product form stationary distributions of stochastic
play

Product form stationary distributions of stochastic reaction - PowerPoint PPT Presentation

Feb 13, 2024 •163 likes •1.44k views

Product form stationary distributions of stochastic reaction networks, and application to constrained averaging of multiscale systems Simon Cotter University of Manchester 19th July 2016 Simon Cotter Product form stationary dist. 0 / 35

  1. QSSA: Simple Example k 2 k 4 / ( k 2 + k 3 ) s k 1 ∅ − − − − → S − − − − − − − − − → ∅ Quantify error by computing error in invariant distribution of S = X 1 + X 2 � k 1 ( k 2 + k 3 ) � S ( ∞ ) ∼ P (QSSA approximation) k 2 k 4 Invariant distribution of full system given by � � k 4 , λ 2 = k 1 ( k 3 + k 4 ) λ 1 = k 1 P k 2 k 4 � � k 1 ( k 2 + k 3 + k 4 ) S ( ∞ ) ∼ P (True invariant distribution) k 2 k 4 QSSA error in Poisson intensity k 1 k 2 Simon Cotter Product form stationary dist. 9 / 35

  2. QSSA: Simple Example k 2 k 4 / ( k 2 + k 3 ) s k 1 ∅ − − − − → S − − − − − − − − − → ∅ Quantify error by computing error in invariant distribution of S = X 1 + X 2 � k 1 ( k 2 + k 3 ) � S ( ∞ ) ∼ P (QSSA approximation) k 2 k 4 Invariant distribution of full system given by � � k 4 , λ 2 = k 1 ( k 3 + k 4 ) λ 1 = k 1 P k 2 k 4 � � k 1 ( k 2 + k 3 + k 4 ) S ( ∞ ) ∼ P (True invariant distribution) k 2 k 4 QSSA error in Poisson intensity k 1 k 2 Simon Cotter Product form stationary dist. 9 / 35

  3. QSSA: Simple Example k 2 k 4 / ( k 2 + k 3 ) s k 1 ∅ − − − − → S − − − − − − − − − → ∅ Quantify error by computing error in invariant distribution of S = X 1 + X 2 � k 1 ( k 2 + k 3 ) � S ( ∞ ) ∼ P (QSSA approximation) k 2 k 4 Invariant distribution of full system given by � � k 4 , λ 2 = k 1 ( k 3 + k 4 ) λ 1 = k 1 P k 2 k 4 � � k 1 ( k 2 + k 3 + k 4 ) S ( ∞ ) ∼ P (True invariant distribution) k 2 k 4 QSSA error in Poisson intensity k 1 k 2 Simon Cotter Product form stationary dist. 9 / 35

  4. Outline Introduction 1 The Quasi Steady-State Assumption 2 The Constrained Approach 3 Product Form Stationary Distributions for Non-Mass Action 4 Kinetics 5 Numerical Example Conclusions 6 Simon Cotter Product form stationary dist. 9 / 35

  5. The Constrained Approach The Constrained philosophy The assumption is that the fast variables equilibrate on a timescale which is negligible with respect to the rate of change of slow species, but that the invariant distribution of the fast variables can be affected by the slow reactions . Slow stoichiometries are not in general perpendicular to the fast stoichiometries! In practice We cannot approximate P ( F | S ) by considering just the fast subsystem, as we need to take into account the effect of the slow reactions on the invariant distribution of the fast variables . This can be approximated by considering the dynamics of the full system, constrained to a particular value of the slow variables . NB: Assymptotically (in the limit of large timescale gaps), this approach is in agreement with the QSSA Simon Cotter Product form stationary dist. 10 / 35

  6. The Constrained Approach The Constrained philosophy The assumption is that the fast variables equilibrate on a timescale which is negligible with respect to the rate of change of slow species, but that the invariant distribution of the fast variables can be affected by the slow reactions . Slow stoichiometries are not in general perpendicular to the fast stoichiometries! In practice We cannot approximate P ( F | S ) by considering just the fast subsystem, as we need to take into account the effect of the slow reactions on the invariant distribution of the fast variables . This can be approximated by considering the dynamics of the full system, constrained to a particular value of the slow variables . NB: Assymptotically (in the limit of large timescale gaps), this approach is in agreement with the QSSA Simon Cotter Product form stationary dist. 10 / 35

  7. The Constrained Approach The Constrained philosophy The assumption is that the fast variables equilibrate on a timescale which is negligible with respect to the rate of change of slow species, but that the invariant distribution of the fast variables can be affected by the slow reactions . Slow stoichiometries are not in general perpendicular to the fast stoichiometries! In practice We cannot approximate P ( F | S ) by considering just the fast subsystem, as we need to take into account the effect of the slow reactions on the invariant distribution of the fast variables . This can be approximated by considering the dynamics of the full system, constrained to a particular value of the slow variables . NB: Assymptotically (in the limit of large timescale gaps), this approach is in agreement with the QSSA Simon Cotter Product form stationary dist. 10 / 35

  8. The Constrained Approach ν ′ R 1 : ν 1 → 1 , ν ′ R 2 : ν 2 → 2 , . . . ν ′ R M : ν M → M , Pick slow variable S = � i a ij X i which is invariant with respect to fast reactions, i.e. perpendicular to fast stoichiometries Form basis with the slow variables, along with chosen fast variables F Define constrained stoichiometric projector P : R d → R d P ([ S , F ]) = [ 0 , F ] Simon Cotter Product form stationary dist. 11 / 35

  9. The Constrained Approach ν ′ R 1 : ν 1 → 1 , ν ′ R 2 : ν 2 → 2 , . . . ν ′ R M : ν M → M , Pick slow variable S = � i a ij X i which is invariant with respect to fast reactions, i.e. perpendicular to fast stoichiometries Form basis with the slow variables, along with chosen fast variables F Define constrained stoichiometric projector P : R d → R d P ([ S , F ]) = [ 0 , F ] Simon Cotter Product form stationary dist. 11 / 35

  10. The Constrained Approach ν ′ R 1 : ν 1 → 1 , ν ′ R 2 : ν 2 → 2 , . . . ν ′ R M : ν M → M , Pick slow variable S = � i a ij X i which is invariant with respect to fast reactions, i.e. perpendicular to fast stoichiometries Form basis with the slow variables, along with chosen fast variables F Define constrained stoichiometric projector P : R d → R d P ([ S , F ]) = [ 0 , F ] Simon Cotter Product form stationary dist. 11 / 35

  11. The Constrained Approach ν ′ R 1 : ν 1 → 1 , ν ′ R 2 : ν 2 → 2 , . . . ν ′ R M : ν M → M , Pick slow variable S = � i a ij X i which is invariant with respect to fast reactions, i.e. perpendicular to fast stoichiometries Form basis with the slow variables, along with chosen fast variables F Define constrained stoichiometric projector P : R d → R d P ([ S , F ]) = [ 0 , F ] Simon Cotter Product form stationary dist. 11 / 35

  12. The Constrained Subsystem ν 1 + P ( ν ′ R 1 : ν 1 → 1 − ν 1 ) = ˆ ν 1 , ν 2 + P ( ν ′ → 2 − ν 2 ) = ˆ R 2 : ν 2 ν 2 , . . . ν M + P ( ν ′ R M : ν M → M − ν M ) = ˆ ν M . Preserves changes to fast variables, removes changes to slow variables Multiply propensities by ✶ X + P ( ν ′ 0 to preserve k − ν k ) ∈ Z d non-negativity Constrains the slow variable to its starting value Invariant distribution a good approximation of P ( F | S ) for full system Invariant distribution of constrained system needs to be found Simon Cotter Product form stationary dist. 12 / 35

  13. The Constrained Subsystem ν 1 + P ( ν ′ R 1 : ν 1 → 1 − ν 1 ) = ˆ ν 1 , ν 2 + P ( ν ′ → 2 − ν 2 ) = ˆ R 2 : ν 2 ν 2 , . . . ν M + P ( ν ′ R M : ν M → M − ν M ) = ˆ ν M . Preserves changes to fast variables, removes changes to slow variables Multiply propensities by ✶ X + P ( ν ′ 0 to preserve k − ν k ) ∈ Z d non-negativity Constrains the slow variable to its starting value Invariant distribution a good approximation of P ( F | S ) for full system Invariant distribution of constrained system needs to be found Simon Cotter Product form stationary dist. 12 / 35

  14. The Constrained Subsystem ν 1 + P ( ν ′ R 1 : ν 1 → 1 − ν 1 ) = ˆ ν 1 , ν 2 + P ( ν ′ → 2 − ν 2 ) = ˆ R 2 : ν 2 ν 2 , . . . ν M + P ( ν ′ R M : ν M → M − ν M ) = ˆ ν M . Preserves changes to fast variables, removes changes to slow variables Multiply propensities by ✶ X + P ( ν ′ 0 to preserve k − ν k ) ∈ Z d non-negativity Constrains the slow variable to its starting value Invariant distribution a good approximation of P ( F | S ) for full system Invariant distribution of constrained system needs to be found Simon Cotter Product form stationary dist. 12 / 35

  15. The Constrained Subsystem ν 1 + P ( ν ′ R 1 : ν 1 → 1 − ν 1 ) = ˆ ν 1 , ν 2 + P ( ν ′ → 2 − ν 2 ) = ˆ R 2 : ν 2 ν 2 , . . . ν M + P ( ν ′ R M : ν M → M − ν M ) = ˆ ν M . Preserves changes to fast variables, removes changes to slow variables Multiply propensities by ✶ X + P ( ν ′ 0 to preserve k − ν k ) ∈ Z d non-negativity Constrains the slow variable to its starting value Invariant distribution a good approximation of P ( F | S ) for full system Invariant distribution of constrained system needs to be found Simon Cotter Product form stationary dist. 12 / 35

  16. The Constrained Subsystem ν 1 + P ( ν ′ R 1 : ν 1 → 1 − ν 1 ) = ˆ ν 1 , ν 2 + P ( ν ′ → 2 − ν 2 ) = ˆ R 2 : ν 2 ν 2 , . . . ν M + P ( ν ′ R M : ν M → M − ν M ) = ˆ ν M . Preserves changes to fast variables, removes changes to slow variables Multiply propensities by ✶ X + P ( ν ′ 0 to preserve k − ν k ) ∈ Z d non-negativity Constrains the slow variable to its starting value Invariant distribution a good approximation of P ( F | S ) for full system Invariant distribution of constrained system needs to be found Simon Cotter Product form stationary dist. 12 / 35

  17. The Constrained Subsystem ν 1 + P ( ν ′ R 1 : ν 1 → 1 − ν 1 ) = ˆ ν 1 , ν 2 + P ( ν ′ → 2 − ν 2 ) = ˆ R 2 : ν 2 ν 2 , . . . ν M + P ( ν ′ R M : ν M → M − ν M ) = ˆ ν M . Preserves changes to fast variables, removes changes to slow variables Multiply propensities by ✶ X + P ( ν ′ 0 to preserve k − ν k ) ∈ Z d non-negativity Constrains the slow variable to its starting value Invariant distribution a good approximation of P ( F | S ) for full system Invariant distribution of constrained system needs to be found Simon Cotter Product form stationary dist. 12 / 35

  18. Constrained Approach: Simple Example Consider the system: k 2 x 1 − − − − → k 1 k 4 x 2 → X 1 X 2 ∅ − − − − ← − − − − − − − − → ∅ k 3 x 2 First construct basis made up of slow/fast variables S = X 1 + X 2 , F = X 1 or F = X 2 We will pick F = X 2 Therefore P ([ X 1 , X 2 ]) = [ − X 2 , X 2 ] Apply to all reactions to arrive at constrained subsystem Simon Cotter Product form stationary dist. 13 / 35

  19. Constrained Approach: Simple Example Consider the system: k 2 x 1 − − − − → k 1 k 4 x 2 → X 1 X 2 ∅ − − − − ← − − − − − − − − → ∅ k 3 x 2 First construct basis made up of slow/fast variables S = X 1 + X 2 , F = X 1 or F = X 2 We will pick F = X 2 Therefore P ([ X 1 , X 2 ]) = [ − X 2 , X 2 ] Apply to all reactions to arrive at constrained subsystem Simon Cotter Product form stationary dist. 13 / 35

  20. Constrained Approach: Simple Example Consider the system: k 2 x 1 − − − − → k 1 k 4 x 2 → X 1 X 2 ∅ − − − − ← − − − − − − − − → ∅ k 3 x 2 First construct basis made up of slow/fast variables S = X 1 + X 2 , F = X 1 or F = X 2 We will pick F = X 2 Therefore P ([ X 1 , X 2 ]) = [ − X 2 , X 2 ] Apply to all reactions to arrive at constrained subsystem Simon Cotter Product form stationary dist. 13 / 35

  21. Constrained Approach: Simple Example Consider the system: k 2 x 1 − − − − → k 1 k 4 x 2 → X 1 X 2 ∅ − − − − ← − − − − − − − − → ∅ k 3 x 2 First construct basis made up of slow/fast variables S = X 1 + X 2 , F = X 1 or F = X 2 We will pick F = X 2 Therefore P ([ X 1 , X 2 ]) = [ − X 2 , X 2 ] Apply to all reactions to arrive at constrained subsystem Simon Cotter Product form stationary dist. 13 / 35

  22. Constrained Approach: Simple Example Consider the system: k 2 x 1 − − − − → k 1 k 4 x 2 → X 1 X 2 ∅ − − − − ← − − − − − − − − → ∅ k 3 x 2 First construct basis made up of slow/fast variables S = X 1 + X 2 , F = X 1 or F = X 2 We will pick F = X 2 Therefore P ([ X 1 , X 2 ]) = [ − X 2 , X 2 ] Apply to all reactions to arrive at constrained subsystem Simon Cotter Product form stationary dist. 13 / 35

  23. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  24. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  25. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  26. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  27. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  28. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  29. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  30. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  31. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  32. Constrained Approach: Simple Example k 2 x 1 − − − − − − − → X 1 X 2 , ← − − − − − − − S = X 1 + X 2 ( k 3 + k 4 ) x 2 Invariant distribution X 2 ∼ B ( S , λ 2 ) = π ( X 2 ) � � ( k 3 + k 4 ) k 2 [ λ 1 , λ 2 ] = k 2 + k 3 + k 4 , steady state solution of mean field k 2 + k 3 + k 4 ODE: k 2 λ 1 = ( k 3 + k 4 ) λ 2 , λ 1 + λ 2 = 1 Compute expectation of the rate of reaction R 4 k 2 k 4 S α 4 = E ( α 4 | S ) = k 4 E ( X 2 | S ) = ˆ k 2 + k 3 + k 4 Simon Cotter Product form stationary dist. 15 / 35

  33. Constrained Approach: Simple Example k 2 x 1 − − − − − − − → X 1 X 2 , ← − − − − − − − S = X 1 + X 2 ( k 3 + k 4 ) x 2 Invariant distribution X 2 ∼ B ( S , λ 2 ) = π ( X 2 ) � � ( k 3 + k 4 ) k 2 [ λ 1 , λ 2 ] = k 2 + k 3 + k 4 , steady state solution of mean field k 2 + k 3 + k 4 ODE: k 2 λ 1 = ( k 3 + k 4 ) λ 2 , λ 1 + λ 2 = 1 Compute expectation of the rate of reaction R 4 k 2 k 4 S α 4 = E ( α 4 | S ) = k 4 E ( X 2 | S ) = ˆ k 2 + k 3 + k 4 Simon Cotter Product form stationary dist. 15 / 35

  34. Constrained Approach: Simple Example k 2 x 1 − − − − − − − → X 1 X 2 , ← − − − − − − − S = X 1 + X 2 ( k 3 + k 4 ) x 2 Invariant distribution X 2 ∼ B ( S , λ 2 ) = π ( X 2 ) � � ( k 3 + k 4 ) k 2 [ λ 1 , λ 2 ] = k 2 + k 3 + k 4 , steady state solution of mean field k 2 + k 3 + k 4 ODE: k 2 λ 1 = ( k 3 + k 4 ) λ 2 , λ 1 + λ 2 = 1 Compute expectation of the rate of reaction R 4 k 2 k 4 S α 4 = E ( α 4 | S ) = k 4 E ( X 2 | S ) = ˆ k 2 + k 3 + k 4 Simon Cotter Product form stationary dist. 15 / 35

  35. Constrained Approach: Simple Example k 2 k 4 / ( k 2 + k 3 + k 4 ) s k 1 ∅ − − − − → S − − − − − − − − − − − → ∅ To compare with QSSA, compute invariant distribution � k 1 ( k 2 + k 3 + k 4 ) � S ( ∞ ) ∼ P (Constrained approximation) k 2 k 4 Invariant distribution identical to true invariant distribution of S = X 1 + X 2 Simon Cotter Product form stationary dist. 16 / 35

  36. Constrained Approach: Simple Example k 2 k 4 / ( k 2 + k 3 + k 4 ) s k 1 ∅ − − − − → S − − − − − − − − − − − → ∅ To compare with QSSA, compute invariant distribution � k 1 ( k 2 + k 3 + k 4 ) � S ( ∞ ) ∼ P (Constrained approximation) k 2 k 4 Invariant distribution identical to true invariant distribution of S = X 1 + X 2 Simon Cotter Product form stationary dist. 16 / 35

  37. Constrained Approach: Simple Example k 2 k 4 / ( k 2 + k 3 + k 4 ) s k 1 ∅ − − − − → S − − − − − − − − − − − → ∅ To compare with QSSA, compute invariant distribution � k 1 ( k 2 + k 3 + k 4 ) � S ( ∞ ) ∼ P (Constrained approximation) k 2 k 4 Invariant distribution identical to true invariant distribution of S = X 1 + X 2 Simon Cotter Product form stationary dist. 16 / 35

  38. Simple Example 0.07 Full generator QSSA generator 0.06 CMA generator 0.05 Probability Mass 0.04 0.03 0.02 0.01 0 0 20 40 60 80 100 S = X 1 + X 2 SLC, “Constrained Approximation of Effective Generators for Multiscale Stochastic Reaction Networks and Application to Conditioned Path Sampling”, in review. Simon Cotter Product form stationary dist. 17 / 35

  39. Outline Introduction 1 The Quasi Steady-State Assumption 2 The Constrained Approach 3 Product Form Stationary Distributions for Non-Mass Action 4 Kinetics 5 Numerical Example Conclusions 6 Simon Cotter Product form stationary dist. 17 / 35

  40. Some Network Properties: Complexes k − → S 3 S 1 + S 2 S 1 + S 2 reactant complex S 3 product complex C number of complexes in network k 1 X 1 ( X 1 − 1 ) − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 C = 4 Simon Cotter Product form stationary dist. 18 / 35

  41. Some Network Properties: Complexes k − → S 3 S 1 + S 2 S 1 + S 2 reactant complex S 3 product complex C number of complexes in network k 1 X 1 ( X 1 − 1 ) − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 C = 4 Simon Cotter Product form stationary dist. 18 / 35

  42. Some Network Properties: Complexes k − → S 3 S 1 + S 2 S 1 + S 2 reactant complex S 3 product complex C number of complexes in network k 1 X 1 ( X 1 − 1 ) − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 C = 4 Simon Cotter Product form stationary dist. 18 / 35

  43. Some Network Properties: Complexes k − → S 3 S 1 + S 2 S 1 + S 2 reactant complex S 3 product complex C number of complexes in network k 1 X 1 ( X 1 − 1 ) − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 C = 4 Simon Cotter Product form stationary dist. 18 / 35

  44. Some Network Properties: Complexes k − → S 3 S 1 + S 2 S 1 + S 2 reactant complex S 3 product complex C number of complexes in network k 1 X 1 ( X 1 − 1 ) − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 C = 4 Simon Cotter Product form stationary dist. 18 / 35

  45. Some Network Properties: Complexes k − → S 3 S 1 + S 2 S 1 + S 2 reactant complex S 3 product complex C number of complexes in network k 1 X 1 ( X 1 − 1 ) − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 C = 4 Simon Cotter Product form stationary dist. 18 / 35

  46. Some Network Properties: Linkage Classes Network can be represented uniquely by graph with C nodes k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 2 S 1 S 2 S 1 ∅ Each connected graph is a linkage class l number of linkage classes (in this example l = 1) Simon Cotter Product form stationary dist. 19 / 35

  47. Some Network Properties: Linkage Classes Network can be represented uniquely by graph with C nodes k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 2 S 1 S 2 S 1 ∅ Each connected graph is a linkage class l number of linkage classes (in this example l = 1) Simon Cotter Product form stationary dist. 19 / 35

  48. Some Network Properties: Linkage Classes Network can be represented uniquely by graph with C nodes k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 2 S 1 S 2 S 1 ∅ Each connected graph is a linkage class l number of linkage classes (in this example l = 1) Simon Cotter Product form stationary dist. 19 / 35

  49. Some Network Properties: Linkage Classes Network can be represented uniquely by graph with C nodes k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 2 S 1 S 2 S 1 ∅ Each connected graph is a linkage class l number of linkage classes (in this example l = 1) Simon Cotter Product form stationary dist. 19 / 35

  50. Some Network Properties: Linkage Classes Network can be represented uniquely by graph with C nodes k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 2 S 1 S 2 S 1 ∅ Each connected graph is a linkage class l number of linkage classes (in this example l = 1) Simon Cotter Product form stationary dist. 19 / 35

  51. Some Network Properties: Stoichiometric Subspace Denote each reaction as: → ν ′ ν m − − − − m Stochiometric subspace given by: S = span m { ν m − ν ′ m } Denote s = dim ( S ) Simon Cotter Product form stationary dist. 20 / 35

  52. Some Network Properties: Stoichiometric Subspace Denote each reaction as: → ν ′ ν m − − − − m Stochiometric subspace given by: S = span m { ν m − ν ′ m } Denote s = dim ( S ) Simon Cotter Product form stationary dist. 20 / 35

  53. Some Network Properties: Stoichiometric Subspace Denote each reaction as: → ν ′ ν m − − − − m Stochiometric subspace given by: S = span m { ν m − ν ′ m } Denote s = dim ( S ) Simon Cotter Product form stationary dist. 20 / 35

  54. Some Network Properties: Deficiency Zero Deficiency δ = C − l − s k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , − − − − → ∅ , S 1 k 2 X 2 δ = 4 − 1 − 2 = 1 Networks with deficiency zero have certain properties and are well studied Simon Cotter Product form stationary dist. 21 / 35

  55. Some Network Properties: Deficiency Zero Deficiency δ = C − l − s k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , − − − − → ∅ , S 1 k 2 X 2 δ = 4 − 1 − 2 = 1 Networks with deficiency zero have certain properties and are well studied Simon Cotter Product form stationary dist. 21 / 35

  56. Some Network Properties: Deficiency Zero Deficiency δ = C − l − s k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , − − − − → ∅ , S 1 k 2 X 2 δ = 4 − 1 − 2 = 1 Networks with deficiency zero have certain properties and are well studied Simon Cotter Product form stationary dist. 21 / 35

  57. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  58. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  59. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  60. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  61. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  62. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  63. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  64. Preliminaries: form of intensity functions The theorem that follows relies on the following form of the intensity/hazard functions d ν mi − 1 � � α m ( x ) = k m θ i ( x i − j ) , i = 1 j = 0 Mass action kinetics if θ is the identity Simon Cotter Product form stationary dist. 23 / 35

  65. Preliminaries: form of intensity functions The theorem that follows relies on the following form of the intensity/hazard functions d ν mi − 1 � � α m ( x ) = k m θ i ( x i − j ) , i = 1 j = 0 Mass action kinetics if θ is the identity Simon Cotter Product form stationary dist. 23 / 35

  66. Preliminaries: form of intensity functions The theorem that follows relies on the following form of the intensity/hazard functions d ν mi − 1 � � α m ( x ) = k m θ i ( x i − j ) , i = 1 j = 0 Mass action kinetics if θ is the identity Simon Cotter Product form stationary dist. 23 / 35

  67. Product Form Stationary Distributions Theorem (Anderson, Craciun, Kurtz, 2010) Let {S , C , R} be a zero deficiency, weakly reversible reaction network. Modeled deterministically with mass action kinetics and rate constants k m the system has complex balanced equilibrium c ∈ R d > 0 . Then, the stochastically modeled system with the specified intensity functions admits the invariant measure on each stoichiometric class d c x i � i π ( x ) ∝ . � x i − 1 j = 0 θ i ( x i − j ) i = 1 The measure π can be normalized to a stationary distribution so long as it is summable. Simon Cotter Product form stationary dist. 24 / 35

  68. Motivating Example for New Result k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 Multiscale system with reversible dimerisation reactions fast Consider the constrained fast subsystem with slow variable S = X 1 + 2 X 2 k 1 X 1 ( X 1 − 1 )+ k 3 ✶ { X 1 > 1 } − − − − − − − − − − − → 2 S 1 S 2 , ← − − − − − − − − − − − ( k 2 + k 4 ) X 2 Intensity functions do not satisfy the assumptions of the existing results Simon Cotter Product form stationary dist. 25 / 35

  69. Motivating Example for New Result k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 Multiscale system with reversible dimerisation reactions fast Consider the constrained fast subsystem with slow variable S = X 1 + 2 X 2 k 1 X 1 ( X 1 − 1 )+ k 3 ✶ { X 1 > 1 } − − − − − − − − − − − → 2 S 1 S 2 , ← − − − − − − − − − − − ( k 2 + k 4 ) X 2 Intensity functions do not satisfy the assumptions of the existing results Simon Cotter Product form stationary dist. 25 / 35

  70. Motivating Example for New Result k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 Multiscale system with reversible dimerisation reactions fast Consider the constrained fast subsystem with slow variable S = X 1 + 2 X 2 k 1 X 1 ( X 1 − 1 )+ k 3 ✶ { X 1 > 1 } − − − − − − − − − − − → 2 S 1 S 2 , ← − − − − − − − − − − − ( k 2 + k 4 ) X 2 Intensity functions do not satisfy the assumptions of the existing results Simon Cotter Product form stationary dist. 25 / 35

  71. Motivating Example for New Result k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 Multiscale system with reversible dimerisation reactions fast Consider the constrained fast subsystem with slow variable S = X 1 + 2 X 2 k 1 X 1 ( X 1 − 1 )+ k 3 ✶ { X 1 > 1 } − − − − − − − − − − − → 2 S 1 S 2 , ← − − − − − − − − − − − ( k 2 + k 4 ) X 2 Intensity functions do not satisfy the assumptions of the existing results Simon Cotter Product form stationary dist. 25 / 35

  72. Motivating Example for New Result k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 Multiscale system with reversible dimerisation reactions fast Consider the constrained fast subsystem with slow variable S = X 1 + 2 X 2 k 1 X 1 ( X 1 − 1 )+ k 3 ✶ { X 1 > 1 } − − − − − − − − − − − → 2 S 1 S 2 , ← − − − − − − − − − − − ( k 2 + k 4 ) X 2 Intensity functions do not satisfy the assumptions of the existing results Simon Cotter Product form stationary dist. 25 / 35

  73. New Structural Assumption Partition all species S = { S 1 , S 2 , . . . , S N } into S 1 and S 2 . S i ∈ S 2 if for some η i ≥ 2 we have that ν ki is always some nonnegative multiple of η i For all other S i ∈ S 1 , let η i = 1. Assume that all intensity functions satisfy: ν mi η i − 1 d � � α m ( x ) = k m θ i ( X i − j η i ) , i = 1 j = 0 where κ k > 0 and θ i ( x i ) = 0 if and only if x i ≤ α i − 1. Simon Cotter Product form stationary dist. 26 / 35

  74. New Structural Assumption Partition all species S = { S 1 , S 2 , . . . , S N } into S 1 and S 2 . S i ∈ S 2 if for some η i ≥ 2 we have that ν ki is always some nonnegative multiple of η i For all other S i ∈ S 1 , let η i = 1. Assume that all intensity functions satisfy: ν mi η i − 1 d � � α m ( x ) = k m θ i ( X i − j η i ) , i = 1 j = 0 where κ k > 0 and θ i ( x i ) = 0 if and only if x i ≤ α i − 1. Simon Cotter Product form stationary dist. 26 / 35

  75. New Structural Assumption Partition all species S = { S 1 , S 2 , . . . , S N } into S 1 and S 2 . S i ∈ S 2 if for some η i ≥ 2 we have that ν ki is always some nonnegative multiple of η i For all other S i ∈ S 1 , let η i = 1. Assume that all intensity functions satisfy: ν mi η i − 1 d � � α m ( x ) = k m θ i ( X i − j η i ) , i = 1 j = 0 where κ k > 0 and θ i ( x i ) = 0 if and only if x i ≤ α i − 1. Simon Cotter Product form stationary dist. 26 / 35

  76. New Structural Assumption Partition all species S = { S 1 , S 2 , . . . , S N } into S 1 and S 2 . S i ∈ S 2 if for some η i ≥ 2 we have that ν ki is always some nonnegative multiple of η i For all other S i ∈ S 1 , let η i = 1. Assume that all intensity functions satisfy: ν mi η i − 1 d � � α m ( x ) = k m θ i ( X i − j η i ) , i = 1 j = 0 where κ k > 0 and θ i ( x i ) = 0 if and only if x i ≤ α i − 1. Simon Cotter Product form stationary dist. 26 / 35

  77. New Structural Assumption Partition all species S = { S 1 , S 2 , . . . , S N } into S 1 and S 2 . S i ∈ S 2 if for some η i ≥ 2 we have that ν ki is always some nonnegative multiple of η i For all other S i ∈ S 1 , let η i = 1. Assume that all intensity functions satisfy: ν mi η i − 1 d � � α m ( x ) = k m θ i ( X i − j η i ) , i = 1 j = 0 where κ k > 0 and θ i ( x i ) = 0 if and only if x i ≤ α i − 1. Simon Cotter Product form stationary dist. 26 / 35

  78. Theorem: Product Form Stationary Distributions Theorem (Anderson, Cotter, 2016) Let {S 1 ∪ S 2 , C , R} be a zero deficiency, weakly reversible reaction network satisfying the above assumption. Modeled deterministically with mass action kinetics and rate constants k m the system has complex balanced equilibrium c ∈ R d > 0 . Then, the stochastically modeled system with the specified intensity functions admits the invariant measure on each stoichiometric class d c X i � i π ( x ) ∝ . � ⌊ X i /η i ⌋− 1 θ i ( X i − j η i ) i = 1 j = 0 The measure π can be normalized to a stationary distribution so long as it is summable. Simon Cotter Product form stationary dist. 27 / 35

Recommend

Agenda Course 02402 Introduction to Statistics 1 Stochastic Variables and Distributions The

Agenda Course 02402 Introduction to Statistics 1 Stochastic Variables and Distributions The

Agenda Course 02402 Introduction to Statistics 1 Stochastic Variables and Distributions The definition of a Stochastic Variable Lecture 2: Discrete Distributions Density function The distribution function 2 Specific Statistical distributions

289 views • 9 slides
Composition of product-form Generalized Stochastic Petri Nets: a modular approach Simonetta

Composition of product-form Generalized Stochastic Petri Nets: a modular approach Simonetta

Composition of product-form Generalized Stochastic Petri Nets: a modular approach Simonetta Balsamo and Andrea Marin Universit` a Ca Foscari di Venezia Dipartimento di Informatica Italy October 2009 Introduction Problems in exact

290 views • 17 slides
Modelling retrial-upon-conflict systems with product-form stochastic Petri nets Simonetta Balsamo

Modelling retrial-upon-conflict systems with product-form stochastic Petri nets Simonetta Balsamo

Modelling retrial-upon-conflict systems with product-form stochastic Petri nets Simonetta Balsamo Gian-Luca Dei Rossi Andrea Marin Dipartimento di Scienze Ambientali, Informatica e Statistica Universit` a Ca Foscari, Venezia ASMTA 13,

371 views • 19 slides
Randomness in Computing L ECTURE 27 Last time Stationary distributions Random walks on

Randomness in Computing L ECTURE 27 Last time Stationary distributions Random walks on

Randomness in Computing L ECTURE 27 Last time Stationary distributions Random walks on graphs Algorithm for - -PATH Today Sublinear algorithms Differential privacy 4/29/2020 Sofya Raskhodnikova;Randomness in

873 views • 34 slides
Quasi-stationary distributions and diffusion models in population dynamics P. Cattiaux, P.

Quasi-stationary distributions and diffusion models in population dynamics P. Cattiaux, P.

Quasi-stationary distributions and diffusion models in population dynamics P. Cattiaux, P. Collet, A. Lambert, S. Martinez, S. Mlard, J. San Martin, Ecole Polytechnique, Universidad de Chile, Universit Paris 6, Universit Toulouse III

892 views • 26 slides
Residence-time distributions as a measure for stochastic resonance WIAS Berlin, Germany

Residence-time distributions as a measure for stochastic resonance WIAS Berlin, Germany

W eierstra-Institut fr Angewandte Analysis und Stochastik Period of Concentration: Stochastic Climate Models MPI Mathematics in the Sciences, Leipzig, 23 May 1 June 2005 Barbara Gentz Residence-time distributions as a measure for

706 views • 25 slides
Spectral Analysis of Stationary Stochastic Process Hanxiao Liu hanxiaol@cs.cmu.edu February 20,

Spectral Analysis of Stationary Stochastic Process Hanxiao Liu hanxiaol@cs.cmu.edu February 20,

Spectral Analysis of Stationary Stochastic Process Hanxiao Liu hanxiaol@cs.cmu.edu February 20, 2016 1 / 16 Outline Stationarity The time-frequency dual Spectral representation Marginal/conditional dependencies Inference 2

336 views • 16 slides
Tail of stationary probability of Stochastic Dynamical systems Gerold Alsmeyer (University of

Tail of stationary probability of Stochastic Dynamical systems Gerold Alsmeyer (University of

Tail of stationary probability of Stochastic Dynamical systems Gerold Alsmeyer (University of Muenster) Sara Brofferio (University Paris Sud) Dariusz Buraczewski (University of Wroclaw) June 2016 Alsmeyer, Brofferio, Buraczewski Stationary

756 views • 46 slides
Stochastic Integration for non-Martingales Stationary Increment Processes Multi-color noise

Stochastic Integration for non-Martingales Stationary Increment Processes Multi-color noise

Introduction Main Result Applications Summary Stochastic Integration for non-Martingales Stationary Increment Processes Multi-color noise approach Daniel Alpay 1 Alon Kipnis 1 1 Department of Mathematics Ben-Gurion University of the Negev

1.13k views • 87 slides
What is accomplished by successful non stationary stochastic prediction? Glenn Shafer, Rutgers

What is accomplished by successful non stationary stochastic prediction? Glenn Shafer, Rutgers

Workshop on Robust Methods in Probability & Finance ICERM, Brown University, June 19 23, 2017 What is accomplished by successful non stationary stochastic prediction? Glenn Shafer, Rutgers University, www.glennshafer.com Answer: It

738 views • 49 slides
Numerical stochastic models of joint non-stationary and non-gaussian time series of weather

Numerical stochastic models of joint non-stationary and non-gaussian time series of weather

Numerical stochastic models of joint non-stationary and non-gaussian time series of weather elements for solving the statistical meteorology problems Ogorodnikov V.A., Khlebnikova E.I., Derenok K.V Institute of Computational Mathematics and

374 views • 36 slides
Lecture 6: Outline Recap the stationary distribution, and the vector ( k ) . The vector ( k

Lecture 6: Outline Recap the stationary distribution, and the vector ( k ) . The vector ( k

02407 Stochastic Processes Outline Outline Lecture 6: Outline Recap the stationary distribution, and the vector ( k ) . The vector ( k ) The vector ( k ) The stationary The stationary Markov chains on finite state spaces

585 views • 6 slides
Exponential Family Distributions CMSC 691 UMBC Exponential Family Form Exponential Family Form

Exponential Family Distributions CMSC 691 UMBC Exponential Family Form Exponential Family Form

Exponential Family Distributions CMSC 691 UMBC Exponential Family Form Exponential Family Form Support function Formally necessary, often irrelevant (e.g., Gaussian distributions), except when it isnt (e.g., Dirichlet

522 views • 35 slides
From stochastic resonance to stationary measures of McKean-Vlasov type equations S. Herrmann

From stochastic resonance to stationary measures of McKean-Vlasov type equations S. Herrmann

From stochastic resonance to stationary measures of McKean-Vlasov type equations S. Herrmann Institut de Mathmatiques de Bourgogne joint work with P. Imkeller (Berlin - D), D. Peithmann (Berlin - D), J. Tugaut (Bielefeld - D) October 6,

157 views • 14 slides
Foundations of Chemical Kinetics Lecture 23: The chemical master equation: Stationary

Foundations of Chemical Kinetics Lecture 23: The chemical master equation: Stationary

Foundations of Chemical Kinetics Lecture 23: The chemical master equation: Stationary distributions Marc R. Roussel Department of Chemistry and Biochemistry The stationary distribution In ordinary chemical kinetics, a closed chemical

533 views • 24 slides
Stationary Distributions of Markov Chains Will Perkins April 4, 2013 Back to Markov Chains

Stationary Distributions of Markov Chains Will Perkins April 4, 2013 Back to Markov Chains

Stationary Distributions of Markov Chains Will Perkins April 4, 2013 Back to Markov Chains Recall a discrete time, discrete space Markov Chain, is a process X n so that Pr[ X n = x n | X n 1 = x n 1 , . . . X 1 = x 1 ] = Pr[ X n = x n |

665 views • 20 slides
EC3062 ECONOMETRICS IDENTIFICATION OF ARMA MODELS A stationary stochastic process can be

EC3062 ECONOMETRICS IDENTIFICATION OF ARMA MODELS A stationary stochastic process can be

EC3062 ECONOMETRICS IDENTIFICATION OF ARMA MODELS A stationary stochastic process can be characterised, equivalently, by its autocovariance function or its partial autocovariance function. It can also be characterised by is spectral density

438 views • 32 slides
Towards an alternative characterisation of Boucherie product form Nigel Thomas University of

Towards an alternative characterisation of Boucherie product form Nigel Thomas University of

Towards an alternative characterisation of Boucherie product form Nigel Thomas University of Durham Contents Existing methods: Boucherie Motivation Mechanisms Characterisations Conditions for simple product form

306 views • 16 slides
American-style options, stochastic volatility, and degenerate parabolic variational inequalities

American-style options, stochastic volatility, and degenerate parabolic variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References American-style options, stochastic

1.2k views • 79 slides
Stochastic Optimization for Regularized Wasserstein Estimators ICML 2020 Francis Bach Quentin

Stochastic Optimization for Regularized Wasserstein Estimators ICML 2020 Francis Bach Quentin

Stochastic Optimization for Regularized Wasserstein Estimators ICML 2020 Francis Bach Quentin Berthet Marin Ballu Wasserstein Distance: a natural geometry for distributions How does one compute the distance between two data distributions? 1

305 views • 29 slides
Rao-Blackwellized Stochastic Gradients for Discrete Distributions Runjing (Bryan) Liu June 11,

Rao-Blackwellized Stochastic Gradients for Discrete Distributions Runjing (Bryan) Liu June 11,

Rao-Blackwellized Stochastic Gradients for Discrete Distributions Runjing (Bryan) Liu June 11, 2019 University of California, Berkeley Objective We fit a discrete latent variable model . Objective We fit a discrete latent variable

525 views • 24 slides
Semi-stationary reflection, stationary reflection and combinatorics Hiroshi Sakai (joint work

Semi-stationary reflection, stationary reflection and combinatorics Hiroshi Sakai (joint work

Semi-stationary reflection, stationary reflection and combinatorics Hiroshi Sakai (joint work with Boban Veli ckovi c) October 28, 2010 1.1 Stationary reflection For a cardinal 2 , the stationary reflection in [ ] is

157 views • 14 slides
Improved Boltzmann sampling for the Hadamard product of distributions Andrea Sportiello CNRS and

Improved Boltzmann sampling for the Hadamard product of distributions Andrea Sportiello CNRS and

Improved Boltzmann sampling for the Hadamard product of distributions Andrea Sportiello CNRS and LIPN, Universit e Paris Nord, Villetaneuse GASCom 2016, La Marana, Corsica June 4th 2016 Andrea Sportiello Sampling the Hadamard product of

617 views • 44 slides
Low Cost 3D Rotational Input Devices: the Stationary Spinball and the Stationary Spinball and

Low Cost 3D Rotational Input Devices: the Stationary Spinball and the Stationary Spinball and

Low Cost 3D Rotational Input Devices: the Stationary Spinball and the Stationary Spinball and the Mobile Soap3D Marcus Tnnis, Florian Echtler, Manuel Huber, Gudrun Klinker Fachgebiet Augmented Reality T Technische Universitt Mnchen,

660 views • 21 slides

More recommend