16,666 research outputs found
On the Spectral Properties of Matrices Associated with Trend Filters
This paper is concerned with the spectral properties of matrices associated
with linear filters for the estimation of the underlying trend of a time
series. The interest lies in the fact that the eigenvectors can be interpreted
as the latent components of any time series that the filter smooths through the
corresponding eigenvalues. A difficulty arises because matrices associated with
trend filters are finite approximations of Toeplitz operators and therefore
very little is known about their eigenstructure, which also depends on the
boundary conditions or, equivalently, on the filters for trend estimation at
the end of the sample. Assuming reflecting boundary conditions, we derive a
time series decomposition in terms of periodic latent components and
corresponding smoothing eigenvalues. This decomposition depends on the local
polynomial regression estimator chosen for the interior. Otherwise, the
eigenvalue distribution is derived with an approximation measured by the size
of the perturbation that different boundary conditions apport to the
eigenvalues of matrices belonging to algebras with known spectral properties,
such as the Circulant or the Cosine. The analytical form of the eigenvectors is
then derived with an approximation that involves the extremes only. A further
topic investigated in the paper concerns a strategy for a filter design in the
time domain. Based on cut-off eigenvalues, new estimators are derived, that are
less variable and almost equally biased as the original estimator, based on all
the eigenvalues. Empirical examples illustrate the effectiveness of the method
Consistent thermodynamic derivative estimates for tabular equations of state
Numerical simulations of compressible fluid flows require an equation of
state (EOS) to relate the thermodynamic variables of density, internal energy,
temperature, and pressure. A valid EOS must satisfy the thermodynamic
conditions of consistency (derivation from a free energy) and stability
(positive sound speed squared). When phase transitions are significant, the EOS
is complicated and can only be specified in a table. For tabular EOS's such as
SESAME from Los Alamos National Laboratory, the consistency and stability
conditions take the form of a differential equation relating the derivatives of
pressure and energy as functions of temperature and density, along with
positivity constraints. Typical software interfaces to such tables based on
polynomial or rational interpolants compute derivatives of pressure and energy
and may enforce the stability conditions, but do not enforce the consistency
condition and its derivatives. We describe a new type of table interface based
on a constrained local least squares regression technique. It is applied to
several SESAME EOS's showing how the consistency condition can be satisfied to
round-off while computing first and second derivatives with demonstrated
second-order convergence. An improvement of 14 orders of magnitude over
conventional derivatives is demonstrated, although the new method is apparently
two orders of magnitude slower, due to the fact that every evaluation requires
solving an 11-dimensional nonlinear system.Comment: 29 pages, 9 figures, 16 references, submitted to Phys Rev
Curriculum Guidelines for Undergraduate Programs in Data Science
The Park City Math Institute (PCMI) 2016 Summer Undergraduate Faculty Program
met for the purpose of composing guidelines for undergraduate programs in Data
Science. The group consisted of 25 undergraduate faculty from a variety of
institutions in the U.S., primarily from the disciplines of mathematics,
statistics and computer science. These guidelines are meant to provide some
structure for institutions planning for or revising a major in Data Science
Generalized structured additive regression based on Bayesian P-splines
Generalized additive models (GAM) for modelling nonlinear effects of continuous covariates are now well established tools for the applied statistician. In this paper we develop Bayesian GAM's and extensions to generalized structured additive regression based on one or two dimensional P-splines as the main building block. The approach extends previous work by Lang und Brezger (2003) for Gaussian responses. Inference relies on Markov chain Monte Carlo (MCMC) simulation techniques, and is either based on iteratively weighted least squares (IWLS) proposals or on latent utility representations of (multi)categorical regression models. Our approach covers the most common univariate response distributions, e.g. the Binomial, Poisson or Gamma distribution, as well as multicategorical responses. For the first time, we present Bayesian semiparametric inference for the widely used multinomial logit models. As we will demonstrate through two applications on the forest health status of trees and a space-time analysis of health insurance data, the approach allows realistic modelling of complex problems. We consider the enormous flexibility and extendability of our approach as a main advantage of Bayesian inference based on MCMC techniques compared to more traditional approaches. Software for the methodology presented in the paper is provided within the public domain package BayesX
Bayesian Analysis for Penalized Spline Regression Using WinBUGS
Penalized splines can be viewed as BLUPs in a mixed model framework, which allows the use of mixed model software for smoothing. Thus, software originally developed for Bayesian analysis of mixed models can be used for penalized spline regression. Bayesian inference for nonparametric models enjoys the flexibility of nonparametric models and the exact inference provided by the Bayesian inferential machinery. This paper provides a simple, yet comprehensive, set of programs for the implementation of nonparametric Bayesian analysis in WinBUGS. Good mixing properties of the MCMC chains are obtained by using low-rank thin-plate splines, while simulation times per iteration are reduced employing WinBUGS specific computational tricks.
B-spline techniques for volatility modeling
This paper is devoted to the application of B-splines to volatility modeling,
specifically the calibration of the leverage function in stochastic local
volatility models and the parameterization of an arbitrage-free implied
volatility surface calibrated to sparse option data. We use an extension of
classical B-splines obtained by including basis functions with infinite
support. We first come back to the application of shape-constrained B-splines
to the estimation of conditional expectations, not merely from a scatter plot
but also from the given marginal distributions. An application is the Monte
Carlo calibration of stochastic local volatility models by Markov projection.
Then we present a new technique for the calibration of an implied volatility
surface to sparse option data. We use a B-spline parameterization of the
Radon-Nikodym derivative of the underlying's risk-neutral probability density
with respect to a roughly calibrated base model. We show that this method
provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch
a Galerkin method with B-spline finite elements to the solution of the partial
differential equation satisfied by the Radon-Nikodym derivative.Comment: 25 page
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