Local Smoothing Method Research Articles

Varying Coefficient Models for Sparse Noise-contaminated Longitudinal Data.

In this paper we propose a varying coefficient model for highly sparse longitudinal data that allows for error-prone time-dependent variables and time-invariant covariates. We develop a new estimation procedure, based on covariance representation techniques, that enables effective borrowing of information across all subjects in sparse and irregular longitudinal data observed with measurement error, a challenge in which there is no adequate solution currently. More specifically, sparsity is addressed via a functional analysis approach that considers the observed longitudinal data as noise contaminated realizations of a random process that produces smooth trajectories. This approach allows for estimation based on pooled data, borrowing strength from all subjects, in targeting the mean functions and auto- and cross-covariances to overcome sparse noisy designs. The resulting estimators are shown to be uniformly consistent. Consistent prediction for the response trajectories are also obtained via conditional expectation under Gaussian assumptions. Asymptotic distribution of the predicted response trajectories are derived, allowing for construction of asymptotic pointwise confidence bands. Efficacy of the proposed method is investigated in simulation studies and compared to the commonly used local polynomial smoothing method. The proposed method is illustrated with a sparse longitudinal data set, examining the age-varying relationship between calcium absorption and dietary calcium. Prediction of individual calcium absorption curves as a function of age are also examined.

SU-E-J-16: Noise Reduction with Detail Preservation for Low-Dose KV CBCT Using Non-Local Means: Simulated Patient Study

Purpose: With the frequent uses of CBCT in IGRT, the cumulative imaging dose to normal tissues may not be insignificant. A lower mAs protocol in CBCT acquisition reduces the dose, but dramatically degrades image quality due to excessive noise. The purpose of this study is to examine the effectiveness of the nonlocal means (NL‐means) denoising algorithm in reducing noise while preserving details in simulated low‐dose patient CBCT. Method and Materials:NL‐means algorithm estimates the true value of a pixel as a weighted average of all pixels in the image, where the weights depend on the similarity between the pixels. Compared with the local smoothing or filtering methods, NL‐means can reduce noise while preserve details. Low‐dose patient CBCTimages were generated by adding noise to the normal‐dose images based on stochastic property of incident photons from the x‐ray source, simulating data acquisition with reduced mAs. The low‐dose images were normalized to be within [0 1] and then fed to the NL‐means for denoising. The optimal parameters for NL‐means were experimentally determined. Results: The simulated low‐dose CBCT had only 6.3% of the patient CBCT dose. NL‐means clearly reduced the noise without obvious blurring, and the images appear to have similar quality as normal‐dose images. The suppressed noise resembled the desired white noise except at sharp edges. The mean signal‐to‐noise ratio in homogeneous regions was increased from 7.6 to 28.1. NL‐means preserved anatomic details as fine as 2 pixels but blurred single‐pixel details. Conclusion: Results on the patient data demonstrated that NL‐means effectively reduced the noise while preserving most fine details in simulated low‐dose patient CBCT, consistent with results on phantom study. A single set of optimal parameters was suitable for various CBCTs. This post‐processing method can be straightforwardly implemented in OBI software.

Nonparametric additive model-assisted estimation for survey data

An additive model-assisted nonparametric method is investigated to estimate the finite population totals of massive survey data with the aid of auxiliary information. A class of estimators is proposed to improve the precision of the well known Horvitz–Thompson estimators by combining the spline and local polynomial smoothing methods. These estimators are calibrated, asymptotically design-unbiased, consistent, normal and robust in the sense of asymptotically attaining the Godambe-Joshi lower bound to the anticipated variance. A consistent model selection procedure is further developed to select the significant auxiliary variables. The proposed method is sufficiently fast to analyze large survey data of high dimension within seconds. The performance of the proposed method is assessed empirically via simulation studies.

Barrier Option Pricing with Model Averaging Methods under Local Volatility Models

In this paper, we propose a method to provide the distribution of option price under local volatility model when market-provided implied volatility data are given. The local volatility model is one of the most widely used smile-consistent models. In local volatility model, the volatility is a deterministic function of the random stock price. Before estimating local volatility surface (LVS), we need to estimate implied volatility surfaces (IVS) from market data. To do this we use local polynomial smoothing method. Then we apply the Dupire formula to estimate the resulting LVS. However, the result is dependent on the bandwidth of kernel function employed in local polynomial smoothing method and to solve this problem, the proposed method in this paper makes use of model averaging approach by means of bandwidth priors, and then produces a robust local volatility surface estimation with a confidence interval. After constructing LVS, we price barrier option with the LVS estimation through Monte Carlo simulation. To show the merits of our proposed method, we have conducted experiments on simulated and market data which are relevant to KOSPI200 call equity linked warrants (ELWs.) We could show by these experiments that the results of the proposed method are quite reasonable and acceptable when compared to the previous works.

Functional Varying Coefficient Models for Longitudinal Data

The proposed functional varying coefficient model provides a versatile and flexible analysis tool for relating longitudinal responses to longitudinal predictors. Specifically, this approach provides a novel representation of varying coefficient functions through suitable auto and cross-covariances of the underlying stochastic processes, which is particularly advantageous for sparse and irregular designs, as often encountered in longitudinal studies. Existing methodology for varying coefficient models is not adapted to such data. The proposed approach extends the customary varying coefficient models to a more general setting, in which not only current but also recent past values of the predictor time course may have an impact on the current value of the response time course. The influence of past predictor values is modeled by a smooth history index function, while the effects on the response are described by smooth varying coefficient functions. The resulting estimators for varying coefficient and history index functions are shown to be asymptotically consistent for sparse designs. In addition, prediction of unobserved response trajectories from sparse measurements on a predictor trajectory is obtained, along with asymptotic pointwise confidence bands. The proposed methods perform well in simulations, especially when compared with commonly used local polynomial smoothing methods for varying coefficient models, and are illustrated with longitudinal primary biliary liver cirrhosis data. The data application and detailed assumptions and proofs can be found in online Supplemental Material.

TU‐D‐204B‐06: Noise Reduction with Detail Preservation for Low‐Dose Kilovoltage CBCT Using Nonlocal Means Algorithm

Purpose: kV‐CBCT has shown enormous benefits for patient setup and treatment verification in radiotherapy. With the frequent uses of CBCT, the cumulative imaging dose to normal tissues may not be insignificant. A lower mAs protocol in CBCT acquisition reduces the dose, but dramatically degrades image quality due to excessive noise. This work studies the effectiveness of a denoising algorithm, namely nonlocal means (NL‐means), in reducing noise while preserving details in low‐dose CBCT images. Method and Materials: NL‐means algorithm estimates the true value of a pixel as a weighted average of all pixels in the image, where the weights depend on the similarity between the pixels. Unlike the local smoothing or filtering methods, NL‐means can reduce noise while preserve details. The parameters of NL‐means were optimized for the low‐dose CBCT. A non‐stationary correction for avoiding blurring out nonrepeated structures was examined. A CatPhan and an anthropomorphic head phantom were used to evaluate its performance. Results: In all images tested, the noise was clearly reduced by the NL‐means algorithm. In the five contrast objects, the contrast‐to‐noise ratios were increased significantly (mean 1.1 to 5.5), and even higher than those in high‐dose CBCT (mean 3.0). Line pairs with spatial resolutions 1 to 5 lp/cm were more distinguishable. Finer lines (6 –7 lp/cm) were blurred out compared to the high‐dose CBCT. For the anthropomorphic head phantom, the suppressed noise resembled the desired white noise. The noise in all structures was reduced while the details and fine structures were well preserved. The image appears to be of similar or even higher quality than high‐dose CBCT. Conclusion: Results on a CatPhan and an anthropomorphic phantom demonstrated that NL‐means effectively reduced the noise while preserved most fine structures in low‐dose CBCT.

Estimation in Single-Index Panel Data Models with Heterogeneous Link Functions

In this paper, we study semiparametric estimation for a single-index panel data model where the nonlinear link function varies among the individuals. We propose using the so-called refined minimum average variance estimation based on a local linear smoothing method to estimate both the parameters in the single-index and the average link function. As the cross-section dimension N and the time series dimension T tend to infinity simultaneously, we establish asymptotic distributions for the proposed parametric and nonparametric estimates. In addition, we provide two real-data examples to illustrate the finite sample behavior of the proposed estimation method in this paper.

Local scales and multiscale image decompositions

This paper is devoted to the study of local scales (oscillations) in images and use the knowledge of local scales for image decompositions. Denote by K t ( x ) = ( e − 2 π t | ξ | 2 ) ∨ ( x ) , t > 0 , the Gaussian (heat) kernel. Motivated from the Triebel–Lizorkin function space F ˙ p , ∞ α , we define a local scale of f at x to be t ( x ) ⩾ 0 such that | S f ( x , t ) | = | t 1 − α / 2 ∂ K t ∂ t ∗ f ( x ) | is a local maximum with respect to t for some α < 2 . For each x, we obtain a set of scales that f exhibits at x. The choice of α and a local smoothing method of local scales via the nontangential control will be discussed. We then extend the work in [J.B. Garnett, T.M. Le, Y. Meyer, L.A. Vese, Image decomposition using bounded variation and homogeneous Besov spaces, Appl. Comput. Harmon. Anal. 23 (2007) 25–56] to decompose f into u + v , with u being piecewise-smooth and v being texture, via the minimization problem inf u ∈ BV { K ( u ) = | u | BV + λ ‖ K t ¯ ( ⋅ ) ∗ ( f − u ) ( ⋅ ) ‖ L 1 } , where t ¯ ( x ) is some appropriate choice of a local scale to be captured at x in the oscillatory part v.

On kernel method for sliced average variance estimation

In this paper, we use the kernel method to estimate sliced average variance estimation (SAVE) and prove that this estimator is both asymptotically normal and root n consistent. We use this kernel estimator to provide more insight about the differences between slicing estimation and other sophisticated local smoothing methods. Finally, we suggest a Bayes information criterion (BIC) to estimate the dimensionality of SAVE. Examples and real data are presented for illustrating our method.

Spherical mapping for processing of 3D closed surfaces

The spherical harmonic (SPHARM) description is a powerful surface modeling technique used by many applications in biomedical imaging. However, earlier SPHARM studies employ a spherical parameterization procedure that can be applied only to voxel surfaces. This paper presents CALD, a new spherical parameterization algorithm that makes the SPHARM model applicable to general triangle meshes. The CALD algorithm starts from an initial mapping and performs novel local and global smoothing methods alternately until a solution is approached. This new algorithm can also be used for processing of 3D closed surfaces in many different areas, including medical image analysis, computer vision, and computer graphics.

Normalization of two-channel microarray experiments: a semiparametric approach

An important underlying assumption of any experiment is that the experimental subjects are similar across levels of the treatment variable, so that changes in the response variable can be attributed to exposure to the treatment under study. This assumption is often not valid in the analysis of a microarray experiment due to systematic biases in the measured expression levels related to experimental factors such as spot location (often referred to as a print-tip effect), arrays, dyes, and various interactions of these effects. Thus, normalization is a critical initial step in the analysis of a microarray experiment, where the objective is to balance the individual signal intensity levels across the experimental factors, while maintaining the effect due to the treatment under investigation. Various normalization strategies have been developed including log-median centering, analysis of variance modeling, and local regression smoothing methods for removing linear and/or intensity-dependent systematic effects in two-channel microarray experiments. We describe a method that incorporates many of these into a single strategy, referred to as two-channel fastlo, and is derived from a normalization procedure that was developed for single-channel arrays. The proposed normalization procedure is applied to a two-channel dose-response experiment.

Online monitoring with local smoothing methods and adaptive ridging

We consider online monitoring of sequentially arising data as e.g. met in clinical information systems. The general focus thereby is to detect breakpoints, i.e. timepoints where the measurement series suddenly changes the general level. The method suggested is based on local estimation. In particular, local linear smoothing is combined by ridging with local constant smoothing. The procedure is demonstrated by examples and compared with other available online monitoring routines.

Application of stress smoothing methods to a finite element pile driving analysis

In finite element analysis of pile driving, the nodes of the finite element mesh are the most important locations for output stresses. Especially at the pile‐soil interface, it is essential to obtain accurate nodal stresses. Several global and local stress smoothing methods available in the literature were reviewed and examined. Global methods are found to be computationally expensive, so results obtained from several local stress smoothing methods are compared. It is shown that accurate nodal stresses can be obtained by approximating the stress distribution inside four‐element patches by a polynomial with order equal to the order of the shape functions. Equally good results can be obtained by approximating the stress distribution inside each element by a bilinear surface. When a method taking into account both equilibrium and boundary conditions was applied, a set of ill‐conditioned matrices was produced for the four‐element patches. Such methods are therefore not recommended.

Kernel and Probit Estimates in Quantal Bioassay

Abstract A kernel method for the estimation of quantal dose-response curves is considered. In contrast to parametric modeling, this local smoothing method does not require any assumptions beyond smoothness of the dose-response curve and, in this sense, is nonparametric. In finite-sample situations, the kernel estimate of the dose-response curve is not necessarily monotone and, therefore, if the additional assumption of monotonicity of the dose-response curve is made, a monotonized version is discussed. Bias, variance, asymptotic normality, and uniform consistency, including rates of convergence of kernel estimates, are derived and applied to establish consistency and limiting distribution of kernel estimates of the EDα. Here, EDα is the effective dose at level α, that is, the dose where 100α% of the subjects show a response. The properties of the kernel estimated EDα are compared with the corresponding properties of the maximum likelihood estimator assuming the probit model. Practical application of the kernel method requires choice of a kernel function and of a bandwidth (smoothing window), and methods for global as well as local bandwidth selection are discussed. Another point of interest is the construction of confidence intervals for the estimated EDα. For the kernel method, two asymptotically consistent approaches are presented. These are compared in a simulation study, where the behavior of kernel and probit estimates of the EDα, α = .01, .05, .1, and .5, and of corresponding confidence intervals is observed for four different models (two probit models, one Weibull model, and one normal mixture model). The choice of different kernels and of local bandwidths was compared in another Monte Carlo study. The observed finite sample behavior of the kernel estimate of the ED50 was considerably better (gain of 40%-70% in terms of mean squared error) than that of the probit method under the two nonsigmoid models, whereas it was not drastically worse under the probit models (loss of 20%-30% in terms of mean squared error). Application of higher-order kernels and local bandwidth choice turned out to yield favorable versions of the kernel method. The results concerning observed coverage probabilities for the confidence intervals based on different approaches were more mixed.