Deriving and Applying Improved Upper Bounds for Multivariate Normal Probability Outside of N-Cubes

1988 ◽  
Author(s):  
Donald R. Hoover
Keyword(s):  
Upper Bounds ◽  
Multivariate Normal ◽  
Normal Probability

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.

Bonferroni bounds revisited

1989 ◽  
Vol 26 (2) ◽  
pp. 233-241 ◽  
Author(s):  
Stratis Kounias ◽  
Kiki Sotirakoglou
Keyword(s):  
Normal Distribution ◽  
Multivariate Normal Distribution ◽  
Upper Bounds ◽  
Multivariate Normal ◽  
Lower And Upper Bounds

Lower and upper bounds of degree m for the probability of the union of n not necessarily exchangeable events are established. These bounds may be constructed to improve the Bonferroni and the Sobel–Uppuluri bounds.An application to equi-correlated multivariate normal distribution is given.

An upper bound for the probability of a union

1976 ◽  
Vol 13 (03) ◽  
pp. 597-603 ◽  
Author(s):  
David Hunter
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Spanning Tree ◽  
Upper Bounds ◽  
Minimal Spanning Tree ◽  
Multivariate Normal ◽  
Tail Probabilities ◽  
Tree Algorithm ◽  
New Family ◽  
The Family

The problem of bounding P(∪ Ai ) given P(A i) and P(A i A j) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.

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