SHOJIMA KojiroThe National Center for University Entrance Examinations

shojima@rd.dnc.ac.jp

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||

15

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 (shojima@rd.dnc.ac.jp)