2,941 research outputs found
Penalized nonparametric mean square estimation of the coefficients of diffusion processes
We consider a one-dimensional diffusion process which is observed at
discrete times with regular sampling interval . Assuming that
is strictly stationary, we propose nonparametric estimators of the
drift and diffusion coefficients obtained by a penalized least squares
approach. Our estimators belong to a finite-dimensional function space whose
dimension is selected by a data-driven method. We provide non-asymptotic risk
bounds for the estimators. When the sampling interval tends to zero while the
number of observations and the length of the observation time interval tend to
infinity, we show that our estimators reach the minimax optimal rates of
convergence. Numerical results based on exact simulations of diffusion
processes are given for several examples of models and illustrate the qualities
of our estimation algorithms.Comment: Published at http://dx.doi.org/10.3150/07-BEJ5173 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Score Function Features for Discriminative Learning: Matrix and Tensor Framework
Feature learning forms the cornerstone for tackling challenging learning
problems in domains such as speech, computer vision and natural language
processing. In this paper, we consider a novel class of matrix and
tensor-valued features, which can be pre-trained using unlabeled samples. We
present efficient algorithms for extracting discriminative information, given
these pre-trained features and labeled samples for any related task. Our class
of features are based on higher-order score functions, which capture local
variations in the probability density function of the input. We establish a
theoretical framework to characterize the nature of discriminative information
that can be extracted from score-function features, when used in conjunction
with labeled samples. We employ efficient spectral decomposition algorithms (on
matrices and tensors) for extracting discriminative components. The advantage
of employing tensor-valued features is that we can extract richer
discriminative information in the form of an overcomplete representations.
Thus, we present a novel framework for employing generative models of the input
for discriminative learning.Comment: 29 page
Solving linear parabolic rough partial differential equations
We study linear rough partial differential equations in the setting of [Friz
and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear
parabolic partial differential equation driven by a deterministic rough path
of H\"older regularity with . Based on a stochastic representation of the solution of the rough
partial differential equation, we propose a regression Monte Carlo algorithm
for spatio-temporal approximation of the solution. We provide a full
convergence analysis of the proposed approximation method which essentially
relies on the new bounds for the higher order derivatives of the solution in
space. Finally, a comprehensive simulation study showing the applicability of
the proposed algorithm is presented
Efficient Localization of Discontinuities in Complex Computational Simulations
Surrogate models for computational simulations are input-output
approximations that allow computationally intensive analyses, such as
uncertainty propagation and inference, to be performed efficiently. When a
simulation output does not depend smoothly on its inputs, the error and
convergence rate of many approximation methods deteriorate substantially. This
paper details a method for efficiently localizing discontinuities in the input
parameter domain, so that the model output can be approximated as a piecewise
smooth function. The approach comprises an initialization phase, which uses
polynomial annihilation to assign function values to different regions and thus
seed an automated labeling procedure, followed by a refinement phase that
adaptively updates a kernel support vector machine representation of the
separating surface via active learning. The overall approach avoids structured
grids and exploits any available simplicity in the geometry of the separating
surface, thus reducing the number of model evaluations required to localize the
discontinuity. The method is illustrated on examples of up to eleven
dimensions, including algebraic models and ODE/PDE systems, and demonstrates
improved scaling and efficiency over other discontinuity localization
approaches
Robust Stability Analysis of Nonlinear Hybrid Systems
We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems
- …