Independent Normal Variables Research Articles

Deformed Normal Distributions

It is well known that the probability distribution of the product of two normal variables itself approaches a normal distribution if one of the factor variables has a low coefficient of variation. A related, but apparently somewhat neglected problem, is to find the conditional distribution of a factor variable when the product of two independent normal variables is known. In the present paper we indicate why this conditional distribution also tends towards a normal distribution under similar conditions, and demonstrate numerically that this is indeed the case. Power series expansions for the mean and variance of the conditional distribution are also presented, which hold some problems of convergence but nevertheless provide good approximations when one coefficient of variation is low. Finally, a simplified version is presented of an actual application in object tracking, yielding an approximation for the probability distribution of the distance out to an object when the relative azimuth is known. The power series expansions shown in this paper are most conveniently developed in a more abstract setting, yielding results about the more general notion of a deformed normal distribution.

On the Sum of Products of Independent Complex Normal Variables: Understanding the Fundamental SNR Gain Limit of MIMO-RIS

On the Sum of Products of Independent Complex Normal Variables: Understanding the Fundamental SNR Gain Limit of MIMO-RIS

Poisson PCA for matrix count data

We develop a dimension reduction framework for data consisting of matrices of counts. Our model is based on the assumption of existence of a small amount of independent normal latent variables that drive the dependency structure of the observed data, and can be seen as the exact discrete analogue of a contaminated low-rank matrix normal model. We derive estimators for the model parameters and establish their limiting normality. An extension of a recent proposal from the literature is used to estimate the latent dimension of the model. The method is shown to outperform both its vectorization-based competitors and matrix methods assuming the continuity of the data distribution in analysing simulated data and real world abundance data.

SIMULATION MODELING OF THE SYSTEM OF RELATIONS BETWEEN THE PARTICIPANTS OF THE TRANSPORT AND LOGISTICS PROCESS AT THE SEA CARGO TERMINAL

In order to analyze the effectiveness of interaction of the cargo terminal operator with customs authorities and participants in foreign economic activity, a four-sector model of the network diagram of the interaction stages between participants in the transport and logistics process at the sea cargo terminal is presented in the paper; the model allows visualizing and controlling the process of aggregated relationships during cargo operations and the provision of customs services in the area of seaports activity. Calculations of works time parameters are performed on the basis of a four-sector model, in which the event is divided into four sectors, where the number of the event is indicated on the top, the early start of work is indicated on the left, the late end of work is marked in the right sector, and the number of the initial event of the previous work is indicated in the lower sector, according to which the critical path passes. At the same time, simulation modeling of the system of relations between the participants in the transport and logistics process at the sea cargo terminal is carried out. When modeling by the Monte Carlo method, a set of parametric values presented in the form of a probability distribution is used. It is noted that the operations performed in the simulation are subject to the normal distribution law, since a large number of participants are involved in the transport and logistics system. The results of the algorithm operation are presented in the form of a data array, which is a sequence of works performed during the interaction of participants in the transport and logistics process at the sea cargo terminal. One of the methods for modeling randomly distributed variables is the Box-Muller generation method. The use of the Box-Muller method in the work makes it possible to apply the fundamental law of probability transformation to obtain two independent normal variables that have a standard normal distribution. Thus, the chosen method gives a set of random numbers that have a normal distribution with zero mean and unit standard deviation. The results of simulation modeling allow us to conclude that the aggregated system of relationship between participants in the transport and logistics process, which uses the technology of preliminary declaration in order to make a decision on the early placement of goods in the inspection zone, is effective.

A copula-based uncertainty propagation method for structures with correlated parametric p-boxes

In the response analysis of uncertain structural models with limited information, probability-boxes can be effectively employed to address the aleatory and epistemic uncertainty together. This paper presents a copula-based uncertainty propagation method which can accurately perform uncertainty propagation analysis with correlated parametric probability-boxes. Firstly, the parameter estimation and Akaike information criterion analysis are utilized to select an optimal copula based on available samples, by which the joint cumulative distribution function is constructed for the correlated input variables. Then, using the obtained joint cumulative distribution function, the correlated parametric probability-boxes are transformed into independent normal variables, and an efficient method based on sparse grid numerical integration is proposed to calculate the bounds on statistical moments of a response function. Finally, numerical examples and an engineering application are provided to verify the effectiveness of the presented method.

Traces of powers of matrices over finite fields

Let $M$ be a random matrix chosen according to Haar measure from the unitary group $\mathrm{U}(n,\mathbb{C})$. Diaconis and Shahshahani proved that the traces of $M,M^2,\ldots,M^k$ converge in distribution to independent normal variables as $n \to \infty$, and Johansson proved that the rate of convergence is superexponential in $n$. We prove a finite field analogue of these results. Fixing a prime power $q = p^r$, we choose a matrix $M$ uniformly from the finite unitary group $\mathrm{U}(n,q)\subseteq \mathrm{GL}(n,q^2)$ and show that the traces of $\{ M^i \}_{1 \le i \le k,\, p \nmid i}$ converge to independent uniform variables in $\mathbb{F}_{q^2}$ as $n \to \infty$. Moreover we show the rate of convergence is exponential in $n^2$. We also consider the closely related problem of the rate at which characteristic polynomial of $M$ equidistributes in `short intervals' of $\mathbb{F}_{q^2}[T]$. Analogous results are also proved for the general linear, special linear, symplectic and orthogonal groups over a finite field. In the two latter families we restrict to odd characteristic. The proofs depend upon applying techniques from analytic number theory over function fields to formulas due to Fulman and others for the probability that the characteristic polynomial of a random matrix equals a given polynomial.

A Time-Variant Reliability Analysis Method Based on the Stochastic Process Discretization under Random and Interval Variables

In practical engineering, it is a cost-consuming problem to consider the time-variant reliability of both random variables and interval variables, which usually requires a lot of calculation. Therefore, a time-variant reliability analysis approach with hybrid uncertain variables is proposed in this paper. In the design period, the stochastic process is discretized into random variables. Simultaneously, the original random variables and the discrete random variables are converted into independent normal variables, and the interval variables are changed into standard variables. Then it is transformed into a hybrid reliability problem of static series system. At different times, the limited state functions are linearized at the most probable point (MPP) and at the most unfavorable point (MUP). The transformed static system reliability problem with hybrid uncertain variables can be solved effectively by introducing random variables. To solve the double-loop nested optimization in the hybrid reliability calculation, an effective iterative method is proposed. Two numerical examples and an engineering example demonstrate the validity of the present approach.

Uncertainty propagation method for probabilistic fatigue crack growth life prediction

Predicting the probability distribution of fatigue crack growth (FCG) life is very important for probabilistic damage tolerance analysis and structural integrity assessment. This paper proposes a novel uncertainty propagation method for probabilistic FCG life prediction, which allows efficiently estimating the probability density function (PDF), and cumulative distribution function (CDF) of the FCG life considering the randomness of the loads, material properties of crack propagation rate, and crack geometries. First, the relationship between the PDF of FCG life and the one of its logarithm is established by a logarithmic analysis. Then, the random variables are uniformly transformed into independent normal variables that obey distributions of N(0,1/2) using variable transformation technique. Thus, the bivariate dimension reduction method can be conveniently applied to calculate the first four moments of the logarithmic FCG life, and the corresponding PDF is further estimated by the maximum entropy principle. Finally, the PDF and CDF of the FCG life are analytically derived from the ones of the logarithmic FCG life. The validity of the proposed method is verified by numerical, experimental and engineering examples. It is demonstrated that the PDF and CDF of FCG life are well predicted by the proposed method, regardless of the strong nonlinearity of FCG life function and the large uncertainties of input random variables. This method outperforms the Monte Carlo simulation (MCS) in terms of computational efficiency, and in the meanwhile it has an acceptable computational accuracy.

Stochastic Data Generation Technique Using Autoregressive Moving Average (ARMA) Model

Data generation mechanisms have been widely applied in hydrology. Models are built for generation of data having the statistical properties of the historical records. The creation of synthetic time series starts with the generation of independent normal variables with average zero and variance one, then adding the time and spatial dependence structure as well as periodic components, whichever necessary. The data generation can be accomplished via the analysis of the historical data to check its suitability for generation, Selection, identification of the form, estimation of parameters, & check of the data generation model and Application of the model & testing of the results. This paper summarizes the required work to be done as per the above steps taking the autoregressive moving average as an example of the data generation model Keywords: Induced Voltage, – Electric Fields, HVDC Transmission, Finite Element Method, Hybrid Transmission Lines. DOI : 10.7176/ISDE/10-1-02

SU(p,q) coherent states and a Gaussian de Finetti theorem

We prove a generalization of the quantum de Finetti theorem when the local space is an infinite-dimensional Fock space. In particular, instead of considering the action of the permutation group on n copies of that space, we consider the action of the unitary group U(n) on the creation operators of the n modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in U(n). Our first result is a complete characterization of this subspace, which turns out to be spanned by a family of generalized coherent states related to the special unitary group SU(p, q) of signature (p, q). More precisely, this construction yields a unitary representation of the noncompact simple real Lie group SU(p, q). We therefore find a dual unitary representation of the pair of groups U(n) and SU(p, q) on an n(p + q)-mode Fock space. The (Gaussian) SU(p, q) coherent states resolve the identity on the symmetric subspace, which implies a Gaussian de Finetti theorem stating that tracing over a few modes of a unitary-invariant state yields a state close to a mixture of Gaussian states. As an application of this de Finetti theorem, we show that the n × n upper-left submatrix of an n × n Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if n3 = O(m).

Design of Exponentially Weighted Moving Average Scheme for Standardized Means

Exponentially Weighted Moving Average (EWMA) scheme in monitoring process mean is favored because it remembers the past information and detects the small shifts in the mean of a sequence of independent normal variates. However the design procedure of the EWMA scheme was complex till Crowder (1989) presents a simplified scheme for monitoring process mean. Currently standardizing the variables and monitoring them jointly is widely practiced and many schemes are presented under joint quality monitoring. Therefore a need arises to design EWMA schemes for monitoring standardized process means which will easier the work of monitoring more than one process variable at a time. In this paper an alternative design of such a scheme is presented. The new scheme is easy to apply and is independent of sample size. The scheme parameters are the same for any process if the in-control average run length (ARL) and the shifts in means to be detected are the same. In industrial application this property is more beneficial for monitoring few variables in one display. DOI: http://dx.doi.org/10.4038/sljastats.v14i2.6221

An Approach to Distribution of the Product of Two Normal Variables

The distribution of product of two normally distributed variables come from the first part of the XX Century. First works about this issue were [1] and [2] showed that under certain conditions the product could be considered as a normally distributed. A more recent approach is [3] that studied approximation to density function of the product using three methods: numerical integration, Monte Carlo simulation and analytical approximation to the result using the normal distribution. They showed as the inverse variation coefficient μ σ increases, the distribution of the product of two independent normal variables tends towards a normal distribution. Our study is focused in Ware and Lad approaches. The objective was studying which factors have more influence in the presence of normality for the product of two independent normal variables. We have considered two factors: the inverse of the variation coefficient value μ σ and the combined ratio (product of the two means divided by standard deviation): μ1μ2 σ for two normal variables with the same variance. Our results showed that for low values of the inverse of the variation coefficient (less than 1) normal distribution is not a good approximation for the product. Another one, influence of the combined ratio value is less than influence of the inverse of coefficients of variation value. 88 A. Seijas-Macias and A. Oliveira

Marginal Distributions of Random Vectors Generated by Affine Transformations of Independent Two-Piece Normal Variables

Marginal probability density and cumulative distribution functions are presented for multidimensional variables defined by nonsingular affine transformations of vectors of independent two-piece normal variables, the most important subclass of Ferreira and Steel's general multivariate skewed distributions. The marginal functions are obtained by first expressing the joint density as a mixture of Arellano-Valle and Azzalini's unified skew-normal densities and then using the property of closure under marginalization of the latter class.

Hybrid regions for post-change mean in a sequence of normal variables

The hybrid bootstrap uses resampling ideas to extend the duality approach to the interval estimation for a parameter of interest when there are nuisance parameters. The confidence region constructed by the hybrid bootstrap may perform much better than the ordinary bootstrap region in a situation where the data provide substantial information about the nuisance parameter, but limited information about the parameter of interest. We apply this method to estimate the post-change mean after a change is detected by a stopping procedure in a sequence of independent normal variables. Since distribution theory in change point problems is generally a challenge, we use bootstrap simulation to find empirical distributions of test statistics and calculate critical thresholds. Both likelihood ratio and Bayesian test statistics are considered to set confidence regions for post-change means in the normal model. In the simulation studies, the performance of hybrid regions are compared with that of ordinary bootstrap regions in terms of the widths and coverage probabilities of confidence intervals.

On the uncertainty and risks of macroeconomic forecasts: combining judgements with sample and model information

Institutions which publish macroeconomic forecasts usually do not rely on a single econometric model to generate their forecasts. The combination of judgements with information from different models complicates the problem of characterizing the predictive density. This article proposes a parametric approach to construct the joint and marginal densities of macroeconomic forecasting errors, combining judgements with sample and model information. We assume that the relevant variables are linear combinations of latent independent two-piece normal variables. The baseline point forecasts are interpreted as the mode of the joint distribution, which has the convenient feature of being invariant to judgments on the balance of risks.

A simple generalization of the Box–Muller method for obtaining a pair of correlated standard normal variables

In the simple and widely used method of Box–Muller [G. Box and M. Muller, A note on the generation of random normal deviates, Ann. Math. Statist. 29 (1958), pp. 610–611], from a pair of uniform and independent random variables in (0,1), a pair of standard and independent normal variables is obtained. In this article, we present a very simple and elegant generalization of this method to obtain a pair of correlated standard normal variables with a given coefficient of correlation. This generalized method, which is computationally very easy, is interpreted in geometric terms, considering a translation of the uniform interval (0,1) and a rotation of a defined angle, both related to the coefficient of correlation. Some numerical results are simulated and statistically analysed, proving that the generalization is extremely simple and powerful.

Computing Bounds on the Expected Maximum of Correlated Normal Variables

We compute upper and lower bounds on the expected maximum of correlated normal variables (up to a few hundred in number) with arbitrary means, variances, and correlations. Two types of bounding processes are used: perfectly dependent normal variables, and independent normal variables, both with arbitrary mean values. The expected maximum for the perfectly dependent variables can be evaluated in closed form; for the independent variables, a single numerical integration is required. Higher moments are also available. We use mathematical programming to find parameters for the processes, so they will give bounds on the expected maximum, rather than approximations of unknown accuracy. Our original application is to the maximum number of people on-line simultaneously during the day in an infinite-server queue with a time-varying arrival rate. The upper and lower bounds are tighter than previous bounds, and in many of our examples are within 5% or 10% of each other. We also demonstrate the bounds’ performance on some PERT models, AR/MA time series, Brownian motion, and product-form correlation matrices.

Dynamics of Mandelbrot cascades

Mandelbrot multiplicative cascades provide a construction of a dynamical system on a set of probability measures defined by inequalities on moments. To be more specific, beyond the first iteration, the trajectories take values in the set of fixed points of smoothing transformations (i.e., some generalized stable laws). Studying this system leads to a central limit theorem and to its functional version. The limit Gaussian process can also be obtained as limit of an “additive cascade” of independent normal variables.

A general dimension reduction technique for derivative pricing

For a trajectory simulated from s standardized independent normal variates ε = (ε1, . . . , εs )1, the payoff of a European option can be represented as max[g(ε), 0], where the function g(ε) is assumed to be differentiable and it relates to the nature of the derivative securities. In this paper, we develop a new simulation technique by introducing an orthogonal class of transformation to ε so that the function g is instead generated from g(Aε), where A is an s-dimensional orthogonal matrix. The matrix A is optimally determined so that the effective dimension of the underlying function is minimized, thereby enhancing the quasi-Monte Carlo (QMC) method. The proposed simulation approach has the advantage of greater generality and has a wide range of applications as long as the problem of interest can be represented by g(ε). The flexibility of our proposed technique is illustrated by applying it to two high-dimensional applications: Asian basket options and European call options with a stochastic volatility model. We benchmark our proposed method against well-known efficient simulation algorithms that have been advocated in these applications. The numerical results demonstrate that our proposed technique can be an extremely powerful simulation method when combined with QMC.

Comparison of two common estimators of the ratio of the means of independent normal variables in agricultural research

This paper addresses the problem of estimating the ratio of the means of independent normal variables in agricultural research. The first part of the research examines the distributional properties of the ratio of independent normal variables, both theoretically and using simulation. The second part of the research evaluates the relative merits of two common estimators of the ratio of the means of independent normal variables in agricultural research, an arithmetic average and a weighted average, via simulation experiments using normal distributions. The results are then tested using research data from rice breeding multi-environment trials in Jilin Province, China, in 1994. These data are used to demonstrate the diagnostic approach developed for assessing the “safe” use of the arithmetic and the weighted average methods for estimating the ratio of the means of independent normal variables.