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In this paper, we present an efficient and robust approach to compute a normalized B-spline-like basis for spline spaces with pieces drawn from extended Tchebycheff spaces. The extended Tchebycheff spaces and their dimensions are allowed to change from interval to interval. The approach works by constructing a matrix that maps a generalized Bernstein-like basis to the B-spline-like basis of interest. The B-spline-like basis shares many characterizing properties with classical univariate B-splines and may easily be incorporated in existing spline codes. This may contribute to the full exploitation of Tchebycheffian splines in applications, freeing them from the restricted role of an elegant theoretical extension of polynomial splines. Numerical examples are provided that illustrate the procedure described.

This paper develops a general theory on rates of convergence of penalized spline estimators for function estimation when the likelihood functional is concave in candidate functions, where the likelihood is interpreted in a broad sense that includes conditional likelihood, quasi-likelihood and pseudo-likelihood. The theory allows all feasible combinations of the spline degree, the penalty order and the smoothness of the unknown functions. According to this theory, the asymptotic behaviors of the penalized spline estimators depends on interplay between the spline knot number and the penalty parameter. The general theory is applied to obtain results in a variety of contexts, including regression, generalized regression such as logistic regression and Poisson regression, density estimation, conditional hazard function estimation for censored data, quantile regression, diffusion function estimation for a diffusion type process and estimation of spectral density function of a stationary time series. For multidimensional function estimation, the theory (presented in the Supplementary Material) covers both penalized tensor product splines and penalized bivariate splines on triangulations.

In many application areas, data are collected on a count or binary response with spatial covariate information. In this article, we introduce a new class of generalized geoadditive models (GGAMs) for spatial data distributed over complex domains. Through a link function, the proposed GGAM assumes that the mean of the discrete response variable depends on additive univariate functions of explanatory variables and a bivariate function to adjust for the spatial effect. We propose a two-stage approach for estimating and making inferences of the components in the GGAM. In the first stage, the univariate components and the geographical component in the model are approximated via univariate polynomial splines and bivariate penalized splines over triangulation, respectively. In the second stage, local polynomial smoothing is applied to the cleaned univariate data to average out the variation of the first-stage estimators. We investigate the consistency of the proposed estimators and the asymptotic normality of the univariate components. We also establish the simultaneous confidence band for each of the univariate components. The performance of the proposed method is evaluated by two simulation studies. We apply the proposed method to analyze the crash counts data in the Tampa-St. Petersburg urbanized area in Florida. Supplementary materials for this article are available online.

Distributed lag non-linear models (DLNMs) are a modelling tool for describing potentially non-linear and delayed dependencies. Here, we illustrate an extension of the DLNM framework through the use of penalized splines within generalized additive models (GAM). This extension offers built-in model selection procedures and the possibility of accommodating assumptions on the shape of the lag structure through specific penalties. In addition, this framework includes, as special cases, simpler models previously proposed for linear relationships (DLMs). Alternative versions of penalized DLNMs are compared with each other and with the standard unpenalized version in a simulation study. Results show that this penalized extension to the DLNM class provides greater flexibility and improved inferential properties. The framework exploits recent theoretical developments of GAMs and is implemented using efficient routines within freely available software. Real-data applications are illustrated through two reproducible examples in time series and survival analysis.

We study additive models built with trend filtering, that is, additive models whose components are each regularized by the (discrete) total variation of their kth (discrete) derivative, for a chosen integer k ≥ 0. This results in kth degree piecewise polynomial components, (e.g., k = 0 gives piecewise constant components, k = 1 gives piecewise linear, k = 2 gives piecewise quadratic, etc.). Analogous to its advantages in the univariate case, additive trend filtering has favorable theoretical and computational properties, thanks in large part to the localized nature of the (discrete) total variation regularizer that it uses. On the theory side, we derive fast error rates for additive trend filtering estimates, and show these rates are minimax optimal when the underlying function is additive and has component functions whose derivatives are of bounded variation. We also show that these rates are unattainable by additive smoothing splines (and by additive models built from linear smoothers, in general). On the computational side, we use backfitting, to leverage fast univariate trend filtering solvers; we also describe a new backfitting algorithm whose iterations can be run in parallel, which (as far as we can tell) is the first of its kind. Lastly, we present a number of experiments to examine the empirical performance of trend filtering.

Background: The APHENA (Air Pollution and Health: A Combined European and North American Approach) study is a collaborative analysis of multicity time-series data on the effect of air pollution on population health, bringing together data from the European APHEA (Air Pollution and Health: A European Approach) and U.S. NMMAPS (National Morbidity, Mortality and Air Pollution Study) projects, along with Canadian data. Objectives: The main objective of APHENA was to assess the coherence of the findings of the multicity studies carried out in Europe and North America, when analyzed with a common protocol, and to explore sources of possible heterogeneity. We present APHENA results on the effects of particulate matter (PM) ≤ 10 μm in aerodynamic diameter (PM₁₀) on the daily number of deaths for all ages and for those < 75 and ≥ 75 years of age. We explored the impact of potential environmental and socioeconomic factors that may modify this association. Methods: In the first stage of a two-stage analysis, we used Poisson regression models, with natural and penalized splines, to adjust for seasonality, with various degrees of freedom. In the second stage, we used meta-regression approaches to combine time-series results across cites and to assess effect modification by selected ecologic covariates. Results: Air pollution risk estimates were relatively robust to different modeling approaches. Risk estimates from Europe and United States were similar, but those from Canada were substantially higher. The combined effect of PM₁₀ on all-cause mortality across all ages for cities with daily air pollution data ranged from 0.2% to 0.6% for a 10-μg/m³ increase in ambient PM₁₀ concentration. Effect modification by other pollutants and climatic variables differed in Europe and the United States. In both of these regions, a higher proportion of older people and higher unemployment were associated with increased air pollution risk. Conclusions: Estimates of the increased mortality associated with PM air pollution based on the APHENA study were generally comparable with results of previous reports. Overall, risk estimates were similar in Europe and in the United States but higher in Canada. However, PM₁₀ effect modification patterns were somewhat different in Europe and the United States.

We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339-360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute kth order discrete derivatives over the input points. Perhaps not surprisingly, trend filtering estimates appear to have the structure of kth degree spline functions, with adaptively chosen knot points (we say \"appear\" here as trend filtering estimates are not really functions over continuous domains, and are only defined over the discrete set of inputs). This brings to mind comparisons to other nonparametric regression tools that also produce adaptive splines; in particular, we compare trend filtering to smoothing splines, which penalize the sum of squared derivatives across input points, and to locally adaptive regression splines [Ann. Statist. 25 (1997) 387-413], which penalize the total variation of the kth derivative. Empirically, we discover that trend filtering estimates adapt to the local level of smoothness much better than smoothing splines, and further, they exhibit a remarkable similarity to locally adaptive regression splines. We also provide theoretical support for these empirical findings; most notably, we prove that (with the right choice of tuning parameter) the trend filtering estimate converges to the true underlying function at the minimax rate for functions whose kth derivative is of bounded variation. This is done via an asymptotic pairing of trend filtering and locally adaptive regression splines, which have already been shown to converge at the minimax rate [Ann. Statist. 25 (1997) 387-413]. At the core of this argument is a new result tying together the fitted values of two lasso problems that share the same outcome vector, but have different predictor matrices.

In the field of cardio-thoracic surgery, valve function is monitored over time after surgery. The motivation for our research comes from a study which includes patients who received a human tissue valve in the aortic position. These patients are followed prospectively over time by standardized echocardiographic assessment of valve function. Loss of follow-up could be caused by valve intervention or the death of the patient. One of the main characteristics of the human valve is that its durability is limited. Therefore, it is of interest to obtain a prognostic model in order for the physicians to scan trends in valve function over time and plan their next intervention, accounting for the characteristics of the data. Several authors have focused on deriving predictions under the standard joint modeling of longitudinal and survival data framework that assumes a constant effect for the coefficient that links the longitudinal and survival outcomes. However, in our case, this may be a restrictive assumption. Since the valve degenerates, the association between the biomarker with survival may change over time. To improve dynamic predictions, we propose a Bayesian joint model that allows a time-varying coefficient to link the longitudinal and the survival processes, using P-splines. We evaluate the performance of the model in terms of discrimination and calibration, while accounting for censoring.

We study trend filtering, a relatively recent method for univariate non-parametric regression. For a given integer r ≥ 1, the rth order trend filtering estimator is defined as the minimizer of the sum of squared errors when we constrain (or penalize) the sum of the absolute rth order discrete derivatives of the fitted function at the design points. For r = 1, the estimator reduces to total variation regularization which has received much attention in the statistics and image processing literature. In this paper, we study the performance of the trend filtering estimator for every r ≥ 1, both in the constrained and penalized forms. Our main results show that in the strong sparsity setting when the underlying function is a (discrete) spline with few “knots,” the risk (under the global squared error loss) of the trend filtering estimator (with an appropriate choice of the tuning parameter) achieves the parametric n −1-rate, up to a logarithmic (multiplicative) factor. Our results therefore provide support for the use of trend filtering, for every r ≥ 1, in the strong sparsity setting.