Martin Schlather

• University of Mannheim

In the last decade, a number of methods have been suggested to deal with large amounts of genetic data in genomic predictions. Yet, steadily growing population sizes and the suboptimal use of computational resources are pushing the practical application of these approaches to their limits. As an extension to the C/CUDA library miraculix, we have de...

In the last two decades, a number of so-called single-step models have been suggested to incorporate both pedigree and genomic data in statistical breeding value estimation for breeding programs. Yet, steadily growing population sizes and the suboptimal use of computational resources are pushing the practical application of these models to their li...

Stochastic models of point patterns in space and time are widely used to issue forecasts or assess risk, and often they affect societally relevant decisions. We adapt the concept of consistent scoring functions and proper scoring rules, which are statistically principled tools for the comparative evaluation of predictive performance, to the point p...

Pseudo cross-variograms appear naturally in the context of multivariate Brown–Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo cross-variogram, and further provide a Schoenberg-type result connecting pseudo cro...

We provide sufficient conditions of Pólya type which guarantee the positive definiteness of an isotropic 2 × 2-matrix-valued function in R and R3. Several isotropic bivariate covariance models have been proposed in literature, where all components of the covariance matrix are of the same parametric family, such as the bivariate Matérn model. Based...

Pseudo-variograms appear naturally in the context of multivariate Brown-Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo-variogram, and further provide a Schoenberg-type result connecting pseudo-variograms and...

Context Breeding programs aim at improving the genetic characteristics of livestock populations with respect to productivity, fitness and adaptation, while controlling negative effects such as inbreeding or health and welfare issues. As breeding is affected by a variety of interdependent factors, the analysis of the effect of certain breeding actio...

Background: Göttingen Minipigs (GMP) is the smallest commercially available minipig breed under a controlled breeding scheme and is globally bred in five isolated colonies. The genetic isolation harbors the risk of stratification which might compromise the identity of the breed and its usability as an animal model for biomedical and human disease....

The R-package MoBPS provides a computationally efficient and flexible framework to simulate complex breeding programs and compare their economic and genetic impact. Simulations are performed on the base of individuals. MoBPS utilizes a highly efficient implementation with bit-wise data storage and matrix multiplications from the associated R-packag...

Since the calculation of a genomic relationship matrix needs a large number of arithmetic operations, fast implementations are of interest. Our fastest algorithm is more accurate and 25x faster than a AVX double precision floating-point implementation.

The R-package MoBPS provides a computationally efficient and flexible framework to simulate complex breeding programs and compare their economic and genetic impact. Simulations are performed on the base of individuals and haplotypes are calculated on-the-fly by only saving founder haplotypes, points of recombination and mutations. MoBPS utilizes a...

The additive genomic variance in linear models with random marker effects can be defined as a random variable that is in accordance with classical quantitative genetics theory. Common approaches to estimate the genomic variance in random-effects linear models based on genomic marker data can be regarded as estimating the unconditional (or prior) ex...

The concept of haplotype blocks has been shown to be useful in genetics. Fields of application range from the detection of regions under positive selection to statistical methods that make use of dimension reduction... The concept of haplotype blocks has been shown to be useful in genetics. Fields of application range from the detection of regions...

Sup‐normalized spectral functions form building blocks of max‐stable and Pareto processes and therefore play an important role in modelling spatial extremes. For one of the most popular examples, the Brown–Resnick process, simulation is not straightforward. In this paper, we generalize two approaches for simulation via Markov chain Monte Carlo meth...

The additive genomic variance in linear models with random marker effects can be defined as a random variable that is in accordance with classical quantitative genetics theory. Common approaches to estimate the genomic variance in random-effects linear models based on genomic marker data can be regarded as the unconditional (or prior) expectation o...

Sup-normalized spectral functions form building blocks of max-stable and Pareto processes and therefore play an important role in modeling spatial extremes. For one of the most popular examples, the Brown-Resnick process, simulation is not straightforward. In this paper, we generalize two approaches for simulation via Markov Chain Monte Carlo metho...

Background: Domestication has led to substantial phenotypic and genetic variation in domestic animals. In pigs, the size of so called minipigs differs by one order of magnitude compared to breeds of large body size. We used biallelic SNPs identified from re-sequencing data to compare various publicly available wild and domestic populations against...

The concept of haplotype blocks has been shown to be useful in genetics. Fields of application range from the detection of regions under positive selection to statistical methods that make use of dimension reduction. We propose a novel approach ("HaploBlocker") for defining and inferring haplotype blocks that focuses on linkage instead of the commo...

Circulant embedding is a powerful algorithm for fast simulation of stationary Gaussian random fields on a rectangular grid in , which works perfectly for compactly supported covariance functions. Cut‐off circulant embedding techniques have been developed for univariate random fields for dimensions up to and rely on the modification of a covariance...

In a recent study, Jenko et al. (2015) proposed to accelerate genetic progress by integrating a genome editing (GE) step in genomic breeding programs. This concept, called 'promotion of alleles by genome editing' (PAGE) was implemented in a simulation study suggesting a substantial extra increase of genetic gain. As an example, editing in each gene...

The Mat\'ern hard-core processes are classical examples for point process models obtained from (marked) Poisson point processes. Points of the original Poisson process are deleted according to a dependent thinning rule, resulting in a process whose points have a prescribed hard-core distance. We present a new model which encompasses recent approach...

The Mat\'ern hard-core processes are classical examples for point process models obtained from (marked) Poisson point processes. Points of the original Poisson process are deleted according to a dependent thinning rule, resulting in a process whose points have a prescribed hard-core distance. We present a new model which encompasses recent approach...

We present a model of a random field on a topological space $M$ that unifies well-known models such as the Poisson hyperplane tessellation model, the random token model, and the dead leaves model. In addition to generalizing these submodels from $\mathbb{R}^d$ to other spaces such as the $d$-dimensional unit sphere $\mathbb{S}^d$, our construction...

A new approach for evaluating time-trends in extreme values accounting also for spatial dependence is proposed. Based on exceedances over a space-time threshold, estimators for a trend function and for extreme value parameters are given, leading to a homogenization procedure for then applying stationary extreme value processes. Extremal dependence...

The integration of physical relationships into stochastic models is of major interest e.g. in data assimilation. Here, a multivariate Gaussian random field formulation is introduced, which represents the differential relations of the two-dimensional wind field and related variables such as streamfunction, velocity potential, vorticity and divergenc...

The integration of physical relationships into stochastic models is of major interest e.g. in data assimilation. Here, a multivariate Gaussian random field formulation is introduced, which represents the differential relations of the two-dimensional wind field and related variables such as streamfunction, velocity potential, vorticity and divergenc...

To improve the forecasts of weather extremes, we propose a joint spatial model for the observations and the forecasts, based on a bivariate Brown-Resnick process. As the class of stationary bivariate Brown-Resnick processes is fully characterized by the class of pseudo cross-variograms, we contribute to the theorical understanding of pseudo cross-v...

For a stochastic process {X t }t∈T with identical one-dimensional margins and upper endpoint τ up its tail correlation function (TCF) is defined through \(\chi ^{(X)}(s,t) = \lim _{\tau \to \tau _{\text {up}}} P(X_{s} > \tau \,\mid \, X_{t} > \tau )\). It is a popular bivariate summary measure that has been frequently used in the literature in orde...

Due to the discrepancy of the high energy demand for rapidly increasing milk production and limited feed intake in the transition period around parturition, dairy cows require considerable metabolic adaptations. We hypothesize that some cows are genetically less suited to cope with these metabolic needs than others, leading to adverse follow-up eff...

A simple variogram model with two parameters is presented that includes the power variogram for fractional Brownian motion, a modified De Wijsian model, the generalized Cauchy model and the multiquadric model. One parameter controls the sample path roughness of the process. The other parameter allows for a smooth transition between bounded and unbo...

We introduce a class of spatial stochastic processes in the max-domain of attraction of familiar max-stable processes. The new class is based on Cox processes and comprises models with short range dependence. We show that statistical inference is possible within the given framework, at least under some reasonable restrictions.

Multivariate data measured in space, such as temperature and pressure or the content of two metals in geological deposits, requires models that allow to incorporate spatial and cross-dependence of observations. We introduce some novel bivariate models, the powered exponential (or stable) covariance model and the Cauchy covariance model with flexibl...

A theorem of the Kolmogorov--Chentsov type is proved for random fields on a Riemannian manifold.

A theorem of the Kolmogorov--Chentsov type is proved for random fields on a Riemannian manifold.

Different variants of a mathematical model for carrier-mediated signal transduction are introduced with focus on the odor dose–electrophysiological response curve of insect olfaction. The latter offers a unique opportunity to observe experimentally the effect of an alteration in the carrier molecule composition on the signal molecule-dependent resp...

Simulation and Analysis of Random Fields

The understanding of non-random association between loci, termed linkage disequilibrium (LD), plays a central role in genomic research. Since causal mutations are generally not included in genomic marker data, LD between those and available markers is essential for capturing the effects of causal loci on localizing genes responsible for traits. Thu...

The ability to predict quantitative trait phenotypes from molecular polymorphism data will revolutionize evolutionary biology, medicine and human biology, and animal and plant breeding. Efforts to map quantitative trait loci have yielded novel insights into the biology of quantitative traits, but the combination of individually significant quantita...

The extremal coefficient function (ECF) of a max-stable process $X$ on some index set $T$ assigns to each finite subset $A\subset T$ the effective number of independent random variables among the collection $\{X_t\}_{t\in A}$. We introduce the class of Tawn-Molchanov processes that is in a 1:1 correspondence with the class of ECFs, thus also provin...

The metabolic adaptation of dairy cows during the transition period has been studied intensively in the last decades. However, until now, only few studies have paid attention to the genetic aspects of this process. Here, we present the results of a gene-based mapping and pathway analysis with the measurements of three key metabolites, (1) non-ester...

Modeling of and inference on multivariate data that have been measured in space, such as temperature and pressure, are challenging tasks in environmental sciences, physics and materials science. We give an overview over and some background on modeling with cross-covariance models. The R package RandomFields supports the simulation, the parameter es...

The tail correlation function (TCF) is a popular bivariate extremal dependence measure to summarize data in the domain of attraction of a max-stable process. For the class of TCFs, being largely unexplored so far, several aspects are contributed: (i) generalization of some mixing max-stable processes (ii) transfer of two geostatistical construction...

Although the extremes of high-frequency financial transaction data have a huge economic impact, basic characteristics of the data have not been addressed up to now. To capture dependence between the tail behavior of inter-transaction returns and the pattern of transaction times, this paper combines marked point process (MPP) theory with extreme val...

[This corrects the article DOI: 10.1371/journal.pone.0126880.].

A simple variogram model with two parameters is presented that includes the power variogram for the fractional Brownian motion, a modified De Wijsian model, the generalized Cauchy model and the multiquadrics model. One parameter controls the smoothness of the process. The other parameter allows for a smooth parametrization between stationary and in...

Mean marks form a versatile toolbox in the analysis of marked point processes (MPPs). For ergodic processes, their definition is straightforward and practical application is well established. In the stationary non-ergodic case, though, different definitions of mark averages are possible and might be practically relevant. In this paper, the classica...

In this paper we provide the basis for new methods of inference for max-stable processes ξ on general spaces that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in...

Estimation of extreme value parameters from observations in the max-domain of attraction of a multivariate max-stable distribution commonly uses aggregated data such as block maxima. Multivariate peaks-over-threshold methods, in contrast, exploit additional information from the non-aggregated ‘large’ observations. We introduce an approach based on...

The key challenge during food-borne disease outbreaks, e.g. the 2011 EHEC/HUS outbreak in Germany, is the design of efficient mitigation strategies based on a timely identification of the outbreak's spatial origin. Standard public health procedures typically use case-control studies and tracings along food shipping chains. These methods are time-co...

The tail correlation function (TCF) is one of the most popular bivariate extremal dependence measures that has entered the literature under various names. We study to what extent the TCF can distinguish between different classes of well-known max-stable processes and identify essentially different processes sharing the same TCF.

Biological pathways provide rich information and biological context on the genetic causes of complex diseases. The logistic kernel machine test integrates prior knowledge on pathways in order to analyze data from genome-wide association studies (GWAS). In this study, the kernel converts the genomic information of 2 individuals into a quantitative v...

In order to incorporate the dependence between the spatial random fields of observed and forecasted maximal wind gusts, we propose to model them jointly by a bivariate Brown-Resnick process. As there is a one-to-one correspondence between bivariate Brown-Resnick processes and pseudo cross-variograms, station- ary Brown-Resnick processes can be char...

We use a Markov chain to model the ligand binding dynamics of a single molecule and show that its stationary distribution coincides with the laws of the Grand Canonical Ensemble. This way of deriving the equilibrium laws has the following advantages: Firstly, the derivation is short and does not require the knowledge of the Microcanonical, Canonica...

The normalized spectral representation of a max-stable process on a compact set is the unique representation where all spectral functions share the same supremum. Among the class of equivalent spectral representations of a process, the normalized spectral representation plays a distinctive role as a solution of two optimization problems in the cont...

Interpolation of spatial data is a very general mathematical problem with various applications. Different ways corresponding to different modeling assumptions have been proposed to tackle it. In geostatistics, it is assumed that the underlying structure of the data is a stochastic pro-cess which leads an interpolation procedure known as kriging. Th...

The expected level of linkage disequilibrium (LD) in a finite ideal population at equilibrium is of relevance for many applications in population and quantitative genetics. Several recursion formulae have been proposed during the last decades, whose derivations mostly contain heuristic parts and therefore remain mathematically questionable. We prop...

Objectives: The logistic kernel machine test (LKMT) is a testing procedure tailored towards high-dimensional genetic data. Its use in pathway analyses of case-control genome-wide association studies results from its computational efficiency and flexibility in incorporating additional information via the kernel. The kernel can be any positive defin...

In the first part of this work we formulated the decoupled sites representation for two different types of ligands and highlighted special properties of the case of n binding sites for ligand L 1 and one binding site for ligand L 2. Moreover, for this case, we identified the microstate constants as unique components all decoupled molecules share. I...

The decoupled sites representation (DSR) for one type of ligand allows to regard complex overall titration curves as sum of classical Henderson-Hasselbalch (HH) titration curves. In this work we transfer this theoretical approach to molecules with different types of interacting ligands (e.g. protons and electrons), prove the existence of decoupled...

Geostatistical approaches to modeling spatiotemporal data rely on flexible space–time covariance models. The area has been in vigorous development recently, and we review constructions of nonseparable spatiotemporal covariance functions that are based on geometric anisotropy, Fourier inversion, completely monotone functions, and dynamic physical mo...

The investigation of thermodynamic properties of ligand binding is a classical field of (bio)chemistry and (bio)physics. Commonly, an algebraic description using polynomials (e.g. the binding polynomial) and rational functions (e.g. titration curves) is used to characterize systems of molecules and their ligand(s). However, the algebraic model is a...

The ubiquitous assumption of normality for modeling spatial and spatio-temporal data can be understood for many reasons. A major one is that the multivariate normal distribution is completely characterized by its first two moments. In addition, the stability of multivariate normal distribution under summation and conditioning offers tractability an...

For a non-stationary or non-ergodic marked point process (MPP) on $\R^d$, the definition of averages becomes ambiguous as the process might have a different stochastic behavior in different realizations (non-ergodicity) or in different areas of the observation window (non-stationarity). We investigate different definitions for the moments, includin...

Covariance functions and variograms are the most important ingredients in the classical approaches to geostatistics. We give an overview over the approaches how models can be obtained. Variant types of scale mixtures turn out to be the most important way of construction. Some of the approaches are closely related to simulation methods of unconditio...

This paper provides the basis for new methods of inference for max-stable processes \xi\ on general spaces that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in s...

Estimation of extreme-value parameters from observations in the max-domain of attraction (MDA) of a multivariate max-stable distribution commonly uses aggregated data such as block maxima. Since we expect that additional information is contained in the non-aggregated, single "large" observations, we introduce a new approach of inference based on a...

We construct matrix-valued covariance functions and in ℝ and ℝ, starting from an arbitrary scalar-valued variogram. It is shown that sufficiently smooth random vector fields (RVFs) with these covariance functions have divergence-free and curl-free sample paths, respectively. Conversely, essentially all models with such sample paths can be obtained...

We focus on two dependency quantities of a max-stable random field $X$ on some space $T$: the extremal coefficient function $\theta$ which we define on finite sets of $T$ and the extremal correlation function $\chi(s,t)=\lim_{x \uparrow \infty} \PP(X_s \geq x \mid X_t \geq x)$. We fully characterize extremal coefficient functions $\theta$ by a prop...

Let $X_{i,n}, n\in\mathbb{N}, 1\leq i \leq n$, be a triangular array of independent $\R^d$-valued Gaussian random vectors with covariance matrices $\Sigma_{i,n}$. We give necessary conditions under which the row-wise maxima converge to some max-stable distribution which generalizes the class of H\"usler-Reiss distributions. In the bivariate case th...

Mean and standard deviation of phenotypic values and of the number of individual records per line. (PDF)

Results of variance component estimation using ASReml for startle response. Different linear models for individual trait records were investigated. (PDF)

Manhattan plot of the estimated SNP effects for starvation resistance for different chromosomes. The SNP effects were estimated using the GBLUP approach and sex-averaged phenotypic values of lines. Vertical lines indicate the significant SNP positions according to the GWAS of [27] using sex-pooled records. (PDF)

We give more details on the formula of [52] for the expected linkage disequilibrium as well as the derivation of the number of independently segregating chromosome segments [9] and the expected accuracy of prediction [20] in the case of D. melanogaster. We also derive the expected value of the genomic relationship matrix of [8] and show that , wh...

Manhattan plot of the estimated SNP effects for startle response for different chromosomes. The SNP effects were estimated using the GBLUP approach and sex-averaged phenotypic values of lines. Vertical lines indicate the significant SNP positions according to the GWAS of [27] using sex-pooled records. (PDF)

Predictive ability of 5-fold CV with GBLUP for starvation resistance using different set of SNPs with different average minor allele frequencies. Each boxplot shows the average predictive abilities for replicates of 5-fold CV using GBLUP and SNPs with different average minor allele frequencies. The different average minor allele frequencies are plo...

Variance components and heritabilities estimated from GBLUP using all lines. Variance components were estimated by maximum likelihood using the R-package “RandomFields” and its function “fitvario.” (PDF)

Results of variance component estimation using ASReml for starvation resistance. Different linear models for individual trait records were investigated. (PDF)

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 resis...

This paper deals with the question of conditional sampling and prediction for the class of stationary max-stable processes which allow for a mixed moving maxima representation. We develop an exact procedure for conditional sampling using the Poisson point process structure of such processes. For explicit calculations we restrict ourselves to the on...

This book arises as the natural continuation of the International Spring School "Advances and Challenges in Space-Time modelling of Natural Events," which took place in Toledo (Spain) in March 2010. This Spring School above all focused on young researchers (Master students, PhD students and post-doctoral researchers) in academics, extra-university...

We study the behavior of a real-valued and unobservable process (Y_t) under an extreme event of a related process (X_t) that is observable. Our analysis is motivated by the well-known GARCH model which represents two such sequences, i.e. the observable log returns of an asset as well as the hidden volatility process. Our results complement the find...

Genomic data provide a valuable source of information for modeling covariance structures, allowing a more accurate prediction of total genetic values (GVs). We apply the kriging concept, originally developed in the geostatistical context for predictions in the low-dimensional space, to the high-dimensional space spanned by genomic single nucleotide...