Polynomial Kernel Regression Research Articles

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

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

Application of machine learning techniques to derive sea water turbidity from Sentinel-2 imagery

Earth Observation (EO) from satellites has the potential to provide comprehensive, rapid and inexpensive information about water bodies, integrating in situ measurements. Traditional methods to retrieve optically active water quality parameters from satellite data are based on semi-empirical models relying on few bands, which often revealed to be site and season specific. The use of machine learning (ML) for remotely sensed water quality estimation has spread in recent years thanks to the advances in algorithm development and computing power. These models allow to exploit the wealth of spectral information through more flexible relationships and are less affected by atmospheric and other background factors. The present study explores the use of Sentinel-2 MultiSpectral Instrument (MSI) Level-1C Top of Atmosphere spectral radiance to derive water turbidity, through application of machine learning techniques. A dataset of 222 combination of turbidity measurements, collected in the North Tyrrhenian Sea – Italy from 2015 to 2021, and values of the 13 spectral bands in the pixel corresponding to the sample location was used. Two regression techniques were tested and compared: a Stepwise Linear Regression (SLR) and a Polynomial Kernel Regression. The two models show accurate and similar performance (R2 = 0.736, RMSE = 2.03 NTU, MAE = 1.39 NTU for the SLR and R2 = 0.725, RMSE = 2.07 NTU, MAE = 1.40 NTU for the Kernel). A band importance analysis revealed the contribution of the different spectral bands and the main role of the red-edge range. The work shows that it is possible to reach a good accuracy in turbidity estimation from MSI TOA reflectance using ML models, fed by the whole spectrum of available bands, although the possible generation of errors related to atmospheric effect in turbidity estimates was not evaluated. Comparison between turbidity estimates obtained from the models with turbidity data from Copernicus CMEMS dataset named ‘Mediterranean Sea, Bio-Geo-Chemical, L3, daily observation’ produced consistent results. Finally, turbidity maps from satellite imagery were produced for the study area, showing the ability of the models to catch extreme events.

MODELING OF LOCAL POLYNOMIAL KERNEL NONPARAMETRIC REGRESSION FOR COVID DAILY CASES IN SEMARANG CITY, INDONESIA

Coronavirus disease 2019 (COVID-19) is an infectious disease caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) which was recently discovered. Coronavirus disease is now a pandemic that occurs in many countries in the world, one of which is Indonesia. One of the cities in Indonesia that has found many COVID cases is Semarang city, located in Central Java. Data on cases of COVID patients in Semarang City which are measured daily do not form a certain distribution pattern. We can build a model with a flexible statistical approach without any assumptions that must be used, namely the nonparametric regression. The nonparametric regression in this research using Local Polynomial Kernel approach. Determination of the polynomial order and optimal bandwidth in Local Polynomial Kernel Regression modeling use the GCV (Generalized Cross Validation) method. The data used this research are data on the number of COVID patients daily cases in Semarang, Indonesia. Based on the results of the application of the COVID patient daily cases in Semarang City, the optimal bandwidth value is 0.86 and the polynomial order is 4 with the minimum GCV is 3179.568 so that the model estimation results the MSE is 2922.22 and the determination coefficient is 97%. The estimation results show the highest number of Corona in the Semarang City at the beginning of July 2020. After the corona case increased in July, while the corona case in August decreased.

Investigating effects of bandwidth selection in local polynomial regression model with applications

Cross-validation (CV) and direct plug-in (DPI) are two commonly used bandwidth selection methods in nonparametric estimation. In this paper, we compare the performance of CV and DPI methods in local polynomial kernel regression through simulation study. We consider continuous response and binary re sponse cases, with local constant, local linear and local quadratic kernel estimators, respectively. Furthermore, we investigate the first derivatives of local quadratic kernel estimators. Our results show that CV and DPI methods excel in different cases, in terms of minimizing the Mean Integrated Squared Errors (MISE); the results are also verified in empirical studies.

Nonparametric functional concurrent regression models

Function‐on‐function regression refers to the situation where both independent and dependent variables in a regression model are of functional nature. Functional concurrent regression is a specific type of function‐on‐function regression that relates the response function at a specific point to the covariate value at that point and the point itself. Standard functional concurrent models are linear (a linear combination of the covariates is used), and often criticized due to their linearity assumption and lack of flexibility. This gives rise to nonparametric functional concurrent regression that models the response function at a specific point using a multivariate nonparametric function of both the point and the covariate value at that point. Such models allow for much more flexibility and predictive accuracy, especially when the underlying relationship is nonlinear. In the past decade, several methods have been proposed to perform estimation, prediction and inference in the nonparametric concurrent models using various methods such as spline smoothing, Gaussian process regression and local polynomial kernel regression. Such models have been shown to be useful tools in functional regression as well as stepping stone for further development. WIREs Comput Stat 2017, 9:e1394. doi: 10.1002/wics.1394This article is categorized under: Statistical and Graphical Methods of Data Analysis > Nonparametric Methods

Longitudinal data analysis based on generalized linear partially varying-coefficient models

ABSTRACTIn this article, we consider a generalized linear partially varying-coefficient model for longitudinal data analysis. A local quasi-likelihood method is proposed to estimate the constant-coefficient and varying-coefficient functions simultaneously based on the local polynomial kernel regression. The corresponding standard error estimates are derived. Large sample properties are investigated. The proposed methodologies are demonstrated by extensive simulation studies and a real example.

Local polynomial quasi-likelihood regression with truncated and dependent data

The local polynomial quasi-likelihood estimation has several good statistical properties such as high minimax efficiency and adaptation of edge effects. In this paper, we construct a local quasi-likelihood regression estimator for a left truncated model, and establish the asymptotic normality of the proposed estimator when the observations form a stationary and α-mixing sequence, such that the corresponding result of Fan et al. [Local polynomial kernel regression for generalized linear models and quasilikelihood functions, J. Amer. Statist. Assoc. 90 (1995), pp. 141–150] is extended from the independent and complete data to the dependent and truncated one. Finite sample behaviour of the estimator is investigated via simulations too.

A Unifying View of Wiener and Volterra Theory and Polynomial Kernel Regression

Volterra and Wiener series are perhaps the best-understood nonlinear system representations in signal processing. Although both approaches have enjoyed a certain popularity in the past, their application has been limited to rather low-dimensional and weakly nonlinear systems due to the exponential growth of the number of terms that have to be estimated. We show that Volterra and Wiener series can be represented implicitly as elements of a reproducing kernel Hilbert space by using polynomial kernels. The estimation complexity of the implicit representation is linear in the input dimensionality and independent of the degree of nonlinearity. Experiments show performance advantages in terms of convergence, interpretability, and system sizes that can be handled.

Kernel spline regression

The authors propose «kernel spline regression,» a method of combining spline regression and kernel smoothing by replacing the polynomial approximation for local polynomial kernel regression with the spline basis. The new approach retains the local weighting scheme and the use of a bandwidth to control the size of local neighborhood. The authors compute the bias and variance of the kernel linear spline estimator, which they compare with local linear regression. They show that kernel spline estimators can succeed in capturing the main features of the underlying curve more effectively than local polynomial regression when the curvature changes rapidly. They also show through simulation that kernel spline regression often performs better than ordinary spline regression and local polynomial regression.

Analysis of lidar fields using local polynomial regression

AbstractLidar (light detection and ranging) is a laser based tool for remote measurement of several atmospheric species of importance. We consider the analysis of a field, consisting of several consecutive measurements, in which the concentrations are proportional to the derivatives in the directions of the light paths. Inference is based on local polynomial kernel regression, both for estimation of the derivatives of the mean‐function and for estimation of the variance‐function. Bivariate bandwidth matrices are selected using the empirical‐bias bandwidth selector (EBBS) adapted to allow for dependent data and to support selection of bivariate bandwidths. The estimation procedure is demonstrated on measurements of atomic mercury from the Solvay industries mercury cell chlor‐alkali plant in Rosignano Solvay, Italy. Copyright © 2005 John Wiley & Sons, Ltd.

Local Polynomial Mixed-Effects Models for Longitudinal Data

We consider a nonparametric mixed-effects model yi(tij)=η(tij)+vi(tij)+ϵi(tij),j=1,2,…,ni;i=1,2,…,n for longitudinal data. We propose combining local polynomial kernel regression and linear mixed-effects (LME) model techniques to estimate both fixedeffects (population) curve η(t) and random-effects curves vi(t). The resulting estimator, called the local polynomial LME (LLME) estimator, takes the local correlation structure of the longitudinal data into account naturally. We also propose new bandwidth selection strategies for estimating η(t) and vi(t). Simulation studies show that our estimator for η(t) is superior to the existing estimators in the sense of mean squared errors. The asymptotic bias, variance, mean squared errors, and asymptotic normality are established for the LLME estimators of η(t). When ni is bounded and n tends to infinity, our LLME estimator converges in a standard nonparametric rate, and the asymptotic bias and variance are essentially the same as those of the kernel generalized estimating equation estimator proposed by Lin and Carroll. But when both ni and n tend to infinity, the LLME estimator is consistent with a slower rate of n1/2 compared to the standard nonparametric rate, due to the existence of within-subject correlations of longitudinal data. We illustrate our methods with an application to a longitudinal dataset.

Nonparametric Function Estimation for Clustered Data When the Predictor is Measured without/with Error

We consider local polynomial kernel regression with a single covariate for clustered data using estimating equations. We assume that at most m < ∞ observations are available on each cluster. In the case of random regressors, with no measurement error in the predictor, we show that it is generally the best strategy to ignore entirely the correlation structure within each cluster and instead pretend that all observations are independent. In the further special case of longitudinal data on individuals with fixed common observation times, we show that equivalent to the pooled data approach is the strategy of fitting separate nonparametric regressions at each observation time and constructing an optimal weighted average. We also consider what happens when the predictor is measured with error. Using the SIMEX approach to correct for measurement error, we construct an asymptotic theory for both the pooled and the weighted average estimators. Surprisingly, for the same amount of smoothing, the weighted average estimators typically have smaller variances than the pooling strategy. We apply the proposed methods to analysis of the AIDS Costs and Services Utilization Survey.

LOCALLY WEIGHTED LEAST SQUARES KERNEL REGRESSION AND STATISTICAL EVALUATION OF LIDAR MEASUREMENTS

The LIDAR technique is an efficient tool in monitoring the distribution of atmospheric species of importance. We study the concentration of atmospheric atomic mercury in an Italian geothermal field and discuss the possibility of using recent results from local polynomial kernel regression theory for the evaluation of the derivative of the DIAL curve. A MISE-optimal bandwidth selector, which takes account of the heteroscedasticity in the regression is suggested. Further, we estimate the integrated amount of mercury in a certain area.

Local Polynomial Kernel Regression for Generalized Linear Models and Quasi-Likelihood Functions

Local Polynomial Kernel Regression for Generalized Linear Models and Quasi-Likelihood Functions

Local Polynomial Kernel Regression for Generalized Linear Models and Quasi-Likelihood Functions

Abstract We investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi-likelihood contexts. In the ordinary regression case, local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity. One noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators. These properties are shown to carry over to generalized linear model and quasi-likelihood settings. We also derive the asymptotic distributions of the proposed class of estimators that allow for straightforward interpretation and extensions of state-of-the-art bandwidth selection methods.