model based dynamic analysis of human … · model based dynamic analysis of human sleep...

MODEL BASED DYNAMIC ANALYSIS

OF HUMAN SLEEP

ELECTROENCEPHALOGRAM

Thesis submitted for the degree of Doctor of

Philosophy at the University of Leicester

by

Yuehe Wang

Engineering Department

Leicester University

March 1997

UMI Number: U090574

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... O sleep, O gentle sleep,Nature’s soft nurse, how have I frighted thee, That thou no more wilt weight my eyelids down and sleep my senses in forgetfulness...

Shakespeare

{it has been considered that he suffered from insomnia)

MODEL BASED DYNAMIC ANALYSIS OF HUMAN SLEEP

ELECTROENCEPHALOGRAM

by

Yuehe Wang

Declaration of Originality

A thesis submitted in fulfilment of the requirements for the degree of Doctor of

Philosophy in the Department of Engineering, The University of Leicester, UK. All

work recorded in this thesis is original unless otherwise acknowledged in the text or

by references. No part of it has been submitted for any degree, either to the

University of Leicester or to any other university.

Yuehe Wang

March 1997

Acknowledgements

Firstly, I would like to express my thanks to Professor Barry Jones for his

kindness and his appreciation of my abilities (capabilities). Deepest thanks go to him

and to Dr. Chris D. Hanning (General Hospital, Leicester, UK.) for their

encouragement, advice and kindly help both academic and financial during my

Ph.D. studies in Leicester University. Without their support it would be impossible

for me to finish.

Particular thanks also go to Dr. John C. Fothergill and Dr. F.S. Schlindwein for

their extremely beneficial advice and supervision of my work.

I would like to express my gratitude to Dr.Chris Idzikowski (NCE Brainwaves,

N. Ireland), Dr. Stephen Roberts (Dept, of Engineering Science, Oxford University)

and James Pardey (Dept, of Engineering Science, Oxford University) for their

kindness in supplying the EEG data used and for their helpful advice. Many thanks

also go to Mrs. Jane Jones (NCE Brainwaves, N. Ireland) for staging sleep recording

used and teaching me the rudiments of EEG recording.

My research was made enjoyable by all members of the Biomedical Engineering

Group in the Dept, of Engineering at Leicester University. Thanks must go to Mr.

Yuhua Li, Dr. Paul Goodyer, Dr. Michael J. Pont, and Dr. Manho Kim, to name but

a few.

Thanks are also due to Dr. Zhonghe Wang, Dr. Lu Xiaoyun, Dr. Sun Wei and

Mr. Lou Zuhua for their valuable comments on my work and their enthusiastic

support.

Finally, very special thanks go to my wife Hao Wang and my parents who so

encouraged and supported me throughout the period of my stay in Leicester. I want

to thank Jeffrey, my son, for being bom.

Abstract

MODEL BASED DYNAMIC ANALYSIS OF

HUMAN SLEEP ELECTROENCEPHALOGRAM

Yuehe Wang Ph.D. thesisEngineering Department March 1997Leicester University

For sleep classification, automatic electroencephalogram (EEG) interpretation techniques are of interest because they are labour saving, in contrast to manual (visual) methods. More importantly, some automatic methods, which offer a less subjective approach, can provide additional information which it is not possible to obtain by manual analysis.

An extensive literature review has been undertaken to investigate the background of automatic EEG analysis techniques. Frequency domain and time domain methods are considered and their limitations are summarised. The weakness in the R & K rules for visual classification and from which most of the automatic systems borrow heavily are discussed.

A new technique — model based dynamic analysis — was developed in an attempt to classify the sleep EEG automatically. The technique comprises of two phases, these are the modelling of EEG signals and the analysis of the model’s coefficients using dynamic systems theory. Three techniques of modelling EEG signals are compared: the implementation of the non-linear prediction technique of Schaffer and Tidd (1990) based on chaos theory; Kalman filters and a recursive version of a radial basis function for modelling and forecasting the EEG signals during sleep. The Kalman filter approach produced good results and this approach was used in an attempt to classify the EEG automatically. For classifying the model’s (Kalman filter’s) coefficients, a new technique was developed by a state- space approach. A ‘state variable’ was defined based on the state changes of the EEG and was shown to be correlated with the depth of sleep. Furthermore it is shown that this technique may be useful for automatic sleep staging. Possible applications include automatic staging of sleep, detection of micro-arousals, anaesthesia monitoring, and monitoring the alertness of workers in sensitive or potentially dangerous environments.

vi

Contents

TABLE OF CONTENTS viLIST OF ABBREVIATIONS AND SYMBOLS ix

1. INTRODUCTION 1

1.1 Background 1

1.2 Signal acquisition 3

1.3 Sleep related signals 8

1.3.1 EEG signals 8

1.3.2 EMG signals 10

1.3.3 EOG signals 11

1.4 Sleep staging techniques 12

1.5 Sleep stage definitions 14

1.5.1 wakefulness 16

1.5.2 NREM sleep 16

1.5.3 REM sleep 18

2. REVIEW OF AUTOMATIC EEG ANALYSIS 25

2.1 Introduction 25

2.2 EEG interpretation 26

2.2.1 Interpretation in the frequency domain 26

2.2.2 Interpretation in the time domain 29

2.2.3 Fractal and deterministic chaos theory in EEG analysis 32

2.3 EEG feature classification 35

2.3.1 Techniques for automatic EEG feature classification 36

vii

2.3.2 EEG feature classification for sleep staging 38

2.4 Controversies over classification rules 39

3. DIFFERENTIAL TOPOLOGY 42

3 1 Basic topology 42

3.2 Quotient space and quotient topology 45

3.3 Tangent Bundles and Tangent Space 46

3.4 Vector fields and solutions 49

4. BACKGROUND THEORIES FOR MODELLING THE EEG 52

4.1 Introduction 52

4.2 Kalman filtering 54

4.2.1 Introduction 54

4.2.2 State-space representations 58

4.2.3 Kalman filter algorithm 61

4.3 Non-linear modelling techniques 70


4.3.2 State space reconstruction (Method of delays) 73

4.3.3 Global prediction techniques 78

4.3.4 Local prediction techniques 79

4.3.5 Radial Basis Functions 81

5. MODELLING OF EEG 84

5.1 Introduction 84

5.2 EEG modelling using local prediction technique 86

5.3 EEG modelling using Kalman filter 91


5.3.2 Model order 92

5.3.3 EEG modelling 96

viii

5.4 EEG modelling using Radial Basis Functions

- Adaptive non-linear modelling by a modified Kalman Filtering

approach 99


5.4.2 Outline of the algorithm 100

5.4.3 EEG modelling 104

5.5 Performance comparison 109

6. A METHOD FOR CLASSIFYING THE COEFFICIENTS OF THE

MODEL 1216.1 Introduction 121

6.2 Embedding the model’s coefficients into their state space 124

6.3 Classifying the model’s coefficients 127

7. RESULTS 140

8. CONCLUSIONS AND FURTHER WORK 172

8.1 EEG interpretation 172

8.2 EEG feature classification 177

8.3 Discussion 178

8.4 Further work 180

REFERENCE

APPENDICES

A. Akaike’s Final Prediction Error criterion

B. Details of the EEG signals used

LIST OF ABBREVIATIONS AND SYMBOLS

ANFIS Adaptive-Network-based Fuzzy Inference System

APs Action potentials

AR Autoregressive model

ARMA Autoregressive moving-average model

ARMAX Autoregresive moving-average model with exogenous

inputs

ARX Autoregressive model with exogenous input

ED Embedding dimension

EEG Electroencephalogram

EKF Extended Kalman filter

EMG Electromyogram

EOG Electrooculogram

FFT F ast F ourier transform

FIR Finite impulse response model

FPE Akaike’s Final Prediction Error

HR

MA

ME

MT

MU

MUAP

NLF

NPE

NREM

PA

PSPs

R & K rules (criteria)

RBF

REM

Infinite impulse response model

Moving-average model

Burg's maximum entropy

Movement time

Motor units

Motor unit action potential

A package of Non-linear Forecasting For Dynamical

Systems

Normalized prediction error

Non-rapid eye movement sleep

Atlas points or points used for embedding

Post-synaptic potentials

Rechtschaffen and Kales scoring system for sleep

Stages of human subjects

Radial Basis Functions

Rapid eye movement sleep

Chapter 1

INTRODUCTION

This chapter contains some historical and technical vignettes, as a modicum of

background knowledge which is necessary in order to show how a small plant has

grown into a remarkable tree.

1.1 Background

The nature of sleep has been a topic of constant interest since antiquity (over

2000 years), but systematic research on sleep and sleep mechanisms began only in

the 19th century and it is only in recent years that the true importance of sleep

studies both in clinical and scientific applications has been recognised.

In 1929, Hans Berger first recorded the electrical activity, termed the

electroencephalogram (EEG), of the human brain. Since then, the application of

EEG in characterizing different levels of sleep by Loomis paved the way to the

Chapter 1. INTRODUCTION 2

discovery by Aserinsky and Kleitman that sleep consisted of two distinct phases

rather than a single one which merely varied along a continuum in depth. In Loomis’

time, EEG patterns were classified from wakefulness to sleep into five stages, (A,

B l, B2, C, D, and E). This classification was widely adopted, until some years later

(in 1953), Aserinsky and Kleitman discovered rapid eye movement (REM) sleep.

Dement and Kleitman then (in 1953) proposed a classification system, in which

REM was differentiated from non-rapid eye movement (NREM) sleep. This system

was modified by Rechtschaffen and Kales in 1968 (R & K rules) and has been the

most widely used since then.

Traditionally, the most important aspect of sleep analysis is sleep staging by

visual assessment of the EEG, electrooculogram (EOG) and electromyogram (EMG)

by trained observers using a set of standardised rules (i.e. R & K rules).

Computerised analysis of sleep recordings was first tried in 1968 (Lacroix, 1984).

Since then, a number of automated systems have been developed for EEG analysis,

but few of them have been designed for routine sleep staging in a clinical

environment (Stanus, 1987). Automatic sleep staging and automatic EEG

interpretation techniques are of interest not only because they are labour saving, but

also because they can be made consistent and quantitative in contrast to manual

methods. More importantly, automatic methods may provide additional information

which is not obtainable by manual analysis


Although a great deal of work has been done on sleep since the 1930s, there is

still much that we do not know and there are many controversies that have not been

resolved.

1.2 Signal acquisition

In order to identify and classify sleep, it is necessary to monitor simultaneously

the electrical activity of three systems: the brain (by means of EEG), the movement

of the eyes (by means of EOG), and the muscles tone (by means of EMG). The

recording of the EEG is a very important technique in studies on sleep. Much of our

current knowledge concerning sleep has been made possible by the use of EEG

recording techniques. The EEG, EMG and EOG have provided an apparently

"objective" basis for the study of sleep and sleep related phenomena.

There are two techniques that can be used for EEG aquisition, invasive or

non-invasive recording. The most widely used is that non-invasive method from the

scalp by means of surface electrodes. Obviously, the advantage of non-invasive

procedures in the sleep environment is that it does not cause significant patient

discomfort, so that there is little risk of disrupting the sleep process.

As the activity recorded differs from one region of the scalp to another, in a full

EEG recording session, up to 20 channels are recorded simultaneously with the

electrodes distributed widely over the head. In contrast, sleep EEG recordings use 1


to 4 channels and are recorded in parallel with EOG and/or EMG recordings. EEG

signals may be measured in three ways: (i) from single electrodes each with

reference to a common electrode (usually on the mastoid or preauricular point); (ii)

between pairs of electrodes; or (iii) from single electrodes with respect to the

average of all the other electrodes.

The first step for the electrode attachment is the measurement of electrode

positions according to the so called international 10-20 system of electrode

placement. Figure 1.1 illustrates the 10-20 placement system and the reliable

recording of EEG relies on accurate measurement of the skull according to this

system.

After measurements are made, the skin or superficial dermis, where the

electrodes will be placed, needs to be degreased and cleansed thoroughly by brisk

rubbing with gauze or cotton wool on which acetone or now, more commonly, skin

prep has embedded, (skin prep contains some impedance reducing electrolyte

material.) This is generally sufficient to ensure adequate conduction when the

electrode is applied. EEG electrodes should be non-polarisable and of the

silver/silver chloride type (or gold type) attached to the scalp with collodion. The

electrolyte under the electrode should be scooped into the electrode before the

electrode is glued in place. While by using Montreal Neurological Institute type of

electrode, which has a hole in the back, the electrolyte can be added after the

electrode is glued tightly on the head. Moreover this electrode is very helpful for

long time, such as over night sleep, recording (10 - 12 hours), as it has a hole in the


back of the electrode through which further electrolyte paste may be added during

the recording. Electrode impedance must be carefully checked before recording. It

should be less than 5 kilohms.

N asion '

P re a u r ic u la rP o in t

In io n

Figure 1.1 Schematic diagram showing measurements for the

international 10-20 electrode placement system.

For routine recording of EOG, the R and K manual recommends referential

recordings from each outer canthus to the ipsilateral ear (or side of neck). The

electrodes should be offset from horizontal, one slightly above and one slightly

below the horizontal plane. These derivations have the advantage of showing

horizontal and vertical eye movements as out-of-phase potentials in the two

channels. However, they have the disadvantage of containing much EEG artifact in


the leads, especially in slow wave sleep when the EEG reaches maximum amplitude.

The eye movement electrodes should be non-polarisable and of stick-on type.

Regular EEG electrodes could be used. The skin need to be cleansed as in

preparation for EEG leads. It is recommended that they are kept in place by

microporous adhesive tape or sticky discs which retains its adhesion well over long

recording. As the upper limit of the EOG frequency band is much lower than that of

EEG, an EEG recording channel can be used without further modification.

The EMG is taken as the potential between two electrodes, one on each side of

the neck beneath the chin over the mylohyoid and digastric muscles. Stick-on

silver/silver chloride electrodes may be used, and regular EEG electrodes could be

used. As in the case of EEG and EOG recording, the skin is thoroughly cleansed

before applying the electrodes. General requirement of impedance for each electrode

need to be less than 5 kilohms. Sticky discs and flexible adhesive tape are

recommended to place the electrodes firmly on the skin. The electrodes are

connected by bipolar linkage to a single channel. It is often adequate for indicating

the presence of muscular activity by using an EEG channel with the highest possible

frequency response (usually 70 - 100 Hz) for recording of EMG activity in most

instances.

In sleep studies, the most commonly used placement of the electrodes for the

EEG, EMG, and EOG recording is shown in Figure 1.2, by which the upper drawing

represents recommendations for placement of electrodes (El, E2, A1 and A2) for


recording eye movements (EOG) and electrodes for recording EMG; lower drawing

shows recommendations for placement of C3/A2, and/or C4/A1 electrodes for

recording EEG. In some laboratories an occipital EEG (usually 01 /A2 or 02/A1) is

record routinely as an adjunct to the central EEG. It is particularly useful for

assessing sleep onset or arousals during sleep.

LEFT EYE - A1

RIGHT EYE - A1

EMG

C4 - A1

Figure 1.2 Electrode placement in sleep research. (From

Rechtschaffen and Kales, 1968)


1.3 Sleep related signals

1.3.1 EEG signals

EEG analysis is concerned with the study of a small, constantly changing

electrical potentials form the brain which can be collected from scalp electrode. The

electrode, with about 100 mm2 in area, converges the averaged electrical activity

from a substantial volume of underlying cortex through the thickness of skull and

meninges. It was originally thought that EEG waves might be made up of summated

action potentials, but because their short duration ( 1 - 2 ms) tend to overlap much

less than do Post-synaptic potentials (PSPs). PSPs are of electrical changes in the

post-synaptic membrane with lower amplitude than the action potential and last

longer, 10 - 250 ms. Enough evidence, exists to state that the EEG on the scalp is

mainly composed by synchronously occurring PSPs, (for example, the research of

simultaneous recordings o f the activity of individual neurones and of the overlying

EEG achieved by O.D. Creutzfeldt, et al. 1966). It has been estimated that

synchronous PSPs in only 1% of cortical neurones would be sufficient to account for

the signals normally seen in the EEG.

The frequency range of the scalp EEG has a fuzzy lower and upper limit. The

major power distributes in the range of 0.5 to 60 Hz in which most EEG studies for

clinical and research purposes are carried out. This may reflect the limitations of the

recording systems rather than the actual range of activity present. There is for


instance evidence that EEG contains information at over 200 Hz. However, the ultra-

slow and ultra-fast frequency components play no significant role in the clinical

analysis. By convention, and partly for historical reasons, the frequency range is

subdivided into four frequency bands which are:

Delta (8) — below 4 Hz;

Theta (0) — not less than 4 but less than 8 Hz;

Alpha (a) — 8 to 13 Hz inclusive;

Beta (p) — More than 13 Hz.

Amplitudes of the scalp EEG range from 10 to lOOpV rarely exceeding 150 pV in a

normal waking subject. EEG amplitudes vary with many factors, such as age,

electrode placement and skull morphology, therefore the precise determination of the

voltage of each wave is unnecessary and should be discouraged.

The EEG signal, which reflects the general functional state of the brain, has

become a standard measurement made in clinical neurophysiology. In sleep studies,

EEG is the core measurement in polysomnography. The four stages of NREM sleep

are distinguished from one another principally along this signal.

The EEG changes continually in a random manner, and in association with

wakefulness and sleep cycle. It also changes gradually over the lifetime of the

individual and shows marked differences between one person and another.


1.3.1 EMG signals

EMG is the study of electrical activity in the muscles. With electrodes placed on

the skin surface, the signal recorded when a muscle contracts is known as the surface

EMG. EMG is one of the largest and most easily measured bioelectrical signals.

When quantified in some way it is very reliable indicator of whether a muscle is

active.

Muscle fibres are organised into functional units within a muscle, which are

called motor units (MU). The fibres belonging to one MU are spread over a certain

area of the muscle cross-section, and are thus intermingled with fibres from several

other MUs. A MU consisting of several muscle fibres is innervated by a single

motor neuron. Roughly at the midpoint along the length of each muscle fibre is an

endplate, where action potentials (APs) are generated after synaptic transmission

from the motor nerve. The muscle fibres as well as the nerves obey the “all-or-

nothing” law, i.e. they have two states: inactive and excited. When the motor neuron

is activated, all muscle fibres in the MU respond, producing a motor unit action

potential (MUAP) which is the temporal and spatial summation of the APs of all

muscle fibres in the MU. At low contraction levels few motor units are active. With

increasing contraction strength, the firing rate of these motor units increases and

new, larger motor units are recruited. Even at moderate contraction levels single

MUAPs tend to overlap, producing a random like signal, the interference (quite

complex) EMG.


The EMG signal is composed of a mixture of different frequency components,

with most of the signal energy falling within the 10 to 1000 Hz range.

EMG recording is essential for the study of certain types of muscle activity

during sleep. In the standard polysomnographic recording, the EMG is used as a

criterion for staging REM sleep. The level of tonic EMG is normally absent in REM

sleep. In addition of NREM sleep, the level of EMG usually decreases from

wakefulness through stages 1, 2, 3 and 4.

1.3.1 EOG signals

The EOG records the electrical potential generated within the eye, and is made

simply to document the presence or absence of eye movements.

The EOG recordings are based on the small electrical potential difference, often

over 200 pV, from the front to the back of the eye. The cornea is positively-charged

with respect to the negatively-charged retina. Therefore, the eye ball acts as a

potential field within a volume conductor in the head. Because of this essentially

constant potential difference between the retina and the cornea, movement of the

eyes can be measured from electrodes placed beside the eyes. An electrode nearest

the cornea will register a positive potential; an electrode nearest the retina will

register a negative potential. As the eye moves, the positions of the cornea and retina

change relative to a fixed position of the electrode, and a potential change will

registered by the electrode.


The EOG has an amplitude of about 20 pV per degree of rotation of the eyeball,

and frequency response up to about 30 Hz is adequate for the recording most of the

rapid eye movements

In sleep research, for the recognition of sleep stages, eye movement recording is

necessary for sleep staging and it is required by the R and K criteria. That is the

rolling eye movements of stage 1 and the rapid eye movement of stage REM. Eye

movement recording by means of EOG is also useful in EEG recording for the

identification of eye movement artifacts.

1.4 Sleep staging techniques

Conventionally, sleep stage is visually assessed from a paper record by an

expert human observer. Three parameters (EEG, EOG and EMG) to assess sleep

according to internationally standardized criteria (R & K. rules) are needed, and

EEG is the core measurement among them. This classification of different sleep

stages is based on patterns of EEG waveforms (i.e., delta waves, K-complexes, theta,

alpha, beta waves, and sleep spindles in the EEG channels), combined with eye

movement in the EOG channel, and the bursts of muscle activity in the EMG

channel when available.

The R. & K. rules provide detailed guidelines and criteria for staging normal

human sleep. When staging a sleep recording, it is necessary and convenient to


divide the chart into suitable segments or epochs and to assign a sleep stage value to

it based on the dominant pattern in that epoch. The most common epoch lengths are

30 sec or 20 seconds. Epochs of longer than 30 or 40 seconds tend to overlook stage

changes of relatively short duration, while those less than 20 seconds involve

excessive work and are considered too tedious by most sleep laboratory in scoring

the record. At the present time, there is some criticism of the use of these epochs as

being artificial, since this system regards the essentially continuous process of sleep

as a set of discrete stages, thus giving the impression of sleep as “stepwise” changes

whereas it is probably a continuum. Furthermore, very small time-scale events tend

to be missing with these fairly large epoch lengths. These events may be

characteristic of micro-arousals or the types of disturbed sleep associated with

certain disorders, and are therefore of particular interest to the clinician.

The R. & K. rules were initially designed for visual sleep staging but most

automatic systems for sleep analysis used today are also based on this system. It is

noticeable that the procedure of visually scoring of sleep contains some major

disadvantages. First of all, it is labour-intensive and time-consuming. Secondly, the

R. and K. rules contain some subjective components, such as segmentation sleep

recording, discretization sleep procedure and scoring some sleep stages based on

very short events (or specific patterns, e.g. K complexes and sleep spindles) rather

then background activity (also see chapter 2, section 2.4). Thirdly, the procedure

suffers from low accuracy and consistency, i. e. the staging results are often different

from one observer to another, even between two assessments of a record by a single


scorer. Therefore, an alternative, rapid and objective assessment of sleep recording is

desirable in a clinical environment.

Automatic sleep staging and/or automatic EEG analysis are becoming important

tools in this field because they are labour saving and can be made consistent and

quantitative in contrast to manual methods. More importantly, automatic methods

may provide additional information which are not obtainable by manual analysis,

such as the detection of brief arousals. Although a number of automated systems

have been developed for EEG analysis during the last 20 years, few of them have

been designed for routine sleep staging in a clinical environment (Stanus, 1987).

The techniques used in visual sleep staging are relatively unchanged from

Rechtschaffen and Kales time (1968) and an overall review can be found in papers

of Hasan (1983), Binnie (1982), and Cox Jr. (1972).

1.5 Sleep stage definitions

Traditionally the most important aspect of sleep analysis is sleep staging. Based

on a collection of physiological parameters (i.e. EEG, EOG, and EMG), two separate

states have been defined within sleep. These are the states of non-rapid eye

movement (NREM) and rapid eye movement (REM), NREM and REM exist

virtually in all mammals and birds.

NREM sleep is conventionally subdivided into four stages (i.e., stage 1, 2, 3 and

4), which are mainly defined along one measure of EEG. The EEG pattern in NREM


sleep is commonly described as synchronous, with such characteristic waveforms as

sleep spindles, K complexes, and high-voltage slow waves. REM sleep generally is

not divided into stages, and it is, by contrast, defined by episodic bursts of rapid eye

movements, muscle atonia, and EEG activation. NREM sleep and REM sleep

continue to alternate through the night in cyclic fashion with a period of about 90 to

110 minutes. The normal adult human enters sleep through NREM sleep (begins

with sleep stage 1), and REM sleep does not occur until about 80 minutes or later.

REM sleep episodes generally become longer across the night. Sleep stages 3 and 4

occupy less time in the second cycle and may disappear altogether from later cycles,

as the sleep stage 2 expands to occupy the NREM portion of the cycle.

The hypnogram, see figure 1.3, is a plot of sleep stage against time. Movement

time (MT) and wakefulness as additional stages may be added into the hypnogram.

According to R. and K. rules, large body movements are associated with high

amplitude EMG activity which commonly also involves EEG and EOG channels.

When the EEG and EOG are obscured by such muscle tension and/or amplifier

blocking artefacts for more than half of an epoch it is impossible to stage it and the

epoch is scored as movement time (MT). For a normal young adult the hypnogram

typically takes the form shown in figure 1.3.

In summary, sleep stages may be defined as follows:


1.5.1 Wakefulness

An overnight recording usually contains a period of wakefulness before sleep

onset in which the individual normally is in a relaxed state with eyes closed. During

this relaxed wakefulness, the EEG is composed predominantly of sinusoidal alpha

activity (8 -13 Hz inclusive) intermixed with lower amplitude irregular beta waves

(more than 13 Hz). Alpha activity is the most important feature though this may be

suppressed by attention and anxiety or blocked if the subject is looking about.

Muscle tone is generally high, and eye movements may be present, such as eyelid

blinks or slow rolling movements. As the subject becomes more drowsy, alpha

activity decreases with accompanying relatively slow activity.

1.5.2 NREM sleep

The NREM state is often called “quiet sleep” because of the slow, regular

breathing, the general absence of body movement, and the slow, regular brain

activity shown in the EEG.

Sleep stage 1

Stage 1 is characterized by the great decrease of alpha activity (less than 50% of

the record), some increase in beta which is a low-amplitude mixed-ffequency signal

and the slower theta ( 4 - 7 Hz) activity. In the EOG channel slow lateral eye

movements may appear. As the subject progresses toward stage 2, the slower


activity predominates and vertex sharp waves may appear with an increase in slow

components especially in younger subjects.

Stage 1 in the normal young adult occupies approximately 2 - 5% of total night

time sleep and often occurs as a transition from wakefulness or body movements

during sleep to other sleep stages.

Sleep stage 2

This is composed of largely a theta and beta background with some low-

amplitude delta components comprising less than 20% of the record, and is

characterized by the appearance of two types of intermittent events; the spindles and

K-complexes. Spindles are brief bursts of rhythmic 12-14 Hz waves, lasting at least

0.5 second. K-complexes are composed of a high-amplitude negative wave followed

by a positive wave. Sometimes brief bursts of low-amplitude 12-14 Hz activity may

be superimposed on the K-complex. It should be noted that in addition to its

spontaneous appearance during stage 2 sleep, the K-complex can occur at other

times in the sleeping person in response to auditory stimuli. The EMG shows some

tonic activity but this is less than in stage 1 and significantly less than in the waking

stage.

Stage 2 occupies the greatest amount of total sleep time in the normal young

adult about 45 - 55%.


Sleep stage 3

This stage is characterized by the appearance in between 20 -50 % of the epoch

of slow wave activity (at less than 2 Hz), which are high amplitude (at least 75 pV

from peak to peak). Collectively, stage 3 and stage 4 are often referred to as slow-

wave sleep.

Sleep stage 4

In stage 4 slow wave activity makes up more than 50% of the epoch. Sleep

spindles may or may not be present during stage 3 or 4.

Stage 3 normally occupies about 3 - 8% and stage 4 occupies 10 - 15% of

normal over-night sleep in the young adult.

1.5.3 REM sleep

REM sleep, which has been called “active sleep” is an entirely different sleep.

A REM sleep period is characterised by three main features. The first is the

presence of conjugate rapid eye movements. The second that is during REM sleep

the EEG returns to a mixed frequency pattern with medium amplitude, and similar to

stage 1 except that vertex sharp waves are not prominent. In contrast to stage 2, there

are no sleep spindles or K-complexes in REM sleep. The third is that the EMG drops

to very low amplitude, indicating the decrease in tone of the submental muscles.


Under regular EEG laboratory conditions, the observation of REM sleep

requires a long waiting period, since the first phase of REM does not appear prior to

60 to 90 minutes after sleep onset in a normal night time sleep.

Figure 1.4 shows the typical EEG patterns, eye movement in right and left

EOG, and chin EMG patterns for different sleep stages and wakefulness.


Wake

MT

REM

10050 150 2500 200 300 350Time (Minutes)

Figure 1.3 The hypnogram for a normal young adult.

MT = Movement Time.

REM = Rapid eye movement.


L. EOG

R. EOG

EMG

1. Sec.

I = 50 p.V WAKE

(a)L. EOG

- - - - - - - - 1-— | —“ * y — * " ' j

R. EOG

EMG , I

^ V V v ^ v 1

1 Sec.

= 50HV STAGE 1

(b)

Figure 1.4. Stages of sleep as recorded on the EOG, EMG and EEG. (a)

Wakefulness; (b) sleep stage 1. (Stages 2, 3 ,4 and REM are shown on the next

pages.)


L EOG

R. EOG

EM G

EEG

= 50 HV STAGE 21 Sec.

(c)

L. EOG

R. EOG

1 Sec.

I = 50 HV STAGE 3

(d)Figure 1.4 continue. Stages of sleep as recorded on the EOG, EMG and EEG. (c)

Sleep stage 2; (d) sleep stage 3. (Stage 4 and REM are shown on the next page.)


R. EOG

EEG

MJ = 50 p.V

1 Sec.

STAGE 4

(e)

L. EOG

R EOG

EMG

EEG *

W VV'WM

J = 50 nV1 Sec.

STAGEREM

(f)

Figure 1.4 continue. Stages of sleep as recorded on the EOG, EMG and

EEG. (e) Sleep stage 4; (f) REM sleep. Note the high EMG and


eye movements during wakefulness, Sleep stage 2 is

characterized by sleep spindles and K-complexes as showed

underline. The EEG is similar during stage 1 and stage REM,

but the EMG is high and REMs are absent in stage 1. Stage 3

and 4 are characterized by slowing of frequency and increase in

amplitude of the EEG. (From Mendelson W.B. et al., 1977.)

Chapter 2

REVIEW OF AUTOMATIC EEG

ANALYSIS

2.1 Introduction

Most of the automatic procedures for sleep EEG staging include feature

extraction from the EEG followed by the use of these features for classification of

the EEG into various stages (EEG classification).

This chapter will present the background and traditional methods of EEG

analysis based on its interpretation and classification in the context of sleep studies.

Frequency domain and time domain methods are considered and their limitations are

summarised briefly. The novel techniques of fractal and deterministic chaos theory

used for EEG analysis are included. Consequently, the EEG feature classification

techniques are reviewed. Finally, the weaknesses in the R & K criteria for visual

classification and from which most of the automatic systems borrow heavily will be

Chapter 2 REVIEW OF AUTOMATIC EEG ANALYSIS 26

discussed. No attempt has been made in this chapter to provide a comprehensive list

of references, but a sufficiently representative sample of the recent literature is

included.

2.2 EEG interpretation

Many methods of EEG interpretation have been developed in the last few

decades. These methods can be broadly grouped into the two main categories of

frequency domain and time domain methods. Frequency domain analysis is based on

the assumption that the EEG can be interpreted as a collection of periodical signals,

whilst in time domain analysis the consecutive EEG waves are treated as a series of

aperiodic phenomena. In addition, time domain methods normally tend to mimic the

non-automatic interpretation process of a human operator.

2.2.1 Interpretation in the frequency domain

The most used frequency domain method in EEG study is spectral analysis,

which is effective in characterizing dominant quasi-periodic rhythms (Lim A. J. and

Winters W. E., 1980). This method is mainly used for the analysis of background

electrical activity and spectra are computed from fixed-length signal segments

(epochs) of about 30s duration. This yields good results when the background

activity is abnormal but the short time structure of the EEG is lost in this approach

(Bodenstein, 1977). Spectral analysis based on parametric (autoregressive


modelling) or non-parametric methods (e.g. fast Fourier transform) have been major

tools for the representation of the EEG signal segments.

In the parametric method based on linear predictive filtering there are several

approaches using autoregressive moving average (ARMA) and autoregressive (AR)

algorithms (Sanderson, 1980; Bodenstein, 1977; Balocchi, 1987; Lopes da Silva,

1981). An evaluation of these simple algorithms was carried out by W. D. Smith

(1986). The AR method, which was reportedly able to provide high resolution

spectral estimates from short time intervals, has already been applied to EEG

spectral estimation, EEG simulation, transient detection, and the detection of

segment boundaries in several ways (Bodenstein, 1977). The use of a Kalman filter

algorithm (which can be treated as an adaptive AR model) for EEG analysis was

originally introduced by Bohlin (1971) and has been employed by several

researchers (Roberts, 1991; Skagen, 1988; Bartoli and Cerutti, 1982). Jansen (Jansen

et al., 1981) found that the Kalman filter coefficients gave a qualitatively better

description of the spectral properties of the EEG over a longer period of time than

other non-adaptive AR models. However, when the EEG signal is highly non-

stationary, such as in the case of the occurrence of a large artifact, the artifact will

influence the filter coefficients for several seconds thereafter and thus produce

inaccurate spectral estimates for the current epoch. The Burg's maximum entropy

(ME) algorithm avoids this problem by using data only in the interval being

analysed. In Jansen's studies a comparison of spectral analyses from a Kalman filter,

a stationary AR model (derived from the Yule-Walker equations) and Burg's method


was described. Because the Yule-Walker approach sometimes results in unstable

models Jansen et al suggested that it should not be used.

Of the non-parametric methods, the Fourier transform, which may have first

been used in EEG analysis by Dietsch (1932), has been most widely applied

(Scheuler, 1990; Pigeau, 1981; Jervis, 1989; Lacroix and Hanus, 1984; Dumermuth,

1983). It has provided good results and has become more easily implemented with

the introduction of the fast Fourier transform (FFT). There are, however, some well-

known drawbacks related to the Fourier analysis of EEG signals, such as the

enhancement of low-frequency components connected with the shape of the epoch

window (Daskalova, 1988). There seems to be a tendency for the use of the FFT to

be superseded to some extent by AR techniques in recent years, especially as their

computation time (about three times that of a comparable FFT) is no longer a

problem with fast modem computers.

Other approaches to non-parametric methods include: the Walsh transformation

(Li, 1990), power spectral density (Torbjom, 1986; Saltzberg, 1985), individual

frequency band analysis (Laurian, 1984; Barcaro, 1983; Scheuler, 1988), coherence

analysis (Sterman, 1977), and EEG variance (Hiroyoshi, 1991). Hao (1992) used

complex demodulation (the Hilbert transform), which enables the amplitude and

phase of particular frequency components to be described as functions of time so

that the instantaneous frequencies of sleep spindles in the EEG may be estimated.

Several of the methods for detection of sleep spindles are in the frequency domain


(Pivik, 1982; Fish, 1988). Recently, Sauter (1991) used an asymptotic local

approach for the detection of spindles together with an AR model.

The analysis methods in the frequency domain have proved very useful. Most

EEG spectra exhibit a distinctive structure in which peaks and valleys can be clearly

distinguished and it is the peaks (corresponding to ’’rhythms") to which most

attention has been devoted.

2.2.2 Interpretation in the time domain

Although spectral analysis of EEG has provided a considerable amount of

information, some of the significant EEG patterns are aperiodic, e.g. K-complexes

and spike waves. The identification of these aperiodic events is critical to both

diagnostic EEG and to sleep stage EEG. To detect these aperiodic waveforms

different data-processing techniques (many in the time domain) may be employed.

The method of visual interpretation of the EEG is considered as a time domain

classification in which the electroencephalographer sees the paper record as a series

of waves of varying duration ("period", "interval", or "wave duration"). As already

mentioned, many methods of automatic EEG interpretation in the time domain

mimic the visual process. Thus, there is a family of techniques of "period analysis".

This family has two groups: the first one defines individual waves by the points at

which the signal passes through a base-line or near zero threshold and is known as


the level-crossing technique; the second defines the waves by peaks and troughs, the

so-called peak detection technique.

One of the first methods of period analysis was proposed by Cohn (1963). He

used a level-crossing technique to obtain a histogram of the number of pulses with

respect to inter-pulse interval. The other approaches at that time were by Legewie

(1969, zero-crossing technique) and Leader (1967, peak detection method). In 1980,

Lim combined zero-crossing and peak detection algorithms for sleep stage analysis.

Methods based on level-crossing detection tend to favour slow waves, while

those based on peak detection tend to favour fast waves (Lim, 1980; Kuwahara,

1988). This family of techniques suffers from problems associated with the

uncertainty in base line position and its fluctuations; these affect the accuracy of the

period measurements. Sometimes a high-pass filter or a complicated method of

finding the waveform midpoints or inflection points is used. Leader (1967) tried to

overcome the problem by the use of waveform midpoint detection. Daskalova

(1988,) used the alternative technique of finding the minima and maxima of

successive waves, which does not depend on the base line position. The method of

period analysis has been further improved by taking into account both the period and

the peak amplitudes of the EEG waves by Carrie (1971), Dascalov (1974), and

Palem (1982).

A novel approach was described by Hjorth (1970), who proposed a description

of the EEG in terms of three normalized slope descriptors: “activity”, “mobility” and


“complexity”. “Activity” is defined as the squared standard deviation of the

amplitude which is closely related to power. “Mobility”, which can be conceived of

as a mean frequency, is calculated from the standard deviation of the slope with

reference to the standard deviation of the amplitude. “Complexity”, a measure of the

'shape' of the signal, compares the rate of change of the slope with that of a sine

wave which has a complexity of 1. The difference between the mobility and the

complexity of a signal reflects the scatter of frequencies present. The descriptors are

comparatively simple to calculate and take much less computer time than the

calculation of the power spectrum. Experience suggests that this method can be used

to characterize different stages of sleep (Layzell, 1973; Binnie, 1982; Harris, 1987).

Kuwahara, in 1988, implemented another approach in which 3 different

recognition systems were used. The first was a modified interval histogram method.

The EEG amplitude was divided by 32 slice lines with an equivalent resolution of

6.25 pV. The period was measured as the time interval between the 2 points at

which the same slice line crosses consecutive positive slopes of EEG signals and the

interval histogram was made for the period collected for each 20s epoch. The second

was a zero-crossing detection algorithm for high voltage delta waves. The third was

a spindle selection algorithm.

The techniques for K-complex detection are in the time domain. The first

automatic detection was achieved by G. Bremer in 1970. Recently (in 1992),

Bankman used a feature-based K-complex waveform detection technology, which


involved neural networks and provided good agreement with visual K-complex

recognition.

Possibly because these so-called time-domain techniques have a less formal

mathematical basis, a multiplicity of different methods has been devised and,

therefore, it is harder to divide them into mathematical categories than it is for those

in the frequency domain.

2.2.3 Fractal and deterministic chaos theory in EEG analysis

From the mid-1980s new methods of signal processing which involve the

techniques of fractal and deterministic chaos theory have emerged. Conventional

signal processing or time series analysis has been limited for many years by the

underlying assumption of linearity. In the real world, of course, this assumption is

often far from reasonable (Kearney, 1992). Thus, there has been an explosion of

interest in non-linear dynamic systems and fractal analysis techniques after the

realization that a very simple non-linear system can lead to extremely complex

behaviours.

The term "fractal" was introduced by B. B. Mandelbrot (1982) to describe

objects (e.g. sets, functions or physical objects) that are too irregular to describe

using traditional geometry. Fractal geometry provides a general framework for the

study of such irregular sets. One of the main parameters used in fractal geometry is

the fractal dimension, which indicates geometrical properties such as scaling


properties and self-similarity. The fractal dimension, dp which is non-integer, relates

a body's volume, V, (assuming it to be a homogeneous solid) to its linear dimension,

L, by V oc Ldf ■ The next highest integer value above ^indicates how many spatial

dimensions would be filled by the object.

Systems in the real world are often non-linear and likely to have several degrees

of freedom and their state space will thus be multi-dimensional. Even a very simple

such system can lead to extremely complex behaviour and its orbits may appear to

move about at random, but always remaining close to a certain set in the multi

dimensional phase space. This particular set is called as an attractor. If this attractor

appears as a fractal, i.e. having a non-integer dimension, it is called a strange

attractor (or a fractal attractor). If a system has a strange attractor then it is exhibits

chaotic behaviour. A "box-counting" method (or other method) can be used to

estimate its fractal dimension from its attractor, this is usually established

approximately by the correlation dimension. The correlation dimension is a very

important parameter since it represents how complex the behaviour of the dynamic

system is. A correlation dimension of unity (or integer) indicates the system is

periodic, a correlation dimension of infinity indicates a truly random (totally

unpredictable) system. Any system with this dimension much larger than about 10

may be indistinguishable from a truly random system (Keamey, 1992).

In signal processing, an analysis, analogous to the linear predictive filtering of

the Kalman filter, may be carried out on a time series describing a parameter of a


system which exhibits deterministic chaos. Such a technique is used to predict the

next value(s) in the time series on the basis of the previous values; the number of

previous values, typically 2 to 8, depends on the complexity (or auto-correlation) of

the series. The state of the system at any time may be represented by a co-ordinate in

multi-dimensional space where each orthogonal axis represents either an

independent parameter or a previous point in the time series. In a chaotic system,

successive points will align themselves in orbits or patterns in this multi

dimensional space. The number of dimensions required, the so-called embedding

dimension, m, is related to the number of degrees of freedom, d, by m>2d+l. The

path of successive points is not possible to predict exactly (unless the system can be

accurately physically modelled) but approximate predictions can be made whose

accuracy deteriorates exponentially with the prediction interval. For any system the

pattern of points is known as a means of detecting the attractor.

Much work has been done to estimate the correlation dimension of the EEG on

the assumption that the brain is a complex dynamic system. Many researchers, such

as Abu-Faraj (1991), Xu and Xu (1988), Jan Pieter Pijn (1991), have used

correlation dimension to implement the characterization of the EEG. The

mammalian brain is certainly one of the most complex systems encountered in

nature. Many researches have shown that the EEG is generated by a complex

dynamical system (a high dimensional system) and has features of deterministic

chaos (e.g. Abu-Faraj, 1991, Doyon, 1992). The correlation dimension is a measure

of the complexity of a dynamic system. Mayer-Kress and layne (1988) used the


correlation dimension to evaluate depth of anaesthesia and discussed problems

associated with this dimensional analysis of the EEG. Babloyantz and Salazer (1985)

used non-linear dynamical methods for the study of brain activity during the sleep

cycle. They found the existence of chaotic attractors for sleep stage four and stage

two, but failed to find them in the awake stage and REM stage. Babloyantz and

Salazer also studied the correlation dimension and found that the correlation

dimension is near 4 when the subject is in deep sleep (stage four), 5 in sleep stage

two and about 6-7 in awakening. Awakening with opened eyes and REM sleep were

difficult to estimate (> 8-9 ?) (Doyon, 1992).

Fractal and deterministic chaos theory have thus made a claim of future clinical

value for characterising of complex or irregular data sets that defy interpretation by

conventional analytic tools. At present, however, it seems unclear whether this claim

is true. It may be that the use of the Kalman filter for predicting values in a time

series would give similar results, in practice, to a deterministic chaos approach even

though the underlying assumptions are not necessarily true (for example see Fowler,

1988).


2.3 EEG feature classification

Initially, automatic interpretation of the EEG was largely based on numerical

procedures that extracted certain features from the EEG segments; these features

were used in subsequent pattern classification stages.

2.3.1 Techniques for automatic EEG feature classification

Many different techniques for automatic EEG classification have been used

including: 1) rule-based systems (Baas, 1984; FFT was employed for spectral

estimation), 2) artificial neural networks (Jando, 1986; with FFT as the numerical

scheme to extract features), 3) fuzzy logic (Hu, 1991; Gath, 1980; using frequency

characteristics from FFT and AR model respectively), and 4) Bayesian filtering

(Lacroix and Hanus, 1984; also FFT was employed).

Most of the systems have had limited success because they have not taken into

account contextual information. This information relates to important spatio-

temporal relationships that exist in intrachannel and interchannel EEG data. In the

analysis of EEG, spatio-temporal information is of considerable importance.

Syntactic analysis (the analysis of temporal and spatial patterns within the EEG) has

been suggested as a possible approach since it can utilise contextual information and

therefore has good potential for EEG analysis (Cohen A., 1986; Gath 1989). Jansen

and Dawant (1989) employed a knowledge-based blackboard-system approach to

automated sleep EEG analysis in which spatio-temporal information was used. The


system consisted of five units: a blackboard, a collection of object descriptions, a set

of specialists, a scheduler and an object detection module. The object detection

module was used to identify what features need to be extracted and to fire

specialized signal processing modules. One of the advantages of this system is that it

achieves an opportunistic approach which allows the extraction of quantitative

information from the EEG signal only when needed by the reasoning processes.

Another approach, which utilised the contextual information, was implemented by

Jagannathan et al. in 1982 with programs which used rule-based logic with backward

chaining and a simple implementation of fuzzy logic in premise clauses comprising

IFTH EN rules.

Groups from the Johns Hopkins University and Hospital, USA, have used

neural networks for EEG waveform classification (Miller 1992). They are

developing this system by using CASENET, a flexible neural network simulation

package (Ebertart et al, 1989). The purpose of this work is to produce a portable

device for spike/seizure detection using low-cost hardware. Principe (1989) and his

colleagues (Chang et al., 1989) used different methods of EEG signal classification

for automatic sleep scoring by means of a rule-based expert system and a neural

network respectively (Miller, 1992).

In recent years, symbolic processing (including expert systems and co-operative

knowledge-based systems) and neural networks have attracted special attention

because of the novel approach; knowledge (sometimes "deep knowledge") of the


system leading to pseudo-intelligent decision making; and because of their success

in other medical fields.

2.3.2 EEG feature classification for sleep staging

Most of automatic sleep staging systems first extract certain features from EEG

segments (as mentioned in section 2.2), and then map these features into a certain

sleep stage. In sleep staging, feature classification techniques can be any of those

described above. As an example, the most recent automatic systems which involve

modem techniques for sleep EEG analysis will be very briefly discussed in the

following section.

In 1992, Roberts and Tarassenko published their work for automatic analysis of

human sleep EEG which mainly employed the Kalman filter as the EEG features

extractor and a self-organising neural network for clustering these high dimensional

features (the Kalman filter coefficients are treated as a vector, the numbers of entries

in this vector can be treated as dimensions of these features) into a high dimensional

space (100-dimensional space in this approach) called the output space or feature

map. The Kalman filter is an adaptive method which updates the initial estimates of

the AR model coefficients based on every new observation of the signal. The self

organised neural network is a modified Kohonen network which is a two layered

network with two-way connections between the layers providing the capability of

self-organisation. In this the weight vectors, associated with the feature map, are

updated according to an adaptive gain parameter (learning rate parameter), as well as


a decreasing function of distance between the selected unit and other units within the

neighbourhood in the feature map. It is implemented such that not all weight vectors

within the neighbourhood around the selected unit are updated equally. In addition,

the neighbourhood is no longer just a decreasing function of time but also decreases

linearly with the number of prior visits to that unit. One of the results is that there are

8 halting states in the feature map which should relate to the time course of the sleep

process itself. None of the states, however, has a one-to-one correspondence with the

6 main stages of sleep according to the R & K rules. Roberts and Tarassenko

suggested that the set of halting states is more closely related to the bulk cortical

action during sleep and argued that a better description of the state of the EEG would

be a probability density function with 8 components, one for each of the halting

states. In order to classify sleep into 6 stages based on standard rules (R & K), a

multi-layer neural network architecture was used in which the modified Kohonen

network works as a hidden-layer. Three likelihoods were generated according to the

probability density of the 8 halting stages and were linearly mapped to the output

layer.

2.4 Controversies over classification rules

Computerised analysis of sleep recordings was first investigated in 1968

(Lacroix, 1984). Since then, numerous attempts have been undertaken (Jansen and


Dawant, 1989; Lim and winters, 1980; Principe et al., 1989; Lacroix, 1984; Chang et

al., 1989; Smith J. R. et al., 1969; etc.), and many of the technical problems have

been solved. It is, however, unlikely that complete agreement between human and

machine sleep scoring can be achieved (Christine, 1989).

There have always been problems associated with computer scored results

because they do not correlate with the results of human analysis. Many advanced

methods have been developed and have considerable potential. The results however

still do not completely correlate with the R & K standard, although the rules which

most of these systems use borrow heavily from the R & K manual. Several

publications have mentioned that the rules contain a number of weaknesses, produce

sub-optimal decisions, and their application breaks down if sleep is disturbed,

abnormal or is in very young or elderly subjects. Serious objections have been raised

against the rules by Lairy (1977) and Kubicki et al. (1982).

The rules in fact contain many subjective components and do not always

provide an unequivocal basis for decision. They regard the essentially continuous

process of sleep as a set of discrete stages, and impose a coarse temporal resolution

(20-3Os). Their definition of these stages relies explicitly upon measures of the

absolute amplitude and frequency of the EEG. (Amplitude is not a true sleep related

variable, since it depends upon such factors as age, electrode placement and skull

morphology.) Scoring some sleep stages is based on very short events rather than on

background activity. The criteria are known to break down when applied to


abnormal, elderly or very young subjects, as mentioned before. All of these may

contribute to the insufficient reliability of automatic sleep staging systems.

Of course there are some other reasons that contribute to the insufficient

reliability of automatic sleep staging. The first is poor understanding of the physical

mechanisms by which the EEG is generated. The second is inter and intra - observer

variation. The third is that the most automated methods are objective but arbitrary.

In spite of these limitations, the R & K rules still find much acceptance of use

among clinicians. They have virtually become a standard all over the world,

enabling easy international comparison of sleep research. It should be mentioned,

however, that, in Europe at least, the obsession with adherence to the visual scoring

format is beginning to wane. Indeed, in 1989 the Commission of the European

Communities set up an initiative with the aim of providing a re-definition of, and

proposals for analysis of the sleep-wake continuum (Roberts, 1991). It is now a good

time to refine the definitions used in sleep research to enable modem automatic

procedures to be used effectively. This may result in an acceleration of research and

provide tools for the advancement of diagnosis and therapy in sleep disorders.

Chapter 3

DIFFERENTIAL TOPOLOGY

The aim of this chapter is to look at some aspects of differential topology which

is a mathematical language widely used in dynamic systems that will be used in the

following chapters. We start by mentioning, without very much detail, some

terminology and ideas in the theory of differential topology. More detailed

explanations can be found in many textbooks on differential topology or differential

manifolds (e.g. Chillingworth, 1976).

3.1 Basic topology

Topological space: Topological space is a set S on which a topological structure

(or just topology) is given. A topological structure on S is a collection of subsets of

S, called open sets, satisfying:

Chapter 3 DIFFERENTIAL TOPOLOGY 43

a. The union of any number of open sets is open.

b. The intersection of any finite number of open sets is open.

c. Both S itself and the empty set 0 are open.

Banach space'. A normed linear space which is also complete when viewed as a

metric space is known as a Banach space. Every finite-dimensional normed linear

space is automatically a Banach space.

Domain and co-domain: A function / i n a set A with values in a set B may be

written a s / A -» B. The subset of A on w hich/is defined is defined as domain, and

the set of all values of / i s the co-domain (range) of /

Injection: Given a function / A —» B and any subset V c B, we denote by

/ _1: V the set of all elements a £ A such that fa e V. / is an injection when

f a = f a x only if a =

Bijection: An injection/ A -> B whose domain is A and whose co-domain is B

is called a bijection.

Isomorphism: A bijective map / : A —» B is called isomorphic if for any

a, a x € A, one has/ ( a +ax) - f a + f a h o r f ( a x a x) = f a x f a x. Thus, A and B can

be called as isomorphic. If a map L : V -> F is a linear isomorphism, the linear

structure of V corresponds precisely to that of F, via L. Thus the two given linear

spaces V and F are indistinguishable as linear spaces and they are called isomorphic.


Homeomorphism: Let S, T be two topological spaces, and suppose/ S -> T is a

bijection. If / is continuous, and at the same time its inverse f ' x: T -> S is

continuous, then / is called a homeomorphism. If there exists a homeomorphism

/ S T then, as far as their topological structure is concerned, S and T are

indistinguishable. We say that S and T are topologically equivalent or, more usually,

homeomorphic.

Diffeomorphism: A differentiable map with differentiable inverse is called a

diffeomorphism. If M, N are two Cr differentiable manifolds and there exists a

diffeomorphism/ : M -> N, the manifolds M and N are said to be diffeomorphic or

differential equivalent and are indistinguishable as far as their topologies and

differentiable structures are concerned.

In general, two Cr vector fields, / and g are said to be Ck equivalent (k < r) if

there exists a Ck diffeomorphism O, such that O / = g O. O is an invertible,

possibly non-linear, change of coordinates, which will do so smoothly though

distorting the flow and will not confuse the order in which the points on the

trajectory are visited.

Embedding: A smooth map / : M N is an embedding if it is a

diffeomorphism from M to a smooth submanifold N. Therefore, an embedding of M

in N can be regarded as a realization of M as a submanifold of N.


3.2 Quotient space and quotient topology

Let S be a topological space, and suppose that S is expressed as the disjoint

union of a family of sets S^. The elements of S are objects which we are attempting

to classify and each represents a collection of objects having a certain property

in common. Now let us regard two objects as the same if they belong to the same S^.

Consider a new set S . The elements of this new set S are themselves the S^, and the

new set S inherits a topology from that of S. For example, given any set W c S let

W denote the subset of S consisting of the union of all those which belong to

W . Then W is open in S if W is open in S. See Figure 3.1. Now if we let R

indicate some equivalence relation defined on S, then R gives a decomposition of S

into disjoint subsets S^. If we denote the set of equivalence classes by S/R, and S/R

corresponds to S , we have a map

^ :S -> S /R

taking each x e S to its equivalence class. Then S/R = S is called the quotient

space and the topology on it is called the quotient topology.


w w

Figure 3.1. Quotient space § is obtained from the classifying of elements

in S based on certain properties.

3.3 Tangent bundles and tangent space

To perform the analysis of an evolving dynamical system which is represented

by a point moving on a manifold M, we are likely to be interested not only in the

position but also in some sense of the velocity of the point. The tangent bundle of M

is the space of all positions and velocities o f points moving on M.

Let E, F be Banach spaces (e.g. Euclidean spaces Rn) and U, V to be open sets in E

and F respectively, if / : U —> V is a differentiable smooth map at p in U, then the


derivative D /(p ):E —» F is characterized by the effect of / on smooth paths in U

based at p. The smooth path in U based at p means a smooth map c : J —» U . J is an

open interval (a, b) with a < 0 < b and c(0) = p. See figure 3.2. By the Chain Rule,

the composition / • c : J -» V is a smooth path in V base at ftp), and

D ( / - c)(0) = D / ( c(0 ) ) -D c(0 ) :R —» F

( / * c )'(0) = D /(p ) -c ' (O )

Thus the derivative D/{p) takes the tangent to the path c at p to the tangent to the

path / • c at f(p).

J c(oyf<p)

Figure 3.2. The derivative D/(p) of a differentiable smooth

map f. U—»V at p in U.

Tangency is an equivalence relation of the set of all paths in U based at p, and

the equivalence classes are called tangency classes at p. We use [c] to denote


tangency class of c, and TpU to denote the set of tangency classes of smooth paths in

U based at p. We can regard TpU as a normed linear space isomorphic to E, and

Tf(P)V as an isomorphic copy of F. To construct an explicit isomorphism requires

choosing a chart around p, but the linear structure induced on TpU does not depend

on the chart. The linear space TpU is that of tangent space to U at the point p.

Elements of TpU can be called tangent vectors to U at p. The linear map

D / (p): E -» F becomes the linear map

T p / : T p U - > T / (p)V<

Now, define the tangent bundle TU of U to be the union of all the linear spaces

TpU as p runs through U. Equivalently, TU is the set of all tangent vectors

everywhere on U. The topology obtained for TU does not depend on the choice of

the chart as well. If U has dimension n then TU has dimension 2n. Any smooth map

/ :U —» V induces a tangent map

Tf : TU TV

defined as TpF on each linear space TpU. We regard TU, TV as UxE, VxF. If f is a

diffeomorphism then so is Tf.

Because TU itself is a smooth open set, it has its own tangent bundle denoted as

T(TU) or T2U of dimension 4n.


The definition of Tangent map and tangent bundle on a smooth manifold are

similar to the above (replace U and V with two manifolds) and it is the formal way

of capturing the idea of velocities of a point moving on the manifold.

3.4 Vector fields and solutions

If a system S is governed by a set of first order autonomous ordinary differential

equations, we can write the system as

i = X(x)

where x=(x1? *2 , x3, ...... *n) ^es *n some °Pen subset U of Rn and X is a map from U

to a set c Rn. The set U is called the phase space of S.

If we have some initial conditions, then we can expect a solution which is a path

d J U satisfying c(0) = p and

c( t ) = X(c ( t ) )

for all t in the interval J if Ms regarded here as a measure of time. It should be

mentioned that X(x) is not an element of Rn in which x lies, but an element of the

tangent space TXU for every x in U, and X is a map of

X : U - » T U = U x R n


Such a map is called as a vector field on U and X-1 is called a natural projection TU

-» U. See figure 3.3.

TUTPU

X(PimageofX

'i

P

Figure 3.3. Vector field X and its inverse — natural

projection X '1.

A non-autonomous equation means a system of first order equations which

contains t explicitly on the right-hand side. It is equivalent to saying that the vector

field on a manifold M is varying with t, and can be written as X,. By introducing

another variable u = t one can interpret the vector fields X, as just one vector field X

on the product manifold M x R. At a point (x, u) of M x R the tangent space to M x

R is TXM x TmR = TXM x R, and the element of X(x,w) in the second factor is 1

since u = 1.


For higher order equations, there is a standard trick for converting an nth order

equation in one variable

d" v , d d""1 ,x = X(x,— jc ,—,------- x)1 - - n - 1d t m dt d t

into a system of n first order equations in n variables. That is to write xl for x, and

x 2 = x i ,x 3 =X 2 ,•••, Xn = F(xi ,X2 ,-**,x«-1 ) , gives a first order system on r ”. It

is equivalent to saying that from the nth order equation on an open interval U in Rn-1

we obtain a vector field (described by first order equations) on T U.

Chapter 4

BACKGROUND THEORIES FOR

MODELLING THE EEG

4.1 Introduction

Given a set of observations of a system, it is often necessary to condense and

summarise the data by fitting it to a model that depends on adjustable parameters.

The models can be a class of functions, and the parameters (or coefficients) come

from some underlying theory that the data are supposed to satisfy. There are many

theories and criteria for the choice of the models and fitting of the appropriate

coefficients, (e.g. FFT, MA model, ARMA model, AR model, etc. are different

models and least-squares criterion, Maximum likelihood, Maximum entropy, etc. are

different parameter estimation theories). It may be reasonable to divide the models

into two groups according to assumption of linearity or nonlinearity of the system.

Chapter 4 BACKGROUND THEORY FOR MODELLING THE EEG 53

Conventionally signal modelling has been limited by the underlying assumption

of linearity for many years. This assumption conveys many advantages, (such as it

makes computation easier and, in some cases, feasible rather than impossible) but in

the real world this assumption is often far from reasonable. The realization that a

very simple non-linear system can show extremely complex or chaotic behaviour has

lead to an explosion of interest in trying to extend our understanding of non-linear

systems and, therefore, lead to a development of the so called deterministic chaos

theory recently. Chaos occurs in many different non-linear mechanical systems and

the observed behaviour appears to be random. However, in principle, it is often

possible to predict chaotic sequences over short timescales if the time series is

deterministic.

The aim of this chapter is to describe some background theories for modelling

stochastic and chaotic processes which are useful in this project for modelling EEG

signals in the context of linear and non-linear assumptions. We will mainly focus on

some state space estimate techniques from stochastic processes. First of all, a linear

model of Kalman filtering will be briefly introduced, which is then followed by a

synopsis of non-linear modelling techniques.


4.2 Kalman filtering

4.2.1 Introduction

In many applications, people are frequently faced with the problems of

measuring a quantity to infer specific information about some phenomenon. The

quantity one measures in practice is often a random signal or stochastic processes

with attached noise. In linear modelling techniques, the random signals are fitted to

linear systems, then spectral representations can be used to describe the process. A

typical linear system is depicted in figure 4.1. It is well known that random inputs

u(t) applied to a linear time invariant casual system with impulse response g(t) yields

convolution and frequency relations as:

oo

y ( t ) = g { t ) * u { t ) = £ g ( / M r - 0;=0

where y(t) is output of the system. Taking the Fourier transform of this relation

gives:

T O ) = G(co)U(o))

and the output spectral density Sy(a>) will be

Sy(co) = G(co)G* O ) Su( co) = | G(co)\2 Su(co)

where Su(co) is input spectral density and G*(co) is the complex conjugate of G(co).


«(/)

U(oj)

y{t) = g{t) * u(t) (Convolution)Y{co) = G{a>) UifiS) (Multiplication)

Figure 4.1 Linear system model in time or frequency domain.

The fundamental result used in modelling stochastic processes is that when a

white noise input is applied to a system, and the system is a linear time-invariant and

asymptotically stable (poles inside the unit circle) with rational pulse transfer

function G(co), then the output spectral density is also rational. Given these

constraints, there exists a rational G such that

Sy(co) = G(co) G* ( a ) .

This means that if we can represent spectral densities in factored form, then all

stationary processes can be thought of as the outputs of dynamical systems with

white noise inputs.

One way to generate (or simulate) such a sequence is by using the input-output

model (or transfer-function model), which primarily operates on the input and output

variables (y(/)} and (w(/)}, that is

C h a p t e r 4 B A C K G R O U N D T H E O R Y F O R M O D E L L IN G T H E E E G 56

G(s) = -------U(s)

where G(s) is the Laplace transform of the impulse response of the system and s is a

complex variable in the Laplace transform domain. Thus G(s) could be termed as a

transfer-function (also called system function) and the models could be transfer-

function models. These models are very familiar to engineers and have the

advantage of being easy to use. In the discrete time case, the transfer function is

given as

rv % BA ( z )

where A and B are the polynomials

A ( z ) = 1 + Y J c i kZ~kk = 1

N h- kB(z) = Y Jb„z

k=0

Going back to the time domain representation we have a difference equation relating

the output sequence {y(ri)} to the input sequence {«(/?)}.

Nby ( n ) - ' £ ia ‘y ( n - i ) = '£lb j u ( n - j ) .

i=1 j =0

If a backward-shift operator (p with the property that <p~ky(n)=y(n-k)is used and

supposing that the system is excited by random inputs with exogenous inputs as


shown in figure 4.2, we have the so-called autoregresive moving-average model with

exogenous inputs (ARMAX):

A(<p~1)y(n) = B(<p~')u(n) + C(<p~])e(n)

where A, B, and C are polynomials and {e(t)} is white noise.

whitenoise e(n)

inputu(n)

1 -A iV 1)

A{VA)y(n) = B{<PA)u{n)+

Figure 4.2 ARMAX input-output model, where (p is the

backward shift operator, A, B, C are polynomials.

ARMAX model represents the general form for popular time-series and digital-

filter models. Kalman filter could be treated as an (adaptive) AR model in which B =

0 and C = 1. Among others, there are HR model (C = 0), FIR model


(A = 1, C = 0), MA model (A = 1, B = 0), ARMA model (B = 0), and ARX model (C

= 1).

4.2.2 State-space representations

The alternative way to generate (or simulate) the sequence is that of the so -

called state-space model or the state-variable method.

There are many theories developed for linear and non-linear representations of a

system in the state variable form. (For non-linear modelling see section 4.3). In

linear representation, a time-variant system, such as the one shown in figure 4.3, is

given by the state equation

A X ( t ) = - A c X « ) + B' U( t ) (4.1)at

with a measurement (or output) equation

y ( t ) = C ' X ( t )

where X, w, and y are the n state, m input, and p output vectors (normally

m —p = 1, that is a single-input, single-output system) and Ac, Bc, and Cc are the (n x

n) system, (n x m) input, and (px n) measurement matrices. It can be shown that the

solution of the state differential equation (4.1) is given by

X ( t ) = ¥(?,?<> ) X( to ) + j ' i , U , r ) B c u ( T ) d r (4.2)to

C h a p t e r 4 B A C K G R O U N D T H E O R Y F O R M O D E L L IN G T H E E E G 59

if we know the initial state X(t0), where the n x n matrix ^(t, t0) is called the state-

transition matrix of the system characterized by the triple (Ac, Bc, Cc). The

corresponding output equation will be given as

iy ( t ) = CcX ( t ) = o ) X( t o ) + JcV V ( t , T)Bcu(T)dz

c

“(f)Y* \

I_____ \

r A-\\

\

H

__ /~ i- L ___ J

/\ ! /

x{f)

y(t)

state variables (Voltages, currents)

state variables

Figure 4.3 State variable description of a system


For sampled continuous systems (or discrete time systems) with the sampling

interval T, we are interested in the solution of the state at particular sampling

instants. If we have the value at t0 = kT, we would like to know the value of the state

vector at time t = (k+ \)T. Thus, substituting for t and t0 in equation 4.2 we have

(* + 1) 7'

X((k+\ )T)='¥[ (k+\ )Tt f ]X(kT)+ ^ [ { k + ^ T ,T]Bcu(t)dTkT

If we assume that u(t) is piecewise constant between sampling instants, i.e.

u ( t ) = u(kT) for kT< t < (k+l)T

we obtain the sampled (or discrete) state space representation

X( ( k + 1 )T) = AaX( k T) + Bau(kT) (4.3)

with

Aa = ¥ [ ( * + l ) r , kT]

and

( * + 1 ) 7

Ba = J'FK* + \ ) T , t ) B c d rkT

therefore the discrete output equation is given by

y(kT) = C( k T) X( k T) (4.4)


The state variables can also be used to reconstruct the input-output description,

thus, in this sense, the state-variable method is equivalent to the input-output

description. However, state space analysis has many advantages over input-output

description, and is, therefore, quite popular.

The classical transfer function approach does not carry all the information about

the system (e.g., neglecting the initial conditions) and typically requires that the

model be linear and time-invariant in order to apply Laplace or z-transform

techniques. State-space representation is easily generalized and competent to deal

with non-linear, time-variant or even nonstationary random systems.

The Kalman filter uses this kind of approach and can be thought as a state

estimator or reconstructor, that is, it reconstructs estimates of the state X(t) from

noisy measurements y{t).

4.2.3 Kalman filter algorithm

The Kalman filter utilizes the state-space representation and measurement

equation to estimated random process in a recursive fashion. Let us begin with

rewriting the discrete state equation (equation 4.3) but simplifying the notation of

letting T = 1 and dropping the subscript, thus giving the state equation as

X ( k + 1) = AX( k ) + Bu(k) (4.5)


where X{k) is state vector at time th A and B are appropriately dimensioned system

matrices, and u{k) is random input or excitation signal. By using a recursive process,

we can obtain the solution of equation 4.5 if we have the initial state X(0), that is

X ( \ ) = AX(0) + Bu(0)

X( 2 ) = AX( l ) + Bu(l) = A2X( 0) + ABu(0) + Bu(l)

X( 3) = AX(2) + Bu(2) = A3X( 0) + A2Bu(0) + ABu(l) + Bu(2)

X { k ) = Ak X{ 0 ) + YJAk-'-> Bu{j ) .7=0

Let 'VQz, 0) = A , the equation 4.6 will be

jt-iX ( k ) = 4»(*.0)AT(0) + £ '* '[ ( * - 1 - 7 ) .0 ]5«(7) (4.7)

7= 0

If Icq represent the initial value of k, the recursive process of equation 4.7 will begin

from X(ko), then we have

k - 1

X ( k ) = ' ¥ ( k , k o )X(ko ) + '£1' ¥ ( k , j + 1 ) Bu( j ) (4.8)j =k o

where ¥(&, k0) could be interpreted as a state-transition matrix from k0 state to k

state. If we let k$ = k -1 , then we get a one-step recursive process as

X( k ) = - 1 ) X( k - 1) + 'F(0)Bi/(* - 1). (4.9)


Because 4^(0) =A°= 1, the equation 4.9 has the form

X( k) = ̂ ( k , k - 1 ) X( k - 1) + Bu(k - 1). (4.10)

We further assume that the excitation is a white sequence with known covariance

structure, thus Bu(k - 1) = w(k - 1), and the system could be a time-variant system,

that is 'i'ik, k - 1) = A(k), then the equation 4.10 could be rewritten as

X{ k ) = A( k) X( k - 1) + w(k - 1).

Further simplifying the notation, we have

X k ~~ A k X k —\ + "Wk-1 (4.11)

Thus, we get a simple one-step recursive state function, and by using this we can

estimate the state Xk at some point in time tk based on all of our knowledge about the

process at time tkA.

Now rewriting the simplified measurement equation (4.4) as

y k = C k X k (4.12)

and we make a further generalization at this point by assuming that the signal yk can

not be measured exactly, but, as shown in figure 4.4, is always associated with

measurement error or noise in observation, therefore, we have

Zk = yk + v*

or

C h a p t e r 4 BACKGROUND THEORY FOR MODELLING THE EEG 64

Z k — C k X k + V k (4.13)

where zk is the observation, vk is the measurement error and assumed to be a white

sequence with known covariance structure and uncorrelated with the wk sequence.

X k

Figure 4.4 Model of Kalman filter.

The covariance matrices for the w k and v* vectors are given by

E[yn wf] = j j *

E[vk vf] =R;

/ = k i * k

i - k i * k

E [ w k vf] = 0, for all A: and i, (4.14)

and assume that both w k and v* have zero mean, that is E[wk] = 0, and E[vjf\ = 0.


Suppose at some point in time th we have an initial estimate of the process

based on all of our knowledge about the process prior to tk. This prior estimate is

A

denoted as X k and is our best estimate when neglecting wk of the state function

(equation 4.11) because it has zero mean and is uncorrelated with the previous w’s,

that is let

Xk = A* Xk-x (4-15)

where x k-\ the updated estimation of XkA. Similarly the estimated observation

has the form

Z k Ct Xk = C t A t Xk-x- (4-16)

To obtain the optimal in some sense of the estimate x k, it is necessary to use the

measurement z k and a coefficient vector K k to improve the x k > that is

X k = x k + ( Zk - Zk)

= A k X k - i + K k ( z k - C k A k X k - 1 ) (4.17)

Substituting equation 4.11 and equation 4.13 in to the above equation, we have

X k = A k X k - i + K k [ C k X k + V k — Ck Ak X k - l ]

= ( I - K k C k ) A k X k -1 + K k C k ( A k X k - i + w*-i ).+ K k V k (4.18)


Now we use minimum mean-square error as the performance criterion and define the

estimation error to be

ek — X k — X k

and substituting equations 4.11 and 4.18 into above equation, we have

A

€ k — Ak X k -1 + Wk - 1 — X k

= ( / - K k C k ) [ A t ( X k -1 - X k -1 ) + WA-i ] - K k V k (4.19)

The associated mean square error matrix (or error covariance matrix) can be written

as

Pk = E [ e t e t T] = E [ ( X k - X „ X I* - X t ) r ]. (4.20)

Let

Pk = E [ ( X k - x k ) ( X k - x k) T] (4.21)

and substituting equations 4.11, 4.14, and 4.15 into the above equation 4.21, we

have

P k = E [ ( A k ek-i + w*_i ) ( A k e k - i + wk-i ) r ]

= A k P k - x A l + Qk-i . (4.22)

By substituting equations 4.14, and 4.19 into equation 4.20, we get the error

covariance matrix as


Pk = ( / - K k C k ) / > * ( / - K k C k f + K k R k K l

= Pk - Kk Ck Pk ~ Pk Cl K l + Kk(Ck Pk Cl + Rk ) K l ■ (4.23)

Because ( CkPkCl + Rk ) is positive definite, it can be expressed as

CkPkCl + Rk = SST,

We let

H = Pk C Tk

and because

Pk = P [ ,

we have

H T = ( P k C l ) = CkPk-

Therefore the equation 4.23 can be rewritten as

Pk = Pk - K k H T - HKl + K t S S T K l

= [ K k S - H ( s T) ' ' U K k S - H ( s tY ' ] T + Pk

- Pk Cl ( Ck P\ Cl + Rk )■' Ck p \ (4.24)

It is obvious that only the first term in the above equation involves Kh and it is

positive semi-definite, thus, to minimize the optimum Kk should satisfy

K k S - H ( s Ty ' = 0

that is


Kk = H ( s T)~l S~l = H ( S S T) ' 1

= Pk c l (Ck p \ Cl + Rk y l (4.25)

This particular Kh namely the one that minimizes the mean square estimation error,

is called the Kalman gain.

The covariance matrix associated with the optimal gain may now be rewritten

as

Pk = P k - Pk Ck (Ck Pk c l + Rk )■' Ck Pk

= Pk - KkCkPk = ( / - Kk Ck ) P \ (4.26)

Now we get the final form of the Kalman filter, and equation 4.15, 4.17, 4.22,

4.25, and 4.26 comprise the Kalman filter recursive equations. The Kalman filter

loop is shown in figure 4.5 and it should be clear that once the loop is entered, it can

be continued ad infinitum. For more ditails please refer Brown R.G. and Hwang

P.Y.C., 1983 and Candy J.V., 1986.


Project ahead

Xk - AkXk-i , Pk -AkP/c.iAk + Qk-i

Kalman gain

Kk = P'kC Tk(CkP'kCl+Rk)

Update estimate

Xk = AkXk-i+Kk(zk-CicAjcXjc-\ )

Initialize Xo = E [X d , Po,

Figure 4.5 Kalman filter loop.


4.3 Non-linear modelling techniques

4.3.1 Introduction

Many techniques exist for the analysis of chaotic systems. Most of them are

algorithms for calculating geometric and dynamical invariance of an underlying

strange attractor such as the largest Lyapunov exponent which gives an indication of

how far into the future reliable predictions can be made. Dimensional analysis (e.g.

correlation) may be used to give an indication of how complex a model must be.

These techniques leave much to be desired from an experimentalist’s point of view

and are of limited practical use in many projects. It is desired, in many cases, to

reconstruct the state space or a predictive model from the time series. If this can be

done consistently, then the underlining dynamic system has in some sense been

modelled and it will be a great advantage for analysing such a system. In this section

the principles behind the techniques of invariant analyses will be omitted, but a brief

discussion of how to construct a predictive model directly from time series data will

be presented.

To reconstruct a predictive model directly form a time series, state space

reconstruction is necessarily the first step, followed by use of the techniques which

essentially involve interpolating or approximating unknown functions from scattered

data points.


It is believed that the past and/or future of a time series x(t) contain information

about unobserved state variables that can be used to define a state at the present time

of the underlining dynamic system. If only the past of the time series is used the

reconstruction is predictive. Consider the dynamic system with a ^-dimensional

strange attractor M formally as:

s = f ( s ) (4.27)

where the vector field / i s in general a non-linear smooth map, s = (sj, s2, ... 5m)

which represents a state of the system, and m is the number of a priori degrees of

freedom of the system (where d < m, as the system evolves the flow contracts

normally onto sets of lower dimension). Given an initial value s0 e M, the solution

at time t will be:

S = g So( t ) (4.28)

with Sq = g$o(0). The time evolution corresponding to an initial position s0 will be

denoted as where \|/,: M -»M. If solutions to all possible initial value

problems for the system are considered, the map \\ft will represent a flow on M.

The time series is related to the dynamical system by a measurement function:

*(0 = A(*4, ( 0 ) . (4.29)


In general, h is a smooth dimension reduction measurement function, and in our case

x(t) is 1-dimensional time series, therefore h: M -> R.

State space reconstruction is the problem of reconstructing the ^-dimensional

manifold M. In most cases, / and h are both unknown, so that it is not possible to

reconstruct state space in its original form. However, it is possible to reconstruct a

state space that is in some sense equivalent to the original. This will be described in

section 4.3.2.

In practice, with a sampled data set {x(t-J}, i = 1,..., N, the re-established states

s'n are sequential in the reconstructed manifold M \ The number in the sequence is

normally finite and is related to N and embedding dimension m' (< N - m', with

m' > 2d+\ to satisfy the Whitney embedding theorem (Whitney H. 1936)). With

these states s 'n what remains is to construct a predictive model / : Rm —> Rm, for

which s 'n+ l = / (s'n), or more general, particularly when the time series is affected by

noise, s 'n+1 « / (s'n). This is a standard problem in approximation theory, and many

suitable interpolation techniques exist. In the case of m ' > 1, the interpolation

problem amounts geometrically to fitting m' smooth functions or “hypersurfaces”

Kjf: Rm —» R through the data points (s'n, ^ s 'n+1), 1 < n< N - m', where 7t} denotes

the projection onto the jth coordinate with j = 1, ..., m'. In practice, / does not

necessarily have to be a smooth function. There are a variety of numerical

techniques for doing this, including Local Prediction Techniques, Radial Basis


Functions (RBF) and Global Prediction Techniques. These techniques are briefly

discussed in sections 4.3.3, 4.3.4 and 4.3.5 respectively. Some of their known

advantages and limitations will be also listed.

In order to evaluate how good / is as a predictor, a normalized prediction error

is defined as

a2 (f) = §cpred{‘) ~ * ( ') f ) / (j|*0) - « 0 » |2) (4.30)

with “< )” means average and “|| ||” denotes a norm. If <?2( / ) = 0, the predictions are

perfect; a 2( / ) = 1 then the performance is no better than a constant predictor

*pred(0=<*(W-

4.3.2 State space reconstruction (method of delays)

It is already clear that much could be learned about the dynamical behaviour of

a system from an analysis of system trajectories in a multi-dimensional state space in

which a single point characterizes the state of the entire system at an instant of time.

However, for most real-world systems it is often difficult to identify all of the state

variables as they are generally embedded in a more complex, higher-dimensional

system. Very often we only have a time series of discrete measurements of a single

observable variable.


In 1980 Packard et al. first suggested that qualitative information about the

dynamics could nevertheless be recovered from the time series. One year latter, a

similar method of state space reconstruction was suggested independently by Takens

(1981). Packard et al. and Takens demonstrated that it is possible to preserve

geometrical invariants, such as the eigenvalues of a fixed point, the fractal

dimension of an attractor, or the Lyapunov exponents of a trajectory. This was

demonstrated numerically by Packard et al. and was proven by Takens.

The basic idea behind state space reconstruction is that the past and future of a

time series contain information about unobserved state variables that can be used to

define a state at the present time. There are a number of techniques currently used

for state space reconstruction which differ in the method of extracting this useful

information from the time series. These techniques may include delay coordinates,

derivative coordinates, and global principal value decomposition and may make a

big difference in the quality of the resulting coordinates. It is not clear, in general,

which method is the best. However, the well-known and the most widely used

method, which we shall discuss below, is the technique of delay coordinates or the

so called “method of delays”.

In fact, state space reconstruction is an embedding of differential manifolds. In

our case, it is an embedding of M in M' and can be regarded as a ‘realization’ of M

as a submanifold of M'. In other words, an embedding is a smooth map, say O, from

the manifold M to M' such that the image <I>(M) = M' is a smooth submanifold and


that O is a diffeomorphism between M and 0(M). Thus one can think of this as

meaning that diffeomorphism O gives an important differentiable equivalence

relation between M and M \ A general existence theorem for embeddings in

Euclidean space was given by Whitney (1936). He proved that any Cr-m-manifold

M may be Cr-embedded in Euclidean space E2m+1. This theorem is the basis of state

space reconstruction from time series. Takens (1981) further developed the

embedding theory by considering that the flow corresponding to a physical process

of the underlying dynamics systems. In the present notation his theorem (theorem 2)

states:

Let M be a compact manifold of dimension d. For pairs (f, h),fa. smooth vector field

and h a smooth function on M, it is a generic property that

a:M —> R2fl!+1, defined by

= (Ks% WWite)), ••• > (4.31)

is an embedding, where vj// is the flow off

It is easy to relate the equation 4.31 to a time series of measurements made on

the system: i = 1,..., N, by makingx(t-J = /z(vj/j(s)). Therefore, in practice state

space reconstruction could be achieved by implementing this delay reconstruction

map <$>f h, and this method is the so called the method of delays. The space which

contains the image of <Dy h will be called the embedding space and its dimension will

be regarded as the embedding dimension. The above discussion is illustrated in


figure 4.6, in which a dynamic system of an experimental system is assumed to be

evolved in an m-dimensional submanifold M of space Rm. A sequence of real-valued

j i I

measurements, h, is used to construct a map of <£y A:M —» R . Based on the

Taken’s theorem, the evolution on M and Oy A(M) is diffeomorphism (C1-

equivalent).

Di ffeomorph ism<P:

Figure 4.6. Illustration of the state space reconstruction using

the method of delays.


It is obvious that in practice, the dimension d of the manifold M is not known a

priori so that the embedding dimension has to be systematically increased, normally

until the trajectories no longer appear to intersect. Therefore the embedding

dimension m' is denoted as m' > 2d+\ to satisfy the Whitney embedding theorem.

Taken’s theorem does not give any clue to what is the “best” delay lag, and the

“best” sampling rate. Clearly, too high the sampling rate and too small the delay lag

will give coordinates which are too strongly correlated. This will introduce an

artificial symmetry into the phase portrait and cause the trajectory to lie close to the

diagonal in the embedding space. For too large a delay lag, particular with the

presence of noise in the time series, the sequences of the trajectory will show no

causal connection. Picking a good lag time is critical in practice.

In brief, one way of embedding a time series, {x(t-J}, i = 1,..., N, in a state space

is by setting coordinates Xj(/) = jc(^), x2(t) = x(tr r), ..., = x(fr (nT-l)r), and,

thus, creating a state vector X(t)

X(t ) = [Xi(0 ,x2( 0 , ........» x m (0]T (4.32)

where r is a time delay. In practice, there will be cases where it will be desirable to

have rmuch larger than the original sampling interval of the time series.


4.3.3 Global prediction techniques

Global prediction utilizes a single function / to predict the behaviour of the

whole time series. In this approach the coordinate functions, 7Zj / : Rm -» R,

j = 1, m', are chosen from a standard function basis, such as a Afh-order

polynomial, and the coefficients are fitted to the data set using a conventional linear

least squares criterion, by minimizing

t,(^jX„+]-7rjT(x„))2 (4 '3 3 -*rt* I J *

Therefore, the predictor / will be a Ath-order polynomial in m' dimensions.

One advantage is that the predictor is in a "standard form". The disadvantage is

that there is a very large number ((k+ m')!/k! m'f) of free parameters which need to

be chosen, therefore it is extremely computational resource consuming when m', k

and the amount of data points are large and may be intractable for very large m \

A related approach is to construct rational predictors by considering the ratio of

two polynomials, where . ^ f is chosen to be a ratio p / q of polynomials, and the

coefficients ofp and q are chosen to minimize:

'£(7TjXn + \ q ( x n) - p ( x n)yn * \


This approach shares the same advantages and disadvantages. However, there

are known to be distinct advantages of rational approximation over polynomial

approximation if m' = 1 (M. Casdagli, 1989).

Global approximation techniques only work well for smooth functions and

higher iterates of chaotic mappings are not smooth.

4.3.4 Local prediction techniques

The technique of so called “local prediction” is a method which first embeds

one part of the time series, as mentioned in 4.3.2, as a manifold in the state space

and then uses only nearby states on the manifold to make predictions for the

remaining data.

As shown in figure 4.7, the prediction of the value of the signal x(M-T), which

could be equal to state vector KjX(t+T) with a given j (e.g. j = 1), is based on the

current position of the state vector X(t) as equation 4.32, the positions of the K

nearest neighbours of X(t), which may be denoted as X(f) where f < t and the

future K values x(t'+T) (equal to 7ixX(t'+T)) corresponding to these neighbours

(where T is the interval over which forecasts are to be made). The region used for

the forecast is defined as a sphere of radius epsilon s about the point to be

predicted. Each neighbour X(f) is regarded by the local predictor n- f̂ \ Rm R (j

= 1) as a point in the domain and, each x(/’+T) as the corresponding point in the

co-domain. A first order polynomial is fitted to the pairs (X(t9), x(t’+Tj). To


ensure stability of the solution it is frequently advantageous to take K > m'+l

(J.D. Farmer and Sidorowich, 1987), which implies that epsilon should not be

chosen to be arbitrarily small. In practice, choosing a larger K may improve the

prediction error a 2(^ j/) . A polynomial with Mi-order (k > 1, but small) could be

used, and in this case K must be at least as big as (k+ m’)!/k! m7 (M. Casdagli,

1989). However, J.D. Farmer and Sidorowich (1987) suggest that there are no

significantly better results obtained by using higher-order polynomials than those

of first order.

M’

m*

Figure 4.7. A schematic diagram of the formation of the state

vectors and neighbourhoods.


As a contrast, local techniques consume much less computational resource

than that of global technique and this feature can be improved even more if data

trees are used to organize nearest neighbour searches. The disadvantage is

obviously that the resulting predictor n^f is in general a discontinuous function,

and is not in any standard form.

4.3.5 Radial Basis Functions

Radial Basis Functions, which have been related to interpolation theory, are a

well-known tool for multivariate approximation and for scattered data interpolating.

This technique is, in fact, a global interpolation technique, but with good localization

properties. For given N different points {Xx i = 1, 2, ... N} in Rm, the radial basis

function interpolation selects 7t}f : Rm -> R as the linear combination of N radial

basis functions, i.e.

Xjftx) = £ a,«K\x - Rra’/= ]

(4.34)

where || • || denotes the Euclidean norm on Rm, and the coefficients A, (i = 1, 2 ... N)

are defined by the interpolation conditions:

x j f ( . X , ) = X j { X M), j = 1 ,2 , . . . ,m'. (4.35)


^ is a fixed function from R+ to R, and it could be one of the expressions of:

<p(r) = r linear

^ (r) = r cubic

(j> (r) = r2log r thin plate spline

(j> (r) = exp(-r2) Gaussian

o “7 i n .<f> (r) = (r + c )' inverse multiquadric

(j) (r) = (r2 + c2)m multiquadric

where c is a positive constant and r > 0. All of the above are radially symmetric,

non-linear (except linear one) basis functions, hence the term radial basis functions.

It has been proved (Micchelli, 1986) that the matrix

Ay = ^ ( || Aj - || ), i,j = l , 2 , . . . ,N

is non-singular if the data points are all distinct, therefore the interpolation

conditions of equation (4.35) define the coefficients {^; i = 1,2, ... , N} uniquely.

The multiquadric and the Gaussian methods are of this type and highly successful

results in practice have been achieved in a number of applications by using these

techniques, thus, multiquadric and Gaussian are often the choice in the engineering

environment.

Radial basis functions, as predictors in non-linear system analysis, provide a

global smooth interpolation of scattered data in an arbitrary dimension. As they have

a linear structure in the parameters, they allow fast convergence and have a general


modelling capability. The advantage of this technique is that it is easy to implement

and can reach very high fitting accuracy. However, disadvantages exist: If the

number of the data points N is large, the standard matrix inversion algorithms have

large memory requirements and are very time consuming. This can be improved by

"localizing" the technique, and sacrificing smoothness.

In addition, strict interpolation does not guarantee high predictive accuracy. For

example, when the matrix A is nearly singular (which occurs if the time series is

roughly cyclical), or if there is some noise, then predictive accuracy will be poor (X.

He and Lapedes, 1993).

Chapter 5

MODELLING OF EEG

5.1 Introduction

Quantitative analysis of the EEG often involves, as the first step, the extraction

of features that adequately describe the state of the underlying system. Two models,

transfer-function and state space models, could be used for extracting useful

information (for details, see Chapter 4, section 4.2). Among the transfer-function

models, Fourier series representation, which contains many frequency components,

has been a very popular technique for modelling EEG signals in the past decades.

However, a large number of Fourier series coefficients will generally be required to

represent a time series, albeit at the cost of losing phase information when a power

spectral density representation is used. This technique (and all of the other transfer-

function methods) is based on linear modelling and does not carry all the

information about a system (e.g., by neglecting the initial conditions). Furthermore,

it starts from the assumption that the underlying systems are time invariant systems

Chapter 5 MODELLING OF EEG 85

or stationary random processes. This is often far from reasonable in the case of EEG

signals.

State space approaches have many advantages over transfer-function methods.

Perhaps it is fair to say that one of the most important advantages is that state space

methods can be easily generalized. Therefore they might be more suitable to

characterize a wider range of systems, such as non-linear systems, time-varying

systems and nonstationary random processes. Thus, they are superior, in principle, to

transfer-function methods for the analysis of nonstationary, time-varying or even

chaotic signals such as the EEG signals.

In this chapter we will investigate the performance of implementations of state

space approaches, linear and non-linear models for modelling and forecasting the

electroencephalogram (EEG) during sleep. Firstly in section 5.2, a non-linear local

prediction technique of Schaffer and Tidd (1990) will be investigated. Secondly in

section 5.3, a linear state space version of the Kalman filtering technique will be

implemented. Thirdly, an adaptive non-linear modelling technique which involves a

modified Kalman filtering approach using Radial Basis functions will be considered

in section 5.4. Finally, all the resulting performances will be compared in section

5.5. No attempt has been made of trying to model the whole night sleep EEG by

using a non-linear global technique as this would obviously be unreasonable.


5.2 EEG modelling using local prediction technique

Many studies have shown that EEG signals exhibit chaotic behaviour (A.

Babloyantz and Salazar 1985; N. Xu and J. Xu 1988; G. Mayer-Kress and Layne

1988; B. Doyon 1992; Jan Pieter Pijn et al 1991; etc.). The realization that

deterministic chaos may be generated from rather simple deterministic dynamics led

to the search for non-linear dynamic systems for EEG analysis. Modelling sleep

EEG based on non-linear prediction was performed with the assumption that the

EEG is generated by a non-linear dynamical system (complex and high

dimensional).

The modelling of sleep EEG by using the non-linear local prediction technique

was obtained by using a package of Non-linear Forecasting For Dynamical Systems,

known as NLF, produced by W. M. Schaffer and C. W. Tidd, 1990. The technique

behind this package is the, so called, local prediction technique which, first, embeds

one part of the time series as a manifold in a state space and then uses only nearby

states on the manifold to make predictions for the remaining data. As more data

become available they are added to the manifold and used for the next prediction, i.e.

future behaviour is predicted from situations when the system was observed to be in

a similar dynamic situation. The principles behind this technique are introduced in

detail in Chapter 4, section 4.3.


There are various parameter settings which need to be selected before the NLF

package (W. M. Schaffer and C. W. Tidd, 1990) can be put into use. The main

parameters of the model are:

• number of Atlas Points (PA) - determines the number of points used to make the

first forecast. A typical value (used as the default in this package) is 20% of the

number of data points.

• embedding dimension (ED) - NLF uses Takens’ (1981) method of delays to

represent a univariate time series as a m -dimensional trajectory. From the

original time series x(t) the program constructs the vectors

X( t ) = x ( t ) , x ( t - T ) , x ( t - 2 r ) , . . . , x [ t - ( m - 1 ) t ] (5.1)

where m ’ is the embedding dimension and t is the time or embedding delay. The

trajectory of the vector X(t) completely characterises the dynamic system

provided that m ’ is large enough.

• embedding delay (x) - The time delay, r in equation 5.1, used in reconstruction

trajectories in the state space.

• prediction epsilon (s) - NLF uses all points in the reconstructed phase space

within a radius epsilon of the point for which a forecast is to be made for its

prediction. Epsilon is normalised on the interval [0,1], so if all the data is to be

used, s = l . When choosing the value of epsilon, the data was re-scaled on the

interval [0,1].


• prediction interval - the interval over which forecasts are to be made. The

default setting is also 1 sample.

In order to evaluate how good the p red ictor / is, we adopt the notation of

normalized prediction error (NPE) defined as in Chapter 4 of equation 4.30 and

rewritten as follows

(where “( )” means average over time and xpred(t) is the predicted value for the

signal), to estimate the accuracy of the predictor / . I f NPE( f ) = 0 the predictions

are all perfect, while NPE( f ) = 1 would indicate that the performance of the

predictor / is no better than that of a constant predictor - for which

x pred ( t ) = ( x ( t ) ) . The results of the application of the NLF to segments of EEG

in different sleep stages is summarised in table 5.1 (10,000 samples or approx. 78

seconds worth of EEG signal with 8 bits resolution were used).

If the normalised prediction error were the only criterion for setting the

parameters of the package the embedding dimension would be only 2 for all sleep

stages (which is far from reasonable) and the embedding delay and prediction

interval would be 1 (NPE increases sharply as the embedding delay and prediction

interval increase). The fact that the error was least for an embedding dimension of 2

NPE( f ) = (5.2)


seems to contradict what was expected (that a higher dimension would produce a

smaller forecasting error). The main reasons for this behaviour are the unstable,

noisy properties of the EEG signals and the limited capacity of the package (which

limits the maximum number of points to 10,000) which results in the state vectors

X( t ) being scattered in the state space. To ensure stability, the number of previous

vector positions considered for the forecast should be greater than the embedding

dimension (J. D. Farmer and Sidorowich, 1987) and to achieve this with so few

samples of the EEG the prediction s has to be set to more than 4% of the peak-to-

peak value of the EEG signal. The greater the value of epsilon, the lower the

prediction accuracy. As the embedded dimension was increased, the vectors

scattered further for the EEG and the prediction error increased. This was not

expected initially.

Based on the evidence from other investigations that the EEG is generated by a

much higher dimensional system, (the correlation dimension is near 4 in sleep stage

four and 5 in sleep stage two according to B. Doyon (1992)), four more trials, with

higher dimension settings, were made, and the results are also listed in table 5.1.

To evaluate the performance of this package further and to obtain a subjective

measure of prediction error and complexity, two simple predictors were

implemented. The first is the zero order interpolator for which

X p r e d ( t ) = x( t - T ) and the second is the straight line predictor for which

Xpred(t) = x( t - T) + [ x ( t - T) — x ( t - 2 T)]. The results are listed in


table 5.1. Figure 5.1 and figure 5.2 show us about 2 seconds of sleep stage 4 and 2

EEG data and its prediction which is immediately after the embedding of EEG time

series into the state space. It is guaranteed that no significant state changes happened

in the EEG at that moment, and the unsatisfactory performance is obvious.

TA BLE 5.1 N orm alized P redictor Error (NPE) for D ifferent Sleep Stages and D ifferent Param eter Settings.

Sleep stage 4 PARAM ETER N P E (x 10~3>

N on-linear P rediction ED=2, PA =2000 7.2



Horizontal Line Prediction

/ 14.3

Straight Line Prediction / 7.3

Sleep stage 2

N on-linear P red iction ED=2, PA =2000 33.

N on-linear P red iction ED=8, PA=4000 88.5

N on-linear P red iction ED=8, PA =8000 69.0

Horizontal Line Prediction

/ 59.2

Straight Line Prediction / 38.6

ED = Embedding Di mens ion , PA = Atlas Points or Points used for Embedding.


5.3 EEG modelling using a Kalman filter

5.3.1 Introduction

Conventionally, linear approximation is an important tool for analysing a non

linear system, By using such an approximation we lose accuracy but gain linearity,

which allows relative ease of calculation and even, in some cases, makes analysis

possible rather than impossible.

For non-linear dynamical systems there are many approximated optimal filters

developed (Mous and Johan, 1993 and references therein). A more traditional

method to estimate parameters and states of a dynamical system is the Kalman

filtering technique. Some research results suggest that Kalman-type filters can

sometimes be quite well behaved when applied to chaotic systems, and they have

been widely applied in various areas to estimate the states of non-linear systems

(e.g., Mous, 1993; Bockman, 1991; Myers et. al., 1992).

Since Bohliris time (1971), Kalman filters have been used for EEG analysis by

many researchers (Roberts, 1991; Bartoli and Cerutti, 1982; Jansen, 1981). Most of

them used Kalman filters for spectral estimation and then used the results of the

spectral estimation for the classification (Skagen, 1988; Jansen 1981; Woolfson,

1991; Bohlin, 1971). Numbers of promising results have been reported since then,

including that the Kalman filters have the ability to reduce muscular noise

superimposed to the EEG signals (F. Bartoli and Cerutti, 1982).


As mentioned in Chapter 2, section 2.1.1., Kalman filters can be treated as

adaptive AR models. Indeed a major advantage of the AR modelling over the

traditional Fourier technique is that it allows the use of short data segments that

fulfill the stationarity requirements. Moreover, as Kalman filters have the ability of

adaptation along changes of signal properties, they are superior, in principle, to other

models (e.g. non-adaptive AR model) in such an unstable EEG environment. But, in

practice, because spectral estimation needs a higher model order and when the order

of the Kalman filter is too high, spurious detail in the spectra may result, the superior

potential of Kalman filter have been given full play in few EEG analyses up to now.

In Chapter 4, section 4.2 we briefly introduced the principles behind the

Kalman filter, and in the following sections we will deal with details about the

choice of model order and other specific issues in EEG modelling.

5.3.2 Model order

Kalman filter modelling requires, as part of the implementation procedure, the

choice of the model order to be used for prediction in the same way as non-adaptive

autoregressive modelling (Schlindwein, 1990). In principle, the performance of the

resulting AR models is evaluated by calculating their "prediction errors" or the

prediction mean square error, defined as

E r2 (AR) = i^Xpnd ( 0 - x ( 0 |2 ) (5-3)


which is a function of the order of the model and monotonically decreases with the

order. This would seem to indicate that the higher the order the better the prediction,

but the reality is not so simple. In most cases, the model order will be influenced

heavily by many different factors. With too high an order, the bias of the estimate

towards the current realisation of the process (and the noise in it) increases.

Therefore, several criteria are used for model order setting. Among them, the

simplest is finding a point where the curve of prediction error (or normalized

prediction error) versus model order becomes “flat”. It is widely believed that when

the curve becomes flat the performance of the AR models will have no remarkable

improvement with increased model order. But most of the time, this criterion often

gives a sub-optimal (or too low) model order for power spectrum estimation. That is

the spectrum, as a classifier, obtained from AR model, on the one hand needs higher

model order. The choice of the model order should be made with great care, and it is

recommended that some experimentation with varying model orders be carried out

for spectrum estimation. There have been extensive investigations of topics which

are very closely related to this subject. Among the objective techniques to find the

optimal AR model order, the most successful and the most widely used, is that of

Akaike's (1970) Final Prediction Error (FPE) criterion.

The principle behind the FPE criterion is to take the change of AR model

coefficients into account to find a higher model order to fit to the time series up to

some upper limit. It was proved that the FPE has the form

Chapter § MODELLING OF EEG 94

M MFPE = Rh + • t e( i y\E[x( t - m) ■ x( t - /)]. (5.4)

m=\ 1=1

Where, the first term, R2M, which is defined as

/ M \ 2 R2m = E [ { x ( t ) - ]T<zO,m) • x ( t - m ) ) ]m = 1

is the variance of the residuals of the Mh-order AR model fit to the realization of a

given stochastic process x(t), if a(x, m) denote the mth AR coefficients. In the sense

of the least mean squared error criterion, R2M is expected to be a minimum with

respect to any other criteria (such as Maximum likelihood, Maximum entropy, and

etc.) and is a monotonically decreasing function with the model order M The second

term in equation 5.4, corresponds to the statistical variation of the AR coefficients

and is increases with the order M. The sum of these two terms should have a

minimum which will indicate a model order for which the bias is not very significant

and at the same time would not produce a too big mean square prediction error. It

may be fair to say, at this stage, that the FPE criterion evaluates a successively

higher-order for an AR model to fit the time series. For more details about FPE

criterion please see Appendix A.

Many remarkable results by using Akaike's FPE criterion for spectrum

estimates have been achieved since then, and many researchers have mentioned that

too high an AR model order compared with Akaike's final prediction error criterion

will introduce spurious details into the spectrum.


It is clear now that the FPE criterion is a kind of "upper limit" for the AR model

order setting. It seems to give optimum model order only in the sense that the

classifier based on spectral estimation gives better results for higher model orders.

Therefore, it is natural to conclude that FPE might not be the indicator of best

model order for modelling the time series in the most efficient way for prediction.

Based on our experience, we suggest that, for Kalman filter modelling, the

preferred order is somewhat lower than the “best order” required for non-adaptive

AR spectral estimation. As Kalman’s approach allows the coefficients to vary with

the signal, the Kalman filter technique seems to perform better with a lower model

order compared to that using the FPE criterion. Jansen (1981) suggested that using

an order higher than that indicated by the FPE criterion may result in spurious peaks

in the spectra, especially when calculated using the Kalman filter method. We

suggest here a different approach for EEG classification which is not based on

spectral estimation and does not favour any particular model order, but is based on

the Kalman coefficients (for details please see Chapter 6). We intended to model the

EEG in the time-domain and to track the behaviour of the signal by using the

shortest segment length possible. This is fundamentally different to the approach

used for obtaining spectrum estimates, where normally a segment corresponding to

around 1 second of data must be used since the spectral resolution is related to the

length of the data frame. In this sense our ‘best’ model order is different to that used

for spectrum estimation since our number of samples is much smaller. A model


order somewhat lower than the “best order” for the spectral estimation criterion was

used therefore.

Figure 5.3. shows the behaviour of the Kalman filter predictor error

against model order M for 1.5 minute sections for sleep EEG of sleep stage 1,2,3,4

and REM. From these results it is clear that there is no great improvement in using

orders higher than M = 5, this value was therefore chosen for this work.

5.3.3 EEG Modelling

For modelling EEG by using Kalman filtering, we consider the discrete

dynamic system (equation 4.11 in Chapter 4) with the state vector X k of dimension

M = 5, which develops according to

Here, A is a M x M matrix which for our purpose may be considered constant, i.e.

A = 1. The vector w kA is a white noise series with zero mean and with covariance

matrix

Er 2 (KALMAN) = <|x pnd ( f ) - x ( 0 |2>

X k — A X k - l + W k- 1 .

i = ki k


The relationship between the state vector Xk and the observation zk from

equation 4.13, Chapter 4, is rewritten as

Zk = CkXk + Vk (5.4)

again, vk is assumed to be a white noise series with zero mean and variance

To apply a Kalman filter to the problem of estimating the coefficient in the AR

model, we let Ck be a vector of the previous M members of measurement zh that is

Ck = [ Z k - 1 , Z k - 2 , Z k - 3 , Z k - 4 , Z k - 5 ]^.

Therefore the state vector Xk will be the AR model’s coefficients, and the equation

5.4 will equivalent to the autoregressive model of order M, that is

Mz( t ) = akz( t - k) + v( t )

k=1

with X = [fli , 0 2 , ••• om ].

To evaluate how well the coefficient vector Xk adapts to non-stationarities, D.

Skagen (1988) derives an expression of the “adaptive ability” for Kalman filtering,

which can be expressed as the relative reduction in error variance due to adaptation


To adjust the state vector X k efficiently, we let Q and R be set at the value of 1 for

all of the further work, based on the work of S. Roberts (1991).

Figure 5.4 and figure 5.5 show us the same segments of EEG time series x(t)

and its prediction xpre£j(t) as figure 5.1 and figure 5.2. Table 5.2. lists normalized

predictor error of sleep stages 2 and 4 for the comparison between the Kalman filter

and the non-linear local prediction techniques.

TABLE 5.2 N orm alized P redictor E rror (NPE) for D ifferent Sleep Stages and D ifferen t P red iction m odels.

Sleep stage 4 PARAM ETER N P E (x lO -3)

N on-linear m odel ED = 5, PA = 4000 11.8

KALMAN filte r 1.5

Sleep stage 2

N on-linear model ED = 8, PA = 4000 88.5

KALMAN filte r Af= 5 6.8

ED = Embedding Dimension, PA = Atlas Points or Points used for Embedding.


5.4 EEG modelling using Radial Basis Functions

—Adaptive non-linear modelling by a modified Kalman filtering approach

5.4.1 Introduction

Due to the non-linear nature of the EEG phenomena, non-linear EEG modelling

has received considerable attention in the past few years. The Kalman filter is a

linear (discrete-time , finite-dimensional, time variant) system, but, due to the fact

that the Kalman filter is described in terms of state space representation difference

equations (in the case of discrete-time systems), non-linearities can be introduced

onto it in a variety of ways. By incorporating some form of nonlinearity in the

structure of the adaptive filter, it is possible to account for the non-linear behaviour

of physical phenomena responsible for generating the data and overcome some

limitations with a hope of broadening of application areas.

The resulting filter by introducing non-linearities onto the Kalman filter is

naturally referred to as the extended Kalman filter (EKF). There are many papers

related to this topic, and most of the EKFs have the form of non-linear vector

difference equation in state space as

X( t ) = a[ X( t - 1)] + b[u( t - 1)] + w( t ) (5.5)

with the corresponding measurement model


z ( t ) = c [X( t ) ] + v ( 0 (5.6)

where #[•], 6[], and/or c[-] are non-linear functions of X and u, and w and v are white

and gaussian. Generally, as a part of the requirement for implementation of non

linear filtering algorithms, explicit non-linear difference equations 5.5 and/or 5.6

characterizing of the underlying system are required. This seems to make no sense in

EEG environment. Therefore in this section, we will describe a new technique of

using Radial Basis Functions with the modified Kalman Filtering approach for non

linear modelling of EEG.

5.4.2 Outline of the algorithm

In linear representations of stochastic processes the EEG time series z(k), z(k -

1), ..., z(k - M), can be represented as a realization of an autoregressive process of

order M in the form

where ah i = 1,2, ..., M, are AR parameters, and v(k) is a white noise process. That

is, z(k) equals a finite linear combination of past values of the process, z(k - i),

i = 1, 2, ..., M, plus an error term under the assumption of linearity. It is equally

acceptable to say that z{k) could be treated as a dependent variable of the previous

values of itself, i.e.

M

z (k) = a , z {k - i) + v(£) (5.7)i=l

z {k) = f ( { z ( k - z),z = 1,2, ••• M}) + v(£) (5.8)


with the linearity imposed on f It is only under this linear assumption that the AR

model can make sense for classical transfer-function techniques. Because we will

use state space analysis rather than the techniques involving convolution, Laplace or

z-transform, it is possible to release this limitation by introducing Radial Basis

Functions into an AR model of equation 5.8.

In the literature of approximation theory, the scattered data interpolation by

using Radial Basis Function interpolation is the following problem. Given M

different points { £ i = 1, 2, . . . , M } in Rn, and A/real numbers {df. i = 1,2, ..., M),

one has to calculate a function / : Rn -> R that satisfies the interpolation conditions

of

/ ( £ , ) = rf/ i = 1 ,2 , .» , A/. (5.9)

As introduced in Chapter 4, section 4.3.5, a Radial Basis Function approximation

has the form

M

d, = - Zj I) i = 1,2, — , M (5.10)j = 1

where (j) is from R+ to R and Xj are the coefficients of the RBF approximation. It is

important that the matrix

V, J = K \ 4 , - 4 j ) ||) i , j = 1, 2 , •••, M (5. 11)


is proved to be non-singular, therefore the set ofcoefficients j = 1, 2, , M]

defined by the interpolation conditions of equation 5.9 is unique. This approach is

considered as providing a highly promising way of dealing with irregularly

positioned data points and is particularly well behaved in multivariable interpolation

problems. It should be an advantage in the EEG environment.

Under our assumption, the EEG signal {*,} is generated from a dynamical

system with a finite degree of freedom. According to Takens Embedding Theorem,

for such a system, there is some finite integer m and a function /"such that

Xi+1 = r ( x i , •••, X/-/.I-1 ) (5.12)

and the dynamics generated by /"is equivalent to the original dynamics which gave

rise to {*,}. The function /"is generally a non-linear map in contrast to traditional

AR models of equation 5.7. Unfortunately, Takens’ theorem merely ensures the

existence of some /"and m, but does not show us how to get the map /" nor the value

of m.

In chapter 4, section 4.3, we considered several techniques to construct the

dynamics /"from the time series {xt}. Among them, one is Radial Basis Functions.

In other words, we want to construct an approximation to the function /" such that

the equation 5.12 hold by using Radial Basis Functions. Thus, the state space

reconstruction is necessarily the first step. Again, we will use the method of delays

to reconstruct the state space. Therefore, with the embedding dimension m’ and the


embedding delay r, the system state will be written as a state vector Sf = (xh x,-.r,

xHm'-i)T) at the moment f,. If we plot the point (x/+1, S,) of all our data points {x„ i = 1,

2, N} concerned, they will lie on a surface in the m'+l dimension space and it is

Radial Basis Functions we will use to approximate this surface. In this case, we let dt

= xi+l of equation 5.10, and S, = £ r Thus we have

MXM = - 5 ; II) (5.13)

j = 1

with M = N - m! - (w '-l)(r-l). In above equation || *S/ - Sj || = r is the Euclidean

norm between two different state vectors and (f> (r) is the so-called Radial Basis

Function. The Aj are unknown parameters and their values could be determined by

the interpolation conditions of equation 5.9.

In practice, however, the time series {xh i = 1, 2, ..., N} are often combined

with additional noise. The above procedure is no help in decomposing a signal from

a noisy environment. Moreover, the Aj have to be calculated once and for all and this

computation can be quite time consuming. Therefore, to overcome this, we consider

here a recursive version of the above scheme by involving the Kalman filtering

technique, treating equation 5.13 as a non-linear AR model and updating the Aj by

using incoming values of the time series xt.


Now for completing the modified Kalman filtering by using a Radial Basis

Function, we began by rewriting the state and measurement function of the Kalman

filter of equation 4.11 and 4.13 in Chapter 4 as

X k = Ak X k - 1 + W k - \ (5.14)

Zk = Ck X k + Vk (5.15)

where w e RM and v e R again are zero mean Gaussian noises with noise intensities

Q and R, respectively, assumed to be uncorrelated. We will let zk be the observation

of :t/+I and

X k = A = [ A 1 , A 2 A m Y

from equation 5.13. Furthermore, we will let

Ck = [*(||5* -5 * | | ) , * ( | | 5 * - 5 * - i | | ) , . . . , * ( | | 5 * - 5 * +i-a#||)].

It is obvious that the measurement function no longer is a linear function about

the time series, but yet is linear in the parameters structure. The Kalman filter can

still be treated as a linear system in this case, thus, all the recursive equations of

4.15, 4.17, 4.22, 4.25, and 4.26 developed in Chapter 4 are still valid with

guaranteed convergence.


5.4.3 EEG Modelling

There are many possible choices for Radial Basis Functions as mentioned in

Chapter 4, section 4.3.5. Among them, the so called multiquadric is often the choice

in the engineering environment. That is

4>{r) = ( r 2 + c )1'2

for some constant c > 0. Gaussian and inverse multiquadric Radial Basis Function

were also under our investigation, but they do not perform so well as the

multiquadric function. Thus, in this section we will mainly discuss the performance

of multiquadric function when modelling EEG signals.

At first glance, we know that the constant c needs to be settled. Our experience

suggest that the constant c shouldn’t be too big nor too small compared with r2.

Selecting too small a value for c, the system tends to be unstable and the prediction

error increases, while too big a c setting will remarkably reduce the influence of the

Euclidean norm r between two state vectors. It is believed also that c could have a

wide range of choice if the system stability is the main consideration. Thus, we

simply select

with “( )” means average.


There are obviously other three parameters which need to be carefully selected

in this scheme. These are the order of the Kalman filter M, the embedding delay r

and the embedding dimension m'. A variety of techniques exist for the estimation of

embedding dimension, among them the most convenient technique could be via the

correlation dimension, which is a measure of the complexity and/or number of

variables required to describe the underlying dynamic system. But with the presence

of noise, this technique may not give an accurate result. Thus, based on the

performance of the scheme, we will use the prediction accuracy to estimate the

embedding dimension m' and the model order M. The mean square prediction error

for sleep stages 2 and 4 of the scheme are listed in table 5.3 and plotted in figure 5.6

and figure 5.7 for different model orders M and embedding dimensions m' settings

with r = 1. The definition of the prediction error is

£ r 2 (RBF) = <|* ( 0 - x ( 0 | 2>.

Two minutes of data, with 12 bits resolution of sleep stage 2 and sleep stage 4, are

used. To make sense of the performance, the linear Kalman filtering prediction

errors of different model order setting are also listed in table 5.3 (III) with the same

segment of EEG signals.

Finally, the performance against the embedding delay is depicted in figure 5.8

with model order M - 5 and embedding dimension m! - 5.


It is interesting to notice from figure 5.6 and figure 5.7 (or from table 5.3) that

the prediction error improves faster with embedding dimension than with model

order. It is not recommended, however, to use too low a model order, although in the

sense of the prediction error, the performance of the scheme is quite good with low

model order and high embedding dimension. That is because with too low a model

order the system tends to be unstable (adjust itself too frequently). Furthermore from

figure 5.8, it appears that the performance of the scheme wouldn’t be improved

along with the increase of the embedding delay r.


Table 5.3 (I). Mean square error of the Radial Basis Function against the model

order M and Embedding dimension m' for sleep stage 2. (See figure 5.6.)

Sleep stage 2

Order Embedding dimension rri

M 2 3 4 5 6 7 8 9 10

2 7617 5778 4596 4328 3887 3722 3691 3667 3664

3 5958 4695 4064 3798 3717 3647 3631 3599 3572

4 5434 4350 4012 3677 3646 3605 3597 3558 3520

5 5044 4374 3911 3682 3637 3610 3588 3542 3488

6 4971 4312 3896 3706 3653 3621 3587 3530 3500

7 : 4807 4245 3899 3709 3655 3620 3580 3540 3501

8 4690 4195 3869 3697 3647 3612 3587 3537 3500

9 4605 4120 3819 3680 3637 3613 3577 3526 3495

10 4468 4029 3768 3645 3652 3600 3560 3514 3489


Table 5.3 (II). Mean square error of the Radial Basis Function against the model

order M and Embedding dimension m! for sleep stage 4. (See figure 5.6.)

Sleep stage 4

Order Embedding dimension m'

M 2 Q W m 4 5 6 7 8 9 10

2 8907 6325 5549 4932 4594 4350 4146 3940 3829

3 6847 5300 4625 4277 4078 3887 3800 3605 3454

4 6081 4848 4320 3996 3847 3722 3639 3478 3419

5 5571 4602 4132 3914 3747 3653 3551 3409 3383

6 5314 4564 4033 3849 3709 3598 3501 3389 3369

7 5151 4452 4051 3837 3682 3574 3487 3389 3369

8 4921 4389 4032 3824 3683 3564 3480 3397 3381

9 4839 4342 3982 3812 3689 3576 3477 3405 3386

10 4765 4291 3978 3812 3732 3605 3496 3417 3403


Table 5.3 (III). Mean error of the linear Kalman filter against the model order M

for sleep stage 2 and stage 4.

Sleep stage 2

Model order M

2 3 4 5 6 1 8 9 10

20076 15318 6471 6420 3707 3359 3153 3072 3045

Sleep stage 4

Model order M

2 3 5 6 8 9 10

300960 13845 13714 6373 5391 4721 4710 4500 4404

5.5 Performance comparison

From table 5.1 and table 5.2, (and figures 5.1 and 5.2,) the poor performance of

the NLF is obvious. In addition to the reasons summarised in section 5.2, further

considerable practical difficulties are worth mentioning here. Firstly, sleep EEG is

Chapter 5 MODELLING OF EEG i l l

widely believed to be unstable. In terms of dynamic systems theories, the sleep EEG

may have multi-attractors, or no definable attractors at all. That means, the

underlying dynamics evolve on the variant manifolds in a changeable dimensional

space. This violates the assumption that the attractor has an ergodic natural invariant

measure. Furthermore, there is not enough evidence for the existence of chaotic

attractors at all in some sleep stages, for example Babloyantz (1985) failed to find

them in the awake stage and the REM stage. In such a case, most non-linear analysis

of EEG signals achieved recently are dimensional (such as correlation dimension,

information dimension and etc.) and Lyapunov exponents analysis. Studies by phase

portraits can only be implemented in very short segments (only about severed tens of

seconds duration), in which the EEG can be treated as stable. Secondly, there are

difficulties of limited computer capacity. On the one hand, for example, by using

package of NLF, it costs about 20 minutes to model 1 minute EEG of sleep stage 2

(ED = 8) on a 386 personal computer. It amounts to showing that about one week

computing time would necessary to model a whole night sleep EEG (say, 8 hours),

and this is obviously unacceptable. On the other hand, memory capacity may

become unmanageable when increasing Embedding Dimension and number of Atlas

Points or using other techniques of non-linear analysis (such as global techniques

which have a very large number of free parameters to be estimated) for precise

modelling of a whole night sleep EEG. Thus it is unrealistic to precisely model a

whole night sleep EEG by using this NLF package. Table 5.2 and figures 5.1, 5.2,


5.4 and 5.5 show that the Kalman filter is much better than the Non-linear

Forecasting package for prediction.

Our results by using Radial Basis Functions are promising. As shown in table

5.3, this scheme has more or less the same prediction error as that of the Kalman

filter. Thus it could be said that the scheme seems to perform as well as the Kalman

filter in the sense of prediction error. It worth mentioning here that the scheme’s

prediction error will improve further if the multiquadric constant c is increased. But

it must be admitted at this stage that this technique is very much in its infancy. Only

the prediction errors are used for evaluation of the performance here and nothing

more. Thus, much more investigation are needed before enough confidence can be

obtained and the scheme can be used in practice.

The results of the comparative study between the non-linear prediction

technique and Kalman filtering showed that, as a predictor, the Kalman filter

approach was superior and this approach is the only choice used for the further

analysis in an attempt to classify the EEG automatically.


150

100

50

0

-50

Observed-100

Predicted

-150500 100 150 200 250

Data points

Figure 5.1. Two seconds of EEG and its prediction by using Non-linear

Forecasting package of NLF for sleep stage 4 with ED=5, PA=8000. The

ordinates are in arbitrary units.


100

-20Observed

-40 Predicted

- 60,100 150 200 250

Data points

Figure 5.2. Two seconds of EEG and its prediction by using Non-linear

Forecasting package of NLF for sleep stage 2 with ED=5, PA=8000. The

ordinates are in arbitrary units.


1.6

1.5

1.4

1.3

1.2

1.1

1

0.9

0.8

0.7

log10 (Er (KALMAN))

Sleep stage 4

Sleep stage 3

Sleep stage 2

Sleep stage 1

REM

, - L p20

Figure 5.3. Kalman filter prediction error against model order

M for 1.5 minute sections of EEG during sleep.


150

100

-50

Observed-100

Predicted

- 150,150

Data points

100 200 250

Figure 5.4. Two seconds of EEG and its prediction by using Kalman

filter modelling for sleep stage 4. The ordinates are in arbitrary units.


40

-20

Observed-40 Predicted

- 60 . 100 150

Data points

200 250

Figure 5.5. Two seconds of EEG and its prediction by using Kalman

filter modelling for sleep stage 2. The ordinates are in arbitrary units.


8000

7000

6000

5000

4000

3000

Model orderEmbedding dimension

Figure 5.6. The mean square prediction error of the modified Kalman

filter by using RBF for 2 minute section of EEG during sleep stage 2

against model order Mand embedding dimension m’ setting with

embedding delay z= 1.


9000

8000

7000

6000

5000

4000

3000

Model orderEmbedding dimension

Figure 5.7. Figure 5.6. The mean square prediction error of the modified

Kalman filter by using RBF for 2 minute section of EEG during sleep

stage 4 against model order M and embedding dimension m' setting with

embedding delay r = 1.


4800

4600

4400

4200

Sleep stage 24000

Sleep stage 4

3800

3600

T (embedding delay)

Figure 5.8. The mean square prediction error of the modified Kalman

filter by using RBF for 2 minute sections of EEG during sleep stage 2

and 4 the embedding delay with model order M - 5 and embedding

dimension rri = 5.

Chapter 6

A METHOD FOR CLASSIFYING THE

COEFFICIENTS OF THE MODEL

6.1 Introduction

As mentioned in previous chapters, the most widely used classifier following a

linear AR model is via spectrum estimation. Another widely used approach is the

use of neural networks.

The spectrum may be obtained by z-transforminf the AR model coefficients,

and all information contained in the coefficients will be transferred into the form of a

spectrum from a theoretical point of view. For sleep EEG analysis, the construction

of such a spectrum does not seem to make the analysis easier in many occasions.

The reason for this is that EEG signals, which are generated from one of the most

complicated dynamic systems, the human brain, are composed of substantial

numbers of averaged post-synaptic potentials. Thousands or even millions of

Chapter 6 A METHOD FOR CLASSIFYING THECOEFFICIENTS OF THE MODEL

122

neurones in the underlying cortex may be involved. There are good reasons to

believe that the EEG contains a vast amount of information and much of it is not just

related to sleep events. To interpret the EEG, the most important and the most

difficult job, is to extract events related information and "filter out" irrelevant

features. It seems that the main reasons for unsuccessful automatic sleep EEG

analysis are due to the failure to extract useful or relevant information and filtering

out the irrelevant features. Spectrum estimation fails to do this and it is widely

believed that no frequency band in the spectrum is actually a true sleep variable.

After spectrum estimation, researchers are still facing how to reduce the amount of

information.

Artificial neural networks seem capable of doing a good job in many practical

applications, and it seems there is a surge of interest among researchers, engineers

and industrialists. This interest is motivated by some important properties of neural

networks, such as: nonlinearity, learning capability, generalization, fault tolerance

and the ability to approximate prescribed input-output mapping of a continuous

nature. However, some of the limitations of the neural networks need to be

considered: First of all, there is much less understanding of the theory behind the

networks. Because neural networks operate as “black boxes”, i.e. their rules of

operation are completely unknown, it sometimes makes it difficult to explain the

results that are obtained from the networks. Secondly for network architecture, there

are no theoretical results nor satisfactory empirical rules suggesting how a network

should be dimensioned to solve a particular problem, such as the overall size of the


123

network, the number of hidden layers and the number of neurons in each of the

hidden layers. Thirdly, convergence may also be a problem. There is no theoretical

proof at all of the convergence of the algorithm to find the coefficients, and in

practice, the most serious problem with the neural networks approach is usually the

speed of its convergence. Finally, the implementation of networks consume large

quantities of computing time as the majority of networks are simulated on sequential

machines. It is particularly true when attempting to increase the size of the networks

that the processing time requirements will increase very rapidly. Because of this

operational problem, present techniques only allow neural networks to solve fairly

small problems. In the EEG environment, there is another concern about the use of

neural networks for the classification of the different sleep stages. As mentioned in

section 2.4 of chapter 2, the criteria (i.e. R & K. rules) used contain many subjective

components and do not always provide an unequivocal basis for decision. It is a fact

that many advanced methods have been developed and have considerable potential,

the results however still do not completely correlate with the R & K standard. Thus,

it is questionable, at the present time, whether the results could be improved

significantly for classification of different sleep stages by using the input-output

mapping property of neural networks directly.

In this chapter a novel technique will be described which can provide additional

information not obtainable by manual analysis or by automatic techniques that

merely mimic the process of visual sleep staging. In this approach, dynamic systems

theory (or state-space methods) is involved for classifying AR model (Kalman filter)


124

coefficients. In section 6.2, the coefficients will be embedded in a reconstructed state

space. Following this a classification of the tangent bundle will be made based on

the state changes in the EEG signals. Finally, the so called “state variable” will be

derived in section 6.4. It is considered that this state variable is generally related to

the sleep stage classified by the R & K rules.

6.2 Embedding the model’s coefficients into their state

space

After applying a Kalman filter to the EEG, it was felt reasonable to believe that

the behaviour of the five coefficients {oL(tJ} along time contained most of the useful

information in the EEG. Each coefficient was treated as a coordinate of a phase

space E = R5 (5th order Kalman filtering) and the output of the Kalman filter was

seen as a dynamic system with a point evolving on a manifold U in E. Suppose this

system is governed by an rth order differential equation, r > 1, then it can be written

as

* = X k (*)

with


125

x e I I E = R 5r r

As a map, this rth order differential equation on U is a vector field

X w : T r_1 U -> T r U

on Tr_1U c R5r, after converting the original equation into a set of first order

differential equations, with T( • ) defined a tangent bundle of ( • ). This differential

equation is certainly non-autonomous as the Kalman filter is just a linear

approximation to the EEG signals and secondly, the EEG itself is an unstable signal.

A variable u e R was introduced as a measure of time to indicate that the vector

field Xu is variable with time. Thus, the differential equation can be rewritten as

X = X ( x , u )

u — 1

or X = X ( x ) (6.1)

with X = ( x , u ) .

As a map, the rth order differential equation on U can be written as a static vector

field

X :T r l U x R —>-T(Tr’ 1U x R ) = T r U x R (6.2)

o n T M U x R c R 5,+l.


126

To obtain information about this dynamic system, the state space reconstruction

is necessarily the first step. As mentioned in section 4.3.2 of chapter 4, with too high

a sampling rate and too small a delay lag the method of time delays will make the

coordinates strongly correlated. This introduces an artificial symmetry into the phase

portrait and causes the trajectory to lie close to the diagonal in the embedding space.

Unfortunately this happens to be the case here. To overcome this, the method

developed in the work of Takens (1981), theorem 3 will be used. In the notation of

section 4.3.2, his theorem states that (for the dynamic systems with one observable):

Let M be a compact manifold of dimension d. For pairs f t h\fa . smooth vector field

and h a smooth function on M, it is a generic property that the map

O / ,h : M -» R2̂ 1, defined by

(5) = (* (* ) , T r W ' i ' ' (* )» . (»))) )i=o d t ,=0

(6.3)

is an embedding, Here vp/ (again) denotes the flow of/; this time, smooth means at

least C2̂ 1.

This theorem tells us that state space reconstruction could be achieved by

implementing this derivative map O / ,h. Thus, in our case, to reconstruct the state

space of the dynamic system of equation 6.1, one can embed the model’s

coefficients {aft)}, in a space of


127

d d2r( « ( t i ) > T T a ( n )>•• >------7 7 a ( ?i ) ) x R ( 6 -4 )At d t 2

with a(^) being a vector of dimension 5. For convenience, the state space will be

represented as the form of tangent bundle. Thus, the state space of equation 6.4 can

qbe written as T UxR with q = 2r.

6.3 Classifying the model’s coefficients

The derivative mapping O gives an important equivalent relation between the

manifold in the state space and the original dynamic system (equation 6.1). Thus, it

is reasonable to say that the state space contains all the information which Kalman

coefficients have if q is fairly high. Therefore it is possible to find some way of

classifying a model’s coefficients in state space according to some state changes of

EEG signals. That is, to find some equivalence relation Q defined on T UxR based

qon some sleep events, then Q gives a decomposition of T UxR into a quotient space

(T ,UxR)/Q.

It was found that the manifold U c E i s kept unchanged along the whole night

sleep, except during the wake stage and movement arousal. However, the system

image in the space of TUxR (i.e. q = 1) or more efficiently in (TUxR)/V does vary


128

with the deepness of the sleep if V denotes the relation of all points p in the manifold

U that are equivalent. Space (TUxR)/V in fact comes from a projection n from TUx

R onto an arbitrary tangent space TpUxR, as shown in figure 6.1 below and it gives

ideas only about the velocity (a vector) of the point moving round in the manifold U.

(TUXR)/V

Projection

71

TUXR

TPUXR

XUXR

Figure 6.1. Space (TUxR)/V is a quotient space of TUxR. V denotes

the equivalent relation of all points p in the manifold U.

Statistics analyses were carried out in the space TU/V for different sleep stages

and it was turn out that the direction of the velocity appear to be random. As an

example, figure 6.2 shows how the system behaved in the space TU/V of dimension

5. One and half minute of data are used in this example and only three projections


129

are plotted which involve all the axes in the space. (All the projections of different

axis combination were studied indeed and the results are the same.) Thus it is the

speed (scalar quantity) rather than the velocity that changes along with the sleep. If

we define the norm in the space (TUxR)/V of dimension 5 as

x =2\ 1/2

where x e (TUxR)/V and let W be the equivalence relation that relates one point x

to another x’ if ||x || = ||x '||, we get another quotient space

(TUxR)/(VxW )

of dimension 2 which makes sense of how turbulent or non-stationary the system is.

The information contained in the quotient space of (TUxR)/(VxW) seems well

correlated with the deepness of the sleep and it seems that no further relevant

qinformation can be obtained from the space of T UxR with q > 1. Figure 6.3a and

figure 6.3b show two system images in the space of (TUxR)/(VxW ) of 10 second

long segments in sleep stages 4 and 2 respectively.

It is found that the main parameter which appears to distinguish the sleep stages

is the intermittence of the peak in the changes of the system image in (TUxR)/(Vx

W) which are the continuous "low" peaks. In order to determine how deep and how

long the intermittences are, an envelope and threshold is applied. Defining a peak as


130

a turning point from a positive to a negative slope, a moving window of 8 data

points is used and the envelope is estimated by averaging the amplitudes of all peaks

within this window. The window is then moved 4 data points at a time and the

procedure repeated. Figure 6.4 shows the envelopes under a threshold for

approximately 10 seconds long segments of sleep stages 1, 2, 3, 4 and wakefulness.

The integrated area between the envelope and a threshold was defined as a

“state variable”. The decision on the level of the threshold is based on the data in

figure 6.5 which shows the averaged state variables over 90 second segments of

EEG against the threshold for different sleep states. From figure 6.5 it is clear that

the differences between sleep stages 1, 2, 3 and 4 do not increase further beyond a

threshold of about 0.08. The difference between sleep stage 1 and wakefulness

however, still increases even beyond the point where threshold = 0.2. For selecting a

threshold that separates the different sleep stages efficiently, the relative difference

of state variables between sleep stage 1 and wakefulness against the averaged state

variable of wakefulness is calculated and plotted in figure 6.6. The maximum point

is about 0.08 and that was chosen as the threshold level.

The state variable is obtained from the integration of the area between the

threshold and the envelope below that threshold in a 2 seconds segment; So that 64

envelope data points are included. Figure 6.7 plots the state variables in 90 second

segments of different sleep stages and the segments were chosen where they are free


131

from noise and artifacts, such as the high frequency noise from EMG channel and

movement artifact.


132

axis2

1.5

1

0.5

0

-0.5

-1

-1.5

-2 -1.5 -1 -0.5 0 0.5 1 a x is l(a)

ax is4


133

axis5

0.6

0.4

0.2

0

- 0.2

-0.4

- 0.6

- 0.8

-1

"1- .̂5 -1 -0.5 o 0.5 1 axis4(c)

Figure 6.2 (a), (b), (c). The system image in the space of TU/V.

(Arbitrary units. 1.5 minute of data.)


134

Data points1200

Figure 6.3a. The system image in the space of

(TUxR)/(Vx W) for sleep stage 4 in arbitrary units. (10 seconds of data.)

Chapter 6 A METHOD FOR CLASSIFYING THE 135COEFFICIENTS OF THE MODEL

0.5

0.4

0.3

Data points

Figure 6.3b. The system images in the space of

(TUxR)/(VxW) for sleep stage 2 in arbitrary units. (10 seconds of data.)


136

0.05Sleep stage 4

100 150 200 250 300

0.05 Sleep stage 3

150100 200 250 300

0.05 Sleep stage 2

0 50 100 150 200 250 300

Sleep stage 1

*0 50 100 150 200 250 300

(DataPoints)/4

Figure 6.4. The envelops under a threshold o f 0.08 for about 10

seconds long segm ents o f d ifferen t sleep stages. (10 seconds o f data.)


137

6 Wakefulness

Sleep stage 1

Sleep stage 2

++ +++ Sleep stage 3

Sleep stage 4

5

4

3

2

1

0.0.1 0.150.050 0.2 0.25

Threshold level

Figure 6.5. Averaged state variable (in arbitrary units) over one and half

minutes long EEG segments against threshold level.


138

0.5

- 0.5

0.05 0.1 0.15 0.2 0.25

Threshold level

Figure 6.6. Relative difference of state variables between

sleep stage 1 and wakefulness in arbitrary units.


139

40

Stage 4

20

0

Stage 4

40

Stage 3

20

0

40

Stage 2

20

0

Stage 1

0 10 20 30 40

40

WakefulnessArtifact

20

0

1.5 minute segments

F ig u re 6 .7 . T he s ta te v a r ia b le s fo r d if f e re n t s leep

s ta g e s . (E ach d a ta is g e n e ra te d from a 2 sec o n d s ep o ch .)

Chapter 7

RESULTS

A pilot study of analysis of the human sleep continuum has been numerically

presented in previous chapters — Model based dynamic analysis, in which the EEG

signal is first characterized by using adaptive AR model (Kalman filter), and then

the model coefficients are analysed dynamically. In figure 6.6 (in chapter 6), the

state variable appears to vary with different sleep stages. To demonstrate the

potential usefulness of the approach, the behaviour of the state variable across a

whole night will be analysed in this chapter.

Figure 7.1 presents the performance of the state variable for the whole night

recording from which the method was developed (i.e. the same sleep EEG of figure

6.6 ). Figure 7.2 indicates the manually scored hypnogram of the same sleep as of

figure 7.1. It is encouraging to observe that the state variable appears to be correlated

with the depth o f the sleep.

A further method of demonstration the correlation is to average the state

variable in each (30 or 20 second) epoch, as shown in figure 7.3, and then to analyse

Chapter 7 RESULTS 141

the distribution of the averaged state variable within each manually scored sleep

stage. To check this pilot study a different record (of subject 2 if the subject 1 was

used for the pilot study) from the same data source is also used for the further studies

(see APPENDIX B for more details about each record). Figure 7.4 shows the

averaged state variable with the whole night sleep of the subject 2 and the visually

scored hypnogram. To analyse the distribution of the state variable within each sleep

stage, the averaged state variable of subject 1 and 2 against the manually scored

sleep stage for the same epoch are plotted in figure 7.5. For a perfect agreement, one

would expect that the points in figure 7.5 would fall on a diagonal line. There are

three reasons why this is not the case. Firstly, the state variable is a continuous

parameter, whilst the manually scored sleep stages are discrete. Thus it is inevitable

that the image in figure 7.5 has an cascade-like structure. Discretization of human

sleep has been considered as one of the major shortcomings of the rules used in sleep

analysis. Secondly, the rules used to score the sleep stages have arbitrarily-defined

thresholds. For example, scoring of sleep stage 2 is mainly based on very short

duration events such as K-complexes and sleep spindles (each of which are about

half second duration). Scoring of stage 3 requires 20 - 50% of the delta activity

during an epoch with a frequency of 2 Hz or lower and a peak-to-peak amplitude of

75 fiV or higher in the EEG channel. A subjective assessment of the EEG is

therefore required when applying the rules and this can lead to unreliable results and

poor agreement between scorers (Kelley et al. 1985). Thirdly, the noise contained in

the EEG channel contributes to the scatter of the points in figure 7.5. Close study


shows that most, if not all, of the outlying peaks and troughs in figure 7.1, 7.3 and

7.4(a) are caused by noise. When a low frequency artifact appeared in the EEG

channel caused by a body movement, there will be a considerable high peak in these

figures. When there is high frequency noise, such as in a case of a micro-arousal, a

significant trough will present. It may also worth to note that the manual observer

tends to assume that a state persists when he/she assesses a sleep record, i.e.

“expected stage” depends upon previous decisions. While, the model based dynamic

analysis makes no such assumption and evaluates each epoch independently. This

may also contribute to the scatter of the points in figure 7.5.

Figure 7.5 indicates also that the state variable has better agreement within

stages of NREM sleep than during REM sleep and MT (Movement Time) stages.

This may happen because REM sleep is mainly characterized by bursts of rapid eye

movement and suppression of EMG activity, whilst the EEG pattern resembles that

of the stage 1. In MT stage the EEG tracing is normally obscured for at least half the

epoch by muscle tension and/or amplifier blocking artifacts associated with

movement of the subject. Thus the state variable is expected to be scattered over a

wide range.

For further analysis of the distribution of the state variable in each sleep stage, a

group of figures are produced. Figure 7.6 and 7.8 demonstrate the distribution, non

normalized and normalized, of the averaged state variable for different NREM sleep

stages (sleep stage 1, 2, 3 and 4) of subject 1 and 2, followed by figure 7.7 and 7.9,


in which the normalized distributions of the averaged state variable are separately

plotted again in different NREM sleep stages. It appears that the means of the state

variable for each sleep stage can be distinguished but there are overlaps of the

distribution among them. It is believed that the arbitrary way the R and K rules are

defined, the subjective assessment of the EEG, as mentioned above, and the noise

contained in it may be among the main reasons for the overlapping. To further

demonstrate the consistency of the technique over different subjects, the state

variable against each subject in each sleep stage are plotted in figure 7.10. It appears

the technique has a good consistency within these two subjects.

Four more overnights’ sleep from different data source have been analysed in

this way to gain more insight into the distribution and overlaps of the state variable.

This makes a total of six night’s sleep altogether and it may be enough to examine

overall values of the technique. The records are divided into two groups based on the

two data sources. The two records in the first group, i.e. subject 1 and 2, are visually

staged in the Department of Engineering Science, Oxford University and 30 second

epochs are used. The second group of records of subject 3 to 6 are manually assessed

by Jane Jones (NCE Brainwaves, N. Ireland) and 20 second segments are employed.

The way the EEG signals are digitized is also different in these two groups (for more

details see Appendix B). In the first group, the EEG signals were digitized using 8

bit resolution and the remainder were digitized in 12 bit. Thus the second group of

data has the following characteristics; it is of better amplitude resolution, better time


resolution and more consistent scoring (four sleeps were staged by one scorer). All

the results are presented in the following figures (figure 7.11- 7.24).

Figure 7.11, 7.12, 7.13, and 7.14 show the averaged state variable for the whole

night sleeps of the 4 additional subjects (subject 3 to 6) and their manually scored

hypnogram s. The averaged state variable of each subject against sleep stages are

plotted in figure 7.15, (the terms “U”, “N” and “C” are described below) and the

distribution and normalized distribution of the state variable for subject 3 to 6 are

presented in figure 7.16, 7.18, 7.20, and 7.22. Again, the normalized distribution in

different NREM sleep stages are separately displayed in figure 7.17, 7.19, 7.21 and

7.23. The consistency of the distribution of the state variable against each subject in

each sleep stage are plotted in figure 7.24. Comparing the two groups of sleep, it

appears that the image of the averaged' state variable in the second group (in figure

7.11(a), 7.12(a), 7.13(a) and 7.14(a)) have more hair-like peaks than that of the first

group (figure 7.3 and 7.4(a)). The main reason is that a 20 second epoch is used in

the second group, whilst a 30 second epoch is used for the first.

A confidence factor is also given in scoring the second group of sleep along

with sleep stages. For each epoch, not only is a sleep stage attached by the scorer,

but also a factor o f confidence. As shown in figure 7.15 the factor involves three

levels, in which U means “uncertain”, N means “normal”, and C means “certain”.

Only the epoch with confident factors of certain are used in the following analysis of

the state variable distribution. Because of the difference in digitized resolution, the


ranges of the state variable therefore are different in the two groups of sleep. This

would not appear to be a problem as only relative values of the state variable are of

interest in this analysis. If many different sources were used then a normalization

procedure on the raw data could be used to make all data comparable.

The results shown in the figures mentioned above suggest that the state variable

behaved in the same manner in the six different sleeps, and it is well correlated with

the depth of the sleep. The distributions of the state variable did not appear to be

significantly different in different sleeps. If there are differences, they may be in the

mean distribution of state variable in stage 3 and the range in stage 2. The mean

distribution in stage 3 seems slightly closer to that of sleep stage 2 in the first group

of sleeps, and the distribution range in stage 2 in second group seems wider than that

in first group. The reason for this is not clear, but it is believed that the reason is the

arbitrarily-defined thresholds on sleep stage 2 and subjective assessment of the EEG

when applying the rules.

It is notable that the sleeps of subject 5 and 6 have very little sleep stage 4 (two

epochs of sleep stage 4 for subject 5 and none for subject 6), Thus these may not be

normal sleeps, but the technique still works well. Since there is not enough stage 4 in

subjects 5 and 6, it is hard to evaluate the consistency of the state variable in this

stage. But based on the previous analysis, it seems to have a good ability to separate

sleep stage 4 from other stages, as shown in figure 7.6, 7.8, 7.16, and 7.18. The

consistency in sleep stage 2 and 3 as shown in figure 7.24 appears good, although it


appears that the state variable has a wider range in the second group of stage 2. The

consistency among the different subjects in sleep stage 1 appears not so good in the

second group of recordings. Obviously, stage 1 is a light sleep, more noise is caused

by movement and arousals in this stage, therefore the distribution of the state

variable scattered by the noise contained in the EEG leads to poor consistency

among different subjects.


State variable

Time (minutes)

F igure 7.1. State variab le for a whole n ig h t’s sleep o f

sub ject 1. (Each data is generated from a 2 seconds

epoch.)


Stage 4

Stage 3

Stage 2

Stage 1

REM

MT

Wake

10050 150Time (minutes)

200 250

F igure 7.2. V isually scored hypnogram o f the same sleep as

figure 7.1. (30 seconds epoch.)


State variable

35

30

25

20

15

10

5

Epoch =

6000.

1000 200 300 400 500

Figure 7.3. Averaged state variable within the same sleep as o f

figure 7.1.


State variable

30

25

20

100 200 300 400 500 600 700 800 900

Sleep stage

Stage 4

Stage 3 -

Stage 2

Stage 1

REM

W ake

Epoch = 30s

900200 300 4000 100 500 600 700 800

Figure 7.4. (a) Averaged state variable for a whole n ig h t’s

sleep of subject 2 and (b) its v isually scored hypnogram .


- State variable0

5

-10

-15

-20

-25

-30

Sleep stage- 35.0 WakeMTStage 4 Stage 3 Stage 2 Stage 1 REM

- State variable

-10

-15

-20

-25

-30

Sleep stage-35

WakeMTStage 4 Stage 3 Stage 2 Stage 1 REM

Figure 7.5. Averaged state variable against sleep stages for subject 1 (a)

and subject 2 (b).


Number45 Stage 2

40

35

30

25

20 Stage 4

Stage 3

10

Stage 1

20 2515 30State variable

Normalized distributionState 20.25

State 3State 1 -I

0.2

State 4

0.15

0.1

0.05

0, 10 2015 255 300 35State variable

Figure 7.6. Distribution (a) and normalized distribution (b) of averaged

state variable for different NREM sleep stages of subject 1.


Normalized distribution0.250.25

0.150.15

0.050.05

30 State variable 0(a)

0.25 r

0.15

0.05

0.25r

0.2

0.15

0.05

Figure 7.7. Normalized distribution of averaged state variable for

sleep stage 1 (a), stage 2 (b), stage 3 (c) and stage 4 (d) of subject 1.


Number

140Stage 2

120

100

80

60

40 -

Stage 4Stage 320

Stage 1v -

10 20 25 30 35State variable

Normalized distribution

Stage 20.35 r

0.3

Stage 30.25

0.2Stage 4

0.15Stage 1

0.05

20 30 35State variable

Figure 7.8. Distribution (a) and normalized distribution (b) of averaged




0.250.25

0.150.15

0.050.05

30 State variable 0(a) <b)

0.25

0.15

0.05

0.25-

0.15

0.05

Figure 7.9. Normalised distribution of averaged state variable for sleep

stage 1 (a), stage 2 (b), stage 3 (c) and stage 4 (d) of subject 2.


State variable

-5 -10

-15 -20

-25 -30 -35

(a)Subject

State variable

-10

-15-20-25-30-35

2 Subject

State variable

-5 -10

-15 -20 -25 -30 -35

(b)Subject

State variable

-5 -10

-15 -20 -25 -30 -35

1 2 Subject

Figure 7.10. Averaged state variables against different subjects in the

first group, (a) sleep stage 1, (b) stage 2, (c) stage 3, (d) stage 4.


State variable

Sleep stage

Stage 4

Stage 3

Stage 2

Stage 1

REM

MT

W ake

0

Epoch = 20s

I]

Epoch = 20s

500(b)

1000 1500

Figure 7.11. (a) Averaged state variable for a whole night’s sleep of

subject 3 and (b) its visually scored hypnogram.


State variable60

1000

Epoch = 20s

1400

Sleep stage

MT

Wake

Epoch = 20s

200 400 600 800(b)

1000 1200 1400

Figure 7.12. (a) A veraged sta te variable for a whole n ig h t’s

sleep of subject 4 and (b) its v isua lly scored hypnogram .


State variable

60 r

50 -

0<------------ 1-------------1------------ 1-------------1------------ 1________ i________ i0 200 400 600 800 1000 1200 1400

(a)Sleep stage

Stage 4

Stage 3

Stage 2

Stage 1

REM

M T

W ake

800200 400 600 10000 1200(b)

Figure 7.13. (a) A veraged sta te variab le for a w hole n ig h t’s


20s

= 20s


State variable

Sleep stage

400 600 800 (a)

1000 1200

Epoch = 20s

1400

Stage 4

Stage 3

Stage 2

Stage 1

REM

MT

Wake

0 200 400 600 800 1000 1200 1400(b)

Figure 7.14. (a) A veraged state variable for a whole n ig h t’s



- State variable

-10 U N C

-20

-30

-40 UncertainNormalCertain

-50

Sleep stage-60

WakeStage 4 Stage 3 Stage 2 Stage 1 REM MT

- State variable

-10

-20

-30

-40U — Uncertain N — Normal C — Certain

-50

Sleep stage-60

WakeStage 4 Stage 3 Stage 2 Stage 1 REM MT

Figure 7.15. Averaged state variable against sleep stages, (a) subject 3, (b) subject 4.

(Subjects 5 and 6 are shown on the next page.)


- State variable

-10

-15

-20

-25

-30

-35 UncertainNormalCertain-40

-45Sleep

-50WakeStage 4 Stage 3 Stage 2 Stage 1 REM MT

- State variable

-10

-20

-30

-40U — Uncertain N — Normal C — Certain-50

-60WakeStage 4 Stage 3 Stage 2 Stage 1 REM MT

Figure 7.15 contiune. Averaged state variable against sleep stages,

subject 4, (c) subject 5, (d) subject 6.


Number60

Stage 250

40

20Stage 3

Stage 4Stage 110

4020 50State variable


0.18

0.16Stage 4Stage 1

0.14

Stage 30.12

0.1Stage 2

0.08

0.06

0.04

0.02

40 50State variable

Figure 7.16. (a) Distribution and (b) normalized distribution of

averaged state variable for different NREM sleep stages of subject 3.


Normalized distribution0.1

0.08

0.06

0.04

0.02

0,0 20 6040

0.15

0.05

State variable

0.2

0.1

00 20 40 60

0.12

0.08

0.06

0.04

0.02

Figure 7.17. Normalized distribution of averaged state variable for sleep


165r e s u l t s _

Chapter 7 _------ .--------- '

Stage 2

45

40

Stage 3Stage 4

Stage 1

State variable

Normalized DistributionStage 3

Stage 4

Stage 2

S tagel

0.08

0.06

0.04

0.02

State variable40

Figure 7.18. (a) Distribution and (2) no

state variable for

irmalized distribution of averaged

different NREM sleep stages of subject 4.



0.1

0.08

0.06

0.04

0.02

00 20 40 60

0.08

0.06

0.04

0.02State variable

0.15

0.1

0.05

00 20 40 60

0.15

0.05




Number60

50 Stage 2

40

30

20

Stage 1Sleep stage 4

Stage 3 / V

4020 50 60(a) State variable

Normalized distributionStage 4

0.9

0.8

0.7

0.6

0.5

0.4

0.3 Stage 1;;Stage 3

0.2

Stage 2

20 30 40 5010 60State variable

Figure 7.20. (a) Distribution and (b) normalized distribution of averaged




0.4 r

0.3

0.2

State avriable

0.08

0.06

0.04

0.02

0.2 1

0.8

0.6

0.4

0.2

00 20 (d) 40 60




Number60 r

Stage 250

40

Stage 1

Stage 3

40 50State variable

NormaHzed distribution

Stage 3

Stage 10.1

0.08

Stage 20.06

0.04

0.02

30 40 50Stage variable

Figure 7.22. (a) Distribution and (b) normalized distribution of averaged




0.06

0.04-

0.02

0.12

0.08

0.06

0.04

0.02State variable

0.12

0.08-

0.06

0.04

0.02

Figure 7.23. Normalized distribution of averaged state variable for

sleep stage 1 (a), stage 2 (b) and stage 3 (c) of subject 6.


State variable0

-10

-20

-30

-40

-50

-60

State variable

4 5(a)

6 Subject

-10

-20

-30

-40

-50

-606 Subject

State variable

Or

-10

-20

-30

-40

-50

-60

State variable

4 5(b)

6 Subject

-10

-20

-30

-40

-50

-606 Subject

Figure 7.24. Averaged state variables against different subject in the second

group, (a) sleep stage 1, (b) stage 2, (c) stage 3, (d) stage 4.

Chapter 8

CONCLUSIONS AND FURTHER WORK

Traditionally, the most important aspect of sleep analysis is sleep staging. For

sleep staging, EEG signals are usually recorded in order to obtain relevant

information about the process of the sleep. However, inevitably, this information is

hidden in noise. These neurophysiological signals are recorded at a rate of over a

hundred samples per second and sometimes over a number of channels together with

the EMG and/or EOG signals. Both data reduction and extraction of the relevant

information are therefore common goals of most automatic analysis methods. To

achieve these, feature extraction from the EEG (EEG interpretation) followed by

features classification are the procedures included in most of the automatic methods.

8.1 EEG interpretation

There are many existing techniques for the interpretation of EEG signals.

Among them, the most commonly used in EEG processing are the frequency domain

Chapter 8 CONCLUSIONS AND FURTHER WORK 173

methods (or transfer-function models). In frequency domain analysis the EEG waves

are treated as a collection of periodic signals, with an assumption that the underlying

system is linear and time invariant system. This is often far from reasonable in the

EEG environment. The realization that deterministic chaos may be generated from

rather simple deterministic dynamics led to the search for non-linear dynamic

systems in the area of EEG analysis.

To release these assumptions imposed on the frequency analysis, a state space

approach (rather than transfer-function modelling) for EEG interpretation has been

used in this project, since state space modelling has many distinct advantages over

transfer-function techniques. One of the most important advantages may be that the

state space methods can be easily generalized. Thus they have the capability to

characterize a non-linear, time varying, or nonstationary random system.

In this project, the performance of three kinds of state space approaches

(described in chapter 5) was investigated, i.e. one linear and two non-linear models

for modelling and forecasting the electroencephalogram (EEG) during sleep. Firstly

in section 5.2, the non-linear local prediction technique of Schaffer and Tidd (1990),

known as NLF, was investigated. This technique treated the EEG signals as

generated from a non-linear but stationary system. Secondly in section 5.3, a linear

state space approach of the so called Kalman filtering technique was implemented.

As Kalman filters have the ability of self-adaptation along changes of signal

properties, they have the capability to characterize time varying systems. Finally, a


recursive version of the Radial Basis function scheme was considered in section 5.4.

In the scheme developed here, the structure of a Kalman filter was adopted, but

Radial Basis Functions were introduced into the measurement function. It could

therefore be treated as a non-linear, time varying approach.

To evaluate the performance of NLF in EEG analysis, the Normalised Predictor

Error (NPE) of NLF with different parameter settings of sleep stage 4 and 2 were

compared in chapter 5 along with two simple predictors (zero and first order

polynomial analysis). It was clear that the NLF, with the embedding dimension set

to 2, is only marginally better than the first order linear predictor. It was expected

that NLF would perform better with higher settings of the embedding dimension as

EEG signals are considered to have high dimensional properties (B. Doyon, 1992).

However, this was not the case as performance worsened as the embedding

dimension was increased. The results also suggest that increasing the Atlas Points

would not be o f any benefit. This behaviour of the NLF is mainly because of the

unstable and noisy properties of the EEG signals. NLF treated the signals as

generated from a stationary system and it had no ability to filter out any noise which

corrupted the signals. As many non-linear system analysis techniques are very

susceptible to noise, it may be the main reason that the success of applying chaos

theory (non-linear system analysis) to most natural phenomena has been much

slower than that of well-controlled and somewhat artificial laboratory experiments

and why many algorithms available for analysis of chaotic time series are of limited

practical use. Among other reasons may be the limited capacity of the NLF package


which limits the maximum number of points to 10000. Compared with Kalman

filtering of order 5 the unsatisfactory performance of the NLF is evident.

Linear approximation is an important tool for analysing a non-linear system.

Kalman filters, as linear state space representations, have an advantage over classical

transfer function methods, i.e. they have the ability of adaptation to changes of

signal properties. Thus they are superior, in principle, to other AR models, can

sometimes be quite well behaved when applied to chaotic systems, and they have

been widely applied in various areas to estimate the states of non-linear systems

(e.g., Mous and Grasman, 1983; Bockman, 1991; Myers et. al., 1992). But it seems

that the superior potential of Kalman filters have not been fully exploited in EEG

analyses. In practice, most researchers have used Kalman filters as a first step toward

spectrum estimation, for spectrum estimation however higher model orders are

needed while Kalman filters are sensitive to the model order and with too high a

model order (indicated by the FPE criterion) Kalman filters may produce spurious

peaks in the spectra. It is suggested in this work that a different approach for EEG

classification can be used which is not based on spectral estimation and does not

favour any particular model order. In this way, a model order somewhat lower than

the “best order” for the spectral estimation approach can be used and it is shown that

such low model order Kalman filters are used for analysing EEG signals, making it

possible to use a short segment length. Thus from an engineering point of view,

there is an advantage in using such a short segment length to deal with unstable


system like EEG and in clinical point of view, it makes possible the unmasking of

micro-arousals.

The use of Radial basis functions for modelling EEG signals was also

considered. In the literature of approximation theory, Radial basis function

techniques are solutions to real multivariable interpolation problems. This approach

to multivariable interpolation provides a highly promising way of dealing with

irregularly positioned data points. One disadvantage of this procedure is that the

parameters have to be calculated once and for all and this was shown to be quite

time consuming. Therefore this algorithm is not particularly suited for real-time

application. Another major disadvantage of this procedure is that it has no ability for

modelling signals with noise superimposed. Therefore, a modified Kalman filtering

approach was proposed in which Radial Basis Functions were introduced and led to

a recursive version of the scheme that continuously updated the parameters using

incoming values of the time series. This resulted in an algorithm much more

appropriate to on-line forecasting. At the same time, the scheme had the potential to

account for some non-linear behaviour of EEG signals, and also to reduce noise.

Although the scheme is a non-linear model, it has a linear parameters’ structure

which allows fast convergence as well as a general modelling capability.

The performance of the modified Kalman filter was considered in terms of

model order, embedding dimension and embedding delay together with the

performance of the linear Kalman filter. It seems that the scheme is well behaved in

the EEG environment in the sense of prediction error and could perform even better


than the linear Kalman filter if the multiquadric constant could be increased further.

It is also concluded that increasing the embedding dimension will reduce the

prediction error more efficiently than increasing the model order, and the results also

suggested that increasing the embedding delay will not be of any benefit.

Since the scheme was only evaluated in the EEG environment and only in the

sense of prediction error, many more investigations are needed before enough

confidence can be obtained so that the algorithm can be used in practice.

8.2 EEG feature classification

After applying a Kalman filter to the EEG, it was felt reasonable to believe that

the behaviour of the coefficients with time contained most of the information in the

EEG (except some high frequency noise). It was also reasonable to believe that

classification of the coefficients via spectrum estimation will be of little help for data

reduction and relevant information extraction, particularly when using such a low

model order. Therefore, a novel technique was developed with the view that it can

easily be used for data reduction and relevant information extraction, and can

provide additional information which is not obtainable by manual analysis or some

automatic techniques.

Each coefficient was treated as a coordinate of Euclidean space of dimension 5.

Therefore the output of the Kalman filter could be seen as a dynamic system with a


point evolving in a manifold U in this phase space. This is a non-autonomous system

with order not smaller than 1. It is always possible to convert an ftth order equation

in one variable into a system of n first order equations in n variables, and

interpreting a non-autonomous equation as a static vector field on a product

manifold. For de-correlating adjacent samples, the method used was that developed

in the work of Takens (1981), theorem 3 for the state space reconstruction for such

an autonomous system (rather than the method of delays), i.e. embedding the system

into the tangent bundle of the product manifold.

The image in the tangent bundle TU x R appears to vary with the depth of sleep.

Further analysis showed that it was the speed (scalar) of the point moving round in

the manifold U that changed with sleep. A quotient space, which contains

information about the speed of the point moving round the manifold U was defined,

and it makes sense of how turbulent or non-stationary the system is. For further

relevant information extraction, a state variable was defined based on state change in

the system.

8.3 Discussion

The results o f the application of Kalman filter coefficients classification using

this new technique and the definition of the state variable to EEG analysis are very

encouraging because: it is clear that the state variable is correlated with sleep stage

as defined by the R & K rules. Further analysis of the distribution of the state


variable shows that the state variable behaves well in NREM sleep (e.g. in sleep

stages 1 to 4). It is believed that the poor agreement in REM sleep, MT and

wakefulness stages is due to the way in which the stages are defined and the noise

contained in the EEG channel. Indeed only the four stages of NREM sleep are

distinguished from one another principally along the EEG signal. The main

advantage of the technique is that it provides a continuous parameter which

correlates with the depth of the sleep. The use o f 2 second epochs (or even shorter),

should permit the fine structure of each sleep stage to be displayed and to detect

transient changes in sleep state.

Possible applications include automatic sleep staging, anaesthesia monitoring,

and monitoring the alertness of workers in sensitive or potentially dangerous

environments. Previous techniques of EEG analysis using the Kalman filter

approach (e.g.: B. H. Jansen, Bourne and Ward, 1981; T. Bohlin, 1971), have relied

on spectral analysis as a part of the classification procedure and, because spectral

estimation needs a high order model, it was not as attractive as the method here

proposed. Another point is that the methods of analysis used make the problem

mathematically tractable under the theory of dynamic systems and the results

obtained are based on the state changes in the EEG. The method makes possible a

less subjective approach in the interpretation of the EEG.

The results of the present study indicate that the technique of model based

dynamic analysis of EEG can be used to obtain useful information. However more

work is needed before automatic sleep staging or anaesthesia monitoring, micro


arousal detecting and monitoring the ‘state of vigilance’ can be made. It is believed

that further studies concerning sleep as a continuous process may contribute to

achieving reliable automatic sleep analysis systems.

8.4 Further work

Before proceeding with any further analysis, some improvements may need to

be considered. This scheme, like many other sleep analysers, is sensitive to both

high and low frequency noise contained in the EEG signals. Thus, a band pass filter

is often necessary to filter out the noise. For doing this, one should take a caution on

the resolution when the signal digitized. When the resolution is low, such as 8 bit

resolution, a high gain in the amplifier section is often employed to reduce digitizing

distortion at low amplitude of the EEG. Thus, the A to D converter is often blocked

by high amplitude EEG signals or artifacts, e.g. when subject is in movement arousal

or in movement time. Filtering out low frequency noise will make the clipped

signals appear similar to that of the delta wave often appearing in sleep stage 3 and

4, and therefore, reducing the accuracy of the method. For this reason, eliminating

the A to D converter blocking may become necessary before filtering out low

frequency noise. One way to do this would be to decrease the gain in the amplifier

section before the A to D converter. However, this would relatively increase the

quantization distortion. It appears that using 8 bits binary numbers to digitize EEG


signals is not accurate enough, thus 12 bit or even 16 bit resolution is recommended

for the further analysis.

The method described in previous chapters appears to be able to provide a

continuous indication of the depth of the sleep. Thus it should be possible to

construct a sleep EEG analyser for sleep staging, micro-arousal detection and

abnormal sleep analysing. As the state variable is continuous and correlated with

depth of sleep, the method could be used for continuous sleep staging and therefore

it may more accurately reflect the process of sleep. If the standard criteria (R & K

rules) are to be used, there are several points that should be kept in mind. These

include, first of all, sleep is a biological phenomenon in which two separate states

have been defined, i.e. REM sleep and NREM sleep. NREM sleep is conventionally

subdivided into four stages (i.e., stage 1, 2, 3 and 4), which characterize the depth of

sleep. REM sleep is not subdivided into stages and appears to be a different

phenomenon to NREM sleep. It is defined mainly on the appearance of rapid eye

movements and the diminution in EMG activity. So, EOG signals are necessary for

separating REM from other sleep stages. The second point is that, according to R

and K rules, sleep stage 2 is separated from sleep stage 1 mainly by very short

events, (i.e. by sleep spindles and K-complexes which are about only 0.5 to 1 second

long) rather than by the background activity of the EEG signals. This may well

explain the results shown particularly in the second group of records that the state

variable have a wider range in stage 2 than other sleep stage. Therefore for sleep

staging in the strict sense of R and K rules, some techniques which can detect sleep


spindles and K-complexes may be useful. But in fact, one may doubt if the sleep

spindles and K-complexes can more accurately reflect the sleep than the state

variable, especially when it is known that sleep spindles and K complexes may be

absent from some subjects' whole night sleep or they may appear during rapid eye

movement. Finally, because the tonic of EEG activity is helpful for scoring of stage

REM, wakefulness and movement time, the EMG recording from muscle areas on

and beneath the chin is recommended.

Fuzzy logic, neural-network or neuro-fuzzy synergisms would be suitable

techniques to achieve the final phase of automatic sleep staging. Since on the one

hand, the (normalized) distribution of the state variable in each stage can be treated

as fuzzy sets, and on the other hand, it would be fairly easy for a neural network to

achieve sleep staging by using the state variable accompanied be the EMG and EOG

signals or even the detected spindles and K-complexes if possible. The essential part

of neuro-fuzzy synergisms comes from a common framework called adaptive

networks, which unifies both neural networks and fuzzy models. The fuzzy models

under the framework of adaptive networks are called Adaptive-Network-based

Fuzzy Inference System (ANFIS), which possesses certain advantages over neural

networks (Jang J. and Sun C-T., 1995).

The potential of the scheme for micro arousal detecting is also promising.

Normally a micro arousal is defined as any clearly visible EEG arousal lasting two

seconds or longer (but not associated with any stage or stage change in the epoch


scoring). The state variable, fortunately, is just obtained in two seconds long

segments and even shorter segments could be used if necessary. Whenever micro

arousal happens, the state variable goes down. But, most of the time, micro arousal

may combined with subject's movement (which could be defined as a movement

arousal). If that happened some artifact (very low frequency noise or clipped signals)

will be in the time series which makes the state variable go up again. Figure 8.1

shows how the state variable varies with some micro-arousals. This is demonstrated

by the scheme’s ability to pinpoint micro-arousals and the micro-arousals are of

particular interest to the clinician because they may used to highlight periods of

severely disturbed sleep caused by certain sleep disorders. Further works may

include analysing the abnormal sleep associated with frequent arousals, drug , sleep

related breathing disorders and etc.. To achieve accurate detecting of micro-arousal

it may be necessary, as a first step, to eliminate the A to D converter blocking and

filter out low frequency noise caused by the body movement.

There is still insufficient evidence to say that the state variable is a true sleep

variable, but it is almost certain that the R and K rules result in a “sleep state” that is

not a “true sleep variable”. The results suggest that the state variable is well

correlated with the depth of sleep as classified using the R & K rules. The methods

of the analysis used make the problem mathematically tractable and are consistent

with methods used in dynamic systems and it is important to note that the result

achieved is based on the state changes in the EEG. It is therefore possible to use less

subjective methods to interpret the sleep EEG. Many analysis schemes include


problems and compromises, for example those associated with the spectrum analysis

of EEG signals and the compromise between the model order setting and the setting

up of the spectrum. The method suggested in this thesis overcomes these

weaknesses.


Artifact

13 minutes long segments

Figure 8.1. The state variable varies with micro arousal (indicated

by the arrows).

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APPENDIX A

Akaike’s Final Prediction Error Criterion

We present here a brief outline of the Akaike’s Final Prediction Error criterion.

The FPE is defined as the mean square prediction error. If y(t) is a realization

for a given AR process, the FPE will be

FPE = E [0 >(t) -y p red W )2]-

Now considering another realization x(t) of the same AR process (that means

the processes x(t) and y(t) are statistically equivalent), and jcpred(t) is generated by

the predictor which is determined by using the process of y(t), then

FPE = E[(x(t) - x prei( t) )2] (A .l)

If let a(x, m) and a(y, m) denote the mth AR coefficients that have been estimated

from realizations of the Mh-order AR processes x(t) and y(t) respectively, then:

M

jcpred (t) = ^ a(y, m) • x(t - m) (A .2)OT = 1

APPENDIX A b

and let

Aa(m) = a (y , m) - a(x, m) (A .3)

Now substituting A.l into A.2 and then a(y, m) by A.3 we get

FPE = ( x(t) - 'Z(Aa(m) + a(x, m)) • x(t - m)

Since Aa(m) is not correlated with neither a(x, m) nor x(t), the FPE reduces to

It is clear that two items contribute to the FPE. The first one is the minimum

expectation of square residual of the Mth-order AR model fit to x(t), it can be

denoted by RM2. It is minimum because a(x, m) is established by means of the least

mean squared error criterion. The second one corresponds to the statistical deviation

of a(y, m) from a(x, m). Normally the first item RM2 decreases when M is increased,

whereas the second item increases. Therefore, in some sense (e.g. for spectrum

estimates), The value of M which minimizes the FPE is the optimum for the setting

of a model order.

FPE = E (x(t) - m) ■ x(t - mj'j + eT^ £ Aa(m) • x(t - m) j

)21 m m

+ S E E[ Aa(m) • Aa(l)] E[x(t - m) • x(t - I)]

APPENDIX A c

In the practical application, according to this principle, Akaike has developed

expressions for an efficient estimate of this minimum criterion. If we call (FPE)e as

estimated FPE, the (FPE)e is shown by Akaike to be

< F P E > ' ■ ( % ! £ ! } ) • ^

where N is the length of the segments. It is equal to say that the factor before RM2

gives some penalty as model order M goes up, and it is the trough of (FPE)e against

M which gives the estimation of a optimum model order, in the sense of spectrum

estimates at least.

APPENDIX B

Details of the EEG signals used

The EEG data and the hypnograms of the first group records (subject 1 and 2)

used were kindly provided by Dr. S.J. Roberts of Oxford University. The EEG data

were obtained from healthy volunteers with no history of sleep disorders and

recorded on analogue cassette tape recorder (Medilog 9000-11 recorder, Oxford

Medical Ltd, Oxford.). The analogue bandwidth of the recorder is 0.5 - 40 Hz with

roll-of at -40 dB/decade. The EEG data was digitalized in 8 bit (0 - 255) with

sampling rate of 128 Hz and a low-pass digital filter of linear-phase was used with

the passband cut-off frequency of 30 Hz, a passband gain of 1±0.01 and a stopband

gain of -50 dB at 50 Hz.

The second group of records (e.i., subjects 3, 4, 5 and 6) were kindly supplied

by Dr. Chris Idzikowski (NCE Brainwaves, N. Ireland). The EEG signals were

obtained from healthy subjects with no history of sleep disorders and recorded on an

analogue recorder (Store 14 DS, Racal Recorders,). The EEG signals were

digitalized in 12 bit resolution with 122 Hz sampling rate and digitally filtered with

APPENDIX B b

a bandpass filter. The centre frequency of the filter is 17.8 Hz with bandwidth of

34.4 Hz and stopband attenuation of -50 dB. Left transition band was set to 0.3 Hz

and the right was set to 5 Hz.

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