ඇਖ਼‫ن‬ੑͷԼͰͷଟมྔઢ‫ܗ‬Ϟσϧʹର͢Δߴ࣍‫ݩ‬઴ۙཧ࿦
੒᪟େֶཧ޻ֶ෦
೔ຊେֶ޻ֶ෦
ඣ໺఩ਓ
ࢁాོߦ
ɹۙ೥ɺϏοάσʔλ͕༷ʑͳ෼໺Ͱ஫໨͞ΕΔΑ͏ʹͳΓɺߴ࣍‫ݩ‬σʔλͷղੳख๏ͷॏཁੑ͕૿͖ͯ͠
͍ͯΔɻͦ͜Ͱɺຊ‫Ͱڀݚ‬͸ଟมྔઢ‫ܗ‬Ϟσϧ
Y = XB + EΣ1/2
(1)
ʹண໨͢Δɻ͜͜ͰɺY ͸ n × p ͷ‫؍‬ଌߦྻɺX ͸ n × k ͷ‫ܭ‬ըߦྻɺB ͸ k × p ͷະ஌ͷ܎਺ߦྻɺ
E = ( 1, . . . ,
n)
͸ n × p ͷ‫͠ͱྻߦࠩޡ‬ɺ 1 , . . . ,
n
͸ͦΕͧΕಠཱͰಉҰͳ෼෍ F ʹै͏΋ͷͱ͠ɺF
ͷฏ‫ۉ‬ϕΫτϧ͸ 0ɺ෼ࢄ‫ڞ‬෼ࢄߦྻ͸ Ip ͱ͢Δɻͦͯ͠ɺn, p ͱ΋ʹे෼େ͖͍ͱ͍͏ঢ়‫گ‬ͷԼͰɺઢ
‫ܗ‬Ծઆ
CB = O
(2)
Λߟ͑Δɻͨͩ͠ɺrank(C) = q ͱ͢Δɻ
͜ͷΑ͏ͳଟมྔઢ‫ܗ‬Ϟσϧ (1) ͷઢ‫ܗ‬Ծઆ (2) ʹର͢Δ‫ݕ‬ఆ౷‫ྔܭ‬ͷߴ࣍‫ݩ‬σʔλղੳ͸͜Ε·Ͱʹ
ଟ͘ͷ‫ʹऀڀݚ‬Αͬͯ‫ߦ͕ڀݚ‬ΘΕ͓ͯΓɺಛʹ‫؍‬ଌσʔλ͕ਖ਼‫ن‬෼෍ʹै͏৔߹ʢͭ·ΓɺF = N (O, I)
ͷ৔߹ʣ͸༷ʑͳ݁Ռ͕ใࠂ͞Ε͍ͯΔ (Fujikoshi et al. (2004, JJSS), Schott (2007, JMVA), Srivastava
(2005, JJSS) ౳)ɻ͔͠͠ɺҰൠʹߴ࣍‫ݩ‬σʔλͷ৔߹ɺσʔλ͕ਖ਼‫ن‬෼෍ʹै͏͔Ͳ͏͔Λௐ΂Δ͜ͱ΋
೉͍ͨ͠Ίɺਖ਼‫ن‬෼෍ʹैΘͳ͍৔߹ͷ‫͕ڀݚ‬ॏཁͱͳΔɻ
͕ͩɺਖ਼‫ن‬෼෍ʹैΘͳ͍৔߹ʹ͸ɺ‫੍͍ڧ‬໿͕ඞཁʹͳΔ৔߹͕ଟ͍ɻྫ͑͹ɺ1 ඪຊͷσʔλ (k = 1)
ͷ৔߹Ͱ͢Βɺ‫ྻߦࠩޡ‬ͷશ੒෼͕ಠཱ·ͨ͸ͦΕʹ͍ۙԾఆ͕͓͔ΕΔ৔߹͕ଟ͍ (Chen et al. (2010,
JASA), Chen and Qin (2010, Ann. Statist.), Srivastava et al. (2011, JMVA) ౳)ɻͦ͜Ͱɺզʑ͸ͦͷ৚
݅Λ؇࿨͢ΔͨΊʹɺਖ਼‫ن‬෼෍͔Βͷဃ཭ͷఔ౓Λද͢ύϥϝʔλ
κ11 = E[( i Σ i )2 ] − 2trΣ2 − (trΣ)2
Λ༻͍Δ͜ͱͱͨ͠ʢਖ਼‫ن‬෼෍ͷԼͰ͸ κ11 = 0 ͱͳΔ (Magnus and Neudecker, (1979, Ann. Statist.))ʣɻ
͜ͷύϥϝʔλΛ༻͍Δ͜ͱʹΑΓɺैདྷͷ‫ڀݚ‬ΑΓ΋޿͍෼෍଒ʹର͠ɺ‫ݕ‬ఆΛߦ͏͜ͱΛՄೳͱͨ͠ɻ
·ͨɺ͜ͷύϥϝʔλͷਪఆྔΛఏҊ͢Δ͜ͱͰɺߴ࣍‫ݩ‬σʔλͷਖ਼‫ن‬ੑΛ‫ݕ‬ఆ͢Δ‫ݕ‬ఆ౷‫ྔܭ‬ͷఏҊΛ
ߦͬͨ (Himeno and Yamada, 2014)ɻ͞ΒʹɺଟඪຊͷԼͰ͢΂ͯͷฏ‫͕ۉ‬౳͍͠ͱ͍͏‫ؼ‬ແԾઆʹର͢
Δ‫ݕ‬ఆ౷‫ྔܭ‬΋ఏҊ͠ɺͦͷ઴ۙ෼෍Λ‫ؼ‬ແԾઆͱରཱԾઆͷԼͰಋग़͠ɺ઴ۙ‫ݕ‬ग़ྗΛ‫ٻ‬Ίɺ͜Ε͕ਖ਼‫ن‬
ੑͷԼͰఏҊͤͨ͞‫ݕ‬ఆ౷‫ྔܭ‬ͷ֦ுͱͳ͍ͬͯΔ͜ͱΛ֬ೝͨ͠ (Yamada and Himeno, 2013)ɻ
ຊൃදͰ͸ɺଟඪຊͷԼͰͷ݁Ռ (Yamada and Himeno (2013)) ΛҰൠͷଟมྔઢ‫ܗ‬Ϟσϧ (1) ʹ֦ு
͢ΔͨΊʹඞཁͳ৚݅Λ۩ମతʹௐ΂ɺͲͷΑ͏ͳ৚݅ͷԼͰ͋Ε͹ɺ‫ؼ‬ແԾઆ (2) ʹର͢Δ‫ݕ‬ఆ౷‫ྔܭ‬
ͷ઴ۙ‫ؼ‬ແ෼෍ͷਖ਼‫ن‬ੑ͕ಘΒΕΔ͔Ͳ͏͔Λٞ࿦͢Δɻ
ࢀߟจ‫ݙ‬
[1] Yamada, T. and Himeno, T. (2013), Testing homogeneity of mean vectors under heteroscedasticity
in high-dimension, TR No. 13-13, Statistical Research Group, Hiroshima University.
[2] Himeno, T. and Yamada, T. (2014), Estimations for some functions of covariance matrix in high
dimension under non-normality and its applications, Journal of Multivariate Analysis, 130, 27-44.

行列 （p.9-10

講演概要

Kinectにより観測された人の動作を説明する確率的言語生成への 取り組み