Bayesian Texture Classification From Indirect Observations Using Fast Sampling

Strong regularity for stationary discrete random fields is discussed. An extension of the classical Beurling's Theorem to functions of several variables is given. Necessary and sufficient conditions for the moving average representation of stationary random fields are obtained. A recipe formula for the best linear extrapolator is also given.

Two kinds of estimation variance functions for estimating a local average (LA) of a stationary (homogeneous) random field (RF) are derived. One is local estimation (LE) and the other is general estimation (GE) of LA. The former is for estimating LA at the observation location, and the latter is for obtaining LA at any arbitrary location within the RF. The geotechnical implications of these two estimations are that LE is for estimating LA of geotechnical parameters at the spot where the investigations are made, whereas GE is for estimating LA at any arbitrary location within the same layer. The behavior of the two estimation variance functions differs greatly, controlled by the number of observations (i.e. sample size) and the normalized layer thickness (layer thickness divided by autocorrelation distance of RF). Based on the derived estimation variance functions, methods for determining reliable characteristic values of geotechnical parameters and necessary sample size are proposed. The methods are based on the same framework as that of the traditional statistical theory, i.e. confidence interval of estimated parameters. However, the assumption of independently and identically distributed (i.i.d.) samples in the traditional statistical theory is replaced by the assumption of correlated samples from a stationary RF. The results obtained from the proposed methods for LE and GE differ from each other as well as from the traditional results, which has significant implications for geotechnical parameter estimation in geotechnical engineering practice.

ObjectiveAlthough metabolic profiles have been associated with chronic disease risk, lack of temporal stability of metabolite levels could limit their use in epidemiological investigations. The present study aims to evaluate the reliability over a two-year period of 158 metabolites and compare reliability over time in fasting and non-fasting serum samples.MethodsMetabolites were measured with the AbsolueIDQp180 kit (Biocrates, Innsbruck, Austria) by mass spectrometry and included acylcarnitines, amino acids, biogenic amines, hexoses, phosphatidylcholines and sphingomyelins. Measurements were performed on repeat serum samples collected two years apart in 27 fasting men from Turin, Italy, and 39 non-fasting women from Utrecht, The Netherlands, all participating in the European Prospective Investigation into Cancer and Nutrition (EPIC) study. Reproducibility was assessed by estimating intraclass correlation coefficients (ICCs) in multivariable mixed models.ResultsIn fasting samples, a median ICC of 0.70 was observed. ICC values were <0.50 for 48% of amino acids, 27% of acylcarnitines, 18% of lysophosphatidylcholines and 4% of phosphatidylcholines. In non-fasting samples, the median ICC was 0.54. ICC values were <0.50 for 71% of acylcarnitines, 48% of amino acids, 44% of biogenic amines, 36% of sphingomyelins, 34% of phosphatidylcholines and 33% of lysophosphatidylcholines. Overall, reproducibility was lower in non-fasting as compared to fasting samples, with a statistically significant difference for 19–36% of acylcarnitines, phosphatidylcholines and sphingomyelins.ConclusionA single measurement per individual may be sufficient for the study of 73% and 52% of the metabolites showing ICCs >0.50 in fasting and non-fasting samples, respectively. ICCs were higher in fasting samples that are preferable to non-fasting.

The problems of model specification in one‐ and two‐dimensional stationary random fields are considered, with particular reference to the class of spatial autoregressions. The role of the autocorrelation function is discussed and it is shown to have a valuable exploratory use, especially in identifying directional bias in random fields. It is recommended that this analysis be supplemented by a likelihood‐ratio test, which permits more precise tests of hypotheses about the form of the underlying model.

: A particular two dimensional model in a stationary random field, which has a wide applications in statistical signal processing and in texture classifications, is considered. We prove the consistency and also obtain the asymptotic distributions of the least squares estimators of the different model parameters. It is observed that the asymptotic distribution of the least squares estimators are multivariate normal. Some numerical experiments are performed to see how the asymptotic results work for finite samples. We propose some open problems at the end.

Geophysical inverse problems are inherently ill-posed due to sparse, noisy, and indirect observations, making Uncertainty Quantification (UQ) a fundamental requirement for reliable subsurface characterization. Bayesian inversion provides a comprehensive probabilistic framework for inferring subsurface parameters by coherently combining prior knowledge with observational data through the likelihood function. However, the practical deployment of Bayesian methods in large-scale geophysical settings is often hampered by the prohibitive computational cost of repeated forward model evaluations. In this context, uncertainty is often not solely driven by observational noise; a substantial and sometimes dominant contribution arises from model error, resulting from simplified physical descriptions, numerical discretization, and uncertain boundary conditions. When these sources of uncertainty are neglected or inadequately represented, Bayesian inversions may yield biased posterior estimates and unrealistically narrow uncertainty bounds. These limitations are particularly acute in deep Earth applications, where complex rheologies, poorly constrained geometries, and computationally intensive forward models coexist.A key challenge is the accurate delineation of the Lithosphere–Asthenosphere Boundary (LAB), which plays a central role in controlling mantle dynamics, lithospheric deformation, and deep geothermal processes. Despite the necessity of relying on Bayesian approaches to estimate the LAB and its associated uncertainties, the high computational cost of repeated evaluations of the forward solver makes this unfeasible within realistic time frames [1]. To address these limitations, this work investigates Reduced-Order Modeling (ROM) techniques to enable efficient Bayesian inversion of LAB geometry in geodynamical Stokes flow models. ROMs construct low-dimensional surrogates of high-fidelity solvers, allowing rapid forward simulations while preserving the dominant physical behavior of mantle flow. By integrating ROMs with Bayesian inference, the proposed framework enables effective and reliable UQ for LAB characterization.Keywords: Geophysical inverse problems; Bayesian inversion; Uncertainty Quantification; Reduced-Order Modeling; Lithosphere–Asthenosphere BoundaryAcknowledgement This research was conducted within the EarthSafe Doctoral Network and has received funding from the European Union’s Horizon Europe research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 101120556.

The problem of sequentially scanning and predicting data arranged in a multidimensional array is considered. We introduce the notion of a scandictor, which is any scheme for the sequential scanning and prediction of such multidimensional data. The scandictability of any finite (probabilistic) data array is defined as the best achievable expected performance on that array. The scandictability of any (spatially) stationary random field on /spl Zopf//sup m/ is defined as the limit of its scandictability on finite boxes (subsets of /spl Zopf//sup m/), as their edges become large. The limit is shown to exist for any stationary field, and essentially be independent of the ratios between the box dimensions. Fundamental limitations on scandiction performance in both the probabilistic and the deterministic settings are characterized for the family of difference loss functions. We find that any stochastic process or random field that can be generated autoregressively with a maximum-entropy innovation process is optimally scandicted the way it was generated. These results are specialized for cases of particular interest. The scandictability of any stationary Gaussian field under the squared-error loss function is given a single-letter expression in terms of its spectral measure and is shown to be attained by the raster scan. For a family of binary Markov random fields (MRFs), the scandictability under the Hamming distortion measure is fully characterized.

We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.

Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.

A central limit theorem for functions of stationary max-stable random fields on [formula omitted

Almost sure limit theorem for stationary Gaussian random fields

We consider the problem of recovering linear image of unknown signal belonging to a given convex compact signal set from noisy observation of another linear image of the signal. We develop a simple generic efficiently computable nonlinear in observations “polyhedral” estimate along with computation-friendly techniques for its design and risk analysis. We demonstrate that under favorable circumstances the resulting estimate is provably near-optimal in the minimax sense, the “favorable circumstances” being less restrictive than the weakest known so far assumptions ensuring near-optimality of estimates which are linear in observations.

Slope reliability analysis has always been a research focus for geotechnical engineers. Currently, most studies use stationary random fields to simulate the spatial variability of soil parameters. However, many field test data show that soil parameters such as cohesion and friction angle often exhibit obvious nonstationary characteristics along the burial depth direction. Moreover, the slope reliability analysis method based on Monte Carlo simulation requires a combination of the FEM and a large number of computational resources, resulting in extremely low computational efficiency. Therefore, this paper proposes a convolutional neural network (CNN)-based surrogate model for slope reliability analysis, emphasizing the nonstationary spatial variability of cohesion and friction angle. First, stationary and nonstationary random fields considering the spatial variability of shear strength parameters are generated, and sample data sets are collected using an automated computational program. Next, a CNN surrogate model is established to predict the factor of safety (FOS), which is trained using nine sample data sets containing stationary and nonstationary random fields. Finally, the model performance is validated using six slope models with different shear strength parameters. The results show that the CNN surrogate model can adapt to the data characteristics of different random fields and also shows good predictive performance in handling unseen sample data sets. The predicted FOS values show a high degree of consistency with the results calculated by the FEM, with the computation time being significantly reduced.

Emission reductions were mandated in the Clean Air Act Amendments of 1990 with the expectation that they would result in major reductions in the concentrations of atmospherically transported pollutants. The emission reductions are intended to reduce public health risks and to protect sensitive ecosystems. To determine whether the emission reductions are having the intended effect on atmospheric concentrations, monitoring data must be analyzed taking into consideration the spatial structure shown by the data. Maps of pollutant concentrations and fluxes are useful over different geopolitical boundaries, to discover when, where, and to what extent the U.S. Nation's air quality is improving or declining. Since the spatial covariance structure shown by the data changes with location, the standard kriging methodology for spatial interpolation cannot be used because it assumes stationarity of the process.We present a new methodology for spatial interpolation of non‐stationary processes. In this method the field is represented locally as a stationary isotropic random field, but the parameters of the stationary random field are allowed to vary across space. A procedure for interpolation is presented that uses an expression for the spectral density at high frequencies. New fitting algorithms are developed using spectral approaches. In cases where the data are distributed exactly or approximately on a lattice, it is argued that spectral approaches have potentially enormous computational benefits compared with maximum likelihood. The methods are extended to interpolation questions using approximate Bayesian approaches to account for parameter uncertainty. We develop applications to obtain the total loading of pollutant concentrations and fluxes over different geo‐political boundaries. Copyright © 2001 John Wiley &amp; Sons, Ltd.

We establish a central limit theorem for partial sums of stationary linear random fields with dependent innovations, and an invariance principle for anisotropic fractional Brownian sheets. Our result is a generalization of the invariance principle for fractional Brownian motions by Dedecker et al. (Bernoulli 17:88–113, 2011) to high dimensions. A key ingredient of their argument, the martingale approximation, is replaced by an \(m\)-approximation argument. An important tool of our approach is a moment inequality for stationary random fields recently established by El Machkouri et al. (Stoch. Process. Appl. 123:1–14, 2013).