Deterministic Quantities Research Articles

Dependence of Metrics and Noise Intensity on Escape Dynamics:Mean First Exit Time and Escape Probability

Abstract This paper focuses on analyzing two deterministic quantities of the dynamics of stochastic 
systems, mean first exit time and escape probability. Through a detailed exploration of the escape phenomenon, we have successfully fitted the complex dynamics of the mean first exit time as a function of interval length (area size) and noise intensity, respectively.
We have found that the mean first exit time tends to increase monotonically with the interval length (area size).
Meanwhile, we also accurately captured the dynamical behavior of the minimum escape probability under various parameter cases. 
The relationship between them displays a range of patterns when we observed different intervals.
Our research highlights the importance of noise effects when considering escape processes, especially in the context of interaction of interval length (area size) and noise intensity.
A concrete error analysis is conducted to ensure the accuracy and generalization ability of the model's predictions.

A Physically Inspired Model for Estimating Clear-Sky Solar Irradiance

Abstract The estimation of solar irradiance under clear sky conditions is still a research topic. Many times, accessible empirical models are applied in practice, even if more uncertainty is expected than in the case of complex models. This paper reports a class of innovative empirical models for estimating clear sky global solar irradiance. Although the models are of an empirical nature, the mathematical equations are inspired by the physics behind the radiative transfer in the atmosphere. The models are developed with high-quality data recorded on the Solar Platform of the West University of Timisoara, Romania. Considering the local character of the empirical models, the development process is described in detail so that it can be replicated in other locations. Having only deterministic quantities as input, the proposed models harmoniously combine the two key requirements for practical applications: accessibility and accuracy.

Online Stochastic DCA With Applications to Principal Component Analysis.

Stochastic algorithms are well-known for their performance in the era of big data. In this article, we study nonsmooth stochastic Difference-of-Convex functions (DC) programs-the major class of nonconvex stochastic optimization, which have a variety of applications in divers domains, in particular, machine learning. We propose new online stochastic algorithms based on the state-of-the-art DC Algorithm (DCA)-a powerful approach in nonconvex programming framework, in the online context of streaming data continuously generated by some (unknown) source distribution. The new schemes use the stochastic approximations (SAs) principle: deterministic quantities of the standard DCA are replaced by their noisy estimators constructed using newly arriving samples. The convergence analysis of the proposed algorithms is studied intensively with the help of tools from modern convex analysis and martingale theory. Finally, we study several aspects of the proposed algorithms on an important problem in machine learning: the expected problem in principal component analysis (PCA).

Stochastic and multi-objective design of photonic devices with machine learning

Compact and highly performing photonic devices are characterized by non-intuitive geometries, a large number of parameters, and multiple figures of merit. Optimization and machine learning techniques have been explored to handle these complex designs, but the existing approaches often overlook stochastic quantities. As an example, random fabrication uncertainties critically determines experimental device performance. Here, we present a novel approach for the stochastic multi-objective design of photonic devices combining unsupervised dimensionality reduction and Gaussian process regression. The proposed approach allows to efficiently identify promising alternative designs and model the statistic of their response. Incorporating both deterministic and stochastic quantities into the design process enables a comprehensive analysis of the device and of the possible trade-offs between different performance metrics. As a proof-of-concept, we investigate surface gratings for fiber coupling in a silicon-on-insulator platform, considering variability in structure sizes, silicon thickness, and multi-step etch alignment. We analyze 86 alternative designs presenting comparable performance when neglecting variability, discovering on the contrary marked differences in yield and worst-case figures for both fiber coupling efficiency and back-reflections. Pareto frontiers demonstrating optimized device robustness are identified as well, offering a powerful tool for the design and optimization of photonic devices with stochastic figures of merit.

Statistical perspectives on the ripple index from a case study of aeolian megaripples

The ripple index (RI) is a common metric in the study of bedforms. In this work we review the application of RI to aeolian megaripples and perform a statistical analysis of field measurements. The research highlights ambiguity in how RI is calculated and used to describe and infer bedform morphometry. The statistical analysis demonstrates that although the prevailing application of RI encourages an interpretation as a deterministic quantity, it is in fact a stochastic variable with large variance and several use cases. RI can be used as an individual descriptive statistic, an aggregated descriptive statistic, or a linear predictor. We develop a simple linear model to investigate the notion of a representative RI value for megaripples. We find a value of 17.3 as a linear predictor for individual megaripples and a value of 17.6 as a linear predictor for aggregated megaripples, but show that the large variance observed in megaripple morphology mean that interpreting these values as representative of all megaripples can be misleading. This empirical finding of large variance in megaripple RI values aligns with recent work showing that megaripples are highly variable bedforms. Broadly, we recommend that all research involving RI should use it in an appropriate individual or aggregated data context and present these data with measures of dispersion.

Transient properties of grazing ecosystem driven by Lévy noise and Gaussian noise

This paper explores how Lévy noise and Gaussian noise affect the ecological grazing system by using the fourth-order Runge–Kutta method to simulate dynamic system and the Janicki-Weron algorithm to produce Lévy noise. Two deterministic quantities, the mean first passage time (MFPT) and the probability density function (PDF) of the first passage time (FPT), are utilized to explore the transient properties of grazing ecosystem. Our research results show that: (i) The transitions between two alternative stable states can be induced by the Gaussian noise intensity, Lévy noise intensity, Lévy stability index and Lévy skewness parameter. (ii) A higher Lévy noise intensity and a larger Lévy stability index or Lévy skewness parameter make the MFPT from desert state to sustainable vegetated state shorter; a higher Gaussian noise intensity makes it longer, which indicates that increased Lévy noise intensity can mitigate ecosystem degradation; increased Gaussian noise intensity will lead to desertification of vegetation. (iii) For larger Lévy noise intensity, the MFPT from sustainable vegetated state to desert state as a function of Gaussian noise intensity exhibits one maximum value when it transformation from the sustainable vegetated state to the desert state, the noise enhanced stability (NES) phenomena of grazing ecosystems are observed.

Statistical analysis of randomized pseudo-first/second order kinetic models. Application to study the adsorption on cadmium ions onto tree fern

Adsorption kinetics are commonly modeled using pseudo-first order (PFO) and pseudo-second order (PSO) rate laws. Both models are formulated via differential equations whose coefficients are commonly treated as deterministic quantities, which are calculated from experimental data using regression techniques. In this paper, we propose a full randomization of both models by assuming that all the parameters of the PFO and PSO models are random variables with arbitrary densities. Then, we probabilistically solve both models by determining semi-explicit expressions of the first probability density functions of the corresponding solution stochastic processes, as well as the densities of the time until a fixed adsorbed amount of reactant has been reached for each model too. The analysis is conducted under very general assumptions and is based on extensive application of the so-called Random Variable Transformation (RVT) technique. Finally, we apply all the aforementioned theoretical findings to model the adsorption of cadmium ions onto tree fern, using real data. We compare the results obtained by the randomized PFO and PSO models, using a Bayesian and a randomized version of the least mean square method to assign reasonable densities to model parameters. After comparing the results, the PSO model is selected, and, we then show how the RVT technique can be applied to obtain further key information, such as the second probability density function, of the solution and the covariance function.

Unification of the first law of quantum thermodynamics

Underlying the classical thermodynamic principles are analogous microscopic laws, arising from the fundamental axioms of quantum mechanics. These define quantum thermodynamic variables such as quantum work and heat and characterize the possible transformations of open quantum systems. The foremost quantum thermodynamic law is a simple statement concerning the conservation of energy. Nevertheless, there exist ambiguity and disagreement regarding the precise partition of a quantum system’s energy change to work and heat. By treating quantum mechanics as a comprehensive theory, applicable to both the micro and macroscopic domains, and employing dynamical symmetries, we bridge the gaps between five popular thermodynamic approaches to the first law. These include both autonomous and semi-classical formulations, which define work in terms of an ensemble average, as well as the single shot paradigm, where work is defined as a deterministic quantity.

Novel closure approximation for prediction of the effective elastic properties of composites with discontinuous reinforcement

The paper introduces a novel closure approximation method for the determination of the fourth-order orientation tensor in terms of known second-order tensor. The fourth-order orientation tensor must be known to predict the effective elastic properties of composites with misaligned fibres but very often only the second-order orientation is available. The proposed method operates on a set of outcomes of optimization based on a genetic algorithm. In consequence, it leads to obtaining distributions of effective elastic constants instead of deterministic quantities. Results obtained by using the novel approach have been compared with results based on exact fourth-order orientation tensor, tensor known from experimental data and determined by using popular closure approximations from literature. Furthermore, the obtained results have been compared with results of finite element analysis based on geometrical models representing the microstructure which contains several hundred fibres. The new method provides high accuracy of elastic constant predictions for all studied fibre orientation distributions, which distinguishes it from the closure approximation methods known from the literature. Moreover, results obtained in form of distributions give an insight into the uncertainty of the predictions.

A Novel Stochastic Model of the Photo-Thermoelasticity Theory of the Non-Local Excited Semiconductor Medium

A novel technique under the effect of stochastic heating due to the thermal effect of the photothermal theory is investigated. Realistically, stochastic processes are taken on the boundary of the non-local semiconductor medium. The interactions between optical, thermal, and mechanical waves in a half-space of the medium are studied according to the photo-thermoelasticity theory. The governing equations are described in one dimension (1D) according to the elastic-electronic deformation. Laplace transforms with short-time approximation are used to analyze the main physical fields in linearity form. To study the problem more realistically, some conditions are taken as random with white noise on the free surface of the elastic medium. The deterministic physical quantities are obtained with a stochastic calculus when a numerical inversion of the Laplace transform is applied. The silicon material is utilized to make the stochastic simulation. The comparisons are carried out between the distributions of deterministic and stochastic (statistically, the mean and variance) of the main physical quantities along different sample paths graphically and discussed for the non-local silicon semiconductor material.

Bias for the Trace of the Resolvent and Its Application on Non-Gaussian and Non-Centered MIMO Channels

The mutual information (MI) of Gaussian multi-input multi-output (MIMO) channels has been evaluated by utilizing random matrix theory (RMT) and shown to asymptotically follow Gaussian distribution, where the ergodic mutual information (EMI) converges to a deterministic quantity. However, with non-Gaussian channels, there is a bias between the EMI and its deterministic equivalent (DE), whose evaluation is not available in the literature. This bias of the EMI is related to the bias for the trace of the resolvent in large RMT. In this paper, we first derive the bias for the trace of the resolvent, which is further extended to compute the bias for the linear spectral statistics (LSS). Then, we apply the above results on non-Gaussian MIMO channels to determine the bias for the EMI. It is also proved that the bias for the EMI is $-0.5$ times of that for the variance of the MI. Finally, the derived bias is utilized to modify the central limit theory (CLT) and calculate the outage probability. Numerical results show that the modified CLT significantly outperforms previous methods in approximating the distribution of the MI and improves the accuracy for the outage probability evaluation.

Uncertainties in internal pressure of oil transmission pipelines and implications for the reliability analysis

This study addresses a significant knowledge gap in the reliability-based fitnessfor-service assessment of pipelines, namely the statistical information of the internal operating pressure. To this end, probabilistic characteristics of the internal pressure of an in-service oil transmission pipeline are derived based on minute-by-minute pressure records collected from the discharge and suction ends of a pump station on the pipeline. The arbitrary-point-in-time discharge and suction pressures are found to follow the Johnson SB distribution; the monthly and annual maximum discharge pressures are found to be well represented by a deterministic quantity equal to 90% of the maximum operating pressure, and the monthly and annual maximum suction pressures follow the beta distributions. The autocorrelation and power spectral density functions of the discharge and suction pressures characterized as stationary stochastic processes are also derived from the pressure records. Furthermore, the statistics of the pressure ranges obtained from the rainflow counting analysis of the pressure records are obtained. Two example pipelines are used to examine the implications of the uncertainty in the internal pressure for the reliability analysis of corroding pipelines.

Fully-Automated On-Chip Multi-Cell Arraying With Deterministic Quantities

Microfluidic devices for cell immobilization have significantly advanced the biological analysis at the single-cell level. However, the previous research on immobilization of multiple single cells, especially with deterministic quantities, is insufficient. In this paper, we proposed a novel microfluidic device based on the passive hydrodynamics and the uniform geometric design principle, which can array different numbers of cells in every capture cavity. The capture cavities could be stretched to accommodate more cells, and the trapping force was adjusted by modifying the related geometric parameters of the inside channel. The whole procedure was monitored and further automatized by integrating computer vision technology under a microscope. On the proposed integrated on-chip platform, we realized full-automated arraying of a single cell, two cells, and three cells on a single chip, achieving success rates up to 95%, 75%, and 72%, respectively. As a primary experimental demonstration, the cell viability test of arraying multiple cells with different quantities showed excellent biocompatibility and no significant association between trapping quantity and cell survivability. We envision that the proposed quantity-controllable, high-efficiency microfluidic devices for multiple cell arraying could be a powerful platform for an in-depth study of cell heterogeneity and cell communication between multiple cells.

A novel stochastic photo-thermoelasticity model according to a diffusion interaction processes of excited semiconductor medium.

A novel technique under the impact of stochastic heating due to the thermal effect of photothermal theory is investigated. Realistically, stochastic processes are taken on the boundary of the semiconductor medium. The interactions between optical, thermal, and mechanical waves in a half-space of the medium are studied according to the photo-thermoelasticity theory. The governing equations are described in one-dimensional elastic-electronic deformation. Laplace transforms with short-time approximation are used to analyze the main physical fields. To study the problem more realistically, some conditions are taken as random with white noise on the free surface of the elastic medium. The deterministic physical quantities are obtained with a stochastic calculus when a numerical inversion of the Laplace transform is applied. The silicon material is utilized to make the stochastic numerical simulation. The comparisons are carried out between the distributions of deterministic and stochastic (statistically, the mean and variance) the main physical quantities along different sample paths graphically and discussed.

On elevating the reliability of the hygienic assessment of water quality of natural sources of drinking water supply

Introduction. As new toxic substances accumulate in water bodies, improving the reliability of the hygienic assessment of water quality is put forward in the first place. For the correct solution of this problem, it is relevant to use the direct comparison of deterministic quantities such as the maximum permissible concentration (MPC) of toxic pollutants with the result of its measurements (C) and methods of approach for evaluating the results of research. Material and methods. To improve the reliability of hygienic assessment of water quality in the proposed method tolerance control, envisaging the elucidation of the implementation type inequality and estimation of the probability of this event: where is the risk of false imprisonment. The number of concurrent measurements of water quality parameters guaranteeing an acceptable level of risk of error of research results is established by the authority of sanitary-epidemiological supervision, receiving a limited level of tolerable error taking into account the possibilities of metrological assurance of conformity assessment/quality of water and its normative level. Results. It is shown that in cases when the compliance of controlled safety indicators is evaluated, the values of which are limited on both sides, as is the case when studying the physiological usefulness of water, it is necessary to establish a two-way range of values of the permissible probability of violation of the specified requirements. Conclusion. First installed: the rule “if it is impossible to argue that controlled water quality indicators do not meet hygienic requirements. Thus requirements are met” is not equivalent to the law “if the indicators meet the requirements, then they can’t be inappropriate”, which more reliably indicates compliance with these requirements.

Probabilistic method for time-varying reliability analysis of structure via variational bayesian neural network

This study proposes a novel framework computing the dynamic reliability and associated uncertainty quantification of structures under time-varying excitation with significantly reduced time complexity. For this purpose, the deep neural network’s power and the Bayesian theory’s probabilistic ability are leveraged, forming a Bayesian neural network data-driven model (BNN). The BNN-based surrogate model can yield a probability distribution of outputs of interest, e.g., a limit state function and its derived statistics such as median value, confidence interval rather than only a deterministic quantity. The effectiveness and correctness of the proposed method are reaffirmed via three case studies involving examples from the literature and a 3D numerical model of a prestressed reinforced concrete bridge structure, showing a reduction in time complexity up to three orders of magnitude compared to the Monte Carlo method only using finite element models. As a result, an 11-year maintenance routine is recommended for a marine and chemically aggressive environment to ensure the high reliability of prestressed bridge structures when accounting for uncertainty estimation.

Entropy as an Objective Function of Optimization Multimodal Transportations.

This article considers the use of the entropy method in the optimization and forecasting of multimodal transport under conditions of risks that can be determined simultaneously by deterministic, stochastic and fuzzy quantities. This will allow to change the route of transportation in real time in an optimal way with an unacceptable increase in the risk at one of its next stages and predict the redistribution of the load of transport nodes. The aim of this study is to develop a mathematical model for the optimal choice of an alternative route, the best for one or more objective functions in real time. In addition, it is proposed to use this mathematical model to estimate the dynamic change in turnover through intermediate transport nodes, forecasting their loading over time under different conditions that also include long-term risks which are significant in magnitude. To substantiate the feasibility of the proposed mathematical model, the analysis and forecast of cargo turnover through the seaports of Ukraine are presented, taking into account and analysing the existing risks.

A deterministic economic production quantity model with partial backordering and outsourcing repair of non-conforming products

In this paper, an economic production quantity model for an imperfect production system with reparation of non-conforming products is studied. The system allows shortages and a proportion of the shortages are backordered, following a last-in/first-out (LIFO) policy. It is assumed that the repair of non-conforming products cannot be carried out in the manufacturer's production system. Therefore, the company decides to have the repairs carried out in a repair centre. Depending on the time when the consignment of reworked items returns to the manufacturer's stocks, this study identifies three cases. The paper shows the derivation of the optimal solution for each case. The required conditions for the initial value of the backordering rate are developed in order to achieve optimal partial backordering. The model is also compared with a modified version that includes not only outsourcing repair of non-conforming products but also full backordering. The numerical examples comparing full backordering with partial backordering indicate that partial backordering is more profitable than full backordering when return of the repaired items to the system leaves a negative inventory level; that is, when there is a shortage.

Visualizing Uncertainty for Non-Expert End Users: The Challenge of the Deterministic Construal Error

There is a growing body of evidence that numerical uncertainty expressions can be used by non-experts to improve decision quality. Moreover, there is some evidence that similar advantages extend to graphic expressions of uncertainty. However, visualizing uncertainty introduces challenges as well. Here, we discuss key misunderstandings that may arise from uncertainty visualizations, in particular the evidence that users sometimes fail to realize that the graphic depicts uncertainty. Instead they have a tendency to interpret the image as representing some deterministic quantity. We refer to this as thedeterministic construal error. Although there is now growing evidence for thedeterministic construal error, few studies are designed to detect it directly because they inform participants upfront that the visualization expresses uncertainty. In a natural setting such cues would be absent, perhaps making the deterministic assumption more likely. Here we discuss the psychological roots of this key but underappreciated misunderstanding as well as possible solutions. This is a critical question because it is now clear that members of the public understand that predictions involve uncertainty and have greater trust when uncertainty is included. Moreover, they can understand and use uncertainty predictions to tailor decisions to their own risk tolerance, as long as they are carefully expressed, taking into account the cognitive processes involved.

The Assouad spectrum of random self-affine carpets

AbstractWe derive the almost sure Assouad spectrum and quasi-Assouad dimension of one-variable random self-affine Bedford–McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely ‘as large as possible’ (a deterministic quantity). This has been verified in conformal and non-conformal settings. In the conformal setting, the Assouad spectrum and quasi-Assouad dimension behave rather differently, tending to almost surely coincide with the upper box dimension. Here we investigate the non-conformal setting and find that the Assouad spectrum and quasi-Assouad dimension generally do not coincide with the box dimension or Assouad dimension. We provide examples highlighting the subtle differences between these notions. Our proofs combine deterministic covering techniques with suitably adapted Chernoff estimates and Borel–Cantelli-type arguments.