Gaussian Random Variables Research Articles

Asymptotic approximations for the distribution of the product of correlated normal random variables

We obtain asymptotic approximations for the probability density function of the product of two correlated normal random variables with non-zero means and arbitrary variances. As a consequence, we deduce asymptotic approximations for the tail probabilities and quantile functions of this distribution, as well as an asymptotic approximation for the widely used risk measures value at risk and tail value at risk.

Improved Variance Estimation From Trimmed Samples

ABSTRACTSeveral methods for the robust estimation of the variance of a normal random variable are based on trimming. For example, the popular estimator is based on trimming the pairs of samples with the largest absolute distance. Another example is least trimmed squares regression, where all unusually large residuals are removed before the estimation of the variance. Here, in this work, we propose two new classes of estimators that use an optimal linear combination of the trimmed samples to achieve a lower mean squared error (MSE). The first class guarantees the smallest variance among all unbiased estimators that are based on linear combinations, while the second class of estimators guarantees the smallest MSE at the cost of some small bias. Through simulations, we demonstrate that our proposed estimators can have considerably smaller MSE than other robust estimators both in the presence and absence of outliers.

Context-Based Meta-Reinforcement Learning With Bayesian Nonparametric Models.

Deep reinforcement learning agents usually need to collect a large number of interactions to solve a single task. In contrast, meta-reinforcement learning (meta-RL) aims to quickly adapt to new tasks using a small amount of experience by leveraging the knowledge from training on a set of similar tasks. State-of-the-art context-based meta-RL algorithms use the context to encode the task information and train a policy conditioned on the inferred latent task encoding. However, most recent works are limited to parametric tasks, where a handful of variables control the full variation in the task distribution, and also failed to work in non-stationary environments due to the few-shot adaptation setting. To address those limitations, we propose MEta-reinforcement Learning with Task Self-discovery (MELTS), which adaptively learns qualitatively different nonparametric tasks and adapts to new tasks in a zero-shot manner. We introduce a novel deep clustering framework (DPMM-VAE) based on an infinite mixture of Gaussians, which combines the Dirichlet process mixture model (DPMM) and the variational autoencoder (VAE), to simultaneously learn task representations and cluster the tasks in a self-adaptive way. Integrating DPMM-VAE into MELTS enables it to adaptively discover the multi-modal structure of the nonparametric task distribution, which previous methods using isotropic Gaussian random variables cannot model. In addition, we propose a zero-shot adaptation mechanism and a recurrence-based context encoding strategy to improve the data efficiency and make our algorithm applicable in non-stationary environments. On various continuous control tasks with both parametric and nonparametric variations, our algorithm produces a more structured and self-adaptive task latent space and also achieves superior sample efficiency and asymptotic performance compared with state-of-the-art meta-RL algorithms.

Stochastic stability of random stacking of blocks

The paper explores the stability of a tower obtained by stacking identical rectangular blocks on top of each other with an inherent randomness due to slight positional offsets between successive blocks. With probabilistic modeling techniques, the diffusive behavior of the stacking process is studied and the collapse is seen as a first passage time problem. In the considered model, alignment errors are idealized as independent Gaussian random variables with zero mean and given standard deviation. We derive expressions for the joint probability density functions of block positions, and analyze their correlation. The study extends to the stochastic stability of a stack of given height, by exploring the statistical characteristics of the center of gravity of the part of tower above each block. Eventually, the probabilistic analysis of collapse is developed to quantify the statistics of the number of blocks that can be heaped up before the tower topples. Although this problem may initially appear playful, it offers an illustrated introduction to first passage problems on a non homogenous process. From a practical standpoint, this analysis offers a simple understanding of the influence of alignment errors on the overall stability of a stack, which may find several fields of application.

Statistical distribution of surface elevation by using quadratic forms of normal random variables

This paper is concerned with the statistical properties of surface elevation, which is crucial for the design and operation of marine and coastal structures and is helpful to the prediction of rogue waves. A semi-analytic quadratic model is proposed to describe the statistical distribution of surface elevation by using the theory of quadratic forms of normal random variables, in which the characteristic function and cumulants are given in analytical form, and the probability density is obtained by the numerical inversion of the characteristic function using the fast Fourier transform algorithm. Compared with Monte Carlo simulations, the quadratic model presents excellent performance in describing the probability density function of surface elevation, especially at the tails, for varying degrees of steepness, bandwidth, and directional spreading in both finite and infinite water depth. The effect of these parameters that characterize sea states on the statistical properties of surface elevation is also investigated. Through a comparative study, the quadratic model is superior to previously existing theoretical models.

A Stein characterisation of the distribution of the product of correlated normal random variables

We obtain a Stein characterisation of the distribution of the product of two correlated normal random variables with non-zero means, and more generally the distribution of the sum of independent copies of such random variables. Our Stein characterisation is shown to naturally generalise a number of other Stein characterisations in the literature. From our Stein characterisation we derive recursive formulas for the moments of the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, which allows for efficient computation of higher order moments.

Capacity analysis over fractional order Rayleigh fading channel under additive white generalized Gaussian noise

AbstractThis study presents an innovative fractional order Rayleigh fading model that can be used for channel capacity estimation in the presence of additive white generalized Gaussian noise. The proposed model assumes that the real and imaginary parts of channel gains are generalized Gaussian random variables, which makes it possible to consider the traditional Rayleigh fading model as a special case of fractional order Rayleigh fading. Compared to the Rayleigh model, the fractional order Rayleigh fading model offers a more precise representation of new real‐world communication, such as integrating terrestrial and underwater networks in sixth‐generation communications channels. The probability density function of the channel gain with additive white generalized Gaussian noise is analyzed here. Furthermore, the ergodic and outage capacities of the channel are determined, taking into account the assumption that the channel state information is only available at the receiver. The ergodic capacity is calculated using Meijer's G‐functions, resulting in a closed‐form expression. Numerical simulations demonstrate the superiority of the fractional order Rayleigh fading model over the Rayleigh channel. Moreover, the impact of ergodic and outage capacities under diverse channel characteristics is assessed.

Absolute moments of the variance-gamma distribution

We obtain exact formulas for the absolute raw and central moments of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. When the skewness parameter is equal to zero (the symmetric variance-gamma distribution), the infinite series reduces to a single term. Moreover, for the case that the shape parameter is a half-integer (in our parameterisation of the variance-gamma distribution), we obtain a closed-form expression for the absolute moments in terms of confluent hypergeometric functions. As a consequence, we deduce new exact formulas for the absolute raw and central moments of the asymmetric Laplace distribution and the product of two correlated zero mean normal random variables, and more generally the sum of independent copies of such random variables.

Chance-constrained programming for drinking water quality management under fuzziness and randomness

ABSTRACT In this paper, an inexact fuzzy random chance-constrained programming (IFRCCP) model was proposed to deal with the randomness and fuzziness in the optimization of booster cost for the water distribution system (WDS). The IFRCCP model was applied to a WDS to analyze the effects of randomness and fuzziness on the booster costs. The booster cost intervals were obtained under various violation levels and confidence levels. The results indicated that the lower and upper booster costs increased with the confidence levels of the lower limits and decreased with the violation levels of the lower limits for both normal and log-normal distribution fuzzy random variables (FRVs). Moreover, the increased magnitude of lower and upper booster costs under normal distribution FRV is a little less than that under log-normal distribution FRV, while the decreased magnitude of lower and upper booster costs under normal distribution FRV is larger than that under log-normal distribution FRV. The same regulation can be found for the FVR with both triangular and trapezoidal distribution FRVs. With the increase of booster number, the increased magnitude of the lower and upper booster costs decreased for both normal and log-normal distribution FRVs.

Importance sampling for counting statistics in one-dimensional systems.

In this paper, we consider the problem of numerical investigation of the counting statistics for a class of one-dimensional systems. Importance sampling, the cornerstone technique usually implemented for such problems, critically hinges on selecting an appropriate biased distribution. While an exponential tilt in the observable stands as the conventional choice for various problems, its efficiency in the context of counting statistics may be significantly hindered by the genuine discreteness of the observable. To address this challenge, we propose an alternative strategy, which we call importance sampling with the local tilt. We demonstrate the efficiency of the proposed approach through the analysis of three prototypical examples: a set of independent Gaussian random variables, Dyson gas, and symmetric simple exclusion process with a steplike initial condition.

Combination of dependent and partially reliable Gaussian random fuzzy numbers

Gaussian random fuzzy numbers are random fuzzy sets generalizing Gaussian random variables and possibility distributions. They define belief functions on the real line that can be conveniently combined by the product-intersection rule under the independence assumption. In this paper, we introduce various extensions of this rule to account for dependence and partial reliability of the pieces of evidence. We first provide formulas for the combination of an arbitrary number of Gaussian random fuzzy numbers whose dependence is described by a correlation matrix, and we introduce a minimum-conflict combination operation. To account for partially reliable evidence, we then introduce two discounting operations called possibilistic and evidential discounting, as well as several combination operators based on different assumptions, each one parameterized by a correlation matrix and a vector of discounting coefficients. We demonstrate the application of these operators to the combination of predictions with different sets of inputs in machine learning, and show that performance can be enhanced by optimizing the parameters of the combination operators.

Causalized Convergent Cross Mapping and Its Implementation in Causality Analysis.

Rooted in dynamic systems theory, convergent cross mapping (CCM) has attracted increased attention recently due to its capability in detecting linear and nonlinear causal coupling in both random and deterministic settings. One limitation with CCM is that it uses both past and future values to predict the current value, which is inconsistent with the widely accepted definition of causality, where it is assumed that the future values of one process cannot influence the past of another. To overcome this obstacle, in our previous research, we introduced the concept of causalized convergent cross mapping (cCCM), where future values are no longer used to predict the current value. In this paper, we focus on the implementation of cCCM in causality analysis. More specifically, we demonstrate the effectiveness of cCCM in identifying both linear and nonlinear causal coupling in various settings through a large number of examples, including Gaussian random variables with additive noise, sinusoidal waveforms, autoregressive models, stochastic processes with a dominant spectral component embedded in noise, deterministic chaotic maps, and systems with memory, as well as experimental fMRI data. In particular, we analyze the impact of shadow manifold construction on the performance of cCCM and provide detailed guidelines on how to configure the key parameters of cCCM in different applications. Overall, our analysis indicates that cCCM is a promising and easy-to-implement tool for causality analysis in a wide spectrum of applications.

The Measurement Errors and Their Effects on the Cumulative Sum Schemes for Monitoring the Ratio of Two Correlated Normal Variables

Monitoring the ratio of two correlated normal random variables is often used in many industrial manufacturing processes. At the same time, measurement errors inevitably exist in most processes, which have different effects on the performance of various charting schemes. This paper comprehensively analyses the impacts of measurement errors on the detection ability of the cumulative sum (CUSUM) charting schemes for the ratio of two correlated normal variables. A thorough numerical assessment is performed using the Monte Carlo simulation, and the results indicate that the measurement errors negatively impact the performance of the CUSUM scheme for the ratio of two correlated normal variables. Increasing the number of measurements per set is not a lucrative approach for minimizing the negative impact of measurement errors on the performance of the CUSUM charting scheme when monitoring the ratio of two correlated normal variables. We consider a food formulation as an example that illustrates the quality control problems involving the ratio of two correlated normal variables in an industry with a measurement error. The results are presented, along with some suggestions for further study.

Gauge Field Marginal of an Abelian Higgs Model

We study the gauge field marginal of an Abelian Higgs model with Villain action defined on a 2D lattice in finite volume. Our first main result, which holds for gauge theories on arbitrary finite graphs and does not assume that the structure group is Abelian, is a loop expansion of the Radon–Nikodym derivative of the law of the gauge field marginal with respect to that of the pure gauge theory. This expansion is similar to the one of Seiler (Gauge theories as a problem of constructive quantum field theory and statistical mechanics, volume 159 of lecture notes in physics, Springer, Berlin, p v+192. https://doi.org/10.1007/3-540-11559-5, 1982) but holds in greater generality and uses a different graph theoretic approach. Furthermore, we show ultraviolet stability for the gauge field marginal of the model in a fixed gauge. More specifically, we show that moments of the Hölder–Besov-type norms introduced in Chevyrev (Commun Math Phys 372(3):1027–1058. https://doi.org/10.1007/s00220-019-03567-5, 2019) are bounded uniformly in the lattice spacing. This latter result relies on a quantitative diamagnetic inequality that in turn follows from the loop expansion and elementary properties of Gaussian random variables.

A fast algorithm for computing product moments of multivariate normal random variables

We provide a simple identity that decomposes a product moment of multivariate normal random variables as a sum of various products of univariate moments of one of the random variables and multivariate moments of the other random variables. The new identity allows for much faster computation of the product moments of multivariate normal random variables than existing methods.

The distribution of the product of independent variance-gamma random variables

Let X and Y be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the product XY is derived. Some basic distributional properties are also derived, including formulas for the cumulative distribution function and the characteristic function, as well as asymptotic approximations for the density, tail probabilities and the quantile function. As special cases, we deduce some key distributional properties for the product of two independent asymmetric Laplace random variables as well as the product of four jointly correlated zero mean normal random variables with a particular block diagonal covariance matrix. As a by-product of our analysis, we deduce some new reduction formulas for the Meijer G-function.

Portfolio optimization with mental accounts under uncertain random environment and butterfly optimization algorithm with adaptive strategies

In complex security markets, uncertainty and randomness may coexist. The investors hold different risk attitudes toward different investment objectives in reality. In order to reflect this phenomenon, this paper applies mental accounts to portfolio optimization in an uncertain random environment. Firstly, considering the influence of transaction costs and diversification degree, this paper establishes an uncertain random mean-absolute semi-deviation-entropy bi-objective optimization model. Then the bi-objective model is converted to two single-objective models through three steps. Secondly, the equivalent forms of the models are deduced when the return rates of random risky securities are assumed to be normal random variables and the return rates of uncertain risky securities are assumed to be linear and zigzag uncertain variables. Furthermore, we propose an improved butterfly optimization algorithm (IBOA) to solve the two single-objective models. Finally, numerical simulations are presented to analyze the practicability and effectiveness of the models with different mental accounts and the IBOA algorithm. The results indicate that the IBOA algorithm is effective and putting money into more mental accounts may gain higher returns.

Evaluation of different DEWMA schemes for monitoring the ratio of two normal random variables

In many fields such as baking industry and cosmetic factory, the ratio of two normal random variables (RZ) is one of the key quality characteristics that needs to be monitored. In this paper, we study the properties of three double exponentially weighted moving average (DEWMA) schemes for monitoring the RZ. The run length (RL) properties, average run length (ARL) and standard deviation of run length (SDRL), and charting parameters of the proposed DEWMA RZ schemes are simulated for different correlation parameter between the two normal variables. These numerous simulation results are presented in tables and figures to make comparisons with the existing exponentially weighted moving average (EWMA) RZ scheme. The comparisons show that the proposed DEWMA RZ schemes are more effective than the EWMA RZ in detecting small changes. Additionally, the proposed three one-sided DEWMA RZ schemes’ properties are compared in detail. A food industry example details the implementation of the proposed RZ schemes.

Gaussian Mutation Based Teaching-Learning Optimization for reactive power dispatch

This paper develops the Gaussian Mutation Based Teaching-Learning Optimization (GMBTLO) to solve different reactive power dispatch problems. In GMBTLO, Gaussian random variables are instituted, which includes both the ‘Teacher phase’ as well as ‘Learner phase’ helps to improve the efficiency of searching. Simultaneously, it ensures more feasibility to obtain the global optimal. Gaussian random variables are presented in GMBTLO, where both the ‘Teacher phase’ as well as ‘Learner phase’ is considered. This advances searching competence and guarantee a great possibility of achieving global optimum without affecting the convergence speed. In this paper, GMBTLO is implemented to enhance solution quality as well as convergence speed. Developed GMBTLO is exploited for obtaining the control variables settings including terminal voltages of a generator, transformer taps, as well as output power of shunt VAR compensators (reactive) for achieving a reduced loss in real power transmission and enhanced voltage profile. GMBTLO is validated by taking IEEE 30-bus, 57-bus along with 118-bus test systems. Also, test outcomes are matched with the results already acquired from other implemented evolutionary techniques. It is concluded that the suggested GMBTLO extends a better solution.

Multiplier subsample bootstrap for statistics of time series

Block-based bootstrap, block-based subsampling and multiplier bootstrap are three common nonparametric tools for statistical inference under dependent observations. Combining the ideas of those three, a novel resampling approach, the multiplier subsample bootstrap (MSB), is proposed. Instead of generating a resample from the observations, the MSB imitates the statistic by weighting the block-based subsample statistics with independent standard Gaussian random variables. Given the asymptotic normality of the statistic, the bootstrap validity is established under some mild moment conditions. Involving the idea of MSB, the other resampling approach, the hybrid multiplier subsampling periodogram bootstrap (HMP), is developed for mimicking frequency-domain spectral mean statistics in the paper. A simulation study demonstrates that both the MSB and HMP achieve good performance.