Integration of the general bivariate Gaussian distribution over an offset circle

DiDonato, A. R.; Jarnagin, M. P.

doi:10.1090/s0025-5718-1961-0129116-8

Cited by 22 publications

(8 citation statements)

References 4 publications

(10 reference statements)

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“…In addition, numerical methods are needed for computing the

H_{B} false(bold-italicθ false)

function in (1), on which the probabilities

π_{j} false(bold-italicθ false)

and the likelihood function are based. For the Thomas process, the inner integral in

H_{B} false(bold-italicθ false)

, that is,

F_{γ, x} false(B false) = \int_{B} f false(t - x false| γ false) d t

, may be computed using an efficient numerical method described in DiDonato and Jarnagin (), which is implemented in, for example, the pmvnEll function in the package shotGroups (Wollschlaeger, ) written for use in r (R Core Team, ). If the point process is a Matérn cluster process,

F_{γ, x} false(B false)

may be computed analytically (Appendix S2).…”

Section: Methodsmentioning

confidence: 99%

Estimating density from presence/absence data in clustered populations

et al. 2020

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Inventories of plant populations are fundamental in ecological research and monitoring, but such surveys are often prone to field assessment errors. Presence/absence (P/A) sampling may have advantages over plant cover assessments for reducing such errors. However, the linking between P/A data and plant density depends on model assumptions for plant spatial distributions. Previous studies have shown, for example, how that plant density can be estimated under Poisson model assumptions on the plant locations. In this study, new methods are developed and evaluated for linking P/A data with plant density assuming that plants occur in clustered spatial patterns. New theory was derived for estimating plant density under Neyman–Scott‐type cluster models such as the Matérn and Thomas cluster processes. Suggested estimators, corresponding confidence intervals and a proposed goodness‐of‐fit test were evaluated in a Monte Carlo simulation study assuming a Matérn cluster process. Furthermore, the estimators were applied to plant data from environmental monitoring in Sweden to demonstrate their empirical application. The simulation study showed that our methods work well for large enough sample sizes. The judgment of what is' large enough’ is often difficult, but simulations indicate that a sample size is large enough when the sampling distributions of the parameter estimators are symmetric or mildly skewed. Bootstrap may be used to check whether this is true. The empirical results suggest that the derived methodology may be useful for estimating density of plants such as Leucanthemum vulgare and Scorzonera humilis. By developing estimators of plant density from P/A data under realistic model assumptions about plants' spatial distributions, P/A sampling will become a more useful tool for inventories of plant populations. Our new theory is an important step in this direction.

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“…In addition, numerical methods are needed for computing the

H_{B} false(bold-italicθ false)

function in (1), on which the probabilities

π_{j} false(bold-italicθ false)

and the likelihood function are based. For the Thomas process, the inner integral in

H_{B} false(bold-italicθ false)

, that is,

F_{γ, x} false(B false) = \int_{B} f false(t - x false| γ false) d t

F_{γ, x} false(B false)

may be computed analytically (Appendix S2).…”

Section: Methodsmentioning

confidence: 99%

Estimating density from presence/absence data in clustered populations

et al. 2020

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“…Several implementations are studied and compared in the first approach (Duchesne and Lafaye de Micheaux, 2010) and the one in Imhof (1961) showing reliable results is used here. A method was designed in the latter approach (DiDonato and Jarnagin, 1961) to speed up the calculation, which was first used in navigation integrity monitoring by Milner and Ochieng (2010). These two implementations to calculate the probability can achieve a pre-defined accuracy, and they are both referred to as the “exact distribution” in this paper.…”

Section: Introductionmentioning

confidence: 99%

A New Approach to Calculate the Horizontal Protection Level

Jiang

Wang

2015

J. Navigation

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A method to compute the minimum Horizontal Protection Level (HPL) using the test statistic of normal distribution, which will exploit advances in computational power to meet the requirement of Time to Alert (TTA), is proposed to improve service availability. To obtain the minimum solution, two approximations used in traditional algorithms need exact solutions: the distribution of the horizontal position error and the determination of the worst case to ensure that the resulting HPL is able to accommodate all possible bias. This is validated with results such that the optimal solution is achieved with a pre-defined accuracy and sufficient computational efficiency. Also, the new HPL is used to determine if current approximated methods are conservative, where one of the methods does not meet the integrity requirement with given test statistics, error model and integrity risk definition. K E Y WO R D S 1. HPL.2. RAIM. GPS

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“…The answers are in closed form only under very restrictive assumptions, and most of the methods documented unavoidably use numerical integration schemes to get answers, see [1][2][3][4][5]. The fast converging series solution presented is highly accurate and can be performed rapidly using the recursion relations for Hermite polynomials.…”

Section: Introductionmentioning

confidence: 99%

Distribution of general noncentral positive definite quadratic form in K-dimensions

Reifler

autor.

Malakian

1989

IEEE Trans. Aerosp. Electron. Syst.

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We present a series solution using Hermite polynomials to a long standing problem of computing the probability P that positive definite noncentral quadratic form dCX') of a Gaussian random vector "1" E RK satisfies dCX') -s: ,2 for any given, E R.This problem bas wide applications in radar, tracking, air traffic control, etc. The fast converging series solution presented Is highly accurate and can be performed rapidly using the recursion relatiollS for Hermite polynomials.

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Integration of the general bivariate Gaussian distribution over an offset circle

Cited by 22 publications

References 4 publications

Estimating density from presence/absence data in clustered populations

Estimating density from presence/absence data in clustered populations

A New Approach to Calculate the Horizontal Protection Level

Distribution of general noncentral positive definite quadratic form in K-dimensions

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