Distance-basedanalysisofdynamicalsystemsandtimeseriesbyoptimaltransport Uitnodiging 0.2 voorhetbijwonenvande openbareverdediging 0.05 vanmijnproefschrift 0.05 0.2 Distance-basedanalysis 0.1 ofdynamicalsystems andtimeseries byoptimaltransport 0.15 0.2 0.05 opdonderdag 11februari2010 om11.15uur indeSenaatskamervan het Academiegebouw, Rapenburg73teLeiden receptienaafloop 0.2 0.05 0.15 0.05 0.05 0.1 MichaelMuskulus 0.2 0.2 Distance-basedanalysisofdynamical systemsandtimeseries WitteSingel37 2311BJLeiden byoptimaltransport NjordsVeg22 7032 Trondheim(Norway) MichaelMuskulus [email protected] MichaelMuskulus 1.0 Paranimfen: 0.8 SanneHouweling ISBN978-90-5335-254-0 0.6 [email protected] 0.4 +31-644024956 0.2 Truepositiverate B2 Accuracy=0.800 Sönke Ahrens A 0.0 [email protected] B1 9 789053 352540 0.0 0.2 0.4 0.6 0.8 1.0 +49-17670018136 Falsepositiverate Distance-based analysis of dynamical systems and time series by optimal transport PROEFSCHRIFT ter verkrijging van de graad van Doctor aan de Universiteit Leiden, op gezag van Rector Magnificus prof.mr. P.F. van der Heijden, volgens besluit van het College voor Promoties te verdedigen op donderdag 11 Februari klokke 11.15 uur door Michael Muskulus geboren te Sorengo, Switzerland in 1974 Promotiecommissie Promotor: prof. dr. S.M. Verduyn Lunel Overige leden: dr. S.C. Hille prof. dr. J.J. Meulman prof. dr. P.J. Sterk (Academisch Medisch Centrum, Universiteit van Amsterdam) prof. dr. P. Stevenhagen prof. dr. S.J. van Strien (University of Warwick) Distance-based analysis of dynamical systems and time series by optimal transport THOMAS STIELTJES INSTITUTE FOR MATHEMATICS Muskulus, Michael, 1974– Distance-based analysis of dynamical systems and time series by optimal transport AMS 2000 Subj. class. code: 37M25, 37M10, 92C50, 92C55, 62H30 NUR: 919 ISBN: 978-90-5335-254-0 Printed by Ridderprint Offsetdrukkerij B.V., Ridderkerk, The Netherlands Cover: Michael Muskulus This work was partially supported by the Netherlands Organization for Scientific Research (NWO) under grant nr. 635.100.006. Copyright © 2010 by Michael Muskulus, except the following chapters: Chapter 8 J. Neurosci. Meth. 183 (2009), 31–41: Copyright © 2009 by Elsevier B.V. DOI: 10.1016/j.jneumeth.2009.06.035 Adapted and reprinted with permission of Elsevier B.V. No part of this thesis may be reproduced in any form without the express written consent of the copyright holders. After I got my PhD, my mother took great relish in introducing me as, “This is my son. He’s a doctor, but not the kind that helps people”. Randy Pausch Für Frank & Ingrid And to the most beautiful neuroscientist in the world Sanne, thank you for our adventures in the past, in the present, and in the future Contents Prologue xv 1 General Introduction 1 1.1 Distance-based analysis ........................... 1 1.2 Reader’s guide ................................. 5 1.3 Major results & discoveries ......................... 9 2 Dynamical systems and time series 11 2.1 Introduction .................................. 11 2.2 Wasserstein distances ............................. 14 2.3 Implementation ................................ 18 2.3.1 Calculation of Wasserstein distances ................ 18 2.3.2 Bootstrapping and binning ..................... 19 2.3.3 Incomplete distance information .................. 19 2.3.4 Violations of distance properties .................. 20 2.4 Analysis .................................... 21 2.4.1 Distance matrices ........................... 21 2.4.2 Reconstruction by multidimensional scaling ........... 21 2.4.3 Classification and discriminant analysis .............. 25 2.4.4 Cross-validation ........................... 26 2.4.5 Statistical significance by permutation tests ............ 27 2.5 Example: The Hénon system ........................ 28 2.5.1 Sample size and self-distances ................... 28 2.5.2 Influence of noise ........................... 29 2.5.3 Visualizing parameter changes ................... 30 2.5.4 Coupling and synchronization ................... 32 2.5.5 Summary ............................... 35 2.6 Example: Lung diseases ........................... 37 vii Contents 2.6.1 Background .............................. 37 2.6.2 Discrimination by Wasserstein distances ............. 39 2.7 Generalized Wasserstein distances ..................... 43 2.7.1 Translation invariance ........................ 44 2.7.2 Rigid motions ............................. 45 2.7.3 Dilations and similarity transformations ............. 46 2.7.4 Weighted coordinates ........................ 47 2.7.5 Residuals of Wasserstein distances ................. 48 2.7.6 Optimization of generalized cost .................. 49 2.7.7 Example: The Hénon system .................... 50 2.8 Nonmetric multidimensional scaling .................... 50 2.9 Conclusions .................................. 52 Applications 55 3 Lung diseases 57 3.1 Respiration ................................... 57 3.2 The forced oscillation technique ....................... 59 3.3 Asthma and COPD .............................. 63 3.3.1 Materials: FOT time series ...................... 64 3.3.2 Artifact removal ........................... 65 3.4 Fluctuation analysis .............................. 65 3.4.1 Power-law analysis .......................... 66 3.4.2 Detrended fluctuation analysis ................... 68 3.5 Nonlinear analysis .............................. 71 3.5.1 Optimal embedding parameters .................. 72 3.5.2 Entropy ................................ 73 3.6 Results ..................................... 74 3.6.1 Statistical analysis .......................... 74 3.6.2 Variability and fluctuation analysis ................. 78 3.6.3 Distance-based analysis ....................... 80 3.6.4 Nonlinear analysis .......................... 83 3.6.5 Entropy analysis ........................... 84 3.7 Discussion ................................... 84 3.7.1 Main findings ............................. 85 3.7.2 Clinical implications ......................... 87 3.7.3 Further directions ........................... 88 3.7.4 Conclusion .............................. 89 viii Contents 4 Structural brain diseases 91 4.1 Quantitative MRI ............................... 91 4.2 Distributional analysis ............................ 93 4.3 Systemic lupus erythematosus ....................... 95 4.3.1 Materials ................................ 96 4.3.2 Histogram analysis .......................... 97 4.3.3 Multivariate discriminant analysis ................. 99 4.3.4 Fitting stable distributions ......................101 4.3.5 Distance-based analysis .......................103 4.3.6 Discussion ...............................104 4.3.7 Tables: Classification accuracies ..................106 4.4 Alzheimer’s disease ..............................107 4.4.1 Materials ................................109 4.4.2 Results .................................110 5 Deformation morphometry 113 5.1 Overview ....................................113 5.2 Introduction ..................................113 5.3 The Moore-Rayleigh test ...........................115 5.3.1 The one-dimensional case ......................117 5.3.2 The three-dimensional case .....................119 5.3.3 Power estimates ............................121 5.4 The two-sample test ..............................125 5.4.1 Testing for symmetry .........................125 5.4.2 Further issues .............................128 5.5 Simulation results ...............................129 5.6 Application: deformation-based morphometry ..............130 5.6.1 Synthetic data .............................130 5.6.2 Experimental data ..........................133 5.7 Discussion ...................................136 6 Electrophysiology of the brain 139 6.1 Introduction ..................................139 6.2 Distance properties ..............................141 6.2.1 Metric properties ...........................141 6.2.2 Embeddability and MDS .......................143 6.2.3 Graph-theoretic analysis .......................146 6.3 Connectivity measures ............................147 6.3.1 Statistical measures ..........................147 6.3.2 Spectral measures ...........................150 6.3.3 Non-linear measures .........................152 6.3.4 Wasserstein distances ........................153 ix Contents 6.4 Example: MEG data during motor performance .............155 6.5 Example: Auditory stimulus processing ..................159 6.6 Conclusion ...................................160 Epilogue 163 Appendices 167 A Distances 169 A.1 Distance geometry ..............................169 A.1.1 Distance spaces ............................169 A.1.2 Congruence and embeddability ...................172 A.2 Multidimensional scaling ..........................175 A.2.1 Diagnostic measures and distortions ................177 A.2.2 Violations of metric properties and bootstrapping ........181 A.3 Statistical inference ..............................184 A.3.1 Multiple response permutation testing ..............185 A.3.2 Discriminant analysis ........................186 A.3.3 Cross-validation and diagnostic measures in classification . 189 A.3.4 Combining classifiers ........................191 B Optimal transportation distances 193 B.1 The setting ...................................193 B.2 Discrete optimal transportation .......................194 B.3 Optimal transportation distances ......................199 C The dts software package 201 C.1 Implementation and installation ......................201 C.2 Reference ....................................202