Nonparametric Component Research Articles

Selection of Optimal Smoothing Parameters in Mixed Estimator of Kernel and Fourier Series in Semiparametric Regression

In this article, we propose a new method of selecting smoothing parameters in semiparametric regression. This method is used in semiparametric regression estimation where the nonparametric component is partially approximated by multivariable Fourier Series and partly approached by multivariable Kernel. Selection of smoothing parameters using the method with Generalized Cross-Validation (GCV). To see the performance of this method, it is then applied to the data drinking water quality sourced from Regional Drinking Water Company (PDAM) Surabaya by using Fourier Series with trend and Gaussian Kernel. The results showed that this method contributed a good performance in selecting the optimal smoothing parameters.

Improving the Reliability of Sub‐Seasonal Forecasts of High and Low Flows by Using a Flow‐Dependent Nonparametric Model

AbstractSub‐seasonal streamflow forecasts are important for a range of water resource management applications, with a distinct practical interest in forecasts of high flows (e.g., for managing flood events) and low flows (e.g., for managing environmental flows). Despite this interest, differences in forecast performance for high and low flow events are not routinely investigated. Our study reveals that while forecasts evaluated over the full flow range can appear reliable, stratification into high/low flow ranges highlights significant under/over‐estimation of forecast uncertainty, respectively. We overcome this challenge by introducing a flow‐dependent (FD) nonparametric component into a post‐processing model of hydrological forecasting errors, the Multi‐Temporal Hydrological Residual Error (MuTHRE) model, yielding the MuTHRE‐FD model. The MuTHRE and MuTHRE‐FD models are compared in a case study with 11 Australian catchments, the GR4J rainfall‐runoff model and post‐processed rainfall forecasts from ACCESS‐S. Through its improved treatment of flow‐dependence, the MuTHRE‐FD model achieves practically significant improvements over the original MuTHRE model in the reliability of forecasted cumulative volumes for: (a) high flows out to 7 days; (b) low flows out to 2 days; and (c) mid flows for majority of lead times. The new MuTHRE‐FD model provides seamless sub‐seasonal forecasts with high quality performance for both high and low flows over a range of lead times. This improvement provides forecast users with increased confidence in using sub‐seasonal forecasts across a wide range of applications.

A semiparametric mixed-effects model for censored longitudinal data.

In longitudinal studies involving laboratory-based outcomes, repeated measurements can be censored due to assay detection limits. Linear mixed-effects (LMEs) models are a powerful tool to model the relationship between a response variable and covariates in longitudinal studies. However, the linear parametric form of linear mixed-effect models is often too restrictive to characterize the complex relationship between a response variable and covariates. More general and robust modeling tools, such as nonparametric and semiparametric regression models, have become increasingly popular in the last decade. In this article, we use semiparametric mixed models to analyze censored longitudinal data with irregularly observed repeated measures. The proposed model extends the censored linear mixed-effect model and provides more flexible modeling schemes by allowing the time effect to vary nonparametrically over time. We develop an Expectation-Maximization (EM) algorithm for maximum penalized likelihood estimation of model parameters and the nonparametric component. Further, as a byproduct of the EM algorithm, the smoothing parameter is estimated using a modified linear mixed-effects model, which is faster than alternative methods such as the restricted maximum likelihood approach. Finally, the performance of the proposed approaches is evaluated through extensive simulation studies as well as applications to data sets from acquired immune deficiency syndrome studies.

Mixed estimator of spline truncated, Fourier series, and kernel in biresponse semiparametric regression model

Regression analysis is a method of analysis to determine the relationship between the response and the predictor variables. There are three approaches in regression analysis, namely the parametric, nonparametric, and semiparametric approaches. Biresponse Semiparametric regression model is a regression model that uses a combination approach between parametric and nonparametric components, where two response variables are correlated with each other. For data cases with several predictor variables, different estimation technique approaches can be used for each variable. In this study, the parametric component is assumed to be linear. At the same time, the nonparametric part is approached using a mixture of three estimation techniques, namely, spline truncated, Fourier series, and the kernel. The unknown data pattern is assumed to follow the criteria of each of these estimation techniques. The spline is used when the data pattern tends to change at certain time intervals, the Fourier series is used when the data pattern tends to repeat itself, and the kernel is used when the data does not have a specific way. This study aims to obtain parameter estimates for the mixed semiparametric regression model of spline truncated, Fourier series, and the kernel on the biresponse data using the Weighted Least Square (WLS) method. The formed model depends on the selection of knot points, oscillation parameters, and optimal bandwidth. The best model is based on the smallest Generalized Cross Validation (GCV).

Partially Linear Generalized Single Index Models for Functional Data (PLGSIMF)

Single-index models are potentially important tools for multivariate non-parametric regression analysis. They generalize linear regression models by replacing the linear combination α0⊤X with a non-parametric component η0α0⊤X, where η0(·) is an unknown univariate link function. In this article, we generalize these models to have a functional component, replacing the generalized partially linear single index models η0α0⊤X+β0⊤Z, where α is a vector in IRd, η0(·) and β0(·) are unknown functions that are to be estimated. We propose estimates of the unknown parameter α0, the unknown functions β0(·) and η0(·) and establish their asymptotic distributions, and furthermore, a simulation study is carried out to evaluate the models and the effectiveness of the proposed estimation methodology.

Robust Inference for Nonstationary Time Series with Possibly Multiple Changing Periodic Structures

Motivated by two examples concerning global warming and monthly total import and export by China, we study time series that contain a nonparametric periodic component with an unknown period, a nonparametric trending behavior and also additive covariate effects. Further, as the amplitude function may change at some known or unknown change-point(s), we extend our model to take this dynamical periodicity into account and introduce two change-point estimators. To the best of knowledge, this is the first work to study such complex periodic structure. A two-step estimation procedure is proposed to estimate accurately the periodicity, trend and covariate effects. First, we estimate the period with the trend and covariate effects being approximated by B-splines rather than being ignored. To achieve robustness we employ a penalized M-estimation method which uses post model selection inference ideas. Next, given the period estimate, we estimate the amplitude, trend and covariate effects. Asymptotic properties of our estimators are derived, including consistency of the period estimator and asymptotic normality and oracle property of the estimated periodic sequence, trend and covariate effects. Simulation studies confirm superiority of our method and illustrate good performance of our change-point estimators. Applications to the two motivating examples demonstrate utilities of our methods.

Efficient estimation of semiparametric varying-coefficient partially linear transformation model with current status data

We consider a varying-coefficient partially linear transformation model with current status data, which extends several semiparametric models for current status data in the literature. Sieve maximum likelihood estimation method is used to obtain an integrated estimate for both the parametric components and nonparametric components in the model, i.e. the linear regression coefficients, the varying-coefficient functions and the baseline survival function. Under some regularity conditions, the proposed parameter estimators are proved to be semiparametrically efficient and asymptotically normal, and the estimators for the nonparametric functions achieve the optimal rate of convergence. Simulation studies assure the theoretical results, and a real data is reanalysed using the proposed method and it yields new findings.

Feedback in panel data models

Much of the analysis of panel data has been based on an assumption of strict exogeneity. Distributions are specified for outcome variables conditional on a latent individual effect and conditional on observed predictor variables at all dates, with the future values of the predictor variables assumed to have no effect on the conditional distribution. The paper relaxes this assumption in order to allow for lagged dependent variables and, more generally, for feedback from lagged dependent variables to current values of the predictor variables. Such feedback would arise in an evaluation study if the treatment variable is randomly assigned only conditional on the individual effect and on previous outcomes.An information bound is derived for a semiparametric regression model with sequential moment restrictions, with the information set increasing over time. The bound is then applied to a model with a (scalar) multiplicative random effect. The mean of the random effect conditional on the predictor variables is not restricted, so that the random effect can control for various omitted variables. This conditional mean is the nonparametric component of the semiparametric regression model. There is a transformation that eliminates the random effect and leads to a set of sequential moment restrictions in which the moment function depends on only a finite-dimensional parameter. The information bound for this simpler problem coincides with that of the original problem. The form of the optimal instrumental variables is derived.The paper also considers the identification problems that arise when the random effect is a vector with two or more components.

Double penalized regularization estimation for partially linear instrumental variable models with ultrahigh dimensional instrumental variables

Under ultrahigh dimensional instrumental variables, we consider the estimation for a class of partially linear models with endogenous covariates. To overcome the difficulty of ultrahigh dimensionality of the instrumental variables, we propose a double penalized regularization estimation procedure for identifying the optimal instrumental variables, and estimating covariate effects of the parametric and nonparametric components. With some regularity conditions, some asymptotic properties of the proposed estimation are derived, such as the consistency of the resulting estimators for parametric and nonparametric components. Lastly, we examine the finite sample performance of the proposed method by some simulation studies and a real data analysis.

Partially linear functional-coefficient dynamic panel data models: sieve estimation and specification testing

We study the nonparametric estimation and specification testing for partially linear functional-coefficient dynamic panel data models, where the effects of some covariates on the dependent variable vary nonparametrically according to a set of low-dimensional variables. Based on the sieve approximation of unknown slope functions, we propose a sieve 2SLS procedure to estimate the model. The asymptotic properties of the estimators of both parametric and nonparametric components are established when sample size N and T tend to infinity jointly. A nonparametric specification test for the constancy of slopes is also proposed. We show that after being appropriately standardized, the test is asymptotically normally distributed under the null hypothesis. The asymptotic properties of the test is also studied under a sequence of local Pitman alternatives and global alternatives. A set of Monte Carlo simulations show that our sieve 2SLS estimators and specification test perform remarkably well in finite samples. We apply our method to study the impact of income on democracy, and find strong evidence of nonlinear/nonconstant effect of income on democracy.

Robust estimates of regional treatment effects in multiregional randomized clinical trials with semiparametric logistic model.

In multiregional randomized clinical trials (MRCTs), determining the regional treatment effect of a new treatment over an existing one is important to both the sponsor and related regulatory agencies. Also of particular interest is to test the null hypothesis that the treatment benefit is the same among all the regions. Existing methods are mainly for continuous endpoint and use parametric models, which are not robust. MRCTs are known for facing increased variation and heterogeneity and a robust model for its design and analysis would be desirable. We consider clinical trials with a binary primary endpoint and propose a robust semiparametric logistic model which has a known parametric and an unknown nonparametric component. The parametric component represents our prior knowledge about the model, and the nonparametric part reflects uncertainty. Compared to the classic logistic model for this problem, the proposed model has the following advantages: robust to model assumption, more flexible and accurate to model the relationship between the response and covariates, and possibly more accurate parameter estimates. The model parameters are estimated by profile maximum likelihood approach, and the null hypothesis of regional treatment difference being the same is tested by the profile likelihood ratio statistic. Asymptotic properties of the estimates are derived. Simulation studies are conducted to evaluate the performance of the proposed model, which demonstrated clear advantages over the classic logistic model. The method is then applied to analyzing a real MRCT.

Nonparametric homogeneity pursuit in functional-coefficient models

ABSTRACT This paper explores the homogeneity of coefficient functions in nonlinear models with functional coefficients and identifies the underlying semiparametric modelling structure. With initial kernel estimates, we combine the classic hierarchical clustering method with a generalised version of the information criterion to estimate the number of clusters, each of which has a common functional coefficient, and determine the membership of each cluster. To identify a possible semi-varying coefficient modelling framework, we further introduce a penalised local least squares method to determine zero coefficients, non-zero constant coefficients and functional coefficients which vary with an index variable. Through the nonparametric kernel-based cluster analysis and the penalised approach, we can substantially reduce the number of unknown parametric and nonparametric components in the models, thereby achieving the aim of dimension reduction. Under some regularity conditions, we establish the asymptotic properties for the proposed methods including the consistency of the homogeneity pursuit. Numerical studies, including Monte-Carlo experiments and two empirical applications, are given to demonstrate the finite-sample performance of our methods.

High-dimensional generalized semiparametric model for longitudinal data

This paper considers the problem of estimation in the generalized semiparametric model for longitudinal data when the number of parameters diverges with the sample size. A penalization type of generalized estimating equation method is proposed, while we use the regression spline to approximate the nonparametric component. The proposed procedure involves the specification of the posterior distribution of the random effects, which cannot be evaluated in a closed form. However, it is possible to approximate this posterior distribution by producing random draws from the distribution using a Metropolis algorithm. Under some regularity conditions, the resulting estimators enjoy the oracle properties, under the high-dimensional regime. Simulation studies are carried out to assess the performance of our proposed method, and two real data sets are analyzed for procedure demonstration.

A Nanoradiomics Approach for Differentiation of Tumors Based on Tumor-Associated Macrophage Burden.

Objective Tumor-associated macrophages (TAMs) within the tumor immune microenvironment (TiME) of solid tumors play an important role in treatment resistance and disease recurrence. The purpose of this study was to investigate if nanoradiomics (radiomic analysis of nanoparticle contrast-enhanced images) can differentiate tumors based on TAM burden. Materials and Methods In vivo studies were performed in transgenic mouse models of neuroblastoma with low (N = 11) and high (N = 10) tumor-associated macrophage (TAM) burden. Animals underwent delayed nanoparticle contrast-enhanced CT (n-CECT) imaging at 4 days after intravenous administration of liposomal-iodine agent (1.1 g/kg). CT imaging-derived conventional tumor metrics (tumor volume and CT attenuation) were computed for segmented tumor CT datasets. Nanoradiomic analysis was performed using a PyRadiomics workflow implemented in the quantitative image feature pipeline (QIFP) server containing 900 radiomic features (RFs). RF selection was performed under supervised machine learning using a nonparametric neighborhood component method. A 5-fold validation was performed using a set of linear and nonlinear classifiers for group separation. Statistical analysis was performed using the Kruskal–Wallis test. Results N-CECT imaging demonstrated heterogeneous patterns of signal enhancement in low and high TAM tumors. CT imaging-derived conventional tumor metrics showed no significant differences (p > 0.05) in tumor volume between low and high TAM tumors. Tumor CT attenuation was not significantly different (p > 0.05) between low and high TAM tumors. Machine learning-augmented nanoradiomic analysis revealed two RFs that differentiated (p < 0.002) low TAM and high TAM tumors. The RFs were used to build a linear classifier that demonstrated very high accuracy and further confirmed by 5-fold cross-validation. Conclusions Imaging-derived conventional tumor metrics were unable to differentiate tumors with varying TAM burden; however, nanoradiomic analysis revealed texture differences and enabled differentiation of low and high TAM tumors.

Estimation of varying coefficient models with measurement error

We propose a semiparametric estimator for varying coefficient models when the regressors in the nonparametric components are measured with error. Varying coefficient models are an extension of other popular semiparametric models, including partially linear and nonparametric additive models, and deliver an attractive solution to the curse-of-dimensionality. We use deconvolution kernel estimation in a two-step procedure and show that the estimator is consistent and asymptotically normally distributed. We do not assume that we know the distribution of the measurement error a priori. Instead, we suppose we have access to a repeated measurement of the noisy regressor and present results using the approach of Delaigle, Hall and Meister (2008) and, for cases when the measurement error may be asymmetric, the approach of Li and Vuong (1998) based on Kotlarski’s (1967) identity. We show that the convergence rate of the estimator is significantly reduced when the distribution of the measurement error is assumed unknown and possibly asymmetric. We study the small sample behaviour of our estimator in a simulation study and apply it to a real dataset. In particular, we consider the role of cognitive ability in augmenting the effect of risk preferences on earnings.

Weak consistency for the estimators in a semiparametric regression model based on negatively associated random errors

In this article, we consider the semiparametric regression model: , which is an important and very useful statistical model. We investigate the parametric component and nonparametric component estimators in a semiparametric regression model based on negatively associated random errors. The weak consistency for the estimators and of and respectively are established under some suitable conditions. In addition, a simulation to study the numerical performance of the consistency for the nearest neighbor weight function estimators is provided and a real data application is then presented.

Double/debiased machine learning for logistic partially linear model.

We propose double/debiased machine learning approaches to infer a parametric component of a logistic partially linear model. Our framework is based on a Neyman orthogonal score equation consisting of two nuisance models for the nonparametric component of the logistic model and conditional mean of the exposure with the control group. To estimate the nuisance models, we separately consider the use of high dimensional (HD) sparse regression and (nonparametric) machine learning (ML) methods. In the HD case, we derive certain moment equations to calibrate the first order bias of the nuisance models, which preserves the model double robustness property. In the ML case, we handle the nonlinearity of the logit link through a novel and easy-to-implement 'full model refitting' procedure. We evaluate our methods through simulation and apply them in assessing the effect of the emergency contraceptive pill on early gestation and new births based on a 2008 policy reform in Chile.

Orthogonality based penalized GMM estimation for variable selection in partially linear spatial autoregressive models

By combining penalized GMM estimation method with the QR decomposition technique, we propose an orthogonal projection-based regularization estimation method for a class of partially linear spatial autoregressive models. The proposed method can select important covariates in the parametric component, and can also identify the significance of spatial effects. Under some conditions, some theoretical properties are studied, such as the consistency of the proposed variable selection procedure and the oracle property of the resulting estimators for parametric and nonparametric components. Furthermore, some simulation studies are carried out to examine the finite sample performances of the proposed regularization estimation method.

Statistical inference in sparse high-dimensional additive models

In this paper, we discuss the estimation of a nonparametric component f1 of a nonparametric additive model Y=f1(X1)+⋯+fq(Xq)+ϵ. We allow the number q of additive components to grow to infinity and we make sparsity assumptions about the number of nonzero additive components. We compare this estimation problem with that of estimating f1 in the oracle model Z=f1(X1)+ϵ, for which the additive components f2,…,fq are known. We construct a two-step presmoothing-and-resmoothing estimator of f1 and state finite-sample bounds for the difference between our estimator and a corresponding smoothing estimator fˆ1(oracle) in the oracle model. In an asymptotic setting, these bounds can be used to show asymptotic equivalence of our estimator and the oracle estimator; the paper thus shows that, asymptotically, under strong enough sparsity conditions, knowledge of f2,…,fq has no effect on estimation accuracy. Our first step is to estimate f1 with an undersmoothed estimator based on near-orthogonal projections with a group Lasso bias correction. In the second step, we construct pseudo responses Yˆ by evaluating this undersmoothed estimator of f1 at the design points and then apply the smoothing method of the oracle estimator fˆ1(oracle) to the nonparametric regression problem with “responses” Yˆ and covariates X1. Our mathematical exposition centers primarily on establishing properties of the presmoothing estimator. We present simulation results demonstrating close-to-oracle performance of our estimator in practical applications.

Uncontrolled Confounders May Lead to False or Overvalued Radiomics Signature: A Proof of Concept Using Survival Analysis in a Multicenter Cohort of Kidney Cancer.

PurposeWe aimed to explore potential confounders of prognostic radiomics signature predicting survival outcomes in clear cell renal cell carcinoma (ccRCC) patients and demonstrate how to control for them.Materials and MethodsPreoperative contrast enhanced abdominal CT scan of ccRCC patients along with pathological grade/stage, gene mutation status, and survival outcomes were retrieved from The Cancer Imaging Archive (TCIA)/The Cancer Genome Atlas—Kidney Renal Clear Cell Carcinoma (TCGA-KIRC) database, a publicly available dataset. A semi-automatic segmentation method was applied to segment ccRCC tumors, and 1,160 radiomics features were extracted from each segmented tumor on the CT images. Non-parametric principal component decomposition (PCD) and unsupervised hierarchical clustering were applied to build the radiomics signature models. The factors confounding the radiomics signature were investigated and controlled sequentially. Kaplan–Meier curves and Cox regression analyses were performed to test the association between radiomics signatures and survival outcomes.Results183 patients of TCGA-KIRC cohort with available imaging, pathological, and clinical outcomes were included in this study. All 1,160 radiomics features were included in the first radiomics signature. Three additional radiomics signatures were then modelled in successive steps removing redundant radiomics features first, removing radiomics features biased by CT slice thickness second, and removing radiomics features dependent on tumor size third. The final radiomics signature model was the most parsimonious, unbiased by CT slice thickness, and independent of tumor size. This final radiomics signature stratified the cohort into radiomics phenotypes that are different by cancer-specific and recurrence-free survival; HR (95% CI) = 3.0 (1.5–5.7), p <0.05 and HR (95% CI) = 6.6 (3.1–14.1), p <0.05, respectively.ConclusionRadiomics signature can be confounded by multiple factors, including feature redundancy, image acquisition parameters like slice thickness, and tumor size. Attention to and proper control for these potential confounders are necessary for a reliable and clinically valuable radiomics signature.