Modelling Spatial Intensity for Replicated Inhomogeneous Point Patterns in Brain Imaging

Abstract

Pharmacological experiments in brain microscopy study patterns of cellular activ- ation in response to psychotropic drugs for connected neuroanatomic regions. A typical ex- perimental design produces replicated point patterns having highly complex spatial variability. Modelling this variability hierarchically can enhance the inference for comparing treatments. We propose a semiparametric formulation that combines the robustness of a nonparametric kernel method with the efficiency of likelihood-based parameter estimation. In the convenient framework of a generalized linear mixed model, we decompose pattern variation by kriging the intensities of a hierarchically heterogeneous spatial point process. This approximation entails discretizing the inhomogeneous Poisson likelihood by Voronoi tiling of augmented point patterns. The resulting intensity-weighted log-linear model accommodates spatial smoothing through a reduced rank penalized linear spline. To correct for anatomic distortion between subjects, we interpolate point locations via an isomorphic mapping so that smoothing occurs relative to common neuroanatomical atlas co-ordinates. We propose a criterion for choosing the degree and spatial locale of smoothing based on truncating the ordered set of smoothing covariates to minimize residual extra-dispersion. Additional spatial covariates, experimental design factors, hierarchical random effects and intensity functions are readily accommodated in the linear predictor, enabling comprehensive analyses of the salient properties underlying replicated point patterns. We illustrate our method through application to data from a novel study of drug effects on neuronal activation patterns in the brain of rats. Copyright 2004 Royal Statistical Society.

Convergence of the Lloyd Algorithm for Computing Centroidal Voronoi Tessellations

“…A centroidal Voronoi tessellation (CVT) is a special Voronoi tessellation of a given set such that the associated generating points are the centroids (centers of mass) of the corresponding Voronoi regions with respect to a predefined density function [7]. CVTs are indeed special as they enjoy very natural optimization properties which make them very popular in diverse scientific and engineering applications that include art design, astronomy, clustering, geometric modeling, image and data analysis, resource optimization, quadrature design, sensor networks, and numerical solution of partial differential equations [1,2,3,4,7,8,9,10,11,13,14,17,15,26,29,30,31,39,44,45]. In particular, CVTs have been widely used in the design of optimal vector quantizers in electrical engineering [25,28,40,43].…”

Two types of Drosophila R7 photoreceptor cells are arranged randomly: A model for stochastic cell‐fate determination

“…However, with recent advances in microscopy and technology, particularly in the biological sciences, rigorous statistical methods are being applied to data consisting of replicated spatial univariate point patterns. These include several applications of statistical methods for replicated spatial point patterns, including mixed models and pseudolikelihood (Bell and Grunwald, 2004); mixed models and kriging for inhomogeneous patterns (Wager et al, 2004); parametric models (Mateu, 2001); a nonparametric K-function ANOVA (Diggle et al, 1991); parametric and nonparametric methods (Diggle et al, 2000); and L-, F-, and G-functions (Baddeley et al, 1993). The rigorous statistical analysis of replicated bivariate point patterns (when there are two types of points), similar to the work we have shown, has a much less well developed methodology.…”

Independent component models for replicated point processes

“…But situations where several replications of a process are available are increasingly common. Among the few papers proposing statistical methods for replicated point processes we can cite Diggle et al (1991), Baddeley et al (1993), Diggle et al (2000), Bell and Grunwald (2004), Landau et al (2004), Wager et al (2004), and Pawlas (2011). However, these papers mostly propose estimators for summary statistics of the processes (the so-called F , G and K statistics) rather than for the intensity functions that characterize the processes.…”