Spatially-adaptive Penalties for Spline Fitting

The paper studies spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. The estimates are p th degree piecewise polynomials with p − 1 continuous derivatives. A large and fixed number of knots is used and smoothing is achieved by putting a quadratic penalty on the jumps of the p th derivative at the knots. To be spatially adaptive, the logarithm of the penalty is itself a linear spline but with relatively few knots and with values at the knots chosen to minimize the generalized cross validation (GCV) criterion. This locally-adaptive spline estimator is compared with other spline estimators in the literature such as cubic smoothing splines and knot-selection techniques for least squares regression. Our estimator can be interpreted as an empirical Bayes estimate for a prior allowing spatial heterogeneity. In cases of spatially heterogeneous regression functions, empirical Bayes confidence intervals using this prior achieve better pointwise coverage probabilities than confidence intervals based on a global-penalty parameter. The method is developed first for univariate models and then extended to additive models.     

Penalized triograms: total variation regularization for bivariate smoothing

Summary.  Hansen, Kooperberg and Sardy introduced a family of continuous, piecewise linear functions defined over adaptively selected triangulations of the plane as a general approach to statistical modelling of bivariate densities and regression and hazard functions. These triograms enjoy a natural affine equivariance that offers distinct advantages over competing tensor product methods that are more commonly used in statistical applications. Triograms employ basis functions consisting of linear 'tent functions' defined with respect to a triangulation of a given planar domain. As in knot selection for univariate splines, Hansen and colleagues adopted the regression spline approach of Stone. Vertices of the triangulation are introduced or removed sequentially in an effort to balance fidelity to the data and parsimony. We explore a smoothing spline variant of the triogram model based on a roughness penalty adapted to the piecewise linear structure of the triogram model. We show that the roughness penalty proposed may be interpreted as a total variation penalty on the gradient of the fitted function. The methods are illustrated with real and artificial examples, including an application to estimated quantile surfaces of land value in the Chicago metropolitan area.     

Flexible Copula Density Estimation with Penalized Hierarchical B‐splines

The paper introduces a new method for flexible spline fitting for copula density estimation. Spline coefficients are penalized to achieve a smooth fit. To weaken the curse of dimensionality, instead of a full tensor spline basis, a reduced tensor product based on so called sparse grids (Notes Numer. Fluid Mech. Multidiscip. Des., 31, 1991, 241‐251) is used. To achieve uniform margins of the copula density, linear constraints are placed on the spline coefficients, and quadratic programming is used to fit the model. Simulations and practical examples accompany the presentation.     

Optimizing minimum information pair-copula using genetic algorithm to select optimal basis functions

Constructing pair-copula using the minimum information approach is an appropriate and flexible way to survey the dependency structure between variables of interest. Minimum information pair-copula method approximates multivariate copula by applying some constraints between desired variables that are elicited from the data itself or experts’ judgment. In minimum information pair-copula, selecting basis constraints is a challenge. In this article, we apply genetic algorithms as a heuristic way to select basis constraints to optimize approximated pair-copula. The results gained show that our method optimizes model selection criteria and lead to better pair-copula approximation. Finally, we apply our proposed method to approximate pair-copula density in real dataset.     

P-splines with derivative based penalties and tensor product smoothing of unevenly distributed data

The P-splines of Eilers and Marx (Stat Sci 11:89–121, 1996) combine a B-spline basis with a discrete quadratic penalty on the basis coefficients, to produce a reduced rank spline like smoother. P-splines have three properties that make them very popular as reduced rank smoothers: (i) the basis and the penalty are sparse, enabling efficient computation, especially for Bayesian stochastic simulation; (ii) it is possible to flexibly ‘mix-and-match’ the order of B-spline basis and penalty, rather than the order of penalty controlling the order of the basis as in spline smoothing; (iii) it is very easy to set up the B-spline basis functions and penalties. The discrete penalties are somewhat less interpretable in terms of function shape than the traditional derivative based spline penalties, but tend towards penalties proportional to traditional spline penalties in the limit of large basis size. However part of the point of P-splines is not to use a large basis size. In addition the spline basis functions arise from solving functional optimization problems involving derivative based penalties, so moving to discrete penalties for smoothing may not always be desirable. The purpose of this note is to point out that the three properties of basis-penalty sparsity, mix-and-match penalization and ease of setup are readily obtainable with B-splines subject to derivative based penalization. The penalty setup typically requires a few lines of code, rather than the two lines typically required for P-splines, but this one off disadvantage seems to be the only one associated with using derivative based penalties. As an example application, it is shown how basis-penalty sparsity enables efficient computation with tensor product smoothers of scattered data.     

Smooth piecewise linear regression splines with hyperbolic covariates

We-propose the use of hyperbolas as covariates in piecewise linear regression splines to fit data exhibiting a multi-phase linear response with smooth transitions between phases. The hyperbolic regression spline model, fitted by non-linear regression, provides an intuitive and easy way to extend to multiple phases the two-phase hyperbolic response model previously proposed by others. The small additional effort required to fit non-linear, as opposed to linear, regression models is particularly worthwhile when investigators are unwilling to assume that the slope of the response changes abruptly at the join points. Furthermore, undue influence on the join point and slope estimates, resulting from points in the transition region, may be avoided by using the hyperbolic regression spline. Two examples illustrate the use of this method.     

Quantile splines with several covariates

We extend univariate regression quantile splines to problems with several covariates. We adopt an ANOVA-type decomposition approach with main effects captured by linear splines and second-order ‘interactions’ modeled by bi-linear tensor-product splines. Both univariate linear splines and bi-linear tensor-product splines are optimal when fidelity to data are balanced by a roughness penalty on the fitted function. The problem of sub-model selection and asymptotic justification for using a smaller sub-space of the spline functions in the approximation are discussed. Two examples are considered to illustrate the empirical performance of the proposed methods.     

Constrained penalized splines

The penalized spline is a popular method for function estimation when the assumption of “smoothness” is valid. In this paper, methods for estimation and inference are proposed using penalized splines under additional constraints of shape, such as monotonicity or convexity. The constrained penalized spline estimator is shown to have the same convergence rates as the corresponding unconstrained penalized spline, although in practice the squared error loss is typically smaller for the constrained versions. The penalty parameter may be chosen with generalized cross‐validation, which also provides a method for determining if the shape restrictions hold. The method is not a formal hypothesis test, but is shown to have nice large‐sample properties, and simulations show that it compares well with existing tests for monotonicity. Extensions to the partial linear model, the generalized regression model, and the varying coefficient model are given, and examples demonstrate the utility of the methods. The Canadian Journal of Statistics 40: 190–206; 2012 © 2012 Statistical Society of Canada     

Component Selection in the Additive Regression Model

Abstract. Similar to variable selection in the linear model, selecting significant components in the additive model is of great interest. However, such components are unknown, unobservable functions of independent variables. Some approximation is needed. We suggest a combination of penalized regression spline approximation and group variable selection, called the group‐bridge‐type spline method (GBSM), to handle this component selection problem with a diverging number of correlated variables in each group. The proposed method can select significant components and estimate non‐parametric additive function components simultaneously. To make the GBSM stable in computation and adaptive to the level of smoothness of the component functions, weighted power spline bases and projected weighted power spline bases are proposed. Their performance is examined by simulation studies. The proposed method is extended to a partial linear regression model analysis with real data, and gives reliable results.     

Self‐modelling regression for longitudinal data with time‐invariant covariates

The authors propose the use of self‐modelling regression to analyze longitudinal data with time invariant covariates. They model the population time curve with a penalized regression spline and use a linear mixed model for transformation of the time and response scales to fit the individual curves. Fitting is done by an iterative algorithm using off‐the‐shelf linear and nonlinear mixed model software. Their method is demonstrated in a simulation study and in the analysis of tree swallow nestling growth from an experiment that includes an experimentally controlled treatment, an observational covariate and multi‐level sampling.     

Asymptotics for penalised splines in generalised additive models

This paper discusses asymptotic theory for penalised spline estimators in generalised additive models. The purpose of this paper is to establish the asymptotic bias and variance as well as the asymptotic normality of the ridge-corrected penalised spline estimator. Furthermore, the asymptotics for the penalised quasi-likelihood fit in mixed models are also discussed.     

Density Estimation by Total Variation Penalized Likelihood Driven by the Sparsity ℓ1 Information Criterion

Abstract. We propose a non‐linear density estimator, which is locally adaptive, like wavelet estimators, and positive everywhere, without a log‐ or root‐transform. This estimator is based on maximizing a non‐parametric log‐likelihood function regularized by a total variation penalty. The smoothness is driven by a single penalty parameter, and to avoid cross‐validation, we derive an information criterion based on the idea of universal penalty. The penalized log‐likelihood maximization is reformulated as an ?₁‐penalized strictly convex programme whose unique solution is the density estimate. A Newton‐type method cannot be applied to calculate the estimate because the ?₁‐penalty is non‐differentiable. Instead, we use a dual block coordinate relaxation method that exploits the problem structure. By comparing with kernel, spline and taut string estimators on a Monte Carlo simulation, and by investigating the sensitivity to ties on two real data sets, we observe that the new estimator achieves good L ₁ and L ₂ risk for densities with sharp features, and behaves well with ties.     

Estimating curves and derivatives with parametric penalized spline smoothing

Accurate estimation of an underlying function and its derivatives is one of the central problems in statistics. Parametric forms are often proposed based on the expert opinion or prior knowledge of the underlying function. However, these strict parametric assumptions may result in biased estimates when they are not completely accurate. Meanwhile, nonparametric smoothing methods, which do not impose any parametric form, are quite flexible. We propose a parametric penalized spline smoothing method, which has the same flexibility as the nonparametric smoothing methods. It also uses the prior knowledge of the underlying function by defining an additional penalty term using the distance of the fitted function to the assumed parametric function. Our simulation studies show that the parametric penalized spline smoothing method can obtain more accurate estimates of the function and its derivatives than the penalized spline smoothing method. The parametric penalized spline smoothing method is also demonstrated by estimating the human height function and its derivatives from the real data.     

A method for constructing asymmetric pair-copula and its application

Pair-copula has become a hot spot in the research of both theory and application of statistics. Therefore, proper construction of pair-copula is an area worthy of study. Considering the asymmetry of variate dependence in practical applications and the disadvantages of the widely used asymmetric copulas, we propose a method to construct asymmetric pair-copula. In our method, we treat the asymmetric bivariate copula constructed by Liebscher's method as a generator, using this generator to construct asymmetric pair-copula. Also, on the basis of our method, we propose and prove a reference for selecting copula family in the construction. To verify the method, we construct asymmetric copulas and asymmetric pair-copulas using the daily runoff data collected at Yichang hydrological station to describe the extreme drought events of Yangtze River. After comparing the models in some aspects, we accept a model we construct, and the result displays the feasibility and practicality of the method we propose.     

Estimating mixed-effects differential equation models

Ordinary differential equations (ODEs) are popular tools for modeling complicated dynamic systems in many areas. When multiple replicates of measurements are available for the dynamic process, it is of great interest to estimate mixed-effects in the ODE model for the process. We propose a semiparametric method to estimate mixed-effects ODE models. Rather than using the ODE numeric solution directly, which requires providing initial conditions, this method estimates a spline function to approximate the dynamic process using smoothing splines. A roughness penalty term is defined using the ODEs, which measures the fidelity of the spline function to the ODEs. The smoothing parameter, which controls the trade-off between fitting the data and maintaining fidelity to the ODEs, can be specified by users or selected objectively by generalized cross validation. The spline coefficients, the ODE random effects, and the ODE fixed effects are estimated in three nested levels of optimization. Two simulation studies show that the proposed method obtains good estimates for mixed-effects ODE models. The semiparametric method is demonstrated with an application of a pharmacokinetic model in a study of HIV combination therapy.     

A COMPARISON OF MIXED MODEL SPLINES FOR CURVE FITTING

Three types of polynomial mixed model splines have been proposed: smoothing splines, P‐splines and penalized splines using a truncated power function basis. The close connections between these models are demonstrated, showing that the default cubic form of the splines differs only in the penalty used. A general definition of the mixed model spline is given that includes general constraints and can be used to produce natural or periodic splines. The impact of different penalties is demonstrated by evaluation across a set of functions with specific features, and shows that the best penalty in terms of mean squared error of prediction depends on both the form of the underlying function and the signal:noise ratio.     

Estimation and Classification of BOLD Responses Over Multiple Trials

In this article, we model functional magnetic resonance imaging (fMRI) data for event-related experiment data using a fourth degree spline to fit voxel specific blood oxygenation level-dependent (BOLD) responses. The data are preprocessed for removing long term temporal components such as drifts using wavelet approximations. The spatial dependence is incorporated in the data by the application of 3D Gaussian spatial filter. The methodology assigns an activation score to each trial based on the voxel specific characteristics of the response curve. The proposed procedure has a capability of being fully automated and it produces activation images based on overall scores assigned to each voxel. The methodology is illustrated on real data from an event-related design experiment of visually guided saccades (VGS).     

MODEL SELECTION FOR PENALIZED SPLINE SMOOTHING USING AKAIKE INFORMATION CRITERIA

Two different forms of Akaike's information criterion (AIC) are compared for selecting the smooth terms in penalized spline additive mixed models. The conditional AIC (cAIC) has been used traditionally as a criterion for both estimating penalty parameters and selecting covariates in smoothing, and is based on the conditional likelihood given the smooth mean and on the effective degrees of freedom for a model fit. By comparison, the marginal AIC (mAIC) is based on the marginal likelihood from the mixed‐model formulation of penalized splines which has recently become popular for estimating smoothing parameters. To the best of the authors' knowledge, the use of mAIC for selecting covariates for smoothing in additive models is new. In the competing models considered for selection, covariates may have a nonlinear effect on the response, with the possibility of group‐specific curves. Simulations are used to compare the performance of cAIC and mAIC in model selection settings that have correlated and hierarchical smooth terms. In moderately large samples, both formulations of AIC perform extremely well at detecting the function that generated the data. The mAIC does better for simple functions, whereas the cAIC is more sensitive to detecting a true model that has complex and hierarchical terms.     

Spline smoothing over difficult regions

Summary. It is occasionally necessary to smooth data over domains in R ² with complex irregular boundaries or interior holes. Traditional methods of smoothing which rely on the Euclidean metric or which measure smoothness over the entire real plane may then be inappropriate. This paper introduces a bivariate spline smoothing function defined as the minimizer of a penalized sum-of-squares functional. The roughness penalty is based on a partial differential operator and is integrated only over the problem domain by using finite element analysis. The method is motivated by and applied to two sample smoothing problems and is compared with the thin plate spline.     

Application of shrinkage estimation in linear regression models with autoregressive errors

In this paper, we consider the shrinkage and penalty estimation procedures in the linear regression model with autoregressive errors of order p when it is conjectured that some of the regression parameters are inactive. We develop the statistical properties of the shrinkage estimation method including asymptotic distributional biases and risks. We show that the shrinkage estimators have a significantly higher relative efficiency than the classical estimator. Furthermore, we consider the two penalty estimators: least absolute shrinkage and selection operator (LASSO) and adaptive LASSO estimators, and numerically compare their relative performance with that of the shrinkage estimators. A Monte Carlo simulation experiment is conducted for different combinations of inactive predictors and the performance of each estimator is evaluated in terms of the simulated mean-squared error. This study shows that the shrinkage estimators are comparable to the penalty estimators when the number of inactive predictors in the model is relatively large. The shrinkage and penalty methods are applied to a real data set to illustrate the usefulness of the procedures in practice.