71,126 research outputs found
An Exploration of the Role of Principal Inertia Components in Information Theory
The principal inertia components of the joint distribution of two random
variables and are inherently connected to how an observation of is
statistically related to a hidden variable . In this paper, we explore this
connection within an information theoretic framework. We show that, under
certain symmetry conditions, the principal inertia components play an important
role in estimating one-bit functions of , namely , given an
observation of . In particular, the principal inertia components bear an
interpretation as filter coefficients in the linear transformation of
into . This interpretation naturally leads to the
conjecture that the mutual information between and is maximized when
all the principal inertia components have equal value. We also study the role
of the principal inertia components in the Markov chain , where and are binary
random variables. We illustrate our results for the setting where and
are binary strings and is the result of sending through an additive
noise binary channel.Comment: Submitted to the 2014 IEEE Information Theory Workshop (ITW
Bounds on inference
Lower bounds for the average probability of error of estimating a hidden
variable X given an observation of a correlated random variable Y, and Fano's
inequality in particular, play a central role in information theory. In this
paper, we present a lower bound for the average estimation error based on the
marginal distribution of X and the principal inertias of the joint distribution
matrix of X and Y. Furthermore, we discuss an information measure based on the
sum of the largest principal inertias, called k-correlation, which generalizes
maximal correlation. We show that k-correlation satisfies the Data Processing
Inequality and is convex in the conditional distribution of Y given X. Finally,
we investigate how to answer a fundamental question in inference and privacy:
given an observation Y, can we estimate a function f(X) of the hidden random
variable X with an average error below a certain threshold? We provide a
general method for answering this question using an approach based on
rate-distortion theory.Comment: Allerton 2013 with extended proof, 10 page
SOM-based algorithms for qualitative variables
It is well known that the SOM algorithm achieves a clustering of data which
can be interpreted as an extension of Principal Component Analysis, because of
its topology-preserving property. But the SOM algorithm can only process
real-valued data. In previous papers, we have proposed several methods based on
the SOM algorithm to analyze categorical data, which is the case in survey
data. In this paper, we present these methods in a unified manner. The first
one (Kohonen Multiple Correspondence Analysis, KMCA) deals only with the
modalities, while the two others (Kohonen Multiple Correspondence Analysis with
individuals, KMCA\_ind, Kohonen algorithm on DISJonctive table, KDISJ) can take
into account the individuals, and the modalities simultaneously.Comment: Special Issue apr\`{e}s WSOM 03 \`{a} Kitakiush
Spacecraft dynamics under the action of Y-dot magnetic control law
The paper investigates the dynamic behavior of a spacecraft when a single magnetic torque-rod is used for achieving a pure spin condition by means of the so-called Y-dot control law. Global asymptotic convergence to a pure spin condition is proven on analytical grounds when the dipole moment is proportional to the rate of variation of the component of the magnetic field along the desired spin axis. Convergence of the spin axis towards the orbit normal is then explained by estimating the average magnetic control torque over one orbit. The validity of the analytical results, based on some simplifying assumptions and approximations, is finally investigated by means of numerical simulation for a fully non-linear attitude dynamic model, featuring a tilted dipole model for Earth׳s magnetic field. The analysis aims to support, in the framework of a sound mathematical basis, the development of effective control laws in realistic mission scenarios. Results are presented and discussed for relevant test cases
Motions about a fixed point by hypergeometric functions: new non-complex analytical solutions and integration of the herpolhode
We study four problems in the dynamics of a body moving about a fixed point,
providing a non-complex, analytical solution for all of them. For the first
two, we will work on the motion first integrals. For the symmetrical heavy
body, that is the Lagrange-Poisson case, we compute the second and third Euler
angles in explicit and real forms by means of multiple hypergeometric functions
(Lauricella, functions). Releasing the weight load but adding the complication
of the asymmetry, by means of elliptic integrals of third kind, we provide the
precession angle completing some previous treatments of the Euler-Poinsot case.
Integrating then the relevant differential equation, we reach the finite polar
equation of a special trajectory named the {\it herpolhode}. In the last
problem we keep the symmetry of the first problem, but without the weight, and
take into account a viscous dissipation. The approach of first integrals is no
longer practicable in this situation and the Euler equations are faced directly
leading to dumped goniometric functions obtained as particular occurrences of
Bessel functions of order .Comment: This is a pre-print of an article published in Celestial Mechanics
and Dynamical Astronomy. The final authenticated version is available online
at: DOI: 10.1007/s10569-018-9837-
Precession of a Freely Rotating Rigid Body. Inelastic Relaxation in the Vicinity of Poles
When a solid body is freely rotating at an angular velocity ,
the ellipsoid of constant angular momentum, in the space , has poles corresponding to spinning about the minimal-inertia and
maximal-inertia axes. The first pole may be considered stable if we neglect the
inner dissipation, but becomes unstable if the dissipation is taken into
account. This happens because the bodies dissipate energy when they rotate
about any axis different from principal. In the case of an oblate symmetrical
body, the angular velocity describes a circular cone about the vector of
(conserved) angular momentum. In the course of relaxation, the angle of this
cone decreases, so that both the angular velocity and the maximal-inertia axis
of the body align along the angular momentum. The generic case of an asymmetric
body is far more involved. Even the symmetrical prolate body exhibits a
sophisticated behaviour, because an infinitesimally small deviation of the
body's shape from a rotational symmetry (i.e., a small difference between the
largest and second largest moments of inertia) yields libration: the precession
trajectory is not a circle but an ellipse. In this article we show that often
the most effective internal dissipation takes place at twice the frequency of
the body's precession. Applications to precessing asteroids, cosmic-dust
alignment, and rotating satellites are discussed.Comment: 47 pages, 1 figur
Tidal End States of Binary Asteroid Systems with a Nonspherical Component
We derive the locations of the fully synchronous end states of tidal
evolution for binary asteroid systems having one spherical component and one
oblate- or prolate-spheroid component. Departures from a spherical shape, at
levels observed among binary asteroids, can result in the lack of a stable
tidal end state for particular combinations of the system mass fraction and
angular momentum, in which case the binary must collapse to contact. We
illustrate our analytical results with near-Earth asteroids (8567) 1996 HW1,
(66391) 1999 KW4, and 69230 Hermes.Comment: 13 pages, 3 figures, published in Icaru
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