A review on anisotropy analysis of spatial point patterns

A spatial point pattern is called anisotropic if its spatial structure depends on direction. Several methods for anisotropy analysis have been introduced in the literature. In this paper, we give an overview of nonparametric methods for anisotropy analysis of (stationary) point patterns in $\mathbf{R}^2$ and $\mathbf{R}^3$. We discuss methods based on nearest neighbour and second order summary statistics as well as spectral and wavelet analysis. All techniques are illustrated on both a clustered and a regular example. Finally, we discuss methods for testing for isotropy as well as for estimating preferred directions in a point pattern.Comment: Submitted to Spatial Statistics -journal's special issue of the Spatial Statistics 2017 conferenc

Hierarchical second-order analysis of replicated spatial point patterns with non-spatial covariates

In this paper we propose a method for incorporating the effect of non-spatial covariates into the spatial second-order analysis of replicated point patterns. The variance stabilizing transformation of Ripley’s K function is used to summarize the spatial arrangement of points, and the relationship between this summary function and covariates is modelled by hierarchical Gaussian process regression. In particular, we investigate how disease status and some other covariates affect the level and scale of clustering of epidermal nerve fibres. The data are point patterns with replicates extracted from skin blister samples taken from 47 subjects.Peer reviewe

Point process models for sweat gland activation observed with noise

The aim of the paper is to construct spatial models for the activation of sweat glands for healthy subjects and subjects suffering from peripheral neuropathy by using videos of sweating recorded from the subjects. The sweat patterns are regarded as realizations of spatial point processes and two point process models for the sweat gland activation and two methods for inference are proposed. Several image analysis steps are needed to extract the point patterns from the videos and some incorrectly identified sweat gland locations may be present in the data. To take into account the errors we either include an error term in the point process model or use an estimation procedure that is robust with respect to the errors.Comment: 27 pages, 12 figure

Statistical modeling of diabetic neuropathy: Exploring the dynamics of nerve mortality

Diabetic neuropathy is a disorder characterized by impaired nerve function and reduction of the number of epidermal nerve fibers per epidermal surface. Additionally, as neuropathy related nerve fiber loss and regrowth progresses over time, the two-dimensional spatial arrangement of the nerves becomes more clustered. These observations suggest that with development of neuropathy, the spatial pattern of diminished skin innervation is defined by a thinning process which remains incompletely characterized. We regard samples obtained from healthy controls and subjects suffering from diabetic neuropathy as realisations of planar point processes consisting of nerve entry points and nerve endings, and propose point process models based on spatial thinning to describe the change as neuropathy advances. Initially, the hypothesis that the nerve removal occurs completely at random is tested using independent random thinning of healthy patterns. Then, a dependent parametric thinning model that favors the removal of isolated nerve trees is proposed. Approximate Bayesian computation is used to infer the distribution of the model parameters, and the goodness-of-fit of the models is evaluated using both non-spatial and spatial summary statistics. Our findings suggest that the nerve mortality process changes behaviour as neuropathy advances

Outlying observations and their influence on maximum pseudo-likelihood estimates of Gibbs point processes

Maximum pseudo-likelihood estimation method is an attractive method to estimate interaction parameters of Gibbs point processes. A drawback of the method is that it tends to overestimate interaction if there is strong repulsion between the points. We assumed that one reason for overestimation is that the method is sensitive to outlying points. Several techniques were used to detect outlying observations for the data of amacrine cells for which overestimation is suspected. Some strategies were then tested to take outliers into account in maximum pseudo-likelihood estimation

Multitype spatial point patterns with hierarchical interactions

Multitype spatial point patterns with hierarchical interactions are considered. Here hierarchical interaction means directionality: Points on a higher level of hierarchy affect the locations of points on the lower levels, but not vice versa. Such relations are common, for example, in ecological communities. Interacting point patterns are often modelled by Gibbs processes with pairwise interactions. However, these models are inherently symmetric, and the hierarchy can be acknowledged only when interpreting the results. We suggest the following trick allowing the inclusion of the hierarchical structure in the model. Instead of regarding the pattern as a realisation of a stationary multivariate point process, we build the pattern one type at a time according to the order of the hierarchy by using non-stationary univariate processes. As interactions connected to points x on a certain level are considered, the effect of the higher levels is interpreted as heterogeneity of the pattern x and the points on the lower levels are neglected owing to the hierarchical structure

Challenges in spatial point pattern analysis (motivate breakout E)

In the early spatial point process literature, point patterns were typically small, observed in 2D, had quite simple interaction structures, and there were no repetitions available. The observed point patterns were assumed to be realizations of stationary and isotropic point processes, and e.g. clustered patterns were typically modelled by assuming conditional independence between the cluster points given the Poisson distributed parents. However, large data sets (with repetitions) observed both in 2D and in 3D have become more and more common and it is less likely that stationarity and/or isotropy assumptions hold and that simple interaction structures are enough for realistic modelling of the data. In this talk, I will describe two examples of more complicated data sets where such a simple set-up is not enough. The first example concerns nerve fibre patterns on the epidermis, the outermost living layer of the skin. The spatial structure of nerves plays an important role in understanding how the nerve structure changes due to some small fibre neuropathy. The termination points of the nerve fibres form clusters around the base points of the nerves and even the parent (base) points tend to be clustered. In addition, the daughter points may not be located independently of each other, nor of the other parent points. The second example concerns air bubbles in polar ice. The air bubble patterns deep down in the ice are not isotropic (and some noise bubbles may occur in the data). Being able to estimate the deformation (anisotropy) can help physicists to determine the age of the ice at different depths.Non UBCUnreviewedAuthor affiliation: Chalmers University of TechnologyFacult

What mark variograms tell about spatial plant interactions

Many if not all data collected in ecology have both a spatial as well as a temporal dimension. This suggests the use of summary characteristics from spatial statistics to gain more refined insight into plant interactions. Spatial tree data can for example be considered as point patterns (formed by tree locations) with attached marks (e.g. tree sizes). If only the pattern of tree locations is of interest one can use for example the pair correlation function. If in addition, the sizes (or some other characteristics) of trees or other plants are of interest, marked summary statistics can be more suitable. In this paper, we propose the so-called mark variogram as a useful tool in ecological studies. This summary characteristic basically indicates how similar two plants within a certain distance from each other are. For example, if two plants are approximately of the same size, the mark variogram has small values, and if their sizes differ somewhat, the mark variogram has large values. Recently, there has been a lot of discussion on how to interpret the shape of mark variograms caused by pairs of plants with different sizes at close proximity. Such variogram shapes exhibiting so-called negative autocorrelation, another expression for high small-scaled size diversity, are assumed to indicate strong competition between plants. In this study, we reconstructed two spatial tree time series where negative autocorrelation has occurred and also simulated four alternative forest development paths in order to experimentally explore the causes of negative autocorrelation. Interestingly the results highlighted that man-made disturbances (e.g. thinnings) often result in a significant number of pairs of large and small trees at close proximity leading to negative autocorrelation. We could also show that whilst negative autocorrelation can be the consequence of natural forest development including competition processes, it can, however, also be the consequence of disturbances and of subsequent colonisation by small trees. Since disturbances play an important role in the development of negative autocorrelation, the mark variogram is a key summary characteristic in disturbance ecology