Abstract
An important aspect of decision support systems involves applying sophisticated and flexible statistical models to real datasets and communicating these results to decision makers in interpretable ways. An important class of problem is the modelling of incidence such as fire, disease etc. Models of incidence known as point processes or Cox processes are particularly challenging as they are ‘doubly stochastic’ i.e. obtaining the probability mass function of incidents requires two integrals to be evaluated. Existing approaches to the problem either use simple models that obtain predictions using plug-in point estimates and do not distinguish between Cox processes and density estimation but do use sophisticated 3D visualization for interpretation. Alternatively other work employs sophisticated non-parametric Bayesian Cox process models, but do not use visualization to render interpretable complex spatial temporal forecasts. The contribution here is to fill this gap by inferring predictive distributions of Gaussian-log Cox processes and rendering them using state of the art 3D visualization techniques. This requires performing inference on an approximation of the model on a discretized grid of large scale and adapting an existing spatial-diurnal kernel to the log Gaussian Cox process context.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, R.P., Murray, I., MacKay, D.J.C.: Tractable nonparametric Bayesian inference in poisson processes with gaussian process intensities. In: Proceedings of the 26th International Conference on Machine Learning (ICML 2009), Montreal, Quebec (2009)
Eatonand, J.W., Bateman, D., Hauberg, S.: GNU Octave Manual Version 3. Network Theory Limited (2008)
Brunsdon, C., Corcoran, J., Higgs, G.: Visualising space and time in crime patterns: A comparison of methods. Computers, Environment and Urban Systems 31(1), 52–75 (2007)
Chiles, J., Delfiner, P.: Geostatistics. Wiley, New York (1999)
Cox, D.R.: Some Statistical Methods Connected with Series of Events. Journal of the Royal Statistical Society. Series B 17(2), 129–164 (1955)
Cressie, N.A.C.: Statistics for spatial data. Wiley series in probability and mathematical statistics: Applied probability and statistics. J. Wiley (1991)
Diggle, P.J.: A kernel method for smoothing point process data. Applied Statistics 34, 138–147 (1985)
Kottas, A., Sansó, B.: Bayesian mixture modeling for spatial poisson process intensities, with applications to extreme value analysis. Journal of Statistical Planning and Inference 137, 3151–3163 (2009)
Møller, J., Syversveen, A.R., Waagepetersen, R.P.: Log Gaussian Cox processes. Scandinavian Journal of Statistics 25, 451–482 (1998)
Moller, J., Vej, F.B.: Structured spatio-temporal shot-noise cox point process models, with a view to modelling forest fires. Scandinavian Journal of Statistics 37(1) (2010)
Moller, J., Pettitt, A.N., Reeves, R.W., Berthelsen, K.K.: An efficient markov chain monte carlo method for distributions with intractable normalising constants. Biometrika 93(2), 451–458 (2006)
Murray, I., Ghahramani, Z.: Bayesian learning in undirected graphical models: Approximate MCMC algorithms. In: Proceedings of the 20th Annual Conference on Uncertainty in Artificial Intelligence (UAI 2004), pp. 392–399. AUAI Press, Arlington (2004)
Murray, I., Ghahramani, Z., MacKay, D.J.C.: MCMC for doubly-intractable distributions. In: Proceedings of the 22nd Annual Conference on Uncertainty in Artificial Intelligence (UAI 2006), pp. 359–366. AUAI Press (2006)
Ramachandran, P., Varoquaux, G.: Mayavi: 3D Visualization of Scientific Data. Computing in Science & Engineering 13(2), 40–51 (2011)
Rasmussen, C.E., Williams, C.K.I.: Gaussian processes for machine learning. Adaptive Computation and Machine Learning. MIT Press (2006)
Waagepetersen, R.: Convergence of posteriors for discretized log gaussian cox processes. Statistics & Probability Letters 66, 229–235 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Rohde, D., Corcoran, J., White, G., Huang, R. (2012). Visualization of Predictive Distributions for Discrete Spatial-Temporal Log Cox Processes Approximated with MCMC. In: Yin, H., Costa, J.A.F., Barreto, G. (eds) Intelligent Data Engineering and Automated Learning - IDEAL 2012. IDEAL 2012. Lecture Notes in Computer Science, vol 7435. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32639-4_35
Download citation
DOI: https://doi.org/10.1007/978-3-642-32639-4_35
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32638-7
Online ISBN: 978-3-642-32639-4
eBook Packages: Computer ScienceComputer Science (R0)