فرآیندهای نقطه گیبس برای مطالعه توسعه فرآیندهای تصادفی فضایی- زمانی

Although many studies of marked point processes analyse patterns in terms of purely spatial relationships, in real life spatial structure often develops dynamically through time. Here we use a specific space–time stochastic process to generate such patterns, with the aim of determining purely spatial summary measures from which we can infer underlying generating mechanisms of space–time stochastic processes. We use marked Gibbs processes in the estimation procedure, since these are commonly used models for point patterns with interactions, and can also be chosen to ensure that they possess similar interaction structure to the space–time processes under study. We restrict ourselves to Strauss-type pairwise interaction processes, as these are simple both to construct and interpret. Our analysis not only highlights the way in which Gibbs models are able to capture the interaction structure of the generating process, but it also demonstrates that very few statistical indicators are needed to determine the essence of the process. This contrasts markedly with the relatively large number of estimators usually needed to characterise a process in terms of spectral, autocorrelation or K-function representations. We show that the Strauss-type procedure is robust, i.e. it is not crucial to know the exact process-generating mechanism. Moreover, if we do possess additional information about the true mechanism, then the procedure becomes even more effective.

Although Whittle's (1954) pioneering paper put the study of spatial processes firmly on the statistical map, subsequent developments have mainly centered on purely spatial relationships. In real life, however, patterns often develop dynamically through time. So although a purely spatial analysis is ideal for achieving a snap-shot summary of spatial structure, such as classifying texture through parameter estimates for a Markov random field, to gain an understanding of the underlying evolutionary processes which generate pattern necessitates a spatial–temporal approach. Bartlett (1975) provided a mechanism for achieving this through the development of the spatial–temporal spectrum, though for some time afterwards there was considerable inertia against conducting space–time statistical analyses. In fairness this was most likely due to the difficulty of collecting appropriate data; modern electronic scanning and video devices have clearly made this far less of a problem. One of the early biological experiments that provided the impetus for change was performed by Ford (1975), who studied the effect of between-plant competition on Tagetes patula L. (marigolds) planted on a regular lattice. A time-lapse film over a three-month period produced a (visual) matrix of continuous-valued variables in continuous time; for the purpose of statistical analysis measurements on plant height were recorded at intervals of 2, 4, 6 and 8 weeks. Ford and Diggle (1981) modelled these data by assuming that competition between individuals was for light. Interaction between neighbours was viewed as a spatial process in relation to differences in plant height, and simulations reproduced both bimodality in the frequency distribution of plant size and an even spatial distribution of large or surviving plants. Renshaw (1984) made theoretical progress by exploiting the underlying lattice structure using a frequency-domain approach. This holds a considerable advantage over space-domain techniques since, unlike sample autocovariances, periodogram values are effectively independent of each other. Further stimulii ( Renshaw and Ford 1983 and Renshaw and Ford 1984; Ford and Renshaw, 1984) are provided by the distinctive nature of various ecologically based spectra, and that use of sliding data windows can highlight the presence of nonstationarity.Since here we are interested in phenomena that can be considered to be realisations of spatial marked point processes, the above system , and is not appropriate since it is lattice based. Indeed, transposing this to the real plane (see Daley and Vere-Jones, 1988, Chap. 11) poses fundamental theoretical problems since it destroys the Fourier properties which give rise to such simple functional spectral structures. This difficulty is compounded if we impose a logistic-type reaction function to ensure that the process remains non-explosive. For the foreseeable future theoretical progress therefore appears unlikely, and so study of such spatial–temporal marked point processes will have to proceed via simulation; our aim is to determine optimal ways of characterising their development. Given recent work by several authors on autocovariance and spectral analysis of pure point and marked point processes, together with earlier work involving distance-based measures (e.g. Ripley's K-function), one possible route would be to see how sample autocovariance, spectral and distance functions change with respect to the reaction and diffusion components of the space–time model. However, each of these are many-valued descriptors, and so the question arises as to whether one can construct just a few summary statistics which are sensitive to change in the underlying process. We shall base these on marked Gibbs processes (see, for example, Stoyan et al., 1995), and study whether the interaction parameters of these purely spatial processes provide a useful and sensible characterisation of our space–time process. The reason for considering Gibbs processes is that they have some similarities with the system , and . They are defined through the probability density function (p.d.f.) where ϕ={[x1;m1],[x2;m2],…,[xn;mn]} is a realisation of the point process Φ consisting of the locations xi and marks mi of the points i=1,…,n, V is a normalising constant, and U denotes an energy (reaction) function (see Stoyan et al., 1995, Chap. 5). Now this representation has two key elements in common with our original space–time process. First, U can take exactly the same form as the right-hand side of Eq. (1.1). Second, both the numerators in and take a similar exponential form; the main structural difference lies in the denominator, with the normalising constant V replacing the function ψ. Our aims are therefore two-fold. First, we determine whether purely spatial Gibbs-type estimation procedures yield quantitative insight into underlying space–time stochastic generating mechanisms when their nature is known, and then unknown. Second, we assess whether parameter estimates relating to a given realisation change markedly over time.

Because of its simplicity, the Strauss-type disc process (2.4) is a natural choice for the interaction process in the Gibbs model which underlies our initial estimation procedure, if there are interactions depending on continuous marks. Equally, it is a natural choice for the interaction component in the space–time stochastic process (3.1). This model can be used, for example, to describe interaction between trees (discs or marks being influence zones of trees; see Goulard et al., 1996) or between territorial animals. We have shown it produces similar bimodality of canopy structure as that observed by Ford and Diggle (1981). Table 1 not only highlights the way in which the parallel Gibbs model catches the interaction structure of the generating process, namely r=0.005, but also demonstrates that very few statistical indicators (here at most 3 or 4) are needed to capture the essence of the process. This is in total contrast to the relatively large number of autocorrelation or periodogram estimators that would be needed to characterise the process in terms of time-series type estimators, and is far more concise than the production of successive graphplots resulting from K-function analyses. Fig. 4 demonstrates the variability exhibited by our indicators over time, and highlights the inherent danger of placing too much credence in snap-shot analyses taken at a single-time point. In our second model, (4.2), the range of interaction does not depend on the marks, but only on the interpoint distance. Furthermore, the strength of interaction relates to mark difference; relevant scenarios include the growth of agricultural crops, colonising plants, and naturally regenerating forests. Even though this process is substantially different from the first one, (3.1), the Strauss-disc model is still able to describe the interaction structure quite well. According to the estimated model, there is interaction up to the distance 0.003(mi+mj), which corresponds to the fixed interaction radius 0.05 (for a large mark inhibiting a small one) used in the simulated realisation. This shows that the Strauss-type procedure is robust, i.e. it is not crucial to know the exact process generating mechanism. However, if we do possess additional information, here (expression (4.5)) that the strength of interaction is based upon mi−mj, and that the interaction distance is constant, then the procedure becomes far more effective. The estimates ( (4.6) and Fig. 6) show a clear cut-off at distance 0.05: as before, note the variability in the estimates View the MathML source. Note that the model (4.5) does not have exactly the same interaction structure as the pattern generating process. In the estimation process the interactions are symmetric, whilst in the generating process they are asymmetric: in (4.5) two interacting marks will result in the larger being enhanced and the smaller being diminished, whilst the estimation procedure assumes that both will be diminished. As hinted earlier, matching the estimation and generating processes necessitates the use of the non-symmetric pair potential function (2.5), which is not admissible under the usual Gibbs process framework. One possible solution might be to utilise the time-dependent nature of the data by adapting the approach of Guyon (1995) (see also Bayomog, 1996; Högmander and Särkkä, 1999) applied to Markov random fields, since this allows non-symmetric interactions through time (as opposed to space). In this we could replace (2.5) by equation(5.1) View the MathML source which means that we are now much closer to the true generating mechanism. However, for most marks mj(t) and mj(t−1) will be broadly similar, whence on inserting (5.1) into (3.5) we see that the sum of the pair potentials φ(xi,xj,mi(t),mj(t−1)) and φ(xi,xj,mi(t−1),mj(t)) lies near to zero, which means that only weak interaction estimates can be obtained. Note that if we relate points at the two times t−s and t, then individuals may die and arrive inbetween times. This does not present any analytic difficulty, but as s increases we diverge away from the original generating mechanism. Using (5.1) with the time delays s=1, 2, 5, 10 and 20 and the two interaction parameters 0–0.05 and 0.05–0.1 produced effectively zero estimates, so an alternative strategy is clearly necessary. A promising line of attack is to acknowledge the reality that the right-hand side of (4.2) drives the first derivative of mi(t). So we could consider the pattern as being a two-mark process in which the pair potential function is based on φ(xi,xj,mi(t),mj(t),mi′(t),mj′(t)) where mk′(t) denotes, for example, mk(t)−mk(t−1). This work is currently under investigation and will be reported elsewhere.