A comparative study of the edge—corrected kernel—based nearest neighbour density estimators for point processes

For a stationary long-range dependent point process N(.) with Palm distribution PO0, the Hurst index H =_sup{h : lim supt-o t-2h var N(0, t] = oo} is related to the moment index K - sup{k : EO(Tk) 3, it being known that equality holds for a stationary renewal process. Thus, a stationary point process for which K < 2 is necessarily long-range dependent with Hurst index greater than :. An extended example of a Wold process shows that a stationary point process can be both long-range count dependent and long-range interval dependent and have finite mean square interval length, i.e., EO(T2) <00oo.

Basic methods of explorative statistical analysis for stationary and isotropic planar point processes are briefly and informally reviewed. At the explorative level, planar point patterns may be characterized in terms of the intensity, the K-function and the pair correlation function. These second-order functions enable one to classify a given point process as completely random, clustering or repulsive. The repulsive behaviour may be quantified by an estimate of the hard-core distance. In the exploratory approach, the statistics are essentially free from model assumptions. Second-order spatial functions have been estimated to characterize genuine planar point processes in the macroscopic domain, for example in forestry, geography and epidemiology. For light microscopy and transmission electron microscopy, two situations are distinguished, which may be summarized as the genuine planar case and the stereological case. In the genuine planar case, a direct interpretation of the results of spatial statistics is feasible. Here, monolayers in cell culture, intramembranous particles on freeze fracture specimens and amacrine cells of the retina are mentioned as examples. In the stereological case, point patterns are generated by sections through 3D structures. Here the observed point patterns may arise as the centres of sectional profiles of particles, or as centres of sectional profiles of spatial fibre processes. In both situations, exploratory spatial point process statistics allow a quantitative characterization of sectional images for the purposes of group comparisons and classification. Moreover, for spatial fibre processes it has recently been shown that the observed pair correlation function of the centres of the fibre profiles is an estimate of the reduced pair correlation function of the fibre process in 3D. Hence for fibre processes a stereological interpretation of point process statistics obtained from sections is an additional option.

Point process statistics in atom probe tomography

Machine Learning Classification of Spatial Patterns of Malignant Cells Reveals Implications in Prognosis and Tumor Microenvironment Composition in Lymphoma

A review on anisotropy analysis of spatial point patterns

The upland mesic rainforests of eastern Australia have been described as a "mesothermal archipelago" where a chain of cool mountain "islands" arise from a warm "sea" of tropical and subtropical lowlands. An endemic freshwater crayfish belonging to the genus Euastacus is found on each of these mountain "islands." The Euastacus are particularly suitable for the study of evolution because each mountain harbors a unique species, there are many taxa present providing replication within the group and, most importantly, their distribution is linear, extending along a south-north axis. This group could have evolved by "simultaneous vicariance" where there was one vicariant separation event of a widespread ancestor, or by "south to north stepping stone dispersal" where there were long distance dispersal events from neighboring mountain islands, starting in the south and proceeding north in a dispersal-colonization wave. We used pairwise genetic distances between nearest geographic neighbors as a novel way to test the two hypotheses. If diversification was due to "south to north stepping stone dispersal," then pairwise genetic distances between nearest geographic neighbors should decrease progressively the farther north the taxon pairs are found, reflecting the decreasing periods of isolation. In this case there should be a negative correlation between the south to north rank order of nearest neighbors and pairwise genetic distances. A Spearman's correlation on 16S mtDNA pairwise genetic distances and geographic rank order was not significant, indicating there was no support for the south to north stepping stone dispersal hypothesis. If simultaneous vicariance was responsible for diversification then all nearest geographic neighbor taxon pairs should have similar genetic distances and, therefore, the variance in nearest neighbor distances should be zero, or close to it. To test if the observed variance was tending towards zero we developed a randomization test where nearest neighbor taxon pairs were assigned random genetic distances and the variances calculated. The observed variance lay in the < 0.05 range of the simulated variances, providing support for the simultaneous vicariance hypothesis. The data also suggest there was simultaneous vicariance of at least two ancestral Queensland lineages. The timing of this vicariant event was probably in the Pliocene, which is consistent with the divergence times reported for other Australian mesic rainforest restricted taxa.

The upland mesic rainforests of eastern Australia have been described as a “mesothermal archipelago” where a chain of cool mountain “islands” arise from a warm “sea” of tropical and subtropical lowlands. An endemic freshwater crayfish belonging to the genus Euastacus is found on each of these mountain “islands.” The Euastacus are particularly suitable for the study of evolution because each mountain harbors a unique species, there are many taxa present providing replication within the group and, most importantly, their distribution is linear, extending along a south-north axis. This group could have evolved by “simultaneous vicariance” where there was one vicariant separation event of a widespread ancestor, or by “south to north stepping stone dispersal” where there were long distance dispersal events from neighboring mountain islands, starting in the south and proceeding north in a dispersal-colonization wave. We used pairwise genetic distances between nearest geographic neighbors as a novel way to test the two hypotheses. If diversification was due to “south to north stepping stone dispersal,” then pairwise genetic distances between nearest geographic neighbors should decrease progressively the farther north the taxon pairs are found, reflecting the decreasing periods of isolation. In this case there should be a negative correlation between the south to north rank order of nearest neighbors and pairwise genetic distances. A Spearman's correlation on 16S mtDNA pairwise genetic distances and geographic rank order was not significant, indicating there was no support for the south to north stepping stone dispersal hypothesis. If simultaneous vicariance was responsible for diversification then all nearest geographic neighbor taxon pairs should have similar genetic distances and, therefore, the variance in nearest neighbor distances should be zero, or close to it. To test if the observed variance was tending towards zero we developed a randomization test where nearest neighbor taxon pairs were assigned random genetic distances and the variances calculated. The observed variance lay in the < 0.05 range of the simulated variances, providing support for the simultaneous vicariance hypothesis. The data also suggest there was simultaneous vicariance of at least two ancestral Queensland lineages. The timing of this vicariant event was probably in the Pliocene, which is consistent with the divergence times reported for other Australian mesic rainforest restricted taxa.

There is considerable interest in cell biology in determining whether, and to what extent, the spatial arrangement of nuclear objects affects nuclear function. A common approach to address this issue involves analyzing a collection of images produced using some form of fluorescence microscopy. We assume that these images have been successfully pre-processed and a spatial point pattern representation of the objects of interest within the nuclear boundary is available. Typically in these scenarios, the number of objects per nucleus is low, which has consequences on the ability of standard analysis procedures to demonstrate the existence of spatial preference in the pattern. There are broadly two common approaches to look for structure in these spatial point patterns. First a spatial point pattern for each image is analyzed individually, or second a simple normalization is performed and the patterns are aggregated. In this paper we demonstrate using synthetic spatial point patterns drawn from predefined point processes how difficult it is to distinguish a pattern from complete spatial randomness using these techniques and hence how easy it is to miss interesting spatial preferences in the arrangement of nuclear objects. The impact of this problem is also illustrated on data related to the configuration of PML nuclear bodies in mammalian fibroblast cells.

In this work we focus on relationships between stationary point process using spectral analysis techniques. The evaluation of these relationships is accomplished with the help of the product ratio of association (PRA), which is based on the cumulant densities of the point processes. The estimation procedure is obtained by smoothing the periodogram statistic, a function of the frequency domain. It is proved that the asymptotic distribution of the square root of the estimated PRA is Normal with a constant variance. Statistical tests for hypotheses concerning the independence of two point processes and the characterization of a Poisson process are proposed. Furthermore, approximate 95% pointwise confidence interval can be obtained for the estimated PRA. These results can be applied on stochastic systems involving as input and output stationary point processes. An illustrative example from the framework of neurophysiology is presented.

In this chapter we review some examples, methods, and recent results involving comparison of clustering properties of point processes. Our approach is founded on some basic observations allowing us to consider void probabilities and moment measures as two complementary tools for capturing clustering phenomena in point processes. As might be expected, smaller values of these characteristics indicate less clustering. Also, various global and local functionals of random geometric models driven by point processes admit more or less explicit bounds involving void probabilities and moment measures, thus aiding the study of impact of clustering of the underlying point process. When stronger tools are needed, directional convex ordering of point processes happens to be an appropriate choice, as well as the notion of (positive or negative) association, when comparison to the Poisson point process is considered. We explain the relations between these tools and provide examples of point processes admitting them. Furthermore, we sketch some recent results obtained using the aforementioned comparison tools, regarding percolation and coverage properties of the germ-grain model, the SINR model, subgraph counts in random geometric graphs, and more generally, U-statistics of point processes. We also mention some results on Betti numbers for Čech and Vietoris-Rips random complexes generated by stationary point processes. A general observation is that many of the results derived previously for the Poisson point process generalise to some “sub-Poisson” processes, defined as those clustering less than the Poisson process in the sense of void probabilities and moment measures, negative association or dcx-ordering.KeywordsPoint ProcessBetti NumberPoisson Point ProcessMoment MeasureCluster ComparisonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Many fresh and coastal waters are becoming increasingly turbid because of human activities, which may disrupt the visually mediated behaviours of aquatic organisms. Shoaling fish typically depend on vision to maintain collective behaviour, which has a range of benefits including protection from predators, enhanced foraging efficiency and access to mates. Previous studies of the effects of turbidity on shoaling behaviour have focussed on changes to nearest neighbour distance and average group‐level behaviours. Here, we investigated whether and how experimental shoals of three‐spined sticklebacks (Gasterosteus aculeatus) in clear (<10 Nephelometric Turbidity Units [NTU]) and turbid (~35 NTU) conditions differed in five local‐level behaviours of individuals (nearest and furthest neighbour distance, heading difference with nearest neighbour, bearing angle to nearest neighbour and swimming speed). These variables are important for the emergent group‐level properties of shoaling behaviour. We found an indirect effect of turbidity on nearest neighbour distances driven by a reduction in swimming speed, and a direct effect of turbidity which increased variability in furthest neighbour distances. In contrast, the alignment and relative position of individuals was not significantly altered in turbid compared to clear conditions. Overall, our results suggest that the shoals were usually robust to adverse effects of turbidity on collective behaviour, but group cohesion was occasionally lost during periods of instability.

Null-recurrent random walks with symmetric step distribution are mapped to give null-recurrent Markov chains on the interval (0, 1). These chains are used to construct stationary Wold processes (point processes with Markov-dependent intervals) with infinite intensity, which processes are simple (i.e., almost surely orderly) but not (analytically) orderly. As examples of point processes with infinite intensity, they are constructed without any intermediate step that relies on using point processes of finite intensity.

On the statistical properties of a stationary process sampled by a stationary point process

The simple stationarity of a previously derived equilibrium process of responses in a renewal inhibited stationary point process is established by deriving the joint distribution of the number of responses in contiguous intervals in the process. For a renewal inhibited Poisson process the variancetime function of the process is obtained; the distribution of an arbitrary between-response interval and the synchronous counting distribution are also derived following analytic justification of the required results. These results strengthen earlier results in the theory of stationary point processes. Three other point processes arising from the interaction are briefly discussed.

We consider spatially homogeneous marked point patterns in an unboundedly expanding convex sampling window. Our main objective is to identify the distribution of the typical mark by constructing an asymptotic $\chi^{2}$-goodness-of-fit test. The corresponding test statistic is based on a natural empirical version of the Palm mark distribution and a smoothed covariance estimator which turns out to be mean square consistent. Our approach does not require independent marks and allows dependences between the mark field and the point pattern. Instead we impose a suitable $\beta$-mixing condition on the underlying stationary marked point process which can be checked for a number of Poisson-based models and, in particular, in the case of geostatistical marking. In order to study test performance, our test approach is applied to detect anisotropy of specific Boolean models.