Recursive Calculation Model for a Special Multivariate Normal Probability of First-Order Stationary Sequence

2020 ◽  
Vol 32 (1) ◽  
pp. 164-171
Author(s):  
Jietao Xie ◽  
Juan Wu
Keyword(s):  
Calculation Model ◽  
Stationary Sequence ◽  
Multivariate Normal ◽  
First Order ◽  
Normal Probability ◽  
Recursive Calculation

Error bounds on multivariate Normal approximations for word count statistics

2002 ◽  
Vol 34 (03) ◽  
pp. 559-586 ◽  
Author(s):  
Haiyan Huang
Keyword(s):  
Covariance Matrix ◽  
Error Bounds ◽  
Multivariate Distribution ◽  
Finite Alphabet ◽  
Multivariate Normal ◽  
Word Count ◽  
First Order ◽  
Normal Approximations ◽  
Necessary And Sufficient ◽  
The Given

Given a sequence S and a collection Ω of d words, it is of interest in many applications to characterize the multivariate distribution of the vector of counts U = (N(S,w 1), …, N(S,w d )), where N(S,w) is the number of times a word w ∈ Ω appears in the sequence S. We obtain an explicit bound on the error made when approximating the multivariate distribution of U by the normal distribution, when the underlying sequence is i.i.d. or first-order stationary Markov over a finite alphabet. When the limiting covariance matrix of U is nonsingular, the error bounds decay at rate O ((log n) / √n) in the i.i.d. case and O ((log n)3 / √n) in the Markov case. In order for U to have a nondegenerate covariance matrix, it is necessary and sufficient that the counted word set Ω is not full, that is, that Ω is not the collection of all possible words of some length k over the given finite alphabet. To supply the bounds on the error, we use a version of Stein's method.

scholarly journals Hyperspectral Anomaly Detection: Comparative Evaluation in Scenes with Diverse Complexity

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Dirk Borghys ◽  
Ingebjørg Kåsen ◽  
Véronique Achard ◽  
Christiaan Perneel
Keyword(s):  
Anomaly Detection ◽  
Hyperspectral Data ◽  
Data Cube ◽  
Detection Methods ◽  
Multivariate Normal ◽  
Normal Probability ◽  
Local Methods ◽  
Anomaly Detector ◽  
The Difference ◽  
Rx Detector

Anomaly detection (AD) in hyperspectral data has received a lot of attention for various applications. The aim of anomaly detection is to detect pixels in the hyperspectral data cube whose spectra differ significantly from the background spectra. Many anomaly detectors have been proposed in the literature. They differ in the way the background is characterized and in the method used for determining the difference between the current pixel and the background. The most well-known anomaly detector is the RX detector that calculates the Mahalanobis distance between the pixel under test (PUT) and the background. Global RX characterizes the background of the complete scene by a single multivariate normal probability density function. In many cases, this model is not appropriate for describing the background. For that reason a variety of other anomaly detection methods have been developed. This paper examines three classes of anomaly detectors: subspace methods, local methods, and segmentation-based methods. Representative examples of each class are chosen and applied on a set of hyperspectral data with diverse complexity. The results are evaluated and compared.

scholarly journals On spherical Monte Carlo simulations for multivariate normal probabilities

2015 ◽  
Vol 47 (03) ◽  
pp. 817-836 ◽  
Author(s):  
Huei-Wen Teng ◽  
Ming-Hsuan Kang ◽  
Cheng-Der Fuh
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Variance Reduction ◽  
Multivariate Normal ◽  
Integration Rule ◽  
Radial Integral ◽  
Normal Probability ◽  
Antithetic Variates ◽  
Number Problem ◽  
Spherical Transformation

The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. In this paper we propose a spherical Monte Carlo method with both theoretical analysis and numerical simulation. We start by writing the multivariate normal probability via an inner radial integral and an outer spherical integral using the spherical transformation. For the outer spherical integral, we apply an integration rule by randomly rotating a predetermined set of well-located points. To find the desired set, we derive an upper bound for the variance of the Monte Carlo estimator and propose a set which is related to the kissing number problem in sphere packings. For the inner radial integral, we employ the idea of antithetic variates and identify certain conditions so that variance reduction is guaranteed. Extensive Monte Carlo simulations on some probabilities confirm these claims.

Numerical Simulation of Micro Flow Field of Micro Injector

2011 ◽  
Vol 327 ◽  
pp. 61-65
Author(s):  
Li Li Mu ◽  
Ning Xue
Keyword(s):  
Numerical Simulation ◽  
Flow Field ◽  
Piezoelectric Ceramic ◽  
Calculation Model ◽  
Dynamic Evolution ◽  
Micro Channel ◽  
First Order ◽  
Micro Nozzle ◽  
Vof Model ◽  
Micro Flow

In order to research the effects of digital micro droplet injected by the piezoelectric ceramic inertial driver, the calculation model of micro flow field of micro injector was established based on the VOF model of multiphase flow. The calculation selected the implicit segregated solver and the standard k-e model was used in turbulence of the micro-nozzle. The governing equation was separated in first order upwind, and solved by PISO algorithm. The flow pattern of the micro channel fluid and the dynamic evolution process of the micro droplet generation in the plus wave driving were researched.

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