Dimension Reduction for Big Data Analysis. Dan Shen. Department of Mathematics & Statistics University of South Florida.
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1 Dimension Reduction for Big Data Analysis Dan Shen Department of Mathematics & Statistics University of South Florida October 24,
2 Outline Multiscale weighted PCA for Image Analysis Human brian artery tree analysis 2
3 Outline Multiscale weighted PCA for Image Analysis Human brian artery tree analysis 3
4 Raw Images Image Analysis Image Reconstruction Image Registration Statistical Analysis Image Segmentation 4
5 Image Analysis Image Reconstruction Fourier Transforms Image Registration Image Segmentation 5
6 Challenges in Image Analysis Large p, small n problem p= = 422,250 x i T =( χ 1,i, χ 2,i,... χ p,i ) X p n = p= = 2,079,152 χ 1,1... χ 1,i... χ 1,n χ 2,1... χ 2,i... χ 2,n... χ p,1... χ p,i... χ p,n 6
7 Principal Component Analysis one data object χ 1,1... χ 1,i... χ 1,n X p n = χ 2,1... χ 2,i... χ 2,n... χ p,1... χ p,i... χ p,n 7
8 Our Main Contribution PCA doesn t work for high dimensional image data, what can we do?? Our multiscale weighted PCA will answer this question 8
9 ADNI Data 9
10 Image Classification Underlying spatial information: features are spatially dependent Dimension reduction becomes important and necessary for image data to improve prediction accuracy and increase classification efficiency 10
11 Limitation of PCA Inconsistency of PCA for large p, small n PCA treats all pixels/voxels equally PCA treats all pixels/voxels independently PCA doesn t consider the association with the outcome 11
12 Motivation Propose Multiscale Weighted PCA (MWPCA) enables a selective treatment of individual features has the ability of utilizing the local spatial information takes into account the association with outcome integrates feature selection, smoothing, feature extraction in a single framework 12
13 13 PCA: Reconstruction Find low dimensional representation of the data through minimizing the reconstruction error k k T k I U U where X is the mean and, ) ~ ( ~ 1 2, 1 1 p j i j j i n i n i i k i a u x a U X X T k u p u U ) ~ ~,..., ( 1
14 PCA: Reconstruction Find low dimensional representation of the data through minimizing the reconstruction error n i 1 X where X is the mean and i X U k a i n i 1 p j 1 ( ~ x i, j U k ( u~,..., u~ 1 p ) u~ T j a i ) 2, U T k U k I k columns of U k are the first k principal component directions 14
15 PCA: Reconstruction Find low dimensional representation of the data through minimizing the reconstruction error n i 1 X where X is the mean and i X U k a i n i 1 p j 1 ( ~ x i, j U k ( u~,..., u~ 1 p ) u~ T j a i ) 2, U T k U k I k columns of U k are the first k principal component directions columns of A ( a 1,..., ) k a n T are principal component scores 15
16 16 Multiscale Weighted PCA Two sets of weights: Global spatial weight: for each pixel/voxel with p j i j j i h j B d j n i a u x h d j w w 1 2, ) ; ( 1 ) ~ )( ~ ;, ( j w p j w j p 1
17 Global Spatial Weight j : measure the association between the j-th pixel and the class information For example: pearson correlation, test statistics, and so on. Define global weight : w j f ( j ) For example: w j p p j 1 j j 17
18 18 Multiscale Weighted PCA Two sets of weights: Global spatial weight: for each pixel/voxel with Local spatial weight: for each pixel/voxel in the neigborhood (with radius ) of pixel/voxel with p j i j j i h j B d j n i a u x h d j w w 1 2, ) ; ( 1 ) ~ )( ~ ;, ( j, d j w p j w j p 1 ) ;, ( h d j w h ) ; ( h j B ) ; ( 1 ) ;, ( h j B d h d j w
19 Local Spatial Weight Weight adaptation Stopping 19
20 Local Spatial Weight Weight adaptation Stopping 20
21 Local Spatial Weight Weight adaptation Stopping 21
22 Local Spatial Weight Weight adaptation Stopping 22
23 Local Spatial Weight Weight adaptation Stopping 23
24 Local Spatial Weight Weight adaptation Stopping 24
25 Local Spatial Weight Weight adaptation Stopping 25
26 Local Spatial Weight Weight adaptation Stopping 26
27 Local Spatial Weight Weight adaptation Stopping 27
28 Local Spatial Weight w( j, d; h) Kloc( D ( j, d) / h) Kst ( D2 ( j, d) / C 1 n where K loc (u) and (u) are two decreasing kernel functions. K st ) Distance kernel K loc (u) : more weights on the closer voxels Similarity kernel K st (u) : more weights on the similar voxels 28
29 Simulation Generate two group simulation images First group contains 40 images, whose true image is from class 0 Second group contains 60 images, whose true image is from class 1 29
30 Simulation Classification Error PCA SPCA WPCA MWPCA K-NN (0.071) (0.050) (0.055) (0.025) SVM (0.078) (0.055) (0.067) (0.026) 30
31 ADNI Data Alzheimer's Disease Neuroimaging Initiative (ADNI) data: 390 subjects (218 normal controls and 172 AD patients) Classification Error PCA SPCA WPCA MWPCA K-NN (0.028) (0.045) (0.052) (0.041) SVM (0.029) (0.043) (0.042) (0.032) 31
32 Outline Multiscale weighted PCA for Image Analysis Human brian artery tree analysis 32
33 Data Background Each Data Point : Tree of Brain Arteries For One Person Collected by Liz Bullitt 33
34 Blood vessel tree data One Person MRI view Single Slice From 3-d Image 34
35 Blood vessel tree data One Person s brain: MRA view Finds blood vessels (show up as white) Track through 3d 35
36 Blood vessel tree data One Person s brain: MRA view Finds blood vessels (show up as white) Track through 3d 36
37 Blood vessel tree data One Person s brain: MRA view Finds blood vessels (show up as white) Track through 3d 37
38 Blood vessel tree data One Person s brain: MRA view Finds blood vessels (show up as white) Track through 3d 38
39 Blood vessel tree data One Person s brain: MRA view Finds blood vessels (show up as white) Track through 3d 39
40 Blood vessel tree data One Person s brain: MRA view Finds blood vessels (show up as white) Track through 3d 40
41 Blood vessel tree data One Person s brain: From MRA Segment tree of vessel segments Using tube tracking Bullitt and Aylward (2002) 41
42 Blood vessel tree data One Person s brain: From MRA Reconstruct trees in 3d Rotate to view 42
43 Blood vessel tree data One Person s brain: From MRA Reconstruct trees in 3d Rotate to view 43
44 Blood vessel tree data One Person s brain: From MRA Reconstruct trees in 3d Rotate to view 44
45 Blood vessel tree data One Person s brain: From MRA Reconstruct trees in 3d Rotate to view 45
46 Blood vessel tree data One Person s brain: From MRA Reconstruct trees in 3d Rotate to view 46
47 Blood vessel tree data One Person s brain: From MRA Reconstruct trees in 3d Rotate to view 47
48 Blood vessel tree data n=98,,..., Statistical goals: 1. Structure of Population (understand variation) 2. Gender difference (Classification) 3. Age difference 4. Build model 48
49 Blood vessel tree data n=98,,..., Statistical goals: 1. Structure of Population (understand variation) 2. Gender difference (Classification) 3. Age difference 4. Build model 49
50 Descendant Correspondence flip this vertex flip this vertex Embed 3-d tree in 2-d More descendants to the left 50
51 Individual Back Tree Descendant Correspondence with Branch Length 51
52 Marron s Back Tree Descendant Correspondence with Branch Length 52
53 Dyck Path Representation Example 1, Assume that we have three following trees Tree 1 Tree 2 Tree 3 53
54 Support Tree: union of trees Tree 1 Tree 2 Tree 3 Tree 1 54
55 Support Tree: union of trees Tree 1 Tree 2 Tree 3 Tree 1,2 55
56 Support Tree: union of trees Tree 1 Tree 2 Tree 3 Tree 1,2,3 56
57 Dyck Path Representation Now, we show how to transform the first tree as a curve. Tree 1/ Support Tree 57
58 Dyck Path Representation Now, we show how to transform the first tree as a curve. Tree 1/ Support Tree 58
59 Dyck Path Representation Now, we show how to transform the first tree as a curve. Tree 1/ Support Tree 59
60 Dyck Path Representation Now, we show how to transform the first tree as a curve. Tree 1/ Support Tree 60
61 Dyck Path Representation Now, we show how to transform the first tree as a curve. Tree 1/ Support Tree 61
62 Dyck Path Representation Now, we show how to transform the first tree as a curve. Tree 1/ Support Tree 62
63 Dyck Path Representation Now, we show how to transform the first tree as a curve. Tree 1/ Support Tree 63
64 Dyck Path Representation Now, we show how to transform the first tree as a curve. Tree 1/ Support Tree 64
65 Dyck Path Representation Now, we show how to transform the first tree as a curve. Tree 1/ Support Tree 65
66 Dyck Path Representation Now, we show how to transform the first tree as a curve. Tree 1/ Support Tree 66
67 Dyck Path Representation Now, we show how to transform the first tree as a curve. Tree 1/ Support Tree 67
68 Dyck Path Representation Now, we show how to transform the second tree as a curve. Tree 2/ Support Tree 68
69 Dyck Path Representation Now, we show how to transform the second tree as a curve. Tree 2/ Support Tree 69
70 Dyck Path Representation Now, we show how to transform the second tree as a curve. Tree 2/ Support Tree 70
71 Dyck Path Representation Now, we show how to transform the second tree as a curve. Tree 2/ Support Tree 71
72 Dyck Path Representation Now, we show how to transform the second tree as a curve. Tree 2/ Support Tree 72
73 Dyck Path Representation Now, we show how to transform the second tree as a curve. Tree 2/ Support Tree 73
74 Dyck Path Representation Now, we show how to transform the second tree as a curve. Tree 2/ Support Tree 74
75 Dyck Path Representation Now, we show how to transform the second tree as a curve. Tree 2/ Support Tree 75
76 Dyck Path Representation Now, we show how to transform the second tree as a curve. Tree 2/ Support Tree 76
77 Dyck Path Representation Now, we show how to transform the second tree as a curve. Tree 2/ Support Tree 77
78 Dyck Path Representation Now, we show how to transform the second tree as a curve. Tree 2/ Support Tree 78
79 Dyck Path Representation Now, we show how to transform the third tree as a curve. Tree 3/ Support Tree 79
80 Dyck Path Representation Now, we show how to transform the third tree as a curve. Tree 3/ Support Tree 80
81 Dyck Path Representation Now, we show how to transform the third tree as a curve. Tree 3/ Support Tree 81
82 Dyck Path Representation Now, we show how to transform the third tree as a curve. Tree 3/ Support Tree 82
83 Dyck Path Representation Now, we show how to transform the third tree as a curve. Tree 3/ Support Tree 83
84 Dyck Path Representation Now, we show how to transform the third tree as a curve. Tree 3/ Support Tree 84
85 Dyck Path Representation Now, we show how to transform the third tree as a curve. Tree 3/ Support Tree 85
86 Dyck Path Representation Now, we show how to transform the third tree as a curve. Tree 3/ Support Tree 86
87 Dyck Path Representation Now, we show how to transform the third tree as a curve. Tree 3/ Support Tree 87
88 Dyck Path Representation Now, we show how to transform the third tree as a curve. Tree 3/ Support Tree 88
89 Dyck Path Representation Now, we show how to transform the third tree as a curve. Tree 3/ Support Tree 89
90 Dyck Path Curves (Back Tree) 90
91 Dyck Path Curves Properties: Flat curve segments correspond to missing branches Rainbow color corresponds to age ranging from magenta (for young) to red (for old) The left part is taller than the right part the descendant correspondence The range of x-value is twice of the branch number every branch is passed twice - Dyck Path 91
92 Principal Component Analysis one data object χ 1,1... χ 1,i... χ 1,n X p n = χ 2,1... χ 2,i... χ 2,n... χ p,1... χ p,i... χ p,n 92
93 Principal Component Analysis 93
94 PCA of the Dyck Path Curves (Back Tree) PC1 Variation 94
95 Tree interpretation of the PC direction 95
96 PC1 Direction (Back Tree) 96
97 PC1 Direction (Back Tree) 97
98 PC1 Direction (Back Tree) 98
99 PC1 Direction (Back Tree) 99
100 PC1 Direction (Back Tree) 100
101 PC1 Direction (Back Tree) 101
102 PC1 Direction (Back Tree) 102
103 PC1 Direction (Back Tree) 103
104 PC1 Direction (Back Tree) 104
105 Tree Interpretation of the PC direction (Back Tree) Summary : Main variation: banches in the right part of the binary trees Reflects the result from the PCA of the Dyck path curves 105
106 Blood vessel tree Analysis n=98,,..., Statistical goals: 1. Structure of Population (understand variation) 2. Gender difference (Classification) 3. Age difference 4. Build model 106
107 Blood vessel tree Analysis Dan Shen, et al (2014). Functional data analysis of tree data objects, (Featured Article) Journal of Computational and Graphical Statistics, 23,
108 Thank You! 108
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