Standard Normal Vector Research Articles

Probabilistic Optimal Power Flow Computation Using Stratified Quadrature Rule

In this paper, the probabilistic optimal power flow (POPF) problem is modeled as a black box uncertainty quantification problem. If distribution functions of POPF inputs are unknown, only samples are available, the multivariate Gaussian mixture model is suggested to recover the joint cumulative distribution function of POPF inputs, and a probability selection method is introduced to relate POPF inputs to an independent standard normal vector. Then, moment matching equations are proposed to characterize the uncertainty effects of POPF inputs on outputs, whereby some properties of the POPF problem are revealed, and a stratified quadrature rule (SQR) is proposed for POPF computation by combining a Hadamard matrix based quadrature rule and a discrete sine transformation matrix based quadrature rule. This newly developed SQR has a computational burden linear with the number of POPF inputs, and performs well in estimating means and standard deviations POPF outputs. Finally, case studies are conducted on a modified IEEE 118-bus system to illustrate the accuracy and efficiency of the proposed method.

Second-order Stein: SURE for SURE and other applications in high-dimensional inference

Stein’s formula states that a random variable of the form z⊤f(z)−divf(z) is mean-zero for all functions f with integrable gradient. Here, divf is the divergence of the function f and z is a standard normal vector. This paper aims to propose a second-order Stein formula to characterize the variance of such random variables for all functions f(z) with square integrable gradient, and to demonstrate the usefulness of this second-order Stein formula in various applications. In the Gaussian sequence model, a remarkable consequence of Stein’s formula is Stein’s Unbiased Risk Estimate (SURE), an unbiased estimate of the mean squared risk for almost any given estimator μˆ of the unknown mean vector. A first application of the second-order Stein formula is an Unbiased Risk Estimate for SURE itself (SURE for SURE): an unbiased estimate providing information about the squared distance between SURE and the squared estimation error of μˆ. SURE for SURE has a simple form as a function of the data and is applicable to all μˆ with square integrable gradient, for example, the Lasso and the Elastic Net. In addition to SURE for SURE, the following statistical applications are developed: (1) upper bounds on the risk of SURE when the estimation target is the mean squared error; (2) confidence regions based on SURE and using the second-order Stein formula; (3) oracle inequalities satisfied by SURE-tuned estimates under a mild Lipschtiz assumption; (4) an upper bound on the variance of the size of the model selected by the Lasso, and more generally an upper bound on the variance of the empirical degrees-of-freedom of convex penalized estimators; (5) explicit expressions of SURE for SURE for the Lasso and the Elastic Net; (6) in the linear model, a general semiparametric scheme to de-bias a differentiable initial estimator for the statistical inference of a low-dimensional projection of the unknown regression coefficient vector, with a characterization of the variance after debiasing; and (7) an accuracy analysis of a Gaussian Monte Carlo scheme to approximate the divergence of functions f:Rn→Rn.

Absorption probabilities for Gaussian polytopes and regular spherical simplices

AbstractThe Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in $\mathbb{R}^d$ . We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point $x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\mathcal P_{n,d}$ contains the point $\sigma X$ , where $\sigma\geq 0$ is constant and X is a standard normal vector independent of $\mathcal P_{n,d}$ . As a by-product, we also compute the expected number of k-faces and the expected volume of $\mathcal P_{n,d}$ , thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function $\Phi(z)$ and its complex version $\Phi(iz)$ . The main tool used in the proofs is the conic version of the Crofton formula.

Sampling the Multivariate Standard Normal Distribution under a Weighted Sum Constraint

Statistical modeling techniques—and factor models in particular—are extensively used in practice, especially in the insurance and finance industry, where many risks have to be accounted for. In risk management applications, it might be important to analyze the situation when fixing the value of a weighted sum of factors, for example to a given quantile. In this work, we derive the (n−1)-dimensional distribution corresponding to a n-dimensional i.i.d. standard Normal vector Z=(Z1,Z2,…,Zn)′ subject to the weighted sum constraint w′Z=c, where w=(w1,w2,…,wn)′ and wi≠0. This law is proven to be a Normal distribution, whose mean vector μ and covariance matrix Σ are explicitly derived as a function of (w,c). The derivation of the density relies on the analytical inversion of a very specific positive definite matrix. We show that it does not correspond to naive sampling techniques one could think of. This result is then used to design algorithms for sampling Z under constraint that w′Z=c or w′Z≤c and is illustrated on two applications dealing with Value-at-Risk and Expected Shortfall.

Sharp MSE Bounds for Proximal Denoising

Denoising has to do with estimating a signal $$\mathbf {x}_0$$x0 from its noisy observations $$\mathbf {y}=\mathbf {x}_0+\mathbf {z}$$y=x0+z. In this paper, we focus on the structured denoising where the signal $$\mathbf {x}_0$$x0 possesses a certain structure and $$\mathbf {z}$$z has independent normally distributed entries with mean zero and variance $$\sigma ^2$$ź2. We employ a structure-inducing convex function $$f(\cdot )$$f(·) and solve $$\min _\mathbf {x}\{\frac{1}{2}\Vert \mathbf {y}-\mathbf {x}\Vert _2^2+\sigma {\lambda }f(\mathbf {x})\}$$minx{12źy-xź22+źźf(x)} to estimate $$\mathbf {x}_0$$x0, for some $$\lambda >0$$ź>0. Common choices for $$f(\cdot )$$f(·) include the $$\ell _1$$l1 norm for sparse vectors, the $$\ell _1-\ell _2$$l1-l2 norm for block-sparse signals and the nuclear norm for low-rank matrices. The metric we use to evaluate the performance of an estimate $$\mathbf {x}^*$$xź is the normalized mean-squared error $$\text {NMSE}(\sigma )=\frac{{\mathbb {E}}\Vert \mathbf {x}^*-\mathbf {x}_0\Vert _2^2}{\sigma ^2}$$NMSE(ź)=Eźxź-x0ź22ź2. We show that NMSE is maximized as $$\sigma \rightarrow 0$$źź0 and we find the exact worst-case NMSE, which has a simple geometric interpretation: the mean-squared distance of a standard normal vector to the $${\lambda }$$ź-scaled subdifferential $${\lambda }\partial f(\mathbf {x}_0)$$źźf(x0). When $${\lambda }$$ź is optimally tuned to minimize the worst-case NMSE, our results can be related to the constrained denoising problem $$\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-\mathbf {x}\Vert _2\}$$minf(x)≤f(x0){źy-xź2}. The paper also connects these results to the generalized LASSO problem, in which one solves $$\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-{\mathbf {A}}\mathbf {x}\Vert _2\}$$minf(x)≤f(x0){źy-Axź2} to estimate $$\mathbf {x}_0$$x0 from noisy linear observations $$\mathbf {y}={\mathbf {A}}\mathbf {x}_0+\mathbf {z}$$y=Ax0+z. We show that certain properties of the LASSO problem are closely related to the denoising problem. In particular, we characterize the normalized LASSO cost and show that it exhibits a phase transition as a function of number of observations. We also provide an order-optimal bound for the LASSO error in terms of the mean-squared distance. Our results are significant in two ways. First, we find a simple formula for the performance of a general convex estimator. Secondly, we establish a connection between the denoising and linear inverse problems.

Preconditioned Krylov Subspace Methods for Sampling Multivariate Gaussian Distributions

A common problem in statistics is to compute sample vectors from a multivariate Gaussian distribution with zero mean and a given covariance matrix $A$. A canonical approach to the problem is to compute vectors of the form $y = S z$, where $S$ is the Cholesky factor or square root of $A$, and $z$ is a standard normal vector. When $A$ is large, such an approach becomes computationally expensive. This paper considers preconditioned Krylov subspace methods to perform this task. The Lanczos process provides a means to approximate $A^{1/2} z$ for any vector $z$ from an $m$-dimensional Krylov subspace. The main contribution of this paper is to show how to enhance the convergence of the process via preconditioning. Both incomplete Cholesky preconditioners and approximate inverse preconditioners are discussed. It is argued that the latter class of preconditioners has an advantage in the context of sampling. Numerical tests, performed with stationary covariance matrices used to model Gaussian processes, illustrate the dramatic improvement in computation time that can result from preconditioning.

Distributed Wireless Sensor Network Localization Algorithm using Space Standard Normal Vector (SSNV)

—In this paper a low cost but effective localization algorithm for wireless sensor networks (WSNs) is introduced. From the proposed scheme unknown nodes only have to listen to three anchor nodes and establish a space standard normal vector (SSNV) to estimate the coordinates of them. Many localization algorithms for WSNs require the exits of extra components, such as a GPS, ultrasonic transceiver, and unidirectional antenna on sensor nodes. Conversely the proposed localization scheme is range-free which means there is no demanding of any extra devices for the sensors. In this scheme, by collecting information from the three nearest anchors a geometric plane surface is built whose standard normal vector is also obtained correspondingly. The estimated location of each unknown sensor is adjusted according to its relative normal vector in the space which belongs to the built plane surface. This paper compares the proposed scheme with the well-known DV-Hop mechanism, Centroid method and the OSSDL algorithm that we proposed before. Simulation results show that the proposed scheme outperforms the other three in localization accuracy, communication cost, and computational complexity. Analysis about the effects of different length of normal vector is also made.

Fast and effective integration of multiple overlapping range images

We present a simple and fast integration method, which can fuse registered multiple, overlapping range images effectively while preserving the detailed features. First, normal vectors of the nodes in each range image are estimated and the registered multiple overlapping range images are combined into one point set. Due to registration errors, the normal vectors of homogeneous nodes in the overlapping areas would differ from each other. Thus, the normal vectors of the nodes are corrected iteratively by weighted averaging their neighboring normal vectors. Second, the combined point set is partitioned into a number of small clusters using the k-means clustering algorithm based on user-defined space interval. The clusters subsequently are subdivided into smaller subclusters by using the standard normal vector deviation as a measure of cluster scatter. Finally, in the cluster, the local maxima model, obtained using the mean shift clustering algorithm, is employed to represent the cluster. The proposed method is fast because it does not need to detect iteratively the overlapping areas, which is usually time consuming. In addition, because of the noisy filtering property of the mean shift clustering algorithm, the novel method can produce a smooth and watertight point surface while preserving the detailed features.

Monitoring Multi-Attribute Processes Based on NORTA Inverse Transformed Vectors

Although multivariate statistical process control has been receiving a well-deserved attention in the literature, little work has been done to deal with multi-attribute processes. While by the NORTA algorithm one can generate an arbitrary multi-dimensional random vector by transforming a multi-dimensional standard normal vector, in this article, using inverse transformation method, we initially transform a multi-attribute random vector so that the marginal probability distributions associated with the transformed random variables are approximately normal. Then, we estimate the covariance matrix of the transformed vector via simulation. Finally, we apply the well-known T 2 control chart to the transformed vector. We use some simulation experiments to illustrate the proposed method and to compare its performance with that of the deleted-Y method. The results show that the proposed method works better than the deleted-Y method in terms of the out-of-control average run length criterion.

THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED ON THE n-SPHERE

There are many instances in the statistical literature in which inference is based on a normalized quadratic form in a standard normal vector, normalized by the squared length of that vector. Examples include both test statistics (the Durbin–Watson statistic) and estimators (serial correlation coefficients). Although much studied, no general closed-form expression for the density function of such a statistic is known. This paper gives general formulae for the density in each open interval between the characteristic roots of the matrix involved. Results are given for the case of distinct roots, which need not be assumed positive, and when the roots occur with multiplicities greater than one. Starting from a representation of the density as a surface integral over an (n − 2)-dimensional hyperplane, the density is expressed in terms of top-order zonal polynomials involving difference quotients of the characteristic roots of the matrix in the numerator quadratic form.

First-order concepts in system reliability

The reliability of structural systems is determined by use of first-order concepts. Probability estimates for the components are calculated by linearization of the boundary of the failure domain after transforming an arbitrary vector of uncertain variables into an independent, standard normal vector. A first-order solution to the multinormal integral occurring in the formulation for elementary series or parallel systems is given and a sequence of certain probability bounds on the union of failure events involving the failure probability of any parallel subsystem and the probability of the intersection of any two or three parallel subsystems is derived. The system failure event is represented by the minimal cut set of componental failure events. The accuracy of approximations is systematically investigated and shown to be sufficient for most practical purposes. The method is illustrated with a simple example.