Title Page

ABSTRACT

Contents

Chapter 1. Introduction 20

Chapter 2. Preliminaries 27

2.1. Principal curves 27

2.2. Riemannian manifolds and centrality on manifold 29

2.3. Principal curves on Riemannian manifolds 33

Chapter 3. Spherical principal curves 34

3.1. Enhancement of principal circle for initialization 35

3.1.1. Principal geodesic and principal circle 35

3.1.2. Exact principal circle 37

3.1.3. Extension to hyperspheres 41

3.2. Proposed principal curves 44

3.2.1. Exact projection step on SD[이미지참조] 44

3.2.2. Expectation step on SD[이미지참조] 48

3.2.3. Algorithm 49

3.2.4. Stationarity of principal curves 50

3.3. Numerical experiments 53

3.3.1. Real data analysis 53

3.3.2. Simulation study 58

3.4. Proofs 64

3.4.1. Justification of the projection steps on SD[이미지참조] 64

3.4.2. Stationarity of principal curves 65

3.5. Concluding remarks 81

Chapter 4. Robust spherical principal curves 83

4.1. The proposed robust principal curves 83

4.1.1. Exact projection step on SD[이미지참조] 84

4.1.2. Median step on SD[이미지참조] 85

4.1.3. L₁-type principal curves 86

4.1.4. Huber-type principal curves 88

4.1.5. Roles and effects of parameters T, q, and c on fitted curves 90

4.2. Stationarity of robust spherical principal curves 91

4.3. Numerical experiments 93

4.3.1. Simulation study on S² 93

4.3.2. Simulation study on S⁴ 97

4.3.3. Real data analysis: motion capture data 98

4.4. Summary and future work 99

Chapter 5. spherepc: An R package for dimension reduction on a sphere 103

5.1. Existing methods 104

5.1.1. Principal geodesic analysis 104

5.1.2. Principal circle 107

5.2. Spherical principal curves 110

5.2.1. Options for spherical principal curves 112

5.3. Local principal geodesics 113

5.4. Application 118

5.5. Conclusions 120

Chapter 6. Local principal curves on Riemannian manifolds 131

6.1. Preliminaries 135

6.2. Local principal geodesics 137

6.2.1. Bias relaxation 139

6.2.2. Overlap of curves and merging 142

6.2.3. Consideration of parameters 142

6.2.4. Connection with existing methods 143

6.2.5. Results of local principal geodesics 143

6.3. Local principal curves 144

6.4. Real data analysis 152

6.5. Further work 152

Chapter 7. Conclusion 158

Appendix A 160

A.1. Appendix for Chapter 3 160

A.2. Appendix for Chapter 4 164

A.3. Appendix for Chapter 6 171

Bibliography 187

국문초록 196

Table 3.1. The values of RE and # proj by the proposed methods and Hauberg's method on the earthquake data 56

Table 3.2. The values of RE and # proj by the proposed methods and Hauberg's method on the motion capture data 58

Table 3.3. Averages of reconstruction errors and their standard deviations in the parentheses by each method 60

Table 3.4. Averages of distinct projection points and their standard deviations in the parentheses 62

Table 3.5. A simulation result of waveform data on S⁴ 64

Table 4.1. Averages of reconstruction errors and their standard deviations in the parentheses by each method with T＝50 96

Table 4.2. Averages of reconstruction errors and their standard deviations in the parentheses by each method with T＝100 in waveform simulated data on S⁴ 98

Table 4.3. Reconstruction errors by each method with T＝300 in the contaminated motion capture data 99

Table 5.1. Arguments of the PGA() 106

Table 5.2. Outputs of the PGA() 106

Table 5.3. Arguments of the PrincipalCircle() 107

Table 5.4. Arguments of the GenerateCircle 108

Table 5.5. Arguments of the SPC() 122

Table 5.6. Outputs of the SPC() 123

Table 5.7. Arguments of the LPG() 126

Table 5.8. Outputs of the LPG() 127

Figure 1.1. Process of data generating is illustrated. The population curve (ground-truth) f. [0, 1] → M is colored in black. A data point Xi is generated by adding a...[이미지참조] 23

Figure 1.2. Data (blue) are distributed on M＝RD and three procedures are il-lustrated for data descriptions (red) based on minimization of least squares. Top...[이미지참조] 26

Figure 2.1. Left. The Euclidean mean (orange) of three points (blue) is not lying on the unit 2-sphere. Right. The extrinsic and intrinsic means (green) of the three... 30

Figure 3.1. Left. Spherical distribution of significant earthquakes (blue) with its intrinsic mean (green), the result (pink) by PGA, and the result (red) by our pro-... 36

Figure 3.2. Top Left. Simulated data points (blue) with the intrinsic mean (0, 0, 1) (green) and the result of the proposed principal circle (red); Top Right. The pro-... 38

Figure 3.3. Illustration of the projection procedure on S². (a) The case that C is projected inside AB, i.e., projAB(C)＝proj(C) and I ≥ 0. The projection of C...[이미지참조] 45

Figure 3.4. Illustration of the projection procedure on SD[이미지참조] 47

Figure 3.5. Top. Earthquake data is distributed in globally and they are visualized by two-dimensional and three-dimensional view, from left to right. Bottom. The... 54

Figure 3.6. Projection results by the proposed extrinsic method (left) and Hauberg's method (right) with T＝77 and q＝0.1. 55

Figure 3.7. The results of the proposed extrinsic method (red) and Hauberg's method (yellow) with T＝100 are presented. The results with q＝0.03, q＝0.05... 57

Figure 3.8. From top left to bottom right. True waveform and noisy data (blue dots), the extrinsic principal curve, the intrinsic principal curve, and the curve by... 59

Figure 3.9. Noisy waveform simulated data are colored in blue. Top left. Extrinsic-type principal curves with q＝0.01 (green) and 0.02 (pink) for fixed T＝500; Top... 63

Figure 3.10. (Left) The projection process of C onto the one-dimensional great circle V ∩ SD (red) in a hypersphere SD ⊂ RD+1. (i) find the projection of C onto V,...[이미지참조] 66

Figure 4.1. Simulated circular data with outliers (blue) and the resulting curves. (a) true circular curve, (b) extrinsic spherical principal curve, (c) L₁-type principal... 94

Figure 4.2. From left to right and top to bottom, contaminated waveform data (blue) and population curve (red), principal geodesic analysis, principal nested sphere,... 101

Figure 4.3. (a) Motion capture real data and a pseudo-true curve obtained by spher-ical principal curve. (b)-(d) Contaminated motion capture data and fitted results by... 102

Figure 5.1. From left to right, half-great circle and S-shaped data (blue) and the results (red) of principal geodesic analysis (PGA). The principal geodesic detects... 106

Figure 5.2. Half-great circle data and circular data (blue) and the results (red) of the principal circle from left to right. The principal circle can identify the relatively... 110

Figure 5.3. Top. The waveform data (blue) and the results (red) of Hauberg's prin-cipal curves (left) and spherical principal curves. Bottom. The noisy waveform data... 121

Figure 5.4. Left. Projection result (black) of SPC with q＝0.1. The spherical principal curve (red) continuously represents the earthquake data (blue). Right.... 123

Figure 5.5. Left. Spiral data (blue) and the result (red) of LPG with scale＝0.06 and ν＝0.1. Right. Noisy spiral data (blue) and the result (red) of LPG with... 124

Figure 5.6. Left. zigzag data (blue); Middle. zigzag data (blue) and the result (red) of with scale＝0.1 and ν＝0.1; Right. Noisy zigzag data (blue) and the result (red)... 124

Figure 5.7. Tree data (blue) and the result (red) of LPG with scale＝0.03 and ν＝0.2. The LPG function captures the complex structures of the data well, provided... 125

Figure 5.8. Left. The distribution of significant (8+ Mb magnitude) earthquakes (colored in blue); Right. The earthquake is represented in three-dimensional visualization. 125

Figure 5.9. Earthquake data (blue) and the results (red) of the principal geodesic analysis and principal circle, from left to right. The principal geodesic fails to find... 127

Figure 5.10. Earthquake data (blue) and implementation results (red) with q＝0.1 of the SPC.Hauberg and SPC functions respectively, from left to right. Both meth-... 128

Figure 5.11. From left to right and top to bottom, Earthquake data (blue) and the results (red) of the SPC with q＝0.15, 0.1, 0.03 and 0.02. The larger the parameter... 129

Figure 5.12. From left to right, earthquake data (blue) and the results of the LPG function with scale＝0.5, ν＝0.2 and scale＝0.4, ν＝0.3 are illustrated. Both the... 130

Figure 6.1. Noisy spiral data (blue) and the consequences (red) of principal circle and principal curves (Hauberg, 2016) initialized by the principal circle for q＝0.07,... 134

Figure 6.2. Left. LPG starts at a point c0 and Σ h (c0) is calculated in the h-neighborhood (gray shade). It forward to the direction v'0 that is the resultant...[이미지참조] 141

Figure 6.3. From top to bottom and left to right, simulated zigzag, spiral, T-shaped, X-shaped, doubly circular data are colored in blue. The LPG consequences of zigzag... 154

Figure 6.4. Left. The borders (red) of plates near the East Sea (source. U.S. geological Survey); Right. The distribution of earthquakes (blue) near the East Sea 155

Figure 6.5. From top to bottom and left to right, the consequences of PGA, principal circle, principal curves of Hauberg, and SPC with q＝0.05, are colored in red. 156

Figure 6.6. From left to right, the consequences of LPG with h＝0.3, ν＝0.05 and with h＝0.11, ν＝0.05 are colored in red. 157

Figure A.1. In the proof of Lemma 7, the configuration of M, TpM, V0 (blue dots) and V (black dots) is illustrated.[이미지참조] 177