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31
Differential Equation Models for Statistical Functions
, 2000
"... Di#erential equations have been used in statistics to define functions such as probability densities. But the idea of using di#erential equation formulations of stochastic models has a much wider scope. The author gives several examples, including simultaneous estimation of a regression model and re ..."
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Di#erential equations have been used in statistics to define functions such as probability densities. But the idea of using di#erential equation formulations of stochastic models has a much wider scope. The author gives several examples, including simultaneous estimation of a regression model and residual density, monotone smoothing, specification of a link function, di#erential equation models of data, and smoothing over complicated multidimensional domains. This paper aims to stimulate interest in this approach to functional estimation problems, rather than provide carefully worked out methods. R ESUM E En statistique, les equations di#erentielles ont ete utilisees entre autres pour definir des fonctions de densite. Ce n'est cependant pas la seule facon dont on peut faire appel a ce type d'equation pour definir des modeles stochastiques. L'auteur le montre au moyen d'exemples concernant l'estimation conjointe des parametres et de la loi des residus d'un modele de regression, le l...
Dynamic Profiling of Online Auctions Using Curve Clustering”, Working
, 2003
"... Electronic commerce, and in particular online auctions, have received an extreme surge of popularity in recent years. While auction theory has been studied for a long time from a gametheory perspective, the electronic implementation of the auction mechanism poses new and challenging research questi ..."
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Cited by 15 (8 self)
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Electronic commerce, and in particular online auctions, have received an extreme surge of popularity in recent years. While auction theory has been studied for a long time from a gametheory perspective, the electronic implementation of the auction mechanism poses new and challenging research questions. Although the body of empirical research on online auctions is growing, there is a lack of treatment of these data from a modern statistical point of view. In this work, we present a new source of rich auction data and introduce an innovative way of modelling and analyzing online bidding behavior. In particular, we use functional data analysis to investigate and scrutinize online auction dynamics. We describe the structure of such data and suggest suitable methods, including data smoothing and curve clustering, that allow one to profile online auctions and display different bidding behavior. We illustrate the methods on a set of eBay auction data and tie our results to the existing literature on online auctions. Key words and phrases: functional data analysis, smoothing, penalized splines, clustering, unsupervised
Varying coefficient regression modeling by adaptive weights smoothing
, 2003
"... The adaptive weights smoothing (AWS) procedure was introduced in Polzehl and Spokoiny (2000) in the context of image denoising. The procedure has some remarkable properties like preservation of edges and contrast, and (in some sense) optimal reduction of noise. The procedure is also fully adaptive a ..."
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Cited by 8 (5 self)
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The adaptive weights smoothing (AWS) procedure was introduced in Polzehl and Spokoiny (2000) in the context of image denoising. The procedure has some remarkable properties like preservation of edges and contrast, and (in some sense) optimal reduction of noise. The procedure is also fully adaptive and dimension free. Simulations with artificial images show that AWS is superior to classical smoothing techniques especially when the underlying image function is discontinuous and can be well approximated by a piecewise constant function. However, the latter assumption can be rather restrictive for a number of potential applications. Here the AWS method is generalized to the case of an arbitrary local linear parametric structure. We also establish some important results about properties of the AWS procedure including the so called “propagation condition ” and spatial adaptivity. The performance of the procedure is illustrated by examples for local polynomial regression in univariate and bivariate situations.
Nonparametric item response function estimates with the EM algorithm
 J. Educational and Behavioral Statistics
, 2002
"... The methods of functional data analysis are used to estimate item response functions (IRFs) nonparametrically. The EM algorithm is used to maximize the penalized marginal likelihood of the data. The penalty controls the smoothness of the estimated IRFs, and is chosen so that, as the penalty is incre ..."
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Cited by 8 (0 self)
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The methods of functional data analysis are used to estimate item response functions (IRFs) nonparametrically. The EM algorithm is used to maximize the penalized marginal likelihood of the data. The penalty controls the smoothness of the estimated IRFs, and is chosen so that, as the penalty is increased, the estimates converge to shapes closely represented by the threeparameter logistic family. The onedimensional latent trait model is recast as a problem of estimating a space curve or manifold, and, expressed in this way, the model no longer involves any latent constructs, and is invariant with respect to choice of latent variable. Some results from differential geometry are used to develop a dataanchored measure of ability and a new technique for assessing item discriminability. Functional dataanalytic techniques are used to explore the functional variation in the estimated IRFs. Applications involving simulated and actual data are included.
Improving Penalized Least Squares Through Adaptive Selection Of Penalty And Shrinkage
, 2001
"... . Estimation of the mean function in nonparametric regression is usefully separated into estimating the means at the observed factor levelsa oneway layout problem and interpolation between the estimated means at adjacent factor levels. Candidate penalized least squares (PLS) estimators for th ..."
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Cited by 7 (5 self)
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. Estimation of the mean function in nonparametric regression is usefully separated into estimating the means at the observed factor levelsa oneway layout problem and interpolation between the estimated means at adjacent factor levels. Candidate penalized least squares (PLS) estimators for the mean vector of a oneway layout are expressed as shrinkage estimators relative to an orthogonal regression basis determined by the penalty matrix. The shrinkage representation of PLS suggests a larger class of candidate monotone shrinkage (MS) estimators. Adaptive PLS and MS estimators choose the shrinkage vector and penalty matrix to minimize estimated risk. The actual risks of shrinkageadaptive estimators depend strongly upon the economy of the penalty basis in representing the unknown mean vector. Local annihilators of polynomials, among them di#erence operators, generate penalty bases that are economical in a range of examples. Diagnostic techniques for adaptive PLS or MS estimators i...
Curve fitting of time series Landsat imagery for characterising a mountain pine beetle infestation disturbance
 International Journal of Remote Sensing
, 2010
"... 10In this technical note we present a new technique using mixed linear models for characterizing a mountain pine beetle (Dendroctonus ponderosaeHopkins) infestation from multiyear satellite imagery. The main benefit of our approach is an ability to determine the statistical significance of each ann ..."
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10In this technical note we present a new technique using mixed linear models for characterizing a mountain pine beetle (Dendroctonus ponderosaeHopkins) infestation from multiyear satellite imagery. The main benefit of our approach is an ability to determine the statistical significance of each annual spectral change. Knowledge of the annual spectral change characteristics can then be used to 15statistically determine if a disturbance event has occurred, the timing of a given disturbance event, as well as to provide information for clustering fitted multitemporal reflectance curves (i.e. spectral trajectories) with a common shape. The spatial clustering of spectral trajectories provides insights upon the temporal process towards understanding the nature of the disturbance and recovery evident 20as imposed by infestation by mountain pine beetle over a 14year period. 1.
ASSIST: A suite of S functions implementing spline smoothing techniques. http://www.pstat.ucsb.edu/faculty/yuedong/software.html[3
, 2004
"... We present a suite of user friendly S functions for fitting various smoothing spline models including (a) nonparametric regression models for independent and correlated Gaussian data, and for independent binomial, Poisson and Gamma data; (b) semiparametric linear mixedeffects models; (c) nonpara ..."
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Cited by 5 (3 self)
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We present a suite of user friendly S functions for fitting various smoothing spline models including (a) nonparametric regression models for independent and correlated Gaussian data, and for independent binomial, Poisson and Gamma data; (b) semiparametric linear mixedeffects models; (c) nonparametric nonlinear regression models; (d) semiparametric nonlinear regression models; and (e) semiparametric nonlinear mixedeffects models. The general form of smoothing splines based on reproducing kernel Hilbert spaces is used to model nonparametric functions. Thus these S functions deal with many different situations in a unified fashion. Some well known special cases are polynomial splines, periodic splines, spherical splines, thinplate splines, lsplines, generalized additive models, smoothing spline ANOVA models, projection pursuit models, multiple index models, varying coefficient models, functional linear models, and selfmodeling nonlinear regression models. These nonparametric/semiparametric linear/nonlinear fixed/mixed models are widely used in practice to analyze data arising in many areas of investigation such as medicine, epidemiology, pharmacokinetics, econometrics and social science. This manual describes technical details behind these S functions and illustrate their applications using several examples. 1
Bayesian modeling of markers of dayspecific fertility
 Journal of the American Statistical Association
, 2003
"... SUMMARY. Cervical mucus hydration increases during the fertile interval prior to ovulation. Since sperm can only penetrate mucus having a high water content, cervical secretions provide a reliable marker of the fertile days of the menstrual cycle. This article develops a Bayesian approach for modeli ..."
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SUMMARY. Cervical mucus hydration increases during the fertile interval prior to ovulation. Since sperm can only penetrate mucus having a high water content, cervical secretions provide a reliable marker of the fertile days of the menstrual cycle. This article develops a Bayesian approach for modeling of daily observations of cervical mucus, and applies the approach to assess heterogeneity among women and cycles from a given woman with respect to the increase in mucus hydration during the fertile interval. The proposed model relates the mucus observations to an underlying normal mucus hydration score, which varies relative to a peak hydration day. Uncertainty in the timing of the peak is accounted for, and a novel weighted mixture model is used to characterize heterogeneity in distinct features of the underlying mean function. Prior information on the mucus hydration trajectory is incorporated, and a Markov chain Monte Carlo approach is developed. Based on data from a study of daily fecundability, there appears to be substantial heterogeneity among women in detected preovulatory increases in mucus hydration but only minimal differences among cycles from a given woman.
Choosing Smoothness Parameters for Smoothing Splines by Minimizing an Estimate of Risk
, 2004
"... ... In this paper, we draw a connection between smoothing splines and REACT estimators that provides motivation for the creation of criteria for choosing the smoothness parameter. The new criteria are compared to three existing methods, namely crossvalidation, generalized crossvalidation, and gene ..."
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... In this paper, we draw a connection between smoothing splines and REACT estimators that provides motivation for the creation of criteria for choosing the smoothness parameter. The new criteria are compared to three existing methods, namely crossvalidation, generalized crossvalidation, and generalization of maximum likelihood criteria, by a Monte Carlo simulation and by an application to the study of circadian patterns. For most of the situations presented in the simulations, including the practical example, the new criteria outperform the three existing criteria
The Theory and Application of Penalized Least Squares Methods or Reproducing Kernel Hilbert Spaces Made Easy
, 1997
"... The popular cubic smoothing spline estimate of a regression function is the minimizer of X j d j (Y j \Gamma ¯(t j )) 2 + Z b a h ¯ 00 (t) i 2 dt; where (Y j ; t j ) are the data and the d j 's are positive weights. However, sometimes the data are related to the function of interes ..."
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Cited by 3 (0 self)
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The popular cubic smoothing spline estimate of a regression function is the minimizer of X j d j (Y j \Gamma ¯(t j )) 2 + Z b a h ¯ 00 (t) i 2 dt; where (Y j ; t j ) are the data and the d j 's are positive weights. However, sometimes the data are related to the function of interest ¯ in another way, i.e., E(Y i ) = F i (¯) for some known F i 's. And sometimes, one may wish to replace R (¯ 00 ) 2 with another expression. This paper discusses the solution for these generalizations, that is, the minimization of X j d j (Y j \Gamma L j (¯)) 2 + Z b a h (L¯)(t) i 2 dt: Here, L is a linear differential operator of order m 1: (L¯)(t) = ¯ (m) (t) + P m\Gamma1 j=0 w j (t)¯ (j) (t). This paper outlines basic theory for this general minimization problem, and provides explicit directions for calculating the minimizer. The minimizer depends on the easily calculated reproducing kernel associated with L. 2 Introduction The cubic smoothing spline, a popular regre...