Gaussian Random Field Model Research Articles

Electronic structure of two-dimensional hexagonal diselenides: Charge density waves and pseudogap behavior

We present theoretical study of electronic structure (spectral functions and Fermi surfaces) for incommensurate pseudogap and charge density wave (CDW) and commensurate CDW phases of quasi two dimensional diselenides 2H-TaSe2 and 2H-NbSe2. Incommensurate pseudogap regime is described within the scenario based on short-range order CDW fluctuations, considered within the static Gaussian random field model. In contrast e.g. to high-Tc cuprates layered dichalcogenides have several different CDW scattering vectors and electronic spectrum with two bands at the Fermi level. To this end we present theoretical background for the description of multiple scattering processes within multiple bands electronic spectrum. Thus obtained theoretical spectral functions and Fermi surfaces are compared with recent ARPES experimental data, demonstrating rather good qualitative agreement.

Application of Bayes linear discriminant functions in image classification

In statistical image classification it is usually assumed that feature observations given class labels are independently distributed. Even in the case when training sample is formed by dependent feature observations, the feature observations to be classified are usually assumed to be independent from training sample. In this paper we propose the original method of the incorporation of spatial information into the per-pixel classifiers. Our approach is based on the retraction of the independence assumption by proposing stationary Gaussian random field (GRF) model for features. The conditional distribution of class label of observation to be classified is assumed to be dependent on its spatial adjacency within the spatial framework of the training sample. For a given training sample, plug-in version of the Bayes discriminant function (PBDF) is proposed for classification. Performance of the proposed PBDF is tested and compared with ones ignoring dependence among feature observations to be classified and training sample. For illustration the image of figure corrupted by the additive GRF is analyzed. The advantage of the proposed classifier against the competing one is shown visually and numerically in the first example. In the second example, three spatial sampling designs for training data are compared on the basis of the actual error rate values of the proposed PBDF. For the remotely sensed image, the advantage of the proposed classification method against popular unsupervised classification method is shown in terms of visual evaluation and empirical errors of misclassification.

Level cut Gaussian random field models for transitions from laminar to turbulent flow

Herein, we develop a model for phenomena characterized by random fluctuations between one of two states; these two states can be interpreted as “off” and “on” or 0 and 1. Random phenomena of this type are abundant in nature and can exhibit temporal and/or spatial dependence. For example, the arrival of customers in a queue or vehicles at an intersection are examples of time-dependent random fluctuations, while the presence of random imperfections within material microstructure exhibits spatial dependence. Further, the transition from laminar to turbulent flow over the surface of a structure is believed to involve random fluctuations between laminar and turbulent flow in both time and space. The proposed model is a random field with piecewise constant samples defined by cuts of a Gaussian random field that exceed a specified level set and can be used to model temporally and spatially varying random fluctuations between one of two states. The model can be calibrated to available information, which we assume consists of (1) the marginal probability that the state of the system is “on” and (2) the average number and density of fluctuations between states that occur within a bounded region. Numerous examples are considered to illustrate the calibration of the level cut Gaussian random field to represent these time and spatially varying random fluctuations, including the formation and propagation of localized bursts of turbulent flow over the surface of a flat plate.

Cross-domain active learning for video concept detection

As video data from a variety of different domains (e.g., news, documentaries, entertainment) have distinctive data distributions, cross-domain video concept detection becomes an important task, in which one can reuse the labeled data of one domain to benefit the learning task in another domain with insufficient labeled data. In this paper, we approach this problem by proposing a cross-domain active learning method which iteratively queries labels of the most informative samples in the target domain. Traditional active learning assumes that the training (source domain) and test data (target domain) are from the same distribution. However, it may fail when the two domains have different distributions because querying informative samples according to a base learner that initially learned from source domain may no longer be helpful for the target domain. In our paper, we use the Gaussian random field model as the base learner which has the advantage of exploring the distributions in both domains, and adopt uncertainty sampling as the query strategy. Additionally, we present an instance weighting trick to accelerate the adaptability of the base learner, and develop an efficient model updating method which can significantly speed up the active learning process. Experimental results on TRECVID collections highlight the effectiveness.

On fixed-domain asymptotics and covariance tapering in Gaussian random field models

Gaussian random fields are commonly used as models for spatial processes and maximum likelihood is a preferred method of choice for estimating the covariance parameters. However if the sample size n is large, evaluating the likelihood can be a numerical challenge. Covariance tapering is a way of approximating the covariance function with a taper (usually a compactly supported function) so that the computational burden is reduced. This article studies the fixed-domain asymptotic behavior of the tapered MLE for the microergodic parameter of a Matérn covariance function when the taper support is allowed to shrink as n→∞. In particular if the dimension of the underlying space is ≤3, conditions are established in which the tapered MLE is strongly consistent and also asymptotically normal. Numerical experiments are reported that gauge the quality of these approximations for finite n.

Image Classification Based on Bayes Discriminant Functions

In statistical image classification, it is usually assumed that feature observations given class labels are independently distributed [1]. Even in the case when training sample is formed by dependent feature observations, the feature observations to be classified are often assumed to be independent from training sample [2], [3]. Proposed approach is based on the retraction of this assumption by considering the stationary Gaussian random field (GRF) model for features [4]. The conditional distribution of class label at unclassified location (pixel) is assumed to be dependent on its location spatial adjacency with spatial framework of training sample. For given training sample, the Bayes discriminant function (BDF) and plug-in BDF (PBDF) are proposed for classification. Performance of the proposed BDF and PBDF are tested and compared with ones ignoring spatial correlation among feature observations to be classified and training sample. In the first example, the restoration of the image of figure corrupted by the GRF is performed by pixels classification. In the numerous parametric structure cases, the advantage of proposed BDF against competing one is shown visually and is confirmed numerically. In the second example, three spatial sampling designs for training data are compared on the basis of actual error rate values of proposed PBDF.

Comparison of linear discriminant functions in image classification

In statistical image classification it is usually assumed that feature observations given labels are independently distributed. We have retracted this assumption by proposing stationary Gaussian random field (GRF) model for features observations. Conditional distribution of label of observation to be classified is assumed to be dependent on its spatial adjacency with training sample spatial framework. Perfomance of the Bayes discriminant function (BDF) and performance of plug-in BDFare tested and are compared with ones ignoring spatial correlation among feature observations.For illustration image of figure corrupted by additive GRF is analyzed. Advantage of proposed BDF against competing ones is shown visually and numerically.

Error rates in spatial classification of Gaussian data with random labeling

In spatial classification it is usually assumed that features observations given labels are independently distributed. We have retracted this assumption by proposing stationary Gaussian random field model for features observations. The label are assumed to follow Disrete Random Field (DRF) model. Formula for exact error rate based on Bayes discriminant function (BDF) is derived. In the case of partial parametric uncertainty (mean parameters and variance are unknown), the approximation of the expected error rate associated with plug-in BDF is also derived. The dependence of considered error rates on the values of range and clustering parameters is investigated numerically for training locations being second-order neighbors to location of observation to be classified.

Stochastic financial models

Portfolio Choice Introduction Utility Mean-variance analysis The Binomial Model One-period model Multi-period model A General Discrete-Time Model One-period model Multi-period model Brownian Motion Introduction Hitting-time distributions Girsanov's theorem Brownian motion as a limit Stochastic calculus The Black-Scholes Model Introduction The Black-Scholes formula Hedging and the Black-Scholes equation Path-dependent claims Dividend-paying assets Interest-Rate Models Introduction Survey of interest-rate models Gaussian random-field model Appendix A: Mathematical Preliminaries Appendix B: Solutions to the Exercises Further Reading References Index Exercises appear at the end of each chapter.

The intracluster magnetic field power spectrum in Abell 665

<i>Aims. <i/>The goal of this work is to investigate the power spectrum of the magnetic field associated with the giant radio halo in the galaxy cluster A665.<i>Methods. <i/>For this, we present new deep Very Large Array total intensity and polarization observations at 1.4 GHz. We simulated Gaussian random three-dimensional turbulent magnetic field models to reproduce the observed radio halo emission. By comparing observed and synthetic radio halo images we constrained the strength and structure of the intracluster magnetic field. We assumed that the magnetic field power spectrum is a power law with a Kolmogorov index and we imposed a local equipartition of energy density between relativistic particles and field.<i>Results. <i/>Under these assumptions, we find that the radio halo emission in A665 is consistent with a central magnetic field strength of about 1.3 <i>μ<i/>G. To explain the azimuthally averaged radio brightness profile, the magnetic field energy density should decrease following the thermal gas density, leading to an averaged magnetic field strength over the central 1 Mpc<sup>3<sup/> of about 0.75 <i>μ<i/>G. From the observed brightness fluctuations of the radio halo, we infer that the outer scale of the magnetic field power spectrum is ~450 kpc, and the corresponding magnetic field auto-correlation length is ~100 kpc.

Identifying significant scale-specific spatial boundaries using wavelets and null models: spruce budworm defoliation in Ontario, Canada as a case study

We combine wavelet analysis and multiple null models to identify significant spatial scales of pattern and spatial boundaries in historical spruce budworm defoliation in Ontario, Canada. Previous analyses of budworm defoliation in Ontario over the last two outbreaks have suggested three distinct zones of defoliation. We asked the following three questions: (1) is there statistical support for the existence of these three zones? (2) Are the locations of these boundaries consistent between outbreak periods? And (3) how does boundary identification depend on the spatial null model used? Defoliation data for the two outbreak periods (1941–1965 and 1966–2001), and the combined period (1941–2001) were analyzed using a 1D continuous wavelet transform. Boundaries were identified through comparison of wavelet power spectra of each outbreak period to reference distributions based on three different spatial null models: (1) a complete spatial randomness model, (2) an autoregressive model, and (3) a Gaussian random field model. The Gaussian random field model identified coarser scales of pattern than the autoregressive model. Locally, the Gaussian random field model found significant boundaries similar to those previously described, whereas the autoregressive model only did so for the first outbreak. These results indicate that the coarse scale spatial factors that influenced defoliation were more consistent between outbreaks relative to fine scale factors, and that previously described boundaries were strongly driven by the first outbreak. Wavelet analysis combined with spatial null models provides a powerful tool for identifying non-arbitrary scales of structure and significant spatial boundaries in non-stationary ecological data.

Approximation of Bayesian Inverse Problems for PDEs

Inverse problems are often ill posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is based on an approach to regularization, employing a Bayesian formulation of the problem, which leads to a notion of well posedness for inverse problems, at the level of probability measures. The stability which results from this well posedness may be used as the basis for quantifying the approximation, in finite dimensional spaces, of inverse problems for functions. This paper contains a theory which utilizes this stability property to estimate the distance between the true and approximate posterior distributions, in the Hellinger metric, in terms of error estimates for approximation of the underlying forward problem. This is potentially useful as it allows for the transfer of estimates from the numerical analysis of forward problems into estimates for the solution of the related inverse problem. It is noteworthy that, when the prior is a Gaussian random field model, controlling differences in the Hellinger metric leads to control on the differences between expected values of polynomially bounded functions and operators, including the mean and covariance operator. The ideas are applied to some non-Gaussian inverse problems where the goal is determination of the initial condition for the Stokes or Navier-Stokes equation from Lagrangian and Eulerian observations, respectively.

Modeling the morphology and mechanical behavior of shape memory polyurethanes based on solid-state NMR and synchrotron SAXS/WAXD

A combination of solid-state proton Wide-line Nuclear Magnetic Resonance (NMR) relaxometry and synchrotron Small-angle (SAXS) and Wide-angle (WAXD) X-ray scattering was used to elucidate the microphase morphology of shape memory thermoplastic multi-block polyurethanes based on poly(e-caprolactone), as switching segment and polyurethane based permanent or hard segments (HS). The polyurethanes are produced from the condensation of 1,4-butanediol (BDO) with hexamethylenediisocyanate (HDI). The morphology – induced by the hard-segment crystallization – converts from dispersed randomly placed hard-segment domains into progressively more periodic, but interconnected HS nanophases with increasing HS content. Irrespective of the actual morphology, the SAXS data could be described satisfactorily by using a clipped Gaussian random field (GRF) model. The NMR data demonstrate that the HS domain fraction corresponds to the chemical feed, pointing at a complete phase separation. The material mechanical behavior during repeated deformation cycles can be explained on morphological grounds and involves a spatially heterogeneous plastic deformation of the hard domains.

Hysteresis in theT=0random-field Ising model: Beyond metastable dynamics

We present a numerical study of the zero-temperature response of the Gaussian random-field Ising model to a slowly varying external field, allowing the system to be trapped in microscopic configurations that are not fully metastable. This modification of the standard single-spin-flip dynamics results in an increase in dissipation (hysteresis) somewhat similar to that observed with a finite driving rate. We then study the distribution of avalanches along the hysteresis loop and perform a finite-size scaling analysis that shows good evidence that the critical exponents associated to the disorder-induced phase transition are not modified.

Quantum stress in chaotic billiards

This paper reports on a joint theoretical and experimental study of the Pauli quantum-mechanical stress tensor T_{alphabeta}(x,y) for open two-dimensional chaotic billiards. In the case of a finite current flow through the system the interior wave function is expressed as psi=u+iv . With the assumption that u and v are Gaussian random fields we derive analytic expressions for the statistical distributions for the quantum stress tensor components T_{alphabeta} . The Gaussian random field model is tested for a Sinai billiard with two opposite leads by analyzing the scattering wave functions obtained numerically from the corresponding Schrödinger equation. Two-dimensional quantum billiards may be emulated from planar microwave analogs. Hence we report on microwave measurements for an open two-dimensional cavity and how the quantum stress tensor analog is extracted from the recorded electric field. The agreement with the theoretical predictions for the distributions for T_{alphabeta}(x,y) is quite satisfactory for small net currents. However, a distinct difference between experiments and theory is observed at higher net flow, which could be explained using a Gaussian random field, where the net current was taken into account by an additional plane wave with a preferential direction and amplitude.

Interpolation of spatial and spatio-temporal Gaussian fields using Gaussian Markov random fields

This paper considers interpolation on a lattice of covariance-based Gaussian Random Field models (Geostatistics models) using Gaussian Markov Random Fields (GMRFs) (conditional autoregression models). Two methods for estimating the GMRF parameters are considered. One generalises maximum likelihood for complete data, and the other ensures a better correspondence between fitted and theoretical correlations for higher lags. The methods can be used both for spatial and spatio-temporal data. Some different cross-validation methods for model choice are compared. The predictive ability of the GMRF is demonstrated by a simulation study, and an example using a real image is considered.

The intracluster magnetic field power spectrum in Abell 2382

The goal of this work is to investigate the power spectrum of the magnetic field associated with the giant radio halo in the galaxy cluster A665. For this, we present new deep Very Large Array total intensity and polarization observations at 1.4 GHz. We simulated Gaussian random three-dimensional turbulent magnetic field models to reproduce the observed radio halo emission. By comparing observed and synthetic radio halo images we constrained the strength and structure of the intracluster magnetic field. We assumed that the magnetic field power spectrum is a power law with a Kolmogorov index and we imposed a local equipartition of energy density between relativistic particles and field. Under these assumptions, we find that the radio halo emission in A665 is consistent with a central magnetic field strength of about 1.3 micro-G. To explain the azimuthally averaged radio brightness profile, the magnetic field energy density should decrease following the thermal gas density, leading to an averaged magnetic field strength over the central 1 Mpc^3 of about 0.75 micro-G. From the observed brightness fluctuations of the radio halo, we infer that the outer scale of the magnetic field power spectrum is ~450 kpc, and the corresponding magnetic field auto-correlation length is ~100 kpc.

Metamodel-assisted distributed genetic algorithms applied to structural shape optimization problems†

This article is concerned with the development of a general optimization tool based on distributed real genetic algorithms (DRGAs) assisted by metamodel evaluation and applied to structural shape optimization problems of general boundary-element models (BEMs). The evaluation fitness function is performed by a surrogate function based on multidimensional Gaussian random field models (MGRFMs) in order to minimize the computational cost of the evolutionary algorithms. The model boundary of a structural system or a mechanical tool is discretized using the BEM, and selected parts of the boundary are modelled using β-spline curves or surfaces in order to facilitate re-meshing and adaptation of the boundary to the external actions. Then a hypercube topology of populations of these models follows a genetic evolution process to determine the optimum shape of the system. The optimum models have minimum weight and satisfy all imposed constraints. A numerical example is presented and discussed in order to show the efficiency and robustness of the developed computational tool. The number of function evaluations is substantially reduced compared with previous versions of the optimization algorithm without the metamodel evaluation technique. †This is an extended and enhanced version of work presented at the mini-symposium on Evolutionary Algorithms: Recent Applications in Engineering and Science organized by Dr William Annicchiarico at the 7th World Congress on Computational Mechanics, Los Angeles, July 2006.

T-distributed Random Fields: A Parametric Model for Heavy-tailedWell-log Data1

Histograms of observations from spatial phenomena are often found to be more heavy-tailed than Gaussian distributions, which makes the Gaussian random field model unsuited. A T-distributed random field model with heavy-tailed marginal probability density functions is defined. The model is a generalization of the familiar Student-T distribution, and it may be given a Bayesian interpretation. The increased variability appears cross-realizations, contrary to in-realizations, since all realizations are Gaussian-like with varying variance between realizations. The T-distributed random field model is analytically tractable and the conditional model is developed, which provides algorithms for conditional simulation and prediction, so-called T-kriging. The model compares favourably with most previously defined random field models. The Gaussian random field model appears as a special, limiting case of the T-distributed random field model. The model is particularly useful whenever multiple, sparsely sampled realizations of the random field are available, and is clearly favourable to the Gaussian model in this case. The properties of the T-distributed random field model is demonstrated on well log observations from the Gullfaks field in the North Sea. The predictions correspond to traditional kriging predictions, while the associated prediction variances are more representative, as they are layer specific and include uncertainty caused by using variance estimates.

The intracluster magnetic field power spectrum in Abell 2255

Aims. The goal of this work is to investigate the power spectrum of the magnetic field associated with the giant radio halo in the galaxy cluster A665. Methods. For this, we present new deep Very Large Array total intensity and polarization observations at 1.4 GHz. We simulated Gaussian random three-dimensional turbulent magnetic field models to reproduce the observed radio halo emission. By comparing observed and synthetic radio halo images we constrained the strength and structure of the intracluster magnetic field. We assumed that the magnetic field power spectrum is a power law with a Kolmogorov index and we imposed a local equipartition of energy density between relativistic particles and field. Results. Under these assumptions, we find that the radio halo emission in A665 is consistent with a central magnetic field strength of about 1.3 μG. To explain the azimuthally averaged radio brightness profile, the magnetic field energy density should decrease following the thermal gas density, leading to an averaged magnetic field strength over the central 1 Mpc 3 of about 0.75 μG. From the observed brightness fluctuations of the radio halo, we infer that the outer scale of the magnetic field power spectrum is ∼450 kpc, and the corresponding magnetic field auto-correlation length is ∼100 kpc.