2013년 11월 29일 한국보건정보통계학회 발표

1. 한국보건정보통계학회 추계학술발표회 2013
“빅” 데이터의 분석적 시각화
Analytic Data Visualization
許 明 會
2013.11.29
고려대학교 통계학과 stat420@korea.ac.kr
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2. Data Visualization
- Descriptive vs Analytic ...
- Small vs Big ...
science
technology
art
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3. Contents
- Scatterplot
- Biplot
- Regression Biplot
- Kernel PCA
- SVM Biplot
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4. Scatterplot: 산점도
- “Lego” for analytic data visualization
- Reflecting the third variable
quakes:
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longitude(=x), latitude(=y), depth(=z)
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5. Scatterplot: 산점도
- For the case of large  (≧  ), over-plotting can produce
serious outcome.
Skin Segmentation Data:  (red) vs.  (green)
      
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6. Scatterplot: 산점도
- For the case of large  (≧  ), alpha channel can be utilized.
Skin Segmentation Data:  (red) vs.  (green)
      
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7. Scatterplot: 산점도
- lowess: A nonparametric regression for bivariate data
cars data: distance vs. speed
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8. Scatterplot: 산점도
- 3D Rotation for three variables
Skin Segmentation Data:  (red),  (green),  (blue)
- ggobi:
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3D Rotation for four or more variables
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9. Biplot of Observations and Variables,
Gabriel (1971)
- The biplot is a graph that shows  observations and  variables.
Protein data (row: 25 nations, column: 9 protein sources)
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10. Biplot of Observations and Variables,
Gabriel (1971)
- Idea: Linear projection
Protein data: variable cereal
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11. Regression Biplot,
Huh and Lee (2013)
- Regression biplot is a graph for  observations of   ⋯    ,
arranged by predicted  .
- Assume that the model fit is determined by a function of linear
combination of   ⋯    . For instance,
   ⋯  ,


 
 
or
log           ⋯    .


- Set the vertical dimension by the direction of regression coefficients
  

  ⋮ ,
or      .
∥∥
  
- Set the horizontal dimension by the direction of principal axis of



  ⋯   ,

where  
denotes the orthogonal component generated from the
projection of   on  .
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12. Regression Biplot,
Huh and Lee (2013)
Example 1. Stack Loss Data (  ;   loss of ammonia,         )
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13. Regression Biplot,
Huh and Lee (2013)
Example 2. Magazine Data (  ;   Subscription (0,1),   )
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14. Kernel PCA,
Scholkopf et al. (1998)
- For  observations    ⋯    ( × ), consider the nonlinear mapping
    ⋯   
to a Hilbert space, in which                      .
- Denoting            , Kernel PCA is obtained from
eigen-decomposing
             .






- Kernel PCA yields a plot of observations by projecting       ⋯      
on 




′  

where 
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
′
   ′  ,
     ,   is an eigenvector of  .


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15. Kernel PCA Diagram (or Kernel Biplot),
Huh (2013)
- Aim: Representation of  variables in Kernel PC plot of observations.
- Proposed Procedure:

1) For each    ⋯    , map         on the plane,
   ⋯   , where    is a constant and     ⋯   ⋯    .
Projection is given by


′  
  ′   
′

 
 
  

 ″    
 ′ ″   
 ″ ″′  .

 ″  
 ″  
 ″  ″′  




2) For each  , link the projection points of   and  
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by an arrow.
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16. Example 1. Arrow diagrams [  ] for kernel PCA of the iris data
with rbf kernel,   
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17. Example 1. Arrow diagrams [  ] for kernel PCA of the iris data
with rbf kernel,   
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18. Example 2. Arrow diagrams [  ] for kernel PCA of the spam data
[      ]
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19. SVM-Guided Biplot as an extension of Regression Biplot
- Idea: Combine Linear/Logistic Regression Biplot and Kernel PCA.
- Classification/Regression Part:
Classified
as
SVM classifier
  -1 or 1 for    ⋯   .
              ,

where
 
      , 


Vertical dimension is set to


  
   
  

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


≧ .




(      ,        ).
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20. SVM-Guided Biplot: Classification
- Kernel PCA Part:
         
 

 
∴


(   
      ′  ),
 ′   ′

   ⋯   .
          ′                  ′   ′   


 ′   ′ ,
  ′   ⋯   .
Hence

 →      (   ) or          .



Horizontal dimension is determined by eigen-decomposing  .

- Perturbation Scheme for Arrow Diagrams.
Define      ,  ×  , where  represents a perturbation of
which the magnitude is controlled by . Then, project   on the first
(vertical) and the second (horizontal) dimension.
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21. Example 1. Iris Data: Versicolor vs. Virginica [sigma=0.1, C=1,   ]
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22. Importance of Variables
(in the case of large
)
- It is necessary to select a small number of variables in determining
the first and second dimensions.
- Measures of Importance (definition)  Length of Arrows
1) in vertical direction,
2) in horizontal direction.
- Plot arrow diagrams for importance variables only.
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23. Example 2. Spam Data [sigma=0.1, C=10,   ],   
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24. SVM-Guided Biplot: Regression
- The same method can be applied to SVM regression.
- Example 3. Aerobic Fitness [       ] for oxygen uptake (=  )
with RBF kernel ( =0.1, C=10,  =0.1,   )
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25. Concluding Remarks
- Biplot method can be extended to be suited for linear regression or
classification (logistic regression).
- Biplot method can be extended to allow nonlinear mapping of
observations and variables, by fully utilizing kernel trick.
http://blog.naver.com/huh4200
금붕어 어항 (on the iPad)
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26. References
Gabriel, K.R. (1971). “The biplot display of matrices with the application to
principal component analysis”. Biometrika, 58. 453-467.
Huh, M.H. (2013). “Arrow diagrams for kernel principal component analysis”.
Communications for Statistical Applications and Methods, 20. 175-184.
Huh, M.H. (2013). “SVM-guided biplot of observations and variables”.
Communications for Statistical Applications and Methods. (to appear)
Huh, M.H. and Lee, Y.G. (2013). “Biplots of multivariate data guided by linear
and/or logistic regression”. Communications for Statistical Applications and
Methods, 20. 129-136.
Scholkopf, B., Smola, A. and Muller, K.R. (1998). Nonlinear component analysis as
a kernel eigenvalue problem. Neural Computation, 10. 1299–1319.
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