We introduce a new class of nonstationary covariance functions for spatial modelling. Nonstationary covariance functions allow the model to adapt to spatial surfaces whose variability changes with location. The class includes a nonstationary version of the Matérn stationary covariance, in which the differentiability of the spatial surface is controlled by a parameter, freeing one from fixing the differentiability in advance. The class allows one to knit together local covariance parameters into a valid global nonstationary covariance, regardless of how the local covariance structure is estimated. We employ this new nonstationary covariance in a fully Bayesian model in which the unknown spatial process has a Gaussian process (GP) distribution with a nonstationary covariance function from the class. We model the nonstationary structure in a computationally efficient way that creates nearly stationary local behavior and for which stationarity is a special case. We also suggest non-Bayesian approaches to nonstationary kriging. To assess the method, we compare the Bayesian nonstationary GP model with a Bayesian stationary GP model, various standard spatial smoothing approaches, and nonstationary models that can adapt to function heterogeneity. In simulations, the nonstationary GP model adapts to function heterogeneity, unlike the stationary models, and also outperforms the other nonstationary models. On a real dataset, GP models outperform the competitors, but while the nonstationary GP gives qualitatively more sensible results, it fails to outperform the stationary GP on held-out data, illustrating the difficulty in fitting complex spatial functions with relatively few observations. The nonstationary covariance model could also be used for non-Gaussian data and embedded in additive models as well as in more complicated, hierarchical spatial or spatio-temporal models. More complicated models may require simpler parameterizations for computational efficiency.

This supplement provides detailed information about the functionalities of the Potters-Wheel toolbox as described in the main text. For further information please use the

This article develops methods for fitting spatial models to line transect data. These allow animal density to be related to topographical, environmental, habitat, and other spatial variables, helping wildlife managers to identify the factors that affect abundance. They also enable estimation of abundance for any subarea of interest within the surveyed region, and potentially yield estimates of abundance from sightings surveys for which the survey design could not be randomized, such as surveys conducted from platforms of opportunity. The methods are illustrated through analyses of data from a shipboard sightings survey of minke whales in the Antarctic.

Generalized additive models (GAM) for modeling nonlinear effects of continuous covariates are now well established tools for the applied statistician. A Bayesian version of GAM’s and extensions to generalized structured additive regression (STAR) are developed. One or two dimensional P-splines are used as the main building block. Inference relies on Markov chain Monte Carlo (MCMC) simulation techniques, and is either based on iteratively weighted least squares (IWLS) proposals or on latent utility representations of (multi)categorical regression models. The approach covers the most common univariate response distributions, e.g. the binomial, Poisson or gamma distribution, as well as multicategorical responses. For the first time, Bayesian semiparametric inference for the widely used multinomial logit model is presented. Two applications on the forest health status of trees and a space-time analysis of health insurance data demonstrate the potential of the approach for realistic modeling of complex problems. Software for the methodology is provided within the public domain package BayesX. Key words: geoadditive models, IWLS proposals, multicategorical response, structured additive predictors, surface smoothing

The smooth backfitting introduced by Mammen, Linton and Nielsen [Ann. Statist. 27 (1999) 1443–1490] is a promising technique to fit additive regression models and is known to achieve the oracle efficiency bound. In this paper, we propose and discuss three fully automated bandwidth selection methods for smooth backfitting in additive models. The first one is a penalized least squares approach which is based on higher-order stochastic expansions for the residual sums of squares of the smooth backfitting estimates. The other two are plug-in bandwidth selectors which rely on approximations of the average squared errors and whose utility is restricted to local linear fitting. The large sample properties of these bandwidth selection methods are given. Their finite sample properties are also compared through simulation experiments. 1. Introduction. Nonparametric

Methods for improving estimation of cetacean abundance from line transect and markrecapture surveys are proposed. Using either generalized linear or generalized additive models (GLMs or GAMs), two approaches are suggested whichallowheterogeneity in the spatial distribution of cetaceans to be modelled from standard line transect data. In the rst approach, the transect lines are divided into smaller discrete units, and the expected number of detections in each unit is modelled using explanatory spatial covariates. In the second approach, the response is derived from the observed waiting times (or distances) between detections. Fitting this model within the usual GLM or GAM framework would require restrictive assumptions, therefore an iterative procedure is formulated which enables a realistic model to be tted. Alternatively, it is shown how this approach can be framed as a point process model, and it is suggested how the likelihood for the observed along-trackline distances could be maximized. The methods are illustrated using line transect data from a survey of Antarctic minke whales. A surface representing the variation in density throughout the survey region is obtained, from which abundance may be estimated by numerical integration. It is also

This paper tackles the problem of model complexity in the context of additive models. Several methods have been proposed to estimate smoothing parameters, as well as to perform variable selection. Nevertheless,

Summary. Multicity time series studies of particulate matter and mortality and morbidity have provided evidence that daily variation in air pollution levels is associated with daily variation in mortality counts.These findings served as key epidemiological evidence for the recent review of the US national ambient air quality standards for particulate matter. As a result, methodological issues concerning time series analysis of the relationship between air pollution and health have attracted the attention of the scientific community and critics have raised concerns about the adequacy of current model formulations. Time series data on pollution and mortality are generally analysed by using log-linear, Poisson regression models for overdispersed counts with the daily number of deaths as outcome, the (possibly lagged) daily level of pollution as a linear predictor and smooth functions of weather variables and calendar time used to adjust for timevarying confounders. Investigators around the world have used different approaches to adjust for confounding, making it difficult to compare results across studies. To date, the statistical properties of these different approaches have not been comprehensively compared.To address these issues, we quantify and characterize model uncertainty and model choice in adjusting for seasonal and long-term trends in time series models of air pollution and mortality. First, we

We propose a novel framework that combines penalization techniques with Partial Least Squares (PLS). We focus on two important applications. (1) We combine PLS with a roughness penalty to estimate high-dimensional regression problems with functional predictors and scalar response. (2) Starting with an additive model, we expand each variable in terms of a generous number of B-Spline basis functions. To prevent overfitting, we estimate the model by applying a penalized version of PLS. We gain additional model flexibility by incorporating a sparsity penalty. Both applications can be formulated in terms of a unified algorithm called Penalized Partial Least Squares, which can be computed virtually as fast as PLS using the kernel trick. Furthermore, we prove a close connection of penalized PLS to preconditioned linear systems. In experiments, we show the benefits of our method to noisy functional data and to sparse nonlinear regression models.

(LVCSR) has been shown to generally be more effective than vocabulary-independent techniques for ranked retrieval of spoken content when one or the other approach is used alone. Tuning LVCSR systems to a topic domain can be costly, however, and the experiments in this paper show that Out-Of-Vocabulary (OOV) query terms can significantly reduce retrieval effectiveness when that tuning is not performed. Further experiments demonstrate, however, that retrieval effectiveness for queries with OOV terms can be substantially improved by combining evidence from LVCSR with additional evidence from vocabulary-independent Ranked Utterance Retrieval (RUR). The combination is performed by using relevance judgments from held-out topics to learn generic (i.e., topic-independent), smooth, non-decreasing transformations from LVCSR and RUR system scores to probabilities of topical relevance. Evaluated using a CLEF collection that includes topics, spontaneous conversational speech audio, and relevance judgments, the system recovers 57 % of the mean uninterpolated average precision that could have been obtained through LVCSR domain tuning for very short queries (or 41 % for longer queries).