Wavelets in Functional Data Analysis

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1 Introduction: Examples of Functional Data .............................. 1
1.1 Experimental Data in Tumor Physiology .............................. 3
1.2 Orthosis Data Analysis ................................................. 5
1.3 Functional MRI Brain Data............................................. 6
1.4 Metabolite Progesterone Curves ....................................... 8
1.5 NMR Spectra and SAA (Sulfur Amino Acids) Deficiency ........... 8
2 Wavelets ....................................................................... 11
2.1 Introduction to Wavelets ................................................ 11
2.2 How Do the Wavelets Work? ........................................... 14
2.2.1 The Haar Wavelet ............................................... 14
2.2.2 General Case .................................................... 15
2.2.3 Periodic Wavelets ............................................... 18
2.2.4 Wavelets in the Language of Signal Processing ............... 19
2.3 2-D and n-D Wavelet Decompositions ................................. 22
2.4 Further Topics ........................................................... 27
2.4.1 Construction of Wavelet Matrix................................ 27
2.4.2 Daubechies–Lagarias Algorithm ............................... 28
2.4.3 Pollen-Type Parameterizations of Wavelet Bases ............. 29
2.4.4 Strang–Fix Condition ........................................... 31
2.4.5 Covariance Wavelet Transform ................................ 31
2.4.6 Wavelet Derivatives............................................. 34
3 Wavelet Shrinkage ........................................................... 37
3.1 Introduction ............................................................. 37
3.2 Choice of Thresholds ................................................... 39
3.3 Shrinkage in Several Dimensions ...................................... 45
4 Wavelet-Based Andrews’ Plots ............................................. 51
4.1 Introduction ............................................................. 51
4.2 Andrews’ Plots .......................................................... 52
4.3 Data Sets................................................................. 54
4.4 Variations of Andrews’ Plots ........................................... 58
4.5 Critique of Andrews’ Plots ............................................. 60
4.6 Wavelet-Based Andrews’ Plots ......................................... 61
4.6.1 Properties........................................................ 62
4.6.2 Examples Revisited ............................................. 65
4.7 Discussion ............................................................... 68
5 Functional ANOVA .......................................................... 71
5.1 Introduction ............................................................. 71
5.2 The Functional Analysis of Variance Models.......................... 75
5.3 The Mixed FANOVA Models .......................................... 80
5.4 An Illustration with the Orthosis Data ................................. 82
5.5 An Application in Cloud/Temperature Mapping ...................... 84
6 Further Topics................................................................ 89
6.1 Introduction ............................................................. 89
6.2 The DWT-CEM Algorithm ............................................. 90
6.3 Functional Regression Models ......................................... 93
6.4 Dimension Reduction ................................................... 95
6.5 Concluding Remarks.................................................... 97

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关键词：Functional Analysis function Wavelets Wavelet

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