nips nips2011 nips2011-101 nips2011-101-reference knowledge-graph by maker-knowledge-mining
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Author: Yee W. Teh, Vinayak Rao
Abstract: Renewal processes are generalizations of the Poisson process on the real line whose intervals are drawn i.i.d. from some distribution. Modulated renewal processes allow these interevent distributions to vary with time, allowing the introduction of nonstationarity. In this work, we take a nonparametric Bayesian approach, modelling this nonstationarity with a Gaussian process. Our approach is based on the idea of uniformization, which allows us to draw exact samples from an otherwise intractable distribution. We develop a novel and efficient MCMC sampler for posterior inference. In our experiments, we test these on a number of synthetic and real datasets. 1
[1] J. F. Lawless and K. Thiagarajah. A point-process model incorporating renewals and time trends, with application to repairable systems. Technometrics, 38(2):131–138, 1996.
[2] John P. Cunningham, Byron M. Yu, Krishna V. Shenoy, and Maneesh Sahani. Inferring neural firing rates from spike trains using Gaussian processes. In Advances in Neural Information Processing Systems 20, 2008.
[3] T. Parsons. Earthquake recurrence on the south Hayward fault is most consistent with a time dependent, renewal process. Geophysical Research Letters, 35, 2008.
[4] V. Paxson and S. Floyd. Wide area traffic: the failure of Poisson modeling. IEEE/ACM Transactions on Networking, 3(3):226–244, June 1995.
[5] C. Wu. Counting your customers: Compounding customer’s in-store decisions, interpurchase time and repurchasing behavior. European Journal of Operational Research, 127(1):109–119, November 2000.
[6] Ryan P. Adams, Iain Murray, and David J. C. MacKay. Tractable nonparametric Bayesian inference in Poisson processes with Gaussian process intensities. In Proceedings of the 26th International Conference on Machine Learning (ICML), 2009.
[7] A. Jensen. Markoff chains as an aid in the study of Markoff processes. Skand. Aktuarietiedskr., 36:87–91, 1953.
[8] V. Rao and Y. W. Teh. Fast MCMC sampling for Markov jump processes and continuous time Bayesian networks. In Proceedings of the International Conference on Uncertainty in Artificial Intelligence, 2011.
[9] D.R. Cox. The statistical analysis of dependencies in point processes. In P.A. Lewis, editor, Stochastic point processes, pages 55–56. New York: Wiley 1972, 1972.
[10] Robert E. Kass and Val´ rie Ventura. A spike-train probability model. Neural Computation, 13(8):1713– e 1720, 2001.
[11] C. E. Rasmussen and C. K. I. Williams. Gaussian Processes for Machine Learning. MIT Press, 2006.
[12] M. Berman. Inhomogeneous and modulated gamma processes. Biometrika, 68(1):143, 1981.
[13] Yosihiko Ogata. On Lewis’ simulation method for point processes. IEEE Transactions on Information Theory, 27(1):23–31, 1981.
[14] Mark Berman and T. Rolf Turner. Approximating point process likelihoods with GLIM. Journal of the Royal Statistical Society. Series C (Applied Statistics), 41(1):pp. 31–38, 1992.
[15] I. Sahin. A generalization of renewal processes. Operations Research Letters, 13(4):259–263, May 1993.
[16] Emery N. Brown, Riccardo Barbieri, Val´ rie Ventura, Robert E. Kass, and Loren M. Frank. The timee rescaling theorem and its application to neural spike train data analysis. Neural computation, 14(2):325– 46, February 2002.
[17] I. Gerhardt and B. L. Nelson. Transforming renewal processes for simulation of nonstationary arrival processes. INFORMS Journal on Computing, 21(4):630–640, April 2009.
[18] Bo Henry Lindqvist. Nonparametric estimation of time trend for repairable systems data. In V.V. Rykov, N. Balakrishnan, and M.S. Nikulin, editors, Mathematical and Statistical Models and Methods in Reliability, Statistics for Industry and Technology, pages 277–288. Birkhuser Boston, 2011.
[19] P. A. W. Lewis and G. S. Shedler. Simulation of nonhomogeneous Poisson processes with degree-two exponential polynomial rate function. Operations Research, 27(5):1026–1040, September 1979.
[20] J. F. C. Kingman. Poisson processes, volume 3 of Oxford Studies in Probability. The Clarendon Press Oxford University Press, New York, 1993. Oxford Science Publications.
[21] J George Shanthikumar. Uniformization and hybrid simulation/analytic models of renewal processes. Oper. Res., 34:573–580, July 1986.
[22] Iain Murray, Ryan Prescott Adams, and David J.C. MacKay. Elliptical slice sampling. JMLR: W&CP;, 9, 2010.
[23] Iain Murray and Ryan Prescott Adams. Slice sampling covariance hyperparameters of latent Gaussian models. In Advances in Neural Information Processing Systems 23, 2010.
[24] B. Y. R. G. Jarrett. A note on the intervals between coal-mining disasters. Biometrika, 66(1):191–193, 1979.
[25] Ariel Rokem, Sebastian Watzl, Tim Gollisch, Martin Stemmler, and Andreas V.M. Herz. Spike-Timing Precision Underlies the Coding Efficiency of Auditory Receptor Neurons. Journal of Neurophysiology, pages 2541–2552, 2006.
[26] Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. CODA: Convergence diagnosis and output analysis for MCMC. R News, 6(1):7–11, March 2006. 9