Kernel Smoothing Research Articles

Smoothing parameter selection method for multiresponse nonparametric regression model using smoothing spline and Kernel estimators approaches

The principle problem in multiresponse nonparametric regression model is how we estimate the regression functions which draw association between some dependent (response) variables and some independent (predictor) variables where there are correlations between responses. There are many techniques used to estimate the regression function. Two of them are spline and kernel smoothing techniques. Speaking about smoothing techniques, not only in uniresponse spline and kernel nonparametric regression models but also in multiresponse spline and kernel nonparametric regression models, the estimations of regression functions depend on smoothing parameters. In the privious researches the covariance matrices were assumed to be known. Matrix of covariance is not assumed known in this research. The goals of this research are selecting of optimal smoothing parameters for the model we consider through spline and kernel smoothing techniques. Optimal smoothing parameters can be obtained by taking the solution to generalized cross validation (GCV) optimization problem. The obtained results of this research are the optimal smoothing parameter for smoothing spline estimator approach and the optimal smoothing parameter namely optimal bandwidth for kernel estimator approach.

Dynamic Changes Of Body Image And Quality Of Life In Breast Cancer Patients.

PurposeDynamic changes of body image and quality of life (QoL) in breast cancer patients were not commonly investigated. We aimed to compare the dynamic changes in QoL and body image of breast cancer survivors receiving breast-conserving surgery or total mastectomy within 5–10 years after surgery.MethodsPatients with non-metastatic breast cancer who received surgery were invited to complete the World Health Organization Quality of Life–Brief (WHOQOL-BREF) questionnaire and the Body Image Scale (BIS) within 10 years after surgery. We applied kernel smoothing methods to capture the dynamic changes of the patients’ QoL and body image within 5 years after surgery. We also constructed multiple linear regression models to identify predictive factors for QoL and body image.ResultsA total of 581 patients were collected, and 211 of them received breast-conserving surgery. There were no statistically significant differences in QoL and body image for breast-conserving surgery versus total mastectomy, but the former showed fluctuating trends. BIS was a predictor of every item and domain in the WHOQOL-BREF in the multiple linear regression model, and explanatory of the trends of dynamic change over time. Patients without lymph node dissection seemed to have less positive feelings but were more satisfied with sexual activities.ConclusionBody image is predictive of the QoL of breast cancer patients. Dynamic changes of body image and QoL would be useful for shared decision-making regarding surgery in breast cancer patients.

Smoothed empirical likelihood inference and variable selection for quantile regression with nonignorable missing response

With nonignorable missing responses, an efficient estimator and a variable selection method for quantile regression coefficient are proposed based on smoothed weighted empirical likelihood (SWEL). To handle the identifiability issue, a part of covariates named as nonresponse instrument is used, which is related to response but unrelated to the propensity conditioned on response and other observed covariates. The generalized method of moments is applied to estimate the parameters in the nonresponse propensity. Based on inverse probability weighting and kernel smoothing approaches, consistency and asymptotic normality of the proposed estimator for quantile regression coefficient are established. The asymptotic properties of the resulting SWEL ratio function and the corresponding confidence regions are derived. Further, a penalized SWEL method and its algorithm for variable selection are investigated. The finite-sample performance of the proposed estimator is studied through simulation, and an application to HIV-CD4 data set is also presented.

Kernel-based Inference in Time-Varying Coefficient Cointegrating Regression

This paper studies nonlinear cointegrating models with time-varying coefficients and multiple nonstationary regressors using classic kernel smoothing methods to estimate the coefficient functions. Extending earlier work on nonstationary kernel regression to take account of practical features of the data, we allow the regressors to be cointegrated and to embody a mixture of stochastic and deterministic trends, complications which result in asymptotic degeneracy of the kernel-weighted signal matrix. To address these complications new local and global rotation techniques are introduced to transform the covariate space to accommodate multiple scenarios of induced degeneracy. Under regularity conditions we derive asymptotic results that differ substantially from existing kernel regression asymptotics, leading to new limit theory under multiple convergence rates. For the practically important case of endogenous nonstationary regressors we propose a fully-modified kernel estimator whose limit distribution theory corresponds to the prototypical pure cointegration case (i.e., with exogenous covariates), thereby facilitating inference using a generalized Wald-type test statistic. These results substantially generalize econometric estimation and testing techniques in the cointegration literature to accommodate time variation and complications of co-moving regressors. Finally, Monte-Carlo simulation studies as well as an empirical illustration to aggregate US data on consumption, income, and interest rates are provided to illustrate the methodology and evaluate the numerical performance of the proposed methods in finite samples.

Enhancing Optical Correlation Decision Performance for Face Recognition by Using a Nonparametric Kernel Smoothing Classification.

Optical correlation has a rich history in image recognition applications from a database. In practice, it is simple to implement optically using two lenses or numerically using two Fourier transforms. Even if correlation is a reliable method for image recognition, it may jeopardize decision making according to the location, height, and shape of the correlation peak within the correlation plane. Additionally, correlation is very sensitive to image rotation and scale. To overcome these issues, in this study, we propose a method of nonparametric modelling of the correlation plane. Our method is based on a kernel estimation of the regression function used to classify the individual images in the correlation plane. The basic idea is to improve the decision by taking into consideration the energy shape and distribution in the correlation plane. The method relies on the calculation of the Hausdorff distance between the target correlation plane (of the image to recognize) and the correlation planes obtained from the database (the correlation planes computed from the database images). Our method is tested for a face recognition application using the Pointing Head Pose Image Database (PHPID) database. Overall, the results demonstrate good performances of this method compared to competitive methods in terms of good detection and very low false alarm rates.

Gimme shelter: The effect of rocks and moonlight on occupancy and activity pattern of an endangered rodent, the garden dormouse Eliomys quercinus

Information on spatial behaviour and temporal activity patterns is paramount for the conservation of animal species. This is particularly true for endangered taxa that are threatened by ongoing climatic and environmental changes. The garden dormouse Eliomys quercinus is a native European rodent (family Gliridae), whose populations are declining throughout the Continent. Notwithstanding this, neither International nor National laws explicitly require mandatory monitoring of populations. As a result, compelling information on the spatiotemporal behaviour of dormouse is lacking. We aimed to fill this gap by investigating occupancy patterns in relation to environmental features and activity rhythms in relation to moonlit nights in an Alpine population of dormouse within the Stelvio National Park, northern Italy. Data were collected between May and October 2015. Twenty camera-traps were deployed in a 500 ha coniferous forest, using a random tessellation approach; camera trap data were analyzed with occupancy models and kernel smoothers. Camera-traps provided a reliable assessment of the presence of the garden dormouse, with only 1 % of false absence and a high detection probability (68 %). The occurrence of the garden dormouse was positively influenced by the percentage of rock coverage on the ground. The species showed a strictly nocturnal behaviour, with an activity peak before midnight, negatively related to moonlit nights. The use of rocky areas for nesting, shelter site and thigmotactic movements and moonlight avoidance may represent adaptations of the garden dormouse to avoid predation risk. Our results shed some light on the habitat requirement of a poorly known, near-threatened species, and provide baseline information for future monitoring and conservation activities.

Averaging principle for stochastic 3D fractional Leray-α model with a fast oscillation

This article investigates the multiscale stochastic 3D fractional Leray-α model. By using the Khasminskii technique, we establish the strong average principle for stochastic 3D fractional Leray-α model with a fast oscillation. This model is the stochastic 3D Navier–Stokes equations regularized through a smoothing kernel of order θ1 in the nonlinear term and a θ2-fractional Laplacian. The main result is applicable to the classical stochastic 3D Leray-α model (), stochastic 3D hyperviscous Navier–Stokes equations () and stochastic 3D critical Leray-α model ().

Consistent shepard interpolation for SPH-based fluid animation

We present a novel technique to correct errors introduced by the discretization of a fluid body when animating it with smoothed particle hydrodynamics (SPH). Our approach is based on the Shepard correction, which reduces the interpolation errors from irregularly spaced data. With Shepard correction, the smoothing kernel function is normalized using the weighted sum of the kernel function values in the neighborhood. To compute the correction factor, densities of neighboring particles are needed, which themselves are computed with the uncorrected kernel. This results in an inconsistent formulation and an error-prone correction of the kernel. As a consequence, the density computation may be inaccurate, thus the pressure forces are erroneous and may cause instabilities in the simulation process. We present a consistent formulation by using the corrected densities to compute the exact kernel correction factor and, thereby, increase the accuracy of the simulation. Employing our method, a smooth density distribution is achieved, i.e., the noise in the density field is reduced by orders of magnitude. To show that our method is independent of the SPH variant, we evaluate our technique on weakly compressible SPH and on divergence-free SPH. Incorporating the corrected density into the correction process, the problem cannot be stated explicitly anymore. We propose an efficient and easy-to-implement algorithm to solve the implicit problem by applying the power method. Additionally, we demonstrate how our model can be applied to improve the density distribution on rigid bodies when using a well-known rigid-fluid coupling approach.

Estimation of fully nonparametric transformation models

Consider the following nonparametric transformation model $\Lambda (Y)=m(X)+\varepsilon $, where $X$ is a $d$-dimensional covariate, $Y$ is a continuous univariate dependent variable and $\varepsilon $ is an error term with zero mean and which is independent of $X$. We assume that the unknown transformation $\Lambda $ is strictly increasing and that $m$ is an unknown regression function. Our goal is to develop two new nonparametric estimators of the transformation $\Lambda $, the first one based on the least squares loss and the second one based on the least absolute deviation loss, and to compare their performance with that of the estimators developed by Chiappori, Komunjer and Kristensen (J. Econometrics 188 (2015) 22–39). Our proposed estimators are based on an estimator of the conditional distribution of $U$ given $X$, where $U$ is an appropriate transformation of $Y$ that is uniformly distributed. The main motivation for working with $U$ instead of $Y$ is that, in transformation models, the response $Y$ is often skewed with very long tails, and so kernel smoothing based on $Y$ does not work well. Hence, we expect to obtain better estimators if we pre-transform $Y$ before applying kernel smoothing. We establish the asymptotic normality of the two proposed estimators. We also carry out a simulation study to illustrate the performance of our estimators, to compare these new estimators with the ones of Chiappori, Komunjer and Kristensen (J. Econometrics 188 (2015) 22–39) and to see under which model conditions which estimators behave the best.

Kernel estimation for panel data with heterogeneous dynamics

Summary This paper proposes nonparametric kernel-smoothing estimation for panel data to examine the degree of heterogeneity across cross-sectional units. We first estimate the sample mean, autocovariances, and autocorrelations for each unit and then apply kernel smoothing to compute their density functions. The dependence of the kernel estimator on bandwidth makes asymptotic bias of very high order affect the required condition on the relative magnitudes of the cross-sectional sample size (N) and the time-series length (T). In particular, it makes the condition on N and T stronger and more complicated than those typically observed in the long-panel literature without kernel smoothing. We also consider a split-panel jackknife method to correct bias and construction of confidence intervals. An empirical application illustrates our procedure.

Application of kernel smoothing to estimate the spatio-temporal variation in risk of STEC O157 in England

Identifying geographical areas with significantly higher or lower rates of infectious diseases can provide important aetiological clues to inform the development of public health policy and interventions designed to reduce morbidity. We applied kernel smoothing to estimate the spatial and spatio-temporal variation in risk of STEC O157 infection in England between 2009 and 2015, and to explore differences between the residential locations of cases reporting travel and those not reporting travel. We provide evidence that the distribution of STEC O157 infection in England is non-uniform with respect to the distribution of the at-risk population; that the spatial distribution of the three main genetic lineages infecting humans (I, II and I/II) differs significantly and that the spatio-temporal risk is highly dynamic. Our results also indicate that cases of STEC O157 reporting travel within or outside the UK are more likely to live in the south/south-east of the country, meaning that their residential location may not reflect the location of exposure that led to their infection. We suggest that the observed variation in risk reflects exposure to sources of STEC O157 that are geographically prescribed. These differences may be related to a combination of changes in the strains circulating in the ruminant reservoir, animal movements (livestock, birds or wildlife) or the behavior of individuals prior to infection. Further work to identify the importance of behaviours and exposures reported by cases relative to residential location is needed.

Semiparametric quasi maximum likelihood estimation of the fractional response model

This paper proposes a new semiparametric estimator of models where the response random variable is a fraction. The estimator is constructed by optimizing a semiparametric quasi-maximum likelihood that utilizes kernel smoothing. Under suitable conditions, the consistency and asymptotic normality of the proposed estimator is established allowing for data-driven bandwidth choices as well as random trimming, and its flexibility and robustness are showcased in a Monte Carlo experiment and an empirical application.

Considerations in testing treatment effects on transient event driven health status changes measured by patient reported outcomes.

Many treatments and drugs are intended to reduce the occurrence of negative events of interest, control the severity of the events, accelerate recovery from the events, or a combination of these effects. While assessing the clinical effect is typically the primary objective of a trial, testing the treatment effect on the health status of patients based on patient reported outcome (PRO) can be a useful component in determining the value of a treatment. Analysis of PROs in this setting, however, face the following challenges: the PRO value immediately after the event occurrence is often not captured, and the effect of the event on health status measured by the PRO is transient as subjects recover over time. Therefore, traditional statistical methods used to assess treatment effects suffer from low power for PROs. In this manuscript, we apply a kernel smoothing technique to estimate before- and after-event PRO values. We also propose new test outcomes based on observed and estimated PRO values and evaluate tests that focus on the tail distributions. We demonstrate that the tail distribution tests using the new outcomes can achieve high power under certain conditions.

Novel direction of arrival estimation using Adaptive Directional Spatial Time-Frequency Distribution

Novel spatial time-frequency distributions and instantaneous frequency estimation scheme are developed for the proposed direction of arrival estimation scheme. A new class of spatial adaptive time-frequency distributions is proposed that are obtained by local optimization of the smoothing kernel for obtaining spatial time-frequency distributions. The instantaneous frequencies of the source signals are estimated using modified Viterbi algorithm that exploits both the spatial diversity (estimated through spatial time-frequency distributions) and continuity of the instantaneous frequency curves for accurate estimation at low Signal-to-Noise-Ratio (SNR) scenarios when the signal components intersect each other in time-frequency domain. In the proposed scheme the estimated instantaneous frequencies are used for the direction of arrival estimation of each source separately, thus results in accurate estimates even in low SNR regime. Experimental results are given that indicate that the proposed direction of arrival estimation algorithm outperforms the traditional directional of arrival estimation schemes.

Doubly robust kernel density estimation when group membership is missing at random

When there are subjects with subpopulation memberships missing, the kernel density estimates of the subpopulations based on the subjects with verified memberships may not be valid unless the missingness of the memberships satisfies the stringent missing completely at random assumption. Under more plausible missing at random (MAR) assumptions, the inverse probability weighting (IPW) kernel smoothing method can be employed if the missing mechanism is well understood. On the other hand, if the memberships can be well predicted by a prediction model, one may replace the memberships (either all or only the missing ones), with the probabilities for the memberships to deal with the missing values. By integrating the information of the missing mechanism and the prediction model, we develop a doubly robust kernel density estimate which is valid as long as either of the models is correctly specified. Asymptotic properties of the new estimates are derived and the asymptotic orders of the mean integrated square errors (MISE) are compared with existing methods. We prove that the doubly robust estimate can be more efficient than the IPW estimate when the model for the missing mechanism is correctly specified. We also prove that the prediction-model based approaches may have smaller MISEs than the complete data when the prediction model is correctly specified, suggesting that those methods can be applied even when there are no missing values in the memberships. Simulation studies are carried out to assess the performances of methods and a real study example is used for illustrative purposes.

A Fast and Robust Nonparametric Monitoring Scheme for Free-Form Surface Scanning Data

The advance of new sensing technologies, such as the 3-D laser scanning, creates a data-rich environment for quality control in modern industries. The free-form surfaces of complex manufactured parts can be quickly scanned, producing thousands of data points. To monitor these large-scale surface scanning data, three major challenges have to be solved simultaneously: 1) simple parametric models are no longer sufficient to describe free-form surfaces; 2) the massive data points need fast computations; and 3) the presence of outliers calls for robust analytics. To fulfill this task, this paper proposes a novel monitoring scheme where the control chart is designed based on a new robust bilateral kernel smoothing method. A fast approximation algorithm is also developed for efficient online monitoring. This fast and robust nonparametric control chart shows significant superiority for surface monitoring in our numerical simulations. Finally, a real case study demonstrates the effectiveness of our proposed scheme in monitoring the stability of a 3-D printing process. Note to Practitioners —Recently, the widespread use of 3-D laser scanners in industries enables engineers to digitize any free-form surface into thousands of data points, which provides an unprecedented opportunity for quality control of complex manufactured products. This paper proposes a monitoring scheme which caters to arbitrary geometric features on a surface. Practitioners need no prior and specific surface mathematical models. A robust estimate of the surface can be quickly derived from the noisy scanning data by an approximation algorithm. Then, a statistic calculating the deviation of this surface from the target one is plotted in a control chart to monitor the stability of the manufacturing process. In practice, this monitoring scheme can be integrated with the 3-D laser scanner as a new production module for automatic quality control and decision making.

Micro-Doppler Curves Extraction Based on High-Order Particle Filter Track-Before Detect

Micro-Doppler (MD) radar signatures characterize rich motion information of the targets and are of great significance in target recognition. In this letter, we propose a novel high-order particle filter track-before-detect (PF-TBD) approach for the MD curves extraction. In the proposed approach, the sinusoidal Doppler frequency curve is treated as the state, whose dynamic model is described as a high-order Markov chain. First, the state equation is divided into two parts, the translational motion part represented as a first-order dynamic process and the micromotion part represented as a high-order dynamic process including static model parameters. Then, a kernel smoothing approach is introduced for the static model parameters estimation, and the auxiliary particle filter (APF) is utilized for the instantaneous Doppler curves extraction. Finally, the experiments on the electromagnetic analysis data are carried out to validate the performance of the proposed method.

Functional parameter measurements in children with ataxia telangiectasia.

To collect preliminary functional data on ataxia telangiectasia and create a disease specific scale: the Ataxia Telangiectasia Functional Scale (ATFS). Retrospective information on patients with ataxia telangiectasia referred to the Assistive Technology Unit was included. Functional mobility scales (the Gross Motor Function Classification System [GMFCS] and the Functional Mobility Scale [FMS]-5, FMS-50, FMS-500) and activities of daily living [ADL] parameters were recorded. We created a 51-point ATFS, that consisted of three ambulation items adapted for ataxia telangiectasia in the frame of FMS (home, school, outdoors), five ADL items, and one schooling item. Twenty-seven participants (17 males, 10 females; mean age 10y 8mo [SD 5y 1mo], range 1y 9mo-25y 6mo), were enrolled; 168 measurements were recorded. Patients walked at a mean age of 1year 4months (SD 5y 4mo) and lost ambulatory capacity at 8years 8months (SD 2y 1mo). GMFCS level and FMS-5, FMS-50, FMS-500 assessments correlated with age (Spearman's correlations r=0.555, -0.617, -0.639, -0.662 respectively, p<0.01 for all), but plateaued after 12years of age. ATFS mean score was 37.46 (SD 7.88) and increased with age (Spearman's correlation r=0.585, p<0.01). The scale showed threestages of disease progression. In this pilot study we show longitudinal functional data of ambulation and ADL skills in ataxia telangiectasia, and created a framework for a functional scale. This functional scale closely approximated disease course, but further validation is required. The Gross Motor Function Classification System and the Functional Mobility Scale are ill-suited for ataxia telangiectasia assessments. Three functional mobility scales (home, school, outdoors) suited to ataxia telangiectasiawere created. The Ataxia Telangiectasia Functional Scale (ATFS) combines mobility and items of activities of daily living. The ATFS closely approximates the three-stage progression of the disorder.

Estimation of relative risk for events on a linear network

Motivated by the study of traffic accidents on a road network, we discuss the estimation of the relative risk, the ratio of rates of occurrence of different types of events occurring on a network of lines. Methods developed for two-dimensional spatial point patterns can be adapted to a linear network, but their requirements and performance are very different on a network. Computation is slow and we introduce new techniques to accelerate it. Intensities (occurrence rates) are estimated by kernel smoothing using the heat kernel on the network. The main methodological problem is bandwidth selection. Binary regression methods, such as likelihood cross-validation and least squares cross-validation, perform tolerably well in our simulation experiments, but the Kelsall–Diggle density-ratio cross-validation method does not. We find a theoretical explanation, and propose a modification of the Kelsall–Diggle method which has better performance. The methods are applied to traffic accidents in a regional city, and to protrusions on the dendritic tree of a neuron.

Local stationarity and time-inhomogeneous Markov chains

A primary motivation of this contribution is to define new locally stationary Markov models for categorical or integer-valued data. For this initial purpose, we propose a new general approach for dealing with time-inhomogeneity that extends the local stationarity notion developed in the time series literature. We also introduce a probabilistic framework which is very flexible and allows us to consider a much larger class of Markov chain models on arbitrary state spaces, including most of the locally stationary autoregressive processes studied in the literature. We consider triangular arrays of time-inhomogeneous Markov chains, defined by some families of contracting and slowly-varying Markov kernels. The finite-dimensional distribution of such Markov chains can be approximated locally with the distribution of ergodic Markov chains and some mixing properties are also available for these triangular arrays. As a consequence of our results, some classical geometrically ergodic homogeneous Markov chain models have a locally stationary version, which lays the theoretical foundations for new statistical modeling. Statistical inference of finite-state Markov chains can be based on kernel smoothing and we provide a complete and fast implementation of such models, directly usable by the practitioners. We also illustrate the theory on a real data set. A central limit theorem for Markov chains on more general state spaces is also provided and illustrated with the statistical inference in INAR models, Poisson ARCH models and binary time series models. Additional examples such as locally stationary regime-switching or SETAR models are also discussed.