Pathwise Sensitivity Analysis in Transient Regimes

  • Georgios Arampatzis
  • Markos A. Katsoulakis
  • Yannis Pantazis
Part of the Mathematical Engineering book series (MATHENGIN)


The instantaneous relative entropy (IRE) and the corresponding instantaneous Fisher information matrix (IFIM) for transient stochastic processes are presented in this paper. These novel tools for sensitivity analysis of stochastic models serve as an extension of the well known relative entropy rate (RER) and the corresponding Fisher information matrix (FIM) that apply to stationary processes. Three cases are studied here, discrete-time Markov chains, continuous-time Markov chains and stochastic differential equations. A biological reaction network is presented as a demonstration numerical example.


Markov Chain Stochastic Differential Equation Relative Entropy Markov Chain Model Fisher Information Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The work of the authors was supported by the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract No. DE-SC0010723 and by the European Union (European Social Fund) and Greece (National Strategic Reference Framework), under the THALES Program, grant AMOSICSS.


  1. 1.
    Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S (2008) Global sensitivity analysis. The primer. Wiley, New YorkMATHGoogle Scholar
  2. 2.
    DiStefano III J (2013) Dynamic systems biology modeling and simulation. Elsevier, New YorkGoogle Scholar
  3. 3.
    Glynn PW (1990) Likelihood ratio gradient estimation for stochastic systems. Commun ACM 33(10):75–84CrossRefGoogle Scholar
  4. 4.
    Nakayama M, Goyal A, Glynn PW (1994) Likelihood ratio sensitivity analysis for Markovian models of highly dependable systems. Stoch Models 10:701–717Google Scholar
  5. 5.
    Plyasunov S, Arkin AP (2007) Efficient stochastic sensitivity analysis of discrete event systems. J Comput Phys 221:724–738CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Kim D, Debusschere BJ, Najm HN (2007) Spectral methods for parametric sensitivity in stochastic dynamical systems. Biophys J 92:379–393CrossRefGoogle Scholar
  7. 7.
    Rathinam M, Sheppard PW, Khammash M (2010) Efficient computation of parameter sensitivities of discrete stochastic chemical reaction networks. J Chem Phys 132(1–13):034103CrossRefGoogle Scholar
  8. 8.
    Anderson David F (2012) An efficient finite difference method for parameter sensitivities of continuous-time Markov chains. SIAM J Numer Anal 50(5):2237–2258CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Sheppard PW, Rathinam M, Khammash M (2012) A pathwise derivative approach to the computation of parameter sensitivities in discrete stochastic chemical systems. J Chem Phys 136(3):034115CrossRefGoogle Scholar
  10. 10.
    Meskine H, Matera S, Scheffler M, Reuter K, Metiu H (2009) Examination of the concept of degree of rate control by first-principles kinetic Monte Carlo simulations. Surf Sci 603(10–12):1724–1730CrossRefGoogle Scholar
  11. 11.
    Baiesi M, Maes C, Wynants B (2009) Nonequilibrium linear response for Markov dynamics I: jump processes and overdamped diffusions. J Stat Phys 137:1094CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Baiesi M, Maes C, Boksenbojm E, Wynants B (2010) Nonequilibrium linear response for markov dynamics, II: Inertial dynamics. J Stat Phys 139:492Google Scholar
  13. 13.
    Pantazis Y, Katsoulakis M (2013) A relative entropy rate method for path space sensitivity analysis of stationary complex stochastic dynamics. J Chem Phys 138(5):054115CrossRefGoogle Scholar
  14. 14.
    Dupuis P, Katsoulakis MA, Pantazis Y, Plecháč P Sesnitivity bounds and error estimates for stochastic models (in Preparation)Google Scholar
  15. 15.
    Arampatzis G, Pantazis Y, Katsoulakis MA Accelerated sensitivity analysis in high-dimensional stochastic reaction networks. Submitted to PLoS ONEGoogle Scholar
  16. 16.
    Kullback S (1959) Information theory and statistics. Wiley, New YorkMATHGoogle Scholar
  17. 17.
    Cover T, Thomas J (1991) Elements of information theory. Wiley, New YorkCrossRefMATHGoogle Scholar
  18. 18.
    Kipnis C, Landim C (1999) Scaling limits of interacting particle systems. Springer, BerlinCrossRefMATHGoogle Scholar
  19. 19.
    Avellaneda M, Friedman CA, Holmes R, Samperi DJ (1997) Calibrating volatility surfaces via relative-entropy minimization. Soc Sci Res NetwGoogle Scholar
  20. 20.
    Liu HB, Chen W, Sudjianto A (2006) Relative entropy based method for probabilistic sensitivity analysis in engineering design. J Mech Des 128:326–336CrossRefGoogle Scholar
  21. 21.
    Limnios N, Oprisan G (2001) Semi-Markov processes and reliability. Springer, BerlinCrossRefMATHGoogle Scholar
  22. 22.
    Abramov RV, Grote MJ, Majda AJ (2005) Information theory and stochastics for multiscale nonlinear systems., CRM monograph series. American Mathematical Society, ProvidenceGoogle Scholar
  23. 23.
    Liptser RS, Shiryaev AN (1977) Statistics of random processes: I & II. Springer, New YorkCrossRefGoogle Scholar
  24. 24.
    Oksendal B (2000) Stochastic differential equations: an introduction with applications. Springer, New YorkGoogle Scholar
  25. 25.
    Tsourtis A, Pantazis Y, Harmandaris V, Katsoulakis MA Parametric sensitivity analysis for stochastic molecular systems using information theoretic metrics. Submitted to J Chem PhysGoogle Scholar
  26. 26.
    Kholodenko BN, Demin OV, Moehren G, Hoek J (1999) Quantification of short term signaling by the epidermal growth factor receptor. J Biol Chem 274(42):30169–30181CrossRefGoogle Scholar
  27. 27.
    Moghal N, Sternberg PW (1999) Multiple positive and negative regulators of signaling by the EGF receptor. Curr Opin Cell Biol 11:190–196CrossRefGoogle Scholar
  28. 28.
    Hackel PO, Zwick E, Prenzel N, Ullrich A (1999) Epidermal growth factor receptors: critical mediators of multiple receptor pathways. Curr Opin Cell Biol 11:184–189CrossRefGoogle Scholar
  29. 29.
    Schoeberl B, Eichler-Jonsson C, Gilles ED, Muller G (2002) Computational modeling of the dynamics of the MAP kinase cascade activated by surface and internalized EGF receptors. Nat Biotechnol 20:370–375CrossRefGoogle Scholar
  30. 30.
    Casella G, Berger RL (2002) Statistical inference. Duxbury advanced series in statistics and decision sciencesThomson Learning, LondonGoogle Scholar
  31. 31.
    Kay SM (1993) Fundamentals of statistical signal processing: estimation theory. Prentice-Hall, Englewood CliffsMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Georgios Arampatzis
    • 1
  • Markos A. Katsoulakis
    • 1
  • Yannis Pantazis
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA

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