7,066 research outputs found
Fast Covariance Estimation for High-dimensional Functional Data
For smoothing covariance functions, we propose two fast algorithms that scale
linearly with the number of observations per function. Most available methods
and software cannot smooth covariance matrices of dimension with
; the recently introduced sandwich smoother is an exception, but it is
not adapted to smooth covariance matrices of large dimensions such as . Covariance matrices of order , and even , are
becoming increasingly common, e.g., in 2- and 3-dimensional medical imaging and
high-density wearable sensor data. We introduce two new algorithms that can
handle very large covariance matrices: 1) FACE: a fast implementation of the
sandwich smoother and 2) SVDS: a two-step procedure that first applies singular
value decomposition to the data matrix and then smoothes the eigenvectors.
Compared to existing techniques, these new algorithms are at least an order of
magnitude faster in high dimensions and drastically reduce memory requirements.
The new algorithms provide instantaneous (few seconds) smoothing for matrices
of dimension and very fast ( 10 minutes) smoothing for
. Although SVDS is simpler than FACE, we provide ready to use,
scalable R software for FACE. When incorporated into R package {\it refund},
FACE improves the speed of penalized functional regression by an order of
magnitude, even for data of normal size (). We recommend that FACE be
used in practice for the analysis of noisy and high-dimensional functional
data.Comment: 35 pages, 4 figure
Reduced-rank spatio-temporal modeling of air pollution concentrations in the Multi-Ethnic Study of Atherosclerosis and Air Pollution
There is growing evidence in the epidemiologic literature of the relationship
between air pollution and adverse health outcomes. Prediction of individual air
pollution exposure in the Environmental Protection Agency (EPA) funded
Multi-Ethnic Study of Atheroscelerosis and Air Pollution (MESA Air) study
relies on a flexible spatio-temporal prediction model that integrates land-use
regression with kriging to account for spatial dependence in pollutant
concentrations. Temporal variability is captured using temporal trends
estimated via modified singular value decomposition and temporally varying
spatial residuals. This model utilizes monitoring data from existing regulatory
networks and supplementary MESA Air monitoring data to predict concentrations
for individual cohort members. In general, spatio-temporal models are limited
in their efficacy for large data sets due to computational intractability. We
develop reduced-rank versions of the MESA Air spatio-temporal model. To do so,
we apply low-rank kriging to account for spatial variation in the mean process
and discuss the limitations of this approach. As an alternative, we represent
spatial variation using thin plate regression splines. We compare the
performance of the outlined models using EPA and MESA Air monitoring data for
predicting concentrations of oxides of nitrogen (NO)-a pollutant of primary
interest in MESA Air-in the Los Angeles metropolitan area via cross-validated
. Our findings suggest that use of reduced-rank models can improve
computational efficiency in certain cases. Low-rank kriging and thin plate
regression splines were competitive across the formulations considered,
although TPRS appeared to be more robust in some settings.Comment: Published in at http://dx.doi.org/10.1214/14-AOAS786 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Identifying Finite-Time Coherent Sets from Limited Quantities of Lagrangian Data
A data-driven procedure for identifying the dominant transport barriers in a
time-varying flow from limited quantities of Lagrangian data is presented. Our
approach partitions state space into pairs of coherent sets, which are sets of
initial conditions chosen to minimize the number of trajectories that "leak"
from one set to the other under the influence of a stochastic flow field during
a pre-specified interval in time. In practice, this partition is computed by
posing an optimization problem, which once solved, yields a pair of functions
whose signs determine set membership. From prior experience with synthetic,
"data rich" test problems and conceptually related methods based on
approximations of the Perron-Frobenius operator, we observe that the functions
of interest typically appear to be smooth. As a result, given a fixed amount of
data our approach, which can use sets of globally supported basis functions,
has the potential to more accurately approximate the desired functions than
other functions tailored to use compactly supported indicator functions. This
difference enables our approach to produce effective approximations of pairs of
coherent sets in problems with relatively limited quantities of Lagrangian
data, which is usually the case with real geophysical data. We apply this
method to three examples of increasing complexity: the first is the double
gyre, the second is the Bickley Jet, and the third is data from numerically
simulated drifters in the Sulu Sea.Comment: 14 pages, 7 figure
Rapid evaluation of radial basis functions
Over the past decade, the radial basis function method has been shown to produce high quality solutions to the multivariate scattered data interpolation problem. However, this method has been associated with very high computational cost, as compared to alternative methods such as finite element or multivariate spline interpolation. For example. the direct evaluation at M locations of a radial basis function interpolant with N centres requires O(M N) floating-point operations. In this paper we introduce a fast evaluation method based on the Fast Gauss Transform and suitable quadrature rules. This method has been applied to the Hardy multiquadric, the inverse multiquadric and the thin-plate spline to reduce the computational complexity of the interpolant evaluation to O(M + N) floating point operations. By using certain localisation properties of conditionally negative definite functions this method has several performance advantages against traditional hierarchical rapid summation methods which we discuss in detail
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.
Approximating Data with weighted smoothing Splines
Given a data set (t_i, y_i), i=1,..., n with the t_i in [0,1] non-parametric
regression is concerned with the problem of specifying a suitable function
f_n:[0,1] -> R such that the data can be reasonably approximated by the points
(t_i, f_n(t_i)), i=1,..., n. If a data set exhibits large variations in local
behaviour, for example large peaks as in spectroscopy data, then the method
must be able to adapt to the local changes in smoothness. Whilst many methods
are able to accomplish this they are less successful at adapting derivatives.
In this paper we show how the goal of local adaptivity of the function and its
first and second derivatives can be attained in a simple manner using weighted
smoothing splines. A residual based concept of approximation is used which
forces local adaptivity of the regression function together with a global
regularization which makes the function as smooth as possible subject to the
approximation constraints
Automation of isogeometric formulation and efficiency consideration
This paper deals with automation of the isogeometric finite element formulation. Isogeometric finite element is implemented in AceGen environment, which enables symbolic formulation of the element code and the expressions are automatically opti- mized. The automated code is tested for objectivity regarding numerical efficiency in a numeric test with the Cooke membrane. This test shows that automatic code generation optimizes the isogeometric quadrilateral element with linear Bezier splines to the degree of only twelve percent overhead against standard displacement quadrilateral element of four nodes. Additionaly, the automated isogeometric element code is tested on a set of standard benchmark test cases to further test the accurancy and efficiency of the pre- sented isogeometric implementation. The isogeometric displacement brick element with quadratic Bezier splines is in all tests compared to a collection of standard displacement element formulations and a
selection of EAS elements. The presented results show su- perior behaviour of the isogeometric displacement brick element with quadratic Bezier splines for coarse meshes and best convergence rate with mesh refinement in most test cases. Despite all optimization of the element code the biggest disadvantage of the isogeo- metric model remains the time cost of the isogeometric analysis. Thus, when considering the ratio between solution error and solution time, the use of stable EAS elements, likeTSCG12, remains preferable
TVL<sub>1</sub> Planarity Regularization for 3D Shape Approximation
The modern emergence of automation in many industries has given impetus to extensive research into mobile robotics. Novel perception technologies now enable cars to drive autonomously, tractors to till a field automatically and underwater robots to construct pipelines. An essential requirement to facilitate both perception and autonomous navigation is the analysis of the 3D environment using sensors like laser scanners or stereo cameras. 3D sensors generate a very large number of 3D data points when sampling object shapes within an environment, but crucially do not provide any intrinsic information about the environment which the robots operate within.
This work focuses on the fundamental task of 3D shape reconstruction and modelling from 3D point clouds. The novelty lies in the representation of surfaces by algebraic functions having limited support, which enables the extraction of smooth consistent implicit shapes from noisy samples with a heterogeneous density. The minimization of total variation of second differential degree makes it possible to enforce planar surfaces which often occur in man-made environments. Applying the new technique means that less accurate, low-cost 3D sensors can be employed without sacrificing the 3D shape reconstruction accuracy
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