時系列解析

4コマ分の講義用資料。時系列解析について

Transcript

1. ࣌ܥྻղੳ 5BJDIJ .VSBZBNB େࡕେֶ ࢈ۀՊֶݚڀॴ 2022/09/28 AI⼈材養成プログラム
2. 2 l࣌ܥྻzͱ͸ʁ
3. 3 ࣌ܥྻͱ͸ ࣌ܥྻ UJNF
   TFSJFT ࣌ؒͱͱ΋ʹෆنଇʹมಈ͢Δݱ৅Λɺ࿈ଓతʹ؍ଌͯ͠ಘΒΕͨ ஋ͷܥྻ 𝑡
   ؍ଌͨ࣌͠ࠁ 𝑦! 
   ࣌ࠁ 𝑡 ʹ͓͚Δ؍ଌ஋ 𝑦" , … , 𝑦# 
   ࣌ܥྻ ଟ͘ͷ࣌ؒͱͱ΋ʹมԽ͍ͯ͘͠σʔλ͸ɺ͜ͷΑ͏ͳදهͰදݱ ͢Δ͜ͱ͕Մೳ 𝑇 𝑌 ྫ
   υϧԁ 2022年7⽉1⽇ 𝑦! = 135
4. 4 ࣌ܥྻͱ͸ ೔ܦฏۉגՁ $07*%ͷײછऀ਺ ೥ฏۉؾԹͷมԽ
5. 5 ࣌ܥྻͱ͸ .PUJPO$BQUVSFͷྫ From: Matsubara, Yasuko, and Yasushi Sakurai. "Regime shifts in streams: Real-time forecasting of co- evolving time sequences." Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. 2016.
6. 6 ࣌ܥྻͱ͸ l ࣌ܥྻ༧ଌ
   কདྷͷมಈΛਪଌ l ࣌ܥྻ෼ྨ
   ༩͑ΒΕͨ࣌ܥྻΛछྨ͝ͱʹ෼ྨ l ҟৗݕ஌ 
   มԽݕ஌
   ࣌ܥྻͷதʹଘࡏ͢ΔҟৗΛൃݟ l ՄࢹԽ 
   ಛ௃நग़
   ਓ͕ؒղऍՄೳͳܗʹ࣌ܥྻΛม׵ l ੍ޚ
   ૢ࡞Մೳͳม਺ΛมԽͤ͞ɺ໨తม਺Λ੍ޚ FUD ࣌ܥྻΛѻͬͨ՝୊ 
   λεΫ
7. 7 ࣌ܥྻͱ͸ l ࣌ܥྻ༧ଌ
   কདྷͷมಈΛਪଌ l ࣌ܥྻ෼ྨ
   ༩͑ΒΕͨ࣌ܥྻΛछྨ͝ͱʹ෼ྨ l ҟৗݕ஌ 
   มԽݕ஌
   ࣌ܥྻͷதʹଘࡏ͢ΔҟৗΛൃݟ l ՄࢹԽ 
   ಛ௃நग़
   ਓ͕ؒղऍՄೳͳܗʹ࣌ܥྻΛม׵ l ੍ޚ
   ૢ࡞Մೳͳม਺ΛมԽͤ͞ɺ໨తม਺Λ੍ޚ FUD ࣌ܥྻΛѻͬͨ՝୊ 
   λεΫ
8. 8 ࣌ܥྻͱ͸ ࣌ܥྻ༧ଌ ？ ݱࡏ কདྷ ֶश
9. 9 ࣌ܥྻͱ͸ ࣌ܥྻ༧ଌ ݱࡏ কདྷ ֶश
10. 10 ࣌ܥྻͱ͸ ࣌ܥྻ෼ྨ from: https://medium.com/@hassanismailfawaz/deep-learning-for-time- series-classification-a-brief-overview-73b58767ed0f ？
11. 11 ࣌ܥྻͱ͸ ҟৗݕ஌ 
    มԽݕ஌ From: https://neptune.ai/blog/anomaly-detection-in-time-series ҟৗʂ ҟৗʂ
12. 12 ࣌ܥྻͱ͸ ಛ௃நग़ %FDPNQPTJUJPO ྫ
    ࣌ܥྻ෼ղ ྫߦྻ෼ղ from: https://www.statsmodels.org/dev/generated/statsmodels.tsa.seasonal.seasonal_decompose.html from: https://www.researchgate.net/figure/Matrix-decomposition-vs-tensor-decomposition- a-low-rank-matrix-decomposition_fig4_359367892
13. 13 ࣌ܥྻͱ͸ ੍ޚ from: https://xtech.nikkei.com/atcl/nxt/news/18/08031/
14. 14 ࣌ܥྻͱ͸ ओͳϞσϦϯάϑϨʔϜϫʔΫ モデル σʔλ ܦݧత஌ࣝ ཧ࿦ ࣌ܥྻ༧ଌ ࣌ܥྻ෼ྨ ҟৗ஋ݕ஌
15. 15 ࠓճͷେ·͔ͳ֓ཁ l ౷ܭతϞσϧ l ࣌ܥྻͷੑ࣭ l ݹయత࣌ܥྻϞσϧ l Ϟσϧͷਪఆɾબ୒ l ਂ૚ֶशϕʔεϞσϧ l ۙ೥ͷਂ૚ֶशͷൃల l ࣌ܥྻ༧ଌ
    ਂ૚ֶश ౷ܭతϞσϧ 
    ਂ૚ֶशϕʔεϞσϧ
16. 16 ࣌ܥྻͷੑ࣭
17. 17 ࣌ܥྻͷੑ࣭ l ౳ִؒ࣌ܥྻ l αϯϓϦϯάظ͕ؒҰఆ l ݚڀͳͲͰѻ͏ݚڀͷ΄ͱΜͲ͸͜Εʹ౰ͨΔ ܽଛ஋ิ׬ͳͲͰ౳ִؒʹ͢Δ৔߹΋ l ෆ౳ִؒ࣌ܥྻ l Πϕϯτ࣌ܥྻͳͲ l αϯϓϦϯάظ͕ؒૄΒ ౳ִؒ࣌ܥྻ 
    ෆ౳ִؒ࣌ܥྻ ࣌ؒ 猫かわいい 1PTU 35 35
18. 18 ࣌ܥྻͷੑ࣭ l Ұมྔ࣌ܥྻ l छྨͷܥྻʹΑͬͯߏ੒ l ଟมྔ࣌ܥྻ l ෳ਺ͷܥྻʹΑͬͯߏ੒ Ұมྔ࣌ܥྻ ଟมྔ࣌ܥྻ
19. 19 ࣌ܥྻͷੑ࣭ l ૬Ճ ฏۉ l ෼ࢄ l ฏۉ͔Βͷ͹Β͖ͭ۩߹Λࣔ͢ࢦඪ l ฏۉ஋͔Βͷࠩ෼Ͱ͋ΔภࠩΛೋ৐ͷฏۉ l ڞ෼ࢄ l ڞ෼ࢄ͕େ͖͍ͱਖ਼ͷڧ͍ؔ܎Λ࣋ͭͳͲɺ ૊ͷରԠ͢Δσʔλͷؔ܎ Λࣔ͢ ఆৗੑඇఆৗੑ 𝐸(𝑦) = µ = 1 𝑛 ) 𝑦! 𝜎" = 𝑉𝐴𝑅 𝑦! = 1 𝑛 ) 𝑦! − µ " Cov(𝑥, 𝑦) = 𝜎#$ = E[ 𝑥 − µ# (𝑦 − µ$ )]
20. 20 ࣌ܥྻͷੑ࣭ l ڧఆৗੑ TUSJDU
    TUBUJPOBSJUZ l ೚ҙͷtͱ𝑘 ೚ҙͷ࣌ؒࠩ ʹରͯ͠ɺ σʔλΛੜ੒͢Δ֬཰աఔ f(𝑦!, … , 𝑦!+,)͕ৗʹಉҰͷಉ࣌෼෍Λ࣋ͭ l ֬཰෼෍͕࣌ؒมԽ͠ͳ͍ l ྫ
    αΠίϩͷग़໨ͷܥྻ΍ίΠϯτεͷ݁ՌͳͲ ఆৗੑඇఆৗੑ f 𝑦% , … , 𝑦& = f 𝑦' , … , 𝑦'(&
21. 21 ࣌ܥྻͷੑ࣭ l ऑఆৗੑ XFBL
    TUBUJPOBSJUZ l ظ଴஋ͱࣗݾڞ෼ࢄ͕࣌఺ʹґଘ͠ͳ͍ l ೚ҙͷtͱ𝑘 ೚ҙͷ࣌ؒࠩ ʹରͯ͠ҎԼ͕੒Γཱͭ ࣌఺ʹґଘ͠ͳ͍ l ࣍ͷϞʔϝϯτ·Ͱ͕ෆม l Ұൠతʹɺʮఆৗੑʯͱݺ͹ΕΔ࣌ɺͪ͜ΒΛࢦ͢ࣄ͕ଟ͍ l ࣗݾ૬͕ؔLͱͱ΋ʹࢦ਺తʹݮਰ ఆৗੑඇఆৗੑ E 𝑦* = E 𝑦*(& , VAR 𝑦* = VAR 𝑦*(& Cov 𝑦* , 𝑦*(& = 𝛾& > 0
22. 22 ࣌ܥྻͷੑ࣭ l ऑఆৗੑ XFBL
    TUBUJPOBSJUZ l ظ଴஋ͱࣗݾڞ෼ࢄ͕࣌఺ʹґଘ͠ͳ͍ l ೚ҙͷtͱ𝑘 ೚ҙͷ࣌ؒࠩ ʹରͯ͠ҎԼ͕੒Γཱͭ ࣌఺ʹґଘ͠ͳ͍ l ࣍ͷϞʔϝϯτ·Ͱ͕ෆม l Ұൠతʹɺʮఆৗੑʯͱݺ͹ΕΔ࣌ɺͪ͜ΒΛࢦ͢ࣄ͕ଟ͍ l ࣗݾ૬͕ؔLͱͱ΋ʹࢦ਺తʹݮਰ ఆৗੑඇఆৗੑ E 𝑦* = E 𝑦*(& , VAR 𝑦* = VAR 𝑦*(& Cov 𝑦* , 𝑦*(& = 𝛾& > 0 ଟ͘ͷ࣌ܥྻ౷ܭతϞσϧ͕ఆৗੑΛલఏͱ͍ͯ͠Δͨ Ίɺ෼ੳ͢Δ࣌ܥྻ͕ఆৗͰ͋Δͱخ͍͠ ఆৗੑΛલఏͱ͠ͳ͍ͱɺฏۉ΍෼ࢄ͕ҰఆͰແ͘ͳΔͨ Ίɺ෼ੳ͕ؤ݈Ͱͳ͘ͳΔ
23. 23 ࣌ܥྻͷੑ࣭ l ऑఆৗ
    ਖ਼ن෼෍ 㱺 ڧఆৗ l ฏۉɺ෼ࢄɺڞ෼ࢄ͕ଘࡏ͢Δڧఆৗ 㱺 ऑఆৗ ˎ ڧఆৗΛຬ͔ͨ͢Βͱ͍ͬͯɺඞͣ͠΋ऑఆৗͷ৚݅Λຬͨ͢ ͱ͸ݶΒͳ͍ FH
    ฏۉɺ෼ࢄ͕ଘࡏ͠ͳ͍෼෍ͳͲ ίʔγʔ෼෍ͳͲͷ੄͕ॏ͘ɾ޿͍෼෍ ఆৗੑඇఆৗੑ
24. 24 ࣌ܥྻͷੑ࣭ l ଟ͘ͷϞσϦϯάͰ͸ɺೖྗ࣌ܥྻ͕ʮઢܗɾఆৗɾਖ਼نੑʯͳ ͲΛԾఆͯ͠ϞσϦϯά l ϞσϦϯάΛ༰қʹ͢ΔͨΊʹೖྗܥྻΛม׵͠ɺύϥϝʔλਪ ఆΛ༰қʹ͢Δ͜ͱ͕໨త l ର਺ม׵ l ֊ࠩม׵ l ฏ׈Խ FUDʜ ఆৗੑ΁ͷม׵
25. 25 ࣌ܥྻͷੑ࣭ l ର਺ม׵ l σʔλมಈͷܹ͍࣌͠ܥྻΛɺτϨϯυΛอͬͨ··খ͞ͳ஋ʹม׵͢Δ ख๏ l ਖ਼ͷΈͷσʔλΛෛͷ஋ΛͱΔΑ͏ʹม׵Մೳ ఆৗੑ΁ͷม׵ log 𝑦* = 𝑧* from: https://datascienceplus.com /time-series-analysis-in-r- part-2-time-series- transformations/
26. 26 ࣌ܥྻͷੑ࣭ l #PY$PYม׵ l σʔλͷ෼෍Λਖ਼ن෼෍ʹ͚ۙͮΔͨΊͷ΂͖৐ܕม׵ l 𝜆͸ύϥϝʔλͰɺ 𝜆ʹΑͬͯ ม׵ͷܗ͕ࣜมΘΔ ఆৗੑ΁ͷม׵ 𝑧* = B $!+% , (𝜆 ≠ 0) ln 𝑦 (𝜆 = 0) from: http://www.kmdatascience.com/2017/07/box-cox-transformations-in-python.html
27. 27 ࣌ܥྻͷੑ࣭ l ϩδοτม׵ l ͔Β·Ͱͷ஋ΛͱΔׂ߹֬཰ͷσʔλΛਖ਼ͱෛͷແݶେͷ஋ʹม׵͢ Δ͜ͱͰ࣌ܥྻϞσϦϯάΛ༰қʹ͢Δม׵ ఆৗੑ΁ͷม׵ 𝑧* = log $" %+$" 0,1 ↔ (−∞, ∞)
28. 28 ࣌ܥྻͷੑ࣭ l ֊ ֊ࠩม׵ l τϨϯυ੒෼ΛऔΓআ͖ɺฏۉʹؔͯ͠ఆৗੑΛຬͨ͢Α͏ʹม׵ ఆৗੑ΁ͷม׵ 𝑧* = 𝑦* − 𝑦*+% from: https://datascienceplus.com/time-series-analysis-in-r-part-2-time-series-transformations/
29. 29 ࣌ܥྻͷੑ࣭ l قઅ ֊ࠩม׵ l قઅ੒෼ʹΑΔӨڹपظੑΛऔΓআ͖ɺฏۉʹؔͯ͠ఆৗੑΛຬͨ͢Α ͏ʹม׵ l 𝑠 ͸पظ ఆৗੑ΁ͷม׵ 𝑧* = 𝑦* − 𝑦*+- = (1 − 𝐵-)𝑦* from: https://datascienceplus.com/time-series-analysis-in-r-part-2-time-series-transformations/
30. 30 ࣌ܥྻͷੑ࣭ l ฏ׈Խ l ϊΠζ΍୹ظతͳมಈΛআ͘ޮՌΛظ଴ͯ͠ɺ׈Β͔ʹͳΔΑ͏ʹม׵ l ୅දྫͱͯ͠ɺҠಈฏۉϑΟϧλ .PWJOH
    "WFSBHF
    'JMUFS ఆৗੑ΁ͷม׵ 𝑧* = 1 M ) ./0 1+% 𝑦. from: https://machinelearningmastery.com/moving-average-smoothing-for-time-series- forecasting-python/
31. 31 ࣌ܥྻͷੑ࣭ l ૬ؔ l पظੑ ࣌ܥྻͷ܏޲ͷཧղ
32. 32 ࣌ܥྻͷੑ࣭ l ࣗݾڞ෼ࢄؔ਺ l ऑఆৗաఔͷࣗݾڞ෼ࢄΛԾఆ l ࣗݾڞ෼ࢄؔ਺͸ҎԼͷܗͰఆٛ͞ΕΔ l ࣗݾڞ෼ࢄؔ਺͸ۮؔ਺
    Cov 𝑦!, 𝑦!+, = 𝛾, = 𝛾-, = Cov 𝑦!, 𝑦!-, l ࣗݾڞ෼ࢄؔ਺ͷઈର஋ͷ࠷େ͸𝛾. 
    𝛾. ≥ 𝛾, l ࣗݾڞ෼ࢄؔ਺ͷਤࣔͱͯ͠ίϨϩάϥϜ͕༻͍ΒΕΔ ࣌ܥྻͷ܏޲ͷཧղ Cov 𝑦% , 𝑦& = Cov 𝑦* , 𝑦*(& = 𝛾& 𝑘͸ϥάΛද͢
33. 33 ࣌ܥྻͷੑ࣭ l ࣗݾ૬ؔؔ਺ l ऑఆৗաఔͷࣗݾ૬ؔΛԾఆ l ҎԼͷܗͰఆٛ͞ΕΔ ࣌ܥྻͷ܏޲ͷཧղ Cor 𝑦* , 𝑦*(& = 234 $",$"#$ 678 $" 678($"#$) = ;$ ;% = 𝑅& 𝑘͸ϥάΛද͢
34. 34 ࣌ܥྻͷੑ࣭ l ࣗݾ૬ؔؔ਺ͷίϨϩάϥϜ l ܥྻ͕ఆৗͳΒɺࣗݾ૬ؔؔ਺͕͸ز Կڃ਺తʹʹऩଋ͢Δ͜ͱΛࣔ͢ l पظੑͷ͋ΔܥྻͳͲͷൃݟ΋༰қʹ ࣌ܥྻͷ܏޲ͷཧղ from: 実証のための計量時系列分析
35. 35 ࣌ܥྻͷੑ࣭ ࣌ܥྻͷ܏޲ͷཧղ from: https://elf-c.he.u-tokyo.ac.jp/courses/379
36. 36 ࣌ܥྻͷੑ࣭ ࣌ܥྻͷ܏޲ͷཧղ from: https://elf-c.he.u-tokyo.ac.jp/courses/379
37. 37 ࣌ܥྻͷੑ࣭ l εϖΫτϧղੳ l ࣌ܥྻσʔλΛߏ੒प೾੒෼ʹ෼ղ͠ɺ֤प೾਺ͱΤωϧΪʔͷؔ܎Λऔ Γग़ͨ͢Ίͷख๏ l Α͘༻͍ΒΕΔͷ͸ɺ཭ࢄϑʔϦΤม׵ ''5 l ෳࡶͳपظΛऔΓग़͢͜ͱΛՄೳʹ͢Δɻपظੑͱ͸𝑝͕पظΛࣔ࣌͢ʹ 𝑦! = 𝑦!-/ ͕੒Γཱͭ͜ͱ ࣌ܥྻͷ܏޲ͷཧղ
    εϖΫτϧղੳ from: https://qiita.com/AnchorB lues/items/5497ee68c3a3 d64875d4 FFT 振動数 振幅
38. 38 ࣌ܥྻͷੑ࣭ l ϑʔϦΤม׵ l प೾਺𝒇ɼपظؔ਺y(t ͱͨ͠ͱ͖ҎԼͷܗʹͳΔ l ෳ਺ͷ೾ͰܥྻΛઆ໌Ͱ͖ΔΑ͏ʹ෼ղ l ٯϑʔϦΤม׵ l ೾਺𝟐𝝅𝒇ͷਖ਼ݭ೾ʹରͯ͠ॏΈΛֻ͚ͯɺੵ෼͢Δ͜ͱͰݩͷܥྻʹ໭͢ ࣌ܥྻͷ܏޲ͷཧղ
    εϖΫτϧղੳ 𝒀 𝒇 = 2 "# # 𝒚(𝒕)𝒆"𝟐𝝅𝒊𝒕𝒇𝒅𝒕 ∗ ∫ "# # 𝒚 𝒕 𝒅𝒕 < ∞Λຬͨ͢ 𝒚 𝒕 = 2 "# # 𝒀(𝒇)𝒆𝟐𝝅𝒊𝒕𝒇𝒅𝒇
39. 39 ࣌ܥྻͷੑ࣭ l ϑʔϦΤม׵ l प೾਺𝒇ɼपظؔ਺y(t ͱͨ͠ͱ͖ҎԼͷܗʹͳΔ l ෳ਺ͷ೾ͰܥྻΛઆ໌Ͱ͖ΔΑ͏ʹ෼ղ l ٯϑʔϦΤม׵ l ೾਺𝟐𝝅𝒇ͷਖ਼ݭ೾ʹରͯ͠ॏΈΛֻ͚ͯɺੵ෼͢Δ͜ͱͰݩͷܥྻʹ໭͢ ࣌ܥྻͷ܏޲ͷཧղ
    εϖΫτϧղੳ ∗ ∫ "# # 𝒚 𝒕 𝒅𝒕 < ∞Λຬͨ͢ ॏΈਖ਼ݭ೾ 𝒀 𝒇 = 2 "# # 𝒚(𝒕)𝒆"𝟐𝝅𝒊𝒕𝒇𝒅𝒕 𝒚 𝒕 = 2 "# # 𝒀(𝒇)𝒆𝟐𝝅𝒊𝒕𝒇𝒅𝒇
40. 40 ࣌ܥྻͷੑ࣭ l ύϫʔεϖΫτϧ l ϑʔϦΤ੒෼𝒀(𝒇)͸೾Ͱ͋Δ𝒆𝒊𝒕𝟐𝝅𝒇ͷৼΕ෯Λද͢ 㱺 𝒀(𝒇) 𝟐͸ΤωϧΪʔ l ܥྻͷதʹؚ·ΕΔप೾਺ͱͦͷύϫʔΛՄࢹԽ ࣌ܥྻͷ܏޲ͷཧղ
    εϖΫτϧղੳ 𝒚 𝒕 = 2 "# # 𝒀(𝒇)𝒆𝟐𝝅𝒊𝒕𝒇𝒅𝒇 ॏΈਖ਼ݭ೾
41. 41 ࣌ܥྻͷੑ࣭ ࣌ܥྻͷ܏޲ͷཧղ
    εϖΫτϧղੳ 𝑓(𝑡) = B 1, 0 < 𝑡 < 𝜏 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 ೖྗܥྻ 𝐹 𝜔 = 2 "# # 𝑓(𝑡)𝑒")*!𝑑𝑡 = 2 + , 𝑓(𝑡)𝑒")*!𝑑𝑡 = 1 −𝑗𝑤 𝑒")*! + , = 1 𝑗𝑤 1 − 𝑒")*, = -!"#$/& ). 𝑒)*,/0 − 𝑒")*,/0 = 𝑒")*,/0 0 * sin *, 0 ϑʔϦΤม׵ 𝐹 𝜔 = 𝑒")*,/0 2 𝜔 sin 𝜔𝜏 2 = 2 𝜔 sin 𝜔𝜏 2 ৼΕ෯εϖΫτϧ ( 𝐹 𝜔 = 𝐹 𝜔 𝑒)1(*) ) ύϫʔεϖΫτϧ 𝐹 𝜔 0 = 2 𝜔 sin 𝜔𝜏 2 0 = 4 𝜔0 sin0 𝜔𝜏 2
42. 42 ࣌ܥྻͷੑ࣭ ࣌ܥྻͷ܏޲ͷཧղ
    εϖΫτϧղੳ 𝑓(𝑡) = B 1, 0 < 𝑡 < 𝜏 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 ೖྗܥྻ 𝐹 𝜔 = 2 "# # 𝑓(𝑡)𝑒")*!𝑑𝑡 = 2 + , 𝑓(𝑡)𝑒")*!𝑑𝑡 = 1 −𝑗𝑤 𝑒")*! + , = 1 𝑗𝑤 1 − 𝑒")*, = -!"#$/& ). 𝑒)*,/0 − 𝑒")*,/0 = 𝑒")*,/0 0 * sin *, 0 ϑʔϦΤม׵ 𝐹 𝜔 = 𝑒")*,/0 2 𝜔 sin 𝜔𝜏 2 = 2 𝜔 sin 𝜔𝜏 2 ৼΕ෯εϖΫτϧ ( 𝐹 𝜔 = 𝐹 𝜔 𝑒)1(*) ) ύϫʔεϖΫτϧ 𝐹 𝜔 0 = 2 𝜔 sin 𝜔𝜏 2 0 = 4 𝜔0 sin0 𝜔𝜏 2
43. 43 ࣌ܥྻͷੑ࣭ l ύϫʔεϖΫτϧ l ϑʔϦΤ੒෼𝒀(𝒇)͸೾Ͱ͋Δ𝒆𝒊𝒕𝟐𝝅𝒇ͷৼΕ෯Λද͢ 㱺 𝒀(𝒇) 𝟐͸ΤωϧΪʔ l ܥྻͷதʹؚ·ΕΔप೾਺ͱͦͷύϫʔΛՄࢹԽ l ΢Οʔφʔɾώϯνϯ 8JFOFS,IJOUDIOF ͷެࣜ
    ࣗݾڞ෼ࢄؔ਺𝛾, ͷϑʔ ϦΤม׵͸ύϫʔεϖΫτϧʹͳΔ ࣌ܥྻͷ܏޲ͷཧղ
    εϖΫτϧղੳ 𝒚 𝒕 = 2 "# # 𝒀(𝒇)𝒆𝟐𝝅𝒊𝒕𝒇𝒅𝒇 ॏΈਖ਼ݭ೾ Cov 𝑦", 𝑦, = 𝐶ov 𝑦!, 𝑦!+, = 𝛾, 𝑘͸ϥάΛද͢ 𝑌 𝑓 = ? !"#$ $ 𝛾!𝑒#%&'!( − 1 2 ≤ 𝑓 ≤ 1 2
44. 44 ࣌ܥྻͷੑ࣭ l ύϫʔεϖΫτϧ l ۮؔ਺ͱίαΠϯม׵ͷੑ࣭͔Β l ྫ
    ୯ৼಈ ࣌ܥྻͷ܏޲ͷཧղ
    εϖΫτϧղੳ 𝑌 𝑓 = ? !"#$ $ 𝛾!𝑒#%&'!( = 𝛾) + 2 ? !"* $ 𝛾!cos (2𝜋𝑘𝑓) 𝛾! = 𝑎% 2 cos(2𝜋𝑓)𝑘) 𝑌 𝑓 = ? !"#$ $ cos(2𝜋𝑓)𝑘)cos(2𝜋𝑓𝑘) = 1 2 ? !"#$ $ cos 2𝜋𝑘 𝑓) − 𝑓 − cos 2𝜋𝑘 𝑓) + 𝑓 = M * %+, 𝑓 = 𝑓) 0, 𝑓 ≠ 𝑓)
45. 45 ࣌ܥྻϞσϧ
46. 46 ࣌ܥྻϞσϧ ࣌ܥྻϞσϧͷߏங ⼊⼒特徴 𝑦!-W … , 𝑦!-" ͳͲ ホワイトノイズ 𝜀! 時系列モデル 時系列 𝑦!
47. 47 ࣌ܥྻϞσϧ l ࣌ؒతʹ૬͕ؔແ͍࣌ܥྻ𝜀! l ࣗݾ૬ؔؔ਺΋ l ֬཰తͳ͹Β͖ͭΛදݱ l FH
    ಠཱʹੜ੒ͨ͠ਖ਼نཚ਺ྻͳͲ ന৭ࡶԻ ϗϫΠτϊΠζ 𝐸 𝜀* = 0, 𝑉𝐴𝑅 𝜀* = 𝜎", 𝑅. = 0 𝑗͸ϥάΛද͢ from: https://machinelearningmastery.com/white-noise-time-series-python/
48. 48 ࣌ܥྻϞσϧ l 𝑝࣍ͷઢܗࠩ෼ํఔࣜʹϗϫΠτϊΠζΛՃ͑ͨϞσϧ
    "3 𝒑 l 𝑝࣍"3Ϟσϧͱݺ͹ΕΔ l աڈͷܥྻ 𝑦!-" , … 𝑦!-/ ͷೖྗͱϗϫΠτϊΠζͰݱࡏͷܥྻΛճؼ͢Δ Ϟσϧ ࣗݾճؼաఔ "3Ϟσϧ 𝑦* = ) !/% P 𝑎! 𝑦*+! + 𝜀*
49. 49 ࣌ܥྻϞσϧ l ௚લͷ𝒒 ݸͷϗϫΠτϊΠζͷՃॏ࿨Ͱఆٛ͞ΕΔ𝒒 ࣍."Ϟσϧ
    ." 𝒒 l աڈͷܥྻ 𝑦!-", … 𝑦!-/ ʹ༩͑ΒΕΔϗϫΠτϊΠζ ϥϯμϜγϣοΫ ͕࣌ܥྻͷকདྷͷ஋ʹ௚઀఻ൖ͢Δ͜ͱΛϞσϦϯά l ϗϫΠτϊΠζͱಉ༷ʹࣗݾڞ෼ࢄ΍ظ଴஋͸࣌఺ʹґଘͤͣɺऑఆৗੑ Λຬͨ͢ Ҡಈฏۉաఔ ."Ϟσϧ 𝑦* = 𝜇 + ) !/0 Q 𝑏! 𝜀*+!
50. 50 ࣌ܥྻϞσϧ l ௚લͷ𝒑 ݸͷܥྻͱ௚લͷ𝒒 ݸͷϗϫΠτϊΠζͷՃॏ࿨ʹΑͬͯߏ੒͞ ΕΔϞσϧ
    "3." 𝒑, 𝒒) l "3Ϟσϧ΋."Ϟσϧ΋"3."Ϟσϧͷಛघͳྫ l "3Ϟσϧ
    𝒒 = 0 l ."Ϟσϧ 𝒑 = 0 l ϗϫΠτϊΠζ
    𝒑 = 0, 𝒒 = 0 l ϗϫΠτϊΠζ𝜀! ͸աڈͷܥྻͱແ૬ؔ
    𝔼 𝜀! , 𝑦!-X = 0, j > 0 ࣗݾճؼҠಈฏۉաఔ "3."Ϟσϧ 𝑦* = ) !/% P 𝑎! 𝑦*+! + ) !/% Q 𝑏! 𝜀*+! + 𝜀*
51. 51 ࣌ܥྻϞσϧ l -BH
    PQFSBUPSΛ༻͍Δ͜ͱͰ"3." 𝒑, 𝒒)ϞσϧΛҎԼͷܗʹॻ͖׵͑Δ ͜ͱ͕Ͱ͖Δ ࣗݾճؼҠಈฏۉաఔ "3."Ϟσϧ 𝑦* = ) !/% P 𝑎! 𝑦*+! + ) !/% Q 𝑏! 𝜀*+! + 𝜀* 𝐿
    -BH
    PQFSBUPS
    𝐿% 𝑦* = 𝑦*+% , 𝐿& 𝑦* = 𝑦*+& , (1 − ) !/% P 𝑎! 𝐿!)𝑦* = (1 − ) !/% Q 𝑏! 𝐿!)𝜀* = 𝑎(𝐿) = 𝑏(𝐿) "3
    0QFSBUPS ."
    0QFSBUPS 𝑎(𝐿)𝑦* = 𝑏(𝐿)𝜀*
52. 52 ࣌ܥྻϞσϧ ࣗݾճؼҠಈฏۉաఔ "3."Ϟσϧ (1 − ) !/% P 𝑎! 𝐿!)𝑦* = (1 − ) !/% Q 𝑏! 𝐿!)𝜀* = 𝑎(𝐿) = 𝑏(𝐿) 𝑎(𝐿)𝑦* = 𝑏(𝐿)𝜀* 𝑦* = 𝑎(𝐿)+%𝑏(𝐿)𝜀* 𝑦* = ) ./0 T 𝑔. 𝐿.𝜀* = ) ./0 T 𝑔. 𝜀*+.
53. 53 ࣌ܥྻϞσϧ ࣗݾճؼҠಈฏۉաఔ "3."Ϟσϧ (1 − ) !/% P 𝑎! 𝐿!)𝑦* = (1 − ) !/% Q 𝑏! 𝐿!)𝜀* = 𝑎(𝐿) = 𝑏(𝐿) 𝑎(𝐿)𝑦* = 𝑏(𝐿)𝜀* 𝑦* = 𝑎(𝐿)+%𝑏(𝐿)𝜀* 𝑦* = ) ./0 T 𝑔. 𝐿.𝜀* = ) ./0 T 𝑔. 𝜀*+. "3."Ϟσϧ͸ແݶ࣍ͷ."Ϟσϧ Ͱදݱ͕Մೳ Πϯύϧε Ԡ౴ؔ਺
54. 54 ࣌ܥྻϞσϧ ΠϯύϧεԠ౴ؔ਺ *3' 𝑦* = ) ./0 T 𝑔. 𝜀*+. l ༩͑ΒΕͨϊΠζ Πϯύϧε ͕𝑗ظޙʹ࣌ܥྻʹͲΕ͘Β͍ӨڹΛ༩͑ Δ͔ʢӨڹྗ͕ͲΕ͚ͩ࢒Δ͔ʣΛࣔͨؔ͠਺ 𝑔0 = 1, 𝑔! = ) ./% P 𝑎. 𝑔!+. − 𝑏! *3'ͷࢉग़ from: https://alexchinco.com/impulse-response-functions-for-vars/
55. 55 ࣌ܥྻϞσϧ l ࣗݾ૬ؔؔ਺ BVUPDPSSFMBUJPO
    GVODUJPO
    
    "$' l ภࣗݾ૬ؔؔ਺ BVUPDPSSFMBUJPO
    GVODUJPO
    
    1"$' l ϥά𝑘ͷࣗݾ૬ؔΛߟྀ͢Δࡍʹɺϥά1, … 𝑘 − 1·Ͱͷܥྻͷࣗݾ૬ؔͷ Өڹ΋ଘࡏ͢Δ l ͜Ε·ͰͷϥάͷӨڹΛআڈͨ͠૬ؔؔ਺ ࣗݾ૬ؔؔ਺ "$' ͱภࣗݾ૬ؔؔ਺ 1"$' Cor 𝑦* , 𝑦*(& = 234 $",$"#$ 678 $" 678($"#$) = ;$ ;% = 𝑅& 𝑘͸ϥάΛද͢ PACF 𝑦* , 𝑦*(& = Cov 𝑦*(& 𝑦*(&+% , … , 𝑦*(% , 𝑦* 𝑦*(&+% , … , 𝑦*(% 𝑉𝑎𝑟 𝑦* 𝑦*(&+% , … , 𝑦*(% 𝑉𝑎𝑟 𝑦*(& 𝑦*(&+% , … , 𝑦*(%
56. 56 ࣌ܥྻϞσϧ l ࣗݾ૬ؔؔ਺ BVUPDPSSFMBUJPO
    GVODUJPO
    
    "$' l ภࣗݾ૬ؔؔ਺ BVUPDPSSFMBUJPO
    GVODUJPO
    
    1"$' l ϥά𝑘ͷࣗݾ૬ؔΛߟྀ͢Δࡍʹɺϥά1, … 𝑘 − 1·Ͱͷܥྻͷࣗݾ૬ؔͷ Өڹ΋ଘࡏ͢Δ l ͜Ε·ͰͷϥάͷӨڹΛআڈͨ͠૬ؔؔ਺ ࣗݾ૬ؔؔ਺ "$' ͱภࣗݾ૬ؔؔ਺ 1"$' Cor 𝑦* , 𝑦*(& = 234 $",$"#$ 678 $" 678($"#$) = ;$ ;% = 𝑅& 𝑘͸ϥάΛද͢ PACF 𝑦* , 𝑦*(& = Cov 𝑦*(& 𝑦*(&+% , … , 𝑦*(% , 𝑦* 𝑦*(&+% , … , 𝑦*(% 𝑉𝑎𝑟 𝑦* 𝑦*(&+% , … , 𝑦*(% 𝑉𝑎𝑟 𝑦*(& 𝑦*(&+% , … , 𝑦*(%
57. 57 ࣌ܥྻϞσϧ l "3Ϟσϧͷ"$'ͱ1"$' ࣗݾ૬ؔؔ਺ "$' ͱภࣗݾ૬ؔؔ਺ 1"$'
58. 58 ࣌ܥྻϞσϧ l ."Ϟσϧͷ"$'ͱ1"$' ࣗݾ૬ؔؔ਺ "$' ͱภࣗݾ૬ؔؔ਺ 1"$'
59. 59 ࣌ܥྻϞσϧ l "3."Ϟσϧͷ"$'ͱ1"$' ࣗݾ૬ؔؔ਺ "$' ͱภࣗݾ૬ؔؔ਺ 1"$'
60. 60 ࣌ܥྻϞσϧ l "3Ϟσϧͷࣗݾ૬ؔͷ௕͞͸ແݶ͕ͩɺภࣗݾ૬ؔͷ௕͞͸"3 ͷ܎਺ l ."Ϟσϧͷภࣗݾ૬ؔͷ௕͞͸ແݶ͕ͩɺࣗݾ૬ؔͷ௕͞͸." ͷ܎਺ l "3."Ϟσϧ͸ࣗݾ૬ؔɺภࣗݾ૬ؔͱ΋ʹݮਰ͍͕ͯ͘͠Ұఆ ʹଘࡏ͢Δ ࣗݾ૬ؔؔ਺ "$' ͱภࣗݾ૬ؔؔ਺ 1"$'
61. 61 ࣌ܥྻϞσϧ ύϫʔεϖΫτϧͱ"3."Ϟσϧ from: https://pyspectrum.readthedocs.io/en/latest/ref_param.html
62. 62 ࣌ܥྻϞσϧ l ฏۉʹؔͯ͠ඇఆৗͳ࣌ܥྻΛL֊ࠩ෼Λߦ͍ɺ"3."ϞσϧΛϑΟο ςΟϯάͨ͠Ϟσϧ
    "3*." 𝒑, 𝒌, 𝒒) l 𝒌 = 𝟎ͷ࣌"3."Ϟσϧ ࣗݾճؼ࿨෼Ҡಈฏۉաఔ "3*."Ϟσϧ 𝑧* = ) !/% P 𝑎! 𝑧*+! + ) !/% Q 𝑏! 𝜀*+! + 𝜀* ∆𝑦* = 𝑦* − 𝑦*+% 𝑧* = ∆&𝑦* L֊ࠩ෼ΛऔΔ͜ͱͰ ఆৗͳ࣌ܥྻʹม׵
63. 63 ࣌ܥྻϞσϧ l قઅੑ֊ࠩΛߟྀͨ͠"3*."Ϟσϧͱɺقઅੑ֊ࠩΛߟྀ͠ͳ͍௨ৗͷ "3*."Ϟσϧͷͭͷ੒෼Λ૊Έ߹Θͨ͠Ϟσϧ
    4"3*."
    
    "3*." 𝒑, 𝒌, 𝒒) º 𝑷, 𝑲, 𝑸)𝑺 4FBTPOBM
    "3*."Ϟσϧ 4"3*." 𝑧* U = ) !/% P 𝑎! U𝑧*+! U + ) !/% Q 𝑏! U𝜀*+! + 𝜀* 𝑧* U = 𝑦* − 𝑦*+U 𝑆͸Ұपظͷ௕͞
64. 64 ࣌ܥྻϞσϧ l ྫ
    𝑆 = 12ͷ4"3*."Ϟσϧ 4"3*."
    
    "3*." 𝟏, 𝟎, 𝟎) º 𝟏, 𝟎, 𝟎)𝟏𝟐 4FBTPOBM
    "3*."Ϟσϧ 4"3*." 1 − 𝑎% 𝐿% 1 − 𝑎% U𝐿%" 𝑦* = 𝜀* 𝑦* = 𝑎% 𝑦*+% + 𝑎% U𝑦*+%" − 𝑎% 𝑎% U𝑦*+%V + 𝜀* 𝐿
    -BH
    PQFSBUPS
    𝐿% 𝑦* = 𝑦*+% , 𝐿& 𝑦* = 𝑦*+& , قઅੑ ඇقઅੑ
65. 65 ࣌ܥྻϞσϧ 4FBTPOBM
    "3*."Ϟσϧ 4"3*." Kumar Barik, Aditya, et al. "Analysis of GHI Forecasting Using Seasonal ARIMA." Data Management, Analytics and Innovation. Springer, Singapore, 2021. 55-69.
66. 66 ࣌ܥྻϞσϧ l ࣌ܥྻΛτϨϯυɺقઅੑͳͲͷෳ਺ͷߏ੒ཁૉʹ෼ղ͢Δख๏ l "EEJUJWF
    EFDPNQPTJUJPO l .VMUJQMJDBUJWF
    EFDPNQPTJUJPO l ୅දతͳख๏ l X11 decomposition l STL decopostion l SEATS decompositon ࣌ܥྻ෼ղ 5JNF
    TFSJFT
    EFDPNQPTJUJPO 𝑦! = 𝑆! ×𝑇! ×𝑅! , log 𝑦! = log 𝑆! + log 𝑇! + log 𝑅! 𝑦! = 𝑆! + 𝑇! + 𝑅!
67. 67 ࣌ܥྻϞσϧ ࣌ܥྻ෼ղ 5JNF
    TFSJFT
    EFDPNQPTJUJPO from: https://otexts.com/fpp2/components.htmlå
68. 68 ࣌ܥྻϞσϧ l ࣌఺𝑡ʹ͓͚Δ𝑛ݸͷมྔ͔ΒͳΔܥྻσʔλΛϕΫτϧͱͯ͠ଊ͑Δ l ϗϫΠτϊΠζ΋ಉ༷ʹϕΫτϧͱͯ͠ଊ͑Δ l 7"3 Q Ϟσϧ͸ҎԼͷࣜͱͳΔ ϕΫτϧࣗݾճؼϞσϧ 7"3Ϟσϧ 𝑌* = (𝑦%,* , 𝑦",* , … , 𝑦',* )W 𝔼 𝜺𝒕 = [0, … , 0]W, 𝔼 𝜺𝒕 𝜺𝒕+𝒌 = [ 𝚺 (𝑘 = 0) 𝟎, (𝑘 ≠ 0) 𝑌* = ) !/% P 𝑨𝒊 𝑌*+! + 𝜺𝒕 𝑨𝒊 = 𝑎! (1,1) ⋯ 𝑎! (1, 𝑛) ⋮ ⋱ ⋮ 𝑎! (𝑛, 1) ⋯ 𝑎! (𝑛, 𝑛)
69. 69 ࣌ܥྻϞσϧ άϨϯδϟʔҼՌ (SBOHFS
    $BVTBMJUZ l ࣌ܥྻؒಉ࢜ͷҼՌؔ܎Λਪఆ͢ΔͨΊͷํ๏Ͱɺܥྻಉ࢜ͷ޲ ͖Λൃݟ͢Δख๏ l 𝑦%,* ͷ༧ଌʹ͓͍ͯ 𝑦%,* ͷաڈͷ஋ʹج͍ͮͨ༧ଌΑΓ΋ɺ 𝑦%,* , 𝑦" * ͷաڈͷ஋Λ༻͍ͨ༧ଌʹΑͬͯ.4&͕খ͘͞ͳΔ৔߹ 𝒚𝟐 𝒕 ͔Β 𝒚𝟏,𝒕 ΁ͷάϨϯδϟʔҼՌੑ͕ଘࡏ͢Δͱ͢Δ 𝑦% 𝑦"
70. 70 ࣌ܥྻϞσϧ άϨϯδϟʔҼՌ (SBOHFS
    $BVTBMJUZ l άϨϯδϟʔҼՌͷൃݟʹ͸'౷ܭྔΛ༻͍Δͷ͕Ұൠత 𝑦% ͷϞσϧʹΑΔ࢒ࠩฏํ࿨
    𝑆𝑆𝑅0 𝑦% , 𝑦" ͷϞσϧʹΑΔ࢒ࠩฏํ࿨
    𝑆𝑆𝑅% '౷ܭྔͷࢉग़
    𝐹 ≡ (UU]% +UU]&)/8 UU]&/(W +'P +%) '౷ܭྔͷ஋ʹج͍ͮͯάϨϯδϟʔҼՌ ͕ଘࡏ͢Δ͔Ͳ͏͔Λݕఆ
71. 71 ࣌ܥྻϞσϧ l ඇఆৗ࣌ܥྻΛϑΟοςΟϯά͢ΔͨΊͷϞσϦϯάख๏ͷ̍ͭ ہॴఆৗ"3Ϟσϧ 
    ϨδʔϜ෼ׂ Matsubara, Yasuko, Yasushi Sakurai, and Christos Faloutsos. "Autoplait: Automatic mining of co-evolving time sequences." Proceedings of the 2014 ACM SIGMOD international conference on Management of data. 2014.
72. 72 ࣌ܥྻϞσϧ l ֤۠෼͸ఆৗͱԾఆͯ͠ɺ֤۠෼͝ͱ ʹ"3ϞσϧͳͲͷ࣌ܥྻϞσϧΛద༻ l ۠෼෼͚ͷมԽ఺͸"*$΍.%-ͳͲΛ ༻͍Δ͜ͱͰൃݟ ہॴఆৗ"3Ϟσϧ 
    ϨδʔϜ෼ׂ
73. 73 ࣌ܥྻϞσϧ l ࣮ࡍͷঢ়ଶΛද͢ม਺ͱ࣮ࡍʹ؍ଌͰ͖Δม਺͕ҟͳΔΑ͏ͳܥΛ਺ࣜͰ දݱͨ͠ϑϨʔϜϫʔΫ l "3ϞσϧͳͲͷઢܗ࣌ܥྻϞσϧ΋औΓѻ͑ΔΑ͏ͳҰൠతͳϑϨʔϜ ϫʔΫ l ඇఆৗϞσϧ΍ϕΠζϞσϧΛදݱ͢Δ͜ͱ͕Մೳ ঢ়ଶۭؒϞσϧ
74. 74 ࣌ܥྻϞσϧ ঢ়ଶۭؒϞσϧ 𝑦! 𝑦!"# 𝑦!$# 𝑥!$# 𝑥! 𝑥!"# 𝑣!"# 𝑣! 𝑣!$# 𝑤!$# 𝑤! 𝑤!"# 𝐹! 𝐺! 𝐻! 𝑥! = 𝐹! 𝑥!$# + 𝐺! 𝑣!$# 𝑦! = 𝐻! 𝑥! + 𝑤! γεςϜϞσϧ ؍ଌϞσϧ 𝑦* 
    𝑛࣍ݩ࣌ܥྻ 𝑥* 
    𝑚࣍ݩঢ়ଶϕΫτϧ 𝑣* 
    𝑘࣍ݩγεςϜϊΠζ 𝑤* 
    𝑛࣍ݩ؍ଌϊΠζ 𝑣* ~ 𝑁 0, 𝑄* , 𝑤* ~ 𝑁 0, 𝑅*
75. 75 ࣌ܥྻϞσϧ "3ϞσϧΛঢ়ଶۭؒϞσϧͰදݱ 𝑥! = 𝐹! 𝑥!$# + 𝐺! 𝑣!$# 𝑦! = 𝐻! 𝑥! + 𝑤! γεςϜϞσϧ ؍ଌϞσϧ 𝑦* = ) !/% P 𝑎! 𝑦*+! + 𝜀* 𝜀* ~ N(0, 𝜎") "3Ϟσϧ 𝑦! 𝑦!-" ⋮ 𝑦!-/+" = 𝑎4 𝑎0 1 ⋯ 𝑎5 ⋮ ⋱ ⋮ ⋯ 1 𝑦!-" 𝑦!-Z ⋮ 𝑦!-/ + 1 0 ⋮ 0 𝜀! 𝑦* = 1 0 … 0 𝑦* 𝑦*+% ⋮ 𝑦*+P(% γεςϜϞσϧ ؍ଌϞσϧ
76. 76 ࣌ܥྻϞσϧ "3ϞσϧΛঢ়ଶۭؒϞσϧͰදݱ 𝑥! = 𝐹! 𝑥!$# + 𝐺! 𝑣!$# 𝑦! = 𝐻! 𝑥! + 𝑤! γεςϜϞσϧ ؍ଌϞσϧ 𝑦* = ) !/% P 𝑎! 𝑦*+! + 𝜀* 𝜀* ~ N(0, 𝜎") "3Ϟσϧ 𝑦! 𝑦!-" ⋮ 𝑦!-/+" = 𝑎4 𝑎0 1 ⋯ 𝑎5 ⋮ ⋱ ⋮ ⋯ 1 𝑦!-" 𝑦!-Z ⋮ 𝑦!-/ + 1 0 ⋮ 0 𝜀! 𝑦* = 1 0 … 0 𝑦* 𝑦*+% ⋮ 𝑦*+P(% 𝐹! 𝐺! 𝐻! 𝑥!
77. 77 ࣌ܥྻϞσϧ "3ϞσϧΛঢ়ଶۭؒϞσϧͰදݱ 𝑥! = 𝐹! 𝑥!$# + 𝐺! 𝑣!$# 𝑦! = 𝐻! 𝑥! + 𝑤! γεςϜϞσϧ ؍ଌϞσϧ 𝑦* = ) !/% P 𝑎! 𝑦*+! + 𝜀* 𝜀* ~ N(0, 𝜎") "3Ϟσϧ 𝑦! 𝑦!-" ⋮ 𝑦!-/+" = 𝑎4 𝑎0 1 ⋯ 𝑎5 ⋮ ⋱ ⋮ ⋯ 1 𝑦!-" 𝑦!-Z ⋮ 𝑦!-/ + 1 0 ⋮ 0 𝜀! 𝑦* = 1 0 … 0 𝑦* 𝑦*+% ⋮ 𝑦*+P(% 𝐹! 𝐺! 𝐻! 𝑥! "3Ϟσϧ͸࣌ෆมͰ؍ଌϊΠζ 𝑤! = 0ͷ ঢ়ଶۭؒϞσϧͰදݱՄೳ
78. 78 ࣌ܥྻϞσϧ "3."ϞσϧΛঢ়ଶۭؒϞσϧͰදݱ 𝜀!~ N(0, 𝜎Z) "3."Ϟσϧ 𝑦! _ 𝑦!+"|!-" ⋮ _ 𝑦!+,-"|!-" = 𝑎" 1 𝑎Z ⋯ ⋮ ⋱ 1 𝑎, ⋯ 𝑦!-" _ 𝑦!|!-Z ⋮ _ 𝑦!+,-Z|!-Z + 1 𝑏" ⋮ 𝑏,-" 𝜀! 𝑦* = 1 0 … 0 𝑦* m 𝑦*(%|*+% ⋮ m 𝑦*(&+%|*+% 𝐹! 𝐺! 𝐻! 𝑥! 𝑦! = b \]" / 𝑎\𝑦!-\ + b \]" ^ 𝑏\𝜀!-\ + 𝜀! _ 𝑦!+X|!-" = b \]X+" / 𝑎\ 𝑦!+X-\ + b \]X ^ 𝑏\ 𝜀!+X-\ 𝑘 = max(𝑝, 𝑞 + 1) 𝑥!-"
79. 79 ࣌ܥྻϞσϧ ঢ়ଶۭؒϞσϧͷਪఆ 𝑦! 𝑦!"# 𝑦!$# 𝑥!$# 𝑥! 𝑥!"# 𝑣!"# 𝑣! 𝑣!$# 𝑤!$# 𝑤! 𝑤!"# 𝐹! 𝐺! 𝐻! 𝑦% 𝑥% 𝑣% ʜ σʔλʹج͍ͮͯঢ়ଶ 𝑥* Λਪఆ͍ͨ͠ 𝑝(𝑥! |𝑦& )
80. 80 ࣌ܥྻϞσϧ ঢ়ଶۭؒϞσϧͷਪఆ 𝑦! 𝑦!"# 𝑦!$# 𝑥!$# 𝑥! 𝑥!"# 𝑣!"# 𝑣! 𝑣!$# 𝑤!$# 𝑤! 𝑤!"# 𝐹! 𝐺! 𝐻! 𝑦% 𝑥% 𝑣% ʜ σʔλʹج͍ͮͯঢ়ଶ 𝑥* Λਪఆ͍ͨ͠ 𝑝(𝑥! |𝑦& ) 𝑗 < 𝑡ͷ৔߹ ༧ଌ
81. 81 ࣌ܥྻϞσϧ ঢ়ଶۭؒϞσϧͷਪఆ 𝑦! 𝑦!"# 𝑦!$# 𝑥!$# 𝑥! 𝑥!"# 𝑣!"# 𝑣! 𝑣!$# 𝑤!$# 𝑤! 𝑤!"# 𝐹! 𝐺! 𝐻! 𝑦% 𝑥% 𝑣% ʜ σʔλʹج͍ͮͯঢ়ଶ 𝑥* Λਪఆ͍ͨ͠ 𝑝(𝑥! |𝑦& ) 𝑗 = 𝑡ͷ৔߹ ϑΟϧλ
82. 82 ࣌ܥྻϞσϧ ঢ়ଶۭؒϞσϧͷਪఆ 𝑦! 𝑦!"# 𝑦!$# 𝑥!$# 𝑥! 𝑥!"# 𝑣!"# 𝑣! 𝑣!$# 𝑤!$# 𝑤! 𝑤!"# 𝐹! 𝐺! 𝐻! 𝑦% 𝑥% 𝑣% ʜ σʔλʹج͍ͮͯঢ়ଶ 𝑥* Λਪఆ͍ͨ͠ 𝑝(𝑥! |𝑦& ) 𝑗 > 𝑡ͷ৔߹ ฏ׈Խ
83. 83 ࣌ܥྻϞσϧ ঢ়ଶۭؒϞσϧͷਪఆ 𝑦! 𝑦!"# 𝑦!$# 𝑥!$# 𝑥! 𝑥!"# 𝑣!"# 𝑣! 𝑣!$# 𝑤!$# 𝑤! 𝑤!"# 𝐹! 𝐺! 𝐻! 𝑦% 𝑥% 𝑣% ʜ σʔλʹج͍ͮͯঢ়ଶ 𝑥* Λਪఆ͍ͨ͠ 𝑝(𝑥! |𝑦& ) ࣌ܥྻϞσϧͷ໬౓ܭࢉ΍௕ظ༧ଌɺܽଛ஋ͷิ׬ ͳͲͷλεΫ͕͜ͷϑϨʔϜϫʔΫͰ࣮ݱͰ͖Δ
84. 84 ࣌ܥྻϞσϧ ઢܗɾΨ΢εܕঢ়ଶۭؒϞσϧ 𝑦! 𝑦!"# 𝑦!$# 𝑥!$# 𝑥! 𝑥!"# 𝑣!"# 𝑣! 𝑣!$# 𝑤!$# 𝑤! 𝑤!"# 𝐹! 𝐺! 𝐻! σʔλʹج͍ͮͯঢ়ଶ 𝑥* Λਪఆ͍ͨ͠ 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& )
85. 85 ࣌ܥྻϞσϧ ઢܗɾΨ΢εܕঢ়ଶۭؒϞσϧ 𝑦! 𝑦!"# 𝑦!$# 𝑥!$# 𝑥! 𝑥!"# 𝑣!"# 𝑣! 𝑣!$# 𝑤!$# 𝑤! 𝑤!"# 𝐹! 𝐺! 𝐻! σʔλʹج͍ͮͯঢ়ଶ 𝑥* Λਪఆ͍ͨ͠ 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) ΧϧϚϯϑΟϧλΛར༻
86. 86 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) l ༧ଌ 𝑥!|!$# = 𝐹! 𝑥!$#|!$# 𝑉!|!$# = 𝐹! 𝑉!$#|!$# 𝐹! ( + 𝐺! 𝑄! 𝐺! ( 𝑥!$# 𝐻!
87. 87 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) l ༧ଌ 𝑥!|!$# = 𝐹! 𝑥!$#|!$# 𝑉!|!$# = 𝐹! 𝑉!$#|!$# 𝐹! ( + 𝐺! 𝑄! 𝐺! ( 𝑥!$# 𝐻! ͭલͷঢ়ଶ 𝑡 − 1 ͔Βݱࡏͷঢ়ଶ 𝑡 Λਪఆ 𝑥!|!$# = 𝔼 𝑥! 𝑦!$# ] = 𝔼 𝐹𝑥!$# + 𝐺! 𝑣! 𝑦!$# ] = 𝐹𝔼 𝑥!$# 𝑦!$# ] = 𝐹𝑥!$#|!$#
88. 88 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) l ༧ଌ 𝑥!|!$# = 𝐹! 𝑥!$#|!$# 𝑉!|!$# = 𝐹! 𝑉!$#|!$# 𝐹! ( + 𝐺! 𝑄! 𝐺! ( 𝑥!$# 𝐻! ͭલͷঢ়ଶޡࠩ ͔Βݱࡏͷঢ়ଶޡࠩΛਪఆ 𝑉!|!$# = 𝔼[ 𝑥!$ 𝑥!|!$# ' (𝑥!$ 𝑥!|!$# )] = 𝔼[(𝐹 𝑥!$#$ 𝑥!$#|!$# + 𝐺𝑣! )' (𝐹 𝑥!$#$ 𝑥!$#|!$# + 𝐺𝑣! )] = 𝐹𝔼 𝑥!$#$ 𝑥!$#|!$# ' 𝑥!$#$ 𝑥!$#|!$# 𝐹' + 𝐺𝔼 𝑣! '𝑣! 𝐺
89. 89 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) l ϑΟϧλ 𝑥!$# K! = 𝑉!|!$# 𝐻! ((𝐻! 𝑉!|!$# 𝐻! ( + 𝑅! )$# 𝑉!|! = (𝐼 − 𝐾! 𝐻! )𝑉!|!$# 𝐻! 𝑥!|! = 𝑥!|!$# + 𝐾! (𝑦! − 𝐻! 𝑥!|!$# )
90. 90 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) l ϑΟϧλ 𝑥!$# K! = 𝑉!|!$# 𝐻! ((𝐻! 𝑉!|!$# 𝐻! ( + 𝑅! )$# 𝑉!|! = (𝐼 − 𝐾! 𝐻! )𝑉!|!$# 𝐻! 𝑥!|! = 𝑥!|!$# + 𝐾! (𝑦! − 𝐻! 𝑥!|!$# ) 𝐾* ΧϧϚϯήΠϯͱݺ͹ΕΔ΋ͷΛࢉग़ 𝑤* 
    𝑛࣍ݩ؍ଌϊΠζ 𝑤* ~ 𝑁 0, 𝑅*
91. 91 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) l ϑΟϧλ 𝑥!$# K! = 𝑉!|!$# 𝐻! ((𝐻! 𝑉!|!$# 𝐻! ( + 𝑅! )$# 𝑉!|! = (𝐼 − 𝐾! 𝐻! )𝑉!|!$# 𝐻! 𝑥!|! = 𝑥!|!$# + 𝐾! (𝑦! − 𝐻! 𝑥!|!$# ) ਪఆޡࠩͱ؍ଌϊΠζ 𝑹𝒕 ͷ࿨ʹ͋ͨΔ෦෼ ΧϧϚϯήΠϯ͸શମͷޡࠩʹ͓͚Δਪఆޡࠩͷׂ ߹ʹ૬౰
92. 92 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) l ϑΟϧλ 𝑥!$# K! = 𝑉!|!$# 𝐻! ((𝐻! 𝑉!|!$# 𝐻! ( + 𝑅! )$# 𝑉!|! = (𝐼 − 𝐾! 𝐻! )𝑉!|!$# 𝐻! 𝑥!|! = 𝑥!|!$# + 𝐾! (𝑦! − 𝐻! 𝑥!|!$# ) ؍ଌ஋ͱࣄલʹਪఆ͞Εͨঢ়ଶͱͷࠩ
93. 93 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) l ϑΟϧλ 𝑥!$# K! = 𝑉!|!$# 𝐻! ((𝐻! 𝑉!|!$# 𝐻! ( + 𝑅! )$# 𝑉!|! = (𝐼 − 𝐾! 𝐻! )𝑉!|!$# 𝐻! 𝑥!|! = 𝑥!|!$# + 𝐾! (𝑦! − 𝐻! 𝑥!|!$# ) ΧϧϚϯήΠϯ͸ॏΈͷ໾ׂͰ؍ଌޡࠩͱਪఆޡࠩ ͷόϥϯεΛऔΓঢ়ଶ 𝑥* Λߋ৽
94. 94 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) l ϑΟϧλ 𝑥!$# K! = 𝑉!|!$# 𝐻! ((𝐻! 𝑉!|!$# 𝐻! ( + 𝑅! )$# 𝑉!|! = (𝐼 − 𝐾! 𝐻! )𝑉!|!$# 𝐻! 𝑥!|! = 𝑥!|!$# + 𝐾! (𝑦! − 𝐻! 𝑥!|!$# ) ؍ଌޡ͕ࠩେ͖͍ͱڞ෼ࢄ͕େ͖͘ͳΔΑ͏ʹߋ৽
95. 95 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλͷஞ࣍ਪఆ from: https://elf-c.he.u-tokyo.ac.jp/courses/388
96. 96 ࣌ܥྻϞσϧ ઢܗɾΨ΢εܕঢ়ଶۭؒϞσϧ 𝑦! 𝑦!"# 𝑦!$# 𝑥!$# 𝑥! 𝑥!"# 𝑣!"# 𝑣! 𝑣!$# 𝑤!$# 𝑤! 𝑤!"# 𝐹! 𝐺! 𝐻! σʔλʹج͍ͮͯঢ়ଶ 𝑥* Λਪఆ͍ͨ͠ 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) ΧϧϚϯϑΟϧλΛར༻
97. 97 ࣌ܥྻϞσϧ ௕ظ༧ଌΛ͍ͨ͠৔߹ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) l ௕ظ༧ଌ 𝑥!")|!$# = 𝐹! 𝑥!$#")|!$# 𝑉!")|!$# = 𝐹! 𝑉!")|!")$# 𝐹! ( + 𝐺! 𝑄! 𝐺! ( 𝑥!$# 𝐻! ϑΟϧλͷॲཧΛলུ͢Δ͜ͱͰ࣮ݱ
98. 98 ࣌ܥྻϞσϧ ௕ظ༧ଌΛ͍ͨ͠৔߹ from: https://elf-c.he.u-tokyo.ac.jp/courses/388
99. 99 ࣌ܥྻϞσϧ ঢ়ଶۭؒϞσϧͷਪఆ 𝑦! 𝑦!"# 𝑦!$# 𝑥!$# 𝑥! 𝑥!"# 𝑣!"# 𝑣! 𝑣!$# 𝑤!$# 𝑤! 𝑤!"# 𝐹! 𝐺! 𝐻! 𝑦% 𝑥% 𝑣% ʜ σʔλʹج͍ͮͯঢ়ଶ 𝑥* Λਪఆ͍ͨ͠ 𝑝(𝑥! |𝑦& ) 𝑗 > 𝑡ͷ৔߹ ฏ׈Խ
100. 100 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ ฏ׈Խ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) 𝑥!$# 𝐻! l ฏ׈Խ 𝑥!|% = 𝑥!|! + 𝐴! (𝑥!"#|( − 𝑥!"#|! ) 𝑉!|( = 𝑉!|! + 𝐴! (𝑉!"#|( − 𝑉!"#|! )𝐴! ( 𝐴! = 𝑉!|! 𝐹!"# ( 𝑉!"#|! $#
101. 101 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ ฏ׈Խ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) 𝑥!$# 𝐻! l ฏ׈Խ 3BVDI5VOH4USJFCFM TNPPUIFS 𝑥!|% = 𝑥!|! + 𝐴! (𝑥!"#|( − 𝑥!"#|! ) 𝑉!|( = 𝑉!|! + 𝐴! (𝑉!"#|( − 𝑉!"#|! )𝐴! ( 𝐴! = 𝑉!|! 𝐹!"# ( 𝑉!"#|! $# 𝐴* 
     ฏۉԽརಘ
102. 102 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ ฏ׈Խ 𝑦! 𝑥! 𝑣! 𝑤! 𝐹! 𝐺! 𝑝 𝑥! 𝑦& ~ N(𝑥!|& , 𝑉!|& ) 𝑥!$# 𝐻! l ฏ׈Խ 3BVDI5VOH4USJFCFM TNPPUIFS 𝑥!|% = 𝑥!|! + 𝐴! (𝑥!"#|( − 𝑥!"#|! ) 𝑉!|( = 𝑉!|! + 𝐴! (𝑉!"#|( − 𝑉!"#|! )𝐴! ( 𝐴! = 𝑉!|! 𝐹!"# ( 𝑉!"#|! $# ฏۉԽརಘ͕ঢ়ଶߋ৽ʹ͓͚ΔॏΈͷ໾ׂ
103. 103 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ ฏ׈Խ from: https://elf-c.he.u-tokyo.ac.jp/courses/388
104. 104 ࣌ܥྻϞσϧ ඇઢܗɾඇΨ΢εܕঢ়ଶۭؒϞσϧ l ઢܗɾΨ΢εܕঢ়ଶۭؒϞσϧ Ͱ͸ରॲͰ͖ͳ͍࣌ܥྻͷ܏޲ ΋ଟ͘ଘࡏ l ߏ଄มԽ l ҟৗ஋ l ඇରশ෼෍ l ඇઢܗੑ
105. 105 ࣌ܥྻϞσϧ ඇઢܗɾඇΨ΢εܕঢ়ଶۭؒϞσϧ 𝑦! 𝑦!"# 𝑦!$# 𝑥!$# 𝑥! 𝑥!"# 𝑣!"# 𝑣! 𝑣!$# 𝑤!$# 𝑤! 𝑤!"# 𝐹! 𝐺! 𝐻! 𝑥! = 𝐹! 𝑥!$# + 𝐺! 𝑣! 𝑦! = 𝐻! 𝑥! + 𝑤! ઢܗɾΨ΢εܕ ඇઢܗɾඇΨ΢εܕ 𝑥! = 𝑓(𝑥!$# , 𝑣! ) 𝑦! = ℎ(𝑥! ) + 𝑤! 𝑣! ~𝑞 𝑣 , 𝑤! ~𝑟(𝑤)
106. 106 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ l ༧ଌ l ϑΟϧλ l ฏ׈Խ 𝑝 𝑥! 𝑌!$# ) = ; $* * 𝑝 𝑥! 𝑥!$# )𝑝 𝑥!$# 𝑌!$# )𝑑𝑥!$# 𝑥! = 𝑓(𝑥!$# , 𝑣! ) 𝑦! = ℎ(𝑥! ) + 𝑤! 𝑝 𝑥! 𝑌! ) = 𝑝 𝑦! 𝑥! )𝑝 𝑥! 𝑌!$# ) 𝑝 𝑦! 𝑌!–# ) 𝑝 𝑥! 𝑌( ) = 𝑝 𝑥! 𝑌! ) ∫ $* * , -!"# -!), -!"# /$) , -!"# /!) 𝑑𝑥!"#
107. 107 ࣌ܥྻϞσϧ ΧϧϚϯϑΟϧλ from: https://elf-c.he.u-tokyo.ac.jp/courses/388
108. 108 ࣌ܥྻϞσϧ ཻࢠϑΟϧλ from: https://www.ieice.org/jpn/books/kaishikiji/2005/200512.pdf l ඇઢܗɾඇΨ΢εঢ়ଶۭؒϞσϧͰ͸ੵ෼ܭࢉ͕ղੳతʹࠔ೉Ͱ ͋Δ͜ͱ͔Β༻͍ΒΕΔख๏ l ղੳతʹදݱͰ͖ͳ͍ಉ࣌෼෍Λʮཻࢠʯ ʹج͍ͮͨཚ਺ϕΫτϧΛੜ੒ɻ ੜ੒ͨ͠ཚ਺ϕΫτϧʹجͮ͘ܦݧ෼෍ ؔ਺ͰٻΊ͍ͨঢ়ଶۭؒͷ෼෍Λۙࣅɻ
109. 109 ࣌ܥྻϞσϧ Ϛϧίϑ࿈࠯ϞϯςΧϧϩ๏ .$.$ from: https://www.ieice.org/jpn/books/kaishikiji/2005/200512.pdf l ཚ਺ൃੜΞϧΰϦζϜͷҰछͰɺੜ੒͞Εͨཚ਺Λ༻͍ͯۙࣅܭ ࢉΛߦ͏ɻ l ࣄޙ෼෍ͷܗͦͷ΋ͷΛਪఆ l ঢ়ଶਪఆ͚ͩͰͳ͘ɺϞσϧͷύϥϝλʔ΋ಉ࣌ʹਪఆ͢Δ
110. 110 ࣌ܥྻϞσϧ ඇઢܗɾඇΨ΢εܕঢ়ଶۭؒϞσϧͷྫ from: https://elf-c.he.u-tokyo.ac.jp/courses/388
111. 111 ࣌ܥྻϞσϧ ඇઢܗɾඇΨ΢εܕঢ়ଶۭؒϞσϧͷྫ from: https://elf-c.he.u-tokyo.ac.jp/courses/388
112. 112 ࣌ܥྻϞσϧ ඇઢܗɾඇΨ΢εܕঢ়ଶۭؒϞσϧͷྫ from: https://elf-c.he.u-tokyo.ac.jp/courses/388
113. 113 Ϟσϧͷਪఆɾબ୒
114. 114 Ϟσϧͷਪఆɾબ୒ ओͳϞσϦϯάϑϨʔϜϫʔΫ モデル σʔλ ܦݧత஌ࣝ ཧ࿦ ࣌ܥྻ༧ଌ ࣌ܥྻ෼ྨ ҟৗ஋ݕ஌ ͲͷΑ͏ͳϞσϧΛֶश͠࠷ద ͳϞσϧΛબ୒͢Ε͹Α͍͔ʁ
115. 115 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷਪఆ
     "3Ϟσϧ l "3Ϟσϧ΍7"3Ϟσϧͷύϥϝʔλͷਪఆͱͯ͠:VMF8BMLFS๏ ΍)PVTFIPMEFS๏ɺ1"3$03๏ͳͲ͕ଘࡏ l ࠓճ͸ҰൠతͳճؼϞσϧͰ΋༻͍ΒΕ͍ͯΔ࠷খೋ৐๏ 0-4 ΍ ࠷໬ਪఆ .-& ʹߜͬͯ঺հ
116. 116 Ϟσϧͷਪఆɾબ୒ l ߏங͞Εͨؔ਺Ϟσϧ𝑓(ɾ)ͱ؍ଌ஋ͱͷؒͷೋ৐ޡࠩ ࢒ࠩ𝑒! ͷೋ ৐ ͕࠷খͱͳΔΑ͏ͳύϥϝʔλΛਪఆ l ֬཰ͷ֓೦͕ݱΕͳ͍ l ճؼ෼ੳͳͲʹ͓͍ͯɺղੳతʹύϥϝʔλಋग़͕Մೳ Ϟσϧͷਪఆ
     ࠷খೋ৐๏ 0-4 𝐸 = b \]" _ (𝑦\ − 𝑓(𝑥\))Z
117. 117 Ϟσϧͷਪఆɾબ୒ l ྫ
     ճؼϞσϧ "3  l ͭͷ܎਺Λ࣋ͭճؼϞσϧ l ೋ৐ޡࠩ Ϟσϧͷਪఆ
     ࠷খೋ৐๏ 0-4 𝐰𝐓 = (𝑤0 , 𝑤% ), 𝐱𝐢 b = (1, 𝑥)ͱఆٛ͢Δͱ 𝑓 𝑥! = v 𝑦! = 𝑤0 + 𝑤% 𝑥! = 𝐰𝐓𝐱 v 𝑦% v 𝑦c = 1, 𝑥% 1, 𝑥d 𝑤0 𝑤% v 𝐲 = 𝐗𝐰 ܭըߦྻ E = (𝐲 − 𝐗𝐰)b(𝐲 − 𝐗𝐰)
118. 118 Ϟσϧͷਪఆɾબ୒ l ྫ
     ճؼϞσϧ "3  l ೋ৐ޡࠩEΛ𝐰Ͱඍ෼ Ϟσϧͷਪఆ
     ࠷খೋ৐๏ 0-4 E = 𝐲 − 𝐗𝐰 b 𝐲 − 𝐗𝐰 = 𝐲𝐓𝐲 − 𝟐𝐰𝐓 𝐗𝐓𝐲 + 𝐰𝐓𝐗𝐓𝐗𝐰 𝜕𝐸 𝜕𝐰 = −𝟐𝐗b𝐲 + 𝐗b𝐗 + 𝐗b𝐗 b 𝐰 = −2𝐗b𝐲 + 𝟐𝐗b𝐗w = 0 w = (𝐗b𝐗)+%𝐗b𝐲
119. 119 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷਪఆ
     ࠷໬ਪఆ l ,VMMCBDL-FJCMFS ,- ৘ใྔ l ͭͷ֬཰෼෍ͷࠩҟΛଌఆ͢Δई౓ l ਪఆ͞ΕΔ༧ଌ෼෍͕ਅͷ෼෍Λଊ͍͑ͯΔ͔ΛධՁ l g(x)͕ਅͷ෼෍ 
     f(x)͕Ϟσϧͷ෼෍ͱͨ࣌͠ɺ,-৘ใྔ͸ҎԼͷܗ l D`a (g| 𝑓 = 0ͷ࣌ɺ g x = f(x) l ࡾ֯ෆ౳ࣜ΍ରশੑͳͲͷڑ཭ͷެཧ͸ຬͨ͞ͳ͍ Def (g||𝑓) = ~ +T T g x log g x f x 𝑑𝑥 ∗ Def (g||𝑓) ≥ 0
120. 120 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷਪఆ
     ࠷໬ਪఆ l ਖ਼ن෼෍ಉ࢜ͷ,-ڑ཭ D`a(g||𝑓) = m -b b g x log g x 𝑑𝑥 − m -b b g x log 𝑓 x 𝑑𝑥 g x ~ N 𝜇c , 𝜎c Z , f x ~ N 𝜇d , 𝜎d Z m -b b g x log 𝑓 x 𝑑𝑥 = − 1 2 log2π𝜎d Z − 𝜎c Z + 𝜇c − 𝜇d Z 2𝜎d Z m -b b g x log g x 𝑑𝑥 = − 1 2 log2π𝜎c Z − 1 2 正規分布の確率密度関数
121. 121 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷਪఆ
     ࠷໬ਪఆ Def (g||𝑓) = ~ +T T g x log g x f x 𝑑𝑥 ∗ Def (g||𝑓) ≥ 0 from: https://qiita.com/ceptree/items/9a473b5163d5655420e8
122. 122 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷਪఆ
     ࠷໬ਪఆ l ϞσϦϯάʹ͓͚ΔKL৘ใྔͷਪఆ l g(x)͕ਅͷ෼෍ 
     f(x)͕Ϟσϧͷ෼෍ Def (g||f) = ~ +T T g x log g x f x 𝑑𝑥 = ~ +T T g x log g x 𝑑𝑥 − ~ +T T g x log f x 𝑑𝑥
123. 123 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷਪఆ
     ࠷໬ਪఆ l ϞσϦϯάʹ͓͚ΔKL৘ใྔͷਪఆ l g(x)͕ਅͷ෼෍ 
     f(x)͕Ϟσϧͷ෼෍ Def (g||f) = ~ +T T g x log g x f x 𝑑𝑥 = ~ +T T g x log g x 𝑑𝑥 − ~ +T T g x log f x 𝑑𝑥 Θ͔Βͳ͍
124. 124 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷਪఆ
     ࠷໬ਪఆ l ϞσϦϯάʹ͓͚ΔKL৘ใྔͷਪఆ l g(x)͕ਅͷ෼෍ 
     f(x)͕Ϟσϧͷ෼෍ Def (g||f) = ~ +T T g x log g x f x 𝑑𝑥 = ~ +T T g x log g x 𝑑𝑥 − ~ +T T g x log f x 𝑑𝑥 Θ͔Βͳ͍ ฏۉର਺໬౓ g(x)͕Θ͔Βͳ͍ͨΊະ஌ ͕ͩɺͲͷ f(x) Ͱ΋Ұఆ
125. 125 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷਪఆ
     ࠷໬ਪఆ l ϞσϦϯάʹ͓͚ΔKL৘ใྔͷਪఆ l g(x)͕ਅͷ෼෍ 
     f(x)͕Ϟσϧͷ෼෍ Def (g||f) = ~ +T T g x log g x f x 𝑑𝑥 = ~ +T T g x log g x 𝑑𝑥 − ~ +T T g x log f x 𝑑𝑥 Θ͔Βͳ͍ ฏۉର਺໬౓ g(x)͕Θ͔Βͳ͍ͨΊະ஌ ͕ͩɺͲͷ f(x) Ͱ΋Ұఆ ૬ରతʹɺϞσϧ෼෍ 𝐟(𝐱)ͷ ༗ޮੑʹ͍ͭͯධՁͰ͖Δʁ
126. 126 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷਪఆ
     ࠷໬ਪఆ ~ +T T g x log f x 𝑑𝑥 l ฏۉର਺໬౓ l ର਺໬౓ ະ஌ 1 𝑛 b \]" W 𝐼(𝑥\) େ਺ͷ๏ଇʹΑΓɺ༩͑ΒΕͨ αϯϓϧσʔλͰۙࣅ ਅͷ໬౓ αϯϓϧσʔλ ͷ໬౓ 𝜃+ 𝜃678 ฏۉର਺໬౓ ର਺໬౓ ℓ = ) !/% ' log f x
127. 127 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷਪఆ
     ࠷໬ਪఆ l ࠷໬ਪఆ .-& l ର਺໬౓Λ༻͍ͨύϥϝʔλਪఆ๏ l ର਺໬౓Λ࠷େԽ͢Δύϥϝʔλ ࠷໬ਪఆ஋ 𝜃fgh ͕ۙࣅతʹ,-৘ใྔ Λ࠷΋࠷খԽ͢ΔύϥϝʔλʹͳΔͱਪఆͰ͖Δ l ର਺໬౓Λ࠷େԽ͢Δύϥϝʔλ ࠷໬ਪఆ஋ 𝜃fgh ͸ؔ਺ℓ(𝜃)͕𝜃Ͱඍ෼ ͯ͠ ij(k) ik = 0ͱͳΔ΋ͷΛਪఆ ਅͷ໬౓ αϯϓϧσʔλ ͷ໬౓ 𝜃+ 𝜃678 ฏۉର਺໬౓ ର਺໬౓ ℓ(𝜃) = ) !/% ' log f x|θ 𝜃1gh = max i ℓ(𝜃)
128. 128 Ϟσϧͷਪఆɾબ୒ ࠷খೋ৐๏ͱ࠷໬ਪఆ l ྫɿճؼϞσϧͷ࠷໬ਪఆɻޡࠩΛਖ਼ن෼෍ͱԾఆ v 𝑦! = 𝑓 𝑥! = 𝑁(𝑤0 + 𝑤% 𝑥! , 𝜎") ℓ(𝜃) = ) !/% ' log 1 2𝜋𝜎" exp(− 𝑦! − (𝑤0 + 𝑤% 𝑥! ) " 2𝜎" ) = − 𝑛 2 log 2𝜋 − 𝑛 2 log 𝜎" − ) !/% ' ( 𝑦! − (𝑤0 + 𝑤% 𝑥! ) " 2𝜎" )
129. 129 Ϟσϧͷਪఆɾબ୒ l ߏங͞Εͨؔ਺Ϟσϧ𝑓(ɾ)ͱ؍ଌ஋ͱͷؒͷೋ৐ޡࠩ ࢒ࠩ𝑒! ͷೋ ৐ ͕࠷খͱͳΔΑ͏ͳύϥϝʔλΛਪఆ l ֬཰ͷ֓೦͕ݱΕͳ͍ l ճؼ෼ੳͳͲʹ͓͍ͯɺղੳతʹύϥϝʔλಋग़͕Մೳ ࠷খೋ৐๏ͱ࠷໬ਪఆ 𝐸 = b \]" _ (𝑦\ − 𝑓(𝑥\))Z
130. 130 Ϟσϧͷਪఆɾબ୒ ࠷খೋ৐๏ͱ࠷໬ਪఆ l ྫɿճؼϞσϧͷ࠷໬ਪఆɻޡࠩΛਖ਼ن෼෍ͱԾఆ v 𝑦! = 𝑓 𝑥! = 𝑁(𝑤0 + 𝑤% 𝑥! , 𝜎") ℓ(𝜃) = ) !/% ' log 1 2𝜋𝜎" exp(− 𝑦! − (𝑤0 + 𝑤% 𝑥! ) " 2𝜎" ) = − 𝑛 2 log 2𝜋 − 𝑛 2 log 𝜎" − ) !/% ' ( 𝑦! − (𝑤0 + 𝑤% 𝑥! ) " 2𝜎" ) ର਺໬౓ͷ࠷େԽ͸࠷খೋ৐๏ͱҰக 㱺 ࠷খೋ৐๏͸࠷໬ਪఆͷಛघྫͱҰக
131. 131 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબఆ 𝜃+ "3  "3*."    ."  ͲͷϞσϧΛ࢖͏ͱ ྑ͍ʁʁʁ
132. 132 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબఆ 𝜃+ "3  "3*."    ."  ͲͷϞσϧΛ࢖͏ͱ ྑ͍ʁʁʁ ର਺໬౓͕࠷΋େ͖͍ Ϟσϧ͕ྑ͍ͷͰ͸ʁ
133. 133 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબఆ l ଟ߲ࣜճؼͷର਺໬౓ from: https://elf-c.he.u-tokyo.ac.jp/courses/382 ࣍਺Λ૿΍͢ͱ ର਺໬౓΋্ঢ
134. 134 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબఆ l ଟ߲ࣜճؼͷର਺໬౓ from: https://elf-c.he.u-tokyo.ac.jp/courses/382 ࣍਺Λ૿΍͢ͱ ର਺໬౓΋্ঢ આ໌ม਺Λ૿΍͠·͘Ε͹ର਺໬౓ͷ؍఺ Ͱ΋ྑ͍ϞσϧΛ࡞ΕΔͷͰ͸ʁʁʁ
135. 135 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબఆ l ࠷΋ྑ͍ର਺໬౓Λ࣋ͭϞσϧΛબ୒ͨ͠৔߹ l աద߹ PWFSGJU ͷ໰୊ l ༩͑ΒΕͨσʔλ͸े෼આ໌Ͱ͖ͨͱͯ͠΋ɺ৽͍͠σʔλΛे෼ʹઆ໌Ͱ ͖Δͱ͸ݶΒͳ͍ from: https://datascience.foundation/sciencewhitepaper/u nderfitting-and-overfitting-in-machine-learning
136. 136 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબఆ l ࠷΋ྑ͍ର਺໬౓Λ࣋ͭϞσϧΛબ୒ͨ͠৔߹ l աద߹ PWFSGJU ͷ໰୊ l ༩͑ΒΕͨσʔλ͸े෼આ໌Ͱ͖ͨͱͯ͠΋ɺ৽͍͠σʔλΛे෼ʹઆ໌Ͱ ͖Δͱ͸ݶΒͳ͍ from: https://datascience.foundation/sciencewhitepaper/u nderfitting-and-overfitting-in-machine-learning ෳࡶ͗͢ͳ͍࠷దͳϞσϧΛ બ୒͢Δ͜ͱ͕ॏཁ
137. 137 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબఆ l ద੾ͳϞσϧબ୒ͷͨΊʹ l ର਺໬౓ʹΑͬͯਪఆ͞ΕΔ𝜃fgh ͱ ਅͷύϥϝʔλ 𝜃. ͷؒʹόΠΞε ͕ଘࡏ l ର਺໬౓ͱฏۉର਺໬౓ͷόΠΞε ΛධՁͯ͠ద੾ʹิਖ਼͢Δ͜ͱͰ ࠷దͳϞσϧΛબ୒͢Δඞཁ
138. 138 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબఆ l ద੾ͳϞσϧબ୒ͷͨΊʹ l 𝜃fgh ʹ͓͚Δର਺໬౓ͱฏۉର਺ ໬౓ͷࠩ 𝐷Λิਖ਼͍ͨ͠ 1 𝑛 ? '"* , log f x|𝜃-./ ͜ͷ෦෼Λ ิਖ਼͍ͨ͠ʂ ظ଴஋ΛͱΔ 𝔼0[𝐷] = t #$ $ g x [D]𝑑𝑥 ฏۉର਺໬౓ ର਺໬౓ 𝐷 = 1 𝑛 ? '"* , log f x|𝜃-./ − t #$ $ g x log f x|𝜃-./ 𝑑𝑥
139. 139 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબఆ l ద੾ͳϞσϧબ୒ͷͨΊʹ l 𝜃fgh ʹ͓͚Δର਺໬౓ͱฏۉର਺ ໬౓ͷࠩ 𝐷Λิਖ਼͍ͨ͠ 1 𝑛 ? '"* , log f x|𝜃-./ ͜ͷ෦෼Λ ิਖ਼͍ͨ͠ʂ ฏۉର਺໬౓ ର਺໬౓ ظ଴஋ΛͱΔ 𝔼0[𝐷] = t #$ $ g x [D]𝑑𝑥 ͜ͷิਖ਼෦෼͕ύϥϝʔλ਺Ͱ͋Δ 𝑝ʹۙࣅͰ͖Δ ิਖ਼ 𝐷 = 1 𝑛 ? '"* , log f x|𝜃-./ − t #$ $ g x log f x|𝜃-./ 𝑑𝑥
140. 140 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબఆ
     "*$ l ద੾ͳϞσϧબ୒ͷͨΊʹ l ࠷େର਺໬౓ όΠΞε 
     𝜃fgh ʹ͓͚Δฏۉର਺໬౓ 1 𝑛 ? '"* , log f x|𝜃-./ ∑!/% ' log f x|𝜃1gh − 𝔼x [𝐷] = 𝑛 ∫ +T T g x log f x|𝜃1gh 𝑑𝑥 ∑!/% ' log f x|𝜃1gh − 𝑝= n ∫ +T T g x log f x|𝜃1gh 𝑑𝑥 𝔼0[𝐷]
141. 141 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબఆ
     "*$ l ੺஑৘ใྔج४ "*$ l ฏۉର਺໬౓ͷ 𝜃fgh ʹ͓͚ΔਪఆྔͰ͋ΔҎԼͷࣜͷࠨลΛº͢Δ͜ͱͰ ౷ܭϞσϧͷࢦඪͱͳΔ l 𝑝͸ࣗ༝ύϥϝʔλ਺ l ͜ͷࢦඪ͕খ͍͞౷ܭϞσϧ͕ྑ͍Ϟσϧͱਪఆ͞ΕΔ l ৘ใྔج४ͷछ ∑!/% ' log f x|𝜃1gh − 𝑝= n ∫ +T T g x log f x|𝜃1gh 𝑑𝑥 AIC = −2 ) !/% ' log f x|𝜃1gh + 2𝑝
142. 142 Ϟσϧͷਪఆɾબ୒ "*$ͷྫ l ଟ߲ࣜճؼͷ"*$ʹΑΔධՁ from: https://elf-c.he.u-tokyo.ac.jp/courses/382
143. 143 Ϟσϧͷਪఆɾબ୒ "*$ͷྫ l ͲͷΑ͏ͳ#JOͷ਺Ͱදݱ͢Δͷ͕ྑ͍ͷ͔ʁ l ͲͷΑ͏ͳଟ߲෼෍ϞσϧʹϑΟοτͤ͞Δͷ͕ྑ͍ͷ͔ʁ
144. 144 Ϟσϧͷਪఆɾબ୒ "*$ͷྫ
145. 145 "*$ͷྫ Ϟσϧͷਪఆɾબ୒
146. 146 Ϟσϧͷਪఆɾબ୒ from: https://elf-c.he.u-tokyo.ac.jp/courses/382 "*$ͷྫ
147. 147 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબఆ
     "*$ l Ϟσϧͷઆ໌ม਺͕গͳ͍ͱɺ#JBT͕େ͖͘ͳΔ ֶशෆ଍ l Ϟσϧͷઆ໌ม਺͕ଟ͍ͱɺ#JBT͸খ͘͞ͳΔ͕ɺ7BSJBODF͕େ ͖͘ͳΓɺաֶशͱͳΔ l "*$ͳͲͷ৘ใྔج४͸ɺόΠΞεͱόϦΞϯεͷͭΛόϥϯε Α͘࠷খʹ͢Δʢظ଴༧ଌޡࠩͷ࠷খʣͷͨΊͷࢦඪ ༧ଌޡࠩ 
     #JBT
     
     7BSJBODF
148. 148 Ϟσϧͷਪఆɾબ୒ "*$ͷར༻ʹ͍ͭͯ l "*$ͷઈର஋ʹ͸ҙຯ͕ͳ͍͕ɺ"*$ͷࠩʹ͸ҙຯ͕͋Δ l "*$͸༧ଌ෼෍ΛධՁ͢Δ΋ͷͰɺઆ໌ม਺ͷޮՌͷݕূ΍ݕఆ ͳͲʹ༻͍Δ͜ͱ͸ద੾Ͱͳ͍ɻ l આ໌ม਺Λ૿΍͢͜ͱͰ"*$͕ৗʹ૿͑ଓ͚Δ৔߹ɺԾఆ͍ͯ͠ ΔϞσϧ଒͕ద͞ͳ͍Մೳੑ͕͋Δ
149. 149 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબ୒
     ଞͷબ୒ํ๏ l ৘ใྔج४ *$ l Ϟσϧͷ౰ͯ͸·Γͷྑ͞ͱɺϞσϧͷෳࡶ͞ʹର͢ΔϖφϧςΟ͔Βߏ ੒͞ΕΔࢦඪͷҰൠܥ l ର਺໬౓ʹج͍ͮͨؔ਺ͱϖφϧςΟؔ਺𝐼(𝑛)ʹΑͬͯߏ੒ l ϖφϧςΟؔ਺ΛͲͷΑ͏ʹઃܭ͢Δ͔Ͱෳ਺ͷࢦඪ͕ଘࡏ l ৘ใྔج४͕খ͍͞Ϟσϧ͕ྑ͍Ϟσϧͷࢦඪ l "*$͸𝐼 𝑛 = 2 ͷϞσϧ IC = −2 ) !/% ' log f x|𝜃1gh + 𝐼(𝑛)𝑝
150. 150 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબ୒
     ଞͷબ୒ํ๏ l ҳ୤౓৘ใྔج४ %*$ l ৘ใྔج४ͷछ l ର਺໬౓ͷࣄޙฏۉΛਪఆྔͱͯ͠ར༻͢ΔϕΠζϕʔεͷࢦඪ l ϖφϧςΟؔ਺͸ࣄલ෼෍ʹΑΔର਺໬౓ͱࣄޙฏۉͷର਺໬౓ͷࠩ 𝐷IC = −2 ~ log f x|𝜃y1 𝑑𝜃y1 + 𝐼 ・ 2 𝐼 ・ =2 ∫ log f x|𝜃y1 𝑑𝜃y1 − 2 ∫ log f x|𝜃 ∏(𝜃|𝑥) 𝑑𝜃 𝜃y1 = ∫ 𝜃 ∏(𝜃|𝑥) 𝑑𝜃
151. 151 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબ୒
     ଞͷબ୒ํ๏ l ϕΠζ৘ใྔج४ #*$ l ৘ใྔج४ͷछ l ,-৘ใྔʹجͮ͘ͷͰ͸ͳ͘पล໬౓ʹج͍ͮͨ৘ใྔ l 𝑛 ≥ 8ͷ࣌ʹ"*$ΑΓ΋ϖφϧςΟ͕ڧ͘ͳΔɻͭ·Γɺ"*$ΑΓ΋ύϥ ϝʔλ਺͕খ͍͞ϞσϧΛબ୒͢Δ܏޲ 𝐵IC = −2 ) !/% ' log f x|𝜃1gh + 𝑝 log 𝑛
152. 152 Ϟσϧͷਪఆɾબ୒ Ϟσϧͷબ୒
     ଞͷબ୒ํ๏ l .JOJNVN
     %FTDSJQUJPO
     -FOHUI
     .%- l σʔλΛϞσϧΛ༻͍ͯѹॖ͢Δࡍͷූ߸௕ͷ࠷খԽʹج͍ͮͨࢦඪ l ৘ใྔج४͸ਅͷ෼෍ͷڑ཭Λߟྀ͢Δ͕ɺ.%-͸σʔλѹॖʹ͍ͭͯ ߟ͑Δ l σʔλ΁ͷద߹ ର਺໬౓ ͱϞσϧίετͷͭͰߏ੒ l 𝐵͕ූ߸௕ʹ͋ͨΔ MDL = − ) !/% ' log f x|𝜃1gh + 𝐵𝑝
153. 153 ਂ૚ֶशͷਐల
154. 154 ਂ૚ֶशͷൃల "MQIB(P;FSP ը૾ੜ੒ 4UBCMF
     %JGGVTJPO ݴޠϞσϧ λϯύΫ࣭ߏ଄༧ଌ "MQIB'PME
155. 155 ਂ૚ֶशͷൃల ҩྍը૾ จ๏ޡΓగਖ਼ ϩϘοτ ෆద੾౤ߘͷݕ஌
156. 156 ਂ૚ֶशͷൃల *NBHF/FU
     $IBMMFOHF
157. 157 ਂ૚ֶशͷൃల *NBHF/FU
     $IBMMFOHF
158. 158 ਂ૚ֶशͷൃల "MFY/FU
159. 159 ਂ૚ֶशͷൃల ਂ૚ֶश l ਂ૚ֶश
     ਂ͍ߏ଄Λ࣋ͭχϡʔϥϧωοτϫʔΫ l χϡʔϥϧωοτϫʔΫ /FVSBM
     /FUXPSL 
     ೴ͷߏ଄ɾػೳͷ໛฿͔ Βൃలͨ͠ϞσϧͰɺਓ޻χϡʔϩϯΛ૊Έ߹ΘͤͨωοτϫʔΫ l Ϟσϧ͸ؔ਺ͷू߹ ⼊⼒ 出⼒ 𝑓% 𝑓" 𝑓V ඍ෼Մೳͳؔ਺
160. 160 ਂ૚ֶशͷൃల ϑΟʔυϑΥϫʔυχϡʔϥϧωοτϫʔΫ Input Layer Hidden Layers Output Layer X = A[0] a[4] A[1] A[3] X Ŷ a[1] n a[2] 1 a[2] 2 a[2] 3 a[2] n a[3] 1 a[3] 2 a[3] 3 a[3] n A[2] A[4] 𝑓% ؔ਺ 𝑓% ͷྫ 𝐴[%] = 𝑓% 𝑥 = 𝑊𝑥 + 𝑏 ઢܗϞσϧ
161. 161 ਂ૚ֶशͷൃల ϑΟʔυϑΥϫʔυχϡʔϥϧωοτϫʔΫ Input Layer Hidden Layers Output Layer X = A[0] a[4] A[1] A[3] X Ŷ a[1] n a[2] 1 a[2] 2 a[2] 3 a[2] n a[3] 1 a[3] 2 a[3] 3 a[3] n A[2] A[4] 𝑓% ؔ਺ 𝑓% ͷྫ 𝐴[%] = 𝑓% 𝑥 = 𝑊𝑥 + 𝑏 ΞϑΟϯม׵ 0.6 0.2 0.8 −0.3 −0.3 0.4 0.1 0.7 ॏΈΛσʔλʹ ߹͏Α͏ʹֶश
162. 162 ਂ૚ֶशͷൃల ϑΟʔυϑΥϫʔυχϡʔϥϧωοτϫʔΫ Input Layer Hidden Layers Output Layer X = A[0] a[4] A[1] A[3] X Ŷ a[1] n a[2] 1 a[2] 2 a[2] 3 a[2] n a[3] 1 a[3] 2 a[3] 3 a[3] n A[2] A[4] 𝑓% 𝐴[%] = 𝑚𝑎𝑥{𝑥, 0} ׆ੑԽؔ਺ ɾ3F-6 𝐴[%] = 1/(1 + 𝑒+#) ɾ4JHNPJE
163. 163 ਂ૚ֶशͷൃల ਂ૚Ϟσϧͷֶश ⼊⼒ 出⼒Ŷ 𝑓% 𝑓" 𝑓V l ଛࣦؔ਺ L 𝑊 = 4 !/% ' ℓ(𝑌! , 𝐹(𝑥! )) ೋ৐ଛࣦ NFBO
     TRVBSFE
     FSSPS ℓ 𝑦, 𝑦€ = (𝑦 − 𝑦€)" ަࠩΤϯτϩϐʔޡࠩ $SPTT
     &OUSPQZ
     MPTT ℓ 𝑦, 𝑦€ = − 4 &/% • 𝑦& log 𝑦& € ͜ΕΛ࠷খԽ͢ΔΑ͏ʹֶश min l L 𝑊
164. 164 ਂ૚ֶशͷൃల ਂ૚Ϟσϧͷֶश l ޡࠩٯ఻೻๏ l ύϥϝʔλߋ৽ͷख๏ l ଛࣦؔ਺ͷඍ෼ΛޙΖ ೖྗํ޲ ʹͲΜͲΜ࿈࠯཯Λ༻͍ͯ఻ൖͤ͞Δ 𝜕𝐿 𝜕𝑤% 𝑥 = 𝜕𝐿 𝜕𝑓V 𝜕𝑓V 𝜕𝑓" 𝜕𝑓" 𝜕𝑓% 𝜕𝑓% 𝜕𝑤% 𝑥 ⼊⼒ 𝑓% 𝑓" 𝑓V 出⼒Ŷ L 𝑊
165. 165 ਂ૚ֶशͷൃల ਂ૚Ϟσϧͷֶश l ޯ഑߱Լ๏ l ଛࣦؔ਺͕࠷খͱͳΔΑ͏ʹ఻ൖ͞ΕͨޡࠩʹԠͯ͡ɺύϥϝʔλΛগ͠ ͣͭߋ৽͍ͯ͘͠ख๏ l ֤ύϥϝʔλͷଛࣦؔ਺ʹର͢Δภඍ෼ Λߦ͍ύϥϝʔλΛߋ৽ 𝑊* = 𝑊*+% − 𝜂 A 𝜕𝐿 𝜕𝑊 ‚/‚"'& From: https://axa.biopapyrus.jp/deep-learning/gradient_descent_method.htm
166. 166 ਂ૚ֶशͷൃల ઢܗϞσϧͱͷҧ͍ a[4] a[4] X ઢܗϞσϧ ֶश ਂ૚ֶशϞσϧ ΧʔωϧϞσϧ ֶश ݻఆ ֶश ֶश
167. 167 ਂ૚ֶशͷൃల ਂ૚ֶशͷಛ௃ l ສೳۙࣅೳྗ l ૚χϡʔϥϧωοτϫʔΫ͸े෼ͳ਺ͷύϥϝʔλ͕͋Ε͹ɺ࿈ଓؔ਺ Λ೚ҙͷਫ਼౓ͰۙࣅՄೳ l ཧ࿦తʹ͸σʔλ͕ແݶʹ͋Γɼૉࢠ਺͕ແݶʹ͋Δχϡʔϥϧωοτ ϫʔΫΛ༻͍Ε͹ͲΜͳ໰୊Ͱ΋ֶशՄೳ from: https://www.slideshare.net/trinmu/ss-226711018
168. 168 ਂ૚ֶशͷൃల ਂ૚ֶशͷಛ௃ l ଟ૚ԽͷԸܙ l χϡʔϥϧωοτϫʔΫͷදݱྗ͸૚ ͷ਺ʹΑͬͯදݱྗ͕ࢦ਺తʹ޲্ l ਂ͍χϡʔϥϧωοτϫʔΫͷදݱೳ ྗΛઙ͍χϡʔϥϧωοτϫʔΫͰද ݱ͠Α͏ͱ͢Δͱେ͖ͳԣ෯ χϡʔ ϩϯ ͕ඞཁ Montufar, Guido F., et al. "On the number of linear regions of deep neural networks." Advances in neural information processing systems 27 (2014). Arora, R., Basu, A., Mianjy, P., & Mukherjee, A. (2016). Understanding deep neural networks with rectified linear units. arXiv preprint arXiv:1611.01491.
169. 169 ਂ૚ֶशͷൃల ਂ૚ֶशͷಛ௃ l εέʔϦϯάଇ l 5SBOTGPSNFSͷੑೳ͸ύϥϝʔλ ਺ɺσʔληοταΠζɺܭࢉ༧ ࢉΛม਺ͱͨ͠γϯϓϧͳ΂͖৐ ଇʹै͏ l ͜ͷ๏ଇ͕ݴޠɾը૾ɾಈըͳͲ ͷ༷ʑͳλεΫʹద༻͞ΕΔ͜ͱ Λࣔ͢ from: Henighan, Tom, Jared Kaplan, Mor Katz, Mark Chen, Christopher Hesse, Jacob Jackson, Heewoo Jun et al. "Scaling laws for autoregressive generative modeling." arXiv preprint arXiv:2010.14701 (2020).
170. 170 ਂ૚ֶशͷൃల ਂ૚ֶशͷಛ௃ l 0WFSQBSBNFUFSJ[BUJPO l ଟ͘ͷେن໛ϞσϧͰͷֶशͰ͸σʔλ਺ 
     ύϥϝʔλ਺ l ύϥϝʔλ਺Λ૿΍͠΋ɺ࣮ࡍͷֶशʹ͓͍ͯաֶश͸ੜ͡ͳ͍ l χϡʔϥϧωοτϫʔΫͰ͸൚Խޡࠩͱ܇࿅ޡࠩͷ͕ࠩখ͘͞ͳ ΔͨΊʁʢཧ࿦త෼ੳ͕ߦΘΕ͍ͯΔʣ Neyshabur, Behnam, et al. "The role of over- parametrization in generalization of neural networks." 7th International Conference on Learning Representations, ICLR 2019. 2019.
171. 171 ࣌ܥྻσʔλͱ ਂ૚ֶश
172. 172 ࣌ܥྻσʔλͱਂ૚ֶश 'FFE'PSXBSE
     //ͱ࣌ܥྻ l ௨ৗͷ'FFE'PSXBE //ʹ࣌ܥྻσʔλΛ ద༻͢Δ͜ͱ͸Մೳ͕ͩɺ࣌ؒͷྲྀΕΛଊ ͑Δ͜ͱ͕Ͱ͖ͳ͍ l ྫ͑͹ɺ༧ଌͷλεΫͰ͸ɺੲͷ෦෼͸ܰ ͘ѻͬͯɺ࠷ۙͷ෦෼͸ॏཁͳΑ͏ʹѻ͍ ͍ͨ l ࣌఺𝑡ͷֶशʹ͓͍ͯকདྷͷ஋𝑡 + 1Λ࢖Θ ͣɺաڈͷ஋͚ͩΛ༻ֶ͍ͯश͍ͨ͠ l ೖྗ௕͕ݻఆͰͳ͘ՄมͰ͋Δ͜ͱͷ΄͏ ͕ɺ࣌ܥྻΛೖྗͷࡍʹخ͍͠
173. 173 ࣌ܥྻσʔλͱਂ૚ֶश 3FDVSSFOU
     /FVSBM
     /FUXPSL 3// ・・・ 𝑥% 𝑥" 𝑥V 𝑥W Feature Vecter l ӅΕঢ়ଶΛ࣋ͬͨχϡʔϥϧωοτϫʔΫ l ࣌ؒ৘ใͷ֓೦Λ࣋ͪɺ࣍ͷ࣌ؒʹӅΕঢ়ଶͷ৘ใΛ͓͘Δ RNN Block RNN Block RNN Block RNN Block
174. 174 ࣌ܥྻσʔλͱਂ૚ֶश 3FDVSSFOU
     /FVSBM
     /FUXPSL 3// 𝑋 Feature Vecter RNN Block from: https://stanford.edu/~shervine/teaching/cs-230/cheatsheet- recurrent-neural-networks 𝑎o!p = 𝑔"(𝑊 qq𝑎o!-"p + 𝑊 qr𝑥o!p + 𝑏q) 𝑦o!p = 𝑔Z(𝑊 sq𝑎o!p + 𝑏s)
175. 175 ࣌ܥྻσʔλͱਂ૚ֶश 3FDVSSFOU
     /FVSBM
     /FUXPSL 3// l ڧΈ l Մม௕ʹରԠ l ϞσϧαΠζ͕ೖྗ௕ʹΑͬͯมԽ͠ͳ͍ l աڈͷ৘ใΛߟྀͨ͠ܭࢉ͕Մೳ l ϞσϧͷॏΈ͕͕࣌ؒมԽͯ͠΋ڞ༗ l ऑΈ l ܭࢉͷ஗͞ l ݱࡏͷঢ়ଶʹରͯ͠কདྷͷೖྗΛߟྀͰ͖ͳ͍ l աڈͷ৘ใ͕࢒͍ͬͯͳ͍Մೳੑ
176. 176 ࣌ܥྻσʔλͱਂ૚ֶश ଞͷ3//ܥ౷ͷϞσϧ l -POH
     4IPSU5FSN
     .FNPSZ
     -45. l ࣌ؒมԽ͢Δ௕ظهԱͷͨΊͷ ϝϞϦ𝑐o!pΛ࣋ͭ l ͭͷήʔτΛ࣋ͭ͜ͱͰաڈͷ৘ใͷ ๨٫΍ɺ৽͍͠ೖྗͷऔΓೖΕΛௐ੔ l Γ1 
     3FMFWBODF
     (BUF l Γ( 
     'PSHFU
     (BUF l Γ2: 6QEBUF
     (BUF l Γ3 
     0VUQVU
     (BUF 𝑐o!p = Γt ⋆ ̃ 𝑐o!p + Γu ⋆ 𝑐o!-"p ̃ 𝑐o!p = tanh(𝑊 v Γw ⋆ 𝑎o!-"p, 𝑥o!p + 𝑏v) 𝑎o!p = Γx ⋆ 𝑐o!p from: https://stanford.edu/~shervine/teaching/cs-230/cheatsheet- recurrent-neural-networks
177. 177 ࣌ܥྻσʔλͱਂ૚ֶश ଞͷ3//ܥ౷ͷϞσϧ l (BUFE
     3FDVSSFOU
     6OJU
     (36 l -45.ΛҰൠԽͨ͠Ϟσϧ l -45.ΑΓ΋ܰྔͱͳ͓ͬͯΓɺͭͷ ήʔτͷΈͰߏ੒͠ܭࢉޮ཰ԽΛ޲্ l Γ1 
     3FMFWBODF
     (BUF l Γ2: 6QEBUF
     (BUF l -45.΋(36΋৽͍͠৘ใΛ֮͑ɺ աڈͷ৘ใΛద౓ʹ๨ΕΔػߏΛ උ͑ͨϞσϧ 𝑐o!p = Γt ⋆ ̃ 𝑐o!p + (1 − Γt) ⋆ 𝑐o!-"p ̃ 𝑐o!p = tanh(𝑊 v Γw ⋆ 𝑎o!-"p, 𝑥o!p + 𝑏v) 𝑎o!p = 𝑐o!p from: https://stanford.edu/~shervine/teaching/cs-230/cheatsheet-recurrent-neural-networks
178. 178 ࣌ܥྻσʔλͱਂ૚ֶश 3FDVSSFOU
     /FVSBM
     /FUXPSL 3// l ڧΈ l Մม௕ʹରԠ l ϞσϧαΠζ͕ೖྗ௕ʹΑͬͯมԽ͠ͳ͍ l աڈͷ৘ใΛߟྀͨ͠ܭࢉ͕Մೳ l ϞσϧͷॏΈ͕͕࣌ؒมԽͯ͠΋ڞ༗ l ऑΈ l ܭࢉͷ஗͞ l ݱࡏͷঢ়ଶʹରͯ͠কདྷͷೖྗΛߟྀͰ͖ͳ͍ l աڈͷ৘ใ͕࢒͍ͬͯͳ͍Մೳੑ
179. 179 ࣌ܥྻσʔλͱਂ૚ֶश ૒ํ޲3//
     #JEJSFDUJPOBM
     3// from: https://stanford.edu/~shervine/teaching/cs-230/cheatsheet-recurrent-neural-networks l ܥྻͷ࠷ॳ͔Βͷॱ఻ൖʹՃ͑ͯɺܥྻ ͷ࠷ޙ͔Βॱʹٯ޲͖ͷ఻ൖɺ྆ํ޲ʹ ߦ͏ωοτϫʔΫ l ࣌ܥྻσʔλʹద༻͞ΕΔ͜ͱ͸ଟ͘ͳ ͍͕ɺݴޠγʔΫΤϯεͳͲʹద༻͞Ε Δ͜ͱ͕ଟ͍
180. 180 ࣌ܥྻσʔλͱਂ૚ֶश 3FDVSSFOU
     /FVSBM
     /FUXPSL 3// l ڧΈ l Մม௕ʹରԠ l ϞσϧαΠζ͕ೖྗ௕ʹΑͬͯมԽ͠ͳ͍ l աڈͷ৘ใΛߟྀͨ͠ܭࢉ͕Մೳ l ϞσϧͷॏΈ͕͕࣌ؒมԽͯ͠΋ڞ༗ l ऑΈ l ܭࢉͷ஗͞ l ݱࡏͷঢ়ଶʹରͯ͠কདྷͷೖྗΛߟྀͰ͖ͳ͍ l աڈͷ৘ใ͕࢒͍ͬͯͳ͍Մೳੑ
181. 181 ࣌ܥྻσʔλͱਂ૚ֶश ޯ഑ফࣦ 7BOJTIJOH
     HSBEJFOU ・・・ 𝑥% 𝑥" 𝑥V 𝑥W Feature Vecter RNN Block RNN Block RNN Block RNN Block 𝜕𝐿 𝜕𝑤% 𝑥% = 𝜕𝐿 𝜕𝑅𝑁𝑁W … 𝜕𝑅𝑁𝑁% 𝜕𝑤% 𝑥% ޡࠩٯ఻೻ʹඞཁͳޯ഑͕ඇৗʹ খ͘͞ͳΓɺֶश੍͕ޚͰ͖ͳ͘ ͳΔ໰୊
182. 182 ࣌ܥྻσʔλͱਂ૚ֶश 3//ͷར༻ํ๏ l 0OFUP.BOZܕ l ࣌ܥྻੜ੒ͳͲ l .BOZUP0OFܕ l ࣌ܥྻ෼ྨ
183. 183 ࣌ܥྻσʔλͱਂ૚ֶश 3//ͷར༻ํ๏ l .BOZUP.BOZܕ  l ҟৗ஋ݕ஌ͳͲ l &ODPEFS%FDPEFSܕ l ࣌ܥྻ༧ଌͳͲ
184. 184 ࣌ܥྻσʔλͱਂ૚ֶश &ODPEFS%FDPEFS
     4FRVFODFUP4FRVFODF l ௕ظ༧ଌɾੜ੒Λߦ͏ͨΊͷϑϨʔϜϫʔΫ l ೖྗܥྻΛ&ODPEFSͱݺ͹ΕΔ3//CBTFEΛ༻͍ͯϕΫτϧʹม׵ &ODPEF ͠ɺ%FDPEFSͱݺ͹ΕΔผͷ3//CBTFEΛ༻͍ͯλʔ ήοτܥྻʹۙ͘ͳΔΑ͏ʹग़ྗ͢ΔϞσϧ
185. 185 ࣌ܥྻσʔλͱਂ૚ֶश %FDPEFS΁ͷೖྗ l ੜ੒͞Εͨ΋ͷΛͦͷ··ೖྗ͢Δͱɺޡ͕ࠩͲΜͲΜੵΈॏͳΓ ֶश͕ෆ҆ఆɾऩଋ͕஗͘ͳΔͱ͍͏໰୊
186. 186 ࣌ܥྻσʔλͱਂ૚ֶश %FDPEFS΁ͷೖྗ l ਖ਼ղΛͦͷ··ೖྗ͢Δख๏ l ֶश͕҆ఆɾऩଋ͕଎͘ͳΔ͕ɺ܇࿅࣌ͱςετ࣌ͷ෼෍͕ҟͳΔͱ ͍͏໰୊΋ 5FBDIFS
     'PSDJOH
187. 187 ࣌ܥྻσʔλͱਂ૚ֶश %FDPEFS΁ͷೖྗ l ֬཰ʹԠͯ͡ੜ੒σʔλ͔ڭࢣσʔλͷೖྗΛมԽͤ͞Δख๏ 4DIFEVMFE
     4BNQMJOH from: Bengio, Samy, et al. "Scheduled sampling for sequence prediction with recurrent neural networks." Advances in neural information processing systems 28 (2015).
188. 188 ࣌ܥྻσʔλͱਂ૚ֶश %FDPEFS΁ͷೖྗ l ("/Λ༻͍ͯੜ੒σʔλͱڭࢣσʔλͷೖྗʹΑΔੜ੒ͷࠩΛখ͞ ͘͢Δ 1SPGFTTPS
     4BNQMJOH from: Lamb, Alex M., et al. "Professor forcing: A new algorithm for training recurrent networks." A neural information processing systems 29 (2016).
189. 189 ࣌ܥྻσʔλͱਂ૚ֶश $POWPMVUJPOBM
     /FVSBM
     /FUXPSL $// l ը૾ͳͲΛଊ͑ΔͨΊʹ༻͍ΒΕΔ/FVSBM
     /FUXPSL l $POWPMVUJPOBM͸৞ΈࠐΈͱݺ͹Εɺબ୒͞ΕͨϑΟϧλʔαΠζʹ Ԡͯ͡ը૾৘ใΛѹॖ͍ͯ͘͠
190. 190 ࣌ܥྻσʔλͱਂ૚ֶश $POWPMVUJPOBM
     /FVSBM
     /FUXPSL $// l ը૾ͳͲΛଊ͑ΔͨΊʹ༻͍ΒΕΔ/FVSBM
     /FUXPSL l $POWPMVUJPOBM͸৞ΈࠐΈͱݺ͹Εɺબ୒͞ΕͨϑΟϧλʔαΠζʹ Ԡͯ͡ը૾৘ใΛѹॖ͍ͯ͘͠ ྖҬΛ৞ΈࠐΜͰѹॖ͢Δख๏Λ ࣌ܥྻσʔλʹԠ༻Ͱ͖ͳ͍͔ʁ
191. 191 ࣌ܥྻσʔλͱਂ૚ֶश %JMBUFE
     $BVTBM
     $POWPMVUJPO from: https://deepmind.com/blog/article/wavenet-generative-model-raw-audio
192. 192 ࣌ܥྻσʔλͱਂ૚ֶश %JMBUFE
     $BVTBM
     $POWPMVUJPO from: Oord, A. V. D., Dieleman, S., Zen, H., Simonyan, K., Vinyals, O., Graves, A., ... & Kavukcuoglu, K. (2016). Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499. l $//Λ༻͍ͨ࣌ؒ೾ܗ Ի੠ੜ੒ ʹର͢Δ֬཰తੜ੒ϞσϧͰ͋Δ 8BWF/FUͰఏҊ͞Εͨߏ੒ཁૉͷͭɻ l ೖྗ૚͔ΒॱʹͦΕͧΕɺɺɺݸͣͭεΩοϓ͠ͳ͕Β৞Έࠐ ΈΛܭࢉɻ৞ΈࠐΈ૚ͷਂ͞ʹԠͯ͡ɺೖྗͰ͖Δܥྻ௕͕ࢦ਺తʹ ૿Ճ͍ͯ͘͠ɻ l શ݁߹//ͷΑ͏ʹύϥϝʔλ͕ଟ͘ͳ͘ɺ3//ͷΑ͏ʹճؼతͳ઀ ଓ͕ແ͍ͷͰɺֶश͕3//ΑΓ଎͘ͳ͍ͬͯΔ l ௕ظͷܥྻΛೖྗͱͯ͠΋ޯ഑ফࣦ͕ੜ͡ͳ͍
193. 193 ࣌ܥྻσʔλͱਂ૚ֶश 5SBOTGPSNFS l l"UUFOUJPO
     JT
     "MM
     :PV
     OFFEzͰఏҊ l ओʹࣗવݴޠॲཧ΍ը૾ॲཧͳͲʹ͓͍ͯߴ͍ਫ਼౓Λୡ੒͍ͯ͠ΔϞ σϧͷҰछ BERT ViT from: Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., ... & Polosukhin, I. (2017). Attention is all you need. Advances in neural information processing systems, 30.
194. 194 ࣌ܥྻσʔλͱਂ૚ֶश 5SBOTGPSNFS l 4FMG"UUFOUJPOͱ'FFE
     'PSXBSEͰओʹߏ੒ l &ODPEFS%FDPEFSϞσϧ l $//ͱൺ΂େҬతͳؔ܎Λଊ͑Δ͜ͱ͕Մೳ
195. 195 ࣌ܥྻσʔλͱਂ૚ֶश 5SBOTGPSNFS l 4FMG"UUFOUJPOͱ'FFE
     'PSXBSEͰओʹߏ੒ l &ODPEFS%FDPEFSϞσϧ l $//ͱൺ΂େҬతͳؔ܎Λଊ͑Δ͜ͱ͕Մೳ l 1PTJUJPOBM
     &ODPEJOHͱ͍͏֤ೖྗͷ Ґஔ৘ใΛϕΫτϧͰදݱ
196. 196 ࣌ܥྻσʔλͱਂ૚ֶश 1PTJUJPOBM
     &ODPEJOH l 5SBOTGPSNFSͷಛੑ্ɺೖྗܥྻͷҐஔ৘ใ͕ࣦΘΕͯ͠· ͏ͨΊҐஔΛ໌ࣔతʹࣔͨ͠ಛ௃ྔΛೖྗʹՃ͑Δ l 4JOVTPJEBM
     QPTJUJPOBM
     FNCFEEJOHT l ೖྗͷ֤UPLFOͷҐஔΛTJOؔ਺ͱDPTؔ਺ʹΑΔϕΫτϧͰදݱ l ֶशແ͠ʹ૬ରతͳҐஔ৘ใΛදݱ
197. 197 ࣌ܥྻσʔλͱਂ૚ֶश 1PTJUJPOBM
     &ODPEJOH from: https://kazemnejad.com/blog/transformer_architecture_positional_encoding/
198. 198 ࣌ܥྻσʔλͱਂ૚ֶश 1PTJUJPOBM
     &ODPEJOH l 5JNF4UBNQ QPTJUJPOBM
     FNCFEEJOH l ࣌ܥྻσʔλΛೖྗͷࡍʹ೥݄೔࣌ؒͳ Ͳͷ৘ใ͕͋Ε͹ར༻Ͱ͖Δ&NCFEEJOH ͷҰछ l *OGPSNFS΍"VUPGPSNFSͳͲͷ࠷৽ͷ 5SBOTGPSNFSϕʔεͷ࣌ܥྻ༧ଌϞσϧ ʹ༻͍ΒΕ͍ͯΔ
199. 199 ࣌ܥྻσʔλͱਂ૚ֶश 5SBOTGPSNFS Transformer Block l 4FMG"UUFOUJPOͱ'FFE
     'PSXBSEͰओʹߏ੒ l &ODPEFS%FDPEFSϞσϧ l $//ͱൺ΂େҬతͳؔ܎Λଊ͑Δ͜ͱ͕Մೳ l 1PTJUJPOBM
     &ODPEJOHͱ͍͏֤ೖྗͷ Ґஔ৘ใΛϕΫτϧͰදݱ l &ODPEFS
     %FDPEFSͱ΋ʹ .VMUJIFBE
     BUUFOUJPOͱ'FFE
     'PSXBSEͰ ߏ੒͞Εͨ5SBOTGPSNFS
     #MPDLͷੵΈॏͶ
200. 200 5SBOTGPSNFS
     #MPDL ࣌ܥྻσʔλͱਂ૚ֶश • &ODPEFSɼ%FDPEFSͱͱ΋ʹ5SBOTGPSNFS
     #MPDLͷੵΈॏͶͰߏ੒ • 5SBOTGPSNFS
     #MPDLͷߏ੒ 
     .VMUJ)FBE
     4FMGBUUFOUJPO 
     3FTJEVBM
     $POOFDUJPO
     ࢒ࠩ઀ଓ 
     -BZFS
     /PSNBMJ[BUJPO
     
     1PTJUJPOXJTF
     'FFE
     'PSXBSE 
     %SPQPVU Transformer Block
201. 201 4FMGBUUFOUJPO ࣌ܥྻσʔλͱਂ૚ֶश ⼊⼒系列の 潜在表現 系列⻑ × 次元数 Key : K Query: Q 𝑊y 𝑊z Value: V Attention Map : M 系列⻑ × 系列⻑ 𝑊{ 𝑊xt! Output 𝑠𝑜𝑓𝑡𝑚𝑎𝑥( 𝑄𝐾# 𝑑 )
202. 202 4FMGBUUFOUJPO ࣌ܥྻσʔλͱਂ૚ֶश Value: V Attention Map : M 系列⻑ × 系列⻑ 𝑊xt! Output 𝑠𝑜𝑓𝑡𝑚𝑎𝑥( 𝑄𝐾# 𝑑 ) ֤࣌఺ͷಛ௃͕ଞͷ࣌఺Λߟྀ ͯ͠࠶ߏ੒͞ΕΔ
203. 203 .VMUJ)FBE
     4FMGBUUFOUJPO ࣌ܥྻσʔλͱਂ૚ֶश • ઌఔͷ4FMGBUUFOUJPOͷܭࢉΛෳ਺ฒߦ ࣮ͯ͠ߦ • ෳ਺ͷϕΫτϧʹ੾Γ෼͚ͯܭࢉ͢Δ͜ͱ Ͱೖྗؒͷଟ༷ͳྨࣅੑΛൃݟ͠ɼΑΓ ଟ༷ͳදݱྗΛ֫ಘ
204. 204 5SBOTGPSNFS
     #MPDL ࣌ܥྻσʔλͱਂ૚ֶश • &ODPEFSɼ%FDPEFSͱͱ΋ʹ5SBOTGPSNFS
     #MPDLͷੵΈॏͶͰߏ੒ • 5SBOTGPSNFS
     #MPDLͷߏ੒ 
     .VMUJ)FBE
     4FMGBUUFOUJPO 
     3FTJEVBM
     $POOFDUJPO
     ࢒ࠩ઀ଓ 
     -BZFS
     /PSNBMJ[BUJPO
     
     1PTJUJPOXJTF
     'FFE
     'PSXBSE 
     %SPQPVU Transformer Block
205. 205 3FTJEVBM
     $POOFDUJPO
     ࢒ࠩ઀ଓ ࣌ܥྻσʔλͱਂ૚ֶश Residual𝑂𝑢𝑡𝑝𝑢𝑡 = 𝑥 + 𝐴𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(𝑥) l ೖྗ𝑥Λ.VMUJ)FBE
     4FMG"UUFOUJPOͷग़ྗʹ Ճ͑Δ͜ͱͰɺֶशͷ҆ఆԽΛਤΔ l 3FT/FUͳͲͷϞσϧͰఏҊ͞Εͨख๏Ͱɺ ޯ഑ফࣦɾരൃ໰୊ΛճආͰ͖Δ
206. 206 5SBOTGPSNFS
     #MPDL ࣌ܥྻσʔλͱਂ૚ֶश • &ODPEFSɼ%FDPEFSͱͱ΋ʹ5SBOTGPSNFS
     #MPDLͷੵΈॏͶͰߏ੒ • 5SBOTGPSNFS
     #MPDLͷߏ੒ 
     .VMUJ)FBE
     4FMGBUUFOUJPO 
     3FTJEVBM
     $POOFDUJPO
     ࢒ࠩ઀ଓ 
     -BZFS
     /PSNBMJ[BUJPO
     
     1PTJUJPOXJTF
     'FFE
     'PSXBSE 
     %SPQPVU Transformer Block
207. 207 -BZFS
     /PSNBMJ[BUJPO
     ࣌ܥྻσʔλͱਂ૚ֶश 𝐿𝑎𝑦𝑒𝑟𝑁𝑜𝑟𝑚 𝑥 = 𝛾 𝑉𝑎𝑟 𝑥 ∗ 𝑥 − 𝜇 𝑥 + 𝛽 l ֤࣍ݩͷग़ྗΛฏۉ෼ࢄʹਖ਼نԽ͢Δख๏ l ޯ഑ফࣦɾޯ഑രൃΛ཈͑Δ໾ׂ
208. 208 5SBOTGPSNFS
     º *OGMVFO[B
     GPSFDBTUJOH ࣌ܥྻσʔλͱਂ૚ֶश Wu, N., Green, B., Ben, X., & O'Banion, S. (2020). Deep transformer models for time series forecasting: The influenza prevalence case. arXiv preprint arXiv:2001.08317.
209. 209 ਂ૚ֶश WT
     ౷ܭతϞσϧ ࣌ܥྻσʔλͱਂ૚ֶश Wu, N., Green, B., Ben, X., & O'Banion, S. (2020). Deep transformer models for time series forecasting: The influenza prevalence case. arXiv preprint arXiv:2001.08317. .σʔληοτͷTUFQ
     BIFBE GPSFDBTUJOHͷਫ਼౓ݕূ
210. 210 ࣌ܥྻλεΫͱ ۙ೥ͷݚڀ
211. 211 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ ࣌ܥྻ༧ଌ ？ ݱࡏ কདྷ ֶश
212. 212 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ ࣌ܥྻ༧ଌ ݱࡏ কདྷ ֶश
213. 213 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ /#FBUT l 'VMMZ
     $POOFDUFEͰߏ੒͞Εͨϒ ϩοΫͷੵΈ্͛ʹΑΔ࣌ܥྻ༧ଌ Ϟσϧ l ϒϩοΫؒͰ͸࢒ࠩػߏʹΑΔ઀ଓ ͱɺϒϩοΫ಺෦Ͱ͸ઢܗ݁߹Ͱߏ ੒͞Ε͓ͯΓɺղऍੑͷߴ͍Ϟσϧ ͱͳ͍ͬͯΔʢͲͷ෦෼Ͱ൓Ԡͯ͠ ͍Δ͔͕෼͔Δʣ from: Oreshkin, Boris N., et al. "N-BEATS: Neural basis expansion analysis for interpretable time series forecasting." International Conference on Learning Representations. 2019.
214. 214 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ ঢ়ଶۭؒϞσϧ º ਂ૚ֶश
     4 l εςοϓҎ্ͷඇৗʹ௕͍γʔέϯεʹରԠͰ͖Δঢ়ଶۭؒϞσ ϧΛ׆༻ͨ͠ख๏ΛఏҊ l ঢ়ଶۭؒϞσϧΛ3FDVSSFOUͱ$POWPMVUJPOBMදݱʹஔ͖׵͑ l )JQQPߦྻΛ3//ʹ૊ΈࠐΉ͜ͱͰ௕ظهԱੑΛ֫ಘ from: Gu, Albert, Karan Goel, and Christopher Re. "Efficiently Modeling Long Sequences with Structured State Spaces." International Conference on Learning Representations. 2021.
215. 215 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ ਂ૚ֶश º ෼෍ग़ྗ from: Salinas, David, et al. "DeepAR: Probabilistic forecasting with autoregressive recurrent networks." International Journal of Forecasting 36.3 (2020): 1181-1191. l %FFQ"3 l ֬཰తͳ༧ଌΛ࣮ݱ͢ΔͨΊʹෛͷೋ߲໬౓ؔ਺ͱϞϯςΧϧϩαϯϓ Ϧϯάͷग़ྗܗࣜͷಋೖ Training Prediction
216. 216 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ ਂ૚ֶश º ෼෍ग़ྗ from: Guen, Vincent Le, and Nicolas Thome. "Probabilistic time series forecasting with structured shape and temporal diversity." arXiv preprint arXiv:2010.07349 (2020). l 453*1& l Ψ΢ε෼෍ͳͲͷҰൠతͳ ෼෍Ͱ͸ͳ͘ɺඇఆৗͳ࣌ ܥྻʹ΋ରԠͨ࣌ؒ͠తɺ ܗঢ়త؍఺Ͱॊೈͳ෼෍༧ ଌ͕Մೳͳ&ODPEFS %FDPEFSϞσϧͷఏҊ
217. 217 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ ਂ૚ֶश º ෼෍ग़ྗ from: de Bézenac, E., Rangapuram, S. S., Benidis, K., Bohlke-Schneider, M., Kurle, R., Stella, L., ... & Januschowski, T. (2020). Normalizing kalman filters for multivariate time series analysis. Advances in Neural Information Processing Systems, 33, 2995-3007. l /PSNBMJ[JOH
     ,BMNBO
     'JMJUFST l ઢܗΨ΢εঢ়ଶۭؒϞσϧΛ/PSNBMJ[JOH
     GMPXͰิڧͯ͠ଟมྔ࣌ܥྻ ͷϞσϦϯάΛ࣮ݱ͢Δ l 3FBM/71ͷΞʔΩςΫνϟΛར༻
218. 218 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ ਂ૚ֶश º ෼෍ग़ྗ from: https://lilianweng.github.io/posts/2018-10-13-flow-models/ l /PSNBMJ[JOH
     'MPX l ୯७ͳ֬཰ม਺𝑧. ʹରͯ͠ඇઢܗม׵𝑓\ ΛॏͶΔ͜ͱʹΑͬͯɺॊೈͳ෼ ෍𝑝y(𝑍y)Λ֫ಘ͢ΔͨΊͷख๏
219. 219 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ ෆ౳ִؒ࣌ܥྻ from: Chen, Ricky TQ, et al. "Neural ordinary differential equations." Advances in neural information processing systems 31 (2018). l 0%&3// l &ODPEFS%FDPEFSϞσϧʹ͓͚Δσίʔμͷજࡏม਺ग़ྗΛ0%&/FUʹ ஔ͖׵͑Δ͜ͱͰෆ౳ִؒ࣌ܥྻʹରԠ
220. 220 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ ෆ౳ִؒ࣌ܥྻ from: Chen, Ricky TQ, et al. "Neural ordinary differential equations." Advances in neural information processing systems 31 (2018). l /FVSBM
     0SEJOBSZ
     %JGGFSFOUJBM
     &RVBUJPO l 3FT/FUͱৗඋ෼ํఔࣜʹྨࣅ఺ʹண໨͠ɺৗඍ෼ํఔࣜͷղ๏Λχϡʔ ϥϧωοτͷදݱʹ༻͍Δख๏ l ͜ΕΛ༻͍Δ͜ͱͰϝϞϦޮ཰ͷߴ͍ ࣌ؒ࿈ଓͳϞσϧΛߏங͢Δ͜ͱ͕Մೳ ͱͳΔ l જࡏදݱ͸0%&
     4PMWFSʹΑͬͯܭࢉՄೳ
221. 221 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ 5SBOTGPSNFS
     º ࣌ܥྻ༧ଌ l *OGPSNFS l 5SBOTGPSNFS͕௕ظܥྻ༧ଌʹ͸ؤ݈Ͱͳ͍͜ͱ΍ɺܭࢉޮ཰͕ѱ͍͜ ͱΛղܾ͢ΔͨΊʹ4FMGBUUFOUJPOͷܭࢉޮ཰޲্ͷςΫχοΫΛఏҊ from: Zhou, Haoyi, et al. "Informer: Beyond efficient transformer for long sequence time-series forecasting." Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 35. No. 12. 2021. Method Complexity per Layer Convolutional 𝑂 𝐾 ~ 𝐷0 ~ 𝐿 Recurrent 𝑂 𝐿 ~ 𝐷0 Self-attention (Transformer) 𝑂 𝐿0 ~ 𝐷 K: the length of filter D: dimensionality of space L: input length N: Number of layers Computational Complexity 𝑶 𝑵×(𝑳𝟐 ‰ 𝑫)
222. 222 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ 5SBOTGPSNFS
     º ࣌ܥྻ༧ଌ l *OGPSNFS ̎ͭͷ$PNQMFYJUZ࡟ݮख๏ΛఏҊ l 1SPC4QBSTF
     ॏཁ౓ͷߴ͍"UUFOUJPO
     .BQͷΈΛར༻ 𝑂 𝐿Z ‹ 𝐷 ˠ 𝑂 𝐿 log 𝐿 ‹ 𝐷 ʹ࡟ݮ l 4FMGBUUFOUJPO
     %JTUJMMJOH
     ηϧϑΞςϯγϣϯ૚Λग़Δ౓ ܥྻͷ௕͕͞൒෼ʹͳΔΑ͏ৠཹ 𝑂 N ‹ ⋯ ˠ 𝑂 2 − 𝜖 ‹ ⋯ ʹ࡟ݮ
223. 223 1SPC4QBSTF ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ ⼊⼒系列の 潜在表現 系列⻑(L) × 次元数(D) Key : K Query: • 𝑄 𝑊y 𝑊z Value: V Attention Map : M 系列⻑ × 系列⻑ 𝑊{ 𝑊xt! Output 𝑠𝑜𝑓𝑡𝑚𝑎𝑥( • 𝑄𝐾# 𝐷 ) 上位u件のQuery のみを利⽤
224. 224 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ 5SBOTGPSNFS
     º ࣌ܥྻ༧ଌ l "VUPGPSNFS l 5SBOTGPSNFS಺෦ʹ࣌ܥྻ෼ղͷػߏΛඋ͑ͨϞσϧ from: Wu, Haixu, et al. "Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting." Advances in Neural Information Processing Systems 34 (2021): 22419-22430.
225. 225 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ Ґஔ৘ใ º ࣌ܥྻ༧ଌ l %JGGVTJPO
     $POWPMVUJPOBM
     3// l ଟมྔ࣌ܥྻͷؔ܎Λάϥϑͱͯ͠ଊ͑ͯ ಓ࿏ͷಓͳͲ ࣌ܥྻ༧ଌΛ ߦ͏ϞσϧΛఏҊ l (SBQI
     /FVSBM
     /FUXPSLͱ3//ͷ૊Έ߹Θͤ from: Li, Yaguang, et al. "Diffusion Convolutional Recurrent Neural Network: Data-Driven Traffic Forecasting." International Conference on Learning Representations. 2018.
226. 226 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ (SBQI
     /FVSBM
     /FUXPSL l άϥϑσʔλͷֶशͷͨΊʹχϡʔϥϧωοτϫʔΫʹద༻ͨ͠Ϟσϧ l άϥϑ͸ϊʔυͱΤοδͰߏ੒͞ΕΔදݱํ๏Ͱɺϊʔυ΍Τοδͷද ݱΛྡ઀ϊʔυͷ৘ใΛ༻͍ͯߋ৽͍ͯ͘͠
227. 227 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ (SBQI
     /FVSBM
     /FUXPSL l छྨͷ(// l (SBQI
     $POWPMVUJPOBM
     //
     $//ͷ৞Έ͜Έͷ֓೦Λάϥϑʹద༻
228. 228 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ (SBQI
     /FVSBM
     /FUXPSL l (SBQI
     $POWPMVUJPOBM
     // l $//ͷ৞Έ͜Έͷ֓೦Λάϥϑʹద༻
229. 229 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ༧ଌ (SBQI
     /FVSBM
     /FUXPSL l (SBQI
     "UUFOUJPO
     // l ϊʔυͷྡ઀ʹ͍ͭͯͷॏΈΛ"UUFOUJPOͰදݱ
230. 230 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ෼ྨ ࣌ܥྻ෼ྨ from: https://medium.com/@hassanismailfawaz/deep-learning-for-time- series-classification-a-brief-overview-73b58767ed0f ？
231. 231 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ෼ྨ 4IBQFMFU l Ϋϥε෼ྨʹ༗ޮͳ೾ܗύλʔϯɻ෼ྨʹ༗ޮͳಛ௃͸࣌ܥྻશମͰ͸ ͳ͘গ਺ͷ෦෼࣌ܥྻʹදΕΔͱ͍͏ΞΠσΞʹجͮ͘ɻ l 4IBQFMFUΛൃݟޙɺߴ଎ʹਪ࿦Ͱ͖Δ from: Grabocka, Josif, et al. "Learning time-series shapelets." Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining. 2014.
232. 232 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ෼ྨ *ODFQUJPO
     5JNF l ը૾෼ྨʹ༻͍ΒΕΔϑϨʔϜϫʔΫͰ͋Δ*ODFQUJPOΛ࣌ܥྻʹԠ༻ l ࣌ܥྻΛ#PUUMFOFDLͰѹॖ͠ɺ$POWPMVUJPOBMΧʔωϧͰֶश from: Ismail Fawaz, Hassan, et al. "Inceptiontime: Finding alexnet for time series classification." Data Mining and Knowledge Discovery 34.6 (2020): 1936-1962.
233. 233 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ෼ྨ 30$,&5 l ཚ਺഑ྻͱ࣌ܥྻσʔλΛ͔͚߹Θͤͨಛ௃ྔΛେྔʹੜ੒͠ɺੜ੒͞ Εͨܥྻͷಛ௃ʢ࠷େ஋ͱ಺ੵ͕ਖ਼Ͱ͋ͬͨλΠϜ΢Οϯυ΢ͷׂ߹ʣ Λ༻͍Δ͜ͱͰ࣌ܥྻ෼ྨΛֶश from: Dempster, Angus, François Petitjean, and Geoffrey I. Webb. "ROCKET: exceptionally fast and accurate time series classification using random convolutional kernels." Data Mining and Knowledge Discovery 34.5 (2020): 1454-1495. 畳み込み Time Window 1.23 2.34 Time Window分の特徴を⽣成 ・・・ 特徴量を作成 最⼤: 2.34 正の割合: 0.21 学習
234. 234 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ෼ྨ 5-PTT l ଛࣦؔ਺Λڑ཭ֶशʹ͢Δ͜ͱͰڭࢣͳ͠Ͱ࣌ܥྻͷಛ௃ྔΛֶश l Ϟσϧࣗମ͸%JMBUFE
     $//Λར༻ Franceschi, Jean-Yves, Aymeric Dieuleveut, and Martin Jaggi. "Unsupervised scalable representation learning for multivariate time series." Advances in neural information processing systems 32 (2019).
235. 235 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ࣌ܥྻ෼ྨ ڑ཭ֶश .FUSJD
     -FBSOJOH l ࣅͨαϯϓϧಉ࢜ͷಛ௃ྔ͸͚ۙͮɺࣅ͍ͯͳ ͍ಛ௃ྔͷڑ཭͸ԕ͚͟Δͱ͍͏ૢ࡞Λଛࣦؔ ਺Λ௨ֶͯ͡श͢Δख๏ l ୅දతͳڑ཭ֶश༻ͷଛࣦؔ਺ͱͯ͠ɺ5SJQMFU
     -PTT΍$POUSBTUJWF
     -PTTͳͲ͕ଘࡏ
236. 236 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ҟৗݕ஌ ҟৗݕ஌ 
     มԽݕ஌ From: https://neptune.ai/blog/anomaly-detection-in-time-series ҟৗʂ ҟৗʂ
237. 237 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ҟৗݕ஌ -PDBM
     0VUMJFS
     'BDUPS
     -0' l ࣗ਎ͷ఺͔Βۙ๣L
     ݸͷ఺ͱ͍͔ʹີ͔Ͱ͋Δ͔ࣔ͢ہॴີ౓ -PDBM
     EFOTJUZ Λ༻͍ͨ֎Ε஋ͷݕग़
238. 238 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ҟৗݕ஌ &ODPEFS%FDPEFSʹΑΔѹॖ from: Li, Shu, et al. "Fair Outlier Detection Based on Adversarial Representation Learning." Symmetry 14.2 (2022): 347.
239. 239 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ҟৗݕ஌ *TPMBUJPO
     'PSFTU from: https://qiita.com/tchih11/items/d76a106e742eb8d92fb4
240. 240 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ҟৗݕ஌ .BUSJY
     1SPGJMF
     from: https://towardsdatascience.com/introduction-to-matrix-profiles-5568f3375d90
241. 241 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ҟৗݕ஌ .BUSJY
     1SPGJMF
     
242. 242 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ϞσϦϯά ϞσϦϯά モデル
243. 243 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ϞσϦϯά 8FCσʔλͷϞσϦϯά from: Murayama, Taichi, Yasuko Matsubara, and Sakurai Yasushi. "Mining Reaction and Diffusion Dynamics in Social Activities." arXiv preprint arXiv:2208.04846 (2022). l ֦ࢄ൓ԠํఔࣜͱχϡϥʔϧωοτϫʔΫͷ૊Έ߹Θͤ
244. 244 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ϞσϦϯά 4/4ؒͷϑϩʔ l ଟมྔ)BXLFTաఔΛ༻͍ͨ4/4ؒͷྲྀΕͷՄࢹԽ from: Zannettou, Savvas, et al. "The web centipede: understanding how web communities influence each other through the lens of mainstream and alternative news sources." Proceedings of the 2017 internet measurement conference. 2017.
245. 245 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ϞσϦϯά 1PJOU
     1SPDFTT
     ఺աఔ l ΠϕϯτσʔλͷϞσϦϯάʹ༻͍ΒΕΔ౷ܭతϞσϧ l Ұఆظؒͷۭؒɾ࣌ؒɾΠϕϯτͷੑ࣭ΛϞσϦϯά l 1PJTTPOաఔ l ۚ༥΍஍਒ͳͲͷ֬཰తʹൃੜ͢Δࣄ৅ͷϞσϦϯάʹ༻͍ΒΕΔ l )BXLFTաఔ l ࣗ෼ͷաڈͷΠϕϯτʹґଘ ͠ TFMGFYDJUJOH ͳੑ࣭Λ࣋ͪɺ 4/4ͷόʔετͳͲΛϞσϦϯά
246. 246 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ϞσϦϯά ਓؒͷߦಈʹ͍ͭͯͷϞσϦϯά l ਓؒͷߦಈɺӡಈɺਭ຾ɺ৯ࣄͳͲΛه࿥͢Δ.PCJMF
     IFBMUI
     BQQMJDBUJPOΛ׆༻ͨ͠ߦಈͷϞσϦϯά l (BVTT෼෍ͷࠞ߹ʹΑΔ*OUFOTJUZͰਓؒͷߦಈ܏޲΍)BXLFT
     1SPDFTT Ͱपظతͳ૬ޓґଘͷؔ܎Λଊ͑Δ from: Kurashima, Takeshi, Tim Althoff, and Jure Leskovec. "Modeling interdependent and periodic real-world action sequences." Proceedings of the 2018 world wide web conference. 2018.
247. 247 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ϞσϦϯά ਓؒͷձ࿩ʹ͍ͭͯͷϞσϦϯά l ʮਓͷౖΕΔஉʯʹ͓͚Δܶதձ࿩ͷϞσϦϯά from: Guo, Fangjian, et al. "The bayesian echo chamber: Modeling social influence via linguistic accommodation." Artificial Intelligence and Statistics. PMLR, 2015.
248. 248 ࣌ܥྻλεΫͱۙ೥ͷݚڀ
     ϞσϦϯά ϑΣΠΫχϡʔεʹ͍ͭͯͷϞσϦϯά l ϑΣΠΫχϡʔεͷ֦ࢄͰ͸ɺ֦ࢄλΠϛϯάͱमਖ਼λΠϛϯάͷͭͷ ظ͕ؒଘࡏͦ͠ΕͧΕόʔετݱ৅͕ੜ͡Δ͜ͱΛ)BXLFT
     1SPDFTTͰ ϞσϦϯά from: Murayama, Taichi, et al. "Modeling the spread of fake news on Twitter." Plos one 16.4 (2021): e0250419.
249. 249 લ൒෦෼ͰओͳϕʔεͱͳͬͯΔॻ੶ l ΢ΥϧλʔΤϯμʔεʮ࣮ূͷͨΊͷܭྔ࣌ܥྻ෼ੳʯ l ๺઒ݯ࢛࿠ ʮ࣌ܥྻղੳೖ໳ʯ l ԭຊཽٛ ʮܦࡁɾϑΝΠφϯεσʔλͷܭྔ࣌ܥྻ෼ੳʯ ࢀߟॻ
250. 250 ͦͷଞ l খ੢ఃଇɼ๺઒ݯ࢛࿠ʮ৘ใྔن४ʯ l അ৔ޱ ొ
     தଜ ࿨ߊʮ৽͍͠৴߸ॲཧͷڭՊॻʯ l Ҫख ߶ ʮೖ໳ ػցֶशʹΑΔҟৗݕ஌ʕ3ʹΑΔ࣮ફΨΠυʯ l ۙߐ ਸ޺
     ໺ଜ ढ़Ұʮ఺աఔͷ࣌ܥྻղੳʯ l "JMFFO
     /JFMTFOʮ࣮ફ ࣌ܥྻղੳ ʕ౷ܭͱػցֶशʹΑΔ༧ ଌʯ l ਺ཧख๏ᶝ
     ࣌ܥྻղੳ
     
     ౦ژେֶʮ਺ཧɾσʔλαΠΤϯεڭ ҭϓϩάϥϜʯ ࢀߟॻ