs of lower MSE values, and for increasingkmax values past 50 the FD estimations only get worse. Thelowest MSE values for each trough correspond to the followingvalues of kmax: 9, 25 and 50. All these values were consideredas candidates for FD estimation using Higuchi’s algorithmwhen applied to the EEG data, as they correspond to similarMSE values. Therefore, all three of them were tested and theresults are presented in section 4.2.

3.2.2. k-NN algorithm. The same procedure as describedfor Higuchi’s algorithm, based on the MSE of estimations,

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Figure 4. MSE of FD estimations using the k-NN algorithm for allpossible (kmin, kmax) combinations of values examined. For (kmin,kmax) = (1, 173) the minimum MSE = 6.5 × 10−4 is achieved.

was applied for estimating (kmin, kmax) for the k-NN algorithm.Again, a wide range of possible (kmin, kmax) values was tested,i.e. kmin was assigned values in the range 1–5 and kmax inthe range 100–250. The MSE was calculated for all possible(kmin, kmax) combinations (and it only increased for values of(kmin, kmax) outside the range presented here). The results arepresented in figure 4. In the case of (kmin, kmax) selection for thek-NN-based estimation, the problem of multiple troughs withsimilar MSEs, such as in figure 3, did not appear. In contrast,there existed a single minimum towards which all MSE valuesconverged, which was achieved for the combination (kmin,kmax) = (1, 173). Therefore, this pair of values was selectedas the optimal and was used for all the calculations in the restof this work, both with synthetic signals and with scalp EEG.

3.3. Evaluation of the accuracy of FD estimation with thedifferent algorithms

In order to assess the accuracy of the three algorithmsunder examination, Weierstrass functions were generated asdescribed in section 3.1. For the estimation of FDs usingHiguchi’s algorithm, kmax = 50 was used, while for thek-NN algorithm, (kmin, kmax) = (1, 173) were used. Figure 5shows the estimated FD values, calculated using each ofthe algorithms and plotted against the corresponding knowntheoretical FD values of the synthetic signals. The resultspresented in figure 5 are discussed in section 3.5.

3.4. Noise sensitivity

The EEG is frequently contaminated by electrophysiologicalpotentials generated by muscle activity (Vergult et al 2007).Muscle artefacts, due to their broad frequency spectrum(Goncharova et al 2003), may be considered as white noise(De Clercq et al 2006). In the current study, we assessed thereliability and robustness of each FD algorithm in the presenceof noise, using Weierstrass signals with added white Gaussiannoise of different signal-to-noise ratios (SNRs). Eleven noiselevels were used, i.e. 30–10 dB with step −2 dB. For eachnoise level, 100 Weierstrass sequences of the same theoreticalFD with additive noise were produced. Using each algorithm,

Figure 5. Estimated FD values of Weierstrass synthetic signals(FDe) plotted against the corresponding theoretical FD values(FDth), using all three FD estimation algorithms.

the FD values of the 100 sequences with a particular theoreticalFD and SNR were estimated, and the mean value and standarddeviation of the estimates were calculated. Again, for theestimation of FDs using Higuchi’s algorithm, kmax = 50 wasused, while for k-NN, (kmin, kmax) = (1, 173) were used.

The results for all different noise levels and FD algorithmsare presented in figures 6(a)–(c). In figure 6(d), the MSEsof all algorithms plotted against decreasing noise power areshown (MSEs estimated as described in section 3.2). Theresults presented in figure 6 are discussed in section 3.5.The calculations of figure 6(b) (Higuchi’s algorithm) wererepeated for kmax = 9 and kmax = 25. The analysis revealedthat, for increasing kmax values, the accuracy of estimationachieved improved, as the estimated points tended to betterapproach the diagonal. This result provided a first indicationthat selection of kmax = 50 might be more appropriate for thecalculations with EEG data. More about this selection followsin section 4.2.

3.5. Comparison of algorithms based on synthetic signalanalysis

The most desirable characteristic of an FD estimationalgorithm is its ability to clearly discriminate among signalsof different complexities. For instance, in the frameworkdescribed in this section, we would ideally expect an algorithmto provide FD estimates that fall onto a straight line of slopeequal to one and going through the axes origin (indicated infigure 5 by a bold line). It can be deduced from figure 5 thatKatz’s algorithm yielded the worst estimation as comparedto Higuchi’s and the k-NN algorithms since it overestimatedthe FD values for the whole range of theoretical FD valuesexamined. Higuchi’s algorithm provided the most accurateestimations for the whole range of theoretical FD values.On the other hand, the k-NN algorithm provided satisfactoryestimations for almost the whole range of theoretical FDvalues, slightly overestimating the lower and higher FDs andunderestimating the middle ones. All algorithms demonstrateda wide dynamic range, in contrast to other FD estimationalgorithms, such as Petrosian’s algorithms (Esteller et al

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(a) (b)

(c) (d)

Figure 6. Mean FD estimates using Katz’s (a), Higuchi’s (b) and the k-NN (c) algorithms using 100 Weierstrass cosine functions withadditive white Gaussian noise of increasing power (FD estimations were averaged for 100 Weierstrass cosine functions for each theoreticalFD value and each noise level). Error bars indicate the standard deviation of the FD estimates. (d) MSE estimates plotted against SNR forall different algorithms.

2001b); therefore, we expect them to discriminate betweensignals of different complexities.

The results presented in figure 6 reveal that, notsurprisingly, the accuracy of estimation for all FD algorithmsdecreases with increasing noise power. For all FD algorithms,their estimates move towards higher values. This is inagreement with results presented in Accardo et al (1997),where noise was directly added to EEG signals. Thiseffect is more prominent for the lower FD values. That isreasonable since noise addition to signals with higher FDvalues, which are already ‘complicated’ enough, would notbe expected to have a decisive impact on their estimatedcomplexity. Katz’s and Higuchi’s algorithms failed to providemonotonically increasing estimates in the range 1.1–1.6 and1.1–1.5, respectively, for SNR = 20 and 10 (figures 6(a) and(b)). The k-NN algorithm, on the other hand, maintained itsability to discriminate among different FD values, even for thelower theoretical FD values and for SNR levels down to 10 dB(figure 6(c)). Additionally, figure 6(d) reveals that the smallerMSEs for all SNR levels are achieved by the k-NN algorithmwhich seems to be the most robust and reliable in the presenceof noise.

4. Results using scalp EEG

4.1. Katz’s algorithm

FD time profiles using Katz’s algorithm were produced asdescribed in section 2.3. Indicative results derived from therecordings of two patients from the training and two patientsfrom the testing data set are depicted in figure 7. The valuesof the FD time profiles away from the seizures have a meanvalue of around 1.9–2.2, which appear to be limiting valuesfor the FD of an EEG signal. This is in agreement with theoverestimations of the true FD values of the synthetic signalsusing Katz’s algorithm (see section 3.3, figure 5). In thecase of patient 1 (figure 7(a)), no important changes during theseizures were observed. During the first two seizures of patient4 (figure 7(b)), a slight drop was evident, but the existence ofthis drop was not consistent for all the seizures of that patient.In the case of patient 7 (figure 7(c)), a slight rise of FD wasrecorded around the seizures (but not strictly localized to theseizure duration), while a similar situation was observed whenpatient 8 was examined (figure 7(d)). Similar FD profiles wereproduced from the analysis of the other four patients.

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(a) (b)

(c) (d)

Figure 7. FD time profiles using Katz’s algorithm for patients 1 (a) and 4 (b) from the training data set and patients 7 (c) and 8 (d) from thetesting data set. Solid vertical lines indicate the start and end of seizures (seizure starts are aligned with t = 0), while dashed vertical linesindicate the times of ‘alarms’. Solid horizontal lines indicate the generic threshold for Katz’s algorithm Tg = 1.48.

Using Katz’s algorithm, application of the seizuredetection methodology described in section 2.3 produced theresults presented in figure 13. Using the generic thresholdvalue Tg = 1.48, no seizures were detected for either thetraining or the testing data set, resulting in a 0% sensitivity.Nevertheless, the FP rates produced were, in four patient cases,close to or above 0.3 h−1. From both visual analysis (figure 7)and numerical results (figure 13), it becomes clear that Katz’salgorithm did not produce FD changes which were pronouncedenough to enable seizure detection.

4.2. Higuchi’s algorithm

As already discussed, based on synthetic signal analysis, theperformance of Higuchi’s algorithm depends on the selectionof the parameter kmax. In section 3.2.1, it was illustrated thatthere are three troughs in the plot of MSE versus kmax (figure 3).The MSE minimum values appeared at kmax = 9, 25, 50, and itwas mentioned that those three values could all be consideredas candidates for FD estimation using Higuchi’s algorithmwhen applied to the EEG data. According to Higuchi’salgorithm, the FD of a curve is estimated by means of a least-squares linear best-fitting procedure. In section 2.2.2, it was

mentioned that if (〈L(k)〉)∝k−D , then the curve is fractal withdimension D and, in that case, the plot of ln(〈L(k)〉) againstln(k) should fall on a straight line with a slope equal to −D.When analysing real data (e.g. EEG), it is possible that thepoints (ln(k), ln(〈L(k)〉)) might not fall on a straight line for thewhole range of k values. In that case, kmax must be selectedappropriately in order for the estimated slope to optimallyapproximate the slope of the linear part of the ln(〈L(k)〉) versusln(k) plot. An example is provided in figure 8, where ln(〈L(k)〉)is plotted against ln(k), as estimated using 2 s of EEG data frompatient 1. In figure 8(b), the least-squares fits for kmax = 9, 25,50 are presented. The straight line that best approximates thelinear part of the ln(〈L(k)〉) versus ln(k) curve for a broad rangeof k values is the one corresponding to kmax = 50. Similar plotswere acquired when using different EEG segments from thesame patient and also from different patients.

FD time profiles using Higuchi’s algorithm were producedas described in section 2.3. Figures 9(a)–(c) depict 2000 s ofHiguchi’s FD time profiles obtained from the first recordingof patient 1 (containing a seizure) and estimated using kmax =9 (figure 9(a)), 25 (figure 9(b)) and 50 (figure 9(c)). It can beobserved that as kmax increases, the estimated FD values alsoincrease. This can be explained by figure 8, where, for the same

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(a)

(b)

Figure 8. (a) 2 s of EEG from patient 1. (b) Least-squares best fit lines calculated from the (ln(k), ln(〈L(k)〉)) points as estimated using theEEG data segment presented in (a), using different values of kmax. Those lines lead to estimates of the FD (with Higuchi’s algorithm) whichare equal to −slope (slope is defined in the legend), for each value of kmax.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9. 2000 s of FD time profiles estimated using Higuchi’s algorithm, taken from the first recording of patient 1 (containing a seizure),and estimated using kmax = 9 (a), 25 (b), 50 (c). Part of FD time profiles depicted in (a)–(c), zoomed around the seizure, using kmax = 9 (d),25 (e), 50 (f) (beginning and end of seizure is indicated by vertical lines).

EEG data window, use of a higher k (increasing kmax values)leads to greater slopes and, therefore, higher FD estimates.As can be seen in figures 9(a)–(c), selection of kmax = 50provided FD values closer to the ones estimated using thek-NN algorithm (as will be shown in figure 11). Moreover,when zooming into the seizure part displayed in figures 9(d)–(f), it is clear that during the seizure, the value kmax = 50(figure 9(f)) is the one which achieves better discriminationof the beginning of the seizure (bigger drop in the FD valuesduring the beginning of the seizure as compared to the timeinterval immediately preceding the seizure) amongst kmax =9, 25, 50. The same was observed for other seizures fromthe same patient and from different patients. Moreover, recallfrom section 3.4 that analysis of synthetic signals with addednoise revealed that the most robust estimates were generatedfor kmax = 50. Based on these observations, the value kmax =50 was selected for the analysis of the EEG data.

Figure 10 illustrates indicative FD time profiles derivedusing Higuchi’s algorithm with kmax = 50, from the recordingsof two patients from the training and two patients from thetesting data set (same as in figure 7). In this case, the valuesof the FD time profiles away from the seizures had a meanvalue of around 1.3–1.7. In the case of patient 1, there was ashort drop in the Higuchi FD profile at the beginning of eachseizure (figure 10(a)), but during the rest of the seizure theFD values ranged at similar levels as in the interictal periods.Similar profiles, displaying a short drop at the beginning ofa seizure, were produced for some seizures of patient 6. Incontrast, the FD values derived from the EEG data of patient4 (figure 10(b)) exhibited a drop during most of the durationof the seizures, for all seizures. Drops were also recordedduring most of the duration of the seizures of patient 2 andduring the second half of the seizure duration of patients 3and 7 (figure 10(c), upper two subplots). However, for some

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(a) (b)

(c) (d)

Figure 10. FD time profiles produced using Higuchi’s algorithm for patients 1 (a) and 4 (b) from the training data set and patients 7 (c) and8 (d) from the testing data set. Solid vertical lines indicate the start and end of seizures (seizure starts are aligned with t = 0), while dashedvertical lines indicate the times of ‘alarms’. Solid horizontal lines indicate the generic threshold for Higuchi’s algorithm Tg = 1.29.

seizures of patients 6 and 7 (figure 10(c), lower subplot) andfor the seizures of patients 5 and 8 (figure 10(d)), either anincrease or no particular change in FD values was observed. Itis interesting to note that this increase was not specific to theseizure and also appeared in other parts of the recording as well(as an example, see the rise of FD after the seizure in the lastseizure data displayed for patient 8 in figure 10(d)). This couldbe attributed to higher frequency components present duringthose recording parts, as discussed in section 5.2. Similarlyto Katz’s algorithm, Higuchi’s algorithm failed to produceconsistent changes of a single character (either a drop or anincrease) that could provide a systematic and specific criterionindicative of a seizure. In addition, in some seizure cases,the drops or increases were of short duration in comparison tothe seizure duration and, therefore, did not provide a distinctcharacteristic of the dynamics of the whole seizure.

Using Higuchi’s algorithm, application of the seizuredetection methodology described in section 2.3 produced theresults presented in figure 13. Using the generic thresholdvalue Tg = 1.29, 24 of 37 seizures were detected in the trainingand 1 of 18 seizures in the testing data set. The FP rates werefor six patients well below 0.3 h−1, while for one patient(patient 3) the FP rate approached 1. However, for patient 8,

the high value of 19.27 h−1 was produced. This was observedbecause, in that case, the FD time profiles away from theseizures had a mean value of around 1.3 (figure 10(d)), i.e.a value very close to Higuchi’s generic threshold Tg = 1.29.Similarly, the use of a generic threshold for seizure detectionin Saab and Gotman (2005) resulted in some patients havinghigher FP rates than others. In order to remedy that, the authorsproposed a threshold-‘tuning’ mechanism. A similar approachcould be applied to our seizure detection methodology and willbe discussed in section 5.3.

4.3. k-NN algorithm

k-NN FD time profiles were produced as described in section2.3. Figure 11 illustrates indicative FD time profiles, derivedusing the k-NN algorithm with (kmin, kmax) = (1, 173). Therecordings of two patients from the training and two patientsfrom the testing data set are depicted (same patients as infigures 7 and 10). In almost all cases, the FD values duringthe seizures were of clearly different mean amplitude incomparison to what happened away from the seizures wherethe mean FD value for all patients was above 1.4. This cleardistinction of the ictal period that the k-NN algorithm achieves

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(a) (b)

(c) (d)

Figure 11. FD time profiles produced using the k-NN algorithm for patients 1 (a) and 4 (b) from the training data set and patients 7 (c) and8 (d) from the testing data set. Solid vertical lines indicate the start and end of seizures (seizure starts are aligned with t = 0), while dashedvertical lines indicate the times of ‘alarms’. Solid horizontal lines indicate the generic threshold for the k-NN algorithm Tg = 1.27.

is very important when it comes to applying a simple thresholdfor seizure detection.

As can be seen in figure 11, the k-NN FD for some datawindows took values greater than 2 (when, theoretically, theFD of a curve can only have values between 1 and 2). Theseoverestimations occurred in cases where the EEG in the timewindow under investigation included some values that were‘outliers’ when regarding the amplitude distribution of the restof the EEG data points in the window (those were mostlydue to artefacts, e.g. chewing activity). An example of anEEG segment causing overestimations in the k-NN FD timeprofile is illustrated in figure 12. It is worth noting that thoseoverestimations did not affect the result of the seizure detectionmethodology, as production of ‘alarms’ was associated withFD points being below the threshold.

Using the k-NN algorithm, application of the seizuredetection methodology described in section 2.3 produced theresults presented in figure 13. Using the generic thresholdvalue Tg = 1.27, all seizures were detected both for the trainingand the testing data set, resulting in 100% sensitivity. Of allseizure detections, 89.09% were achieved within the first thirdof the corresponding seizure duration. Only 6 of 55 seizureswere detected ‘late’, 5 of which were of patient 3 and 1 of

patient 4. Relatively low mean FP rates were produced bothfor the training and the testing data set.

5. Discussion

5.1. General comments on the contribution of the currentwork

In epileptogenesis, according to most of the theoriescommonly accepted today, neuronal synchronization isconsidered to be decisive (Mormann et al 2000 and referencestherein). In epileptic EEG, seizures are usually characterizedby rhythmic patterns (Meier et al 2008). This fact points to thepossible usefulness of fractal analysis in the context of seizuredetection. Synchronized, rhythmic activity during the seizuresis expected to lead to a reduction in complexity, as compared tothe more ‘disorganized’ interictal activity. This reduction can,in principle, be quantified utilizing nonlinear measures suchas the FD, which is expected to show a drop in values duringthe seizure period, in comparison to the interictal period.

As presented in the introduction, the idea of analysingEEG recordings using FD methods for epileptic seizuredetection has been examined in a few past studies (Esteller et al

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(a)

(b)

Figure 12. (a) EEG data segment of patient 1 contaminated with muscle artefact due to chewing. (b) Overestimations (FD k-NN>2) in thek-NN FD time profile. A star indicates the FD value which is estimated based on the EEG values of the 2 s data window which starts at thepoint in time that the star lies. Dashed vertical lines in (a) mark the time window which produced the first overestimated FD value indicatedby a dashed vertical line in (b). A second overestimated value follows in the next window.

(a)

(b)

(c)

Figure 13. Performance of the seizure detection methodology using Katz’s, Higuchi’s and the k-NN FD estimation algorithms.(a) Sensitivity achieved for each FD algorithm using the generic threshold values Tg = 1.48 (Katz’s algorithm), 1.29 (Higuchi’s algorithm)and 1.27 (k-NN algorithm). In the bars, the number of detected seizures over the total number of seizures for each patient is depicted. (b)Mean values and standard deviations of DDs for the patients in the training and testing data sets. (c) FP rates for the patients in the trainingand testing data sets. Mean values for sensitivity, DD and FP rate for both the training and the testing data set are presented at the right endof each bar chart.

2001b, Accardo et al 1997, Bullmore et al 1994, and referencestherein). Nevertheless, to the best of our knowledge, noextensive study has been published to date on an FD estimationalgorithm for seizure onset detection applied on multi-dayscalp or intracranial EEG, including all different physiologicalstates that could exist in a long-term EEG monitoring setting(e.g. eating, sleeping, etc) as well as several possible artefacts.The main objective of this study was to examine and comparethe ability of different FD estimation algorithms to detect the

onset of epileptic seizures in such a context, achieving bothhigh sensitivity and specificity, accompanied by low DDs. Tothis end, three waveform FD estimation algorithms were used.

In order to pursue the goal of seizure onset detection,a careful and extensive evaluation of the algorithms utilizedusing synthetic data of known FD was found to be necessary.This evaluation was the second objective of the current study,and it was twofold. On the one hand, parameters involvedin the calculation of Higuchi’s and the k-NN algorithms were

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estimated based on minimization of the MSE of the FD usingthe synthetic data. To the best of our knowledge, such anapproach for selecting the parameters for both algorithms hasnot been presented in the literature before. The results of thecurrent study, however, emphasize the importance of theseparameters’ selection. On the other hand, the accuracy of thealgorithms and their robustness in the presence of noise wasassessed. A similar comparison between Katz’s and Higuchi’salgorithms, based on synthetic data, has been attempted in thepast (Esteller et al 2001b), but the current work is enhancedwith the inclusion of the k-NN algorithm in the comparison,which was found to be the most robust in the presence of noise.

The results of the current work indicate that fractalanalysis can indeed be useful for epileptic seizure onsetdetection with high sensitivity and specificity, but only whenusing the k-NN algorithm. Katz’s and Higuchi’s algorithmsfailed to produce systematic and specific changes in FD timeprofiles and showed non-satisfactory numerical results. Thereason that seizure onset detection was feasible in the currentstudy using a simple threshold (Tg) on the k-NN FD timeprofiles is twofold. On the one hand, the k-NN FD valuesstarted dropping immediately after the seizure onset in mostseizures. On the other hand, the separation between the k-NNFD values corresponding to seizures and those correspondingto non-seizure periods was big enough and allowed us to usea simple threshold to distinguish between the two, without theproduction of an unacceptable number of FPs.

Utilizing the k-NN algorithm as a time series analysismethodology in the current work brought out its potentialusefulness for other time series analysis applications. Its highdiscriminatory power, illustrated in the context of consistentlyidentifying the dynamics of the seizure state, indicates that itcan be applied to other signals besides the EEG (e.g. financialtime series) in order to provide a characterization of possiblydifferent states of the system generating the signal.

5.2. Comparison of FD algorithms based on scalp EEGanalysis

The main purpose of our work was to compare the suitabilityof different FD methods for detecting seizures of MTL origin.In that vein we provided evidence supporting the superiorityof the k-NN method, application of which resulted in 100%sensitivity accompanied by relatively low DD times andrelatively low FP rates. On the other hand, Katz’s methodproduced relatively low mean FP rates in both the testingand the training data sets, but failed to detect any seizures.Higuchi’s method did not detect all the seizures in the trainingdata set and detected only one seizure in the testing, and atthe same time produced a higher mean FP rate and DD ascompared to the results of the k-NN method in both the trainingand the testing data set. The success of the k-NN method couldbe attributed to its ability to produce lower FD values during theseizures even in the presence of higher frequency activity, incontrast to Katz’s and Higuchi’s methods. Therefore, the k-NNmethod is expected to be successful in detecting seizures withhigh-frequency content, such as those of neocortical origin(Worrell et al 2004).

As already mentioned, both Katz’s and Higuchi’salgorithms failed to produce consistent changes of a singletype (either a drop or an increase) that could provide asystematic and specific criterion indicative of seizures. It isworth noting that Katz’s algorithm, in general, produced lessdistinct changes in FD time profiles as compared to Higuchi’salgorithm. In quite a few cases, no clear change would beobserved using Katz’s algorithm. This probably explains thefailure of Katz’s algorithm to detect any seizures both in thetraining and the testing data set.

The FD time profiles produced based on both Katz’sand Higuchi’s methods, though, seemed to follow a similarunderlying trend: during seizures where Higuchi’s FD timeprofile would show a drop (or increase), something similar(but less pronounced) would happen for Katz’s FD time profile(figures 7 and 10). This could be attributed to the commonapproach behind FD estimation used in both algorithms, whichis estimation of the length of a curve. The superiority ofHiguchi’s algorithm might be related to a more accurate curve–length estimation or to its superior robustness in the presenceof noise.

On the other hand, the k-NN algorithm exhibited a FDdrop which was evident during most seizures of all patientsand was also specific to the seizures. This could be attributed tothe different underlying approach that the k-NN algorithm usesfor FD estimation, which is a fixed-mass approach, instead ofa curve–length estimation approach. The fixed-mass approachfor estimation might also be the reason that, when usingsynthetic signals contaminated with noise, the k-NN algorithmproved to be superior in terms of robustness of estimation, asillustrated in section 3.4 (figure 6).

Why would the FD time profiles using Katz’s andHiguchi’s algorithms show in some seizure cases an increaseinstead of a drop? Investigation of the frequency content of theEEG signal during those seizures revealed that this increasemight be associated with the presence of higher frequencycomponents in the EEG (above the alpha EEG rhythm range),which were not present during seizures for which a drop wasobserved. This observation is in agreement with a previousstudy using a line-length metric for seizure detection (Estelleret al 2001a), according to which the line-length metric grewas the data sequence frequency or magnitude increased. Suchhigher frequency components also appeared in our EEG dataset away from seizures, for instance due to artefacts or muscleactivity.

5.3. Seizure onset detection methodology

Due to the very different nature of scalp and intracranial EEG,we could only compare the results of the present study to thoseof seizure onset detection studies based on scalp EEG (Meieret al 2008, Saab and Gotman 2005, Shoeb et al 2004, Qu andGotman 1997). The system of Saab and Gotman (2005) aimedat detecting the onset of epileptic seizures in scalp EEG, basedon wavelet decomposition and Bayesian probabilities. Usinga threshold-‘tuning’ mechanism, they reported in their testingdata set (360 h of scalp EEG, which included 69 seizuresin 16 patients suffering from various epilepsy types) 76%

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sensitivity, a FP rate of 0.34 h−1 and a median DD of 10 s.The results without ‘tuning’ were 77.9%, 0.86 h−1 and 9.8 s,respectively. Our methodology based on the k-NN algorithm,applied to our testing data set, seems to compare favourablyin terms of sensitivity (100%), FP rate (0.42 h−1) and (mean)DD (8.82 s). However, a direct comparison might not beappropriate since, on the one hand, we only included seizuresof MTL origin in our study, and, on the other hand, we did notuse any techniques for automatic artefact rejection.

Saab and Gotman (2005), in addition to filtering thedata between 3 and 30 Hz, applied various artefact rejectiontechniques prior to automatic detection production, to confrontcommon sources of FP for their system, such as alpha EEGactivity, EMG and electrode failure of different kinds. Inour work, no artefact rejection techniques were applied, otherthan filtering the data between 3 and 30 Hz and makinguse of the parameter w. The most common sources ofFPs using our methodology were artefacts mainly causedby chewing, movement of the reference electrode or otherelectrode artefacts, and bursts of rhythmic EEG activity. In afew cases, sleep rhythmic EEG activity and activity of epilepticorigin caused some FPs. The latter, nevertheless, were notactual epileptic seizures. Therefore, they were categorizedas FPs. Inclusion of some artefact rejection techniques mightimprove the performance of the seizure detection methodologyof the current study.

In a recent study (Meier et al 2008), the authors’ attentionwas focused on detecting different seizure morphologies,rather than just seizures originating from different epilepsytypes. Using 91 seizures, representing the most commonictal morphologies, from 57 patients, they reported FP rates<0.5 h−1 (for specific ictal morphologies even <0.25 h−1),with average sensitivity >96% and very short DDs, of about1.6 ± 2.8 s. Nevertheless, they defined the seizure onset as ‘thebeginning of the first observable seizure pattern in the EEG’rather than ‘the time point at which the first EEG changesoccur which lead to a clear seizure discharge’ as defined in thecurrent study. Thus, direct comparisons in terms of DDs maynot be appropriate.

The authors in Qu and Gotman (1997) designed a systembased on a seizure template for each patient and achieved aseizure onset detection rate of 100%, with an average delayof 9.35 s after onset, accompanied by an FP rate of 0.02 h−1

(method evaluated in 12 patients with a total of 47 seizures). Intheir system, at least one seizure, as well as a broad variety ofthe patient’s background EEG patterns had to be available fortuning the method on each patient separately before actuallyapplying the method. In Shoeb et al (2004), the authorsutilized wavelet decomposition and support vector machines todetect 131 out of 139 studied seizures of different types (94%sensitivity) within 8.0 ± 3.2 s of seizure onset. In 60 h of EEG,15 false detections were declared (FP rate of 0.25 h−1). Thoseresults were fairly satisfactory, but their system required 2–4seizures to be a priori available for each patient, in addition tonon-seizure EEG segments separating the seizure occurrencesfor each patient.

The proposed seizure detection methodology does notrequire a priori information about the morphology of the

seizures analysed or about the interictal content of the EEGrecords of a patient. After an initial calculation of a genericthreshold Tg from the training data, the same threshold canbe applied to any patient and the value of this threshold isthe only information needed for producing ‘alarms’ using oneEEG channel. Nevertheless, the selection of the appropriateEEG channel to be used, as described in section 2.3, requiresthe recording of at least one seizure. Note though that thisdoes not affect the way the FD time profiles are generated.It only affects the post-processing of the FD time profiles for‘alarms’.

Saab and Gotman (2005) described a seizure detectionmethodology which was also based on the application of ageneric threshold. Application of their methodology to scalpEEG revealed that some patients had higher false detectionrates than others. In order to improve performance for thosepatients in a clinical setting, the authors described the ideaof properly ‘tuning’ the threshold in order not to exceed apredefined FP rate (see above for their results). This ‘tuning’could be applied to our seizure detection methodology. Thereasoning behind the applicability of a ‘tuning’ mechanism isthe following: first, different seizures, even if they all originatefrom MTL, can exhibit a wide range of morphologies. If theseizures of a patient are characterized by high rhythmicity,the corresponding FD values can be significantly lower ascompared to the interictal FD values and could, therefore,be detected using relatively low threshold values. However,seizures of not enough rhythmic content could be missed.Secondly, during the interictal period, there may exist differentrhythmic EEG patterns related or unrelated to epilepsy, whichcan lead, for some patients, to interictal FD values closer tothe generic threshold, thus causing a high number of FPs. The‘tuning’ mechanism can provide a means for ‘correcting’ forthe different characteristics of each patient.

It should be pointed out that direct comparison betweendifferent seizure onset detection algorithms would beappropriate only if the algorithms were to be applied on thesame data set. Differences in the length of data and typeof epilepsies under investigation, the variability of seizurepatterns, the presence of different uncontrollable technicalartefacts in the EEG, and even different recording settingsduring data acquisition could make a direct comparisonunfeasible.

5.4. Future work

For future work towards automatic seizure onset detection,application of our method to EEG data from more patients andof more epilepsy types is needed. Appropriate modificationof the k-NN FD estimation methodology as to includeinformation from multiple channels would address the issue ofinitialization which requires the selection of a single electrodechannel for the analysis to be performed. Utilization ofmultiple channels would also facilitate the inclusion of someartefact removal techniques. Additionally, analysing the EEGdata using different frequency bands could prove useful forsome types of seizure morphologies. Finally, since the k-NNFD estimation algorithm, using a simple thresholding

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operation, was shown to be quite a robust detector of seizures,its inclusion as a feature in more sophisticated seizure detectionsystems using, for instance, artificial neural networks, supportvector machines or other artificial intelligence tools (e.g. expertsystems) could possibly lead to improved performance ofseizure detection methodologies.

Acknowledgment

The work of GEP was supported by the Hellenic StateScholarships Foundation.

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