Second-order Distributions Research Articles

Simulation Modeling of QS with Hyper-Erlang and Erlang Distributions

The article is devoted to the construction of simulation models of customer delay in the queue in the form of a queuing system (QS) described by both ordinary and right-shifted hyper-Erlang and second-order Erlang distributions. This article is a logical continuation of previous works devoted to the construction of numerical-analytical QS models with shifted distribution laws. In the article, the Erlang distribution is considered as a special case of the more general Gamma distribution law, in contrast to the normalized Erlang distribution. These two forms of the Erlang distribution differ in numerical characteristics, except for the coefficient of variation. To solve the problem, the system of discrete-event modeling GPSS WORLD was used.

Features of Analytical Modeling of QS with Hypererlang and Erlang Distributions

The article is devoted to the construction of a mathematical model for delaying claims in a queue in the form of a queuing system (QS) described by right-shifted hyper-Erlang and second-order Erlang distributions. This article is a logical continuation of the previous works of the author. In the article, the Erlang distribution is considered as a special case of the more general Gamma distribution law, in contrast to the normalized Erlang distribution. These two forms of the Erlang distribution differ in numerical characteristics, except for the coefficient of variation. To solve the problem, we used the method of spectral solution of the Lindley integral equation.

Equilibrium mean–variance reinsurance and investment strategies for a general insurance company under smooth ambiguity

This work investigates the equilibrium investment and reinsurance strategies for a general insurance company under smooth ambiguity. The general insurance company holds shares of an insurance company and a reinsurance company. The claims of the insurer follow a compound Poisson process. The insurer can divide part of the insurance risk to the reinsurer. Besides, the insurer and reinsurer both participate in the financial market and invest in cash and stock. However, the general insurance company is ambiguous about the insurance and financial risks and is an ambiguity-averse manager (AAM). The uncertainties over the insurance and financial risks are described by second-order distributions. The AAM aims to maximize the average performance of the weighted sum surplus process of the insurer and reinsurer under the mean–variance criterion and smooth ambiguity. We present the extended Hamilton–Jacobi–Bellman (HJB) system for the optimization problem combining the mean–variance criterion and smooth ambiguity. In the case that the second-order distributions are Gaussian, we obtain the closed-forms of the equilibrium reinsurance and investment strategies. At the end of this work, sensitivity analyses are presented to show the economic behaviors of the AAM.

Delay model based on shifted hyperexponential and Erlang distributions

This article is devoted to the derivation of results for the average delay of requests in the queue for a queuing system formed by two flows with the laws of interval distributions in the form of second-order hyperexponential and Erlang distributions shifted to the right. In queuing theory, studies of G/G/1 systems are relevant due to the fact that there is no solution in the final form for the general case. Therefore, in the study of such systems, various particular distribution laws are used as an arbitrary distribution law for G. In this case, the use of the hyperexponential distribution law ensures the coefficient of variation of the input flow intervals is large units, and the Erlang distribution is less than one. To solve the problem posed, the method of spectral decomposition of the solution of the integral Lindley equation was used, which plays an important role in the queueing theory. This method made it possible to obtain a solution for the average delay of requests in the queue for the system under consideration in a closed form. As is known, the remaining characteristics of the queuing system are derived from the average delay of requests.

Short-duration exposure of 3-µm polystyrene microplastics affected morphology and physiology of watermilfoil (sp. roraima)

Microplastics are one of the most widely discussed environmental issues worldwide. Several studies have shown the effect of microplastic exposure on the marine environment; however, studies on freshwater systems are lacking. This study was conducted to investigate the effect of microplastics on hydroponically growing emergent freshwater macrophytes, watermilfoil (sp. roraima) under controlled environmental conditions. Plants were exposed to 0mg L-1 (control), 0.05mg L-1, 0.25mg L-1, 1.25mg L-1, and 6mg L-1 of 3-µm polystyrene microspheres for 7days. The oxidative stress, antioxidant response, pigmentations, Fv/Fm, and growth parameters in the above-water and below-water parts were analyzed separately. Microscopic observations were performed to confirm the tissue absorbance of the microplastics. Exposure to microplastics altered some parameters; however, growth was not affected. The effect of microplastics was not linear with the exposure concentration for most of the parameters and between 1.25 and 6mg L-1 concentrations. The response trends mostly followed the second-order polynomial distributions. Under the 1.25mg L-1 exposure, there were significant changes in root length, H2O2 content, catalase activity, anthocyanin content, and Fv/Fm. There were differences in parameters between the above-water and below-water parts, and the responses of the microplastics followed different trends. Microscopic observations confirmed the attachment of microplastic particles onto newly formed roots, except for older roots or shoot tissues.

DEVELOPMENT OF A MODEL FOR THE DYNAMICS OF PROBABILITIES OF STATES OF SEMI-MARKOV SYSTEMS

The subject is the study of the dynamics of probability distribution of the states of the semi-Markov system during the transition process before establishing a stationary distribution. The goal is to develop a technology for finding analytical relationships that describe the dynamics of the probabilities of states of a semi-Markov system. The task is to develop a mathematical model that adequately describes the dynamics of the probabilities of the states of the system. The initial data for solving the problem is a matrix of conditional distribution laws of the random duration of the system's stay in each of its possible states before the transition to some other state. Method. The traditional method for analyzing semi-Markov systems is limited to obtaining a stationary distribution of the probabilities of its states, which does not solve the problem. A well-known approach to solving this problem is based on the formation and solution of a system of integral equations. However, in the general case, for arbitrary laws of distribution of the durations of the stay of the system in its possible states, this approach is not realizable. The desired result can only be obtained numerically, which does not satisfy the needs of practice. To obtain the required analytical relationships, the Erlang approximation of the original distribution laws is used. This technique significantly increases the adequacy of the resulting mathematical models of the functioning of the system, since it allows one to move away from overly obligatory exponential descriptions of the original distribution laws. The formal basis of the proposed method for constructing a model of the dynamics of state probabilities is the Kolmogorov system of differential equations for the desired probabilities. The solution of the system of equations is achieved using the Laplace transform, which is easily performed for Erlang distributions of arbitrary order. Results. Analytical relations are obtained that specify the desired distribution of the probabilities of the states of the system at any moment of time. The method is based on the approximation of the distribution laws for the durations of the stay of the system in each of its possible states by Erlang distributions of the proper order. A fundamental motivating factor for choosing distributions of this type for approximation is the ease of their use to obtain adequate models of the functioning of probabilistic systems. Conclusions. A solution is given to the problem of analyzing a semi-Markov system for a specific particular case, when the initial distribution laws for the duration of its sojourn in possible states are approximated by second-order Erlang distributions. Analytical relations are obtained for calculating the probability distribution at any time.

Egeria densa Allelopathy on Microcystis aeruginosa Under Different Light Intensities and Preliminary Insight into Inter-Parameter Relationships

Egeria densa release allelochemicals harmful to other species, including cyanobacteria. Under natural conditions, E. densa and Microcystis aeruginosa can be found in various aquatic systems with different light levels. The effect that the coexistence of E. densa has on M. aeruginosa was tested under different photosynthetically active radiation (PAR) intensities. The growth, chlorophyll-a, oxidative stress, and antioxidant activities of M. aeruginosa were quantified after 7 days of coexistence with E. densa. A control experiment without E. densa was conducted for M. aeruginosa under the same conditions to distinguish the effect of the PAR intensity from the allelopathic effect. The allelopathy of E. densa on M. aeruginosa is significantly influenced by the PAR intensity. PAR intensities lower or higher than the favorable range further suppressed M. aeruginosa under coexistence. Research has revealed that the allelopathic effect of E. densa can be utilized to biologically suppress M. aeruginosa under various light conditions. Strong second-order polynomial distributions between the optical density (OD) and cellular H2O2 content of M. aeruginosa were found for both the control and coexistence conditions. In the practical application of E. densa allelopathy, OD can be adopted as a convenient method to obtain an approximate stress status on M. aeruginosa.

Quantitative Research and Characterization of the Loess Microstructure in the Bai Lu Tableland, Shaanxi Province, China

Loess is a special geotechnical material with strong structural properties, and the microstructural characteristics of loess significantly influence its macroscopic physical features, mechanical properties, and catastrophic behavior. In this paper, serial samples were extracted from the continuous loess and paleosol strata of the Bai Lu tableland; with these samples, the optical microscopy‐based serial sectioning method was adopted to study the quantitative characterization and variation in the loess microstructure. Three‐dimensional characteristics and quantitative parameters of the particles, pores, and throats of the loess were obtained. The results indicate that the volume, Eq‐Radius, and major‐minor axis ratio of the loess particles satisfy third‐order, third‐order, and second‐order Gaussian distributions, respectively. The Eq‐Radius of the loess pores and throats satisfies a first‐order Gaussian distribution, and the throat channel length satisfies a gamma distribution. With increasing stratum depth, the particles become more flattened, the throat radii become larger and the pore channels become slenderer. The variation in fitting parameters and the correlations between the macrophysical and mechanical properties of loess were then explored. The study of the microstructure of loess contributes to a better understanding of the catastrophic behavior of loess and the physical mechanism of geologic hazards in this area.

Methodology of probabilistic analysis of state dynamics of multi­dimensional semi­Markov dynamic systems

The problem of probabilistic analysis of a complex dynamic system, which in the process of functioning passes from one state to another at random times, is considered. The methodology for calculating the conditional probabilities of the system getting into a given state at a given time t, provided that at the initial time the system was in any of the possible states is proposed. The initial data for analysis are a set of experimentally obtained values of the duration of the system stay in each of the states before transition to another state. Approximation of the resulting histograms using the Erlang distribution gives a set of distribution densities of the duration of the system stay in possible states before transition to other states. At the same time, the choice of the proper Erlang distribution order provides an adequate description of the semi-Markov processes occurring in the system. The mathematical model that relates the obtained distribution densities to the functions determining the probabilistic dynamics of the system is proposed. The model describes a random process of system transitions from any possible initial state to any other state during a given time interval. Using the model, a system of integral equations for the desired functions describing the probabilistic transition process is obtained. To solve these equations, the Laplace transform is used. As a result of solving the system of integral equations, functions are obtained that specify the probability distribution of the system states at any time t. The same functions also describe the asymptotic probability distribution of states. An illustrative example of solving the problem for the case when the distribution densities of the lengths of the system stay in possible states are described by the second-order Erlang distributions is given. The solution procedure is described in detail for the most natural special case, when the initial state is H 0

Imprecise Uncertain Reasoning: A Distributional Approach.

The contribution proposes to model imprecise and uncertain reasoning by a mental probability logic that is based on probability distributions. It shows how distributions are combined with logical operators and how distributions propagate in inference rules. It discusses a series of examples like the Linda task, the suppression task, Doherty's pseudodiagnosticity task, and some of the deductive reasoning tasks of Rips. It demonstrates how to update distributions by soft evidence and how to represent correlated risks. The probabilities inferred from different logical inference forms may be so similar that it will be impossible to distinguish them empirically in a psychological study. Second-order distributions allow to obtain the probability distribution of being coherent. The maximum probability of being coherent is a second-order criterion of rationality. Technically the contribution relies on beta distributions, copulas, vines, and stochastic simulation.

Two-site jumps in dimethyl sulfone studied by one- and two-dimensional 17O NMR spectroscopy

Polycrystalline dimethyl sulfone is studied using central-transition oxygen-17 exchange NMR. The quadrupolar and chemical shift tensors are determined by combining quantum chemical calculations with line shape analyses of rigid-lattice spectra measured for stationary and rotating samples at several external magnetic fields. Quantum chemical computations predict that the largest principal axes of the chemical shift anisotropy and electrical field gradient tensors enclose an angle of about 73°. This prediction is successfully tested by comparison with absorption spectra recorded at three different external magnetic fields. The experimental one-dimensional motionally narrowed spectra and the two-dimensional exchange spectrum are compatible with model calculations involving jumps of the molecules about their two-fold symmetry axis. This motion is additionally investigated by means of two-time stimulated-echo spectroscopy which allows for a determination of motional correlation functions over a wider temperature range than previously reported using carbon and deuteron NMR. On the basis of suitable second-order quadrupolar frequency distributions, sin-sin stimulated-echo amplitudes are calculated for a two-site model in the limit of vanishing evolution time and compared with experimental findings. The present study thus establishes oxygen-17 NMR as a powerful method that will be particularly useful for the study of solids and liquids devoid of nuclei governed by first-order anisotropies.

Fitting Matrix Geometric Distributions by Model Reduction

A novel algorithmic method is proposed to fit matrix geometric distributions of desired order to empirical data or arbitrary discrete distributions. The proposed method effectively combines two existing approaches from two different disciplines: well-established model reduction methods used in system theory and moment matching methods of applied probability that employ second-order discrete phase-type distributions. The proposed approach is validated with exhaustive numerical examples including well-known statistical data.

Second-Order Semiclassical Perturbation Theory for Diffractive Scattering from a Surface

A second-order semiclassical perturbation theory is developed and applied to the elastic scattering of an atom from a corrugated surface. Analytical expressions for the diffraction pattern in the momentum space are obtained based on a sine corrugation function and a Morse potential for the interaction of the particle with the surface. The theory is implemented for a model of the in-plane scattering of Ar atoms from a LiF(100) surface. The resulting diffraction intensities are compared with second-order perturbation theory classical distributions and close-coupling results for two incident energies of 300 and 700 meV. The previous first-order perturbation theory predicts a symmetric diffraction pattern about the elastic peak, while the second-order semiclassical perturbation theory accounts correctly for the asymmetry in the diffraction pattern.

Implementing Second-Order Decision Analysis: Concepts, Algorithms, and Tool

We present implemented concepts and algorithms for a simulation approach to decision evaluation with second-order belief distributions in a common framework for interval decision analysis. The rationale behind this work is that decision analysis with interval-valued probabilities and utilities may lead to overlapping expected utility intervals yielding difficulties in discriminating between alternatives. By allowing for second-order belief distributions over interval-valued utility and probability statements these difficulties may not only be remedied but will also allow for decision evaluation concepts and techniques providing additional insight into a decision problem. The approach is based upon sets of linear constraints together with generation of random probability distributions and utility values from implicitly stated uniform second-order belief distributions over the polytopes given from the constraints. The result is an interactive method for decision evaluation with second-order belief distributions, complementing earlier methods for decision evaluation with interval-valued probabilities and utilities. The method has been implemented for trial use in a user oriented decision analysis software.

An analytical simplified model to characterize focused aperture antennas

This paper presents an analytical model to characterize the radiation pattern of focused aperture antennas. The model is based on the classic parabolic on pedestal distribution for amplitude, but in this work the focusing phase term is considered and applied in the Fresnel region. The model is useful for millimeter and submillimeter wave imaging radar systems that usually work in the Fresnel region of the antenna. Analytical closed expressions are developed to predict the available resolution (transversal beamwidth) and operating range (axial beamwidth) of such systems. The effects of the first- and second-order phase distributions on the aperture have also been also studied in order to show the scanning effect, the axial refocusing, and the astigmatic beam degradation.

Knowledge Fusion for Probabilistic Generative Classifiers with Data Mining Applications

If knowledge such as classification rules are extracted from sample data in a distributed way, it may be necessary to combine or fuse these rules. In a conventional approach this would typically be done either by combining the classifiers' outputs (e.g., in form of a classifier ensemble) or by combining the sets of classification rules (e.g., by weighting them individually). In this paper, we introduce a new way of fusing classifiers at the level of parameters of classification rules. This technique is based on the use of probabilistic generative classifiers using multinomial distributions for categorical input dimensions and multivariate normal distributions for the continuous ones. That means, we have distributions such as Dirichlet or normal-Wishart distributions over parameters of the classifier. We refer to these distributions as hyperdistributions or second-order distributions. We show that fusing two (or more) classifiers can be done by multiplying the hyperdistributions of the parameters and derive simple formulas for that task. Properties of this new approach are demonstrated with a few experiments. The main advantage of this fusion approach is that the hyperdistributions are retained throughout the fusion process. Thus, the fused components may, for example, be used in subsequent training steps (online training).

Second-Order Risk Constraints in Decision Analysis

Recently, representations and methods aimed at analysing decision problems where probabilities and values (utilities) are associated with distributions over them (second-order representations) have been suggested. In this paper we present an approach to how imprecise information can be modelled by means of second-order distributions and how a risk evaluation process can be elaborated by integrating procedures for numerically imprecise probabilities and utilities. We discuss some shortcomings of the use of the principle of maximising the expected utility and of utility theory in general, and offer remedies by the introduction of supplementary decision rules based on a concept of risk constraints taking advantage of second-order distributions.

Uncertainty Levels of Second-Order Probability

Since second-order probability distributions assign probabilities to probabilities there is uncertainty on two levels. Although different types of uncertainty have been distinguished before and corresponding measures suggested, the distinction made here between first- and second-order levels of uncertainty has not been considered before. In this paper previously existing measures are considered from the perspective of first- and second-order uncertainty and new measures are introduced. We conclude that the concepts of uncertainty and informativeness needs to be qualified if used in a second-order probability context and suggest that from a certain point of view information can not be minimized, just shifted from one level to another

Second-Order Weight Distributions

A fundamental property of codes, the second-order weight distribution, is proposed to solve the problems such as computing second moments of weight distributions of linear code ensembles. A series of results, parallel to those for weight distributions, is established for second-order weight distributions. In particular, an analogue of MacWilliams identities is proved. The second-order weight distributions of regular LDPC code ensembles are then computed. As easy consequences, the second moments of weight distributions of regular LDPC code ensembles are obtained. Furthermore, the application of second-order weight distributions in random coding approach is discussed. The second-order weight distributions of the ensembles generated by a so-called 2-good random generator or parity-check matrix are computed, where a 2-good random matrix is a kind of generalization of the uniformly distributed random matrix over a finite filed and is very useful for solving problems that involve pairwise or triple-wise properties of sequences. It is shown that the 2-good property is reflected in the second-order weight distribution, which thus plays a fundamental role in some well-known problems in coding theory and combinatorics. An example of linear intersecting codes is finally provided to illustrate this fact.

Expected Utility from Multinomial Second-order Probability Distributions

We consider the problem of maximizing expected utility when utilities and probabilities are given by discrete probability distributions so that expected utility is a discrete stochastic variable. As for discrete second-order distributions, that is probability distributions where the variables are themselves probabilities, the multinomial family is a reasonable choice at least if first-order probabilities are interpreted as relative frequencies. We suggest a decision rule that reflects the uncertainty present in distribution-based probabilities and utilities and we show an example of this rule in action with multinomial second-order distributions. Index Terms—Imprecise probability. second-order probability, discrete probability distributions, multinomial distributions, expected utilty.