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Two-stage regression spline modeling based on local polynomial kernel regression

COMPUTATIONAL STATISTICS 2025年第1期40卷 383-403页

作者： Mraoui, Hamid El-Alaoui, Ahmed Bechrouri, Souad Mohaoui, Nezha Monir, Abdelilah Moulay Ismail Univ Meknes Fac Sci Meknes Morocco Mohammed First Univ Fac Sci Oujda Morocco

This paper introduces a new nonparametric estimator of the regression based on local quasi-interpolation spline method. This model combines a B-spline basis with a simple local polynomial regression, via blossoming approach, to produce a reduced rank spline like smoother. Different coefficients functionals are allowed to have different smoothing parameters (bandwidths) if the function has different smoothness. In addition, the number and location of the knots of this estimator are not fixed. In practice, we may employ a modest number of basis functions and then determine the smoothing parameter as the minimizer of the criterion. In simulations, the approach achieves very competitive performance with P-spline and smoothing spline methods. Simulated data and a real data example are used to illustrate the effectiveness of the method proposed in this paper.

关键词： local polynomial regression Bandwidth B-spline Blossoming Quasi-interpolation

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local polynomial regression for symmetric positive definite matrices

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JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY 2012年第4期74卷 697-719页

作者： Yuan, Ying Zhu, Hongtu Lin, Weili Marron, J. S. Univ North Carolina Chapel Hill Chapel Hill NC USA Univ North Carolina Chapel Hill Gillings Sch Global Publ Hlth Dept Biostat Chapel Hill NC 27599 USA

local polynomial regression has received extensive attention for the non-parametric estimation of regression functions when both the response and the covariate are in Euclidean space. However, little has been done when the response is in a Riemannian manifold. We develop an intrinsic local polynomial regression estimate for the analysis of symmetric positive definite matrices as responses that lie in a Riemannian manifold with covariate in Euclidean space. The primary motivation and application of the methodology proposed is in computer vision and medical imaging. We examine two commonly used metrics, including the trace metric and the log-Euclidean metric on the space of symmetric positive definite matrices. For each metric, we develop a cross-validation bandwidth selection method, derive the asymptotic bias, variance and normality of the intrinsic local constant and local linear estimators, and compare their asymptotic mean-square errors. Simulation studies are further used to compare the estimators under the two metrics and to examine their finite sample performance. We use our method to detect diagnostic differences between diffusion tensors along fibre tracts in a study of human immunodeficiency virus.

关键词： Bandwidth Cross-validation Intrinsic conditional expectation local polynomial regression Symmetric positive definite matrix

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local polynomial regression with correlated errors in random design and unknown correlation structure

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BIOMETRIKA 2018年第3期105卷 681-690页

作者： De Brabanter, K. Cao, F. Gijbels, I. Opsomer, J. Iowa State Univ Dept Stat 2419 Snedecor Hall Ames IA 50011 USA Katholieke Univ Leuven Dept Math Celestijnenlaan 200B B-3001 Leuven Belgium Colorado State Univ Dept Stat Ft Collins CO 80523 USA

Automated or data-driven bandwidth selection methods tend to break down in the presence of correlated errors. While this problem has previously been studied in the fixed design setting for kernel regression, the results were applicable only when there is knowledge about the correlation structure. This article generalizes these results to the random design setting and addresses the problem in situations where no prior knowledge about the correlation structure is available. We establish the asymptotic optimality of our proposed bandwidth selection criterion based on kernels K satisfying K(0) = 0.

关键词： Autocorrelation Correlated errors Crossvalidation local polynomial regression

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Medical image segmentation using rough set and local polynomial regression

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MULTIMEDIA TOOLS AND APPLICATIONS 2015年第6期74卷 1885-1914页

作者： Xie, Cong-Hua Liu, Yong-Jun Chang, Jin-Yi Changshu Inst Technol Sch Comp Sci & Engn Suzhou 215500 Jiangsu Peoples R China

Rough-set based multimodal histogram thresholding technique is effective for medical image segmentation. However, it is difficult to obtain the significant peaks and valleys of the roughness measure. Moreover, it is sensitive to the noise for medical image. In this paper, we proposed a new medical image segmentation method using rough set theory and local polynomial regression model to address those disadvantages. Firstly, compute histogram of image intensity information and histon of image intensity and spatial information. Secondly, use the local polynomial regression model to smooth the histogram and histon. The smoothed histogram correlates with lower approximation and the smoothed histon correlates with upper approximation. Lastly, rough measure is calculated with the lower and upper approximations. And then, multimodal thresholding method is applied to medical image segmentation. The local polynomial regression model can obtain a smooth rough measure and has two advantages: first, it is easy to find the real peaks and valleys of the smoothed roughness measure to segment medical image;second, the local polynomial regression reduces the effect of noise and can find the thresholds correctly. The proposed approach is compared with the histogram based approach, histon based approach, and rough set with the histogram and histon based approach. Experimental results demonstrate that our approach can find the real peaks and valleys more easily and yields better segmentation results than those of other three methods.

关键词： Medical image segmentation Rough set local polynomial regression Thresholding segmentation

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On bandwidth selection in local polynomial regression analysis and its application to multi-resolution analysis of non-uniform data

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JOURNAL OF SIGNAL PROCESSING SYSTEMS FOR SIGNAL IMAGE AND VIDEO TECHNOLOGY 2008年第3期52卷 263-280页

作者： Zhang, Z. G. Chan, S. C. Ho, K. L. Ho, K. C. Univ Hong Kong Dept Elect & Elect Engn Hong Kong Hong Kong Peoples R China Univ Missouri Dept Elect & Comp Engn Columbia MO 65211 USA

This paper studies adaptive bandwidth selection method for local polynomial regression (LPR) and its application to multi-resolution analysis (MRA) of non-uniformly sampled data. In LPR, the observations are modeled locally by a polynomial using least-squares criterion with a kernel having a certain support or bandwidth so that a better bias-variance tradeoff can be achieved. In this paper, two bandwidth selection methods, namely the Fan and Gijbels's bandwidth selection (FGBS) method (Fan and Gijbels, local polynomial Modelling and Its Applications, Chapman and Hall, London, 1996;Fan and Gijbels, Stat Sin 57:371-394, 1995) in the statistical community and the intersection of confidence intervals (ICI) method commonly used in the signal and image processing communities, are reviewed and compared in terms of their performance and implementation complexity using standard testing data sets. Furthermore, using the result of Stankovi (IEEE Trans Signal Proc 52:1228-1234, 2004), a new refined ICI-based adaptive bandwidth selection method for LPR and its associated reliability analysis are proposed. In addition, recursive implementations of LPR with the two classes of bandwidth selection methods are considered for online applications. Simulation results show that the performances of the FGBS method and the refined ICI method are comparable for the data sets tested. Since LPR with adaptive bandwidths can be naturally applied to non-uniformly sampled noisy observations, we propose to use it as a pre-processing step to a conventional MRA so that a MRA of non-uniformly sampled data can be realized. Simulation results show that the proposed LPR-based MRA gives better results than conventional linear interpolation of the data.

关键词： adaptive bandwidth selection intersection of confidence intervals local polynomial regression non-uniformly sampled data analysis wavelet

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Solution of fifth order boundary value problems by using local polynomial regression

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APPLIED MATHEMATICS AND COMPUTATION 2007年第2期186卷 952-956页

作者： Caglar, Hikmet Caglar, Nazan Istanbul Kultur Univ Fac Econ & Adm Sci TR-34156 Istanbul Turkey Istanbul Kultur Univ Dept Math Comp TR-34156 Istanbul Turkey

In this paper, we present a novel method based on the local polynomial regression for solving of fifth order boundary value problems. The method is tested on numerical example to demonstrate its usefulness. The method presented in this paper is also compared with those developed by Siddiqi and Akram [Solution of fifth order boundary value problems using nonpolynomial spline technique, Appl. Math. Comput. 175 (2006) 1575-1581], as well and is observed to be better. (c) 2006 Elsevier Inc. All rights reserved.

关键词： boundary value problems local polynomial regression kernel functions

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Smoothing time series with local polynomial regression on time

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2008年第6期37卷 959-971页

作者： Ledolter, Johannes Univ Iowa Dept Stat & Actuarial Sci Dept Management Sci Iowa City IA 52242 USA

Time series smoothers estimate the level of a time series at time t as its conditional expectation given present, past and future observations, with the smoothed value depending on the estimated time series model. Alternatively, local polynomial regressions on time can be used to estimate the level, with the implied smoothed value depending on the weight function and the bandwidth in the local linear least squares fit. In this article we compare the two smoothing approaches and describe their similarities. Through simulations, we assess the increase in the mean square error that results when approximating the estimated optimal time series smoother with the local regression estimate of the level.

关键词： local polynomial regression structural time series model time series smoothing trend model weighted least squares

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Identifying transient patterns of in vivo muscle behaviors during isometric contraction by local polynomial regression

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BIOMEDICAL SIGNAL PROCESSING AND CONTROL 2016年 24卷 93-102页

作者： Chen, Xin Wen, Huiying Li, Qiaoliang Wang, Tianfu Chen, Siping Zheng, Yong-Ping Zhang, Zhiguo Shenzhen Univ Sch Med Natl Reg Key Technol Engn Lab Med Ultrasound Guangdong Key Lab Biomed Measurements & Ultrasoun Shenzhen Peoples R China Hong Kong Polytech Univ Interdisciplinary Div Biomed Engn Kowloon Hong Kong Peoples R China Nanyang Technol Univ Sch Chem & Biomed Engn Singapore 639798 Singapore Nanyang Technol Univ Sch Elect & Elect Engn Singapore 639798 Singapore

polynomial regression is the most common method to estimate the relationship between muscle signals and torque during muscle contraction, but it is not capable of characterizing important transient patterns in the signal-torque relationship that only exist during short bursts of torque but may convey detailed information of muscle behavior. In this study, we proposed an integrated data analysis approach based on local polynomial regression (LPR) to identify transient patterns in the signal-torque relationship. For each subject, the LPR method can represent electromyography (EMG), mechanomyography (MMG) and ultrasonography (US) features as nonlinear functions of torque and can further estimate the derivatives of these signal-torque nonlinear functions. Further, a number of break points can be detected from the derivatives of the signal-torque relationships at the group level, and they can segment the signal-torque relationships into several stages, where multimodal features change with torque in different dynamic manners. Eight subjects performed isometric ramp contraction of knee up to 90% of the maximal voluntary contraction (MVC). EMG, MMG and US were simultaneously recorded from the rectus femoris muscle. Results showed that, for each feature, the whole torque range were clearly segmented into several distinct stages by the proposed method and the feature-torque relationship could be approximately described by a piecewise linear function with different slopes at different stages. A critical break-point of 20% MVC was detected during the isometric contraction for all muscle signals. As compared with the conventional regression methods, the proposed LPR-based data analysis approach can effectively identify stage-dependent transient patterns in the feature-torque relationships, providing deeper insights into the motor unit activation strategy. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Electromyography Mechanomyography Ultrasonography Isometric contraction local polynomial regression Transient muscle behavior

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Statistical Downscaling Using local polynomial regression for Rainfall Predictions - A Case Study

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WATER RESOURCES MANAGEMENT 2016年第1期30卷 183-193页

作者： George, Jany Janaki, Letha Gomathy, Jairaj Parameswaran Coll Engn Trivandrum Dept Civil Engn Thiruvananthapuram Kerala India Cochin Univ Sci & Technol Cochin 682016 Kerala India

This article presents a methodology for statistical downscaling using local polynomial regression for obtaining the future projections of rainfall in a catchment. local polynomial regression offers a method to catch the nonlinearities in the input-output relationship compared to traditional regression by identifying the nearest neighbors of the predictor point for a specified band width. It fits a low degree polynomial model to the subset of the data at each point by weighted least squares. The local regression fit is complete when the regression function values are calculated for all the data points. A smooth curve through the data points is obtained by this method. Mean sea level pressure, geopotential height 500 mb, air temperature, relative humidity and wind speed are identified as the potential predictors for predicting the rainfall. Monthly data on the predictors for nine grid points around the study area are obtained from National Centre for Environmental Prediction (NCEP)/ National Centre for Atmospheric Research (NCAR) Re-analysis data. The model was applied to forecast the rainfall in the catchment of Idukky reservoir in Kerala, India. The model performance was compared with that of multiple linear regression and artificial neural network models. It is seen that the local polynomial regression model gives a better performance in forecasting the rainfall. The new methodology adopted is computationally simple, easy to implement and it captures the linear and non linear features in the data set preserving the dynamics of the atmosphere and the properties of the historical series.

关键词： Statistical downscaling local polynomial regression Nearest neighbors Principal component analysis

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More efficient local polynomial regression with random-effects panel data models

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ECONOMETRIC REVIEWS 2018年第7期37卷 760-776页

作者： Yang, Ke Univ Hartford Barney Sch Business Dept Econ Hartford CT 06117 USA

We propose a modification on the local polynomial estimation procedure to account for the within-subject correlation presented in panel data. The proposed procedure is rather simple to compute and has a closed-form expression. We study the asymptotic bias and variance of the proposed procedure and show that it outperforms the working independence estimator uniformly up to the first order. Simulation study shows that the gains in efficiency with the proposed method in the presence of within-subject correlation can be significant in small samples. For illustration purposes, the procedure is applied to explore the impact of market concentration on airfare.

关键词： Kernel regression local polynomial regression Nonparametric method Random-effects panel data model Variance reduction

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