Kernel Smoothing Research Articles

Private land conservation has landscape‐scale benefits for wildlife in agroecosystems

AbstractPrivate lands contain much of the world's biodiversity. Conservation of private land, especially agricultural land, is urgent yet challenging because of the diverse priorities of landowners. Local effects of farmland conservation programmes have been evaluated thoroughly, but population‐level response to these programmes may depend on effects that extend beyond targeted land parcels. We investigated the landscape‐scale effects of a grassland conservation initiative, the Conservation Reserve Enhancement Program (CREP), on a socially and economically important game bird, the Northern BobwhiteColinus virginianus.Barriers to assessing population‐level response to conservation include determining the spatial scale at which a species responds to environmental change (the scale of effect) and untangling density‐dependent processes. We performed point counts over 6 years at 247 sites with similar local CREP density but varying landscape‐scale CREP density. We used an open‐population distance sampling model to evaluate population response to landscape‐level CREP density and to forecast population densities under differing re‐enrolment scenarios. Our model included kernel smoothing techniques to estimate scale of effect and an estimator of the strength of density dependence.Density dependence moderated the effectiveness of the CREP, but overall populations responded positively to increasing landscape‐scale CREP density. We estimated that at least 5% of the landscape needs to be in CREP to meet population density goals of 0.25 bobwhite/ha. Conservatively, we recommend 10% of the landscape to be in CREP. Our percent cover recommendations are based on a distance‐weighted average of CREP around focal sites.Landscape‐scale effects diminished with distance. For example, assuming all else is equal, a CREP field 3,000 m away had 88% less of an effect on local abundance than a field 1,000 m away. Fields farther than 5,000 m away had no effect on local abundance.Synthesis and applications. Our study underscores the importance of a landscape‐scale approach to farmland conservation. Benefits of these programmes to wildlife can extend beyond the local scale, but their importance to local populations diminishes with distance. Estimating this relationship and incorporating it into a decision framework could help practitioners target land enrolment to meet broadscale population objectives.

Efficiency of the QR class estimator in semiparametric regression models to combat multicollinearity

ABSTRACTThere are some classes of biased estimators for solving the multicollinearity among the predictor variables in statistical literature. In this research, we propose a modified estimator based on the QR decomposition in the semiparametric regression models, to combat the multicollinearity problem of design matrix which makes the data to be less distorted than the other methods. We derive the properties of the proposed estimator, and then, the necessary and sufficient condition for the superiority of the partially generalized QR-based estimator over partially generalized least-squares estimator is obtained. In the biased estimators, selection of shrinkage parameters plays an important role in data analysing. We use generalized cross-validation criterion for selecting the optimal shrinkage parameter and the bandwidth of the kernel smoother. Finally, the Monté-Carlo simulation studies and a real application related to bridge construction data are conducted to support our theoretical discussion.

Fast Kernel Smoothing by a Low-Rank Approximation of the Kernel Toeplitz Matrix

Kernel smoothing methods, including the bilateral filter, are commonly used in data processing/modeling and edge-aware image smoothing. Due to their nonlinear nature, these filters require significant computational time. In this paper, we address this problem by studying a practical case in which the data to be processed are integers. The basic idea is to use eigendecomposition to approximate the kernel matrix which is a real symmetric Toeplitz matrix. This approximation leads to more efficient computation. We study the distribution of its eigenvalues and show that the upper bounds of the eigenvalues can be expressed analytically in terms of the Fourier transform of the kernel function. This result not only captures the relationship between the order of the low-rank approximation of the kernel matrix and the filtering quality, but also shows that among the three kernel functions considered in this work, the Gaussian kernel can be most efficiently approximated. We have applied the proposed fast algorithm to implement the bilateral filter. By taking advantage of a property of the Gaussian kernel, we have also proposed another algorithm with even faster speed. Experimental results show that the performance of the proposed algorithms is competitive with those state-of-the-art algorithms in terms of speed and quality.

Modified estimators in semiparametric regression models with right-censored data

ABSTRACTIn this work we introduce different modified estimators for the vector parameter and an unknown regression function g in semiparametric regression models when censored response observations are replaced with synthetic data points. The main idea is to study the effects of several covariates on a response variable censored on the right by a random censoring variable with an unknown probability distribution. To provide the estimation procedure for the estimators, we extend the conventional methodology to censored semiparametric regression using different smoothing methods such as smoothing spline (SS), kernel smoothing (KS), and regression spline (RS). In addition to estimating the parameters of the semiparametric model, we also provide a bootstrap technique to make inference on the parameters. A simulation study is carried out to show the performance and efficiency properties of the estimators and analyse the effects of the different censoring levels. Finally, the performance of the estimators is evaluated by a real right-censored data set.

Data-driven decision making under uncertainty integrating robust optimization with principal component analysis and kernel smoothing methods

This paper proposes a novel data-driven robust optimization framework that leverages the power of machine learning and big data analytics for decision-making under uncertainty. By applying principal component analysis to uncertainty data, correlations between uncertain parameters are effectively captured, and latent uncertainty sources are identified. These data are then projected onto each principal component to facilitate extracting distributional information of latent uncertainties using kernel density estimation techniques. To explicitly account for asymmetric distributions, we introduce forward and backward deviation vectors into the data-driven uncertainty set, which are further incorporated into novel data-driven static and adaptive robust optimization models. The proposed framework not only significantly ameliorates the conservatism of robust optimization, but also enjoys computational efficiency and wide-ranging applicability. Three applications on optimization under uncertainty, including model predictive control, batch production scheduling, and process network planning, are presented to demonstrate the applicability of the proposed framework.

Time-dynamic varying coefficient models for longitudinal data

A new varying coefficient model that relates functional response to functional predictors is proposed and studied. The model accommodates the influence of the functional predictors on the time-varying coefficient functions. A powerful kernel smoothing technique is developed for estimating the model with longitudinal observations of the functional response and predictors. The method involves a backfitting iteration that is based on alternating conditional expectation. The convergence of the algorithm is established and the asymptotic distribution of the coefficient function estimators is derived. It is shown that the method works well for finite sample sizes via simulation studies. The proposed model and method are also applied to analyzing an air quality dataset.

A hybrid denoising algorithm based on shearlet transform method and Yaroslavsky’s filter

In this paper, we have proposed a hybrid denoising algorithm based on combining of the shearlet transform method, as a pre-processing step, with the Yaroslavsky’s filter, as a kernel smoother, on a wide class of images with various properties such as thin features and textures. In the other word, proposed algorithm is a two-step algorithm, where in the first step the image is filtered by shearlet transform method and in the second step the weighted Yaroslavsky’s filter is applied on result of first step. The weight coefficients of the Yaroslavsky’s filter are achieved by pixel similarities in the denoised image from the first step. The theoretical results are confirmed via simulations for 2D images corrupted by additive white Gaussian noise. Experimental results illustrate that proposed hybrid method has good effect on suppressing the pseudo-Gibbs and shearlet-like artifacts can obtain better performance in terms of mean square error (MSE), peak signal to noise ratio (PSNR) and structural similarity (SSIM) index rather than existing state-of-the-art methods.

Effect of Spatial Smoothing on Task fMRI ICA and Functional Connectivity.

Spatial smoothing is a widely used preprocessing step in functional magnetic resonance imaging (fMRI) data analysis. In this work, we report on the spatial smoothing effect on task-evoked fMRI brain functional mapping and functional connectivity. Initially, we decomposed the task fMRI data into a collection of components or networks by independent component analysis (ICA). The designed task paradigm helps identify task-modulated ICA components (highly correlated with the task stimuli). For the ICA-extracted primary task component, we then measured the task activation volume at the task response foci. We used the task timecourse (designed) as a reference to order the ICA components according to the task correlations of the ICA timecourses. With the re-ordered ICA components, we calculated the inter-component function connectivity (FC) matrix (correlations among the ICA timecourses). By repeating the spatial smoothing of fMRI data with a Gaussian smoothing kernel with a full width at half maximum (FWHM) of {1, 3, 6, 9, 12, 15, 20, 25, 30, 35} mm, we measured the spatial smoothing effects. Our results show spatial smoothing reveals the following effects: (1) It decreases the task extraction performance of single-subject ICA more than that of multi-subject ICA; (2) It increases the task volume of multi-subject ICA more than that of single-subject ICA; (3) It strengthens the functional connectivity of single-subject ICA more than that of multi-subject ICA; and (4) It impacts the positive-negative imbalance of single-subject ICA more than that of multi-subject ICA. Our experimental results suggest a 2~3 voxel FWHM spatial smoothing for single-subject ICA in achieving an optimal balance of functional connectivity, and a wide range (2~5 voxels) of FWHM for multi-subject ICA.

Investigating light curve modulation via kernel smoothing

Context.Recent studies have revealed a hitherto unknown complexity of Cepheid pulsations by discovering irregular modulated variability using photometry, radial velocities, and interferometry.Aims.We aim to perform a statistically rigorous search and characterization of such phenomena in continuous time, applying it to 53 classical Cepheids from the OGLE-III catalog.Methods.We have used local kernel regression to search for both period and amplitude modulations simultaneously in continuous time and to investigate their detectability. We determined confidence intervals using parametric and non-parametric bootstrap sampling to estimate significance, and investigated multi-periodicity using a modified pre-whitening approach that relies on time-dependent light curve parameters.Results.We find a wide variety of period and amplitude modulations and confirm that first overtone pulsators are less stable than fundamental mode Cepheids. Significant temporal variations in period are more frequently detected than those in amplitude. We find a range of modulation intensities, suggesting that both amplitude and period modulations are ubiquitous among Cepheids. Over the 12-year baseline offered by OGLE-III, we find that period changes are often nonlinear, sometimes cyclic, suggesting physical origins beyond secular evolution. Our method detects modulations (period and amplitude) more efficiently than conventional methods that are reliant on certain features in the Fourier spectrum, and pre-whitens time series more accurately than using constant light curve parameters, removing spurious secondary peaks effectively.Conclusions.Period and amplitude modulations appear to be ubiquitous among Cepheids. Current detectability is limited by observational cadence and photometric precision: detection of amplitude modulation below 3 mmag requires space-based facilities. Recent and ongoing space missions (K2, BRITE, MOST, CoRoT) as well as upcoming ones (TESS, PLATO) will significantly improve detectability of fast modulations, such as cycle-to-cycle variations, by providing high-cadence high-precision photometry. High-quality long-term ground-based photometric time series will remain crucial to study longer-term modulations and to disentangle random fluctuations from secular evolution.

Process convolution approaches for modeling interacting trajectories

Gaussian processes are a fundamental statistical tool used in a wide range of applications. In the spatiotemporal setting, several families of covariance functions exist to accommodate a wide variety of dependence structures arising in different applications. These parametric families can be restrictive and are insufficient in some situations. In contrast, process convolutions represent a flexible, interpretable approach to defining the covariance of a Gaussian process and have modest requirements to ensure validity. We introduce a generalization of the process convolution approach that employs multiple convolutions sequentially to form a “process convolution chain”. In our proposed multistage framework, complex dependencies that arise from a combination of different interacting mechanisms are decomposed into a series of interpretable kernel smoothers. We demonstrate an application of process convolution chains to model killer whale movement, in which the paths taken by multiple individuals are not independent but reflect dynamic social interactions within the population. Our proposed model for dependent movement provides inference for the latent dynamic social structure in the study population. Additionally, by leveraging the positive dependence among individual paths, we achieve a reduction in uncertainty for the estimated locations of the killer whales, compared with a model that treats paths as independent.

A high plasma: red blood cell transfusion ratio during liver transplantation is associated with decreased blood utilization.

During massive transfusion, the volume ratio of administered plasma (PL Vol) to red blood cell (RBC Vol) appears to be associated with reduced blood utilization and improved survival. The aim of this study was to evaluate the optimal component ratio in the setting of liver transplantation. This is a retrospective chart review of patients who underwent liver transplantation and received at least 500 ml of red blood cells from January 2013 through December 2015. Kernel smoothing analysis determined the proper component ratios to evaluate were a ≥0·85:1 ratio (high) to a ≤0·85:1 ratio (low). Two groups, plasma volume to RBC volume (PL Vol/RBC Vol) and plasma contained in the platelet units added to the plasma calculation [PL + PLT (platelet)] Vol/RBC Vol, were used to evaluate the component ratios. A total of 188 patients were included in the analysis. In the PL Vol/RBC Vol evaluation, a low ratio revealed that 1238 ml (977-1653 ml) (P < 0·0001) and 1178 ml (747-1178) (P < 0·0001) of RBC were used in excess compared to the high ratio, in the univariable and multivariable analysis, respectively. In the PL +PLT Vol/RBC Vol evaluation, a low ratio used 734 ml (193-1275) (P = 0·008) and 886 ml (431-1340) (P < 0·0001) of RBC in excess when compared to high ratio in the univariable and multivariable analysis, respectively. In patients undergoing liver transplantation, the transfusion of plasma to RBC ratio ≥0·85 was associated with decreased need of RBC transfusions.

Jackknife empirical likelihood-based inferences for Lorenz curve with kernel smoothing

ABSTRACTThe Lorenz curve describes the wealth proportion for an income-ordered population. In this paper, we introduce a kernel smoothing estimator for the Lorenz curve and propose a smoothed jackknife empirical likelihood method for constructing confidence intervals of Lorenz ordinates. Extensive simulation studies are conducted to evaluate finite sample performances of the proposed methods. A real dataset of Georgia professor’s income is used to illustrate the proposed methods.

Nonparametric Estimation and Inference for Panel Data Models

This chapter surveys nonparametric methods for estimation and inference in a panel data setting. Methods surveyed include profile likelihood, kernel smoothers, as well as series and sieve estimators. The practical application of nonparametric panel-based techniques is less prevalent that, say, nonparametric density and regression techniques. It is our hope that the material covered in this chapter will prove useful and facilitate their adoption by practitioners.

KVLMM: A Trajectory Prediction Method Based on a Variable-Order Markov Model With Kernel Smoothing

With the dramatic proliferation of global positioning system (GPS) devices, a rich range of research has been conducted on the analysis of GPS trajectories. Research on trajectory prediction uses historical trajectory data to forecast future positions. The typical method is to use a statistical model based on the Markov chain. However, existing models are inefficient in two aspects. The methods of using lower-order Markov models use only current information and ignore historical information, degrading the prediction accuracy. In contrast, higher-order Markov models can improve the prediction accuracy but incur increased time and space complexity. Here, we propose the kernel variable length Markov model (KVLMM), a variable-order Markov model based on kernel smoothing, which combines sequence analysis with the Markov statistical model. The KVLMM can adaptively train trajectory data and learn rules from the training results. When training a large data sample, the KVLMM can rapidly execute training in linearly complex time and space. Moreover, this model uses kernel smoothing when training fewer data samples. In other words, the KVLMM improves the prediction accuracy and reduces the overhead of the data process. Our experimental results show that KVLMM has a lower algorithm complexity and a higher prediction accuracy.

Kernel Estimation for Panel Data with Heterogeneous Dynamics

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 and Monte Carlo simulations illustrate our procedure in finite samples.

Relative Optimality Conditions and Algorithms for Treespace Fréchet Means

Recent interest in treespaces as well-founded mathematical domains for phylogenetic inference and statistical analysis for populations of anatomical trees has motivated research into efficient and rigorous methods for optimization problems on treespaces. A central problem in this area is computing an average of phylogenetic trees, which is equivalently characterized as the minimizer of the Fr\'echet function. The Fr\'echet mean can be used for statistical inference and exploratory data analysis: for example it can be leveraged as a test statistic to compare groups via permutation tests, or to find trends in data over time via kernel smoothing. By analyzing the differential properties of the Fr\'echet function along geodesics in treespace we obtained a theorem describing a decomposition of the derivative along a geodesic. This decomposition theorem is used to formulate optimality conditions which are used as a logical basis for an algorithm to verify relative optimality at points where the Fr\'echet function gradient does not exist.

Indirect Inference With a Non-Smooth Criterion Function

Indirect inference requires simulating realisations of endogenous variables from the model under study. When the endogenous variables are discontinuous functions of the model parameters, the resulting indirect inference criterion function is discontinuous and does not permit the use of derivative-based optimisation routines. Using a change of variables technique, we propose a novel simulation algorithm that alleviates the discontinuities inherent in such indirect inference criterion functions, and permits the application of derivative-based optimisation routines to estimate the unknown model parameters. Unlike competing approaches, this approach does not rely on kernel smoothing or bandwidth parameters. Several Monte Carlo examples that have featured in the literature on indirect inference with discontinuous outcomes illustrate the approach, and demonstrate the superior performance of this approach over existing alternatives.

Nonparametric model checks of single-index assumptions

Semiparametric single-index assumptions are convenient and widely used dimen\-sion reduction approaches that represent a compromise between the parametric and fully nonparametric models for regressions or conditional laws. In a mean regression setup, the SIM assumption means that the conditional expectation of the response given the vector of covariates is the same as the conditional expectation of the response given a scalar projection of the covariate vector. In a conditional distribution modeling, under the SIM assumption the conditional law of a response given the covariate vector coincides with the conditional law given a linear combination of the covariates. Several estimation techniques for single-index models are available and commonly used in applications. However, the problem of testing the goodness-of-fit seems less explored and the existing proposals still have some major drawbacks. In this paper, a novel kernel-based approach for testing SIM assumptions is introduced. The covariate vector needs not have a density and only the index estimated under the SIM assumption is used in kernel smoothing. Hence the effect of high-dimensional covariates is mitigated while asymptotic normality of the test statistic is obtained. Irrespective of the fixed dimension of the covariate vector, the new test detects local alternatives approaching the null hypothesis slower than $n^{-1/2}h^{-1/4},$ where $h$ is the bandwidth used to build the test statistic and $n$ is the sample size. A wild bootstrap procedure is proposed for finite sample corrections of the asymptotic critical values. The small sample performances of our test compared to existing procedures are illustrated through simulations.

Smoothing parameter values in automatic choice procedure and in acceptable interval in the kernel density estimation

Automatic procedure for determining the parameters of kernel method, allows the simultaneous selection of two method parameters: kernel function and smoothing parameter. This approach simplifies the procedure for parameters selection and at the same time provides a good properties of kernel estimators. The second procedure regarded in the paper is the acceptable interval of values of smoothing parameter, allowing for a much more generalized approach in choosing the smoothing parameter in the kernel estimation. The results of the smoothing parameter values comparison, where these values are set in the automatic procedure and the procedure of the acceptable interval of smoothing parameters values in the estimation of density function, are presented in the paper. Comparison of these values is made basing on the results of applying the simulation methods. Basing on simulation studies results new intervals of values of smoothing parameter are proposed and analyzed.

Comparisons between the Local Polynomial Kernel estimators and the Cubic Smoothing Splines

The term Nonparametric regression comes to signify the absence of parameters in the regression model. The scope of nonparametric regression is very broad nowadays, ranging from "smoothing" the relationship between two variables in a scatter plot to multiple-regression analysis. Extraordinary, only several years ago, methods of nonparametric regression analysis have been used widely because of advances in statistical techniques and related computing facilities, and are now a serious alternative to many traditional parametric regression modeling. This paper aims to provide using two methods of the nonparametric regression, namely; the Local Polynomial Kernel (LPK) estimator and Cubic Smoothing Splines (CSS) estimator with emphasis on the role of selecting the smoothing parameter (bandwidth) in each estimator. We have investigated the performance of these two smoothing estimators (LLK and CSS) under two different settings, fixed or random designs. This was achieved through conducting many simulation studies using four different example regression functions; two different distributions of errors term either normal or mixture, and different sample sizes. Two study the estimation of the derivative functions using those two smoothing estimators (Local Quadratic/Cubic Kernel and Cubic Smoothing Splines) in the process of smoothing the 1st / 2nd derivative functions respectively also been considered through conducting several simulation studies as well. Applications on real datasets also been considered. Lastly, the advantages and disadvantages of these two nonparametric regression estimators have been discussed.