522 research outputs found
Self-Similar Factor Approximants
The problem of reconstructing functions from their asymptotic expansions in
powers of a small variable is addressed by deriving a novel type of
approximants. The derivation is based on the self-similar approximation theory,
which presents the passage from one approximant to another as the motion
realized by a dynamical system with the property of group self-similarity. The
derived approximants, because of their form, are named the self-similar factor
approximants. These complement the obtained earlier self-similar exponential
approximants and self-similar root approximants. The specific feature of the
self-similar factor approximants is that their control functions, providing
convergence of the computational algorithm, are completely defined from the
accuracy-through-order conditions. These approximants contain the Pade
approximants as a particular case, and in some limit they can be reduced to the
self-similar exponential approximants previously introduced by two of us. It is
proved that the self-similar factor approximants are able to reproduce exactly
a wide class of functions which include a variety of transcendental functions.
For other functions, not pertaining to this exactly reproducible class, the
factor approximants provide very accurate approximations, whose accuracy
surpasses significantly that of the most accurate Pade approximants. This is
illustrated by a number of examples showing the generality and accuracy of the
factor approximants even when conventional techniques meet serious
difficulties.Comment: 22 pages + 11 ps figure
Identification of Systems
Quasilinearization for system identification and programming strategie
A Sums-of-Squares Extension of Policy Iterations
In order to address the imprecision often introduced by widening operators in
static analysis, policy iteration based on min-computations amounts to
considering the characterization of reachable value set of a program as an
iterative computation of policies, starting from a post-fixpoint. Computing
each policy and the associated invariant relies on a sequence of numerical
optimizations. While the early research efforts relied on linear programming
(LP) to address linear properties of linear programs, the current state of the
art is still limited to the analysis of linear programs with at most quadratic
invariants, relying on semidefinite programming (SDP) solvers to compute
policies, and LP solvers to refine invariants.
We propose here to extend the class of programs considered through the use of
Sums-of-Squares (SOS) based optimization. Our approach enables the precise
analysis of switched systems with polynomial updates and guards. The analysis
presented has been implemented in Matlab and applied on existing programs
coming from the system control literature, improving both the range of
analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure
The Multivariate Watson Distribution: Maximum-Likelihood Estimation and other Aspects
This paper studies fundamental aspects of modelling data using multivariate
Watson distributions. Although these distributions are natural for modelling
axially symmetric data (i.e., unit vectors where \pm \x are equivalent), for
high-dimensions using them can be difficult. Why so? Largely because for Watson
distributions even basic tasks such as maximum-likelihood are numerically
challenging. To tackle the numerical difficulties some approximations have been
derived---but these are either grossly inaccurate in high-dimensions
(\emph{Directional Statistics}, Mardia & Jupp. 2000) or when reasonably
accurate (\emph{J. Machine Learning Research, W. & C.P., v2}, Bijral \emph{et
al.}, 2007, pp. 35--42), they lack theoretical justification. We derive new
approximations to the maximum-likelihood estimates; our approximations are
theoretically well-defined, numerically accurate, and easy to compute. We build
on our parameter estimation and discuss mixture-modelling with Watson
distributions; here we uncover a hitherto unknown connection to the
"diametrical clustering" algorithm of Dhillon \emph{et al.}
(\emph{Bioinformatics}, 19(13), 2003, pp. 1612--1619).Comment: 24 pages; extensively updated numerical result
Smooth Parametrizations in Dynamics, Analysis, Diophantine and Computational Geometry
Smooth parametrization consists in a subdivision of the mathematical objects
under consideration into simple pieces, and then parametric representation of
each piece, while keeping control of high order derivatives. The main goal of
the present paper is to provide a short overview of some results and open
problems on smooth parametrization and its applications in several apparently
rather separated domains: Smooth Dynamics, Diophantine Geometry, Approximation
Theory, and Computational Geometry.
The structure of the results, open problems, and conjectures in each of these
domains shows in many cases a remarkable similarity, which we try to stress.
Sometimes this similarity can be easily explained, sometimes the reasons remain
somewhat obscure, and it motivates some natural questions discussed in the
paper. We present also some new results, stressing interconnection between
various types and various applications of smooth parametrization
Time Delay and Noise Explaining Cyclical Fluctuations in Prices of Commodities
This paper suggests to model jointly time delay and random effects in economics and finance. It proposes to explain the random and often cyclical fluctuations in commodity prices as a consequence of the interplay between external noise and time delays caused by the time between initiation of production and delivery. The proposed model is formulated as a stochastic delay differential equation. The typical behavior of a commodity price index under this model will be discussed. Methods for parameter estimation and the evaluation of functionals will be proposed.commodity prices; stochastic delay differential equation; cyclical behavior; scenario simulation; parameter estimation; autocorrelation function
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