EEG Signal Processing for Epilepsy

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EEG Signal Processing for Epilepsy
Carlos Guerrero-Mosquera1, Armando Malanda Trigueros2
and Angel Navia-Vazquez1
1University Carlos III of Madrid, Signal Theory and Communications Department
Avda, Universidad, 30 28911 Leganes
2Public University of Navarre, Electrical and Electronic Engineering Department
Campus Arrosadia, 31006 Pamplona
Spain
1. Introduction
Neural activity in the human brain starts from the early stages of prenatal development. This
activity or signals generated by the brain are electrical in nature and represent not only the
brain function but also the status of the whole body.
At the present moment, three methods can record functional and physiological changes within
the brain with high temporal resolution of neuronal interactions at the network level: the
electroencephalogram (EEG), the magnetoencephalogram (MEG), and functional magnetic
resonance imaging (fMRI); each of these has advantages and shortcomings. MEG is not
practical for experimental work when subjects may move freely, because of the large size
of magnetic sensors. For image sequences, fMRI has a time resolution very low and many
types of EEG activities, brain disorders and neurodegenerative diseases cannot be recorded.
On the other hand the spatial resolution of the EEG is limited to the number of electrodes, as
described in Ebersole & Pedley (2003); Sanei & Chambers (2007).
Much effort has been made to integrate information of multiple modalities during the
same task in an attempt to establish an alternative high-resolution spatiotemporal imaging
technique. The EEG provides an excellent tool for the exploration of network activity in
the brain associated to synchronous changes of the membrane potential of neighboring
neurons. Understanding of neuronal functions and neurophysiological properties of the brain
together with the mechanisms underlying the generation of biosignals and their recordings is
important in the detection, diagnosis, and treatment of brain disorders.
Cerebral sources of electroencephalography potentials are three-dimensional volumes of
cortex. These sources produce three-dimensional potential fields within the brain. From
the surface of the scalp, these can be recorded as two-dimensional fields of time-varying
voltage. The physical and functional factors that determine the voltage fields that these
sources produce could be appreciated in order to locate and characterize cortical generators
of the EEG.
Electroencephalography enables clinician to study and analyze electrical fields of brain
activity recorded with electrodes placed on the scalp, directly on the cortex (e.g., with subdural
electrodes), or within the brain (with depth electrodes). For each type of recordering, the
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specialist attempts to determine the nature and location of EEG patterns and whether they
correspond to normal or abnormal neural activity.
In this chapter will introduce several typical methods in which EEG signal pre-processing
and processing in EEG signals with epilepsy. The chapter is organized as follows: Section
2 presents a brief outline of electroencephalography, Section 3 introduces to EEG waveform
analysis, Section 4 is an overview of different alternatives in EEG signal modeling and feature
extractions, Section 5 presents the state of art in EEG epilepsy detection and classification,
Section 6 shows different methods to dimensionality reduction for EEG signals and Section 7
gives a summary and conclusions of this chapter.
2. Outline of electroencephalography
The nervous system is an organ system containing a network of specialized cells called
neurons that gathers, communicates, and processes information from the body and send
out both internal and external instructions that are handled rapidly and accurately. In most
animals the nervous system is divided in two parts, the central nervous system (CNS) and
the peripheral nervous system (PNS). CNS contain the brain and the spinal cord, and the
PNS consists of sensory neurons, grouping of neurons called ganglia, and nerves cells that are
interconnected and also connect to the CNS. The two systems are closely integrated because
sensory input from the PNS is processed by the CNS, and responses are sent by the PNS to the
organs of the body. Neurons transmit electrical potentials to other cells along thin fibers called
axons, which cause chemicals called neurotransmitters that permit the neuronal function
called synapses. These electrical potentials, called as “action potentials” is the information
transmitted by a nerve that, in one cell, cause the production of action potentials in another
cell at the synapse. A potential of 60-70 mV with some polarity may be recorded under the
membrane of the cell body. This potential changes with variations in the synaptic process.
In this sequence, the first cell to produce actions potentials is called the presynaptic cell,and
the second cell, which responds to the first cell across the synapse, is called the postsynaptic
cell. Presynaptic cells are typically neurons, and postsynaptic cells are typically other neurons,
muscle cells, or gland cells. A cell that receives a synaptic signal may be inhibited, excited or
otherwise modulated. The Fig.1 shows the synaptic activities schematically.
The CNS is a major site for processing information, initiating responses, and integrating
mental processes. It is analogous to a highly sophisticated computer with the ability to
receive inputs, process and store information, and generate responses. Additionally, it can
produce ideas, emotions, and other mental processes that are not automatic consequences of
the information input.
2.1 Neural activities
Cells of the nervous system include neurons and nonneural cells. Neurons or nerve cell
communicate information to and from the brain. They are organized to form complex
networks that perform the functions of the nervous systems. All nerve cells are collectively
referred to as neurons although their size, shape, and functionality may differ widely.
Neurons can be classified with reference to morphology or functionality. Using the latter
classification scheme, three types of neurons can be defined: sensory neurons, connected to
sensory receptors, motor neurons, connected to muscles, and interneurons, connected to other
neurons.
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EEG Signal Processing for Epilepsy 3
Fig. 1. Presynaptic and postsynaptic activities in the neurons. An action potential that travels
along the fibre ends in an excitatory synapse. This process causes an excitatory postsynaptic
potential in the following neuron.
The cell body is called the soma, from which two types of structures extend: the dendrites
and the axon. Dendrites are short and consist of as many as several thousands of branches,
with each branch receiving a signal from another neuron. The axon is usually a single branch
which transmits the output signal of the neuron to various parts of the nervous system. Each
axon has a constant diameter and can vary in size from a few millimeters to more than 1 m
in length; the longer axons are those which run from the spinal cord to the feet. Dendrites
are rarely longer than 2 mm. and are connected to either the axons or dendrites of other cells.
These connexions receive impulses from other nerves or relay the signals to other nerves.
The human brain has approximately 10,000 connexions between one nerve and other nerves,
mostly through dendritic connections.
Neurons are, of course, not working in splendid isolation, but are interconnected into
different circuits (“neural networks”), and each circuit is tailored to process a specific type
of information.
2.2 Cerebral cortex
The cerebral cortex constitutes the outermost layer of the cerebrum and physically it is a
structure within the brain that plays an important role in memory, perceptual awareness,
attention, thought, consciousness and language. Normally, it is called “grey matter” for its
grey color and it is formedby neurons and “gray fibers” coveredby a dielectric called myelin.
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Myelinated axons are white in appearance, this characteristic is the origin of the name “white
matter,” and it is localized below the grey matter of the cortex. Their composition is formed
predominantly by myelinated axons interconnecting different regions of the nervous central
system.
Fig. 2. Cerebral cortex and its four lobes.
The human cerebral cortex is 2-4 mm thick. The cortical surface is highly convoluted by
ridges and valleys of varying sizes and thus increases the neuronal area; the total area is as
large as 2.5 m2and includes more than 1010 neurons. The cortex consists of two symmetrical
hemispheres–left and right–which are separated by the deep longitudinal fissure (the central
sulcus). Each cerebral hemisphere is divided into lobes, which are named for the skull bones
overlying each one: the frontal lobe, involved with decision-making, motor speech, problem
solving, and planning; temporal lobe, involved with memory, sensory speech, emotion,
hearing, and language; the parietal lobe, involved in the reception, reading comprehension
and processing of sensory information from the body; and the occipital lobe, involved with
vision, see Fig.2.
3. Introduction to the EEG waveform analysis
Most of the brain disordersare diagnosed by visual inspection of EEGsignals and the analysis
is a rational and systematic process requiring a series of orderly steps characterizing the
recorded electrical activity in terms of specific descriptors or features and measurements as
viewed in Table.1.
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EEG Signal Processing for Epilepsy 5
1. Frequency or wavelength
2. Voltage
3. Waveform.
4. Regulation
a. Frequency
b. Voltage
5. Manner of occurrence (random, serial, continuous)
6. Locus
7. Reactivity (eye opening, mental calculation, sensory stimulation,
movement, affective state)
8. Interhemispheric coherence (homologous areas)
a. Symmetry
i. Frequency
ii. Voltage
b. Synchrony
i. Wave
ii. Burst
Table 1. Essential features of EEG analysis described in Ebersole & Pedley (2003).
For example, an EEG from an 8 year old child, some 2 Hz waves are identified in the awake
EEG. This activity must then be characterized according to their location, voltage, waveform,
manner of occurrence, frequency, amplitude modulation, synchrony and symmetry. A change
in any of these features might entirely change the significance of the 2 Hz waves finding
this difference as abnormal. Some clinical information is required before the EEG analysis
is begun, by example the patient’s age and state. Both age and birth date should be part of
the EEG record. For example, there are clearly defined differences between the EEG of a
premature infant with a conceptional age of 36 weeks, but there are no important or sharply
delineated differencesbetween the EEG of a 3 year old child and that 4 year old child described
in Ebersole & Pedley (2003).
The clinical experts in the fields are familiar with manifestation of brain rhythms in the
EEG signals and it is important to recognize that the identification of a particular activity
or phenomenon may depend on its “reactivity” (see Table.1). An important element of
the recording and its analysis is the testing of the reactions, or responses, of the various
components of the EEG to certain physiological changes.
Specification of the reactivity of a given activity, rhythm or pattern is essential for the
identification and subsequent analysis of the activity and may clearly differentiate it from
another activity with similar characteristics. For example, in healthy adults, the amplitudes
and frequencies of brain rhythms change from one state of the human to another, such as
wakefulness and sleep. Similarly, a series of rhythmic, high voltage 3 to 4 Hz waves in the
prefrontal leads (just over the eyes) occurring in association with arousal in a young child
may be normal, but a similar burst occurring spontaneously and not associated with arousal
may be abnormal.
3.1 Brain rhythms and waveforms
The electrical activity of the cerebral cortex is often called as rhythm because this recorded
signals exhibit oscillatory, repetitive behavior. The diversity of EEG rhythms is enormous and
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depends, among many other things, on the mental state of the subject, such as the degree of
attentiveness, waking, and sleeping. The rhythms usually are conventionally characterized
by their frequency range and relative amplitude.
On the other hand, there are five brain waves characterized by their frequency bands. These
frequency ranges are alpha (α), theta (θ), beta (β), delta (δ), and gamma (γ)andtheir
frequencies range from low to high frequencies respectively. The alpha and beta waves were
introduced in 1929 by Berger. In 1938, Jasper and Andrews found waves above 30 Hz that
labeled as “ gamma” waves. A couple years before, in 1936, Walter introduced the delta
rhythm to designate all frequencies below the alpha range and he also introduced theta waves
as those frequencies within the range of 4-7.5 Hz. In 1944 the definition of a theta wave was
introduced by Wolter and Dovey in Sanei & Chambers (2007).
Alpha waves are over the occipital region of the brain and appear in the posterior half of
the head. The normal range for the frequency of the occipital alpha rhythm in adults is
usually given as 8 to 13 Hz, and commonly appears as a sinusoidal shaped signal. However,
sometimes it may manifest itself as sharp waves. In such cases, the alpha wave consist of a
negative and positive component that appears to be sharp and sinusoidal respectively. In fact,
this wave is very similar to the morphology of the brain wave called rolandic mu (μ)rhythm.
Delta waves lie within the range of 0.5-4 Hz. These waves appear during deep sleep and have
a large amplitude. It is usually not encountered in the awake, normal adult, but is indicative
of, e.g., cerebral damage or brain disease.
Theta waves are the electrical activity of the brain varying the range of 4-7.5 Hz and its name
might be chosen to origin assumption from thalamic region. The theta rhythm occurs during
drowsiness and in certain stages of sleep or consciousness slips towards drowsiness. Theta
waves are related to access to unconscious material, creative inspiration and associated to
deep meditation.
Beta waves are within the range of 14-26 Hz and consists in a fast rhythm with low amplitude,
associated with an activated cortex and observed during certain sleep stages. This rhythm is
mainly present in the frontal and central regions of the scalp.
Gamma waves (sometimes called the fast beta waves) are those frequencies above 30 Hz
(mainly up to 45 Hz) related to a state of active information processing of the cortex. The
observation of gamma rhythm during finger movement is done simply by using an electrode
located over the sensorimotor area and connected to a high-sensitivity recording system.
Other waves frequencies much higher than the normal activity range of EEG have been found
in the range of 200-300 Hz. The localization of these frequencies take place in cerebellar
structures of animals, but they do not play any role in clinical neurophysiology. Most of the
above rhythms may persist up to several minutes, while others occur only for a few seconds,
such as the gamma rhythm.
Fig.3 shows typical normal brain waves. There are also less common rhythms introduced by
researchers such as Phi (ϕ), Kappa (κ), Sigma (σ), Tau (τ), Chi (χ), Lambda (λ) and transient
waveforms associated to two sleep states, commonly referred to as non-REM (Rapid Eye
Movement) and REM sleep: vertex waves, sleep spindles, and K complexes described in Sanei
& Chambers (2007); Sörnmo & Laguna (2005).
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EEG Signal Processing for Epilepsy 7
Fig. 3. Typical normal brain waves in the EEG
It is often difficult to understand and detect the brain rhythms and waves from the scalp
EEGs, even with trained eyes. New applications in advanced signal processing tools, however,
should enable analysis and separation of the desired waveforms from the EEGs. Definitions
such as foreground and background EEG are very subjective and totally depends on the
abnormalities and applications. Possibly it is more useful to divide the EEG signal into
two general categories: the spontaneous brain activity (the “background EEG”); and brain
potentials which are evoked by various sensory and cognitive stimuli (evoked potentials,
EPs).
3.2 Artifacts
Analysis of EEG activity usually raises the problem of differentiating between genuine EEG
activity and that which is introduced through a variety of external influence. These artifacts
may affect the outcome of the EEG recording. Artifacts originate from a variety of sources
such as eyes movement, the heart, muscles and line power. Their recognition, identification,
and eventual elimination are a primary responsibility of the EEG expert. Even the most
experienced neurophysiologist cannot always eliminate all artifacts in EEG records. However,
it is always a major goal to identify the artifactual activity and be sure that it is not of cerebral
origin and should not be misinterpreted as such.
Following Ebersole & Pedley (2003); Fisch (1999), artifacts are generally divided into two
groups: physiological and non-physiological. Physiological artifacts usually arise from
generator sources within the body but not necessarily the brain, for example, eye movements;
electrocardiographic and electromyographic artifacts, galvanic skin response and so on.
Biological generators present in the body may produce artifacts when an EEG recording is
made directly from the surface of the brain. Nonphysiological artifacts come from a variety
of sources such as instrumental and digital artifacts (electronic components, line power,
inductance, etc.), electrode artifacts, environment, etc.
As technology expands and additional equipment is developedand put into clinical use, novel
artifacts will apper. Then, a correct artifact filtering strategy should on the one hand eliminate
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unnecessary amount of information that has to be eliminated, and on the other hand maintain
or ensure that the resulting information is not affected by undetected artifacts. Sometimes
visual artifacts inspections could be a good alternative in cases when the artifacts are relatively
easy detected by the EEG experts. However, there is the possibility that duringthe analysis of
EEG databases these patterns from artifacts cause serious misinterpretation and then reduce
the clinical usefulness of the EEG recordings.
3.3 Abnormal EEG patterns
Any variation in EEG patterns for certain states of the subject indicate abnormality. This may
be due to many causes such as distortion and loss of normal patterns, increased occurrence
of abnormal patterns, or disappearance of all patterns. In most abnormal EEGs, the abnormal
EEG patterns do not entirely replace normal activity: they appear only intermittently, only in
certain head regions, or only superimposed on a normal background.
An EEG is considered abnormal if it contains (a) generalized intermittent slow wave
abnormalities, commonly associated in the delta wave range and brain dysfunctions, (b)
bilateral persistent EEG, often associated with impaired conscious cerebral reactions, and (c)
focal persistent EEG usually associated with focal cerebral disturbance.
The classification of the three categories presented before is not easy and needs to be
extended to several neurological diseases and any other available information. A precise
characterization of the abnormal patterns leads to a clearer insight into some specific
neurodegenerative diseases such as epilepsy, Parkinson, Alzheimer, dementia and sleep
disorders, or specific disease processes, for example Creutzfeldt-Jakob disease (CJD) described
in Sanei & Chambers (2007). However, following Fisch (1999), recent studies have
demonstrated that there is correlation between abnormal EEG patterns, general cerebral
pathology and specific neurological diseases.
4. Modelling and segmentation
4.1 Modelling the EEG signals
Modelling the brain activities is not an easy task as compared with modelling any other organ.
First literature related to EEG signal generation includes physical model such as the model
proposed by Hodgkin and Huxley, linear models such as autoregressive (AR) modelling, AR
moving average (ARMA), multivariate AR (MVAR), Prony methods and so on. There are
also methods based on no-linear models such as autoregressive conditional heteroskedasticity
(GARCH), Wiener modeling and local EEG model method (LEM). More details about the
methods described above can be found in Celka & Colditz (2002); Sanei & Chambers (2007).
Following Senhadji & Wendling (2002), other model relates a sampled EEG signal X(n)with
relevant activities as elementary waves, background activity, noise and artifacts as:
X(n)=F(n)+
np
∑
i=1
Pi(n−tpi)+
na
∑
j=1
Rj(n−taj)+B(n)(1)
where F(n)is the background activity; the Piterms represent brief duration potentials
corresponding to abnormal neural discharges; the Rjterms are related to artifacts ( discussed
later in section 3.2) and B(n)is the measurement noise which is modeled as a stationary
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EEG Signal Processing for Epilepsy 9
process. This model shows all the EEG information including the abnormal EEG signal.
This is a mathematical model rather than an EEG generation signal model, but facilitates the
manipulation of concepts that are introduced in the next sections.
4.2 Signal segmentation
Signal segmentation is a process that divides the EEG signal by segments of similar
characteristics that are particularly meaningful to EEG analysis. Traditional techniques of
signal analysis, for example, spectrum estimation techniques, assume time-invariant signals
but in practice, this is not true because the signals are time-varying and parameters such as
amplitude, frequency and phase change over time. Furthermore, the presence of short time
events in the signal causes a nonstationarity effect.
Non-stationary phenomena are present in EEG usually in the form of transient events, such
as sharp waves, spikes or spike-wave discharges which are characteristic for the epileptic
EEG, or as alternation of relatively homogenous intervals (segments) with different statistical
features (e.g., with different amplitude or variance). The transient phenomena have specific
patterns which are relatively easy to identify by visual inspection in most cases, whereas the
identification of the homogeneous segments of EEG, known as quasi-stationary, requires a
certain theoretical basis. Usually each quasi-stationary segment is considered statistically
stationary with similar time and frequency statistics. This eventually leads to a dissimilarity
measurement denoted as d(m)between the adjacent EEG frames, where mis a integer value
indexing the frame and the difference iscalculated between the mand (m−1)th (consecutive)
signal frames.
There are different dissimilarity measures such as autocorrelation, high-order statistics,
spectral error, autoregressive (AR) modelling and so on, presented in Sanei & Chambers
(2007). These methods are effective in EEG analysis but can not be efficient for detection
of certain abnormalities due to the impossibility of obtaining segments completely stationary.
It is then necessary to take into account a different group of methods potentially useful for
detecting and analyzing non-stationary EEG signals where the segmentation does not play a
fundamental role such as the time-frequency distributions (TFDs).
4.3 Denoising and filtering
Biomedical signals in general, but more particularly EEG signals, are subject to noise and
artifacts which are introduced through a variety of external influences. These undesired
signals may affect the outcome of the recording procedure, being necessary a method that
appropriately eliminates then without altering original brain waves. EEG denoising methods
try to reject artifacts originated in the brain or body such as ocular movements, muscle
artifacts, ECG etc.
Filtering is a signal processing operation whose objective is to process a signal in order to
manipulate the information contained in the signal. In other words, a filter is a device
that maps an input signal to an output signal facilitating the extraction (or elimination) of
information (or noise) contained in the input signal. In our context, the filtering process is
oriented to eliminate electrical noise generated by electrical power line or extracting certain
frequency bands.
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4.3.1 Lowpass filtering
Most frequently EEG signals contain neuronal information below 100 Hz, for example,
epileptic waves lie below 30 Hz, it is possible to remove frequency components above this
value simply using lowpass filters. In the cases where the EEG data acquisition system is
unable to remove electrical noise as 50 or 60 Hz line frequency, it is necessary to use a notch
filter to remove it. Although digital filters could introduce nonlinearities or distortions to the
signal in both of amplitude and phase, there are digital EEG process that allow corrections of
these distortions using commercial hardware devices. However, it should be better to know
the characteristics of the internal and external noises that affect the EEG signals but these
information usually is not available.
4.4 Independent component analysis (ICA)
ICA is of interest to scientists and engineers because it is a mathematical tool able to reveal the
driving forces which underlie a set of observed phenomena. These phenomena may well
be the firing of a set of neurons, mobile phone signals, brain images such as fMRI, stock
prices, or voices, etc. In each case, a set of complex signals are measured, and it is known
that each measured signal depends on several distinct underlying factors, which provide the
driving forces behind the changes in the measured signals. These factors or source signals
(that are primary interest) are buried within a large set of measured signals or signal mixtures.
Following Stone (2004), ICA can be used to extract the source signals underlying a set of
measured signal mixtures.
ICA belongs to a class of blind source separation (BSS) methods for estimating or separating
data into underlying informational components. The term “blind” is intented to imply
that such methods can separate data into source signals using only the information of their
mixtures observed at the recordingchannels. BSS in acoustics is well explained in the “cocktail
party problem,” which aims to separate individual sounds from a number of recordings in an
uncontrolled environment such as a cocktail party. So, simply knowing that each voice is
statistically unrelated to the others suggests a strategy for separating individual voices from
mixtures of voices. The property of being unrelated is of fundamental importance, because
it can be generalized to separate not only mixtures of sounds, but mixtures of other kind of
signals such as biomedical signals, images, radio signals and so on.
The informal notion of unrelated signals can be associated to the more precise concept of
statistical independence. If two or more signals are statistically independent of each other then
the value of one signal provides no information regarding the value of the other signals. ICA
works under this assumption and this concept plays a crucial role in separating and denoising
the signals.
4.4.1 ICA fundamentals
The basic BSS problem that ICA attempts to solve assumes a set of mmeasured data points at
time instant t,x(t)=[x1(t),x2(t), ..., xm(t)]Tto be a combination of nunknown underlying
sources s(t)=[s1(t),s2(t), ..., sn(t)]T. The combination of the sources is generally assumed to
be linear and fixed, and the mixing matrix describing the linear combination of s(t)is given
by the full rank n×mmatrix Asuch that
x(t)=As(t)(2)
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ICA
Electrodes signals
Separated
sources
Brain signals sources
s1s2
s3
s4
s5
x1
x2
x3
x4
x5
ˆ
s1
ˆ
s2
ˆ
s3
ˆ
s4
ˆ
s5
Fig. 4. General ICA process applied to EEG signals
It is also generally assumed that the number of underlying sources is less than or equal to the
number of measurement channels (n≤m).
The task of the ICA algorithms is to recover the original sources s(t)from the observations
x(t)and this is generally equivalent to that of finding a separating (de-mixing matrix) Wsuch
that
ˆ
s(t)=Wx(t)(3)
given the set of observed values in x(t)and where ˆ
s(t)are the resulting estimates of the
underlying sources. This idealistic representation of the ICA problem is described in Fig.4.
In reality the basic mixing model assumed in Eq.2 is simplistic and assumed for the ease of
implementation. In fact, a perfect separation of the signals requires taking into account some
assumptions and the structure of the mixing process:
•Linear mixing: The first traditional assumption for ICA algorithms is that of linear mixing,
a realistic model can be formulated as
x(t)=As(t)+n(t)(4)
where Ais the linear mixing matrix described earlier and n(t)is additive sensor noise
corrupting the measurements x(t)(generally assumed to be i.i.d. spatially and temporally
white noise, or possibly temporally colored noise), as described in James & Hesse (2005).
In a biomedical signal context, linear mixing assumes (generally instantaneous) mixing of
the sources using simple linear superposition of the attenuated sources at the measurement
channel.
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•Noiseless mixing: If observations x(t)are noiseless (or at least the noise term n(t)is
negligible) then Eq.4 reduces to Eq.2. Whilst this is probably less realistic in practical
terms, it allows ICA algorithms to separate sources of interest even if the separate sources
themselves remain contaminated by the measurement noise.
•Square mixing matrix: So far it has been assumed that the mixing matrix Amay be
non-square (n×m); in fact most classical ICA algorithms assume a square-mixing matrix,
i.e. m=n, this makes the BSS problem more tractable. From a biomedical signal analysis
perspective the square-mixing assumption is sometimes less than desirable, particularly
in situations where high-density measurements are made over relatively short periods of
time such as in most MEG recordings or fMRI.
•Stationary mixing: Another common assumption is that the statistics of the mixing matrix
Ado not change with time. In terms of biomedical signals this means that the physics of
the mixing of the sources as measured by the sensors is not changing.
•Statistical independence of the sources: The most important assumption in ICA is that the
sources are mutually independent. Two random variables are statistically independent if
there is a joint distribution of functions of these variables. This means, for example, that
independent variables are uncorrelated and have no higher order correlations. In the case
of time-series data, it is assumed that each source is generated by a random process which
is independent of the random processes generating the other sources.
4.5 Feature extraction
Feature extraction consist in finding a set of measurements or a block of information with
the objective of describing in a clear way the data or an event presents in a signal. These
measurements or features are the fundamental basis for detection, classification or regression
tasks in biomedical signal processing and is one of the key steps in the data analysis process.
These features constitute a new form of expressing the data, and can be binary, categoricals or
continuous, and also represent attributes or direct measurements of the signal. For example,
features may be age, health status of the patient, family history, electrode position or EEG
signal descriptors (amplitude, voltage, phase, frequency, etc.).
More formally, feature extraction assumes we have for Nsamples and Dfeatures, a matrix
N×D,whereDrepresents the dimension of the feature matrix. That means, at the sample n
from the feature matrix, we could obtain an unidimensional vector x=[x1,x2, ... , xD]called
as “pattern vector.” Several methods in EEG feature extractions can be found in the literature,
see Guyon et al. (2006).
More specifically in EEG detection and classification sceneries, features based on power
spectral density are introduced in Lehmanna et al. (2007); Lyapunov exponents are introduced
in Güler & Übeyli (2007); wavelet transform are described in Hasan (2008); Lima et al.
(2009); Subasi (2007) and Xu et al. (2009); sampling techniques are used in Siuly & Wen
(2009) and time frequency analysis are presented on Boashash (2003); Guerrero-Mosquera,
Trigueros, Franco & Navia-Vazquez (2010); Tzallas et al. (2009) and Boashash & Mesbah
(2001). Other approach in feature extraction based in the fractional Fourier transform is
described in Guerrero-Mosquera, Verleysen & Navia-Vazquez (2010). It is important to add
that features extracted are directly dependent on the application and also to consider that there
are important properties of these features to have into account, such as noise, dimensionality,
time information, nonstationarity, set size and so on (Lotte et al. (2007)).
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This section emphasizes methods oriented to frequency analysis, without excluding the
time domain that permits to justify the importance of the frequency analysis and their
shortcomings in front of nonstationary signals like the EEG.
4.5.1 Classical signal analysis tools
A signal could be represented in different forms being for example in time and frequency.
While time domain indicates how a signal changes over time, frequencydomain indicates how
often such changes take place. For example, let us consider a signal with a linear frequency
modulation varying from 0 to 0.5 Hz and with constant amplitude (see Fig.5). Looking at
the time domain representation (Fig.5 upper) it is not easy to say what kind of modulation
is contained in the signal; and from the frequency domain representation (see Fig.5 bottom),
nothing can be said about the evolution in time of the frequency domain characteristics of the
signal.
0 20 40 60 80 100 120 140
−1
0
1
Time [s]
Amplitude
Signal in time
−0.5 0 0.5
0
200
400
Normalized frequency
Magnitude
Signal in frequency
Fig. 5. Chirp signal using time domain (upper) and frequency domain (bottom).
The two representations are related by the Fourier transform (FT) as:
X(ω)=∞
−∞x(t)e−jωtdt (5)
or by the inverse Fourier transform (IFT) as:
x(t)=∞
−∞X(ω)e−jωtdω(6)
Eq.6 indicates that signal x(t)can be expressedas the sum of complex exponentials of different
frequencies, whose amplitudes are the complex quantities X(ω)defined by Eq.5.
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The squared magnitude of the Fourier transform , |X(ω)|2, is often taken as the frequency
representation of the signal x(t), which allows in some sense easier interpretation of the signal
nature than its time representation.
Better interpretation is obtained using a domain that directly represents frequency content
while still keeping the time description parameter. This characteristic is the aim of time
frequency analysis. To illustrate this, let us represent the chirp signal explained above using
the spectrogram (more details about this in the following). Note how it is possible to see the
linear progression with time of the frequency components, from 0 to 0.5 (Fig.6).
−0.5
0
0.5
1
Real part
Signal in time
SP, Lh=16, Nf=64, lin. scale, imagesc, Threshold=5%
Time [s]
Frequency [Hz]
20 40 60 80 100 120
0
0.1
0.2
0.3
0.4
0.5
Fig. 6. Spectrogram representation of the chirp
4.5.2 Time-frequency distributions (TFD)
In a series of papers (Akay (1996); Cohen (1995)), Cohen generalized the definition of
time-frequency distributions (TFDs) in such a way that a wide variety of distributions could
be included in the same framework. Specifically the TFD of a real signal x(n)is computed as:
P(t,ω)= 1
2π∞
−∞∞
−∞A(θ,τ)Φ(θ,τ)e-jθt-jωτdθdτ(7)
where,
A(θ,τ)= 1
2π∞
−∞x(u+τ
2)x∗(u−τ
2)ejθudu (8)
is the so-called ambiguity function and the weighting function Φ(θ,τ)is a function called the
kernel of the distribution that, in general, may depend on time and frequency.
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EEG Signal Processing for Epilepsy 15
If Φ(θ,τ)=1inEq.(7),wehave
P(t,w)= 1
2π∞
−∞∞
−∞x(u+τ
2)x∗(u−τ
2)e−jωτ
1
2π∞
−∞e−jθ(t-u)dθdudτ(9)
where
1
2π∞
−∞e-jθ(t-u)dθ=δ(t−u)(10)
and we know that
∞
−∞x(u+τ
2)x∗(u−τ
2)δ(t−u)du =x(t+τ
2)x∗(t−τ
2)(11)
If we substitute the Eq.(10) and Eq.(11) in Eq.(9), then we have the Wigner-Ville distribution
(WV) defined as:
WV(ω,t)= 1
2π∞
−∞x(t+τ
2)x∗(t−τ
2)e−jωτdτ(12)
Following Hammond & White (1996), the recurrent problem of the WV is the so-called
crossterm interference, due to bilinear nature of its definition. These crossed terms tend to
be located mid-way between the two auto terms and are oscillatory in nature.
When Φ(θ,τ)=1, we have the Wigner-Ville distribution WV(t,ω). The Smooth
Pseudo Wigner-Ville (SPWV) distribution is obtained by convolving the WV(t,ω)with a
two-dimensional filter in tand ω. This transform incorporates smoothing by independent
windows in time and frequency, namely Ww(τ)and Wt(t):
SPWV(t,ω)=∞
−∞Ww(τ)∞
−∞Wt(u−t)x(u+τ
2)
x∗(u−τ
2)due−jωτdτ(13)
Eq.(13) provides great flexibility in the choice of time and frequency smoothing, but the length
of the windows should be determined empirically according to the type of signal analyzed
and the required cross term suppression, discussed in Afonso & Tompkins (1995).
As proved in Hlawatsch & Boudreaux-Bartels (1992), the SPWV in Eq.(13) does not satisfy
the marginal properties, that is, the frequency and time integrals of the distribution do not
correspond to the instantaneous signal power and the spectral energy density, respectively.
However, it is still possible for a distribution to give the correct value for the total energy
without satisfying the marginals, described in Cohen (1989; 1995). Therefore the total energy
can be a good feature to detect signal events in the SPWV representation because the energy
in EEG seizure is usually larger than the one during normal activity.
The TFDs offer the possibility of analyzing relatively long continuous segments of EEG data
even when the dynamics of the signal are rapidly changing. Taking the most of these, it can
extract features from the time frequency plane such as ridges energy, frequency band values
and so on. However, three considerations have to be taken, presented in Cohen (1989; 1995)
and Durka (1996):
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- A TFD will need signals as clean as possible for good results.
- A good resolution both in time and frequency is necessary and as the “uncertainty
principle” states, it is not possible to have a good resolution in both variables
simultaneously.
- It is also required to eliminate the spurious information (i.e. cross-term artifacts) inherent
in the TFDs.
The first consideration implies a good pre-processing stage to eliminate artifacts and noise.
Second and third considerations have motivated the TFD selection or design, then it is
important and necessary to choose a suitable TFD for seizure detection in EEG signals as well
as for a correct estimation of frequencies on the time-frequency plane. Indeed, it is desirable
that the TFD has both low cross-terms and high resolution. Choosing a distribution depends
on the information to be extracted and demands a good balance between good performance,
low execution time, good resolution and few and low-amplitude cross terms.
One consideration before using the TFD is to convert each EEG segment into its analytic
signal for a better time-frequency analysis. The analytic signal is defined to give an identical
spectrum to positive frequencies and zero for the negative frequencies, and shows an
improved resolution in the time-frequency plane, discussed in Cohen (1989). It associates
agivensignalx(n)to a complex valued signal y(n)defined as: y(n)=x(n)+jHT{x(n)},
where y(n)is the analytic signal and HT{.}is the Hilbert transform.
4.5.3 Wavelet coefficients
The EEG signals can be considered as a superposition of different structures occurring on
different time scales at different times. As presented in Latka & Was (2003), the Wavelet
Transform (WT) provides a more flexible way of time-frequency representation of a signal
by allowing the use of variable size windows and can constitute the foundation of a relatively
simple yet effective detection algorithm. Selection of appropriate wavelets and the number of
decomposition levels is very important in the analysis of signals using the WT. The number of
decomposition levels is chosen based on the dominant frequency components of the signals.
Large windows are used to get a finer low-frequency information and short windows are
used to get high-frequency resolution. Thus, WT gives precise frequency information at low
frequencies and precise time information at high frequencies. This makes the WT suitable for
EEG analysis of spikes patterns or epileptic seizures.
Wavelets overcome the drawback of a fixed time-frequency resolution of short time Fourier
transforms. The WT performs a multiresolution analysis, WΨf(a,b)of a signal, x(n)by
convolution of the mother function Ψ(n)with the signal, as given in Latka & Was (2003),
and Mallat (2009) as:
WΨx(b,a)=
N−1
∑
n=0
x(n)Ψ∗n−b
a(14)
Ψ(t)∗denote the complex conjugate of Ψ(n)(basis function), athe scale coefficient, bthe shift
coefficient and a,b∈,a=0.
Wavelets overcome the drawback of a fixed time-frequency resolution of short time Fourier
transforms. The WT performs a multiresolution analysis, WΨf(a,b)of a signal, x(n)by
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EEG Signal Processing for Epilepsy 17
convolution of the mother function Ψ(n)with the signal, as given in Latka & Was (2003),
and Mallat (2009) as:
WΨx(b,a)=
N−1
∑
n=0
x(n)Ψ∗n−b
a(15)
Ψ(t)∗denote the complex conjugate of Ψ(n)(basis function), athe scale coefficient, bthe shift
coefficient and a,b∈,a=0.
In the procedure of multiresolutiondecomposition of a signal x(n), each stage consists of two
digital filters and two downsamplers by 2. The bandwidth of the filter outputs are half the
bandwidth of the original signal, which allows for the downsampling of the output signals by
two without loosing any information according to the Nyquist theorem. The downsampled
signals provide detail D1 and approximation A1 of the signal, this procedure is described in
Hasan (2008).
Once the mother wavelet is fixed, it is possible to analyze the signal at every possible scale
aand translation b.IfthebasisfunctionΨ(n)is orthogonal, then the original signal can be
reconstructed from the resulting wavelet coefficients accurately and efficiently without any
loss of information. The Daubechies’ family of wavelets is one of the most commonly used
orthogonal wavelets to non-stationary EEG signals presenting good properties and allowing
reconstruction of the original signal from the wavelet coefficients, as described Mallat (2009).
4.5.4 Fractional Fourier transform
Fourier analysis is undoubtedly one of the most used tools in signal processing and other
scientific disciplines and this technique uses harmonics for the decomposition of signals with
time-varying periodicity. Similarly, TFDs are very frequently used in signal analysis especially
when it is necessary to eliminate the windowing dependence on non-stationary signals.
In 1930, Namias employed the fractional Fourier transform (FrFT) to solve partial differential
equations in quantum mechanics from classical quadra-tic Hamiltonians1.Theresultswere
later improved by McBride and Kerr in Tao et al. (2008). They developed operational calculus
to define the FRFT. The FrFT is a new change in the representation of the signal which is
an extension of the classical Fourier transform. When fractional order gradually increases, the
FrFT of a signal can offer much more information represented in an united representation than
the classical Fourier transform and it provides a higher concentration than TFDs, avoiding the
cross terms components produced by quadratics TFDs.
FrFT has established itself as a potential tool for analyzing dynamic or time-varying signals
with changes in very short time and it can be interpreted as the representation of a signal
in neutral domain by means of the rotation of the signal by the origin in counter-clockwise
direction with rotational angle αin time-frequency domain as shown in Fig.7. The FrFT with
1A development based on a concept called fractional operations. For example, the n-th derivative of f(x)
can be expressed as dnf(x)/dxnfor any positive integer n. If another value derived is required, i.e. the
0.5-th derivative, it is necessary to define the operator daf(x)/dxa, where the value acould be an any
real value. The function [f(x)]0.5 is the square root of the function f(x).Butd0.5 f(x)/dx0.5 is the 0.5-th
derivative of f(x)(a=0.5), (df(x)/dx)0.5 being the square root of the derivative operator d/dx.Asit
can be seen, fractional operations is a concept that goes from the whole of an entity to its fractions.
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u=tcosα +wsinα
v=−tsinα +wcosα
α
Time (t)
α
Frequency (w)
v
u
Fig. 7. The relation of fractional domain (u,v)with traditional time-frequency plane (t,w)
rotated by an angle α.
angle αof a signal x(t), denoted as Xα(u)is defined in Almeida (1994) as:
Xα(u)=∞
−∞x(t)Kα(t,u)dt (16)
where Kα(u,t)is a linear kernel function continuous in the angle α, which satisfies the basic
conditions for being interpretable as a rotation in the time-frequency plane. The kernel has
the following properties
Kα(t,u)=Kα(u,t)(17)
K−α(t,u)=K∗
−α(t,u)(18)
Kα(−t,u)=Kα(t,−u)(19)
∞
−∞Kα(t,u)Kβ(u,z)du =Kα+β(t,z)(20)
∞
−∞Kα(t,u)K∗
α(t,u)dt =δ(u−u)(21)
The FrFT is given by
Xα(u)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
(1−jcotα
2πeju2
2cotα∞
−∞x(t)ejt2
2cotαejutcscαdt,
if αisnotamultipleofπ
x(t),ifαis multiple of 2π
x(−t),ifα+πis multiple of 2π
More detailed definitions, proof and further properties of the kernel can be found in Almeida
(1994).
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EEG Signal Processing for Epilepsy 19
In summary, the FrFT is a linear transform, continuous in the angle α, which satisfies the basic
conditions for being interpretable as a rotation in the time-frequency plane.
5. The detection problem in EEG signals
Epilepsy is considered the disease with major prevalence within disorders with neurological
origin. The recurrent and sudden incidence of seizures can lead to dangerous and possibly
life-threatening situations. Since disturbance of consciousness and sudden loss of motor
control often occur without any warning, the ability to predict epileptic seizures would reduce
patients’ anxiety, thus improving quality of life and safety considerably.
Intractable epilepsy is one of the most physically and emotionally destructive neurological
disorders affecting population of all ages. It is generally accepted that surgical rejection of
epileptic foci is the best solution. However, before conducting neurosurgery, it is necessary to
study the presence of epileptiform activity, which is distinct from background EEG activity.
The analysis of EEG data and the extraction of information is not an easy task. EEG recording
may be contaminated by extraneous biologically generated (human body) and externally
generated signals (power line, electrode movement etc.). The presence of this kind of noise
or “artifacts” makes it difficult to discriminate between original brain waves and noise. This
problem motivates a preprocessing step to obtain clean signals before the detection task.
Another important problem in EEG processing is to figure out which kind of information
or “patterns” we want to extract from the signal. This procedure is known as feature
extraction. Extracted features depend considerably on the method used, which are usually
transformations to other domains that permit the extraction of hidden information in the
signal. Care has to be taken not to extract similar or irrelevant features that could reduced
the detector performance or increase the computational load. Therefore, a feature selection
procedure is also necessary to complement the features extraction procedure.
Other important task in the medical environment to diagnose, classify or detect abnormalities,
is to obtain ictal and interictal patterns. This usually involves monitoring of the patient during
several weeks. Continuous observation or patient monitoring is a care activity that requires
time and expensive work, being necessary specialized personnel for alerting of possible
changes that a patient may have. When information is stored, there is another activity equally
important: the analysis of the EEG registers. The specialists have to analyze waveforms,
spectrum and peaks, and based on this analysis try to determine the pathology that the patient
suffers. Usually they use a video unit. In many instances, there are disagreements among
specialists about the same record due to the subjective nature of the analysis.
The introduction of new techniques and mathematical algorithms in the EEG analysis can be
helpful to design new supporting methods in medical decision and diagnosis, thus avoiding
tedious analysis of long-term records and doubts about the brain pathology that a patient
suffers.
Nowadays there are many published studies about neurological diseases detection but
these results are very focused on private institutional databases or rely on impractical
numerical methods which are difficult to implement in a hospital environment. Therefore,
the implementation and design of practical and reliable detection systems are very important
in hospitals. This doctoral thesis, tries to narrow the gap that exists between EEG signal theory
and practical implementation for the medical practice.
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5.1 Summary of previous work in epileptic detection on EEG signals
Some methods of seizure detection were based on detecting strong rhythmic movements
of the patient, but these methods had a limitation: seizures do not always present strong
movements. This limitation led the detection problem to methods based on EEG signal
analysis, for example, detection of large seizures discharges in several EEG channels by
amplitude discrimination was described by J.R. Ives & Woods (1974); T.L. Babb & Crandall
(1974) designed an electronic circuit for seizures detection from intracraneal electrodes.
However, some seizures do not present EEG changes, therefore seizure detection only based
on EEG analysis was not at all reliable and it was necessary to combine it with other
methods. For example, P.F. Prior & Maynard (1973) identified on the EEG signal a large
increase followed by a clear decrease in the amplitude and at the same time by large
electromyogram (EMG) activity; A.M.Murro&Meador (1991)described a method based on
spectral parameters and discriminant analysis.
New alternatives for this detection problem are addressed from the point of view of pattern
recognition. Gotman (1982) presented an automatic detection system based on seizure
patterns. The drawback of this method is the necessity of traditional visual inspection of
the patterns, being necessary a careful examination of them by a specialist.
Presently, EEG epileptic detectors have evolved including new techniques such as neural
networks, non-linear models, independent component analysis (ICA), Bayesian methods,
support vector machines and variance-based methods, as described in Guerrero-Mosquera,
Trigueros, Franco & Navia-Vazquez (2010). Other group of methods potentially useful
for detecting and analyzing non-stationary signals are time-frequency distributions (TFDs)
Cohen (1995). These methods allow us to visualize the evolution of the frequency behavior
during some non-stationary event by mapping a one dimensional (1-D) time signal into a
two-dimensional (2-D) function of time and frequency. Therefore, from the time-frequency
(TF) plane it is possible to extract relevant information using methods such as peak matching,
filter banks, energy estimation, etc.
On the other hand, most of the detection methods proposed in the literature assume a
clean EEG signal free of artifacts or noise, leaving the preprocessing problem open to any
denoising algorithm such as digital filters, independent component analysis (ICA) or adaptive
schemes using the electrooculogram (EOG) as reference signal, as in Guerrero-Mosquera &
Navia-Vazquez (2009).
5.2 Classification algorithms for EEG signals
Unlike many theoretical approaches that solve certain problems using some model or formula,
many classifiers are based on statistical learning. In such cases the system should be trained to
obtain a good classifier taking into account that, under the following considerations described
in Sanei & Chambers (2007), classification algorithms do not perform efficiently when:
• the number of features is high,
• there is limited execution time for a classification task,
• the classes or labels from feature matrix are unbalanced,
• there are nonlinearities between inputs and outputs,
• data distribution is unknow,
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EEG Signal Processing for Epilepsy 21
• there is no convergence guarantee to best solution (problem not convex or monotonic).
Up-today, several algorithms in EEG signal classification and detection have been propose in
the literature. For example, Multiple signal classification (MUSIC) combining EEG and MEG
for EEG source localization described in Mosher & Leahy (1998); classification of patients with
Alzheimer using Support Vector Machine (SVM) and neural networks (NNs) described in
Lehmanna et al. (2007); Güler & Übeyli (2007) introduced the multiclass SVM for EEG. Lotte
et al. (2007) describes several applications for BCI using methods such as Hidden Markov
Modelling (HMM), Linear Discriminant Analysis (LDA) and fuzzy logic; Chiappa & Barber
(2006) used the Bayes’s rule to discriminate mental tasks; detection of ERPs using SVM
described in Thusalidas et al. (2006); Fuzzy SVM (FSVM) is utilized in Xu et al. (2009); Fisher’s
discriminant is introduced in Müller et al. (2003). Applications in epilepsy classification such
as Artificial Neural Networks (ANN) described in Subasi (2007); k-NN classifier and logistic
regression with TFDs are used in Tzallas et al. (2009); Least Square SVM (LS-SVM) in Siuly
& Wen (2009); Learning Vector Quantization with NN (LVQ-NN) described in Hasan (2008);
Mixture of Experts (ME) and Multilayered Perceptron (MLP) in Subasi (2007); an automatic
EEG signal classification using Relevance Vector Machine (RVM) is proposed by Lima et al.
(2009).
As showed Guyon et al. (2006), SVM and its variants have more applications in different
classification scenarios and are powerful approach for pattern recognition, showing to
be a good alternative for EEG signal classification due to their high performance, good
generalization and adaptability in stationary and nonstationary environments compared to
other methods such as NN.
6. Dimensionality reduction for EEG signals
After feature extraction, it is necessary to select the subset of features that present better
performance or are most useful for a problem at hand, such as regression, classification or
detection. The data acquisition in environments such as biomedical signals leads to define
each problem by hundreds or thousands of measurements leading to obtain high dimensional
data with high computational cost.
As discussed in Guyon & Elisseeff (2003), feature selection is based on the principle
that choosing a smaller number of variables among the original ones, leads to an easier
interpretation. In fact, under the assumption that reducing the training data2might improve
the performance task, the feature selection methods also allows a better data understanding
and visualization together with reduction in data measurement and storage.
Feature selection could be summarized in two main tasks: choosing the relevant features and
searching the best feature subset. The first one tries to solve the question: is a feature (or subset
of features) relevant for the problem? And the second one tries to search the best feature subset
among all the possible subsets extracted from the initial task3. The application of these two
2Concept related to the fact of using a data set (also called data points, samples, patterns or observations)
in order to gain knowledge, learn a task associated with desired outcomes.
3Although feature extraction and feature selection are different aspects of the pattern recognition
process, it is important to distinguish the difference between them. The first one aims at building a
good feature representation based on several measurements, and the second one tries to reduce the
feature matrix by selecting subsets of features more useful in determined tasks.
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tasks to high dimensional data causes a reduction in the data dimension, process known as
dimensionality reduction.
Besides feature selection, there is another set of methods known as projection methods that
perform the same task but in practice could retain the problems suffered by high dimensional
data, presented in Rossi et al. (2007; 2006). Typical projection algorithms are Principal
Component Analysis (PCA), Sammon’s Mapping, Kohonen maps, Linear Discriminant
Analysis (LDA), Partial Least Squares (PLS) or Projection pursuit, amongst others ( see Duda
et al. (2009)).
6.1 Subset relevant assessment
This step is mainly based on a relevance criterion that has to be chosen by some measurement.
The best choice for the criterion is certainly to estimate the performances of the model itself,
i.e., an individual feature ranking could be appropriate at scenarios where the features provide
a good performance by itself and there is the possibility of choosing features associated to high
ranks.
The idea of the “individual relevance ranking” can be clarified by the following example:
Fig.8 shows a situation where the feature X2is more relevant individually to predict
the output Ythan the feature Y1. Notice the importance of choosing the right features
to improve the performance of a task, which in this example is related to prediction of
Y. There are different alternatives in relevance criteria, such as the Pearson correlation
0 0.5 1
0
0.2
0.4
0.6
0.8
1
X
1
Y
0 0.5 1
0
0.2
0.4
0.6
0.8
1
X
2
Y
Fig. 8. Simple prediction problem. The horizontal axis represents the feature and the vertical
axis the output. It can see that feature X2(left) is more relevant individually than feature X1
(right) in this simple prediction problem.
coefficient, mutual information (MI) and wrapper methodology. Although each method
has its advantages and disadvantages, mutual information has proven to be an appropriate
measure in several applications such as selection of spectral variables, spectrometric nonlinear
modelling and functional data classification, see Gomez-Verdejo et al. (2009); Rossi et al. (2007;
2006). Moreover, as discussed in Cover & Thomas (1991), correlation does not measure
nonlinear relations among features and wrapper approach presents a high computational
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EEG Signal Processing for Epilepsy 23
load. Furthermore, MI could be seen as a correlation measure applied to determine the
nonlinearity among features.
Next section focuses on the well-known concept of MI and shows why this relevance criterion
is applicable for feature selection.
6.2 Mutual information (MI)
Mutual information (MI) measures the relevance between a group of features Xand the
variable or output Y. This relationship is not necessarily linear. As described in Cover &
Thomas (1991), the mutual information between two variables is the amount of uncertainty
(or entropy) that is lost on one variable when the other is known, and vice-versa. The variables
Xand Ycould be multidimensional, solving the drawback in correlation measurements that
are based on individual variables.
Let pX(x)and pY(y)be the marginal of probability density function (pdf) of Xand Y
respectively, and the joint probability density function of Xand Yis pX,Y(x,y).IfXhas X
alphabets, the entropy of X is defined as
H(X)=−∑
x∈X
pX(x)log pX(x)(22)
The base of the logarithm determines the units in which information is measured. Particularly,
if the logarithm is base 2 the entropy is expressedin bits.
The joint entropy H(X,Y)of a pair of discrete random variables (X,Y)with a joint distribution
pX,Y(x,y)is defined as
H(X,Y)=−∑
x∈X
∑
y∈Y
pX,Y(x,y)log pY|X(y|x)(23)
And the MI between two variables is calculated as
I(X,Y)= ∑
x∈X
∑
y∈Y
pX,Y(x,y)log pX,Y(x,y)
pX(x)pY(y)(24)
Eq.24 gives the relation between Xand Y, meaning that I(X,Y)is large (small) the variables
are closely (not closely) related. The MI and entropy have the following relation, see Cover &
Thomas (1991):
I(X,Y)=H(Y)−H(Y|X)(25)
For continuous variables, the entropy and MI are defined as
H(X)=−∞
−∞pX(x)log pX(x)dx (26)
I(X,Y)=∞
−∞∞
−∞pX,Y(x,y)log pX,Y(x,y)
pX(x)pY(y)dxdy (27)
Note in Eq.24 and Eq.27 that it is necessary to know the exact pdf’s for estimating the MI and
this is the most sensitive part in the MI estimation. Several methods have been proposed in
the literature to estimate such joint densities, see Duda et al. (2009); Lotte et al. (2007).
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7. Summary and conclusions
In this chapter the fundamental concepts in the nervous system and different tools for
EEG signal processing have been briefly explained. Several concepts in visual analysis of
the EEG, brain rhythms, artifacts and abnormal EEG patterns, including EEG applications
such as epilepsy detection, EEG modelling, EEG feature extraction, epilepsy detection and
classification with methods oriented to dimensionality reduction have been reviewed. The
chapter also provides key references for further reading in the field of EEG signal processing.
Although all methods have been described in a brief way, they are introduced to give a good
theoretical grounding in EEG processing and to better understand the methods proposed
and their performance. Signal processing algorithms for EEG applications have specific
requirements in filtering, feature extractions and selections.
EEG is widely used as a diagnostic tool in clinical routine with an increasing develop of both
analytical and practical methods. Its simplicity, low cost and higher temporal resolution of
EEG maintains this tool to be considered in applications such as epilepsy seizures detection,
sleep disorders and BCI.
Future work implies the design of new EEG artifacts elimination methods, feature extraction
to obtain possible hidden information and dimensionality data reduction.
In medical environment, the steps that we follow in a classification problem are: (i) denoising
and artifacts removal, (ii) LFE features extractions, (iii) detection/classification using these
features and their combinations, (iv) if we do not have conclusive results, we add features
from wavelets and fractional Fourier transform, (v) detection/classification using these
features and their combinations, (vi) we apply dimensionality reduction if necessary.
An additional potential field of research is the EEG Integration with other techniques such
as fMRI. The principal drawback of the EEG is its low spatial resolution because it depends
on the number of electrodes. MEG has a better temporal resolution than EEG but suffers the
same disadvantage. fMRI solves this problem and its spatial resolution is on the order of
milimeters. The integration of these techniques is of vital importance in neuroscience studies
because this could improve the detection of other neurodegenerative diseases like Alzheimer,
Parkinson, depression or dementia.
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