Kernel Smoothing Research Articles

A Fast-Converging Kernel Density Estimator for Dispersion in Horizontally Homogeneous Meteorological Conditions

Kernel smoothers are often used in Lagrangian particle dispersion simulations to estimate the concentration distribution of tracer gasses, pollutants etc. Their main disadvantage is that they suffer from the curse of dimensionality, i.e., they converge at a rate of 4/(d+4) with d the number of dimensions. Under the assumption of horizontally homogeneous meteorological conditions, we present a kernel density estimator that estimates a 3D concentration field with the faster convergence rate of a 1D kernel smoother, i.e., 4/5. This density estimator has been derived from the Langevin equation using path integral theory and simply consists of the product between a Gaussian kernel and a 1D kernel smoother. Its numerical convergence rate and efficiency are compared with that of a 3D kernel smoother. The convergence study shows that the path integral-based estimator has a superior convergence rate with efficiency, in mean integrated squared error sense, comparable with the one of the optimal 3D Epanechnikov kernel. Horizontally homogeneous meteorological conditions are often assumed in near-field range dispersion studies. Therefore, we illustrate the performance of our method by simulating experiments from the Project Prairie Grass data set.

Nonparametric estimation of directional highest density regions

Highest density regions (HDRs) are defined as level sets containing sample points of relatively high density. Although Euclidean HDR estimation from a random sample, generated from the underlying density, has been widely considered in the statistical literature, this problem has not been contemplated for directional data yet. In this work, directional HDRs are formally defined and plug-in estimators based on kernel smoothing and associated confidence regions are proposed. We also provide a new suitable bootstrap bandwidth selector for plug-in HDRs estimation based on the minimization of an error criteria that involves the Hausdorff distance between the boundaries of the theoretical and estimated HDRs. An extensive simulation study shows the performance of the resulting estimator for the circle and for the sphere. The methodology is applied to analyze two real data sets in animal orientation and seismology.

Measuring the smearing of the Galactic 511-keV signal: positron propagation or supernova kicks?

Abstract We use 15 years of γ-ray data from INTEGRAL/SPI in a refined investigation of the morphology of the Galactic bulge positron annihilation signal. Our spatial analysis confirms that the signal traces the old stellar population, revealing for the first time that it traces the boxy bulge and nuclear stellar bulge, while disfavouring the presence of additional dark matter components. Using a 3D smoothing kernel, we find that the signal is smeared out over a characteristic length scale of 150 ± 50 pc, suggesting either annihilation in situ at astrophysical sources kicked at formation or positron propagation away from sources. The former is disfavoured by its requiring kick velocities different between the Galactic nucleus (≳ 50 km s−1) and wider bulge (≲ 15 km s−1) source. Positron propagation prior to annihilation can explain the overall phenomenology of the 511 keV signal for positrons injection energies ≲ 1.4 MeV, suggesting a nucleosynthesis origin.

Spatiotemporal dynamics and risk factors of oak decline and Mortality in the Missouri Ozarks of the United States based on repeatedly measured FIA data

During recent decades there has been widespread oak decline and mortality in the Missouri Ozarks, USA. We extracted data of 18,403 oak trees measured during 1999–2019 from the US Forest Service’s Forest Inventory and Analysis (FIA) database to examine the spatiotemporal patterns of oak decline and mortality and associated risk factors. The Missouri Ozarks were classified into three spatial clusters with low (<1.0%), moderate (1.0–1.5%) and high (>1.5%) annual oak mortality rates using kernel smoothing and the Jenks natural breaks method. Oaks within each spatial cluster were further divided into four, five and three risk groups respectively with differentiated annual mortalities (0.35%–3.05%) using a classification and regression tree model. The Kaplan-Meier survival analyses showed that oak trees from individual spatial clusters and risk groups responded differently to droughts, a major regional or sub-regional inciting factor of oak decline. Droughts could have a three to nine year lagged impact on tree mortality in the high risk groups, whereas had little effect on oak mortality in the low risk groups. Tree species, crown class, and stand age were the regionwide predisposing factors of oak decline. However, localized disturbances and geographic/vegetation conditions (e.g., drought, fire, insects, aspect, elevation, ecoregions, historical forest types) could either incite or alleviate the decline process and the fate of declining trees depending on the differentiated spatial clusters or risk groups. Resource managers can base associated risk factors and spatial clusters to rank risk levels to prioritize and plan management activities to mitigate oak decline and mortality.

How Important is the Choice of Bandwidth in Kernel Equating?

Kernel equating uses kernel smoothing techniques to continuize the discrete score distributions when equating test scores from an assessment test. The degree of smoothness of the continuous approximations is determined by the bandwidth. Four bandwidth selection methods are currently available for kernel equating, but no thorough comparison has been made between these methods. The overall aim is to compare these four methods together with two additional methods based on cross-validation in a simulation study. Both equivalent and non-equivalent group designs are used and the number of test takers, test length, and score distributions are all varied. The results show that sample size and test length are important factors for equating accuracy and precision. However, all bandwidth selection methods perform similarly with regards to the mean squared error and the differences in terms of equated scores are small, suggesting that the choice of bandwidth is not critical. The different bandwidth selection methods are also illustrated using real testing data from a college admissions test. Practical implications of the results from the simulation study and the empirical study are discussed.

Relative error prediction: Strong uniform consistency for censoring time series model

This article considers an adaptive method based on the relative error criteria to estimate the regression operator by a kernel smoothing. It is assumed that the variable of interest is subject to random right censoring and that the observations are from a stationary α-mixing process. The uniform almost sure consistency over a compact set with rate where we highlighted the covariance term is established. The simulation study indicates that the proposed approach has better performance in the presence of high level censoring and outliers in data to an existing classical method based on the least squares. An experiment prediction shows the quality of the relative error predictor.

Stochastic smoothing of point processes for wildlife-vehicle collisions on road networks

Wildlife-vehicle collisions on road networks represent a natural problem between human populations and the environment, that affects wildlife management and raise a risk to the life and safety of car drivers. We propose a statistically principled method for kernel smoothing of point pattern data on a linear network when the first-order intensity depends on covariates. In particular, we present a consistent kernel estimator for the first-order intensity function that uses a convenient relationship between the intensity and the density of events location over the network, which also exploits the theoretical relationship between the original point process on the network and its transformed process through the covariate. We derive the asymptotic bias and variance of the estimator, and adapt some data-driven bandwidth selectors to estimate the optimal bandwidth. The performance of the estimator is analysed through a simulation study under inhomogeneous scenarios. We present a real data analysis on wildlife-vehicle collisions in a region of North-East of Spain.

Numerical Simulation on the Progressive Failure Processes of Foundation Pit Excavation Based on a New Particle Failure Method

The predictions of failure zone during the foundation excavations will provide important guidance for the safety constructions of engineering structures. Based on this background, the smoothing kernel function in the traditional SPH method has been improved. The failure mark η is introduced into the program to realize the failure characteristics of particles at meso–scale. The “Killing Particle Method” has also been proposed, which can realize the simulations of complex excavation processes. The whole progressive failure processes of the excavation of a foundation pit are numerically simulated and the results show that (1) the failure zone of the excavated foundation pit without retaining walls appears at the corner and then gradually develops into the deep. However, the failure zone of the excavated foundation pit with retaining walls only develops longitudes along the retaining wall. (2) The stiffness of retaining wall has a great impact on the failure zone of foundation pit excavation. The greater the stiffness of retaining wall, the greater the damage degree. (3) The rationality of the proposed method is verified by the comparisons of the simulation results of the proposed method with the ABAQUS numerical examples and the engineering practices. Future research directions should focus on developing the 3D parallel IKSPH programs. The research results can provide some references for the applications of SPH method into predicting the failure zone of foundation pit excavations and ensuring the safety of engineering constructions.

Consistent second-order discrete kernel smoothing using dispersed Conway–Maxwell–Poisson kernels

The histogram estimator of a discrete probability mass function often exhibits undesirable properties related to zero probability estimation both within the observed range of counts and outside into the tails of the distribution. To circumvent this, we formulate a novel second-order discrete kernel smoother based on the recently developed mean-parametrized Conway–Maxwell–Poisson distribution which allows for both over- and under-dispersion. Two automated bandwidth selection approaches, one based on a simple minimization of the Kullback–Leibler divergence and another based on a more computationally demanding cross-validation criterion, are introduced. Both methods exhibit excellent small and large sample performance. Computational results on simulated datasets from a range of target distributions illustrate the flexibility and accuracy of the proposed method compared to existing smoothed and unsmoothed estimators. The method is applied to the modelling of somite counts in earthworms, and the number of development days of insect pests on the Hura tree.

Environmental Mixture Methods: Recommendations to Promote Robust Results in the Presence of Random Sampling

BACKGROUND AND AIM: Statistical learning is being increasingly used in environmental epidemiology, including in health analyses of environmental mixtures. Statistical learning methods, such as shrinkage methods or kernel smoothing methods, can perform well in instances with complex or high-dimensional data—settings in which traditional statistical methods fail. These novel methods, however, often include random sampling which may induce variability in results. Best practices in data science can help to ensure robustness. METHODS: We used data from the National Health and Nutrition Examination Survey (NHANES) 2001–2002 cycle to evaluate the potential variability in the estimated association between 18 persistent organic pollutants (POPs) and leukocyte telomere length (LTL) among 1,003 US adults, using four statistical learning models that have been applied previously to analyze the relationship between environmental mixtures and health outcomes. We included two penalized regression methods borrowed from machine learning, lasso and group lasso, and two statistical learning methods designed for environmental health data, weighted quantile sum (WQS) regression and Bayesian kernel machine regression (BKMR). We ran each model across 100 initializing values for random number generation (‘seeds’) and assessed variability in resulting estimation and inference. RESULTS:All methods exhibited some seed-dependent variability in results. The degree of variability differed across methods and POPs. Regardless of variability, all methods repeatedly identified furan 2,3,4,7,8-pncdf and PCB 126 as bad actors in the mixture. Both WQS and BKMR consistently estimated a harmful overall mixture effect. CONCLUSIONS:Any statistical learning method reliant on a random seed will exhibit some degree of seed sensitivity. We recommend that researchers repeat their analysis with various seeds as a sensitivity analysis when implementing these methods to enhance interpretability and robustness of results. KEYWORDS: Mixtures, Mixtures analysis, Modeling, Environmental epidemiology, Epidemiology

Modeling conditional reference regions: Application to glycemic markers.

Many clinical decisions are taken based on the results of continuous diagnostic tests. Usually, only the results of one single test is taken into consideration, the interpretation of which requires a reference range for the healthy population. However, the use of two different tests, can be necessary in the diagnosis of certain diseases. This obliges a bivariate reference region be available for their interpretation. It should also be remembered that reference regions may depend on patient variables (eg, age and sex) independent of the suspected disease. However, few proposals have been made regarding the statistical modeling of such reference regions, and those put forward have always assumed a Gaussian distribution, which can be rather restrictive. The present work describes a new statistical method that allows such reference regions to be estimated with no insistence on the results being normally distributed. The proposed method is based on a bivariate location-scale model that provides probabilistic regions covering a specific percentage of the bivariate data, dependent on certain covariates. The reference region is estimated nonparametrically and the nonlinear effects of continuous covariates via polynomial kernel smoothers in additive models. The bivariate model is estimated using a backfitting algorithm, and the optimal smoothing parameters of the kernel smoothers selected by cross-validation. The model performed satisfactorily in simulation studies under the assumption of non-Gaussian conditions. Finally, the proposed methodology was found to be useful in estimating a reference region for two continuous diagnostic tests for diabetes (fasting plasma glucose and glycated hemoglobin), taking into account the age of the patient.

Peculiar velocities in the local Universe: comparison of different models and the implications for H0 and dark matter

ABSTRACT When measuring the value of the Hubble parameter, H0, it is necessary to know the recession velocity free of the effects of peculiar velocities. In this work, we study different models of peculiar velocity in the local Universe. In particular, we compare models based on density reconstruction from galaxy redshift surveys and kernel smoothing of peculiar velocity data. The velocity field from the density reconstruction is obtained using the 2M++ galaxy redshift compilation, which is compared to two adaptive kernel-smoothed velocity fields: the first obtained from the 6dF Fundamental Plane sample and the other using a Tully–Fisher catalogue obtained by combining SFI++ and 2MTF. We highlight that smoothed velocity fields should be rescaled to obtain unbiased velocity estimates. Comparing the predictions of these models to the observations from a few test sets of peculiar velocity data, obtained from the Second Amendment Supernovae catalogue and the Tully–Fisher catalogues, we find that 2M++ reconstruction provides a better model of the peculiar velocity in the local Universe than the kernel-smoothed peculiar velocity models. We study the impact of peculiar velocities on the measurement of H0 from gravitational waves and megamasers. In doing so, we introduce a probabilistic framework to marginalize over the peculiar velocity corrections along the line of sight. For the megamasers, we find H0 = 70.1 ± 2.9 km s−1 Mpc−1 using the 2M++ velocity field. We also study the peculiar velocity of the galaxy NGC 1052-DF2, concluding that a short ∼13 Mpc distance is not a likely explanation of the anomalously low dark matter fraction of that galaxy.

A cautionary note on the variance of empirical pair correlation function

Point processes are popular models for data consisting of a random configuration of points, such as positions of trees of a given species in a forest stand, locations of earthquakes' epicentres or positions of proteins in a cell membrane. Pair correlation function is a functional summary characteristic quantifying the interactions between pairs of points of the process at different distances. In practice, kernel estimators of the pair correlation function are mostly used, which require the user to choose the bandwidth of the smoothing kernel. When looking for the optimal bandwidth, higher order terms of the variance of the estimator are often neglected. We present the detailed formulas for the variance and argue that the higher order terms contribute the most to the variance and should not be ignored. Furthermore, we adapt a well‐known variance approximation formula to accommodate the popular translation edge correction factor. We compare the original and the adjusted approximation formula to the exact variance obtained by numerical integration and to the empirical variance obtained from simulated realizations. We conclude that the variance approximations do not perform well and that modern cross‐validation approaches to bandwidth selection should be preferred in practice.

Geospatial constrained optimization to simulate and predict spatiotemporal trends of air pollutants

For exposure estimation of air pollutants, data measurement errors and modeling uncertainty may lead to estimation bias and abnormal predictions and relationships between variables. This paper proposes a method of geospatial constrained optimization and deep learning to reliably simulate and predict spatiotemporal trends of air pollutants. In the proposed method, k-nearest neighbors (k-NN) was first used to retrieve the nearest samples to spatialize regular local temporal basis functions at each target location or subregion; then, a convolutional neural network (CNN) was used to extrapolate temporal basis functions for prediction. Domain and empirical knowledge was embedded in extensive constrained optimization to obtain reasonable simulations and predictions. Bootstrapping was used to estimate the uncertainty of constrained optimized values. The method reduced the bias in the point estimates and obtained robust predictions and their uncertainty estimates of spatiotemporal trend for each spatial target location. In the location-based validations of NO2, NOx and PM2.5 in California, even with limited noise input, the proposed method captured the primary spatiotemporal variability (correlation with measured values: 0.75–0.91; explaining 55–84% of the variance). In addition, compared with generalized additive spatiotemporal model, kernel smoother and CNN, the proposed method made one-year reliable spatiotemporal forecasts of weekly averages. The proposed method has important implications for reducing the estimation bias and predicting trends in the air pollutant spatiotemporal fields.

Penalized kernel quantile regression for varying coefficient models

In nonparametric models, numerous penalization methods using a nonparametric series estimator have been developed for model selection and estimation. However, a penalization has been poorly understood combined with kernel smoothing. This can be attributed to the intrinsic technical and computational difficulties, which leads to different treatments from the developments of the penalized series estimators. Kernel smoothing is a popular and useful nonparametric estimation method; thus it is desirable to establish theoretical and computational analyses for penalized kernel smoothing. In this paper, we develop a novel penalized kernel quantile regression that has nice theoretical and computational properties in varying coefficient models. We show that the proposed method consistently identifies the partially linear structure of the varying coefficient model even when the number of covariates is allowed to increase with a sample size. We develop an efficient algorithm using the alternating direction method of multipliers with a computational convergence guarantee. We derive the plug-in bandwidth selection using high-dimensional kernel regression theory and the penalty parameter is selected by the proposed Bayesian Information Criterion. Our developments require novel high-dimensional kernel regression and computational analyses. The various simulations and real data analyses conducted in this study demonstrate the effectiveness and numerically verify the proposed method.

High-order consistent SPH with the pressure projection method in 2-D and 3-D

Mesh-free methods such a smoothed particle hydrodynamics (SPH) have advantages over mesh-based methods for flow in complex domains but attaining consistent high-order accurate solutions with the conventional form of SPH has yet to be resolved. The high-order smoothing error dominates the SPH error until increasingly fine resolutions cause the low-order discretisation error to dominate; this is related to discretising a volume integral into a summation. In this paper, a high-order consistency correction based on modified SPH (MSPH) and the modified finite particle method (FPM) is proposed for improving the order of the limiting discretisation error. The new technique is an arbitrarily high-order extension of these schemes where the complexity of the consistency correction and the required computations are reduced by using simplified versions of the smoothing kernel derivatives. Tested in Eulerian form, the proposed high-order consistent SPH technique (HOCSPH) is combined with new high-order SPH kernel functions, designed to improve the order of the SPH smoothing error, and the resulting hybrid technique is shown to converge according to the smoothing error initially before converging according to the HOCPSH error once the latter becomes dominant. The initial high-order convergence lowers the computational effort required to achieve higher accuracy with hybrid HOCSPH in comparison to HOCSPH with second-order smoothing accuracy kernel functions. However, for highly irregular distributions it is found that the use of kernels with second-order smoothing accuracy provides more consistent convergence properties. A number of flows are simulated in 2-D and 3-D using the new HOCSPH technique in combination with the pressure projection method, and the results show that the method is accurate and able to model highly complex flow patterns. Some issues with stability of the projection method related to pressure–velocity collocation are identified, and several remedies are proposed. While effective for the test cases herein, these remedies are new in this context and require further attention for generalisation in future studies.

Debt-by-Price Ratio, End-of-Year Economic Growth, and Long-Term Prediction of Stock Returns

With the prominent role of government debt in economic growth in recent decades, one would expect that government debt alongside economic growth to be a risk factor priced in the time series of stock returns. In this paper, this idea is investigated by applying a nonparametric model, namely, a local-linear kernel smoother with the aim of forecasting long-term stock returns where the model and smoothing parameters are chosen by cross-validation. While a wide range of predictive variables are examined, we find that our newly introduced debt-by-price ratio and the third to fourth quarter economic growth are robust predictors of stock returns, beating the well-known predictive variables in the literature by a significant difference. The combination of these two covariates can explain almost 30% variation of stock returns at a one-year horizon. This is very crucial considering the difficulty in capturing even a small proportion of movements in stock returns.

A Nonlinear Anisotropic Diffusion Equation for Image Restoration with Forward-backward Diffusivities

Background: In the image processing area, deblurring and denoising are the most challenging hurdles. The deblurring image by a spatially invariant kernel is a frequent problem in the field of image processing. Methods: For deblurring and denoising, the total variation (TV norm) and nonlinear anisotropic diffusion models are powerful tools. In this paper, nonlinear anisotropic diffusion models for image denoising and deblurring are proposed. The models are developed in the following manner: first multiplying the magnitude of the gradient in the anisotropic diffusion model, and then apply priori smoothness on the solution image by Gaussian smoothing kernel. Results: The finite difference method is used to discretize anisotropic diffusion models with forward- backward diffusivities. Conclusion: The results of the proposed model are given in terms of the improvement.

Semi-Parametric Models for Longitudinal Data Analysis

Longitudinal studies are widely used in various fields, such as public health, clinic trials and financial data analysis. A major challenge for longitudinal studies is the repeated measurements from each subject, which cause time dependent correlations within subjects. Generalized Estimating Equations (GEE) can deal with correlated outcomes for longitudinal data through marginal effect. Our proposed model will be based on GEE, with a semi-parametric approach, to provide a flexible structure for regression models: coefficients for parametric covariates will be estimated and nuisance covariates will be fitted in kernel smoothers for the non-parametric part. The profile kernel estimator and the seemingly unrelated kernel estimator (SUR) will be used to obtain consistent and efficient semi-parametric estimators. We provide simulation results for estimating semi-parametric models with one or multiple non-parametric terms. Financial market data is a major component of data analysis; thus, we focus on the financial market in the application part. Credit card loan data will be used with the payment information for each customer across six months to investigate whether gender, income, age, or other factors will influence payment status significantly. Furthermore, we propose model comparisons to evaluate whether different models should be fitted for different subgroups of consumers, such as male and female.

A Comparative Study of Nearest Neighbor Regression and Nadaraya Watson Regression

Two non-parametric statistical methods are studied in this work. These are the nearest neighbor regression and the Nadaraya Watson kernel smoothing technique. We have proven that under a precise circumstance, the nearest neighborhood estimator and the Nadaraya Watson smoothing produce a smoothed data with a same error level, which means they have the same performance. Another result of the paper is that nearest neighborhood estimator performs better locally, but it graphically shows a weakness point when a large data set is considered on a global scale.