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Data-Informed Parameter Synthesis for Population Markov Chains

Part of the Lecture Notes in Computer Science book series (LNBI,volume 11705)

Abstract

Stochastic population models are widely used to model phenomena in different areas such as chemical kinetics or collective animal behaviour. Quantitative analysis of stochastic population models easily becomes challenging, due to the combinatorial propagation of dependencies across the population. The complexity becomes especially prominent when model’s parameters are not known and available measurements are limited. In this paper, we illustrate this challenge in a concrete scenario: we assume a simple communication scheme among identical individuals, inspired by how social honeybees emit the alarm pheromone to protect the colony in case of danger. Together, n individuals induce a population Markov chain with n parameters. In addition, we assume to be able to experimentally observe the states only after the steady-state is reached. In order to obtain the parameters of the individual’s behaviour, by utilising the data measurements for population, we combine two existing techniques. First, we use the tools for parameter synthesis for Markov chains with respect to temporal logic properties, and then we employ CEGAR-like reasoning to find the viable parameter space up to desired coverage. We report the performance on a number of synthetic data sets.

TP’s research is supported by the Ministry of Science, Research and the Arts of the state of Baden-Württemberg, and the DFG Centre of Excellence 2117 ‘Centre for the Advanced Study of Collective Behaviour’ (ID: 422037984), MH’s research is supported by Young Scholar Fund (YSF), project no. \(P83943018 FP 430\_/18\). MN’s research is supported by the Mentorship grant from the Zukunftskolleg. DŠ’s research is supported by the Czech Grant Agency grant no. GA18-00178S.

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Fig. 1.
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Fig. 4.

Notes

  1. 1.

    In our case, ‘help’ does not involve interaction between agents, - it is simultaneously broadcasted from an agent to all the others.

  2. 2.

    In general, the reachability probabilities for a pMC can be expressed by rational functions; In our case study, polynomials will suffice because the underlying transition system is acyclic.

  3. 3.

    If the coverage is not set below 50%.

References

  1. Alistarh, D., Gelashvili, R., Vojnović, M.: Fast and exact majority in population protocols. In: Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, pp. 47–56. ACM (2015)

    Google Scholar 

  2. Aspnes, J., Ruppert, E.: An introduction to population protocols. In: Garbinato, B., Miranda, H., Rodrigues, L. (eds.) Middleware for Network Eccentric and Mobile Applications, pp. 97–120. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-540-89707-1_5

    CrossRef  Google Scholar 

  3. Backenköhler, M., Bortolussi, L., Wolf, V.: Generalized method of moments for stochastic reaction networks in equilibrium. In: Bartocci, E., Lio, P., Paoletti, N. (eds.) CMSB 2016. LNCS, vol. 9859, pp. 15–29. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45177-0_2

    CrossRef  MATH  Google Scholar 

  4. Backenkohler, M., Bortolussi, L., Wolf, V.: Moment-based parameter estimation for stochastic reaction networks in equilibrium. IEEE/ACM Trans. Comput. Biol. Bioinf. 15(4), 1180–1192 (2018)

    CrossRef  Google Scholar 

  5. Bartocci, E., Bortolussi, L., Nenzi, L., Sanguinetti, G.: System design of stochastic models using robustness of temporal properties. Theor. Comput. Sci. 587, 3–25 (2015)

    MathSciNet  CrossRef  Google Scholar 

  6. Bortolussi, L., Cardelli, L., Kwiatkowska, M., Laurenti, L.: Approximation of probabilistic reachability for chemical reaction networks using the linear noise approximation. In: Agha, G., Van Houdt, B. (eds.) QEST 2016. LNCS, vol. 9826, pp. 72–88. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-43425-4_5

    CrossRef  MATH  Google Scholar 

  7. Bortolussi, L., Hillston, J., Latella, D., Massink, M.: Continuous approximation of collective system behaviour: a tutorial. Perform. Eval. 70(5), 317–349 (2013)

    CrossRef  Google Scholar 

  8. Bortolussi, L., Sanguinetti, G.: Learning and designing stochastic processes from logical constraints. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 89–105. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40196-1_7

    CrossRef  Google Scholar 

  9. Bortolussi, L., Silvetti, S.: Bayesian statistical parameter synthesis for linear temporal properties of stochastic models. In: Beyer, D., Huisman, M. (eds.) TACAS 2018. LNCS, vol. 10806, pp. 396–413. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-89963-3_23

    CrossRef  Google Scholar 

  10. Brim, L., Češka, M., Dražan, S., Šafránek, D.: Exploring parameter space of stochastic biochemical systems using quantitative model checking. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 107–123. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39799-8_7

    CrossRef  Google Scholar 

  11. Češka, M., Dannenberg, F., Paoletti, N., Kwiatkowska, M., Brim, L.: Precise parameter synthesis for stochastic biochemical systems. Acta Informatica 54(6), 589–623 (2017)

    MathSciNet  CrossRef  Google Scholar 

  12. Daca, P., Henzinger, T.A., Křetínský, J., Petrov, T.: Faster statistical model checking for unbounded temporal properties. ACM Trans. Comput. Log. (TOCL) 18(2), 12 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Daws, C.: Symbolic and parametric model checking of discrete-time Markov Chains. In: Liu, Z., Araki, K. (eds.) ICTAC 2004. LNCS, vol. 3407, pp. 280–294. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-31862-0_21

    CrossRef  MATH  Google Scholar 

  14. de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_24

    CrossRef  Google Scholar 

  15. Dehnert, C., et al.: PROPhESY: a PRObabilistic ParamEter SYnthesis tool. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 214–231. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21690-4_13

    CrossRef  Google Scholar 

  16. Dehnert, C., Junges, S., Katoen, J.-P., Volk, M.: A Storm is coming: a modern probabilistic model checker. In: Majumdar, R., Kunčak, V. (eds.) CAV 2017. LNCS, vol. 10427, pp. 592–600. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63390-9_31

    CrossRef  Google Scholar 

  17. Dorigo, M., Birattari, M., Blum, C., Clerc, M., Stützle, T., Winfield, A.: Ant Colony Optimization and Swarm Intelligence, vol. 5217. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  18. Kluyver, T., et al.: Jupyter notebooks - a publishing format for reproducible computational workflows. In: Positioning and Power in Academic Publishing: Players, Agents and Agendas, pp. 87–90. IOS Press (2016)

    Google Scholar 

  19. Giacobbe, M., Guet, C.C., Gupta, A., Henzinger, T.A., Paixão, T., Petrov, T.: Model checking the evolution of gene regulatory networks. Acta Informatica 54(8), 765–787 (2017)

    MathSciNet  CrossRef  Google Scholar 

  20. Giardina, I.: Collective behavior in animal groups: theoretical models and empirical studies. HFSP J. 2(4), 205–219 (2008)

    CrossRef  Google Scholar 

  21. Hansen, L.P.: Large sample properties of generalized method of moments estimators. Econometrica 50, 1029–1054 (1982)

    MathSciNet  CrossRef  Google Scholar 

  22. Hillston, J.: Challenges for quantitative analysis of collective adaptive systems. In: Abadi, M., Lluch Lafuente, A. (eds.) TGC 2013. LNCS, vol. 8358, pp. 14–21. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-05119-2_2

    CrossRef  Google Scholar 

  23. Jansen, N., et al.: Accelerating parametric probabilistic verification. In: Norman, G., Sanders, W. (eds.) QEST 2014. LNCS, vol. 8657, pp. 404–420. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10696-0_31

    CrossRef  Google Scholar 

  24. Katoen, J.-P.: The probabilistic model checking landscape. In: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, pp. 31–45. ACM (2016)

    Google Scholar 

  25. Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22110-1_47

    CrossRef  Google Scholar 

  26. Loreti, M., Hillston, J.: Modelling and analysis of collective adaptive systems with CARMA and its tools. In: Bernardo, M., De Nicola, R., Hillston, J. (eds.) SFM 2016. LNCS, vol. 9700, pp. 83–119. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-34096-8_4

    CrossRef  Google Scholar 

  27. Mai, M., et al.: Monitoring pre-seismic activity changes in a domestic animal collective in central Italy. In: EGU General Assembly Conference Abstracts, vol. 20, p. 19348 (2018)

    Google Scholar 

  28. Nouvian, M., Reinhard, J., Giurfa, M.: The defensive response of the honeybee Apis mellifera. J. Exp. Biol. 219(22), 3505–3517 (2016)

    CrossRef  Google Scholar 

  29. Daca, P., Henzinger, T.A., Křetínský, J., Petrov, T.: Faster statistical model checking for unbounded temporal properties. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 112–129. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49674-9_7

    CrossRef  Google Scholar 

  30. Polgreen, E., Wijesuriya, V.B., Haesaert, S., Abate, A.: Data-efficient Bayesian verification of parametric Markov Chains. In: Agha, G., Van Houdt, B. (eds.) QEST 2016. LNCS, vol. 9826, pp. 35–51. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-43425-4_3

    CrossRef  MATH  Google Scholar 

  31. Quatmann, T., Dehnert, C., Jansen, N., Junges, S., Katoen, J.-P.: Parameter synthesis for Markov models: faster than ever. In: Artho, C., Legay, A., Peled, D. (eds.) ATVA 2016. LNCS, vol. 9938, pp. 50–67. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-46520-3_4

    CrossRef  MATH  Google Scholar 

  32. Schnoerr, D., Sanguinetti, G., Grima, R.: Approximation and inference methods for stochastic biochemical Kinetics–a tutorial review. J. Phys. A: Math. Theor. 50(9), 093001 (2017)

    MathSciNet  CrossRef  Google Scholar 

  33. Shorter, J.R., Rueppell, O.: A review on self-destructive defense behaviors in social insects. Insectes Soc. 59(1), 1–10 (2012)

    CrossRef  Google Scholar 

  34. Sokolova, A., de Vink, E.P.: Probabilistic automata: system types, parallel composition and comparison. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 1–43. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24611-4_1

    CrossRef  Google Scholar 

  35. Stoelinga, M.: An introduction to probabilistic automata. Bull. EATCS 78(176–198), 2 (2002)

    MathSciNet  MATH  Google Scholar 

  36. Česka, M., Šafránek, D., Dražan, S., Brim, L.: Robustness analysis of stochastic biochemical systems. PLoS ONE 9(4), 1–23 (2014)

    CrossRef  Google Scholar 

  37. Wu, S.-H., Smolka, S.A., Stark, E.W.: Composition and behaviors of probabilistic I/O automata. Theor. Comput. Sci. 176(1–2), 1–38 (1997)

    MathSciNet  CrossRef  Google Scholar 

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Correspondence to Tatjana Petrov .

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A Performance Comparison

A Performance Comparison

Table 2. Runtime comparison for parameter space refinement using our algorithm – Alg1-3. Computation times for respective population size, two/multi param case, data, algorithm, and setting. All models used in the comparison are semisynchronous and the results were computed using all-in-one approach. Computation time in seconds.
Table 3. Runtime comparison of our algorithm, PRISM, and Storm. Our algorithm, Alg3, and PRISM were using semisynchronous models and Storm used asynchronous models. Rows for Alg3 contain the time used to compute polynomials \(F(k)\) (done by PRISM), plus the time needed to refine the space. By N/A we denote cases when Storm was unable to return a result due to a technical problem. We used Data set 1.

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Hajnal, M., Nouvian, M., Šafránek, D., Petrov, T. (2019). Data-Informed Parameter Synthesis for Population Markov Chains. In: Češka, M., Paoletti, N. (eds) Hybrid Systems Biology. HSB 2019. Lecture Notes in Computer Science(), vol 11705. Springer, Cham. https://doi.org/10.1007/978-3-030-28042-0_10

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