M A S A R Y K O V A U N I V E R Z I T A
P Ř Í R O D O V Ě D E C K Á F A K U L T A
INSTITUT BIOSTATISTIKY A ANALÝZ
Diplomová práce
BRNO 2016 RADOMÍR KŮS
(Ml
Co
I
M A S A R Y K O V A U N I V E R Z I T A
P Ř Í R O D O V Ě D E C K Á F A K U L T A
INSTITUT BIOSTATISTIKY A ANALÝZ L F A P Ř F M U
C E N T R U M PRO V Ý Z K U M TOXICKÝCH L Á T E K V PROSTŘEDÍ
Morphometry of magnetic
resonance brain images
based on patterns
Diplomová práce
Radomír Kus
Vedoucí práce: doc. Ing. Daniel Schwarz, Ph.D. Brno 2016
Bibliographie Entry
Author: Be. Radomír Kůs
Faculty of Science, Masaryk University
Institute of Biostatistics and Analyses M U
Research Centre for Toxic Compounds in the Environment
Title of Thesis: Morphometry of magnetic resonance brain images based on
patterns
Degree Programme: Experimental Biology
Field of Study: Mathematical Biology
Supervisor: doc. Ing. Daniel Schwarz, Ph.D.
Academic Year: 2015/2016
Number of Pages:
Keywords:
89 + 7
classification; computer-aided diagnosis, schizophrenia;
K - S V D ; machine learning; pattern recognition; magnetic
resonance imaging
Bibliografický Záznam
Autor: Bc. Radomír Kůs
Přírodovědecká fakulta, Masarykova univerzita
Institut biostatistiky a analýz L F a PřF M U
Centrum pro výzkum toxických látek v prostředí
Název práce: Morfometrie obrazů mozku z magnetické rezonance založená na
vzorech
Studijní program: Experimentální Biologie
Studijní obor: Matematická Biologie
Vedoucí práce: doc. Ing. Daniel Schwarz, Ph.D.
Akademický rok: 2015/2016
Počet stran:
Klíčová slova:
89 + 7
klasifikace; počítačově podporovaná diagnostika; schizofrenie;
K - S V D ; strojové učení; rozpoznávání; zobrazování pomocí
magnetické rezonance
Abstract
The thesis deals with machine learning and brain morphometry techniques serving as
a tool for computer-aided diagnosis of neuro-psychiatric disorders based on neuroimaging
data. The existing techniques are reviewed and put in a broader context of a classification
pipeline. Various feature extraction and selection methods are incorporated into a newly
proposed classification scheme and implemented in M A T L A B . Parameters of the methods
are explored and tuned on a set of first-episode schizophrenia patients and healthy controls
pre-processed by the means of voxel-based and deformation-based morphometry. The
obtained results are discussed evaluating the classification performance of the methods
and morphometry approaches, with the highest classification accuracy reaching 70%. The
source code of the proposed and implemented algorithms is attached on C D .
Abstrakt
Tato práce se zabývá metodami strojového učení a morfometrie mozku, sloužícími
jako nástroj pro počítačově podporovanou diagnostiku neuropsychiatrických poruch na
základě dat z neurozobracovacích technik. Součástí práce je přehled existujících metod,
které jsou dány do širšího kontextu klasifikačních algoritmů. Dále je navrženo nové klasifikační
schéma, obsahující různé metody extrakce a selekce příznaků, jež je posléze implementováno
v programovém prostředí M A T L A B . Jsou prozkoumány parametry těchto
metod s cílem vyladit jejich nastavení na souboru pacientů s první epizodou schizofrenie
a zdravých dobrovolníků, který je předzpracován podle metodik morfometrie založené na
voxelech a morfometrie založené na deformacích. Získané výsledky jsou vyhodnoceny a
porovnány napříč metodami z hlediska jejich klasifikačních schopností, přičemž nejvyšší
přesnost klasifikace dosahuje 70 %. Zdrojový kód navržených a realizovaných algoritmů
je přiložen na C D .
/ ^ \ \ Centrum pro výzkum m
mstitut i . i ,
( • J J i toxických látek mm bíostatístíky I SMI l
V ^ i t / / v prostředí I D / I a analýz ^ S r
Kamenice 126/3, 625 00 Brno, fax : +420 549 49 2840, www.recetox.cz
ZADÁNÍ DIPLOMOVÉ PRÁCE
Jméno studenta/ky: Bc. Radomír Kůs UČO: 376214
Studijní program/Obor: Experimentální biologie/Matematická biologie
Název práce: Morfometrie obrazů mozku z magnetické rezonance založená na vzorech
Název práce anglicky: Morphometry of magnetic resonance brain images based on
patterns
Vedoucí: Ing. Daniel Schwarz, Ph.D.
Konzultant(i):
Datum zadání: 18.12.2013
Datum odevzdání: květen 2015
Předpokládaný rozsah: 50 - 70 stran
Postup a zásady pro vypracování:
Cílem práce je seznámit se s pokročilou analýzou obrazů z magnetické rezonance pro
hodnocení morfologie mozku pacientů s neuropsychiatrickými poruchami a spojit tyto
metody zpracování a analýzy obrazů mozku s algoritmy strojového učení pro klasifikaci či
rozpoznávání. Konkrétně bude student řešit následující úkoly:
- Rešerše odborné literatury relevantní k problematice morfologických analýz obrazů
mozku včetně metod VBM (voxel-based morphometry), DBM (deformation-based
morphometry) a SBM (source-based morphometry) a PBM (pattern-based morphometry).
- Prostudování klasifikačních algoritmů vhodných pro MRI data a pro příznaky extrahované
z těchto dat morfometrickými metodami.
- Návrh algoritmu pro výběr příznaků a klasifikaci a dále jeho implementace a otestování
včetně kvantitativního zhodnocení na reálných datech pořízených pro výzkum první
epizody schizofrenie.
- Praktické části diplomové práce budou řešeny ve vývojovém prostředí MATLAB s využitím
toolboxů pro zpracování MRI dat SPM (Statistical Parametric Mapping).
Text práce bude vypracován v anglickém jazyce.
(©)
Centrum pro výzkum
toxických látek
v prostředí IBA
Institut
biostatistiky
a analýz
Kamenice 126/3, 625 00 Brno, fax : +420 549 49 2840, www.recetox.cz
Doporučená literatura:
• MECHELLI, A., PRICE, C. 1, FRISTON, K. J., ASHBURNER, J. Voxel-Based
Morphometry of the Human Brain: Methods and Applications. Current Medical
Imaging Reviews, 2005, vol. 1, no. 2, p. 105 - 113.
• ASHBURNER, 1, HUTTON, C , FRACKOWIAK, R., JOHNSRUDE, I., PRICE, C ,
FRISTON, K. Identifying global anatomical differences: deformation-based
morphometry. Human Brain Mapping, 1998, vol. 6, no. 5 - 6, p. 348 - 357.
• FRACKOWIAK, R. S. 1, ASHBURNER, J. T., PENNY, W. D., ZEKI, S. Human Brain
Function, Second Edition, Academic Press, 2004.
• GASER, C , NENADIC, I., BUCHSBAUM, B. R., HAZLETT, E. A., BUCHSBAUM, M. S.
Deformation-based morphometry and its relation to conventional volumetry of
brain lateral ventricles in MRI. Neurolmage, 2001, vol. 13, no. 6 , p. 1140 - 1145.
• LEPORE, N., BRUN, C , CHOU, Y. -Y., CHIANG, M. -C, DUTTON, R. A., HAYASHI,
K. M., LUDERS, E., LOPEZ, O. L, AIZENSTEIN, H. J., TOGA, A. W., BECKER, J. T.,
THOMPSON, P. M. Generalized Tensor-Based Morphometry of HIV/AIDS Using
Multivariate Statistics on Deformation Tensors. IEEE Transactions on Medical
Imaging, 2008, vol. 27, no. 1, p. 129 - 141.
• XU, L, LIU, 1, ADALI, T., CALHOUN, V. D. Source based morphometry using
structural MRI phase images to identify sources of gray matter and white matter
relative differences in schizophrenia versus controls. In IEEE International
Conference on Acoustics, Speech and Signal Processing, 2008. Las Vegas, Nevada
NV (USA), 2008, p. 533 - 536.
• CALHOUN, V. D., ADALI, T , PEARLSON, G. D., PEKAR, J. J. A method for making
group inferences from functional MRI data using independent component analysis.
Human Brain Mapping, 2001, vol. 14, no. 3, p. 140 - 151.
• PANTAZIS, D., LEAHY, R. M., NICHOLS, T. E., STYNE, M. Statistical surface-based
morphometry using a non-parametric approach .In 2004 2nd IEEE International
Symposium on Biomedical Imaging: Macro to Nano. Arlington, VA (USA), vol. 2,
2004, p. 1283 - 1286.
• GAONKAR, B., POHL, K., DAVATZIKOS, C. Pattern based morphometry. Medical
image computing and computer-assisted intervention 2011, vol. 14, no. 2, pp. 459
- 466.
V Brne dne: 18.12.2013
Podpis vedoucího práce:
Podpis studenta/ky:
Podpis garanta oboru:
doc. RNDr. Ladislav Dušek, Ph.D.
Acknowledgement
First of all, I would like to thank doc. Ing. Daniel Schwarz, Ph.D. for his brilliant
guidance, his time and the experience he shared with me during his supervision of my
work, for fruitful discussions and his critical reading without which this manuscript would
not have been written. I also want to express my gratitude towards my sister, RNDr.
Kateřina Kusová, Ph.D., for her priceless advice and grammar corrections that helped me
to form this work. Last but not least, I would like to thank my parents and friends who
supported me during my studies.
Declaration of Honour
This thesis is a presentation of my original research work. Wherever contributions of
others are involved, every effort is made to indicate this clearly, with due reference to the
literature, and acknowledgement of collaborative research and discussions.
Brno, 1 January 2016
Radomír Kůs
MU
MASARYKOVA UNIVERZITA
IBA INSTITUT BIOSTATISTIKY A ANALÝZ
Institut biostatistiky a analýz Lékařské a Přírodovědecké
fakulty Masarykovy univerzity spolupracuje na organizačním
zajištění výuky studijního oboru Matematická biologie
s Centrem pro výzkum toxických látek v prostředí
Přírodovědecké fakulty MU.
Table of Contents
List of Figures xi
List of Tables xii
List of Abbrevations xiii
Chapter 1. Introduction 1
Chapter 2. Schizophrenia 4
2.1 Occurrence and Causes 4
2.2 Diagnostics of Schizophrenia 5
2.3 Schizophrenia Through Optics of Neuroimaging 6
Chapter 3. Machine Learning in Neuroscience 9
3.1 Applications of Machine Learning 9
3.1.1 Brain Mapping 10
3.1.2 Computer-Aided Diagnosis 10
Chapter 4. Classification Pipeline 13
4.1 M R I Data Acquisition and Preprocessing 15
4.1.1 Magnetic Resonance Imaging 16
4.1.2 M R I Preprocessing 19
4.2 Feature Extraction and Selection 20
4.2.1 Univariate Approaches of Brain Morphometry 23
4.2.2 From Univariate to Multivariate Techniques 28
4.2.3 Multivariate Approaches of Machine Learning 29
4.3 Setting up Classification Rule 38
4.3.1 Classification Algorithms Based on Partitioning of Feature Space . . . 39
4.4 Classifier Validation 41
4.5 Evaluation of Classifier Performance 43
Chapter 5. Aims of The Thesis 45
Chapter 6. Datasets 46
6.1 G M Densities 48
6.2 Volume Changes 48
-ix-
Chapter 7. Classification Schemes 49
7.1 Mann-Whitney Testing 51
7.2 Inter-subject P C A 52
7.3 K - S V D 53
7.3.1 Pattern-based Morphometry 54
7.3.2 Concatenated Dictionaries 55
Chapter 8. Results 56
8.1 Preliminary Experiments and Parameters Tuning 56
8.1.1 Synthetic Dataset 57
8.1.2 Mann-Whitney Testing 57
8.1.3 Inter-subject P C A 58
8.1.4 K - S V D 60
8.2 Final Results 66
8.2.1 G M Densities 67
8.2.2 Volume Changes 68
Chapter 9. Discussion 69
Chapter 10. Conclusion 73
References 75
Appendices 90
A Appendix 1 90
B Appendix 2 91
C Appendix 3 92
D Appendix 4 93
E Appendix 5 94
F Appendix 6 95
List of Figures
4.1 Classification Pipeline 14
4.2 Difference Between Tl-and T2-weighted Image 17
4.3 Spatial Normalization 19
4.4 VBM Flow Diagram 25
4.5 DBM Flow Diagram 27
4.6 Comparison of Univariate and Multivariate Approaches 30
4.7 Classification Algorithm (Training and Testing Phase) 39
4.8 10-fold Cross-validation 43
6.1 Scheme of Creation of the Datasets 47
7.1 General Classification Scheme 51
7.2 Generation of a Difference Images Matrix 54
8.1 Classification Results for Various Parameter c Settings (GM Densities) . . 59
8.2 Classification Results for Various Parameter c Settings (Volume Changes) 59
8.3 Classification Results for Various Parameter i Settings (A & Q) 61
8.4 Classification Results for Various Parameter s Settings (A & O ) 62
8.5 Classification Results for Various Parameter m Settings (A&Q)) 62
8.6 Classification Results for Various Parameter m Settings (Real Datasets) . 63
8.7 PBM Classification Results for Various Parameter m Settings (Real Datasets) 65
-xi-
List of Tables
8.1 Summary of Parameters to Be Tuned 56
8.2 Classification Results for Various Parameter t Settings (GM Densities) . . 57
8.3 Classification Results for Various Parameter t Settings (Volume Changes) 58
8.4 Classification Results for Various Parameter p Settings (A& Q)) 64
8.5 KSVD-CD Classification Results for Various Parameter m Settings With
Different Classification Rules (Real Datasets) 66
8.6 Final Settings of Parameters 67
8.7 Final Classification Results (GM Densities) 68
8.8 Final Classification Results (Volume Changes) 68
-xii-
List of Abbrevations
A D Alzheimers's Disease
BPRS Brief Psychiatric Rating Scale
C A D Computer-Aided Diagnosis
C D Concatenated Dictionary
CSF Cerebrospinal Fluid
D A L Y Disability-Adjusted Life Year
D B M Deformation-Based Morphometry
D S M Diagnostic nad Statistical Manual
F D R False Discovery Rate
FES First-Episode Schizophrenia patient(s)
fMRI functional Magnetic Resonance Imaging
FRS First Rank Symptoms
F W E R FamilyWise Error Rate
F W H M Full Width at Half Maximum
G A F Global Assessment of Functioning
G M Gray Matter
G M L General Linear Model
H C Healthy Control(s)
ICA Independent Component Analysis
ICD International statistical Classification of Disease and related health problems
IQR InterQuartile Range
isPCA inter-subject Principal Component Analysis
L D A Linear Discriminant Analysis
L O O Leave-One-Out
L P O Leave-Pair-Out
M C I Mild Cognitive Impairment
M D Major Depression
M O D Method of Optimal Directions
M R Magnetic Resonance
M R I Magnetic Resonance Imaging
M W Mann-Whitney (testing)
N N Nearest Neighbours
OA Overall Accuracy
OCR Optical Character Recognition
O M P Orthogonal Matching Pursuit
PANSS Positive And Negative Syndrome Scale
P B M Pattern-Based Morphometry
-xiii-
List of Abbrevations xiv
P C A Principal Component Analysis
PD Parkinson's Disease
ROI Region Of Interest
RP Random Projections
S B M Source-Based Morphometry
SENS Sensitivity
SfBM Surface-Based Morphometry
SPEC Specificity
S P M Statistical Parametric Map
S V D Singular Value Decomposition
S V M Support Vector Machine
T L C scale for the assessment of Thought, Language, and Communication
V B M Voxel-Based Morphometry
W H O World Health Organization
W M White Matter
Chapter 1
Introduction
As reflected at the Preamble to the Constitution of the World Health Organization (WHO),
health is defined as "... a state of complete physical, mental and social well-being and not
merely the absence of disease or infirmity" (WHO, 1948). Likewise, mental health, as
an integral part of health, is not only an absence of a mental disorder but also "... a state
of successful performance of mental function resulting in productive activities, fulfilling
relationships with people and the ability to adapt to change and to cope with adversity "
(U.S. Department of Health and Human Services, 1999).
From the perspective of positive psychology, mental health can be seen as an unstable
continuum and possibly be operationalized "as a syndrome of symptoms of an individual's
subjective well-being" (Keyes, 2002). Such well-being is combined from emotional, psychological
and social well-being with dimensions of self-acceptance, positive relations
with others, purpose in life, etc. and is characterized by individual's perceptions and evaluations
of their lives. The presence of subjective well-being is described as flourishing
whereas living a hollow and empty life is characterized as languishing (Keyes, 2002).
No matter what definition we use, it can be seen that mental health is a ubiquitous part
of our everyday lives along with all its downsides. For example, W H O (2013) states that
in 2004 depression alone accounted for 4.3% of total global burden of disease and all mental
disorders took up to 13%, where almost three quarters of the global burden occurred in
low- and middle- income countries (WHO, 2015a). Regarding D A L Y (Disability-Adjusted
Life Year1
), mental disorders are the leading condition in terms of years lived with disability,
with unipolar depressive disorder, alcohol use disorders and schizophrenia contributing
to the global burden the most (Bloom et al., 2012).
Not only mental health influences lives of individuals, it also has an impact on their
closest relatives and on the whole society and its economy. In that manner it was estimated
that, in 2010, the lost economic output caused by mental disorders amounted to almost
1
Disability-adjusted life year is the number of years lost due to ill-health, disability or early death.
-1-
Chapter 1. Introduction 2
US$ 2.5 trillion when considered worldwide. In future, the costs are projected to grow
(Bloom etal.,2012).
Ordinarily, people with mental health conditions have troubles finding job, attending
school, they are subjected to restrictions of their civil rights or to discrimination (WHO,
2015a). Additionally, it was pointed out by W H O (2013) that people suffering from major
depression and schizophrenia have approximately by half greater chance of dying than
general population since they either neglect their physical health problems to an extent
inconsistent with life or commit suicide. Among young people, suicide takes either the
second or the third place in terms of death causes. Worldwide, in 2014 the suicide rate
was estimated to be 11.4 per 100,000 people (WHO, 2015b).
Speaking of mental disorders, one might ask a question what such a disorder stems
from. During the 20t h
century, two different points of view started the categorization of
mental disorders into organic (pathologically-based) and functional (experience-based)
phenomena. These two approaches continued to diverge and nowadays are known as neurology,
looking for the causes of disorders in the physiology of central and peripheral
nervous system, and psychiatry, seeing disorders as an illness of mind (Martin, 2002).
Indeed, where a neurologist search for pathology in neural connections, a psychiatrist interviews
patient for his family history. Nevertheless, many neurological diseases have
psychiatric manifestations (e.g. as Alzheimer's disease progresses depression, dementia
and cognitive defects often appear) and vice-versa. Therefore the modern scientific community
finds the separation of mental disorders into neurological and psychiatric rather
artificial (Yudofsky and Hales, 2002).
As a consequence of such discrepancies, an interdisciplinary field of biological psychology
has started to gain supporters. Its development is closely connected to the understanding
of neuroanatomy and neurochemistry of the limbic system and closely connected
structures, since it is related to emotional aspects of life and formation of memories (Trimble
and George, 2010). Naturally, other interdisciplinary fields, for instance neuroscience
with neuroimaging, psychopathology or psychopharmacology, accompanied by other well
established fields such as genetics, molecular biology, etc. are inevitably linked to the research
of mental disorders as well.
Be it as it may, mental disorders are of grave importance and open up a challenge not
only for neurologists and psychiatrists but to the whole society altogether. Therefore it is
essential for all the specialists to put their heads together and conjointly try to understand
how a disease of the brain results in an illness of the mind, and to reduce or mitigate
the impact of biological factors such as genes, overwhelming stressful situations or other
perils which surround us in our everyday lives. The fact that such an initiative, trying to
tackle the problem of mental health on a global scale, already exists (Patel and Prince,
2010) only stresses how important an issue mental disorders are and that every piece of
information capable of bringing new light to the field is welcomed.
Chapter 1. Introduction 3
Also this thesis contributes to global health and public well-being by assembling pieces
of information about a recognized psychiatric disorder, schizophrenia, and interconnecting
them with distant fields of statistics and machine learning in order to reduce the load
on physicians and the whole society by facilitating its early detection and hastening its
treatment response.
Chapter 2
Schizophrenia
Schizophrenia is a chronic disabling mental disorder, falling within the scope of psychotic
illnesses, which means that the patients suffering from schizophrenia at some point
develop symptoms that lead them to misinterpret reality (Picchioni and Murray, 2008;
Tandon et al., 2008). Lack of insight, auditory hallucinations (e.g. hearing voices) and
delusions belong to the most common symptoms of schizophrenia (Picchioni and Murray,
2008), but other symptoms (such as suspiciousness, flatness of affect, social withdrawal,
emotional blunting, illogical speech, cognitive difficulties, etc.) can be observed
in schizophrenia patients as well (Picchioni and Murray, 2008; Tandon et al., 2009).
2.1 Occurrence and Causes
In general, men are more prone to develop schizophrenia than women and rates of
schizophrenia are higher in urban regions than in rural catchment areas (McGrath et al.,
2004). Schizophrenia incidence peaks from the age of 10 to 25 for men and between 25
and 35 for women (Rajji et al., 2009). The severity of symptoms differs between individuals
with early-onset schizophrenia and late-onset schizophrenia (Rajji et al., 2009),
where clinical, cognitive, genetic and imaging data suggested larger deficits in cognitive
measures in the former group (Vyas et al., 2011).
Although science has gone a long way in studying schizophrenia over the past century,
its causes and pathogenesis still remain unclear. It is believed that 80% of the liability
for developing schizophrenia is related to a positive family history, i.e. heritability (Riley
and Kendler, 2006). For instance, in Kendler et al. (1993), it is shown that the risk of
schizophrenia is slightly over 6% in the first-degree relatives of the affected patients and
in the case of being a monozygotic twin, according to Cardno et al. (1999), the risk rises
over 40%, whereas in the general population the risk does not exceed 1% (McGrath et al.,
2008). Environmental factors such as cannabis use, prenatal infection, social interaction
-4-
Chapter 2. Schizophrenia 5
or malnutrition contribute to the liability as well (Tandon et al., 2008), but it is the geneenvironment
interactions which compose the overall risk (Riley and Kendler, 2006).
2.2 Diagnostics of Schizophrenia
Two international standardised systems for diagnosing schizophrenia were released,
International Statistical Classification of Disease and Related Health Problems (ICD), created
by W H O , and The Diagnostic and Statistical Manual (DSM), established by American
Psychiatric Association (Picchioni and Murray, 2008; Tandon et al., 2009).
Depending on the balance of the symptoms, several subtypes of schizophrenia are
recognised: simple (loss of personal drive coupled with a decline in social, academic and
employment performance), paranoid (accompanied by fears of persecution), hebephrenic
(lack of goal directed behaviour), catatonic (sustained evidence of abnormal motor behaviour),
etc. (Picchioni and Murray, 2008), although in practice, the borders between
the subtypes are blurred and the classification itself is not unified (compare ICD-10 to
DSM-V).
Nowadays, the diagnostics of schizophrenia is based on a statement by a psychiatrist
or other licensed mental health professional who goes through several sittings with the
patient. The final verdict is partly based on observing patient's actions and noting the
constellation of patient's symptoms, partly on psychiatric rating systems and diagnostic
classification and rating scales which were introduced to psychiatry in order to establish
measurement techniques that are standardised and well-characterised (Lawrie et al., 2011).
For instance, the presence of FRS (First Rank Symptoms, Saddichha et al., 2010),
originally defined by Schneider and now included in both ICD-10 and D S M - V manuals,
is often sufficient for diagnosing a patient with schizophrenia (Lawrie et al., 2011).
Furthermore, one of the scales commonly used in practice for measuring symptom
severity is PANSS (Positive and Negative Syndrome Scale, Kay et al., 1987) which is a
brief, 30-item interview assigning a score ranging from 1 to 7 for each item. The items
are divided into 3 groups, positive scale for positive symptoms, negative scale for negative
symptoms and general psychopathology scale. Unlike positive symptoms, which are not
observed in healthy people and can be loosely defined as losing touch with reality such as
hallucinations and disorganized thinking, negative symptoms are associated with deficits
in normal life behaviour such as passive withdrawal and emotional blunting. After the
interview, the sums of scores within each group are evaluated separately and a diagnosis
can be given.
Undoubtedly, FRS and PANSS are far from being the only leads for schizophrenia
diagnosing. Several other scales, such as BPRS (Brief Psychiatric Rating Scale, Overall
and Gorham, 1962), T L C (Scale for the Assessment of Thought, Language, and Communication,
Andreasen, 1986) or G A F (Global Assessment of Functioning), which is a scale
Chapter 2. Schizophrenia 6
ranging from 1 to 100 representing a very sick patient and a asymptomatic person respectively
(Hall, 1995), could be used for schizophrenia diagnostics as well (e.g. Andreasen
and Grove, 1986).
However, it should be stated that, since psychiatry deals with mental states of patients,
the aforementioned measurement techniques evaluate general symptoms common to variety
of mental disorders rather than the specific ones. Moreover, given the fact that the
diagnostic criteria are not entirely unified and that schizophrenia hand in hand with all the
psychiatry are both very complex, the issue of subjectivity of schizophrenia diagnosing
arises. In other words, most of the diagnoses are dependent not only on the symptoms and
clinical history of the patient, but also on the subjective perspective of a psychiatrist or
other specialist testing the patient (Lawrie et al., 2011). Thus, more sophisticated methods
which would take into account more aspects than a naked eye does are desired.
Regarding the fact that schizophrenia is a mental disorder, one can presume that patients
with schizophrenia would have some schizophrenia related neural manifestations
or vice versa that the abnormalities in neural structure would manifest in cognitive disturbances
in schizophrenia patients. A n important part of methods offering unparalleled
opportunity to explore the neural system of schizophrenia patients and compare their structure
and functionality with the ones of people who are considered as non-suffering from
psychotic illnesses is called neuroimaging.
2.3 Schizophrenia Through Optics of Neuroimaging
Neuroimaging falls within the scope of clinical non-invasive techniques which have
gained popularity among both doctors and patients since they serve as a useful tool for
identification of physiological mechanisms and for disease diagnosing and, potentially,
treatment whilst still being gentle to human body. Even though the advent of neuroimaging
dates back to 19t h
century (Sandrone et al., 2014), most of the modern methods as we know
them today have not been brought to light until the 1970s (Filler, 2009).
In general, the term neuroimaging refers to producing images of the whole nervous
system although customarily it is perceived as a technique to gain images of brain in
particular. The readout of the images can be either structural, i.e. dealing with a brain
structure mainly on a large scale (ranging from gross structure to the scale of molecular
structure), or functional, used to reveal brain functions and metabolic processes occurring
over short timescales (from fractions of second up to minutes). However, the distinction
between the categories might be blurred (Wise, 2013).
Neuroimaging offers a great deal of modalities, which are typically categorized as
structural, concentrating on the morphology of brain, e.g. M R I (Magnetic Resonance
Imaging), DT-MRI (diffusion tensor MRI) and C T (Computerized Tomography), and
functional, revealing brain functions, such as fMRI (functional MRI) and PET (Positron
Chapter 2. Schizophrenia 7
Emission Technology) (Crosson et al., 2010; Wise, 2013) . Yet, none of the techniques
is suited to addressing every question, i.e. each has different time and spatial resolution
characteristics. Thus, jointly, they can offer a complete overview of how a brain works
(Wise, 2013).
From the above-mentioned list, M R I (see 4.1) will be in the main focus of this thesis
since nowadays it is more and more used to explore the structure of brain in order
to understand the neurobiology of brain disorders such as schizophrenia (Shenton et al.,
2010).
Even though in terms of neurology and psychiatry M R I usually serves as a tool of
differential diagnosis (e.g. Yekhlef et al., 2003), its development has gone far enough to
distinguish between some of the disorders exclusively. For instance, M R I can be successfully
used to diagnose mupltiple sclerosis (Polman et al., 2011), progressive supranuclear
palsy (Oba et al., 2005), Alzheimer's or Parkinson's disease (Orru et al., 2012), etc. (for
more on this topic, please refer to 3.1.2) . Nevertheless, the understanding of structural
changes in brains of schizophrenia patients is not yet sufficient to distinguish schizophrenia
from other disorders by means of M R I and therefore, as stated in 2.2, an interview
along with the rating systems and scales stays in the role of its leading diagnosing tool.
As a consequence of development of MRI, a lot of studies have attempted to find
the connection between schizophrenia neuropathology and brain structure. The results
are not completely consistent, which is shown in reviews trying to compare such studies
(Antonova et a l , 2004; Ellison-Wright et al., 2008; Honea et al., 2005; Lawrie and Abukmeil,
1998; Shenton et a l , 2001; Shepherd et al., 2012; Wright et al., 2000). Nevertheless,
even though the general consensus upon what brain structures are affected in schizophrenia
has not been achieved, it has been revealed that structural changes happen in both
gray (Shepherd et al., 2012) and white matter (Antonius et al., 2011) of a schizophrenia
patient's brain and that these changes are very complex and complicated, since they are
not bound to a specific region of the brain but rather they are distributed with an unknown
pattern.
Studies have also shown that the structure of a brain changes with the progression of
the disease and the medication intake. These then happen to be confounding factors when
interpreting brain abnormalities in schizophrenia patients (Haren et al., 2012; Vita et al.,
2012). Therefore, it is reasonable to base the studies upon patients with the first episode of
schizophrenia (i.e. those whose symptoms appeared for the first time and remained longer
than one month) to reduce the confounding factors to a minimum (Schwarz and Kašpárek,
2014).
In conclusion, diagnosis of schizophrenia based on merely relating structural changes
to subtypes of schizophrenia is not possible. Moreover, no exact test for diagnosing
schizophrenia has been introduced yet and therefore the diagnosis is based on doctors'
1
Also, many other modalities such as S P E C T (Single Positron Emission Computed Tomography), M E G
(Magnetoencephalography), NIRS (Near Infrared Spectroscopy), etc. exist.
Chapter 2. Schizophrenia 8
observations of patients' actions, which makes the diagnosis itself rather subjective, not
to mention time-consuming. Thus, methods which would help to make the process of the
diagnosing faster, more precise and more exact to improve the treatment of schizophrenia
are called for.
Luckily, it has been shown in many studies (e.g. in Cocosco et al., 2003; Schwarz
and Kašpárek, 2014; Wilson et al., 2009a; Zacharaki et al., 2009a) that neuroimaging data
can be to a lesser or greater extent successfully used as an input data to machine learning
techniques which attempt to reduce or completely eliminate the need for human intuition
in data analysis. It also allows researchers to analyse the brain images further by using
advanced mathematical and statistical methods making the analysis more objective. In
other words, the combination of neuroimaging and machine learning algorithms could be
possibly exploited to create a fully or at least semi-automated approach (later on in 3.1.2
referred to as computer-aided diagnosis) capable of diagnosing schizophrenia patients.
Moreover, such a diagnosis would be possible at individual level.
Chapter 3
Machine Learning in Neuroscience
Although the term machine learning has been established relatively recently, its ideas
had existed for decades and in some fields for more than a century (Wernick et al., 2010).
Originally, machine learning emerged from computer science as a study of algorithms
capable of learning from data and possibly making data-driven decisions. In engineering,
such a study is called pattern recognition, where a pattern refers to a regularity in data. As
stated in the textbook on pattern recognition and machine learning (Bishop, 2006a), both
of these terms can be seen as two facets of one single field although sometimes pattern
recognition is considered solely as a part of machine learning. Be it as it may, since
there is a controversy in opinions and an obvious overlap between pattern recognition and
machine learning, for the purposes of clarity this thesis will use only the latter term.
Generally, machine learning deals with the development of algorithms searching for
patterns existing in data along with quantifying relationships within them and with making
predictions based on new data according to these patterns (Wernick et al., 2010). Typically,
the predictions are based on finding a new representation of the data, where the predictions
are easier to be made. It is essential that such a representation and the whole algorithm
generalizes well, i.e. given an independent dataset, its performance does not change. This
machine learning pipeline will be further discussed in detail in 4.
3.1 Applications of Machine Learning
Applications of the aforementioned approach already play an important role in everyday
situations such as spam filtering (Guzella and Caminhas, 2009). Other examples
could be found in computer science where O C R (Optical Character Recognition) is used
to convert mechanically created text into its electronic representation (Junker and Hoch,
1998), in banking where machine learning algorithms are used for targeted marketing or
credit scoring (Baesens et al., 2003), in risk management either as fraud detection or even
- 9 -
Chapter 3. Machine Learning in Neuroscience 10
for detection of oil spills in satellite radar images (Kubat et al., 1998) implemented in
Canada.
In the field of medical imaging, two most noteworthy applications of machine learning
are brain mapping and computer-aided diagnosis (CAD) (Wernick et al., 2010).
3.1.1 Brain Mapping
Brain mapping slightly deviates from the general machine learning pipeline (4). Instead
of making predictions, it aims at pattern identification solely. In the case of functional
brain mapping, these patterns are components of the signal bearing information
about brain functions and neural connectivity of the brain. Moreover each pattern correspond
to a specific part of brain. Thus, once the patterns are revealed, they can jointly
create a spatial representation of the brain (functional map, connectome,...) and, provided
suitable data are input, cast light on functions and disease processes in various brain regions
(Wernick et al., 2010). Functional brain mapping has been quite successfully used
with for instance Alzheimer's disease (e.g. Buckner et al., 2009, 2005; Supekar et al.,
2008).
In the case of schizophrenia, as mentioned in 2.3, countless studies have attempted
to discover what parts of brain are schizophrenia-related but the consensus is yet to be
reached. The majority of the brain mapping studies of schizophrenia anatomy reported
enlargement of the lateral ventricles (e.g. Shenton et al., 2001) along with deficits in the
medial temporal lobe, particularly in the gyrus parahippocampalis, amygdala, hippocampus
and thalamus (e.g. Ellison-Wright et al., 2008; Wright et al., 2000), and superior temporal
gyrus (e.g. Honea et al., 2005), and they also suggest deterioration of the prefronal
cortex function (Meyer-Lindenberg, 2010).
A noteworthy piece of information results from Moorhead et al. (2013), where authors
reported that deficits in the medial temporal lobe are also evident in population at
enhanced risk of schizophrenia (e.g. relatives or people with intellectual impairment), i.e.
the changes can be detected even before the onset of the disease, which provide opportunity
for early detection and intervention.
3.1.2 Computer-Aided Diagnosis
On the other hand, C A D precisely follows the work flow of the machine learning
pipeline (4). It aims at making predictions pertaining to patients in order to, as its name
suggests, help a physician to give a proper diagnosis of the patient. The rationale stems
from a simple assumption that two heads know more than one. Therefore, the computer
output is often characterized as a "second opinion" for the physician (Shiraishi et al.,
2011).
Chapter 3. Machine Learning in Neuroscience 11
Since the idea of computers being involved in diagnosing was introduced for the first
time, scientists have been playing with the idea to entirely replace physicians in detecting
abnormalities with computers. Such an approach is called automated computer diagnosis
or shortly computer diagnosis. The substantial difference between automated computer
diagnosis and C A D is that when a computer alone is supposed to give a diagnose, its
performance needs to be better than that by a physician, whereas in the case of C A D this
superiority is unnecessary (Doi, 2007).
In regard to C A D , it has already been introduced into the routine clinical work. Breast
cancer detection and diagnosis with mammography serves as its typical example (Tang
et al., 2009). Other applications of computers involved in tumour detection could be
found in screening examinations, e.g. for detecting polyps in colon with C T colonography
(Yoshida et al., 2002) or lung nodule detection (Armato et al., 2002). MRI-based C A D has
also been tested for brain tumour type classification where the algorithm was capable of
distinguishing metastases from gliomas and high-grade from low-grade neoplasms at the
same time (Zacharaki et al., 2009b).
Up to now, the discussed C A D applications were from the field of oncology. Nevertheless,
attempts to use machine learning for the purposes of C A D can also been found
in neurology and psychiatry in spite of the fact that detecting mental disorders might be
harder than detection of cancerous symptoms.
Relative to neurological disorders, the most of the conducted studies concentrated on
mild cognitive impairment (MCI) and probable dementia of Alzheimer type also known
as Alzheimer's disease (AD). Among others, one can find disorders such as idiopathic
Parkinson's disease (PD), major depression (MD), progressive supranuclear palsy, autistic
disorders and many others (Orru et al., 2012). More specifically, the studies attempt to
create a machine learning approach exploiting neuroimaging data such as MRI, fMRI,
SPECT, etc. images to classify between two groups (typically between probable patients
and healthy controls). The assessment of the algorithm is a percentage of individuals
classified correctly to the group they belong to, later in 4.5 defined as accuracy1
.
In general, the studies of M C I and A D yield the best results where both Fan et al.
(2008) and Grana et al. (2011) reached 100% accuracy and other many studies exceeded
90%, e.g. Gerardin et al. (2009) (94%), Magnin et al. (2008) (94%), Lopez et al. (2009)
(92%), Padilla et al. (2012) (91%). Very high accuracies were also achieved with primary
progressive aphasia (Wilson et al., 2009b) (92%) and P D (Feis et al., 2015) (96%),
although usually the classification performance of P D is inferior (e.g. Bharti Rana, 2015
(87%)). Some other examples of C A D in neuropsychiatry are Fu et al. (2008) (depression,
86%) Hahn T et al. (2011) (MD, 83%), Ingalhalikar et al. (2010) (autistic disorder, 90%)
and Sun et al. (2009) (psychosis, 86%). Interestingly, machine learning has also been used
for lie detection (Davatzikos et al., 2005) with 88% accuracy achieved. The accuracies
mentioned above were achieved in machine learning studies with the use of data from
various neuroimaging modalities.
'More precisely as overall accuracy, sometimes referred to as cross-validation accuracy
Chapter 3. Machine Learning in Neuroscience 12
Ordinarily, studies of schizophrenia do not achieve as good results as those of A D . The
explanation can be that the changes in schizophrenic brains are more complex than those
of A D since the degenerative processes in the former are more distributed over a patient's
brain. Nevertheless, some of the studies pertaining to schizophrenia have reached classification
performance above 90%, e.g. Ardekani et al. (2011) (96%), Costafreda et al. (2011)
(92%), Karageorgiou et al. (2011) (92%) and Ingalhalikar et al. (2010) (91%). Notwithstanding
these high accuracies, normally the percentage of correctly classified individuals
is lower, e.g. Schwarz and Kašpárek (2014) (88%), Shen et al. (2010) (84%), Janoušová
et al. (2015) (82%), Ota et al. (2012) (74%), Zanetti et al. (2013) (73%), Kašpárek et al.
(2011) (72%), Arribas et al. (2010) (70%) and Nieuwenhuis et al. (2012) (70%).
Chapter 4
Classification Pipeline
From the above-mentioned examples of C A D applications (see 3.1.2), one can derive a
question of what do all of them have in common. The answer is simply that all of them
lead to the so-called classification problem, which has already been touched in 3, where a
classifier partitions a set of observations into distinctive subsets1
. The goal is to, as accurately
as possible, differentiate between the subsets in, depending on the aim of analysis,
the spatial or temporal or spectral domain in order to contrast the different brain states
(Lemm et al., 2011). In the beginning, this chapter provides a general overview of this
classification pipeline. The following sections then elaborate on each of the main steps
of the pipeline.
The number of the steps, into which the pipeline can be divided, depends on how
coarsely or finely one wants to look under the hood of the pipeline. The following pipeline
overview, along with its level of detail, was created especially for the purposes of this
thesis and, if not explicitly states otherwise, it was compiled from Bishop (2006b), Holcfk
(2012), Lemm et al. (2011), Pereira et al. (2009) and (Zarogianni et al., 2013). A schematic,
coarse level depiction of the pipeline is in Figure 4.1.
Before we start processing data through the pipeline, we need to acquire them according
to a chosen paradigm that suits our purposes the best. Once the data are acquired, they
are usually full of unsolicited noise which spoils the signal containing the information
we are in search for. That is where data preprocessing takes place. The purpose of this
step is to clean the data of noise and other artefacts whilst retaining as much of the useful
information as possible. Both of the steps are examined in 4.1.
Hereafter, another problem is encountered. The data usually exhibit an adverse ratio
between its dimensionality and the number of acquired samples which oftentimes poses a
threat to the ensuing classifier, typically because it does not scale well to high-dimensional
data and therefore it has difficulties classifying based on samples so sparsely distributed
1
Also referred to and in 4.3 defined as classes.
-13-
Chapter 4. Classification Pipeline 14
Data Acquisition
&
Preprocessing
Phase A
Feature Extraction
&
Feature Selection
T
Classification Rule
fsef up)
Classification Rule
(use)
Validation
Phase B
Performance
Evaluation
Phase C
Figure 4.1: Classification Pipeline. The pipeline is divided into three phases: A) real
world attributes are measured and transformed for specific purposes of classification; B)
classification algorithm is schemed; C) the usefulness of the algorithm is evaluated in
terms of generalization and performance and the output is further analysed. The main steps
of the whole pipeline are: data acquisition, preprocessing, feature extraction and selection,
setting up a classification rule, classifier validation and evaluation of its performance.
over a given space. The problem stems from the fact that adding extra dimensions to
the space is associated with exponential increase in its volume. This is referred to as the
curse of dimensionality (Bellman, 1961). Also, in terms of computation, the higher the
dimension of the data, the more time- and memory-demanding the algorithm is.
Despite these inconveniences, the curse of dimensionality can be dealt with. The effective
dimensionality of real data is typically lower than its originally acquired dimension.
Therefore, we should be able to reduce the dimension and retain almost all the underlying
information at the same time. The choice of the new dimensions should be made in a
way that the new representation bears the most discriminative features. By afeature2
, we
understand an observed measurable property of the data. Thus, the reduced (or in another
way transformed) space is called a feature space. Every object given as a point in the Ddimensional
feature space is represented by a D-dimensional vector consisting of features,
i.e. by afeature vector, also referred to as a pattern.
2
A l s o called a descriptor or, in statistics, a variable.
Chapter 4. Classification Pipeline 15
Some of the techniques which could be used to reduce the dimension of the data are,
for example, cluster analysis, regularization (Rasmussen and Williams, 2005), converting
data to a sparse representation, variety of orthogonal transformations and many others.
Since by the reduction we determine what feature space would a classifier be based upon,
we call this step feature extraction or feature selection. More elaborated discussion on
this topic will be mentioned further in 4.2.
After proper features have been opted out, we face another quandary. A classification
rule must be set up. Again, a great number of possible candidates for a classification
algorithm exist. Since this thesis is not capable of providing their exhaustive listing, only
a general overview of classification algorithms, along with a more detailed description
of one representative classifier3
, will be discussed further in 4.3. Providing we know
what classifier suits our purposes the best, feature vectors, representing the objects in our
dataset, could be used to train the classifier. The term training is characterized as setting
algorithm parameters to the values with which the classifier discriminates between the
classes the best.
Subsequently, after the choice of the algorithm has been made and the parameters have
been estimated, i.e. the classification rule has been set, we can proceed to start applying
the classification rule for new subjects, which is commonly know as a testing phase. This
can be either viewed as an individual-level prediction of a new object's affiliation to the
groups or as an evaluation of the classifier performance, providing we know to what
subset the new object should belong to. Techniques to do the latter will be reviewed in
4.5.
One of the pitfalls of the classification problem is known as overfitting. It refers to
a state when a classifier is overadjusted to the data it has been trained on and therefore
it describes more noise than the underpinning relationship. Should the classifier be used
for different data, it would fail to determine its affiliation to subsets properly. Hence, the
generalization property of the classifier would deteriorate. For such reasons, a whole set
of validation techniques have been developed. Some of them will be described later in
4.4.
Up till now the classification pipeline was described from a coarse point of view. Here,
it should be noted that the implementation of the pipeline varies with specific applications,
although the general work flow remains untouched. In the rest of the chapter, a more
detailed description of each step will be discussed with the special focus on brain M R I
data and schizophrenia classification.
4.1 MRI Data Acquisition and Preprocessing
In regard to schizophrenia, several conceptually distinctive theories endeavour to explain
causation of developing schizophrenia on different levels of abstraction. Two of
3
The term classifier refers to an algorithm that implements classification.
Chapter 4. Classification Pipeline 16
them were already mentioned in 1, neurology as a study of neurobiological aspects (brain)
and psychiatry as a study of phenomenological experience (mind). Another point of view
is taken by pharmacology, attempting to assign the causes to chemical substances (such
as antipsychotic medication) and organic compounds (such as dopamine level) (Kapur,
2003). Since heritability plays its role in schizophrenia development, genetics (e.g. Riley
and Kendler, 2006) can not be left out from the enumeration as well.
Accepting the presumption that schizophrenia can be revealed by structural changes in
brain morphology, we henceforward turn our focus to neuroimaging, particularly to structural
MRI. A brief introduction to the field of neuroimaging was given in 2.3. Further text
of this section will be dedicated to how magnetic resonance works and how the images are
acquired together with misrepresentation errors of tissue structures, in signal processing
referred to as artefacts, and how to subsequently deal with such artefacts in order to get
rid of the undesirable noise.
4.1.1 Magnetic Resonance Imaging
In this thesis, only the basic principles of M R I will be discussed. For further reading
on this topic please refer to Reimer et al. (2010).
Since a human body mostly consists of water and fat, it is full of hydrogen atoms. Each
of these atoms have a single proton. This facilitates the observation of its spin, which is
associated with a magnetic moment of the proton. Therefore, when tissue is exposed to
an external static magnetic field, all its hydrogen protons align either parallelly or antiparallelly
with the direction of the field (usually referred to as z-axis) and thus, when we
average the spins, tissue magnetization on a macroscopic level can be measured.
An application of another magnetic field, perpendicular to the original one (i.e. in
the so-called x-y plane), with various amplitudes and durations can turn the macroscopic
magnetization around any angle. This can be achieved providing the frequency of the
signal inducing the extra field, referred to as a radiofrequency (RF) pulse, matches the
frequency4
of the spins in the main field. In fact, when such a process occurs, the R F
pulse causes some of the atoms to resonate. This phenomenon gave magnetic resonance
its name.
These physical principles (and many more) are nowadays incorporated in MRI. The
readout of M R I is a three-dimensional (3-D) image of a selected region. The smallest
spatial dimension of the image is called a voxel. Its size is dependent on the settings of
magnetic resonance (MR) scanner. A l l voxels with the same value on an axis compound
a slice. When the axis is the z-axis, we call the image axial. If we slice in the direction of
the jc-axis, the resulting image is sagittal, and in the case the y-axis, we come out with a
coronal image.
4
This frequency is called Larmor frequency and for hydrogen nuclei it is 42.58 M H z / T .
Chapter 4. Classification Pipeline 17
In terms of M R scanner implementation and usage, MR scanner is composed of a main
magnet, which induces static magnetic field (along the z-axis), gradient coils, strengthening
the main field and serving as signal receivers, and a R F transmitter.
Once a patient is placed inside the main magnet, R F transmitter pulses a current
through the patient causing the protons in the tissues to flip. When this happens for the
first time, the amplitude of the signal emitted from each voxel is proportional to the number
of protons (PD5
) involved in the excitation process. Images resulting from this step
are called PD-weighted scans. Once the R F current has ceased, the protons start to realign
back to their equilibrium state. This process is characterized as relaxation. Subsequently,
relaxation times can be measured. Firstly, T l (the so-called spin-lattice) relaxation time
associated with the spins getting back to their original positions and, secondly, T2 (the
so-called spin-spin) relaxation time pertaining to dephasing of the macroscopic magnetization.
The resulting images are then called, correspondingly, Tl- and T2-weighted scans.
In practice, to be able to measure the T l and T2 relaxation times properly, a sequence of
R F pulses rather than a single pulse must be emitted.
The reason why such a complicated process has been implemented is that the physical
and chemical characteristics of each tissue in a human body is different. Therefore, also
a ratio of protons and their relaxation times differ throughout the tissues, which enables a
tissue specific visualization. Particularly, M R scanners are suited for soft tissues and nonbony
parts. Also, the contrast of the tissues varies with the weighting model. For instance,
on Tl-weighted scans, cerebrospinal fluid (CSF) appears to be dark whereas short T l
relaxation time of fat causes its bright appearance in the images. On T2-weighted scans,
white matter shows up as hypointense whereas CSF results in voxels with high intensities.
A n example of the different contrasting is depicted in Figure 4.2.
Figure 4.2: Difference Between Tl - and T2-weighted Image. A n example of an axial slice
of brain on Tl-weighted (left) and T2-weighted (right) M R I scan. Whereas fat (e.g. in
white matter) appears bright and water dark on the former, on the latter it is the other way
around. Adapted from (UC San Diego Center for Functional MRI, 2015).
5
Proton density.
Chapter 4. Classification Pipeline 18
Artefacts in MRI Images
In order to obtain an M R I image, a patient must remain very still during the acquisition
of the image. Otherwise the resulting image would be blurred. This is only one of many
difficulties M R I faces. Not only the interaction of a patient with an M R scanner cause
troubles. The imaging process itself produces artefacts in the images which have to be
dealt with. Since the problem domain is broad, only a general overview of the artefacts
will be mentioned (Krupa and Bekiesihska-Figatowska, 2015; van der Graaf et al., 2014;
Vargas et a l , 2009):
• Pulsation artefacts - are related to pulsation of CSF or blood flow. Protons flowing
at a high velocity can disturb the homogeneity of the magnetic field.
• Magnetic susceptibility artefacts - pertain to a distortion in the image. Foreign
bodies (typically metallic objects) cause local magnetic field inhomogeneities. Some
of the most easy-to-eliminate objects are, e.g. make-up and hair ties. Other foreign
bodies could be found in the field of medicine, ranging from dental implants and
orthodontic braces all the way to surgical clips and left-behind screws. Magnetic
susceptibility artefacts usually worsen with stronger magnetic fields6
.
• Motion artefact (Ghosting) - an artefact induced by patient's movement. In the
image it appears as double contours. Ghosting can be reduced by patient immobi-
lization.
• RF-noise artefact - is provoked by an interference of the M R scanner with other
device emitting electromagnetic field, such as another medical device or television.
Once encountered, it usually appears in all images as a regular striped pattern. This
artefact can be eliminated by isolating the M R scanner from the external signal.
• Chemical shift artefacts - typically occur at interfaces between fat and fluid-filled
structures (e.g. eye balls in the orbits). They cause either signal cancellation or shift
some of the voxels in the reconstructed image.
• Truncation (Dark rim) artefact - also called Gibbs ringing artefact appears as
several alternating bright and dark lines and occurs at the intersection of sharp highcontrast
boundaries. This artefact can be reduced by increasing spatial resolution.
• Aliasing - occurs when some of the parts being mapped are outside of the area
covered by the homogeneous magnetic field. These parts are then projected onto
another side of the image.
• Spike noise artefacts - are caused by static electricity from the clothes and in the
image they appear as a checkered pattern.
6
Nowadays, in clinical practice, 1.5 or 3 T M R I scanners are used, whereas in research the scanners
reach the strength of the magnetic field up to 7 or 9 T.
Chapter 4. Classification Pipeline 19
4.1.2 MRI Preprocessing
As mentioned above, before we proceed further to extracting and selecting the features
for the subsequent classification, we need to eliminate noise and the effect of artefacts on
the data since they can seriously degrade the diagnostic quality of the acquired images.
Most of the time, it is in hands of a radiologist operating the M R scanner, since basically
all the aforementioned artefacts (4.1.1) can be mitigated by appropriate scanner settings
(van der Graaf et a l , 2014).
Unfortunately, for classification, one more issue arises since a classifier does not evaluate
only one image, but rather works with groups. Let us imagine a situation, when a
family comes to have M R I scans of their brains acquired. Each family member is perfectly
immobilized, the scanner parameters are set correctly and all the images come out
clear. But when they subsequently want to compare their results, they find out that father's
image is much bigger than the one of his daughter and that mother's is strangely askew
when compared to the others. In other words, although all the artefacts on an individual
level have been taken care of, the straightforward comparison of the images is not possible
because no matter what, since every person has a different size and shape of head and a
positioning inside of the scanner. This is obviously something that can not be accounted
for by a radiologist no matter how hard he tries.
Thus, to ensure the comparability of the images, a preprocessing step of spatial normalization
must be carried out before proceeding further on. Spatial normalization is
performed with the use of image registration and resampling to iso voxel7
. What usually
happens is that the images are registered in accordance with an appropriate brain template
making them end up in the so-called stereotactic space. Once the images are transformed
in such a way, the differences in positioning and brain sizes should disappear (see Figure
4.3).
Figure 4.3: Spatial Normalization. Acquired M R I images are co-registered to a chosen
template using one of the many registration techniques and resampled in order to account
for different brain shapes and sizes and positioning inside a scanner thus facilitating a
following analysis. (The I C B M 152 template image was adapted after Fonov et al., 2009.)
sef of raw
T1-weighted images
ICBM152 T1 template
spatially normalized
images
7
A n iso voxel is a voxel of the same size, typically 1 mm, in all three directions.
Chapter 4. Classification Pipeline 20
Hitherto, several brain templates and brain atlases, which can be used as a stereotactic
space, have been introduced. For instance, Gholipour et al. (2007) presented MNI305
(Evans et al., 1993), ICBM152 (Mazziotta et al., 2001) and Atlas of Talairach and Tournoux
(Talairach, 1988) as the most common ones, but other variants are used as well.
Once a template for registration is chosen, the search for a proper transformation starts.
Typically, the optimal transformation is found by maximizing the mutual information of
the image and the template (e.g. Maes et al. 2003). Also, minimising the residual squared
difference between the image and the template can be employed (Mechelli et al., 2005).
Other cost functions can be based on normalized cross-correlation or information theory
divergence measures (Lepore et al., 2008). Moreover, a Bayesian scheme prior knowledge
can be incorporated into the registration to obtain more robust transformations (Ashburner
et a l , 1997).
Registration can also be assessed from a different point of view. Features, such as
points, curves and surface landmarks, from both the template and the image, can be compared
in a new intermediate parameter space in which the actual estimation of anatomical
correspondences based on these features takes place (Joshi et al., 2009).
However, it should be noted that the registration is not meant to match all the images
perfectly, but only to correct the global differences in them. Doing so would make all
the subtle differences insignificant and we would lose the essential information, thus preventing
the classifier from distinguishing between the groups properly (Mechelli et al.,
2005).
4.2 Feature Extraction and Selection
The reasoning of why feature extraction and feature selection are important has
already been introduced in the general overview of the classification pipeline (4). Now,
knowing that our data are in the form of 3-D brain M R I images, we are able to elaborate
more on the topic.
Let us pretend we have an M R I image with a spatial resolution o f l 6 0 x 5 1 2 x 5 1 2
voxels. Considering we reorganize the voxels into one big vector, we end up with a feature
vector with almost one million features8
per image in total. Handling such large data
is computationally expensive. However, if one aims to create a C A D technique, the computation
should be efficient in order not to delay common patient flow in clinics. Thus,
it is advisable to reduce the size of the data to a minimum without losing the information
embedded in the data, making the computation faster yet precise.
Additionally, since brain M R I is quite a costly procedure (e.g. Pasalic et al., 2015), the
studies are usually not able to cover more than tens or, in some cases (e.g. Nieuwenhuis
et al., 2012), hundreds of individual scans. Without dimensionality reduction, this would
8
Every voxel of the image can be looked at as a possible feature for the ensuing classification.
Chapter 4. Classification Pipeline 21
lead to at maximum hundreds of data samples in a million-dimensional feature space. Not
only would the computation take long, also the classification algorithms would not be able
to set up the classification rule good enough to distinguish new scans as correctly as it is
supposed to be in a C A D . In fact, the search for the appropriate setting can be compared to
looking for a needle in a haystack speaking both figuratively and literally. This assertion
can be demonstrated on the following toy example.
Imagine a situation with two haystacks in the field, both next to each other looking
alike. The only difference is that one of them has a needle hidden inside. In the morning
a farmer comes to look for the needle. When his stare rests upon the haystacks, he sees no
difference. The only way for him to find the needle is to choose one haystack and go piece
by piece through all of it. If the needle is not found, he must proceed to the other haystack
and repeat the whole process. Depending on the sizes of the haystacks, he can spend the
whole day searching. The very next day the situation repeats. But today he knows his wife
had hidden the needle to a part of a haystack facing the south at the very bottom of it. He
comes to the first haystack, putting his hand inside the indicated region. When he does not
feel the needle, he instantly moves to the other haystack and he finds the needle in a blink
of an eye.
This situation can be compared to a classifier looking at two brains. On a macroscopic
level they both look similar although somewhere inside there is a part indicating that one
of the brains belongs to a schizophrenia patient. If the classifier has no beforehand knowledge
about the schizophrenia-specific region, the chances of succeeding in labelling the
schizophrenic brain are slim. Therefore, on one hand, having the region predefined would
increase the chances several-fold since the classifier could disregard the macroscopic differences
and rather concentrate on the subtle ones with higher discriminative power. On
the other hand, if another brain had to be assessed by a classifier concentrating only on
the specified region, the schizophrenia-specific part of the new brain would need to be
enclosed in that region otherwise the classifier would not find it.
A few lessons, summarizing the merits and pitfalls of an appropriate choice of classification
features, can be drawn from the examples. Namely:
• When the dimensionality of a feature space is reduced, the computation is less time
and memory consuming.
• Dimensionality reduction ought to retain as much of the underlying information as
possible, loosely speaking, it should not erase what we are looking for.
• Reduction closely corresponds to feature selection.
• Some of the features can be disregarded while positively affecting the results of the
classification (e.g. noise filtering).
• Some of the features can be disregarded without negatively affecting the results of
the classification.
Chapter 4. Classification Pipeline 22
• Disregarding some of the features could negatively affect the results of the classification
(e.g. information loss or overfitting).
• Only the features with the best discriminative power should be chosen for the clas-
sification.
• The optimal number of the chosen features is not generally known and must be
estimated for each classification problem separately.
• There exists no classifier, and no algorithm in general, that is optimal for all classes
of problems (Wolpert and Macready, 1997).
In regards to M R I and schizophrenia, our ambition is to identify what brain parts are
prone to the schizophrenia-related structural changes, more precisely upon which voxels
should the classification rule be based. Unfortunately, classification rules by themselves
are not constructed with the aim of finding the appropriate features and therefore it is
needed to use other techniques created specially for this problem.
The advancement of M R I facilitated the development of brain volumetry which is a
technique used to assess brain volume changes in order to support disease diagnostics,
understand mechanisms of the disease and monitor clinical progression of the disease and
a treatment effect (Giorgio and De Stefano, 2013). Especially in the past, the gold standard
for finding the features was based on manual tracking of the so-called region of interest
(ROI). Nevertheless, at the dawn of the 20t h
century, McCarley et al. (1999) claimed that
the ROI identification was a labour-intensive and subjective task compared it to "monks
hand-copying manuscripts before the invention of the printing press". Simultaneously,
the authors foresaw that automated methods for brain division and following statistical
analysis needed to be developed. Nowadays, manual volumetric methods are still in use,
but beside them a variety of semi and completely automated methods exist as well (Giorgio
and De Stefano, 2013). One shared aspect of these techniques is that examination of
a brain can be assessed in an unbiased and evenhanded manner (Mechelli et al., 2005),
which is a demanded characteristics for schizophrenia diagnosis (as mentioned in 2.2).
For the sake of completeness, the terms feature extraction and feature selection should
be introduced. Whereas feature extraction involves transforming the feature space to a new
one while retaining all the features, feature selection selects a subset of already existing
features and thus it only reduces the feature space dimensionality (König, 2000).
For instance, by acquiring an M R I image we discretize a real brain structure into a
3-D matrix of voxels where each voxel represents a specific region of the brain. Also, the
value of each voxel contains only a small fraction (such as PD, T l relaxation times, etc.)
of the original information. As we will see later on in the section, other mathematical
transformations can follow. Applying the terminology from the text above, such processes
are called feature extraction.
Chapter 4. Classification Pipeline 23
Once the transformations of a feature space are finished, we end up with a set of
features, i.e. a pattern. Now, by solely selecting (either randomly or based on mathematical
criteria) some of the features, we only reduce the size of the pattern without changing the
meaning each of the features carries.
Nevertheless, the techniques mentioned further on often cover both of these steps and
therefore the line between when we complete the extraction part and start with the selection
is often fuzzy, which is the reason why in this thesis they are together in one chapter. The
rest of the chapter will be organized as follows. Firstly, brain morphometry techniques,
repeatedly found across studies related to schizophrenia classification, will be discussed
as a concept of assessing brains in an objective manner. Furthermore, the limitations of
their approach will be mentioned. Finally, machine learning algorithms will be presented
as an instrument to overcome some of the limitations.
4.2.1 Univariate Approaches of Brain Morphometry
Generally speaking, morphometry is a concept of analysing sizes and shapes. Consequently,
brain morphometry is concerned with the structure of a brain and its changes
over time (evolution, ageing, disease, etc.) (Kašpárek and Schwarz, 2011). Two main concepts
used for assessing M R I brain scans in schizophrenia research are voxel-based morphometry
(VBM) and deformation-based morphometry (DBM) (e.g. Gaser et al., 2001).
Both of them proceed from the preprocessing part of registration (4.1.2) with the aim to
find the most discriminative features between the groups of schizophrenia patients and
healthy controls and, as a by-product, to create a statistical parametric map (SPM) of
schizophrenia-related features, i.e. to identify which voxels, in terms of statistics, significantly
distinguish between the groups (Flandin and Friston, 2008).
SPMs resulting from V B M or D B M show structural changes such as what part of the
brain is larger in a schizophrenic brain than in a normal one. When fMRI images are
used instead of M R I scans, the SPMs are closely related to brain functions or reactions to
external impulses (please refer to brain mapping mentioned in 3.1.1).
Creating a S P M can be viewed as the objective of the whole analysis and thus it could
be its final step. Nevertheless, the features derived from V B M and D B M often serve for a
subsequent classification.
Voxel-Based Morphometry
The most important role in the discovery of statistically significant voxels in V B M is
played by the general linear model (GLM) (Mechelli et al., 2005). The G L M (Friston
et al., 1994) models a relationship between a response variable y y and a linear combination
of covariates xij as:
ytj = XÍ, i ßi j +x
i,ißi,j + •••+x
i,Kßxj + ßij i e {1,...,/}, 7 e {1,...,/};
Chapter 4. Classification Pipeline 24
where / is the number of M R I scans and 7 is the number of voxels in a scan. The j are K
unknown parameters and j are independent and identically normally distributed errors
with the zero mean and a constant variance. In matrix form, G L M can be defined as:
Y=Xp+£,
where Y is a data matrix with each 3-D scan reshaped into one row composed of all its
voxels. The matrix X is referred to as a design matrix.
In general, the G L M is a flexible framework for various types of statistical testing. For
instance, it can take into account interactions among variables and adjust for the confounding
effects of no interest such as age or sex of a patient (Friston et al., 1994). Nevertheless,
since it presumes variables to be normally distributed, the registered images must undergo
another preprocessing.
After registration 4.1.2 is finished, the normalised images are segmented into several
homogeneous, nonoverlapping regions which are meaningful for a subsequent analysis.
Typically, M R I images are segmented according to specific tissue types into gray matter
(GM), white matter (WM) and CSF using one of the many image segmentation techniques
(Despotovic et al., 2015). Since different tissues typically exhibit different values of M R I
attributes such as intensity a simple thresholding can be employed to differentiate between
the tissues. However, usually more sophisticated methods are used. Their overview can be
found for example in Balafar et al. (2010). Recently, the unified segmentation framework
(Ashburner and Friston, 2005), which combines both image registration and segmentation
into a single generative model, has gained popularity and has been implemented into
several neuroimaging toolboxes.
Thereafter, smoothing of gray and white matter images by convolving them with a
Gaussian kernel is employed. The rationale behind the smoothing is to average the amount
of G M or W M from around the voxel, thus reducing the impact of the spatial normalization
imperfections. On the other hand, it removes the finescale structure of the images
(Penny et al., 2011). Therefore, it is essential to use a proper width of the kernel, since
too wide kernels would reduce the localization accuracy (Mechelli et al., 2005). Particularly,
the kernel width should be in correspondence with the expected spatial differences
between the studied groups (Jones et al., 2005). By the central limit theorem, smoothing
conveniently renders data more towards the normal distribution and thus it increases the
credibility of the G L M testing (Mechelli et al., 2005).
It should be noted that registration may lead to unbalanced changes in volumes of
different parts of a brain. Depending on the final interpretation of results, the images
can be modulated in order to correct for the unbalanced changes, or left non-modulated.
Another way to minimize the error caused by registration is to perform it on G M and W M
volumes separately. Such an approach is referred to as optimised V B M . The comparison
between standard and optimised V B M is depicted in Figure 4.4 (Mechelli et al., 2005).
After all the preprocessing has been carried out, statistical analysis based on the G L M
is employed. The outcome of the statistical testing is an S P M (Penny et al., 2011). To gain
Chapter 4. Classification Pipeline 25
Standard VBM Optimized VBM
Modulation
(optional)
T
Smoothing
t
Statistical Testing
Figure 4.4: VBM Flow Diagram. Comparison of standard and optimized V B M flow diagrams
(Mechelli et al., 2005). In the case of standard V B M , the images are first spatially
normalized, then segmented and further processed. In the case of optimized V B M , they
are first segmented and, before the images are further processed, the spatial normalization
is employed on gray matter and white matter images separately. In other words, whereas
with standard V B M spatial normalization precedes segmentation, in the case of optimized
V B M it is the other way around. (Template, G M and W M images adapted after Fonov
et a l , 2009.)
such a map, a great amount of statistical hypotheses testing has to be done, providing the
multiple comparison problem is being encountered.
Since the voxels are highly correlated, a simple familywise error rate (FWER) procedure
that aims at controlling the probability of making one or more false discoveries,
the Bonferroni correction (Dunn, 1959) is not applicable. Moreover, F W E R corrections
in general are often too conservative and therefore some other techniques must be used
(Mechelli et al., 2005). For instance, false discovery rate (FDR, Benjamini and Hochberg,
1995) correction which, instead of restraining the probability of false discovery among
all the tested voxels, controls for the proportion of false discoveries, provides a less stringent
alternative to type I errors controlling. Yet, sometimes F D R can be too strict as well.
In such cases, the p-value threshold is set manually, most often as 0.001 (Kašpárek and
Schwarz, 2011).
Chapter 4. Classification Pipeline 26
Deformation-Based Morphometry
Unlike in the case of V B M , which consists of several steps, deformation-based morphometry
(DBM) analyses are based on values resulting straight from registration. In
other words, transformations used for warping the images to match the template are directly
used for a subsequent analysis. Therefore, it is essential that the registration is very
precise (Scanlon et al., 2011). Whereas registration as we defined above (4.1.2) consists
of low-dimensional transformations (e.g. affine or so-called rigid transformations) in order
to account for a general shift, orientation and overall size adjustments, rather than being
related to actual differences between the structures of brains (Mechelli et al., 2005), D B M
registration is highly non-linear and high-dimensional in order to capture not only macroscopic
differences but mesoscopic and very subtle ones as well (Scanlon et al., 2011).
Also, since D B M omits segmentation, the analysis takes place on whole brains.
The idea behind the D B M approach is that since D B M registration aims to precisely
superimpose corresponding parts of brain on top of each other, voxels in the images must
be deformed according to a non-linear function. Applying such non-linear deformations
results in the so-called displacement fields, which contain the information about the displacement
needed to warp the brain to match its template (Gaser et al., 2001).
Since we deal with 3-D images, displacement fields have three displacement components
u = (ux,uy,uz) along each axis of the image. Then a deformation function at a
position p = (x,y,z) can be defined as (Chung et al., 2001)
5(p) =p + u(p).
From the deformation 8 a Jacobian matrix 7 can be derived at each voxel. Since each voxel
can be considered a unit-cube, the displacement fields can be transformed into scalar fields
by computing the local Jacobian determinants det(7).
The determinant measures the change in the volume after registration at each voxel
and thus it restricts the information to local volume changes of the brain structure only,
i.e. the shrinkage or enlargement caused by warping. The Jacobian determinant is given
by the formula (Chung et al., 2001)
det(7) =
The usage of the Jacobian determinant follows a simple reasoning. If all the displacement
components point outward from a voxel, the voxel needs to be displaced outward to
match the template. From that, one can infer that the current region surrounding the voxel
is smaller than the one of the corresponding template (Gaser et al., 2001). For this situation,
the determinant generates a value larger than 1. And otherwise, having the value of
the determinant smaller than 1 implies that the region of the voxel needs local shrinkage
Chapter 4. Classification Pipeline 27
to match the template (Lepore et al., 2008). Thus, the relative volume of a brain region is
directly encoded in one value per voxel.
The statistical analysis of the resulting feature vectors is very straightforward. Univariate
statistic methods such as Mest can be used to find the voxels with significantly
different local volume changes between the groups (Gaser et al., 2001). Again, the techniques
to deal with the multiple comparison problem must be used. For better illustration,
the whole D B M is delineated in Figure 4.5.
Spatial Normalization for D B M
template
Spatial Normalization
m
raw images
displacement fields
u(x,y,z)
_ L
Conversion to
Scalar Values
det(J)
Statistical
Analysis
Figure 4.5: DBM Flow Diagram. Raw M R I images are first registered with the use
of affine transformation to control for macroscopic differences. Subsequently a highly
non-linear registration with a deformation function starts, resulting in displacement field.
Those are later transformed to scalar values using the Jacobian determinant and further
analysed. Adapted from (Schwarz and Kašpárek, 2014).
Interestingly, another method emerged as a derivative of D B M . Its difference from the
previous deformation-based approach is that the morphometric measures are derived from
geometric models of the cortical surface, i.e. from an inherent 2-D structure instead of a
3-D field. The rationale is as follows. As a brain changes over time (or over schizophrenia
progression), so does its cortical local surface area, its thickness and curvature. Thus, this
method was given a name surface-based morphometry (SfBM) (Chung et al., 2003):
• The first step of SfBM is extraction of the cortical surface from M R I images. Cortical
surface is derived as a part of a brain between pial surface9
and the boundary
between G M and W M . Additionally, it is modelled as a smooth triangular mesh at
each vertex1 0
(Greve, 2012).
9
The boundary between G M and pia mater and/or CSF.
°The place where the corners of the triangles meet.
Chapter 4. Classification Pipeline 28
• From its nature, cortical surface has gyri and sulci. Such a structure allows the surface
to be inflated into a 3-D structure as a hot air balloon. To measure how sharply
the cortex is folded at each vertex, curvature as a metric is introduced. Cortical
surfaces from all the images are then transformed into a sphere and, in this space,
spatially normalised. The outcome of such normalisation are spherical deformation
fields which are subsequently applied on another measure of cortical surface showing
how much of G M is contained in each location, thickness, to generate a common
group space in which a statistical analysis can take place (Chung et al., 2003; Greve,
2012).
• In reality, one more metric, the area dilatation rate (Chung et al., 2003), is needed to
successfully normalise all the pictures and the process itself gets then more complicated.
Nevertheless, the explanation provided above is sufficient for the purposes of
this thesis.
4.2.2 From Univariate to Multivariate Techniques
No matter how big a step VBM and DBM from manual ROI identification are, they
still have their limitations. In the case of V B M , the issues are related to registration and
segmentation of atypical brains (such as dealing with tumours and artero-venous malformations),
robustness of standard parametric tests requiring normal data distribution, accuracy
of localisation, sensitivity to non-linear differences and difficulties with interpretation
(Mechelli et al., 2005).
For instance, since smoothing removes finescale structure of the images, it increases
the sensitivity of V B M to differences that are expressed at a larger spatial scale. If a too
wide kernel was used, it would blur the images out enough to make correct statistical
inference impossible (Penny et al., 2011). Unfortunately, no gold standard on what width
of the kernel is too large has been introduced. Moreover, modulation of the images is an
optional step with again no standardized suggestions on when to use it or not. However,
a recent study (Radua et al., 2014) suggests that employing modulation decreases V B M
sensitivity to mesoscopic brain volume abnormalities.
Many of the limitations of V B M are overcome by D B M . Since D B M includes neither
segmentation, nor smoothing and modulation, the problem of finding a gold standard for
the method settings disappears. In fact, the preciseness of D B M depends solely on the
preceding registration. Therefore, it is essential to choose a registration technique that
suits our purposes the best. However, since D B M registration is harder to implement, a
majority of studies prefers V B M over D B M (Scanlon et al., 2011).
Another limitation, which V B M and D B M have in common, is that a statistical analysis
of M R I images or deformation fields involves testing hypotheses for a large number
of elements and thus it has to be controlled for the multiple comparison problem, which
requires several assumptions. In practice, such conditions are often hard to be met. Therefore,
non-parametric methods which rely on minimal assumptions (for instance, they are
Chapter 4. Classification Pipeline 29
often distribution free, and are adaptive to correlation patterns in the data) can be used
to exhibit statistically significant morphological variation between the groups of interest
while controlling the risk of false positives (Pantazis et al., 2004).
Probably the strongest critique of both of these methods is that they deal with brain
images on a voxel-to-voxel basis, which makes them highly localised in space. Therefore,
the Mest-based voxel-wise comparison fails to account for multivariate group differences
such as interactions between several voxels (Gaonkar et al., 2011). To make inferences
about the differences between brain morphology while taking such interactions into consideration,
other approaches, which will be described further in 4.2.3, have to be applied.
We will end this short discussion about the limitations of V B M and D B M with a quotation
from Zarogianni et al. (2013), which underlines the attitude of a scientific community
towards the problem of MRI-based C A D of schizophrenia.
"Taking into account the fact that brain alterations in schizophrenia expand over a
widely distributed network of brain regions, univariate analysis methods may not be the
most suited choice for imaging data analysis. To address these limitations, the neuroimaging
community has turned to machine learning methods both because of their ability to
examine voxels jointly and their potential for making inferences at a single-subject level."
(Zarogianni et al., 2013)
4.2.3 Multivariate Approaches of Machine Learning
Unlike univariate statistical approaches which result in feature selection, multivariate
approaches are capable of reducing data dimensionality while retaining all the features,
i.e. feature extraction, which is typically achieved by transforming the data to a representation
in other domain (such as frequency, time or statistical domain) (Soman et al., 2009).
Thus, the curse of dimensionality typical for M R I data can be overcome by transforming
the data to a new set of axes or basis functions where the data are more dense. These axes
represent classification features from a new perspective, such as a linear combination of
the original ones. Furthermore, feature selection might take place also in the new domain
reducing the dimensionality of the data even more. The comparison of univariate and multivariate
approaches is diagrammatic ally shown in Figure 4.6. Related to schizophrenia,
it is therefore desired that the new axes are constructed in a way to facilitate the classification,
i.e. the new domain offers the most discriminative features and the collinearity
between the features is removed.
From the view of mathematics, the feature extraction transformations can be categorized
to linear and non-linear ones and, from the view of signal processing, they are
divided to either lossy or lossless11
. Whereas with lossless feature extraction we indeed
retain all the information, lossy transformation discard a part of it (Soman et al., 2009).
1 1
Typically, these terms are related to digital data compression such as image compression, audio compression,
etc.
Chapter 4. Classification Pipeline 30
dimensionality
reduction
selected voxels
Feature
Selection
x3, x7, ... x„„,
Univariate Approaches
Multivariate Approaches
Further
Analysis
Feature
Extraction
transformation
to another domain
new bases
Feature
Selection
Further
Analysis
selected bases
dimensionality
reduction
Figure 4.6: Comparison of Univariate and Multivariate Approaches. Whereas in the case
of univariate approaches, where statistical analysis is based directly on the voxels selected
regardless of any voxel-to-voxel interaction, the multivariate approaches transform the
images to another domain, with a more dense representation of the images. Such a representation
takes into account possible interconnections among voxels and facilitates the
employment of more sophisticated analyses. Adapted after (Xu et al., 2008).
Last, some of the transformations might be fixed to a predefined set of axes or they might
be adaptive to data. For schizophrenia classification the latter one is more suitable since
not enough a-priori information about what features are the best has been gathered so far.
Principal Component Analysis and Singular Value Decomposition
One of the most common dimensionality reduction technique using orthogonal transformations
to convert a given space into a new one is the principal component analysis
(PCA) which gained its name from principal components, i.e. the newly constructed linearly
uncorrected axes (König, 2000). This first part of the P C A is regarded as a lossless
transformation since all the variability in the data is retained. Thus, applying what was
mentioned above, up to now the P C A corresponds to feature extraction.
The idea behind the dimensionality reduction is that the principal components are
sorted by the amount of data variability they capture. Thus, the last ones capture only
a little piece of information and as such they can be discarded without destroying the
signal carried by the data (Soman et al., 2009). Nevertheless, the components with the
greatest amount of retained data variability may not bear the most discriminative features
pertaining to schizophrenia and, vice-versa, the last components may correspond to subtle
morphological brain changes and therefore be crucial for schizophrenia classification.
Consequently, it is advisable to sort the component, rather than explained variance, according
to how well they distinguish between the classes. At this point we do not any
Typically used when the problem is well-known such as J P E G compression.
Chapter 4. Classification Pipeline 31
more transform the space, we solely get rid of some features that are not important for the
further classification. In other words, we select a subspace spanned by the most discriminative
features which was our main goal in the first place.
Formally, the P C A is computed from the sample covariance matrix
j N ^ N
Q = Jj E ( x
' ~ /*) (*'" ~ trf* with /I = - £ x t ,
i= 1 ('= 1
which corresponds to the covariance among the data samples x;-. Such a matrix is square,
symmetric and positive semi-definite, which are demanded characteristics for manipulating
with the matrix. It facilitates the transformation to an orthogonal space, which can be
easily achieved by computing the eigenvectors of the matrix and using them as the bases
for the new space. The orthogonalization comprises of a matrix decomposition (Bingham
and Mannila, 2001)
Q = LALT
,
where A is a diagonal matrix of the eigenvalues and L are the so-called loadings13
. The
data can be subsequently projected onto the orthogonal space by a simple multiplication
LT
X = S,
where X is the data matrix. The matrix S then consists of the so-called component scores.
Moreover, if the eigenvectors are sorted according to their corresponding eigenvalues
in the ascending order, the data can be reduced by projecting them onto only first k axes as
XPCA — L[x.
We implicitly rely on the presumption that using k small enough to greatly reduce the
dimensionality does not erase the signal we are in the search for in our data.
Nevertheless, in practice the covariance matrix is often too large to be used in computations.
For instance, recalling the example of 160 x 512 x 512 voxels per image, the case
of a million-dimensional feature vector yields a million x million covariance matrix which
is a trillion of values to be stored in a computer memory. Fortunately, the P C A can be
computed by another matrix factorisation technique, the singular value decomposition
(SVD) (Shlens, 2014).
In general, the S V D also utilizes eigenvectors to decompose a matrix. Unlike with
the P C A , the decomposition can be performed on an arbitrary matrix, i.e. the matrix is
not necessarily square and symmetric. Therefore, the computation of a sample covariance
matrix Q can be avoided and the decomposition can be performed straight on the data.
Loadings are the weight by which each standardized original variable should be multiplied to get the
component score.
Chapter 4. Classification Pipeline 32
The mathematical form of the factorization is
X = UZVT
,
where the matrix E contains singular values1 4
of X on its diagonal and the matrices U
and VT
are orthogonal containing, respectively, left1 5
and right1 6
singular vectors of X.
Geometrically, E corresponds to scaling and U and VT
to rotations (Shlens, 2014).
Again, the data can be projected to the new orthogonal space by multiplying
UT
X = S
and the dimensionality reduction can be achieved by projecting the data onto a subspace
spanned by selected k left singular vectors corresponding the largest singular values (Bingham
and Mannila, 2001) as
XSVD — U{X.
One of the great attributes of the SVD, leading to a more efficient implementation,
is that when performing the reduction, it is not necessary to compute all the matrices
explicitly but it is sufficient to compute only some of the singular vectors and singular
values. Yet, the computations are still quite expensive (Bingham and Mannila, 2001).
In order to avoid heavy computation, various P C A modifications have been introduced,
mainly in the field of computer vision with the need to reduce the size of images. Some
of the examples are two-dimensional P C A (2DPCA, Yang et al., 2004) or two-directional
two-dimensional P C A ((2D)2PCA, Zhang and Zhou, 2005). Interestingly, there has also
been tendencies to use P C A for sparse data representations1 7
, e.g. sparse P C A via regularized
S V D (sPCA-rSVD, Shen and Huang, 2008).
In the field of neuroimaging, the computation can be reduced by performing P C A only
a preselected set of voxels (e.g. Karageorgiou et al., 2011), which, on the other hand,
brings up the above-mentioned pitfalls of choosing ROI (see 4.2). Another concept how to
avoid the computation of the Q matrix was introduced by Thomaz et al. (2007) and later
in Janousova et al. (2015) it was named inter-subject PCA. The rationale of this method
stems from the fact that the sample covariance matrix can be calculated on the basis of
covariances among the data samples instead of the voxels, thus reducing the dimension of
the matrix to n x n, while preserving all the variability captured in the data.
Independent Component Analysis
The idea of the independent component analysis (ICA) is somewhat similar to the
idea of PCA, i.e. to represent the data in a transformed space in which the data samples
1 4
Singular values of X are the square roots of the eigenvalues of XT
X and XXT
.
1 5
Left singular vectors of X are eigenvectors of XXT
.
1 6
Right singular vectors of X are eigenvectors of XT
X.
1 7
M o r e elaborated text on sparsity will follow later in the section about K - S V D .
Chapter 4. Classification Pipeline 33
would be more dense. However, unlike P C A , which minimizes the mean square error, with
the I C A we assume the signal to be composed of several additive components, loosely
speaking the signal is assumed to come from several different sources1 8
, yielding the main
ICA assumption that the sources are statistically independent1 9
(Soman et al., 2009).
Consequently, each data sample x;- can be expressed as a linear combination of m
sources Sj, referred to as latent variables, which can be written as (Hyvarinen and Oja,
2000)
m
x
i=Y<ai
Js
j i e { l , . . . , / » } ,
where a; J are the mixing weights and n is the number of the data samples. In matrix form,
the data matrix X can be expressed as
X=AS.
In order to recover the sources S, we need to invert the matrix A and multiply it with
the data,
S = A~1
X = WX.
Unfortunately, the mixing matrix A, and naturally its inverse W as well, is unknown and
therefore it needs to be estimated. Adding a presumption that the components have nonGaussian
distributions, A can be estimated by maximization of non-Gaussianity (e.g. measured
by kurtosis) or by minimization of mutual information, where both of these techniques
are perceived as the measures of the sources independence, or by the maximum
likelihood estimation (Hyvarinen and Oja, 2000). However, these techniques do not always
result in a completely orthogonal space as it was the case with P C A and thus some
of the components stay correlated.
Thereafter, some of the estimated sources can be selected as new features and utilized
for the forthcoming classification. In fact, the idea to apply the I C A on the M R I scans of
schizophrenia patients appeared in X u et al. (2008) as a new approach to brain morphometry,
the so-called source-based morphometry (SBM):
• S B M originates from the V B M preprocessing steps, where the images are spatially
normalised, segmented and smoothed (see 4.2.1). Rather than applying univariate
statistics separately on white and gray matter images, S B M conjugates the images to
construct one complex variable with G M as its real part and W M as the imaginary
1 8
Therefore, it belongs among the so-called blind source separation methods. Very often, digital signal
processing deals with the problem of how can be an unknown mixture of audio signals decomposed, which
is sometimes referred to as the cocktail party problem.
1 9
Random variables y\ and yi are statistically independent i f and only i f their simultaneous probability
density function can be factorized as p(y\,yi) = P\{y\)pi{yi)-
Chapter 4. Classification Pipeline 34
one. The phase of this variable does not any more represent the relative volume
changes in G M and W H , but it reflects their relative partition information of a certain
brain volume instead.
• Subsequently, ICA is applied to those so-called phase images to reduce the problem
dimensionality and to obtain the components along with the degrees of to what extent
each component contributes to each image and voxel and, reciprocally, to what
extent each voxel and each image contribute to a given component. Optionally, the
components can be adjusted for confounding effects such as age or sex. The following
statistical testing reveals in what region of the brain the ratio of W M over G M is
increased or decreased.
K-SVD
Since M R I data are very high-dimensional, new means of reducing their dimensionality,
and potentially yielding more robust results, are sought-after in order to reduce the
computation time yet offer features suitable for classification. With P C A , in order to derive
the features we utilize mathematical properties known from linear algebra to compute
eigenvectors and eigenvalues, i.e. the system is determined and thus only one solution exists.
On the other hand, in many techniques both the new bases and the coefficients are
unknown (see I C A above) and thus we face the problem of finding both. Such an optimization
problem is also encountered in the following techniques that attempt to reduce the
data dimensionality by finding its sparse representation, i.e. they endeavour to transform
the data matrix X into such a domain where it can be expressed as
X = 3>A,
where <I>, by the terminology of sparse representations, is a dictionary of the so-called
atoms (the new bases) and A is a sparse coefficient matrix (Elad et al., 2010).
Formally, a signal x = (xi ,X2, ...,XK) € M.K
is sparse if
|1 <k<K\xk^0\ <£,
i.e. if the most of its elements are equal to zero. Similarly, a signal is sparse in a dictionary
4> = (0i, 02,<j>j) if it can be expressed as a linear combination of its atoms <j>j, i.e.
J
x = ^ a
$j =
; = i
where 7 e R and the most of the coefficients aj e K. are zero. The number of non-zero
elements S in the vector a is expressed as the /°-norm of the vector, i.e.
| | a | | 0 = 5.
Typically, the goal is to find a representation with the lowest S (Elad et al., 2010) resulting
in an overcomplete (redundant) dictionary, i.e. some of the atoms in the dictionary can
Chapter 4. Classification Pipeline 35
be expressed as a linear combination of the others and thus they can be removed without
losing information.
In practice, the overcompleteness of a dictionary is a demanded characteristics, since
it leads to more stable and robust results (Balan et al., 2006). On the other hand, the usage
of overcomplete dictionaries in computations results in the finding solutions to underdetermined
systems20
. In order to find such a solution, extra conditions on the system must
be imposed (Donoho, 2006).
In the case of sparse representations, the extra condition is related to the sparsity of
the coefficients a. Theoretically, given a dictionary d> and an image x, we attempt to
find the sparsest possible representation of the image, i.e. to minimize the /°-norm of the
coefficients, given by
mini I alio subject to llx —dpalk < <5,
a
with a small error 8 (Elad et al., 2010)2 1
. Such a formulation of the problem is referred to
as the error-constrained sparse coding problem (Ron Rubinstein, 2008). Alternatively, the
problem can be expressed as the sparsity-constrained sparse coding problem as follows
minllx —4>a||2 subject to llallo < T,
a
with T being the upper threshold for the number of non-zero elements in a.
Unfortunately, since such a problem is considered to be NP-hard, the optimal solution
can not be efficiently discovered. However, when the optimal solution is sparse, it is
sufficient to approximate the /°-norm with ^-norm making the problem convex and thus
the solution traceable (Donoho and Tanner, 2005). In other words, in practice, the solution
can be found by relaxing the condition yet yielding sparse coefficients.
The process of finding the sparsest representation is characterized as sparse coding
and the optimizing algorithm itself is referred to as a pursuit algorithm (Aharon et al.,
2006). In the field of machine learning, many pursuit algorithm have been introduced
such as matching pursuit (Mallat and Zhang, 1993), orthogonal matching pursuit (OMP,
Pati et al., 1993), method of optimal directions (MOD, Engan et al., 1999) or basis pursuit
(Donoho, 2006).
Nevertheless, in order to retrieve the sparsity coefficients, in the sparse coding stage
both the data and the dictionary must be known. Since the data are our M R I images, an
appropriate dictionary remains to be chosen. Conventionally, the dictionaries are fixed
to a predefined set of axes or basis functions such as in the case of the discrete cosine
transformation or wavelets, very often discussed as the methods of image compression
(Yu et al., 2011). However, for the purposes of M R I it is demanded that the dictionaries
are adaptive to the data, as it was also the case with both P C A and ICA.
2 0
I n general, underdetermined systems have infinite number of solutions.
21
11 • 112 is the /2
-norm, widely known as the Euclidean norm.
Chapter 4. Classification Pipeline 36
A possible way to create a data-driven dictionary and estimate the sparse coefficients
at the same time is to divide the optimization into two phases, where in a sparse coding
phase the dictionary will be fixed and the pursuit algorithm will be employed to find the
optimal coefficients and in the other one, a dictionary update phase, the coefficients will
be treated as final and only the dictionary will be optimized. Such an approach has been
introduced in Aharon et al. (2006) and it is known as K-SVD. It can be summarized as
follows.
The aim of K - S V D is to find the best sparse representation of the images x;- captured
in X by solving
min||X —4>A||2 subject to \ \oci\\o < 7b,
where i G { 1 , n ) and 11 • | \F
is the Frobenius norm2 2
. Firstly, the dictionary is initialized
with /2-normalized columns. The subsequent optimization process iteratively alternates
between the sparse coding phase, when the optimization of each a ; takes place, and the
dictionary update phase, when for every atom of the dictionary an error matrix, representing
the error of discarding the atom from the dictionary, is computed, restricted to the
columns that correspond to non-zero sparse coefficients and finally it is decomposed using
SV D . The update of both the dictionary and the coefficients is dependable on the matrices
resulting from the S V D factorization and since such a step occurs for each of K atoms and
coefficients the designation of the method is obvious.
Moreover, an interesting characteristic of S V D can be used to reduce the problem
dimensionality even more. When dealing with sparse data representations, not only the
computational complexity of S V D decreases, the complexity of the whole optimization
process might be reduced since the sparsity properties facilitates to employ random projections
which sufficiently approximate the data in the lower dimensions (Bingham and
Mannila, 2001).
Since the advent of K - S V D , tweaks to improve the classification capacity of the algorithm
have emerged. For instance, introducing the minimal classification error as an additional
constraint to the objective function resulted in the so-called discriminative K - S V D
(D-KSVD, Zhang and L i , 2010). A similar approach was taken in label consistent K - S V D
( L C - K S V D , Jiang et al., 2013), where class labels were associated with each atom during
the optimizing process. Moreover, another framework, where the authors successfully
applied also the aforementioned random projections, to create class-specific dictionaries
was employed in Guha and Ward (2012).
Another dimensionality reduction technique applicable to M R I images is called compressed
sensing. With K - S V D , they share a common idea to create a sparse representation
of the images. However, compressed sensing takes the compression one step further than
K - S V D . First, an image x is represented as a product of the so-called sparsity bases
2 2
T h e Frobenius norm of a given matrix Y is defined as |\Y\\F
= * / T y F ^ .
Chapter 4. Classification Pipeline 37
and coefficients c. Afterwards a sensing matrix Q. is used to sample out and compress the
signal into a new vector y as
y = £Wc,
ensuing a way to recover the signal from a lower number of samples than stated by the
Shannon-Nyquist sampling theorem. Thus, compressed sensing is used at the stage of
acquiring M R I images to reduce the acquisition time (Yu et al., 2011).
The sparsity bases in compressed sensing are traditionally fixed. However, Yu et al.
(2011) showed that also data-driven bases discovered with S V D can be used as well. At
this point, compressed sensing and K - S V D start appearing very much alike. Indeed, in
Rubinstein et al. (2010b) the idea to incorporate dictionaries that are sparse by themselves
into K - S V D was proposed. Such dictionaries are created as a product of two matrices,
where one consists of predefined sparse atoms and the other serves for selecting the appropriate
atoms adaptively to the data. Recently, a slightly revised version of the idea
was published in Pourkamali Anaraki and Hughes (2013), giving the method a new name,
compressive K - S V D .
For the purposes of schizophrenia classification, in parallel with ICA, K - S V D has
already been exploited. The method is known as pattern-based morphometry (PBM,
Gaonkar et al., 2011). In contrast with other morphometry methods, P B M does not attempt
to infer about the group differences by directly testing whether images in one group differ
from the images of the other one, but instead it assumes that by a simple subtraction
of image matrices from different groups it can actually generate those differences. The
subtracted images are referred to as difference images. The whole method work flow is as
follows:
• Providing the images have undergone V B M preprocessing steps, the generation of
difference images commences. For each image of the first group, its 7?-nearest23
neighbours (the neighbours are chosen only from the images belonging to the second
group) are found. A l l these neighbours are then subtracted from every image of the
first group to obtain the set of difference images.
• As the second step, the K - S V D algorithm is employed to extract a dictionary of
patterns from the difference images. The sparsity constraint is applied to prohibit
the algorithm from disintegrating patterns into smaller local ones and thus complex
global patterns of schizophrenia-related brain morphology differences can be uncovered
and captured in the dictionary. Also, a sparse coefficient matrix, ranking
the importance of each pattern, i.e. loadings, is output from K - S V D .
• Lastly, the loadings are evaluated to determine how many of discriminative patterns
are statistically significant and thus how many patterns capable of distinguishing
between the groups are present in the data.
The distance is measured using an Euclidean metric.
Chapter 4. Classification Pipeline 38
4.3 Setting up Classification Rule
Once the most discriminative features have been opted out, we finally proceed to the
classification itself. Generally, the task of classification is to find a rule according to which
an observation is assigned to one of several classes.
Traditionally, a classification problem is categorized into supervised learning, where
a supervisor knows what the correct classification is, and unsupervised learning, which
aims to find the regularities in the input data without a supervision. In the field of machine
learning, the former one is simply called classification and the latter one is referred to as
clustering (Alpaydin, 2010). Since in the case of schizophrenia we know what person an
M R I scan belongs to and we assume also its diagnosis is known, henceforward we will
use the term classification in the former sense.
At this stage, we divide the classification into two phases:
• Training phase - newly selected features of M R I brain scans in the form of feature
vectors, here referred to as a training set, are used to train the classifier in order to
find an optimal classification mapping.
• Testing phase - the mapping is used to predict a membership of a new person to
either a group of schizophrenia patients or to a group of healthy controls, i.e. to
diagnose the person.
Formally, a classifier is a function
c = f(y) c e { l , . . . , C } ,
that maps a given object y to its class label c belonging to a finite set2 4
of C classes2 5
.
During the training phase, given a set of data instances and their classes, the whole
function is found or, providing we know its general form, its parameters are set. In the
testing phase, the function is evaluated for a new object outputting its label. Subsequently,
the output label can be compared to the true one to evaluate the performance of the classifier,
which will be elaborated on in 4.5. The whole process is schematically depicted in
Figure 4.7.
On account of what classification rule is used, classification can be based on:
• discriminant functions - For each of the classes a discriminant function exists.
After a new object is separately input into each of the functions, it belongs to the
2 4
I n the case of classification, variable c can be categorical or nominal. With c being a continuous variable,
we deal with an analogous problem known as regression.
2 5
I n our case, the number of classes equals 2, a scan can be either of a schizophrenia patient or a healthy
control.
Chapter 4. Classification Pipeline 39
Training
Dataset
Testing
Dataset
Feature
1 Extraction
&
1 Feature
Selection
Tapplication of
Training Data I a bo s
Testing Data
Classification
Rule (set up)
Classification
Rule (set up)
application of
transformations
I
classification rule
, I , Classification
Rule (use)
, I , Classification
Rule (use)
Classification
Rule (use)
Classification
Rule (use) vs
Predicted True
Labels Labels
Performance
Evaluation
Figure 4.7: Classification Algorithm (Training and Testing Phase). During the training
phase, transformation and reduction functions for feature extraction and feature selection
are found and applied on the training dataset. Those, along with a vector of labels determining
an affiliation of each image to given classes, are subsequently used to set up the
parameters (train) of a classification rule. In the testing phase, the testing dataset is transformed
according to the rules trained in the training phase. A true classification of the
testing images is then compared to the one predicted with the classification rule in order
to evaluate the classifier performance.
class corresponding to the function with the highest output value. A n example of a
discriminant functions based classifier is the naive Bayes classifier (e.g. Pernkopf,
2005).
• distance or similarity - Every class is represented by an etalon or their set. A class
membership of a new object then depends on the distance or similarity to the etalon.
Usually the object is assigned to the class with the least distance to the etalon or to
the most similar one. A well known member of this class of methods is for example
^-nearest neighbours classifier (&-NN, e.g. Pernkopf, 2005).
• boundaries - A feature space is partitioned into disjoint subspaces with each subspace
representing one class. A new object is then projected to the feature space and
assigned to the corresponding class.
4.3.1 Classification Algorithms Based on Partitioning of Feature Space
In the case of schizophrenia M R I scans, every feature vector can be projected as a point
in the feature space. Thus, classification rules based on the boundaries in the feature
space, such as linear discriminant analysis (LDA, e.g. Alpaydin, 2010) and support vector
machine algorithm, are often used. In this thesis, solely the latter will be described in
detail.
Chapter 4. Classification Pipeline 40
Support Vector Machine
In the schizophrenia research a widely used and respected classification algorithm is
support vector machine ( S V M , Cortes and Vapnik, 1995). For instance, as already mentioned
in 3.1.2, it has been successfully used in Ardekani et al. (2011), Ingalhalikar et al.
(2010), Schwarz and Kašpárek (2014), etc. to distinguish between the healthy controls and
schizophrenia patients. The algorithm can be described as follows.
Given an n-dimensional feature space with training data of two classes, S V M aims at
constructing an n — 1-dimensional hyperplane, a boundary, separating the data samples
into the classes with the lowest generalization error, i.e. by the largest perpendicular distance,
the so-called margin, to the closest data samples on both sides of the hyperplane,
referred to as support vectors.
Formally, the hyperplane is defined as a decision function
/(x) = w-x + Z? = 0,
where all the training samples xř- with /(x( ) > 0 belong to one class and the samples with
/(x,-) < 0 to the other class. The distance of a given training sample from the hyperplane
is given by
||w||
where 11 • 11 is a norm of a vector assigning the vector its length .
In order to find the optimal hyperplane, we need to determine the optimal weights wo,
which orientate the hyperplane, and a constant b, determining the hyperplane offset from
the origin.
If the data is linearly separable, since more hyperplanes separating the classes might
exist, the largest margin condition,
w • x; + b > +1 when y; = +1
w-Xi + b<— 1 wheny,- = —1,
where yř- G { — 1,1} are class labels for the corresponding training vectors xř-, is introduced.
The support vectors are those samples lying on the very edge of the margin, i.e. for the
support vectors the inequalities are altered to equalities.
Thus, the optimal margin, or more precisely twice the margin
1 1 2
11 Wo 11 11 Wo 11 11 Wo 11'
2 6
I n Euclidean space the length of x is captured by the formula | |x| | = -y/x-x.
Chapter 4. Classification Pipeline 41
needs to be maximized, which leads to an optimization problem. In practice, for the
convenience of optimization, the problem is given a different formulation,
for any i— 1, ...,n. Moreover, it can be shown that the optimal weights wo can be written
as a linear combination of support vectors.
A complication arises when the classes can not be separated by a linear hyperplane
without misclassifying some of the data samples. For such purposes, instead of the maximum
margin criterion, the soft margin method has been introduced. Apart from the largest
margin condition, the so-called slack variables measuring the classification error for
given data samples x;-, are added to the optimization. Consequently, the search for the
optimal hyperplane turns into looking for the trade-off between a large margin width and
a small number of misclassification errors.
Linearly non-separable problems can be also resolved by applying the so-called kernel
trick which exploits kernel transformations in order to find a separating hyperplane, instead
of in the original n-dimensional space, in a different space of higher dimension where the
problem suddenly becomes separable by means of a linear hyperplane (Boser et al., 1992).
The advantage of using S V M as a maximum margin classifier is that, since it builds the
classification rule upon the support vectors only, it eliminates the need of supervising for
outliers and atypical patterns (Boser et al., 1992). Also, it is quite robust to model variance
and bias (Meyer et al., 2003). However, as already mentioned in 4.2, no algorithm outperforms
the others in all classes of problems (Wolpert and Macready, 1997). Therefore, the
choice of the classification algorithm is not straightforward and it depends mainly on our
purposes and demands.
For the purposes of C A D , it is essential that a classifier is not overfitted , i.e. it should
perform well on new data. To evaluate whether the classifier generalizes well, it is advisable
that the testing set does not overlap with the training set in order to avoid data
circularity. The best way to assess the generalization property is when a completely independent
dataset is used (Lewis, 2000). Particularly, new M R I brain scans are acquired,
the diagnoses are predicted by the classification algorithm and then compared to the true
ones. Unfortunately, researchers usually struggle with the lack of data at the first place and
therefore, in terms of time and expenses, it is not feasible to obtain a new dataset. Hence,
validation techniques have been developed.
In general, validation techniques emulate the training and testing set by partitioning the
original data. Thereafter, it is very important to incorporate the data partitioning former
2 7
T h e problem of overfitting has been introduced at the beginning of this chapter (4).
4.4 Classifier Validation
Chapter 4. Classification Pipeline 42
to the stage of the feature extraction and selection, otherwise the classifier rule would
still be based on the features from the testing subject. Such a validation scheme does
not mimic the situation with previously unseen data sufficiently and the results are often
overoptimistic (Bermingham et al., 2015). In the text below, some of the most common
validation techniques will be introduced.
Holdout (true validation)
Holdout is a validation technique which takes the existing dataset and samples a part
of its data out. This part is used in the testing phase whereas the rest is retained as the
training set. Commonly, two thirds of the data are retained. Thus, the downside of this
technique is that an already small dataset is reduced by one third and the performance of
the classification is negatively influenced (Kohavi, 1995).
Cross-validation
Cross-validation techniques also exploit the idea to divide an existing dataset into two
complementary parts which are subsequently used for training and testing. The difference
from the previously mentioned holdout is that cross-validation is performed several times
using different partitioning schemes in order to reduce the influence of chance.
Ways to distribute the data into subsets are manifold. &-fold cross-validation which
divides the dataset into k equally sized subsets is often used in practice. Consequently,
in each iteration, one of the subsets is used as a testing set and the rest k—1 subsets are
retained for the classifier training. The overall accuracy is then computed as an average
across the iterations (Kohavi, 1995). From Figure 4.8, where a 10-fold variant of the &-fold
cross-validation is depicted, it should be noted that it is still important to enclose feature
extraction and selection into each iteration separately.
The most extreme case of the &-fold cross-validation is when k equals the number of
data samples. Such a variant is referred to as leave-one-out (LOO) cross-validation. L O O
cross-validation is especially attractive to be adopted for limited datasets as it reduces the
perturbations to the training data to the minimum (Cawley and Talbot, 2003). On the other
hand, when the subsets do not comprise of the same number of data samples from each
group, the L O O estimates can be biased (Kohavi, 1995).
Besides, the iterative approach makes cross-validation computationally expensive. Nevertheless,
the computational complexity can be reduced by the proper implementation
(Cawley and Talbot, 2003). Fortunately, for the purposes of C A D the cross-validation is
needed to be performed only during the creation of the classifier. Once the classifier is
validated, i.e. it has been shown that it performed well on independent data, it is no more
necessary to validate it. Therefore, the computation time is reduced to evaluation of a
single new individual to be diagnosed.
Chapter 4. Classification Pipeline 43
10-fold cross-validation loop
Dataset
10 times 10 times
Training Dataset i
(90% samples)
10 times
Feature
Extraction
&
Feature
Classification
i
Extraction
&
Feature
Rule (set up)
Selection
tapplication of
transformations
Testing Dataset
(10% samples)
application of
classification rule
Classification
Rule (use)
Performance
Evaluation
Figure 4.8: 10-fold Cross-validation. The same dataset is 10 times divided into a training
set consisting of 90% of images, upon which feature extraction and selection is employed
and a classification rule is set up, and the rest 10% of images, which are used to evaluate
the performance of the classification algorithm. Since the algorithm is evaluated 10 times,
the influence of chance is reduced which is a useful characteristics if the algorithm is to be
used on new data.
Bootstrap
Another method, allowing the estimation of accuracy measures, is bootstrap which relies
on random sampling with replacement, i.e. n samples are uniformly chosen from a
dataset of size n with the possibility of selecting the same sample more than once. Those
samples then serve as the training set and the not chosen ones2 8
are left for testing. The
whole process is repeated many times in order to obtain an estimate of the classifier accuracy
(Kohavi, 1995).
4.5 Evaluation of Classifier Performance
In the text above, the term classification accuracy has been mentioned several times.
In this section, it will be defined formally along with other measures of the classifier
performance.
Generally, the classifier is tested during the testing phase by comparing its predictions
to reality. For instance, if the tested M R I scan belongs to a schizophrenia patient and the
classifier assigns the scan to the group of healthy controls, it would perform badly and
vice-versa. Provided the prediction have been carried out on more scans, the results can
be summarized in a 2 x 2 contingency matrix also known as a confusion matrix (Fawcett,
2006):
On average approximately 37% are not sampled.
Chapter 4. Classification Pipeline 44
schizophrenia patient healthy control
classified as TP FP
schizophrenia patient (true positive) (false positive)
classified as F N T N
healthy control (false negative) (true negative)
The elements of the matrix represents:
• TP - the number of correctly classified schizophrenic brain scans;
• FP - the number of healthy brain scans misclassified as schizophrenic ones;
• FN - the number of schizophrenic brain scans misclassified as healthy ones;
• 77V - the number of correctly classified healthy brain scans.
The confusion matrix serves as the basis for many measures (Baldi et al., 2000; Powers,
2011) of classifier performance such as:
• overall accuracy - commonly referred to as accuracy, i.e. the percentage of correctly
classified individuals
TP + TN
OA = ;
TP + FP + FN+TN
• sensitivity - the probability of correctly predicting a schizophrenia patient
TP
SENS =
TP + FN'
specificity - the probability of correctly predicting a healthy control
TN
SPEC
FP + TN'
• other measures:
- positive predictive value (precision) PPV — Tp^FP',
- negative predictive value (precision) NPV — F™TN;
- misclassification rate (error) ERR — TP+FP+FN+TNUltimately,
it should be stated that the performance of classification is dependent on
the implementation of the pipeline as a whole, ranging from the characteristics of M R
scanner and data acquisition, trough preprocessing and a choice of the features, to selecting
a proper classifier, validation technique and the measure of accuracy itself. Therefore,
a plethora of possible implementations with myriad of subtle settings exist and nobody
is capable of exploring all of them. It is thus necessary for each researcher to concentrate
only on a very specific bit of the methods and put them together later on with the
rest of the scientific community and so, jointly, find the best solution to computer-aided
schizophrenia diagnostics.
Chapter 5
Aims of The Thesis
The primary focus of this thesis is to propose and implement an algorithm for feature extraction
and selection from M R I images and to use the obtained features on the data of firstepisode
schizophrenia patients (FES) and healthy controls ( H Q for C A D of schizophrenia.
In other words, the primary objective of the thesis is to implement and evaluate a classification
algorithm for schizophrenia diagnosis from imaging data. Especially, the thesis
concentrates on methods of feature extraction and selection and their comparison.
In regards to schizophrenia classification, the foregoing chapters briefly introduced the
problem of brain disorders with a special focus on schizophrenia and the field of neuroimaging
as a concept of visualising and objectively assessing the brain morphology and
its changes during the progression of a disease. Subsequently, a motivation of why to
concentrate on the field of machine learning and how it is applied to neuroscience was
presented. Ultimately, a classification pipeline was described with the main emphasis on
brain morphometry techniques and machine learning transformations as two particular
ways to perform the feature extraction and selection step.
The following chapters will concentrate on the description of the dataset upon which
the ensuing classification will be performed and the methods of opting out the most discriminative
features. Furthermore, the performance of the classification algorithm will
be evaluated and compared across several feature extraction and selection methods. The
classification implementation itself will be carried out in M A T L A B (MathWorks, 2015).
-45-
Chapter 6
Datasets
The dataset consists of 104 individual Tl-weighted M R I whole-head scans, where exactly
one half of the scans belongs to 52 FES who were recruited at the Department of Psychiatry,
University Hospital Brno. The patients were all male with the mean age of 24 years
(±5.1). The diagnosis was based on diagnostic interviews regarding patients history, substance
abuse, etc. and evaluated using PANSS (see 2.2). A senior psychiatrist reviewed the
tests and, in compliance with ICD-10, established the diagnosis. Additionally, the patients
were physically examined and, given specific criteria such as suffering from other neurological
disease, substance dependence, etc. were met, excluded from the study. The other
52 scans were acquired from volunteering H C whose mean age (24 ±3.1) and handedness
matched with the patients.
The images were obtained using the 1.5 T M R scanner with a resolution of 160 x 512 x
512 voxels per scan and subsequently, using the V B M 8 toolbox (VBM8, 2015) available
in the SPM8 Matlab software package, they were corrected for bias-field inhomogeneity
and spatially normalized by affine co-registration to the standard S P M T l template.
At this point, we create two datasets from the data with regards to the information they
provide, i.e. both offering a different perspective on the data. At first, we focus on T l
densities which straightforwardly yield the local G M volumes (in the following referred
to as G M Densities, see also 6.1) and, secondly, on the information about the local volume
changes represented by the Jacobian determinants (the so-called Volume Changes, see also
6.2). Ensuing from what was mentioned in 4.2.1 such approaches correspond to the V B M
and D B M approaches, respectively, and therefore they represent a way to extract features
from M R I data.
In the rest of this chapter, the two datasets resulting from applying the V B M and D B M
approaches on the acquired data, will be briefly described. For better illustration, the
datasets generation is schematically depicted in Figure 6.1. A more detailed description
of the datasets and their preprocessing can be found in Schwarz and Kašpárek (2014).
-46-
Chapter 6. Datasets 47
Logarithmic
Transformation
Volume Changes
Dataset
Figure 6.1: Scheme of Creation of the Datasets. Acquired images of 52 FES and 52
H C were corrected for bias-field inhomogeneity and spatially normalized by affine coregistration.
Next, the G M tissue densities dataset was created by applying D A R T E L on
the preprocessed images while retaining gray matter segments and by employing 8 mm
F W H M Gaussian kernel smoothing. Following the D B M flow diagram, the Jacobian
determinants were calculated to convert displacement fields, generated from the preprocessed
images applying the high-dimensional deformable registration technique, to scalar
values ranging from 0 upwards. In order to generate the whole-brain volume changes
dataset, logarithmic transformations were applied on the scalars to make the distribution
of the values symmetric around zero, where values larger than 0 correspond to expansions
in volume and values lower than 0 represent volume compressions. Adapted from
(Schwarz and Kašpárek, 2014).
GM Densities
Dataset
Chapter 6. Datasets
6.1 GM Densities
48
In order to create the G M densities dataset, additional steps needed to be performed
following the V B M approach. After the affine co-registration of the Tl-weighted images,
the images were non-linearly registered using D A R T E L (diffeomorphic image registration
algorithm, Ashburner, 2007). Resulting G M tissue segments were modulated with
the determinant of Jacobian matrices of the deformations to account for registration related
changes in local volumes. Subsequently, the modulated G M segment images were
smoothed with 8 mm F W H M (Full Width at Half Maximum) Gaussian kernel to enable
inter-subject comparisons.
6.2 Volume Changes
As the Volume Changes dataset follows the D B M approach, it is based on an additional
spatial normalization which outputs displacement fields referring to volume adjustments
needed for each image to match the template. Thus, after the images were normalized
to the same stereotactic space, a high-dimensional deformable registration technique
(Schwarz et al., 2007), which attempts to maximize the normalized mutual information between
the images and the I C B M template, was performed. The obtained 3-D displacement
fields were converted to scalars by computing the Jacobian determinants at each voxel.
Additionally, the scalar values were logarithmically transformed in order to distribute the
values symmetrically around zero instead of an asymmetric distribution of solely positive
values which the determinant of Jacobian matrix normally yields.
Chapter 7
Classification Schemes
A plethora of different possibilities how to combine machine learning and other methods
to create a classification scheme capable of processing M R I data exists. Since it is not
feasible to explore all of them, this thesis concentrates mainly on the comparison of the
extraction and selection methods important for determining what features yield the best
discriminative power in terms of classifying between FES and H C .
The creation of datasets described in 6 represents two different ways of extracting
features from data, i.e. the raw images were processed according to the V B M and D B M
approaches in order to generate a set of data matrices upon which the classification can be
based. Thus, those datasets may lead to different results in classification and definitely to
different interpretation of the results.
Next, working with each dataset separately, we can either apply univariate statistics to
select the most significant features (voxels) to base a classification rule upon as a typical
univariate brain morphometry method would do or we can utilize machine learning techniques
to reduce the dimensionality of the problem by transforming the data into another
domain with a more dense data representation and perform the classification in the newly
created feature space.
Thereafter, by testing classifiers on new or cross-validated subjects, the performance
of various classification schemes can be evaluated. Considering the fact we are aiming at
exploring and comparing the aforementioned feature extraction and selection methods, it
is convenient to fix the remaining classification steps following a simple reasoning - since
everything except for the features chosen for the classification is fixed, the variability in
the results is directly related to the feature extraction and selection step allowing for their
straightforward comparison.
The data acquisition and preprocessing step is fixed by evaluating the datasets separately.
Moreover, we limit the evaluation to measuring solely overall accuracy, sensitivity
-49-
Chapter 7. Classification Schemes 50
and specificity. Likewise, a choice of a classifier is limited to linear S V M solely since
it is a widely used and robust method enabling us to also compare the results with other
studies.
The last question is what validation technique is the most suitable for our purposes.
Theoretically, one of the datasets can be viewed as a way of validating the methods of the
other one and vice-versa. Nevertheless, this assertion is somewhat vague since the datasets
are only different modalities of the same subjects. Therefore we avoid regarding them as
independent datasets and rather comment on the distinctive perspectives they provide.
Since the datasets already exhibit a very inconvenient ratio1
of the number of subjects
over the length of descriptors2
it is desirable to employ a validation scheme with the most
subjects retained for a training phase. The most promising candidates therefore are leaveone-out
(LOO) cross-validation, with solely one testing subject per cross-validation cycle,
and 52-fold cross-validation, where only 2 subjects (1 FES and 1 HC) are used in each
iteration as a testing set, which is referred to as leave-pair-out (LPO). However, in order to
perform the latter cross-validation scheme fully3
, one needs to recompute the classification
scheme 2704 times contrasting mere 104 iterations needed with LOO. Although in practice
the number of iterations that should be sufficient to perform L P O correctly is lower, it still
requires a substantial amount of computational power in comparison with LOO. Thus,
and also since L O O is probably the most commonly employed cross-validation technique
again allowing us for an across-studies comparison, we implement it as our fixed validation
step.
In the following text, we will describe our proposed classification schemes for distinguishing
between FES and H C and we will elaborate on parameters of the schemes.
In this thesis, 3 methods of feature extraction and/or selection will be implemented.
At the beginning, a simple univariate statistics (Mann-Whitnney testing) will be exploited
to select voxels which are significantly different for each group, stemming from the basic
brain morphometry approaches. Nevertheless, since this thesis mainly focuses on the multivariate
machine learning transformations as a way of conjointly assessing more voxels
at the same time, the univariate approach will play the role of a basic comparator to more
complex methods.
Secondly, P C A will play the part of a classic multivariate feature extraction and selection
procedure. Although, instead of the common PCA, inter-subject P C A (isPCA) will be
employed since it entails lower computational demands. Ultimately, K - S V D as a recently
established technique utilizing sparse data representations will be applied to extract the
features for the ensuing classification.
1
104 subjects over 581,828 features.
2
The length of a feature vector, i.e. the number of features describing a subject.
3
To cover all the possible combinations of dividing 2 subjects from different classes into a testing subset.
Chapter 7. Classification Schemes 51
Since K - S V D is not a standard procedure, we will also focus on tweaks facilitating
the computation or, theoretically, schizophrenia classification. Namely, we will be dealing
with random projections as a way of reducing the computation requirements, with P B M
as a multivariate approach created especially for the purposes of M R I data classification
and with extracting class-specific features enabling to easily classify into more than 2
classes. In the last mentioned case, we will also divert from the general classification
scheme, depicted in Figure 7.1, since for the purposes of exploring the full capabilities of
the method apart from S V M we implement a different classification rule as well.
leave-one-out cross-validation
GM Densities
or
Volume Changes
104 times
Training Dataset
(51 + 52 samples)
104 times
Testing Dataset
(1 sample)
feature extraction
and selection step
MW testing
isPCA
K-SVD
PBM RP
K-SVD-CD
application of
transformations
104 times
SVM
(set up)
application of
classification rule
SVM
(use)
Performance
Evaluation
Figure 7.1: General Classification Scheme. Leave-one-out cross-validation scheme is
used to assess the performance of the S V M classifier based on features extracted and/or
selected from the G M densities or the Volume changes dataset. Three main methods for
opting out the feature are: Mann-Whitney testing ( M W testing), inter-subject principal
component analysis (isPCA) and K - S V D . The last algorithm can be either employed in its
simple form (K-SVD), or as a part of pattern-based morphometry (PBM) or implementing
a class-specific (the so-called concatenated) dictionary (K-SVD-CD). Additionally,
random projections (RP) may be used to reduce the dimensionality of the features. It is
noteworthy that the feature extraction and selection step is performed in each iteration of
the cross-validation.
7.1 Mann-Whitney Testing
Mann-Whitney testing (from now on abbreviated as M W ) is a simple univariate method
for testing whether the tested variables come from the same distribution. Since we aim to
construct a robust method potentially exploitable for any M R I data, M W was chosen over
a typically used Mest because it serves as its non-parametric alternative and as such it can
be applied on various unknown distributions.
Applying M W on each voxel, we select those voxels which statistically belong to
different populations, i.e. they should be important for distinguishing between FES and
H C . In general, when testing multiple hypotheses at the same time, one should correct
for the number of false discoveries either with F W E R or F D R corrections. However,
it should be noted that statistical significance does not necessarily imply discriminative
Chapter 7. Classification Schemes 52
power. Therefore, we regard the resulting p-values as a selection criterion rather than a
level of significance. In other words, we can choose technically any threshold t for pvalues
dividing the voxels to those which are to be incorporated into classification and
which are to be disregarded.
Thus, the threshold t plays a role of the only parameter of this classification scheme.
Recalling what was already stated in 4.2.1, Kašpárek and Schwarz (2011) claim that both
F W E R and F D R corrections can be too stringent for M R I data and thus they often select
no significant voxel at all. Therefore, the threshold is set manually. The appropriate setting
for our datasets will be discussed further in 8.1. The result of applying the threshold on
the p-values corresponding to each voxel in the image will be, in each L O O iteration, a
binary mask selecting a set of voxels as features for ensuing classification.
7.2 Inter-subject PCA
As described in 4.2.3, P C A is a classic multivariate procedure seeking a transformation
converting data to a set of orthogonal principal components ordered according to the
amount of variance they explain in the original data. Unfortunately, P C A requires a covariance
matrix of descriptors to be computed which, in the case of our data, is not feasible
since the number of voxels in each image is over a half of a million.
Luckily, it has been proven (Demirci et al., 2008) that the eigenvectors Vy, corresponding
to new components, can be computed from the eigenvectors Wy of covariance matrix
of subjects4
as
X T
W y
greatly reducing the demands on computation. The matrix X r
represents a transposed data
matrix containing N subjects and qj are the eigenvalues of the inter-subject covariance
matrix. Such a method, the so-called inter-subject P C A (isPCA, Janoušová et al., 2015),
allows us to preserve all the dataset variability using solely N — 1 eigenvectors.
By disregarding some of the eigenvectors, we can reduce the feature space dimensionality
even more. Since the eigenvectors are sorted in an ascending order of explained data
variance, at first sight it may be tempting to get rid of the last ones. However, the amount of
explained variance does not necessarily imply schizophrenia-related differences between
FES and H C and therefore just as the first component can be for instance related to differences
in liquor the last component might be crucial for recognizing the proper affiliation
of the subject.
Thus, it is favourable to sort the components according to their discriminative power.
One way to accomplish that is to again exploit Mann-Whitney testing to compute p-values
4
The covariance matrix of subjects is only 104 x 104 for the whole dataset and only 103 x 103 when
taking into account a L O O cross-validation scheme.
Chapter 7. Classification Schemes 53
as measures of how much FES differ from HC. Only this time the hypotheses testing is
performed with the subjects projected into the new feature space spanned by the components.
Subsequently, the dimensionality reduction can be performed by retaining first c
components yielding the most discriminative patterns.
7.3 K-SVD
Another matrix factorization technique is the K - S V D algorithm (see 4.2.3). It serves
for finding sparse representations of data via creating a data-driven dictionary of atoms and
a sparse coefficient matrix. To do so, it iteratively alternates between a dictionary update
phase and a sparse coding phase, seeking optimal solution to matrix decomposition. However,
since the problem is non-convex, in practice, the solutions are only approximated and
thus the algorithm does not guarantee finding a global optimum (Rubinstein et al., 2010a).
Whereas in the dictionary update phase S V D is exploited to find the most suitable
dictionary, a pursuit algorithm seeks optimal sparse coefficients in the sparse coding stage.
To M A T L A B , K - S V D has been implemented in K S V D - B o x which, for the sparse coding
stage, requires OMP-Box implementing O M P as a simple variant of a pursuit algorithm
to be installed (Ron Rubinstein, 2008). The implemented ksvd function requires several
parameters to be set.
First, since we deal with an optimization problem, the number of iterations i must be
set. Secondly, the upper threshold s for the number of non-zero elements in each column
of a sparse matrix is needed, constraining the sparsity of the matrix.
Last but not least, the number of atoms m in a dictionary to be learned is required.
By nature, K - S V D dictionary should be overcomplete. The overcompleteness factor is
characterized as a ratio of m over d, given a data matrix X decomposition
XdxN — ^dxm AmxNi
where /V is the number of subjects and d is the length of descriptors. We will elaborate on
the settings of the parameters later on in 8.1.4.
Unlike in the case of PCA, K - S V D does not provide atoms sorted in any meaningful
order. Thus, if we wanted to reduce the dimensionality of the dictionary ad-hoc, we
would need to sort the atoms first. Again, Mann-Whitney testing can be performed on
data projected to the feature space spanned by atoms, assigning the atoms their discriminative
power. However, it is not clear whether this step is demanded or rather artificial and
therefore it will be tested in 8.1.4.
In regard to K - S V D , we will mention another set of techniques that proved to be useful
for turning overcompleteness ratio in our favour. They are called random projections,
since by multiplying a data matrix with a random matrix R, they reduce dimensionality as
Chapter 7. Classification Schemes 54
follows
YpxN — RpxdXdxN,
where p < d. In other words, they are capable of reducing the length of descriptors while
allowing for sufficient approximations. Again, a proper setting of p will be examined later
in 8.1.4.
According to Achlioptas (2003), it is not indispensable that the matrix R is generated
using Gaussian distribution. Instead he proposed a way to generate the matrix from a
simpler distribution as
( 1 with probability I
0 with probability 1
— 1 with probability \
for each element r of the matrix.
7.3.1 Pattern-based Morphometry
One of the methods utilizing K - S V D for classification purposes is P B M (see 4.2.3).
Unlike in the above-mentioned case where the dictionary is built upon data matrix of images,
P B M introduces the idea of generating atoms from difference images. The dictionary
is then used in the same manner as with K - S V D . The generation of a difference images
matrix is diagrammatically depicted in Figure 7.2.
Dataset
(104 samples)
A = 52* FES
•- 52* HC
V<3
k-NN(4,.ß) ->/V,|
k-NN(Bfc,A) —»• Ab-
v*,v;i{f,...,t}
D[=Alt-N[,\. DA
[DA,DB]=X
Figure 7.2: Generation of a Difference Images Matrix. For each image, using Euclidean
metrics, we find a set of its ^-nearest images with a different affiliation. In other words,
for an image a belonging to the group A (FES), we search for its k most similar images
belonging to the group B (HC) and vice-versa. Subsequently, we subtract the images from
their neighbours N. In the end, we put the resulting difference images D together into a
single matrix X. Assuming the images are in columns, the new matrix will have &-times
more columns than the original matrix.
Chapter 7. Classification Schemes 55
7.3.2 Concatenated Dictionaries
Second utilization of K - S V D can be found in Guha and Ward (2012), where the goal
was to create a dictionary as a concatenation of a set of class-specific dictionaries. For our
purposes, a pair of dictionaries related to FES and H C will be learned separately and subsequently
concatenated into one. We will refer this method to as K S V D - C D (concatenated
dictionaries).
An interesting feature of this approach is that it can be used for as many classes as
required without the need to recompute the whole dictionary. In fact, a new class-specific
dictionary can be simply concatenated with the existing one.
Ultimately, to explore this approach more closely, apart from S V M , we will employ a
different classifier as well. Since the dictionary is created as a concatenation, its atoms can
be assigned to the group they came from. Thus, by running a pursuit algorithm for a new
subject, we come up with a coefficient vector with the first half5
of elements belonging to
FES and the second half corresponding to HC. Since the vector is sparse, most of its elements
are equal to zero. Therefore the non-zero elements show what atoms are important
for the decomposition and since we know their affiliation we can base a classifying rule
on the number of non-zero elements in each group. From now on, the classification rule
will be referred to as Coeff.
5
In order to make a training set divisible into halves, in each iteration one subject from the other group
is randomly omitted.
Chapter 8
Results
8.1 Preliminary Experiments and Parameters Tuning
The preceding chapter (7) provided an overview of algorithm schemes to recognize
FES from H C employed in this thesis. A summary of the parameters to be set for each
algorithm and its tweaks are delineated in Table 8.1. This section will elaborate on how to
set the parameters for our purposes. As suggested before, M A T L A B , as a computational
environment offering all the implementations of the aforementioned algorithms, will be
used for tuning the parameters and for the classification itself.
Algorithm Parameter Token
M W p-values threshold t
isPCA number of retained components c
number of atoms learned m
K - S V D sparsity constraint s
number of iterations i
RP reduced length of descriptors P
P B M number of nearest neighbours k
Table 8.1: Summary of Parameters to Be Tuned. For each algorithm listed at least one
parameter with an unknown setting exists. Those are summarized in the table. Note
that K S V D - C D is not included in the table, since the number of atoms learned for each
dictionary separately stems from parameter m. However, as long as we want to divide m
in two exact halves, m must be kept even.
Since each of the algorithm along with its parameter setting requires computation of
104 iterations before its evaluation can be performed, we execute preliminary experiments
on a synthetic dataset yielding lower computation demands. On the other hand, such a
dataset may be insufficient yo genuinely emulate the real datasets and therefore in cases
where we are not convinced about the results, we tune the parameters on the real datasets.
-56-
Chapter 8. Results 57
In this section, for simplicity, the evaluation will be provided only in terms of OA,
although SENS and SPEC are required for a proper evaluation as well. Therefore, they
will be included in the next section (8.2), where the final results will be put forward once
all the parameters are tuned.
8.1.1 Synthetic Dataset
To reduce the amount of computation, a synthetic dataset for tuning the parameters
of the algorithms will consist of artificial 2-D images instead of 3-D M R I images. The
dataset consists of 20 subjects where exactly 10 images are of hand-drawn triangles and
the other 10 of hand-drawn circles. Therefore, sometimes when referring to the synthetic
dataset we will use ideographic symbols A & OEach
image contains 50,184 pixels with values ranging from 0 to 255. A n example
of a triangle and a circle is attached in Appendix A . Unfortunately, those images are very
simplified, for instance most of the pixels are related to image background and therefore
the resulting parameter setting is sometimes unrealistic.
8.1.2 Mann-Whitney Testing
The only parameter to be tuned in the case of M W is t, the threshold for p-values
selecting voxels for subsequent classification. The usage of the parameter is necessary
since both F W E R and F D R proved to be too stringent, i.e. they marked all voxels to be
insignificant. The algorithm was tested with various settings of t. The results, showing O A
and numbers and percentages of voxels selected from the synthetic dataset can be found
in Appendix B . They confirm the suspicion that the A & O dataset might be in some
cases too simple to mimic real situations, since the optimal threshold was chosen to be
0.2, whereas we would expect values between 0.001 and 0.05.
Thus, the parameter was also estimated for the G M Densities and Volume Changes
datasets. The results are depicted in Table 8.2 and Table 8.3, respectively. For G M Densities,
the optimal threshold is 0.01 and for Volume Changes we chose t — 0.05.
Value oft OA [%] Number of Selected Voxels (Percentage)
0.001 50.96 7,900(1.36%)
0.01 67.31 41,250 (7.09%)
0.05 66.35 109,250(18.79 %)
Table 8.2: Classification Results for Various Parameter t Settings (GM Densities). The
table summarizes overall accuracies (OA) and numbers of selected voxels (with the percentage
out of 581,828) for M W classification with a threshold t performed on the G M
Densities dataset.
Chapter 8. Results 58
Value of t OA [%] Number of Selected Voxels (Percentage)
0.001 59.62 2,600 (0.45 %)
0.01 64.42 17,400 (2.99 %)
0.05 66.35 58,700 (10.09 %)
raZ?/e &3: Classification Results for Various Parameter t Settings (Volume Changes). The
table summarizes overall accuracies (OA) and numbers of selected voxels (with the percentage
out of 581,828) for M W classification with a threshold t performed on the Volume
Changes dataset.
The explanation of such a big difference between the real datasets and the synthetic
one can be explained by histogram of p-values for each dataset (see Appendix B), since
in the case of the latter only a few p-values are under 0.1 and therefore the number of
features selected for classification is insufficient for t lower than 0.05. Percentages of
voxels selected for classification suggest that it is favourable to select approximately 10%
of all voxels. Although in the case of G M Densities, the number is slightly lower.
8.1.3 Inter-subject PCA
In the case of isPCA, the maximal number of non-zero eigenvectors is M — 1, where,
since isPCA is performed on a training dataset, M is lower by one1
than the original
number of images in a dataset. Therefore, for A & O 18 eigenvectors can be found and
102 eigenvectors exist for the real datasets.
At first, we attempted to reduce the number of components sorted according to the variance
of original data they explain2
. Nevertheless, it is advisable to sort the components according
to their discriminative power, enhancing their importance for classification. Doing
so renders the behaviour of the curve representing the dependency of classification performance
on the number of retained components c to be more monotonie although it does not
completely remove sudden drops and rises in OA, which suggests that adding or removing
a component can both worsen and improve classification since the components carry also
undesirable noise. The comparison of the curves sorted either by explained variability
or their discriminative power for A & O is depicted in Appendix C. The results suggest
that for the real datasets we should employ the latter, i.e. we first sort the components in
terms of M W p-values and subsequently gradually remove them in order to choose the
best setting for the parameter c.
Figure 8.1 delineates classification results for G M Densities when components are
sorted relative to their discriminative power and one by one disregarded until only the most
discriminative one is left. In general, the shape of the curve is not smooth and as such it
is difficult to properly choose the best c. Nevertheless, on a coarse level we can claim
' i n the case of the L O O cross-validation, the original dataset is divided into one testing subject and the
rest of subjects designated for training.
2
In other words, they are sorted according to eigenvalues from the largest to the lowest.
Chapter 8. Results. .59
that the best results are achieved when approximately 1 to 20 components are retained,
then they drop down rapidly until 40 retained components after which the trend slightly
increases back even though not all the way up.
OA vs Number of components (ordered by MW p-values)
70 80 90 100
Number of retained components
Figure 8.1: Classification Results for Various Parameter c Settings (GM Densities). The
graph shows relation between O A and the number of isPCA-learned components retained
for classification. The components are sorted according to their discriminative power measured
here by Mann-Whitney p-values.
Surprisingly, inference from the curve for Volume Changes (Figure 8.2) is practically
the other way around. The best classification results are achieved for 75 to 102 (all) components
retained whilst lower parameter c settings deteriorate them.
OA vs Number of components (ordered by MW p-values)
40 50 60 70
Number of retained components
Figure 8.2: Classification Results for Various Parameter c Settings (Volume Changes).
The graph shows relation between O A and the number of isPCA-learned components
retained for classification. The components are sorted according to their discriminative
power measured here by Mann-Whitney p-values.
Chapter 8. Results _
8.1.4 K-SVD
60
Tuning K - S V D parameters poses bigger challenge than with the previous algorithms
since combinations of parameters must be estimated, making the parameter space more
complex. Also, K - S V D is very recent and thus the parameter space has not been profoundly
explored yet. Luckily, some studies utilizing K - S V D have already been published
providing us a lead to ground our exploration on.
Moreover, by nature, K - S V D is an optimization problem meaning it might not converge
to the global optimum and thus the results might be influenced by chance.
The rest of the section will be structured as follows. At first, we introduce our findings
for various combinations of the parameters for A & O- Namely, we concentrate on the
number of iterations i and the sparsity constraint s. Subsequently, an analysis of the dictionary
dimensionality parameter m will be provided along with the influence of the random
projections parameter p on classification results. In the end, P B M and K S V D - C D settings
will be commented on, concluding the preliminary experiments part.
In order to avoid heavy computations, O A with various parameters settings were calculated
for the synthetic dataset. Namely, we included 9 equidistantly chosen settings of m,
3 different settings of i and 14 settings of s distributed over the parameter space in a way
to capture a trend the parameter setting yields. To visualize the results, we summarized
O A for a given parameter across all values of the other two parameters.
Concentrating on the number of iterations, the summarized O A are shown in Figure
8.3. Implicitly KSVD-Box sets the parameter to be 10. However, Guha and Ward
(2012) clearly state to use 20 iterations for K - S V D learning. Therefore, we included those
two options into the analysis and added the third option (15) to see what the trend is. As
we can see, the classification results slightly improve with 20 iterations set and thus the
rest of experiments will be calculated with that value.
Regarding the upper threshold of the number of non-zero coefficients in each column
of the sparse coefficient matrix, Guha and Ward (2012) suggest s to be approximately JQ
of the length of descriptors. However, in our case it would mean loosening the constraint
up to 5,000 non-zero values. Since the implementation of K - S V D in KSVD-Box does not
allow for dictionary to consist of more atoms than the number of subjects in training data, it
is not reasonable to set s higher than 19 which is the case when every atom might influence
the decomposition, i.e. the corresponding column of a sparse matrix would contain no zero
element.
Nevertheless, since the matrix should be sparse, it is advisable to set s even lower. In
Gaonkar et al. (2011), utilizing K - S V D for a successful classification of M R I data of A D ,
the value of the parameter ranged from 1 to 7, compliant with our findings.
Chapter 8. Results 61
OA vs Number of iterations
Number of iterations
Figure 8.3: Classification Results for Various Parameter i Settings (A & Q>). The influence
of the number of iterations on the K - S V D classification performance for the synthetic
dataset. A boxplot for each i represents the median (red horizontal line), interquartile range
(IQR) (edged of the box) and the most extreme values not exceeding 1.5 IQR (whiskers)
of overall accuracies of classifications for all settings of other two parameters.
Thus, we conducted the analysis to mainly cover s between 1 and 19 although we also
included settings up to 600 to estimate how it influences OA. The results are depicted in
Figure 8.4. The boxplots suggest no visible trend with the values over 20 although they
suggest a slight increase with the lowest settings of s. Nevertheless such an increase might
be an artefact of the synthetic dataset. Indeed, Gaonkar et al. (2011) states that for real
datasets "experiments with several other parameter values does not change the results
greatly'. Since our real datasets contain roughly 5 x more subjects than A & Q, i.e. also
the maximal number of atoms in dictionary is 5x higher, we manually set the sparsity
constraint to be 5 3
with the belief that the influence of setting a different values would be
minor.
Finally, we proceed to tuning the dimensionality of the dictionary parameter m. The
summarized OA, shown in Figure 8.5, are in correspondence with the ones resulting from
isPCA (see Appendix C), i.e. the more atoms are learned, the better the classification.
However, in the case of K - S V D the order of atoms is somewhat random. In order to
evaluate whether sorting them by their discriminative power enhances the classification,
another experiment was performed. At first, the classification was performed for m— 19
with s — 5 and i — 20 and then the atoms were one by one cumulatively disregarded.
Afterwards the same classification was carried out with the atoms sorted according to
their M W p-values. The results, which can be found in Appendix D, suggest that the latter
ordering might enhance classification when only several atoms are retained. The reasoning
is as follows. As the atoms are implicitly in random order, the first ones might (but also
3
The sparsity constraint is intentionally set to be odd, which is the feature exploited later on when classspecific
dictionaries are learned.
Chapter 8. Results 62
might not) be insufficient for a successful classification whereas once they are sorted, the
most suitable ones are pushed forward in the ranking leading to better results.
2 0.6
a)
O
OA vs Sparsity constraint
T
Sparsity constraint
Figure 8.4: Classification Results for Various Parameter s Settings (A & Q). The influence
of the sparsity constraint parameter on the K - S V D classification performance for
the synthetic dataset. A boxplot for each s represents the median (red horizontal line),
interquartile range (IQR) (edged of the box) and the most extreme values not exceeding
1.5 IQR (whiskers) of overall accuracies of classifications for all settings of other two
parameters.
OA vs Number of learned atoms
7 9 11 13 15 17
Number of learned atoms
Figure 8.5: Classification Results for Various Parameter m Settings (A & Q)). The influence
of dictionary dimensionality on the K - S V D classification performance for the
synthetic dataset. A boxplot for each s represents the median (red horizontal line), interquartile
range (IQR) (edged of the box), the most extreme values not exceeding 1.5 IQR
(whiskers) and outliers (+) of overall accuracies of classifications for all settings of other
two parameters.
Chapter 8. Results 63
Nevertheless, since K - S V D is an optimization problem, disregarding some atoms adhoc
is not advisable. Moreover, according to (Wright et al., 2009), when sparsity is properly
harnessed in a classification scheme, feature selection criterion is no longer critical.
Therefore, we will not attempt to make such a selection by any means and rather rely on
the implicit output we have in hands.
In Figure 8.6, relations of O A and various settings of m for the real datasets are delineated.
Unfortunately, in the case of the real datasets, computation time is very limiting and
therefore the results are sparsely distributed, i.e. not all the possibilities of m were tested.
Favourably, the results again resemble those from isPCA (Figure 8.1 and Figure 8.2) whereas
as atoms are added to dictionary the results worsen for G M Densities, in the case
of Volume Changes the most promising results are achieved with m set to largest possible
values.
OA vs Number of atoms learned (GM Densities) OA vs Number of atoms learned (Volume Changes)
Number of atoms learned
20 40 60 80 100 120
Number of atoms learned
Figure 8.6: Classification Results for Various Parameter m Settings (Real Datasets). The
graphs show relation between O A and the number of K-SVD-learned atoms (ordered
implicitly by KSVD-Box) for the G M Densities dataset (left) and the Volume Changes
dataset (right) with s — 5 and i — 20.
After all the K - S V D parameters were examined, we get to the point of commenting on
the tweaks. As defined in 7.3, the number of atoms in a dictionary ought to be larger than
the length of descriptors of data to be decomposed. According to Guha and Ward (2012),
the overcompleteness factor should be 2 or 4. Nevertheless, as we already touched when
tuning the parameters in the text above, our datasets do not allow us to set m to be so large.
Thus, the only way to increase the ratio of m over the features per subject is to reduce them
in number.
Such reduction can be done by pre-selecting some of the features either manually (e.g.
ROI tracking, see 4.2) or by the means of statistical or machine learning methods. Since
this thesis deals with multivariate machine learning algorithms, and especially with sparse
representations, we employed random projections with the use of Achlioptas matrix (7.3).
Table 8.4 summarizes the results with various settings of the parameter p.
Chapter 8. Results 64
Length of Descriptors Ratio of Features Retained OA [%]
10 0.0002 50
20 0.0004 45
50 0.0010 55
100 0.0020 60
250 0.0050 80
600 0.0120 80
1,000 0.0199 90
2,000 0.0400 80
5,000 0.0996 90
20,000 0.3985 90
50,184 1.0000 90
Table 8.4: Classification Results for Various Parameter p Settings (A & Q). The table
summarizes the influence of random projections (by Achlioptas matrix) on K - S V D
(m = 19, s — 5 and i — 20) classification performance. In general, reducing the length of
descriptors gradually deteriorates overall accuracy (OA). However, when the ratio of features
retained over original features is above ^ the results are nearly as good as without
employing R R
At first it is noteworthy that the multiplication of Achlioptas matrix and data matrix
with no reduction at all does not change classification performance. Another promising
conclusion is that while retaining more than ^ of the original features, the outcome is
still satisfactory. In Bingham and Mannila (2001) where the authors explored RP in detail
they were able to reduce the length of descriptors 250 x without losing underlying information.
Unfortunately, in our case although the results are promising, it does not allow
us to reduce the length enough to turn the overcompleteness ratio towards the demanded
values completely.
Nevertheless, the reason to employ RP is not only to reduce the number of rows of K S
V D dictionary but also to diminish computational time which was successfully achieved.
Thus, the rest of the experiments (PBM and K S V D - C D ) will undergo dimensionality reductions
by RP with Achlioplast matrix and the number of retained features p set as
of the original ones. The rationale why we chose the value to be lower than the one in our
findings is to ensure ourselves not to deteriorate the results.
Pattern-based morphometry
For the P B M approach, it is essential to select the number of neighbours k for generating
difference images. For our classification, we will adopt the setting according to
Gaonkar et al. (2011), since he successfully dealt with a problem similar to ours. Thus,
we set k — 3.
Chapter 8. Results. 65
As far as P B M utilizes K - S V D as its part, we must find its appropriate parameters
values as well. Since k — 3, also the matrix of training data to be decomposed consists of
3x more subjects. Thus we conducted an experiment with different values of m. For the
synthetic dataset, the resulting O A were higher with higher values of m4
, following the
trend found with the original K - S V D and also with isPCA.
Likewise, the results from the real datasets suggest the trends discovered by experiments
above are applicable to P B M . Whereas the classification performance worsens with
increasing m for G M Densities, it improves in the case of Volume Changes, although the
curves themselves slightly differ. The trends are depicted in Figure 8.7, suggesting the
best settings are the lowest possible m for the former dataset and the largest m for the latter
one.
The rest of the K - S V D parameters, the number of iterations i and the sparsity constraint
s, and the length of descriptors p required for RP were set as discussed above.
PBM: OA vs Number of atoms learned (GM Densities) PBM: OA vs Number of atoms learned (Volume Changes)
150 200 25C
Number of atoms learned
150 200 250
Number of atoms learned
Figure 8.7: PBM Classification Results for Various Parameter m Settings (Real Datasets).
The graphs show relation between O A and the number of K-SVD-learned atoms
(ordered implicitly by KSVD-Box) for P B M classification of G M Densities (left)
and Volume Changes (right) with s — 5 and i — 20 for the K - S V D part and p —
Y^Q of original descriptors length for the R P part.
Concatenated Dictionaries
As long as K S V D - C D basically consists of two K - S V D applied separately on each
class, we will again set i — 20, s — 5 and p (for RP) as discussed above. The value
of s is intentionally chosen to be odd, since when the Coeff classification rule (7.3.2) is
employed, the number of non-zero of values must be different for FES and HC. However,
since s represents an upper threshold, a pursuit algorithm might find less than s non-zero
4
The highest O A , 90%, was achieved with m = 57 which is the highest possible value considering 3
difference images per each of 19 training subjects.
Chapter 8. Results 66
coefficients, possibly leading to the situation when the numbers of non-zero elements are
the same for both classes. For such situations, we created additional rule, taking into
account the values of the non-zero elements as well, i.e. the higher the value, the more
important the element is for the classification.
What is left to be set is again the parameter m. Surprisingly, the best results were
achieved with the highest possible values of m for both real datasets when S V M was
applied. Whereas the trend in the Volume Changes dataset stayed untouched, for G M
Densities no visible trend was found. Nevertheless, since we conducted the experiment
with only several parameter settings, we are not capable of drawing definitive conclusions.
Unfortunately, our experiments with Coeff proved to be rather poor. On one hand, it
agrees with the findings by Guha and Ward (2012) where the authors claimed that S V M
performance exceeds the one of Coeff. On the other hand, since the O A were around 50%,
it rises questions whether the classification rule is inappropriate for our data or whether
we did not utilize it to its full potential.
The results for both datasets and both classification rules are summarized in Table 8.5.
OA [%]
Number of Atoms GM Densities Volume Changes
S V M Coeff S V M Coeff
1 65.38 50.96 64.42 47.12
2 64.42 44.23 66.35 51.92
5 66.35 50.96 67.31 48.08
40 65.38 44.23 63.46 52.88
51 67.31 48.08 70.19 50.96
Table 8.5: KSVD-CD Classification Results for Various Parameter m Settings With Different
Classification Rules (Real Datasets). The table overviews classification performance
for both classification rules employed on both real datasets with various settings of the
parameter m. The results suggest that S V M greatly outperforms Coeff in both cases. Nevertheless,
this does not stem from the fact that S V M yields such high overall accuracies
(OA), but from the fact that the performance of the Coeff algorithm is very low.
8.2 Final Results
In the previous chapter (7), classification schemes this thesis employed for the purposes
of schizophrenia classification were introduced along with parameters each method
requires to be set before the classification starts. Subsequently, the parameters of our algorithms
were tuned (8.1). Table 8.6 summarizes their final settings.
Chapter 8. Results 67
Algorithm Token
Value
Algorithm Token
GM Densities Volume Changes
M W t 0.01 0.05
isPCA c l ; 11 102;84
K - S V D s 5 5
i 20 20
m 1 103
P B M s 5 5
i 20 20
m 1 309
k 3 3
K S V D - C D s 5 5
i 20 20
m 51; 5 51; 40
RP P 5,819 5,819
Table 8.6: Final Settings of Parameters. The table summarizes values of the parameters
(second column) of each algorithm (first column) set separately for G M Densities (third
column) and for Volume Changes (fourth column). In the case of isPCA, we provide two
different options for c. The first one allows us to mimic the situation when either only one
or all components were chosen for the classification as was the case with both K - S V D and
P B M where such settings were the best options. The second one is a setting yielding the
best classification results for isPCA. In the case of K S V D - C D , the first value yields the
best results when S V M is employed whereas the second value is the best option for the
Coeff algorithm.
The following text will provide an overview of classification results for the algorithms,
separately for G M Densities and Volume Changes, allowing us to, later on in 9, discuss
the topic of which of the feature extraction and selection algorithms are suitable
for schizophrenia classification, what are their upsides and downsides and how they stand
compared to each other and compared to other studies.
8.2.1 GM Densities
Mostly, the classification results were already mentioned in 8.1, when the parameters
were tuned. Nevertheless, here we enrich the aforementioned O A with the percentages
of sensitivities and specificities, providing us with more information about our algorithms
performance. In Table 8.7 we provide an overview of the results for the G M Densities
dataset with the final parameters settings for each algorithm.
Chapter 8. Results 68
Algorithm OA [%] SENS [%] SPEC [%]
M W 67.31 63.46 71.15
isPCA(l) 67.31 67.31 67.31
i s P C A ( l l ) 68.27 63.46 73.08
K - S V D 67.31 67.31 67.31
K - S V D - R P 65.38 63.46 67.31
P B M - R P 64.42 63.46 65.38
K S V D - C D - S V M 66.35 61.54 71.15
K S V D - C D - S V M - R P 67.31 63.46 71.15
KSVD-CD-Coeff-RP 50.96 50.00 51.92
Table 8.7: Final Classification Results (GM Densities). The comparison of final results
for the G M Densities dataset, characterized by O A (overall accuracy), SENS (sensitivity)
and SPEC (specificity) for each algorithm.
8.2.2 Volume Changes
At last, the final results for the Volume Changes dataset are overviewed in Table 8.8.
Algorithm OA [%] SENS [%] SPEC [%]
M W 66.35 65.38 67.3
isPCA(102) 65.38 63.46 67.31
isPCA(84) 69.23 71.15 67.31
K - S V D 66.35 63.46 69.23
K - S V D - R P 69.23 69.23 69.23
P B M - R P 70.19 69.23 71.15
K S V D - C D - S V M 69.23 65.38 67.31
K S V D - C D - S V M - R P 70.19 69.23 71.15
KSVD-CD-Coeff-RP 52.88 67.31 38.46
Table 8.8: Final Classification Results (Volume Changes). The comparison of final results
for the Volume Changes dataset, characterized by O A (overall accuracy), SENS (sensitivity)
and SPEC (specificity) for each algorithm.
Chapter 9
Discussion
Having all the experiments performed and analysed (8.1) and the results presented (8.2),
we proceed to evaluation of the feature extraction and selection methods in a broader
context. At first, we will concentrate on the algorithms themselves along with a discussion
about the difference between data modalities we utilized. Subsequently, we will focus on
their applicability into clinical practice and compare the results with other studies dealing
with the same problem domain.
The main distinction we would like to stress is the difference in results for G M Densities
and Volume Changes. Considering the datasets are two modalities of the same data,
we are able to evaluate the difference between the V B M and D B M approaches.
In the case of the G M Densities dataset, isPCA, with K - S V D , K S V D - C D - S V M - R P
and M W coming next, slightly outperformed other methods although the actual difference
in the performance in comparison with the other algorithms was subtle. The most noteworthy
piece of information stems from the parameters settings indicating the number of
features needed for the classification. At most cases, the best classification results were
achieved with a minimum of features retained.
On the contrary, the Volume Changes dataset yielded the best results when the number
of features was set at its highest values1
. Moreover, multivariate approaches outperformed
univariate M W serving for mere selection. Such a behaviour indicates that Volume
Changes conceals more sophisticated patterns than it can be discovered disregarding
voxel-to-voxel interactions.
Consequently, our results confirm that whereas V B M serves mainly for extracting information
about changes on a local scale, D B M preserves information from a wider region.
Nevertheless, neither of the techniques is capable of harnessing patterns distributed over
'Note that also the p-value threshold for M W selection is higher leading to more features to be selected,
although as a univariate statistics it is hardly comparable to the other methods as such.
-69-
Chapter 9. Discussion 70
whole brain entirely and thus other modalities, either morphometric or neuroimaging ones,
might improve performance of subsequent machine learning algorithms.
Findings from Janousova et al. (2015), where isPCA components are evaluated in more
detail, claim that the most discriminative component calculated from intensities of G M by
itself captures roughly 4.5 x more variance of the original data than the one calculated
from covariance matrix corresponding to deformations. Our datasets are in correspondence
with such a behaviour although the ratio of variance captured by the most discriminative
component is different. In our case, when comparing components with the most
variance explained the ratio was approximately 2.5 whereas when regarding the most discriminative
components the ratio rose up to 12.5. Also, the classification performance
of only one component taken is much greater in the case of G M Densities where such a
setting produces almost the best results.
Related to patterns hidden in the data, they can be revealed by visualizing a set of voxels
selected for classification. We employed such a visualization for the synthetic dataset
as follows. Sets of selected voxels were cumulated across all iterations of the L O O crossvalidation
loop. The cumulated patterns (see Appendix E) suggest that M W dismantles
subjects to incoherent regions where the underpinning patterns are still visible but not
captured entirely. For the multivariate algorithms, we displayed only the most discriminative
pattern. Whereas isPCA captures voxels corresponding to almost all subjects, i.e.
it regards all subjects as equals, K - S V D puts stress on one subject mainly although the
other subjects are taken into account as well. In the case of P B M , the cumulated patterns
clearly depict several difference images superimposed on top of each other. Unfortunately,
to evaluate the patterns found for the real datasets is above the scope of this thesis and it is
a subject for further research.
Next, commenting on K - S V D as a new approach for M R I data classification, the
biggest unanswered question is related to overcompleteness factor. Since in our implementation
we did not succeed in turning it above 1 as it would be demanded, its full
potential has not been probably reached. The limiting factor is the number of subjects
included in the analysis. Unfortunately, this problem is typically met when dealing with
M R I data and as such it is hard to tackle.
One way to overcome the problem was introduced by P B M which enlarges a dataset
by computing difference images. However, since the length of descriptors is a great deal
higher than the number of subjects enlarging a dataset is not sufficient. Therefore, another
way to tackle the problem are RP which reduce the length of descriptors in number.
Indeed, the best OA, above 70%, was reached when both P B M and RP were applied at
the same time. Another possibility to improve the overcompleteness ratio is to pre-select
features beforehand, reducing the length of descriptors even more.
Chapter 9. Discussion 71
In general, RP also reduce computation time. For instance, each iteration of P B M
without RP would take almost 15 x longer2
than when RP were employed. Moreover, if
not overdone RP do not deteriorate classification results. Interestingly, when RP was used
with the K - S V D algorithm, it improved classification performance for Volume Changes
even though for G M Densities the resulting O A were lower. The reasoning might be
found in the dimensions of a feature space that ensuing S V M was based on. Unlike in
the case of G M Densities when only one atom was utilized for the S V M classification,
the feature space dimensionality for Volume Changes was full. Consequently, RP yielded
more dense representation of the data in the feature space, facilitating the classification.
Finally, we will comment on a possibility of extracting class-specific features offered
by the last implemented algorithm of the thesis, K S V D - C D . With S V M as its classifier
it achieved O A over 70% equalizing in performance P B M with RP. However, another
classifier taking into account the number of non-zero sparse coefficients for each class
performed poorly. Be it as it may, the approach provides the opportunity to incorporate
schizophrenia subtypes into its computer-aided classification. Thus, the capabilities of the
approach should be further examined, for instance by, as suggested in Guha and Ward
(2012), comparing histograms of the coefficients instead of their 1° norms.
Now we get to the point, where suitability of the algorithms not only for schizophrenia
research but also for its C A D potentially used in practice will be discussed. The best
result achieved was O A = 70.19% with S E N S = 69.23% and S P E C = 71.15% leading to a
sad conclusion that the performance of the algorithms is insufficient for clinical practice.
Moreover, its generalization ability is debatable since it has not been examined on an
independent dataset although L O O cross-validation was employed.
According to Kohavi (1995), L O O can lead to biased estimations of classification performance.
Therefore it is advisable to rather use L P O instead, where both classes consist
of the same number of subjects during all the classification process. Nevertheless, when
we compared L P O results with L O O results for our datasets, the results were almost identical.
Therefore, we gave priority to computationally less demanding L O O in order to gain
more space for parameters tuning and additional experiments.
On the other hand, even though the results could be better, they are adequately good in
comparison with other studies considering what a dataset we employed the classification
algorithms on. As stated in 2.3, schizophrenia aggravates as the disease progresses. Thus,
since we are dealing with FES, the progression of the disease is not yet manifested in the
patients brains enough to discover the abnormalities perfectly. For that reason, we will
mention only studies working the patients with the first episode of schizophrenia.
For instance, Zanetti et al. (2013) reached O A = 73.4% (SENS= 79.0% and SPEC=
67.7%) with the application of S V M using the L O O cross-validation scheme on a dataset
2
O n a 2.5 G H z two-processor unit with 64 G B of memory it took around 240 seconds to compute one
iteration of P B M and around 3,600 seconds to compute it with the use of RP, which is the reason why the
former approach was not carried out entirely.
Chapter 9. Discussion 72
containing 62 FES and 62 matched HC. Furthermore, Mourao-Miranda et al. (2012) reported
O A = 67% (SENS= 71% and S P E C = 61%) on first-episode patients with a continuous
progression of the disease versus H C reached by the S V M algorithm and O A = 54%
(SENS= 64% and S P E C = 43%) for those patients who had only an episodic form of
schizophrenia.
In Kašpárek et al. (2011), another classification rule, L D A , was used to distinguish 39
FES from 39 H C with O A = 72.0% (SENS= 66.7% and S P E C = 76.9%). Regarding other
neuroimaging modalities, Yoon et al. (2012) exploited data from fMRI acquisition of FES
brains with O A = 58.8% (SENS= 62.7% and S P E C = 54.9%).
In the end, those values suggest that the results of the thesis are comparable to state-ofthe-art
studies. Moreover, our algorithms usually yield more stable performance in terms
of SENS and SPEC whose values do not vary to the extent they do in the aforementioned
studies.
Since the classification results are typically much better for chronic patients (see 3.1.2)
than for FES, whereas the difference between the algorithms themselves is not so immense,
it seems that proper data acquisition and pre-processing part along with a way of
capturing the underlying information, i.e. neuroimaging and brain morphometry modalities,
should be of the greatest importance.
Indeed, recent studies attempted to combine two or more brain morphometry modalities
resulting in better performance of schizophrenia classification. More specifically,
Janoušová et al. (2015) ensembled results from three modalities ( G M intensities, deformations
and whole-brain intensities) leading to O A = 81.6% (SENS= 75.5% and SPEC=
87.8%) while classifying FES from H C . On top of that, Schwarz and Kašpárek (2014)
who combined G M intensities with deformations to classify FES and HC, achieved O A =
83.7% (SENS= 84.6% and S P E C = 82.7%) with the use of S V M and L O O and O A =
87.5% (SENS= 88.5% and S P E C = 86.5%) when the 11-NN algorithm was employed.
Therefore, we conclude the discussion with a positive frame of mind believing that the
findings can lead, in the future, to successful C A D of schizophrenia, relieving the patients
from the burden they are suffering from nowadays and alleviating load on physicians,
dealing with those patients, yielding more space to tackle other health problems of the
world of today.
Chapter 10
Conclusion
In this thesis, several classification schemes were proposed for diagnosing patients with
the first episode of schizophrenia. The theoretical part briefly introduced the disease,
mainly through the optics of neuroimaging as a mean to objectively assess brain morphology
and the abnormalities appearing with the offset of schizophrenia. Subsequently, a
methodology how to process magnetic resonance images in order to create a computeraided
classification tool for schizophrenia was reviewed, focusing on brain morphometry
techniques and machine learning methods.
In the practical part, the methodology was applied to distinguish between first-episode
patients and healthy controls on the basis of magnetic resonance images of their brains.
The proposed classification schemes varied in the feature extraction and selection steps.
Namely, Man-Whitney testing was implemented as a simple univariate approach playing
the role of a comparator to multivariate methods such as inter-subject principal component
analysis, K - S V D along with an alternative method creating class-specific dictionaries and
pattern based morphometry. At first, each method was thoroughly examined in order to
explore its parameters and their influence on the classification. Subsequently, the methods
were evaluated in terms of classification performance and the results were discussed.
The source code of the algorithms implemented in M A T L A B is attached on CD. A brief
description of the scripts can be found in Appendix F.
In order to evaluate the performance, the algorithms were applied on two datasets,
T l densities of local gray matter volumes and whole-brain local volume deformations.
Since those are different modalities gained from the same subjects, the algorithms were
also compared across the datasets, evaluating the distinction between the voxel-based and
deformation-based morphometry approaches.
The results obtained from both the datasets almost reached on the state-of-the-art level
when compared to studies concentrating on first-episode schizophrenia classification performed
on one magnetic resonance modality. Moreover, they outperformed the other stud-
-73-
Chapter 10. Conclusion 74
ies in terms of difference between sensitivity and specificity values. Nevertheless, they do
not reach the level needed to be established as a tool for computer-aided diagnosis in clinical
practice. Thus, information from various neuroimaging modalities as well as different
morphometry and machine learning techniques should be exploited combining them in a
robust approach capable of overcoming pitfalls that are left untackled when one utilizes,
although correctly, solely one modality curtailing the information captured in data.
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Appendices
A Appendix 1
100 120 140 160 18 100 120 140 160 180 200
Figure A.l: Example of Images From the Synthetic Dataset (A & Q).
-90-
Appendices 91
B Appendix 2
Value oit OA [%] Percentage of Selected Voxels (Percentage)
0.01 60.00 110(0.22)
0.05 85.00 1,000(1.99)
0.1 90.00 1,900(3.79)
0.2 100.00 4,800 (9.56)
0.3 95.00 6,900(13.75)
Table B.l: Classification Results for Various Parameter t Settings (A & Q)). The table
summarizes overall accuracies (OA) and numbers of selected voxels (with the percentage
out of 50,184) for M W classification with a threshold t performed on the synthetic dataset
of triangles and circles.
Figure B.l: Histograms of p-values for A & O an
d GM Densities. Whereas in the case
of G M Densities (right), p-values lay most frequently between 0 and 0.1, they are usually
much higher in the case of the synthetic dataset (left).
Appendices 92
C Appendix 3
OA vs Number of components (ordered by EA) OA vs Number of components (ordered by MW p-values)
0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18
N u m b e r of retained c o m p o n e n t s N u m b e r of retained c o m p o n e n t s
Figure C.l: Comparison of Classification Results for Various Parameter c Settings With
Different Ordering of Components (A & Q). The graphs show relation between O A and
the number of isPCA-learned components retained for classification. The components are
sorted in ascending order according to the amount of variance they explain in original
data (left) and according to their discriminative power measured here by Mann-Whitney
p-values (right). (EA = explained variance).
Appendices 93
D Appendix 4
OA vs Number of atoms learned OA vs Number of atoms learned (ordered by MW p-values)
Figure D.l: Comparison of Classification Results for Various Parameter m Settings With
Different Ordering of Atoms (A & Q). The graphs show relation between O A and the
number of K-SVD-learned atoms retained for classification. The atoms are sorted as output
from K S V D - B o x (left) and according to their discriminative power measured here by
Mann-Whitney p-values (right).
Appendices 94
E Appendix 5
MW: Set of selected voxels (cumulated across CV) isPCA; The most discriminative pattern (cumulated across CV}
20 40 60 SO 100 120 140 160 190 200 20 40 60 SO 100 120 140 160 130 200
K-SVD: The most discriminative pattern (cumulated across CV} PBM: The most discriminative pattern (cumulated across Cv)
20 40 60 SO 100 120 140 160 180 200 20 40 80 8 0 100 120 140 160 180 200
Figure E.l: Comparison of Patterns Discovered By Various Feature Extraction And Selection
Methods (A & Q). The patterns are learned in each iteration of the L O O crossvalidation
loop and cumulated as it progresses further on across all the loop. The cumulated
patterns suggest that M W (top-left) dismantles subjects to incoherent regions where
the underpinning patterns are still visible but not captured entirely. For the multivariate algorithms,
only the most discriminative patterns are displayed. Whereas isPCA (top-right)
captures voxels corresponding to almost all subjects, i.e. it regards all subjects as equals,
K - S V D (bottom-left) puts stress on one subject mainly although the other subjects are
taken into account as well. In the case of P B M (bottom-right), the cumulated patterns
clearly depict several difference images superimposed on top of each other.
Appendices 95
F Appendix 6
The following lists describe the structure of files of the scripts implementing the classification
algorithms proposed in this thesis. The scripts themselves are implemented in
M A T L A B and, along with a dataset of triangles and circles and prerequisite toolboxes and
functions, they can be found on attached CD.
A l l files are divided into the following folders:
• Data - contains only a toy example represented by the synthetic dataset, since the
real datasets can not be published,
• A u x i l i a r y - functions and toolboxes needed for scripts to work,
• Algorithms - implementation of the algorithms proposed in the thesis.
Folder Data:
• xxx.mat:
- 'mask' - contains information about brain mask,
- 'data' - data themselves in an object as:
* .labels - 1 FES, 0 HC,
* .images - original . n i i images (121x145x121),
* .brains - vectorized images after masking (581282 voxels);
• synthetic.mat:
- a toy example of data with no masking included,
- results are different from those from the real datasets.
Folder A u x i l i a r y :
• ksvdboxl3 - K S V D - B o x by Ron Rubinstein to perform K - S V D ,
• ompboxlO - OMP-Box by Ron Rubinstein to compute the O M P pursuit algorithm,
• generate_diff .images .m - function generating difference images given two sets
of images and the number of nearest neighbours to find are input (for PBM),
• normcols .m, m u l t c o l s .m - those functions must be explicitly in M A T L A B path
in order to run O M P manually (for K S V D - C D ) .
Appendices 96
Folder Algorithms:
• MW_SVM_L00.m,
• isPCA_SVM_L00.m,
• KSVD_SVM_L00.m,
• KSVD_SVM_RP_L00.m,
• PBM_SVM_RP_L00.m,
• KSVDCD_SVM_L00.m,
• KSVDCD_SVM_RP_LOO.m,
• KSVDCD_Coeff_RP_LOO.m;
• legend:
- MW = Man-Whitney testing,
- isPCA = inter-subject principal component analysis,
- KSVD = the K - S V D algorithm,
- PBM = pattern based morphometry,
- KSVDCD = K - S V D with concatenated dictionaries,
- RP = random projections,
- SVM = support vector machine,
- Coef f = classification rule based on,
- LOO = leave-one-out cross-validation.