Bounds for the Probability of a Union, with Applications

Kounias, Eustratios G.

doi:10.1214/aoms/1177698049

Cited by 214 publications

(75 citation statements)

References 4 publications

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“…Denoting the failure probability of two failure modes failing simultaneously as = , the failure probability of the systems , according to the narrow bound [38,40], is…”

Section: Ditlevsen's Bounds Methodsmentioning

confidence: 99%

“…Since the physical mechanism of the two errors affecting the hydrologic model is unclear, an unknown performance function is first considered. In reliability theory, the wide and narrow bounds methods [38][39][40] were used to calculate reliability indexes of serial and parallel structural systems. Here, the serial or parallel structured error represent the relationship to the reliability of the hydrological model, where the forecasted rainfall and flood are both used to calculate the forecast error.…”

Section: Unknown Performance Functionmentioning

confidence: 99%

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Performance Assessment of Hydrological Models Considering Acceptable Forecast Error Threshold

Dong

Lü

2015

Water

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Abstract:It is essential to consider the acceptable threshold in the assessment of a hydrological model because of the scarcity of research in the hydrology community and errors do not necessarily cause risk. Two forecast errors, including rainfall forecast error and peak flood forecast error, have been studied based on the reliability theory. The first order second moment (FOSM) and bound methods are used to identify the reliability. Through the case study of the Dahuofang (DHF) Reservoir, it is shown that the correlation between these two errors has great influence on the reliability index of hydrological model. In particular, the reliability index of the DHF hydrological model decreases with the increasing correlation. Based on the reliability theory, the proposed performance evaluation framework incorporating the acceptable forecast error threshold and correlation among the multiple errors can be used to evaluate the performance of a hydrological model and to quantify the uncertainties of a hydrological model output.

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“…Denoting the failure probability of two failure modes failing simultaneously as = , the failure probability of the systems , according to the narrow bound [38,40], is…”

Section: Ditlevsen's Bounds Methodsmentioning

confidence: 99%

Section: Unknown Performance Functionmentioning

confidence: 99%

Performance Assessment of Hydrological Models Considering Acceptable Forecast Error Threshold

Dong

Lü

2015

Water

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“…The bound is a suboptimal instance of the following bound by Kounias [16], which is generally infeasible to determine exactly…”

Section: A Stepwise Algorithm Lower Boundmentioning

confidence: 99%

“…This bound-which is due to Kounias [16]-is expressed in terms of only the individual (P (A i )'s) and pairwise (P (A i \ A j )'s) event probabilities. It is obtained as an optimal bound over a family of bounds which is indexed by the set of all subsets of f1; 2; .…”

Section: Introductionmentioning

confidence: 99%

Tight error bounds for nonuniform signaling over AWGN channels

Kuai

Alajaji

Takahara

2000

IEEE Trans. Inform. Theory

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Abstract-We consider a Bonferroni-type lower bound due to Kounias on the probability of a finite union. The bound is expressed in terms of only the individual and pairwise event probabilities; however, it suffers from requiring an exponentially complex search for its direct implementation. We address this problem by presenting a practical algorithm for its evaluation. This bound is applied together with two other bounds, a recent lower bound (the KAT bound) and a greedy algorithm implementation of an upper bound due to Hunter, to examine the symbol error ( ) and bit error ( ) probabilities of an uncoded communication system used in conjunction with -ary phase-shift keying (PSK)/quadrature amplitude (QAM) (PSK/QAM) modulations and maximum a posteriori (MAP) decoding over additive white Gaussian noise (AWGN) channels. It is shown that the bounds-which can be efficiently computed-provide an excellent estimate of the error probabilities over the entire range of the signal-tonoise ratio (SNR). The new algorithmic bound and the greedy bound are particularly impressive as they agree with the simulation results even during very severe channel conditions. Index Terms-Additive white Gaussian noise (AWGN) channels, bit error rate (BER), lower bound, maximum a posteriori (MAP) and maximumlikelihood (ML) decoding, probability of a union, phase-shift keying (PSK) and quadrature amplitude (QAM) modulations, symbol error rate, upper bound.

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“…An attractive approach is, therefore, to investigate more sophisticated bounds that yield accurate error estimates. Three such bounds on the probability of a finite union of arbitrary events were recently studied in [8] and [9]: a lower bound (the Kuai-Alajaji-Takahara (KAT) bound) [8], a practical algorithmic stepwise lower bound [9] originating from Kounias [6], and a greedy algorithmic implementation [9] of an upper bound due to Hunter [5]. In this letter, we apply these bounds, which are only expressed in terms of the individual event probabilities and the pairwise event probabilities , to estimate the performance of nonuniform signaling under Gray mapping over memoryless Rayleigh fading channels.…”

mentioning

confidence: 99%

Error Analysis for Nonuniform Signaling Over Rayleigh Fading Channels

Zhong¹,

Alajaji²,

Takahara³

2005

IEEE Trans. Commun.

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Abstract-We investigate the error performance of a communication system where a nonuniform memoryless binary source is transmitted via Gray-mapped -ary phase-shift keying or quadrature amplitude modulation over memoryless Rayleigh fading channels, and demodulated via optimal maximum a posteriori detection. Using recently derived upper and lower bounds on the probability of a general union of events, which are tight and can be efficiently computed, the system symbol-error ( ) and bit-error ( ) rates are evaluated for a wide range of channel conditions. Since for nonuniform signaling, Gray mapping is not necessarily optimal for minimizing or (as was recently shown by Takahara et al.), we also evaluate the system performance under the map obtained by Takahara et al. and compare it with a Gray-mapped system. Index Terms-Bit and symbol error rates, Gray mappings, lower and upper bounds, maximum a posteriori (MAP) decoding, nonuniform sources, phase-shift keying (PSK) and quadrature amplitude modulation (QAM) modulations, probability of a union, Rayleigh fading channels.

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Bounds for the Probability of a Union, with Applications

Cited by 214 publications

References 4 publications

Performance Assessment of Hydrological Models Considering Acceptable Forecast Error Threshold

Performance Assessment of Hydrological Models Considering Acceptable Forecast Error Threshold

Tight error bounds for nonuniform signaling over AWGN channels

Error Analysis for Nonuniform Signaling Over Rayleigh Fading Channels

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