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图数据与不规则多维数据的多尺度方法AI 解读

Summary For regularly spaced one-dimensional data, wavelet shrinkage has proven to be a compelling method for non-parametric function estimation. We create three new multiscale methods that provide wavelet-like transforms both for data arising on graphs and for irregularly spaced spatial data in more than one dimension. The concept of scale still exists within these transforms, but as a continuous quantity rather than dyadic levels. Further, we adapt recent empirical…

基于小波或提升格式的缺失值插补方法AI 解读

Summary The paper proposes a new approach to imputation using the expected sparse representation of a surface in a wavelet or lifting scheme basis. Our method incorporates a Bayesian mixture prior for these wavelet coefficients into a Gibbs sampler to generate a complete posterior distribution for the variable of interest. Intuitively, the estimator operates by borrowing strength from those observed neighbouring values to impute at the unobserved sites. We demonstrate the…

摘要暂缺

在函数型主成分分析中纳入参数效应AI 解读

SUMMARY The ideas of functional principal component analysis are extended to deal with data that are hybrids of ‘functional’ and ‘parametric’ effects. The parametric effects may be more general than just the addition of a multiple of a given function. A detailed development is given in the case of shifts of the time axis for functions observed on a periodic interval, and some remarks are made for the extension to a far more general case. Given data, a Procrustes fitting…

曲线数据的典型相关分析AI 解读

SUMMARY It is not immediately straightforward to extend canonical correlation analysis to the context of functional data analysis, where the data are themselves curves or functions. The obvious approach breaks down, and it is necessary to use a method involving smoothing in some way. Such a method is introduced and discussed with reference to a data set on human gait. The breakdown of the unsmoothed method is illustrated in a practical context and is demonstrated…

Reinsch 样条与 Speckman 样条的比较AI 解读

In the context of nonparametric regression, spline smoothers can be used both for the estimation of the curve itself and for the estimation of the error variance. We compare two approaches that have been suggested: the Reinsch spline, i.e. roughness penalty, and the Speckman spline, minimax. The comparison is made on the basis of the summed mean squared error for curve estimation and on the basis of a closely related criterion for variance estimation. We show in both…

Reinsch 与 Speckman 样条的比较AI 解读

In the context of nonparametric regression, spline smoothers can be used both for the estimation of the curve itself and for the estimation of the error variance. We compare two approaches that have been suggested: the Reinsch spline, i.e. roughness penalty, and the Speckman spline, minimax. The comparison is made on the basis of the summed mean squared error for curve estimation and on the basis of a closely related criterion for variance estimation. We show in both…

曲线数据均值与协方差结构的非参数估计AI 解读

SUMMARY We develop methods for the analysis of a collection of curves which are stochastically modelled as independent realizations of a random function with an unknown mean and covariance structure. We propose a method of estimating the mean function non-parametrically under the assumption that it is smooth. We suggest a variant on the usual form of cross-validation for choosing the degree of smoothing to be employed. This method of cross-validation, which consists of…

非参数回归中的残差方差估计AI 解读

A wide class of estimators of the residual variance in nonparametric regression is considered, namely those that are quadratic in the data, unbiased for linear regression, and always nonnegative. The minimax mean squared error estimator over a natural class of regression functions is derived. This optimal estimator has an interesting structure and is closely related to a minimax estimator of the regression curve itself.

摘要暂缺

非参数回归中残差方差的估计AI 解读

A wide class of estimators of the residual variance in nonparametric regression is considered, namely those that are quadratic in the data, unbiased for linear regression, and always nonnegative. The minimax mean squared error estimator over a natural class of regression functions is derived. This optimal estimator has an interesting structure and is closely related to a minimax estimator of the regression curve itself.

分支曲线的非参数估计AI 解读

The spline smoothing approach to nonparametric regression is extended to cover the estimation of curves that branch at certain specified points in time. This extension is motivated and illustrated by data from an experiment on sunflowers. Apart from providing insight into this particular data-analytic problem, the work described demonstrates the power of the roughness penalty approach in coping with curve estimation of a nonstandard kind. Particularly in the context of…

Bootstrap：是否应当平滑？AI 解读

The bootstrap and smoothed bootstrap are considered as alternative methods of estimating properties of unknown distributions such as the sampling error of parameter estimates. Criteria are developed for determining whether it is advantageous to use the smoothed bootstrap rather than the standard bootstrap. Key steps in the argument leading to these criteria include the study of the estimation of linear functionals of distributions and the approximation of general functionals…

两部关于密度估计的著作AI 解读

here K is a kernel function (usually a symmetric probability density function) and h is a smoothing parameter or bandwidth, the value of which determines how much the data are smoothed to produce the density estimate. The first published paper specifically on density estimation was Rosenblatt (1956); however, density estimates were suggested several years earlier by Fix and Hodges (1951) in a technical report. It is regrettable that this report was never published, since it…

非参数回归曲线拟合中样条平滑方法的若干方面AI 解读

SUMMARY Non-parametric regression using cubic splines is an attractive, flexible and widely-applicable approach to curve estimation. Although the basic idea was formulated many years ago, the method is not as widely known or adopted as perhaps it should be. The topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology. Particular subjects covered include the basic principles of the…

样条平滑：等价变带宽核方法AI 解读

The spline smoothing approach to nonparametric regression and curve estimation is considered. It is shown that, in a certain sense, spline smoothing corresponds approximately to smoothing by a kernel method with bandwidth depending on the local density of design points. Some exact calculations demonstrate that the approximation is extremely close in practice. Consideration of kernel smoothing methods demonstrates that the way in which the effective local bandwidth behaves in…

样条回归平滑参数选择的快速交叉验证方法AI 解读

The spline smoothing approach to nonparametric regression is considered, with particular reference to the problem of choosing how much to smooth. Both computational and statistical aspects of the method of cross-validation are discussed. An approximate cross-validation method is proposed that requires a small linear amount of computer time. A simulation study indicates that the method has very good statistical properties. Some mathematical justification and motivation are…

用最大惩罚似然方法估计概率密度函数AI 解读

A class of probability density estimates can be obtained by penalizing the likelihood by a functional which depends on the roughness of the logarithm of the density. The limiting case of the estimates as the amount of smoothing increases has a natural form which makes the method attractive for data analysis and which provides a rationale for a particular choice of roughness penalty. The estimates are shown to be the solution of an unconstrained convex optimization problem,…

多类型点的空间过程分析方法AI 解读

Summary Two approaches are described to the analysis of spatial patterns consisting of several types of points. The first approach uses a method of asymptotically unbiased estimation of the second moment distribution; the second uses methods based on regions of empty space in the patterns. Some distributional results are given in each case, and a method of Monte Carlo testing conditional on the marginal structure is described. The methods are illustrated by being applied to…

Summary A technique for using kernel density estimates to investigate the number of modes in a population is described and discussed. The amount of smoothing is chosen automatically in a natural way.