Properties and applications of Fisher distribution on the rotation group

Sei, Tomonari; Shibata, Hiroki; Takemura, Akimichi; Ohara, Katsuyoshi; Takayama, Nobuki

doi:10.1016/j.jmva.2013.01.010

Cited by 39 publications

(76 citation statements)

References 12 publications

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“…Concerning the treatment of the normalizing constant, recently in [11] we proposed a new method, called the holonomic gradient decent (HGD), for evaluating the normalizing constant of the exponential family and for computing MLE. As in the subsequent works ( [5], [13]), we show that HGD works well also for the case of exponential-polynomial distribution.…”

Section: Introductionsupporting

confidence: 85%

Estimation of exponential-polynomial distribution by holonomic gradient descent

Hayakawa

Takemura

2016

Communications in Statistics - Theory and Methods

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We study holonomic gradient decent for maximum likelihood estimation of exponentialpolynomial distribution, whose density is the exponential function of a polynomial in the random variable. We first consider the case that the support of the distribution is the set of positive reals. We show that the maximum likelihood estimate (MLE) can be easily computed by the holonomic gradient descent, even though the normalizing constant of this family does not have a closed-form expression and discuss determination of the degree of the polynomial based on the score test statistic. Then we present extensions to the whole real line and to the bivariate distribution on the positive orthant.

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Section: Introductionsupporting

confidence: 85%

Estimation of exponential-polynomial distribution by holonomic gradient descent

Hayakawa

Takemura

2016

Communications in Statistics - Theory and Methods

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“…In the remainder of the paper, we will focus on evaluating C(θ, γ ) as in (3) where θ has l distinct values. Hashiguchi et al (2013), Sei et al (2013), Koyama (2011), Koyama et al (2014) and Koyama et al (2012) …”

Section: Degenerate Casesmentioning

confidence: 99%

“…However, recently there has been a renewed interest in this problem with the implementation of the holonomic gradient method (HGM), which in theory is exact since the problem of calculating C is mathematically characterized via a solution of an ODE (see e.g. Nakayama et al 2011;Hashiguchi et al 2013;Sei et al 2013;Koyama 2011;Koyama and Takemura 2016;Koyama et al 2014 andKoyama et al 2012).…”

Section: Introductionmentioning

confidence: 99%

On the exact maximum likelihood inference of Fisher–Bingham distributions using an adjusted holonomic gradient method

Kume

Sei

2017

Stat Comput

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Holonomic function theory has been successfully implemented in a series of recent papers to efficiently calculate the normalizing constant and perform likelihood estimation for the Fisher-Bingham distributions. A key ingredient for establishing the standard holonomic gradient algorithms is the calculation of the Pfaffian equations. So far, these papers either calculate these symbolically or apply certain methods to simplify this process. Here we show the explicit form of the Pfaffian equations using the expressions from Laplace inversion methods. This improves on the implementation of the holonomic algorithms for these problems and enables their adjustments for the degenerate cases. As a result, an exact and more dimensionally efficient ODE is implemented for likelihood inference.

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“…Furthermore, the three-sphere can be considered as S 3 = {x, −x | x ∈ RP 3 }, and x T Bx is an even function of B. Applying these to (23),…”

Section: B Case Ii: Large Initial Uncertaintymentioning

confidence: 99%

Global unscented attitude estimation via the matrix Fisher distributions on SO(3)

Lee

2016

2016 American Control Conference (ACC)

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Abstract-This paper is focused on probabilistic estimation for the attitude dynamics of a rigid body on the special orthogonal group. We select the matrix Fisher distribution to represent the uncertainties of attitude estimates and measurements in a global fashion without need for local coordinates. Several properties of the matrix Fisher distribution on the special orthogonal group are presented, and an unscented transform is proposed to approximate a matrix Fisher distribution by selected sigma points. Based on these, an intrinsic, global framework for Bayesian attitude estimation is developed. It is shown that the proposed approach can successfully deal with large initial estimator errors and large uncertainties over complex maneuvers to obtain accurate estimates of the attitude.

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Properties and applications of Fisher distribution on the rotation group

Cited by 39 publications

References 12 publications

Estimation of exponential-polynomial distribution by holonomic gradient descent

Estimation of exponential-polynomial distribution by holonomic gradient descent

On the exact maximum likelihood inference of Fisher–Bingham distributions using an adjusted holonomic gradient method

Global unscented attitude estimation via the matrix Fisher distributions on SO(3)

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