Local diffusion models for stochastic reacting systems: Estimation issues in equation-free numerics

Molecular Simulation

Volumn 33, Issue 9-10, 2007, Pages 713-731

  Calderon, C P ^a  

^a Princeton University   (United States)

Author keywords

Equation free computation; Local diffusion process estimation; Maximum likelihood; Stochastic reaction networks

Indexed keywords

COMPUTER SIMULATION; MATHEMATICAL MODELS; MAXIMUM LIKELIHOOD; RANDOM PROCESSES;

EQUATION-FREE COMPUTATION; LOCAL DIFFUSION PROCESS ESTIMATION; STOCHASTIC REACTION NETWORKS;

DIFFUSION;

EID: 34548306780     PISSN: 08927022     EISSN: 10290435     Source Type: Journal    
DOI: 10.1080/08927020701344740     Document Type: Conference Paper

References (64)

1. Y. Aït-Sahalia. Closed-form likelihood expansions for multivariate diffusions. Technical. Report (2001).
2. Y. Aït-Sahalia. Maximum-likelihood estimation of discretelysampled diffusions: a closed-form approximation approach. Econometrica, 70, 223 (2002).
3. Y. Aït-Sahalia, R. Kimmel. Estimating affine multifactor term structure models using closed-form likelihood expansions. NBER Technical Working Papers 0286, National Bureau of Economic Research, Inc., December. Available at http://ideas.repec.org/p/nbr/ nberte/0286.html (2002).
4. A. Amadei, A.B.M. Luissen, H.J.C. Berendsen. Essential dynamics of proteins. Proteins, 17, 412 (1993).
5. G. Bakshi, N. Ju. A refinement to Aït-Sahalia's (2002) "Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach". J. Business, 78, 2037 (2005).
6. G. Bakshi, N. Ju, H. Ou-Yang. Estimation of continuous-time models with an application to equity volatility dynamics. J. Finan. Econ., 82, 227 (2006).
7. K. Ball, T. Kurtz, L. Popovic, G. Rempala. Asymptotic analysis of multiscale approximations to reaction networks. http: //www. citebase .org/abstract?id = oai :arXiv.org:math/0508015>(2005).
8. B.M. Bibby, M. Sørensen. Martingale estimation functions for discretely observed diffusion processes. Bernoulli, 1, 17 (1995).
9. P.J. Bickel, KJ. Doksum. Mathematical Statistics: Basic Ideas and Selected Topics, Prentice Hall, Upper Saddle River, NJ (2001).
10. M. Bradnt, P. Santa-Clara. Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets. J, Finan, Econ., 63, 161 (2002).
11. C.P. Calderón. Fitting effective diffusion models to data associated with a "glassy potential": estimation, classical inference procedures and some heuristics. SIAM MMS, (to appear).
12. C.P. Calderon, G.A. Tsekouras, A. Provata, I.G. Kevrekidis. Coarse graining the Cyclic Lotka-Volterra Model: SSA and local maximum likelihood estimation. In Proceedings of Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, A. Gorban, N. Kazantzis, I. Kevrekidis, H. Ottinger, C Theodoropoulos (Eds.), Workshop at U. Leicester, Leicester, UK, August 24-26, Springer-Verlag, Heidelberg (2005).
13. C.P. Calderon. On the use of local diffusion models in potential of mean force computations. J. Chem. Phys., 126, 084106 (2007).
14. Y. Cao, D.T. Gillespie, L.R. Petzold. The slow-scale stochastic simulation algorithm. J. Chem. Phys., 122, 0141.16 (2005).
15. A.S. Dalalyan, Y.A. Kutoyants. Asymptotically efficient estimation of the derivative of the invariant density. Stat. Inference Stoch. Process., 6, 89 (2003).
16. F.X, Diebold, T. Gunther, A. Tay. Evaluating density forecasts with applications to financial risk management. Int. Econ. Rev., 39, 863 (1998).
17. W.E.D. Liu, E. Vanden-Eijnden. Analysis of multiscale methods for stochastic differential equations. Corran, Pure Appl. Math., 58, 1544 (2005).
18. W.E.D. Liu, E. Vanden-Eijnden. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem, Phys., 123, 194107 (2005).
19. E. Vanden-Eijnden. Numerical techniques for multi-scale dynami-cal systems with stochastic effects. Comm. Pure Appl. Math., 1, 385 (2003).
20. M. El-Ansary, H. Khalil. On the interplay of singular perturbations and wide-band stochastic fluctuations. SMM J. Control Optim., 24, 83 (1986).
21. S. Ethier, T. Kurtz. Markov Processes: Characterization and Convergence, Wiley-Interscience, New York (1986).
22. E. Fournié, J.L. Lasry, P.L. Lions, J. Lebuchoux, N. Touzi. Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch., 3, 391 (1999).
23. E. Fournié, J.L. Lasry, J. Lebuchoux, P.L. Lions. Applications of Malliavin calculus to Monte Carlo methods in finance i.i. Finance Stoch., 5, 201 (2001).
24. A.R. Gallant, G. Tauchen. Which moments to match? Econometric Theory, 12, 657 (1996).
25. C.W. Gardiner. Handbook of Stochastic Models, Springer-Verlag, Berlin (1985).
26. C.W. Gear. Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Upper Saddle River, NJ (1971).
27. C.W. Gear, T.J. Kaper, I.G. Kevrekidis, A. Zagaris. Projecting to a slow manifold: singularly perturbed systems and legacy codes. SIAM J. Appl. Dyn. Syst., 4, 711 (2005).
28. D.T. Gillespie. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81, 2340 (1.977).
29. D.T. Gillespie, L.R. Petzold. Improved leap-size selection for accelerated stochastic simulation. J. Chem. Phys., 119,8229 (2003).
30. J. Gullingsrud, R. Braun, K. Schulten. Reconstructing potentials of mean force through time series analysis of steered molecular dynamics simulations. J. Comp. Phys., 151, 190 (1.999).
31. J.D. Hamilton. Time Serles Analysis, Princeton University Press, Princeton, NJ (1994).
32. E.L. Haseltine, J.B. Rawlings. Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys., 117, 6959 (2002).
33. Y. Hong, H. Li. Nonparametric specification testing for continuoustime models with applications to term structure of interest rates. Rev. Finan. Stud., 1.8, 37 (2005).
34. G. Hummer, I.G. Kevrekidis. Coarse molecular dynamics of a peptide fragment: free energy, kinetics, and long-time dynamics computations. J. Chem. Phys., 118, 10762 (2003).
35. P. Jeganathan. Some aspects of asymptotic theory with applications to time series models. Econometric Theory, 11, 818 (1995).
36. B. Jensen, R. Poulsen. Transition densities of diffusion processes: numerical comparison of approximation techniques. J. Derivatives, 9, 18 (2002).
37. I.G. Kevrekidis, C.W. Gear, J.M. Hyman, P.G. Kevrekidis, O. Runborg, K. Theodoropoulos. Equation-free coarse-grained multiscale computation: enabling microscopic simulators to perform system-level tasks. Comm. Math. Sci., 1, 715 (2003).
38. I.G. Kevrekidis, C.W. Gear, G. Hummer. Equation-free: the computer-aided analysis of complex multiscale systems. AIChE J., 50, 474 (2004).
39. P. Kloeden, E. Platen. Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin (1992).
40. D.I. Kopelevich, A.Z. Panagiotopoulos, I.G. Kevrekidis. Coarsegrained kinetic computations for rare events: application to micelle formation. J. Chem. Phys., 122, 044908 (2005).
41. S. Kullback, R.A. Leibler. On information and sufficiency. Ann..Math. Stat., 22, 79(1951).
42. R. Kupferman, A.M.. Stuart. Fitting sde models to nonlinear Kac-Zwanzig heat bath models. Phys. D, 199, 279 (2004).
43. H. Kushner. Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhauser, Boston (1990).
44. J.D. Lambert. Numerical Methods for Ordinary Differential Systems: The Initial Value Problem, Wiley, New York (199.1).
45. L. Le Cam, G.L. Yang. Asymptotics in Statistics: Some Basic Concepts, Springer-Verlag, New-York (2000).
46. P.K. Maini, M.A. Burke, J.D. Murray. On the quasi-steady-state assumption applied to Michaelis-Menten and suicide substrate reactions with, diffusion. Phil. Trans. R. Soc., 337A, 299 (1.991).
47. A. Makeev, D. Maroudas, I.G. Kevrekidis. Coarse stability and bifurcation analysis using stochastic simulators: kinetic Monte Carlo examples. J. Chem. Phys., 116, 10083 (2002).
48. A. Makeev, D. Maroudas, A.Z. Panagiotopoulos, I.G. Kevrekidis. Coarse bifurcation analysis of kinetic Monte Carlo simulations: a lattice-gas model with lateral interactions. J Chem. Phys., 117, 8229 (2002).
49. J.D. Murray. Mathematical Biology I: An Introduction, SpringerVerlag, New-York (2004).
50. D. Nualart. The Malliavin Calculus and Related Topics, Springer, New York (1995).
51. S. Park, K. Schulten. Calculating potentials of mean force from. steered molecular dynamics simulations. J. Chem. Phys., 120, 5946 (2004).
52. G.A. Pavliotis, A.M. Stuart. Parameter estimation for multiscale diffusions, http: //www.citebase.org/cgi-bin/cita-tions?id= oai-.arXiv. org¡math/0603668 (2006).
53. A.R. Pedersen. A. new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Stat., 22, 55 (1995).
54. P.E. Protter. Stochastic Integration and Differential Equations, Springer, Berlin (2005).
55. C. Rao, A. Arkin. Stochastic chemical, kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm. J. Chem. Phys., 118, 4999 (2003).
56. R. Rico-Martinez, C.W. Gear, I.G. Kevrekidis. Coarse projective kMC integration: forward/reverse initial and boundary value problems. J. Comput. Phys., 196, 474 (2004).
57. H. Risken. The Fokker-Planck Equation, Springer-Verlag, Berlin (1984).
58. E. Rusli, TO. Drews, D.L. Ma, R.C. Alkire, R.D. Braatz. Robust nonlinear feedback-feedforward control of a coupled kinetic Monte Carlo finite-difference simulation. J. Process Control, 16, 409 (2006).
59. E. Schutz, N. Hartmann, Y. Kevrekidis, R. Imbihl. Microchemical engineering of catalytic reactions. Catal. Lett., 54, 181 (1998).
60. R.U. Seydel. Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Springer-Verlag, New York (1,994).
61. J.M.T. Thompson, H.B. Stewart. Nonlinear Dynamics and Chaos, Wiley, West Sussex, UK (2002).
62. G.M. Torrie, J.P. Valleau. Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling. J. Comp. Phys., 23, 187 (1977).
63. A. van der Vaart. Asymptotic Statistics, Cambridge University Press, New York (1.998).
64. H. White. Maximum likelihood estimation of misspecified models. Econometrica, 50, 1 (1982).