On low-dimensional projections of high-dimensional distributions

Let P be a probability distribution on q -dimensional space. The so-called Diaconis-Freedman effect means that for a fixed dimension d<scale mixture of spherically symmetric Gaussian distributions. The present paper provides necessary and sufficient conditions for this phenomenon in a suitable asymptotic framework with increasing dimension q . It turns out, that the conditions formulated by Diaconis and Freedman (1984) are not only sufficient but necessary as well. Moreover, letting P ^ be the empirical distribution of n independent random vectors with distribution P , we investigate the behavior of the empirical process n √ (P ^ −P) under random projections, conditional on P ^ .

Global Optimization with Sparse and Local Gaussian Process Models

We present a novel surrogate model-based global optimization framework allowing a large number of function evaluations. The method, called SpLEGO, is based on a multi-scale expected improvement (EI) framework relying on both sparse and local Gaussian process (GP) models. First, a bi-objective approach relying on a global sparse GP model is used to determine potential next sampling regions. Local GP models are then constructed within each selected region. The method subsequently employs the standard expected improvement criterion to deal with the exploration-exploitation trade-off within selected local models, leading to a decision on where to perform the next function evaluation(s). The potential of our approach is demonstrated using the so-called Sparse Pseudo-input GP as a global model. The algorithm is tested on four benchmark problems, whose number of starting points ranges from 102 to 104. Our results show that SpLEGO is effective and capable of solving problems with large number of starting points, and it even provides significant advantages when compared with state-of-the-art EI algorithms.

Enriching plant diversity in grasslands by large-scale experimental sward disturbance and seed addition along gradients of land-use intensity

Aims The relationship between biodiversity and ecosystem functioning is among the most active areas of ecological research. Furthermore, enhancing the diversity of degraded ecosystems is a major goal in applied restoration ecology. In grasslands, many species may be locally absent due to dispersal or microsite limitation and may therefore profit from mechanical disturbance of the resident vegetation. We established a seed addition and disturbance experiment across several grassland sites of different land use to test whether plant diversity can be increased in these grasslands. Additionally, the experiment will allow us testing the consequences of increased plant diversity for ecosystem processes and for the diversity of other taxa in real-world ecosystems. Here we present details of the experimental design and report results from the first vegetation survey one year after disturbance and seed addition. Moreover, we tested whether the effects of seed addition and disturbance varied among grassland depending on their land use or pre-disturbance plant diversity. Methods A full-factorial experiment was installed in 73 grasslands in three regions across Germany. Grasslands were under regular agricultural use, but varied in the type and the intensity of management, thereby representing the range of management typical for large parts of Central Europe. The disturbance treatment consisted of disturbing the top 10 cm of the sward using a rotavator or rotary harrow. Seed addition consisted of sowing a high-diversity seed mixture of regional plant species. These species were all regionally present, but often locally absent, depending on the resident vegetation composition and richness of each grassland. Important findings One year after sward disturbance it had significantly increased cover of bare soil, seedling species richness and numbers of seedlings. Seed addition had increased plant species richness, but only in combination with sward disturbance. The increase in species richness, when both seed addition and disturbance was applied, was higher at high land-use intensity and low resident diversity. Thus, we show that at least the early recruitment of many species is possible also at high land-use intensity, indicating the potential to restore and enhance biodiversity of species-poor agricultural grasslands. Our newly established experiment provides a unique platform for broad-scale research on the land-use dependence of future trajectories of vegetation diversity and composition and their effects on ecosystem functioning.