The Fisher-Rao Distance between Multivariate Normal Distributions: Special Cases, Boundsand Applications

The Fisher–Rao distance is a measure of dissimilarity between probability distributions, which, under certain regularity conditions of the statistical model, is up to a scaling factor the unique Riemannian metric invariant under Markov morphisms. It is related to the Shannon entropy and has been used to enlarge the perspective of analysis in a wide variety of domains such as image processing, radar systems, and morphological classification. Here, we approach this metric considered in the statistical model of normal multivariate probability distributions, for which there is not an explicit expression in general, by gathering known results (closed forms for submanifolds and bounds) and derive expressions for the distance between distributions with the same covariance matrix and between distributions with mirrored covariance matrices. An application of the Fisher–Rao distance to the simplification of Gaussian mixtures using the hierarchical clustering algorithm is also presented.

Closeness to singularities of robotic manipulators measured by characteristic angles

SUMMARYSingularities have a great influence on kinematics and dynamics of both serial and parallel robots. In order to prevent a robot from entering singular configurations, it needs to measure the “distance” between the robot current configuration and the singular configuration. This paper presents a novel approach based on characteristic angles to measure closeness to singularities. For the problem of inconsistent dimensions in the scalar product of screws, the physical meanings of twists and wrenches are reinterpreted. For the problem of the metric invariant to origin selection, the origin of the screw frame is required to coincide with the origin of the robotic tool frame. The major merit of the proposed metric lies in the identical result of measuring similar mechanisms with different sizes. Moreover, the measurement is insensitive to screw magnitude, since the metric expression is dimensionless. Furthermore, the geometrical meaning of the determinant of a screw matrix is clarified.

On rationally ergodic and rationally weakly mixing rank-one transformations

We study the notions of weak rational ergodicity and rational weak mixing as defined by J. Aaronson [Rational ergodicity and a metric invariant for Markov shifts.Israel J. Math. 27(2) (1977), 93–123; Rational weak mixing in infinite measure spaces.Ergod. Th. & Dynam. Sys.2012, to appear.http://arxiv.org/abs/1105.3541]. We prove that various families of infinite measure-preserving rank-one transformations possess or do not posses these properties, and consider their relation to other notions of mixing in infinite measure.

Structure and predictive power of intraindividual variability in health and activity measures

Intraindividual variability has been shown in many studies to be well structured over time and capable of predicting a wide range of behaviors. This study provides analyses of weekly variations of health and activity measurements in a sample (N = 52) of elderly persons (mean age = 77.5, SD = 7.2). The three main goals and results entailed: 1. Assessing the reliable nature of a health and an activity factor through a metric invariant confirmatory factor model; 2. Assessing the independence of intraindividual variability from level information and showing that the amount of intra-person variability during the first and second half of data collection correlate very highly; 3. Showing that intraindividual variability information of health and activity scores is superior to level information in predicting mortality.