Interindividual variance of male reproductive success (MRS) contributes to genetic drift, which in turn interacts with selection and migration to determine the short-term response of populations to rapid changes in their environment. Individual relative MRS can be estimated through paternity analysis and can be further dissected into fecundity and spatial components. Existing methods to achieve this decomposition either rely on the strong assumption of a random distribution of pollen donors (TwoGener) or estimate only the part of the variance of male fecundity that is explained by few covariates. We developed here a method to estimate jointly the whole variance of male fecundity and the pollen dispersal curve from the genotypic information of sampled seeds and their putative fathers and geographical information of all individuals in the study area. We modelled the relative individual fecundities as a log-normally distributed random effect. We used a Bayesian approach, well suited to the hierarchical nature of the model, to estimate these fecundities. When applied to Sorbus torminalis, the estimated variance of male fecundity corresponded to an effective density of trees 13 times lower than the observed density (d(obs)/d(ep ) approximately 13). This value is between the value (approximately 2) estimated with a classical mating model including three covariates (neighbourhood density, diameter, flowering intensity) that affect fecundity and the value (approximately 30) estimated with TwoGener. The estimated dispersal kernel was close to previous results. This approach allows fine monitoring of ongoing genetic drift in natural populations, and quantitative dissection of the processes contributing to drift, including human actions.

Standard geostatistical models assume second order stationarity of the underlying Random Function. In some instances, there is little reason to expect the spatial dependence structure to be stationary over the whole region of interest. In this paper, we introduce a new model for second order non-stationary Random Functions as a convolution of an orthogonal random measure with a spatially varying random weighting function. This new model is a generalization of the common convolution model where a non-random weighting function is used. The resulting class of non-stationary covariance functions is very general, flexible and allows to retrieve classes of closed-form nonstationary covariance functions known from the literature, for a suitable choices of the random weighting functions family. Under the framework of a single realization and local stationarity, we develop parameter inference procedure of these explicit classes of non-stationary covariance functions. From a local variogram non-parametric kernel estimator, a weighted local least-squares approach in combination with kernel smoothing method is developed to estimate the parameters. Performances are assessed on two real datasets: soil and rainfall data. It is shown in particular that the proposed approach outperforms the stationary one, according to several criteria. Beyond the spatial predictions, we also show how conditional simulations can be carried out in this non-stationary framework.

Random function models and kriging constitute the core of the geostatistical methods created by Georges Matheron in the 1960s and further developed at the research center he created in 1968 at Ecole des Mines de Paris, Fontainebleau. Initially developed to avoid bias in the estimation of the average grade of mining panels delimited for their exploitation, kriging received progressively applications in all domains of natural resources evaluation and earth sciences, and more recently in completely new domains, for example, the design and analysis of computer experiments (DACE). While the basic theory of kriging is rather straightforward, its application to a large diversity of situations requires extensions of the random function models considered and sound solutions to practical problems. This chapter presents the origins of kriging as well as the development of its theory and its applications along the last fifty years. More details are given for methods presently in development to efficiently handle kriging in situations with a large number of data and a nonstationary behavior, notably the Gaussian Markov random field (GMRF) approximation and the stochastic partial differential (SPDE) approach, with a synthetic case study concerning the latter. IntroductionThe creation of the IAMG is a landmark of year 1968, which motivates the present book. Another important event of this year is the foundation of a research center of Ecole des Mines de Paris dedicated to geostatistics and mathematical morphology, two disciplines created by Georges Matheron. Concerning geostatistics, this research center was about to develop the applications of kriging, invented by Matheron several years earlier. The theory of kriging seems so straightforward that