ᯏ᪾ቇ⠌ℂ⺰ߦၮߠߊ㋕㍑ࡊࡠ࠮ࠬߦ߅ߌࠆ⇣Ᏹᬌ⍮ߣ⇣Ᏹ࿃ሶหቯ⎇ⓥઍ⴫⠪ᄢ㒋ᄢቇ↥ᬺ⑼ቇ⎇ⓥᚲ ഥᢎᴡේ ศિ㧝㧚✜⸒ ㋕㍑↢↥ߦ߅ߌࠆ㜞↢↥࡮቟ቯ↢↥ࠍᡰ߃ࠆߚ߼ߦߪ㧘ࡊࡠ࠮ࠬߩ⇣Ᏹ⸻ᢿᛛⴚ߇ਇนᰳߢ޽ࠆ㧚ᓥ᧪ߪ㧘ࡊࡠ࠮ࠬߩ⸃ᨆ ‛ℂ ࡕ࠺࡞ࠍ↪޿ࠆṶ➈⊛ߥ⇣Ᏹ⸻ᢿᚻᴺ߇⋓ࠎߦ⎇ⓥ࡮㐿⊒ߐࠇߡ߈ߚ㧚ߒ߆ߒ㧘㋕㍑ࡊࡠ࠮ࠬߪᭂ߼ߡᄢⷙᮨ࡮ⶄ㔀ߢ޽ࠆߣหᤨߦ㧘ਇ⏕ታߥⷐ࿃ࠍᄙಽߦ฽߻ߚ߼㧘ߘ߽ߘ߽੐೨⍮⼂ߩߺ߆ࠄᱜ⏕ߥࡕ࠺࡞ࠍ✂⟜⊛ߦ₪ᓧߔࠆ੐ߪ࿎㔍ߢ޽ࠆ㧚ᓥߞߡ㧘ࡊࡠ࠮ࠬ߆ࠄᓧࠄࠇࠆ࠺࡯࠲ࠍ೑↪ߒ㧘ࠪࠬ࠹ࡓߦ㑐ߔࠆᄢၞ⊛ߥᖱႎࠍ⋙ⷞߔࠆ੐ߦࠃߞߡ⇣Ᏹ⸻ᢿࠍታⴕߔࠆ㧘Ꮻ⚊⊛ߥࠕࡊࡠ࡯࠴߇․ߦ᦭ലߢ޽ࠆߣ⠨߃ࠄࠇࠆ㧚ߥ߅㧘ߎࠇࠄ㧞ߟߩࠕࡊࡠ࡯࠴ߪ⋧෻ߔࠆ߽ߩߢߪߥߊ㧘⋧⵬⊛ߥ㑐ଥߦ޽ࠆ㧚 ߎߩࠃ߁ߥ⢛᥊ߩ߽ߣ㧘⎇ⓥઍ⴫⠪ߪ㧘ㆊ෰ᢙᐕߦࠊߚࠅ ,(' ᛛ⎇㧔ᒰᤨ㧕ߣߩ౒ห⎇ⓥߦ߅޿ߡ㧘ߎߩࠃ߁ߥᏫ⚊⊛ߥࠕࡊࡠ࡯࠴ߩ㋕㍑ࡊࡠ࠮ࠬ߳ߩታ↪ൻࠍ೨ឭߣߒߚᛛⴚ㐿⊒ࠍⴕߞߡ߈ߚ㧚ߎߩ౒ห⎇ⓥߢߪ㧘޽ࠆ․ቯߩࡊࡠ࠮ࠬࠍኻ⽎ߦౕ૕⊛ߥ໧㗴⸃᳿ߩߚ߼ߩታ࠺࡯࠲⸃ᨆࠍਛᔃߦ⎇ⓥࠍㅴ߼ߡ߈ߚ⚻✲߇޽ࠆ㧚ᧄ⎇ⓥߦ߅޿ߡߪ㧘ߎߩ౒ห⎇ⓥߣߩ዁᧪⊛ߥ᦭ᯏ⊛ߥⲢวߦࠃࠆታ↪ൻ߳ߩᔕ↪ࠍᔨ㗡ߦ⟎߈ߟߟ߽㧘ࠃࠅၮ␆⊛ߥℂ⺰⎇ⓥߦὶὐࠍวࠊ
ߖⴕࠊࠇߚ㧚․ߦ㧘ㄭᐕߩ㋕㍑↢↥ࡊࡠ࠮ࠬߩᕆㅦߥ㜞ᐲൻ࡮ⶄ㔀ൻࠍ㐓ߺ㧘ᰴߩ㧞ߟߩ࠹࡯ࡑࠍਛᔃߦ㧘ᄢⷙᮨ࡮ⶄ㔀ߥᎿቇࠪࠬ࠹ࡓߦኻߒߡㆡ↪น⢻ߥ⇣Ᏹ⸻ᢿߩ৻⥸⊛ᨒ⚵ߺߩ᭴▽ߣࠕ࡞ࠧ࡝࠭ࡓߩ㐿⊒ࠍⴕߞߚ㧔࿑㧝ࠍෳᾖ㧕㧚⺖㗴㧭 ㅙᰴኒᐲᲧផቯߦၮߠߊᄌൻᬌ಴㋕㍑ࡊࡠ࠮ࠬߥߤ߆ࠄᓧࠄࠇࠆ᷹ⷰ࠺࡯࠲ߪ㧘ࠪࠬ࠹ࡓ߇ᱜᏱ⁁ᘒߦ޽ࠆ႐วߦߪ㧘޽ࠆ৻ቯߩࡕ࠺࡞ ⏕₸ኒᐲಽᏓ ߆ࠄ↢ᚑߒߡ޿ࠆߣ⠨߃ࠄࠇࠆ㧚ᓥߞߡ㧘⃻࿷ߦㄭ޿৻ቯ඙㑆ߣㆊ෰ߩ඙㑆ߩ᷹ⷰ࠺࡯࠲ߩ♽೉ࠍ↢ᚑߔࠆ⏕₸ኒᐲಽᏓࠍㅙᰴᲧセߔࠆ੐ߢ㧘ࠪࠬ࠹ࡓߩᄌൻ ⇣Ᏹ ߩᐲว޿ࠍ⹏ଔߔࠆ੐߇ߢ߈ࠆ㧚ߎߩ⠨߃ᣇߦၮߠ߈㧘 ߟߩ⏕₸ኒᐲಽᏓߩᲧࠍㅙᰴផቯߔࠆ੐ߦࠃࠅ㧘ࡊࡠ࠮ࠬߩ⇣Ᏹࠍᬌ಴ߔࠆᣇᴺߩ㐿⊒ࠍⴕߞߚ㧚⺖㗴㧮 ࿃ሶⷐ࿃ߩផቯᴺ㋕㍑ࡊࡠ࠮ࠬࠍߪߓ߼ߣߒߚᎿቇࠪࠬ࠹ࡓߩ⇣Ᏹ⸻ᢿߦ߅޿ߡߪ㧘⇣Ᏹߩᬌ⍮ߦട߃㧘ߘߩ⇣Ᏹߩⷐ࿃ࠍផቯߒ㧘ේ࿃ⓥ᣿ࠍⴕ߁੐߇㊀ⷐߣߥࠆ㧚ᧄ⎇ⓥߢߪ㧘․ᓽㆬᛯ߿࿃ᨐផ⺰ߣ๭߫ࠇࠆᯏ᪾ቇ⠌ߩᨒ⚵ߺߦၮߠ߈㧘⇣Ᏹ߳ߩฦ᷹ⷰ㊂ߩነਈᐲࠍ᷹ⷰ࠺࡯࠲߆ࠄ⛔⸘⊛ߦ⹏ଔߔࠆᣇᴺߦߟ޿ߡ⎇ⓥࠍⴕߞߚ㧚࿑㧝ᧄ⎇ⓥߩో૕௝

㧙㧙㧞㧚⎇ⓥታᣉౝኈ㧞.㧝 ㅙᰴኒᐲᲧផቯߦၮߠߊᄌൻᬌ಴ ࠪࠬ࠹ࡓ߆ࠄᓧࠄࠇࠆ࠺࡯࠲ࠍ↪޿ߚ⇣Ᏹ⸻ᢿߦ߅ߌࠆᦨ߽㊀ⷐߥ࠲ࠬࠢߩ৻ߟߪ㧘⇣Ᏹߩᬌ⍮㧘ߔߥࠊߜ࠺࡯࠲ࠍ↢ᚑߔࠆࠪࠬ࠹ࡓౝߩ૗ࠄ߆ߩᄌ⺞ࠍߢ߈ࠆߛߌᣧᦼߦᬌ಴ߔࠆ੐ߢ޽ࠆ㧚⇣Ᏹᬌ⍮໧㗴ߪ㧘޽ࠆᤨὐࠍႺߣߒߚ೨ᓟߩ৻ቯᤨ඙㑆ߩ࠺࡯࠲ࠍ↢࿑㧞㧚ᄌൻὐᬌ⍮ߩ᭎ᔨ࿑
ᚑߒߡ޿ࠆࡕ࠺࡞ ⏕₸ಽᏓ ߇หߓߢ޽ࠆߩ߆㧫޽ࠆ޿ߪ㆑߁߽ߩߥߩ߆㧫ࠍ࠺࡯࠲ߩߺ߆ࠄㅙᰴផቯߔࠆ໧㗴ߣߒߡ⠨߃ࠆ੐߇ߢ߈ࠆ ࿑㧞ෳᾖ㧕㧚ߎߩࠃ߁ߥ໧㗴ߪ㧘ᯏ᪾ቇ⠌߿⛔⸘ಽ㊁ߦ߅޿ߡߪᄌൻὐᬌ⍮໧㗴ߣߒߡ⍮ࠄࠇ㧘ℂ⺰⊛ᨒ⚵ߺߩ᭴▽߇ฎߊ߆ࠄⴕࠊࠇߡ޿ࠆ㧚ᧄ⎇ⓥߢߪ㧘ㄭᐕᯏ᪾ቇ⠌ಽ㊁ߢឭ᩺ߐࠇߚኒᐲᲧផቯߣ๭߫ࠇࠆේℂߦၮߠ߈㧘ߎࠇࠍᄌൻᬌ಴߳ߣㆡ↪ߔࠆ੐ߦࠃࠅ㧘ේℂ⊛ߦᕈ⢻߇଻⸽ߐࠇ㧘߆ߟታ↪⊛ߦ㜞޿♖ᐲࠍᜬߞߚᣇᴺߩ᭴▽ࠍⴕߞߚ ᢥ₂  㧚 ೨ㅀߩࠃ߁ߦ㧘ᄌൻὐᬌ⍮໧㗴ߪ㧘㧞ߟߩ࠺࡯࠲ࠍ↢ᚑߔࠆಽᏓࠍᲧセߔࠆ໧㗴ߣߒߡߣࠄ߃ࠆ੐߇ߢ߈ࠆ㧚੹㧘ࡊࡠ࠮ࠬ߆ࠄᓧࠄࠇߚᤨ♽೉࠺࡯࠲ࠍ Y (t ) [ y (t ) T , y (t  1) T , ..., y (t  k  1) T ]T ߣߔࠆ㧚ߎߩߣ߈㧘ߎߎߢߪ㧘ᄌൻὐᬌ⍮໧㗴ࠍᰴߩࠃ߁ߥ㧞ߟߩ઒⺑ߩߤߜࠄ߇ࠃࠅ⏕߆߆ࠍᲧセߔࠆ໧㗴ߣߒߡ⠨߃ࠆ㧔⃻ᤨೞࠍ t ߣߒߡ㧘ኻ⽎ߣߔࠆᤨ඙㑆ߪ t rf d i d t 㧕㧦

H 0 : p(Y(i ))prf (Y(i )) for t rf d i d t

H 1 : p(Y(i ))prf (Y(i )) for t rf d i d t tepte (Y(i )) for t te d i d tp(Y(i ))ߟ߹ࠅ㧘 H 0 ߪోᤨ඙㑆ߢࡕ࠺࡞ ⏕₸ኒᐲಽᏓ ߇ prf ߢ޽ࠅ㧘 H 1 ߪᄌൻὐ߆ߤ߁߆ࠍ⺞ߴߚ޿ᤨೞࠍႺߦࡕ࠺࡞߇ prf ߆ࠄ pte ߳ߣᄌൻߔࠆߣ޿߁઒⺑ߢ޽ࠆ㧚ߎߩߤߜࠄߩ઒⺑߇ࠃࠅ⏕߆ࠄߒ޿߆ߪ㧘ߘߩዕᐲᲧࠍ⸘▚ߔࠆ੐ߢⴕࠊࠇࠆ㧚n/n

3 i rf1 prf (Yrf (i ))3 i rf1 pte (Yte (i ))n3 i rf1 prf (Yrf (i )) 3 inte1 prf (Yte (i ))਄ᑼ߆ࠄಽ߆ࠆࠃ߁ߦ㧘ߎߩዕᐲᲧߩ⸘▚ߪ㧘㧞ߟߩ⏕₸ኒᐲಽᏓ prf ߣ pte ߩᲧ ኒᐲᲧ㧕w(Y)pte (Y) prf (Y)ߩߺߦଐሽߒߡ޿ࠆ੐߇ಽ߆ࠆ㧚ᓥߞߡ㧘⏕₸ಽᏓߪᧂ⍮ߥ⸶ߢ޽ࠆ߇㧘ߎߩᲧߐ߃⸘▚ߢ߈ࠇ߫਄⸥ߩ್ቯࠍⴕ߁੐߇ߢ߈ࠆ㧚ߟ߹ࠅᦨ⚳⊛ߦߪ㧘ฦᤨೞߦ߅޿ߡ㧘w ࠍዕᐲᲧߩᑼߦઍ౉ߒߡᓧࠄࠇࠆᰴߩᜰᮡࠍ㧘࠺࡯࠲߆ࠄ⸘▚ߐࠇߚኒᐲᲧࠍ↪޿ߡ⹏ଔߔࠆ੐ߦࠃࠅᄌൻὐᬌ⍮ࠍታⴕߔࠆ੐߇น⢻ߣߥࠆ㧚Sntepte (Y(i ))¦ ln p (Y(i))
i 1rfฝ࿑㧟ߪ㧘ੱᎿ࠺࡯࠲ࠍ↪޿ߡ㧘ߎߩᜰᮡࠍ⸘▚ߒߚᤨߩ଀ߢ޽ࠆ㧚਄࿑ߪਈ߃ߚ࠺࡯࠲ߢ㧘ᤨೞ 1000 ߩᤨὐߢ㧘࠺࡯࠲ࠍ↢ᚑߔࠆࡄ࡜ࡔ࡯࠲ࠍᄌᦝߒߡੱᎿ⊛ߦᄌൻὐࠍ૞ߞߡ޿ࠆ㧚ਅ㧙㧙࿑㧟ᄌൻὐᬌ⍮ߩ଀࿑ߪ㧘⸘▚ߐࠇߚᜰᮡ S ߢ㧘ታ㓙ߩᄌൻὐߩ⋥ᓟ߆ࠄ୯߇ᕆỗߦᄢ߈ߊߥߞߡ޿ࠆ੐߇ಽ߆ࠆ㧚৻ቯߩ㑣୯ࠍਈ߃ߡߎߩჇടࠍᬌ⍮ߔࠆ੐ߦࠃࠅ㧘࠺࡯࠲ࠍ↢ᚑߒߡ޿ࠆ᭴ㅧߩᄌൻࠍߣࠄ߃ࠆ੐߇น⢻ߣߥࠆ㧚ኒᐲᲧ w ߩ⸘▚ߪ㧘ㄭᐕឭ᩺ߐࠇߚ⋥ធផቯߩᨒ⚵ߺߦၮߠߊ੐ߢน⢻ߣߥࠆ ᢥ₂=? 㧚ߎߩᨒ⚵ߺߢߪ㧘ኒᐲᲧ w ࠍᰴᑼߩࠃ߁ߥࠞ࡯ࡀ࡞㑐ᢙߩ✢ᒻ๺ߣߒߡቯ⟵ߒ㧘࠺࡯࠲߆ࠄࡄ࡜ࡔ࡯࠲(㊀ߺ)ࠍផቯߔࠆ㧚w¦ D KV (Y, Y (l ))ntei 1lteߎߎߢ㧘 K V ߪࠞ࡯ࡀ࡞㑐ᢙߢ޽ࠅ㧘଀߃߫ᰴߩࠃ߁ߦቯ⟵ߐࠇࠆ㧚|| Y  Y ||2
K V (Y , Y ) exp()2V 2ߒ߆ߒᓥ᧪ឭ᩺ߐࠇߚኒᐲᲧផቯߩᣇᴺߪ㧘ࡃ࠶࠴⊛ߥ⸘▚ ࠺࡯࠲ࠍ৻᜝ߢਈ߃ࠆᔅⷐ࿑㧠ㅙᰴኒᐲᲧផቯࠕ࡞ࠧ࡝࠭ࡓߩ⇼ૃࠦ࡯࠼߇޽ࠆ ࠍᔅⷐߣߒ㧘࡝ࠕ࡞࠲ࠗࡓߢታⴕߔ
ࠆᔅⷐ߇޽ࠆ⇣Ᏹ⸻ᢿߦ߅޿ߡߪ⸘▚߇࿎㔍ߣߥࠆ㧚ߘߎߢᧄ⎇ⓥߢߪ㧘ฦᤨೞߢᣂߒ޿࠺࡯࠲߇ᓧࠄࠇࠆᐲߦ㧘ࡄ࡜ࡔ࡯࠲ Į ࠍᦝᣂߒߡ޿ߊ੐߇น⢻ߥㅙᰴဳߩࠕ࡞ࠧ࡝࠭ࡓࠍዉ಴ߒߚ㧚࿑㧠ߪߘߩ⇼ૃࠦ࡯࠼ߢ޽ࠅ㧘࿑ਛ㧘 nrf ߅ࠃ߮ nte ߪฦ‫ޘ‬ෳᾖ඙㑆෸߮⹏ଔ඙㑆ߦ޽ࠆ࠺࡯࠲ߩࠨࡦࡊ࡞
ᢙ㧘K ߿ O ߪࡄ࡜ࡔ࡯࠲ߩᦝᣂ₸ࠍቯ߼ࠆࡄ࡜ࡔ࡯࠲ߣߥߞߡ޿ࠆ㧚 ᓧࠄࠇߚࠕ࡞ࠧ࡝࠭ࡓߩታ㛎⊛ߥᬌ⸽ߪᢥ₂=?ߦ⹦ߒ޿߇㧘ᧄ⎇ⓥߢߪ㧘޽ࠆ㋕㍑ࡊࡠ࠮ࠬߦ
ኻߒߡ߽ឭ᩺ᚻᴺߩㆡ↪ࠍⴕߞߡ޿ࠆ㧚࿑㧡ߪߘߩ৻଀ߢ޽ࠆ㧚਄߆ࠄ㧟ߟ⋡ߩࠣ࡜ࡈ߇ታ㓙ߩ࠺࡯࠲ߢ޽ࠅ ৻⇟ਅߩࠣ࡜ࡈߪߎࠇࠍ๟ᵄᢙᲤߦಽ⸃ߒߚ߽ߩߢ޽ࠆ 㧘ឭ᩺ߒߚᚻᴺࠍㆡ↪ߒߚ⚿ᨐߪ਄߆ࠄ㧞ߟ⋡ߩࠣ࡜ࡈߦߥࠆ㧚࿑ਛ㧘ᤊශ߇ᬌ⍮ߐࠇߚᄌൻὐߢ޽ࠆ㧚߹ߚ৻⇟਄ߩࠣ࡜ࡈߪ
એ೨ឭ᩺ߒߚᚻᴺ ㇱಽⓨ㑆ᴺ㧘ᢥ₂=? ࠍㆡ↪ߒߚ⚿ᨐߢ޽ࠆ㧚ߎߩᚻᴺߪ㧘ࡕ࠺࡞ࠍ᣿␜⊛ߦਈ߃ࠆᚻᴺߢ޽ࠅᧄᚻᴺߦᲧߴ࠺࡯࠲ߦኻߔࠆᨵエᕈ߇ഠࠆ௑ะ߇޽ࠆ߇㧘㧞ߟߩᚻᴺߩㆡ↪ߦࠃࠅᬌ⍮ߐࠇߚᤨὐ߇⇣ߥߞߡ޿ࠆㇱಽߪߎࠇߦ⿠࿃ߔࠆߣ⠨߃ࠄࠇࠆ㧚ㇱಽⓨ㑆ᴺឭ᩺ᚻᴺ࿑㧡޽ࠆ㋕㍑ࡊࡠ࠮ࠬ߆ࠄᓧࠄࠇߚ࠺࡯࠲߳ߩㆡ↪଀㧙㧙㧞.㧞 ࿃ሶⷐ࿃ߩផቯᴺ ⇣Ᏹߩᣧᦼ⊒⷗ߣ౒ߦ㧘⇣Ᏹ⸻ᢿߦ߅޿ߡ߽߁৻ߟߩ㊀ⷐߥ࠲ࠬࠢߪ㧘ߘߩ࿃ሶࠍ޿߆ߦផቯߔࠆ߆ߣ޿߁໧㗴ߢ޽ࠆ㧚ᧄ⎇ⓥߢߪ㧘ᰴߩੑߟߩࠕࡊࡠ࡯࠴߆ࠄ㧘ߎߩ໧㗴ߦᔅⷐߣᕁࠊࠇࠆၮ␆ᛛⴚߦߟ޿ߡ⎇ⓥࠍⴕߞߚ㧚Ԙ ࿃ᨐ᭴ㅧߩផቯߦࠃࠆࠕࡊࡠ࡯࠴ԙ ․ᓽㆬᛯߦࠃࠆࠕࡊࡠ࡯࠴
ߎࠇࠄߦ߅޿ߡ౒ㅢߩ⠨߃ᣇߪ㧘․ቯߩ⌕⋡ߔࠆ᷹ⷰᄌᢙߦᓇ㗀ࠍਈ߃ߡ޿ࠆ㧘ߘࠇએᄖߩᄌᢙࠍ․ቯߔࠆ੐ߦ޽ࠆ㧚ߎࠇߦࠃࠅ㧘޽ࠆ᷹ⷰᄌᢙߦ⇣Ᏹ߇⷗ࠄࠇߚ႐ว㧘ߘࠇߦᓇ㗀ࠍਈ߃ߡ޿ࠆᄌᢙ߆ࠄ㧘ߘߩේ࿃ࠍⓥ᣿ߔࠆ੐߇น⢻ߣߥࠆ㧚એਅ㧘ฦ‫ߡ޿ߟߦޘ‬ㅀߴࠆ㧚Ԙ࿃ᨐ᭴ㅧߩផቯߦࠃࠆࠕࡊࡠ࡯࠴ ᷹ⷰ࠺࡯࠲߇ⶄᢙߩᄌᢙ߆ࠄ᭴ᚑߐࠇࠆ႐ว㧘⊒↢ߒߚ⇣Ᏹේ࿃ߩⓥ᣿ߦߪ㧘ᄌᢙ㑆ߩ࿃ᨐ㑐ଥࠍផቯߔࠆ੐߇᦭↪ߢ޽ࠆߣ⸒߃ࠆ㧚৻⥸ߦ㧘㋕㍑ࡊࡠ࠮ࠬߩࠃ߁ߥࠪࠬ࠹ࡓ߆ࠄขᓧߐ
ࠇࠆߩߪᤨ♽೉࠺࡯࠲ߢ޽ࠆߚ߼㧘ᧄ⎇ⓥߢߪ㧘ᤨ♽೉࠺࡯࠲ y (t ) : ( y1 (t ), ..., y p (t ))T ߩᄌᢙ㑆ߩ࿃ᨐ㑐ଥࠍផቯߔࠆᣇᴺߦ㑐ߒߡ⎇ⓥࠍⴕߞߚ㧚ߟ߹ࠅ᷹ⷰ࠺࡯࠲߆ࠄ㧘ᤨ♽೉ߩฦ᷹ⷰᄌ
ᢙ㑆ߩේ࿃ߣ⚿ᨐߩ㑐ଥ yi  y j ࠍផቯߔࠆ໧㗴ߦߟ޿ߡߩᬌ⸛ࠍⴕߞߚ㧚㋕㍑ࡊࡠ࠮ࠬߩࠃ߁ߥ‛ℂ⊛ߥࠪࠬ࠹ࡓࠍ⴫ߔᦨ߽৻⥸⊛ߥᤨ♽೉ࡕ࠺࡞ߪ㧘⥄Ꮖ࿁Ꮻ⒖േ
ᐔဋ #4/# ࡕ࠺࡞ߣ๭߫ࠇࠆࡕ࠺࡞ߢ޽ࠆ㧚y (t )¦ ) y (t  i)  İ(t )  ¦ 4 İ(t  j ) qpi 1ij 1iİ(t ) ߪࡁࠗ࠭ߢ޽ࠅ㧘) i ߅ࠃ߮ 4 j ߪࡕ࠺࡞ߩࡄ࡜ࡔ࡯࠲ߢ޽ࠆ㧚ᧄ⎇ⓥߢߪ㧘ߎߩࡕ࠺࡞ߦၮߠ߈㧘ᦝߦ࠺࡯࠲ߩ㕖ᱜⷙᕈࠍ↪޿ߡ㧘ᄌᢙ㑆ߩ࿃ᨐ㑐ଥࠍផቯߔࠆᣇᴺࠍዉ಴ߒߚ㧚࠺࡯࠲߇ᜬߟ㕖ᱜⷙᕈߪ㧘ㄭᐕ࿃ᨐ㑐ଥߩផቯߦ߅޿ߡᭂ߼ߡ᦭↪ߢ޽ࠆ੐߇⊒⷗ߐࠇ㧘ߎߩ㕖ᱜⷙᕈࠍ↪޿ߡ㕖ᓴⅣဳߩ᦭ะࠣ࡜ࡈࠍផቯߔࠆᣇᴺߪ .K0)#/ ᴺߣߒߡ⍮ࠄࠇߡ޿ࠆ ᢥ₂=? 㧚.K0)#/ ᴺߢߪ㧘࠺࡯࠲߇ᜬߟ㕖ᱜⷙᕈࠍ೑↪ߒߡ㧘⁛┙ᚑಽಽᨆ +%# ࠍㆡ↪ߔࠆ੐ߦࠃࠅᄌᢙ㑆ߩ㑐ଥࠍផቯߔࠆ㧚ᧄ⎇ⓥߢߪߎߩ .K0)#/ ᴺࠍ㧘#4/# ߦၮߠߊᤨ♽೉ᄌᢙ㑆ߩ㑐ଥߩផቯߦㆡ↪ߔࠆ㧚ߎߩ㓙㊀ⷐߥ੐ߪ㧘ᤨ♽೉ߦ߅޿ߡߪ㧘㧞⒳㘃ߩᤨ㑆⊛ߥ⋧㑐ࠍ⠨ᘦߔࠆᔅⷐ߇޽
ࠆߣ޿߁ὐߢ޽ࠆ㧚ߟ߹ࠅ㧘৻⥸ߦ࠺࡯࠲ࠍขᓧߔࠆᤨ㑆㑆㓒ߪ৻ቯߢ޽ࠆ߇㧘ߘࠇߦኻߒߡ㧘ࠃࠅ㜞ㅦߥ ⍍㑆⊛ߥ ᓇ㗀ߣૐㅦߥ ㄉᗧ㑐࡜ࠣߩ޽ࠆ ᓇ㗀ࠍ⠨ᘦߔࠆᔅⷐ߇޽ࠆ㧚#4/# ࡕ࠺࡞ߪᓟ⠪ߩߺߒ߆⴫⃻ߢ߈ߥ޿ߩߢ㧘ߎߎߢߪ೨⠪߽฽߻ᰴߩࠃ߁ߥࡕ࠺࡞ࠍ⠨߃ࠆ㧚y (t )¦ < y(t  i)  e(t )  ¦ : e(t  j ) qpi 0ߎߎߢࡁࠗ࠭ e(t ) ߪ㕖ᱜⷙߢ޽ࠆߣ
઒ቯߒ㧘<i ߅ࠃ߮ : j ߪࡕ࠺࡞ߩࡄ࡜ࡔ࡯࠲ߢ޽ࠆ㧚#4/# ࡕ࠺࡞ߣߩ㆑޿ߪ㧘⍍㑆⊛ߥ㗄 i 0 ࠍ⠨ᘦߔࠆߣ޿߁ὐߩߺߢ޽ࠆ㧚ߎߩࡕ࠺࡞ߩផቯ
ߪ㧘߹ߕ  ࠺࡯࠲ࠍ↪޿ߡㅢᏱߩ
#4/# ࡕ࠺࡞ࠍផቯߒ㧘ߘߒߡ  ᓧࠄࠇ ߚ ࡁ ࠗ ࠭ ߩ ♽ ೉ İ(t ) ߦ ኻ ߒ ߡLiNGAM ᴺࠍㆡ↪ߔࠆ੐ߢ㧘ᦨ⚳⊛ߦ਄ߩ㧞ߟߩᑼߩ㑐ଥ߆ࠄࡄ࡜ࡔ࡯࠲ࠍផቯߔࠆ੐߇น⢻ߣߥࠆ㧚 ࿑㧢ߪ㧘޽ࠆ‛ℂࠪࠬ࠹ࡓ߆ࠄᓧࠄࠇߚ࠺࡯࠲  ᰴర ߦኻߒߡ㧘ዉ

ij 1

i

0.55

-1.08

1.13

0.77
1.68

1.71

0.52

3.13

1.49
4.53

3.29

1.48
-5.73

࿑㧢#4/#.K0)#/ ᴺߢផቯߒߚᄌᢙ㑆ߩ㑐ଥ㧙㧙಴ߒߚ #4/#.K0)#/ ᴺࠍㆡ↪ߒߚ଀ߢ޽ࠆ㧚⍍㑆⊛ߦߪ㧘ᄌᢙ㧝߆ࠄ㧟㧘㧞߆ࠄ㧠ߩᓇ㗀߇ሽ࿷ߒ㧘ߘߒߡ․ߦᄌ
ᢙ㧝߆ࠄ㧠߳ߩᄢ߈ߥૐㅦߩᓇ㗀߇ሽ࿷ߔࠆ੐߇ផቯߐࠇߡ޿ࠆ㧚ߎߩࠃ߁ߥᤨ♽೉ߩᄌᢙ㑆ߩ⋧㑐ࠍផቯߔࠆ੐ߢ㧘㋕㍑ࡊࡠ࠮ࠬߩࠃ߁ߥⶄᢙߩ᷹ⷰ࠴ࡖࡦࡀ࡞ࠍᜬߟ࠺࡯࠲ߩ࿃ᨐ᭴ㅧࠍផቯߒ㧘⇣Ᏹ⊒↢ᤨߩ࿃ሶផቯߦ೑↪ߢ߈ࠆߣ⠨߃ࠄࠇࠆ㧚ԙ․ᓽㆬᛯߦࠃࠆࠕࡊࡠ࡯࠴  ․ᓽㆬᛯߣߪ㧘ⶄᢙߩ․ᓽ㧔ዻᕈ ࠍᜬߞߚ࠺࡯࠲࿑㧣ࠞ࠶࠹ࠖࡦࠣࡊ࡟ࠗࡦᴺߩ᭎ᔨ࿑߆ࠄ㧘⚿ᨐߦᒝߊነਈߒߡ޿ࠆ․ᓽࠍ⥄േ⊛ߦ⷗ߟߌ಴ߔᛛⴚߢ޽ࠅ㧘ᯏ᪾ቇ⠌ಽ㊁ߦ߅ߌࠆਥⷐ⺖㗴ߩ৻ߟߢ޽ࠆ㧚޽ࠆ౉ജ x   n ߣ಴ജ y   ߣߩ㑐ଥ y f (x) ࠍ࠺࡯࠲߆ࠄផቯߔࠆ႐ว㧘౉ജ x ߩᰴర߇ᄢ߈޿ߣ߈㧘ోߡߩᰴరࠍ↪޿ࠆࠃࠅ㧘㊀ⷐߥ ߎߩ㑐ଥߦᧄ⾰⊛ߥᓎഀࠍᨐߚߔ x ߩ৻ㇱߩߺࠍ↪޿ߚᣇ߇ࠃࠅផቯߩ♖ᐲ߇ะ਄ߔࠆ႐ว߇ᄙ޿㧚ߎߩࠃ߁ߥ㧘ࠃࠅ㊀ⷐߥ x ߩᰴరࠍ⷗ߟߌ಴ߔ໧㗴߇․ᓽㆬᛯߢ޽ࠆ㧚⇣Ᏹ࿃ሶ․ቯߩ໧㗴ߪ㧘ߎߩ․ᓽㆬᛯߩ໧㗴ߣߒߡᝒ߃ࠄࠇࠆ㧚੹㧘౉ജߩฦᰴరࠍⷐ⚛ߣߔࠆࠃ߁ߥ㓸ว V {x1 , ..., xn } ࠍ⠨߃ࠆ㧚ߎߩߣ߈․ᓽㆬᛯ໧㗴ߪ㧘㓸ว V ߇ਈ߃ࠄࠇߚߣ߈㧘૗ࠄ߆ߩ⹏ଔ㑐ᢙ F (S ) ࠍᦨㆡߦߔࠆㇱಽ㓸ว S  V ࠍ⷗ߟߌࠆ໧㗴ߣߒߡቯᑼൻߔࠆ੐߇ߢ߈ࠆ㧚ߎߩ໧㗴ߦኻߒߡߪ㧘ߎࠇ߹ߢ᭽‫ߥޘ‬ᣇᴺ߇ឭ᩺ߐࠇߡ޿ࠆ߇㧘৻⥸ߦᦨ߽ࠃߊ↪޿ࠄࠇࠆᣇᴺߪ㧘(QTYCTF5GNGEVKQP ᴺ ޽ࠆ޿ߪ⽽᰼ᴺߣ߽๭߫ࠇࠆ ߣ๭߫ࠇࠆ߽ߩߢ޽ࠆ㧚ߎߩᣇᴺߪ㧘න⚐ߦ৻ߟߕߟ㊀ⷐߥᰴరࠍㆬࠎߢ޿ߊߣ޿߁߽ߩߢ޽ࠆ㧚߹ߕᦨ߽ F ({xi }) ߇⦟ߊߥࠆ xi ࠍㆬᛯߒ㧘ᰴߦ F ({xi , x j }) ߇ᦨ߽⦟ߊߥࠆ x j ࠍㆬᛯߒ㧘ߣ޿߁ࠃ߁ߦߢ޽ࠆ㧚ߎߩᣇᴺߪන⚐ߢߪ޽ࠆ߇㧘ᄙ㗄ᑼᤨ㑆ࠕ࡞ࠧ࡝࠭ࡓ 㓸วⷐ⚛ᢙ n ߦኻߒߡ㧘✢ᒻߢ⸘▚ᤨ㑆߇Ⴧടߔࠆࠕ࡞ࠧ࡝࠭ࡓ ߩਛߢ㧘ᦨ߽⦟޿ㄭૃࠍਈ߃ࠆ੐߇⍮ࠄࠇߡ޿ࠆ㧚ߒ߆ߒ㧘޽ߊ߹ߢ߽ㄭૃ⊛ߥᚻᴺߢ޽ࠅ㧘⇣Ᏹ⸻ᢿߦ߅ߌࠆಽᨆߩࠃ߁ߦ㧘ߘࠇ⒟⸘▚ߦ߅ߌࠆല₸ᕈࠍᔅⷐߣߒߥ޿႐วߦߪ㧘ᄙዋ⸘▚ࠦࠬ࠻߇㜞ߊߡ߽ࠃࠅ෩ኒߦᦨㆡߥ⸃ߦㄭ޿߽ߩࠍ᳞߼ࠆ੐߇㊀ⷐߣߥࠆ㧚ߚߛߒ㧘਄⸥ߩ໧㗴ߪ㧘෩ኒߦ⸃ߊ੐ߪᭂ߼ߡ࿎㔍ߢ޽ࠆ੐߇⍮ࠄࠇߡ߅ࠅ 02࿎㔍໧㗴ߣߒߡ⍮ࠄࠇߡ޿ࠆ 㧘ߎࠇ߹ߢ෩ኒߥ⸃ࠍ᳞߼ࠆߚ߼ߩ⎇ⓥߪ޽߹ࠅ⷗ࠄࠇߥ޿㧚ᧄ⎇ⓥߢߪ㧘෩ኒߦ޽ࠆᜰᮡࠍᦨㆡߦߔࠆࠃ߁ߥ⸃ࠍ⷗ߟߌࠆ੐߇น⢻ߥ㧘ഠࡕࠫࡘ࡜ࠞ࠶࠻ᴺߣ޿߁ᣇᴺࠍឭ᩺ߒߚ ᢥ₂=? 㧚ߎߩᣇᴺߪ㧘ഠࡕࠫࡘ࡜ᕈߣ๭߫ࠇࠆ㔌ᢔ⊛ߥ᭴ㅧࠍ೑↪
ߒ㧘ᢛᢙ⸘↹໧㗴ߥߤߦᚑഞⵣߦ↪޿ࠄࠇߡ޿ࠆࠞ࠶࠹ࠖࡦࠣ࡮ࡊ࡟ࠗࡦᴺߦၮߠ޿ߡ㧘ല₸࿑㧤⽽᰼ᴺߩ⸃߆ࠄߩᡷༀᐲว࿑㧥⸘▚ߐࠇࠆ਄⇇࡮ਅ⇇ߩ଀㧙㧙⊛ߦ⸃ࠍត⚝ߔࠆ੐ࠍน⢻ߣߔࠆ㧚ߎߩᣇᴺߩၮᧄ⊛ߥ⠨߃ᣇߪ㧘㓸ว㑐ᢙߩᕈ⾰ߢ޽ࠆഠࡕࠫࡘ࡜ᕈߣ㧘ㅪ⛯㑐ᢙߩᕈ⾰ߢ޽ࠆಲ㑐ᢙߣߩ㑐ଥࠍ↪޿ࠆὐߦ޽ࠆ㧚߹ߚ࿑㧣ߪࠞ࠶࠹ࠖࡦࠣ࡮ࡊ࡟
ࠗࡦᴺߩࠗࡔ࡯ࠫߢ޽ࠅ㧘ត⚝ⓨ㑆 ࿑ਛߢߪᄙ㕙૕ 2 ߦ߅޿ߡ㧘⸃ࠍᡷༀߒߥ޿㗔ၞ ࿑ਛߢߪ* ࠍ೥㒰ߔࠆࠃ߁ߥ⿥ᐔ㕙 ࿑ਛ * ࠍ෻ᓳ⊛ߦ᳞߼ࠆ੐ߢ㧘ᦨ⚳⊛ߦᱷࠆὐߣߒߡᦨㆡ⸃ࠍ⸘▚ߔࠆ㧚࿑㧤ߪ޽ࠆታ࠺࡯࠲ߦኻߒߡ㧘ឭ᩺ߒߚᣇᴺࠍㆡ↪ߒߚ㓙ߩ㧘(QTYCTF5GNGEVKQP ᴺߩ⸃߆ࠄߩᡷༀᐲว㧔㧑㧕ࠍ⴫ߒߚ߽ߩߢ޽ࠆ ⹦⚦ߪᢥ₂=?ෳᾖ 㧚వߦㅀߴߚࠃ߁ߦ㧘(QTYCTF5GNGEVKQPᴺߪ⦟޿ㄭૃ⸃ࠍਈ߃ࠆ੐߇⍮ࠄࠇߡ޿ࠆ߇㧘ഠࡕࠫࡘ࡜ࠞ࠶࠻ᴺߦࠃࠅᢙࡄ࡯࠮ࡦ࠻ߩᡷༀ߇น⢻ߣߥࠆ㧚߹ߚഠࡕࠫࡘ࡜ࠞ࠶࠻ᴺߪ㧘෩ኒ⸃ߩ⸘▚ߩߺߢߥߊ㧘⸘▚ߩㅜਛߢ߽⸃ߩ♖ᐲ଻⸽߇น⢻ߢ޽ࠅ㧘࿑㧥ߪ⸘▚ߐࠇߚ਄⇇࡮ਅ⇇ߩ଀ߢ޽ࠆ㧚ߎߩ਄⇇࡮ਅ⇇ߩ᏷߇㧘ℂ⺰⊛ߦ଻⸽ߐࠇࠆ⸃ߩ♖ᐲߣߥࠆ㧚㧟㧚⚿⸒ ᧄ⎇ⓥߢߪ㧘ᄢⷙᮨ࡮ⶄ㔀ࠪࠬ࠹ࡓߢ޽ࠆ㋕㍑ࡊࡠ࠮ࠬߦ߅ߌࠆ⇣Ᏹ⸻ᢿߦᔅⷐߣ⠨߃ࠄࠇࠆℂ⺰⊛ᨒ⚵ߺߩ᭴▽ࠍ⋡⊛ߣߒߡ㧘․ߦ  ኒᐲᲧផቯߦၮߠߊᄌൻᬌ಴ᴺ㧘෸߮  ࿃ሶⷐ࿃ߩផቯᴺߦ㑐ߒߡ⎇ⓥࠍⴕߞߚ㧚  ߦ㑐ߒߡߪ㧘࠺࡯࠲ࠍ↢ᚑߒߡ޿ࠆࡊࡠ࠮ࠬߩᄌ⺞ߩᬌ⍮ࠍኒᐲᲧផቯ໧㗴ߣߒߡቯᑼൻߒ㧘ㄭᐕឭ᩺ߐࠇߚ⋥ធኒᐲᲧផቯᴺߩᨒ⚵ߺࠍࠃࠅല₸ൻߒߚࠕ࡞ࠧ࡝࠭ࡓࠍዉ಴࡮ㆡ↪ߔࠆᣇᴺߦߟ޿ߡ⎇ⓥࠍⴕߞߚ㧚߹ߚ  ߦ㑐ߒߡߪ㧘࿃ᨐಽᨆߦࠃࠆࠕࡊࡠ࡯࠴ߣ㧘․ᓽㆬᛯߦࠃࠆࠕࡊࡠ࡯࠴ߦ㑐ߒߡ㧘⇣Ᏹ⸻ᢿߦᔅⷐߣ⠨߃ࠄࠇࠆၮ␆⊛ߥࠕ࡞ࠧ࡝࠭ࡓߩ㐿⊒ࠍⴕߞߚ㧚੹ᓟ㧘ࠃࠅታ↪⊛ߥⷐ⚛ࠍ⚵ߺ౉ࠇ㧘ታ㓙ߩ㋕㍑ࡊࡠ࠮ࠬߩ࠺࡯࠲߳ㆡ↪ߔࠆߥߤߩᬌ⸽ࠍⴕ޿㧘ߎࠇࠄߩၮ␆⊛ℂ⺰߇⇣Ᏹ⸻ᢿᛛⴚߩታ↪ൻ㧘ᦝߦߪᜬ⛯⊛ߥᚑ㐳ࠍᡰ߃߳ߣߟߥߍߡ੐߇ᦼᓙߐࠇࠆ㧚㧠㧚⻢ㄉ ᧄ⎇ⓥߪ㧘 ⽷ ,('㧞㧝਎♿⽷࿅ߩഥᚑߦࠃࠅⴕࠊࠇߚ߽ߩߢ޽ࠆ੐ࠍઃ⸥ߒ㧘⻢ᗧࠍ⴫ߔࠆ㧚㧡㧚ᢥ₂
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