Analysis of functional magnetic resonance imaging time series by independent component analysis [Elektronische Ressource] / von Mandy Sohr

Sujets

• Mathematics
• Gesundheit

Informations

• Publié par: otto-von-guericke-universitat_magdeburg
• Publié le: 01 janvier 2007
• Nombre de lectures: 33
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Analysis of Functional Magnetic Resonance Imaging
Time Series by Independent Component Analysis
Dissertation
zur Erlangung des akademischen Grades
doctor rerum naturalium
(Dr. rer. nat)
von Dipl.-Statistikerin MandySohr
geb. am 20.02.1980 in Lutherstadt Wittenberg
genehmigt durch die Fakult¨at fur¨ Mathematik
der Otto-von-Guericke-Universit¨at Magdeburg
Gutachterin: Frau Prof. Dr. Waltraud Kahle
Gutachter: Herr Priv.Doz. Dr. Siegfried Kropf
eingereicht am: 26.02.2007
Verteidigung am: 22.06.2007Contents
List of Figures iv
List of Tables vi
Index of Symbols vii
Index of Abbreviations x
1 Introduction 1
2 Fundamentals 6
2.1 Probability Spaces, Random Variables, and Stochastic Processes . . . . . . . 6
2.2 Independence and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Measures of Independence and Nongaussianity . . . . . . . . . . . . . . . . . 11
2.3.1 Measuring Independence by Information-Theoretic Functions . . . . . 11
2.3.2 Measuring Nongaussianity by Kurtosis . . . . . . . . . . . . . . . . . 18
3 Functional Magnetic Resonance Imaging 20
3.1 Functional Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . . . . 20
3.2 FMRI Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 FMRI Time Series Regarded as Stochastic Processes . . . . . . . . . . . . . 23
3.4 Analyzing fMRI Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Classical Methods for Analyzing fMRI Time Series 26
4.1 General Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Time Series Analyzing Methods . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3.1 Stationary Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
iContents
4.3.2 Autocovariance and Autocorrelation Function . . . . . . . . . . . . . 32
4.3.3 Test for White-Noise Process . . . . . . . . . . . . . . . . . . . . . . 32
4.3.4 Test for Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . 33
4.3.5 Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.6 Histograms and Probability Density Estimation . . . . . . . . . . . . 35
5 Independent Component Analysis 37
5.1 Definition of ICA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.1.1 Identification and Restriction of ICA Algorithms . . . . . . . . . . . . 39
5.1.2 Preprocessing the Data . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Relation between PCA and ICA . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Algorithms for ICA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3.1 Jutten-H´erault Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3.2 Algorithms for Maximum Likelihood Estimation . . . . . . . . . . . . 46
5.3.3 ICA by Minimization of Information . . . . . . . . . . . . . . . . . . 48
5.3.4 The Infomax Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3.5 The FastICA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3.6 Molgedey and Schuster Approach . . . . . . . . . . . . . . . . . . . . 53
5.3.7 Nonparametric ICA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3.8 Further ICA Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.4 Performance of ICA Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.5 ICA Applied to fMRI Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6 Simulation Studies 63
6.1 Modelling the Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.1.1 Modelling the Hemodynamic Response Function . . . . . . . . . . . . 64
6.1.2 Variations in the Hemodynamic Response Function . . . . . . . . . . 66
6.1.3 Further Contributing Signals . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Performing the Programming . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.3 Illustrative Results of ICA Decomposition . . . . . . . . . . . . . . . . . . . 73
6.4 Simulation Studies with Variations in the HRF. . . . . . . . . . . . . . . . . 76
6.5 Over- and Underestimation of the Number of Independent Components . . . 85
6.6 Comparing the results of GLM analysis of mixed signals with and without
included ICA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.7 Illustrative Results of Time Series Decomposition . . . . . . . . . . . . . . . 91
iiContents
7 An Auditory Working Memory fMRI Study and ICA-Results 96
7.1 Material and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.3.1 Behavioral Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.3.2 ICA-Results and Time Series Analysis . . . . . . . . . . . . . . . . . 103
7.4 Comparing ICA Time Courses to HRF Time Courses in Correlation Analysis 109
7.5 Discussing the Shape of BOLD Responses . . . . . . . . . . . . . . . . . . . 111
8 Conclusions 113
Bibliography 114
A Properties of Information-Theoretic Functions I
A.1 Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
A.2 Differential Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
A.3 Negentropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV
A.4 Approximation of Information-Theoretic Functions . . . . . . . . . . . . . . VI
iiiList of Figures
2.1 Independent versus uncorrelated random variables . . . . . . . . . . . . . . . 10
3.1 Time course of hemodynamic response function . . . . . . . . . . . . . . . . 22
6.1 Hemodynamic response model . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2 Time course of the hypothetical hemodynamic response function. . . . . . . 66
6.3 Variation of hemodynamic response function (two alternating signal ampli-
tudes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.4 Variation of hemodynamic response function (signal increase over the session) 68
6.5 Variation of hemo response (signal decrease) . . . . . . . . 68
6.6 Variation of hemodynamic response function (dynamic signal decrease) . . . 69
6.7 Variation of hemo response (signal with increasing noise) . 69
6.8 Variation of hemodynamic response function (temporal shift) . . . . . . . . . 70
6.9 MATLAB graphical user interface for simulation studies . . . . . . . . . . . 72
6.10 Time courses of four source signals and four mixed signals. . . . . . . . . . . 73
6.11 Estimated signals by Bell andSejnowski Infomax algorithm. . . . . . . . 75
¨6.12 Est signals by Hyvarinen FastICA algorithm . . . . . . . . . . . . . 75
6.13 Estimated signals by Maximum Likelihood estimation . . . . . . . . . . . . . 75
6.14 Est signals by nonparametric ICA estimation . . . . . . . . . . . . . . 75
6.15 Estimated signals by Molgedey andSchuster algorithm . . . . . . . . . 75
6.16 Est signals by principal component analysis . . . . . . . . . . . . . . . 75
6.17 Error indices for 1000 simulations . . . . . . . . . . . . . . . . . . . . . . . . 76
6.18 Error Indices of 500 simulations (variation of κ and σ) . . . . . . . . . . . . 78a
6.19 Error of 500 simulations (variation of κ and σ) . . . . . . . . . . . 79a2
6.20 Error Indices of 500 simulations (variation of κ and σ) . . . . . . . . . . . . 79m
6.21 Time courses of mixed signals and estimated independent signals . . . . . . . 80
ivList of Figures
6.22 Error Indices of 500 simulations (variation of dynamic signal decrease within
blocks and noise σ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.23 Error Indices of 500 simulations for HRF with increasing noise and variation
of additional noise σ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.24 Error Indices of 500 simulations (variation of κ ) . . . . . . . . . . . . . . . . 83c
6.25 Error Indices of 500 simulations (variation of number of observations N and
noise σ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.26 Error Indices of 500 simulations (variation of number of phases κ and noisep
σ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.27 Underestimating the number of independent components . . . . . . . . . . . 86
6.28 Overestimating the number of independent components . . . . . . . . . . . . 88
6.29 Time series decomposition of observed signals . . . . . . . . . . . . . . . . . 92
6.30 Autocorrelation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.31 Fast Fourier Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.32 Estimated probability density functions . . . . . . . . . . . . . . . . . . . . . 95
7.1 Plot of a frequency modulated tone . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Experimental paradigm of fMRI experiment. . . . . . . . . . . . . . . . . . . 97
7.3 Targets in experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.4 Brodmann areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.5 Hits and false responses of subjects . . . . . . . . . . .