Narrow Results

In this research, a novel family of learning rules called Beta Hebbian Learning (BHL) is thoroughly investigated to extract information from high-dimensional datasets by projecting the data onto low-dimensional (typically two dimensional) subspaces, improving the existing exploratory methods by providing a clear representation of data’s internal structure. BHL applies a family of learning rules derived from the Probability Density Function (PDF) of the residual based on the beta distribution. This family of rules may be called Hebbian in that all use a simple multiplication of the output of the neural network with some function of the residuals after feedback. The derived learning rules can be linked to an adaptive form of Exploratory Projection Pursuit and with artificial distributions, the networks perform as the theory suggests they should: the use of different learning rules derived from different PDFs allows the identification of “interesting” dimensions (as far from the Gaussian distribution as possible) in high-dimensional datasets. This novel algorithm, BHL, has been tested over seven artificial datasets to study the behavior of BHL parameters, and was later applied successfully over four real datasets, comparing its results, in terms of performance, with other well-known Exploratory and projection models such as Maximum Likelihood Hebbian Learning (MLHL), Locally-Linear Embedding (LLE), Curvilinear Component Analysis (CCA), Isomap and Neural Principal Component Analysis (Neural PCA).

The probability density distribution of the traffic density is analyzed based on the empirical data. It is found that the beta distribution can fit the result obtained from the measured traffic density perfectly. Then a modified traffic model is proposed to simulate the microscopic traffic flow, in which the probability density distribution of the traffic density is taken into account. The model also contains the behavior of drivers’ speed adaptation by taking into account the driving behavior difference and the dynamic headway. Accompanied by presenting the flux-density diagrams, the velocity evolution diagrams and the spatial-temporal profiles of vehicles are also given. The synchronized flow phase and the wide moving jam phase are indicated, which is the challenge for the cellular automata traffic model. Furthermore the phenomenon of the high speed car-following is exhibited, which has been observed in the measured data previously. The results set demonstrate the effectiveness of the proposed model in detecting the complicated dynamic phenomena of the traffic flow.

In this paper, the solutions of an extended form of the Fractional-order Neutron Point Kinetics (FNPK) equation in terms of Caputo-time derivatives of the same order are investigated. Instead of using a Caputo derivative, a distributed-order fractional derivative in the Caputo sense was employed in the term of the FNPK equation which is multiplied by the reactivity. This term plays an important role in the description of neutron kinetics during the start-up, shutdown, and steady-state processes in nuclear reactors. The extended (DFNPK) model was solved using the beta, normal, bimodal and Dirac delta distributions to investigate their effect on the transient state solutions of the neutron density. Regardless of the distribution used, the most significant finding is that a destabilizing effect on the neutron density is induced when the mode (or the instant of application of the Dirac delta) of the distribution tends to one while maintaining the orders of the Caputo-time derivatives constant. What defines the destabilizing effect are large magnitude oscillations, a rapid decay, and an oscillation-free steady state with a monotonic increase that is parallel to but somewhat above the trend determined by the FNPK equation. The extended model is anticipated to be effective for modeling neutron density dispersion in a highly heterogeneous medium that may be described using distributed derivatives.

Straightforward methods to evaluate risks arising from several sources are specially difficult when risk components are dependent and, even more if that dependence is strong in the tails. We give an explicit analytical expression for the probability distribution of the sum of non-negative losses that are tail-dependent. Our model allows dependence in the extremes of the marginal beta distributions. The proposed model is flexible in the choice of the parameters in the marginal distribution. The estimation using the method of moments is possible and the calculation of risk measures is easily done with a Monte Carlo approach. An illustration on data for insurance losses is presented.

We provide the exact expression of the reliability of a system under a Bayesian approach, using beta distributions as both native and induced priors at the system level, and allowing uncertainties in sampling, expressed under the form of misclassifications, or noises, that can affect the final posterior distribution. Exact 100(1-α)% highest posterior density credible intervals, for system reliability, are computed, and comparisons are made with results from approximate methods proposed in the literature.

In this paper, we show the Basel problem via the beta distributions, which include the free Poisson distribution and positive arcsine law. This is a generalization of Ref. 2 by Fujita. We also obtain special values of an extension of generalized Hurwitz–Lerch zeta function, which was introduced by Gang, Jain and Kalla.

The authors have previously studied multiplicative renormalization method (MRM) for generating functions of orthogonal polynomials. In particular, they have determined all MRM-applicable measures for renormalizing functions h(x) = e^x, h(x) = (1 - x)^-κ, κ = 1/2, 1, 2. For the cases h(x) = e^x and (1 - x)^-1, there are very large classes of MRM-applicable measures. For the other two cases κ = 1/2, 2, MRM-applicable measures belong to special classes of a certain kind of beta distributions. In this paper, we determine all MRM-applicable measures for h(x) = (1 - x)^-κ with κ ≠ 0, 1, 1/2.

Duration and convexity are important measures in fixed-income portfolio management. In this paper, we analyze this measure of the bonds by applying the beta model. The general usefulness of the beta probability distribution enhances its applicability in a wide range of reliability analyses, especially in the theory and practice of reliability management. We estimate the beta density function of the duration/convexity. This estimate is based on two important and simple models of short rates, namely, Vasicek and CIR (Cox, Ingersoll, and Ross CIR). The models are described and then their sensitivity of the models with respect to changes in the parameters is studied. We generate the stochastic interest rate on the duration and convexity model. The main results show that the beta probability distribution can be applied to model each phase of the risk function. This distribution approved its effectiveness, simplicity and flexibility. In this paper, we are interested in providing a decision-making tool for the manager in order to minimize the portfolio risk. It is helpful to have a model that is reasonably simple and suitable to different maturity of bonds. Also, it is widely used by investors for choosing bond portfolio immunization through the investment strategy. The finding also shows that the probability of risk measured by the reliability function is to highlight the relationship between duration/convexity and different risk levels. With these new results, this paper offers several implications for investors and risk management purposes.

Let $X_1,\dots ,X_N$, $N>n,$ be independent random points in ${ℝ}^n$, distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension $n$ tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated.

This study presents the application of a new method for generating synthetic accelerograms based on statistical distributions for Fourier phase differences and Fourier amplitudes as functions of earthquake magnitude, hypocentral distance and site geology. Two important characteristics of the methodology are that it requires a small number of input parameters and that ground motion time histories can be simulated without any specific modulation function. Two areas with different tectonic patterns (North-Eastern and Central Italy) were selected for the application. The results of our analysis are reliable in the case of Central Italy because the data set is large and quite uniformly distributed, while for North-Eastern Italy our results should not be used for distances greater than 30 km.

Gene regulatory networks (GRNs) control the production of proteins in cells. It is well-known that this process is not deterministic. Numerous studies employed a non-deterministic transition structure to model these networks. However, it is not realistic to expect state-to-state transition probabilities to remain constant throughout an organism’s lifetime. In this work, we focus on modeling GRN state transition (edge) variability using an ever-changing set of propensities. We suspect that the source of this variation is due to internal noise at the molecular level and can be modeled by introducing additional stochasticity into GRN models. We employ a beta distribution, whose parameters are estimated to capture the pattern inherent in edge behavior with minimum error. Additionally, we develop a method for obtaining propensities from a pre-determined network.

Assume that $n$ mobile sensors are thrown uniformly and independently at random with the uniform distribution on the unit interval. We study the expected sum over all sensors $i$ from $1$ to $n,$ where the contribution of the $i^{\mathrm{\text{th}}}$ sensor is its displacement from the current location to the anchor equidistant point $t_i=\frac{i}{n}-\frac{1}{2n},$ raised to the $a^{\mathrm{\text{th}}}$ power, when $a$ is an odd natural number.

As a consequence, we derive the following asymptotic identity. Fix $a$ positive integer. Let $X_{i:n}$ denote the $i^{\mathrm{\text{th}}}$ order statistic from a random sample of size $n$ from the Uniform$(0,1)$ population. Then

The Bouc model describes the behavior of hysteretic materials using three-coupled differential equations. Under white noise excitation, the probability density of the state variables can be approximated using closure methods. Here, a new closure approach is presented that utilizes the beta distribution for hysteretic softening together with a nonlinear mapping using the inverse hyperbolic tangent. The beta distribution properly models the distribution of the force-proportional displacement, the required nonlinear relation between displacement, velocity, and force-proportional displacement is, however, not entirely captured for large excitation levels leading to deviations from the Monte-Carlo simulations.

We consider a financial contract that delivers a single cash flow given by the terminal value of a cumulative gains process. The problem of modelling such an asset and associated derivatives is important, for example, in the determination of optimal insurance claims reserve policies, and in the pricing of reinsurance contracts. In the insurance setting, aggregate claims play the role of cumulative gains, and the terminal cash flow represents the totality of the claims payable for the given accounting period. A similar example arises when we consider the accumulation of losses in a credit portfolio, and value a contract that pays an amount equal to the totality of the losses over a given time interval. An expression for the value process of such an asset is derived as follows. We fix a probability space, together with a pricing measure, and model the terminal cash flow by a random variable; next, we model the cumulative gains process by the product of the terminal cash flow and an independent gamma bridge; finally, we take the filtration to be that generated by the cumulative gains process. An explicit expression for the value process is obtained by taking the discounted expectation of the future cash flow, conditional on the relevant market information. The price of an Arrow Debreu security on the cumulative gains process is determined, and is used to obtain a closed-form expression for the price of a European-style option on the value of the asset at the given intermediate time. The results obtained make use of remarkable properties of the gamma bridge process, and are applicable to a wide variety of financial products based on cumulative gains processes such as aggregate claims, credit portfolio losses, defined benefit pension schemes, emissions and rainfall.

In this chapter, we introduce a variety of discrete probability distributions and continuous probability distributions that are commonly used in economics and finance. Examples of discrete probability distributions include Bernoulli, Binomial, Negative Binomial, Geometric and Poisson distributions. Examples of continuous probability distributions include Beta, Cauchy, Chi-square, Exponential, Gamma, generalized Gamma, normal, lognormal, Weibull, and uniform distributions. The properties of these distributions as well as their applications in economics and finance are discussed. We also show some important techniques of obtaining moments and MGF's for various probability distributions.

In this paper we proposed new estimators of parameters for a Naive Bayes Classifier based on Beta Distributions. Equations were obtained for these estimators using an EM-like algorithm and they provide numerical estimates for those parameters. Furthermore, two forms for that Naive Bayes Classifier were presented.