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 ൪ͷมʹݻ༗ͳ୯มྔ৴߸ e(i) ͔ΒજࡏతͳμΠφϛΫεΛ நग़͢Δ. (b) ଟ࣍ݩ࣌ܥྻσʔλࠞ߹ߦྻ W −1 ͱ d ݸͷࣗݾۦಈҼࢠ {D(1) , ..., D(d) } ʹΑͬͯදݱ͞ΕΔɽ . 1 ModePlait Ϟσϧ ຊઅͰఏҊϞσϧͷৄࡉΛड़ΔɽॳΊʹඞཁͳ֓೦ͷఆ ٛʹ͍ͭͯઆ໌͢Δɽ ఆٛ 1 (ݻ༗৴߸ɿE). ඇΨεʹै͏ d ݸͷ૬ޓʹಠཱ ͨ͠ཁૉΛ࣋ͭ৴߸ E = {e(i) }d i=1 Λݻ༗৴߸ͱݺͿɽͨͩ͠ɼ (i) = {e(i) (1), ..., e(i) (t)} i ൪ͷ୯มྔ࣌ܥྻͰ͋Δɽ͜ Ϋτϧɼݩͷঢ়ଶͱඍಉ૬ͳμΠφϛΫεΛ࣋ͭ͜ͱ͕อ ূ͞ΕΔɽײతʹઆ໌Λ͢Δͱɼ͜ͷ࠶ߏݩͷྗֶܥͷ ಛੑΛཧతʹอͭɼͭ·Γɼϋϯέϧߦྻ H(i) ͷղੳΛ௨ ͯ͡ɼݩͷσʔλ͔Βநग़Ͱ͖ͳ͍ॏཁͳಛΛ໌Β͔ ʹ͢Δ͜ͱΛՄೳʹ͢Δɽଟ͘ͷ߹ɼඍಉ૬ࣸ૾Λ٘ਜ਼ʹ ͢Δ͜ͱͳ͘ຒΊࠐΈ࣍ݩΛબͰ͖Δɽ ͜͜Ͱɼi ൪ͷݻ༗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৽ͨͳύϥϝʔλͰ͋Δߦ ೖ͢Δɽ͜Εɼd ݸͷݻ༗৴߸ؒͷؔੑΛදݱ ͷಛఆͷͨΊʹॏཁͳׂΛՌͨ͢ɽW ͔Β B ͨΊͷΞϧΰϦζϜʹ͍ͭͯ 4. 2. 3 અʹͯઆ໌͢ ΔͱɼҎԼΛಘΔɽ ఆٛ 3 (ϨδʔϜɿθ). θ = {W , D(1) , ..., D(d) } Λ දݱ͢Δύϥϝʔλू߹ͱ͢Δɽ͜͜ͰɼW Ҽ B Λੜ͢ΔͨΊͷج൫ͱͳΔཁૉͰ͋Δɽ