<title>1. Introduction</title> <p>Epilepsy is one of the most common
human brain disorders. It is often accompanied by disturbances in behavior,
brain dysfunction, and cognitive impairment. According to the World Health
Organization, 0.7% to 1% of the world's population suffer from epilepsy and
this generally peaks in childhood and advanced age, meaning that a large
proportion of patients have this chronic disease for most of their lives [##UREF##0##1##].
This supports the importance of identifying this population as early as
possible such that the clinician can prescribe the necessary medication to stop
its progression.</p> <p>Various studies have been carried out to promote
our understanding on the development of this disease and on how epileptic
subjects differ from normal subjects. Most of the work [##UREF##1##2##–##REF##7635101##7##] involves the analysis of
the electroencephalogram (EEG) which has given a boost to the diagnosis of
epilepsy. Complementary studies focusing on the analysis of the heart rate variability
have also been presented [##REF##12028480##8##].</p> <p>Literature indicates that various parametric [##REF##1473824##9##]
and nonparametric [##REF##11528297##10##] techniques have been applied to the analysis of epilepsy.
Feature extraction and detection methods for EEG signals range from frequency,
time-frequency, ARMA based models, to complexity systems modeling, neural networks, and expert systems [##UREF##2##11##–##REF##10337566##16##].
When analyzing EEG signals in the time domain, abnormal patterns such as spikes
and sharp waves are detected; while in the frequency domain, features from the
power spectrum are extracted [##UREF##1##2##, ##REF##16221562##3##, ##REF##7635101##7##].
When comparing seven state-of-the-art approaches for early seizure detection, Jerger
et al. showed that there are no significant differences between linear and
nonlinear methods [##REF##11528297##10##].</p> <p>Both in adult epilepsy and child epilepsy, most of
the published work is focussed on the seizure itself or on related events such
as the ictal, preictal, interictal, postictal parts of the
seizure and spikes [##UREF##6##17##].
Willoughby et al. [##UREF##1##2##] used
the spectral power at 1 Hz interval from 1 Hz to 100 Hz to show whether there are
significant differences in the interictal EEG of a group of patients with
partial generalized epilepsy. Their results showed that there is a persistent
increase in gamma EEG in the absence of epileptic discharges. Jerger et al. [##REF##11528297##10##]
have also focussed on interictal spike patterns to predict and ultimately
control seizure activity. Using twelve intracranially recorded seizures from
four patients, they compared the results of seven linear and nonlinear methods
including the analysis of power spectra, cross-correlation, principal
components, phase, wavelets, correlation integral, and mutual prediction and
showed that all methods were successful in indicating seizure onset before the
neurologist, for all but a few seizures. Blanco et al. [##UREF##2##11##]
applied a time-frequency analysis using the Gabor transform to analyze how the
traditional frequency rhythms of an EEG signal during an epileptic seizure
progress in time. By processing the intracranial recordings obtained for
different channels, frequency evolution series were obtained. A systematic calculation of the linear correlation of these series then allowed the
possibility of extracting information on how the EEG signals across different
regions are related. This together with visual assessment of the EEG and known
clinical patient history can aid in identifying the epileptic focus and provide
further insight on seizure dynamics.</p> <p>Patients suffering from epilepsy are most often
under the control of antiepileptic drugs (AEDs). The effect of these drugs can
also be a reason behind the resulting significant differences between epileptic
patients and controls. Salinsky et al. [##REF##12887435##6##]
and Tuunainen et al. [##REF##7635101##7##]
have both analyzed the effect of adult patients taking AEDs on the spectral
analysis of EEG. Salinsky used four occipital EEG measures including the peak
frequency, median frequency, relative theta, and delta power to study a group
of patients with low-seizure
frequency who were either a starting or a stopping AED therapy. A set of
cognitive tests and a structured EEG were performed before the AED change and
12–16 weeks after.
These results were compared with those of a healthy control group and with
patients receiving continuous long-term AED monotherapy. The results showed
that the EEG changes of the latter group were not significantly different from
the control group. The peak frequency was the most sensitive feature to AED
effect for those stopping or starting AEDs as compared to the healthy subjects.
For those stopping AEDs, the median frequency and the percentage theta power
also gave significant differences. Similarly, Tuunainen et al. used the absolute and relative power as well as the peak
power frequency at left occipital brain lobes as features to identify
differences between the patient and the control groups. Four-second long, artefact-free EEG epochs of subjects who were instructed to stay awake with their eyes open
were analyzed. In this case the results showed that the occipital peak alpha
frequency was significantly lower in patients than in controls. Furthermore,
the absolute power of the patient group over all electrode sites was significantly
higher at baseline than in controls for delta, theta, beta, and total activity.
Absolute alpha power was also found to be higher but this result was insignificant.</p> <p>The discrepancy between the EEGs of epileptic
subjects as compared to controls has been studied mostly in adults. Most of the
work focuses on the alpha band which is the dominant frequency in the human
scalp EEG of adults [##REF##10209231##5##].
Larsson and Kostov [##REF##16221562##3##], for example, considered three 10-second segments, at the beginning, after
hyperventilation, and at the end of their artefact-free EEG and analyzed the
alpha frequency in 18 epileptic patients as compared to 10 controls. In
particular, the peak alpha frequency (PAF) and alpha variations were used as
measures to differentiate between the two subject groups. When analyzing
children, however, one must keep in mind that the frequency spectrum might not
yet be well developed. It is well known that the alpha frequency increases
nonlinearly from early childhood to puberty and then starts to decline with
age. A seven years old child, for example, will probably show two different
peaks in the alpha and theta bands, respectively. This is because at such a
young age the alpha peak is still not well defined [##REF##10209231##5##]. Very few literature
works have applied the traditional techniques for detecting epilepsy on
children. Amongst them is the work of Hongou et al. [##REF##8517979##4##] who investigated the
development of the background EEG of 150 epileptic children by using spectral
analysis on the recordings from occipital regions. As compared to normal
children, their results reflected a significant increase in delta and theta
powers together with a decrease of upper alpha power in epileptic children. Different
types of epilepsies also resulted in different EEG development [##REF##8517979##4##].
It is therefore quite important to develop methods of high sensitivity and
specificity for early detection and categorization of various forms of epilepsy
especially in young children.</p> <p>In this paper, we address the possibility of
identifying changes in epileptic subjects versus control subjects at an early
stage, when just a few seizures occurred in the past. The epileptic population
consists of children selected from the pool of paediatric neurology outpatient
clinics of two hospitals in Heraklion, Crete,
where they were diagnosed and followed at regular intervals. It should be noted
that they were diagnosed with no psychological findings, they were not
suffering from severe epileptic syndromes and the visual inspection of their EEG
was normal. These children, referred to as controlled epileptic, were put under
scrutiny because of their early symptoms, without any detected brain damage; they
had one or more epileptic seizures in the past and some of them were under monotherapy
with drugs in low doses, without clinical side-effects.</p> <p>The EEG study of such children compared with
matched age controls is important from both the clinical and technical
perspectives. Thus, the purpose of this paper is twofold. First, we address the
question of whether controlled-epileptic children exhibit spectral differences
in their EEGs in comparison to an age-matched control group during the
performance of a control task and a mental task. Second, we address the
development of a sensitive and reliable measure for discrimination between the
two groups. According to our knowledge, such an analysis has not been carried
out so far. We compare two different approaches of localizing activity
differences and retrieving relevant information for classifying the two
populations. In particular, we elaborate the differences in classification results
obtained when using a nonparametric signal representation approach such as Fourier
transform or wavelets and a parametric signal modeling approach such as autoregressive
moving average (ARMA).</p> <p>The paper proceeds as follows. <xref ref-type="sec" rid="sec2">Section 2</xref> provides all related clinical
information for the patients being tested, details on the experimental
protocols used for the control and mental tasks considered, and a description
of the method used for extracting biomarkers which can be used for classifying
children with controlled epilepsy and controls. <xref ref-type="sec" rid="sec3">Section 3</xref> presents the results
obtained using a signal representation and a modeling approach and finally, <xref ref-type="sec" rid="sec4">Section
4</xref> presents the discussion of the findings and <xref ref-type="sec" rid="sec5">Section 5</xref> concludes the paper.</p>
<title>2. Subjects and Methods</title> <title>2.1. Subjects</title> <p>The studied population consisted of twenty children aged 9–13 (9 boys, 11
girls) with controlled epileptic seizures, but without any clinical or
laboratory findings of brain dysfunction, and twenty (age and sex) matched
controls on a volunteer basis. Inclusion criteria for patients and controls
consisted of the following: (a) age of 9–13 years old; (b)
normal intellectual potential (assessed with WISC-III); (c) absence of
neurological damage documented by
neurological evaluation for patients and controls and by brain CT and/or MRI
scan for patients; and (d) absence of psychiatric problems (based on parent's
interview). It should be noted that the EEG signals recorded in both groups were
visually evaluated as normal; and detailed clinical, laboratory, and
neuropsychological findings could not indicate any population differences; the
only clinical indication for the epileptic population was the medical diagnosis
of repeated epileptic seizures in the past (the last epileptic event was
diagnosed between a few weeks to 1 and half years before this study). These
children were treated using common antiepileptic medication only after they
exhibit at least two seizures or absences. The types of seizures diagnosed were the most common
ones in childhood: Rolandic epilepsy (4 children), idiopathic
generalized seizures (5 children), focal seizures (3 children), focal secondary
generalized seizures without detectable brain damage (6 children), and absence seizures (2 children). More specifically, the children with absences were
free from seizures from the beginning of the treatment with Depakine. The other forms such as
generalized tonic-clonic seizures or those with rolandic spikes had a history
of two to five episodes, which were prevented after the treatment with common therapeutic (low) dosages of Tegretol. Especially
in the case of the absences the treatment is effective from the very beginning
and these children were monitored, while treated, for one to two years. During
this period no seizures were identified. Absences and idiopathic tonic-clonic
seizures are generated from the brainstem, while rolandic seizures are generated from
the rolandic area [##UREF##7##18##].
Apart from eight children (4 generalized and 4 rolandic), the remaining twelve
children were treated using Tegretol or Depakine, in small doses without
clinical side effects, only after they exhibit at least two seizures or
absences. It should be noted that the dosages used for therapeutic purposes are
not linked to any known side-effects [##REF##16563808##19##]
and the related clinical reports did not diagnose any problems related with
visible anatomical damages.</p> <p>Patients and controls, all right-handed, were individually evaluated in
the clinical neurophysiology laboratory, at the Medical School of the University of Crete. 
All parents of children involved in the study signed a written consent form,
after having been informed about the study's purpose and the required
procedures. The study was approved by the Local Ethics Committee.</p> <title>2.2. Recordings</title> <p>Continuous EEGs were recorded in an electrically
shielded, sound and light attenuated room while participants sat in a reclined
chair. The EEG signals were recorded from 30 electrodes placed according to the
10/20 international system, referred to linked A1+A2 electrodes. This electrode
montage is shown in ##FIG##0##Figure 1##. The signals were amplified using a set of
Contact Precision Instrument amplifiers (Cambridge, Mass, USA, <ext-link ext-link-type="uri" xlink:href="http://www.psylab.com">http://www.psylab.com</ext-link>), filtered online
with a band pass between 0.1 and 200 Hz, and digitized at 400 Hz and 12 Bits. Offline,
the recorded data were carefully reviewed for technical and biogenic artefacts,
so that only artefact free epochs of 10.24-second duration were further investigated.
Experience obtained from our laboratory and many related to the field
publications suggest that this time interval is enough to extract the desired
features. Mathematical studies also accept durations ranging from 8 to 12
seconds as indicative of the underlying cognitive task. Artefacts were treated
visually by an expert, since many automated artefact removal algorithmic
methodologies, even if they are successful in removing certain types of
artefacts, they fail to leave physiological EEG intact. Thus, only signal
segments without visible artefacts (EOG,
EMG, movements) were preserved. For each subject only one representative epoch
is included in the data, following the process of intensive visual scrutiny.</p> <title>2.3. Test Description</title> <p>In this study, two different tasks were analyzed to identify differences
in brain dysfunction under tasks with different brain operations. During the
control task (Task 1) subjects were at rest and had their eyes fixed on a point
displayed on a computer screen to reduce eye artefacts. The second task was a
mathematical task (Task 2) involving the subtraction of two digit numbers [##UREF##8##20##],
displayed on an LCD screen located in front of the participants. Such a mental
task is considered to be difficult for the studied age group. Stimuli were
presented on an LCD screen. Vertical/horizontal eye movements and blinks were
monitored through a bipolar montage from the supraorbital ridge and the lateral
canthus. The analyzed epochs were acquired during the intensive calculation
phase.</p> <title>2.4. Methods</title> <p>The goal of this analysis is to find discriminating features between
epileptic and control children that result in high classification scores. Ideally,
a preprocessing step is used to filter out irrelevant data and enhance the
discriminating features of the signal. A subsequent spectral analysis step
could then be applied to extract those suitable biomarkers. This nonparametric approach,
which is labelled as approach (1) in ##FIG##1##Figure 2##, has its limitations. This is
mainly due to the fact that the nature and form of the most relevant
discriminative features that are to be preserved during the preprocessing step
are not known beforehand. In order to avoid degrading the useful part of the
signal, our process starts directly from the spectral analysis stage. As an
alternative to this scheme, we propose a parametric method that encapsulates
both the pre-processing stage and the spectral analysis stage (approach (2)).
Using an ARMA model, it is possible to model the dynamics of the EEG signal,
without necessarily preserving its detail. Thus, while the first approach
performs signal representation, the second approach performs signal modeling.</p> <p>In order to compare these different approaches in classifying the two
subject groups, both methods were implemented and the results obtained were similarly
analyzed. For the nonparametric approach both a global Fourier Transform (FT)
and wavelets were used for the spectral analysis stage and the biomarkers
extracted from each method were compared. The FT gives an average spectral plot
over the time period considered. On the other hand, wavelets are mathematical functions that divide the data into different
frequency components and then analyze each component with a resolution matched
to their scale. Thus, instead of working on a single time or frequency scale,
they work on a multiscale basis [##REF##15809098##21##].
Wavelets offer a tradeoff between time and frequency resolution but they are
superior to traditional FFT methods when it comes to analyzing data that
contains discontinuities and sharp spikes. In addition, the time-windowed
version of the wavelets offer a scheme that allows for further refinement of
the method in cases where time locked events might be important. When wavelets were
compared to the STFT technique [##REF##15809098##21##],
the results showed that the STFT is computationally faster but wavelets give
more accurate results especially in the detection of epileptic seizures and in
EEG signal classification. For these reasons we opted to use wavelets in
addition to the application of a global FT to extract power spectral features
within predefined frequency bands which are then used for classification
purposes.</p> <p>For the second approach, a time-frequency spectrum
is generated using the estimated parameters of the ARMA model derived from each
EEG signal. Parametric models are known to enhance the time-frequency
resolution of power spectra estimation [##REF##1473824##9##, ##REF##15000382##22##, ##UREF##9##23##]
as they suppress the leakage effect resulting from the used window function [##REF##15000382##22##]. Another
advantage of using such a technique is that the ARMA parameters are being
estimated at each time instant, thus allowing a more accurate representation of
nonstationarities. Once the time-frequency spectra are obtained, spectral features
are extracted and fed to the classifier to discriminate between the two subject
groups.</p> <title>2.4.1. Nonparametric Techniques</title> <p>
<statement id="head1"><title>Fourier transform (FT)</title><p>The FT transforms a signal in the time domain into
its frequency domain representation. By definition, a signal <italic>y</italic>(<italic>t</italic>) has a discrete Fourier transform <italic>Y</italic>(<italic>k</italic>) which is given by [##UREF##10##24##] The power spectral density <italic>S</italic> for such a signal is then estimated as shown
in (##FORMU##1##2##)
This spectrum is then used to extract biomarkers which are then fed
to a classifier to distinguish between the two populations. Biomarkers are found by calculating the total energy for each of six predefined frequency
bands which are the delta (0–4 Hz), theta (4–8 Hz), 
alpha (8–13 Hz), beta
(13–30 Hz), gamma1 (30–45 Hz), 
and gamma2 (45–90 Hz): where <italic>M</italic>
<sub><italic>B</italic></sub> is the biomarker at frequency band <italic>B</italic> and <italic>f</italic>
<sub><italic>s</italic></sub> is the sampling frequency.</p></statement>
</p> <p>
<statement id="head2"><title>Wavelet transform (WT)</title><p>Over the past decade, the WT has been developed into an important tool
for analysis of time series that contain nonstationary power at many different
frequencies (such as the EEG signal), and it has proved to be a powerful
feature extraction method [##UREF##11##25##, ##REF##16988882##26##].
In particular, it has been observed that the epileptic recruitment rhythm
during seizure development is well described in terms of the relative wavelet
energies [##REF##16675027##27##].
The WT is more suitable for analyzing transient signals because both frequency
(scales) and time information can be obtained in good resolution.</p><p>The continuous wavelet transform (CWT) was preferred in this work, so
that the time and scale parameters can be considered as continuous variables.
In the CWT, the notion of scale <italic>s</italic> is
introduced as an alternative to frequency, leading to the so-called time-scale
representation. The CWT of a discrete sequence <italic>x</italic>
<sub><italic>n</italic></sub> with time spacing <italic>δ</italic>
<italic>t</italic> and <italic>N</italic> data points (<italic>n</italic> = 0, 1,…, <italic>N</italic> − 1) is defined as the convolution of <italic>x</italic>
<sub><italic>n</italic></sub> with consecutive scaled
and translated versions of the wavelet function <italic>ψ</italic>
<sub>0</sub>(<italic>η</italic>):
 where <italic>η</italic> and <italic>ω</italic>
<sub>0</sub> indicate nondimensional
“time” and “frequency” parameters, respectively,  and (∗) indicates the complex conjugate. In
our application, <italic>ψ</italic>
<sub>0</sub>(<italic>η</italic>) describes the most commonly used
wavelet type for spectral analyses, that is, the normalized complex Morlet
wavelet as given in (##FORMU##4##5##). The frequency parameter <italic>ω</italic>
<sub>0</sub> is selected equal to 6 since it is a good tradeoff
between time and frequency localization for the Morlet wavelet. The wavelet
function <italic>ψ</italic>
<sub>0</sub> is normalized to
have unit energy at each scale, so that each scale is directly comparable to
each other [##UREF##11##25##].
The power spectrum of the WT is defined by the square of coefficients in
(##FORMU##3##4##) of the wavelet series as |<italic>W</italic>
<sub><italic>n</italic></sub>(<italic>s</italic>)|<sup>2</sup>. As previously noted, there exists a concrete relationship between each scale
and an equivalent set of Fourier frequencies, which for the Morlet wavelet used
in this study is given by <italic>f</italic> = 1/(1.03<italic>s</italic>) [##UREF##12##28##].
The scale set used is given by <italic>s</italic>
<sub><italic>j</italic></sub> = <italic>s</italic>
<sub>0</sub>2<sup><italic>j</italic><italic>δ</italic><italic>j</italic></sup>, <italic>j</italic> = 0,…, <italic>J</italic>,
where <italic>s</italic>
<sub>0</sub> = 2<italic>δ</italic>
<italic>t</italic> is the smallest scale chosen 
and <italic>δ</italic>
<italic>j</italic> specifies the width of the wavelet
function (in our case <italic>δ</italic>
<italic>j</italic> = 0.25,
meaning that there is a scale resolution of four suboctaves per octave). The
largest scale is determined by the value of <italic>J</italic> (in our case <italic>J</italic> = 29, which wraps all
six frequency bands of interest).</p><p>The first stage of the feature extraction method is
based on capturing the time-averaged power spectrum  for each electrode and scale, which is computed by averaging the power
spectrum |<italic>W</italic>
<sub><italic>n</italic></sub>(<italic>s</italic>)|<sup>2</sup> over time: Further averaging in scale is performed, in order to map a single
feature per frequency band of interest. Thus, the time-scale-averaged power
spectrum  is defined as the weighted sum of the time averaged wavelet power spectrum
(##FORMU##7##6##) over scales <italic>s</italic>
<sub><italic>j</italic><sub>1</sub></sub> to <italic>s</italic>
<sub><italic>j</italic><sub>2</sub></sub>: where <italic>C</italic>
<sub><italic>δ</italic></sub> is a
constant, scale independent factor used for the exact reconstruction of a <italic>δ</italic>(·) function from its wavelet transform
(for the Morlet wavelet it equals to 0.776) [##UREF##12##28##].
Finally, the time-scale-averaged power spectrum  for each of the six frequency bands specified earlier was then
calculated as a biomarker, as shown in (##FORMU##11##8##): where <italic>B</italic> indicates the
selected band.</p></statement>
</p> <title>2.4.2. Parametric Techniques</title> <p>
<statement id="head3"><title>Autoregressive moving average (ARMA) modeling</title><p>The autoregressive moving average (ARMA) or Box-Jenkins model is a
parametric model where the estimate of the time series at a time instant
depends on its past values (deterministic part) and on a random disturbance
(stochastic part) [##UREF##13##29##].
Parametric ARMA models might not result in exact signal reconstruction, but can
effectively capture the dynamics of the input process within their time-varying
parameters.</p><p>A parametric method can provide adequate spectral estimates only when the correct model
order is chosen. Various techniques have been developed to estimate the optimal
order, the most renowned being the Akaike's information criterion (AIC) [##UREF##14##30##]; however all techniques
are based on specific constraints and hence the choice of model order remains
questionable. In this analysis, a set
of preliminary tests were carried out to find the suitable order which
adequately discriminates between the two subject groups. Orders (5,2), (12,3),
(18,5), and (24,6) were tested. In ##FIG##2##Figure 3##, plots of the log power spectral
values at each frequency and time instant were plotted for the four different
model orders. For ARMA (5,2), the resolution of the spectral power is very low
(##FIG##2##Figure 3(a)##) and this results in poor discrimination between the two subject
groups. When the model order is increased to (12,3), more features become
apparent in the spectral plot (##FIG##2##Figure 3(b)##) and very low classification of the
two groups can be achieved. Higher performance was found with a model order of
(18,5). As shown in ##FIG##2##Figure 3(c)##, the resolution is substantially improved and
superior classification can be achieved, as will be shown later on in this
paper. Increasing the model order further gives no significant improvement and this can be seen in ##FIG##2##Figure 3(c)## where the spectral plot at (24,6) is very close to that obtained for an order of (18,5).</p><p>An ARMA (<italic>m</italic>, <italic>n</italic>) model [##REF##15000382##22##] is
used to model the EEG signals <italic>y</italic>(<italic>t</italic>) recorded at particular electrodes on
the scalp, which can be defined as where <italic>a</italic>
<sub><italic>t</italic></sub>
<sup>(<italic>j</italic>)</sup> and <italic>b</italic>
<sub><italic>t</italic></sub>
<sup>(<italic>k</italic>)</sup> are the AR and MA parameters at time instant <italic>t</italic>, respectively, <italic>m</italic> is the number of poles, <italic>n</italic> is the number of zeros, and <italic>e</italic> is a
white noise Gaussian process representing the observation error.</p><p>Let <italic>θ</italic>
<sub><italic>t</italic></sub> be the vector of ARMA parameters and let <italic>ψ</italic>
<sub><italic>t</italic></sub> be the regression vector made up of the <italic>m</italic> past signal values and the <italic>n</italic> past observation error values: The ARMA model in (##FORMU##12##9##) can then be rewritten as If random walk is allowed, the update of the parameter vector can be
defined as where <italic>ω</italic>
<sub><italic>t</italic></sub> is a normally distributed white noise process
with zero mean and covariance matrix <italic>Q</italic>.
The set of (##FORMU##14##11##) and (##FORMU##15##12##) represents the structure of a linear state-space
formulation, where the model parameters <italic>θ</italic>
<sub><italic>t</italic></sub> are also referred to as the states of the system. A Kalman smoother [##REF##15000382##22##] is
then used to find an optimal estimate of the time-varying model parameter
vector <italic>θ</italic>
<sub><italic>t</italic></sub>. The advantage of using a smoother rather than a filter is that since data is
not being processed in real time, future measurements can be used to find a
more accurate estimate of the system parameters at time <italic>t</italic>.</p><p>Once an estimate of the ARMA parameters <italic>a</italic>
<sub><italic>t</italic></sub>
<sup>(<italic>j</italic>)</sup> and <italic>b</italic>
<sub><italic>t</italic></sub>
<sup>(<italic>k</italic>)</sup> is available, an estimate of the power
spectral density can be found using the following equation [##REF##15000382##22##]: where <italic>σ</italic>
<sub><italic>ε</italic></sub>
<sup>2</sup>(<italic>t</italic>) is the prediction error variance, which in our
implementation is assumed to be equal to 1 and <italic>f</italic>
<sub><italic>s</italic></sub> is the sampling frequency. In this analysis,
the frequency resolution is set to 1 Hz and frequencies from 1 Hz up to 90 Hz were
analyzed. As a biomarker, the total energy over the entire time period for each
of the six frequency bands, respectively, is then calculated as shown in where <italic>M</italic>
<sub><italic>B</italic></sub> represents the biomarker for frequency band <italic>B</italic>, <italic>f</italic>
<sub>1</sub>, and <italic>f</italic>
<sub>2</sub> represent the range of frequencies falling
within band <italic>B</italic>, <italic>t</italic> = 1,…, <italic>T</italic> covers the entire length of data available, <italic>P</italic>
<sub><italic>t</italic></sub>(<italic>f</italic>) represents the power spectral density for
frequency <italic>f</italic> and time instant <italic>t</italic>, and <italic>f</italic>
<sub><italic>s</italic></sub> is the sampling frequency. These biomarkers are then used to discriminate between children with controlled epilepsy and control
subjects for each of the tasks performed.</p></statement>
</p> <title>2.5. Feature Selection</title> <p>This study proposes a statistical method for mining
the most significant lobes, resembling the way many clinical neurophysiological
studies evaluate the brain activation patterns. Since the goal is to find
significant differences between two groups, the independent two-sample <italic>t</italic>-test is used on the set of biomarkers selected before. The <italic>t</italic>-test assesses whether the means of two populations are
statistically different from each other and as a parametric test it assumes
that (i) data comes from normally distributed populations, (ii) data is measured
at least at the interval level (distance between points
on the scale is equal to all parts along the scale), (iii) variances of the populations involved are
homogenous; and (iv) all observations are mutually independent [##UREF##15##31##]. In
this analysis, the feature vectors for control subjects (<italic>F</italic>
<sub><italic>C</italic></sub>) and for 
epileptic subjects (<italic>F</italic>
<sub><italic>E</italic></sub>) consist of the biomarkers <italic>M</italic>
<sub><italic>B</italic></sub> 
which are the log-transformed values of the power
within a specific frequency band <italic>B</italic> for a particular channel ch. Thus,
the feature vectors are formed as where <italic>M</italic>
<sub><italic>B</italic>,ch</sub>
<sup><italic>C</italic>1</sup> represents the biomarker for control subject 1
(<italic>C</italic>1), within frequency band <italic>B</italic>, and for a particular channel ch. By using the D’Agostino Pearson test
[##UREF##15##31##]
or Kolmogorov-Smrinov's test [##UREF##16##32##],
the features were found to have a normal distribution, thus satisfying
assumption (i). Distance between points along the scale of the possible feature
values was equal at all parts of the scale, thus ensuring that data is measured
at least at the interval level (assumption (ii)). Homogeneity of variances was
tested using Levene's test based on the <italic>F</italic>-statistic
[##UREF##16##32##]
and in this case it was found that the features from the two populations did
not have equal variances. As this violates one of the above assumptions, the <italic>t</italic>-test had to be applied assuming
unequal variances (Behrens-Fisher problem). Finally,
since the biomarkers in <italic>F</italic>
<sub><italic>C</italic></sub> 
and <italic>F</italic>
<sub><italic>E</italic></sub> 
are coming from two independent populations (controls and epileptics) assumption (iv) is reasonable.</p> <p>The former statistical analysis technique was used to identify which
channels and frequency bands give significant differences between the epileptic
subject group and the control group for both the signal representation approaches
and the signal modeling approach.</p> <title>2.6. Classification</title> <p>In this study, the epileptic and control groups were classified by
using a strictly linear discriminant analysis (LDA) classifier based on a linear
discriminant function that fits a multivariate normal density to each group,
with a pooled estimate of covariance as implemented in the MATLAB statistics
toolbox [##UREF##17##33##].
This assumes that the groups can be separated by a linear combination of
features, where in the case of two features the boundaries between groups are
lines.</p>
<title>3. Results</title> <p>The capabilities of the methods described in <xref ref-type="sec" rid="sec2.4">Section 2.4</xref> were previously
tested on a simulated environment [##UREF##18##34##],
where the spectral content of a simulated signal with known spectral activity
was estimated. All methods successfully detected the characteristics of the
signal, thus proving that they are appropriate for the analysis of real
band-limited signals.</p> <p>During the tasks performed, neurological examinations showed that there
are no differences in achievement between children younger than 11 years old
and children in the age of 11 and
above. Therefore, in subsequent analyses the subjects were not divided into
different age groups. Both nonparametric and parametric approaches were then
applied to the real EEG data, where each signal was initially set to zero mean
and unit variance. In each case, we compute the channel/band significance, as
well as the corresponding classification scores with sensitivity-specificity
measures.</p> <p>
##FIG##3##Figure 4## illustrates the topographic maps of the log-transformed <italic>p</italic>-values between the two populations,
obtained for each method, task, and frequency band. Cells which have been left
blank indicate no significant difference at the 90% confidence interval (i.e., <italic>p</italic> &gt; 0.1). All shaded channel locations
represent a <italic>p</italic>-value less than 0.1,
with shades of red indicating the lowest <italic>p</italic>-values.
From these topographic maps, it is clear that for the control task (Task 1),
the ARMA model has effectively modeled the EEG signals as
to be able to extract spectral features that depict significant differences
between the epileptic children and the control group. Even if these differences
are attributed to drugs, they are still highly significant, with <italic>p</italic>-values being in the order of 10<sup>−19</sup>.
The topographic plots show that the gamma2 band (45–90 Hz) has <italic>p</italic>-values closer to the threshold, and
posterior channels are not found to be significant. In all other bands
significant differences are distributed over the entire brain region. From these latter frequency bands, the
occipital region reflects the least discrimination power (higher <italic>p</italic>-values) between the groups from all
significant channels. This is also illustrated in the topographic maps of
classification scores shown in ##FIG##4##Figure 5## where occipital channels gave the
lowest scores from all other significant channels. This figure also shows that
the nonparametric approaches, as compared to the ARMA approach, gave lower
classification scores. The global FT approach resulted in a larger number of
channels showing significant differences between the populations, whereas the
WT approach seems to be more selective (see ##FIG##3##Figure 4##). Left and right brain
areas dominated in most frequency bands but much weaker significance levels
were obtained when compared to the ARMA approach. In most cases frontal
channels were also found to be significant but one must keep note that frontal
channels may be affected by eye movements and can thus result in sporadic
discrimination. For this task and for the type of biomarker considered here, the
WT features in most bands do not show much significant discrimination. The
largest difference was found in frontal channels and in some posterior channels
within the alpha band. The global FT was also able to identify the same regions
as the WT did, but with higher <italic>p</italic>-values (much weaker significance levels) than the ARMA
counterpart. These results indicate the weakness of the nonparametric methods
over their parametric counterpart.</p> <p>The results for Task 2 show less discriminative differences, especially
when using the ARMA approach. WT succeeds in identifying weak spectral
differences within the alpha band (8–13 Hz) for a
number of channels in the left frontal area. Notice that in all cases the
discrimination levels achieved by either method are weak and do not support any
significant differentiation between the studied populations in this task. The
global FT method has also found the left brain area to show significant
differences between the epileptic and the control group especially in the alpha
and beta band, but once again the classification levels (##FIG##4##Figure 5##) are quite
low.</p> <p>The classification scores for both Tasks 1 and 2 are shown in ##FIG##4##Figure 5## as
topographic maps; the complete results in the form of graph bars are presented
in Figures ##FIG##5##6## and ##FIG##6##7##. A linear discriminant analysis (LDA) classifier with the
leave-one-out cross-validation scheme was implemented to derive the number of
correctly classified subjects. The topographic maps depicted in ##FIG##4##Figure 5## present
classification scores above 65% only. The topographic maps for classification
scores present strong similarities with those for statistical significance,
justifying that better discrimination of populations results in increased
classification ability. The results for Task 1 obtained using the ARMA approach
show that the highest scores occur over the left brain area for most bands and shift
more towards the posterior for the beta and gamma1 band. For the WT approach,
the topographic maps of classification scores show a concentration similar to
that for the <italic>p</italic>-values, where only the
alpha band shows a discriminating brain area between the two populations. FT
gives more channels within the different bands with scores greater than 65% but
similar to the WT, these scores are still significantly lower than those of
ARMA. For Task 2, the classification scores for both the nonparametric and the
parametric are sporadic. FT gave the largest number of locations within each
band that can possibly discriminate between the two populations but the
classification scores are still low and hence no conclusions can be derived
concerning population differences for this more complex mathematical task.</p> <p>Both Figures ##FIG##5##6## and ##FIG##6##7## show the scores for all channels together with the
sensitivity and specificity measures for the two approaches. These results indicate
more clearly that for the control task the classification scores for the ARMA parametric
approach are significantly higher than those of the nonparametric approaches
for all channels within the delta to the gamma1 band. With this technique,
classification scores up to 100% were achieved on most channels (except gamma2 band). The
sensitivity and specificity values for the ARMA technique are close to 100%
implying that the ARMA approach can detect practically all epileptic and
control subjects.</p> <p>For Task 2, the classification scores are much lower and the differences
between the parametric and nonparametric approaches are not as clear as for
Task 1. The highest classification score of 80% was obtained by the WT approach
over the frontal channels within the alpha band, particularly over Fp1, which
was also found to be significant (<italic>p</italic> &lt; 0.1)
for this approach. The average classification score over all channels in the
different frequency bands was in the range of 50%–60% but as shown
earlier in the topographic maps and classification scores, the results for this
task are quite random and hence nonconclusive.</p> <p>In order to identify further differences in the feature distributions of
the two subject groups, probability density estimates of the feature values of
the patients and controls over different frequency bands and channel locations
were also computed. ##FIG##7##Figure 8## shows the distributions of the feature values for
Task 1 obtained over channel Cz when considering the delta band. The density
plots for the control and epileptic subjects, for both the FT approach (##FIG##7##Figure 8(a)##) and the WT approach (##FIG##7##Figure 8(b)##), respectively, overlap significantly;
hence the low-classification scores obtained using these nonparametric
techniques. The results for ARMA, however, reached 100% classification over
this channel and this is marked by the biomarker distribution plots (##FIG##7##Figure 8(c)##)
which in this case are clearly separable. These density plots also show that the
control group has a lower power spectral mean than the epileptic group. When the
ARMA biomarkers obtained for the control task for the two subject
groups were averaged across channels and compared, the results showed that
epileptic patients have higher spectral power in all bands except the gamma2
band where the result was found to be insignificant.</p> <p>The set of
plots shown in ##FIG##8##Figure 9## describe the variations in the probability density
functions of epileptics and controls, respectively, across the six different
frequency bands considered. For brevity, only the results across channel FCz for
Task 1 are shown. The plots for the parametric technique (Figures ##FIG##8##9(e)## and ##FIG##8##9(f)##)
show that the biomarker for the epileptic subjects is generally of higher value
than that of the control subjects. Furthermore the spread of the density
estimates is also slightly higher for epileptics than controls. For the nonparametric techniques (Figures ##FIG##8##9(a)##–##FIG##8##9(d)##) this trend is
not clearly visible and within practically all bands there is a substantial
overlap between the plots of the two populations. This distribution of
biomarker values explains the classification scores discussed earlier.</p>
<title>4. Discussion</title> <p>This work considers methods for the discrimination of two
groups of age-matched children, that is, controls and children with controlled
epilepsy. Initial clinical and psychological examinations, as well as visual
EEG inspection, do not provide any information leading to possible differences.
On the original EEG data we apply two types of methodologies, one based on direct
signal representation (through nonparametric techniques, mainly the Fourier transform
and the wavelet transform) and the other at modeling the signal dynamics
(through a parametric ARMA model). The spectral features extracted by these
methods in each channel and spectral band are examined through significance
tests, classification accuracies, and statistical distributions of biomarkers. This
work indicates that parametric modeling of the EEG dynamics provides better
representation of the significant EEG content than nonparametric
techniques for feature extraction. The features extracted by the ARMA model for
the control task provide higher discrimination power than those extracted by
the Fourier transform and wavelet approaches. The results for the Fourier transform
have shown to be slightly superior than those of the wavelet transform in this case where the biomarker is an
average of the spectral power over the whole period of data. This may be the
cause of artefacts introduced by the windowing leakage effect of wavelets which
is less dominant in the global Fourier transform approach where a single window
was considered. In other situations
where the temporal resolution is taken into consideration, it is expected that
wavelets outperform the Fourier transform technique.</p> <p>Comparing the
control and math tasks, the methods derive significant differences during the
control (rest) task, but they are unable to identify any consistent differences
during the more demanding mathematical task where the discrimination of the specific
brain dysfunction seems more difficult.</p> <p>The potential clinical benefit of this work is the analysis of EEG data
towards the identification of children with mild epilepsy at early stages, where
classical, neurological, and clinical examinations and detailed psychological
and neuropsychological testing are unable
to identify any signs of brain dysfunction. The ARMA results show that
epileptic children during the control task have higher activity in frequency
bands up to the gamma1 band, but this activity becomes similar in both groups
for frequencies within the gamma2 band. When analyzing an adult-patient group,
Tuunainen et al. [##REF##7635101##7##]
have also found higher spectral activity in epileptic patients. In particular,
their results showed that the absolute power of the epileptic group
over all electrode sites was significantly higher at baseline than in controls
for delta, theta, beta, and total activity. Absolute alpha power was also
higher but this was not found to be significant. In a similar
study on adult patients, Willoughby et al. [##UREF##1##2##]
illustrated that patients with partial generalized epilepsy have higher
power at 3–7 Hz centrally,
15–17 Hz anteriorly
and over 25 Hz in all channels. Finally, in a preliminary study on children,
increased spectral power was also found in the theta and alpha bands [##UREF##19##35##]. The increased spectral power can be attributed either to the age-group being analyzed , or the type of epilepsy and the
level of brain dysfunction, or to drug effects. AEDs are known to result in
higher power within the lower-frequency
bands [##UREF##1##2##]. But
since in this analysis the drugs provided were in low dosage, it is most
probable that the differences are signs of brain dysfunction for such a typical
child population, which in turn fade out when
intense mental thinking (during the mathematical task) takes over. Furthermore,
if AEDs do influence the EEG signal structure (because
of high dosages), they will be identified as a consistent pattern apparent in every electrode. No such effect
is known to occur with the AEDs used in our case and, if any, it would be
expected to be diffused and affecting only slow waves [##UREF##20##36##].</p>
<title>5. Conclusion</title> <p>This work involved the study of children with mild epilepsy who had
epileptic seizures in the past but who did not exhibit any clinical,
physiological, or visible neurophysiological symptoms during the study. The
goal of this analysis was to develop reliable techniques to test if such
controlled epileptic conditions induce related spectral differences in the
EEG. The results show that parametric
ARMA modeling techniques extract more reliable biomarkers than the
nonparametric Fourier and wavelet transform techniques implemented here. For
the control task, the ARMA technique led to classification scores up to 100%
across all channels for frequency bands ranging from the delta to the gamma1 band.</p> <p>Diagnosis of epilepsy was here conducted by considering biomarkers on an average of the spectral power over the whole 10.24-second period of
data available. Future work will investigate whether taking into account the
temporal information enhances these classification scores specifically for the math
task where the complexity of the task made it difficult to capture any brain
dysfunction through global biomarkers. In the latter case it is expected that wavelets
will outperform the Fourier transform technique and lead to results which are
comparable to its parametric counterpart.</p>
<p>Recommended by Hiroshige Takeichi</p> <p>There is an important evidence of differences in the EEG frequency spectrum of control subjects as compared to epileptic subjects. In particular, the study of children presents difficulties due to the early stages of brain development and the various forms of epilepsy indications. In this study, we consider children that developed epileptic crises in the past but without any other clinical, psychological, or visible neurophysiological findings. The aim of the paper is to develop reliable techniques for testing if such controlled epilepsy induces related spectral differences in the EEG. Spectral features extracted by using nonparametric, signal representation techniques (Fourier and wavelet transform) and a parametric, signal modeling technique (ARMA) are compared and their effect on the classification of the two groups is analyzed. The subjects performed two different tasks: a control (rest) task and a relatively difficult math task. The results show that spectral features extracted by modeling the EEG signals recorded from individual channels by an ARMA model give a higher discrimination between the two subject groups for the control task, where classification scores of up to 100% were obtained with a linear discriminant classifier.</p>
<title>Acknowledgment</title> <p>This work was supported in part by the EC-IST project Biopattern, Contract
no. 508803, and by the internal research grant of the University of Malta LBA-73-967.</p>
<fig id="fig1" position="float"><label>Figure 1</label><caption><p>Electrode montage consisting of 30 electrodes placed according to the 10/20
international system.</p></caption></fig> <fig id="fig2" position="float"><label>Figure 2</label><caption><p>(1) Nonparametric and (2) parametric approaches for feature extraction and classification.</p></caption></fig> <fig id="fig3" position="float"><label>Figure 3</label><caption><p>Time-frequency plots of log-power distribution for ARMA model orders (5,2), (12,3), (18,5), and (24,6) are shown.</p></caption></fig> <fig id="fig4" position="float"><label>Figure 4</label><caption><p>Topographic maps showing the <italic>p</italic>-values for Task 1. The black dots in each image
represent the channel locations. Lower <italic>p</italic>-values are indicated in shades of
red while <italic>p</italic>-values close to the threshold of 0.1 are indicated in shades of
blue. Blank areas within each topographic map indicate that the features
extracted from that particular channel do not give significant differences
between the two populations (<italic>P</italic> &gt; 0.1).</p></caption></fig> <fig id="fig5" position="float"><label>Figure 5</label><caption><p>Topographic maps showing the classification scores above 65% over all 27 channels.</p></caption></fig> <fig id="fig6" position="float"><label>Figure 6</label><caption><p>Classification scores, sensitivity, and specificity results for Task 1.</p></caption></fig> <fig id="fig7" position="float"><label>Figure 7</label><caption><p>Classification scores, sensitivity and specificity results for Task 2.</p></caption></fig> <fig id="fig8" position="float"><label>Figure 8</label><caption><p>Power spectral feature distributions
for control and epileptic subjects over channel Cz. Features were extracted
from the delta band when the subjects were performing the control (rest)
Task 1: (a) shows the results for the FT approach, (b) shows the results
for the WT approach, and (c) shows the results for the ARMA approach.</p></caption></fig> <fig id="fig9" position="float"><label>Figure 9</label><caption><p>Power spectral feature distributions
for epileptics and controls comparing the differences between the different
frequency bands. (a) and (b) show the results for FT, (c) and (d) show the
results for wavelets, and (e) and (f) show the results for ARMA.</p></caption></fig>
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<disp-formula id="eq3"><label>(3)</label><mml:math id="M3"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mrow><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo></mml:mrow><mml:mrow><mml:mn>10</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo minsize="1em" maxsize="1em">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:mi>S</mml:mi><mml:mo>⋅</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>S</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo minsize="1em" maxsize="1em">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula> <disp-formula id="eq4"><label>(4)</label><mml:math id="M4"><mml:mrow><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mtd><mml:mtd><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mstyle><mml:mrow><mml:munderover><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo minsize="1.5em" maxsize="1.5em">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>δ</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo minsize="1.5em" maxsize="1.5em">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo minsize="1.5em" maxsize="1.5em">[</mml:mo><mml:mrow><mml:mo minsize="1em" maxsize="1em">(</mml:mo><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo> </mml:mo><mml:mo>−</mml:mo><mml:mo> </mml:mo><mml:mi>n</mml:mi><mml:mo minsize="1em" maxsize="1em">)</mml:mo><mml:mfrac><mml:mrow><mml:mi>δ</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo minsize="1.5em" maxsize="1.5em">]</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mstyle><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula> <disp-formula id="eq5"><label>(5)</label><mml:math id="M5"><mml:mrow><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mtd><mml:mtd><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mi>η</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula> <inline-formula><mml:math id="M6"><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msqrt></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M7"><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>2</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula> <disp-formula id="eq6"><label>(6)</label><mml:math id="M8"><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>2</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mo minsize="1.5em" maxsize="1.5em">(</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo minsize="1.5em" maxsize="1.5em">)</mml:mo><mml:munderover><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mrow><mml:msup><mml:mrow><mml:mo minsize="1em" maxsize="1em">|</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo minsize="1em" maxsize="1em">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula> <inline-formula><mml:math id="M9"><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>2</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula> <disp-formula id="eq7"><label>(7)</label><mml:math id="M10"><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>2</mml:mi></mml:mrow></mml:msubsup><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mo minsize="1.5em" maxsize="1.5em">(</mml:mo><mml:mfrac><mml:mrow><mml:mi>δ</mml:mi><mml:mi>j</mml:mi><mml:mi>δ</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>δ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo minsize="1.5em" maxsize="1.5em">)</mml:mo><mml:mstyle><mml:mrow><mml:munderover><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:mrow><mml:mo minsize="1.5em" maxsize="1.5em">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo minsize="1em" maxsize="1em">|</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo minsize="1em" maxsize="1em">(</mml:mo><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo minsize="1em" maxsize="1em">)</mml:mo></mml:mrow><mml:mo minsize="1em" maxsize="1em">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo minsize="1.5em" maxsize="1.5em">)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula> <inline-formula><mml:math id="M11"><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>2</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula> <disp-formula id="eq8"><label>(8)</label><mml:math id="M12"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mrow><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo></mml:mrow><mml:mrow><mml:mn>10</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo minsize="1em" maxsize="1em">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>2</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo minsize="1em" maxsize="1em">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula> <disp-formula id="eq9"><label>(9)</label><mml:math id="M13"><mml:mrow><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mo>−</mml:mo><mml:mstyle><mml:mrow><mml:munderover><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mrow></mml:mstyle><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:mstyle><mml:mrow><mml:munderover><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:msubsup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mrow></mml:mstyle><mml:msub><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula> <disp-formula id="eq10"><label>(10)</label><mml:math id="M14"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mo minsize="1em" maxsize="1em">[</mml:mo><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo minsize="1em" maxsize="1em">]</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mo minsize="1em" maxsize="1em">[</mml:mo><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo minsize="1em" maxsize="1em">]</mml:mo><mml:mn>.</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula> <disp-formula id="eq12"><label>(11)</label><mml:math id="M15"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>ψ</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msubsup><mml:mi>θ</mml:mi><mml:mi>t</mml:mi><mml:mi>T</mml:mi></mml:msubsup><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula> <disp-formula id="eq13"><label>(12)</label><mml:math id="M16"><mml:mrow><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo> </mml:mo><mml:mo>+</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula> <disp-formula id="eq14"><label>(13)</label><mml:math id="M17"><mml:mrow><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo minsize="1.25em" maxsize="1.25em">|</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:msubsup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>k</mml:mi><mml:mi>f</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo minsize="1.25em" maxsize="1.25em">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo minsize="1.25em" maxsize="1.25em">|</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:msubsup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>j</mml:mi><mml:mi>f</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo minsize="1.25em" maxsize="1.25em">|</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula> <disp-formula id="eq19"><label>(14)</label><mml:math id="M18"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mrow><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo></mml:mrow><mml:mrow><mml:mn>10</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo minsize="1.75em" maxsize="1.75em">[</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mstyle><mml:mrow><mml:munderover><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:mrow><mml:mstyle><mml:mrow><mml:munderover><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>⋅</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:mrow></mml:mstyle></mml:mrow><mml:mo minsize="1.75em" maxsize="1.75em">]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula> <disp-formula id="eq16"><label>(15)</label><mml:math id="M19"><mml:mrow><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>C</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mo minsize="1em" maxsize="1em">[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mtext>ch</mml:mtext></mml:mrow><mml:mrow><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mtext>ch</mml:mtext></mml:mrow><mml:mrow><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mtext>ch</mml:mtext></mml:mrow><mml:mrow><mml:mi>C</mml:mi><mml:mn>20</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo minsize="1em" maxsize="1em">]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo> </mml:mo><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mo minsize="1em" maxsize="1em">[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mtext>ch</mml:mtext></mml:mrow><mml:mrow><mml:mi>E</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mtext>ch</mml:mtext></mml:mrow><mml:mrow><mml:mi>E</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mtext>ch</mml:mtext></mml:mrow><mml:mrow><mml:mi>E</mml:mi><mml:mn>20</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo minsize="1em" maxsize="1em">]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula>
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[{"label": ["1"], "surname": ["Olesen", "Baker", "Freund"], "given-names": ["J", "MG", "T"], "article-title": ["Consensus document of European brain research"], "italic": ["Journal of Neurology, Neurosurgery, and Psychiatry"], "year": ["2006"], "volume": ["77"], "issue": ["supplement 1"], "fpage": ["1"], "lpage": ["49"]}, {"label": ["2"], "surname": ["Willoughby", "Fitzgibbon", "Pope"], "given-names": ["JO", "SP", "KJ"], "article-title": ["Persistent abnormality detected in the non-ictal electroencephalogram in primary generalised epilepsy"], "italic": ["Journal of Neurology, Neurosurgery, and Psychiatry"], "year": ["2003"], "volume": ["74"], "issue": ["1"], "fpage": ["51"], "lpage": ["55"]}, {"label": ["11"], "surname": ["Blanco", "Kochen", "Rosso", "Salgado"], "given-names": ["S", "S", "OA", "P"], "article-title": ["Applying time-frequency analysis to seizure EEG activity"], "italic": ["IEEE Engineering in Medicine and Biology Magazine"], "year": ["1997"], "volume": ["16"], "issue": ["1"], "fpage": ["64"], "lpage": ["71"]}, {"label": ["12"], "surname": ["Williams", "Zaveri", "Sackellares"], "given-names": ["WJ", "HP", "JC"], "article-title": ["Time-frequency analysis of electrophysiology signals in epilepsy"], "italic": ["IEEE Engineering in Medicine and Biology Magazine"], "year": ["1995"], "volume": ["14"], "issue": ["2"], "fpage": ["133"], "lpage": ["143"]}, {"label": ["13"], "surname": ["Kalayci", "Ozdamar"], "given-names": ["T", "O"], "article-title": ["Wavelet preprocessing for automated neural network detection of EEG spikes"], "italic": ["IEEE Engineering in Medicine and Biology Magazine"], "year": ["1995"], "volume": ["14"], "issue": ["2"], "fpage": ["160"], "lpage": ["166"]}, {"label": ["15"], "surname": ["Shamsollahi", "Senhadji", "Le Bouquin-Jeannes"], "given-names": ["MB", "L", "R"], "article-title": ["Detection and localization of complex SEEG patterns in epileptic seizures using time-frequency analysis"], "conf-name": ["In: Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis"], "conf-date": ["June 1996"], "conf-loc": ["Paris, France"], "fpage": ["105"], "lpage": ["108"]}, {"label": ["17"], "surname": ["Niedermeyer", "Niedermeyer", "Lopes da Silva"], "given-names": ["E", "E", "F"], "article-title": ["Epileptic seizure disorders"], "italic": ["Electroencephalography: Basic Principles, Clinical Applications, and Related Fields"], "year": ["1998"], "edition": ["4th edition"], "publisher-loc": ["Philadelphia, Pa, USA"], "publisher-name": ["Lippincott Williams & Wilkins"], "fpage": ["476"], "lpage": ["585"]}, {"label": ["18"], "surname": ["Wolf"], "given-names": ["P"], "italic": ["Epileptic Seizures and Syndromes"], "year": ["1994"], "publisher-loc": ["London, UK"], "publisher-name": ["John Libbey & Company"]}, {"label": ["20"], "surname": ["von Aster", "Weinhold"], "given-names": ["M", "M"], "italic": ["Neuropsychologische Testbatterie f\u00fcr Zahlenverarbeitung und Rechnen bei Kindern (ZAREKI)"], "year": ["2002"], "publisher-loc": ["Mannedorf, Switzerland"], "publisher-name": ["Swets Test Services"]}, {"label": ["23"], "surname": ["Aboy", "McNames", "Marquez", "Hornero", "Thong", "Goldstein"], "given-names": ["M", "J", "OW", "R", "T", "B"], "article-title": ["Power spectral density estimation and tracking of nonstationary pressure signals based on Kalman filtering"], "conf-name": ["In: Proceedings of the 26th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEMBC '04), vol. 1"], "conf-date": ["September 2004"], "conf-loc": ["San Francisco, Calif, USA"], "fpage": ["156"], "lpage": ["159"]}, {"label": ["24"], "surname": ["Bracewell"], "given-names": ["RN"], "italic": ["The Fourier Transform and Its Applications"], "year": ["2000"], "edition": ["3rd edition"], "publisher-loc": ["New York, NY, USA"], "publisher-name": ["McGraw-Hill"], "comment": ["Electrical Engineering Series."]}, {"label": ["25"], "surname": ["Burrus", "Copinath", "Gao"], "given-names": ["CS", "RA", "H"], "italic": ["Introduction to Wavelets and Wavelet Transforms: A Primer"], "year": ["1998"], "publisher-loc": ["Englewood-Cliffs, NJ, USA"], "publisher-name": ["Prentice-Hall"]}, {"label": ["28"], "surname": ["Torrence", "Compo"], "given-names": ["C", "GP"], "article-title": ["A practical guide to wavelet analysis"], "italic": ["Bulletin of the American Meteorological Society"], "year": ["1998"], "volume": ["79"], "issue": ["1"], "fpage": ["61"], "lpage": ["78"]}, {"label": ["29"], "comment": ["September 2006, "], "ext-link": ["http://www.documents.wolfram.com/applications/timeseries/UsersGuidetoTimeSeries/1.2.1.html"]}, {"label": ["30"], "surname": ["Priestley"], "given-names": ["MB"], "italic": ["Spectral Analysis and Time Series Vol. 1"], "year": ["1981"], "publisher-loc": ["New York, NY, USA"], "publisher-name": ["Academic Press"], "series": ["Univariate Series"]}, {"label": ["31"], "surname": ["Zar"], "given-names": ["JH"], "italic": ["Biostatistical Analysis"], "year": ["1999"], "publisher-loc": ["Englewood Cliffs, NJ, USA"], "publisher-name": ["Prentice-Hall"]}, {"label": ["32"], "surname": ["Field"], "given-names": ["AP"], "italic": ["Discovering Statistics Using SPSS"], "year": ["2005"], "edition": ["2nd edition"], "publisher-loc": ["London, UK"], "publisher-name": ["SAGE"]}, {"label": ["33"], "surname": ["Krzanowski"], "given-names": ["WJ"], "italic": ["Principles of Multivariate Analysis"], "year": ["1988"], "publisher-loc": ["Oxford, UK"], "publisher-name": ["Oxford University Press"]}, {"label": ["34"], "surname": ["Sakkalis", "Zervakis", "Bigan"], "given-names": ["V", "M", "C"], "article-title": ["Validation of time-frequency and ARMA feature extraction methods in classification of \nmild epileptic signal patterns"], "conf-name": ["In: Proceedings of the International Conference on Information Technology in Biomedicine (ITAB '06)"], "conf-date": ["October 2006"], "conf-loc": ["Ioannina, Greece"]}, {"label": ["35"], "surname": ["Martin-Fiori"], "given-names": ["E"], "article-title": ["Thalamo-cortical Oscillation during Brain Development and in Epilepsy"], "comment": ["Web presentation of a project funded by University Children's Hospital, \nOlga-Mayenfisch Foundation, "], "ext-link": ["http://www.forschungsportal.ch/unizh/media/pdf/p8421.pdf"]}, {"label": ["36"], "surname": ["Macdonald"], "given-names": ["RL"], "article-title": ["Antiepileptic drug actions"], "italic": ["Epilepsia"], "year": ["1989"], "volume": ["30"], "issue": ["supplement 1"], "fpage": ["S19"], "lpage": ["S28"], "comment": ["discussion S64-S68"]}]
Comput Intell Neurosci. 2008 Aug 4; 2008:462593