Extrapolating and Interpolating Spatial Patterns
Observations of a spatial pattern are typically conﬁned to a bounded region of space, while the original pattern of interest can often be imagined to extend outside. Much attention has been paid to statistical inference for models of the pattern given only the partial observations in the sampling window. Less attention has been given to prediction or extrapolation of the process (i.e. of the same realisation of the process) beyond the window of observation, conditional on the partially observed realisation. A motivating example is the charting of geological faults encountered during coal mining (Baddeley et al. 1993, Chile`s 1989). It is of interest to predict the likely presence of geological faults outside the region mined so far, and thereby to choose between various mining strategies. Other examples may be found in image processing, for instance the problem of replicating a texture beyond the region where it has been observed as in the editing of a video image so that a foreground object is removed and replaced seamlessly by the background texture (De Bonet 1997). Partial observation of a spatial pattern may also include eﬀects such as aggregation by administrative regions, deletion of part of the pattern, and the unobservability of a related pattern. Recovery of full information in this context might be called interpolation; it resembles a missing data problem. In the mining problem discussed above, mapped charts represent only those parts of geological faults which were physically encountered. Gaps may arise because the mined region is not convex both at its outer boundary and within this boundary, because pillars of unmined material remain. Hence it is of interest to join observed line segments together and to interpret them as part of the same continuous fault zone, a process that is known as ‘interpretation’ by geologists. As another example, geostatistics deals with predicting values of a spatial random process (e.g. precipitation or pollution measurements) from observations at known locations (e.g. Journel and Huijbregts 1978, Cressie 1993, Stein 1999) and interpolation techniques
have been developed under the name of conditional simulation for Gaussian and other second-order random ﬁelds, as well as for discrete Markov random ﬁeld models. Relatively few conditional simulation techniques have been developed for spatial processes of geometric features such as points, line segments and ﬁlled shapes. Those that exist are based largely on Poisson processes and the associated Boolean models (Lantue´joul 1997, Kendall and Tho¨nnes 1999, van Lieshout and van Zwet 2001). A major obstacle is the scarcity of spatial models that are both realistic and tractable for simulation. Some exceptions are the following. There has been much interest in the conditional simulation of oil-bearing reservoirs given data obtained from one or more exploration wells (Haldorsen 1983, Chessa 1995). The wells are essentially linear transects of the spatial pattern of reservoir sand bodies. Typically the sand bodies are idealised as rectangles with horizontal and vertical sides of independent random lengths, placed at random locations following a Poisson point process. For line segment processes, Chile`s (1989) presents some stochastic models with particular application to modelling geological faults (based largely on Poisson processes), geostatistical inference, and possibilities for conditional simulation; Hjort and Omre (1994) describe a pairwise interaction point process model for swarms of small faults in a fault zone, and Stoica et al. (2000,2001) study a line segment process for extracting linear networks from remote sensing images. Some of these authors have correctly noted the sampling bias eﬀect attendant on observing a spatial pattern of geometric features within a bounded window (analogous to the ‘bus paradox’). Techniques from stochastic geometry need to be enlisted to check the validity of simulation algorithms. Extrapolation or interpolation of a spatial pattern entails ﬁtting a stochastic model to the observed data, and computing properties of the conditional distribution of this model given the observed data. We will discuss a variety of stochastic models for patterns of geometric objects, and treat typical issues such as edge eﬀects, occlusion and prediction in some generality. Subsequently, we shall focus on the problem of identifying clusters in a spatial point pattern, which can be regarded as interpolation of a two-type point pattern from observations of points of one type only, the points of the other type being the cluster centres (Baddeley and van Lieshout 1993, Lawson 1993b, van Lieshout 1995, van Lieshout and Baddeley 1995). Applications may be found in epidemiology, forestry, archaeology, coal mining, animal territory research, and the removal of land mines.