when coherent stochastic resonance appears we consider a system driven by a combination of gaussian white noise and a periodic signal it is demonstrated when the mean exit time for this system necessarily exhibits a minimum as a function of the frequency of the periodic bias and gives rise to the phenomenon known as coherent stochastic resonance this reasoning begins by calculating an exact expression for the mean exit time of a system driven by noise and a telegraph signal and showing that no resonancelike behavior arises this unexpected result lets us identify the sufficient conditions for resonance to appear the origin of coherent stochastic resonance lies in the different behavior of low frequency and high frequency periodic signals

first passage time problems and resonant behavior on a fluctuating lattice abstract we study the mean residence time of a random walker on a line segment of a lattice which randomly switches between two states in each of the states only motions in a one way direction are allowed when the particle can leave the line segment through both of its ends the residence time is a monotonous increasing function of the switching rate and does not depend on the initial state of the lattice nevertheless if one of the boundaries of the segment is reflecting it goes through a minimum for a given value of the flipping rate showing a resonancelike behavior the results are compared with previous studies of first passage time problems in time dependent fields

boundary problems for diffusion in a fluctuating potential abstract using a newly developed formalism adapted for diffusion in the presence of partially absorbing boundaries we treat the overdamped motion of a particle in a fluctuating potential and in a restricted space the potential is assumed to be linear with a slope switching between two possible values the switching process is taken as the standard markov alternating process we first construct the free dynamics propagator no boundaries which controls the evolution of the joint probability for both the particle s position and the state of the potential secondly at a given spatial point we affix a two channel boundary a given channel absorbs the trajectories arriving at the boundary with the potential in a given state we demonstrate the asymmetry of the filling process for the individual channels and we discuss their asymptotic occupation depending on the mean switching frequency finally we study the motion in a domain restricted by two boundaries of the above two channel type and the corresponding first passage time the simple problem setting allows for an explicit analysis of the correlation between the particle s motion and the potential switching process

second order reliability method based inverse reliability analysis using hessian update for accurate and efficient reliability based design optimization summary first order reliability method form has been mostly utilized for solving reliability based design optimization rbdo problems efficiently however second order reliability method sorm is required in order to estimate a probability of failure accurately in highly nonlinear performance functions despite accuracy of sorm its application to rbdo is quite challenging due to unaffordable numerical burden incurred by a hessian calculation for reducing the numerical efforts a quasi newton approach to approximate the hessian is introduced in this study instead of calculating the true hessian the proposed sorm with the approximated hessian requires computations only used in form leading to very efficient and accurate reliability analysis the proposed sorm also utilizes a generalized chi squared distribution in order to achieve better accuracy furthermore sorm based inverse reliability method is proposed in this study an accurate reliability index corresponding to a target probability of failure is updated using the proposed sorm two approaches in terms of finding an accurate most probable point using the updated reliability index are proposed the proposed sorm based inverse analysis is then extended to rbdo in order to obtain a reliability based optimum design satisfying probabilistic constraints more accurately even for a highly nonlinear system the numerical study results show that the proposed reliability analysis and rbdo achieve efficiency of form and accuracy of sorm at the same time copyright 2014 john wiley sons ltd

bayesian prediction of deterministic functions with applications to the design and analysis of computer experiments abstract this article is concerned with prediction of a function y t over a multidimensional domain t given the function values at a set of sites t 1 t 2 t n in t and with the design that is with the selection of those sites the motivating application is the design and analysis of computer experiments where t determines the input to a computer model of a physical or behavioral system and y t is a response that is part of the output or is calculated from it following a bayesian formulation prior uncertainty about the function y is expressed by means of a random function y which is taken here to be a gaussian stochastic process the mean of the posterior process can be used as the prediction function t and the variance can be used as a measure of uncertainty this kind of approach has been used previously in bayesian interpolation and is strongly related to the kriging methods used in geostatistics here emphasis is placed on product linear and product cubic correlation func

enriched performance measure approach for reliability based design optimization an enriched performance measure approach is presented for reliability based design optimization to substantially improve computational efficiency when applied to large scale applications in the enriched performance measure approach four improvements are made over the original performance measure approach as a way to launch reliability based design optimization at a deterministic optimum design as a new enhanced hybrid mean value method as an efficient probabilistic feasibility check and as a fast reliability analysis under the condition of design closeness it is found that deterministic design optimization helps improve numerical efficiency by reducing some reliability based design optimization iterations in reliability based design optimization a computational burden on the feasibility check of constraints can be significantly reduced by using a mean value first order method and by carrying out the refined reliability analysis using the enhanced hybrid mean value method for e active and violated constraints the enhanced hybrid mean value method is developed to handle nonlinear and or nonmonotonic constraints in reliability analysis the fast reliability analysis method is proposed to efficiently evaluate probabilistic constraints under the condition of design closeness moreover two numerical examples are provided to compare the enriched performance measure approach to existing reliability based design optimization methods from a numerical efficiency and stability point of view

the mechanism of stochastic resonance it is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance peak in the power spectrum which is absent when either the forcing or the perturbation is absent

coherent stochastic resonance in a sextic double well potential abstract we study the response of a stochastic system described by the one dimensional fokker planck equation with a sextic double well potential u 6 x c 4 x 4 c 6 x 6 to a weak sinusoidal periodic signal two types of boundary conditions have been investigated i absorbing boundary conditions and ii natural boundary conditions the unperturbed propagators are calculated by the eigenfunction expansion method we find that for case i the mean survival time shows the coherent stochastic resonance and for case ii the signal to noise ratio exhibits the stochastic resonance the results are compared with those of the bistable quartic double well potential u 4 x 1 2 x 2 1 4 x 4

moments of the first passage time under external driving a general theory is derived for the moments of the first passage time of a one dimensional markov process in the presence of a weak time dependent forcing the linear corrections to the moments can be expressed by quadratures of the potential and of the time dependent probability density of the unperturbed system or equivalently by its laplace transform if none of the latter functions is known the derived formulas may still be useful for specific cases including a slow driving or a driving with power at only small or large times in the second part of the paper explicit expressions for the mean and variance of the first passage time are derived for the cases of a linear or a parabolic potential and an exponentially decaying driving force the analytical results are found to be in excellent agreement with computer simulations of the respective first passage processes the particular examples furthermore demonstrate that already the effect of a simple exponential driving can be fairly involved implying a nontrivial nonmonotonic behavior of mean and variance as functions of the driving s time scale

estimating population size by recapture sampling a design called recapture debugging is proposed to get extra information to estimate the original number of faults n in a reliability system the system is observed for a fixed amount of time r and the failures are assumed to follow an exponential law with an unknown common parameter a the maximum likelihood and moment estimators of n and a are derived and compared with the existing estimators the recapture sampling provides substantially more information and improved estimators compared to the standard experiment if r is not small significance tests for testing hypotheses about n and a are developed finally for testing the system one can consider r to be determined sequentially and we present a suitable stopping rule

a transition in a noisy linear system driven by a periodic signal we consider a one dimension linear random walk between two trapping points in which the transition probabilities vary periodically in time an earlier analysis of this system showed that the mean time to trapping of a particle in this system exhibits a minimum when considered as a function of frequency in this note we show that this parameter makes a transition in behavior from a monotonic decrease with increasing amplitude of the periodic term to a monotonic increase with this parameter depending on the frequency a physical argument is suggested to explain this behavior confirmation of this crossover can also be derived from a diffusion model

structural reliability optimization using an efficient safety index calculation procedure the objective of this paper is to conduct reliability based structural optimization in a multidisciplinary environment an efficient reliability analysis is developed by expanding the limit functions in terms of intermediate design variables the design constraints are approximated using multivariate splines in searching for the optimum the reduction in computational cost realized in safety index calculation and optimization are demonstrated through several structural problems this paper presents safety index computation analytical sensitivity analysis of reliability constraints and optimization using truss frame and plate examples

sensitivity analyses of form based and drm based performance measure approach pma for reliability based design optimization rbdo in a gradient based design optimization it is necessary to know sensitivities of the constraint with respect to the design variables in a reliability based design optimization rbdo the constraint is evaluated at the most probable point mpp and called the probabilistic constraint thus it requires the sensitivities of the probabilistic constraints at mpp this paper presents the rigorous analytic derivation of the sensitivities of the probabilistic constraint at mpp for both first order reliability method form based performance measure approach pma and dimension reduction method drm based pma numerical examples are used to demonstrate that the analytic sensitivities agree very well with the sensitivities obtained from the finite difference method fdm however since the sensitivity calculation at the true drm based mpp requires the second order derivatives and additional mpp search the sensitivity derivation at the approximated drmbased mpp which does not require the second order derivatives and additional mpp search to find the drm based mpp is proposed in this paper a convergence study illustrates that the sensitivity at the approximated drm based mpp converges to the sensitivity at the true drm based mpp as the design approaches the optimum design hence the sensitivity at the approximated drm based mpp is proposed to be used for the drm based rbdo to enhance the efficiency of the optimization

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

coherent stochastic resonance in one dimensional diffusion with one reflecting and one absorbing boundaries it is shown that the single step periodic signal periodic telegraph signal can not produce coherent stochastic resonance for diffusion on a segment with one absorbing and one reflecting end points while the multi step periodic signal does the general features of this process are exihibited the resonant frequency is found to decrease and the mean first passage time at resonant frequency increases linearly as we increase the length of the medium the cycle variable is shown to be the proper argument to express the first passage probability at resonance a formula for first passage probability at resonance is derived in terms of two universal functions which clearly isolates its dependence on the length of the medium

inverse analysis method using mpp based dimension reduction for reliability based design optimization of nonlinear and multi dimensional systems abstract there are two commonly used analytical reliability analysis methods linear approximation first order reliability method form and quadratic approximation second order reliability method sorm of the performance function the reliability analysis using form could be acceptable in accuracy for mildly nonlinear performance functions whereas the reliability analysis using sorm may be necessary for accuracy of nonlinear and multi dimensional performance functions even though the reliability analysis using sorm may be accurate it is not as much used for probability of failure calculation since sorm requires the second order sensitivities moreover the sorm based inverse reliability analysis is rather difficult to develop this paper proposes an inverse reliability analysis method that can be used to obtain accurate probability of failure calculation without requiring the second order sensitivities for reliability based design optimization rbdo of nonlinear and multi dimensional systems for the inverse reliability analysis the most probable point mpp based dimension reduction method drm is developed since the form based reliability index is inaccurate for the mpp search of the nonlinear performance function a three step computational procedure is proposed to improve accuracy of the inverse reliability analysis probability of failure calculation using constraint shift reliability index update and mpp update using the three steps a new drm based mpp is obtained which estimates the probability of failure of the performance function more accurately than form and more efficiently than sorm the drm based mpp is then used for the next design iteration of rbdo to obtain an accurate optimum design even for nonlinear and or multi dimensional system since the drm based rbdo requires more function evaluations the enriched performance measure approach pma with new tolerances for constraint activeness and reduced rotation matrix is used to reduce the number of function evaluations

problems with likelihood estimation of covariance functions of spatial gaussian processes summary maximum likelihood has frequently been suggested as a way to estimate covariance parameters in spatial gaussian processes the results of some simple examples based on simulated data are discussed it is shown by an example with real data that the profile likelihood can be multimodal and that the global maximum may not correspond to a sensible value of the parameters mardia marshall 1984 worked in the following framework we have a gaussian process z x i x e x with mean given by a regression equation e z x f x t for a p dimensional vector the covariance function cov z x z y c x y 0 is also specified by a q dimensional vector 0 this process is observed at n points xi xn and from these observations we wish to estimate 3 and 0 let zn z xi z xn t be the vector of observations and k c xi xj 0 be its covariance matrix finally let f be the matrix with rows f xi t then zn has a multivariate normal distribution with mean f18 dispersion matrix k and the log likelihood is up to an additive constant

an objective approach to prior mass functions for discrete parameter spaces we present a novel approach to constructing objective prior distributions for discrete parameter spaces these types of parameter spaces are particularly problematic as it appears that common objective procedures to design prior distributions are problem specific we propose an objective criterion based on loss functions instead of trying to define objective probabilities directly we systematically apply this criterion to a series of discrete scenarios previously considered in the literature and compare the priors the proposed approach applies to any discrete parameter space making it appealing as it does not involve different concepts according to the model supplementary materials for this article are available online

advanced probabilistic structural analysis method for implicit performance functions in probabilistic structural analysis the performance or response functions usually are implicitly defined and must be solved by numerical analysis methods such as finite elemen t methods in such cases the commonly used probabilistic analysis tool is the mean based second moment method which provides only the first two statistical moments this paper presents an advanced mean based method which is capable of establishing the full probability distributions to provide additional information for reliability design the method requires slightly more computations than the mean based second moment method but is highly efficient relative to the other alternative methods several examples are presented to demonstrate the method in particular the examples show that the new mean based method can be used to solve problems involving nonmonotonic functions that result in truncated distributions

reduction of ordering effect in reliability based design optimization using dimension reduction method in reliability based design optimization problems with correlated input variables a joint cumulative distribution function needs to be used to transform the correlated input variables into independent standard gaussian variables for the inverse reliability analysis to obtain a true joint cumulative distribution function a very large number of data if not infinite needs to be used which is impractical in industry applications in this paper a copula is proposed to model the joint cumulative distribution function using marginal cumulative distribution functions and correlation parameters obtained from samples using the joint cumulative distribution function modeled by the copula the transformation and the first order reliability method can be carried out however the first order reliability method may yield different reliability analysis results for different transformation ordering of input variables thus the most probable point based dimension reduction method which is more accurate than the first order reliability method and more efficient than the second order reliability method is proposed for the inverse reliability analysis to reduce the effect of transformation ordering

nonconventional stochastic resonance it is argued on the basis of linear response theory lrt that new types of stochastic resonance sr are to be anticipated in diverse systems quite different from the one most commonly studied to date which has a static double well potential and is driven by a net force equal to the sum of periodic and stochastic terms on this basis three new nonconventional forms of sr are predicted sought found and investigated both theoretically and by analogue electronic experiment a in monostable systems b in bistable systems with periodically modulated noise and c in a system with coexisting periodic attractors in each case it is shown that lrt can provide a good quantitative description of the experimental results for sufficiently weak driving fields it is concluded that sr is a much more general phenomenon than has hitherto been appreciated

stochastic resonance and symmetry breaking in a one dimensional system we derive and discuss properties of an exact solution for the average time for trapping of a brownian particle driven by a random asymmetric but unbiased telegraph signal the particle moves along a line segment terminated by either two traps or a trap and a reflecting point numerical results suggest that stochastic resonance defined as a nonmonotonic behavior of the mean trapping time is absent in the first case but present in the second this generalizes a result obtained earlier by doering and gadoua phys rev lett 69 2318 1992 and implies that symmetry breaking alone does not necessarily create stochastic resonance

estimating the number of faults in a system abstract we consider a reliability system with an unknown number of faults which is observed for a fixed length of time failures are assumed to follow an exponential distribution with an unknown mean and we concentrate on estimation of the number of faults v the likelihood function considered as a function of v has its maximum at one of the boundary points the observed number of faults or infinity with substantial probability likelihood interval estimates and improved point estimates are discussed the harmonic mean of the endpoints of the likelihood interval turns out to be a reasonable point estimate we comment on some extensions of this problem suggested in the software reliability literature

model robust response surface designs scaling two level factorials summary response surface methods use simple graduating functions to study the relationship between an experimental response variable and a set of continuous explanatory variables in designing a response surface study an experimenter must decide how far apart to set the levels of each factor i e how to scale the design a good choice should be sensitive to the fact that the graduating function is only an approximation to the true response function a bayesian model is proposed that makes explicit assumptions about the inadequacy of an assumed model and a design criterion based on the model leads to reasonable choices of scale for two level factorial designs the choice of scale is found to be insensitive to the prior distributions in the model

initial size estimation for the linear pure death process summary consider a linear pure death process in which the common death rate and the initial size n of the population are both unknown an estimate of n is required based on the observed times of the firstj deaths i e with censored sampling it is shown that the only reliable non bayesian method is to use interval estimates for all the standard point estimation techniques are liable to fail appropriate interval estimators are derived and discussed the corresponding truncated sampling model is briefly considered the linear pure death process may be characterized as representing a dying population in which the lifetimes are independently and identically distributed each with probability density ae t t 0 for some a 0 we assume that both the initial population size n and the rate parameter a are unknown and our aim is to estimate n for many years work on the estimation of population size such as by johnson 1962 hoel 1968 and marcus blumenthal 1974 assumed a lifetime distribution which was general but completely specified recently interest has been shown in lifetime distributions with unknown parameters in particular in papers by sanathanan 1977 who gives asymptotic results for maximum likelihood estimation and blumenthal marcus 1975 who consider precisely the above problem we consider mainly the censored sampling model under which an estimate of n is required immediately after the jth death where j is a preassigned integer known not to exceed n in 2 we derive an invariantly sufficient statistic and its distribution in 3 we discuss point estimation of n unbiased estimators do not exist while the methods of moments and of maximum likelihood are both liable to fail in that there is positive probability sometimes close to 2 that the equation for the estimator has no finite solution in 4 we derive an interval estimator and discuss its properties while an interval estimate is always available it is in some circumstances quite vague this brings out clearly that for the present problem it is more than usually inadvisable to rely on point

first passage time statistics for diffusion processes with an external random force we present exact equations and expressions for the first passage time statistics of dynamical systems that are a combination of a diffusion process and a random external force modeled as dichotomous markov noise we prove that the mean first passage time for this system does not show any resonantlike behavior textcopyright 1996 the american physical society

inference and prediction for a general order statistic model with unknown population size abstract suppose that the first n order statistics from a random sample of n positive random variables are observed where n is unknown this the general order statistic model has been applied to the study of market penetration capture recapture burn in in repairable systems software reliability growth the estimation of the number of individuals exposed to radiation and the estimation of the number of unseen species inference is to be made about the unknown parameters especially n and future observations are to be predicted a bayes empirical bayes approach to inference is presented this permits the comparison of competing perhaps nonnested models for the distribution of the random variables in a natural way it also provides easily implemented inference and prediction procedures that avoid the difficulties of non bayesian methods one such difficulty is that the maximum likelihood estimator of n may be infinite results are given for the case in which vague prior information about the model

incorporating uncertainty in mechanical properties for finite element based evaluation of bone mechanics finite element fe models of bone developed from computed tomography ct scan data are used to evaluate stresses and strains load transfer and fixation of implants and potential for fracture the experimentally derived relationships used to transform ct scan data in hounsfield unit to modulus and strength contain substantial scatter the scatter in these relationships has potential to impact the results and conclusions of bone studies the objectives of this study were to develop a computationally efficient probabilistic fe based platform capable of incorporating uncertainty in bone property relationships and to apply the model to a representative analysis variability in stresses and fracture risk was predicted in five proximal femurs under stance loading conditions based on published variability in strength and modulus relationships derived in the proximal femur the probabilistic analysis predicted the distributions of stress and risk for the five femurs analyzed the 1 and 99 percentile bounds varied by an average of 17 3 mpa for stress and by 0 28 for risk in each femur the predicted variability in risk was greater than 50 of the mean risk calculated with obvious implications for clinical assessment results using the advanced mean value amv method required only seven analysis trials 1 h and differed by less than 2 when compared to a 1000 trial monte carlo simulation 400 h the probabilistic modeling platform developed has broad applicability to bone studies and can be similarly implemented to investigate other loading conditions structures sources of uncertainty or output measures of interest r 2007 elsevier ltd all rights reserved

reliability analysis of stiffened aircraft panels using adjusting mean value method with regard to real world engineering structures an adjusting mean value called the self adjusting mean value smv is proposed for improving the efficiency and robustness of a maximum performance

curvilinear fatigue crack reliability analysis by stochastic boundary element method in this paper the stochastic boundary element method which combines the mixed boundary integral equations method explored in reference 1 with the first order reliability method is developed to study probabilistic fatigue crack growth due to the high degree of complexity and non linearity of the response direct differentiation coupied with the response surface method is employed to determine the response gradient three random processes the mode i and mode ii stress intensity factors and the crack direction angle are included in the expression of the response gradient the sensitivity of these random processes is determined using a first order response model an iteration scheme based on the hl rf method2 is applied to locate the most probable failure point on the limit state surface the accuracy and efficiency of the stochastic boundary element method are evaluated by comparing the cumulative distribution function of the fatigue life obtained with monte carlo simulation the reliability index and the corresponding probability of failure are calculated for a fatigue crack growth problem with randomness in the crack geometry defect geometry fatigue parameters and external loads the response sensitivity of each primary random variable at the design point is determined to show its role in the fatigue failure the variation of each primary random variable at the design point with the change of probability of failure is also presented in numerical examples

objective priors for discrete parameter spaces this article considers the development of objective prior distributions for discrete parameter spaces formal approaches to such development such as the reference prior approach often result in a constant prior for a discrete parameter which is questionable for problems that exhibit certain types of structure to take advantage of structure this article proposes embedding the original problem in a continuous problem that preserves the structure and then using standard reference prior theory to determine the appropriate objective prior four different possibilities for this embedding are explored and applied to a population size model the hypergeometric distribution the multivariate hypergeometric distribution the binomial beta distribution and the binomial distribution the recommended objective priors for the first third and fourth problems are new

maximum likelihood estimation of models for residual covariance in spatial regression we describe the maximum likelihood method for fitting the linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model containing unknown parameters observations are assumed to be gaussian we give conditions which ensure consistency and asymptotic normality of the estimators our main concern is with the analysis of spatial data and in this context we describe some simulation experiments to assess the small sample behaviour of estimators we also discuss an application of the spectral approximation to the likelihood for processes on a lattice

stopping rules and estimation for recapture debugging with unequal failure rates summary a theoretical optimal stopping rule based on the minimization of the testing cost and expected penalty due to the unremoved bugs is derived without the equal failure rate assumption under nayak s 1988 recapture debugging procedure two adaptive stopping rules suitable for practical use are constructed and tested by simulation the estimation of the number of undetected bugs is also considered via remaining failure rate estimation