PC expansion for material parameters using artificial data and statistical methods

This paper investigates the uncertainty of a hyper-elastic model by random material parameters as stochastic variables. For its stochastic discretization a polynomial chaos (PC) is used to expand the coefficients into deterministic and stochastic parts. Then, from experimental data in combination with artificial data for elastomers the distribution of the force-displacement curves are known. In the numerical example the PC-based stochastic and the deterministic parameter identification are used for generation of the distribution of Ogden's material parameters. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

SFEM for rubber-like materials at large deformations

This paper investigates the uncertainty of hyper-elastic model, by random material parameters as stochastic variables. For its stochastic discretization a polynomial chaos (PC) is used to expand the coefficients into deterministic and stochastic parts. Then, from experimental data in combination with artificial data for elastomers the distribution of the force-displacement curves are obtained. In the numerical example the PC-based Stochastic Finite Element Method (SFEM) and the deterministic parameter identification are used for generation of the distribution of Ogden's material parameters. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal Open-Loop Feedback Control of Robots under uncertainty

The aim of this presentation is to construct an optimal open-loop feedback controller for robots, which takes into account stochastic uncertainties. This way, optimal regulators being insensitive with respect to random parameter variations can be obtained. Usually, a precomputed feedback control is based on exactly known or estimated model parameters. However, in practice, often exact informations about model parameters, e.g. the payload mass, are not given. Supposing now that the probability distribution of the random parameter variation is known, in the following, stochastic optimisation methods will be applied in order to obtain robust open-loop feedback control. Taking into account stochastic parameter variations, the method works with expected cost functions evaluating the primary control expenses and the tracking error. The expectation of the total costs has then to be minimized. Corresponding to Model Predictive Control (MPC), here a sliding horizon is considered. This means that, instead of minimizing an integral from a starting time point t₀ to the final time t_f, the future time range [t; t+T], with a small enough positive time unit T, will be taken into account. The resulting optimal regulator problem under stochastic uncertainty will be solved by using the Hamiltonian of the problem. After the computation of a H-minimal control, the related stochastic two-point boundary value problem is then solved in order to find a robust optimal open-loop feedback control. The performance of the method will be demonstrated by a numerical example, which will be the control of robot under random variations of the payload mass. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A functional limit theorem for random graphs with applications to subgraph count statistics

We consider a random graph that evolves in time by adding new edges at random times (different edges being added at independent and identically distributed times). A functional limit theorem is proved for a class of statistics of the random graph, considered as stochastic processes. the proof is based on a martingale convergence theorem. the evolving random graph allows us to study both the random graph model K_{n, p}, by fixing attention to a fixed time, and the model K_{n, N}, by studying it at the random time it contains exactly N edges. in particular, we obtain the asymptotic distribution as n → ∞ of the number of subgraphs isomorphic to a given graph G, both for K_{n, p} (p fixed) and K_{n, N} (N/(ⁿ₂)→ p). the results are strikingly different; both models yield asymptotically normal distributions, but the variances grow as different powers of n (the variance grows slower for K_n, N; the powers of n usually differ by 1, but sometimes by 3). We also study the number of induced subgraphs of a given type and obtain similar, but more complicated, results. in some exceptional cases, the limit distribution is not normal.     

Strong Consistency of Bayes Estimates in Stochastic Regression Models

Under minimum assumptions on the stochastic regressors, strong consistency of Bayes estimates is established in stochastic regression models in two cases: (1) When the prior distribution is discrete, the p.d.f.fof i.i.d. random errors is assumed to have finite Fisher informationI=∫^∞_−∞(f′)²/f dx<∞; (2) for general priors, we assumefis strongly unimodal. The result can be considered as an application of a theorem of Doob to stochastic regression models.     

Parametric Estimation of a Boolean Segment Process with Stochastic Restoration Estimation

Abstract

We propose a stochastic restoration estimation (SRE) algorithm to estimate the parameters of the length distribution of a boolean segment process. A boolean segment process is a stochastic process obtained by considering the union of independent random segments attached to random points independently scattered on the plane. Each iteration of the SRE algorithm has two steps: first, censored segments are restored; second, based on these restored data, parameter estimations are updated. With a usually straightforward implementation, this algorithm is particularly interesting when censoring effects are difficult to take into account. We illustrate this method in two situations where the parameter of interest is either the mean of the segment length distribution or the variance of its logarithm. Its application to vine shoot length distribution estimation is presented.     

Probabilistic modeling and global sensitivity analysis for CO2 storage in geological formations: a spectral approach

This work focuses on the simulation of CO₂ storage in deep underground formations under uncertainty and seeks to understand the impact of uncertainties in reservoir properties on CO₂ leakage. To simulate the process, a non-isothermal two-phase two-component flow system with equilibrium phase exchange is used. Since model evaluations are computationally intensive, instead of traditional Monte Carlo methods, we rely on polynomial chaos (PC) expansions for representation of the stochastic model response. A non-intrusive approach is used to determine the PC coefficients. We establish the accuracy of the PC representations within a reasonable error threshold through systematic convergence studies. In addition to characterizing the distributions of model observables, we compute probabilities of excess CO₂ leakage. Moreover, we consider the injection rate as a design parameter and compute an optimum injection rate that ensures that the risk of excess pressure buildup at the leaky well remains below acceptable levels. We also provide a comprehensive analysis of sensitivities of CO₂ leakage, where we compute the contributions of the random parameters, and their interactions, to the variance by computing first, second, and total order Sobol’ indices.     

Storage capacity of a dam with gamma type inputs

Summary Consider mutually independent inputsX ₁,...,X _n onn different occasions into a dam or storage facility. The total input isY=X ₁+...+X _n. This sum is a basic quantity in many types of stochastic process problems. The distribution ofY and other aspects connected withY are studied by different authors when the inputs are independently and identically distributed exponential or gamma random variables. In this article explicit exact expressions for the density ofY are given whenX ₁,...,X _n are independent gamma distributed variables with different parameters. The exact density is written as a finite sum, in terms of zonal polynomials and in terms of confluent hypergeometric functions. Approximations whenn is large and asymptotic results are also given.     

Some results on the residual life at random time

In this paper, we consider the residual life at random time, i.e.X _Y =X−Y\X>Y, whereX andY are non-negative random variables. We establish a number of stochastic comparison properties forX _Y under various assumptions ofX andY. Under the assumption thatY has decreasing reverse hazard rate (DRHR), we show that ifX is in any one of the classes IFR, DFR, DMRL or IMRL thenX _Y is in the same class asX. We also obtain some useful bounds for the distribution and the moment ofX _Y . Because the idle time in classicalGI/G/1 queuing system can be regarded as the residual life at random time, the results obtained in this paper have applications in the study of such system. This work is supported by the National Natural Science Foundation of China.     

General shock models with random threshold

A general shock model associated with a correlated pair (X _n ,Y _n ) of renewal sequences is considered. The system fails when the magnitude of the shock exceeds a random threshold Zfollowing exponential law. The distribution of the system failure time T _Z is found and first two moments of T _Z are derived. A class of correlated cumulative shock models is also studied. As an application stochastic clearing system is studied in detail.     

A Unified Derivation of the Complementary Waiting Time Distribution in Sequential Occupancy

Occupancy distributions are defined on the stochastic model of random allocation of balls to a specific number of distinguishable urns. The reduction of the joint distribution of the occupancy numbers, when a specific number of balls are allocated, to the joint conditional distribution of independent random variables given their sum, when the number of balls allocated is unspecified, is a powerful technique in the study of occupancy distributions. Consider a supply of balls randomly distributed into n distinguishable urns and assume that the number X of balls distributed into any specific urn is a random variable with probability function P(X = x) = q _x , x = 0, 1,.... The probability function of the number L _r of occupied urns until r balls are placed into previously occupied urns is derived in terms of convolutions of q _x , x = 0, 1,... and their finite differences. Further, using this distribution, the minimum variance unbiased estimator of the parameter n, based on a suitable sequential sampling scheme, is deduced. Finally, some illustrating applications are discussed.      

Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions: II. The Heterogeneous Case

We consider the problem of discriminating between two independent multivariate normal populations, N_p(μ₁, Σ₁) and N_p(μ₂, Σ₂), having distinct mean vectors μ₁ and μ₂ and distinct covariance matrices Σ₁ and Σ₂. The parameters μ₁, μ₂, Σ₁, and Σ₂ are unknown and are estimated by means of independent random training samples from each population. We derive a stochastic representation for the exact distribution of the “plug-in” quadratic discriminant function for classifying a new observation between the two populations. The stochastic representation involves only the classical standard normal, chi-square, and F distributions and is easily implemented for simulation purposes. Using Monte Carlo simulation of the stochastic representation we provide applications to the estimation of misclassification probabilities for the well-known iris data studied by Fisher (Ann. Eugen.7 (1936), 179–188); a data set on corporate financial ratios provided by Johnson and Wichern (Applied Multivariate Statistical Analysis, 4th ed., Prentice–Hall, Englewood Cliffs, NJ, 1998); and a data set analyzed by Reaven and Miller (Diabetologia16 (1979), 17–24) in a classification of diabetic status.     

Stochastic programming with incomplete information:a surrey of results on postoptimization and sensitivity analysis

The possibility of successful applications of stochastic programming decision models has been limited by the assumed complete knowledge of the distribution Fof the random parameters as well as by the limited scope of the existing numerical procedures.

We shall give a survey of selected methods which can be used to deal with the incomplete knowledge of the distribution F, namely to study robustness of the optimal solution and the optimal value of the objective function relative to small changes of the underlying distribution and to get error bounds in approximation schemes.     

Mann-Whitney test for associated sequences

Let {X ₁, ...,X _m } and {Y ₁, ...,Y _n } be two samples independent of each other, but the random variables within each sample are stationary associated with one dimensional marginal distribution functionsF andG, respectively. We study the properties of the classical Wilcoxon-Mann-Whitney statistic for testing for stochastic dominance in the above set up.     

Degree distribution of the greatest common divisor of polynomials over 𝔽q

We study the degree distribution of the greatest common divisor of two or more random polynomials over a finite field ??_q. We provide estimates for several parameters like number of distinct common irreducible factors, number of irreducible factors counting repetitions, and total degree of the gcd of two or more polynomials. We show that the limiting distribution of a random variable counting the total degree of the gcd is geometric and that the distributions of random variables counting the number of common factors (with and without repetitions) are very close to Poisson distributions when q is large. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Adaptive control of stochastic systems with unknown disturbance distribution: discounted criteria

We consider a class of discrete-time stochastic control systems, with Borel state and action spaces, and possibly unbounded costs. The processes evolve according to the equation x _{t +1}=F(x _t , a _t , ξ _t ), t=0, 1, ..., where the ξ _t are i.i.d. random vectors whose common distribution is unknown. Assuming observability of {ξ _t }, we use the empirical estimator of its distribution to construct adaptive policies which are asymptotically discounted cost optimal .AMS Subject Classification (2000) 93E10, 90C40     

Stochastic Galerkin techniques for random ordinary differential equations

Over the last decade the stochastic Galerkin method has become an established method to solve differential equations involving uncertain parameters. It is based on the generalized Wiener expansion of square integrable random variables. Although there exist very sophisticated variants of the stochastic Galerkin method (wavelet basis, multi-element approach) convergence for random ordinary differential equations has rarely been considered analytically. In this work we develop an asymptotic upper boundary for the L ₂-error of the stochastic Galerkin method. Furthermore, we prove convergence of a local application of the stochastic Galerkin method and confirm convergence of the multi-element approach within this context.     

Distribution Functions of Poisson Random Integrals: Analysis and Computation

We want to compute the cumulative distribution function of a one-dimensional Poisson stochastic integral I(g) = ò₀^T g(s) N(ds)I(g) = \displaystyle \int_0^T g(s) N(ds), where N is a Poisson random measure with control measure n and g is a suitable kernel function. We do so by combining a Kolmogorov–Feller equation with a finite-difference scheme. We provide the rate of convergence of our numerical scheme and illustrate our method on a number of examples. The software used to implement the procedure is available on demand and we demonstrate its use in the paper.     

Spectral theory of minimax estimation

The set ofS ₁-estimates of solutions of systems of linear equations with random parameters is found. It is proved that the maximal eigenvalue in the goodness criterion is not simple. For the purpose of finding estimates from theS ₁ set, the perturbation formulas for eigenvalues and formulas for distribution density of random matrices are used.     

Statistical analysis of hierarchical stochastic models: Examples and approaches

This paper introduces and illustrates the concept of hierarchical or random parameter stochastic process models. These models arise when members of a population each generate a stochastic process governed by certain parameters and the values of the parameters may be viewed as single realizations of random variables. The paper treats the estimation of the individual parameter values and the parameters of the superpopulation distribution. Examples from system reliability, pharmacokinetic compartment models, and criminal careers are introduced; a reliability (Poisson process-exponential interval) process is examined in greater detail. An explicit, approximate, robust estimator of individual (log) failure rates is presented for the case of a long-tailed (Studentt) superpopulation. This estimator exhibits desirable limited shrinkage properties, refusing to borrow unjustified strength. Numerical properties of such estimators are described more fully elsewhere.