On a Geometric Notion of Quantiles for Multivariate Data

Chaudhuri, Probal

doi:10.1080/01621459.1996.10476954

Cited by 296 publications

(234 citation statements)

References 49 publications

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“…This kind of multivariate quantiles is developed by Abdous and Theodorecu (1992) and Chaudhuri (1996). This extension corresponds to the following characteristic of the univariate p th‐quantile (see Ferguson, 1967):…”

Section: Multivariate Quantiles In Statistical Literaturementioning

confidence: 68%

Multivariate quantiles in hydrological frequency analysis

2011

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Abstract2 Several hydrological phenomena are described by two or more correlated characteristics. 3These dependent characteristics should be considered jointly to be more representative of the 4 multivariate nature of the phenomenon. Consequently, probabilities of occurrence cannot be 5 estimated on the basis of univariate frequency analysis (FA). The quantile, representing the value 6 of the variable(s) corresponding to a given risk, is one of the most important notions in FA. The 7 estimation of multivariate quantiles has not been specifically treated in the hydrological FA 8 literature. In the present paper, we present a new and general framework for local FA based on a 9 multivariate quantile version. The multivariate quantile offers several combinations of the 10 variable values that lead to the same risk. A simulation study is carried out to evaluate the 11 performance of the proposed estimation procedure and a case study is conducted. Results show 12 that the bivariate estimation procedure has an analogous behaviour to the univariate one with 13 respect to the risk and the sample size. However, the dependence structure between variables is 14 ignored in the univariate case. The univariate estimates are obtained as special combinations by 15 the multivariate procedure and with equivalent accuracy. 16 17 18 3

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Section: Multivariate Quantiles In Statistical Literaturementioning

confidence: 68%

Multivariate quantiles in hydrological frequency analysis

2011

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“…Geometric quantiles were used for calculating the multidimensional median of each bin as the centroid and for estimating the standard deviation from MAD. The geometric quantile, Q, is defined as the data point that minimizes the following target function as described by Chaudhuri :

f true({\overset{Q}{true}}^{true(m true)} true) = \sum_{i = 1}^{n} true{true| {\overset{X}{true}}_{i} - {\overset{Q}{true}}^{true(m true)} true| + \vec{u} \cdot true({\overset{X}{true}}_{i} - {\overset{Q}{true}}^{true(m true)} true) true}

in which n is the number of data points in each bin;

{\overset{X}{true}}_{i}

is the data point, and

{\overset{Q}{true}}^{true(m true)}

is the quantile of the m th iteration;

u = 2 α - 1

, where α is fractional quantile. For example,

α = 0.5

for median (50% quantile) and the target function reduces to

f true({\overset{Q}{true}}^{true(m true)} true) = \sum_{i = 1}^{n} {| {\vec{X}}_{i} - {\vec{Q}}^{(m)} |}

.…”

Section: Methodsmentioning

confidence: 99%

FAST: Rapid determinations of antibiotic susceptibility phenotypes using label‐free cytometry

2018

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Sepsis, a life-threatening immune response to blood infections (bacteremia), has a ∼30% mortality rate and is the 10th leading cause of US hospital deaths. The typical bacterial loads in adult septic patients are ≤100 bacterial cells (colony forming units, CFU) per ml blood, while pediatric patients exhibit only ∼1000 CFU/ml. Due to the low numbers, bacteria must be propagated through ∼24-hours blood cultures to generate sufficient CFUs for diagnosis and further analyses. Herein, we demonstrate that, unlike other rapid post-blood culture antibiotic susceptibility tests (ASTs), our phenotypic approach can drastically accelerate ASTs for the most common sepsis-causing gram-negative pathogens by circumventing long blood culture-based amplification. For all blood isolates of multi-drug resistant pathogens investigated (Escherichia coli, Klebsiella pneumoniae, and Acinetobacter nosocomialis), effective antibiotic(s) were readily identified within the equivalent of 8 hours from initial blood draw using <0.5 mL of adult blood per antibiotic. These methods should drastically improve patient outcomes by significantly reducing time to actionable treatment information and reduce the incidence of antibiotic resistance. © 2018 International Society for Advancement of Cytometry.

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“…For further motivation and theoretical considerations of this approach, see [7,10]. Note that, because R d , d > 1 has no total ordering, there are many other notions of multivariate quantiles (see, e.g., [4,[29][30][31][32]). However, we do not consider these further here.…”

Section: Definition 1 ([9]mentioning

confidence: 99%

Confidence Regions for Multivariate Quantiles

Coblenz

Dyckerhoff

Grothe

2018

Water

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Multivariate quantiles are of increasing importance in applications of hydrology. This calls for reliable methods to evaluate the precision of the estimated quantile sets. Therefore, we focus on two recently developed approaches to estimate confidence regions for level sets and extend them to provide confidence regions for multivariate quantiles based on copulas. In a simulation study, we check coverage probabilities of the employed approaches. In particular, we focus on small sample sizes. One approach shows reasonable coverage probabilities and the second one obtains mixed results. Not only the bounded copula domain but also the additional estimation of the quantile level pose some problems. A small sample application gives further insight into the employed techniques.

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On a Geometric Notion of Quantiles for Multivariate Data

Cited by 296 publications

References 49 publications

Multivariate quantiles in hydrological frequency analysis

Multivariate quantiles in hydrological frequency analysis

FAST: Rapid determinations of antibiotic susceptibility phenotypes using label‐free cytometry

Confidence Regions for Multivariate Quantiles

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