Biased estimators on quotient spaces

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14328
DOI : http://dx.doi.org/10.1007/978-3-319-25040-3_15You do not have permission to access embedded form.

Résumé

Usual statistics are defined, studied and implemented on Euclidean spaces. But what about statistics on other mathematical spaces, like manifolds with additional properties: Lie groups, Quotient spaces, Stratified spaces etc? How can we describe the interaction between statistics and geometry? The structure of Quotient space in particular is widely used to model data, for example every time one deals with shape data. These can be shapes of constellations in Astronomy, shapes of human organs in Computational Anatomy, shapes of skulls in Palaeontology, etc. Given this broad field of applications, statistics on shapes -and more generally on observations belonging to quotient spaces- have been studied since the 1980’s. However, most theories model the variability in the shapes but do not take into account the noise on the observations themselves. In this paper, we show that statistics on quotient spaces are biased and even inconsistent when one takes into account the noise. In particular, some algorithms of template estimation in Computational Anatomy are biased and inconsistent. Our development thus gives a first theoretical geometric explanation of an experimentally observed phenomenon. A biased estimator is not necessarily a problem. In statistics, it is a general rule of thumb that a bias can be neglected for example when it represents less than 0.25 of the variance of the estimator. We can also think about neglecting the bias when it is low compared to the signal we estimate. In view of the applications, we thus characterize geometrically the situations when the bias can be neglected with respect to the situations when it must be corrected.

Biased estimators on quotient spaces

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Usual statistics are defined, studied and implemented on Euclidean spaces. But what about statistics on other mathematical spaces, like manifolds with additional properties: Lie groups, Quotient spaces, Stratified spaces etc? How can we describe the interaction between statistics and geometry? The structure of Quotient space in particular is widely used to model data, for example every time one deals with shape data. These can be shapes of constellations in Astronomy, shapes of human organs in Computational Anatomy, shapes of skulls in Palaeontology, etc. Given this broad field of applications, statistics on shapes -and more generally on observations belonging to quotient spaces- have been studied since the 1980’s. However, most theories model the variability in the shapes but do not take into account the noise on the observations themselves. In this paper, we show that statistics on quotient spaces are biased and even inconsistent when one takes into account the noise. In particular, some algorithms of template estimation in Computational Anatomy are biased and inconsistent. Our development thus gives a first theoretical geometric explanation of an experimentally observed phenomenon. A biased estimator is not necessarily a problem. In statistics, it is a general rule of thumb that a bias can be neglected for example when it represents less than 0.25 of the variance of the estimator. We can also think about neglecting the bias when it is low compared to the signal we estimate. In view of the applications, we thus characterize geometrically the situations when the bias can be neglected with respect to the situations when it must be corrected.

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Nina Miolane – Biased Estimators on Quotient Spaces Biased estimators in Quotient spaces Statistical properties of the Template in Computational Anatomy Geometric Sciences of Information 2015 Nina Miolane (1,2), Xavier Pennec (1) (1) INRIA (Asclepios Team), (2) Stanford University (Statistics) Data base ⇒ Nina Miolane – Biased Estimators on Quotient Spaces 2 Model & Analysis of the variability of human anatomical shapes Computational Anatomy [Kurtek and al, 2011] [McLeod and al, 2013][Darmante and al, 2014] Landmarks Medical images 1-D signals Surfaces and manifolds [Lorenzi and al, 2010] Nina Miolane – Biased Estimators on Quotient Spaces 3 Model & Analysis of the variability of human anatomical shapes Computational Medicine Automatic diagnosis Treatment Computational Anatomy Compare patient's shape with mean shape of healthy data base, i.e. with the template New patient [Kurtek and al, 2011] [McLeod and al, 2013][Darmante and al, 2014] Landmarks Medical images 1-D signals Surfaces and manifolds [Lorenzi and al, 2010] Nina Miolane – Biased Estimators on Quotient Spaces 4 Model & Analysis of the variability of human anatomical shapes Computational Medicine Automatic diagnosis Treatment Computational Anatomy Compare patient's shape with mean shape of healthy data base, i.e. with the template New patient [Kurtek and al, 2011] [McLeod and al, 2013][Darmante and al, 2014] Landmarks Medical images 1-D signals Surfaces and manifolds [Lorenzi and al, 2010] Nina Miolane – Biased Estimators on Quotient Spaces Template computed as the empirical mean anatomical shape of