Excursion Set Research Articles

A supermartingale approach to Gaussian process based sequential design of experiments

Gaussian process (GP) models have become a well-established framework for the adaptive design of costly experiments, and notably of computer experiments. GP-based sequential designs have been found practically efficient for various objectives, such as global optimization (estimating the global maximum or maximizer(s) of a function), reliability analysis (estimating a probability of failure) or the estimation of level sets and excursion sets. In this paper, we study the consistency of an important class of sequential designs, known as stepwise uncertainty reduction (SUR) strategies. Our approach relies on the key observation that the sequence of residual uncertainty measures, in SUR strategies, is generally a supermartingale with respect to the filtration generated by the observations. This observation enables us to establish generic consistency results for a broad class of SUR strategies. The consistency of several popular sequential design strategies is then obtained by means of this general result. Notably, we establish the consistency of two SUR strategies proposed by Bect, Ginsbourger, Li, Picheny and Vazquez (Stat. Comput. 22 (2012) 773–793) – to the best of our knowledge, these are the first proofs of consistency for GP-based sequential design algorithms dedicated to the estimation of excursion sets and their measure. We also establish a new, more general proof of consistency for the expected improvement algorithm for global optimization which, unlike previous results in the literature, applies to any GP with continuous sample paths.

Identification and isotropy characterization of deformed random fields through excursion sets

Abstract A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.

Minkowski Tensors in Three Dimensions: Probing the Anisotropy Generated by Redshift Space Distortion

Abstract We apply the Minkowski tensor statistics to three-dimensional Gaussian random fields. Minkowski tensors contain information regarding the orientation and shape of excursion sets that is not present in the scalar Minkowski functionals. They can be used to quantify globally preferred directions and also provide information on the mean shape of the subsets of a field. This makes them ideal statistics to measure the anisotropic signal generated by redshift space distortion in the low-redshift matter density field. We review the definition of the Minkowski tensor statistics in three dimensions, focusing on two coordinate invariant quantities, and . We calculate the ensemble average of these 3 × 3 matrices for an isotropic Gaussian random field, finding that they are proportional to products of the identity matrix and a corresponding scalar Minkowski functional. We show how to numerically reconstruct and from discretely sampled fields and apply our algorithm to isotropic Gaussian fields generated from a linear ΛCDM matter power spectrum. We then introduce anisotropy by applying a linear redshift space distortion operator to the matter density field and find that both and exhibit a distinct signal characterized by inequality between their diagonal components. We discuss the physical origin of this signal and how it can be used to constrain the redshift space distortion parameter ϒ ≡ f/b.

Articulated Estimator Random Field and Geometrical Approach Applied in System Identification

A new point of view of robust statistics based on a geometrical approach is tackled in this paper. Estimation procedures are carried out from a new robust cost function based on a chaining of elementary convex norms. This chain is randomly articulated in order to treat more efficiently natural outliers in data-set. Estimated parameters are considered as random fields and each of them, named articulated estimator random field (AERF) is a manifold or stratum of a stratified space with Riemannian geometry properties. From a high level excursion set, a probability distribution model Msta is presented and a system model validation geometric criterion (SYMOVAGEC) for system model structures Msys based on Riccian scalar curvatures is proposed. Numerical results are drawn in a context of system identification.

Kinematic formula for heterogeneous Gaussian related fields

We provide a generalization of the Gaussian Kinematic Formula (GKF) in Taylor et al. (2006) for multivariate, heterogeneous Gaussian-related fields. The fields under consideration, f=F∘y, are non-Gaussian fields built out of smooth, independent Gaussian fields y=(y1,y2,..,yK) with heterogeneity in distribution amongst the individual building blocks. Our motivation comes from potential applications in the analysis of Cosmological Data (CMB). Specifically, future CMB experiments will be focusing on polarization data, typically modeled as isotropic vector-valued Gaussian related fields with independent, but non-identically distributed Gaussian building blocks; this necessitates such a generalization. Extending results (Taylor et al., 2006) to these more general Gaussian relatives with distributional heterogeneity, we present a generalized Gaussian Kinematic Formula (GKF). The GKF in this paper decouples the expected Euler characteristic of excursion sets into Lipschitz Killing Curvatures (LKCs) of the underlying manifold and certain Gaussian Minkowski Functionals (GMFs). These GMFs arise from Gaussian volume expansions of ellipsoidal tubes as opposed to the usual tubes in the Euclidean volume of tube formulae. The GMFs form a main contribution of this work that identifies this tubular structure and a corresponding volume of tubes expansion in which the GMFs appear.

Detection of z ∼ 2.3 Cosmic Voids from 3D Lyα Forest Tomography in the COSMOS Field

Abstract We present the most distant detection of cosmic voids (z ∼ 2.3) and the first detection of three-dimensional voids in the Lyα forest. We used a 3D tomographic map of the absorption with an effective comoving spatial resolution of 2.5 h −1 Mpc and a volume of 3.15 × 105 h −3 Mpc3, which was reconstructed from moderate-resolution Keck I/LRIS spectra of 240 background Lyman-break galaxies and quasars in a 0.16 deg2 footprint in the COSMOS field. Voids were detected using a spherical overdensity finder calibrated from hydrodynamical simulations of the intergalactic medium (IGM). This allows us to identify voids in the IGM corresponding to voids in the underlying matter density field, yielding a consistent volume fraction of voids in both data (19.5%) and simulations (18.2%). We fit excursion set models to the void radius function and compare the radially averaged stacked profiles of large voids (r > 5 h −1 Mpc) to stacked voids in mock observations and the simulated density field. Comparing with 432 coeval galaxies with spectroscopic redshifts in the same volume as the tomographic map, we find that the tomography-identified voids are underdense in galaxies by 5.95σ compared to random cells.

Confidence Regions for Spatial Excursion Sets From Repeated Random Field Observations, With an Application to Climate

ABSTRACTThe goal of this article is to give confidence regions for the excursion set of a spatial function above a given threshold from repeated noisy observations on a fine grid of fixed locations. Given an asymptotically Gaussian estimator of the target function, a pair of data-dependent nested excursion sets are constructed that are sub- and super-sets of the true excursion set, respectively, with a desired confidence. Asymptotic coverage probabilities are determined via a multiplier bootstrap method, not requiring Gaussianity of the original data nor stationarity or smoothness of the limiting Gaussian field. The method is used to determine regions in North America where the mean summer and winter temperatures are expected to increase by mid-21st century by more than 2 degrees Celsius.

An Analytic Formulation of the 21 cm Signal from the Early Phase of the Epoch of Reionization

Abstract We present an analytic formulation to model the fluctuating component of the H i signal from the epoch of reionization during the phase of partial heating. During this phase, we assume self-ionized regions, whose size distribution can be computed using excursion set formalism, to be surrounded by heated regions. We model the evolution of the heating profile around these regions (near zone) and their merger into the time-dependent background (far zone). We develop a formalism to compute the two-point correlation function for this topology, taking into account the heating autocorrelation and heating-ionization cross-correlation. We model the ionization and X-ray heating using four parameters: efficiency of ionization, ζ; number of X-ray photons per stellar baryon, N heat; spectral index of X-ray photons, α; and minimum frequency of X-ray photons, ν min. We compute the H i signal in the redshift range 10 < z < 20 for the ΛCDM model for a set of these parameters. We show that the H i signal for a range of scales 1–8 Mpc shows a peak strength of 100–1000 (mK)2 during the partially heated era. The redshift at which the signal makes a transition to a uniformly heated universe depends on the modeling parameters; e.g., if ν min is changed from 100 eV to 1 keV, this transition moves from z ≃ 15 to z ≃ 12. This result, along with the dependence of the H i signal on the modeling parameters, is in reasonable agreement with existing results from N-body simulations.

Intracell Interference Characterization and Cluster Interference for D2D Communication

The homogeneous spatial Poisson point process (SPPP) is widely used for spatial modeling of mobile terminals (MTs). This process is characterized by a homogeneous distribution, complete spatial independence, and constant intensity measure. However, it is intuitive to understand that the locations of MTs are neither homogeneous, due to inhomogeneous terrain, nor independent, due to homophilic relations. Moreover, the intensity is not constant due to mobility. Therefore, assuming an SPPP for spatial modeling is too simplistic, especially for modeling realistic emerging device-centric frameworks such as device-to-device (D2D) communication. In this paper, assuming inhomogeneity, positive spatial correlation, and random intensity measure, we propose a doubly stochastic Poisson process, a generalization of the homogeneous SPPP, to model D2D communication. To this end, we assume a permanental Cox process (PCP) and propose a novel Euler-Characteristic-based approach to approximate the nearest-neighbor distribution function. We also propose a threshold and spatial distances from an excursion set of a chi-square random field as interference control parameters for different cluster sizes. The spatial distance of the clusters is incorporated into a Laplace functional of a PCP to analyze the average coverage probability of a cellular user. A closed-form approximation of the spatial summary statistics is in good agreement with empirical results, and its comparison with an SPPP authenticates the correlation modeling of D2D nodes.

Quantile-based spatiotemporal risk assessment of exceedances

Structural characteristics of random field excursion sets defined by threshold exceedances provide meaningful indicators for the description of extremal behaviour in the spatiotemporal dynamics of environmental systems, and for risk assessment. In this paper a conditional approach for analysis at global and regional scales is introduced, performed by implementation of risk measures under proper model-based integration of available knowledge. Specifically, quantile-based measures, such as Value-at-Risk and Average Value-at-Risk, are applied based on the empirical distributions derived from conditional simulation for different threshold exceedance indicators, allowing the construction of meaningful dynamic risk maps. Significant aspects of the application of this methodology, regarding the nature and the properties (e.g. local variability, dependence range, marginal distributions) of the underlying random field, as well as in relation to the increasing value of the reference threshold, are discussed and illustrated based on simulation under a variety of scenarios.

The Excursion set approach: Stratonovich approximation and Cholesky decomposition

The excursion set approach is a framework for estimating how the number density of nonlinear structures in the cosmic web depends on the expansion history of the universe and the nature of gravity. A key part of the approach is the estimation of the first crossing distribution of a suitably chosen barrier by random walks having correlated steps: The shape of the barrier is determined by the physics of nonlinear collapse, and the correlations between steps by the nature of the initial density fluctuation field. We describe analytic and numerical methods for calculating such first up-crossing distributions. While the exact solution can be written formally as an infinite series, we show how to approximate it efficiently using the Stratonovich approximation. We demonstrate its accuracy using Monte-Carlo realizations of the walks, which we generate using a novel Cholesky-decomposition based algorithm, which is significantly faster than the algorithm that is currently in the literature.

Halo mass and weak galaxy-galaxy lensing profiles in rescaled cosmological N-body simulations

We investigate 3D density and weak lensing profiles of dark matter haloes predicted by a cosmology-rescaling algorithm for $N$-body simulations. We extend the rescaling method of Angulo & White (2010) and Angulo & Hilbert (2015) to improve its performance on intra-halo scales by using models for the concentration-mass-redshift relation based on excursion set theory. The accuracy of the method is tested with numerical simulations carried out with different cosmological parameters. We find that predictions for median density profiles are more accurate than $\sim 5\,\%$ for haloes with masses of $10^{12.0} - 10^{14.5} h^{-1}\,M_{\odot}$ for radii $0.05 < r/r_{200\text{m}} < 0.5$, and for cosmologies with $\Omega_\text{m} \in [0.15,\,0.40]$ and $\sigma_8 \in [0.6,\,1.0]$. For larger radii, $0.5 < r/r_{200\text{m}} < 5$, the accuracy degrades to $\sim20\,\%$, due to inaccurate modelling of the cosmological and redshift dependence of the splashback radius. For changes in cosmology allowed by current data, the residuals decrease to $\lesssim2\,\%$ up to scales twice the virial radius. We illustrate the usefulness of the method by estimating the mean halo mass of a mock galaxy group sample. We find that the algorithm's accuracy is sufficient for current data. Improvements in the algorithm, particularly in the modelling of baryons, are likely required for interpreting future (dark energy task force stage IV) experiments.

Effect of dark energy perturbation on cosmic voids formation

In this paper, we present the effects of dark energy perturbation on the formation and abundance of cosmic voids. We consider dark energy to be a fluid with a negative pressure characterised by a constant equation of state w and speed of sound $c^2_s$. By solving fluid equations for two components, namely, dark matter and dark energy fluids, we quantify the effects of dark energy perturbation on the sizes of top-hat voids. We also explore the effects on the size distribution of voids based on the excursion set theory. We confirm that dark energy perturbation negligibly affects the size evolution of voids;$c^2_s = 0$ varies the size only by 0.1% as compared to the homogeneous dark energy model. We also confirm that dark energy perturbation suppresses the void size when w < -1 and enhances the void size when $w > -1$ (Basse et al. 2011). In contrast to the negligible impact on the size, we find that the size distribution function on scales larger than 10 Mpc/h highly depends on dark energy perturbation; compared to the homogeneous dark energy model, the number of large voids of radius 30Mpc is 25% larger for the model with $w = -0.9$ and $c^2_s = 0$ while they are 20% less abundant for the model with $w = -1.3$ and $c^2_s = 0$.

Minkowski Tensors in Two Dimensions: Probing the Morphology and Isotropy of the Matter and Galaxy Density Fields

Abstract We apply the Minkowski tensor statistics to two-dimensional slices of the three-dimensional matter density field. The Minkowski tensors are a set of functions that are sensitive to directionally dependent signals in the data and, furthermore, can be used to quantify the mean shape of density fields. We begin by reviewing the definition of Minkowski tensors and introducing a method of calculating them from a discretely sampled field. Focusing on the statistic —a 2 × 2 matrix—we calculate its value for both the entire excursion set and individual connected regions and holes within the set. To study the morphology of structures within the excursion set, we calculate the eigenvalues λ 1, λ 2 for the matrix of each distinct connected region and hole and measure their mean shape using the ratio . We compare both and β for a Gaussian field and a smoothed density field generated from the latest Horizon Run 4 cosmological simulation to study the effect of gravitational collapse on these functions. The global statistic is essentially independent of gravitational collapse, as the process maintains statistical isotropy. However, β is modified significantly, with overdensities becoming relatively more circular compared to underdensities at low redshifts. When applying the statistics to a redshift-space distorted density field, the matrix is no longer proportional to the identity matrix, and measurements of its diagonal elements can be used to probe the large-scale velocity field.

Estimating Orthant Probabilities of High-Dimensional Gaussian Vectors with An Application to Set Estimation

ABSTRACTThe computation of Gaussian orthant probabilities has been extensively studied for low-dimensional vectors. Here, we focus on the high-dimensional case and we present a two-step procedure relying on both deterministic and stochastic techniques. The proposed estimator relies indeed on splitting the probability into a low-dimensional term and a remainder. While the low-dimensional probability can be estimated by fast and accurate quadrature, the remainder requires Monte Carlo sampling. We further refine the estimation by using a novel asymmetric nested Monte Carlo (anMC) algorithm for the remainder and we highlight cases where this approximation brings substantial efficiency gains. The proposed methods are compared against state-of-the-art techniques in a numerical study, which also calls attention to the advantages and drawbacks of the procedure. Finally, the proposed method is applied to derive conservative estimates of excursion sets of expensive to evaluate deterministic functions under a Gaussian random field prior, without requiring a Markov assumption. Supplementary material for this article is available online.

Scalable streaming tools for analyzing N-body simulations: Finding halos and investigating excursion sets in one pass

Cosmological $N$-body simulations play a vital role in studying models for the evolution of the Universe. To compare to observations and make a scientific inference, statistic analysis on large simulation datasets, e.g., finding halos, obtaining multi-point correlation functions, is crucial. However, traditional in-memory methods for these tasks do not scale to the datasets that are forbiddingly large in modern simulations. Our prior paper proposes memory-efficient streaming algorithms that can find the largest halos in a simulation with up to $10^9$ particles on a small server or desktop. However, this approach fails when directly scaling to larger datasets. This paper presents a robust streaming tool that leverages state-of-the-art techniques on GPU boosting, sampling, and parallel I/O, to significantly improve performance and scalability. Our rigorous analysis of the sketch parameters improves the previous results from finding the centers of the $10^3$ largest halos to $\sim 10^4-10^5$, and reveals the trade-offs between memory, running time and number of halos. Our experiments show that our tool can scale to datasets with up to $\sim 10^{12}$ particles while using less than an hour of running time on a single GPU Nvidia GTX 1080.

The median density of the Universe

Despite the fact that the mean matter density of the universe has been measured to an accuracy of a few percent within the standard $\Lambda$CDM paradigm, its median density is not known even to order of magnitude. Typical points lie in low-density regions and are not part of a collapsed structure of any scale. Locally, the dark matter distribution is then simply a stretched version of that in the early universe. In this single-stream regime, the distribution of unsmoothed density is sensitive to the initial power spectrum on all scales, in particular on very small scales, and hence to the nature of the dark matter. It cannot be estimated reliably using conventional cosmological simulations because of the enormous dynamic range involved, but a suitable excursion set procedure can be used instead. For the Planck cosmological parameters, a 100 GeV WIMP, corresponding to a free-streaming mass $\sim 10^{-6}M_\odot$, results in a median density of $\sim 4\times 10^{-3}$ in units of the mean density, whereas a 10 $\mu$eV axion with free-streaming mass $\sim 10^{-12}M_\odot$ gives $\sim 3\times 10^{-3}$, and Warm Dark Matter with a (thermal relic) mass of 1 keV gives $\sim 8\times 10^{-2}$. In CDM (but not in WDM) universes, single-stream regions are predicted to be topologically isolated by the excursion set formalism. A test by direct N-Body simulations seems to confirm this prediction, although it is still subject to finite size and resolution effects. Unfortunately, it is unlikely that any of these properties is observable and so suitable for constraining the properties of dark matter.

Limit theorems for multidimensional renewal sets

Consider the multiple sums Sn of i.i.d. random variables with a positive expectation on the d-dimensional integer grid. We prove the strong law of large numbers, the law of the iterated logarithm and the distributional limit theorem for random sets which appear as upper excursion sets of the interpolated multiple sums, that is, as the set of all arguments $${x\in\mathbb{R}_+^d}$$ such that the interpolated multiple sums Sx exceed t. The results are expressed in terms of set inclusions and using distances between sets.

Prediction of the 21-cm signal from reionization: comparison between 3D and 1D radiative transfer schemes

Three-dimensional radiative transfer simulations of the epoch of reionization can produce realistic results, but are computationally expensive. On the other hand, simulations relying on one-dimensional radiative transfer solutions are faster but limited in accuracy due to their more approximate nature. Here, we compare the performance of the reionization simulation codes grizzly and C2-ray which use 1D and 3D radiative transfer schemes respectively. The comparison is performed using the same cosmological density fields, halo catalogues and source properties. We find that the ionization maps, as well as the 21-cm signal maps from these two simulations are very similar even for complex scenarios which include thermal feedback on low mass halos. The comparison between the schemes in terms of the statistical quantities such as the power spectrum of the brightness temperature fluctuation agree with each other within 10% error throughout the entire reionization history. grizzly seems to perform slightly better than the semi-numerical approaches considered in Majumdar et al. (2014) which are based on the excursion set principle. We argue that grizzly can be efficiently used for exploring parameter space, establishing observations strategies and estimating parameters from 21-cm observations.

Geometry and scaling laws of excursion and iso-sets of enstrophy and dissipation in isotropic turbulence

ABSTRACTMotivated by interest in the geometry of high intensity events of turbulent flows, we examine the spatial correlation functions of sets where turbulent events are particularly intense. These sets are defined using indicator functions on excursion and iso-value sets. Their geometric scaling properties are analysed by examining possible power-law decay of their radial correlation function. We apply the analysis to enstrophy, dissipation and velocity gradient invariants Q and R and their joint spatial distributions, using data from a direct numerical simulation of isotropic turbulence at Reλ ≈ 430. While no fractal scaling is found in the inertial range using box-counting in the finite Reynolds number flow considered here, power-law scaling in the inertial range is found in the radial correlation functions. Thus, a geometric characterisation in terms of these sets’ correlation dimension is possible. Strong dependence on the enstrophy and dissipation threshold is found, consistent with multifractal behaviour. Nevertheless, the lack of scaling of the box-counting analysis precludes direct quantitative comparisons with earlier work based on multifractal formalism. Surprising trends, such as a lower correlation dimension for strong dissipation events compared to strong enstrophy events, are observed and interpreted in terms of spatial coherence of vortices in the flow.