Kernel Smoothing Technique Research Articles

The smooth Colonel meets the Reverend

Kernel smoothing techniques have attracted much attention and some notoriety in recent years. The attention is well deserved as kernel methods free researchers from having to impose rigid parametric structure on their data. The notoriety arises from the fact that the amount of smoothing (i.e., local averaging) that is appropriate for the problem at hand is under the control of the researcher. In this study we provide a deeper understanding of kernel smoothing methods for discrete data by leveraging the unexplored links between hierarchical Bayes models and kernel methods for discrete processes. Several potentially useful results are thereby obtained, including bounds on when kernel smoothing can be expected to dominate non-smooth (e.g., parametric) approaches in mean squared error and suggestions for thinking about the appropriate amount of smoothing.

Statistical estimation in varying coefficient models with surrogate data and validation sampling

Varying coefficient error-in-covariables models are considered with surrogate data and validation sampling. Without specifying any error structure equation, two estimators for the coefficient function vector are suggested by using the local linear kernel smoothing technique. The proposed estimators are proved to be asymptotically normal. A bootstrap procedure is suggested to estimate the asymptotic variances. The data-driven bandwidth selection method is discussed. A simulation study is conducted to evaluate the proposed estimating methods.

Nonparametric spatial analysis to detect high-risk regions for schistosomiasis in Guichi, China

Schistosomiasis control in China is facing a new challenge due to the rebound of epidemics in many areas and the unsustainable effects of the chemotherapy-based control strategy. Identifying high-risk regions for schistosomiasis is an important first step for an effective and sustainable strategy. Direct surveillance of snail habitats to detect high-risk regions is costly and no longer a desirable approach, while indirect monitoring of acute schistosomiasis may be a satisfactory alternative. To identify high-risk regions for schistosomiasis, we jointly used multiplicative and additive models with the kernel smoothing technique as the main approach to estimate the relative risk (RR) and excess risk (ER) surfaces by analyzing surveillance data for acute schistosomiasis. The feasibility of detecting high-risk regions for schistosomiasis through nonparametric spatial analysis was explored and confirmed in this study, and two significant high-risk regions were identified. The results provide useful hints for improving the national surveillance network for acute schistosomiasis and possible approaches to utilizing surveillance data more efficiently. In addition, the commonly used epidemiological indices, RR and ER, are examined and emphasized from the spatial point of view, which will be helpful for exploring many other epidemiological indices.

Free-roaming and surrendered dogs and cats submitted to a humane shelter in Wellington, New Zealand, 1999–2006

AIMS: To describe submissions of dogs and cats to an animal welfare shelter over a 6-year period, and to evaluate the association between the numbers of dogs and cats submitted per square kilometre, human population density, and standardised measures of socioeconomic deprivation. METHODS: Details of free-roaming and surrendered dogs and cats presented to the Wellington shelter of the Society for the Prevention of Cruelty to Animals (SPCA) for the period 01 July 1999 to 28 February 2006 were recorded in a relational database. Data were plotted as counts of animals presented to the shelter each month as a function of calendar time. Kernel-smoothing techniques were used to describe the spatial distribution of capture location. The association between the number of dogs and cats submitted per square kilometre, human population density (both estimated at the mesh-block level), and mesh-block socioeconomic deprivation index were quantified using the Pearson product-moment correlation coefficient (r). RESULTS: The data comprised submission details for 3,709 dogs and 13,563 cats. There was a progressive decline in the number of dogs and cats submitted to the shelter over the study period, and a marked seasonal variation in submissions of cats. Submission density of dogs and cats was positively associated with human population density (Pearson's r=0.12; 95% CI=0.06–0.20 and 0.39; 95% CI=0.33–0.45, respectively) and socioeconomic deprivation (Pearson's r=0.12; 95% CI=0.05–0.19 for both dogs and cats). CONCLUSIONS: Active approaches to free-roaming cat control by the Wellington SPCA should focus on the period June to September, prior to the feline breeding season, in population-dense and socioeconomically deprived areas. Composite analyses of submission details from all animal shelters in the Wellington region should allow factors associated with the distribution of free-roaming dogs and cats to be more precisely estimated.

Density and hazard rate estimation for censored and α-mixing data using gamma kernels

In this paper, we consider the non-parametric estimation for a density and hazard rate function for right censored α-mixing survival time data using kernel smoothing techniques. As survival times are positive with potentially high concentration at zero, one has to take into account the bias problems when the functions are estimated in the boundary region. In this paper, gamma kernel estimators of the density and the hazard rate function are proposed. The estimators use adaptive weights depending on the point in which we estimate the function, and they are robust to the boundary bias problem. For both estimators, the mean-squared error properties, including the rate of convergence, the almost sure consistency, and the asymptotic normality, are investigated. The results of a simulation study demonstrate the performance of the proposed estimators.

An improved state-parameter analysis of ecosystem models using data assimilation

Much of the effort spent in developing data assimilation methods for carbon dynamics analysis has focused on estimating optimal values for either model parameters or state variables. The main weakness of estimating parameter values alone (i.e., without considering state variables) is that all errors from input, output, and model structure are attributed to model parameter uncertainties. On the other hand, the accuracy of estimating state variables may be lowered if the temporal evolution of parameter values is not incorporated. This research develops a smoothed ensemble Kalman filter (SEnKF) by combining ensemble Kalman filter with kernel smoothing technique. SEnKF has following characteristics: (1) to estimate simultaneously the model states and parameters through concatenating unknown parameters and state variables into a joint state vector; (2) to mitigate dramatic, sudden changes of parameter values in parameter sampling and parameter evolution process, and control narrowing of parameter variance which results in filter divergence through adjusting smoothing factor in kernel smoothing algorithm; (3) to assimilate recursively data into the model and thus detect possible time variation of parameters; and (4) to address properly various sources of uncertainties stemming from input, output and parameter uncertainties. The SEnKF is tested by assimilating observed fluxes of carbon dioxide and environmental driving factor data from an AmeriFlux forest station located near Howland, Maine, USA, into a partition eddy flux model. Our analysis demonstrates that model parameters, such as light use efficiency, respiration coefficients, minimum and optimum temperatures for photosynthetic activity, and others, are highly constrained by eddy flux data at daily-to-seasonal time scales. The SEnKF stabilizes parameter values quickly regardless of the initial values of the parameters. Potential ecosystem light use efficiency demonstrates a strong seasonality. Results show that the simultaneous parameter estimation procedure significantly improves model predictions. Results also show that the SEnKF can dramatically reduce the variance in state variables stemming from the uncertainty of parameters and driving variables. The SEnKF is a robust and effective algorithm in evaluating and developing ecosystem models and in improving the understanding and quantification of carbon cycle parameters and processes.

From laser point clouds to surfaces: Statistical nonparametric methods for three-dimensional reconstruction

This paper presents a new method for the reconstruction of three-dimensional objects surveyed with a terrestrial laser scanner. The method is a 2.5D surface modelling technique which is based on the application of statistical nonparametric regression methods for point cloud regularization and mesh smoothing, specifically the kernel-smoothing techniques. The proposed algorithm was tested in a theoretical model-simulations being carried out with the aim of evaluating the ability of the method to filter random noise and oscillations related to the acquisition of data during the fieldwork-and the results were satisfactory. The method was then applied, as a reverse engineering tool, to real-world data in the field of naval construction. A precise solution to the problem of obtaining realistic surfaces and sections of large industrial objects from laser 3D point clouds is provided, which has proved to be efficient in terms of computational time.

Statistical inferences for functional data

With modem technology development, functional data are being observed frequently in many scientific fields. A popular method for analyzing such functional data is first, then estimation. That is, statistical inference such as estimation and hypothesis testing about functional data is conducted based on the substitution of the underlying individual functions by their reconstructions obtained by one smoothing technique or another. However, little is known about this substitution effect on functional data analysis. In this paper this problem is investigated when the local polynomial kernel (LPK) smoothing technique is used for individual function reconstructions. We find that under some mild conditions, the substitution effect can be ignored asymptotically. Based on this, we construct LPK reconstruction-based estimators for the mean, covariance and noise variance functions of a functional data set and derive their asymptotics. We also propose a GCV rule for selecting good bandwidths for the LPK reconstructions. When the mean function also depends on some time-independent covariates, we consider a functional linear model where the mean function is linearly related to the covariates but the covariate effects are functions of time. The LPK reconstruction-based estimators for the covariate effects and the covariance function are also constructed and their asymptotics are derived. Moreover, we propose a L 2 -norm-based global test statistic for a general hypothesis testing problem about the covariate effects and derive its asymptotic random expression. The effect of the bandwidths selected by the proposed GCV rule on the accuracy of the LPK reconstructions and the mean function estimator is investigated via a simulation study. The proposed methodologies are illustrated via an application to a real functional data set collected in climatology.

Correcting Home‐Range Models for Observation Bias

Abstract: Home‐range models implicitly assume equal observation rates across the study area. Because this assumption is frequently violated, we describe methods for correcting home‐range models for observation bias. We suggest corrections for 3 general types of home‐range models including those for which parameters are estimated using least‐squares theory, models utilizing maximum likelihood for parameter estimation, and models based on kernel smoothing techniques. When applied to mule deer (Odocoileus hemionus) location data, we found that uncorrected estimates of the utilization distribution were biased low by as much as 18.4% and biased high by 19.2% when compared to corrected estimates. Because the magnitude of bias is related to several factors, future research should determine the relative influence of each of these factors on home‐range bias.

Improved estimates of incident radiation and heat load using non‐ parametric regression against topographic variables

AbstractQuestion: Can non‐parametric multiplicative regression (NPMR) improve estimates of potential direct incident radiation (PDIR) and heat load based on topographic variables, as compared to least‐squares multiple regression against trigonometric transforms of the predictors?Methods: We used a multiplicative kernel smoothing technique to interpolate between tabulated values of PDIR, using a locally linear model and a Gaussian kernel, with slope, aspect, and latitude as predictors. Heat load was calculated as a 45 degree rotation of the PDIR response surface.Results: This method yielded a fit to a complex response surface with R2 > 0.99 and eliminated the areas of poor fit given by a previously published method based on least squares multiple regression with trigonometric functions of the predictors.Conclusions: Improved estimates of PDIR and heat load based on topographic variables can be obtained by using non‐parametric multiplicative regression (NPMR). The main drawback to the method is that it requires reference to the data tables, since those data are part of the model.

逐次モンテカルロ法に基づく宇宙機異常診断法

We propose a new framework for spacecraft fault diagnosis based on combined parameter and mode online estimation using a sequential Monte Carlo method. Our method can detect and diagnose faults as parameter changes and hence can be considered as a probabilistic approach for the parameter-estimation-based fault diagnosis method which is one of the methodologies on quantitative model-based diagnosis. We derive an algorithm for spacecraft fault diagnosis by describing the parameter-estimation-based fault diagnosis method as a probabilistic inference problem and applying a modified sequential Monte Carlo method, obtained by incorporating fault-modes, risk-sensitivities on modes and kernel-smoothing techniques into the original method, to the problem. The proposed fault-diagnosis algorithm was applied to an artificial data simulating malfunctions of thrusters in rendezvous maneuver of spacecraft, and the feasibility of the method was confirmed.

Improved estimates of incident radiation and heat load using non- parametric regression against topographic variables

Question: Can non-parametric multiplicative regression (NPMR) improve estimates of potential direct incident radiation (PDIR) and heat load based on topographic variables, as compared to least-squares multiple regression against trigonometric transforms of the predictors? Methods: We used a multiplicative kernel smoothing technique to interpolate between tabulated values of PDIR, using a locally linear model and a Gaussian kernel, with slope, aspect, and latitude as predictors. Heat load was calculated as a 45 degree rotation of the PDIR response surface. Results: This method yielded a fit to a complex response surface with R2 > 0.99 and eliminated the areas of poor fit given by a previously published method based on least squares multiple regression with trigonometric functions of the predictors. Conclusions: Improved estimates of PDIR and heat load based on topographic variables can be obtained by using non-parametric multiplicative regression (NPMR). The main drawback to the method is that it requires reference to the data tables, since those data are part of the model.

Sequential Kernel Estimation of the Conditional Intensity of Nonstationary Point Processes

This paper develops adaptive nonparametric methods for analyzing seismic data. Kernel smoothing techniques are suitable for space–time point processes; however, they must be adapted to deal with the nonstationarity of earthquakes. By this we mean changes in the spatial and temporal pattern of point occurrences. A class of recursive kernel density and regression estimators are proposed to study the space–time evolution of earthquakes. Their smoothing coefficients can be designed in an optimal way with prediction error criteria. The entire method is naturally oriented to forecasting. An extensive application to the Northern California Earthquake Catalog (NCEC) data-set, and a simulation experiment illustrates and checks the approach.

Density and Hazard Rate Estimation for Censored and A-Mixing Data Using Gamma Kernels

In this paper we consider the nonparametric estimation for a density and hazard rate function for right censored α-mixing survival time data using kernel smoothing techniques. Since survival times are positive with potentially a high concentration at zero, one has to take into account the bias problems when the functions are estimated in the boundary region. In this paper, gamma kernel estimators of the density and the hazard rate function are proposed. The estimators use adaptive weights depending on the point in which we estimate the function, and they are robust to the boundary bias problem. For both estimators, the mean squared error properties, including the rate of convergence, the almost sure consistency and the asymptotic normality are investigated. The results of a simulation demonstrate the excellent performance of the proposed estimators.

Nonparametric Inference for Local Extrema with Application to Oligonucleotide Microarray Data in Yeast Genome

Identifying local extrema of expression profiles is one primary objective in some cDNA microarray experiments. To study the replication dynamics of the yeast genome, for example, local peaks of hybridization intensity profiles correspond to putative replication origins. We propose a nonparametric kernel smoothing (NKS) technique to detect local hybridization intensity extrema across chromosomes. The novelty of our approach is that we base our inference procedures on equilibrium points, namely those locations at which the first derivative of the intensity curve is zero. The proposed smoothing technique provides both point and interval estimation for the location of local extrema. Also, this technique can be used to test for the hypothesis of either one or multiple suspected locations being the true equilibrium points. We illustrate the proposed method on a microarray data set from an experiment designed to study the replication origins in the yeast genome, in that the locations of autonomous replication sequence (ARS) elements are identified through the equilibrium points of the smoothed intensity profile curve. Our method found a few ARS elements that were not detected by the current smoothing methods such as the Fourier convolution smoothing.

Some Interpolation Estimators in Environmental Risk Assessment for Spatially Misaligned Health Data

Ecological regression studies are widely used in geographical epidemiology to assess the relationships between health hazard and putative risk factors. Very often, health data are measured at an aggregate level because of confidentiality restrictions, while putative risk factors are measured on a different grid, i.e., independent (exposure) variable and response (counts) variable are spatially misaligned. To perform a regression of risk on exposure, one needs to realign the spatial support of the variables. Bayesian hierarchical models constitute a natural approach to the problem because of their ability to model the exposure field and the relationship between exposure and relative risk at different levels of the hierarchy, taking proper account of the variability induced by the covariate estimation. In the current paper, we propose two fully Bayesian solutions to the problem. The first one is based on the kernel-smoothing technique, while the second one is built on the tessellation of the study region. We illustrate our methods by assessing the relationship between exposure to uranium in drinkable waters and cancer incidence, in South Carolina (USA).

Weighted Estimators for Proportional Hazards Regression With Missing Covariates

Missing covariate data are common in epidemiologic studies and disease prevention trials. In this article regression parameter estimation in the Cox proportional hazards model is considered when certain covariates are observed for all study subjects and other covariate data are collected only for a subset. The article presents both simple weighted and kernel-assisted fully augmented weighted estimators that use the partially incomplete data nonparametrically. We use nonparametric methods to estimate selection probabilities in the simple weighted estimating functions. We also use nonparametric kernel smoothing techniques to estimate certain conditional expectations in fully augmented weighted estimating functions. The proposed methods are nonparametric in the sense that they require neither a model for the missing-data mechanism nor specification of the conditional distribution of missing covariates given observed covariates. These estimators allow the missing-data mechanism to depend on outcome variables and observed covariates, and they are applicable to various cohort sampling procedures, including case-cohort and nested case-control designs. We show that the simple and the kernel-assisted fully augmented weighted estimators are typically consistent and asymptotically normal. Moreover, the proposed estimators are more efficient than the simple weighted estimator with the inverse of true selection probability as weight. They also correct the bias of estimates from analysis of the complete data alone when the missing-data mechanism depends on outcome variables. In addition, when covariates are time-independent, certain simple weighted estimators are shown to be asymptotically equivalent to the kernel-assisted fully augmented weighted estimators. Moderate sample size performance of the estimators is examined via simulation and by application to two real datasets.

Using Scale Mixtures Of Normals To Model Continuously Compounded Returns

A new method for estimating the parameters of scale mixtures of normals (SMN) is introduced and evaluated. The new method is called UNMIX and is based on minimizing the weighted square distance between exact values of the density of the scale mixture and estimated values using kernel smoothing techniques over a specified grid of x-values and a grid of potential scale values. Applications of the method are made in modeling the continuously compounded return, CCR, of stock prices. Modeling this ratio with UNMIX proves promising in comparison with other existing techniques that use only one normal component, or those that use more than one component based on the EM algorithm as the method of estimation.

A Perturbation Technique for Sample Moment Matching in Kernel Density Estimation

Summary The fundamental idea of kernel smoothing technique can be recognized as one-parameter data perturbation with a smooth density. The usual kernel density estimates might not match arbitrary sample moments calculated from the unsmoothed data. A technique based on two-parameter data perturbation is developed for sample moment matching in kernel density estimation. It is shown that the moments calculated from the resulting tuned kernel density estimate can be made arbitrarily close to the raw sample moments. Moreover, the pointwise rate of MISE of the resulting density estimates remains optimal. Relevant simulation studies are carried out to demonstrate the usefulness and other features of this technique.

Connecting species richness, abundance and body size in deep‐sea gastropods

ABSTRACTAim This paper examines species richness, abundance, and body size in deep‐sea gastropods and how they vary over depth, which is a strong correlate of nutrient input. Previous studies have documented the empirical relationships among these properties in terrestrial and coastal ecosystems, but a full understanding of how these patterns arise has yet to be obtained. Examining the relationships among macroecological variables is a logical progression in deep‐sea ecology, where patterns of body size, diversity, and abundance have been quantified separately but not linked together.Location 196–5042 m depth in the western North Atlantic.Method Individuals analysed represent all Vetigastropoda and Caenogastropoda (Class Gastropoda) with intact shells, excluding Ptenoglossa, collected by the Woods Hole Benthic Sampling Program (3424 individuals representing 80 species). Biovolume was measured for every individual separately (i.e. allowing the same species to occupy multiple size classes) and divided into log2 body size bins. Analyses were conducted for all gastropods together and separated into orders and depth regions (representing different nutrient inputs). A kernel smoothing technique, Kolmogorov‐Smirnov test of fit, and OLS and RMA were used to characterize the patterns.Results Overall, the relationship between the number of individuals and species is right skewed. There is also a positive linear relationship between the number of individuals and the number of species, which is independent of body size. Variation among these relationships is seen among the three depth regions. At depths inferred to correspond with intermediate nutrient input levels, species are accumulated faster given the number of individuals and shift from a right‐skewed to a log‐normal distribution.Conclusion A strong link between body size, abundance, and species richness appears to be ubiquitous over a variety of taxa and environments, including the deep sea. However, the nature of these relationships is affected by the productivity regime and scale at which they are examined.