Planar Point Patterns Research Articles

Resample-smoothing of Voronoi intensity estimators

Voronoi estimators are non-parametric and adaptive estimators of the intensity of a point process. The intensity estimate at a given location is equal to the reciprocal of the size of the Voronoi/Dirichlet cell containing that location. Their major drawback is that they tend to paradoxically under-smooth the data in regions where the point density of the observed point pattern is high, and over-smooth where the point density is low. To remedy this behaviour, we propose to apply an additional smoothing operation to the Voronoi estimator, based on resampling the point pattern by independent random thinning. Through a simulation study we show that our resample-smoothing technique improves the estimation substantially. In addition, we study statistical properties such as unbiasedness and variance, and propose a rule-of-thumb and a data-driven cross-validation approach to choose the amount of smoothing to apply. Finally we apply our proposed intensity estimation scheme to two datasets: locations of pine saplings (planar point pattern) and motor vehicle traffic accidents (linear network point pattern).

Spatial analysis of data frequency and severity of rural accidents

In this paper, to study the frequency and severity of traffic accidents, a combination of geographic information system and spatial analysis has been used. For this purpose, methods of planar and network statistics including kernel density estimation and nearest neighbor distance analysis were applied for three types of accidents including fatal, injury, and material property damages. In addition, drawbacks of the traditional planar point pattern and the benefits of network analysis are offered. The network statistics are executed through SANET version 4 and ArcMap on the Semnan–Garmsar road in Iran. The nearest neighbor output analysis is consist of four curves, and regarding the fact that the amounts for accidents resulting in fatalities, injuries, and damages, the observed value curve is above the 5% confidence interval; accidents in the study region are more clustered than would be expected by random chance.

Effect of sample plot size and shape on estimates of structural indices: A case study in mature silver birch (Betula pendula Roth) dominating stand in Järvselja

Abstract Structural indices and characteristics have often been employed for assessing different aspects of forest stands. In order to assess the performance of such measures in forest sample plots of different size and shape, the current study is based on a data set for a relatively large (1 ha) silver birch dominated stand, and five simulated planar point patterns (a Poisson process, two clustered and two regular patterns) with similar intensity of points distributed on an area equal to the silver birch stand as reference stands. The analyses was based on repeated samplings of the stands, with randomly placed circular or square shaped plots of different sizes, with area ranging from 0.007 to 0.636 ha. Similar centre positions were considered for plots of different shapes, but the locations of the plot centres differed with size, so that plots of different sizes were able to cover the maximum possible area of reference stands. Some structural indices accounting for one or four nearest neighbours, and also the Point-pair correlation function was then quantified for each random sample plot. We used standard deviation and the differences of the mean estimates from the expected values to assess the sensitivity of the structural measures to the size and shape of sampling plots. The results indicated the significant effect of plot size and shape on structural indices. The correlation between plot size and estimates precision was positive, and small plots, regardless of their shapes, showed big variation among them while circular plots often provided more accurate estimates. Even though, as expected, expanding the plots increased the precision of the neighbourhood-based indices’ estimates (spatial measures), and also resulted in inaccurate estimates for some reference stands, depending on the arrangement of trees/points within that stand.

Assessing minimum contrast parameter estimation for spatial and spatiotemporal log‐Gaussian Cox processes

The univariate log‐Gaussian Cox process (LGCP) has shown considerable potential for the flexible modelling of the spatial, and more recently, spatiotemporal, intensity functions of planar point patterns within a restricted region in space. Its flexibility and mathematical tractability are partly offset by the need to acquire sensible estimates of the parameters controlling the dependence structure of the Gaussian field given the observed data. The method of minimum contrast, which compares theoretical descriptors of the process with their non‐parametric counterparts in order to obtain the required estimates, is arguably the most popular in practice to date. This article provides a comprehensive set of simulation studies focused on gauging the performance and impact of minimum contrast methods for parameter estimation of these processes. Results indicate that concerns over arbitrariness of implementation of minimum contrast give way to satisfactory practical performance.

A brief introduction to computer-intensive methods, with a view towards applications in spatial statistics and stereology

Computer-intensive methods may be defined as data analytical procedures involving a huge number of highly repetitive computations. We mention resampling methods with replacement (bootstrap methods), resampling methods without replacement (randomization tests) and simulation methods. The resampling methods are based on simple and robust principles and are largely free from distributional assumptions. Bootstrap methods may be used to compute confidence intervals for a scalar model parameter and for summary statistics from replicated planar point patterns, and for significance tests. For some simple models of planar point processes, point patterns can be simulated by elementary Monte Carlo methods. The simulation of models with more complex interaction properties usually requires more advanced computing methods. In this context, we mention simulation of Gibbs processes with Markov chain Monte Carlo methods using the Metropolis-Hastings algorithm. An alternative to simulations on the basis of a parametric model consists of stochastic reconstruction methods. The basic ideas behind the methods are briefly reviewed and illustrated by simple worked examples in order to encourage novices in the field to use computer-intensive methods.

Improved Approximation Bounds for Planar Point Pattern Matching

We analyze the performance of simple algorithms for matching two planar point sets under rigid transformations so as to minimize the directed Hausdorff distance between the sets. This is a well studied problem in computational geometry. Goodrich, Mitchell, and Orletsky presented a very simple approximation algorithm for this problem, which computes transformations based on aligning pairs of points. They showed that their algorithm achieves an approximation ratio of 4. We introduce a modification to their algorithm, which is based on aligning midpoints rather than endpoints. This modification has the same simplicity and running time as theirs, and we show that it achieves a better approximation ratio of roughly 3.14. We also analyze the approximation ratio in terms of a instance-specific parameter that is based on the ratio between diameter of the pattern set to the optimum Hausdorff distance. We show that as this ratio increases (as is common in practical applications) the approximation ratio approaches 3 in the limit. We also investigate the performance of the algorithm by Goodrich et al. as a function of this ratio, and present nearly matching lower bounds on the approximation ratios of both algorithms.

Uniform network transformation for points pattern analysis on a non-uniform network

In the real world, there are many kinds of phenomena that are represented by points on a network, such as traffic accidents on a street network. To analyse these phenomena, the basic point pattern methods (i.e. the nearest neighbour distance method, the quadrat method, the K-function method and the clumping method) defined on a plane (referred to as the planar basic point pattern methods) are extended to the basic point pattern methods on a network (referred to as the network basic point pattern methods). However, like the planar basic point pattern methods, the network basic point pattern methods assume a uniform network and this assumption is hard to accept when analysing actual phenomena. To overcome this limitation, this paper formulates a transformation, called the uniform network transformation, that transforms a non-uniform network into a uniform network. This transformation provides a simple procedure for analysing point patterns on non-uniform networks: first, a given non-uniform network is transformed into a uniform network; second, the network basic point pattern methods (which assume a uniform network) are applied to this transformed uniform network. No modification to the network basic point pattern methods is necessary. The paper also shows an actual application of this transformation to traffic accidents in Chosei, Japan.

Explorative statistical analysis of planar point processes in microscopy

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.

Morphological Characterization of Point Patterns

A triplet of function s for the statistical characterization of planar point patterns is introduced. They are related to the integral-geometric quantities area, boundary length and Euler number of patterns of discs centred at the given points. These functions are able to give information on the distribution of a given point pattern which the traditional summary statistics of point process theory do not offer and so can lead to an improved statistical description. The paper describes the statistical estimation of the new characteristics. Some examples illustrate their application in the exploratory analysis of point patterns of tree positions in forests, in comparison to results obtained by means of second-order and distance characteristics.

Statistical analysis of intramembranous particles using freeze fracture specimens.

We studied the point processes of intramembranous particles of mitochondrial membranes from HeLa cells using the freeze fracture technique. Three groups - under normal conditions, after exposition with rotenone, and after exposition with sodium acid - were compared. First, we used several summary statistics in order to study the two-dimensional point patterns of intramembranous particles within each group. Then, we compared the patterns in different groups by bootstrap tests using the K-function and the nearest neighbour distance function G(r). Estimation of the G-function provided significant results but no significant differences between groups were found using the classical K-function; estimation of G(r) should therefore not be omitted when studying observed planar point patterns.

Measuring homogeneity of planar point-patterns by using kurtosis

Kurtosis is generally associated with measurements of peakedness of a distribution. In this paper, we suggest a method where kurtosis can be used as a measure of homogeneity of any quantifiable property on a planar surface. A 2-dimensional, continuous and uniform distribution has kurtosis equal to 5.6. This value is also the limiting value for a discrete uniform distribution defined on a regular, rectangular grid when the number of grid points tend to infinity. Measurements of a planar surface, taken at regular grid points, are considered as realizations of random fields. These are associated with 2-dimensional random variables from which the value of kurtosis can be computed and used as a measure of the homogeneity of the field. A deviation from 5.6 indicates that the stochastic variable is not uniformly distributed and that the corresponding random field is not homogeneous. The model is applied on the spatial variation of the roughness on the surface of newsprint, an application where homogeneity is very important.

Uniform LAN condition of planar Gibbsian point processes and optimality of maximum likelihood estimators of soft-core potential functions

A parametric model of planar point patterns in a bounded region is constructed using grand canonical Gibbsian point processes with soft-core potential functions. A simple and explicit condition that this model becomes a uniform locally asymptotic normal (ULAN) family will be given. From this result we can conclude that the maximum likelihood estimator of the potential function is asymptotically efficient for a wide class of loss functions.

Models for neighbor dependency in planar point patterns

Random processes are defined that generate planar point patterns in which the points have a tendency to cluster, or in which clustering is inhibited. Processes are also defined for labeling a given point pattern in such a way that neighboring points tend to have, or not to have, the same labels.

Recognition and recovery of the three-dimensional orientation of planar point patterns

A model-based approach is developed for recovering 3-dimensional surface orientation from a single image of an arbitrary planar point pattern or polygon. We define an iterative procedure which efficiently computes the correct surface orientation as well as solving the correspondence problem between the set of model features and the set of image features. There are no restrictions on the types of model point patterns and no a priori knowledge about the correspondence is assumed. The number of possible matches considered is linear in the number of points in the worst case, although experimental results show that using our heuristic for ordering candidate correspondences results in only a small constant number of matches that must be tried. The iterative procedure is also robust in that model tolerances and image distortions can be handled by relaxing the termination criterion; the resulting error in orientation is usually of the same order of magnitude as the distortion. Thus the method is of practical value because it uses arbitrary point sets and works on real, noisy data.

Recognition and recovery of the three-dimensional orientation of planar point patterns

Recognition and recovery of the three-dimensional orientation of planar point patterns

A Cautionary Example on the Use of Second-Order Methods for Analyzing Point Patterns

A popular approach to the analysis of mapped planar point patterns is through secondorder methods, where the object of attention is the variance of the number of points falling in a test set of a given size and shape, or the behaviour of all distances between pairs of points in the pattern. Standard references for the analysis of spatial data are works by Ripley (1981, Ch. 8) and Diggle (1983). Many of the methods for point-pattern analysis given in the literature are second-order in nature. The analysis of mapped point patterns is important in a variety of biological applications, ranging from ecology to physiology, and some examples are given in these books by Ripley and by Diggle. Suppose that an observed point pattern can be considered as a realization of a random point-process model which is stationary and isotropic. Of course this is a strong assumption which may not be justified in practice, but it is nevertheless the assumption under which much of the existing point-process methodology has been developed. The property of the underlying random process elicited, directly or indirectly, by second-order methods is the second-moment cumulative function K(t), which satisfies the following properties under suitable regularity conditions (see Ripley, 1977, p. 150): (i) if X is the intensity of the process, then XK(t) is the expected number of further points within distance t of a typical point of the process; (ii) specifying K(t) for all t is equivalent to specifying the variance of the number of points falling in any given set. The function K(t) defined by Ripley (1977) is an edge-corrected version of the empirical distribution function of all interpoint distances in the observed pattern, and provides an approximately unbiased estimator of K(t). The estimate K(t) can be used to construct tests of the hypothesis that an observed pattern is consistent with the 'completely random' Poisson point-process model, and, more importantly, to quantify the apparent deviation