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SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp

Asymmetric von Mises Scaling. SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp. Purpose of Research. Development of an asymmetric multidimensional scaling (MDS) method using a technique from directional statistics

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SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp

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  1. Asymmetric von Mises Scaling SHOJIMA Kojiro The National Center for University Entrance Examinations shojima@rd.dnc.ac.jp

  2. Purpose of Research • Development of an asymmetric multidimensional scaling (MDS) method using a technique from directional statistics • Asymmetric von Misesscaling (AMISESCAL)

  3. Directional Statistics (c.f., Mardia & Jupp, 2000) • A branch of statistics dealing with angles, courses, and directions as data • Magnetic field analysis, animalmigration, disease transmission route, etc.

  4. Slider von Mises distribution • Normal distribution in directional statistics μ: mean direction κ: concentration

  5. Model ||xi－xj|| xj δij θji Person j μj κj κi μi xi Person i πji=f(θji|μj, κj) θij πij=f(θij|μi,κi) 1 5

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

  7. 7 7 7 7 7 1 1 1 7 7 7 1 1 1 1 1 7 7 7 7 1 1 1 1 7 7 1 1 1 1 B C C A D A D B A C B

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

  9. Problem xj δij Person j xi πij=f(θij|μi,κi) Person i 1 5

  10. 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

  11. Result 7 7 7 7 C C 1 1 7 1 1 7 1 1 D A A D B B

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

  13. Result

  14. 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)

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