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![Page 1: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/1.jpg)
SHOJIMA KojiroThe National Center for University Entrance Examinations
Asymmetric von Mises Scaling
![Page 2: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/2.jpg)
Purpose of Research
• Development of an asymmetric multidimensional scaling (MDS) method using a technique from directional statistics
• Asymmetric von Mises scaling (AMISESCAL)
![Page 3: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/3.jpg)
• A branch of statistics dealing with angles, courses, and directions as data– Magnetic field analysis, animal migration, disease
transmission route, etc.
Directional Statistics (c.f., Mardia & Jupp, 2000)
![Page 4: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/4.jpg)
von Mises distribution
• Normal distribution in directional statistics
μ: mean direction κ: concentration
SliderSlider
![Page 5: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/5.jpg)
θij
θjiμjκj
μi
κi
Person i
Person j
Proximity (Data)
RowCol i j
Person i - gij
Person j gji -
Proximity (Model)
RowCol i j
Person i - ξij
Person j ξji -
Proximity (Model)
RowCol i j
Person i - ξij=(1 - πij)δij
Person j ξji=(1 - πji)δij -
δij
||xi -xj||
15
πji=f(θji|μj, κj)
πij=f(θij|μi, κi)
Modelxj
xi
![Page 6: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/6.jpg)
Stress Function
• Optimization– 1st Stage: Genetic Algorithm (GA)
– 2nd Stage: Steepest Descent Method (SDM)
![Page 7: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/7.jpg)
→ A B C DABCD
→ A B C DABCD
→ A B CABC 1 7
1 717
11 7
711
7 7 171 1
7 7 771
7 1111 71
A
B
CAB
C
DA
B
C
D
![Page 8: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/8.jpg)
Omnidirectional and more loveOmnidirectional and less love
Omnidirectionality and the Amount ofOne-sided Love
Reduces to the conventional von Mises distribution when ω=1/(2π)
![Page 9: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/9.jpg)
Proximity (Data)
RowCol i j
Person i - gij
Person j gji -
Proximity (Model)
RowCol i j
Person i - ξij=(1-πij)δij
Person j ξji=(1-πji)δji -
15
Problemxj
xi
Person i
Person jδij
πij=f(θij|μi, κi)
![Page 10: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/10.jpg)
Stress Function (2)
• Adding Penalty Function U– Reward when there are one-sided love targets in the
direction of heavy density
– Penalty when there is no target in the direction of heavy density
• Optimization– GA+SDM
![Page 11: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/11.jpg)
A
B
C
D→ A B C DA 7 7 7B 1 1 7C 1 7 1D 1 1 7
7 7 77
77
1 11 11 1
A
B
C
D
Result
![Page 12: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/12.jpg)
Sociometric Data (Chino, 1997, p.13, Revised)
→ 1 2 3 4 5 6 7 8 9 101 5 4 5 3 3 2 4 1 12 6 4 2 1 2 3 3 4 33 4 4 3 3 4 4 5 4 34 4 1 2 1 1 4 2 4 35 7 1 2 1 1 2 2 2 36 4 3 4 2 3 4 4 4 47 4 3 4 4 5 5 2 4 28 6 4 4 4 3 4 3 4 49 2 3 3 3 3 2 3 3 210 4 4 4 5 4 4 4 4 4
![Page 13: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/13.jpg)
Result
![Page 14: SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp Asymmetric von Mises Scaling.](https://reader035.fdocuments.in/reader035/viewer/2022062422/56649e8e5503460f94b91724/html5/thumbnails/14.jpg)
Future Tasks
• Dealing with diagonal elements
• Expansion to 3D model space
• Expansion to 2 mode (multi-group or longitudinal) data
Thank you for your attention.Kojiro Shojima ([email protected])