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曲线样本结构均值的非参数最大似然估计AI 解读

A random sample of curves can be usually thought of as noisy realisations of a compound stochastic process X(t) = Z{W(t)}, where Z(t) produces random amplitude variation and W(t) produces random dynamic or phase variation. In most applications it is more important to estimate the so-called structural mean μ(t) = E{Z(t)} than the crosssectional mean E{X(t)}, but this estimation problem is difficult because the process Z(t) is not directly observable. In this paper we propose…

自建模变形函数AI 解读

Summary The paper introduces a semiparametric model for functional data. The warping functions are assumed to be linear combinations of q common components, which are estimated from the data (hence the name ‘self-modelling’). Even small values of q provide remarkable model flexibility, comparable with nonparametric methods. At the same time, this approach avoids overfitting because the common components are estimated combining data across individuals. As a convenient…

随机曲线分布众数的非参数估计AI 解读

Summary Motivated by the need to develop meaningful empirical approximations to a ‘typical’ data value, we introduce methods for density and mode estimation when data are in the form of random curves. Our approach is based on finite dimensional approximations via generalized Fourier expansions on an empirically chosen basis. The mode estimation problem is reduced to a problem of kernel-type multivariate estimation from vector data and is solved using a new recursive…

曲线结构平均的渐近与 Bootstrap 置信界AI 解读

For analyzing samples of curves, Kneip and Gasser proposed a “structural analysis” for estimating an average curve which gives the average amplitude and dynamics of the sample curves. An important step of their method is the estimation of the warping functions in order to eliminate the differences in dynamics between different curves. It is of interest to compute confidence bounds for the structural average curve. First, we derive the necessary asymptotic results to obtain…

利用动态时间规整对齐曲线AI 解读

When studying some process or development in different subjects or units--be it biological, chemical or physical--we usually see a typical pattern, common to all curves. Yet there is variation both in amplitude and dynamics between curves. Following some ideas of structural analysis introduced by Kneip and Gasser, we study a method--dynamic time warping with a proper cost function--for estimating the shift or warping function from one curve to another to align the two…

局部多项式的有限样本方差：分析与解决方法AI 解读

Fitting local polynomials in nonparametric regression has a number of advantages. The attractive theoretical features are in a partial contradiction to variance properties for random design and to practical experience over a broad range of situations. No upper bound can be given for the conditional variance. The unconditional variance is infinite when using optimal weights with compact support. Properties are better for Gaussian weights. We analyze local polynomials for…

在曲线样本中寻找结构特征AI 解读

The shape of a regression curve can to a large extent be characterized by the succession of structural features like extrema, inflection points, and so on. When analyzing a sample of regression curves, it is often important to know at an early stage of data analysis which structural features are occurring consistently in each curve of the sample. Such a definition is usually not easy due to substantial interindividual variation both in the x and the y axis and due to the…

二元核回归的带宽选择器AI 解读

SUMMARY For two and higher dimensional kernel regression, currently available bandwidth selection procedures are based on cross-validation or related penalizing ideas. However, these techniques have been shown to suffer from high sample variability and, in addition, can sometimes be difficult to implement when a vector of bandwidths needs to be selected. In this paper we propose a selector based on an iterative plug-in approach for bivariate kernel regression. It is shown to…

核回归估计的局部自适应带宽选择AI 解读

Kernel estimators with a global bandwidth are commonly used to estimate regression functions. On the other hand, it is obvious that the choice of a local bandwidth can lead to better results, because a larger class of kernel estimators is available. Evidently, this may in turn affect variability. The optimal bandwidths depend essentially on the regression function itself and on the residual variance, and it is desirable to estimate them from the data. In this article, a…

重新审视残差方差的非参数估计AI 解读

Several difference-based estimators of residual variance are compared for finite sample size. Since the introduction of a rather simple estimator by Gasser, Sroka & Jennen-Steinmetz (1986) other proposals have been made. Here the one given by Hall, Kay & Titterington (1990) is of particular interest. It minimizes the asymptotic variance. Unfortunately it has severe problems with finite sample bias, and the estimator of Gasser et al. (1986) proves still to be a good choice. A…

重新审视残差方差的非参数估计AI 解读

Several difference-based estimators of residual variance are compared for finite sample size. Since the introduction of a rather simple estimator by Gasser, Sroka & Jennen-Steinmetz (1986) other proposals have been made. Here the one given by Hall, Kay & Titterington (1990) is of particular interest. It minimizes the asymptotic variance. Unfortunately it has severe problems with finite sample bias, and the estimator of Gasser et al. (1986) proves still to be a good choice. A…

残差相关时核回归的带宽选择AI 解读

Standard techniques for selecting the bandwidth of a kernel estimator from the data in a nonparametric regression model perform badly when the errors are correlated. In this paper we propose a modified version of an existing plug-in bandwidth selector. The method is generalized to stationary error variables by estimating a functional of the residual covariance function. The proposed bandwidth selector shows good properties in asymptotic theory and in simulations without…

用于分析曲线样本数据的统计工具AI 解读

The paper is concerned with data representing a sample of smooth curves which can be considered as independent realizations of an underlying biological (chemical, $\ldots$) process. Such samples of curves often possess the following features: There is a typical structural pattern common to all curves of the sample. On the other hand, individual realizations of the typical shape show different dynamics and intensity. In particular, typical peaks are shifted from individual to…

残差相关时核回归的带宽选择AI 解读

Standard techniques for selecting the bandwidth of a kernel estimator from the data in a nonparametric regression model perform badly when the errors are correlated. In this paper we propose a modified version of an existing plug-in bandwidth selector. The method is generalized to stationary error variables by estimating a functional of the residual covariance function. The proposed bandwidth selector shows good properties in asymptotic theory and in simulations without…

灵活快速的自动平滑方法AI 解读

The choice of smoothing parameter or bandwidth is crucial when applying nonparametric regression estimators such as kernel estimators. The optimal choice depends on the data at hand. A data-driven bandwidth selection, close to the optimal one, would make these curve estimators objective, more reliable, and easier to use. Minimizing residual mean squared error criteria, such as cross-validation, have been frequently proposed to estimate the optimal smoothing parameter.…

核回归估计中的权重选择AI 解读

For kernel regression estimation a weighting scheme due to Nadaraya and Watson has been associated with random design, and a convolution type weighting scheme with fixed design. Based on integrated mean square error, none of the estimators is uniformly optimal in either design. However, the convolution type weights are minimax optimal. Further advantages of this estimator can be seen in the structure of the bias.

核回归估计中的权重选择AI 解读

For kernel regression estimation a weighting scheme due to Nadaraya and Watson has been associated with random design, and a convolution type weighting scheme with fixed design. Based on integrated mean square error, none of the estimators is uniformly optimal in either design. However, the convolution type weights are minimax optimal. Further advantages of this estimator can be seen in the structure of the bias.

非参数回归估计的统一方法AI 解读

There exists a variety of nonparametric regression estimators, with cubic smoothing splines, k-nearest-neighbor (k-NN) estimators, and various types of kernel estimators among the most popular. A class of kernel estimators with local bandwidth depending on the density of the design points is introduced, where the degree of design adaptation may be expressed by a single parameter α in [0, 1]. The adaptation to the design is such that the bandwidth is made larger, to a degree…

自建模非线性回归的收敛与一致性结果AI 解读

This paper is concerned with parametric regression models of the form $Y_{ij} = f(t_{ij}, \theta_i) + \text{error}, i = 1, \ldots, n, j = 1, \ldots, T_i$, where the continuous function $f$ may depend nonlinearly on the known regressors $t_{ij}$ and the unknown parameter vectors $\theta_i$. The assumption of an a priori known $f$ is dropped and replaced by the requirement that qualitative information about the structure of the model is available or can be generated by a…

含噪信号的可变潜伏期：脑电位数据的估计与检验AI 解读

The electric response of the brain related to some event, i.e. stimulus, is usually estimated by repeated stimulation and subsequent averaging of the activity recorded time-locked to the stimulus. The present paper deals with the model that the single responses may have varying latency, i.e. arrival time, after stimulus onset. For estimating the single latencies, a maximum likelihood approach employing iterative Fisher-scoring is developed. Further a statistic for testing on…

含噪信号的可变潜伏期：脑电位数据中的估计与检验AI 解读

The electric response of the brain related to some event, i.e. stimulus, is usually estimated by repeated stimulation and subsequent averaging of the activity recorded time-locked to the stimulus. The present paper deals with the model that the single responses may have varying latency, i.e. arrival time, after stimulus onset. For estimating the single latencies, a maximum likelihood approach employing iterative Fisher-scoring is developed. Further a statistic for testing on…

非线性回归中的残差方差与残差模式AI 解读

A nonparametric estimator of residual variance in nonlinear regression is proposed. It is based on local linear fitting. Asymptotically the estimator has a small bias, but a larger variance compared with the parametric estimator in linear regression. Finite sample properties are investigated in a simulation study, including a comparison with other nonparametric estimators. The method is also useful for spotting heteroscedasticity and outliers in the residuals at an early…

非线性回归中的残差方差与残差模式AI 解读

A nonparametric estimator of residual variance in nonlinear regression is proposed. It is based on local linear fitting. Asymptotically the estimator has a small bias, but a larger variance compared with the parametric estimator in linear regression. Finite sample properties are investigated in a simulation study, including a comparison with other nonparametric estimators. The method is also useful for spotting heteroscedasticity and outliers in the residuals at an early…

生长曲线的非参数回归分析AI 解读

In recent years, nonparametric curve estimates have been extensively explored in theoretical work. There has, however, been a certain lack of convincing applications, in particular involving comparisons with parametric techniques. The present investigation deals with the analysis of human height growth, where longitudinal measurements were collected for a sample of boys and a sample of girls. Evidence is presented that kernel estimates of acceleration and velocity of height,…