DEIM2025 時間変化する因果関係の抽出に基づいた高速将来予測

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1. 時間変化する因果関係の抽出に 基づいた⾼速将来予測 千原 直⼰†,‡ 松原 靖⼦† 藤原 廉†,‡ 櫻井 保志† † ⼤阪⼤学産業科学研究所 ‡ ⼤阪⼤学⼤学院情報科学研究科 KDD Paper Source code 発表ID: 4D-03
2. 題⽬ q 研究背景 q 提案⼿法 q 評価実験 q まとめ DEIM2025 © 2025 Naoki Chihara et al. 2
3. 題⽬ q 研究背景 q 提案⼿法 q 評価実験 q まとめ DEIM2025 © 2025 Naoki Chihara et al. 3
4. 研究背景：時系列データ Ø 時系列データは様々な分野において⽇々⽣成されている DEIM2025 Motion analysis Epidemiology Web activity © 2025 Naoki Chihara et al. 4
5. 研究背景：時系列データ Ø 時系列データは様々な分野において⽇々⽣成されている Ø 現実の環境では，これらのデータは迅速かつ継続的に⽣成される DEIM2025 Epidemiology © 2025 Naoki Chihara et al. 5
6. 研究背景：依存関係 Ø 時系列データ中の変数間の関係性は重要な特徴量の１つ v e.g., 相関関係，因果関係，独⽴性など DEIM2025 © 2025 Naoki Chihara et al. 6
7. Ø 時系列データ中の変数間の関係性は重要な特徴量の１つ v e.g., 相関関係，因果関係，独⽴性など 研究背景：依存関係 DEIM2025 q 原因と結果の関係を意味する q ⻑年研究の対象となっている © 2025 Naoki Chihara et al. 7
8. 研究背景：依存関係 Ø 時系列データ中の変数間の関係性は重要な特徴量の１つ v e.g., 相関関係，因果関係，独⽴性など Ø 因果探索のための⼤半の既存⼿法は，因果関係がデータ内で変動し ないことを仮定している v 複雑な実世界への応⽤のためには，因果構造の時間変化を考慮 することが不可⽋である DEIM2025 © 2025 Naoki Chihara et al. 8 😢
9. 研究背景：依存関係 Ø 時系列データ中の変数間の関係性は重要な特徴量の１つ v e.g., 相関関係，因果関係，独⽴性など Ø 因果探索のための⼤半の既存⼿法は，因果関係がデータ内で変動し ないことを仮定している v 複雑な実世界への応⽤のためには，因果構造の時間変化を考慮 することが不可⽋である v この時間依存性はさまざまな影響による環境の移り変わりに従 うと考えられる DEIM2025 © 2025 Naoki Chihara et al. 9 😢
10. 研究背景：依存関係 Ø 時系列データ中の変数間の関係性は重要な特徴量の１つ v e.g., 相関関係，因果関係，独⽴性など Ø 因果探索のための⼤半の既存⼿法は，因果関係がデータ内で変動し ないことを仮定している v 複雑な実世界への応⽤のためには，因果構造の時間変化を考慮 することが不可⽋である v この変化は特徴的な時系列パターンによって引き起こされる DEIM2025 時間変化する因果関係の抽出および将来予測を同時に 達成するための最新⼿法である ModePlait を提案 © 2025 Naoki Chihara et al. 10
11. （𝑡! : 現在時刻） 研究背景：問題定義 Ø Given: 時系列データストリーム 𝑿 = {𝒙 1 , … , 𝒙 𝑡! , … } Ø Goals: 以下の重要な課題を全て達成する v 特徴的な時系列パターン（レジーム）の発⾒ v 時間変化する因果関係の抽出 v 𝑙" ステップ先の値の予測 DEIM2025 半無限⻑な時系列データ © 2025 Naoki Chihara et al. 11
12. 題⽬ q 研究背景 q 提案⼿法 q 評価実験 q まとめ DEIM2025 © 2025 Naoki Chihara et al. 12
13. 提案⼿法：概要 Ø 提案⼿法は構造⽅程式モデル [Pearl 2009] に基づいて設計した DEIM2025 = + © 2025 Naoki Chihara et al. 13 ⼀般的な構造⽅程式モデル (SEM) 観測変数 因果隣接⾏列 外⽣変数 e present our proposed model. The symbols we use described in Table 2. Here, before introducing the rie￿y describe the principles and concepts of M￿￿￿￿ n our proposed model based on the structural equa- ) [44], which is written as ^sem = Hsem^sem+Ksem, he observed variables, Hsem is the causal adjacency m is a set of mutually independent exogenous vari- -Gaussian distribution. Note that we assume that n, we present our proposed model. The symbols we use are described in Table 2. Here, before introducing the e brie￿y describe the principles and concepts of M￿￿￿￿ sign our proposed model based on the structural equa- EM) [44], which is written as ^sem = Hsem^sem+Ksem, s the observed variables, Hsem is the causal adjacency Ksem is a set of mutually independent exogenous vari- non-Gaussian distribution. Note that we assume that present our proposed model. The symbols we use escribed in Table 2. Here, before introducing the ￿y describe the principles and concepts of M￿￿￿￿ our proposed model based on the structural equa- 44], which is written as ^sem = Hsem^sem+Ksem, observed variables, Hsem is the causal adjacency s a set of mutually independent exogenous vari- Gaussian distribution. Note that we assume that we present our proposed model. The symbols we use re described in Table 2. Here, before introducing the brie￿y describe the principles and concepts of M￿￿￿￿ ign our proposed model based on the structural equa- M) [44], which is written as ^sem = Hsem^sem+Ksem, the observed variables, Hsem is the causal adjacency em is a set of mutually independent exogenous vari- on-Gaussian distribution. Note that we assume that we present our proposed model. The symbols we use re described in Table 2. Here, before introducing the brie￿y describe the principles and concepts of M￿￿￿￿ gn our proposed model based on the structural equa- M) [44], which is written as ^sem = Hsem^sem+Ksem, the observed variables, Hsem is the causal adjacency em is a set of mutually independent exogenous vari- on-Gaussian distribution. Note that we assume that esent our proposed model. The symbols we use cribed in Table 2. Here, before introducing the y describe the principles and concepts of M￿￿￿￿ r proposed model based on the structural equa- ], which is written as ^sem = Hsem^sem+Ksem, bserved variables, Hsem is the causal adjacency a set of mutually independent exogenous vari- ussian distribution. Note that we assume that
14. 提案⼿法：概要 Ø 提案⼿法は構造⽅程式モデル [Pearl 2009] に基づいて設計した DEIM2025 = + 各変数に固有な成分 © 2025 Naoki Chihara et al. 14 関係があれば 関係が無ければ 観測変数 因果隣接⾏列 外⽣変数 e present our proposed model. The symbols we use described in Table 2. Here, before introducing the rie￿y describe the principles and concepts of M￿￿￿￿ n our proposed model based on the structural equa- ) [44], which is written as ^sem = Hsem^sem+Ksem, he observed variables, Hsem is the causal adjacency m is a set of mutually independent exogenous vari- -Gaussian distribution. Note that we assume that n, we present our proposed model. The symbols we use are described in Table 2. Here, before introducing the e brie￿y describe the principles and concepts of M￿￿￿￿ sign our proposed model based on the structural equa- EM) [44], which is written as ^sem = Hsem^sem+Ksem, s the observed variables, Hsem is the causal adjacency Ksem is a set of mutually independent exogenous vari- non-Gaussian distribution. Note that we assume that present our proposed model. The symbols we use escribed in Table 2. Here, before introducing the ￿y describe the principles and concepts of M￿￿￿￿ our proposed model based on the structural equa- 44], which is written as ^sem = Hsem^sem+Ksem, observed variables, Hsem is the causal adjacency s a set of mutually independent exogenous vari- Gaussian distribution. Note that we assume that we present our proposed model. The symbols we use re described in Table 2. Here, before introducing the brie￿y describe the principles and concepts of M￿￿￿￿ ign our proposed model based on the structural equa- M) [44], which is written as ^sem = Hsem^sem+Ksem, the observed variables, Hsem is the causal adjacency em is a set of mutually independent exogenous vari- on-Gaussian distribution. Note that we assume that we present our proposed model. The symbols we use re described in Table 2. Here, before introducing the brie￿y describe the principles and concepts of M￿￿￿￿ gn our proposed model based on the structural equa- M) [44], which is written as ^sem = Hsem^sem+Ksem, the observed variables, Hsem is the causal adjacency em is a set of mutually independent exogenous vari- on-Gaussian distribution. Note that we assume that esent our proposed model. The symbols we use cribed in Table 2. Here, before introducing the y describe the principles and concepts of M￿￿￿￿ r proposed model based on the structural equa- ], which is written as ^sem = Hsem^sem+Ksem, bserved variables, Hsem is the causal adjacency a set of mutually independent exogenous vari- ussian distribution. Note that we assume that
15. 提案⼿法：概要 Ø 提案⼿法は構造⽅程式モデル [Pearl 2009] に基づいて設計した DEIM2025 = + © 2025 Naoki Chihara et al. 15 の原因となる変数 e.g., の場合 観測変数 因果隣接⾏列 外⽣変数 e present our proposed model. The symbols we use described in Table 2. Here, before introducing the rie￿y describe the principles and concepts of M￿￿￿￿ n our proposed model based on the structural equa- ) [44], which is written as ^sem = Hsem^sem+Ksem, he observed variables, Hsem is the causal adjacency m is a set of mutually independent exogenous vari- -Gaussian distribution. Note that we assume that n, we present our proposed model. The symbols we use are described in Table 2. Here, before introducing the e brie￿y describe the principles and concepts of M￿￿￿￿ sign our proposed model based on the structural equa- EM) [44], which is written as ^sem = Hsem^sem+Ksem, s the observed variables, Hsem is the causal adjacency Ksem is a set of mutually independent exogenous vari- non-Gaussian distribution. Note that we assume that present our proposed model. The symbols we use escribed in Table 2. Here, before introducing the ￿y describe the principles and concepts of M￿￿￿￿ our proposed model based on the structural equa- 44], which is written as ^sem = Hsem^sem+Ksem, observed variables, Hsem is the causal adjacency s a set of mutually independent exogenous vari- Gaussian distribution. Note that we assume that we present our proposed model. The symbols we use re described in Table 2. Here, before introducing the brie￿y describe the principles and concepts of M￿￿￿￿ ign our proposed model based on the structural equa- M) [44], which is written as ^sem = Hsem^sem+Ksem, the observed variables, Hsem is the causal adjacency em is a set of mutually independent exogenous vari- on-Gaussian distribution. Note that we assume that we present our proposed model. The symbols we use re described in Table 2. Here, before introducing the brie￿y describe the principles and concepts of M￿￿￿￿ gn our proposed model based on the structural equa- M) [44], which is written as ^sem = Hsem^sem+Ksem, the observed variables, Hsem is the causal adjacency em is a set of mutually independent exogenous vari- on-Gaussian distribution. Note that we assume that esent our proposed model. The symbols we use cribed in Table 2. Here, before introducing the y describe the principles and concepts of M￿￿￿￿ r proposed model based on the structural equa- ], which is written as ^sem = Hsem^sem+Ksem, bserved variables, Hsem is the causal adjacency a set of mutually independent exogenous vari- ussian distribution. Note that we assume that + + + + =
16. 提案⼿法：概要 Ø 提案⼿法は構造⽅程式モデル [Pearl 2009] に基づいて設計した DEIM2025 = + 関係があれば 関係が無ければ © 2025 Naoki Chihara et al. 16 各変数に固有な成分 観測変数 因果隣接⾏列 外⽣変数 e present our proposed model. The symbols we use described in Table 2. Here, before introducing the rie￿y describe the principles and concepts of M￿￿￿￿ n our proposed model based on the structural equa- ) [44], which is written as ^sem = Hsem^sem+Ksem, he observed variables, Hsem is the causal adjacency m is a set of mutually independent exogenous vari- -Gaussian distribution. Note that we assume that n, we present our proposed model. The symbols we use are described in Table 2. Here, before introducing the e brie￿y describe the principles and concepts of M￿￿￿￿ sign our proposed model based on the structural equa- EM) [44], which is written as ^sem = Hsem^sem+Ksem, s the observed variables, Hsem is the causal adjacency Ksem is a set of mutually independent exogenous vari- non-Gaussian distribution. Note that we assume that present our proposed model. The symbols we use escribed in Table 2. Here, before introducing the ￿y describe the principles and concepts of M￿￿￿￿ our proposed model based on the structural equa- 44], which is written as ^sem = Hsem^sem+Ksem, observed variables, Hsem is the causal adjacency s a set of mutually independent exogenous vari- Gaussian distribution. Note that we assume that we present our proposed model. The symbols we use re described in Table 2. Here, before introducing the brie￿y describe the principles and concepts of M￿￿￿￿ ign our proposed model based on the structural equa- M) [44], which is written as ^sem = Hsem^sem+Ksem, the observed variables, Hsem is the causal adjacency em is a set of mutually independent exogenous vari- on-Gaussian distribution. Note that we assume that we present our proposed model. The symbols we use re described in Table 2. Here, before introducing the brie￿y describe the principles and concepts of M￿￿￿￿ gn our proposed model based on the structural equa- M) [44], which is written as ^sem = Hsem^sem+Ksem, the observed variables, Hsem is the causal adjacency em is a set of mutually independent exogenous vari- on-Gaussian distribution. Note that we assume that esent our proposed model. The symbols we use cribed in Table 2. Here, before introducing the y describe the principles and concepts of M￿￿￿￿ r proposed model based on the structural equa- ], which is written as ^sem = Hsem^sem+Ksem, bserved variables, Hsem is the causal adjacency a set of mutually independent exogenous vari- ussian distribution. Note that we assume that
17. 提案⼿法：概要 Ø 達成すべき課題は以下のとおり v 固有信号を数式的に記述する v 因果関係を考慮して時系列データを表現する v 時間変化する因果関係を考慮した時系列データストリームの要約 DEIM2025 © 2025 Naoki Chihara et al. 17
18. 提案⼿法：概要 Ø 達成すべき課題は以下のとおり v 固有信号を数式的に記述する v 因果関係を考慮して時系列データを表現する v 時間変化する因果関係を考慮した時系列データストリームの要約 DEIM2025 © 2025 Naoki Chihara et al. 18 1. 固有信号の潜在的な時間ダイナミクス 2. 特徴的な時系列パターン 3. レジームの動的な遷移
19. Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ 固有信号の潜在的な時間ダイナミクス Ø 独⽴性により単変量時系列 𝒆 # を表現する必要がある v ⼀般的に単次元データでは情報が不⼗分 DEIM2025 © 2025 Naoki Chihara et al. 19 😢
20. 固有信号の潜在的な時間ダイナミクス Ø 独⽴性により単変量時系列 𝒆 # を表現する必要がある v ⼀般的に単次元データでは情報が不⼗分 v 時間遅れ埋め込みにより状態空間の次元を拡張する DEIM2025 © 2025 Naoki Chihara et al. 20 Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ 😢 ͷ໰୊఺Λิ͏ͨΊʹɼঢ়ଶۭؒͷ֦ுख๏Λ׆༻͢Δɽಛ ʹɼຊݚڀͰ͸ඇઢܗͳμΠφϛΫεͷநग़ʹ༗ޮͳ࣌ؒ஗ ΕຒΊࠐΈΛ࠾༻͢Δɽ۩ମతʹ͸ɼ͜Ε͸Ұൠతͳ؍ଌྔ g(e(i) (t)) := (e(i) (t), e(i) (t − 1), ..., e(i) (t − h + 1)) ∈ Rh ʹج ͍͓ͮͯΓɼඇઢܗγεςϜͷΞτϥΫλΛزԿֶతʹ࠶ߏ੒ ͢ΔͨΊͷཱ֬͞Εͨख๏Ͱ͋Δɽͨͩ͠ɼh ͸ຒΊࠐΈ࣍ݩ Ͱ͋Δɽ্هͷ g(·) Λ༻͍ͯϋϯέϧߦྻΛܗ੒͢Δɽ H(i) = ⎡ ⎢ ⎢ ⎣ | | | g(e(i) (h)) g(e(i) (h + 1)) · · · g(e(i) (t)) | | | ⎤ ⎥ ⎥ ⎦ (1) ࣜ (1) ͷͱ͓Γɼ֤ঢ়ଶϕΫτϧ͸աڈ৘ใΛ෇༩֦ͯ͠ு͞ Ε͍ͯΔɽ͞ΒʹɼTakens ͷຒΊࠐΈఆཧ [48] ʹΑΕ͹ɼಛ ఆͷ৚݅Լʹ͓͍ͯɼ࣌ؒ஗ΕຒΊࠐΈʹΑͬͯੜ੒͞ΕΔϕ
21. 固有信号の潜在的な時間ダイナミクス Ø 独⽴性により単変量時系列 𝒆 # を表現する必要がある v ⼀般的に単次元データでは情報が不⼗分 v 時間遅れ埋め込みにより状態空間の次元を拡張する DEIM2025 © 2025 Naoki Chihara et al. 21 Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ 😢 ͷ໰୊఺Λิ͏ͨΊʹɼঢ়ଶۭؒͷ֦ுख๏Λ׆༻͢Δɽಛ ʹɼຊݚڀͰ͸ඇઢܗͳμΠφϛΫεͷநग़ʹ༗ޮͳ࣌ؒ஗ ΕຒΊࠐΈΛ࠾༻͢Δɽ۩ମతʹ͸ɼ͜Ε͸Ұൠతͳ؍ଌྔ g(e(i) (t)) := (e(i) (t), e(i) (t − 1), ..., e(i) (t − h + 1)) ∈ Rh ʹج ͍͓ͮͯΓɼඇઢܗγεςϜͷΞτϥΫλΛزԿֶతʹ࠶ߏ੒ ͢ΔͨΊͷཱ֬͞Εͨख๏Ͱ͋Δɽͨͩ͠ɼh ͸ຒΊࠐΈ࣍ݩ Ͱ͋Δɽ্هͷ g(·) Λ༻͍ͯϋϯέϧߦྻΛܗ੒͢Δɽ H(i) = ⎡ ⎢ ⎢ ⎣ | | | g(e(i) (h)) g(e(i) (h + 1)) · · · g(e(i) (t)) | | | ⎤ ⎥ ⎥ ⎦ (1) ࣜ (1) ͷͱ͓Γɼ֤ঢ়ଶϕΫτϧ͸աڈ৘ใΛ෇༩֦ͯ͠ு͞ Ε͍ͯΔɽ͞ΒʹɼTakens ͷຒΊࠐΈఆཧ [48] ʹΑΕ͹ɼಛ ఆͷ৚݅Լʹ͓͍ͯɼ࣌ؒ஗ΕຒΊࠐΈʹΑͬͯੜ੒͞ΕΔϕ (P1) ͸ɼ֎ੜม਺ΛجఈϕΫτϧʢϞʔυʣͷॏͶ߹ΘͤͰ දݱ͢Δɽͦͯ͠ɼ্هͷཁૉΛ૊ΈΘͤͯ (P2) Λଊ͑Δɽ 3. 1. 1 ݻ༗৴߸தͷજࡏతͳ࣌ؒμΠφϛΫε (P1) ॳΊʹɼi ൪໨ͷݻ༗৴߸ e(i) = {e(i) (1), ..., e(i) (t)} ͔Βજ ࡏతͳ࣌ؒμΠφϛΫεΛଊ͑Δํ๏ʹ͍ͭͯઆ໌͢Δɽ໰ ୊఺ͱͯ͠͸ɼγεςϜ಺ͷજࡏతͳμΠφϛΫε͕Ұൠʹ ଟ࣍ݩͰ͋ΔͨΊɼγεςϜΛे෼ʹදݱ͢ΔͨΊʹ͸ɼ୯࣍ ݩͳσʔλͰ͸͠͹͠͹ෆे෼Ͱ͋Δ͜ͱ͕ڍ͛ΒΕΔɽ͜ ͷ໰୊఺Λิ͏ͨΊʹɼঢ়ଶۭؒͷ֦ுख๏Λ׆༻͢Δɽಛ ʹɼຊݚڀͰ͸ඇઢܗͳμΠφϛΫεͷநग़ʹ༗ޮͳ࣌ؒ஗ ΕຒΊࠐΈΛ࠾༻͢Δɽ۩ମతʹ͸ɼ͜Ε͸Ұൠతͳ؍ଌྔ g(e(i) (t)) := (e(i) (t), e(i) (t − 1), ..., e(i) (t − h + 1)) ∈ Rh ʹج ͍͓ͮͯΓɼඇઢܗγεςϜͷΞτϥΫλΛزԿֶతʹ࠶ߏ੒ ͢ΔͨΊͷཱ֬͞Εͨख๏Ͱ͋Δɽͨͩ͠ɼh ͸ຒΊࠐΈ࣍ݩ ͜ Ͱ s ݻ Φ e ఆ Λ ͱ μ
22. (P1) ͸ɼ֎ੜม਺ΛجఈϕΫτϧʢϞʔυʣͷॏͶ߹ΘͤͰ දݱ͢Δɽͦͯ͠ɼ্هͷཁૉΛ૊ΈΘͤͯ (P2) Λଊ͑Δɽ 3. 1. 1 ݻ༗৴߸தͷજࡏతͳ࣌ؒμΠφϛΫε (P1) ॳΊʹɼi ൪໨ͷݻ༗৴߸ e(i) = {e(i) (1), ..., e(i) (t)} ͔Βજ ࡏతͳ࣌ؒμΠφϛΫεΛଊ͑Δํ๏ʹ͍ͭͯઆ໌͢Δɽ໰ ୊఺ͱͯ͠͸ɼγεςϜ಺ͷજࡏతͳμΠφϛΫε͕Ұൠʹ ଟ࣍ݩͰ͋ΔͨΊɼγεςϜΛे෼ʹදݱ͢ΔͨΊʹ͸ɼ୯࣍ ݩͳσʔλͰ͸͠͹͠͹ෆे෼Ͱ͋Δ͜ͱ͕ڍ͛ΒΕΔɽ͜ ͷ໰୊఺Λิ͏ͨΊʹɼঢ়ଶۭؒͷ֦ுख๏Λ׆༻͢Δɽಛ ʹɼຊݚڀͰ͸ඇઢܗͳμΠφϛΫεͷநग़ʹ༗ޮͳ࣌ؒ஗ ΕຒΊࠐΈΛ࠾༻͢Δɽ۩ମతʹ͸ɼ͜Ε͸Ұൠతͳ؍ଌྔ g(e(i) (t)) := (e(i) (t), e(i) (t − 1), ..., e(i) (t − h + 1)) ∈ Rh ʹج ͍͓ͮͯΓɼඇઢܗγεςϜͷΞτϥΫλΛزԿֶతʹ࠶ߏ੒ ͢ΔͨΊͷཱ֬͞Εͨख๏Ͱ͋Δɽͨͩ͠ɼh ͸ຒΊࠐΈ࣍ݩ ͜ Ͱ s ݻ Φ e ఆ Λ ͱ μ 固有信号の潜在的な時間ダイナミクス Ø 独⽴性により単変量時系列 𝒆 # を表現する必要がある v ⼀般的に単次元データでは情報が不⼗分 v 時間遅れ埋め込みにより状態空間の次元を拡張する DEIM2025 © 2025 Naoki Chihara et al. 22 Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ 過去データ 😢 ͷ໰୊఺Λิ͏ͨΊʹɼঢ়ଶۭؒͷ֦ுख๏Λ׆༻͢Δɽಛ ʹɼຊݚڀͰ͸ඇઢܗͳμΠφϛΫεͷநग़ʹ༗ޮͳ࣌ؒ஗ ΕຒΊࠐΈΛ࠾༻͢Δɽ۩ମతʹ͸ɼ͜Ε͸Ұൠతͳ؍ଌྔ g(e(i) (t)) := (e(i) (t), e(i) (t − 1), ..., e(i) (t − h + 1)) ∈ Rh ʹج ͍͓ͮͯΓɼඇઢܗγεςϜͷΞτϥΫλΛزԿֶతʹ࠶ߏ੒ ͢ΔͨΊͷཱ֬͞Εͨख๏Ͱ͋Δɽͨͩ͠ɼh ͸ຒΊࠐΈ࣍ݩ Ͱ͋Δɽ্هͷ g(·) Λ༻͍ͯϋϯέϧߦྻΛܗ੒͢Δɽ H(i) = ⎡ ⎢ ⎢ ⎣ | | | g(e(i) (h)) g(e(i) (h + 1)) · · · g(e(i) (t)) | | | ⎤ ⎥ ⎥ ⎦ (1) ࣜ (1) ͷͱ͓Γɼ֤ঢ়ଶϕΫτϧ͸աڈ৘ใΛ෇༩֦ͯ͠ு͞ Ε͍ͯΔɽ͞ΒʹɼTakens ͷຒΊࠐΈఆཧ [48] ʹΑΕ͹ɼಛ ఆͷ৚݅Լʹ͓͍ͯɼ࣌ؒ஗ΕຒΊࠐΈʹΑͬͯੜ੒͞ΕΔϕ
23. (P1) ͸ɼ֎ੜม਺ΛجఈϕΫτϧʢϞʔυʣͷॏͶ߹ΘͤͰ දݱ͢Δɽͦͯ͠ɼ্هͷཁૉΛ૊ΈΘͤͯ (P2) Λଊ͑Δɽ 3. 1. 1 ݻ༗৴߸தͷજࡏతͳ࣌ؒμΠφϛΫε (P1) ॳΊʹɼi ൪໨ͷݻ༗৴߸ e(i) = {e(i) (1), ..., e(i) (t)} ͔Βજ ࡏతͳ࣌ؒμΠφϛΫεΛଊ͑Δํ๏ʹ͍ͭͯઆ໌͢Δɽ໰ ୊఺ͱͯ͠͸ɼγεςϜ಺ͷજࡏతͳμΠφϛΫε͕Ұൠʹ ଟ࣍ݩͰ͋ΔͨΊɼγεςϜΛे෼ʹදݱ͢ΔͨΊʹ͸ɼ୯࣍ ݩͳσʔλͰ͸͠͹͠͹ෆे෼Ͱ͋Δ͜ͱ͕ڍ͛ΒΕΔɽ͜ ͷ໰୊఺Λิ͏ͨΊʹɼঢ়ଶۭؒͷ֦ுख๏Λ׆༻͢Δɽಛ ʹɼຊݚڀͰ͸ඇઢܗͳμΠφϛΫεͷநग़ʹ༗ޮͳ࣌ؒ஗ ΕຒΊࠐΈΛ࠾༻͢Δɽ۩ମతʹ͸ɼ͜Ε͸Ұൠతͳ؍ଌྔ g(e(i) (t)) := (e(i) (t), e(i) (t − 1), ..., e(i) (t − h + 1)) ∈ Rh ʹج ͍͓ͮͯΓɼඇઢܗγεςϜͷΞτϥΫλΛزԿֶతʹ࠶ߏ੒ ͢ΔͨΊͷཱ֬͞Εͨख๏Ͱ͋Δɽͨͩ͠ɼh ͸ຒΊࠐΈ࣍ݩ ͜ Ͱ s ݻ Φ e ఆ Λ ͱ μ 固有信号の潜在的な時間ダイナミクス Ø 独⽴性により単変量時系列 𝒆 # を表現する必要がある v ⼀般的に単次元データでは情報が不⼗分 v 時間遅れ埋め込みにより状態空間の次元を拡張する DEIM2025 © 2025 Naoki Chihara et al. 23 Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ 過去データ Takens’ theorem に基づいている 😢 ͷ໰୊఺Λิ͏ͨΊʹɼঢ়ଶۭؒͷ֦ுख๏Λ׆༻͢Δɽಛ ʹɼຊݚڀͰ͸ඇઢܗͳμΠφϛΫεͷநग़ʹ༗ޮͳ࣌ؒ஗ ΕຒΊࠐΈΛ࠾༻͢Δɽ۩ମతʹ͸ɼ͜Ε͸Ұൠతͳ؍ଌྔ g(e(i) (t)) := (e(i) (t), e(i) (t − 1), ..., e(i) (t − h + 1)) ∈ Rh ʹج ͍͓ͮͯΓɼඇઢܗγεςϜͷΞτϥΫλΛزԿֶతʹ࠶ߏ੒ ͢ΔͨΊͷཱ֬͞Εͨख๏Ͱ͋Δɽͨͩ͠ɼh ͸ຒΊࠐΈ࣍ݩ Ͱ͋Δɽ্هͷ g(·) Λ༻͍ͯϋϯέϧߦྻΛܗ੒͢Δɽ H(i) = ⎡ ⎢ ⎢ ⎣ | | | g(e(i) (h)) g(e(i) (h + 1)) · · · g(e(i) (t)) | | | ⎤ ⎥ ⎥ ⎦ (1) ࣜ (1) ͷͱ͓Γɼ֤ঢ়ଶϕΫτϧ͸աڈ৘ใΛ෇༩֦ͯ͠ு͞ Ε͍ͯΔɽ͞ΒʹɼTakens ͷຒΊࠐΈఆཧ [48] ʹΑΕ͹ɼಛ ఆͷ৚݅Լʹ͓͍ͯɼ࣌ؒ஗ΕຒΊࠐΈʹΑͬͯੜ੒͞ΕΔϕ
24. 固有信号の潜在的な時間ダイナミクス Ø 𝑖 番⽬の固有信号 𝒆 # は以下のように表現される DEIM2025 © 2025 Naoki Chihara et al. 24 Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ (b) ϨδʔϜ (i.e., θ = {W , D(1) , ..., D(d) }) 2: ModePlait ͷϞσϧ֓ཁਤ: (a) ಈతγεςϜʹै͏ i ൪໨ͷม਺ʹݻ༗ͳ୯มྔ৴߸ e(i) ͔ΒજࡏతͳμΠφϛΫεΛ ग़͢Δ. (b) ଟ࣍ݩ࣌ܥྻσʔλ͸ࠞ߹ߦྻ W −1 ͱ d ݸͷࣗݾۦಈҼࢠ {D(1) , ..., D(d) } ʹΑͬͯදݱ͞ΕΔɽ 1 ModePlait Ϟσϧ Ϋτϧ͸ɼݩͷঢ়ଶͱඍ෼ಉ૬ͳμΠφϛΫεΛ࣋ͭ͜ͱ͕อ ： 観測空間への射影 (ℂ#! → ℝ) ： 𝑘$ 次元の潜在空間 ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a ermining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of e mode, and ⇤(8) is a set of :8 eigenvalues. the decay r the absolut 3.1.2 Dyn how to mo inherent si tackle the n namical pa ity betwee combine th erating the a set of 3 la extend Mo tivariate ti ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a termining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of e mode, and ⇤(8) is a set of :8 eigenvalues. the decay the absolu 3.1.2 Dyn how to mo inherent si tackle the namical pa ity betwee combine th erating the a set of 3 l extend Mo tivariate ti ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a ermining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of the decay r the absolut 3.1.2 Dyn how to mo inherent si tackle the n namical pa ity betwee combine th erating the a set of 3 la extend Mo ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a ermining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of addition, n the decay r the absolu 3.1.2 Dyn how to mo inherent si tackle the n namical pa ity betwee combine th erating the a set of 3 l extend Mo 次元拡張 潜在状態 固有信号 時間遅れ 埋め込み 固有値⾏列 モード 固有ダイナミクス集合 ͕ڍ͛ΒΕΔɽ͜ ๏Λ׆༻͢Δɽಛ ग़ʹ༗ޮͳ࣌ؒ஗ ͸Ұൠతͳ؍ଌྔ h + 1)) ∈ Rh ʹج زԿֶతʹ࠶ߏ੒ ɼh ͸ຒΊࠐΈ࣍ݩ Λܗ੒͢Δɽ | ⎤ ⎥ ݻ༗஋ Λ(i) ∈ Cki×ki ͕࣌ؒμΠφϛΫε Φ(i) ∈ Ch×ki ͓Αͼ g−1(·) ͸࣌ࠁ t ʹ͓͚Δ e(i) (t) Λੜ੒͢ΔͨΊͷࣹӨΛࣔ͢ɽ·ͱΊ ఆٛ 2 (ݻ༗μΠφϛΫεू߹ɿD(i) ). Ϟʔ Λ(i) ʹΑΔू߹ D(i) = {Φ(i) , Λ(i) } Λݻ༗ ͱݺͿɽ͜Ε͸ɼi ൪໨ͷ୯มྔݻ༗৴߸ e( μΠφϛΫεΛදݱ͢Δɽ 3. 1. 2 ୯ҰϨδʔϜ಺ͷಛ௃తͳ࣌ܥྻ ଓ͍ͯɼ࣌ܥྻσʔλετϦʔϜதͷ࣌ؒ
25. 固有信号の潜在的な時間ダイナミクス Ø 𝑖 番⽬の固有信号 𝒆 # は以下のように表現される DEIM2025 © 2025 Naoki Chihara et al. 25 Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ (b) ϨδʔϜ (i.e., θ = {W , D(1) , ..., D(d) }) 2: ModePlait ͷϞσϧ֓ཁਤ: (a) ಈతγεςϜʹै͏ i ൪໨ͷม਺ʹݻ༗ͳ୯มྔ৴߸ e(i) ͔ΒજࡏతͳμΠφϛΫεΛ ग़͢Δ. (b) ଟ࣍ݩ࣌ܥྻσʔλ͸ࠞ߹ߦྻ W −1 ͱ d ݸͷࣗݾۦಈҼࢠ {D(1) , ..., D(d) } ʹΑͬͯදݱ͞ΕΔɽ 1 ModePlait Ϟσϧ Ϋτϧ͸ɼݩͷঢ়ଶͱඍ෼ಉ૬ͳμΠφϛΫεΛ࣋ͭ͜ͱ͕อ ： 観測空間への射影 (ℂ#! → ℝ) ： 𝑘$ 次元の潜在空間 ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a ermining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of e mode, and ⇤(8) is a set of :8 eigenvalues. the decay r the absolut 3.1.2 Dyn how to mo inherent si tackle the n namical pa ity betwee combine th erating the a set of 3 la extend Mo tivariate ti ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a termining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of e mode, and ⇤(8) is a set of :8 eigenvalues. the decay the absolu 3.1.2 Dyn how to mo inherent si tackle the namical pa ity betwee combine th erating the a set of 3 l extend Mo tivariate ti ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a ermining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of the decay r the absolut 3.1.2 Dyn how to mo inherent si tackle the n namical pa ity betwee combine th erating the a set of 3 la extend Mo ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a ermining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of addition, n the decay r the absolu 3.1.2 Dyn how to mo inherent si tackle the n namical pa ity betwee combine th erating the a set of 3 l extend Mo 次元拡張 潜在状態 固有信号 固有値⾏列 時間遅れ 埋め込み モード 固有ダイナミクス集合 ͕ڍ͛ΒΕΔɽ͜ ๏Λ׆༻͢Δɽಛ ग़ʹ༗ޮͳ࣌ؒ஗ ͸Ұൠతͳ؍ଌྔ h + 1)) ∈ Rh ʹج زԿֶతʹ࠶ߏ੒ ɼh ͸ຒΊࠐΈ࣍ݩ Λܗ੒͢Δɽ | ⎤ ⎥ ݻ༗஋ Λ(i) ∈ Cki×ki ͕࣌ؒμΠφϛΫε Φ(i) ∈ Ch×ki ͓Αͼ g−1(·) ͸࣌ࠁ t ʹ͓͚Δ e(i) (t) Λੜ੒͢ΔͨΊͷࣹӨΛࣔ͢ɽ·ͱΊ ఆٛ 2 (ݻ༗μΠφϛΫεू߹ɿD(i) ). Ϟʔ Λ(i) ʹΑΔू߹ D(i) = {Φ(i) , Λ(i) } Λݻ༗ ͱݺͿɽ͜Ε͸ɼi ൪໨ͷ୯มྔݻ༗৴߸ e( μΠφϛΫεΛදݱ͢Δɽ 3. 1. 2 ୯ҰϨδʔϜ಺ͷಛ௃తͳ࣌ܥྻ ଓ͍ͯɼ࣌ܥྻσʔλετϦʔϜதͷ࣌ؒ
26. 固有信号の潜在的な時間ダイナミクス Ø 𝑖 番⽬の固有信号 𝒆 # は以下のように表現される DEIM2025 © 2025 Naoki Chihara et al. 26 Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ (b) ϨδʔϜ (i.e., θ = {W , D(1) , ..., D(d) }) 2: ModePlait ͷϞσϧ֓ཁਤ: (a) ಈతγεςϜʹै͏ i ൪໨ͷม਺ʹݻ༗ͳ୯มྔ৴߸ e(i) ͔ΒજࡏతͳμΠφϛΫεΛ ग़͢Δ. (b) ଟ࣍ݩ࣌ܥྻσʔλ͸ࠞ߹ߦྻ W −1 ͱ d ݸͷࣗݾۦಈҼࢠ {D(1) , ..., D(d) } ʹΑͬͯදݱ͞ΕΔɽ 1 ModePlait Ϟσϧ Ϋτϧ͸ɼݩͷঢ়ଶͱඍ෼ಉ૬ͳμΠφϛΫεΛ࣋ͭ͜ͱ͕อ ： 観測空間への射影 (ℂ#! → ℝ) ： 𝑘$ 次元の潜在空間 ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a ermining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of e mode, and ⇤(8) is a set of :8 eigenvalues. the decay r the absolut 3.1.2 Dyn how to mo inherent si tackle the n namical pa ity betwee combine th erating the a set of 3 la extend Mo tivariate ti ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a termining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of e mode, and ⇤(8) is a set of :8 eigenvalues. the decay the absolu 3.1.2 Dyn how to mo inherent si tackle the namical pa ity betwee combine th erating the a set of 3 l extend Mo tivariate ti ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a ermining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of the decay r the absolut 3.1.2 Dyn how to mo inherent si tackle the n namical pa ity betwee combine th erating the a set of 3 la extend Mo ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a ermining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of addition, n the decay r the absolu 3.1.2 Dyn how to mo inherent si tackle the n namical pa ity betwee combine th erating the a set of 3 l extend Mo 次元拡張 潜在状態 固有信号 固有値⾏列 時間遅れ 埋め込み モード 固有ダイナミクス集合 ͕ڍ͛ΒΕΔɽ͜ ๏Λ׆༻͢Δɽಛ ग़ʹ༗ޮͳ࣌ؒ஗ ͸Ұൠతͳ؍ଌྔ h + 1)) ∈ Rh ʹج زԿֶతʹ࠶ߏ੒ ɼh ͸ຒΊࠐΈ࣍ݩ Λܗ੒͢Δɽ | ⎤ ⎥ ݻ༗஋ Λ(i) ∈ Cki×ki ͕࣌ؒμΠφϛΫε Φ(i) ∈ Ch×ki ͓Αͼ g−1(·) ͸࣌ࠁ t ʹ͓͚Δ e(i) (t) Λੜ੒͢ΔͨΊͷࣹӨΛࣔ͢ɽ·ͱΊ ఆٛ 2 (ݻ༗μΠφϛΫεू߹ɿD(i) ). Ϟʔ Λ(i) ʹΑΔू߹ D(i) = {Φ(i) , Λ(i) } Λݻ༗ ͱݺͿɽ͜Ε͸ɼi ൪໨ͷ୯มྔݻ༗৴߸ e( μΠφϛΫεΛදݱ͢Δɽ 3. 1. 2 ୯ҰϨδʔϜ಺ͷಛ௃తͳ࣌ܥྻ ଓ͍ͯɼ࣌ܥྻσʔλετϦʔϜதͷ࣌ؒ
27. 固有信号の潜在的な時間ダイナミクス Ø 𝑖 番⽬の固有信号 𝒆 # は以下のように表現される DEIM2025 © 2025 Naoki Chihara et al. 27 固有ダイナミクス集合 ： 観測空間への射影 (ℂ#! → ℝ) ： 𝑘$ 次元の潜在空間 ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a ermining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of e mode, and ⇤(8) is a set of :8 eigenvalues. the decay r the absolut 3.1.2 Dyn how to mo inherent si tackle the n namical pa ity betwee combine th erating the a set of 3 la extend Mo tivariate ti ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a termining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of e mode, and ⇤(8) is a set of :8 eigenvalues. the decay the absolu 3.1.2 Dyn how to mo inherent si tackle the namical pa ity betwee combine th erating the a set of 3 l extend Mo tivariate ti ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a ermining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of the decay r the absolut 3.1.2 Dyn how to mo inherent si tackle the n namical pa ity betwee combine th erating the a set of 3 la extend Mo ent signal e(8). This activity is a latent vector s(8) (C) 2 h is :8-dimensional complex-valued latent vector at time where :8 is the number of modes. This vector plays a ermining the 8-th inherent signal 4(8) (C) at time point C. ntly, the dynamical system for the8-th univariate inherent can be described with the following equations: 1. Let s(8) (C) be the latent vector at time point C. The equations govern the 8-th univariate inherent signal e(8), s(8) (C + 1) = ⇤(8)s(8) (C) 4(8) (C) = g 1( (8)s(8) (C)) (2) (·) is the inverse of the observables g(·), each column of addition, n the decay r the absolu 3.1.2 Dyn how to mo inherent si tackle the n namical pa ity betwee combine th erating the a set of 3 l extend Mo 次元拡張 Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ (b) ϨδʔϜ (i.e., θ = {W , D(1) , ..., D(d) }) 2: ModePlait ͷϞσϧ֓ཁਤ: (a) ಈతγεςϜʹै͏ i ൪໨ͷม਺ʹݻ༗ͳ୯มྔ৴߸ e(i) ͔ΒજࡏతͳμΠφϛΫεΛ ग़͢Δ. (b) ଟ࣍ݩ࣌ܥྻσʔλ͸ࠞ߹ߦྻ W −1 ͱ d ݸͷࣗݾۦಈҼࢠ {D(1) , ..., D(d) } ʹΑͬͯදݱ͞ΕΔɽ 1 ModePlait Ϟσϧ Ϋτϧ͸ɼݩͷঢ়ଶͱඍ෼ಉ૬ͳμΠφϛΫεΛ࣋ͭ͜ͱ͕อ 時間遅れ 埋め込み 潜在状態 固有信号 モード 固有値⾏列 ͕ڍ͛ΒΕΔɽ͜ ๏Λ׆༻͢Δɽಛ ग़ʹ༗ޮͳ࣌ؒ஗ ͸Ұൠతͳ؍ଌྔ h + 1)) ∈ Rh ʹج زԿֶతʹ࠶ߏ੒ ɼh ͸ຒΊࠐΈ࣍ݩ Λܗ੒͢Δɽ | ⎤ ⎥ ݻ༗஋ Λ(i) ∈ Cki×ki ͕࣌ؒμΠφϛΫε Φ(i) ∈ Ch×ki ͓Αͼ g−1(·) ͸࣌ࠁ t ʹ͓͚Δ e(i) (t) Λੜ੒͢ΔͨΊͷࣹӨΛࣔ͢ɽ·ͱΊ ఆٛ 2 (ݻ༗μΠφϛΫεू߹ɿD(i) ). Ϟʔ Λ(i) ʹΑΔू߹ D(i) = {Φ(i) , Λ(i) } Λݻ༗ ͱݺͿɽ͜Ε͸ɼi ൪໨ͷ୯มྔݻ༗৴߸ e( μΠφϛΫεΛදݱ͢Δɽ 3. 1. 2 ୯ҰϨδʔϜ಺ͷಛ௃తͳ࣌ܥྻ ଓ͍ͯɼ࣌ܥྻσʔλετϦʔϜதͷ࣌ؒ
28. Time-evolving Causality over Data Streams ￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the variate inherent signal 4(8) (C) at time point C, e(C) be the 3- onal inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), be the 3-dimensional estimated vector at time point C. The g equations govern our single regime, s(8) (C + 1) = ⇤(8)s(8) (C) (1  8  3) 4(8) (C) = g 1( (8)s(8) (C)) (1  8  3) v(C) = ] 1e(C) (3) l 2, we require an additional parameter, demixing matrix ], ! Time " ! Regi " 特徴的な時系列パターン Ø 単⼀の時系列パターンは以下のとおり DEIM2025 © 2025 Naoki Chihara et al. 28 Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ (b) ϨδʔϜ (i.e., θ = {W , D(1) , ..., D(d) }) ਤ 2: ModePlait ͷϞσϧ֓ཁਤ: (a) ಈతγεςϜʹै͏ i ൪໨ͷม਺ʹݻ༗ͳ୯มྔ৴߸ e(i) ͔ΒજࡏతͳμΠφϛΫεΛ நग़͢Δ. (b) ଟ࣍ݩ࣌ܥྻσʔλ͸ࠞ߹ߦྻ W −1 ͱ d ݸͷࣗݾۦಈҼࢠ {D(1) , ..., D(d) } ʹΑͬͯදݱ͞ΕΔɽ . 1 ModePlait Ϟσϧ ຊઅͰ͸ఏҊϞσϧͷৄࡉΛड़΂ΔɽॳΊʹඞཁͳ֓೦ͷఆ ٛʹ͍ͭͯઆ໌͢Δɽ ఆٛ 1 (ݻ༗৴߸ɿE). ඇΨ΢ε෼෍ʹै͏ d ݸͷ૬ޓʹಠཱ ͨ͠ཁૉΛ࣋ͭ৴߸ E = {e(i) }d i=1 Λݻ༗৴߸ͱݺͿɽͨͩ͠ɼ (i) = {e(i) (1), ..., e(i) (t)} ͸ i ൪໨ͷ୯มྔ࣌ܥྻͰ͋Δɽ͜ Ϋτϧ͸ɼݩͷঢ়ଶͱඍ෼ಉ૬ͳμΠφϛΫεΛ࣋ͭ͜ͱ͕อ ূ͞ΕΔɽ௚ײతʹઆ໌Λ͢Δͱɼ͜ͷ࠶ߏ੒͸ݩͷྗֶܥͷ ಛੑΛཧ࿦తʹอͭɼͭ·Γɼϋϯέϧߦྻ H(i) ͷղੳΛ௨ ͯ͡ɼݩͷσʔλ͔Β͸௚઀நग़Ͱ͖ͳ͍ॏཁͳಛ௃Λ໌Β͔ ʹ͢Δ͜ͱΛՄೳʹ͢Δɽଟ͘ͷ৔߹ɼඍ෼ಉ૬ࣸ૾Λ٘ਜ਼ʹ ͢Δ͜ͱͳ͘ຒΊࠐΈ࣍ݩΛબ୒Ͱ͖Δɽ ͜͜Ͱɼi ൪໨ͷݻ༗৴߸ e(i) ͷಈతγεςϜͷͨΊʹɼki Time-evolving Causality over Data Streams ￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the variate inherent signal 4(8) (C) at time point C, e(C) be the 3- nal inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), be the 3-dimensional estimated vector at time point C. The g equations govern our single regime, s(8) (C + 1) = ⇤(8)s(8) (C) (1  8  3) 4(8) (C) = g 1( (8)s(8) (C)) (1  8  3) v(C) = ] 1e(C) (3) ! Time ! Reg " Time-evolving Causality over Data Streams ￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the variate inherent signal 4(8) (C) at time point C, e(C) be the 3- onal inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), ) be the 3-dimensional estimated vector at time point C. The g equations govern our single regime, s(8) (C + 1) = ⇤(8)s(8) (C) (1  8  3) 4(8) (C) = g 1( (8)s(8) (C)) (1  8  3) v(C) = ] 1e(C) (3) ! Time ! Reg " 𝒅 個の固有ダイナミクス集合 KDD ’25, August 3–7, 2025, Toronto, ON, Canada M￿￿￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the univariate inherent signal 4(8) (C) at time point C, e(C) be the 3- ensional inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), v(C) be the 3-dimensional estimated vector at time point C. The owing equations govern the single regime, KDD ’25, August 3–7, 2025, Toronto, ON, Canada (8) (C) be the :8-dimensional latent vector for the erent signal 4(8) (C) at time point C, e(C) be the 3- nt signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), dimensional estimated vector at time point C. The KDD ’25, August 3–7, 2025, Toronto, ON, Canada ￿￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the nivariate inherent signal 4(8) (C) at time point C, e(C) be the 3- sional inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), (C) be the 3-dimensional estimated vector at time point C. The 混合⾏列 推定値 レジーム s(i) (t + 1) = Λ(i) s(i) (t) (1 = i = d) e(i) (t) = g−1(Φ(i) s(i) (t)) (1 < = i < = d) v(t) = W −1e(t) Ϟσϧ 2 ͷͨΊʹɼ৽ͨͳύϥϝʔλͰ͋Δ෼཭ߦ ೖ͢Δɽ͜Ε͸ɼd ݸͷݻ༗৴߸ؒͷؔ܎ੑΛදݱ ܎ͷಛఆͷͨΊʹॏཁͳ໾ׂΛՌͨ͢ɽW ͔Β B ͨΊͷΞϧΰϦζϜʹ͍ͭͯ͸ 4. 2. 3 અʹͯઆ໌͢ ΔͱɼҎԼΛಘΔɽ ఆٛ 3 (ϨδʔϜɿθ). θ = {W , D(1) , ..., D(d) } Λ දݱ͢Δύϥϝʔλू߹ͱ͢Δɽ͜͜ͰɼW ͸Ҽ B Λੜ੒͢ΔͨΊͷج൫ͱͳΔཁૉͰ͋Δɽ
29. Time-evolving Causality over Data Streams ￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the variate inherent signal 4(8) (C) at time point C, e(C) be the 3- onal inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), be the 3-dimensional estimated vector at time point C. The g equations govern our single regime, s(8) (C + 1) = ⇤(8)s(8) (C) (1  8  3) 4(8) (C) = g 1( (8)s(8) (C)) (1  8  3) v(C) = ] 1e(C) (3) l 2, we require an additional parameter, demixing matrix ], ! Time " ! Regi " 特徴的な時系列パターン Ø 単⼀の時系列パターンは以下のとおり DEIM2025 © 2025 Naoki Chihara et al. 29 Time-evolving Causality over Data Streams ￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the variate inherent signal 4(8) (C) at time point C, e(C) be the 3- nal inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), be the 3-dimensional estimated vector at time point C. The g equations govern our single regime, s(8) (C + 1) = ⇤(8)s(8) (C) (1  8  3) 4(8) (C) = g 1( (8)s(8) (C)) (1  8  3) v(C) = ] 1e(C) (3) ! Time ! Reg " Time-evolving Causality over Data Streams ￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the variate inherent signal 4(8) (C) at time point C, e(C) be the 3- onal inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), ) be the 3-dimensional estimated vector at time point C. The g equations govern our single regime, s(8) (C + 1) = ⇤(8)s(8) (C) (1  8  3) 4(8) (C) = g 1( (8)s(8) (C)) (1  8  3) v(C) = ] 1e(C) (3) ! Time ! Reg " 𝒅 個の固有ダイナミクス集合 混合⾏列 推定値 Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ (b) ϨδʔϜ (i.e., θ = {W , D(1) , ..., D(d) }) ਤ 2: ModePlait ͷϞσϧ֓ཁਤ: (a) ಈతγεςϜʹै͏ i ൪໨ͷม਺ʹݻ༗ͳ୯มྔ৴߸ e(i) ͔ΒજࡏతͳμΠφϛΫεΛ நग़͢Δ. (b) ଟ࣍ݩ࣌ܥྻσʔλ͸ࠞ߹ߦྻ W −1 ͱ d ݸͷࣗݾۦಈҼࢠ {D(1) , ..., D(d) } ʹΑͬͯදݱ͞ΕΔɽ . 1 ModePlait Ϟσϧ ຊઅͰ͸ఏҊϞσϧͷৄࡉΛड़΂ΔɽॳΊʹඞཁͳ֓೦ͷఆ ٛʹ͍ͭͯઆ໌͢Δɽ ఆٛ 1 (ݻ༗৴߸ɿE). ඇΨ΢ε෼෍ʹै͏ d ݸͷ૬ޓʹಠཱ ͨ͠ཁૉΛ࣋ͭ৴߸ E = {e(i) }d i=1 Λݻ༗৴߸ͱݺͿɽͨͩ͠ɼ (i) = {e(i) (1), ..., e(i) (t)} ͸ i ൪໨ͷ୯มྔ࣌ܥྻͰ͋Δɽ͜ Ϋτϧ͸ɼݩͷঢ়ଶͱඍ෼ಉ૬ͳμΠφϛΫεΛ࣋ͭ͜ͱ͕อ ূ͞ΕΔɽ௚ײతʹઆ໌Λ͢Δͱɼ͜ͷ࠶ߏ੒͸ݩͷྗֶܥͷ ಛੑΛཧ࿦తʹอͭɼͭ·Γɼϋϯέϧߦྻ H(i) ͷղੳΛ௨ ͯ͡ɼݩͷσʔλ͔Β͸௚઀நग़Ͱ͖ͳ͍ॏཁͳಛ௃Λ໌Β͔ ʹ͢Δ͜ͱΛՄೳʹ͢Δɽଟ͘ͷ৔߹ɼඍ෼ಉ૬ࣸ૾Λ٘ਜ਼ʹ ͢Δ͜ͱͳ͘ຒΊࠐΈ࣍ݩΛબ୒Ͱ͖Δɽ ͜͜Ͱɼi ൪໨ͷݻ༗৴߸ e(i) ͷಈతγεςϜͷͨΊʹɼki KDD ’25, August 3–7, 2025, Toronto, ON, Canada M￿￿￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the univariate inherent signal 4(8) (C) at time point C, e(C) be the 3- ensional inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), v(C) be the 3-dimensional estimated vector at time point C. The owing equations govern the single regime, KDD ’25, August 3–7, 2025, Toronto, ON, Canada (8) (C) be the :8-dimensional latent vector for the erent signal 4(8) (C) at time point C, e(C) be the 3- nt signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), dimensional estimated vector at time point C. The KDD ’25, August 3–7, 2025, Toronto, ON, Canada ￿￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the nivariate inherent signal 4(8) (C) at time point C, e(C) be the 3- sional inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), (C) be the 3-dimensional estimated vector at time point C. The レジーム s(i) (t + 1) = Λ(i) s(i) (t) (1 = i = d) e(i) (t) = g−1(Φ(i) s(i) (t)) (1 < = i < = d) v(t) = W −1e(t) Ϟσϧ 2 ͷͨΊʹɼ৽ͨͳύϥϝʔλͰ͋Δ෼཭ߦ ೖ͢Δɽ͜Ε͸ɼd ݸͷݻ༗৴߸ؒͷؔ܎ੑΛදݱ ܎ͷಛఆͷͨΊʹॏཁͳ໾ׂΛՌͨ͢ɽW ͔Β B ͨΊͷΞϧΰϦζϜʹ͍ͭͯ͸ 4. 2. 3 અʹͯઆ໌͢ ΔͱɼҎԼΛಘΔɽ ఆٛ 3 (ϨδʔϜɿθ). θ = {W , D(1) , ..., D(d) } Λ දݱ͢Δύϥϝʔλू߹ͱ͢Δɽ͜͜ͰɼW ͸Ҽ B Λੜ੒͢ΔͨΊͷج൫ͱͳΔཁૉͰ͋Δɽ
30. Time-evolving Causality over Data Streams ￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the variate inherent signal 4(8) (C) at time point C, e(C) be the 3- onal inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), be the 3-dimensional estimated vector at time point C. The g equations govern our single regime, s(8) (C + 1) = ⇤(8)s(8) (C) (1  8  3) 4(8) (C) = g 1( (8)s(8) (C)) (1  8  3) v(C) = ] 1e(C) (3) l 2, we require an additional parameter, demixing matrix ], ! Time " ! Regi " 特徴的な時系列パターン Ø 単⼀の時系列パターンは以下のとおり DEIM2025 © 2025 Naoki Chihara et al. 30 レジーム Time-evolving Causality over Data Streams ￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the variate inherent signal 4(8) (C) at time point C, e(C) be the 3- nal inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), be the 3-dimensional estimated vector at time point C. The g equations govern our single regime, s(8) (C + 1) = ⇤(8)s(8) (C) (1  8  3) 4(8) (C) = g 1( (8)s(8) (C)) (1  8  3) v(C) = ] 1e(C) (3) ! Time ! Reg " Time-evolving Causality over Data Streams ￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the variate inherent signal 4(8) (C) at time point C, e(C) be the 3- onal inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), ) be the 3-dimensional estimated vector at time point C. The g equations govern our single regime, s(8) (C + 1) = ⇤(8)s(8) (C) (1  8  3) 4(8) (C) = g 1( (8)s(8) (C)) (1  8  3) v(C) = ] 1e(C) (3) ! Time ! Reg " 推定値 Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ (b) ϨδʔϜ (i.e., θ = {W , D(1) , ..., D(d) }) ਤ 2: ModePlait ͷϞσϧ֓ཁਤ: (a) ಈతγεςϜʹै͏ i ൪໨ͷม਺ʹݻ༗ͳ୯มྔ৴߸ e(i) ͔ΒજࡏతͳμΠφϛΫεΛ நग़͢Δ. (b) ଟ࣍ݩ࣌ܥྻσʔλ͸ࠞ߹ߦྻ W −1 ͱ d ݸͷࣗݾۦಈҼࢠ {D(1) , ..., D(d) } ʹΑͬͯදݱ͞ΕΔɽ . 1 ModePlait Ϟσϧ ຊઅͰ͸ఏҊϞσϧͷৄࡉΛड़΂ΔɽॳΊʹඞཁͳ֓೦ͷఆ ٛʹ͍ͭͯઆ໌͢Δɽ ఆٛ 1 (ݻ༗৴߸ɿE). ඇΨ΢ε෼෍ʹै͏ d ݸͷ૬ޓʹಠཱ ͨ͠ཁૉΛ࣋ͭ৴߸ E = {e(i) }d i=1 Λݻ༗৴߸ͱݺͿɽͨͩ͠ɼ (i) = {e(i) (1), ..., e(i) (t)} ͸ i ൪໨ͷ୯มྔ࣌ܥྻͰ͋Δɽ͜ Ϋτϧ͸ɼݩͷঢ়ଶͱඍ෼ಉ૬ͳμΠφϛΫεΛ࣋ͭ͜ͱ͕อ ূ͞ΕΔɽ௚ײతʹઆ໌Λ͢Δͱɼ͜ͷ࠶ߏ੒͸ݩͷྗֶܥͷ ಛੑΛཧ࿦తʹอͭɼͭ·Γɼϋϯέϧߦྻ H(i) ͷղੳΛ௨ ͯ͡ɼݩͷσʔλ͔Β͸௚઀நग़Ͱ͖ͳ͍ॏཁͳಛ௃Λ໌Β͔ ʹ͢Δ͜ͱΛՄೳʹ͢Δɽଟ͘ͷ৔߹ɼඍ෼ಉ૬ࣸ૾Λ٘ਜ਼ʹ ͢Δ͜ͱͳ͘ຒΊࠐΈ࣍ݩΛબ୒Ͱ͖Δɽ ͜͜Ͱɼi ൪໨ͷݻ༗৴߸ e(i) ͷಈతγεςϜͷͨΊʹɼki KDD ’25, August 3–7, 2025, Toronto, ON, Canada M￿￿￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the univariate inherent signal 4(8) (C) at time point C, e(C) be the 3- ensional inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), v(C) be the 3-dimensional estimated vector at time point C. The owing equations govern the single regime, KDD ’25, August 3–7, 2025, Toronto, ON, Canada (8) (C) be the :8-dimensional latent vector for the erent signal 4(8) (C) at time point C, e(C) be the 3- nt signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), dimensional estimated vector at time point C. The KDD ’25, August 3–7, 2025, Toronto, ON, Canada ￿￿￿ 2. Let s(8) (C) be the :8-dimensional latent vector for the nivariate inherent signal 4(8) (C) at time point C, e(C) be the 3- sional inherent signals at time point C (i.e., e(C) = {4(8) (C)}3 8=1 ), (C) be the 3-dimensional estimated vector at time point C. The 𝒅 個の固有ダイナミクス集合 混合⾏列 s(i) (t + 1) = Λ(i) s(i) (t) (1 = i = d) e(i) (t) = g−1(Φ(i) s(i) (t)) (1 < = i < = d) v(t) = W −1e(t) Ϟσϧ 2 ͷͨΊʹɼ৽ͨͳύϥϝʔλͰ͋Δ෼཭ߦ ೖ͢Δɽ͜Ε͸ɼd ݸͷݻ༗৴߸ؒͷؔ܎ੑΛදݱ ܎ͷಛఆͷͨΊʹॏཁͳ໾ׂΛՌͨ͢ɽW ͔Β B ͨΊͷΞϧΰϦζϜʹ͍ͭͯ͸ 4. 2. 3 અʹͯઆ໌͢ ΔͱɼҎԼΛಘΔɽ ఆٛ 3 (ϨδʔϜɿθ). θ = {W , D(1) , ..., D(d) } Λ දݱ͢Δύϥϝʔλू߹ͱ͢Δɽ͜͜ͰɼW ͸Ҽ B Λੜ੒͢ΔͨΊͷج൫ͱͳΔཁૉͰ͋Δɽ
31. レジームの動的な遷移 DEIM2025 © 2025 Naoki Chihara et al. 31 𝑑 Data stream 𝑿 Time 𝑡 Ø 時系列データストリーム中の時系列パターンの動的変化 v レジームセット 𝚯 = 𝜃%, 𝜃&, … , 𝜃' 𝜃# = 𝑾, 𝒟(%) , … , 𝒟(*)
32. レジームの動的な遷移 Ø 時系列データストリーム中の時系列パターンの動的変化 v レジームセット 𝚯 = 𝜃%, 𝜃&, … , 𝜃' 𝜃# = 𝑾, 𝒟(%) , … , 𝒟(*) DEIM2025 © 2025 Naoki Chihara et al. 32 𝑑 Data stream 𝑿 Time 𝑡
33. アルゴリズム 提案アルゴリズムは以下の４要素で構成されている Ø ModeEstimator Ø RegimeCreation Ø ModeGenerator Ø RegimeUpdater DEIM2025 © 2025 Naoki Chihara et al. 33 “Time-evolving” causality ! "! "" # Future values (unknown) Time " Mode Estimator Regime Creation Current regime: $! ∈ & ! insert new regime Forecast !! -steps-ahead future value i.e., !(#! + %" ) "! + (# V! !! Regime set Θ " "" # Causal adjacency matrix Current window: "" ! #!" ##! ##" "(") "($) * , , ⋮ , #"! ##! ##" estimate モデルパラメータ集合： モデル候補： モデル更新⽤パラメータ：
34. アルゴリズム 提案アルゴリズムは以下の４要素で構成されている Ø ModeEstimator v モデルパラメータ集合 ℱ と モデル候補 𝒞 を逐次的に 推定する Ø RegimeCreation Ø ModeGenerator Ø RegimeUpdater DEIM2025 © 2025 Naoki Chihara et al. 34 “Time-evolving” causality ! "! "" # Future values (unknown) Time " Mode Estimator Regime Creation Current regime: $! ∈ & ! insert new regime Forecast !! -steps-ahead future value i.e., !(#! + %" ) "! + (# V! !! Regime set Θ " "" # Causal adjacency matrix Current window: "" ! #!" ##! ##" "(") "($) * , , ⋮ , #"! ##! ##" estimate
35. アルゴリズム 提案アルゴリズムは以下の４要素で構成されている Ø ModeEstimator Ø RegimeCreation v 未知のパターンが現れた場合 カレントウィンドウ 𝑿% から レジーム 𝜽𝒄 を推定し レジームセット 𝚯 に追加する Ø ModeGenerator Ø RegimeUpdater DEIM2025 © 2025 Naoki Chihara et al. 35 “Time-evolving” causality ! "! "" # Future values (unknown) Time " Mode Estimator Regime Creation Current regime: $! ∈ & ! insert new regime Forecast !! -steps-ahead future value i.e., !(#! + %" ) "! + (# V! !! Regime set Θ " "" # Causal adjacency matrix Current window: "" ! #!" ##! ##" "(") "($) * , , ⋮ , #"! ##! ##" estimate
36. アルゴリズム 提案アルゴリズムは以下の４要素で構成されている Ø ModeEstimator Ø RegimeCreation Ø ModeGenerator v モデル候補 𝒞 から因果隣接 ⾏列 𝑩 を⽣成し、𝑙' ステップ 先の値を予測する Ø RegimeUpdater DEIM2025 © 2025 Naoki Chihara et al. 36 “Time-evolving” causality ! "! "" # Future values (unknown) Time " Mode Estimator Regime Creation Current regime: $! ∈ & ! insert new regime Forecast !! -steps-ahead future value i.e., !(#! + %" ) "! + (# V! !! Regime set Θ " "" # Causal adjacency matrix Current window: "" ! #!" ##! ##" "(") "($) * , , ⋮ , #"! ##! ##" estimate
37. アルゴリズム 提案アルゴリズムは以下の４要素で構成されている Ø ModeEstimator Ø RegimeCreation Ø ModeGenerator Ø RegimeUpdater v 最新の値 𝒙 𝑡% を⽤いて レジーム 𝜽𝒄 = 𝑾, 𝒟()) , … , 𝒟(+) を更新する DEIM2025 © 2025 Naoki Chihara et al. 37 Ø 分離⾏列 𝑾 の更新 v 適応フィルタに基づいた更新 v 計算時間 / メモリの観点から効率的 Ø 固有ダイナミクス集合 𝒟(") の更新 Details in paper ͼϞσϧީิ C Λਪఆ͢Δɽ • ModeGenerator: Ϟσϧީิ C ΑΓɼls εςοϓઌͷ ஋ v(tc + ls) Λ༧ଌ͠ɼҼՌྡ઀ߦྻ B Λநग़͢Δɽ • RegimeUpdater: ݱࡏͷߋ৽༻ύϥϝʔλ ωc ͱ࠷৽ͷ ஋ x(tc) Λ༻͍ͯɼݱࡏͷϨδʔϜ θc Λߋ৽͢Δɽ 4. 2. 2 ModeEstimator ݱࡏͷ࣌ࠁ tc ͷ؍ଌ஋ x(tc) ͕༩͑ΒΕͨͱ͖ɼ࠷ॳʹ Ϟσϧύϥϝʔλू߹ F ͓ΑͼΧϨϯτ΢Οϯυ΢ Xc Λ ࠷΋දݱ͢ΔϞσϧީิ C Λஞ࣍తʹߋ৽͢Δɽ͜͜Ͱɼ f(Xc; Sc 0 , θc) ͸ΧϨϯτ΢Οϯυ΢ Xc ͱਪఆ΢Οϯυ΢ V c ͷޡࠩΛ࠷খԽ͢Δ͜ͱʹΑͬͯɼ࠷దͳϞσϧύϥϝʔ λू߹Λࢉग़͢Δʢi.e., f(Xc; Sc 0 , θc) = tc t=tm+h−1 ||x(t) − v(t)||ʣɽࣜ (3) ʹجͮ͘ͱɼSc 0 Λܭࢉ͢Δ࠷΋؆ศͳํ๏͸ {Φ† (i) g(e(i) (tm + h − 1))}d i=1 Λ༻͍Δ͜ͱͰ͋Δɽ͔͠͠ɼա ౓ͳϊΠζؚ͕·Εͨॳظ஋͸ద੾ͳ༧ଌ͕ୡ੒Ͱ͖ͳ͍ɽ͜ ΕΛରॲ͢ΔͨΊʹɼLM (Levenberg-Marquardt) ΞϧΰϦ ζϜ [50] Λ࢖༻ͯ͠ Sc 0 Λ࠷దԽ͠ɼ؍ଌʹ͓͚ΔϊΠζͷӨ ڹΛআڈ͢Δɽ·ͱΊΔͱɼModeEstimator ͸࣍ͷखॱʹ ৽ɼ͓Αͼ (ii) ݻ༗μΠφϛΫεू߹ D ͷߋ৽ɼͷ 2 ͭͷख ॱ͔Βߏ੒͞ΕΔɽखॱ (i) Ͱ͸ɼదԠϑΟϧλʹج͍ͮͨΞ ϧΰϦζϜΛ࢖༻͢Δ [51,52]ɽ͜Ε͸ɼܭࢉͱϝϞϦͷ྆ํ ͷ؍఺Ͱඇৗʹޮ཰తͰ͋Δɽߋ৽खॱ͸ҎԼͷͱ͓ΓͰ͋Δɽ I. ݱࡏ࣌ࠁ tc ʹ͓͍ͯɼߋ৽લͷ W ͷ i ൪໨ͷߦϕΫ τϧ wi ʹ x(tc) ΛࣹӨ͢Δ͜ͱͰɼi ൪໨ͷݻ༗৴߸ g(e(i) (tc)) Λࢉग़͢Δɽ II. g(e(i) (tc)) Λ༻͍ͯɼ෮ݩޡ͓ࠩΑͼΤωϧΪʔ ϵ(i) Λ ܭࢉ͢Δɽ III. ޡ͓ࠩΑͼΤωϧΪʔ ϵ(i) Λ༻͍ͯ wi Λߋ৽͢Δɽ ҰํͰɼखॱ (ii) Ͱ͸ҎԼͷ࠶ؼࣜΛ༻͍Δɽ Anew (i) = Aprev (i) + (g(e(i) (tc)) − Aprev (i) g(e(i) (tc − 1)))γ(i) γ(i) = g(e(i) (tc − 1))⊤P prev (i) µ + g(e(i) (tc − 1))⊤P prev (i) g(e(i) (tc − 1)) P new (i) = 1 µ (P prev (i) − P prev (i) g(e(i) (tc − 1))γ(i) ) (5) ͜͜ͰɼΦ(i) ͓Αͼ Λ(i) ͸ͦΕͧΕ A(i) ͷݻ༗ϕΫτϧɼݻ
38. 理論的分析 DEIM2025 Details in paper © 2025 Naoki Chihara et al. 38 ModePlait は以下の優れた特性を有する Ø 定理 2. ModePlait における因果探索は，ModeGenerator での因果 隣接⾏列 𝑩 の抽出と同値である． v 提案⼿法は真の因果関係を推定可能である Ø 定理 3. （定理１に基づくと）各プロセスにおける ModePlait の計 算時間量は少なくとも 𝑂(𝑁 ∑# 𝑘# + 𝑑ℎ&) であり，⾼々 𝑂(𝑅𝑁 ∑# 𝑘# + 𝑁 𝑑& + ℎ& + 𝑘&) である． v 計算時間が時系列データストリームの⻑さに依存しない v 半無限⻑の⻑さを持つデータに対して実⽤的である
39. 題⽬ q 研究背景 q 提案⼿法 q 評価実験 q まとめ DEIM2025 © 2025 Naoki Chihara et al. 39
40. 検証項⽬ 以下の全ての疑問に対して回答する Ø Q1. 有効性 時間変化する因果関係の抽出に対する提案⼿法の有効性 Ø Q2. 正確性 各課題に対する提案⼿法の精度の検証 Ø Q3. 計算時間 時系列データストリームの予測に対する計算時間の検証 DEIM2025 © 2025 Naoki Chihara et al. 40
41. 実験設定 DEIM2025 © 2025 Naoki Chihara et al. 41 Ø １２種類の⽐較⼿法 v CASPER v DARING v NoCurl v NO-MLP v NOTEARS v LiNGAM v GES v TimesNet v PatchTST v DeepAR v OrbitMap v ARIMA 因果探索のための ７つのモデル 時系列予測のための ５つのモデル Ø ９つのデータセット q ⼈⼯データ v 因果探索の定量的な評価に使⽤ v ５種類の異なるパターン q 実データ v 幅広い分野のデータを使⽤ • COVID-19の感染者数データ • Web検索数データ • モーションキャプチャによる センサデータ
42. Q1. 有効性 DEIM2025 Ø 疫病データストリームに対する出⼒例 v 主要５カ国のCOVID-19の感染者数によって構成されたデータセット ใՊֶݚڀՊ৘ใγεςϜ޻ֶઐ߈ ത࢜લظ՝ఔमֶ࢜Ґ࿦จ ൃදձࢿྉ 2025 ೥ 2 ݄ 7 ೔ ʔλετϦʔϜʹ͓͚Δকདྷ༧ଌͷͨΊͷ࣌ؒมԽ͢ΔҼՌؔ܎ͷநग़ ઍݪ ௚ݾʢ஌ೳσʔλՊֶߨ࠲ʣ ͸ɼ෯޿͍෼໺ʹ͓͍ͯ೔ʑେྔʹੜ੒͞ ɼ࣮ੈքʹ͓͍ͯ͜ΕΒͷσʔλ͸࣌ؒൃ ͔ͭܧଓతʹੜ੒͞Εଓ͚ΔͨΊɼ͜ΕΒ ϦʔϛϯάํࣜͰॲཧ͢Δॏཁੑ͕ۙ೥ߴ ͨɼ࣌ܥྻσʔλͷ؍ଌ஋ؒͷؔ܎ੑ͸ॏཁ ଟछଟ༷ͳԼྲྀλεΫͷਫ਼౓޲্ͷͨΊʹ ɽຊݚڀͰ͸ͦͷதͰ΋ಛʹɼ࣌ܥྻσʔ ʹؚ·ΕΔ࣌ؒมԽ͢ΔҼՌؔ܎ʹண໨͢ ؔ܎ੑ͸ɼ৽نͷݪҼͷൃݟ͓Αͼকདྷ༧ ํࣜͰਖ਼֬ʹ༧ଌ͢ΔͨΊʹඇৗʹॏཁͰ ײછ঱ͷྲྀߦͰ͸ɼ͋ΔࠃͰ৽͍͠มҟג ͦͷࠃ಺Ͱͷײછऀ਺͕ٸ૿͢Δ͚ͩͰͳ ΔҠಈͳͲͱ͍ͬͨಛఆͷ׆ಈ͕ଞࠃͷײ Ҿ͖ى͜͢Մೳੑ͕ଘࡏ͢Δɽ·ͨɼ͜ͷ Δࠃ͸࣌ؒͷܦաͱͱ΋ʹมԽ͢Δɽ CN IT ZA JP US (a-i) January 8, 2021 CN IT ZA JP US (a-ii) May 19, 2022 (a) ҟͳΔ࣌ࠁʹ͓͚ΔҼՌؔ܎ (b-i) September 27, 2021 (b-ii) June 5, 2022 (b) 10 ೔ઌͷ༧ଌͷεφοϓγϣ οτ ਤ 1: ӸපσʔλετϦʔϜʹର͢Δ MODEPLAIT ͷग़ྗྫ 南アフリカ保健省が最新の 変異株 501.V2 の発⾒を発表 上海における過去最⼤の ロックダウン 各⽮印の基部は原因を 先端は結果に対応 時系列パターンの遷移を 考慮した⾼精度な将来予 測を達成 © 2025 Naoki Chihara et al. 42
43. Q1. 有効性 DEIM2025 Ø 疫病データストリームに対する出⼒例 v 主要５カ国のCOVID-19の感染者数によって構成されたデータセット ใՊֶݚڀՊ৘ใγεςϜ޻ֶઐ߈ ത࢜લظ՝ఔमֶ࢜Ґ࿦จ ൃදձࢿྉ 2025 ೥ 2 ݄ 7 ೔ ʔλετϦʔϜʹ͓͚Δকདྷ༧ଌͷͨΊͷ࣌ؒมԽ͢ΔҼՌؔ܎ͷநग़ ઍݪ ௚ݾʢ஌ೳσʔλՊֶߨ࠲ʣ ͸ɼ෯޿͍෼໺ʹ͓͍ͯ೔ʑେྔʹੜ੒͞ ɼ࣮ੈքʹ͓͍ͯ͜ΕΒͷσʔλ͸࣌ؒൃ ͔ͭܧଓతʹੜ੒͞Εଓ͚ΔͨΊɼ͜ΕΒ ϦʔϛϯάํࣜͰॲཧ͢Δॏཁੑ͕ۙ೥ߴ ͨɼ࣌ܥྻσʔλͷ؍ଌ஋ؒͷؔ܎ੑ͸ॏཁ ଟछଟ༷ͳԼྲྀλεΫͷਫ਼౓޲্ͷͨΊʹ ɽຊݚڀͰ͸ͦͷதͰ΋ಛʹɼ࣌ܥྻσʔ ʹؚ·ΕΔ࣌ؒมԽ͢ΔҼՌؔ܎ʹண໨͢ ؔ܎ੑ͸ɼ৽نͷݪҼͷൃݟ͓Αͼকདྷ༧ ํࣜͰਖ਼֬ʹ༧ଌ͢ΔͨΊʹඇৗʹॏཁͰ ײછ঱ͷྲྀߦͰ͸ɼ͋ΔࠃͰ৽͍͠มҟג ͦͷࠃ಺Ͱͷײછऀ਺͕ٸ૿͢Δ͚ͩͰͳ ΔҠಈͳͲͱ͍ͬͨಛఆͷ׆ಈ͕ଞࠃͷײ Ҿ͖ى͜͢Մೳੑ͕ଘࡏ͢Δɽ·ͨɼ͜ͷ Δࠃ͸࣌ؒͷܦաͱͱ΋ʹมԽ͢Δɽ CN IT ZA JP US (a-i) January 8, 2021 CN IT ZA JP US (a-ii) May 19, 2022 (a) ҟͳΔ࣌ࠁʹ͓͚ΔҼՌؔ܎ (b-i) September 27, 2021 (b-ii) June 5, 2022 (b) 10 ೔ઌͷ༧ଌͷεφοϓγϣ οτ ਤ 1: ӸපσʔλετϦʔϜʹର͢Δ MODEPLAIT ͷग़ྗྫ 上海における過去最⼤の ロックダウン 時系列パターンの遷移を 考慮した⾼精度な将来予 測を達成 南アフリカ保健省が最新の 変異株 501.V2 の発⾒を発表 各⽮印の基部は原因を 先端は結果に対応 © 2025 Naoki Chihara et al. 43
44. Q1. 有効性 DEIM2025 Ø 疫病データストリームに対する出⼒例 v 主要５カ国のCOVID-19の感染者数によって構成されたデータセット ใՊֶݚڀՊ৘ใγεςϜ޻ֶઐ߈ ത࢜લظ՝ఔमֶ࢜Ґ࿦จ ൃදձࢿྉ 2025 ೥ 2 ݄ 7 ೔ ʔλετϦʔϜʹ͓͚Δকདྷ༧ଌͷͨΊͷ࣌ؒมԽ͢ΔҼՌؔ܎ͷநग़ ઍݪ ௚ݾʢ஌ೳσʔλՊֶߨ࠲ʣ ͸ɼ෯޿͍෼໺ʹ͓͍ͯ೔ʑେྔʹੜ੒͞ ɼ࣮ੈքʹ͓͍ͯ͜ΕΒͷσʔλ͸࣌ؒൃ ͔ͭܧଓతʹੜ੒͞Εଓ͚ΔͨΊɼ͜ΕΒ ϦʔϛϯάํࣜͰॲཧ͢Δॏཁੑ͕ۙ೥ߴ ͨɼ࣌ܥྻσʔλͷ؍ଌ஋ؒͷؔ܎ੑ͸ॏཁ ଟछଟ༷ͳԼྲྀλεΫͷਫ਼౓޲্ͷͨΊʹ ɽຊݚڀͰ͸ͦͷதͰ΋ಛʹɼ࣌ܥྻσʔ ʹؚ·ΕΔ࣌ؒมԽ͢ΔҼՌؔ܎ʹண໨͢ ؔ܎ੑ͸ɼ৽نͷݪҼͷൃݟ͓Αͼকདྷ༧ ํࣜͰਖ਼֬ʹ༧ଌ͢ΔͨΊʹඇৗʹॏཁͰ ײછ঱ͷྲྀߦͰ͸ɼ͋ΔࠃͰ৽͍͠มҟג ͦͷࠃ಺Ͱͷײછऀ਺͕ٸ૿͢Δ͚ͩͰͳ ΔҠಈͳͲͱ͍ͬͨಛఆͷ׆ಈ͕ଞࠃͷײ Ҿ͖ى͜͢Մೳੑ͕ଘࡏ͢Δɽ·ͨɼ͜ͷ Δࠃ͸࣌ؒͷܦաͱͱ΋ʹมԽ͢Δɽ CN IT ZA JP US (a-i) January 8, 2021 CN IT ZA JP US (a-ii) May 19, 2022 (a) ҟͳΔ࣌ࠁʹ͓͚ΔҼՌؔ܎ (b-i) September 27, 2021 (b-ii) June 5, 2022 (b) 10 ೔ઌͷ༧ଌͷεφοϓγϣ οτ ਤ 1: ӸපσʔλετϦʔϜʹର͢Δ MODEPLAIT ͷग़ྗྫ 上海における過去最⼤の ロックダウン 各⽮印の基部は原因を 先端は結果に対応 時系列パターンの遷移を 考慮した⾼精度な将来予 測を達成 南アフリカ保健省が最新の 変異株 501.V2 の発⾒を発表 © 2025 Naoki Chihara et al. 44
45. ใՊֶݚڀՊ৘ใγεςϜ޻ֶઐ߈ ത࢜લظ՝ఔमֶ࢜Ґ࿦จ ൃදձࢿྉ 2025 ೥ 2 ݄ 7 ೔ ʔλετϦʔϜʹ͓͚Δকདྷ༧ଌͷͨΊͷ࣌ؒมԽ͢ΔҼՌؔ܎ͷநग़ ઍݪ ௚ݾʢ஌ೳσʔλՊֶߨ࠲ʣ ͸ɼ෯޿͍෼໺ʹ͓͍ͯ೔ʑେྔʹੜ੒͞ ɼ࣮ੈքʹ͓͍ͯ͜ΕΒͷσʔλ͸࣌ؒൃ ͔ͭܧଓతʹੜ੒͞Εଓ͚ΔͨΊɼ͜ΕΒ ϦʔϛϯάํࣜͰॲཧ͢Δॏཁੑ͕ۙ೥ߴ ɼ࣌ܥྻσʔλͷ؍ଌ஋ؒͷؔ܎ੑ͸ॏཁ ଟछଟ༷ͳԼྲྀλεΫͷਫ਼౓޲্ͷͨΊʹ ɽຊݚڀͰ͸ͦͷதͰ΋ಛʹɼ࣌ܥྻσʔ ʹؚ·ΕΔ࣌ؒมԽ͢ΔҼՌؔ܎ʹண໨͢ ؔ܎ੑ͸ɼ৽نͷݪҼͷൃݟ͓Αͼকདྷ༧ ํࣜͰਖ਼֬ʹ༧ଌ͢ΔͨΊʹඇৗʹॏཁͰ ײછ঱ͷྲྀߦͰ͸ɼ͋ΔࠃͰ৽͍͠มҟג ͦͷࠃ಺Ͱͷײછऀ਺͕ٸ૿͢Δ͚ͩͰͳ ΔҠಈͳͲͱ͍ͬͨಛఆͷ׆ಈ͕ଞࠃͷײ Ҿ͖ى͜͢Մೳੑ͕ଘࡏ͢Δɽ·ͨɼ͜ͷ Δࠃ͸࣌ؒͷܦաͱͱ΋ʹมԽ͢Δɽ CN IT ZA JP US (a-i) January 8, 2021 CN IT ZA JP US (a-ii) May 19, 2022 (a) ҟͳΔ࣌ࠁʹ͓͚ΔҼՌؔ܎ (b-i) September 27, 2021 (b-ii) June 5, 2022 (b) 10 ೔ઌͷ༧ଌͷεφοϓγϣ οτ ਤ 1: ӸපσʔλετϦʔϜʹର͢Δ MODEPLAIT ͷग़ྗྫ Q1. 有効性 DEIM2025 Ø 疫病データストリームに対する出⼒例 v 主要５カ国のCOVID-19の感染者数によって構成されたデータセット 上海における過去最⼤の ロックダウン 各⽮印の基部は原因を 先端は結果に対応 時系列パターンの遷移を 考慮した⾼精度な将来予 測を達成 南アフリカ保健省が最新の 変異株 501.V2 の発⾒を発表 © 2025 Naoki Chihara et al. 45
46. Q2. 正確性：因果探索 Ø 提案⼿法の因果探索に関する正確性 v 評価指標：SHD, SID（低いほど良） DEIM2025 © 2025 Naoki Chihara et al. 46 KDD ’25, August 3–7, 2025, Toronto, Canada Chihara, et al. Table 3: Causal discovering results with multiple temporal sequences to encompass various types of real-world scenarios. Models M￿￿￿P￿￿￿￿ CASPER DARING NoCurl NO-MLP NOTEARS LiNGAM GES Metrics SHD SID SHD SID SHD SID SHD SID SHD SID SHD SID SHD SID SHD SID 1, 2, 1 3.82 4.94 5.58 7.25 5.75 8.58 6.31 9.90 6.36 8.74 5.03 9.95 7.13 8.23 7.49 11.7 1, 2, 3 4.48 6.51 5.97 8.44 5.81 9.17 6.13 9.51 6.44 8.77 5.69 9.56 6.79 7.33 7.03 10.1 1, 2, 2, 1 4.32 5.88 5.41 8.41 6.54 9.17 6.69 10.0 6.55 8.72 5.23 9.54 7.12 8.65 7.08 9.77 1, 2, 3, 4 4.21 5.76 6.22 8.33 6.12 9.58 6.10 9.61 6.62 8.87 5.73 10.1 7.10 8.50 7.29 11.3 1, 2, 3, 2, 1 4.50 6.11 6.02 8.28 5.45 7.77 6.20 9.83 6.56 8.83 5.57 9.11 7.46 8.05 7.74 12.1 Table 4: Multivariate forecasting results for both synthetic and real-world datasets. We used forecasting steps ;B 2 {5, 10, 15}. Models M￿￿￿P￿￿￿￿ TimesNet PatchTST DeepAR OrbitMap ARIMA Metrics RMSE MAE RMSE MAE RMSE MAE RMSE MAE RMSE MAE RMSE MAE #0 synthetic 5 0.722 0.528 0.805 0.578 0.768 0.581 1.043 0.821 0.826 0.567 0.962 0.748 10 0.829 0.607 0.862 0.655 0.898 0.649 1.073 0.849 0.896 0.646 0.966 0.752
47. Q2. 正確性：時系列予測 Ø 提案⼿法の時系列予測に関する正確性 v 評価指標：RMSE, MAE（低いほど良） DEIM2025 © 2025 Naoki Chihara et al. 47 KDD ’25, August 3–7, 2025, Toronto, Canada Chihara, et al. Table 3: Causal discovering results with multiple temporal sequences to encompass various types of real-world scenarios. Models M￿￿￿P￿￿￿￿ CASPER DARING NoCurl NO-MLP NOTEARS LiNGAM GES Metrics SHD SID SHD SID SHD SID SHD SID SHD SID SHD SID SHD SID SHD SID 1, 2, 1 3.82 4.94 5.58 7.25 5.75 8.58 6.31 9.90 6.36 8.74 5.03 9.95 7.13 8.23 7.49 11.7 1, 2, 3 4.48 6.51 5.97 8.44 5.81 9.17 6.13 9.51 6.44 8.77 5.69 9.56 6.79 7.33 7.03 10.1 1, 2, 2, 1 4.32 5.88 5.41 8.41 6.54 9.17 6.69 10.0 6.55 8.72 5.23 9.54 7.12 8.65 7.08 9.77 1, 2, 3, 4 4.21 5.76 6.22 8.33 6.12 9.58 6.10 9.61 6.62 8.87 5.73 10.1 7.10 8.50 7.29 11.3 1, 2, 3, 2, 1 4.50 6.11 6.02 8.28 5.45 7.77 6.20 9.83 6.56 8.83 5.57 9.11 7.46 8.05 7.74 12.1 Table 4: Multivariate forecasting results for both synthetic and real-world datasets. We used forecasting steps ;B 2 {5, 10, 15}. Models M￿￿￿P￿￿￿￿ TimesNet PatchTST DeepAR OrbitMap ARIMA Metrics RMSE MAE RMSE MAE RMSE MAE RMSE MAE RMSE MAE RMSE MAE #0 synthetic 5 0.722 0.528 0.805 0.578 0.768 0.581 1.043 0.821 0.826 0.567 0.962 0.748 10 0.829 0.607 0.862 0.655 0.898 0.649 1.073 0.849 0.896 0.646 0.966 0.752 15 0.923 0.686 0.940 0.699 0.973 0.706 1.137 0.854 0.966 0.710 0.982 0.765 #1 covid19 5 0.588 0.268 0.659 0.314 0.640 0.299 1.241 0.691 1.117 0.646 1.259 0.675 10 0.740 0.361 0.841 0.410 1.053 0.523 1.255 0.693 1.353 0.784 1.260 0.687 15 0.932 0.461 1.026 0.516 1.309 0.686 1.265 0.690 1.351 0.792 1.277 0.718 #2 web-search 5 0.573 0.442 0.626 0.469 0.719 0.551 1.255 1.024 0.919 0.640 1.038 0.981 10 0.620 0.481 0.697 0.514 0.789 0.604 1.273 1.044 0.960 0.717 1.247 1.037 15 0.646 0.505 0.701 0.527 0.742 0.571 1.300 1.069 0.828 0.631 1.038 0.795 #3 chicken-dance 5 0.353 0.221 0.759 0.490 0.492 0.303 0.890 0.767 0.508 0.316 2.037 1.742 10 0.511 0.325 0.843 0.564 0.838 0.535 0.886 0.753 0.730 0.476 1.863 1.530 15 0.653 0.419 0.883 0.592 0.972 0.654 0.862 0.718 0.903 0.565 1.792 1.481 #4 exercise 5 0.309 0.177 0.471 0.275 0.465 0.304 0.408 0.290 0.424 0.275 1.003 0.748 10 0.501 0.309 0.630 0.381 0.789 0.518 0.509 0.382 0.616 0.377 1.104 0.814 15 0.687 0.433 0.786 0.505 1.147 0.758 0.676 0.475 0.691 0.434 1.126 0.901 • (#2) web-search: consists of web-search counts collected over ten years related to beer queries on Google [3]. Shanghai). Figure 1 (c) shows stream forecasting results. There has been multiple distinct patterns (e.g., a rapid decrease in infections
48. Q2. 正確性：アブレーション研究 Ø 提案⼿法における因果探索の機能が予測性能に与える影響を検証 v 評価指標：RMSE, MAE（低いほど良） DEIM2025 © 2025 Naoki Chihara et al. 48 Modeling Time-evolving Causality over Data Streams KDD ’25, August 3–7, 2025, Toronto, Canada Table 5: Ablation study results with forecasting steps ;B 2 {5, 10, 15} for both synthetic and real-world datasets. Datasets #0 synthetic #1 covid19 #2 web-search #3 chicken-dance #4 exercise Metrics RMSE MAE RMSE MAE RMSE MAE RMSE MAE RMSE MAE M￿￿￿P￿￿￿￿ (full) 5 0.722 0.528 0.588 0.268 0.573 0.442 0.353 0.221 0.309 0.177 10 0.829 0.607 0.740 0.361 0.620 0.481 0.511 0.325 0.501 0.309 15 0.923 0.686 0.932 0.461 0.646 0.505 0.653 0.419 0.687 0.433 w/o causality 5 0.759 0.563 0.758 0.374 0.575 0.437 0.391 0.262 0.375 0.218 10 0.925 0.696 0.848 0.466 0.666 0.511 0.590 0.398 0.707 0.433 15 1.001 0.760 1.144 0.583 0.708 0.545 0.821 0.537 0.856 0.533 on linear-log scales. Our method consistently outperformed its competitors in terms of computation time thanks to our incremen- tal update, which aligns with the discussion presented in Lemma 3. OrbitMap was competitive, but it estimates model parameters
49. Q3. 計算時間 10 0.829 0.607 0.740 0.361 0.620 15 0.923 0.686 0.932 0.461 0.646 w/o causality 5 0.759 0.563 0.758 0.374 0.575 10 0.925 0.696 0.848 0.466 0.666 15 1.001 0.760 1.144 0.583 0.708 Figure 4: Scalability of M￿￿￿P￿￿￿￿: (left) Wall clock time vs. data stream length C2 and (right) average time consumption for (#4) exercise. The vertical axis of these graphs is a loga- rithmic scale. M￿￿￿P￿￿￿￿ is superior to its competitors. It is shown on its compe cremental Lemma 3. rameters v algorithm Other met cause they 6 Con DEIM2025 データストリーム全体の⻑さに対して定数時間で計算可能 各時刻における計算時間 平均計算時間 © 2025 Naoki Chihara et al. 49 10 0.829 0.607 0.740 0.361 15 0.923 0.686 0.932 0.461 w/o causality 5 0.759 0.563 0.758 0.374 10 0.925 0.696 0.848 0.466 15 1.001 0.760 1.144 0.583 Figure 4: Scalability of M￿￿￿P￿￿￿￿: (left) Wall clock time vs. data stream length C2 and (right) average time consumption for (#4) exercise. The vertical axis of these graphs is a loga- rithmic scale. M￿￿￿P￿￿￿￿ is superior to its competitors. It is 1,800x
50. 題⽬ q 研究背景 q 提案⼿法 q 評価実験 q まとめ DEIM2025 © 2025 Naoki Chihara et al. 50
51. まとめ ModePlait は以下の優れた特性を全て満たす Ø Effective v 時系列パターンの遷移に基づいて時間変化する因果関係を抽出可能 Ø Accurate v 理論的に因果関係を抽出し，正確に将来値を予測する v 最新の⽐較⼿法を上回る精度を達成 Ø Scalable v 計算コストが時系列データストリームの⻑さに依存しない DEIM2025 © 2025 Naoki Chihara et al. 51 KDD Paper Source code
52. Appendix DEIM2025 DEIM2024 © 2025 Naoki Chihara et al. 52
53. 関連研究 Ø 以下の５つの観点において ModePlait は優れている DEIM2025 © 2025 Naoki Chihara et al. 53
54. 関連研究 Ø ARIMA [Box and Jenkins 1976] v 古典的な将来予測⼿法 v 時系列間の⾮線形な関係性を考慮できない Ø OrbitMap [Matsubara and Sakurai 2019] v ストリーム⽅式に対応した将来予測を実現 v 時間変化する因果関係に⾮対応 DEIM2025 © 2025 Naoki Chihara et al. 54 😢 😢
55. 関連研究 Ø ⼀般的な因果探索⼿法 v CASPER [Liu et al. 2023] 等 v 時系列データストリームに⾮対応 Ø 深層学習ベースの将来予測⼿法 v TimesNet [Wu et al. 2023] 等 v モデルの学習に膨⼤な時間を要するため、最新の情報を考慮し た⾼速かつ連続的な将来予測が困難 DEIM2025 © 2025 Naoki Chihara et al. 55 😢 😢
56. 提案⼿法：モデル概要図 Ø 提案モデルの概要図は以下のとおり DEIM2025 © 2025 Naoki Chihara et al. 56 Im Re frequency decay rate ! ! ∠ !! eigenvalue !! Interpretability of modes (1 ≤ $ ≤ %! ) Inherent signal !(") !(") " − ℎ + 1 ℎ "(") ℎ '" '" '" #(") , Self-dynamics factor set ((") " ⇓ Hankel matrix ) ⇓ (a) ݻ༗μΠφϛΫεू߹ (i.e., D(i) = {Φ(i) , Λ(i) }) Multivariate time series ! " # ⋮ !(") !($) Regime parameter set $ , # # "%$ Mixing matrix , # # "%$ Inherent signals % #($) #(") " ⋮ Causal relationship $ #(&) ⇓ ⇓ (b) ϨδʔϜ (i.e., θ = {W , D(1) , ..., D(d) }) ਤ 2: ModePlait ͷϞσϧ֓ཁਤ: (a) ಈతγεςϜʹै͏ i ൪໨ͷม਺ʹݻ༗ͳ୯มྔ৴߸ e(i) ͔ΒજࡏతͳμΠφϛΫεΛ நग़͢Δ. (b) ଟ࣍ݩ࣌ܥྻσʔλ͸ࠞ߹ߦྻ W −1 ͱ d ݸͷࣗݾۦಈҼࢠ {D(1) , ..., D(d) } ʹΑͬͯදݱ͞ΕΔɽ
57. 評価実験：評価指標 因果探索の実験では SHD, SID の２種類の評価指標を採⽤した Ø structural Hamming distance (SHD) v 因果隣接⾏列の差異を定量化する指標 v ⽋落した辺，余分な辺，逆転した辺の数 Ø structural intervention distance (SID) v 因果探索の精度評価に特に適した評価指標 v 推定された因果隣接⾏列を使⽤した場合に， 介⼊分布 𝑝 𝑥+ | do(𝑋# = ̅ 𝑥) が誤って検出される (𝑖, 𝑗) の組の数 DEIM2025 © 2025 Naoki Chihara et al. 57
58. 評価実験：評価指標 時系列予測の実験では RMSE, MAE の２種類の評価指標を採⽤した Ø ⼆乗平均誤差 (RMSE) … 外れ値の多さに焦点が当たる 𝑅𝑀𝑆𝐸 = 1 𝑁 H #,% - 𝑦# − K 𝑦# Ø 平均絶対誤差 (MAE) … 全体的な誤差の⼤きさを算出 𝑀𝐴𝐸 = 1 𝑁 H #,% - 𝑦# − K 𝑦# DEIM2025 © 2025 Naoki Chihara et al. 58
59. Ø ⼈⼯データセットには複数のクラスタが存在する v １つのクラスタに１つの因果関係が対応 v 因果隣接⾏列 𝑩 は Eröds-Rényi (ER) モデルに基づいて⽣成 v エッジ密度 𝑝 = 0.5，観測変数の数 𝑑 = 5 評価実験：⼈⼯データセット DEIM2025 © 2025 Naoki Chihara et al. 59 𝑑 Data stream 𝑿 Time 𝑡 ⼈⼯データセット “1, 2, 3“ の例 クラスタ１ クラスタ２ クラスタ３ 実際に存在するエッジの数の割合