We propose a new measure to estimate the direction of information flux in multivariate time series from complex systems. This measure, based on the slope of the phase spectrum (Phase Slope Index) has invariance properties that are important for applications in real physical or biological systems: (a) it is strictly insensitive to mixtures of arbitrary independent sources, (b) it gives meaningful results even if the phase spectrum is not linear, and (c) it properly weights contributions from different frequencies. Simulations of a class of coupled multivariate random data show that for truly unidirectional information flow without additional noise contamination our measure detects the correct direction as good as the standard Granger causality. For random mixtures of independent sources Granger Causality erroneously yields highly significant results whereas our measure correctly becomes non-significant. An application of our novel method to EEG data (88 subjects in eyes-closed condition) reveals a strikingly clear front-to-back information flow in the vast majority of subjects and thus contributes to a better understanding of information processing in the brain.

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BackgroundGraphical Gaussian models are popular tools for the estimation of (undirected) gene association networks from microarray data. A key issue when the number of variables greatly exceeds the number of samples is the estimation of the matrix of partial correlations. Since the (Moore-Penrose) inverse of the sample covariance matrix leads to poor estimates in this scenario, standard methods are inappropriate and adequate regularization techniques are needed. Popular approaches include biased estimates of the covariance matrix and high-dimensional regression schemes, such as the Lasso and Partial Least Squares.ResultsIn this article, we investigate a general framework for combining regularized regression methods with the estimation of Graphical Gaussian models. This framework includes various existing methods as well as two new approaches based on ridge regression and adaptive lasso, respectively. These methods are extensively compared both qualitatively and quantitatively within a simulation study and through an application to six diverse real data sets. In addition, all proposed algorithms are implemented in the R package "parcor", available from the R repository CRAN.ConclusionIn our simulation studies, the investigated non-sparse regression methods, i.e. Ridge Regression and Partial Least Squares, exhibit rather conservative behavior when combined with (local) false discovery rate multiple testing in order to decide whether or not an edge is present in the network. For networks with higher densities, the difference in performance of the methods decreases. For sparse networks, we confirm the Lasso's well known tendency towards selecting too many edges, whereas the two-stage adaptive Lasso is an interesting alternative that provides sparser solutions. In our simulations, both sparse and non-sparse methods are able to reconstruct networks with cluster structures. On six real data sets, we also clearly distinguish the results obtained using the non-sparse methods and those obtained using the sparse methods where specification of the regularization parameter automatically means model selection. In five out of six data sets, Partial Least Squares selects very dense networks. Furthermore, for data that violate the assumption of uncorrelated observations (due to replications), the Lasso and the adaptive Lasso yield very complex structures, indicating that they might not be suited under these conditions. The shrinkage approach is more stable than the regression based approaches when using subsampling.

The efficiency of marker-assisted prediction of phenotypes has been studied intensively for different types of plant breeding populations. However, one remaining question is how to incorporate and counterbalance information from biparental and multiparental populations into model training for genome-wide prediction. To address this question, we evaluated testcross performance of 1652 doubled-haploid maize (Zea mays L.) lines that were genotyped with 56,110 single nucleotide polymorphism markers and phenotyped for five agronomic traits in four to six European environments. The lines are arranged in two diverse half-sib panels representing two major European heterotic germplasm pools. The data set contains 10 related biparental dent families and 11 related biparental flint families generated from crosses of maize lines important for European maize breeding. With this new data set we analyzed genome-based best linear unbiased prediction in different validation schemes and compositions of estimation and test sets. Further, we theoretically and empirically investigated marker linkage phases across multiparental populations. In general, predictive abilities similar to or higher than those within biparental families could be achieved by combining several half-sib families in the estimation set. For the majority of families, 375 half-sib lines in the estimation set were sufficient to reach the same predictive performance of biomass yield as an estimation set of 50 full-sib lines. In contrast, prediction across heterotic pools was not possible for most cases. Our findings are important for experimental design in genome-based prediction as they provide guidelines for the genetic structure and required sample size of data sets used for model training. IN the context of quantitative trait locus (QTL) mapping, multiparental populations have been suggested to be advantageous over biparental families due to their greater allelic diversity and the possibility of evaluating allelic effects in multiple genetic backgrounds (Muranty 1996;Xu 1998;Verhoeven et al. 2006). Especially if the multiparental population consists of several families connected by common parents, they can provide greater power of QTL detection and better resolution of QTL localization compared to individual families (Rebai and Goffinet 1993;Jannink and Jansen 2001;Blanc et al. 2006;Yu et al. 2008;Bardol et al. 2013;Mackay et al. 2014). In the context of genome-based prediction (Meuwissen et al. 2001), accuracies achieved within large biparental families are assumed to be the maximum that can be obtained with a given sample size (Crossa et al. 2014), because of medium allele frequencies, absence of genetic substructure, and equal linkage phases between markers and functional polymorphisms. However, prediction accuracies of newly generated progenies from different crosses will be poor. This is especially true if the respective germplasm exhibits broad allelic diversity and is unrelated to the biparental family from which single nucleotide polymorphism (...