Spatial Median and Directional Data

Ducharme, Gilles R.; Milasevic, Philip

doi:10.2307/2336038

Cited by 5 publications

(18 citation statements)

References 2 publications

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“…, X n ∈ R p include, for example, the Euclidean median (also known as the Weber point (Bajaj 1988)), the mediancentre (Gower 1974), or the projection median (Durocher and Kirkpatrick 2009). For directional and spherical data, exemplary estimators include the circular median (Mardia 1972), the normalized spatial median (Ducharme and Milasevic 1987) and the Fisher median (Fisher 1985) also known as the spherical median. Chan and He (1993) compare the performance of the normalized spatial median, an L 1 estimator on the sphere by He and Simpson (1992) and the Fisher median for the central direction for spherical data following the von Mises-Fisher distribution.…”

Section: The Projected Medianmentioning

confidence: 99%

“…Depending on the scientific literature, the approaches can be quite different. The topic of location estimation has received considerable attention for directional data on circles or spheres (see Fisher 1953;Karcher 1977;Khatri and Mardia 1977;Fisher 1985;Ducharme and Milasevic 1987;Bajaj 1988;Liu and Singh 1992;Chan and He 1993;Mardia and Jupp 2000), but less is known about estimator properties with rotation data. As a compounding factor, several current approaches to estimating S have arisen out of literatures having differing statistical and geometrical emphases.…”

Section: Introductionmentioning

confidence: 99%

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Point Estimation of the Central Orientation of Random Rotations

2013

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Data as three-dimensional rotations have application in computer science, kinematics, and materials sciences, among other areas. Estimating the central orientation from a sample of such data is an important problem, which is complicated by the fact that several different approaches exist for this, motivated by various geometrical and decision-theoretical considerations. However, little is known about how such estimators compare, especially on common distributions for location models with random rotations. We examine four location estimators, three of which are commonly found in different literatures and the fourth estimator (a projected median) is newly introduced. Our study unifies existing literature and provides a detailed numerical investigation of location estimators for three commonly used rotation distributions in statistics and materials science. While the data-generating model influences the best choice of an estimator, the proposed projected median emerges as an overall good performer, which can be suggested without particular distributional assumptions. We illustrate the estimators and our findings with data from a materials science study by approximating the central orientation of cubic crystals on the microsurface of a metal. Accompanying supplementary materials are available online.

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Section: The Projected Medianmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Point Estimation of the Central Orientation of Random Rotations

2013

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“…, X n ∈ R p include, for example, the Euclidean median (also known as the Weber point (Bajaj, 1988)), the mediancentre (Gower, 1974) or the projection median (Durocher and Kirkpatrick, 2009). For directional and spherical data, exemplary estimators include the circular median (Mardia, 1972), the normalized spatial median (Ducharme and Milasevic, 1987) and the Fisher median (Fisher, 1985) also known as the spherical median. Chan and He (1993) compare the performance of the normalized spatial median, an L 1 estimator on the sphere by He and Simpson (1992) A modification of the estimator from Section 2.3.1 in line with these proposals is obtained by replacing the squared distances in (2.5) with absolute distances, leading to a median-type estimator defined as…”

Section: The Projected Medianmentioning

confidence: 99%

“…Another important measure of estimator robustness is efficiency. The efficiency of the spatial median, or mediancentre (Gower, 1974), and the normalized spatial median were considered by Brown (1983) and Ducharme and Milasevic (1987), respectively. The spatial median is defined as the vectorm n that minimizes…”

Section: Literature Reviewmentioning

confidence: 99%

“…The topic of location estimation has received considerable attention for directional data on circles or spheres (see Fisher 1953;Karcher 1977;Khatri and Mardia 1977;Fisher 1985;Ducharme and Milasevic 1987;Bajaj 1988;Liu et al 1992;Chan and He 1993;Mardia and Jupp 2000), but less is known about estimator properties with rotation data. As a compounding factor, several current approaches to estimating S have arisen out of literatures having differing statistical and geometrical emphases.…”

Section: Introductionmentioning

confidence: 99%

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Statistical methods for random rotations

Stanfill

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Data as three-dimensional rotations have application in computer science, kinematics and materials sciences, among other areas. Estimating the central orientation from a sample of such data is an important problem, which is complicated by the fact that several different approaches exist for this, motivated by various geometrical and decision-theoretic considerations. However, little is known about how such estimators compare, especially on common distributions for location models with random rotations. We examine four location estimators, three of which are commonly found in different literatures and the fourth estimator (a projected median) is newly introduced. Our study unifies existing literature and provides a detailed numerical investigation of location estimators for three commonly used rotation distributions in statistics and materials science. While the data-generating model influences the best choice of an estimator, the proposed projected median emerges as an overall good performer, which can be suggested without particular distributional assumptions. We illustrate the estimators and our findings with data from a materials science study by approximating the central orientation of cubic crystals on the micro-surface of a metal. Accompanying supplementary materials are available online.

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Robust estimation of location and concentration parameters for the von Mises–Fisher distribution

Kato

Eguchi

2014

Stat Papers

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Robust estimation of location and concentration parameters for the von MisesFisher distribution is discussed. A key reparametrisation is achieved by expressing the two parameters as one vector on the Euclidean space. With this representation, we first show that maximum likelihood estimator for the von Mises-Fisher distribution is not robust in some situations. Then we propose two families of robust estimators which can be derived as minimisers of two density power divergences. The presented families enable us to estimate both location and concentration parameters simultaneously. Some properties of the estimators are explored. Simple iterative algorithms are suggested to find the estimates numerically. A comparison with the existing robust estimators is given as well as discussion on difference and similarity between the two proposed estimators. A simulation study is made to evaluate finite sample performance of the estimators. We consider a sea star dataset and discuss the selection of the tuning parameters and outlier detection.

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Spatial Median and Directional Data

Cited by 5 publications

References 2 publications

Point Estimation of the Central Orientation of Random Rotations

Point Estimation of the Central Orientation of Random Rotations

Statistical methods for random rotations

Robust estimation of location and concentration parameters for the von Mises–Fisher distribution

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