Conditional simulation of a Cox process using multiple sample supports

This paper deals with the conditional simulation of a Cox process in a family of target blocks given samples of various supports. Here it has been assumed that the distribution of the potential over all samples and blocks factorize according to the following graphical model. A node is associated to each sample and target block. Each sample node is connected to the node of the smallest sample, or block, that contains it. All block nodes are joined by edges. This graphical model possesses conditional independence relationships that can be exploited by a metropolized version of the Gibbs sampler to produce fast conditional simulations. This model can be seen as a generalisation of the discrete gaussian model traditionally used for congruent samples. INTRODUCTION Multiple sample support is extremely common within the mine ral resource industry as different sampling campaigns are often designed with dif ferent objectives, resulting in sample data with different support sizes, shap es and configurations. Thus to incorporate all data for both estimation and uncerta inty exercises is a challenging problem. The present paper deals with the conditional simulation of a Cox process (1955) in a family of target blocks. Provided that the random intensit y function, orpotential, of the samples and the blocks satisfy some conditional indep endence relationships, an iterative simulation algorithm can be set up to accommoda te all conditioning data. A graphical model (Lauritzen, 2001; Jordan, 2004) is i ntroduced to specify these independence relationships in a fully consistent way . This model can be seen as a generalisation of the discrete gaussian model traditio nally used in geostatistics for the non-linear estimation of local reserves starting fr om congruent samples (Matheron, 1976; Rivoirard, 1994; Chilès and Delfiner, 199 9; Emery, 2007). G. BROWN, J. FERREIRA AND C. LANTÚEJOUL This paper starts with a summary of the salient features of th e Cox process, then presents a graphical model that factorizes the joint condit ional distribution of the potential of the blocks and samples, leading to a simple a nd f st conditional simulation algorithm of the Cox process. The relationships between this graphical model and the discrete gaussian model are then established. The proposed methodology is finally demonstrated using data emanating fr om a diamond placer deposit to estimate confidence limits for block concentrati ons. METHODOLOGY Presentation of the problem A Cox process is a Poisson point process with a random intensi ty function, or potential. This potential reflects the propensity for some regions to c ontain more points than others. Figure 1 shows two realisations of a Cox p r cess with their underlying potential. Figure 1:Two realisations of a Cox process, demonstrating the differ ing location of clusters, which contrasts to the standard Poisson point process. Let Z = (Zx, x ∈ IRd) be the random function that denotes the potential of the Cox process. The potential associated to each domain v is denoted by Zv = ∫ v Zx dx v ⊂ IR d The Cox process is characterized by the following condition al property. Given Z, the number of points within pairwise disjoint domains v1, ...,vn are mutually independent Poisson variables with respective parameters Zv1, ...,Zvn . These does not mean that these variables are effectively independent b cause the potential conveys its own structure to the Cox process. For instance, t he covariance between the number of points in two domains v andw is the sum of two terms Cov{Nv,Nw} = Cov{Zv,Zw}+ E{Zv∩w} The first one is derived from the covariance of the potential w hereas the second one stems from the Poisson seeding of the points. The main concern of this paper is the conditional simulation of the number of points (Nb,b ∈ B) in a family of pairwise disjoint target blocks given the numb er of points 2 GEOSTATS 2008, Santiago, Chile CONDITIONAL SIMULATION OF A COX PROCESS (Ns = ns,s ∈ S) in a population of samples. To address this problem, it is con venient first to generate the potential of the target blocks conditio nal to the content 1 of the samples, and second the content of the target blocks give n their potential. As the second step is straightforward (it merely amounts to sim ulating independent Poisson distributions), only the first step is discussed. In principle, the first step only requires the conditional di stribution of the potential of the target blocks to be considered. In practice, however, it is more advantageous to consider the joint conditional distribution of the poten tial of the target blocks together with the samples. The probability density functio n (pdf) of this distribution can be written as f (zB∪S | nS) ∝ f (zB) f (zS\B | zB) p(nS | zS) (1) Note that this pdf is specified up to a constant, the typical si tuation where the Metropolis-Hasting algorithm is required for simulati on. Note also that the conditional generation of the sample potential may be diffic ult, especially if the samples are numerous or have different supports. The graphi c l model that is to be introduced in the next section brings significant simplifica tions. A graphical model for the potential Let us consider a family of samples and blocks, as shown in Fig ure 2. b2 b3 s3