### Penalized Spline Estimation in the Partially Linear Model

, 2012

"... family, and friends who have all supported me throughout these years. ..."

### Massively Parallel Nonparametric Regression, with an Application to Developmental Brain Mapping

, 2012

"... Massively parallel nonparametric regression, with an application to developmental brain mapping ..."

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Massively parallel nonparametric regression, with an application to developmental brain mapping

### Parsimonious Classification Via Generalized Linear Mixed Models

, 2010

"... Abstract: We devise a classification algorithm based on generalized linear mixed model (GLMM) technology. The algorithm incorporates spline smoothing, additive model-type structures and model selection. For reasons of speed we employ the Laplace approximation, rather than Monte Carlo methods. Tests ..."

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Abstract: We devise a classification algorithm based on generalized linear mixed model (GLMM) technology. The algorithm incorporates spline smoothing, additive model-type structures and model selection. For reasons of speed we employ the Laplace approximation, rather than Monte Carlo methods. Tests on real and simulated data show the algorithm to have good classification performance. Moreover, the resulting classifiers are generally interpretable and parsimonious.

### Twenty years of P-splines

"... Abstract P-splines first appeared in the limelight twenty years ago. Since then they have become popular in applications and in theoretical work. The combination of a rich B-spline basis and a simple difference penalty lends itself well to a variety of generalizations, because it is based on regres ..."

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Abstract P-splines first appeared in the limelight twenty years ago. Since then they have become popular in applications and in theoretical work. The combination of a rich B-spline basis and a simple difference penalty lends itself well to a variety of generalizations, because it is based on regression. In effect, P-splines allow the building of a "backbone" for the "mixing and matching" of a variety of additive smooth structure components, while inviting all sorts of extensions: varying-coefficient effects, signal (functional) regressors, two-dimensional surfaces, non-normal responses, quantile (expectile) modelling, among others. Strong connections with mixed models and Bayesian analysis have been established. We give an overview of many of the central developments during the first two decades of P-splines. MSC: 41A15, 41A63, 62G05, 62G07, 62J07, 62J12.

### Penalized wavelets: embedding wavelets

, 2011

"... Abstract: We introduce the concept of penalized wavelets to facilitate seamless embedding of wavelets into semiparametric regressionmodels. In particular, we show that penalized wavelets are analogous to penalized splines; the latter being the established approach to function estimation in semiparam ..."

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Abstract: We introduce the concept of penalized wavelets to facilitate seamless embedding of wavelets into semiparametric regressionmodels. In particular, we show that penalized wavelets are analogous to penalized splines; the latter being the established approach to function estimation in semiparametric regression. They differ only in the type of penal-ization that is appropriate. This fact is not borne out by the existing wavelet literature, where the regression modelling and fitting issues are overshadowed by computational issues such as efficiency gains afforded by the Discrete Wavelet Transform and partially obscured by a tendency to work in the wavelet coefficient space. With penalized wavelet structure in place, we then show that fitting and inference can be achieved via the same general approaches used for penalized splines: penalized least squares, maximum like-lihood and best prediction within a frequentist mixed model framework, and Markov chain Monte Carlo and mean field variational Bayes within a Bayesian framework. Pe-nalized wavelets are also shown have a close relationship with wide data (“p n”) re-gression and benefit from ongoing research on that topic.

### doi:http://dx.doi.org/10.5705/ss.2012.230 A NOTE ON A NONPARAMETRIC REGRESSION TEST THROUGH PENALIZED SPLINES

"... Abstract: We examine a test of a nonparametric regression function based on pe-nalized spline smoothing. We show that, similarly to a penalized spline estimator, the asymptotic power of the penalized spline test falls into a small-K or a large-K scenarios characterized by the number of knots K and t ..."

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Abstract: We examine a test of a nonparametric regression function based on pe-nalized spline smoothing. We show that, similarly to a penalized spline estimator, the asymptotic power of the penalized spline test falls into a small-K or a large-K scenarios characterized by the number of knots K and the smoothing parameter. However, the optimal rate of K and the smoothing parameter maximizing power for testing is different from the optimal rate minimizing the mean squared error for estimation. Our investigation reveals that compared to estimation, some under-smoothing may be desirable for the testing problems. Furthermore, we compare the proposed test with the likelihood ratio test (LRT). We show that when the true function is more complicated, containing multiple modes, the test proposed here may have greater power than LRT. Finally, we investigate the properties of the test through simulations and apply it to two data examples. Key words and phrases: Goodness of fit, likelihood ratio test, nonparametric re-gression, partial linear model, spectral decomposition. 1.

### A Bayesian Multivariate Functional Dynamic Linear Model

, 2015

"... 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 compo-nents—we extend hierarchical dynamic linear models for multivariate time series to the functional ..."

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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 compo-nents—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, pro-vides 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. Supple-mentary materials, including R code and the multi-economy yield curve data, are available online.