2,480 research outputs found
A Bayesian Multivariate Functional Dynamic Linear Model
We present a Bayesian approach for modeling multivariate, dependent
functional data. To account for the three dominant structural features in the
data--functional, time dependent, and multivariate components--we extend
hierarchical dynamic linear models for multivariate time series to the
functional data setting. We also develop Bayesian spline theory in a more
general constrained optimization framework. The proposed methods identify a
time-invariant functional basis for the functional observations, which is
smooth and interpretable, and can be made common across multivariate
observations for additional information sharing. The Bayesian framework permits
joint estimation of the model parameters, provides exact inference (up to MCMC
error) on specific parameters, and allows generalized dependence structures.
Sampling from the posterior distribution is accomplished with an efficient
Gibbs sampling algorithm. We illustrate the proposed framework with two
applications: (1) multi-economy yield curve data from the recent global
recession, and (2) local field potential brain signals in rats, for which we
develop a multivariate functional time series approach for multivariate
time-frequency analysis. Supplementary materials, including R code and the
multi-economy yield curve data, are available online
Trivariate Spline Representations for Computer Aided Design and Additive Manufacturing
Digital representations targeting design and simulation for Additive
Manufacturing (AM) are addressed from the perspective of Computer Aided
Geometric Design. We discuss the feasibility for multi-material AM for B-rep
based CAD, STL, sculptured triangles as well as trimmed and block-structured
trivariate locally refined spline representations. The trivariate spline
representations support Isogeometric Analysis (IGA), and topology structures
supporting these for CAD, IGA and AM are outlined. The ideas of (Truncated)
Hierarchical B-splines, T-splines and LR B-splines are outlined and the
approaches are compared. An example from the EC H2020 Factories of the Future
Research and Innovation Actions CAxMan illustrates both trimmed and
block-structured spline representations for IGA and AM.Comment: 30 pages, 14 figures. This project has received funding from the
European Union's Horizon 2020 research and innovation programme under grant
agreement No 68044
Functional Regression
Functional data analysis (FDA) involves the analysis of data whose ideal
units of observation are functions defined on some continuous domain, and the
observed data consist of a sample of functions taken from some population,
sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the
development of this field, which has accelerated in the past 10 years to become
one of the fastest growing areas of statistics, fueled by the growing number of
applications yielding this type of data. One unique characteristic of FDA is
the need to combine information both across and within functions, which Ramsay
and Silverman called replication and regularization, respectively. This article
will focus on functional regression, the area of FDA that has received the most
attention in applications and methodological development. First will be an
introduction to basis functions, key building blocks for regularization in
functional regression methods, followed by an overview of functional regression
methods, split into three types: [1] functional predictor regression
(scalar-on-function), [2] functional response regression (function-on-scalar)
and [3] function-on-function regression. For each, the role of replication and
regularization will be discussed and the methodological development described
in a roughly chronological manner, at times deviating from the historical
timeline to group together similar methods. The primary focus is on modeling
and methodology, highlighting the modeling structures that have been developed
and the various regularization approaches employed. At the end is a brief
discussion describing potential areas of future development in this field
Adaptive shape optimization with NURBS designs and PHT-splines for solution approximation in time-harmonic acoustics
Geometry Independent Field approximaTion (GIFT) was proposed as a
generalization of Isogeometric analysis (IGA), where different types of splines
are used for the parameterization of the computational domain and approximation
of the unknown solution. GIFT with Non-Uniform Rational B-Splines (NUBRS) for
the geometry and PHT-splines for the solution approximation were successfully
applied to problems of time-harmonic acoustics, where it was shown that in some
cases, adaptive PHT-spline mesh yields highly accurate solutions at lower
computational cost than methods with uniform refinement. Therefore, it is of
interest to investigate performance of GIFT for shape optimization problems,
where NURBS are used to model the boundary with their control points being the
design variables and PHT-splines are used to approximate the solution
adaptively to the boundary changes during the optimization process.
In this work we demonstrate the application of GIFT for 2D acoustic shape
optimization problems and, using three benchmark examples, we show that the
method yields accurate solutions with significant computational savings in
terms of the number of degrees of freedom and computational time
Computer model calibration with large non-stationary spatial outputs: application to the calibration of a climate model
Bayesian calibration of computer models tunes unknown input parameters by
comparing outputs with observations. For model outputs that are distributed
over space, this becomes computationally expensive because of the output size.
To overcome this challenge, we employ a basis representation of the model
outputs and observations: we match these decompositions to carry out the
calibration efficiently. In the second step, we incorporate the non-stationary
behaviour, in terms of spatial variations of both variance and correlations, in
the calibration. We insert two integrated nested Laplace
approximation-stochastic partial differential equation parameters into the
calibration. A synthetic example and a climate model illustration highlight the
benefits of our approach
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