Nadaraya-Watson Kernel Estimator Research Articles

Prediction of leptospirosis outbreaks by hydroclimatic covariates: a comparative study of statistical models.

Leptospirosis, the infectious disease caused by a spirochete bacteria, is a major public health problem worldwide. In Argentina, some regions have climatic and geographical characteristics that favor the habitat of bacteria of the Leptospira genus, whose survival strongly depends on climatic factors, enhanced by climate change, which increase the problems associated with people’s health. In order to have a method to predict leptospirosis cases, in this paper, five time series forecasting methods are compared: two parametric (autoregressive integrated moving average and an alternative one that allows covariates, ARIMA and ARIMAX, respectively), two nonparametric (Nadaraya-Watson Kernel estimator, one and two kernels versions, NW-1 K and NW-2 K), and one semiparametric (semi-functional partial linear regression, SFPLR) method. For this, the number of cases of leptospirosis registered from 2009 to 2020 in three important cities of northeastern Argentina is used, as well as hydroclimatic covariates related to the presence of cases. According to the obtained results, there is no method that improves considerably the rest and can be recommended as a unique tool for leptospirosis prediction. However, in general, the NW-2 K method gets a better performance. This work, in addition to using a long-term high-quality time series, enriches the area of applications of statistical models to epidemiological leptospirosis data by the incorporation of hydroclimatic variables, and it is recommended directing further efforts in this line of research, under the context of current climate change.

Asymptotic normality of the regression mode in the nonparametric random design model for censored data

In the nonparametric regression model, where the regression function is given by for a measurable function estimation of the location (mode) of a unique maximum of by the location of a maximum of the Nadaraya-Watson kernel estimator for the curve is considered. Within the setting of the censored data, we obtain the consistency with rate and the asymptotic normality results for under mild local smoothness assumptions on the regression and the design density of Simulation results are performed to illustrate the performance of the procedure.

Kernel Averaging Estimators

The issue of bandwidth selection is a fundamental model selection problem stemming from the uncertainty about the smoothness of the regression. In this article, we advocate a model averaging approach to circumvent the problem caused by this uncertainty. Our new approach involves averaging across a series of Nadaraya-Watson kernel estimators each under a different bandwidth, with weights for these different estimators chosen such that a least-squares cross-validation criterion is minimized. We prove that the resultant combined-kernel estimator achieves the smallest possible asymptotic aggregate squared error. The superiority of the new estimator over estimators based on widely accepted conventional bandwidth choices in finite samples is demonstrated in a simulation study and a real data example.

A Comparison Between New Modification of ANWK and Classical ANWK Methods in Nonparametric Regression

Nonparametric kernel estimators are mostly used in a variety of statistical research fields. Nadaraya-Watson kernel estimator (NWK) is one of the most important nonparametric kernel estimator that is often used in regression models with a fixed bandwidth. In this article, we consider the four new Proposed Adaptive Nadaraya-Watson Kernel Regression Estimators (Interquartile Range, Standard Deviation, Mean Absolute Devotion, and Median Absolute Deviation) rather than (Fixed Bandwidth, Adaptive Geometric, Adaptive Mean, Adaptive Range, and Adaptive Median). The outcomes in both simulation and actual data in Leukemia Cancer show that the four new ANW Kernel Estimators (Interquartile Range, Standard Deviation, Mean Absolute devotion, and Median Absolute Deviation) is more effective than the kernel estimations with fixed bandwidth in previous studies using Mean Square Error (MSE) Criterion.

Estimation of the bandwidth parameter in Nadaraya-Watson kernel non-parametric regression based on universal threshold level

This paper proposes a new improvement of the Nadaraya-Watson kernel non-parametric regression estimator and the bandwidth of this new improvement is obtained depending on universal threshold level with wavelet of kernel function instead of using fixed bandwidth and variable bandwidth for geometric, arithmetic mean, range and median measurements. A simulation study is presented, including comparisons between the proposed method and five others Nadaraya-Watson kernel estimators (classical methods), as well as using real data depending on a program written in MATLAB language which was designed for this purpose. It was concluded that the proposed method is more accurate than all classical methods for all simulations and real data based on MSE criterion.

An Upper Bound of the Bias of Nadaraya-Watson Kernel Regression under Lipschitz Assumptions

The Nadaraya-Watson kernel estimator is among the most popular nonparameteric regression technique thanks to its simplicity. Its asymptotic bias has been studied by Rosenblatt in 1969 and has been reported in several related literature. However, given its asymptotic nature, it gives no access to a hard bound. The increasing popularity of predictive tools for automated decision-making surges the need for hard (non-probabilistic) guarantees. To alleviate this issue, we propose an upper bound of the bias which holds for finite bandwidths using Lipschitz assumptions and mitigating some of the prerequisites of Rosenblatt’s analysis. Our bound has potential applications in fields like surgical robots or self-driving cars, where some hard guarantees on the prediction-error are needed.

Some asymptotic properties of kernel regression estimators of the mode for stationary and ergodic continuous time processes.

In the present paper, we consider the nonparametric regression model with random design based on (mathbf{X}_mathrm{t},mathbf{Y}_mathrm{t})_{mathrm{t}ge 0} a mathbb {R}^{d}times mathbb {R}^{q}-valued strictly stationary and ergodic continuous time process, where the regression function is given by m(mathbf{x},psi ) = mathbb {E}(psi (mathbf{Y}) mid mathbf{X} = mathbf{x})), for a measurable function psi : mathbb {R}^{q} rightarrow mathbb {R}. We focus on the estimation of the location {varvec{Theta }} (mode) of a unique maximum of m(cdot , psi ) by the location widehat{{varvec{Theta }}}_mathrm{T} of a maximum of the Nadaraya–Watson kernel estimator widehat{m}_mathrm{T}(cdot , psi ) for the curve m(cdot , psi ). Within this context, we obtain the consistency with rate and the asymptotic normality results for widehat{{varvec{Theta }}}_mathrm{T} under mild local smoothness assumptions on m(cdot , psi ) and the design density f(cdot ) of mathbf{X}. Beyond ergodicity, any other assumption is imposed on the data. This paper extends the scope of some previous results established under the mixing condition. The usefulness of our results will be illustrated in the construction of confidence regions.

Standard Errors for Nonparametric Regression

This paper proposes five pointwise consistent and asymptotic normal estimators of the asymptotic variance function of the Nadaraya-Watson kernel estimator for nonparametric regression. The proposed estimators are constructed based on the first-stage nonparametric residuals, and their asymptotic properties are established under the assumption that the same bandwidth sequences are used throughout, which mimics what researchers do in practice while making derivations more complicated instead. A limited Monte Carlo experiment demonstrates that the proposed estimators possess smaller pointwise variability in small samples than the pair and wild bootstrap estimators which are commonly used in practice.

Uniform consistency and uniform in bandwidth consistency for nonparametric regression estimates and conditional U-statistics involving functional data

ABSTRACTW. Stute [(1991), Annals of Probability, 19, 812–825] introduced a class of so-called conditional U-statistics, which may be viewed as a generalisation of the Nadaraya–Watson estimates of a regression function. Stute proved their strong pointwise consistency to We apply the methods developed in Dony and Mason [(2008), Bernoulli, 14(4), 1108–1133] to establish uniformity in and in bandwidth consistency (i.e. , where at some specific rate) to of the estimator proposed by Stute when Y and covariates X are functional taking value in some abstract spaces. In addition, uniform consistency is also established over for a suitably restricted class . The theoretical uniform consistency results, established in this paper, are (or will be) key tools for many further developments in functional data analysis. Applications include the Nadaraya–Watson kernel estimators and the conditional distribution function. Our theorems allow data-driven local bandwidths for these statistics.

Modification of the adaptive Nadaraya-Watson kernel method for nonparametric regression (simulation study)

In this research, a new improvement of the Nadaraya-Watson kernel non parametric regression estimator is proposed and the bandwidth of this new improvement is obtained depending on the three different statistical indicators: robust mean, median and harmonic mean of kernel function instead of using geometric and arithmetic mean, or R. Simulation study is presented, including comparisons with four others Nadaraya-Watson kernel estimators (classical methods). The proposed estimator in the case of harmonic mean is more accurate than all classical methods for all simulations based on MSE criteria.

LQD-RKHS-based distribution-to-distribution regression methodology for restoring the probability distributions of missing SHM data

Data loss is a critical problem in structural health monitoring (SHM). Probability distributions play a highly important role in many applications. Improving the quality of distribution estimations made using incomplete samples is highly important. Missing samples can be compensated for by applying conventional missing data restoration methods; however, ensuring that restored samples roughly follow underlying distributions of true missing data remains a challenge. Another strategy involves directly restoring the probability density function (PDF) for a sensor when samples are missing by leveraging distribution information from another sensor with complete data using distribution regression techniques; existing methods include the conventional distribution-to-distribution regression (DDR) and distribution-to-warping function regression (DWR) methods. Due to constraints on PDFs and warping functions, the regression functions of both methods are estimated from the Nadaraya-Watson kernel estimator with relatively low degrees of precision. This article proposes a new indirect distribution-to-distribution regression method in the context of functional data analysis for restoring distributions of missing SHM data. PDFs are transformed to ordinary functions residing in a Hilbert space via the newly proposed log-quantile-density (LQD) transformation; the regression for distributions is realized in the transformed space via a functional regression model constructed based on the theory of Reproducing Kernel Hilbert Space (RKHS), corresponding result is subsequently mapped back to the density space through the inverse LQD transformation. Test results using field monitoring data indicate that the new method significantly outperforms conventional methods in general cases; however, in extrapolation cases, the new method is inferior to the distribution-to-warping function regression method.

Comparison of noise reduction methods in radiometric correlation measurements of two-phase liquid-gas flows

Two-phase liquid-gas flows occur frequently in the mining, energy, chemical, and petrochemical industries. One of non-contact methods used to analyse these flows is the gamma ray absorption method. However, the signals received from radiation detectors contain a significant stochastic noise, which makes them difficult to analyse. The article describes four methods of noise reduction in cross-correlation measurements of water-air mixture flows in a horizontal pipeline. In addition to the classical method of digital filtering of signals, the methods consisting in signal spectrum filtering, discrete wavelet transformation, and Nadaraya-Watson kernel estimator are described. Sample results of the measurements carried out in the horizontal pipe having the inner diameter of 30 mm for the air bubbles velocity ranging from 0.7 to 1.4 m/s are presented. In the research, the absorption set composed of two linear Am-241 gamma-ray sources and two scintillation NaI(Tl) probes was used. It was found that the lowest measurement uncertainty of the dispersed phase flow velocity is obtained when the cross-correlation distributions of the recorded signals are smoothed using the Discrete Wavelet Transform or the Nadaraya-Watson kernel estimator.

On the Adaptive Nadaraya-Watson Kernel Estimator for the Discontinuity in the Presence of Jump Size

In this paper, we studied an Adaptive Nadaraya Watson kernel estimator to check the bias effect on both side of the discontinuity in the presence of jump size for regression discontinuity model. We have proposed the modified Adaptive Nadaraya Watson kernel estimator and derived its normality and variance. We have also compared with the asymptotic normality of the Mean Integrated Square Error (MISE) of Adaptive Nadaraya Watson kernel estimator and Nadaraya Watson kernel estimator. The results obtained from the simulation study have showed that Adaptive Nadaraya Watson estimator has better performance than the Nadaraya Watson Kernel estimator.

Modified kernel regression method for the denoising of X-ray pulsar profiles

In this paper, a modified kernel regression algorithm is proposed to reduce the noise of pulsar profiles autonomously. Taking advantage of the classical autonomous kernel regression the presented algorithm based on the second-order derivative compensation is developed to improve the performance of the Nadaraya-Watson kernel estimator. The periodic extension technique is introduced to eliminate the boundary issue inherent in kernel regression means. Four indexes are utilized to explore the performance of the proposed method via both emulated and real data. Additionally, other widely accepted denoising methods based on wavelet transformation and empirical model decomposition are simulated to make a comparison. The experimental results have shown that the proposed algorithm achieves a higher quality profile than the compared methods, which will help to discover the emission mechanism of pulsars. According to our experiments, we would like to point out that a smooth profile cannot guarantee an accurate measurement (time of arrival, TOA), which indicates that it is not necessary to denoise the epoch folding profile for estimating the TOA information in X-ray pulsar-based navigation systems.

Nonparametric estimations of the sea state bias for a radar altimeter

To estimate the sea state bias (SSB) for radar altimeter, two nonparametric models, including a Nadaraya-Watson (NW) kernel estimator and a local linear regression (LLR) estimator, are studied based on the Jason-2 altimeter data. Selecting from different combinations of the Gaussian kernel function, spherical Epanechnikov kernel function, a fixed bandwidth and a local adjustable bandwidth, it is observed that the LLR method with the spherical Epanechnikov kernel function and the local adjustable bandwidth is the optimal nonparametric model for the SSB estimation. The comparisons between the nonparametric and parametric models are conducted and the results show that the nonparametric model performs relatively better at high-latitudes of the Northern Hemisphere. This method has been applied to the HY-2A altimeter as well and the same conclusion can be obtained.

Variable Selection in Kernel Regression Using Measurement Error Selection Likelihoods

ABSTRACTThis article develops a nonparametric shrinkage and selection estimator via the measurement error selection likelihood approach recently proposed by Stefanski, Wu, and White. The measurement error kernel regression operator (MEKRO) has the same form as the Nadaraya–Watson kernel estimator, but optimizes a measurement error model selection likelihood to estimate the kernel bandwidths. Much like LASSO or COSSO solution paths, MEKRO results in solution paths depending on a tuning parameter that controls shrinkage and selection via a bound on the harmonic mean of the pseudo-measurement error standard deviations. We use small-sample-corrected AIC to select the tuning parameter. Large-sample properties of MEKRO are studied and small-sample properties are explored via Monte Carlo experiments and applications to data. Supplementary materials for this article are available online.

Nonparametric Regression with Common Shocks

This paper considers a nonparametric regression model for cross-sectional data in the presence of common shocks. Common shocks are allowed to be very general in nature; they do not need to be finite dimensional with a known (small) number of factors. I investigate the properties of the Nadaraya-Watson kernel estimator and determine how general the common shocks can be while still obtaining meaningful kernel estimates. Restrictions on the common shocks are necessary because kernel estimators typically manipulate conditional densities, and conditional densities do not necessarily exist in the present case. By appealing to disintegration theory, I provide sufficient conditions for the existence of such conditional densities and show that the estimator converges in probability to the Kolmogorov conditional expectation given the sigma-field generated by the common shocks. I also establish the rate of convergence and the asymptotic distribution of the kernel estimator.

Regression models with correlated errors based on functional random design

This paper deals with the study of the estimation of the functional regression operator when the explanatory variable takes its values in some abstract space of functions. The main goal of this paper is to establish the exact rate of convergence of the mean squared error of the functional version of the Nadaraya–Watson kernel estimator when the errors come from a stationary process under long or short memory and based on random functional data. Moreover, these theoretical results are checked through some simulations with regular (smooth) and irregular curves and then with real data.

Nonlinear normal modes for damage detection

The sensitivity of the nonlinear dynamic response to damage is investigated in simply supported beams by computing the nonlinear normal modes (NNMs) and exploiting the rich information they unfold about the nonlinear system behavior. Damage is introduced as a reduction of the flexural stiffness within a small segment of the beam span. The problem is formulated in a piece-wise fashion and analytically tackled by using the method of multiple scales to compute the NNMs and the backbone curves. The latter describe the frequency versus free oscillation amplitude for each mode and as such they represent the skeleton of the nonlinear frequency response around each beam modal frequency. The bending of the backbones is regulated by the so-called effective nonlinearity coefficients associated with each mode. The comparison between the damaged and undamaged beams shows a sensitivity of the effective nonlinearity coefficients higher than the sensitivity of the linear natural frequencies. Moreover, the nonlinear frequency trends unfold an interesting dependence of the nonlinear free response on the stiffness reduction at the damage site. Such dependence yields useful information about the damage position. An effective identification strategy for the damage position is proposed by computing the damage-induced discontinuities in the second derivative of the NNMs (i.e., first order estimate of the bending curvature), without resorting to a baseline model of the undamaged beam. This goal is achieved by computing a high order derivative of the Nadaraya–Watson kernel estimator, which is evaluated from a finite set of sampled beam deflections associated with each individual NNM. The results show that the second derivatives of higher NNMs are more sensitive to damage-induced discontinuities than the linear modal curvatures. The robustness against noise of the damage identification process is also discussed.

Non-parametric Variance Estimation Using Donor Imputation Method

The main objective of this study is to investigate the relative performance of donor imputation method in situations that are likely to occur in practice and to carry out numerical comparative study of estimators of variance using Nadaraya-Watson kernel estimators and other estimators. Nadaraya-Watson kernel estimator can be viewed as a non-parametric imputation method as it leads to an imputed estimator with negligible bias without requiring the specification of a parametric imputation model. Simulation studies were carried out to investigate the performance of Nadaraya-Watson kernel estimators in terms of variance. From the results, it was found out that Nadaraya-Watson kernel estimator has negligible bias and its variance is small. When compared with Naïve, Jackknife and Bootstrap estimators, Nadaraya-Watson kernel estimator was found to perform better than bootstrap estimator in linear and non-linear populations.