481 research outputs found
Non-equispaced B-spline wavelets
This paper has three main contributions. The first is the construction of
wavelet transforms from B-spline scaling functions defined on a grid of
non-equispaced knots. The new construction extends the equispaced,
biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new
construction is based on the factorisation of wavelet transforms into lifting
steps. The second and third contributions are new insights on how to use these
and other wavelets in statistical applications. The second contribution is
related to the bias of a wavelet representation. It is investigated how the
fine scaling coefficients should be derived from the observations. In the
context of equispaced data, it is common practice to simply take the
observations as fine scale coefficients. It is argued in this paper that this
is not acceptable for non-interpolating wavelets on non-equidistant data.
Finally, the third contribution is the study of the variance in a
non-orthogonal wavelet transform in a new framework, replacing the numerical
condition as a measure for non-orthogonality. By controlling the variances of
the reconstruction from the wavelet coefficients, the new framework allows us
to design wavelet transforms on irregular point sets with a focus on their use
for smoothing or other applications in statistics.Comment: 42 pages, 2 figure
Splines and vector splines
The thrust of this report concerns spline theory and some of the background to spline theory and follows the development in (Wahba, 1991). We also review methods for determining hyper-parameters, such as the smoothing parameter, by Generalised Cross Validation. Splines have an advantage over Gaussian Process based procedures in that we can readily impose atmospherically sensible smoothness constraints and maintain computational efficiency. Vector splines enable us to penalise gradients of vorticity and divergence in wind fields. Two similar techniques are summarised and improvements based on robust error functions and restricted numbers of basis functions given. A final, brief discussion of the application of vector splines to the problem of scatterometer data assimilation highlights the problems of ambiguous solutions
Splines and local approximation of the earth's gravity field
Bibliography: pages 214-220.The Hilbert space spline theory of Delvos and Schempp, and the reproducing kernel theory of L. Schwartz, provide the conceptual foundation and the construction procedure for rotation-invariant splines on Euclidean spaces, splines on the circle, and splines on the sphere and harmonic outside the sphere. Spherical splines and surface splines such as multi-conic functions, Hardy's multiquadric functions, pseudo-cubic splines, and thin-plate splines, are shown to be largely as effective as least squares collocation in representing geoid heights or gravity anomalies. A pseudo-cubic spline geoid for southern Africa is given, interpolating Doppler-derived geoid heights and astro-geodetic deflections of the vertical. Quadrature rules are derived for the thin-plate spline approximation (over a circular disk, and to a planar approximation) of Stokes's formula, the formulae of Vening Meinesz, and the L₁ vertical gradient operator in the analytical continuation series solution of Molodensky's problem
An approach to construct a three-dimensional isogeometric model from µ-CT scan data with an application to the bridge of a violin
We present an algorithm to build a ready to use isogeometric model from scan data gained by a µ-CT scan. Based on a three-dimensional multi-patch reference geometry, which includes the major topological features, we fit the outline, then the cross-section and finally the three-dimensional geometry. The key step is to fit the outline, where a non-linear least squares problem is solved with a Gauss-Newton approach presented by Borges and Pastva (2002). We extend this approach by a regularisation and a precise interpolation of selected data points. The resulting NURBS geometry is ready for applying isogeometric analysis tools for efficient numerical simulations. As a particular example we examine the scan data of a violin bridge and present the complete workflow from the µ-CT scan up to the numerical simulation based on isogeometric mortar methods. We illustrate the relevance of the constructed geometry with a vibro-acoustical application
Penalized Likelihood and Bayesian Function Selection in Regression Models
Challenging research in various fields has driven a wide range of
methodological advances in variable selection for regression models with
high-dimensional predictors. In comparison, selection of nonlinear functions in
models with additive predictors has been considered only more recently. Several
competing suggestions have been developed at about the same time and often do
not refer to each other. This article provides a state-of-the-art review on
function selection, focusing on penalized likelihood and Bayesian concepts,
relating various approaches to each other in a unified framework. In an
empirical comparison, also including boosting, we evaluate several methods
through applications to simulated and real data, thereby providing some
guidance on their performance in practice
Splinets 1.5.0 -- Periodic Splinets
Periodic splines are a special kind of splines that are defined over a set of
knots over a circle and are adequate for solving interpolation problems related
to closed curves. This paper presents a method of implementing the objects
representing such splines and describes how an efficient orthogonal basis can
be obtained. The proposed orthonormalized basis is called a periodic splinet in
the reference to earlier work where analogous concepts and tools have been
introduced for splines on an interval. Based on this methodology, the periodic
splines and splinets are added to the earlier version of the R package
"Splinets". Moreover, the developed computational tools have been applied to
functionally analyze a standard example of functional circular data of wind
directions and speeds.Comment: 16 page
Structured penalties for functional linear models---partially empirical eigenvectors for regression
One of the challenges with functional data is incorporating spatial
structure, or local correlation, into the analysis. This structure is inherent
in the output from an increasing number of biomedical technologies, and a
functional linear model is often used to estimate the relationship between the
predictor functions and scalar responses. Common approaches to the ill-posed
problem of estimating a coefficient function typically involve two stages:
regularization and estimation. Regularization is usually done via dimension
reduction, projecting onto a predefined span of basis functions or a reduced
set of eigenvectors (principal components). In contrast, we present a unified
approach that directly incorporates spatial structure into the estimation
process by exploiting the joint eigenproperties of the predictors and a linear
penalty operator. In this sense, the components in the regression are
`partially empirical' and the framework is provided by the generalized singular
value decomposition (GSVD). The GSVD clarifies the penalized estimation process
and informs the choice of penalty by making explicit the joint influence of the
penalty and predictors on the bias, variance, and performance of the estimated
coefficient function. Laboratory spectroscopy data and simulations are used to
illustrate the concepts.Comment: 29 pages, 3 figures, 5 tables; typo/notational errors edited and
intro revised per journal review proces
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