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Nonparametric likelihood based estimation of linear filters for point processes

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Abstract

We consider models for multivariate point processes where the intensity is given nonparametrically in terms of functions in a reproducing kernel Hilbert space. The likelihood function involves a time integral and is consequently not given in terms of a finite number of kernel evaluations. The main result is a representation of the gradient of the log-likelihood, which we use to derive computable approximations of the log-likelihood and the gradient by time discretization. These approximations are then used to minimize the approximate penalized log-likelihood. For time and memory efficiency the implementation relies crucially on the use of sparse matrices. As an illustration we consider neuron network modeling, and we use this example to investigate how the computational costs of the approximations depend on the resolution of the time discretization. The implementation is available in the R package ppstat.

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References

  • Andersen, P.K., Borgan, Ø., Gill, R.D., Keiding, N.: Statistical models based on counting processes. Springer Series in Statistics. Springer, New York (1993)

  • Anderson, D., Kurtz, T.: Continuous time markov chain models for chemical reaction networks. In: Koeppl, H., Setti, G., di Bernardo, M., Densmore, D. (eds.) Design and analysis of biomolecular circuits, pp. 3–42. Springer, New York (2011). doi:10.1007/978-1-4419-6766-41

  • Berlinet, A., Thomas-Agnan, C.: Reproducing kernel Hilbert spaces in probability and statistics. Kluwer Academic Publishers, Boston (2004)

  • Bishop, C.M.: Pattern recognition and machine learning (information science and statistics). Springer, Secaucus (2006)

    Google Scholar 

  • Bowsher, C.G.: Stochastic kinetic models: dynamic independence, modularity and graphs. Ann. Stat. 38(4), 2242–2281 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • Brémaud, P., Massoulié, L.: Stability of nonlinear Hawkes processes. Ann. Probab. 24(3), 1563–1588 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Burnham, K.P., Anderson, D.R.: Model selection and multimodel inference. A practical information-theoretic approach, Second edn. Springer, New York (2002)

  • Claeskens, G., Hjort, N.L.: Model selection and model averaging. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  • Gjessing, H.K., Røysland, K., Pena, E.A., Aalen, O.O.: Recurrent events and the exploding Cox model. Lifetime Data Anal 16(4), 525–546 (2010)

    Google Scholar 

  • Hansen, N.R.: Penalized maximum likelihood estimation for generalized linear point processes pp. 1–33 (2013). http://arxiv.org/abs/1003.0848

  • Hastie, T., Tibshirani, R., Friedman, J.: The elements of statistical learning, Data mining, inference, and prediction, second edn. Springer Series in Statistics. Springer, New York (2009). doi:10.1007/978-0-387-84858-7

  • Hautsch, N.: Modelling irregularly spaced financial data, Lecture Notes in Economics and Mathematical Systems, vol. 539. Springer-Verlag, Berlin (2004)

  • Hawkes, A.G.: Spectra of some self-exciting and mutually exciting point processes. Biometrika 58(1), 83–90 (1971). http://www.jstor.org/stable/2334319

    Google Scholar 

  • Hofmann, T., Schölkopf, B., Smola, A.J.: Kernel methods in machine learning. Ann. Stat. 36(3), 1171–1220 (2008). doi:10.1214/009053607000000677

    Article  MATH  Google Scholar 

  • Jacobsen, M.: Point process theory and applications. Probability and its applications. Birkhäuser Boston Inc., Boston, (2006). Marked point and piecewise deterministic processes

  • Pillow, J.W., Shlens, J., Paninski, L., Sher, A., Litke, A.M., Chichilnisky, E.J., Simoncelli, E.P.: Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature 454, 995–999 (2008)

    Article  Google Scholar 

  • Scholkopf, B., Smola, A.J.: Learning with Kernels: support vector machines, regularization, optimization, and beyond. MIT Press, Cambridge (2001)

    Google Scholar 

  • van der Vaart, A.W.: Asymptotic statistics. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  • Wahba, G.: Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics., vol. 59. Society for Industrial and Applied Mathematics (SIAM) (1990)

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Acknowledgments

The neuron spike data were provided by Associate Professor, Rune W. Berg, Department of Neuroscience and pharmacology, University of Copenhagen.

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  1. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 , Copenhagen, Denmark

    Niels Richard Hansen

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  1. Niels Richard Hansen
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Correspondence to Niels Richard Hansen.

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Hansen, N.R. Nonparametric likelihood based estimation of linear filters for point processes. Stat Comput 25, 609–618 (2015). https://doi.org/10.1007/s11222-014-9452-6

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  • DOI: https://doi.org/10.1007/s11222-014-9452-6

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