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A framework of spectral representation-based dimension reduction for simulating multivariate non-stationary stochastic ground motion processes is addressed in this paper. By means of introducing random functions serving as constraints correlating with the orthogonal random variables in the original spectral representation scheme, the high-dimensional randomness degree involved in the multivariate stochastic processes can thus be reduced substantially. To this aim, three random function forms considering the combination of trigonometric functions and orthogonal polynomials are constructed for simulation purpose. Accordingly, the accurate representation of the original stochastic processes is realized with merely three elementary random variables, overcoming the principal challenge of numerous random variables faced by the Monte Carlo simulation method. Also, the consistency of the statistics between the sample functions of stochastic ground motions and strong motion records is established with update of the ground motion model parameters in stochastic simulation techniques. Numerical investigations involving the comparisons with the Monte Carlo simulation method and the validation based on the strong motion records are presented to demonstrate the superiority and effectiveness of the proposed methodology in practical engineering applications. [ABSTRACT FROM AUTHOR]

This paper develops a hybrid approach of spectral representation and random function for simulating stationary stochastic vector processes. In the proposed approach, the high-dimensional random variables, included in the original spectral representation (OSR) formula, could be effectively reduced to only two elementary random variables by introducing the random functions that serve as random constraints. Based on this, a satisfactory simulation accuracy can be guaranteed by selecting a small representative point set of the elementary random variables. The probability information of the stochastic excitations can be fully emerged through just several hundred of sample functions generated by the proposed approach. Therefore, combined with the probability density evolution method (PDEM), it could be able to implement dynamic response analysis and reliability assessment of engineering structures. For illustrative purposes, a stochastic turbulence wind velocity field acting on a frame-shear-wall structure is simulated by constructing three types of random functions to demonstrate the accuracy and efficiency of the proposed approach. Careful and in-depth studies concerning the probability density evolution analysis of the wind-induced structure have been conducted so as to better illustrate the application prospects of the proposed approach. Numerical examples also show that the proposed approach possesses a good robustness. [ABSTRACT FROM AUTHOR]

Conventional Karhunen-Loeve expansions for simulation of stochastic processes often encounter the challenge of dealing with hundreds of random variables. For breaking through the barrier, a random function embedded Karhunen-Loeve expansion method is proposed in this paper. The updated scheme has a similar form to the conventional Karhunen-Loeve expansion, both involving a summation of a series of deterministic orthonormal basis and uncorrelated random variables. While the difference from the updated scheme lies in the dimension reduction of Karhunen-Loeve expansion through introducing random functions as a conditional constraint upon uncorrelated random variables. The random function is expressed as a single-elementary-random-variable orthogonal function in polynomial format (non-Gaussian variables) or trigonometric format (non-Gaussian and Gaussian variables). For illustrative purposes, the simulation of seismic ground motion is carried out using the updated scheme. Numerical investigations reveal that the Karhunen-Loeve expansion with random functions could gain desirable simulation results in case of a moderate sample number, except the Hermite polynomials and the Laguerre polynomials. It has the sound applicability and efficiency in simulation of stochastic processes. Besides, the updated scheme has the benefit of integrating with probability density evolution method, readily for the stochastic analysis of nonlinear structures. [ABSTRACT FROM AUTHOR]

In conjunction with the formulation of random functions, a family of renewed spectral representation schemes is proposed. The selected random function serves as a random constraint correlating the random variables included in the spectral representation schemes. The objective stochastic process can thus be completely represented by a dimension-reduced spectral model with just few elementary random variables, through defining the high-dimensional random variables of conventional spectral representation schemes (usually hundreds of random variables) into the low-dimensional orthogonal random functions. To highlight the advantages of this scheme, orthogonal trigonometric functions with one and two random variables are constructed. Representative-point set of the dimension-reduced spectral model is derived by employing the probability-space partition techniques. The complete set with assigned probabilities of points gains a low-number-sample stochastic process. For illustrative purposes, the stochastic modeling of seismic acceleration processes is proceeded, of which the stationary and non-stationary cases are investigated. It is shown that the spectral acceleration of simulated processes matches well with the target spectrum. Stochastic seismic response analysis, moreover, and reliability assessment of a framed structure with Bouc-Wen behaviors are carried out using the probability density evolution method. Numerical results reveal the applicability and efficiency of the proposed simulation technique. [ABSTRACT FROM AUTHOR]

A random function of a variabletinT(or, a random function onT) is known as a functionfwhose values are random variables all defined on a common probability space, whereTis an arbitrary set. A random function is also called a stochastic (random) process. In this work, we base ourselves on a random function ofE-process type and such a function is also called a random function, briefly. In this approach, the domain of such a random function isor an interval of, and the set of values of this random function is considered as a special probabilistic metric (PM) space (more precisely, anE-space) of metric space-valued random variables, and all our definitions and results are presented using the tools of PM spaces. In this context, we introduce the concept of an exhaustive family of such random functions, which is a natural generalization of equicontinuity, and we investigate its basic properties. We also examine some of the properties related to the continuous convergence in probability for a sequence of such random functions and certain conditions which give rise to the continuity in probability of the limit of a sequence of such random functions. [ABSTRACT FROM PUBLISHER]

In shallow geothermal systems, the main equivalent underground thermal properties are commonly calculated with a thermal response test (TRT). This is a borehole heat exchanger production test where the temperature of a heat transfer fluid is recorded over time at constant power heat injection/extraction. The equivalent thermal parameters (thermal conductivity, heat capacity) are simply deduced from temperature data regression analysis that theoretically is a logarithmic function in the time domain, or else a linear function in the log-time domain. By interpreting the recorded temperatures as a regionalized variable whose drift is the regression function, in both cases the formal problem is a linear estimation of the mean. If the autocorrelation function (variogram, covariance) of residuals is known, coefficient variance can be directly deduced. Coefficient estimates are independent of the drift form adopted, and the residuals are the same in the same points. The random function is different in the time domain, however, and in the log-time domain. In fact, residual variograms are different due to the transformation of the coordinate space. This paper uses a TRT case study to examine the consequences of coordinate space transformation for a random function, namely its variogram. The specific question addressed is the choice of coordinate space and variogram. [ABSTRACT FROM AUTHOR]

We introduce a new approach for ground motion relations (GMR) in the probabilistic seismic hazard analysis (PSHA), being influenced by the extreme value theory of mathematical statistics. Therein, we understand a GMR as a random function. We derive mathematically the principle of area equivalence, wherein two alternative GMRs have an equivalent influence on the hazard if these GMRs have equivalent area functions. This includes local biases. An interpretation of the difference between these GMRs (an actual and a modeled one) as a random component leads to a general overestimation of residual variance and hazard. Beside this, we discuss important aspects of classical approaches and discover discrepancies with the state of the art of stochastics and statistics (model selection and significance, test of distribution assumptions, extreme value statistics). We criticize especially the assumption of logarithmic normally distributed residuals of maxima like the peak ground acceleration (PGA). The natural distribution of its individual random component (equivalent to exp( ε) of Joyner and Boore, Bull Seism Soc Am 83(2):469-487, ) is the generalized extreme value. We show by numerical researches that the actual distribution can be hidden and a wrong distribution assumption can influence the PSHA negatively as the negligence of area equivalence does. Finally, we suggest an estimation concept for GMRs of PSHA with a regression-free variance estimation of the individual random component. We demonstrate the advantages of event-specific GMRs by analyzing data sets from the PEER strong motion database and estimate event-specific GMRs. Therein, the majority of the best models base on an anisotropic point source approach. The residual variance of logarithmized PGA is significantly smaller than in previous models. We validate the estimations for the event with the largest sample by empirical area functions, which indicate the appropriate modeling of the GMR by an anisotropic point source model. The constructed distances like the Joyner-Boore distance do not work well for event-specific GMRs. We discover also a strong relation between magnitude and the squared expectation of the PGAs being integrated in the geo-space for the event-specific GMRs. One of our secondary contributions is the simple modeling of anisotropy for a point source model. [ABSTRACT FROM AUTHOR]

Self-stochasticity theory, established for certain classes of deterministic finite-dimensional dynamical systems, is extended to the continuous time difference equations [image omitted] with [image omitted] and f being continuous interval map. For such equations, self-stochasticity lies in the existence of eventually unpredictable solutions (as smooth as desired) whose long-term behaviour is described with some stochastic processes. [ABSTRACT FROM AUTHOR]

This paper continues our earlier investigations into the inversion of random functions in a general (abstract) setting. In Section 2, we investigate a concept of invertibility and the invertibility of the composition of random functions defined on finite sets. In Section 3, we resolve some questions concerning the number of samples required to ensure the accuracy of maximum likelihood estimation (MLE) in the presence of ‘nuisance’ parameters. A direct application to phylogeny reconstruction is given. [ABSTRACT FROM AUTHOR]