Chunlin Ji & Mike West Department of Statistical Science Duke University

1. JSM 2009, Washington, DC Aug. 4, 2009 Dynamic spatial mixture modelling and its application in Bayesian tracking for cell fluorescent microscopic imaging ChunlinJi & Mike West Department of Statistical ScienceDuke University Department of Statistical Science, Duke University

2. Dynamic spatial point processes Dynamic spatial inhomogeneous point processes Multiple extended targets tracking. Exploratory questions: -Characterizing Intensity dynamic -Quantify drifts in intensity Single-level cell fluorescence microscopic image. (Wang et al. 2009) Department of Statistical Science, Duke University

3. Spatial Poisson point process • Point process over S Intensity function • Density • Realized locations • Likelihood • Flexible nonparametric model for characterizing spatial heterogeneity in • Dirichletprocess mixture for density function (Kottas & Sanso 07; Ji et al 09 ) Department of Statistical Science, Duke University

4. Dynamic spatial DP mixture • DP Mixture at each time point • Time evolution of mixture model parameters induces dynamic model for time-varying intensity function Dynamic spatial point process Intensity function Parameters of DPMs Dependent DP mixture with Generalized Polya Urn (Caron et al., 2007) Department of Statistical Science, Duke University

5. Dynamic spatial mixture modelling • System equation -- • Observation equation • Initial information --Dependent Dirichlet process --Likelihood of spatial Poisson point process --Dirichlet process prior Department of Statistical Science, Duke University

6. Time propagation models • Generalized Polya Urn (GPU) scheme for random partition • Time propagation models for cluster means • Time propagation models for covariances (Caron et al. 2007) --physically attractive dynamic model --discount factor-based stochastic model (Carvalho & West, 2008) Department of Statistical Science, Duke University

7. SMC for Dirichlet process mixtures • Previous work • SMC for nonparametric Bayesian models (Liu, 1996; MacEachern, et al. 1999) • Particle filter for mixtures (Fearnhead, 2004; Fearnhead & Meligkotsidou, 2007) • Particle learning for mixtures (Carvalho, et al., 2009) • Key point • Marginalization of ; propagated and updated only for • SMC for dependent DP mixtures • SMC for time-varying DP mixtures (Caron et al., 2007) --no marginalization, very low effective sample size (ESS) Department of Statistical Science, Duke University

8. SMC for dynamic (spatial) DP mixtures Rao-Blackwellized Particle filter (Caron et al., 2007) (Escobar & West ,1995) Department of Statistical Science, Duke University

9. Simulation study for synthetic data b) Estimation of the intensity of the spatial point processes--image plots a) Synthetic multi-target tracking scenario ESS= c) Estimation of the intensity function--3D mesh plots Department of Statistical Science, Duke University

10. Simulation study of cell fluorescence images Movie of estimated intensity based on the SMC output-DP mixtures. Human cell fluorescence microscopic image Spatial point pattern generated by image segmentation Department of Statistical Science, Duke University

11. Thank You Department of Statistical Science, Duke University