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B3_Seminar_05
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kakubari
February 24, 2017
Technology
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73
B3_Seminar_05
ビックデータ解析入門3
kakubari
February 24, 2017
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Transcript
Ԭٕज़Պֶେֶ ిؾిࢠใֶ՝ఔ ֶ෦ɹ֯ுཽ ࣗવݴޠݚڀࣨ ɹ#̏θϛ ʙୈճʙ ϏοΫσʔλղੳೖᶅ
目次 ˔ͱϞʔϝϯτ ˔͖ͱϞʔϝϯτ ˔͍ҙຯͰͷ͖ ˔ཁ౷ܭྔ ˔౷ܭྔͷਪఆ
分布と統計量 ˔ूஂͷதͷͲͷཁૉબΕΔ֬Λಉ͡ʹ͢Δ ɹແ࡞ҝநग़๏ɺϥϯμϜαϯϓϦϯά ˔ಘΒΕͨඪຊ ɹແ࡞ҝඪຊɺϥϯμϜαϯϓϧ நग़ͨ͠ඪຊͷ࣮ଌʹج͍ͮͯɺ ूஂɺฏۉɺࢄΛਪఆ͢Δɻ
શମ͔ΒภΓͳ͘औΓग़ͨ͠Ұ෦͔ΒશମͷಛੑΛΔ
分布とモーメント ˔Ϟʔϝϯτͱ ɹฏۉࢄͷΑ͏ʹΛಛ͚Δྔ ྫ͑ʜ ਖ਼نฏۉͱࢄ͕༩͑ΒΕΕɺ࠶ݱ͕Մೳ ฏۉࢄͦΕͧΕ̍࣍ɺ̎࣍ͷϞʔϝϯτ ɹฏۉɿ ɹࢄɿ
µ = E[x]= x ⋅ f (x)dx ∫ σ 2 = E[(x −µ)2 ]= (x −µ)2 ⋅ f (x)dx ∫
分布とモーメント ˔Ұൠతͳʹରͯ͠ ɹฏۉࢄΑΓߴ࣍ͷϞʔϝϯτ·Ͱߟ͑Δ͜ͱͰ Λಛ͚Δ ˔֬ʹ͓͍ͯ ɹЋΛத৺ͱͨ͠ҰൠԽ͞ΕͨϞʔϝϯτͷఆٛ
E[(x −α)n ]= (x −α)n ⋅ f (x)dx ∫
べき分布とモーメント ˔ϞʔϝϯτʹΑΔͷಛ͚ͮ ཧ্औΓѻ͍͍͢ ࣮ࡍʹଟ͘ͷ౷ܭख๏ͰҊʹԾఆ͞Ε͍ͯΔ ଟ͘ͷ߹ɺਖ਼نࢦͰ͋Δɻ
Ὃ ݱ࣮ͰɺҟͳΔʹै͏֬ม͕͋Δɻ ͦͷΑ͏ͳ֬มʹै͍ͬͯΔσʔλͰɺ ؍ଌ͞Εͨʹରͯ͠౷ܭख๏͕దͰ͋Δ͔ҙ ͖తͳΛ࣋ͭ
べき分布とモーメント ˔͖ͱʜ ɹɾҝସՁ֨ࠩͷ ɹɾॴಘ͕େ͖͍ྖҬͰͷݸਓॴಘͷ ɹɾจষதͷ୯ޠͷස ࣾձݱɺࣗવݱ
べき分布とモーメント P(≥ x) = Ax−α Լهͷྦྷੵؔʹै͏Λ͖ͱ͍͏ɻ
"ن֨Խఆ ͖͕ͦ͢ް͍͜ͱΛಛͱ͢Δ ʢۃʹେ͖ͳΛ࣋ͭݱ͕ਖ਼نΑΓى͜Γ͍͢ʣ (x ≥ A 1 α ) (1)
べき分布とモーメント ˔͖ͷੑ࣭ ଟ͘ͷখ͞ͳͱগͳ͍ܻҧ͍ʹେ͖ͳΛͱΔ ͷΛؚΉ ਖ਼نΑΓߴ͍֬Ͱܻҧ͍ʹେ͖ͳΛͱΔ ྦྷੵؔΛ྆ରͰϓϩοτ͢ΔͱઢʹͳΔɻ ઢͷ͖͖ࢦЋͰ͋Δɻ
Ћ㱡̎ͰࢄɺЋ㱡̍Ͱฏۉ͕ଘࡏ͠ͳ͍ɻʢЋ࣍ Ҏ্ͷϞʔϝϯτ͕ଘࡏ͠ͳ͍ʣ Ћ̍Ͱɺ࠷େͷγΣΞ͕αϯϓϧ/ˠ㱣Ͱ̌ ʹͳΒͳ͍ɻ ɹ S max = max(x 1 , x 2 ,!, x N ) x i k=1 N ∑
べき分布とモーメント ಛʹࡾͭͷੑ࣭ɺσʔλͷ͕͖ʹै͍ͬͯΔ ͔͔֬ΊΔͨΊʹσʔλղੳʹ͏ɻ σʔλͷྦྷੵΛॻ͘ɻ ྆ରϓϩοτ͠ɺઢͰ͋Δ͜ͱΛ͔֬ΊΔɻ ۙࣅઢΛٻΊɺࢦЋΛٻΊΔɻ
広い意味でのべき分布 ͕ࣜݫີʹΓཱͭ͜ͱݱ࣮ʹͳ͍ɻ Ὃ ͷઈର͕େ͖͍ྖҬͰɺ ͖ؔͰۙࣅͰ͖ΔΑ͏ͳ
͖ͷΛ࣋ͭͱ͍͍ɺ૯͖ͯ͡ͱݺͿɻ ྫʣɾٯΨϯϚ ɹɹɾθʔλ ɹɹଞʹଟ͋Δɻ
要約統計量 ˔ཁ౷ܭྔͱ ɹඪຊͷ࣋ͭੑ࣭Λఆྔతʹಛ͚Δྔ ɾҐஔʹؔ͢Δཁ౷ܭྔ ඪຊฏۉɺதԝ ɾईʹؔ͢Δཁ౷ܭྔ ࢄɺඪ४ภࠩ
統計量の推定 ˔ϏοΫσʔλͷॲཧ ؍ଌ͞Εͨσʔλ͔Βཁ౷ܭྔΛ༻͍ͯɺ ɾͦͷ֬ີؔͷύϥϝʔλΛٻΊΔ ɾσʔλ͕ै͏ํఔࣜͷύϥϝʔλΛٻΊΔ ੳʹΑΓɺूஂ͕࣋ͭະͷύϥϝʔλΛಘΔ ඪຊ͔Βਪఆ͢Δ ਪఆํ๏ʹɺ࠷ਪఆ࠷খೋਪఆ͕͋Δɻ
参考文献 ˔ߴ҆ඒࠤࢠฤஶɺాଜޫଠɾࡾӜߤஶɺ ɹʮֶੜɾٕज़ऀͷͨΊͷϏοΫσʔλղੳೖʯ ʢୈ̏ষʣɺ ɹגࣜձࣾຊධࣾɺ݄