Mari Myllymäki

Global envelope tests for spatial processes

Summary Envelope tests are a popular tool in spatial statistics, where they are used in goodness-of-fit testing. These tests graphically compare an empirical function T(r) with its simulated counterparts from the null model. However, the type I error probability α is conventionally controlled for a fixed distance r only, whereas the functions are inspected on an interval of distances I. In this study, we propose two approaches related to Barnards Monte Carlo test for building global envelope tests on I: ordering the empirical and simulated functions on the basis of their r-wise ranks among each other, and the construction of envelopes for a deviation test. These new tests allow the a priori choice of the global α and they yield p-values. We illustrate these tests by using simulated and real point pattern data.

Analysis of spatial structure of epidermal nerve entry point patterns based on replicated data

Epidermal nerve fiber (ENF) density and morphology are used to diagnose small fiber involvement in diabetic, HIV, chemotherapy induced, and other neuropathies. ENF density and summed length of ENFs per epidermal surface area are reduced, and ENFs may appear clustered within the epidermis in subjects with small fiber neuropathy compared to healthy subjects. Therefore, it is important to understand the spatial behaviour of ENFs in healthy and diseased subjects. This work investigates the spatial structure of ENF entry points, which are locations where the nerves enter the epidermis (the outmost living layer of the skin). The study is based on suction skin blister specimens from two body locations of 25 healthy subjects. The ENF entry points are regarded as a realization of a spatial point process and a second‐order characteristic, namely Ripley’s K function, is used to investigate the effect of covariates (e.g. gender) on the degree of clustering of ENF entry points. First, the effects of covariates are evaluated by means of pooled K functions for groups and, secondly, the statistical significance of the effects and individual variation are characterized by a mixed model approach. Based on our results the spatial pattern of ENFs in samples taken from calf is affected by the covariates but not in samples taken from foot.

Multiple Monte Carlo testing, with applications in spatial point processes

The rank envelope test (Myllymäki 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.

Development and evaluation of spatial point process models for epidermal nerve fibers

We propose two spatial point process models for the spatial structure of epidermal nerve fibers (ENFs) across human skin. The models derive from two point processes, Φb and Φe, describing the locations of the base and end points of the fibers. Each point of Φe (the end point process) is connected to a unique point in Φb (the base point process). In the first model, both Φe and Φb are Poisson processes, yielding a null model of uniform coverage of the skin by end points and general baseline results and reference values for moments of key physiologic indicators. The second model provides a mechanistic model to generate end points for each base, and we model the branching structure more directly by defining Φe as a cluster process conditioned on the realization of Φb as its parent points. In both cases, we derive distributional properties for observable quantities of direct interest to neurologists such as the number of fibers per base, and the direction and range of fibers on the skin. We contrast both models by fitting them to data from skin blister biopsy images of ENFs and provide inference regarding physiological properties of ENFs.

Bayesian inference for Gaussian excursion set generated Cox processes with set-marking

This work considers spatial Cox point processes where the random intensity is defined by a random closed set such that different point intensities appear in the two phases formed by the random set and its complement. The point pattern is observed as a set of point coordinates in a bounded region W⊂ℝd together with the information on the phase of the location of each point. This phase information, called set-marking, is not a representative sample from the random set, and hence it cannot be directly used for deducing properties of the random set. Excursion sets of continuous-parameter Gaussian random fields are applied as a flexible model for the random set. Fully Bayesian method and Markov chain Monte Carlo (MCMC) simulation is adopted for inferring the parameters of the model and estimating the random set. The performance of the new approach is studied by means of simulation experiments. Further, two forestry data sets on point patterns of saplings are analysed. The saplings grow in a clear-cut forest area where, before planting and natural seeding, the soil has been mounded forming a blotched soil structure. The tree densities tend to be different in the tilled patches and in the area outside the patches. The coordinates of each sapling have been measured and it is known whether this location is in a patch or outside. This example has been a motivation for the study.