ߴ࣍‫ݩ‬ͷ৔߹ʹ͓͚Δ࣍‫ݩ‬ͷਪఆ๏ͱ઴ۙతੑ࣭
ਡ๚౦‫ژ‬ཧՊେɾ‫ڞ‬௨‫ڭ‬ҭηϯλʔ
޿ౡେɾཧɾ໊༪‫ڭ‬त
ᓎҪɹ఩࿕
౻ӽɹ߁ॕ
ຊใࠂͰ͸, ଟมྔճ‫ؼ‬෼ੳ, ൑ผ෼ੳ, ਖ਼४૬ؔ෼ੳʹ͓͚Δ࣍‫ݩ‬ͷਪఆʹ͍ͭͯߟ࡯͢
Δ. ͜ΕΒͷ࣍‫ݩ‬ͷਪఆ͸, ଟมྔճ‫Ͱؼ‬͸ճ‫ྻߦ਺܎ؼ‬ͷϥϯΫ, ൑ผ෼ੳͰ͸༗༻ͳ൑
ผؔ਺ͷ‫਺ݸ‬, ਖ਼४૬ؔ෼ੳͰ͸θϩͰͳ͍ਖ਼४૬ؔ܎਺ͷ‫਺ݸ‬Λ೺Ѳ͢Δͷʹ࢖ΘΕΔ.
࣍‫ݩ‬Λਪఆ͢Δํ๏ͱͯ͠, AIC ‫ن‬४΍ Cp ‫ن‬४Λ༻͍Δํ๏͕ఏҊ͞Ε͍ͯΔ. ྫ͑͹,
ඪຊ਺ n, ม਺ͷ਺ p, ‫܈‬ͷ‫ ਺ݸ‬q + 1 ͷ൑ผ෼ੳͰ͸࣍ͷ‫Ͱܗ‬༩͑ΒΕΔ.
q
Aj = n log
(1 + i ) − 2(p − j)(q − j),
j = 0, . . . , q − 1,
Aq = 0,
i=j+1
Cj = n
q
i − 2(p − j)(q − j),
j = 0, . . . , q − 1,
Cq = 0
i=j+1
͜͜Ͱ, 1 > · · · > q > 0 ͸‫಺܈‬ฏํ࿨ੵ࿨ߦྻ Se ͱ‫ؒ܈‬ฏํ࿨ੵ࿨ߦྻ Sh ͔ΒͳΔߦ
ྻ Sh S−1
e ͷθϩͰͳ͍‫ݻ‬༗஋Ͱ͋Δ. ͜ͷͱ͖, Aj ·ͨ͸ Cj ͷ஋͕࠷খͱͳΔ j Λ࣍‫ͱݩ‬
jA = arg minj Aj , ˆjC = arg minj Cj ͱͯ͠༩͑ΒΕΔ. ࡢ೥౓ͷ࿈
ͯ͠ਪఆ͢Δ. ͭ·Γ, ˆ
߹େձʹ͓͍ͯ, ߨԋऀͷҰਓͰ͋Δ౻ӽ͸, ଟมྔճ‫ؼ‬෼ੳ, ൑ผ෼ੳ, ਖ਼४૬ؔ෼ੳʹ͓
͚Δ࣍‫ݩ‬ͷਪఆʹؔͯ͠, AIC ‫ن‬४ͱ Cp ‫ن‬४ʹΑΔਪఆ๏ͷߴ࣍‫ݩ‬ੑ࣭ʹ͍ͭͯൃදͨ͠.
ͱ͘ʹ, ม਺ͷ਺ͱඪຊ਺ͷൺ͕Ұఆ஋ʹ͖ۙͮͳ͕Βେ͖͘ͳΔͱ͖, ͋ΔछͷԾఆͷ΋
ͱͰҰகੑΛ΋ͭ͜ͱΛࣔͨ͠.
͜͜Ͱ͸, ·ͣ BIC ‫ن‬४ʹؔͯ͠ߟ࡯͢Δ. ͜Ε͸࣍ͷΑ͏ʹ༩͑ΒΕΔ.
Bj = n log
q
(1 + i ) − (log n)(p − j)(q − j),
j = 0, . . . , q − 1,
Bq = 0
i=j+1
jB = arg minj Bj ͷߴ࣍‫ݩ‬ੑ࣭ʹ͍ͭͯௐ΂ͨ. ‫ݻ‬༗஋ i ʹରԠ͢
͜ͷͱ͖, ࣍‫ݩ‬ͷਪఆྔ ˆ
Δ฼ूஂ‫ݻ‬༗஋Λ ωi Ͱද͢ͱ, ͜ΕΒ͸ඪຊ਺ n ΍ม਺ͷ਺ p ʹґଘ͢Δ. ͜͜Ͱ͸ਅͷ࣍
‫ݩ‬Λ j0 (≤ q) ͱͯ࣍͠ͷ 2 ͭͷ৔߹Λߟ͑Δ.
(i) ωi = O(n),
i = 1, . . . , j0 ,
(ii) ωi = O(np),
i = 1, . . . , j0
ม਺ͷ਺ͱඪຊ਺ͷൺ͕Ұఆ஋ʹ͖ۙͮͳ͕Βେ͖͘ͳΔͱ͖, (i) ͷ৔߹Ͱ͸ BIC ‫ن‬४͸
ҰகੑΛ΋ͨͳ͍͕, (ii) ͷ৔߹Ͱ͸ BIC ‫ن‬४͸ҰகੑΛ΋ͭ͜ͱΛࣔ͢. Ұํ, AIC ΍ Cp
͸, (i), (ii) ͷ৔߹ʹ͓͍ͯ͋ΔछͷԾఆͷԼͰҰகੑΛ΋ͭ͜ͱ͕ࣔ͞Ε͍ͯΔ.
͜ΕΒͷ݁Ռ͸, ม਺ͷ਺͸େ͖͘ͳΔ͕, ඪຊ਺ΑΓখ͍͞৔߹ͷ݁ՌͰ͋Δ. ຊใࠂͰ
͸, ม਺ͷ਺͕ඪຊ਺ΑΓେ͖͍৔߹ʹ͍ͭͯ΋ߟ࡯͢Δ. ͜ͷͱ͖, ‫ڞ‬෼ࢄߦྻͷ௨ৗͷਪ
ఆྔͷ‫͕ྻߦٯ‬ଘࡏ͠ͳ͘ͳΔ. ରԠࡦͷҰͭͱͯ͠, ‫ڞ‬෼ࢄߦྻΛϦοδ‫Ͱܕ‬ਪఆͨ͠΋
ͱͰͷ AIC ΍ Cp Λ༻͍Δํ๏͕͋Δ. ·ͨ Moore-Penrose ͷҰൠԽ‫ྻߦٯ‬Λ༻͍Δํ๏
΋ߟ͑ΒΕΔ. ଞͷํ๏ͱͯ͠, ଟมྔճ‫ؼ‬ϞσϧͰఏҊ͞Εͨ Bunea, She and Wegkamp
(2011) ʹΑΔ RSC Λద༻͢Δ͜ͱ΋ߟ͑ΒΕΔ. ͜ΕΒͷํ๏ʹ͍ͭͯͷߴ࣍‫ݩ‬ੑ࣭ʹͭ
͍ͯཧ࿦త਺஋త݁Ռʹ͍ͭͯ΋ใࠂ͢Δ.
ࢀߟจ‫ݙ‬
1. Bunea, F., She, Y. and Wegkamp, M. H. (2011). Optimal selection of reduced
rank estimators of high-dimensional matrices. Ann. Statist., 39, 1282–1309.