Explorative statistical analysis of planar point processes in microscopy

Martingale techniques play a fundamental role in the analysis of point processes on \([0,\infty )\). In particular, the compensator of a point process uniquely determines and is determined by its distribution, and an explicit formula involving conditional interarrival distributions is well-known. In two dimensions there are many possible definitions of a point process compensator and we focus here on the one that has been the most useful in practice: the so-called *-compensator. Although existence of the *-compensator is well understood, in general it does not determine the law of the point process and it must be calculated on a case-by-case basis. However, it will be proven that when the point process satisfies a certain property of conditional independence (usually denoted by (F4)), the *-compensator determines the law of the point process and an explicit regenerative formula can be given. The basic building block of the planar model is the single line process (a point process with incomparable jump points). Its law can be characterized by a class of avoidance probabilities that are the two-dimensional counterpart of the survival function on \([0,\infty )\). Conditional avoidance probabilities then play the same role in the construction of the *-compensator as conditional survival probabilities do for compensators in one dimension.

Analysis of massive marked point patterns with stochastic partial differential equations

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).

In the present study, 53 cases of squamous cell carcinomas of the head and neck were characterized by a quantitative histological texture analysis based on principles of spatial statistics. A planar tessellation of the epithelial tumour component was generated by a skeletonization algorithm. The size distribution of the virtual cells of this planar tessellation, and the size distribution of the profiles of the tumour cell nuclei were estimated in terms of area and boundary length. The intensity, the reduced second moment function (K-function) and the pair correlation function of the point process of the centroids of the profiles of the tumour cell nuclei were also estimated. For both purposes, it is necessary to correct for edge effects, which we consider in this paper in some detail. Specifically, the point patterns of the tumour cell nuclei were considered as realizations of a point process, where the points exist only in the epithelial tumour component (the permitted phase) and not in the stroma (the forbidden phase). The methods allow to characterize each individual tumour by a series of summary statistics. The total set of cases was then partitioned into two groups: 19 cases without lymph node metastases (pN0), and 34 nodal positive cases (pN1 or pN2). Statistical analysis showed no significant differences between the intensities, the mean K-functions and the mean pair correlation functions of the tumour cell nucleus profiles of the two groups. However, there were some significant differences between the sizes of the virtual cells and of the nucleus profiles of the nodal negative cases as compared to the nodal positive cases. In a logistic regression analysis, one of the quantitative nuclear size variables (mean nuclear area) was found to be a significant predictor of lymph node metastasis, in addition to tumour stage. The study shows the potential of methods of spatial statistics for objective quantitative grading of squamous cell carcinomas of the head and neck, and provides an example for modelling histological point patterns as realizations of planar point processes occupying a reference phase which is only a partial component of the total tissue.

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

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.

This paper provides a rigorous foundation for the second-order analysis of stationary point processes on general spaces. It illuminates the results of Bartlett on spatial point processes, and covers the point processes of stochastic geometry, including the line and hyperplane processes of Davidson and Krickeberg. The main tool is the decomposition of moment measures pioneered by Krickeberg and Vere-Jones. Finally some practical aspects of the analysis of point processes are 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.

The pair correlation function is a fundamental spatial point process\ncharacteristic that, given the intensity function, determines second order\nmoments of the point process. Non-parametric estimation of the pair correlation\nfunction is a typical initial step of a statistical analysis of a spatial point\npattern. Kernel estimators are popular but especially for clustered point\npatterns suffer from bias for small spatial lags. In this paper we introduce a\nnew orthogonal series estimator. The new estimator is consistent and\nasymptotically normal according to our theoretical and simulation results. Our\nsimulations further show that the new estimator can outperform the kernel\nestimators in particular for Poisson and clustered point processes.\n

Point process statistics in atom probe tomography

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.

The rank envelope test (Myllymaki et al. in J R Stat Soc B, doi: 10.1111/rssb.12172 , 2016) is proposed as a solution to the multiple testing problem for Monte Carlo tests. Three different situations are recognized: (1) a few univariate Monte Carlo tests, (2) a Monte Carlo test with a function as the test statistic, (3) several Monte Carlo tests with functions as test statistics. The rank test has correct (global) type I error in each case and it is accompanied with a p-value and with a graphical interpretation which determines subtests and distances of the used test function(s) which lead to the rejection at the prescribed significance level of the test. Examples of null hypotheses from point process and random set statistics are used to demonstrate the strength of the rank envelope test. The examples include goodness-of-fit test with several test functions, goodness-of-fit test for a group of point patterns, test of dependence of components in a multi-type point pattern, and test of the Boolean assumption for random closed sets. A power comparison to the classical multiple testing procedures is given.

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.

Methods for the statistical analysis of stationary spatial point process data are now well established, methods for nonstationary processes less so. One of many sources of nonstationary point process data is a case-control study in environmental epidemiology. In that context, the data consist of a realization of each of two spatial point processes representing the locations, within a specified geographical region, of individual cases of a disease and of controls drawn at random from the population at risk. In this article, we extend work by Baddeley, Møller, and Waagepetersen (2000, Statistica Neerlandica54, 329-350) concerning estimation of the second-order properties of a nonstationary spatial point process. First, we show how case-control data can be used to overcome the problems encountered when using the same data to estimate both a spatially varying intensity and second-order properties. Second, we propose a semiparametric method for adjusting the estimate of intensity so as to take account of explanatory variables attached to the cases and controls. Our primary focus is estimation, but we also propose a new test for spatial clustering that we show to be competitive with existing tests. We describe an application to an ecological study in which juvenile and surviving adult trees assume the roles of controls and cases.

Suppose there is known the set of points visited by a transient random walk on the real line, but that there is no information concerning the order in which these points are visited. Such a set of points may aptly be called a random walk point process (RWPP), and the object of this paper is to study the class of such processes. This class arises as a generalization of certain one-dimensional clustering processes, and while this is how one of us (D.J.D.) originally came to consider them, it is not the only reason for persevering in this study. Like renewal processes, which are a special sub-class of RWPP's, any random walk point process can be specified by means of the step distribution of the underlying random walk. On the other hand our study does not show whether or not the converse question of determining the step distribution from the RWPP is always feasible. The more demanding question of determining the random walk itself from the RWPP is obviously possible only for very special step distributions. Those results which we obtain below in closed form appear to depend on assuming that at least one tail of the step distribution is exponential. However such results have immediate relevance to the problem of inferring the step distribution from the point process. In the simplest case considered (both tails exponential) we are led to a new model for a stationary point process on the real line that is typically highly over-dispersed. And a final raison d'etre for our study is to exhibit connections with other stochastic processes, notably Poisson cluster processes and, perhaps surprisingly, Markov branching processes and birth and death processes. In Section 2 we introduce our notation and quote and discuss results needed in the sequel from the theory of stationary point processes and the theory of random walks. Integral equations for the one-dimensional counting distribution of the RWPP are written down in Section 3, and their solution is derived when the step distribution has a double exponential form. In Section 4 the second order counting properties are considered and an explicit solution for the second order intensity is derived for the double exponential case. The inter-point properties of the process are considered in Section 5, explicit results being obtained when the dominant tail of the step distribution is exponential. The connection with birth and death processes is shown in that Section, and in the concluding Section 6, the connections with Poisson cluster processes and Markov branching processes are mentioned.