SHOJIMA Kojiro The 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 Slider

θ ij θ ji μjμj κjκj μiμi κiκi Person i Person j Proximity (Data) Row  Col ij Person i -g ij Person j g ji - Proximity (Model) Row  Col ij Person i -ξ ij Person j ξ ji - Proximity (Model) Row  Col ij Person i - ξ ij =(1 － π ij )δ ij Person j ξ ji =(1 － π ji )δ ij - δ ij ||x i － x j || 1 5 π ji =f(θ ji |μ j, κ j ) π ij =f(θ ij |μ i, κ i ) Model xjxj xixi

Stress Function Optimization – 1st Stage: Genetic Algorithm (GA) – 2nd Stage: Steepest Descent Method (SDM)

→ABCD A B C D →ABCD A B C D →ABC A B C A B C A B C D A B C D

Omnidirectional and more loveOmnidirectional and less love Omnidirectionality and the Amount of One-sided Love Reduces to the conventional von Mises distribution when ω=1/(2π)

Proximity (Data) Row  Col ij Person i -g ij Person j g ji - Proximity (Model) Row  Col ij Person i -ξ ij =(1-π ij )δ ij Person j ξ ji =(1-π ji )δ ji Problem xjxj xixi Person i Person j δ ij π ij =f(θ ij |μ i, κ i )

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 →ABCD A 777 B 117 C 171 D A B C D Result

Sociometric Data (Chino, 1997, p.13, Revised) →

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