Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Han Liu MedicReS World Congress 2015

1,406 views

Published on

From High Dimensional Data to Big Data Presentation to MedicReS 5th World Congress on October 19,25,2015 in New York by Han Liu

Published in: Data & Analytics
  • Be the first to comment

  • Be the first to like this

Han Liu MedicReS World Congress 2015

  1. 1. From  High  Dimensional   Data  to  Big  Data Han  Liu
  2. 2. Acknowledgement 2 Ethan Fang Princeton University Fang Han JHU Biostatistics Cun-hui Zhang Rutgers Statistics
  3. 3. Big Data Movement 3
  4. 4. Big Data Movement 3
  5. 5. Big Data Movement 3
  6. 6. Big Data Movement 3
  7. 7. Big Data Movement 3
  8. 8. Big Data Movement 3
  9. 9. Outline 4 • • •
  10. 10. High Dimensional Multivariate Analysis 5
  11. 11. High Dimensional Multivariate Analysis 5 · · ·
  12. 12. High Dimensional Multivariate Analysis 5 · · ·
  13. 13. High Dimensional Multivariate Analysis 5 µ, · · ·
  14. 14. High Dimensional Multivariate Analysis 5 µ, · · ·
  15. 15. Gaussian Graphical Model 6 (µ, ) = ( , . . . , )
  16. 16. Gaussian Graphical Model 6 (µ, ) = ( , . . . , ) = ( , )
  17. 17. Gaussian Graphical Model 6 (µ, ) = ( , . . . , ) = ( , )
  18. 18. Gaussian Graphical Model 6 (µ, ) = ( , . . . , ) | = ( , )
  19. 19. Gaussian Graphical Model 6 (µ, ) = ( , . . . , ) ( ) =| = ( , )
  20. 20. Gaussian Graphical Model 6 (µ, ) = ( , . . . , ) ( ) =| = ( , ) min log | | + , | |
  21. 21. Gaussian Graphical Model 6 (µ, ) = ( , . . . , ) ( ) =| = ( , ) min log | | + , | |
  22. 22. Gaussian Graphical Model 6 (µ, ) = ( , . . . , ) ( ) =| = ( , ) min log | | + , | |
  23. 23. Gaussian Graphical Model 6 (µ, ) = ( , . . . , ) ( ) =| = ( , ) min log | | + , | |
  24. 24. Sparse Principal Component Analysis 7 = ( , . . . , )
  25. 25. Sparse Principal Component Analysis 7 = ( , . . . , )
  26. 26. Sparse Principal Component Analysis 7 = ( , . . . , ) max x = x x (x) k
  27. 27. Sparse Principal Component Analysis 7 = ( , . . . , ) max x = x x (x) k
  28. 28. High Dimensional Theory 8
  29. 29. High Dimensional Theory 8 = ( )
  30. 30. High Dimensional Theory 8 = ( )
  31. 31. High Dimensional Theory 8 = ( ) P ( ) = ( ) ( )
  32. 32. High Dimensional Theory 8 = ( ) P ( ) = ( ) ( )
  33. 33. Theoretical Foundations 9 R4 R3 R2 R5 R1 µ µ max = log max = log = sup v |v ( )v| v = log inf v |v v| v >
  34. 34. Real Data are non-Gaussian 10 ! "## $## (> K) (> K) > K
  35. 35. Real Data are non-Gaussian 10 ! "## $## (> K) (> K) > K
  36. 36. Real Data are non-Gaussian 10 ! "## $## (> K) (> K) > K
  37. 37. Transelliptical Distribution 11
  38. 38. Transelliptical Distribution 11 = ( , . . . , ) (µ, , { } = , ) , . . . , = ( ), . . . , ( ) | | / (y µ) (y µ)
  39. 39. Transelliptical Distribution 11 = ( , . . . , ) (µ, , { } = , ) , . . . , = ( ), . . . , ( ) | | / (y µ) (y µ)
  40. 40. Visualization 12
  41. 41. Special Cases 13 (µ, , { } = , )
  42. 42. Special Cases 13 (µ, , { } = , ) ( ) = ,
  43. 43. Special Cases 13 (µ, , { } = , ) (µ, , ) ( ) = ,
  44. 44. Special Cases 13 (µ, , { } = , ) (µ, , ) ( ) = , ( ) =
  45. 45. Special Cases 13 (µ, , { } = , ) (µ, , ) (µ, , { } = ) ( ) = , ( ) =
  46. 46. Special Cases 13 (µ, , { } = , ) (µ, , ) (µ, , { } = ) ( ) = , ( ) =
  47. 47. 14 Identifiability Conditions ( ), . . . , ( ) (µ, , )
  48. 48. 14 Identifiability Conditions ( ), . . . , ( ) (µ, , )
  49. 49. 14 Identifiability Conditions ( ), . . . , ( ) (µ, , ) µ = ( ) = I
  50. 50. 14 Identifiability Conditions ( ), . . . , ( ) (µ, , ) µ = ( ) = I E = E ( ) = µ Var( ) = Var[ ( )] =
  51. 51. 14 Identifiability Conditions ( ), . . . , ( ) (µ, , ) µ = ( ) = I E = E ( ) = µ Var( ) = Var[ ( )] =
  52. 52. 15 Hierarchical Representation
  53. 53. 15 Hierarchical Representation ( , . . . , ) (µ, , { } = , )
  54. 54. 15 Hierarchical Representation ( , . . . , ) (µ, , ) = ( ) ( , . . . , ) (µ, , { } = , )
  55. 55. 15 Hierarchical Representation ( , . . . , ) (µ, , ) = ( ) ( , . . . , ) (µ, ) = µ + ( µ ) ( , . . . , ) (µ, , { } = , )
  56. 56. 15 Hierarchical Representation ( , . . . , ) (µ, , ) = ( ) ( , . . . , ) (µ, ) = µ + ( µ ) (·) ( , . . . , ) (µ, , { } = , )
  57. 57. 15 Hierarchical Representation ( , . . . , ) (µ, , ) = ( ) ( , . . . , ) (µ, ) = µ + ( µ ) (·) ( , . . . , ) (µ, , { } = , )
  58. 58. 15 Hierarchical Representation ( , . . . , ) (µ, , ) = ( ) ( , . . . , ) (µ, ) = µ + ( µ ) (·) ( , . . . , ) (µ, , { } = , )
  59. 59. 15 Hierarchical Representation ( , . . . , ) (µ, , ) = ( ) ( , . . . , ) (µ, ) = µ + ( µ ) (·) ( , . . . , ) (µ, , { } = , )
  60. 60. Transelliptical Graphical Model 16 (µ, , { } = , )
  61. 61. Transelliptical Graphical Model 16 (µ, , { } = , )
  62. 62. Transelliptical Graphical Model 16 (µ, , { } = , )
  63. 63. Transelliptical Graphical Model 16 ( ) = (µ, , { } = , )
  64. 64. Transelliptical Graphical Model 16 ( ) = (µ, , { } = , )
  65. 65. Transelliptical Graphical Model 16 ( ) = (µ, , { } = , )
  66. 66. Transelliptical Graphical Model 16 ( ) = (µ, , { } = , )
  67. 67. Transelliptical Graphical Model 16 ( ) = (µ, , { } = , )
  68. 68. Transelliptical Graphical Model 16 ( ) = (µ, , { } = , )
  69. 69. Transelliptical Graphical Model 16 ( ) = (µ, , { } = , )
  70. 70. Semiparametric Inference 17 x , . . . , x (µ, , { } = , )
  71. 71. Semiparametric Inference 17 x , . . . , x (µ, , { } = , )
  72. 72. Semiparametric Inference 17 x , . . . , x (µ, , { } = , )
  73. 73. Semiparametric Inference 17 x , . . . , x (µ, , { } = , )
  74. 74. Semiparametric Inference 17 x , . . . , x (µ, , { } = , ) µ, { } =
  75. 75. Semiparametric Inference 17 x , . . . , x (µ, , { } = , ) µ, { } = µ,
  76. 76. Semiparametric Inference 17 x , . . . , x (µ, , { } = , ) µ, { } = µ,
  77. 77. Semiparametric Inference 17 x , . . . , x (µ, , { } = , ) µ, { } = µ,
  78. 78. Technical Requirements 18 R4 R3 R2 R5 R1 µ µ max = log max = log = µ, µ, sup v |v ( )v| v = log inf v |v v| v >
  79. 79. Estimating Mean 19
  80. 80. Estimating Mean 19 µ = · (x µ ) =
  81. 81. Estimating Mean 19 µ = · (x µ ) =
  82. 82. Estimating Mean 19 µ = · (x µ ) =
  83. 83. Estimating Mean 19 µ = · (x µ ) = ( ) = log + + /
  84. 84. Estimating Mean 19 µ = · (x µ ) = ( ) = log + + / µ µ max = log E < R1
  85. 85. Estimating Covariance 20
  86. 86. Estimating Covariance 20 = · ·
  87. 87. Estimating Covariance 20 = · ·
  88. 88. Estimating Covariance 20 = · · = sin
  89. 89. Estimating Covariance 20 = ( ) < (x x ) · (x x ) = · · = sin
  90. 90. Estimating Covariance 20 = ( ) < (x x ) · (x x ) = · · E < R2 max = log = sin
  91. 91. 21 max max = log
  92. 92. 21 max max = log = sin E
  93. 93. 21 max max = log = sin E
  94. 94. 21 max max = log = sin E
  95. 95. 21 max max = log = sin E
  96. 96. 21 max max max E = log max = log = sin E
  97. 97. 21 max max max E = log max = log = sin E
  98. 98. Transelliptical Theory 22
  99. 99. Transelliptical Theory 22 • • •
  100. 100. Transelliptical Theory 22 = • • •
  101. 101. Transelliptical Theory 22 = • • • • • •
  102. 102. Transelliptical Theory 22 = = s • • • • • •
  103. 103. Transelliptical Theory 22 = = s • • • • • •
  104. 104. Proof of the Theory 23
  105. 105. Proof of the Theory 23 R4 R3 R2 R5 R1 µ µ max = log max = log sup v |v ( )v| v = log inf v |v v| v > = log
  106. 106. Proof of the Theory 23 R4 R3 R2 R5 R1 µ µ max = log max = log sup v |v ( )v| v = log inf v |v v| v > = log
  107. 107. Proof of the Theory 23 R4 R3 R2 R5 R1 µ µ max = log max = log sup v |v ( )v| v = log inf v |v v| v > = log
  108. 108. Proof of the Theory 23 R4 R3 R2 R5 R1 µ µ max = log max = log sup v |v ( )v| v = log inf v |v v| v > = log
  109. 109. Proof of the Theory 23 R4 R3 R2 R5 R1 µ µ max = log max = log sup v |v ( )v| v = log inf v |v v| v > = log
  110. 110. Proof of the Theory 23 R4 R3 R2 R5 R1 µ µ max = log max = log sup v |v ( )v| v = log inf v |v v| v > = log
  111. 111. Empirical Results 24
  112. 112. Empirical Results 24 ( , ) = , =
  113. 113. Empirical Results 24 true graph transelliptical glasso ( , ) = , =
  114. 114. Empirical Results 24 true graph transelliptical glasso FP FN ( , ) = , =
  115. 115. Empirical Results 24 true graph transelliptical glasso FP FN ( , ) = , =
  116. 116. Efficiency Loss 25
  117. 117. Efficiency Loss 25 ( , ) = , =
  118. 118. Efficiency Loss 25 ROC curve for graph recovery 1-FP 1-FN ( , ) = , =
  119. 119. Efficiency Loss 25 ROC curve for graph recovery 1-FP 1-FN ( , ) = , =
  120. 120. Efficiency Loss 25 ROC curve for graph recovery 1-FP 1-FN ( , ) = , =
  121. 121. Efficiency Loss 25 ROC curve for graph recovery 1-FP 1-FN ( , ) = , =
  122. 122. 26 Summary
  123. 123. 26 Summary • • •
  124. 124. 26 Summary • • •

×