Martin Schlather

Models for Stationary Max-Stable Random Fields

Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.

Stochastic Models That Separate Fractal Dimension and the Hurst Effect

Fractal behavior and long-range dependence have been observed in an astonishing number of physical, biological, geological, and socioeconomic systems. Time series, profiles, and surfaces have been characterized by their fractal dimension, a measure of roughness, and by the Hurst coefficient, a measure of long-memory dependence. Both phenomena have been modeled and explained by self-affine random functions, such as fractional Gaussian noise and fractional Brownian motion. The assumption of statistical self-affinity implies a linear relationship between fractal dimension and Hurst coefficient and thereby links the two phenomena. This article introduces stochastic models that allow for any combination of fractal dimension and Hurst coefficient. Associated software for the synthesis of images with arbitrary, prespecified fractal properties and power-law correlations is available. The new models suggest a test for self-affinity that assesses coupling and decoupling of local and global behavior.

Stationary max-stable fields associated to negative definite functions

Let Wi, i∈ℕ, be independent copies of a zero-mean Gaussian process {W(t), t∈ℝd} with stationary increments and variance σ2(t). Independently of Wi, let ∑i=1∞δUi be a Poisson point process on the real line with intensity e−y dy. We show that the law of the random family of functions {Vi(⋅), i∈ℕ}, where Vi(t)=Ui+Wi(t)−σ2(t)/2, is translation invariant. In particular, the process η(t)=⋁i=1∞Vi(t) is a stationary max-stable process with standard Gumbel margins. The process η arises as a limit of a suitably normalized and rescaled pointwise maximum of n i.i.d. stationary Gaussian processes as n→∞ if and only if W is a (nonisotropic) fractional Brownian motion on ℝd. Under suitable conditions on W, the process η has a mixed moving maxima representation.

Using Whole-Genome Sequence Data to Predict Quantitative Trait Phenotypes in Drosophila melanogaster

Predicting organismal phenotypes from genotype data is important for plant and animal breeding, medicine, and evolutionary biology. Genomic-based phenotype prediction has been applied for single-nucleotide polymorphism (SNP) genotyping platforms, but not using complete genome sequences. Here, we report genomic prediction for starvation stress resistance and startle response in Drosophila melanogaster, using ∼2.5 million SNPs determined by sequencing the Drosophila Genetic Reference Panel population of inbred lines. We constructed a genomic relationship matrix from the SNP data and used it in a genomic best linear unbiased prediction (GBLUP) model. We assessed predictive ability as the correlation between predicted genetic values and observed phenotypes by cross-validation, and found a predictive ability of 0.239±0.008 (0.230±0.012) for starvation resistance (startle response). The predictive ability of BayesB, a Bayesian method with internal SNP selection, was not greater than GBLUP. Selection of the 5% SNPs with either the highest absolute effect or variance explained did not improve predictive ability. Predictive ability decreased only when fewer than 150,000 SNPs were used to construct the genomic relationship matrix. We hypothesize that predictive power in this population stems from the SNP–based modeling of the subtle relationship structure caused by long-range linkage disequilibrium and not from population structure or SNPs in linkage disequilibrium with causal variants. We discuss the implications of these results for genomic prediction in other organisms.

Fast and Exact Simulation of Large Gaussian Lattice Systems in ℝ2: Exploring the Limits

The circulant embedding technique allows for the fast and exact simulation of stationary and intrinsically stationary Gaussian random fields. The method uses periodic embeddings and relies on the fast Fourier transform. However, exact simulations require that the periodic embedding is nonnegative definite, which is frequently not the case for two-dimensional simulations. This work considers a suggestion by Michael Stein, who proposed nonnegative definite periodic embeddings based on suitably modified, compactly supported covariance functions. Theoretical support is given to this proposal, and software for its implementation is provided. The method yields exact simulations of planar Gaussian lattice systems with 106 and more lattice points for wide classes of processes, including those with powered exponential, Matérn, and Cauchy covariances.

Analogies and correspondences between variograms and covariance functions

Variograms and covariance functions are key tools in geostatistics. However, various properties, characterizations, and decomposition theorems have been established for covariance functions only. We present analogous results for variograms and explore the connections with covariance functions. Our findings include criteria for covariance functions on intervals, and we apply them to exponential models, fractional Brownian motion, and locally polynomial covariances. In particular, we characterize isotropic locally polynomial covariance functions of degree 3.

On the second-order characteristics of marked point processes

This paper aims to combine the central limit theorem with the limit theorems in extreme value theory through a parametrized class of limit theorems where the former ones appear as special cases. To this end the limit distributions of suitably centered and normalized

Some covariance models based on normal scale mixtures

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Predicting Genetic Values: a Kernel-Based Best Linear Unbiased Prediction with Genomic Data

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Interpolation of spatial data – A stochastic or a deterministic problem?

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