Deterministic Quantities Research Articles

Numerical algorithms for mean exit time and escape probability of stochastic systems with asymmetric Lévy motion

For non-Gaussian stochastic dynamical systems, mean exit time and escape probability are important deterministic quantities, which can be obtained from integro-differential (nonlocal) equations. We develop an efficient and convergent numerical method for the mean first exit time and escape probability for stochastic systems with an asymmetric Lévy motion, and analyze the properties of the solutions of the nonlocal equations. The discretized equation has Toeplitz structure that enables utilization of fast Fourier transform in numerical simulations. We also investigate the effects of different system factors on the mean exit time and escape probability, including the skewness parameter, the size of the domain, the drift term and the intensity of Gaussian and non-Gaussian noises. We find that the behavior of the mean exit time and the escape probability has dramatic difference at the boundary of the domain when the index of stability crosses the critical value of one.

On the Ergodic Rate Lower Bounds With Applications to Massive MIMO

A well-known lower bound widely used in the massive MIMO literature hinges on channel hardening , i.e., the phenomenon for which, thanks to the large number of antennas, the effective channel coefficients resulting from beamforming tend to deterministic quantities. If the channel hardening effect does not hold sufficiently well, this bound may be quite far from the actual achievable rate. In recent developments of massive MIMO, several scenarios where channel hardening is not sufficiently pronounced have emerged. These settings include, for example, the case of small scattering angular spread, yielding highly correlated channel vectors, and the case of cell-free massive MIMO . In this short contribution, we present two new bounds on the achievable ergodic rate that offer a complementary behavior with respect to the classical bound: while the former performs well in the case of channel hardening and/or when the system is interference-limited (notably, in the case of finite number of antennas and conjugate beamforming transmission), the new bounds perform well when the useful signal coefficient does not harden but the channel coherence block length is large with respect to the number of users, and in the case where interference is nearly entirely eliminated by zero-forcing beamforming. Overall, using the most appropriate bound depending on the system operating conditions yields a better understanding of the actual performance of systems where channel hardening may not occur, even in the presence of a very large number of antennas.

A stochastic perturbation finite element-least square point interpolation method for the analysis of uncertain structural-acoustics problems with random variables

In this paper, a novel stochastic perturbation finite element-least square point interpolation method (SP-FE-LSPIM) is introduced to improve the calculation accuracy for analyzing structural-acoustics problems. Inherited the element-compatibility of finite element method and the quadratic polynomial completeness of LSPIM, the present method obtains the global shape function for partition of unity (PU) and the least square point interpolation for local approximation. Besides, a first-order perturbation technique is also introduced into this theory for probabilistic analyzing. Thus, the response of the coupled systems can be expressed simply as a linear function of all pre-defined input variables by using the change-of-variable techniques. Due to the linear relationships between variables and the response, the probability density function and the cumulative probability density function of response can be obtained based on a simple mathematical transformation of probability theory. So the proposed approach not only improves the numerical accuracy of deterministic output quantities with respect to a given random variable, but also handles the randomness well in the systems. One numerical example for frequency response analysis of random structural-acoustics is presented and verified by Monte Carlo (MC) simulation and stochastic perturbation finite element method (SP-FEM) to demonstrate the effectiveness of the present method.

Effects of Probability Function on the Performance of Stochastic Programming

Stochastic programming is a valuable optimization tool where used when some or all of the design parameters of an optimization problem are defined by stochastic variables rather than by deterministic quantities. Depending on the nature of equations involved in the problem, a stochastic optimization problem is called a stochastic linear or nonlinear programming problem. In this paper,a stochastic optimization problem is transformed intoan equivalent deterministic problem,which can be solved byany known classical methods (interior penalty method is applied here).The paper mainly focuseson investigatingthe effect of applying various probability functions distributions(normal, gamma, and exponential) for design variables. The following basic required equations to solve nonlinear stochastic problems with various probability functionsfor random variables are derived and sensitivity analyses to studythe effects of distribution function typesand input parameterson the optimum solution are presented as graphs and in tables by studyingtwoconsidered test problems. It is concluded that thedifference between probabilistic and deterministic solutions toa problem, when the normal distribution ofrandom variables isused, is very different fromthe results when gamma and exponential distribution functions are used. Finally, it is shownthat the rate of solution convergence tothe normal distribution is faster than the other distributions.

Structural System Reliability Analysis Based on Improved Explicit Connectivity BNs

The study proposes an improved Explicit Connectivity Bayesian Networks (ECBNs) for system reliability assessment. The framework combines Analytic Hierarchy Process (AHP) with the traditional ECBNs. AHP is adopted to consider probabilistic dependencies between system components. Judgment matrix is constructed and weight vector, which meets the requirement of a random consistency check, is extracted to describe a conditional probability information of BNs intermediate nodes. The framework is especially suitable for the system with rare damage filed data. In addition, considering the multiple failure modes for the system component, the multi-dimensional Performance Limit State (PLS) function is proposed to estimate the marginal probability for the BNs root nodes. PLSs are properly modeled as interdependent random variables instead of deterministic quantities. Failing to properly account for the dependencies between PLSs, the non-conservative failure probability results will be obtained. Finally, the system reliability can be calculated through the BNs Junction Tree forward inference algorithm, and the most vulnerable components in the system can be identified through backward diagnose inference. The improved ECBNs theory is first applied to the system reliability evaluation for a reinforced concrete bridge to verify its validity.

The performance of split and integrated types of air-source heat pump water heaters in South Africa

Renewable energy technologies that can provide optimum and cost-effective energy savings to mitigate global warming, energy crisis and to achieve energy efficiency continue to be of paramount importance. The present study focused on identifying critical parameters such as the volume of hot water drawn off; ambient temperature; relative humidity; refrigerant temperatures at the inlet and outlet of the compressor and condenser; and deterministic quantities such as time used, power consumption and coefficient of performance (COP) as indicators to benchmark the performance of both the split and integrated types of air-source heat pump (ASHP) water heaters. The basis for analysis was on two predominant scenarios: first-hour heating rating and the heating cycle due to controlled volume of hot water drawn-off wherein both the integrated and split types ASHP water heaters experienced vapour compression refrigeration cycles. A data acquisition system was constructed and implemented to monitor the performance of both systems. The results obtained during summer season showed that, under the scenario of 150 L hot water withdrawal, the average COP of the systems was 3.18 and 2.85 for the split and integrated types respectively. The average power consumed was 1.29 (split type) and 0.85 kW (integrated type). The times of operation were 84 minutes (split type) and 138 minutes (integrated type).

Monte Carlo simulation of induction time and metastable zone width; stochastic or deterministic?

The induction time and metastable zone width (MSZW) measured for small samples (say 1 mL or less) both scatter widely. Thus, these two are observed as stochastic quantities. Whereas, for large samples (say 1000 mL or more), the induction time and MSZW are observed as deterministic quantities. The reason for such experimental differences is investigated with Monte Carlo simulation. In the simulation, the time (under isothermal condition) and supercooling (under polythermal condition) at which a first single crystal is detected are defined as the induction time t and the MSZW ΔT for small samples, respectively. The number of crystals just at the moment of t and ΔT is unity. A first crystal emerges at random due to the intrinsic nature of nucleation, accordingly t and ΔT become stochastic. For large samples, the time and supercooling at which the number density of crystals N/V reaches a detector sensitivity (N/V)det are defined as t and ΔT for isothermal and polythermal conditions, respectively. The points of t and ΔT are those of which a large number of crystals have accumulated. Consequently, t and ΔT become deterministic according to the law of large numbers. Whether t and ΔT may stochastic or deterministic in actual experiments should not be attributed to change in nucleation mechanisms in molecular level. It could be just a problem caused by differences in the experimental definition of t and ΔT.

Hybrid manufacturing/remanufacturing lot-sizing and supplier selection with returns, under carbon emission constraint

This paper addresses a lot-sizing problem in manufacturing/remanufacturing systems. The studied system is a single manufacturing line where both regular manufacturing and returns remanufacturing processes are carried out, with different set-up costs for each process. We consider also a returns collection phase from customers/distributors with deterministic returns quantities at each period of the planning horizon. The environmental aspect is assumed in this study by considering a carbon emission constraint for the manufacturing, remanufacturing and transportation activities. A mixed integer programming model to minimise the management cost and meet the customer’s needs under different manufacturing constraints is proposed. Otherwise, An adaptation of the well-known Silver and Meal (SM) heuristic and two hybrid method approaches (HM1 and HM2) providing approximates solutions are developed. The mixed integer model was tested on Cplex (Software optimizer), and the obtained results were compared with the ones provided by the adapted heuristic SM and the hybrid methods. The numerical analyses show that hybrid methods provide good-quality solutions in a moderate computational time. The proposed model establishes a collegial and an integrated process that sets values, goals, decisions and priorities along the considered supply chain while taking into account the environmental aspect.

Posterior uncertainty, asymptotic law and Cramér-Rao bound

In a globally identifiable Bayesian system identification problem, the uncertainty of model parameters can be quantified by their “posterior covariance matrix” calculated for a particular data set. When the data is modeled to be distributed as the likelihood function (i.e., no modeling error), a statistical law analogous to the law of large numbers results, where the posterior covariance matrix is asymptotic to a deterministic quantity that depends on the “information content” of data rather than its particular (stochastic) details. This was referred as the “uncertainty law” in a recent study of the achievable precision of modal parameters in operational modal analysis (OMA). Deriving the uncertainty law involves asymptotics techniques and leveraging on the mathematical structure of the likelihood function, which was found to be tedious. As a sequel to the development, this work shows that for long data and up to a Gaussian approximation of the posterior distribution, the uncertainty law is asymptotic to the inverse of the Fisher information matrix, which coincides with the tightest Cramér-Rao bound in classical statistics. A parametric study is presented to illustrate the theoretical results in the context of OMA. As a direct application with practical relevance, the relationship provides a systematic means for deriving the uncertainty laws in OMA. Applied and interpreted properly, the posterior covariance matrix (for given data), uncertainty law, and Cramér-Rao bound can provide a powerful means for quantifying and managing the uncertainties in structural health monitoring.

A generalized probabilistic edge-based smoothed finite element method for elastostatic analysis of Reissner–Mindlin plates

Probabilistic analysis is becoming more important in mechanical science and real-world engineering applications. In this work, a novel generalized stochastic edge-based smoothed finite element method is proposed for Reissner–Mindlin plate problems. The edge-based smoothing technique is applied in the standard FEM to soften the over-stiff behavior of Reissner–Mindlin plate system, aiming to improve the accuracy of predictions for deterministic response. Then, the generalized nth order stochastic perturbation technique is incorporated with the edge-based S-FEM to formulate a generalized probabilistic ES-FEM framework (GP_ES-FEM). Based upon a general order Taylor expansion with random variables of input, it is able to determine higher order probabilistic moments and characteristics of the response of Reissner–Mindlin plates. The significant feature of the proposed approach is that it not only improves the numerical accuracy of deterministic output quantities with respect to a given random variable, but also overcomes the inherent drawbacks of conventional second-order perturbation approach, which is satisfactory only for small coefficients of variation of the stochastic input field. Two numerical examples for static analysis of Reissner–Mindlin plates are presented and verified by Monte Carlo simulations to demonstrate the effectiveness of the present method.

A stochastic perturbation edge-based smoothed finite element method for the analysis of uncertain structural-acoustics problems with random variables

Among the current methods in predicting the response of structural-acoustics problems in mid-frequency regime, some problems such as low accuracy and inability to deal with the uncertainties still need to be solved. To eliminate these issues, a novel stochastic perturbation edge-based smoothed FEM method (SP-ES-FEM) is proposed for the analysis of structural-acoustics problems in this work. The edge-based smoothing technique is applied in the standard FEM approach to soften the over-stiff behavior of structural-acoustics problems aiming to improve the accuracy of deterministic response predictions. Then, this approach, for the first time, intends to introduce the first-order perturbation technique into the edge-based smoothed FEM theory frame especially for the probabilistic analysis of structural-acoustics problems. The response of the coupled systems can be expressed simply as a linear function of all the pre-defined input variables by using the change of variable techniques. Due to the linear relationships of variables and response, the probability density function and cumulative probability density function of the response can be obtained based on the simple mathematical transformation of probability theory. The proposed approach not only improves the numerical accuracy of deterministic output quantities with respect to a given random variable, but also can handle the randomness well in the systems. Two numerical examples for frequency response analysis of random structural-acoustics are presented and verified by Monte Carlo simulation, to demonstrate the effectiveness of the present method.

Interactions of a co-rotating vortex pair at multiple offsets

Two NACA0012 vanes at various lateral offsets were investigated by wind tunnel testing to observe the interactions between the streamwise vortices. The vanes were separated by nine chord lengths in the streamwise direction to allow the upstream vortex to impact on the downstream geometry. These vanes were evaluated at an angle of incidence of 8° and a Reynolds number of 7×104 using particle image velocimetry. A helical motion of the vortices was observed, with rotational rate increasing as the offset was reduced to the point of vortex merging. Downstream meandering of the weaker vortex was found to increase in magnitude near the point of vortex merging. The merging process occurred more rapidly when the upstream vortex was passed on the pressure side of the vane, with the downstream vortex being produced with less circulation and consequently merging into the upstream vortex. The merging distance was found to be statistical rather than deterministic quantity, indicating that the meandering of the vortices affected their separations and energies. This resulted in a fluctuation of the merging location. A loss of circulation associated with the merging process was identified, with the process of achieving vortex circularity causing vorticity diffusion, however all merged cases maintained higher circulation than a single vortex condition. The presence of the upstream vortex was found to reduce the strength of the downstream vortex in all offsets evaluated.

On the Sensitivity of the Lasso to the Number of Predictor Variables

The Lasso is a computationally efficient regression regularization procedure that can produce sparse estimators when the number of predictors $(p)$ is large. Oracle inequalities provide probability loss bounds for the Lasso estimator at a deterministic choice of the regularization parameter. These bounds tend to zero if $p$ is appropriately controlled, and are thus commonly cited as theoretical justification for the Lasso and its ability to handle high-dimensional settings. Unfortunately, in practice the regularization parameter is not selected to be a deterministic quantity, but is instead chosen using a random, data-dependent procedure. To address this shortcoming of previous theoretical work, we study the loss of the Lasso estimator when tuned optimally for prediction. Assuming orthonormal predictors and a sparse true model, we prove that the probability that the best possible predictive performance of the Lasso deteriorates as $p$ increases is positive and can be arbitrarily close to one given a sufficiently high signal to noise ratio and sufficiently large $p$. We further demonstrate empirically that the amount of deterioration in performance can be far worse than the oracle inequalities suggest and provide a real data example where deterioration is observed.

Mean exit time and escape probability for the anomalous processes with the tempered power-law waiting times

This paper discusses two deterministic quantities, mean first exit time and escape probability, for the anomalous processes having the tempered Lévy stable waiting times with the tempering index and the stability index . We derive the nonlocal elliptic partial differential equations (PDEs) governing the mean first exit time and escape probability. Based on the analysis of the derived PDEs, some interesting phenomena are observed.

Transitions in a genetic transcriptional regulatory system under Lévy motion.

Based on a stochastic differential equation model for a single genetic regulatory system, we examine the dynamical effects of noisy fluctuations, arising in the synthesis reaction, on the evolution of the transcription factor activator in terms of its concentration. The fluctuations are modeled by Brownian motion and α-stable Lévy motion. Two deterministic quantities, the mean first exit time (MFET) and the first escape probability (FEP), are used to analyse the transitions from the low to high concentration states. A shorter MFET or higher FEP in the low concentration region facilitates such a transition. We have observed that higher noise intensities and larger jumps of the Lévy motion shortens the MFET and thus benefits transitions. The Lévy motion activates a transition from the low concentration region to the non-adjacent high concentration region, while Brownian motion can not induce this phenomenon. There are optimal proportions of Gaussian and non-Gaussian noises, which maximise the quantities MFET and FEP for each concentration, when the total sum of noise intensities are kept constant. Because a weaker stability indicates a higher transition probability, a new geometric concept is introduced to quantify the basin stability of the low concentration region, characterised by the escaping behaviour.

On the deterministic and stochastic use of hydrologic models

AbstractEnvironmental simulation models, such as precipitation‐runoff watershed models, are increasingly used in a deterministic manner for environmental and water resources design, planning, and management. In operational hydrology, simulated responses are now routinely used to plan, design, and manage a very wide class of water resource systems. However, all such models are calibrated to existing data sets and retain some residual error. This residual, typically unknown in practice, is often ignored, implicitly trusting simulated responses as if they are deterministic quantities. In general, ignoring the residuals will result in simulated responses with distributional properties that do not mimic those of the observed responses. This discrepancy has major implications for the operational use of environmental simulation models as is shown here. Both a simple linear model and a distributed‐parameter precipitation‐runoff model are used to document the expected bias in the distributional properties of simulated responses when the residuals are ignored. The systematic reintroduction of residuals into simulated responses in a manner that produces stochastic output is shown to improve the distributional properties of the simulated responses. Every effort should be made to understand the distributional behavior of simulation residuals and to use environmental simulation models in a stochastic manner.

Stochastic basins of attraction for metastable states.

Basin of attraction of a stable equilibrium point is an effective concept for stability analysis in deterministic systems; however, it does not contain information on the external perturbations that may affect it. Here we introduce the concept of stochastic basin of attraction (SBA) by incorporating a suitable probabilistic notion of basin. We define criteria for the size of the SBA based on the escape probability, which is one of the deterministic quantities that carry dynamical information and can be used to quantify dynamical behavior of the corresponding stochastic basin of attraction. SBA is an efficient tool to describe the metastable phenomena complementing the known exit time, escape probability, or relaxation time. Moreover, the geometric structure of SBA gives additional insight into the system's dynamical behavior, which is important for theoretical and practical reasons. This concept can be used not only in models with small noise intensity but also with noise whose amplitude is proportional or in general is a function of an order parameter. As an application of our main results, we analyze a three potential well system perturbed by two types of noise: Brownian motion and non-Gaussian α-stable Lévy motion. Our main conclusions are that the thermal fluctuations stabilize the metastable system with an asymmetric three-well potential but have the opposite effect for a symmetric one. For Lévy noise with larger jumps and lower jump frequencies ( α=0.5) metastability is enhanced for both symmetric and asymmetric potentials.

An Algebraic Approach to Unital Quantities and their Measurement

Abstract The goals of this paper fall into two closely related areas. First, we develop a formal framework for deterministic unital quantities in which measurement unitization is understood to be a built-in feature of quantities rather than a mere annotation of their numerical values with convenient units. We introduce this idea within the setting of certain ordered semigroups of physical-geometric states of classical physical systems. States are assumed to serve as truth makers of metrological statements about quantity values. A unital quantity is presented as an isomorphism from the target system’s ordered semigroup of states to that of positive reals. This framework allows us to include various derived and variable quantities, encountered in engineering and the natural sciences. For illustration and ease of presentation, we use the classical notions of length, time, electric current and mean velocity as primordial examples. The most important application of the resulting unital quantity calculus is in dimensional analysis. Second, in evaluating measurement uncertainty due to the analog-to-digital conversion of the measured quantity’s value into its measuring instrument’s pointer quantity value, we employ an ordered semigroup framework of pointer states. Pointer states encode the measuring instrument’s indiscernibility relation, manifested by not being able to distinguish the measured system’s topologically proximal states. Once again, we focus mainly on the measurement of length and electric current quantities as our motivating examples. Our approach to quantities and their measurement is strictly state-based and algebraic in flavor, rather than that of a representationalist-style structure-preserving numerical assignment.

Direction-of-arrival estimation in a mixture of K-distributed and Gaussian noise

We address the problem of estimating the directions-of-arrival (DoAs) of multiple signals received in the presence of a combination of a strong compound-Gaussian external noise and weak internal white Gaussian noise. Since the exact distribution of the mixture is not known, we get an insight into optimum procedure via a related model where we consider the texture of the compound-Gaussian component as an unknown and deterministic quantity to be estimated together with DoAs or a basis of the signal subspace. Alternate maximization of the likelihood function is conducted and it is shown that it operates a separation between the snapshots with small/large texture values with respect to the additive noise power. The modified Cramér–Rao bound is derived and a prediction of the actual mean-square error is presented, based on separation between external/internal-noise dominated samples. Numerical simulations indicate that the suggested iterative DoA estimation technique comes close to the introduced bound and outperform a number of existing routines.

Discussion: Deterioration of performance of the lasso with many predictors

Oracle inequalities provide probability loss bounds for the lasso estimator at a deterministic choice of the regularization parameter and are commonly cited as theoretical justification for the lasso and its ability to handle high-dimensional settings. Unfortunately, in practice, the regularization parameter is not selected to be a deterministic quantity, but is instead chosen using a random, data-dependent procedure, often making these inequalities misleading in their implications. We discuss general results and demonstrate empirically for data using categorical predictors that the amount of deterioration in performance of the lasso as the number of unnecessary predictors increases can be far worse than the oracle inequalities suggest, but imposing structure on the form of the estimates can reduce this deterioration substantially.