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Data-Informed Parameter Synthesis for Population Markov Chains

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Hybrid Systems Biology (HSB 2019)

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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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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%.

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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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