Bernard W. Silverman

Nonparametric regression and generalized linear models

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The Stationary Wavelet Transform and some Statistical Applications

Wavelets are of wide potential use in statistical contexts. The basics of the discrete wavelet transform are reviewed using a filter notation that is useful subsequently in the paper. A ‘stationary wavelet transform’, where the coefficient sequences are not decimated at each stage, is described. Two different approaches to the construction of an inverse of the stationary wavelet transform are set out. The application of the stationary wavelet transform as an exploratory statistical method is discussed, together with its potential use in nonparametric regression. A method of local spectral density estimation is developed. This involves extensions to the wavelet context of standard time series ideas such as the periodogram and spectrum. The technique is illustrated by its application to data sets from astronomy and veterinary anatomy.

The contribution of species richness and composition to bacterial services

Bacterial communities provide important services. They break down pollutants, municipal waste and ingested food, and they are the primary means by which organic matter is recycled to plants and other autotrophs. However, the processes that determine the rate at which these services are supplied are only starting to be identified. Biodiversity influences the way in which ecosystems function, but the form of the relationship between bacterial biodiversity and functioning remains poorly understood. Here we describe a manipulative experiment that measured how biodiversity affects the functioning of communities containing up to 72 bacterial species constructed from a collection of naturally occurring culturable bacteria. The experimental design allowed us to manipulate large numbers of bacterial species selected at random from those that were culturable. We demonstrate that there is a decelerating relationship between community respiration and increasing bacterial diversity. We also show that both synergistic interactions among bacterial species and the composition of the bacterial community are important in determining the level of ecosystem functioning.

Wavelet thresholding via a Bayesian approach

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion that is common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relationship between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relationship gives insight into the meaning of the Besov space parameters. Moreover, the relationship established makes it possible in principle to incorporate prior knowledge about the functions regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a functions regularity properties might be difficult to elicit; with this in mind, we propose a standard choice of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set that was collected in an anaesthesiological study.

Weak and strong uniform consistency of kernel regression estimates

SummaryWe study the estimation of a regression function by the kernel method. Under mild conditions on the “window”, the “bandwidth” and the underlying distribution of the bivariate observations {(Xi, Yi)}, we obtain the weak and strong uniform convergence rates on a bounded interval. These results parallel those of Silverman (1978) on density estimation and extend those of Schuster and Yakowitz (1979) and Collomb (1979) on regression estimation.

Flexible Parsimonious Smoothing and Additive Modeling

A simple method is presented for fitting regression models that are nonlinear in the explanatory variables. Despite its simplicity—or perhaps because of it—the method has some powerful characteristics that cause it to be competitive with and often superior to more sophisticated techniques, especially for small data sets in the presence of high noise.

Empirical Bayes selection of wavelet thresholds

This paper explores a class of empirical Bayes methods for level-dependent threshold selection in wavelet shrinkage. The prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy-tailed density. The mixing weight, or sparsity parameter, for each level of the transform is chosen by marginal maximum likelihood. If estimation is carried out using the posterior median, this is a random thresholding procedure; the estimation can also be carried out using other thresholding rules with the same threshold. Details of the calculations needed for implementing the procedure are included. In practice, the estimates are quick to compute and there is software available. Simulations on the standard model functions show excellent performance, and applications to data drawn from various fields of application are used to explore the practical performance of the approach. By using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence, overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of Besov classes, covering also the case of f of bounded variation. The rates obtained are optimal for any value of the parameter p in (0, ∞], simultaneously for a wide range of loss functions, each dominating the Lq norm of the σth derivative, with σ ≥ 0 and 0 < q ≤ 2. Attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points, and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points. Results for all relevant combinations of these scenarios are obtained. In some cases a key feature of the theory is a particular boundary-corrected wavelet basis, details of which are discussed. Overall, the approach described seems so far unique in combining the properties of fast computation, good theoretical properties and good performance in simulations and in practice. A key feature appears to be that the estimate of sparsity adapts to three different zones of estimation, first where the signal is not sparse enough for thresholding to be of benefit, second where an appropriately chosen threshold results in substantially improved estimation, and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate.

Needles and straw in haystacks: Empirical bayes estimates of possibly sparse sequences

An empirical Bayes approach to the estimation of possibly sparse sequences observed in Gaussian white noise is set out and investigated. The prior considered is a mixture of an atom of probability at zero and a heavy-tailed density y, with the mixing weight chosen by marginal maximum likelihood, in the hope of adapting between sparse and dense sequences. If estimation is then carried out using the posterior median, this is a random thresholding procedure. Other thresholding rules employing the same threshold can also be used. Probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall risk bounds over classes of signal sequences of length n, allowing for sparsity of various kinds and degrees. The signal classes considered are nearly black sequences where only a proportion η is allowed to be nonzero, and sequences with normalized p norm bounded by η, for η > 0 and 0 1. Simulations show excellent performance. For appropriately chosen functions y, the method is computationally tractable and software is available. The extension to a modified thresholding method relevant to the estimation of very sparse sequences is also considered.

The Discrete Wavelet Transform in S

Abstract The theory of wavelets has recently undergone a period of rapid development. We introduce a software package called wavethresh that works within the statistical language S to perform one- and two-dimensional discrete wavelet transforms. The transforms and their inverses can be computed using any particular wavelet selected from a range of different families of wavelets. Pictures can be drawn of any of the one- or two-dimensional wavelets available in the package. The wavelet coefficients can be presented in a variety of ways to aid in the interpretation of data. The packages wavelet transform “engine” is written in C for speed and the object-oriented functionality of S makes wavethresh easy to use. We provide a tutorial introduction to wavelets and the wavethresh software. We also discuss how the software may be used to carry out nonlinear regression and image compression. In particular, thresholding of wavelet coefficients is a method for attempting to extract signal from noise and wavethresh i...

Short distances, flat triangles and Poisson limits

Motivated by problems in the analysis of spatial data, we prove some general Poisson limit theorems for the U -statistics of Hoeffding (1948). The theorems are applied to tests of clustering or collinearities in plane data; nearest neighbour distances are also considered.