Marked Spatial Point Patterns Research Articles

Reliability of metal additive manufactured materials from modeling the microstructure at different length scales

As metal additive manufacturing transitions from a prototypical manufacturing process to producing in-service parts, there is an urgent need for methodologies that are capable of predicting material response and reliability. There is a large body of research that examines physics-based methodologies and, more recently, data-driven models to simulate the relationships surrounding the process–structure–properties paradigm of additive manufactured materials. Both approaches share the same objective; namely, for a given set of process parameters, predict the resulting micro-structure and/or mechanical response of the printed material. Physics-based methodologies come with a high computational cost as the physical phenomena behind the manufacturing process spans multiple length and time scales. Conversely, data-driven models have a low computational cost at the expense of by-passing the physical mechanisms and instead finding correlation between diverse input and output data sets. As a result, the interpretation of data-driven models can be difficult and they have the potential to be bias and brittle. This paper presents a novel workflow that takes into consideration process parameters, build strategy and data obtained from standard material science tests to develop realistic, statistical, finite element models of material micro-structure that can be used to predict reliability without the need to explicitly model the physics behind the manufacturing process. Using micro-structural information extracted from as-fabricated material builds, we show that porosity can be modeled using marked spatial point patterns and scan track morphology can be idealized as concentric, multi-phase, half cylindrical volumes assembled according to the build strategy specified for the material. Furthermore, we show that the finite element model can be tuned to experimental test data, consistent with the concurrent or top-down approach in virtual materials testing. To illustrate this process, we apply the workflow to data obtained from various sources in the literature for as-fabricated AlSi10Mg manufactured using selective laser melting (SLM), a powder bed additive manufacturing process.

A BAYESIAN MARK INTERACTION MODEL FOR ANALYSIS OF TUMOR PATHOLOGY IMAGES.

With the advance of imaging technology, digital pathology imaging of tumor tissue slides is becoming a routine clinical procedure for cancer diagnosis. This process produces massive imaging data that capture histological details in high resolution. Recent developments in deep-learning methods have enabled us to identify and classify individual cells from digital pathology images at large scale. Reliable statistical approaches to model the spatial pattern of cells can provide new insight into tumor progression and shed light on the biological mechanisms of cancer. We consider the problem of modeling spatial correlations among three commonly seen cells observed in tumor pathology images. A novel geostatistical marking model with interpretable underlying parameters is proposed in a Bayesian framework. We use auxiliary variable MCMC algorithms to sample from the posterior distribution with an intractable normalizing constant. We demonstrate how this model-based analysis can lead to sharper inferences than ordinary exploratory analyses, by means of application to three benchmark datasets and a case study on the pathology images of 188 lung cancer patients. The case study shows that the spatial correlation between tumor and stromal cells predicts patient prognosis. This statistical methodology not only presents a new model for characterizing spatial correlations in a multitype spatial point pattern conditioning on the locations of the points, but also provides a new perspective for understanding the role of cell-cell interactions in cancer progression.

Prevalence of Inter-Tree Competition and Its Role in Shaping the Community Structure of a Natural Mongolian Scots Pine (Pinus sylvestris var. mongolica) Forest

Inter-tree competition is considered one of the most important ecological processes of forest development. However, its importance in structuring the spatial patterns of plant communities remains controversial. We collected observational data from two plots in a natural Mongolian Scots pine forest to study the contribution of competition to tree growth, mortality, and size inequality. We used the nearest neighbour method to determine the presence of competition, and unmarked and marked spatial point pattern analyses to test the density-dependent mortality effects and the spatial autocorrelation of tree size. We identified significant positive correlations between tree canopy diameter and nearest neighbour distance in both plots, which were more evident in the denser plot. The pair correlation functions of both plots indicated regular distribution patterns of living trees, and trees living in more crowded environments were more likely to die. However, the mark differentiation characteristics showed weak evidence of a negative spatial autocorrelation in tree size, particularly in the high-density plot. The high mortality rate of suppressed trees and weak asymmetric competition may have accounted for the lack of dissimilarity in tree size. This study showed that inter-tree competition is an important determinant of the development of Mongolian Scots pine forests.

Deviation test construction and power comparison for marked spatial point patterns

Deviation tests play an important role in testing distributional hypotheses in point process statistics. Such tests are based on differences between empirical summary functions and their theoretical counterparts, which depend on a distance variable r in a user-specified interval I. These differences are summarized to a single number, which serves then as the test statistic u. Statistical experience indicates that different distances r have different influence on u. We propose scalings of the differences to equalize the influence of the distances and show that the power of deviation tests can be improved by them. We further study how the power is affected by the other elements of deviation tests, which are the choice of the summary function, the deviation measure and the interval I. We consider in detail the construction of deviation tests for the particular case of testing the random labeling hypothesis, i.e. independence of the marks of a marked point process. By a large simulation study we come to clear statements about the role of the test elements. Furthermore, we demonstrate the potential of scaling by a data example from the literature.

ANALYSIS OF MINNESOTA COLON AND RECTUM CANCER POINT PATTERNS WITH SPATIAL AND NONSPATIAL COVARIATE INFORMATION.

Colon and rectum cancer share many risk factors, and are often tabulated together as "colorectal cancer" in published summaries. However, recent work indicating that exercise, diet, and family history may have differential impacts on the two cancers encourages analyzing them separately, so that corresponding public health interventions can be more efficiently targeted. We analyze colon and rectum cancer data from the Minnesota Cancer Surveillance System from 1998-2002 over the 16-county Twin Cities (Minneapolis-St. Paul) metro and exurban area. The data consist of two marked point patterns, meaning that any statistical model must account for randomness in the observed locations, and expected positive association between the two cancer patterns. Our model extends marked spatial point pattern analysis in the context of a log Guassian Cox process to accommodate spatially referenced covariates (local poverty rate and location within the metro area), individual-level risk factors (patient age and cancer stage), and related interactions. We obtain smoothed maps of marginal log-relative intensity surfaces for colon and rectum cancer, and uncover significant age and stage differences between the two groups. This encourages more aggressive colon cancer screening in the inner Twin Cities and their southern and western exurbs, where our model indicates higher colon cancer relative intensity.

Bayesian modeling of continuously marked spatial point patterns

Many analyses of continuously marked spatial point patterns assume that the density of points, with differing marks, is identical. However, as noted in the seminal paper of Goulard et al. (Scand J Stat 23:365---379, 1996), such an assumption is not realistic in many situations. For example, a stand of forest may have many more small trees than large, hence the model should allow for a higher density of points with small marks. In addition, as suggested by Ogata and Tanemura (Biometrics 41:421---433, 1985), the interaction between points should be a function of their mark, allowing, for example, the range of interaction for large trees to exceed that of smaller trees. The aforementioned articles use frequentist inferential techniques, but interval estimation presents difficulties due to the extremely complex distributional properties of the estimates; it might be possible, however, to use parametric bootstrap methodology for such inferences (Baddeley et al. in J Roy Stat Soc Ser B 67:617---666, 2005). We suggest the use of Bayesian inferential techniques. Although a Bayesian approach requires a complex, computational implementation of (reversible jump) Markov Chain Monte Carlo methodology, it enables a wide variety of inferences (including interval estimation). We demonstrate our approach by analyzing the well known Norway spruce dataset.

Spatial patterns on the sagebrush steppe/Western juniper ecotone

Analysis of the spatial patterns of woody plants is important to better understand the ecological processes that govern the worldwide expansion of woody plants across semi-arid ecosystems. Second-order characteristics of a marked spatial point pattern of western juniper (Juniperus occidentalis subsp. occidentalis) were analyzed using Ripley’s K-functions and the pair-correlation function g. The marked point process of crown diameters was produced via two-dimensional wavelet analysis of a fine scale aerial photograph at the woodland-steppe ecotone in the Reynolds Creek watershed in the Owyhee Mountains, southwestern Idaho. Colonization of J. occidentalis stems from mature juniper trees growing in rocky, fire resistant areas. Although these areas introduce components of natural heterogeneity within the landscape, the selected study area is situated within a single soil type, and we modeled the expansion of juniper plants into previously juniper-free sagebrush steppe as a homogeneous point process with constant intensity. Through this research we have identified two statistically significant spatial scales characteristic of J. occidentalis on the woodland/steppe ecotone: (1) We observed inhibition between J. occidentalis plants at distances <15 m, resulting in a regular pattern, rather than clumped or random. This short-distance inhibition can be attributed to competition for water and other resources. Recruitment of young J. occidentalis occurs significantly more often in a direction away from older plants, maximizing the utilization of water and light resources, and perpetuating the spread of the species into previously juniper-free shrub-steppe. (2) J. occidentalis on the ecotone exhibits significant clustering within a 30–60 m radius. Bivariate point pattern analyses provide evidence that, within a distance of 50–70 m, there is a spatial dependence in tree size such that medium trees are more likely than small trees to be close to large trees. We attribute these phenomena to the fact that juniper seeds are commonly dispersed by berry-eating birds with small territories (0.3–1 ha). Beyond a distance of 50–70 m, juniper plants are randomly distributed, suggesting that additional long-range seed dispersal processes are at work. We further acknowledge the importance of including a reference to spatial scale when formulating hypotheses in statistical analysis of spatio-temporal point patterns.

Spatial point-process statistics: concepts and application to the analysis of lead contamination in urban soil

This article explores the use of spatial point-process analysis as an aid to describe topsoil lead distribution in urban environments. The data used were collected in Glebe, an inner suburb of Sydney. The approach focuses on the locations of punctual events defining a point pattern, which can be statistically described through local intensity estimates and between-point distance functions. F-, G- and K-surfaces of a marked spatial point pattern were described and used to estimate nearest distance functions over a sliding band of quantiles belonging to the marking variable. This provided a continuous view of the point pattern properties as a function of the marking variable. Several random fields were simulated by selecting points from random, clustered or regular point processes and diffusing them. Recognition of the underlying point process using variograms derived from dense sampling was difficult because, structurally, the variograms were very similar. Point-event distance functions were useful complimentary tools that, in most cases, enabled clear recognition of the clustered processes. Spatial sampling quantile point pattern analysis was defined and applied to the Glebe data set. The analysis showed that the highest lead concentrations were strongly clustered. The comparison of this data set with the simulation confidence limits of a Poisson process, a short-radius clustered point process and a geostatistical simulation showed a random process for the third quartile of lead concentrations but strong clustering for the data in the upper quartile. Thus the distribution of topsoil lead concentrations over Glebe may have resulted from several contamination processes, mainly from regular or random processes with large diffusion ranges and short-range clustered processes for the hot spots. Point patterns with the same characteristics as the Glebe experimental pattern could be generated by separate additive geostatistical simulation. Spatial sampling quantile point patterns statistics can, in an easy and accurate way, be used complementarily with geostatistical methods. Copyright © 2005 John Wiley & Sons, Ltd.

A Space-Time Survival Point Process for a Longleaf Pine Forest in Southern Georgia

Abstract 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.

A Space-Time Survival Point Process for a Longleaf Pine Forest in Southern Georgia

Abstract 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.

Estimation of Interaction Potentials of Marked Spatial Point Patterns Through the Maximum Likelihood Method

The likelihood procedure in estimating interaction potentials of spatial point patterns is developed for a set of point locations with attached marks, where the potential functions depend on the marks. Approximate log likelihood functions are derived under the assumption that the point patterns do not deviate much from the Poisson pattern in some sense. Some methods examining this assumption are provided. Analyses of ecological data sets are discussed through a model selection procedure.