1,720,992 research outputs found
Topics in Information Geometry
No full textWe introduce first some of the background ideas on information theory and its role in studying
analytic models for stochastic processes and the geometrization of families of measure functions.
This is then used to present the geometry of important examples of the Riemannian manifolds that
arise. Next, we obtain the proof of two theorems that characterise the metric neighbourhoods of the
two distinguished fundamental states: randomness and independence. These methods have had
applications in modelling cryptographic attacks, cosmological void distributions, porous media,
clustering of: galaxies, communications, and amino acids along protein chains in genomes
An inhomogeneous stochastic rate process for evolution from states in an information geometric neighbourhood of uniform fitness
Full text linkThis study elaborates some examples of a simple evolutionary stochastic rate process
where the population rate of change depends on the distribution of properties---so
different cohorts change at different rates. We investigate
the effect on the evolution arising from parametrized perturbations of
uniformity for the initial inhomogeneity. The information geometric
neighbourhood system yields also solutions
for a wide range of other initial inhomogeneity distributions,
including approximations to truncated Gaussians of arbitrarily small variance
and distributions with pronounced extreme values.
It is found that, under quite
considerable alterations in the shape and variance of the initial distribution of inhomogeneity
in unfitness, the decline of the mean does change markedly with the variation in starting conditions,
but the net population evolution seems surprisingly stable
A short review on Landsberg spaces
Full text linkThis short review is concerned with real finite-dimensional Finsler manifolds
(M,F) with Finsler structures F:TM-->[0,infty) that
satisfy the Landsberg conditions. In particular this includes the case of
Berwald manifolds since their Chern connections on the pullback of TM are fibre-independent.
The aim is to provide an annotated collection of references to geometric
results that seem important in the study of Landsberg spaces and to suggest some
areas for further work in this context
An inhomogeneous stochastic rate process for evolution from states in an information geometric neighbourhood of uniform fitness
No full textThis study elaborates some examples of a simple evolutionary stochastic rate processwhere the population rate of change depends on the distribution of properties---sodifferent cohorts change at different rates. We investigatethe effect on the evolution arising from parametrized perturbations ofuniformity for the initial inhomogeneity. The information geometricneighbourhood system yields also solutionsfor a wide range of other initial inhomogeneity distributions,including approximations to truncated Gaussians of arbitrarily small varianceand distributions with pronounced extreme values.It is found that, under quiteconsiderable alterations in the shape and variance of the initial distribution of inhomogeneityin unfitness, the decline of the mean does change markedly with the variation in starting conditions,but the net population evolution seems surprisingly stable
An approach to protein structure using information geometry
Full text linkIn the light of recent structural developments in DNA structural diversity crystallographic studies and the Protein Data Bank*, this note is intended to draw attention to an interesting feature of the ordering of amino acids along protein chains. They all exhibited clustering compared to a random
distribution, so there is a stable long range ordering that is unexpected. To date
we have no clear explanation of why this should be the case.
* https://doi.org/10.1016/j.jbc.2021.100553
Information geometry for control of some stochastic processes
No full textA basic requirement in control systems is a metric that measures discrepancies between actual and desired states.
For statistically influenced systems information geometric methods provide natural
Riemannian metrics on smooth spaces of states; such manifolds arise in minimum-phase
linear systems and multi-input systems with known stochastic noise.
Commonly recurring practical situations
are `nearly' Poisson or `nearly' Uniform with
a complementarity in the geometry of these two; another involves multivariate Gaussians
and their mixtures.
Similarly we
encounter `nearly' independent Poisson, and `nearly' independent Gaussian processes. For such cases we have information geometric results and examples.
Some of these methods are applicable to control systems for statistically influenced processes, such as monitoring essential features in continuous
production of threads, films, foils and
fibre networks, and batch processing of stochastic textures
On the entropy flows to disorder
Full text linkGamma distributions, which contain the exponential as a
special case, have a distinguished place in the representation of
near-Poisson randomness for statistical processes; typically, they represent
distributions of spacings between events or voids among objects.
Here we look at the properties of the
Shannon entropy function and calculate its corresponding flow curves, relating
them to examples of constrained degeneration from ordered processes.
We consider also univariate and bivariate gamma, as well as Weibull distributions
since these include exponential distributions
Information geometry and entropy in a stochastic epidemic rate process
Full text linkA commonly recurring approximation to real rate processes is of the form:
dN/dt = -m N
where m is some positive rate constant and N(t) measures the current value of some
property relevant to the process---radioactive decay is our typical student example.
The simplest stochastic version addresses the situation where N(t) is the size of
the current population and the rate constant depends on the distribution of properties
in the population---so different sections decay at different rates. Then the interest
lies in the evolution of the distribution of properties and of the related statistical
features like entropy, mean and variance, for given initial distribution. We show that there
is a simple closed solution for an example of an epidemic in which the latency
and infectivity are distributed properties controlled by a bivariate gamma distribution
Information distance estimation between mixtures of multivariate Gaussians
Full text linkThere are efficient software programs for
extracting from image sequences certain mixtures of distributions, such as
multivariate Gaussians, to
represent the important features needed for accurate document retrieval from databases.
This note describes a method to use information geometric methods to measure distances
between distributions in mixtures of multivariate Gaussians.
There is no general analytic solution for the information
geodesic distance between two k-variate Gaussians,
but for many purposes the absolute information distance is not essential and comparative
values suffice for proximity testing.
For two mixtures of multivariate Gaussians
we must resort to approximations to incorporate the weightings.
In practice, the relation between
a reasonable approximation and a true geodesic distance is likely to be monotonic, which
is adequate for many applications. Here we compare several choices for the incorporation of
weightings in distance estimation and provide illustrative results from simulations of
differently weighted mixtures of multivariate Gaussians
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