Viktor Beneš

First-spike latency in the presence of spontaneous activity

A new statistical method for the estimation of the response latency is proposed. When spontaneous discharge is present, the first spike after the stimulus application may be caused by either the stimulus itself, or it may appear due to the prevailing spontaneous activity. Therefore, an appropriate method to deduce the response latency from the time to the first spike after the stimulus is needed. We develop a nonparametric estimator of the response latency based on repeated stimulations. A simulation study is provided to show how the estimator behaves with an increasing number of observations and for different rates of spontaneous and evoked spikes. Our nonparametric approach requires very few assumptions. For comparison, we also consider a parametric model. The proposed probabilistic model can be used for both single and parallel neuronal spike trains. In the case of simultaneously recorded spike trains in several neurons, the estimators of joint distribution and correlations of response latencies are also introduced. Real data from inferior colliculus auditory neurons obtained from a multielectrode probe are studied to demonstrate the statistical estimators of response latencies and their correlations in space.

Stereology of Extremes; Bivariate Models and Computation

The extremal shape factor of spheroidal particles is studied in the stereological context using the extreme value theory. The domain of attraction is invariant with respect to the transformation between spatial characteristics and planar sections characteristics. It is shown that for the Farlie-Gumbel-Morgenstern bivariate distribution of size and shape factor one can estimate the normalizing constants of shape factor conditioned by unknown particle size. The theoretical solution is followed by a detailed simulation study which demonstrates the use of estimation techniques developed. The method is useful for engineering applications in materials science, where microstructural extremes correlate with the properties of materials.

On second-order formulas in anisotropic stereology

Formulas for anisotropic stereology of fibre and surface processes are presented. They concern the relation between second-order quantities of the original process and its projections and sections. Various mathematical tools for handling these formulas are presented, including stochastic optimization. Finally applications in stereology are discussed, relating to intensity estimators using anisotropic sampling designs. Variances of these estimators are expressed and evaluated for processes with the Poisson property.

Functionals of spatial point processes having a density with respect to the Poisson process

U-statistics of spatial point processes given by a density with respect to a Poisson process are investigated. In the first half of the paper general relations are derived for the moments of the functionals using kernels from the Wiener-Ito chaos expansion. In the second half we obtain more explicit results for a system of U-statistics of some parametric models in stochastic geometry. In the logaritmic form functionals are connected to Gibbs models. There is an inequality between moments of Poisson and non-Poisson functionals in this case, and we have a version of the central limit theorem in the Poisson case.

Sliced inverse regression and independence in random marked sets with covariates

Dimension reduction of multivariate data was developed by Y. Guan for point processes with Gaussian random fields as covariates. The generalization to fibre and surface processes is straightforward. In inverse regression methods, we suggest slicing based on geometrical marks. An investigation of the properties of this method is presented in simulation studies of random marked sets. In a refined model for dimension reduction, the second-order central subspace is analyzed in detail. A real data pattern is tested for independence of a covariate.

Anisotropy estimation properties for microstructural models

The paper concerns the evaluation of anisotropy of planar fibre systems. A general method of the estimation error quantification and a test of anisotropy are suggested based on the Prokhorov distance between the theoretical and estimated rose of directions. By means of simulations, the exact distribution of the test statistics can be obtained for various microstructural models, given test systems, and estimation methods. The approach is demonstrated on real data from material research.

3D reconstruction of grains in polycrystalline materials using a tessellation model with curved grain boundaries

A compact and tractable representation of the grain structure of a material is an extremely valuable tool when carrying out an empirical analysis of the material’s microstructure. Tessellations have proven to be very good choices for such representations. Most widely used tessellation models have convex cells with planar boundaries. Recently, however, a new tessellation model — called the generalised balanced power diagram (GBPD) — has been developed that is very flexible and can incorporate features such as curved boundaries and non-convexity of cells. In order to use a GBPD to describe the grain structure observed in empirical image data, the parameters of the model must be chosen appropriately. This typically involves solving a difficult optimisation problem. In this paper, we describe a method for fitting GBPDs to tomographic image data. This method uses simulated annealing to solve a suitably chosen optimisation problem. We then apply this method to both artificial data and experimental 3D electron backscatter diffraction (3D EBSD) data obtained in order to study the properties of fine-grained materials with superplastic behaviour. The 3D EBSD data required new alignment and segmentation procedures, which we also briefly describe. Our numerical experiments demonstrate the effectiveness of the simulated annealing approach (compared to heuristic fitting methods) and show that GBPDs are able to describe the structures of polycrystalline materials very well.

Bias of a length density estimator based on vertical projections

Attention is paid to the stereological estimator of the length density of lineal features in three‐dimensional space. In Gokhale (J. Microsc. (1990) 159, 133–141), the estimator based on measurements performed on a projection of the content of vertical slices with a given thickness Δ was derived. The aim of this paper is to show that using five vertical slices with systematically chosen orientations yields a reliable result, i.e. the bias of the estimator is smaller than 5%. In the case of a choice of the vertical axis such that most lineal features are not perpendicular or nearly perpendicular to the vertical axis, a reliable result can be obtained using only three systematic orientations of vertical slices.

Space-Time Models in Stochastic Geometry

Space-time models in stochastic geometry are used in many applications. Mostly these are models of space-time point processes. A second frequent situation are growth models of random sets. The present chapter aims to present more general models. It has two parts according to whether the time is considered to be discrete or continuous. In the discrete-time case we focus on state-space models and the use of Monte Carlo methods for the inference of model parameters. Two applications to real situations are presented: a) evaluation of a neurophysiological experiment, b) models of interacting discs. In the continuous-time case we discuss space-time Levy-driven Cox processes with different second-order structures. Besides the wellknown separable models, models with separable kernels are considered. Moreover fully nonseparable models based on ambit processes are introduced. Inference for the models based on second-order statistics is developed.

A BAYESIAN ESTIMATE OF THE RISK OF TICK-BORNE DISEASES*

The paper considers the problem of estimating the risk of a tick-borne disease in a given region. A large set of epidemiological data is evaluated, including the point pattern of collected cases, the population map and covariates, i.e. explanatory variables of geographical nature, obtained from GIS.The methodology covers the choice of those covariates which influence the risk of infection most. Generalized linear models are used and AIC criterion yields the decision. Further, an empirical Bayesian approach is used to estimate the parameters of the risk model. Statistical properties of the estimators are investigated. Finally, a comparison with earlier results is discussed from the point of view of statistical disease mapping.