TY - JOUR

T1 - Diffusion means in geometric spaces

AU - Eltzner, Benjamin

AU - Hansen, Pernille E.H.

AU - Huckemann, Stephan F.

AU - Sommer, Stefan

N1 - Publisher Copyright: © 2023 ISI/BS.

PY - 2023

Y1 - 2023

N2 - We introduce a location statistic for distributions on non-linear geometric spaces, the diffusion mean, serving as an extension and an alternative to the Fréchet mean. The diffusion mean arises as the generalization of Gaussian maximum likelihood analysis to non-linear spaces by maximizing the likelihood of a Brownian motion. The diffusion mean depends on a time parameter t, which admits the interpretation of the allowed variance of the diffusion. The diffusion t-mean of a distribution X is the most likely origin of a Brownian motion at time t, given the end-point distribution X. We give a detailed description of the asymptotic behavior of the diffusion estimator and provide sufficient conditions for the diffusion estimator to be strongly consistent. Particularly, we present a smeary central limit theorem for diffusion means and we show that joint estimation of the mean and diffusion variance rules out smeariness in all directions simultaneously in general situations. Furthermore, we investigate properties of the diffusion mean for distributions on the sphere Sm. Experimentally, we consider simulated data and data from magnetic pole reversals, all indicating similar or improved convergence rate compared to the Fréchet mean. Here, we additionally estimate t and consider its effects on smeariness and uniqueness of the diffusion mean for distributions on the sphere.

AB - We introduce a location statistic for distributions on non-linear geometric spaces, the diffusion mean, serving as an extension and an alternative to the Fréchet mean. The diffusion mean arises as the generalization of Gaussian maximum likelihood analysis to non-linear spaces by maximizing the likelihood of a Brownian motion. The diffusion mean depends on a time parameter t, which admits the interpretation of the allowed variance of the diffusion. The diffusion t-mean of a distribution X is the most likely origin of a Brownian motion at time t, given the end-point distribution X. We give a detailed description of the asymptotic behavior of the diffusion estimator and provide sufficient conditions for the diffusion estimator to be strongly consistent. Particularly, we present a smeary central limit theorem for diffusion means and we show that joint estimation of the mean and diffusion variance rules out smeariness in all directions simultaneously in general situations. Furthermore, we investigate properties of the diffusion mean for distributions on the sphere Sm. Experimentally, we consider simulated data and data from magnetic pole reversals, all indicating similar or improved convergence rate compared to the Fréchet mean. Here, we additionally estimate t and consider its effects on smeariness and uniqueness of the diffusion mean for distributions on the sphere.

KW - Diffusion mean

KW - generalized Fréchet mean

KW - geometric statistics

KW - maximum likelihood estimation

KW - spherical statistics

UR - http://www.scopus.com/inward/record.url?scp=85161632699&partnerID=8YFLogxK

U2 - 10.3150/22-BEJ1578

DO - 10.3150/22-BEJ1578

M3 - Journal article

AN - SCOPUS:85161632699

VL - 29

SP - 3141

EP - 3170

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 4

ER -