237 research outputs found
A Bayes interpretation of stacking for M-complete and M-open settings
In M-open problems where no true model can be conceptualized, it is common to
back off from modeling and merely seek good prediction. Even in M-complete
problems, taking a predictive approach can be very useful. Stacking is a model
averaging procedure that gives a composite predictor by combining individual
predictors from a list of models using weights that optimize a cross-validation
criterion. We show that the stacking weights also asymptotically minimize a
posterior expected loss. Hence we formally provide a Bayesian justification for
cross-validation. Often the weights are constrained to be positive and sum to
one. For greater generality, we omit the positivity constraint and relax the
`sum to one' constraint.
A key question is `What predictors should be in the average?' We first verify
that the stacking error depends only on the span of the models. Then we propose
using bootstrap samples from the data to generate empirical basis elements that
can be used to form models. We use this in two computed examples to give
stacking predictors that are (i) data driven, (ii) optimal with respect to the
number of component predictors, and (iii) optimal with respect to the weight
each predictor gets.Comment: 37 pages, 2 figure
Pseudo generators of spatial transfer operators
Metastable behavior in dynamical systems may be a significant challenge for a
simulation based analysis. In recent years, transfer operator based approaches
to problems exhibiting metastability have matured. In order to make these
approaches computationally feasible for larger systems, various reduction
techniques have been proposed: For example, Sch\"utte introduced a spatial
transfer operator which acts on densities on configuration space, while Weber
proposed to avoid trajectory simulation (like Froyland et al.) by considering a
discrete generator.
In this manuscript, we show that even though the family of spatial transfer
operators is not a semigroup, it possesses a well defined generating structure.
What is more, the pseudo generators up to order 4 in the Taylor expansion of
this family have particularly simple, explicit expressions involving no
momentum averaging. This makes collocation methods particularly easy to
implement and computationally efficient, which in turn may open the door for
further efficiency improvements in, e.g., the computational treatment of
conformation dynamics. We experimentally verify the predicted properties of
these pseudo generators by means of two academic examples
The Law of Large Numbers in a Metric Space with a Convex Combination Operation
We consider a separable complete metric space equipped with a convex combination operation. For such spaces, we identify the corresponding convexification operator and show that the invariant elements for this operator appear naturally as limits in the strong law of large numbers. It is shown how to uplift the suggested construction to work with subsets of the basic space in order to develop a systematic way of proving laws of large numbers for such operations with random set
Set-Valued Analysis
This Special Issue contains eight original papers with a high impact in various domains of set-valued analysis. Set-valued analysis has made remarkable progress in the last 70 years, enriching itself continuously with new concepts, important results, and special applications. Different problems arising in the theory of control, economics, game theory, decision making, nonlinear programming, biomathematics, and statistics have strengthened the theoretical base and the specific techniques of set-valued analysis. The consistency of its theoretical approach and the multitude of its applications have transformed set-valued analysis into a reference field of modern mathematics, which attracts an impressive number of researchers
Generalized Lebesgue integral
AbstractA new definition of integral-like functionals exploiting the ideas of the Lebesgue integral construction and extending the idea of pan-integrals is given. Some convergence theorems for sequence of measurable functions are discussed. As a result, a theoretical basis for applications of the generalized Lebesgue integral is provided. Several types of integrals known from the literature are shown to be special cases of generalized Lebesgue integral
Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning
One of the main challenges in molecular dynamics is overcoming the ‘timescale barrier’: in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behaviour on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory, as well as the algorithmic development, from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in molecular dynamics. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject
Approximation Theory and Related Applications
In recent years, we have seen a growing interest in various aspects of approximation theory. This happened due to the increasing complexity of mathematical models that require computer calculations and the development of the theoretical foundations of the approximation theory. Approximation theory has broad and important applications in many areas of mathematics, including functional analysis, differential equations, dynamical systems theory, mathematical physics, control theory, probability theory and mathematical statistics, and others. Approximation theory is also of great practical importance, as approximate methods and estimation of approximation errors are used in physics, economics, chemistry, signal theory, neural networks and many other areas. This book presents the works published in the Special Issue "Approximation Theory and Related Applications". The research of the world’s leading scientists presented in this book reflect new trends in approximation theory and related topics
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