Local Smoothing Method Research Articles

Efficient Estimation of the Nonparametric Mean and Covariance Functions for Longitudinal and Sparse Functional Data

ABSTRACTWe consider the estimation of mean and covariance functions for longitudinal and sparse functional data by using the full quasi-likelihood coupling a modification of the local kernel smoothing method. The proposed estimators are shown to be consistent, asymptotically normal, and semiparametrically efficient in terms of their linear functionals. Their superiority to the competitors is further illustrated numerically through simulation studies. The method is applied to analyze AIDS study and atmospheric study. Supplementary materials for this article are available online.

Two-step estimation of time-varying additive model for locally stationary time series

In the analysis of locally stationary process, a time-varying additive model (tvAM) can effectively capture the dynamic feature of regression function. In combination with the strengths of tensor product of B-spline smoothing and local linear smoothing method, a two-step estimation method is proposed. It is shown that the proposed estimator is uniformly consistent and asymptotically oracle efficient as if the other component functions were known. Furthermore, a nonparametric bootstrap procedure is proposed to test the time-varying property of regression function. Simulation studies investigate the finite-sample performance of the proposed methods and validate the asymptotic theory. An environmental dataset illustrating the proposed method is also considered.

Spatiotemporal incidence rate data analysis by nonparametric regression.

To monitor the incidence rates of cancers, AIDS, cardiovascular diseases, and other chronic or infectious diseases, some global, national, and regional reporting systems have been built to collect/provide population-based data about the disease incidence. Such databases usually report daily, monthly, or yearly disease incidence numbers at the city, county, state, or country level, and the disease incidence numbers collected at different places and different times are often correlated, with the ones closer in place or time being more correlated. The correlation reflects the impact of various confounding risk factors, such as weather, demographic factors, lifestyles, and other cultural and environmental factors. Because such impact is complicated and challenging to describe, the spatiotemporal (ST) correlation in the observed disease incidence data has complicated ST structure as well. Furthermore, the ST correlation is hidden in the observed data and cannot be observed directly. In the literature, there has been some discussion about ST data modeling. But, the existing methods either impose various restrictive assumptions on the ST correlation that are hard to justify, or ignore partially or entirely the ST correlation. This paper aims to develop a flexible and effective method for ST disease incidence data modeling, using nonparametric local smoothing methods. This method can properly accommodate the ST data correlation. Theoretical justifications and numerical studies show that it works well in practice.

Nonparametric testing for smooth structural changes in panel data models

Detecting and modeling structural changes in time series models have attracted great attention. However, relatively little effort has been paid to the testing of structural changes in panel data models despite their increasing importance in economics and finance. In this paper, we propose a new approach to testing structural changes in panel data models. Unlike the bulk of the literature on structural changes, which focuses on detection of abrupt structural changes, we consider smooth structural changes for which model parameters are unknown deterministic smooth functions of time except for a finite number of time points. We use nonparametric local smoothing method to consistently estimate the smooth changing parameters and develop two consistent tests for smooth structural changes in panel data models. The first test is to check whether all model parameters are stable over time. The second test is to check potential time-varying interaction while allowing for a common trend. Both tests have an asymptotic N(0,1) distribution under the null hypothesis of parameter constancy and are consistent against a vast class of smooth structural changes as well as abrupt structural breaks with possibly unknown break points alternatives. Simulation studies show that the tests provide reliable inference in finite samples and two empirical examples with respect to a cross-country growth model and a capital structure model are discussed.

The role of media in preventing student suicides: A Hong Kong experience

IntroductionIn view of the surge of student suicides in March 2016, a large-scale media involvement was engaged to minimize copycat effects and to extensively spread more preventive information. MethodsTrend differences between student suicides in overall as well as youth suicides between 2003 and 2016 were examined. Impacts of media involvement were examined as to the changes of their intensities in reporting student suicides in different manners. Local polynomial smoothing method was used to estimate the intensities throughout the study period. ResultsThe intensity of students’ suicides has been slowly increasing since 2006, which was in contrast to the decreasing trend of overall suicide intensity. After our engagement with the media, acute descriptive-reporting in student suicide news has sharply dropped and remained low since then, whereas preventive-reporting has increased sharply in March and April 2016, but dropped back to normal afterwards. The higher intensity of preventive-reporting seemed to have a protective effect on student suicide. LimitationsData of student suicides in 2016 were substantially extracted from the local news, which may not contain complete information. The analyses suggested, but could not confirm, a causal relationship between the changes of student suicide news-reporting and the incidents of suicides themselves. ConclusionsThe recent surge of student suicides in Hong Kong was statistically abnormal. Media engagement has changed local suicide news-reporting to become more preventive, which could have contributed to the drop of student suicides. It is essential to sustain the media's coverage of preventive reporting to maintain its enduring effect.

Real-time local smoothing for five-axis linear toolpath considering smoothing error constraints

Abstract Five-axis linear toolpaths (or G 01 blocks) are widely used in CNC machine tools. The tangential and curvature discontinuities at the corners of linear toolpath result in feed fluctuation and deteriorate the machining efficiency and quality. Several methods have been proposed to locally smooth the corners for three-axis toolpath, but local smoothing for five-axis toolpath is still challenging due to two main difficulties: control of the tool orientation smoothing error and parameter synchronization between tool position and tool orientation. This paper proposes a real-time G 2 continuous local smoothing method by replacing the corners of tool position and tool orientation paths with cubic B-splines. The two difficulties in five-axis local smoothing are both resolved in simple and analytical ways. With a two-step method, the tool orientation smoothing error is directly and analytically constrained. By converting the remaining linear segments into B-splines, the C 1 continuity of the smooth tool position and smooth tool orientation paths is achieved. The parameter synchronization is realized by sharing the parameter of the tool orientation with that of the tool position. Compared with the existing analytical methods, the proposed method has a higher computation efficiency and a tighter tolerance in control of the orientation smoothing error. We have developed an open-source benchmark which validates the computation efficiency and error control ability of the proposed method. After smoothing and synchronization, the inserted B-spline of tool position is traversed with constant feedrate and the feedrate of the remaining linear segment is planned with jerk-bounded trajectory profile. The proposed smoothing method can be implemented on-line and has been integrated into a self-developed open-architecture CNC system. Its effectiveness for on-line generating smooth motion has been validated via simulations and experiments.

Adaptive quantile regression with precise risk bounds

An adaptive local smoothing method for nonparametric conditional quantile regression models is considered in this paper. Theoretical properties of the procedure are examined. The proposed method is fully adaptive in the sense that no prior information about the structure of the model is assumed. The fully adaptive feature not only allows varying bandwidths to accommodate jumps or instantaneous slope changes, but also allows the algorithm to be spatially adaptive. Under general conditions, precise risk bounds for homogeneous and heterogeneous cases of the underlying conditional quantile curves are established. An automatic selection algorithm for locally adaptive bandwidths is also given, which is applicable to higher dimensional cases. Simulation studies and data analysis confirm that the proposed methodology works well.

A new local estimation method for single index models for longitudinal data

Single index models are natural extensions of linear models and overcome the so-called curse of dimensionality. They are very useful for longitudinal data analysis. In this paper, we develop a new efficient estimation procedure for single index models with longitudinal data, based on Cholesky decomposition and local linear smoothing method. Asymptotic normality for the proposed estimators of both the parametric and nonparametric parts will be established. Monte Carlo simulation studies show excellent finite sample performance. Furthermore, we illustrate our methods with a real data example.

Robust Hessian Locally Linear Embedding Techniques for High-Dimensional Data

Recently manifold learning has received extensive interest in the community of pattern recognition. Despite their appealing properties, most manifold learning algorithms are not robust in practical applications. In this paper, we address this problem in the context of the Hessian locally linear embedding (HLLE) algorithm and propose a more robust method, called RHLLE, which aims to be robust against both outliers and noise in the data. Specifically, we first propose a fast outlier detection method for high-dimensional datasets. Then, we employ a local smoothing method to reduce noise. Furthermore, we reformulate the original HLLE algorithm by using the truncation function from differentiable manifolds. In the reformulated framework, we explicitly introduce a weighted global functional to further reduce the undesirable effect of outliers and noise on the embedding result. Experiments on synthetic as well as real datasets demonstrate the effectiveness of our proposed algorithm.

Composite quantile regression for varying-coefficient single-index models

ABSTRACTThe varying-coefficient single-index model (VCSIM) is a very general and flexible tool for exploring the relationship between a response variable and a set of predictors. Popular special cases include single-index models and varying-coefficient models. In order to estimate the index-coefficient and the non parametric varying-coefficients in the VCSIM, we propose a two-stage composite quantile regression estimation procedure, which integrates the local linear smoothing method and the information of quantile regressions at a number of conditional quantiles of the response variable. We establish the asymptotic properties of the proposed estimators for the index-coefficient and varying-coefficients when the error is heterogeneous. When compared with the existing mean-regression-based estimation method, our simulation results indicate that our proposed method has comparable performance for normal error and is more robust for error with outliers or heavy tail. We illustrate our methodologies with a real example.

Moving Least Squares Regression for High-Dimensional Stochastic Simulation Metamodeling

Simulation metamodeling is building a statistical model based on simulation output as an approximation to the system performance measure being estimated by the simulation model. In high-dimensional metamodeling problems, larger numbers of design points are needed to build an accurate and precise metamodel. Metamodeling techniques that are functions of all of these design points experience difficulties because of numerical instabilities and high computation times. We introduce a procedure to implement a local smoothing method called Moving Least Squares (MLS) regression in high-dimensional stochastic simulation metamodeling problems. Although MLS regression is known to work well when there are a very large number of design points, current procedures are focused on two- and three-dimensional cases. Furthermore, our procedure accounts for the fact that we can make replications and control the placement of design points in stochastic simulation. We provide a bound on the expected approximation error, show that the MLS predictor is consistent under certain conditions, and test the procedure with two examples that demonstrate better results than other existing simulation metamodeling techniques.

Theoretical Analysis on the Measurement Errors of Local 2D DIC: Part II Assessment of Strain Errors of the Local Smoothing Method–Approaching an Answer to the Overlap Question

AbstractIn this paper, the strain error of subset‐based two‐dimensional digital image correlation (DIC) is theoretically derived. Analytical solutions are provided to estimate the strain error. A dimensionless factor is proposed, namely the overlap magnifier, which reveals the dependency of the strain error on the DIC regularisation parameters, that is, subset size, step size and strain window size. The derived equations are validated numerically and experimentally. The estimated random strain error is in good accordance with the experimental data. The proposed derivation can be readily extended to stereo DIC.

Weighted Local Linear Smoothing Method for Randomly Right Censored Varying Coefficient Models

Weighted Local Linear Smoothing Method for Randomly Right Censored Varying Coefficient Models

Fuzzy nonparametric regression based on an adaptive neuro-fuzzy inference system

In this paper, a system, namely the adaptive neuro-fuzzy inference system (ANFIS), is investigated and used for fuzzy nonparametric regression function prediction with crisp input and fuzzy output. The fuzzy least squares problem based on Diamond׳s distance is proposed to optimize the consequent parameters in the hybrid algorithm of the adaptive neuro-fuzzy inference system method. Also, an algorithm is proposed to reduce bias and the boundary effect of the estimates of the underlying regression function. Various examples are used to illustrate and test the performance of this approach. The proposed method is compared with the local linear smoothing method for investigating the accuracy of the approach. The results demonstrate that the proposed method not only gives less biased estimates of the center line, the lower and the upper limit lines of underlying fuzzy regression function but also reduces bias and the boundary effect of the estimates of the underlying regression function by using the proposed algorithm significantly.

Varying-coefficient mean–covariance regression analysis for longitudinal data

By considering within-subject correlation among repeated measures over time, we propose a new and efficient estimation of varying-coefficient models for longitudinal data. Based on a modified Cholesky decomposition, the within-subject covariance matrix is decomposed into a unit triangular matrix involving generalized autoregressive coefficients and a diagonal matrix involving innovation variances. Local polynomial smoothing method is used to estimate the unknown varying coefficient functions of marginal mean and innovation variances. A method is also developed to estimate the autoregressive coefficients. All the resulting estimators are shown to be consistent and asymptotically normal. The proposed estimator of varying coefficient functions are asymptotically more efficient than the ones which ignore the within-subject correlation structure. Simulations are conducted to demonstrate finite sample behaviors of the proposed estimators, and a real example is given to illustrate the value of the proposed methodology.

Statistical inference in the partial linear models with the double smoothing local linear regression method

Statistical inference about a partial linear model includes two parts, one linear and the other nonparametric. The double smoothing method proposed by He and Huang (2009) is a progressive local smoothing method for nonparametric curve estimation. In this paper, we discuss its extension to partial linear models, accompanied with difference-based estimation method for the linear part. Asymptotic theory of the proposed method is developed. The results of simulation studies and real data examples demonstrate that our approach is effective even for data with moderate sample sizes.

On Nonparametric Image Registration

Image registration (IR) aims to map one image to another of a same scene. It is a fundamental task in many imaging applications. Most existing IR methods assume that the mapping transformation has a parametric form or satisfies certain regularity conditions (e.g., it is a smooth function with continuous first or higher order derivatives). They often estimate the mapping transformation globally by solving a global minimization/maximization problem. Such global smoothing methods usually cannot preserve singularities (e.g., discontinuities) and other features of the mapping transformation well. Further, the ill-posed nature of the IR problem, namely, the mapping transformation is not well defined at certain places, including the place where the true image intensity surface is straight, is not handled properly by such methods. In this article, we suggest solving the IR problem locally, by first studying the local properties of a mapping transformation. To this end, some concepts for describing such local properties are suggested, and a local smoothing method for estimating the mapping transformation is proposed. Because of the flexibility of local smoothing, our method does not require any parametric form or other global regularity conditions on the mapping transformation. Both theoretical and numerical studies show that it is effective in various applications. Supplementary materials are available online.

Attenuation of Ground-Motion Spectral Amplitudes and Its Variations across the Central Alborz Mountains

We selected 136 earthquakes recorded at stations of the Institute of Geo- physics of the University of Tehran (IGUT) during 1996-2007, in order to estimate the average attenuation parameters and variations of station corrections across the central Alborz mountains, the northern extension of the Alpine-Himalayan orogeny in wes- tern Asia. The selected events provide 1260 high signal-to-noise ratio (SNR) records with relatively good spatial coverage. We graphically find the distances at which the nature of geometrical spreading attenuation (R b ) changes significantly using a locally weighted scatterplot smoothing (LOWESS, local regression smoothing method) called robust LOWESS. A trilinear function with hinges at distances of about 80 and 160 km describes the geometric spreading attenuation with distance. By regressing to the hinged trilinear function, we found that b1 � −1:15 � 0:21, b2 � 0:09 � 0:31, and a fixed b3 � −0:5 minimize the average absolute value of the Fourier spectrum amplitude residuals. Using an anelastic attenuation coefficient at different frequencies, the direct quality factor Q in the central Alborz region is obtained as Q � 109 � 2f 0:64� 0:04 . The geographical distribution of station corrections at 1 Hz can be clearly separated along a dividing line connecting the North Tehran, Mosha, and Attari faults into two regions: the northern low-attenuating and the southern high attenuating regions. The stations with strong positive station corrections clearly align along the dividing line. This suggests the existence of a higher attenuation zone south of the dividing line. The presence of thick Tertiary-Quaternary sedimentary rocks with thick layers of salt in the southern region may explain the observed greater attenuation of seismic waves.

Local Mixed-Effects Fitting for Detecting Reproductive Hormone Surge Times

Hormone secretion processes remain part of the biological mystery as they are highly regulated and individual specific. When hormone trajectories from multiple subjects are under investigation, both population-average mechanism and subject-specific deviations are of great interest. In particular, statistical methodologies that enable us not only to identify surge times and surge magnitudes but also to make inference on these biological features is in need. In this paper, we propose a local kernel smoothing method to perform the analysis of multiple hormone curves using the nonparametric mixed-effects model. We develop a local quadratic mixed-effects (LQME) fitting procedure that detects local maxima of the population-average profile curve and the individual profile curves. Related statistical inference is established to carry out a hypothesis test for the local surge and to construct a confidence interval for a detected surge feature. This method is illustrated by simulation studies and a reproductive hormone data analysis.

A Lasso-Type Approach for Estimation and Variable Selection in Single Index Models

The single index model is a natural extension of the linear regression model for applications in which linearity does not hold. In this article, we propose a penalized local linear smoothing method, called sim-lasso, for estimation and variable selection under the single index model. The sim-lasso method penalizes the derivative of the link function and thus can be considered an extension of the usual lasso. Computational algorithms are developed for calculating the sim-lasso estimates and solution paths. The properties of the solution paths are also investigated. Simulation study and real data application demonstrate the excellent performance of the sim-lasso method. Supplemental materials for the article are available online.