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Hybrid Adaptive Splines
 Journal of the American Statistical Association
, 1995
"... . An adaptive spline method for smoothing is proposed which combines features from both regression spline and smoothing spline approaches. One of its advantages is the ability to vary the amount of smoothing in response to the inhomogeneous "curvature" of true functions at different locati ..."
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Cited by 61 (6 self)
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. An adaptive spline method for smoothing is proposed which combines features from both regression spline and smoothing spline approaches. One of its advantages is the ability to vary the amount of smoothing in response to the inhomogeneous "curvature" of true functions at different locations. This method can be applied to many multivariate function estimation problems, which is illustrated in this paper by an application to smoothing temperature data on the globe. The performance of this method in a simulation study is found to be comparable to the Wavelet Shrinkage methods proposed by Donoho and Johnstone. The problem of how to count the degrees of freedom for an adaptively chosen set of basis functions is addressed. This issue arises also in the MARS procedure proposed by Friedman and other adaptive regression spline procedures. Key words and phrases: Smoothing, spatial adaptability, splines, stepwise regression, the inflated degrees of freedom for an adaptively chosen basis functi...
Finding Chaos in Noisy Systems
, 1991
"... In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is comp ..."
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Cited by 49 (1 self)
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In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is compared for two similar initial conditions, this exponent is related to the rate at which the subsequent trajectories diverge. A bounded system with a positive LE is one operational definition of chaotic behavior. Most methods for determining the LE have assumed thousands of observations generated from carefully controlled physical experiments. Less attention has been given to estimating the LE for biological and economic systems that are subjected to random perturbations and observed over a limited amount of time. Using nonparametric regression techniques (Neural Networks and Thin Plate Splines) it is possible to consistently estimate the LE. The properties of these methods have been studied using simulated data and are applied to a biological time series: marten fur returns for the Hudson Bay Company (18201900). Based on a nonparametric analysis there is little evidence for lowdimensional chaos in these data. Although these methods appear to work well for systems perturbed by small amounts of noise, finding chaos in a system with a significant stochastic component may be difficult.
Spatial Process Estimates as Smoothers
, 1999
"... Contents Chapter 1 Spatial Process Estimates as Smoothers 1.1 Introduction Spatial statistics has a diverse range of applications and refers to the class of models and methods for data collected over a region. This region might be a mineral field for a geologist, a quadrat in a forest for an ecolo ..."
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Cited by 8 (0 self)
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Contents Chapter 1 Spatial Process Estimates as Smoothers 1.1 Introduction Spatial statistics has a diverse range of applications and refers to the class of models and methods for data collected over a region. This region might be a mineral field for a geologist, a quadrat in a forest for an ecologist or a geographic region for an environmental scientist. A typical problem is to predict values of a measurement at places where it is not observed, or, if the measurements are observed with error, to estimate a smooth spatial process ( i.e. a surface) from the data. A family of techniques, known loosely as Kriging, were originally developed in geostatistics and are now being widely used for spatial prediction. The goals of Kriging sound very much like nonparametric regression and the understanding of spatial estimates can be enriched through their interpretation as smoothing estimates. In the other direction, the random process model that is at th
Model Fitting and Testing for NonGaussian Data with Large Data Sets
, 1996
"... We consider the application of the smoothing spline to the generalized linear model in large data set situations. First we derive a Generalized Approximate Cross Validation function (GACV ), which is an approximate leaveoutone cross validation function used to choose smoothing parameters. In order ..."
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Cited by 5 (2 self)
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We consider the application of the smoothing spline to the generalized linear model in large data set situations. First we derive a Generalized Approximate Cross Validation function (GACV ), which is an approximate leaveoutone cross validation function used to choose smoothing parameters. In order to apply the GACV function to a large data set situation, we propose a corresponding randomized version of it. To reduce the computational intensity of calculating the smoothing spline estimate, we suggest an approximate solution and a clustering method to choose a subset of the basis functions. Combining randomized GACV with this approximate solution, we apply it to binary response data from the Wisconsin Epidemiological Study of Diabetic Retinopathy in order to establish the accuracy of the model when applied to a large data set.
Getting better contour plots with S and GCVPACK
, 1990
"... Abstract: We show how to obtain esthetically pleasing contour plots using New S and GCVPACK. With these codes, thin plate splines can easily be used to interpolate “exact ” data, and to produce smoothly varying contour plots, with none of the jagged corners that plague many other interpolation metho ..."
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Cited by 2 (0 self)
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Abstract: We show how to obtain esthetically pleasing contour plots using New S and GCVPACK. With these codes, thin plate splines can easily be used to interpolate “exact ” data, and to produce smoothly varying contour plots, with none of the jagged corners that plague many other interpolation methods. It is noted that GCVPACK can also be used to interpolate data on the sphere and in Euclidean three space. We observe that a larger class of global interpolation methods (including the thin plate spline) have a Bayesian interpretation, and GCVPACK can be used to compute them.
Testing the Generalized Linear Model Null Hypothesis versus `Smooth' Alternatives
, 1995
"... We consider y i ; i = 1; :::n independent observations from an exponential family with canonical parameter j(x i ), where the predictor variable x is in some index set and j is a `smooth' function of x. The usual GLIM models suppose that j has a parametric form j(x) = P p =1 fi OE (x) where ..."
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Cited by 2 (1 self)
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We consider y i ; i = 1; :::n independent observations from an exponential family with canonical parameter j(x i ), where the predictor variable x is in some index set and j is a `smooth' function of x. The usual GLIM models suppose that j has a parametric form j(x) = P p =1 fi OE (x) where the OE are given. This paper is concerned with testing the hypothesis that j is in the span of a given (low dimensional) set of OE versus general `smooth' alternatives. In the Gaussian case, studied by Cox, Koh, Wahba and Yandell(1988), test statistics are available whose distributions are independent of the nuisance fi , whereas in general this is not the case. We propose a symmetrized KullbackLeibler (SKL) distance test statistic, based on comparing a smoothing spline (penalized likelihood) fit and a GLIM fit, for testing the hypothesis j `parametric' vs j `smooth', in the nonGaussian situation. The spline fit uses a smoothing parameter obtained from the data via either the unbiased risk ...
SMOOTHING IN MAGNETIC RESONANCE IMAGE ANALYSIS AND A HYBRID LOSS FOR SUPPORT VECTOR MACHINE By
, 2005
"... i This thesis will focus on applying smoothing splines to magnetic resonance image (MRI) analysis. Some additional work on support vector machine with a hybrid loss function will be discussed. We apply smoothing splines to both the structural MRI and functional MRI. For the structural MRI, we fit th ..."
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Cited by 1 (0 self)
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i This thesis will focus on applying smoothing splines to magnetic resonance image (MRI) analysis. Some additional work on support vector machine with a hybrid loss function will be discussed. We apply smoothing splines to both the structural MRI and functional MRI. For the structural MRI, we fit thin plate splines to overlapping blocks of the image with different configurations of knots. The optimal configurations are found by the generalized cross validation with a constant factor (Luo and Wahba, 1997). The fitted splines with the optimal configurations are then blended to get a smoothed image of the brain. Thresholds are found along the way with kmeans algorithm and are blended as well. By thresholding the blended image we obtained, we get the boundaries between gray matter, white matter, cerebrospinal fluid, and others. The combination of smoothing and thresholding gives us very good results in terms of segmentation. For the functional magnetic resonance image analysis, we propose a partial spline model for the model fitting and hypothesis testing. Simulation are done to test the theoretical properties of the model. It appears that the partial spline model can compete with the commonly used smoothing+GLM paradigm. A support vector machine with a new hybrid loss is studied in the thesis. We propose a loss function that is a hybrid of the hinge loss and the logistic ii loss, with the aim to achieve the nice properties of these two loss functions, i.e., giving sparse solutions and being able to estimate the conditional probabilities at the same time. Our results and theoretical derivation show that the new loss function has the properties we expected and serves as a nice loss function for classification as well. iii
THINPLATE SPLINES FOR PRINTER DATA INTERPOLATION
"... Thinplate spline models have been used extensively for datainterpolation in several problem domains. In this paper, we present a tutorial overview of their theory and highlight their advantages and disadvantages, pointing out specific characteristics relevant in printer data interpolation applicat ..."
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Cited by 1 (0 self)
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Thinplate spline models have been used extensively for datainterpolation in several problem domains. In this paper, we present a tutorial overview of their theory and highlight their advantages and disadvantages, pointing out specific characteristics relevant in printer data interpolation applications. We evaluate the accuracy of thinplate splines for printer data interpolation and discuss how available knowledge of printer’s physical characteristics may be beneficially exploited to improve performance. 1.
RSplines for Response Surface Modeling
, 2000
"... Rsplines are introduced as splines fit with a polynomial null space plus the sum of radial basis functions. Thin plate splines are a special case of Rsplines. By this broader definition of an Rspline, however, it includes splines in which 2m \Gamma d 0 where the traditional roughness penalty is ..."
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Cited by 1 (1 self)
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Rsplines are introduced as splines fit with a polynomial null space plus the sum of radial basis functions. Thin plate splines are a special case of Rsplines. By this broader definition of an Rspline, however, it includes splines in which 2m \Gamma d 0 where the traditional roughness penalty is not guaranteed to be nonnegative definite. This papers discusses a modification of the roughness penalty that allows for the fitting of reduced polynomial null spaces. A series of examples are used to demonstrate the behavior of this modified roughness penalty. Keywords: Thin plate spline, nonparametric, response surfaces, roughness penalty, DemmlerReinsch basis functions. 1 Introduction Thin plate splines are nonparametric method for fitting response surfaces. A thin plate spline is a smooth surface constructed by adding a base, or null space, polynomial component, and a sum of radial basis functions. The thin plate smoothing splines are a solution to a minimization problem and can be int...
SAS/STAT ® 12.1 User’s Guide The TPSPLINE Procedure (Chapter)
, 2012
"... For a Web download or ebook: Your use of this publication shall be governed by the terms established by the vendor at the time you acquire this publication. The scanning, uploading, and distribution of this book via the Internet or any other means without the permission of the publisher is illegal ..."
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For a Web download or ebook: Your use of this publication shall be governed by the terms established by the vendor at the time you acquire this publication. The scanning, uploading, and distribution of this book via the Internet or any other means without the permission of the publisher is illegal and punishable by law. Please purchase only authorized electronic editions and do not participate in or encourage electronic piracy of copyrighted materials. Your support of others ’ rights is appreciated. U.S. Government Restricted Rights Notice: Use, duplication, or disclosure of this software and related documentation by the