Jump to ContentJump to Main Navigation
Show Summary Details
More options …

The International Journal of Biostatistics

Ed. by Chambaz, Antoine / Hubbard, Alan E. / van der Laan, Mark J.

2 Issues per year

IMPACT FACTOR 2016: 0.500
5-year IMPACT FACTOR: 0.862

CiteScore 2016: 0.42

SCImago Journal Rank (SJR) 2016: 0.488
Source Normalized Impact per Paper (SNIP) 2016: 0.467

Mathematical Citation Quotient (MCQ) 2016: 0.09

See all formats and pricing
More options …

Parameter Estimation of a Two-Colored Urn Model Class

Line Chloé Le Goff / Philippe Soulier
Published Online: 2017-03-25 | DOI: https://doi.org/10.1515/ijb-2016-0029


Though widely used in applications, reinforced random walk on graphs have never been the subject of a valid statistical inference. We develop in this paper a statistical framework for a general two-colored urn model. The probability to draw a ball at each step depends on the number of balls of each color and on a multidimensional parameter through a function, called choice function. We introduce two estimators of the parameter: the maximum likelihood estimator and a weighted least squares estimator which is less efficient, but is closer to the calibration techniques used in the applied literature. In general, the model is an inhomogeneous Markov chain and this property makes the estimation of the parameter impossible on a single path, even if it were infinite. Therefore we assume that we observe i.i.d. experiments, each of a predetermined finite length. This is coherent with the usual experimental set-ups. We apply the statistical framework to a real life experiment: the selection of a path among pre-existing channels by an ant colony. We performed experiments, which consisted of letting ants pass through the branches of a fork. We consider the particular urn model proposed by J.-L. Deneubourg et al. in 1990 to describe this phenomenon. We simulate this model for several parameter values in order to assess the accuracy of the MLE and the WLSE. Then we estimate the parameter from the experimental data and evaluate confident regions with Bootstrap algorithms. The findings of this paper do not contradict the biological literature, but give statistical significance to the values of the parameter found therein.

Keywords: ant behavior; asymptotic proprieties of estimators; Deneubourg’s choice function; reinforced random walk parameter estimation; urn model parameter estimation


  • 1.

    Pólya G. Sur quelques points de la théorie des probabilités. Ann IHP 1931;1:117–61.Google Scholar

  • 2.

    Deneubourg JL, Aron S, Goss S, Pasteels J. The self-organizing exploratory pattern of the argentine ant. J Insect Behav 1990;3:159–68.CrossrefGoogle Scholar

  • 3.

    Tarrès P. Localization of reinforced random walks, 2011. Available at 5536 arXiv:1103.5536 http://arxiv.org/abs/1103.Google Scholar

  • 4.

    Davis B. Reinforced random walk. Probab Theory Related Fields 1990;84:203–29.Google Scholar

  • 5.

    Vittori K, Talbot G, Gautrais J, Fourcassié V, Araujo A, Theraulaz G. Path efficiency of ant foraging trails in an artificial network. J Theor Biol 2006;239:507–15.Google Scholar

  • 6.

    Garnier S, Guérécheau A, Combe M, Fourcassié V, Theraulaz G. Path selection and foraging efficiency in argentine ant transport networks. Behav Ecol Sociobiol 2009;63:1167–79.CrossrefWeb of ScienceGoogle Scholar

  • 7.

    Thienen W, Metzler D, Choe DH, Witte V. Pheromone communication in ants: a detailed analysis of concentration-dependent decisions in three species. Behav Ecol Sociobiol 2014;68:1611–27.Web of ScienceCrossrefGoogle Scholar

  • 8.

    Khanin K, Khanin R. A probabilistic model for the establishment of neuron polarity. J Math Biol 2001;42:26–40.CrossrefGoogle Scholar

  • 9.

    Jeanson R, Ratnieks F, Deneubourg JL. Pheromone trail decay rates on different substrates in the pharaohs ant, monomorium pharaonis. Physiol Entomol 2003;28:192–8.CrossrefGoogle Scholar

  • 10.

    Pemantle R. A survey of random processes with reinforcement. Probab Surv 2007;4:1–79.CrossrefGoogle Scholar

  • 11.

    Arganda S, Nicolis S, Perochain A, Péchabadens C, Latil G, Dussutour A. Collective choice in ants: The role of protein and carbohydrates ratios. J Insect Physiol 2014;69:19–26.Web of ScienceCrossrefGoogle Scholar

  • 12.

    Aron S, Deneubourg JL, Goss S, Pasteels J. Functional self-organisation illustrated by inter-nest traffic in ants : the Case of the Argentine ant In: Alt W, Hoffmann G, editors. Biological Motion: Proceedings of a Workshop held in Königswinter, Germany, March 16–19, 1989. Berlin, Heidelberg: Springer; 1990:533–47.

  • 13.

    Beckers R, Deneubourg JL, Goss S. Modulation of trail laying in the ant lasius niger (hymenoptera: formicidae) and its role in the collective selection of a food source. J Insect Behav 1993;6:751–9.CrossrefGoogle Scholar

  • 14.

    Dussutour A, Deneubourg JL, Fourcassié V. Amplification of individual preferences in a social context: the case of wall-following in ants. Proc R Soc London Ser B 2005;272:705–14.Google Scholar

  • 15.

    Nicolis S, Deneubourg JL. Emerging patterns and food recruitment in ants: an analytical study. J Theor Biol 1999;198:575–92.Google Scholar

  • 16.

    Nicolis S, Self-organization Dussutour A., collective decision making and resource exploitation strategies in social insects. Eur Phys J B 2008;65:379–85.CrossrefWeb of ScienceGoogle Scholar

  • 17.

    Freedman D. Bernard Friedman’s urn. Ann Math Stat 1965;36:956–70.CrossrefGoogle Scholar

  • 18.

    Beckers R, Deneubourg JL, Goss S. Trails and u-turns in the selection of a path by the ant lasius niger. J Theor Biol 1992;159:397–415.Google Scholar

  • Pasteels J, Deneubourg JL, Goss S. Transmission and amplification of information in a changing environment: the case of insect societies. In: Prigogine I, Sanglier M, editors. Laws of nature and human conduct: Specificities and unifying themes. Brussels: Gordes, Brussels; 1987.Google Scholar

  • Wasserman L. All of statistics: a concise course in statistical inference. New York: Springer; 2004.Google Scholar

  • 21.

    Van Vorhis Key S, Baker T. Trail-following responses of the argentine ant, iridomyrmex humilis (mayr), to a synthetic trail pheromone component and analogs. J Chem Ecol 1982;8:3–14.CrossrefGoogle Scholar

  • 22.

    Aron S, Pasteels J, Deneubourg JL. Trail-laying behaviour during exploratory recruitment in the argentine ant, iridomyrmex humilis (mayr). Biol Behav 1989;14:207–17.Google Scholar

  • Le Goff LC, Raimond O. Vertex reinforced non-backtracking random walks: an example of path formation, 2015. Available at: http://arxiv.org/abs/1506.01239, arXiv:1506.01239.

  • 24.

    Dacunha-Castelle D, Duflo M. Probability and statistics. vol. II. New York: Springer; 1986.Google Scholar

About the article

Published Online: 2017-03-25

Citation Information: The International Journal of Biostatistics, ISSN (Online) 1557-4679, DOI: https://doi.org/10.1515/ijb-2016-0029.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston. Copyright Clearance Center

Comments (0)

Please log in or register to comment.
Log in