31,465 research outputs found
Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix
In this work we construct an optimal shrinkage estimator for the precision
matrix in high dimensions. We consider the general asymptotics when the number
of variables and the sample size so
that . The precision matrix is estimated
directly, without inverting the corresponding estimator for the covariance
matrix. The recent results from the random matrix theory allow us to find the
asymptotic deterministic equivalents of the optimal shrinkage intensities and
estimate them consistently. The resulting distribution-free estimator has
almost surely the minimum Frobenius loss. Additionally, we prove that the
Frobenius norms of the inverse and of the pseudo-inverse sample covariance
matrices tend almost surely to deterministic quantities and estimate them
consistently. At the end, a simulation is provided where the suggested
estimator is compared with the estimators for the precision matrix proposed in
the literature. The optimal shrinkage estimator shows significant improvement
and robustness even for non-normally distributed data.Comment: 26 pages, 5 figures. This version includes the case c>1 with the
generalized inverse of the sample covariance matrix. The abstract was updated
accordingl
Evolution of the social network of scientific collaborations
The co-authorship network of scientists represents a prototype of complex
evolving networks.
By mapping the electronic database containing all relevant journals in
mathematics and neuro-science for an eight-year period (1991-98), we infer the
dynamic and the structural mechanisms that govern the evolution and topology of
this complex system.
First, empirical measurements allow us to uncover the topological measures
that characterize the network at a given moment, as well as the time evolution
of these quantities.
The results indicate that the network is scale-free, and that the network
evolution is governed by preferential attachment, affecting both internal and
external links.
However, in contrast with most model predictions the average degree increases
in time, and the node separation decreases.
Second, we propose a simple model that captures the network's time evolution.
Third, numerical simulations are used to uncover the behavior of quantities
that could not be predicted analytically.Comment: 14 pages, 15 figure
An Improvement in a Local Observer Design for Optimal State Feedback Control: The Case Study of HIV/AIDS Diffusion
The paper addresses the problem of an observer design for a nonlinear system for which a preliminary linear
state feedback is designed but the full state is not measurable. Since a linear control assures the fulfilment of
local approximated conditions, usually a linear observer is designed in these cases to estimate the state with
estimation error locally convergent to zero. The case in which the control contains an external reference, like
in regulations problems, is studied, showing that the solution obtained working with the linear approximation
to get local solutions produces non consistent results in terms of local regions of convergence for the system
and for the observer. A solution to this problem is provided, proposing a different choice for the observer
design which allows to obtain all conditions locally satisfied on the same local region in the neighbourhood of
a new equilibrium point. The case study of an epidemic spread control is used to show the effectiveness of the
procedure. The linear control with regulation term is present in this case because the problem is reconducted to
a Linear Quadratic Regulation problem. Simulation results show the differences between the two approaches
and the effectiveness of the proposed on
Breakdown of the standard Perturbation Theory and Moving Boundary Approximation for "Pulled" Fronts
The derivation of a Moving Boundary Approximation or of the response of a
coherent structure like a front, vortex or pulse to external forces and noise,
is generally valid under two conditions: the existence of a separation of time
scales of the dynamics on the inner and outer scale and the existence and
convergence of solvability type integrals. We point out that these conditions
are not satisfied for pulled fronts propagating into an unstable state: their
relaxation on the inner scale is power law like and in conjunction with this,
solvability integrals diverge. The physical origin of this is traced to the
fact that the important dynamics of pulled fronts occurs in the leading edge of
the front rather than in the nonlinear internal front region itself. As recent
work on the relaxation and stochastic behavior of pulled fronts suggests, when
such fronts are coupled to other fields or to noise, the dynamical behavior is
often qualitatively different from the standard case in which fronts between
two (meta)stable states or pushed fronts propagating into an unstable state are
considered.Comment: pages Latex, submitted to a special issue of Phys. Rep. in dec. 9
Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows
Published versio
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