CONVERGENCE RATES IN THE STRONG LAWS OF NONSTATIONARYρ~*-MIXING RANDOM FIELDS

By using a Rosenthal type inequality established in this paper, the complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are discussed for *-mixing random fields.

An Interpolation Method for Karhunen-Loève Expansion of Random Field Discretization

In the context of global mean square error concerning the number of random variables in the representation,the Karhunen–Loève(KL)expansion is the optimal series expansion method for random field discretization.The computational efficiency and accuracy of the KL expansion are contingent upon the accurate resolution of the Fredholm integral eigenvalue problem(IEVP).The paper proposes an interpolation method based on different interpolation basis functions such as moving least squares(MLS),least squares(LS),and finite element method(FEM)to solve the IEVP.Compared with the Galerkin method based on finite element or Legendre polynomials,the main advantage of the interpolation method is that,in the calculation of eigenvalues and eigenfunctions in one-dimensional random fields,the integral matrix containing covariance function only requires a single integral,which is less than a two-folded integral by the Galerkin method.The effectiveness and computational efficiency of the proposed interpolation method are verified through various one-dimensional examples.Furthermore,based on theKL expansion and polynomial chaos expansion,the stochastic analysis of two-dimensional regular and irregular domains is conducted,and the basis function of the extended finite element method(XFEM)is introduced as the interpolation basis function in two-dimensional irregular domains to solve the IEVP.

Complete Convergence Properties of the Sums for -mixing Sequences of Random Variables

This paper discusses complete convergence properties of the sums of -mixing random sequences.As a result,we improve the corresponding results of Wu Qunying(2001). And extended the Baum and Katz complete convergence to the case of -mixing random sequences by moment inequality and truncating without necessarily adding any extra conditions.

Some Convergence Results for Sequences of -mixing Random Variables

In this paper, we will present some strong convergence results for sequences of ψ-mixing random variables. The results for sequences of ψ-mixing random variables generalize the corresponding results for independent random variable sequences without any extra conditions.

Characterizing large-scale weak interlayer shear zones using conditional random field theory

The shear behavior of large-scale weak intercalation shear zones(WISZs)often governs the stability of foundations,rock slopes,and underground structures.However,due to their wide distribution,undulating morphology,complex fabrics,and varying degrees of contact states,characterizing the shear behavior of natural and complex large-scale WISZs precisely is challenging.This study proposes an analytical method to address this issue,based on geological fieldwork and relevant experimental results.The analytical method utilizes the random field theory and Kriging interpolation technique to simplify the spatial uncertainties of the structural and fabric features for WISZs into the spatial correlation and variability of their mechanical parameters.The Kriging conditional random field of the friction angle of WISZs is embedded in the discrete element software 3DEC,enabling activation analysis of WISZ C2 in the underground caverns of the Baihetan hydropower station.The results indicate that the activation scope of WISZ C2 induced by the excavation of underground caverns is approximately 0.5e1 times the main powerhouse span,showing local activation.Furthermore,the overall safety factor of WISZ C2 follows a normal distribution with an average value of 3.697.

Strong Law of Large Numbers and Complete Convergence for Sequences of -Mixing Random Variables

We give some theorems of strong law of large numbers and complete convergence for sequences of φ-mixing random variables. In particular, Wittmann＇s strong law of large numbers and Teicher＇s strong law of large nnumbers for independent random variables are generalized to the case of φ -minxing random variables.

Impact of random and scattered coincidences from outside of field of view on positron emission tomography/computed tomography imaging with different reconstruction protocols

Image quality in positron emission tomography(PET)is affected by random and scattered coincidences and reconstruction protocols.In this study,we investigated the effects of scattered and random coincidences from outside the field of view(FOV)on PET image quality for different reconstruction protocols.Imaging was performed on the Discovery 690 PET/CT scanner,using experimental configurations including the NEMA phantom(a body phantom,with six spheres of different sizes)with a signal background ratio of 4:1.The NEMA phantom(phantom I)was scanned separately in a one-bed position.To simulate the effect of random and scatter coincidences from outside the FOV,six cylindrical phantoms with various diameters were added to the NEMA phantom(phantom II).The 18 emission datasets with mean intervals of 15 min were acquired(3 min/scan).The emission data were reconstructed using different techniques.The image quality parameters were evaluated by both phantoms.Variations in the signal-to-noise ratio(SNR)in a 28-mm(10-mm)sphere of phantom II were 37.9%(86.5%)for ordered-subset expectation maximization(OSEM-only),36.8%(81.5%)for point spread function(PSF),32.7%(80.7%)for time of flight(TOF),and 31.5%(77.8%)for OSEM+PSF+TOF,respectively,indicating that OSEM+PSF+TOF reconstruction had the lowest noise levels and lowest coefficient of variation(COV)values.Random and scatter coincidences from outside the FOV induced lower SNR,lower contrast,and higher COV values,indicating image deterioration and significantly impacting smaller sphere sizes.Amongst reconstruction protocols,OSEM+PSF+TOF and OSEM+PSF showed higher contrast values for sphere sizes of 22,28,and 37 mm and higher contrast recovery coefficient values for smaller sphere sizes of 10 and 13 mm.

Complete Convergence and Weak Law of Large Numbers for <i><span style="text-decoration:overline;">ρ</span></i>-Mixing Sequences of Random Variables

In this paper, the complete convergence and weak law of large numbers are established for ρ-mixing sequences of random variables. Our results extend and improve the Baum and Katz complete convergence theorem and the classical weak law of large numbers, etc. from independent sequences of random variables to ρ-mixing sequences of random variables without necessarily adding any extra conditions.

Probabilistic stability analyses of undrained slopes by 3D random fields and finite element methods

A long slope consisting of spatially random soils is a common geographical feature. This paper examined the necessity of three-dimensional(3 D) analysis when dealing with slope with full randomness in soil properties. Although 3 D random finite element analysis can well reflect the spatial variability of soil properties, it is often time-consuming for probabilistic stability analysis. For this reason, we also examined the least advantageous(or most pessimistic) cross-section of the studied slope. The concept of"most pessimistic" refers to the minimal cross-sectional average of undrained shear strength. The selection of the most pessimistic section is achievable by simulating the undrained shear strength as a 3 D random field. Random finite element analysis results suggest that two-dimensional(2 D) plane strain analysis based the most pessimistic cross-section generally provides a more conservative result than the corresponding full 3 D analysis. The level of conservativeness is around 15% on average. This result may have engineering implications for slope design where computationally tractable 2 D analyses based on the procedure proposed in this study could ensure conservative results.

An extended stochastic response surface method for random field problems

An efficient and accurate uncertainty propagation methodology for mechanics problems with random fields is developed in this paper. This methodology is based on the stochastic response surface method （SRSM） which has been previously proposed for problems dealing with random variables only. This paper extends SRSM to problems involving random fields or random processes fields. The favorable property of SRSM lies in that the deterministic computational model can be treated as a black box, as in the case of commercial finite element codes. Numerical examples are used to highlight the features of this technique and to demonstrate the accuracy and efficiency of the proposed method. A comparison with Monte Carlo simulation shows that the proposed method can achieve numerical results close to those from Monte Carlo simulation while dramatically reducing the number of deterministic finite element runs.

Fast segmentation approach for SAR image based on simple Markov random field

Traditional image segmentation methods based on MRF converge slowly and require pre-defined weight. These disadvantages are addressed, and a fast segmentation approach based on simple Markov random field （MRF） for SAR image is proposed. The approach is firstly used to perform coarse segmentation in blocks. Then the image is modeled with simple MRF and adaptive variable weighting forms are applied in homogeneous and heterogeneous regions. As a result, the convergent speed is accelerated while the segmentation results in homogeneous regions and boarders are improved. Simulations with synthetic and real SAR images demonstrate the effectiveness of the proposed approach.

Detection and characterization of regulatory elements using probabilistic conditional random field and hidden Markov models

By altering the electrostatic charge of histones or providing binding sites to protein recognition molecules, Chromatin marks have been proposed to regulate gene expression, a property that has motivated researchers to link these marks to cis-regulatory elements. With the help of next generation sequencing technologies, we can now correlate one specific chromatin mark with regulatory elements (e.g. enhancers or promoters) and also build tools, such as hidden Markov models, to gain insight into mark combinations. However, hidden Markov models have limitation for their character of generative models and assume that a current observation depends only on a current hidden state in the chain. Here, we employed two graphical probabilistic models, namely the linear conditional random field model and multivariate hidden Markov model, to mark gene regions with different states based on recurrent and spatially coherent character of these eight marks. Both models revealed chromatin states that may correspond to enhancers and promoters, transcribed regions, transcriptional elongation, and low-signal regions. We also found that the linear conditional random field model was more effective than the hidden Markov model in recognizing regulatory elements, such as promoter-, enhancer-, and transcriptional elongation-associated regions, which gives us a better choice.

Optimization by Estimation of Distribution with DEUM Framework Based on Markov Random Fields

This paper presents a Markov random field （MRP） approach to estimating and sampling the probability distribution in populations of solutions. The approach is used to define a class of algorithms under the general heading distribution estimation using Markov random fields （DEUM）. DEUM is a subclass of estimation of distribution algorithms （EDAs） where interaction between solution variables is represented as an undirected graph and the joint probability of a solution is factorized as a Gibbs distribution derived from the structure of the graph. The focus of this paper will be on describing the three main characteristics of DEUM framework, which distinguishes it from the traditional EDA. They are： 1） use of MRF models, 2） fitness modeling approach to estimating the parameter of the model and 3） Monte Carlo approach to sampling from the model.

Spin-1 Blume-Capel model with longitudinal random crystal and transverse magnetic fields:A mean-field approach

The spin-1 Blume–Capel model with transverse and longitudinal external magnetic fields h, in addition to a longitudinal random crystal field D, is studied in the mean-field approximation. It is assumed that the crystal field is either turned on with probability p or turned off with probability 1 p on the sites of a square lattice. Phase diagrams are then calculated on the reduced temperature crystal field planes for given values of γ=Ω/J and p at zero h. Thus, the effect of changing γ and p are illustrated on the phase diagrams in great detail and interesting results are observed.

Neumann stochastic finite element method for calculating temperature field of frozen soil based on random field theory

To study the effect of uncertain factors on the temperature field of frozen soil, we propose a method to calculate the spatial average variance from just the point variance based on the local average theory of random fields. We model the heat transfer coefficient and specific heat capacity as spatially random fields instead of traditional random variables. An analysis for calculating the random temperature field of seasonal frozen soil is suggested by the Neumann stochastic finite element method, and here we provide the computational formulae of mathematical expectation, variance and variable coefficient. As shown in the calculation flow chart, the stochastic finite element calculation program for solving the random temperature field, as compiled by Matrix Laboratory （MATLAB） sottware, can directly output the statistical results of the temperature field of frozen soil. An example is presented to demonstrate the random effects from random field parameters, and the feasibility of the proposed approach is proven by compar- ing these results with the results derived when the random parameters are only modeled as random variables. The results show that the Neumann stochastic finite element method can efficiently solve the problem of random temperature fields of frozen soil based on random field theory, and it can reduce the variability of calculation results when the random parameters are modeled as spatial- ly random fields.

Prediction of Potential Disease-Associated MicroRNAs Based on Hidden Conditional Random Field

MicroRNAs( miRNAs) are reported to be associated with various diseases. The identification of disease-related miRNAs would be beneficial to the disease diagnosis and prognosis. However,in contrast with the widely available expression profiling, the limited knowledge of molecular function restrict the development of previous methods based on network similarity measure. To construct reliable training data,the decision fusion method is used to prioritize the results of existing methods. After that,the performance of decision fusion method is validated. Furthermore,in consideration of the long range dependencies of successive expression values,Hidden Conditional Random Field model( HCRF) is selected and applied to miRNA expression profiling to infer disease-associated miRNAs. The results show that HCRF achieves superior performance and outperforms the previous methods. The results also demonstrate the power of using expression profiling for discovering disease-associated miRNAs.

Some Limit Theorems for Weighted Sums of Random Variable Fields

Let{Xn^-,n^-∈N^d}be a field of Banach space valued random variables, 0 〈r〈p≤2 and{an^-,k^-, （n^-,k^-） ∈ N^d × N^d ,k^-≤n^-} a triangular array of real numhers, where N^d is the d-dimensional lattice （d≥1 ）. Under the minimal condition that {｜｜Xn^-｜｜ r,n^- ∈N^d} is {｜an^-,k^-｜^r,（n^-,k^-）} ∈ N^d ×N^d,k^-≤n^-}-uniformly integrable, we show that ∑（k^-≤n^-）an^-,k^-,Xk^-^（L^r（or a,s,）→0 as ｜n^-｜→∞ In the above, if 0〈r〈1, the random variables are not needed to be independent. If 1≤r〈p≤2, and Banach space valued random variables are independent with mean zero we assume the Banaeh space is of type p. If 1≤r≤p≤2 and Banach space valued random variables are not independent we assume the Banach space is p-smoothable.

Image Dehazing by Incorporating Markov Random Field with Dark Channel Prior

As one of the most simple and effective single image dehazing methods, the dark channel prior(DCP) algorithm has been widely applied. However, the algorithm does not work for pixels similar to airlight(e.g., snowy ground or a white wall), resulting in underestimation of the transmittance of some local scenes. To address that problem, we propose an image dehazing method by incorporating Markov random field(MRF) with the DCP. The DCP explicitly represents the input image observation in the MRF model obtained by the transmittance map. The key idea is that the sparsely distributed wrongly estimated transmittance can be corrected by properly characterizing the spatial dependencies between the neighboring pixels of the transmittances that are well estimated and those that are wrongly estimated. To that purpose, the energy function of the MRF model is designed. The estimation of the initial transmittance map is pixel-based using the DCP, and the segmentation on the transmittance map is employed to separate the foreground and background, thereby avoiding the block effect and artifacts at the depth discontinuity. Given the limited number of labels obtained by clustering, the smoothing term in the MRF model can properly smooth the transmittance map without an extra refinement filter. Experimental results obtained by using terrestrial and underwater images are given.

Rockhead profile simulation using an improved generation method of conditional random field

Rockhead profile is an important part of geological profiles and can have significant impacts on some geotechnical engineering practice,and thus,it is necessary to establish a useful method to reverse the rockhead profile using site investigation results.As a general method to reflect the spatial distribution of geo-material properties based on field measurements,the conditional random field(CRF)was improved in this paper to simulate rockhead profiles.Besides,in geotechnical engineering practice,measurements are generally limited due to the limitations of budget and time so that the estimation of the mean value can have uncertainty to some extent.As the Bayesian theory can effectively combine the measurements and prior information to deal with uncertainty,CRF was implemented with the aid of the Bayesian framework in this study.More importantly,this simulation procedure is achieved as an analytical solution to avoid the time-consuming sampling work.The results show that the proposed method can provide a reasonable estimation about the rockhead depth at various locations against measurement data and as a result,the subjectivity in determining prior mean can be minimized.Finally,both the measurement data and selection of hyper-parameters in the proposed method can affect the simulated rockhead profiles,while the influence of the latter is less significant than that of the former.

Power entity recognition based on bidirectional long short-term memory and conditional random fields

With the application of artificial intelligence technology in the power industry,the knowledge graph is expected to play a key role in power grid dispatch processes,intelligent maintenance,and customer service response provision.Knowledge graphs are usually constructed based on entity recognition.Specifically,based on the mining of entity attributes and relationships,domain knowledge graphs can be constructed through knowledge fusion.In this work,the entities and characteristics of power entity recognition are analyzed,the mechanism of entity recognition is clarified,and entity recognition techniques are analyzed in the context of the power domain.Power entity recognition based on the conditional random fields (CRF) and bidirectional long short-term memory (BLSTM) models is investigated,and the two methods are comparatively analyzed.The results indicated that the CRF model,with an accuracy of 83%,can better identify the power entities compared to the BLSTM.The CRF approach can thus be applied to the entity extraction for knowledge graph construction in the power field.