Kernel Smoothing Research Articles

A Boundary Corrected Non-Parametric Regression Estimator for Finite Population Total

This study explores the estimation of finite population total. For many years design-based approach dominated the scene in statistical inference in sample surveys. The scenario has since changed with emergence of the other approaches (Model-Based, Model-Assisted and the Randomization-Assisted), which have proved to rival the conventional approach. This paper focuses on a model based approach. Within this framework a nonparametric regression estimator for finite population total is developed. The nonparametric technique has been found from previous studies to be advantageous than its parametric counterpart in terms of robustness and flexibility.  Kernel smoother has been used in construction of the estimator. The challenge of the boundary problem encountered with the Nadaraya-Watson estimator has been addressed by modifying it using reflection technique. The performance of the proposed estimator has been compared to the design-based Horvitz Thompson estimator and the model –based nonparametric regression estimator proposed by (Dorfman, 1992) and the ratio estimator using simulated data.

Fingerprint Classification Using Kernel Smoothing Technique and Generalized Regression Neural Network and Probabilistic Neural Network

Fingerprint classification is a significant process by which identification procedure can be accelerated. Feature extraction might be afflicted with rotation. Thus, all images get through an introduced criterion to rectify rotated images. The core point of fingerprints is utilized widely in both classification and recognition process. In some cases, however, inaccurate location of it might contribute to incorrect categorization. Therefore, the common point is initiated for the purpose of better performance. Features are extracted according to the way ridges’ angles are distributed across images. Plus, kernel smoothing technique is used to enhance the process. Generalized regression neural network (GRNN) and Probabilistic neural network (PNN) are employed to classify fingerprints in four categories. Fingerprint verification competition (FVC) database is used to evaluate and train the networks. The simulation is performed by MATLAB and 97.4% accuracy is achieved for both GRNN and PNN.

Solving 2D Poisson-type equations using meshless SPH method

Abstract In the present study, 2D Poisson-type equation is solved by a meshless Symmetric Smoothed Particle Hydrodynamics (SSPH) method. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. Compared with other kernel functions, the cubic B-spline kernel function shows good capacity to reproduce a complicated function value and its first and second derivatives, when the smoothing length is chosen as 1.1 times the particles’ distance. Several types of Poisson equations are solved by SSPH method, and the numerical results exhibit a very good accuracy when comparing with the analytical solutions.

Smooth Bayesian network model for the prediction of future high-cost patients with COPD

IntroductionThe clinical course of chronic obstructive pulmonary disease (COPD) is marked by acute exacerbation events that increase hospitalization rates and healthcare spending. The early identification of future high-cost patients with COPD may decrease healthcare spending by informing individualized interventions that prevent exacerbation events and decelerate disease progression. Existing studies of cost prediction of other chronic diseases have applied regression and machine-learning methods that cannot capture the complex causal relationships between COPD factors. Thus, the exploration of these factors through nonlinear, high-dimensional but explainable modeling is greatly needed. ObjectivesWe aimed to develop a machine-learning model to identify future high-cost patients with COPD. Such a model should incorporate expert knowledge about causal relationships, and the method for estimating the model could provide more accurate predictions than other machine learning methods. MethodsWe used the 2011–2013 medical insurance data of patients with COPD in a large city. The data set included demographic information and admission records. Leveraging on developments in graphical modeling methods, we proposed a smooth Bayesian network (SBN) model for the prediction of high-cost individuals using medical insurance data. The modeling method incorporated some expert knowledge about causal relationships (i.e., about the Bayesian network structure). We employed a smoothing kernel based on the weighted nearest neighborhood method in the SBN model to address overfitting, case-mix effect, and data sparsity (i.e., using data about “similar patients”). ResultsThe proposed SBN achieved the area under curve (AUC) of 0.80 and showed considerable improvement over the baseline machine-learning methods. Besides confirming the known factors from the literature, we found “region” (i.e., a suburban or urban area) to be a significant factor, and that in a 3-tier system with primary, secondary and tertiary hospitals, COPD patients who had been admitted to primary hospitals were more likely to develop into future high-cost patients than patients who had been admitted to tertiary hospitals. ConclusionThe proposed SBN model not only obtained higher prediction accuracy and stronger generalizability than a number of benchmark machine-learning methods, but also used the Bayesian network to capture the complex causal relationships between different predictors by incorporating expert knowledge. Furthermore, a framework was developed to establish the relationships between exposure to historical trajectory and future outcome, which can also be applied to other temporal data to model different trajectory information and predict other outcomes.

Moving least squares particle hydrodynamics method for Burgers’ equation

In this paper, a meshless method named Moving Least Squares Particle Hydrodynamics (MLSPH) method is applied to obtain numerical solutions of the Burgers’ equation with a set of initial and boundary conditions. The influence of kernel function and smoothing length on the accuracy of MLSPH method is investigated. Several various forms of one- and two- dimensional Burgers’ equations are successfully calculated. It is concluded that the proposed method can obtain second-order accuracy. The way of non-uniform distribution is applied to present the desired performance near the shocks. The numerical results obtained by the proposed method are compared with the analytical solutions and the results obtained by some other methods. The satisfactorily small errors indicate the high precision of the proposed method to solve this kind of partial differential equation.

Local Least Product Relative Error Estimation for Varying Coefficient Multiplicative Regression Model

In this article, we consider the varying coefficient multiplicative regression model, which is very useful to model the positive response. The criterion of least product relative error (LPRE) is extended to the varying coefficient multiplicative regression model by kernel smoothing techniques. Consistency and asymptotic normality of the proposed estimator are established. Some numerical simulations are carried out to assess the performance of the proposed estimator. As an illustration, the ethanol data is analyzed.

Maximum likelihood method for bandwidth selection in kernel conditional density estimate

This paper discusses the kernel estimator of conditional density. A significant problem of kernel smoothing is bandwidth selection. The problem consists in the fact that optimal bandwidth depends on the unknown conditional and marginal density. This is the reason why some data-driven method needs to be applied. In this paper, we suggest a method for bandwidth selection based on a classical maximum likelihood approach. We consider a slight modification of the original method—the maximum likelihood method with one observation being left out. Applied to two types of conditional density estimators—to the Nadaraya–Watson and local linear estimator, the proposed method is compared with other known methods in a simulation study. Our aim is to compare the methods from different points of view, concentrating on the accuracy of the estimated bandwidths, on the final model quality measure, and on the computational time.

Systematic analysis of bias and variability of morphologic features for lung lesions in computed tomography.

We propose to characterize the bias and variability of quantitative morphology features of lung lesions across a range of computed tomography (CT) imaging conditions. A total of 15 lung lesions were simulated (five in each of three spiculation classes: low, medium, and high). For each lesion, a series of simulated CT images representing different imaging conditions were synthesized by applying three-dimensional blur and adding correlated noise based on the measured noise and resolution properties of five commercial multislice CT systems, representing three dose levels ( of 1.90, 3.75, 7.50mGy), three slice thicknesses (0.625, 1.25, 2.5mm), and 33 clinical reconstruction kernels from five clinical scanners. The images were segmented using three segmentation algorithms and each algorithm was evaluated by computing a Sørensen-Dice coefficient between the ground truth and the segmentation. A series of 21 shape-based morphology features were extracted from both "ground truth" (i.e., preblur without noise) and "image rendered" lesions (i.e., postblur and with noise). For each morphology feature, the bias was quantified by comparing the percentage relative error in the morphology metric between the imaged lesions and the ground-truth lesions. The variability was characterized by calculating the average coefficient of variation averaged across repeats and imaging conditions. The active contour segmentation had the highest average Dice coefficient of 0.80 followed by 0.63 for threshold, and 0.39 for fuzzy c-means. The bias of the features was segmentation algorithm and feature-dependent, with sharper kernels being less biased and smoother kernels being more biased in general. The feature variability from simulated images ranged from 0.30% to 10% for repeats of the same condition and from 0.74% to 25.3% for different lesions in the same spiculation class. In conclusion, the bias of morphology features is dependent on the acquisition protocol in combination with the segmentation algorithm used and the variability is primarily dependent on the segmentation algorithm.

SArKS: de novo discovery of gene expression regulatory motif sites and domains by suffix array kernel smoothing.

We set out to develop an algorithm that can mine differential gene expression data to identify candidate cell type-specific DNA regulatory sequences. Differential expression is usually quantified as a continuous score-fold-change, test-statistic, P-value-comparing biological classes. Unlike existing approaches, our de novo strategy, termed SArKS, applies non-parametric kernel smoothing to uncover promoter motif sites that correlate with elevated differential expression scores. SArKS detects motif k-mers by smoothing sequence scores over sequence similarity. A second round of smoothing over spatial proximity reveals multi-motif domains (MMDs). Discovered motif sites can then be merged or extended based on adjacency within MMDs. False positive rates are estimated and controlled by permutation testing. We applied SArKS to published gene expression data representing distinct neocortical neuron classes in Mus musculus and interneuron developmental states in Homo sapiens. When benchmarked against several existing algorithms using a cross-validation procedure, SArKS identified larger motif sets that formed the basis for regression models with higher correlative power. https://github.com/denniscwylie/sarks. Supplementary data are available at Bioinformatics online.

Analysis of longitudinal data with semiparametric varying-coefficient mean–covariance models

The efficient estimation of regression coefficients in the longitudinal data analysis requires a correct specification of the covariance structure. Existing approaches usually focus on modeling the mean with specification of certain covariance structures, which may lead to inefficient or biased estimators of parameters in the mean if misspecification occurs. In this article, we propose a novel data-driven approach based on semiparametric varying-coefficient models to model the mean and the covariance simultaneously, motivated by the modified Cholesky decomposition. An iterative estimation method is proposed, consisting of an orthogonality-based technique for parameters, an adaptive jump-preserving estimation method for varying coefficients, a modification of local linear smoothing technique for the autoregressive coefficient function, and a kernel smoothing technique for the variance function. Theoretical properties of the resulting estimators including uniform consistency and asymptotic normality are explicitly studied under certain mild conditions. Simulation studies are carried out to evaluate the efficacy of the proposed methods, and an analysis of a real data example is provided for illustration.

A new method for reconstructing past-climate trends using tree-ring data and kernel smoothing

Mediterranean high-relief karst areas are very vulnerable to changes in temporal patterns of precipitation and temperature. Understanding climate change in these areas requires current climate trends to be assessed within the context of the variability of rainfall and temperature trends in the recent past. A major difficulty is that the instrumental record in these high-relief areas is very limited and the use of data from paleoclimatic proxies, such as tree-ring data, is required to infer past climate variability. Furthermore, for complex relationships between tree-ring data and climatic variables, it is almost impossible to infer past inter-annual variations in temperature or precipitation, and the inference is limited to the reconstruction of low-frequency variability (i.e., the trend). To do so, in this work, we propose a new method based on detecting trends (by kernel smoothing) in tree variables that show maximum correlation with the trends (also estimated by kernel smoothing) of climate variables. This enables a standard regression framework to be established to reconstruct past climate. We have used tree-ring proxy data from Abies pinsapo to evaluate past climate trends in the Sierra de las Nieves karst massif in Southern Spain. Our analysis has found that during the last three hundred years the smoothed mean annual rainfall steadily decreased until the beginning of the 20th century and thereafter it remained more or less constant until the end of the century. On the other hand, the smoothed mean annual temperature has steadily increased since the beginning of the 18th century until recent times. These trends are also suggested by the climate projections for the latter part of the current 21st century. As the study area is a high-relief karst massif of significant hydrologic and ecologic interest, the implications of these trends should be taken into account when formulating effective action plans to mitigate the impact of climate change.

A Box Regularized Particle Filter for state estimation with severely ambiguous and non-linear measurements

The first stage in any control system is to be able to accurately estimate the system’s state. However, some types of measurements are ambiguous (non-injective) in terms of state. Existing algorithms for such problems, such as Monte Carlo methods, are computationally expensive or not robust to such ambiguity. We propose the Box Regularized Particle Filter (BRPF) to resolve these problems. Based on previous works on box particle filters, we present a more generic and accurate formulation of the algorithm, with two innovations: a generalized box resampling step and a kernel smoothing method, which is shown to be optimal in terms of Mean Integrated Square Error. Monte Carlo simulations demonstrate the efficiency of BRPF on a severely ambiguous and non-linear estimation problem, that of Terrain Aided Navigation. BRPF is compared to the Sequential Importance Resampling Particle Filter (SIR-PF), Monte Carlo Markov Chain (MCMC), and the original Box Particle Filter (BPF). The algorithm outperforms existing methods in terms of Root Mean Square Error (e.g., improvement up to 42% in geographical position estimation with respect to the BPF) for a large initial uncertainty. The BRPF reduces the computational load by 73%and 90% for SIR-PF and MCMC, respectively, with similar RMSE values. This work offers an accurate (in terms of RMSE) and robust (in terms of divergence rate) way to tackle state estimation from ambiguous measurements while requiring a significantly lower computational load than classic Monte Carlo and particle filtering methods.

Kernel density estimation for hierarchical data

Multistage sampling is a common sampling technique employed in many studies. In this setting, observations are identically distributed but not independent, thus many traditional kernel smoothing techniques, which assume that the data are independent and identically distributed (i.i.d.), may not produce reasonable density estimates. In this paper, we sample repeatedly with replacement from each cluster, create multiple i.i.d. samples containing one observation from each cluster, and then create a kernel density estimate from each i.i.d. sample. These estimates will then be combined to form an estimate of the marginal probability density function of the population.

An Efficient Nonparametric Estimate for Spatially Correlated Functional Data

Functional data are often generated by modern biomedical technologies where features related to the pathophysiology and pathogenesis of a disease are interrogated repeatedly over time and at multiple spatially interdependent regions. To reduce model complexity and simplify the resulting inference, possible spatial correlation among neighboring regions is often neglected. In this article, we propose a weighted kernel smoothing estimate of the mean function that leverages the spatial and temporal correlation. We also address the companion problem of developing a simultaneous prediction method for individual curves using discrete samples. We establish the asymptotic properties of the proposed estimate, including its unique maximum efficiency achieving minimum asymptotic variance. The proposed method improves estimation and prediction in the presence of sparse observations, and therefore, is advantageous to biomedical applications that utilize markers to identify features intrinsic to a particular disease at multiple interdependent sites within an organ. Our simulation and case studies show that the proposed method outperforms conventional approaches for characterizing the dynamic functional imaging data, with the maximum benefit achieved in the presence of a small number of repeated scans.

Ergodicity of 3D Leray-α model with fractional dissipation and degenerate stochastic forcing

By using the asymptotic coupling method, the asymptotic log-Harnack inequality is established for the transition semigroup associated to the 3D Leray-[Formula: see text] model with fractional dissipation driven by highly degenerate noise. As applications, we derive the asymptotic strong Feller property and ergodicity for the stochastic 3D Leray-[Formula: see text] model with fractional dissipation, which is the stochastic 3D Navier–Stokes equation regularized through a smoothing kernel of order [Formula: see text] in the nonlinear term and a [Formula: see text]-fractional Laplacian. The main results can be applied to the classical stochastic 3D Leray-[Formula: see text] model ([Formula: see text]), stochastic 3D hyperviscous Navier–Stokes equation ([Formula: see text]) and stochastic 3D critical Leray-[Formula: see text] model ([Formula: see text]).

Kernel Estimation of the Baseline Function in the Cox Model

Survival analysis is the analysis of time-to-event data. Two important functions in the analysis of survival data are the survival function and the hazard function. The Kaplan-Meier method is widely used to estimate the survival function. One of the objectives of the analysis of survival data might be to examine whether survival times are related to other features. A popular regression model for the analysis of survival data is the Cox proportional hazard regression model. The most commonly used approaches, for the baseline survival function, are the Breslow and Kalbfleisch-Prentice methods. These methods provide a step function estimate of the survivor function and in many instances a continuous estimate would be preferable. For these reason, in this paper we proposed a kernel smoothing technique for baseline estimator, based on Kalbfleisch-Prentice method. We start with kernel smoothing of baseline hazard function, based on Kalbfleisch-Prentice estimator and epanechnikov kernel, than we use it to calculate the baseline survival function. To evaluate the usefulness of the kernel estimator of the baseline function, in the case of right censoring, based on KalbfleischPrentice estimator we conduct simulation studies across a range of conditions, by varying the sample size and censoring rate. We compare it with the smoothing of the Breslow estimator regarding bias.

Composite Coefficient of Determination and Its Application in Ultrahigh Dimensional Variable Screening

ABSTRACT In this article, we propose to measure the dependence between two random variables through a composite coefficient of determination (CCD) of a set of nonparametric regressions. These regressions take consecutive binarizations of one variable as the response and the other variable as the predictor. The resulting measure is invariant to monotonic marginal variable transformation, rendering it robust against heavy-tailed distributions and outliers, and convenient for independent testing. Estimation of CCD could be done through kernel smoothing, with a consistency rate of root-n. CCD is a natural measure of the importance of variables in regression and its sure screening property, when used for variable screening, is also established. Comprehensive simulation studies and real data analysis show that the newly proposed measure quite often turns out to be the most preferred compared to other existing methods both in independence testing and in variable screening. Supplementary materials for this article are available online.

Insights from kernel conditional-probability estimates into female labour force participation decision in the UK

The female labour force participation decision in the UK is a well-researched topic in empirical economics and econometrics. In this paper, using data from the UK Labour Force Survey in 2007, we contribute to the rich body of the literature by estimating a model using the kernel mixed-data-type conditional-probability (KMDTCP) estimator. We extend the analysis by comparing the predicted probabilities of the KMDTCP estimator with the estimates of a multinomial logit (MNL) model. Additionally, kernel smoothing specification tests are applied in order to compare different functional forms for the independent variables in the MNL model against the KMDTCP estimates. Our results demonstrate that the difference between the predictions from the KMDTCP estimator and the MNL is reduced when the functional form of continuous independent variables in the MNL model is relaxed to a semi-parametric form using dummy variables and interaction terms. The properties that are captured by the KMDTCP estimator show clearly that the propensity to work decreases with the increase in the number of dependent children. Furthermore, our analyses show that females older than 40 years of age have a high propensity to work irrespective of the number of dependent children.

Adjusted time-varying population attributable hazard in case-control studies.

Population attributable fraction is a widely used measure for quantifying the disease burden associated with a modifiable exposure of interest at the population level. It has been extended to a time-varying measure, population attributable hazard function, to provide additional information on when and how the exposure's impact varies over time. However, like the classic population attributable fraction, the population attributable hazard is generally biased if confounders are present. In this article, we provide a natural definition of adjusted population attributable hazard to take into account the effects of confounders, and its alternative that is identifiable from case-control studies under the rare disease assumption. We propose a novel estimator, which combines the odds ratio estimator from logistic regression model, and the conditional density function estimator of the exposure and confounding variables distribution given the failure times of cases or the current times of controls from a kernel smoother. We show that the proposed estimators are consistent and asymptotically normal with variance that can be estimated empirically from the data. Simulation studies demonstrate that the proposed estimators perform well in finite sample sizes. Finally, we illustrate the method by an analysis of a case-control study of colorectal cancer. Supplementary materials for this article are available online.

Numerical equivalence between SPH and probabilistic mass transfer methods for Lagrangian simulation of dispersion

Several Lagrangian methodologies have been proposed in recent years to simulate advection-dispersion of solutes in fluids as a mass exchange between numerical particles carrying the fluid. In this paper, we unify these methodologies, showing that mass transfer particle tracking (MTPT) algorithms can be framed within the context of smoothed particle hydrodynamics (SPH), provided the choice of a Gaussian smoothing kernel whose bandwidth depends on the dispersion and the time discretization. Numerical simulations are performed for a simple dispersion problem, and they are compared to an analytical solution. Based on the results, we advocate for the use of a kernel bandwidth of the size of the characteristic dispersion length ℓ=2DΔt, at least given a “dense enough” distribution of particles, for in this case the mass transfer operation is not just an approximation, but in fact the exact solution, of the solute’s displacement by dispersion in a time step.