Development and validation of statistical models of femur geometry for use with parametric finite element models

Statistical models from a previous study that predict male and female femur geometry as functions of age, body mass index (BMI), and femur length were updated as part of an effort to develop lower-extremity finite element models with geometries that are parametric with subject characteristics. The process for updating these models involved extracting femur geometry from clinical CT scans of an additional 8 men and 36 women (previous models used CT scans from 62 men and 36 women for a new total of 70 men and 72 women), using all of the scans for fitting a template finite element femur mesh to the surface geometry of each patient, and then programmatically determining thickness at each nodal location. Principal component analysis was then performed on the thickness and geometry nodal coordinates, and linear regression models were developed to predict principal component scores as functions of age, BMI, and femur length. The results from the updated models were compared to the previous study, and the only improvement was in the R2 value for the female models (0.74 to 0.82). The largest differences between the original models and the previous models occurred in the ends of the femur, where the largest errors in model predictions occurred.National Highway Traffic Safety Administrationhttp://deepblue.lib.umich.edu/bitstream/2027.42/116208/1/103222.pdfDescription of 103222.pdf : Final repor

Algebraic statistical models

Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the parameter space of a reference model with nice properties, such as for example a regular exponential family. This observation leads to the definition of an `algebraic exponential family'. This new definition provides a unified framework for the study of statistical models with algebraic structure. In this paper we review the ingredients to this definition and illustrate in examples how computational algebraic geometry can be used to solve problems arising in statistical inference in algebraic models

Tropical Geometry of Statistical Models

This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are coordinates of a point in the variety, and the sum-product algorithm is an efficient tool for evaluating specific coordinates. The question addressed here is how the solutions to various inference problems depend on the model parameters. The proposed answer is expressed in terms of tropical algebraic geometry. A key role is played by the Newton polytope of a statistical model. Our results are applied to the hidden Markov model and to the general Markov model on a binary tree.Comment: 14 pages, 3 figures. Major revision. Applications now in companion paper, "Parametric Inference for Biological Sequence Analysis

Nonparametric Information Geometry

The differential-geometric structure of the set of positive densities on a given measure space has raised the interest of many mathematicians after the discovery by C.R. Rao of the geometric meaning of the Fisher information. Most of the research is focused on parametric statistical models. In series of papers by author and coworkers a particular version of the nonparametric case has been discussed. It consists of a minimalistic structure modeled according the theory of exponential families: given a reference density other densities are represented by the centered log likelihood which is an element of an Orlicz space. This mappings give a system of charts of a Banach manifold. It has been observed that, while the construction is natural, the practical applicability is limited by the technical difficulty to deal with such a class of Banach spaces. It has been suggested recently to replace the exponential function with other functions with similar behavior but polynomial growth at infinity in order to obtain more tractable Banach spaces, e.g. Hilbert spaces. We give first a review of our theory with special emphasis on the specific issues of the infinite dimensional setting. In a second part we discuss two specific topics, differential equations and the metric connection. The position of this line of research with respect to other approaches is briefly discussed.Comment: Submitted for publication in the Proceedings od GSI2013 Aug 28-30 2013 Pari

Dual Connections in Nonparametric Classical Information Geometry

We construct an infinite-dimensional information manifold based on exponential Orlicz spaces without using the notion of exponential convergence. We then show that convex mixtures of probability densities lie on the same connected component of this manifold, and characterize the class of densities for which this mixture can be extended to an open segment containing the extreme points. For this class, we define an infinite-dimensional analogue of the mixture parallel transport and prove that it is dual to the exponential parallel transport with respect to the Fisher information. We also define {\alpha}-derivatives and prove that they are convex mixtures of the extremal (\pm 1)-derivatives