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Automatic, computer aided geometric design of free-knot, regression splines
A new algorithm for Computer Aided Geometric Design of least squares (LS) splines with variable knots, named GeDS, is presented. It is based on interpreting functional spline regression as a parametric B-spline curve, and on using the shape preserving property of its control polygon. The GeDS algorithm includes two major stages. For the first stage, an automatic adaptive, knot location algorithm is developed. By adding knots, one at a time, it sequentially "breaks" a straight line segment into pieces in order to construct a linear LS B-spline fit, which captures the "shape" of the data. A stopping rule is applied which avoids both over and under fitting and selects the number of knots for the second stage of GeDS, in which smoother, higher order (quadratic, cubic, etc.) fits are generated. The knots appropriate for the second stage are determined, according to a new knot location method, called the averaging method. It approximately preserves the linear precision property of B-spline curves and allows the attachment of smooth higher order LS B-spline fits to a control polygon, so that the shape of the linear polygon of stage one is followed. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline regression functions. The GeDS algorithm is very fast, since no deterministic or stochastic knot insertion/deletion and relocation search strategies are involved, neither in the first nor the second stage. Extensive numerical examples are provided, illustrating the performance of GeDS and the quality of the resulting LS spline fits. The GeDS procedure is compared with other existing variable knot spline methods and smoothing techniques, such as SARS, HAS, MDL, AGS methods and is shown to produce models with fewer parameters but with similar goodness of fit characteristics, and visual quality
Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation
In this paper, we present the optimization formulation of the Kalman
filtering and smoothing problems, and use this perspective to develop a variety
of extensions and applications. We first formulate classic Kalman smoothing as
a least squares problem, highlight special structure, and show that the classic
filtering and smoothing algorithms are equivalent to a particular algorithm for
solving this problem. Once this equivalence is established, we present
extensions of Kalman smoothing to systems with nonlinear process and
measurement models, systems with linear and nonlinear inequality constraints,
systems with outliers in the measurements or sudden changes in the state, and
systems where the sparsity of the state sequence must be accounted for. All
extensions preserve the computational efficiency of the classic algorithms, and
most of the extensions are illustrated with numerical examples, which are part
of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure
Fast Ensemble Smoothing
Smoothing is essential to many oceanographic, meteorological and hydrological
applications. The interval smoothing problem updates all desired states within
a time interval using all available observations. The fixed-lag smoothing
problem updates only a fixed number of states prior to the observation at
current time. The fixed-lag smoothing problem is, in general, thought to be
computationally faster than a fixed-interval smoother, and can be an
appropriate approximation for long interval-smoothing problems. In this paper,
we use an ensemble-based approach to fixed-interval and fixed-lag smoothing,
and synthesize two algorithms. The first algorithm produces a linear time
solution to the interval smoothing problem with a fixed factor, and the second
one produces a fixed-lag solution that is independent of the lag length.
Identical-twin experiments conducted with the Lorenz-95 model show that for lag
lengths approximately equal to the error doubling time, or for long intervals
the proposed methods can provide significant computational savings. These
results suggest that ensemble methods yield both fixed-interval and fixed-lag
smoothing solutions that cost little additional effort over filtering and model
propagation, in the sense that in practical ensemble application the additional
increment is a small fraction of either filtering or model propagation costs.
We also show that fixed-interval smoothing can perform as fast as fixed-lag
smoothing and may be advantageous when memory is not an issue
FFTPL: An Analytic Placement Algorithm Using Fast Fourier Transform for Density Equalization
We propose a flat nonlinear placement algorithm FFTPL using fast Fourier
transform for density equalization. The placement instance is modeled as an
electrostatic system with the analogy of density cost to the potential energy.
A well-defined Poisson's equation is proposed for gradient and cost
computation. Our placer outperforms state-of-the-art placers with better
solution quality and efficiency
Penalized Clustering of Large Scale Functional Data with Multiple Covariates
In this article, we propose a penalized clustering method for large scale
data with multiple covariates through a functional data approach. In the
proposed method, responses and covariates are linked together through
nonparametric multivariate functions (fixed effects), which have great
flexibility in modeling a variety of function features, such as jump points,
branching, and periodicity. Functional ANOVA is employed to further decompose
multivariate functions in a reproducing kernel Hilbert space and provide
associated notions of main effect and interaction. Parsimonious random effects
are used to capture various correlation structures. The mixed-effect models are
nested under a general mixture model, in which the heterogeneity of functional
data is characterized. We propose a penalized Henderson's likelihood approach
for model-fitting and design a rejection-controlled EM algorithm for the
estimation. Our method selects smoothing parameters through generalized
cross-validation. Furthermore, the Bayesian confidence intervals are used to
measure the clustering uncertainty. Simulation studies and real-data examples
are presented to investigate the empirical performance of the proposed method.
Open-source code is available in the R package MFDA
Parametrization and penalties in spline models with an application to survival analysis
In this paper we show how a simple parametrization, built from the definition of cubic
splines, can aid in the implementation and interpretation of penalized spline models, whatever
configuration of knots we choose to use. We call this parametrization value-first derivative
parametrization. We perform Bayesian inference by exploring the natural link between quadratic
penalties and Gaussian priors. However, a full Bayesian analysis seems feasible only for some
penalty functionals. Alternatives include empirical Bayes methods involving model selection
type criteria. The proposed methodology is illustrated by an application to survival analysis
where the usual Cox model is extended to allow for time-varying regression coefficients
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