5,723 research outputs found
Direct Ensemble Estimation of Density Functionals
Estimating density functionals of analog sources is an important problem in
statistical signal processing and information theory. Traditionally, estimating
these quantities requires either making parametric assumptions about the
underlying distributions or using non-parametric density estimation followed by
integration. In this paper we introduce a direct nonparametric approach which
bypasses the need for density estimation by using the error rates of k-NN
classifiers asdata-driven basis functions that can be combined to estimate a
range of density functionals. However, this method is subject to a non-trivial
bias that dramatically slows the rate of convergence in higher dimensions. To
overcome this limitation, we develop an ensemble method for estimating the
value of the basis function which, under some minor constraints on the
smoothness of the underlying distributions, achieves the parametric rate of
convergence regardless of data dimension.Comment: 5 page
Nonparametric estimation of infinitely divisible distributions based on variational analysis on measures
The paper develops new methods of non-parametric estimation a compound
Poisson distribution. Such a problem arise, in particular, in the inference of
a Levy process recorded at equidistant time intervals. Our key estimator is
based on series decomposition of functionals of a measure and relies on the
steepest descent technique recently developed in variational analysis of
measures. Simulation studies demonstrate applicability domain of our methods
and how they positively compare and complement the existing techniques. They
are particularly suited for discrete compounding distributions, not necessarily
concentrated on a grid nor on the positive or negative semi-axis. They also
give good results for continuous distributions provided an appropriate
smoothing is used for the obtained atomic measure
On Bayesian based adaptive confidence sets for linear functionals
We consider the problem of constructing Bayesian based confidence sets for
linear functionals in the inverse Gaussian white noise model. We work with a
scale of Gaussian priors indexed by a regularity hyper-parameter and apply the
data-driven (slightly modified) marginal likelihood empirical Bayes method for
the choice of this hyper-parameter. We show by theory and simulations that the
credible sets constructed by this method have sub-optimal behaviour in general.
However, by assuming "self-similarity" the credible sets have rate-adaptive
size and optimal coverage. As an application of these results we construct
-credible bands for the true functional parameter with adaptive
size and optimal coverage under self-similarity constraint.Comment: 11 pages, 2 figure
Sieve Inference on Semi-nonparametric Time Series Models
The method of sieves has been widely used in estimating semiparametric and nonparametric models. In this paper, we first provide a general theory on the asymptotic normality of plug-in sieve M estimators of possibly irregular functionals of semi/nonparametric time series models. Next, we establish a surprising result that the asymptotic variances of plug-in sieve M estimators of irregular (i.e., slower than root-T estimable) functionals do not depend on temporal dependence. Nevertheless, ignoring the temporal dependence in small samples may not lead to accurate inference. We then propose an easy-to-compute and more accurate inference procedure based on a "pre-asymptotic" sieve variance estimator that captures temporal dependence. We construct a "pre-asymptotic" Wald statistic using an orthonormal series long run variance (OS-LRV) estimator. For sieve M estimators of both regular (i.e., root-T estimable) and irregular functionals, a scaled "pre-asymptotic" Wald statistic is asymptotically F distributed when the series number of terms in the OS-LRV estimator is held fixed. Simulations indicate that our scaled "pre-asymptotic" Wald test with F critical values has more accurate size in finite samples than the usual Wald test with chi-square critical values.Weak dependence, Sieve M estimation, Sieve Riesz representor, Irregular functional, Misspecification, Pre-asymptotic variance, Orthogonal series long run variance estimation, F distribution
Electronic Spectra from TDDFT and Machine Learning in Chemical Space
Due to its favorable computational efficiency time-dependent (TD) density
functional theory (DFT) enables the prediction of electronic spectra in a
high-throughput manner across chemical space. Its predictions, however, can be
quite inaccurate. We resolve this issue with machine learning models trained on
deviations of reference second-order approximate coupled-cluster singles and
doubles (CC2) spectra from TDDFT counterparts, or even from DFT gap. We applied
this approach to low-lying singlet-singlet vertical electronic spectra of over
20 thousand synthetically feasible small organic molecules with up to eight
CONF atoms. The prediction errors decay monotonously as a function of training
set size. For a training set of 10 thousand molecules, CC2 excitation energies
can be reproduced to within 0.1 eV for the remaining molecules. Analysis
of our spectral database via chromophore counting suggests that even higher
accuracies can be achieved. Based on the evidence collected, we discuss open
challenges associated with data-driven modeling of high-lying spectra, and
transition intensities
Fluctuation theorems for stochastic dynamics
Fluctuation theorems make use of time reversal to make predictions about
entropy production in many-body systems far from thermal equilibrium. Here we
review the wide variety of distinct, but interconnected, relations that have
been derived and investigated theoretically and experimentally. Significantly,
we demonstrate, in the context of Markovian stochastic dynamics, how these
different fluctuation theorems arise from a simple fundamental time-reversal
symmetry of a certain class of observables. Appealing to the notion of Gibbs
entropy allows for a microscopic definition of entropy production in terms of
these observables. We work with the master equation approach, which leads to a
mathematically straightforward proof and provides direct insight into the
probabilistic meaning of the quantities involved. Finally, we point to some
experiments that elucidate the practical significance of fluctuation relations.Comment: 48 pages, 2 figures. v2: minor changes for consistency with published
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