Discrete Random Fields Research Articles

On Lamperti transformation and AR(1) type characterisations of discrete random fields

In this article we characterise discrete time stationary fields by difference equations involving stationary increment fields and self-similar fields. This gives connections between stationary fields, stationary increment fields and, through Lamperti transformation, self-similar fields. Our contribution is a natural generalisation of recently proved results covering the case of stationary processes.

Stochastic Periodic Microstructures for Multiscale Modelling of Heterogeneous Materials

Plurigaussian simulation is a method of discrete random field generation that can be used to generate many complex geometries depicting real world structures. Whilst it is commonly applied at larger scales to represent geological phenomena, the highly flexible approach is suitable for generating structures at all scales. Here, an extension of plurigaussian simulation to periodic plurigaussian simulation (P-PGS) is presented, such that the resulting fields are periodic in nature. By using periodic Gaussian random fields as components of the method, periodicity is enforced in the generated structures. To substantiate the use of P-PGS in capturing complex heterogeneities in a physically meaningful way, the pore-scale microstructure of cement paste was represented such that its effective properties can be calculated through a computational homogenisation approach. The finite element method is employed to model the diffusion of heat through the medium under dry and saturated pore conditions, where numerical homogenisation is conducted to calculate the effective thermal conductivity of the medium. Comparison of the calculated values with experimental observations indicated that the generated microstructures are suitable for pore-scale representation, given their close match. A maximal error of 1.38% was observed in relation to the numerically determined effective thermal conductivity of mortar paste with air filled pores, and 0.41% when considering water filled pores. As the assumption of a periodic domain is often an underlying feature of numerical homogenisation, this extension of plurigaussian simulation enables a path for its integration into such computational schemes.Article HighlightsIntegrating P-PGS into numerical homogenisation frameworks enhances complex heterogeneous material representationThe flexibility of P-PGS enables a wide range of material microstructures to be represented accuratelyUse of the generated structures allows material properties to be estimated accurately through numerical homogenisation

Investigation on wave-induced responses of heterogeneous seabed around a buried pipeline based on nonstationary random field model

The instability of the seabed poses a threat to the structural integrity and functionality of submarine pipelines. This numerical study investigates wave-induced responses in a heterogeneous seabed around a buried pipeline, considering spatial variability in soil properties through nonstationary random field models. Three scenarios of spatial variability for soil parameters changing with depth are simulated using both stationary and nonstationary random field models via the Karhunen-Loève expansion method. The discrete random fields are validated in terms of correlation structures and statistical characteristics. The stochastic finite element method, incorporating random fields and a poroelastic seabed model through a LiveLink code, is employed to examine probabilistic outcomes of dynamic soil responses and liquefaction zone distributions within a probabilistic framework. Contrastive analyses are conducted for seabed models with different random field scenarios, in addition to deterministic analyses. The results of the stochastic analysis show that the increasing trend of soil strength with depth has significant effects on the seabed responses and transient liquefaction characteristics around a pipeline.

On the relation between Fourier and Walsh–Rademacher spectra for random fields

We discuss relations between the expansion coefficients of a discrete random field when analyzed with respect to different hierarchical bases. Our main focus is on the comparison of two such systems: the Walsh–Rademacher basis and the trigonometric Fourier basis. In general, spectra computed with respect to one basis will look different in the other. In this paper, we prove that, in a statistical sense, the rate of spectral decay computed in one basis can be translated to the other. We further provide explicit expressions for this translation on quadrilateral meshes. The results are illustrated with numerical examples for deterministic and random fields.

CORRELATION FUNCTIONS AND QUASI-DETERMINISTIC SIGNALS

When processing data on random functions, they are most often limited to constructing an empirical correlation function. In this regard, the problem arises of constructing a random function (a quasi-deterministic signal) determined by a finite set of random variables and having a given correlation function. Moreover, a random function can often be considered Gaussian, since in many cases a random signal is obtained at the output of the system, which is fairly well approximated by a Gaussian. For stationary random processes and for random fields, this problem has been considered. For random sequences and discrete random fields, as well as for non-stationary random signals, the problem remained open. The article considers the problem of restoring a random sequence from known mathematical expectation and correlation function. Such a model random sequence is constructed, in which the mathematical expectation and correlation function coincide with the given ones. The mathematical expectation and the correlation function are the simplest probabilistic numerical characteristics, but they do not uniquely determine the corresponding set of probability distribution densities that satisfy the conditions of normalization and consistency, provided that for each fixed integer value of the parameter, the random sequence is a continuous random variable. The article considers the restoration of a quasi-deterministic signal in stationary and non-stationary cases. For the stationary case, three examples are given for constructing a quasi-deterministic discrete signal EMBED Equation.DSMT4 , provided that the spectral density has three different forms. For the non-stationary case, the corresponding quasi-deterministic signal was obtained for various cases of the spectrum. The use of a random function model determined by a finite number of parameters makes it possible to significantly simplify the analysis of applied problems, the solution of which is associated with differential equations with random coefficients, which are such quasi-deterministic signals. In this case, there is no need to use a complex apparatus of stochastic differential equations, since the solution of such an equation simply depends on random variables as on parameters.

Hysteresis in a One-Dimensional 3-state Clock Model at Zero Temperature with Absorbing State

We employ probabilistic method to obtain exact solutions of the zero temperature hysteresis loops of a 3-state clock model with quenched random fields in the presence of an absorbing state and in the limit of zero frequency of the applied external field. We consider discrete random fields drawn from a uniform distribution and present exact solutions of the model in one dimension. The analytical results are in excellent agreement with the simulation data of the model.

Infinite Volume Gibbs States and Metastates of the Random Field Mean-Field Spherical Model

For the discrete random field Curie–Weiss models, the infinite volume Gibbs states and metastates have been investigated and determined for specific instances of random external fields. In general, there are not many examples in the literature of non-trivial limiting metastates for discrete or continuous spin systems. We analyze the infinite volume Gibbs states of the mean-field spherical model, a model of continuous spins, in a general random external field with independent identically distributed components with finite moments of some order larger than four and non-vanishing variances of the second moments. Depending on the parameters of the model, we show that there exist three distinct phases: ordered ferromagnetic, ordered paramagnetic, and spin glass. In the ordered ferromagnetic and ordered paramagnetic phases, we show that there exists a unique infinite volume Gibbs state almost surely. In the spin glass phase, we show the existence of chaotic size dependence, provide a construction of the Aizenman–Wehr metastate, and consider both the convergence in distribution and almost sure convergence of the Newman–Stein metastates. The limiting metastates are non-trivial and their structure is universal due to the presence of Gaussian fluctuations and the spherical constraint.

Sharp phase transition for Cox percolation

We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence, satisfies a local boundedness condition and can be constructed from a discrete iid random field, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction that allows us to compare different notions of influence.

A Robust Authentication Algorithm for Medical Images Based on Fractal Brownian Model and Visual Cryptography

In this paper, we aimed to discuss the security authentication requirements of medical images in the medical network, and a security authentication method is designed based on fractal and visual cryptography. Based on the discrete fractal Brownian random field model, the gray-level statistical information and spatial structure information of medical images is fully mined. The gray distribution of medical images is expressed in the form of fractal features. By using the spatial data mining methods, the data of fractal structure space is analyzed, and by using the stability of the energy structure, the authentication features are formed. Using the visual cryptography (VC), the robustness of the authentication method is further enhanced. Through the centralized test of common medical images and the comparison analysis with existing methods, it is further verified that the method is effective against common attacks such as JPEG compression, scaling, rotation operation, clipping, added noise, filtering, and blurring.

Discrete convolutional CRF networks for depth estimation from monocular infrared images

Predicting the depth of a scene from monocular infrared images, which plays a crucial role in understanding three-dimensional structures, is one of the challenging tasks in machine learning and computer vision. Considering the lack of texture and color information in infrared images, a novel discrete convolutional conditional random field network is proposed for depth estimation. The proposed method inherits several merits of conditional random fields and deep learning. First, the pairwise features are automatically extracted and optimized through deep architectures. Second, the monocular-images-based depth regression is converted into a multi-class classification, in which the order information of different levels of depths is considered in the loss function. Our experiments demonstrate that this conversion achieves much higher accuracy and faster conversion. Third, to obtain fine-grained level details, we have further proposed a multi-scale discrete convolutional conditional random field network that computes the pairwise features of the discrete conditional random field at different spatial levels. Extensive experiments on the infrared image dataset NUSTMS demonstrate that the proposed method outperforms other depth estimation methods. Specifically, for the proposed method, the mean relative error is 0.181, the mean log10 error is 0.072, and the accuracy with a threshold (t = 1.253) is 95.3%.

Innovative methods for random field establishment and statistical parameter inversion exemplified with 6082-T6 aluminum alloy

This paper aims to eliminate the disharmony between simulation and experiment, and takes the mechanical properties of 6082-T6 Al alloy as an example. In order to obtain the equivalent distribution of material properties after considering the randomicity of materials, a new inversion method combining with stochastic finite element method (SFEM) is proposed. Besides, the discrete random field in SFEM is established by an innovative method to overcome some discretization difficulties in conventional methods. In summary, the generic methods proposed in this study can give a new solution for the correlation of meso-structure and macro-performance in computational materials science.

Time-optimal algorithms focused on the search for random pulsed-point sources

The article describes methods and algorithms related to the analysis of dynamically changing discrete random fields. Time-optimal strategies for the localization of pulsed-point sources having a random spatial distribution and indicating themselves by generating instant delta pulses at random times are proposed. An optimal strategy is a procedure that has a minimum (statistically) average localization time. The search is performed in accordance with the requirements for localization accuracy and is carried out by a system with one or several receiving devices. Along with the predetermined accuracy of localization of a random pulsed-point source, a significant complicating factor of the formulated problem is that the choice of the optimal search procedure is not limited to one-step algorithms that end at the moment of first pulse generation. Moreover, the article shows that even with relatively low requirements for localization accuracy, the time-optimal procedure consists of several steps, and the transition from one step to another occurs at the time of registration of the next pulse by the receiving system. In this case, the situation is acceptable when during the process of optimal search some of the generated pulses are not fixed by the receiving system. The parameters of the optimal search depending on the number of receiving devices and the required accuracy of localization are calculated and described in the paper.

Foot intrinsic muscle forces impact on internal stresses and strain on diabetic foot subjects

This paper presents the analysis of the statistical size effect in steel structural elements using the stochastic finite element (FE) method. The three-dimensional random field theory is applied to simulate the uncertainty of a nonlinear material. The proposed approach, which combines the discrete random field describing the randomness of the material properties via Monte Carlo simulations, is used for computing the bearing capacity variability of the steel structures. In addition, the mapping interpolation method makes it possible to separate the random field mesh and FE model mesh. The three-dimensional random field for the material properties is established according to the yield strength with a lognormal distribution based on reasonable assumptions, and the corresponding random field parameters are studied. Finally, the results of experimental investigation and simulations are compared based on the tensile and bending tests, and they indicate that the statistical size effect cannot be overlooked in S235JR steel structural elements and that the use of stochastic FE in the analysis of the statistical size effect has tremendous potential.

Interpolation for second order stationary random fields: time domain recipe

We consider a discrete time second order stationary random field and provide a time domain recipe for the interpolation based on the southwest and northeast corners. Our method is based on Salehi's approach, applying Von Neumann's celebrated alternative projection for-\break \noindent mula, but making a short cut by interpolating the innovations in the forward and backward moving average representations. We provide explicit expressions for the interpolator and error terms for the moving average random fields of finite order; for the MA($\boldsymbol {1}$) spatial model, we express the interpolator in terms of the observed values and the coefficients of the model. Following Kohli and Pourahmadi, we also derive the covariances between the present values and interpolation errors.

Quantifying geographic regions of excess stillbirth risk in the presence of spatial and spatio-temporal heterogeneity

Motivated by population-based geocoded data for Iowa stillbirths and live births delivered during 2005–2011, we sought to identify spatio-temporal variation of stillbirth risk. Our high-quality data consisting of point locations of these delivery events allows use of a Bayesian Poisson point process approach to evaluate the spatial pattern of events. With this large epidemiologic dataset, we implemented the integrated nested Laplace approximation (INLA) to fit the conditional formulation of the point process via a Bayesian hierarchical model and empirically showed that INLA, compared to Markov chain Monte Carlo (MCMC) sampling, is an attractive approach. Furthermore, we modeled the temporal variability in stillbirth to better understand how stillbirths are geographically linked over the seven-year study period and demonstrate the similarity between the conditional formulation of the spatio-temporal model and a log Gaussian Cox process governed by discrete space-time random fields. After controlling for important features of the data, the Bayesian temporal relative risk maps identified areas of increasing and decreasing stillbirth risk over the birth period, which may warrant further public health investigation in the regions identified.

Experimental and numerical investigations of statistical size effect in S235JR steel structural elements

Abstract This paper presents the analysis of the statistical size effect in steel structural elements using the stochastic finite element (FE) method. The three-dimensional random field theory is applied to simulate the uncertainty of a nonlinear material. The proposed approach, which combines the discrete random field describing the randomness of the material properties via Monte Carlo simulations, is used for computing the bearing capacity variability of the steel structures. In addition, the mapping interpolation method makes it possible to separate the random field mesh and FE model mesh. The three-dimensional random field for the material properties is established according to the yield strength with a lognormal distribution based on reasonable assumptions, and the corresponding random field parameters are studied. Finally, the results of experimental investigation and simulations are compared based on the tensile and bending tests, and they indicate that the statistical size effect cannot be overlooked in S235JR steel structural elements and that the use of stochastic FE in the analysis of the statistical size effect has tremendous potential.

Sampling Strategies for Uncertainty Reduction in Categorical Random Fields: Formulation, Mathematical Analysis and Application to Multiple-Point Simulations

The task of optimal sampling for the statistical simulation of a discrete random field is addressed from the perspective of minimizing the posterior uncertainty of non-sensed positions given the information of the sensed positions. In particular, information theoretic measures are adopted to formalize the problem of optimal sampling design for field characterization, where concepts such as information of the measurements, average posterior uncertainty, and the resolvability of the field are introduced. The use of the entropy and related information measures are justified by connecting the task of simulation with a source coding problem, where it is well known that entropy offers a fundamental performance limit. On the application, a one-dimensional Markov chain model is explored where the statistics of the random object are known, and then the more relevant case of multiple-point simulations of channelized facies fields is studied, adopting in this case a training image to infer the statistics of a non-parametric model. In both contexts, the superiority of information-driven sampling strategies is proved in different settings and conditions, with respect to random or regular sampling.

Edge extraction of mineralogical phase based on fractal theory

In order to understand the microstructure of pellets and improve the metallurgical properties of pellets, fractal theory is introduced to extract the edge features of pellets. Firstly, the original mineralogical phase is obtained by experiment, which is preprocessed by the histogram equalization to enhance the overall contrast. Based on the discrete Fractional Brownian Random Field Model, the algorithm is redesigned to calculate the dimension of each pixel, map the gray space of image into the dimension space, select the appropriate window size, transform and edge extraction. Comparing the algorithm in this paper with Canny operator and Laplace–Gauss operator, it is concluded that the algorithm in this paper has certain advantages in mineralogical phase edge extraction. Then, Gauss noise is added to the original gray image, and Canny operator and this algorithm are used to extract the edges of the noisy image. The numerical results of peak signal to noise ratio and root mean square error are obtained. Finally, the comparison proves that the algorithm can extract more complete edges, and has a stronger noise immunity.

The systematic effect of streambed conductivity heterogeneity on hyporheic flux and residence time

Abstract A systematic understanding of hyporheic flux (HF) and residence times (RT) is important as they are a major control of biogeochemical processing in streambeds. Previous studies addressing the effect of heterogeneity in streambed hydraulic conductivity (K) on HF and RT have come to deviating conclusions depending on the specific study design and the selection of heterogeneity cases being investigated. To more systematically evaluate the effects of conductivity heterogeneity on HF and RT, we simulated hyporheic exchange induced by idealized streambed ripples over a large range of heterogeneities. Conductivity heterogeneity was represented in the simulations in terms of 10,000 different heterogeneity realizations from a geostatistical model based on continuous Gaussian and discrete indicator random fields. We demonstrate that any isotropic homogeneous K-field, as an average of a heterogeneous K-field, can only match RT or HF of the respective heterogeneous K-field, but never both. We found exponential correlations of RT and HF with the variance of heterogeneous conductivity. Based on these correlations, an equivalent anisotropic homogeneous conductivity tensor K can be derived. This equivalent anisotropic K efficiently accounts for the effects of small scale heterogeneity on HF and RT. It can be calculated from the median and variance of the hydraulic conductivity distribution of the targeted heterogeneous sediment, without explicitly characterizing the sediment texture.

Fourier transform infrared spectroscopy microscopic imaging classification based on spatial–spectral features

The discrete fractional Brownian incremental random (DFBIR) field is used to describe the irregular, random, and highly complex shapes of natural objects such as coastlines and biological tissues, for which traditional Euclidean geometry cannot be used. In this paper, an anisotropic variable window (AVW) directional operator based on the DFBIR field model is proposed for extracting spatial characteristics of Fourier transform infrared spectroscopy (FTIR) microscopic imaging. Probabilistic principal component analysis first extracts spectral features, and then the spatial features of the proposed AVW directional operator are combined with the former to construct a spatial–spectral structure, which increases feature-related information and helps a support vector machine classifier to obtain more efficient distribution-related information. Compared to Haralick’s grey-level co-occurrence matrix, Gabor filters, and local binary patterns (e.g. uniform LBPs, rotation-invariant LBPs, uniform rotation-invariant LBPs), experiments on three FTIR spectroscopy microscopic imaging datasets show that the proposed AVW directional operator is more advantageous in terms of classification accuracy, particularly for low-dimensional spaces of spatial characteristics.