shojima kojiro the national center for university entrance examinations [email protected]...
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SHOJIMA KojiroThe National Center for University Entrance Examinations
Asymmetric von Mises Scaling
Purpose of Research
• Development of an asymmetric multidimensional scaling (MDS) method using a technique from directional statistics
• Asymmetric von Mises scaling (AMISESCAL)
• 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)
von Mises distribution
• Normal distribution in directional statistics
μ: mean direction κ: concentration
SliderSlider
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||
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πji=f(θji|μj, κj)
πij=f(θij|μi, κi)
Modelxj
xi
Stress Function
• Optimization– 1st Stage: Genetic Algorithm (GA)
– 2nd Stage: Steepest Descent Method (SDM)
→ 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
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)
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)
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
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
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
Result
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])