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A marked spatial point pattern of trees and their diameters is the result of a dynamic biological process that takes place over time as well as space. Such patterns can be modeled as realizations of marked space-time survival point processes, where trees are born at some random location and time and then live, grow, and produce offspring in a random fashion. A model for a marked space-time survival point process is fit to data from a longleaf pine ( Pinus palustris ) forest in southern Georgia. The space-time survival point process is divided into three components: a birth process, a growth process, and a survival process. Each of the component processes is analyzed individually, from which conclusions regarding the dynamic ecological processes can be made. By using this reductionist approach, questions concerning each individual process can be addressed that might not have been answerable otherwise.

Content uploaded by Stephen Rathbun

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All content in this area was uploaded by Stephen Rathbun on Apr 29, 2015

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... To provide context for the changes in forest structure and spatial pattern from selection treatments in second growth forests, we also incorporate data from a well-studied 200-ha old-growth longleaf pine woodland on the Wade Tract in Thomasville GA (30.75 • N 84.00 • W; Platt et al., 1988;Platt and Rathbun, 1993;Rathbun and Cressie, 1994). Although the Wade Tract experienced grazing and other minor disturbances (Engstrom et al., in press), recent records indicate no extensive clearing or plowing, and limited selective logging or salvaging of some lightning struck trees (Platt et al., 1988). ...

... Thus, the Wade Tract is thought to be a representative example of old-growth longleaf pine woodlands of the Tallahassee Red Hills. We use a publicly available dataset of the Wade Tract analyzed in Rathbun and Cressie (1994) available in the spatstat package in R (Baddeley and Turner, 2005). This dataset includes point locations and dbh for 584 trees in a 4 ha area ca. ...

... To determine how selection treatments altered the spatial pattern of overstory trees, we used a point process-based approach (Rathbun and Cressie, 1994). We tested whether trees within each treatment were distributed randomly, uniformly, or aggregated by using the distancedependent univariate pair correlation function, g(r) (Dale and Fortin, 2014). ...

Natural disturbance-based silviculture emphasizes harvest methods that emulate the timing and structural changes of natural disturbances. Longleaf pine woodlands are ecologically important ecosystems of the southeastern U.S. that support high biodiversity. Options for multi-aged silviculture include individual tree and group selection methods to promote regeneration—the latter method may be modified by retention of reserve trees. To explore the extent to which selection methods are congruent with natural disturbance regimes, we evaluated how treatments in mature second-growth longleaf pine woodlands affected overstory structure, pattern, and dynamics, and we made comparisons to an old-growth longleaf stand. In 2010, stands were harvested using individual tree selection, group selection, or group selection with reserves. We compared treatment effects on spatial pattern of residual trees, recruitment of trees into the 10 cm diameter class (hereafter “recruitment”), and tree mortality 8 years after harvest. Basal area and residual volume were similar among treatments, but the individual tree selection treatment had lower density and a unimodal rather than bimodal diameter distribution compared to other treatments. Group selection and group selection with reserves increased spatial aggregation, compared to individual tree selection which reduced aggregation. Recruitment was similar across treatments but usually occurred near existing trees in the group selection treatments and was further from existing trees in the individual tree selection treatment. Tree mortality primarily occurred as single trees rather than tree groups for all treatments. These results indicate that natural disturbance-based treatments vary in their effects on overstory spatial pattern and alter forest dynamics. In longleaf pine woodlands, the rationale for individual tree selection emphasizes maintaining a continuous input of needles as fine fuels for control of resprouting hardwoods with prescribed fire. This method, may simplify overstory spatial structure and alter forest dynamics after initial harvest. Maintenance of vertical and horizontal complexity is a central tenet of natural disturbance-based management, thus attention to spatial pattern must be given when individual tree selection methods are used. In longleaf pine woodlands, natural disturbance-based techniques such as group selection with reserves may better mimic spatial patterns seen in old-growth stands while preserving continuity of fine fuels.

... We next consider treatment decisions for egos in a social network, where each individual may have more than one neighbour. In particular, we consider five types of fixed network: (i) a ring consisting of points on a circle, (ii) a square lattice, (iii) an Erdős-Rényi (ER) network realization, (iv) a Barabási-Albert (BA) network realization, and (v) the Longleaf Pines (Rathbun & Cressie, 1994) spatial network. The ring and the square lattices are simple network structures with a constant degree. ...

Precision medicine describes health care where patient‐level data are used to inform treatment decisions. Within this framework, dynamic treatment regimes (DTRs) are sequences of decision rules that take individual patient information as input data and then output treatment recommendations. DTR estimation from observational data typically relies on the assumption of no interference: i.e., the outcome of one individual is unaffected by the treatment assignment of others. However, in many social network contexts, such as friendship or family networks, and for many health concerns, such as infectious diseases, this assumption is questionable. We investigate the DTR estimation method of dynamic weighted ordinary least squares (dWOLS), which boasts of easy implementation and the so‐called double‐robustness property, but relies on the assumption of no interference. We define a network propensity function and build on it to establish an implementation of dWOLS that remains doubly robust under interference associated with network links. The method's properties are demonstrated via simulation and applied to data from the Population Assessment of Tobacco and Health (PATH) study to investigate cigarette dependence within two‐person household networks. La médecine de précision est une discipline médicale qui vise à dresser le portrait de soins de santé d'un patient en se fiant à ses données spécifiques pour prendre une décision éclairée quant aux traitement et soins à lui prodiguer. Dans ce contexte, les régimes de traitement dynamiques (RTD) sont des suites de règles de décision qui se servent des données individuelles du patient pour produire des recommandations de traitement personnalisé. L'estimation de RTD à partir de données d'observation repose généralement sur l'hypothèse de non‐interférence, dans le sens que, le résultat d'un patient n'est pas affecté par le traitement prescrit à d'autres patients. Cependant, dans de nombreux contextes de réseaux sociaux, tels que les réseaux d'amis ou de proches, et pour de nombreux problèmes de santé, tels que les maladies infectieuses, cette hypothèse est questionable. Les auteurs de ce travail étudient la méthode éstimation de RTD par les moindres carrés ordinaires pondérés dynamiques (dWOLS), méthode qui repose sur l'hypothèse de non‐interférence tout en étant facile à mettre en œuvre et jouit de la propriété dite de double robustesse. Pour ce faire, ils définissent une fonction de propension de réseau sur laquelle ils s'appuient pour établir une mise en œuvre des dWOLS qui reste doublement robuste même en présence ínterférence associée aux liens réseaux. Les propriétés de la méthode sont illustrées par simulation, et appliquées aux données de l'étude PATH (Population Assessment of Tobacco and Health) pour étudier la dépendance à la cigarette au sein de réseaux de ménages de deux personnes.

... We illustrate the above on a real dataset as suggested in Example 5.3 of Taddy and Kottas (2012). The suggested dataset, longleaf is part of the R package spatstat (Baddeley and Turner 2005) and a detailed space-time survival analysis based on this was developed in Rathbun and Cressie (1994). The observations are locations of 584 pine trees in a 200 × 200 square and the marks are diameters of the trees at breast height (only for trees having this diameter greater than 2 cm). ...

Predictive recursion (PR) is a fast, recursive algorithm that gives a smooth estimate of the mixing distribution under the general mixture model. However, the PR algorithm requires evaluation of a normalizing constant at each iteration. When the support of the mixing distribution is of relatively low dimension, this is not a problem since quadrature methods can be used and are very efficient. But when the support is of higher dimension, quadrature methods are inefficient and there is no obvious Monte Carlo-based alternative. In this paper, we propose a new strategy, which we refer to as PRticle filter, wherein we augment the basic PR algorithm with a filtering mechanism that adaptively reweights an initial set of particles along the updating sequence which are used to obtain Monte Carlo approximations of the normalizing constants. Convergence properties of the PRticle filter approximation are established and its empirical accuracy is demonstrated with simulation studies and a marked spatial point process data analysis.

... Given the importance of longleaf pine ecosystems, a thorough understanding of stand dynamics of the species seems to be needed for its restoration. Research on the spatial patterns of trees in longleaf pine forests has been limited (Platt et al., 1988;Rathbun and Cressie 1994). Spatial patterns of trees would give many ideas on the stand dynamics of longleaf pine forests. ...

... Cox-process models (Møller et al., 1998), and these are difficult and time-consuming to fit (Teng et al., 2017). A CAR model as a random effect on a grid is a fast approximation for the spatial point process intensity surface (Rathbun and Cressie, 1994), and still provides an MCMC sample from the posterior distribution (Besag, 1994). A sensitivity analysis could be performed (Kéry and Royle, 2016, p. 415) on hexagon size, and ultimately, the coarsest-scale grid that meets objectives should be adopted as the fastest method. ...

With the advent of technology for data‐gathering and storage, opportunistic citizen‐science data are proliferating. Species distribution models (SDMs) aim to use species occurrence or abundance for ecological insights, prediction, and management. We analyzed a massive opportunistic data set with over 100,000 records of incidental shipboard observations of marine mammals. Our overall goal was to create maps of species density from massive opportunistic data by using spatial regression for count data with an effort offset. We illustrate the method with two marine mammals in the Gulf of Alaska and Bering Sea. We counted the total number of animals in 11,424 hexagons based on presence‐only data. To decrease bias, we first estimated a spatial density surface for ship‐days, which was our proxy variable for effort. We used spatial considerations to create pseudo‐absences, and left some hexagons as missing values. Next, we created SDMs that used modeled effort to create pseudo‐absences, and included the effort surface as an offset in a second stage analysis of two example species, northern fur seals and Steller sea lions. For both effort and species counts, we used spatial count regression with random effects that had a multivariate normal distribution with a conditional autoregressive (CAR) covariance matrix, providing 2.5 million Markov chain Monte Carlo (MCMC) samples (1000 were retained) from the posterior distribution. We used a novel MCMC scheme that maintained sparse precision matrices for observed and missing data when batch sampling from the multivariate normal distribution. We also used a truncated normal distribution to stabilize estimates, and used a look‐up table for sampling the autocorrelation parameter. These innovations allowed us to draw several million samples in just a few hours. From the posterior distributions of the SDMs, we computed two functions of interest. We normalized the SDMs and then applied an overall abundance estimate obtained from the literature to derive spatially explicit abundance estimates, especially within subsetted areas. We also created “certain hotspots” that scaled local abundance by standard deviation and using thresholds. Hexagons with values above a threshold were deemed as hotspots with enough evidence to be certain about them.

... See, for example, Besag [4,5] and Besag [6], for some early studies. For some recent studies over the past three decades, see, for example, Rathban and Cressie [14], Heagerty and Lele [9], Lin and Clayton [11], and Ainsworth et al. [1]. As far as the model for spatial binary data is concerned, these existing studies mostly used the CAR [6], BPM [9], and WCQL [11] models to accommodate spatial binary correlations. ...

There is a long history of spatial regression analysis where it is important to accommodate the spatial correlations among the responses from neighboring locations for any valid inferences. Among numerous modeling approaches, the so-called spatial auto-regression (SAR) model in a linear setup, and the conditional auto-regression (CAR) model in a binary setup, are widely used. For spatial binary analysis, there exists two other competitive approaches, namely the bivariate probit models (BPM) based composite likelihood approach using local lattices; and a ‘Working’ correlations based QL (quasi-likelihood) (WCQL) approach. These correlation models, however, fail to accommodate both within and between correlations among spatial families, where a spatial family is naturally formed within a threshold distance of a selected location, and the member locations between two neighboring families may also be correlated. In this paper, we exploit this latter two-ways, within and between correlations among spatial families and develop a unified correlation model for all exponential family based such as linear, count or binary data. We further exploit the proposed correlation structure based generalized quasi-likelihood (GQL) and method of moments (MM) approaches for model parameters estimation. As far as the estimation properties are concerned, because in practice one encounters a large spatial sample, we make sure that the proposed GQL and MM estimators are consistent.

There exist many studies on regression analysis for spatial binary data, espsecially in ecological, environmental and socio-economic setups, where spatial responses from neighboring locations within a given threshold distance are correlated. However, in some of these studies, it could be more natural to consider a spatial regression analysis for categorical response data with more than two categories, as an improvement over the spatial binary analysis. But, this type of regression analysis for spatial categorical/multinomial data is not adequately addressed in the literature. One of the main reasons is the difficulty of modeling the spatial familial correlations for categorical data, where a spatial family is generated within the threshold distance for each of the two selected neighboring locations. Also, some of the locations from two families may be pair-wise correlated. Unlike the existing studies, in this paper we propose a familial random effects based multinomial logits mixed (MLM) effects model which accommodates both within and between familial correlations for spatial multinomial data. In this context, the proposed spatial multinomial correlations are contrasted with existing longitudinal multinomial correlations so that the longitudinal correlation models are avoided for spatial multinomial data. Both regression effects and the random effects influence parameters are estimated using the generalized quasi-likelihood approach, whereas the random effects variance and correlation parameters are estimated by the well known method of moments. The large sample properties such as consistency of the proposed estimators are studied analytically. The asymptotic normality of the regression estimators is also studied for the convenience of constructing the confidence intervals when needed. The devirations and proofs are given in details, as opposed to conducting a limited simulation study, to justify the validity and convergence properties of the proposed estimators. The estimating equations those produced consistent estimates are clearly formulated for the computational benefit to the practitioners.

The Beta family owes its privileged status within unit interval distributions to several relevant features such as, for example, easiness of interpretation and versatility in modeling different types of data. However, the flexibility of its density at the endpoints of the support is poor enough to prevent from properly modeling the data portions having values next to zero and one. Such a drawback can be overcome by resorting to the class of the Non‐central Beta distributions. Indeed, the latter allows the density to take on arbitrary positive and finite limits which have a really simple form. Nevertheless, the analytical and mathematical complexity of this distribution poses strong limitations on its use as a model for data on the Real interval (0,1). That said, an in‐depth study of a newly found analogue of the Non‐central Beta distribution is carried out in this paper. The latter preserves the applicative potential of the standard Non‐central Beta class but with the advantage of showing a more straightforward and easily handleable density.

We introduce a new class of spatial Cox processes driven by a Hilbert-valued random log-intensity. We adopt a parametric framework in the spectral domain, to estimate its spatial functional correlation structure. Specifically, we consider a spectral functional, approach based on the periodogram operator, inspired on Whittle estimation methodology. Strong consistency of the parametric estimator is proved in the linear case. We illustrate this property in a simulation study under a Gaussian first-order Spatial Autoregressive Hilbertian scenario for the log-intensity model. Our method is applied to the spatial functional prediction of respiratory disease mortality in the Spanish Iberian Peninsula, in the period 1980–2015.

In an old-growth longleaf pine population in which all trees of at least 2 cm in dbh were mapped and tagged, the population was of uneven age and size; tree size correlated positively with tree age. Large or old trees were only loosely aggregated, forming a background matrix that filled the forest. Juvenile trees were highly aggregated, located in areas of low adult densities. Recruitment thus occurs primarily within open spaces created by the deaths of large trees. Variable time lags may occur before the colonization of open spaces, however, because of temporal variation in seed production and occurrence of summer ground fires. Recruitment within the mapped plot has occurred frequently for at least the past 250 yr. Temporal variation in adult mortality and recruitment into open spaces, coupled with strong negative interactions between cohorts of different ages, appears likely to produce alternating phases of population growth and decline that are highly variable in length and magnitude. An upper bound to population size occurs when all available space is filled with trees; but no lower bound exists, and extinction probabilities may be increased at very low densities. The population is buffered from declines to very low densities, however, by the tendency for small trees to recruit into openings created by the deaths of adults. Longleaf pine possibly maintains the environment in an open state suitable for its own regeneration by transmuting a localized disturbance (lightning) into a widespread disturbance (ground fires). Fire facilitation results in an extended, but indefinite, increase in the persistence of environmental conditions in which longleaf pine, but no other tree species, can survive and reproduce. -from Authors

In previous papers (1976), (1977) limit theorems were obtained for the classical Ising model in the absence of an external magnetic field, thereby providing a basis for asymptotic inference. The present paper extends these results to arbitrary external magnetic fields. Statistical inference for this model is important because its nearest-neighbour interactions provide a natural first approximation to spatial interaction among binary variables located on square lattices.
The most interesting behaviour occurs in zero field and at or beyond the critical point. In this case, the central limit result for nearest-neighbour interactions requires an unusual norming, the limiting variances may depend on the nature of the boundary conditions, and there cannot be any central limit result for external magnetic field. The implications of these phenomena for statistical inference are also discussed. In particular, the maximum likelihood estimator of magnetic field is not consistent. Rather it appears to have a non-trivial asymptotic distribution.

A definition of the Markovian property is given for a lattice process and a Gaussian Markovian lattice process is constructed on a torus lattice. From this a Gaussian Markovian process is constructed for a lattice in the plane and its properties are studied.

From certain points of view, the range of probability models currently available for describing the joint behaviour of two point processes is rather limited. In this paper we explore the structure of some further models and apply our results to the statistical analysis of bivariate spatial point patterns.

B. Kaufmann's exact characterization of the partition function for the classical Ising model is used to obtain limit theorems for the sample correlation between nearest neighbors in the non-critical case. This provides a basis for the asymptotic testing and estimation (by confidence intervals) of the correlation between nearest neighbors.

In Pickard (1976) limit theorems were obtained for the classical Ising model at non-critical points. These determined the asymptotic distribution of the sample nearest-neighbour correlation, thereby providing a basis for statistical inference by confidence intervals. In this paper, these limit theorems are extended to the statistically significant case of different vertical and horizontal interactions. Results at critical points are also obtained. Critical points clearly have the potential to seriously distort statistical inferences, especially in their immediate neighbourhoods. For our Ising model it turns out that such distortion is relatively minor. Surprisingly, in the two-parameter case the correlation between the sufficient statistics exhibits peculiar asymptotic behaviour resulting in a singular covariance matrix at critical points in the central limit theorem. Finally, at critical points, unusual norming constants are required for the central limit theorem, and our results are much more sensitive to the relative rate at which m,n tend to infinity.

In previous papers (1976), (1977) limit theorems were obtained for the classical Ising model in the absence of an external magnetic field, thereby providing a basis for asymptotic inference. The present paper extends these results to arbitrary external magnetic fields. Statistical inference for this model is important because its nearest-neighbour interactions provide a natural first approximation to spatial interaction among binary variables located on square lattices.
The most interesting behaviour occurs in zero field and at or beyond the critical point. In this case, the central limit result for nearest-neighbour interactions requires an unusual norming, the limiting variances may depend on the nature of the boundary conditions, and there cannot be any central limit result for external magnetic field. The implications of these phenomena for statistical inference are also discussed. In particular, the maximum likelihood estimator of magnetic field is not consistent. Rather it appears to have a non-trivial asymptotic distribution.

Cluster point processes are defined and studied using the probability generating functional. Necessary and sufficient conditions for the existence of a cluster process are proved and applied to particular cases. A result on mixing in cluster processes is established.

Recent yearly bole growth of individual trees, as estimated from height and annual growth ring measurements, is considered as a function of the number, distance and size of neighbours in a young Pinus rigida stand in New Jersey. Results are consistent with a model in which the growth of an individual is inversely related to the total effect of interference, and the contribution of each neighbour to this effect is additive in proportion to its size and inversely proportional to the square of its distance. While results show, as expected, that the effect of a neighbour decreases with its distance, they do not allow one to distinguish between alternative formulations with confidence, but a modified version of the model in which the effect of a neighbour decreases with its distance always resulted in a slightly improved fit over the original formulation in which a neighbour's effect decreases with the square of its distance. -from Author