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The probability analysis of ground deformation is becoming a trend to estimate and control the risk brought by shield tunnelling. The gap parameter is regarded as an effective tool to estimate the ground loss of tunnelling in soft soil. More specifically, ω, which is a gap parameter component defined as the over (or insufficient) excavation due to the change in the posture of the shield machine, may contribute more to the uncertainty of the ground loss. However, the existing uncertainty characterization methods for ω have several limitations and cannot explain the uncertain correlations between the relevant parameters. Along these lines, to better characterize the uncertainty of ω, the multivariate probability distribution was developed in this work and a dynamic prediction was proposed for it. To attain this goal, 1 523 rings of the field data coming from the shield tunnel between Longqing Road and Baiyun Road in Kunming Metro Line 5 were utilized and 44 parameters including the construction, stratigraphic, and posture parameters were collected to form the database. According to the variance filter method, the mutual information method, and the value of the correlation coefficients, the original 44 parameters were reduced to 10 main parameters, which were unit weight, the stoke of the jacks (A, B, C, and D groups), the pressure of the pushing jacks (A, C groups), the chamber pressure, the rotation speed, and the total force. The multivariate probability distribution was constructed based on the Johnson system of distributions. Moreover, the distribution was satisfactorily verified in explaining the pairwise correlation between ω and other parameters through 2 million simulation cases. At last, the distribution was used as a prior distribution to update the marginal distribution of ω with any group of the relevant parameters known. The performance of the dynamic prediction was further validated by the field data of 3 shield tunnel cases.

We consider a discrete search problem to detect a hidden target on one of the M-intersected real lines at the origin. The target position is a vector of independent random variables with a known multivariate probability distribution that is symmetric around the origin. From the available information about the probability of the target, we can consider one bounded interval as a search space on each line. To maximize the detection probability, we divide each interval into a number of small subintervals. Some subintervals have low target probabilities; thus, we delete them and distribute their probabilities among the searched subintervals. The problem has a discrete version where each line contains a set of independent search subintervals. The search effort is bounded by a normal distribution. More than obtaining the maximum detection probability, we also need to get the minimum search effort to detect the target. After studying the stability of the minimum search effort, we present an illustrative example to show the effectiveness of our model. [ABSTRACT FROM AUTHOR]

In this paper, a new bivariate absolutely continuous probability distribution is introduced. The new distribution, which is called the bivariate unit-sinh-normal (BVUSHN) distribution, arises by applying a transformation to the bivariate Birnbaum–Saunders distribution (BVBS). The main properties of the new proposal are studied in detail. In addition, from the new distribution, the BVUSHN regression model is also introduced. For both the bivariate probability distribution and the respective associated regression model, parameter estimation is conducted from a classical approach by using the maximum likelihood method together with the two-step estimation method. A small Monte Carlo simulation study is carried out to evaluate the behavior of the used estimation method and the properties of the estimators. Finally, for illustrative purposes, two applications with real data are presented in which the usefulness of the proposals is evidenced. [ABSTRACT FROM AUTHOR]

This paper assumes that the particle jumps randomly (Guassian jumps) from one point to another along one of the imaginary lines inside the interactive medium. Since this study was done in the space, we consider that the position of the particle at any time t has a multivariate distribution. The random waiting time of the particle for each Gaussian jump depends on its length. An identical set of programed nanosensors (with unit speed) were used to track this particle. Each line has a sensor that starts the tracking process from the origin. The existence of the necessary conditions which give the optimal search plan and the minimum expected value of the particle detection has been proven. This study is supported by a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, a new bivariate absolutely continuous probability distribution is introduced. The new distribution, which is called the bivariate unit-sinh-normal (BVUSHN) distribution, arises by applying a transformation to the bivariate Birnbaum–Saunders distribution (BVBS). The main properties of the new proposal are studied in detail. In addition, from the new distribution, the BVUSHN regression model is also introduced. For both the bivariate probability distribution and the respective associated regression model, parameter estimation is conducted from a classical approach by using the maximum likelihood method together with the two-step estimation method. A small Monte Carlo simulation study is carried out to evaluate the behavior of the used estimation method and the properties of the estimators. Finally, for illustrative purposes, two applications with real data are presented in which the usefulness of the proposals is evidenced.

Probabilistic site characterization is the cornerstone of data-centric geotechnics, which underpins the digitalization in geotechnical engineering. The essential aspect of probabilistic site characterization is to reasonably infer the probability density distributions of geotechnical parameters, based on field and laboratory data. In this paper, a clay property database, SZ-CLAY/11/5130, was established by collecting geotechnical investigation reports from 52 engineering projects in Shenzhen city. This SZ-CLAY/11/5130 database was first compared with the global clay database CLAY/10/7490 and the Shanghai clay database SH-CLAY/11/4051, in terms of their sample ranges, correlations, and scatter plots. Then, the multivariate probability distribution of eight clay parameters was established using the training records selected from SZ-CLAY/11/5130, to characterize the generic correlations between multiple clay parameters. Based on Bayesian theory, the posterior distributions of clay mechanical parameters from a construction site in Shenzhen city were updated using both the site-specific correlations and the generic correlations given by SZ-CLAY/11/5130. The constructed multivariate probability density distribution is necessary for this updating step. [ABSTRACT FROM AUTHOR]

We address the inverse Frobenius–Perron problem: given a prescribed target distribution ρ, find a deterministic map M such that iterations of M tend to ρ in distribution. We show that all solutions may be written in terms of a factorization that combines the forward and inverse Rosenblatt transformations with a uniform map; that is, a map under which the uniform distribution on the d-dimensional hypercube is invariant. Indeed, every solution is equivalent to the choice of a uniform map. We motivate this factorization via one-dimensional examples, and then use the factorization to present solutions in one and two dimensions induced by a range of uniform maps.

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In this contribution we propose a method for determining the type of computer component whose replacement minimizes repair costs depending on the warranty period. Components that most often cause a dysfunctional state of a computer are first identified using records of warranty repairs from the customer service department. Repair costs covered by warranty are then calculated using a semi-analytic method and simulation of the most critical component lifetime in different user environments. It was assumed that the use of a cheaper critical component generally results in higher costs for computer warranty services due to shorter failure-free runs. For this reason, the company should choose a compromise between price and quality. Input data was taken from the records supplied by the computer assembling company (times between failures) and from the records from time tracking software (computer up-time). [ABSTRACT FROM AUTHOR]

Bayesian belief nets (BBNs) have become a popular tool for specifying high-dimensional probabilistic models. Commercial tools with an advanced graphical user interface that support BBNs construction and inference are available. Thus, building and working with BBNs is very efficient as long as one is not forced to quantify complex BBNs. A high assessment burden of discrete BBNs is often caused by the discretization of continuous variables. Until recently, continuous BBNs were restricted to the joint normal distribution. We present the ‘copula–vine’ approach to continuous BBNs. This approach is quite general and allows traceable and defendable quantification methods, but it comes at a price: these BBNs must be evaluated by Monte Carlo simulation. Updating such a BBN requires re-sampling the whole structure. The advantages of fast updating algorithms for discrete BBNs are decisive. A hybrid method advanced here samples the continuous BBN once, and then discretizes this so as to enable fast updating. This combines the reduced assessment burden and modelling flexibility of the continuous BBNs with the fast updating algorithms of discrete BBNs. Sampling large complex structures only once can still involve time consuming numerical calculations. Therefore a new sampling protocol based on normal vines is developed. Normal vines are used to realize the dependence structure specified via (conditional) rank correlations on the continuous BBN. We will emphasize the advantages of this method by means of examples. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In radio wave propagation and wireless communication fields, it is known that rainfall attenuation (or rainfall intensity) in an area with small reception power or time ratio under a fading environment can be well approximated by a Gamma distribution (Nakagami-m distribution). To investigate the selection diversity method, a joint probability function that takes correlation into consideration for multiple random variables must be used to quantitatively evaluate its results. In this paper, the authors derive an exact representation according to an infinite series for a general multivariate (M-dimensional) Gamma distribution function with correlation. In addition, besides investigating the numerical behavior of the exact representation, they obtain an extremely effective approximate representation for practical use. That approximation, which is a revision of the probability distribution with no correlation according to the determinant of a matrix having the square roots of the correlation coefficients as its elements, not only is easy to handle, but also provides an excellent physical interpretation. © 2004 Wiley Periodicals, Inc. Electron Comm Jpn Pt 1, 87(8): 1–12, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecja.10224 [ABSTRACT FROM AUTHOR]

In the present work a general kernel density model (KDM) has been introduced and assessed for the analytic representation of any empirical distribution function (univariate or multivariate) of metocean parameters. This model is based on the concept of kernel density function, which has been first introduced in the context of non-parametric discriminant analysis by Fix and Hodges [Nonparametric discrimination: consistency properties, 1951], and generalised to the multivariate case by Cacoulos [Ann Inst Math 18 (1966) 178]. In its standard form, the kernel density estimators are applied to given samples of observations, producing analytical (yet non-parametric) estimations of the unknown (underlying) probability density functions. Motivated by the fact that, in many practical applications, metocean data are available only in the form of histograms (grouped data), the present KDM is implemented in a way permitting to obtain analytical estimates of the underlying probability distributions on the basis of grouped data. The main features of the proposed KDM are: (i) it can treat multivariate data with very reasonable computational cost, (ii) it is flexible enough to represent quite general distribution patterns both in the univariate and in the multivariate case, (iii) with an appropriate selection of the bandwidth, results in very satisfactory representations, avoiding local pathologies, (iv) it is marginally consistent, i.e. any marginal distribution calculated by integration from the multivariate model is identical with the corresponding marginal KDM, obtained directly from the marginal data. First numerical results are presented herein, showing that the performance of the present model is very satisfactory for all the wave parameters studied, univariate, bivariate, the trivariate (HS,Tm,Θm), and conditionals. A more detailed investigation, also including applications to other metocean parameters, will be presented as a second part. [Copyright &y& Elsevier]

A vine is a new graphical model for dependent random variables. Vines generalize the Markov trees often used in modeling multivariate distributions. They differ from Markov trees and Bayesian belief nets in that the concept of conditional independence is weakened to allow for various forms of conditional dependence. A general formula for the density of a vine dependent distribution is derived. This generalizes the well-known density formula for belief nets based on the decomposition of belief nets into cliques. Furthermore, the formula allows a simple proof of the Information Decomposition Theorem for a regular vine. The problem of (conditional) sampling is discussed, and Gibbs sampling is proposed to carry out sampling from conditional vine dependent distributions. The so-called ‘canonical vines’ built on highest degree trees offer the most efficient structure for Gibbs sampling. [ABSTRACT FROM AUTHOR]

• We provide a structure-based risk framework for design and risk assessment. • It relies on the multivariate distribution of hydrological load and system failure. • The method is exemplified for flood mitigation offline reservoirs. • Both simplified routing and Monte Carlo detailed hydrodynamic simulation are used. • Roles of design parameters are disentangled, providing indications for practitioners. The design of hydraulic structures for water control is based on the principle of accepting that structures might fail to fulfill their intended purpose as consequence of extreme, yet rare events producing damages. Traditionally, structures are sized to resist to a "design event", i.e. a hyetograph or hydrograph associated to a fixed probability of occurrence. This implicitly assumes that the probability of structure failure equals that of the hydrological load, although unjustified under general conditions. Indeed, structure failure also depends on the failure mechanism – which is a characteristic of the specific structure and problem at hand – ruling the interaction between the hydrological load and the structure itself. To overcome the limitations of the hydrological design event approach, we provide here a general, consistent, "structure-based" risk framework, for design or risk assessment. The framework combines the multivariate statistical distribution of the hydrological process and the hydraulic response of the structure. Remarkably, it reduces the potential complexity of the problem to the statistical analysis of the univariate probability distribution of the damage, i.e. the consequence of structure failure. The approach is exemplified for an offline reservoir for flood mitigation, by adopting both a simplified routing model and a Monte Carlo procedure based on a detailed hydrodynamic simulation. Results demonstrate the potential of the framework, regardless of the accuracy of the modeling approach adopted for its implementation. Specifically, the framework allows disentangling the roles of the different design parameters involved, by providing clear indications for practitioners. [ABSTRACT FROM AUTHOR]

We address the inverse Frobenius–Perron problem: given a prescribed target distribution ρ , find a deterministic map M such that iterations of M tend to ρ in distribution. We show that all solutions may be written in terms of a factorization that combines the forward and inverse Rosenblatt transformations with a uniform map; that is, a map under which the uniform distribution on the d-dimensional hypercube is invariant. Indeed, every solution is equivalent to the choice of a uniform map. We motivate this factorization via one-dimensional examples, and then use the factorization to present solutions in one and two dimensions induced by a range of uniform maps. [ABSTRACT FROM AUTHOR]

To assess the impact of implementing energy efficiency and renewable energy measures, urban building energy models are emerging. In these models, due to the lack of data, the natural variability of the existing building stock is often highly underestimated and uncertainty on the simulated energy use arises. Therefore, this work proposes a probabilistic building characterization method to model the variability of the existing residential building stock. The method estimates realistic distributions of five input variables: U-values of the floor, external walls, windows and roof as well as window-to-wall ratio, based on known data (location, geometry and construction year). First, quantile regression has been implemented to generate the uncorrelated distributions based on the Flemish energy performance certificates database. The accuracy of the marginal distributions is good, as the empirical coverage on the 50%, 80%, 90% and 98% prediction interval deviates 0.6% at most. However, it is needed to include the correlations between these variables. Hence, three main methods to build multivariate distributions from marginal distributions and to draw correlated samples are implemented and extensively compared. The Gaussian copula method is put forward as the preferred method. Considering the mean-maximum discrepancy (MMD), this method performs eight times better than the uncorrelated case (MMD of 0.0027 versus 0.0228). [ABSTRACT FROM AUTHOR]