Generalized Spatial Dirichlet Process Models

Abstract

SUMMARYMany models for the study of point-referenced data explicitly introduce spatial random effects to capture residual spatial association. These spatial effects are customarily modelled as a zeromean stationary Gaussian process. The spatial Dirichlet process introduced by Gelfand et al. (2005) produces a random spatial process which is neither Gaussian nor stationary. Rather, it varies about a process that is assumed to be stationary and Gaussian. The spatial Dirichlet process arises as a probability-weighted collection of random surfaces. This can be limiting for modelling and inferential purposes since it insists that a process realization must be one of these surfaces. We introduce a random distribution for the spatial effects that allows different surface selection at different sites. Moreover, we can specify the model so that the marginal distribution of the effect at each site still comes from a Dirichlet process. The development is offered constructively, providing a multivariate extension of the stick-breaking representation of the weights. We then introduce mixing using this generalized spatial Dirichlet process. We illustrate with a simulated dataset of independent replications and note that we can embed the generalized process within a dynamic model specification to eliminate the independence assumption.

Hierarchical Bayesian nonparametric models with applications

“…The visual texture and color at site i are then generated by a draw from the distribution θ * * kt i . To obtain a spatially dependent Pitman-Yor process, Sudderth and Jordan (2009) adapt an idea of Duan et al (2007), who used a latent collection of Gaussian processes to define a spatially dependent set of draws from a Dirichlet process. In particular, to each index t we associate a zero-mean Gaussian process, u t .…”

The Dirichlet labeling process for clustering functional data

“…The updates of parameters α, τ, φ θ , σ θ are standard, see ,e.g., Duan et al (2007). For canonical curves, under a Gaussian process, the prior for vector θ * j = (θ * j (x 1 ), .…”

“…The key distinction between these and our work is that the Dirichlet labeling process allows distributional specification of labeling realizations over continuous domain without the need for kernel basis specification. More similar to our approach is the work of Duan et al (2007). This work also specifies a generalized DP mixture model using the view of hybrid species curves.…”

“…In certain applications (such as image modeling), a canonical curve might simply be a (random) constant function that represents a corresponding level set in the image. The notion of hybrid species curves has been explored in various contexts, including text analysis and genetics (Blei et al, 2003;Pritchard et al, 2000), as well as in the context of spatial and functional data (Duan et al, 2007;Petrone et al, 2009). In particular, our approach is based on the hybrid Dirichlet process mixture model first introduced by Petrone et al (2009).…”

“…Our labeling process is arguably more computationally tractable, especially for high-dimensional D and large m due to the exploitation of spatial structure in the model that yields accurate conditional probability approximation. A number of recent papers introduce various constructions based on the Sethuraman's stick-breaking representation, with varying weights assigned for different locations (e.g., Griffin and Steel (2006); Dunson and Park (2008); Duan et al (2007); Sudderth et al (2008)). The work of Griffin and Steel (2006) and Dunson and Park (2008) exemplify several distinct proposals for constructing spatially dependent DP mixture marginals.…”