Marginal Probability Distribution Functions Research Articles

An iterative multi-fidelity scheme for simulating multi-dimensional non-Gaussian random fields

This paper presents an efficient multi-fidelity scheme to simulate multi-dimensional non-Gaussian random fields that are specified by covariance functions and marginal probability distribution functions. To develop the multi-fidelity scheme, two numerical algorithms are proposed in turn. The first algorithm is used to simulate random samples of the random field. In this algorithm, initial random samples are first generated to meet the marginal distribution and an iterative procedure is adopted to change the ranking of the random samples to match the target covariance function. By using the above algorithm, random samples satisfying the target covariance function and the target marginal distribution are obtained, but it is computationally intensive and is not suitable to simulate large-scale random fields. By taking advantage of Karhunen–Loève expansion, a multi-fidelity algorithm is then proposed to reduce the computational effort. The random variables in Karhunen–Loève expansion are calculated via performing the first algorithm on the low-fidelity model and the deterministic functions in Karhunen–Loève expansion are solved on the high-fidelity model. In this way, the proposed method has low computational effort and a high fidelity simultaneously. Three numerical examples, including two- and three-dimensional non-Gaussian random fields, are used to verify the good performance of the proposed algorithms.

Maximum entropy copula for bivariate drought analysis

Copula functions are widely used to derive multivariate probability distributions in hydrometeorology. One of the key steps in the copula method is the derivation of marginal distributions of individual variables which can be accomplished using the principle of maximum entropy where the distribution parameters are estimated from the specified constraints. This study, investigated two drought variables (severity and duration) by coupling the principle of maximum entropy with parametric and empirical copulas. So, homogeneous climatic zones were first identified by applying the fuzzy clustering method to data from 39 synoptic stations in Iran and then drought severity and duration were determined with the standardized precipitation index. These two variables were scaled and their marginal probability distribution functions were derived using the principle of maximum entropy as well as empirically. Then, the joint probability distribution of drought severity and duration was determined using maximum entropy-copula, and parametric and empirical copulas. Thereafter, bivariate conditional return periods were determined for each homogeneous region. Results showed that 1) univariate and bivariate distributions can be obtained by maximizing entropy; 2) the dependence structure via Spearman's rho, which directly affects the Lagrange parameters of entropy copula, was a controlling factor to optimize the objective function; 3) for a given set of constraints, the maximum entropy copula is independent of the types of marginals; 4)The entropy-entropy copula (with entropy marginals) is considered a better method than alternatives, because it has a similar result to the parametric methods while it only needs to fit a single model.

Use of S-transform and a new multi-taper S-transform for spectral and coherence estimates: With application to characterize and simulate multivariate non-Gaussian wind pressure coefficient

Obtaining a long duration time series of wind pressure coefficient from wind tunnel tests may not be economically feasible for some projects, and simulated synthetic time series of wind pressure coefficient can be used. In this study, we propose a new multi-taper S-transform (ST), where the discrete prolate spheroidal sequences are used for tapering, and ST is applied to each tapered time series. The proposed multi-taper ST is simple and efficient. The simulation results indicate that the proposed transform provides an unbiased estimate of the power spectral density (PSD) function with small variability. It is also shown that the use of the temporally-averaged results from ST or the proposed multi-taper ST leads to an unbiased estimate of the coherence function. As an application, we use the obtained PSD and coherence, and marginal probability distribution function (PDF) from the time series of the wind pressure coefficient as the targets to simulate univariate and multivariate processes. For the simulation of the non-Gaussian processes, we use the iterative power and amplitude correction but include a digital filter to consider sample-to-sample variability in the PSD function. We show that the PSD and coherence functions and the marginal PDF from the samples match their targets.

Directional wind characteristics analysis in the mountainous area based on field measurement

Wind characteristics are one of the most important factors when designing long-span bridges. In a typical deep-cutting gorge, the wind field is influenced by local terrain and shows dependence on wind direction according to preliminary numerical simulations. Field measurement is then employed to evaluate the impact of wind direction further. Instead of the common method, which splits samples into sectors by referring to the canyon direction, a qualitative computation approach is adopted. Specifically, marginal probability distribution functions (PDFs) of various wind characteristic parameters are explored initially, followed by binary and ternary joint PDFs associated with wind direction using the modified Bernstein copula. The directional wind characteristics are then examined utilizing conditional PDFs and mathematical expectations for all wind directions. The results show that both the distribution and mean value of wind parameters in severe winds exhibit a substantial association with wind direction. The statistical properties change significantly and cannot be captured using the common method when airflows enter from downstream of the canyon and are blocked by the ridge near the bridge. When designing long-span bridges sensitive to wind action, the impact of wind direction should be carefully factored, especially when local terrain blocks the frequent incoming flow.

Use of transform pairs to represent and simulate nonstationary non-Gaussian process with applications

We extend the use of the iterative power and amplitude correction (IPAC) algorithm to simulate nonstationary non-Gaussian processes by considering five transform pairs, including the redundant and non-redundant transform pairs that could provide time–frequency or time-scale decomposition. We compare the performance of the IPAC algorithm to that of the spectral representation method (SRM) in terms of matching the prescribed power spectral density function (PSDF) and the marginal probability distribution function. We also extend and examine the IPAC algorithm to simulate concatenated nonstationary non-Gaussian processes. It is emphasized that the IPAC algorithm can be used with a power or energy distribution defined in a transform domain corresponding to a selected transform pair and without directly relying on the definition of the evolutionary process. Unlike SRM, the use of the IPAC algorithm to simulate the nonstationary non-Gaussian process does not require evaluating the underlying Gaussian PSDF. This is especially advantageous for cases when one is interested in estimating the reliability or performance of a structural system during its design life that is subjected to a number of nonstationary non-Gaussian excitations, each defined by a different PSDF.

A framework for conditional simulation of nonstationary non-Gaussian random field and multivariate processes

We propose a conditional simulation framework for the nonstationary/nonhomogeneous non-Gaussian/Gaussian field and multivariate processes. There are three distinct stages within the proposed framework. The first one is to sample directly the unconditional nonstationary non-Gaussian field or multivariate processes without resorting to the underlying Gaussian characteristics. The second stage maps the unconditional samples and the conditions (i.e., observations) to the Gaussian space and uses them to find the conditional mean and covariance. In the final stage, the conditional samples by considering the effect of the conditioning are generated in Gaussian space and mapped back to the original non-Gaussian field. The efficiency of the framework relies on the efficient unconditional simulation algorithms (i.e., the iterative rank-dependent shuffling algorithm, and the iterative power and amplitude correction algorithm), and the reuse of the unconditional samples for the conditional simulation. The conditional samples match the conditional mean and conditional variance–covariance. The framework can directly apply to the random field and multivariate processes defined by the marginal probability distribution function and the covariance-variance information, or the time–frequency dependent PSD function. The framework is illustrated using several examples.

Anomalous diffusion in branched elliptical structure

Diffusion in narrow curved channels with dead-ends as in extracellular space in the biological tissues, e.g., brain, tumors, muscles, etc. is a geometrically induced complex diffusion and is relevant to different kinds of biological, physical, and chemical systems. In this paper, we study the effects of geometry and confinement on the diffusion process in an elliptical comb-like structure and analyze its statistical properties. The ellipse domain whose boundary has the polar equation with 0 < e < 1, θ ∈ [0,2π], and b as a constant, can be obtained through stretched radius r such that Υ = r ρ (θ) with r ∈ [0,1]. We suppose that, for fixed radius r = R, an elliptical motion takes place and is interspersed with a radial motion inward and outward of the ellipse. The probability distribution function (PDF) in the structure and the marginal PDF and mean square displacement (MSD) along the backbone are obtained numerically. The results show that a transient sub-diffusion behavior dominates the process for a time followed by a saturating state. The sub-diffusion regime and saturation threshold are affected by the length of the elliptical channel lateral branch and its curvature.

Estimation of Hydropower Potential Using Bayesian and Stochastic Approaches for Streamflow Simulation and Accounting for the Intermediate Storage Retention

Hydropower is the most widely used renewable power source worldwide. The current work presents a methodological tool to determine the hydropower potential of a reservoir based on available hydrological information. A Bayesian analysis of the river flow process and of the reservoir water volume is applied, and the estimated probability density function parameters are integrated for a stochastic analysis and long-term simulation of the river flow process, which is then used as input for the water balance in the reservoir, and thus, for the estimation of the hydropower energy potential. The stochastic approach is employed in terms of the Monte Carlo ensemble technique in order to additionally account for the effect of the intermediate storage retention due to the thresholds of the reservoir. A synthetic river flow timeseries is simulated by preserving the marginal probability distribution function properties of the observed timeseries and also by explicitly preserving the second-order dependence structure of the river flow in the scale domain. The synthetic ensemble is used for the simulation of the reservoir water balance, and the estimation of the hydropower potential is used for covering residential energy needs. For the second-order dependence structure of the river flow, the climacogram metric is used. The proposed methodology has been implemented to assess different reservoir volume scenarios offering the associated hydropower potential for a case study at the island of Crete in Greece. The tool also provides information on the probability of occurrence of the specific volumes based on available hydrological data. Therefore, it constitutes a useful and integrated framework for evaluating the hydropower potential of any given reservoir. The effects of the intermediate storage retention of the reservoir, the marginal and dependence structures of the parent distribution of inflow and the final energy output are also discussed.

Stochastic field model for the residual radius along the length of naturally and artificially corroded rebars

This paper examines the random variation of the residual radius along the length of both naturally corroded and artificially corroded rebars. The rebars are washed to remove rust, and their residual area is measured by the water volume method for every 20-mm long segment of each rebar. Tests are carried out to determine the stationarity or non-stationarity of the stochastic field used to model the residual radius of the specimens. Then, the autocorrelation function, power spectrum and marginal probability distribution function of each specimen are estimated. It is found that the residual radius field of both naturally and artificially corroded rebars with mean corrosion levels below 20% is stationary, while the situation can change as the mean corrosion level increases and concrete cover spalling occurs. The residual radius field along the rebar length of stationary specimens is found to be white noise for both naturally and artificially corroded rebars. The marginal probability distribution function of the residual radius is determined to be skewed and can be fitted with a Beta distribution for naturally corroded rebars and with a Weibull distribution for artificially corroded rebars. This is the first paper studying and comparing both naturally and artificially corroded rebars.

Simulating nonstationary non-Gaussian vector process based on continuous wavelet transform

In the present study, we propose a new iterative algorithm based on discretized continuous wavelet transform (CWT) to simulate nonstationary non-Gaussian vector process for prescribed marginal probability distribution functions and the time-scale power spectral density function matrix. The algorithm is designed for a CWT pair within the wavelet analysis framework. It can be used with analytical wavelets satisfying the admissibility condition and can cope with time-dependent and time-independent coherence. The proposed algorithm is applied to generate nonstationary downburst winds and seismic ground motions at multiple sites within the wavelet analysis framework. The numerical validation analysis confirms that the proposed algorithm can lead to the simulated records matching the prescribed marginal probability distribution, marginal time-scale power spectral density function and coherence function.

An algorithm to simulate nonstationary and non-Gaussian stochastic processes

We proposed a new iterative power and amplitude correction (IPAC) algorithm to simulate nonstationary and non-Gaussian processes. The proposed algorithm is rooted in the concept of defining the stochastic processes in the transform domain, which is elaborated and extend. The algorithm extends the iterative amplitude adjusted Fourier transform algorithm for generating surrogate and the spectral correction algorithm for simulating stationary non-Gaussian process. The IPAC algorithm can be used with different popular transforms, such as the Fourier transform, S-transform, and continuous wavelet transforms. The targets for the simulation are the marginal probability distribution function of the process and the power spectral density function of the process that is defined based on the variables in the transform domain for the adopted transform. The algorithm is versatile and efficient. Its application is illustrated using several numerical examples.

Simulation of stationary Gaussian/non-Gaussian stochastic processes based on stochastic harmonic functions

A new model is proposed to represent and simulate Gaussian/non-Gaussian stochastic processes. In the proposed model, stochastic harmonic function (SHF) is extended to represent multivariate Gaussian process firstly. Compared with the conventional spectral representation method (SRM), the SHF based model requires much fewer variables and Cholesky decompositions. Then, SHF based model is further extended to univariate/multivariate non-Gaussian stochastic process simulation. The target non-Gaussian process can be obtained from the corresponding underlying Gaussian processes by memoryless nonlinear transformation. For arbitrarily given marginal probability distribution function (PDF), the covariance function of the underlying multivariate Gaussian process can be determined easily by introducing the Mehler’s formula. And when the incompatibility between the target non-Gaussian power spectral density (PSD) or PSD matrix and marginal PDF exists, the calibration of the target non-Gaussian spectrum will be required. Hence, the proposed model can be regarded as SRM to efficiently generate Gaussian/non-Gaussian processes. Finally, several numerical examples are addressed to show the effectiveness of the proposed method.

Linear prediction and z-transform based CDF-mapping simulation algorithm of multivariate non-Gaussian fluctuating wind pressure

Methods for stochastic simulation of non-Gaussian wind pressure have increasingly addressed the efficiency and accuracy contents to offer an accurate description of the extreme value estimation of the long-span and high-rise structures. This paper presents a linear prediction and z-transform (LPZ) based Cumulative distribution function (CDF) mapping algorithm for the simulation of multivariate non-Gaussian fluctuating wind pressure. The new algorithm generates realizations of nonGaussian with prescribed marginal probability distribution function (PDF) and prescribed spectral density function (PSD). The inverse linear prediction and z-transform function (ILPZ) is deduced. LPZ is improved and applied to non-Gaussian wind pressure simulation for the first time. The new algorithm is demonstrated to be efficient, flexible, and more accurate in comparison with the FFT-based method and Hermite polynomial model method in two examples for transverse softening and longitudinal hardening non-Gaussian wind pressures.

A novel approach for reliability analysis with correlated variables based on the concepts of entropy and polynomial chaos expansion

Correlated random variables are common in industry field. In reliability analysis community, Nataf transformation is considered as a powerful tool for handling correlated random variables, since it only requires the marginal probability distribution functions of input random variables. However, when accurate marginal probability distributions are unavailable, Nataf transformation cannot be used. This paper presents an alternative method for transforming correlated random variables into independent ones based on the maximum entropy principle and the polynomial chaos expansion. The proposed method only requires the first-several statistical moments of input random variables but not the probability distribution functions. Based on the proposed method for handling correlated random variables, the statistical moments of performance functions can be calculated. In order to predict the failure probability, the fractional moment-based maximum entropy method (FM-MEM) is employed due to its accuracy. However, the FM-MEM is sensitive to the initial point of its outer loop and also requires too much CPU time. Thus, an improved version is developed to enhance the performance of the algorithm. To verify the validity of the proposed method, three numerical examples and one engineering example are tested. The results show that the proposed method is a good choice for reliability analysis with correlated random variables, especially when only the statistical moment information of input random variables is available.

Telescope method for characterizing the spatial structure of a pine-oak mixed forest in the Xiaolong Mountains, China

ABSTRACTForest structure is largely determined by nearest neighbor interactions, and reveals detailed interpretations of the current situation and development potential of the stand. A structure-based method is needed to analyze the relationship of nearest neighbor tree groups and therefore to characterize the forest spatial heterogeneity in a comprehensively and systematically way. A natural mixed forest of Quercus aliena var. acutiserrata and Pinus tabulaeformis in the Xiaolong Mountains, China was taken as an example to demonstrate the telescope method (i.e. N-variate distributions). Each tree with DBH ≥ 5 cm was investigated and located, and the frequencies of N-variate distributions flexibly combined with four structure parameters, uniform angle index, mingling, dominance and crowding were calculated using Excel pivot tables and the Winkelmass software. The telescope method could systematically interpret the forest structural characteristics at different resolutions. Especially, the quadrivariate distribution provides the most detailed and comprehensive spatial structure information. Based on the vertical projection dimension reduction and marginal probability distribution function, the continuous recursive process well revealed the inherently quantitative relationships among different distributions. This could be conducive to flexibly optimizing and reconstructing forest structure, and effectively selecting the trees to be removed.

Analysis of Maximum Runoff Volumes with Different Time Durations of Flood Waves: A Case Study on Topl’a River in Slovakia

In applied hydrology, it is problematic to assign the flood wave volume values with a certain probability of exceedance to given corresponding T-year discharges. This dependence is highly irregular, and requires knowledge the flood wave course of the given probability. For this reason, this work deals with the determination of the annual maximum discharge volumes on the Topl’a River for the time duration of 2-, 5-, 10-, and 15-days (Vtmax). The series of 84 years (1931-2015) mean daily discharges of the Topl’a River at Hanušovce above Topl’a station was used as input data to calculate the maximum annual volumes of runoff of the Topl’a River. Subsequently, the theoretical curves of exceedance of the maximal discharge volumes Vtmax were determined by the Log-Pearson distribution of the Type III. This type of probability distribution is used to estimate maximum (extreme) values across a range of natural processes. The results showed relatively small differences in estimated T-year volumes when compared to other types of theoretical distribution functions used in hydrological extreme analyses in Slovakia (Gamma, Log-normal, etc.). The second part of our work was focused on the bivariate analysis of the relationship between T-year maximum volumes with different duration and peak discharges by the three Archimedean copula functions (Clayton, Gumbel-Hougaard and Frank). The LPIII distribution was used as marginal probability distribution function. Subsequently joint and conditional return periods of the T-year maximum annual flows and T-year maximum volumes with different time duration were calculated. The first one defines joint return periods as the return periods using one random variable equalling or exceeding a certain magnitude and/or using another random variable equalling or exceeding another certain magnitude. The second one is conditional return periods for one random variable, given that another random variable equals or exceeds a specific magnitude.

Force distributions in frictional granular media.

We report a joint experimental and theoretical investigation of the probability distribution functions (PDFs) of the normal and tangential (frictional) forces in amorphous frictional media. We consider both the joint PDF of normal and tangential forces together, and the marginal PDFs of normal forces separately and tangential forces separately. A maximum entropy formalism is utilized for all these cases after identifying the appropriate constraints. Excellent agreements with both experimental and simulation data are reported. The proposed joint PDF predicts giant slip events at low pressures, again in agreement with observations.

Laplace Autoregressive Model for Cascaded Systems

System modeling problems, such as the channel of a single-hop relay communication system in a flat-fading environment, require a cascade of two or more autoregressive (AR) processes to capture the entire system characteristics. However, for the purpose of system simulation and parameter estimations, it is more convenient if the entire system is modeled by a single AR model. In this paper, we consider a cascade system whose statistical characteristics of its subsystems is represented by two independent ${p}$ th-order complex Gaussian AR processes, and model it by a single ${p}$ th-order Laplace AR process. In our analysis, we first demonstrate that the marginal probability distribution functions of the real and imaginary components of a system described by a cascade of the two complex Gaussian AR processes are Laplace distributed. Then, to model the cascaded system, we develop a single complex Laplace AR process whose parameters are configured to match other statistical characteristics of the cascaded system. Specifically, we show that the autocorrelation of the developed Laplace AR process satisfies Yule–Walker type of equations and derive the steps for the design of its parameters through autocorrelation matching.

Evaluation of rainfall spatial correlation effect on rainfall-runoff modeling uncertainty, considering 2-copula

Lack of accuracy of rainfall-runoff simulation (RRS) remains critical for some applications. Among various sources of uncertainty, precipitation plays a particular role. Rainfall rates as the main input data of RRS are of the first factors controlling the accuracy. In addition to the depth, spatial and temporal distributions of rainfall impact the flood discharge. Most of the previous studies on RRS uncertainty have ignored rainfall spatial distribution, where in large catchments, it is necessary to be modeled explicitly. Karoon III is one most important basin of the Iran because of the Karoon III dam in the outlet. In the present work, effect of spatial correlation of rainfall on HEC-HMS (SMA) continuous RRS uncertainty is evaluated using 2variate copula (2copula). Monte Carlo simulation (MCS) approach was used to consider the rainfall spatial dependence. To reduce the computational expense, sampling efficiency and convergence for MCS, Latin hypercube sampling (LHS) was used. Copula functions consider wide range of marginal probability distribution functions (PDFs), eliminating limits of regular join PDFs. For this aim, two scenarios were investigated. In the first scenario, sub-basin rainfall was considered independent, and in the second scenario, 2copula was adopted to model spatial correlation of rainfall. Dimensionless rainfall depths were calculated for each sub-basin, and the PDFs were determined. The generated random dimensionless rainfalls were reweighted and multiplied by watershed’s mean rainfall value. Stochastic Climate Library was used to generate continuous daily rainfalls. Sampling from dimensionless rainfalls using LHS algorithm, 100 runs of calibrated model-simulated 100 flows for each day following MCS, and 80 % certainty bound was calculated. Results showed that considering dependence decreased 18 % of the maximum uncertainty bound width, so the methodology could be recommended for decreasing predicted runoff error.

Statistical reconstruction and Karhunen–Loève expansion for multiphase random media

SummaryTermed as random media, rocks, composites, alloys and many other heterogeneous materials consist of multiple material phases that are randomly distributed through the medium. This paper presents a robust and efficient algorithm for reconstructing random media, which can then be fed into stochastic finite element solvers for statistical response analysis. The new method is based on nonlinear transformation of Gaussian random fields, and the reconstructed media can meet the discrete‐valued marginal probability distribution function and the two‐point correlation function of the reference medium. The new method, which avoids iterative root‐finding computation, is highly efficient and particularly suitable for reconstructing large‐size random media or a large number of samples. Also, benefiting from the high efficiency of the proposed reconstruction scheme, a Karhunen–Loève (KL) representation of the target random medium can be efficiently estimated by projecting the reconstructed samples onto the KL basis. The resulting uncorrelated KL coefficients can be further expressed as functions of independent Gaussian random variables to obtain an approximate Gaussian representation, which is often required in stochastic finite element analysis. Copyright © 2015 John Wiley &amp; Sons, Ltd.