Decompounding on Compact Lie Groups

Said, Salem; Lageman, Christian; Bihan, Nicolas Le; Manton, Jonathan H.

doi:10.1109/tit.2010.2046216

Cited by 13 publications

(24 citation statements)

References 29 publications

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“…This provides the technical basis for Section V, with Proposition 7 extending the classical Kac's theorem. This paper complements a previous one [1], with both generalizing the results of [2]. In [1], a solution is provided for a problem of nonparametric estimation for compound Poisson processes in a compact Lie group.…”

Section: Introductionmentioning

confidence: 60%

“…This paper complements a previous one [1], with both generalizing the results of [2]. In [1], a solution is provided for a problem of nonparametric estimation for compound Poisson processes in a compact Lie group. Brownian processes and compound Poisson processes are the two main types of Lévy processes in a Lie group [3].…”

Section: Introductionmentioning

confidence: 60%

“…Brownian processes are outlined in [14]. For a more self contained version of Section II-B, see [1]; a detailed and highly readable reference is [15] and a variety of engineering applications are discussed in [16]. A recent and comprehensive treatment of stochastic processes in Lie groups and their connection to information theory can be found in [17].…”

Section: Mathematical Backgroundmentioning

confidence: 99%

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Extrinsic Mean of Brownian Distributions on Compact Lie Groups

Said

Manton

2012

IEEE Trans. Inform. Theory

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Abstract-This paper studies Brownian distributions on compact Lie groups. These are defined as the marginal distributions of Brownian processes and are intended as a natural extension of the well-known normal distributions to compact Lie groups. It is shown that this definition preserves key properties of normal distributions. In particular, Brownian distributions transform in a nice way under group operations and satisfy an extension of the central limit theorem. Brownian distributions on a compact Lie group belong to one of two parametric families and -and a positive-definite symmetric matrix. In particular, the parameter appears as a location parameter. An approach based on the extrinsic mean for estimation of the parameters and is studied in detail. It is shown that is the unique extrinsic mean for a Brownian distribution or . Resulting estimates are proved to be consistent and asymptotically normal. While they may also be used to simultaneously estimate and , it is seen this requires that be embedded into a higher dimensional matrix Lie group. Going beyond Brownian distributions, it is shown the extrinsic mean can be used to recover the location parameter for a wider class of distributions arising more generally from Lévy processes. The compact Lie group structure places limitations on the analogy between normal distributions and Brownian distributions. This is illustrated by the study of multivariate Brownian distributions. These are introduced as Brownian distributions on some product group-e.g., . This paper describes their covariance structure and considers its transformation under group operations.Index Terms-Brownian motion, central limit theorem, compact Lie group, extrinsic and intrinsic mean, noncommutative harmonic analysis, Lévy process.

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Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 60%

Section: Mathematical Backgroundmentioning

confidence: 99%

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Extrinsic Mean of Brownian Distributions on Compact Lie Groups

Said

Manton

2012

IEEE Trans. Inform. Theory

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“…Namely, we consider the problem of multiple scattering for particules in a random media and model the output distribution of the direction of propagation of the particule as a mixture of vMF distribution. This model can be compared with the one proposed in [5,6] where multiple scattering is described as a Compound Poisson Process (CPP) on the rotation group SO(3). In [5], it is shown that this model allows to describe forward multiple scattering, and its accuracy is high when the number of diffusion events is low.…”

Section: Introductionmentioning

confidence: 99%

“…It allows us to give the Fourier expansion of this approximate distribution. This makes possible the use of a method of moment (MoM) based estimation technique similar to decompounding [6]. We present results of such estimation procedure to identify the parameter of the vMF distribution.…”

Section: Introductionmentioning

confidence: 99%

Von Mises-Fisher approximation of multiple scattering process on the hypersphere

Chatelain

Bihan

2013

2013 IEEE International Conference on Acoustics, Speech and Signal Processing

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This paper presents a "method of moments" estimation technique for the study of multiple scattering on the hypersphere. The proposed model is similar to a compound Poisson process evolving on a special manifold: the unit hypersphere. The presented work makes use of an approximation result for multiply convolved von Mises-Fisher distributions on hyperspheres. Comparison with other approximations show the accuracy of the proposed model to provide estimators for the mean free path and concentration parameters when studying a multiple scattering process. Such a process is classically used to model the propagation of waves or particules in random media.Index Terms-von Mises-Fisher distribution, random walk on hypersphere, multiple scattering, Method of moments estimation.

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Directional Statistics, II

Jupp

2005

Encyclopedia of Statistical Sciences

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Decompounding on Compact Lie Groups

Cited by 13 publications

References 29 publications

Extrinsic Mean of Brownian Distributions on Compact Lie Groups

Extrinsic Mean of Brownian Distributions on Compact Lie Groups

Von Mises-Fisher approximation of multiple scattering process on the hypersphere

Directional Statistics, II

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