Estimation for generalized partially functional linear additive regression model

Checking the adequacy of functional linear quantile regression model

Statistical inference for partially linear additive spatial autoregressive models

Functional linear regression models have been widely used in the gene association analysis of complex traits. These models retain all the genetic information in the data and take full advantage of spatial information in genetic variation data, which leads to brilliant detection power. However, the significant association signals identified by the high-power methods are not all the real causal SNPs, because it is easy to regard noise information as significant association signals, leading to a false association. In this paper, a method based on the sparse functional data association test (SFDAT) of gene region association analysis is developed based on a functional linear regression model with local sparse estimation. The evaluation indicators CSR and DL are defined to evaluate the feasibility and performance of the proposed method with other indicators. Simulation studies show that: (1) SFDAT performs well under both linkage equilibrium and linkage disequilibrium simulation; (2) SFDAT performs successfully for gene regions (including common variants, low-frequency variants, rare variants and mix variants); (3) With power and type I error rates comparable to OLS and Smooth, SFDAT has a better ability to handle the zero regions. The Oryza sativa data set is analyzed by SFDAT. It is shown that SFDAT can better perform gene association analysis and eliminate the false positive of gene localization. This study showed that SFDAT can lower the interference caused by noise while maintaining high power. SFDAT provides a new method for the association analysis between gene regions and phenotypic quantitative traits.

Our work focuses on the functional linear model given by \(Y=\langle\theta,X\rangle+\epsilon,\) where Y and e are real random variables, X is a zero-mean random variable valued in a Hilbert space \((\mathcal{H},\langle\cdot,\cdot\rangle)\), and \(\theta\in\mathcal{H}\) is the fixed model parameter. Using an initial sample \(\{(X_i,Y_i)\}_{i=1}^n\), a bootstrap resampling \(Y_i^{*}=\langle\hat{\theta},X_i\rangle+\hat{\epsilon}_i^{*}\), \(i=1,\ldots,n\), is proposed, where \(\hat{\theta}\) is a general pilot estimator, and \(\hat{\epsilon}_i^{*}\) is a naive or wild bootstrap error. The obtained consistency of bootstrap allows us to calibrate distributions as \(P_X\{\sqrt{n}(\langle\hat{\theta},x\rangle-\langle\theta,x\rangle)\leq y\}\) for a fixed x, where P X is the probability conditionally on \(\{X_i\}_{i=1}^n\). Different applications illustrate the usefulness of bootstrap for testing different hypotheses related with θ, and a brief simulation study is also presented.

This paper focuses on robust estimation for varying coefficient partially functional linear regression model (VCPFLM) without specification of the error distributions. The proposed estimators of the nonparametric coefficients and slope function are based on B-spline approximation and rank regression, which demonstrates that the proposed estimation is robust for non-normal error distributions compared with that of least square estimate. In addition, the optimal convergence rates of the estimators are established under some mild regularity conditions. Finally, simulation study and a real data analysis are conducted to illustrate the finite sample performance of the proposed method.

On rates of convergence in functional linear regression

Functional linear regression model with randomly censored data: Predicting conversion time to Alzheimer ’s disease

In this paper, we consider rank estimation for partial functional linear regression models based on functional principal component analysis. The proposed rank-based method is robust to outliers in the errors and highly efficient under a wide range of error distributions. The asymptotic properties of the resulting estimators are established under some regularity conditions. A simulation study conducted to investigate the finite sample performance of the proposed estimators shows that the proposed rank method performs well. Furthermore, the proposed methodology is illustrated by means of an analysis of the Berkeley growth data.

We consider the functional linear regression model where the explanatory variable is a random surface and the response is a real random variable, in various situations where both the explanatory variable and the noise can be unbounded and dependent. Bivariate splines over triangulations represent the random surfaces. We use this representation to construct least squares estimators of the regression function with a penalisation term. Under the assumptions that the regressors in the sample span a large enough space of functions, bivariate splines approximation properties yield the consistency of the estimators. Simulations demonstrate the quality of the asymptotic properties on a realistic domain. We also carry out an application to ozone concentration forecasting over the USA that illustrates the predictive skills of the method.

Inference in functional linear quantile regression

As an important part of modern health care, medical imaging data, which can be regarded as densely sampled functional data, have been widely used for diagnosis, screening, treatment, and prognosis, such as finding breast cancer through mammograms. The aim of this paper is to propose a functional linear regression model for using functional (or imaging) predictors to predict clinical outcomes (e.g., disease status), while addressing missing clinical outcomes. We introduce an exponential tilting semiparametric model to account for the nonignorable missing data mechanism. We develop a set of estimating equations and its associated computational methods for both parameter estimation and the selection of the tuning parameters. We also propose a bootstrap resampling procedure for carrying out statistical inference. Under some regularity conditions, we systematically establish the asymptotic properties (e.g., consistency and convergence rate) of the estimates calculated from the proposed estimating equations. Simulation studies and a real data analysis are used to illustrate the finite sample performance of the proposed methods.

In this paper, we forecast ground level ozone concentrations over the USA, using past spatially distributed measurements and the functional linear regression model. We employ bivariate splines defined over triangulations of the relevant region of the USA to implement this functional data approach in which random surfaces represent ozone concentrations. We compare the least squares method with penalty to the principal components regression approach. Moderate sample sizes provide good quality forecasts in both cases with little computational effort. We also illustrate the variability of forecasts owing to the choice of smoothing penalty. Finally, we compare our predictions with the ones obtained using thin‐plate splines. Predictions based on bivariate splines require less computational time than the ones based on thin‐plate splines and are more accurate. We also quantify the variability in the predictions arising from the variability in the sample using the jackknife, and report that predictions based on bivariate splines are more robust than the ones based on thin‐plate splines. Copyright © 2012 John Wiley & Sons, Ltd.

Marker segregation distortion is a common natural phenomenon. However, relatively little is known about utilizing distorted markers for detecting quantitative trait loci (QTL). Therefore, in this study we proposed a multi-QTL mapping approach that uses distorted markers. First, the information from all markers, including distorted markers, was used to detect segregation distortion loci (SDL). Second, the information from the detected SDL was used to correct the conditional probabilities of the QTL genotypes conditional on marker information, and these corrected probabilities were then incorporated into a multi-QTL mapping methodology. Finally, the proposed approach was validated by both Monte Carlo simulation studies and real data analysis. The results from the simulation studies show that as long as one or two SDL are placed around the simulated QTL, there are no differences between the new method and the ordinary interval mapping method in terms of the power of QTL detection or the estimates of the position and dominant effects of the QTL. However, the power of QTL detection is higher under the dominant genetic model of SDL than under the additive genetic model, and the estimate for the additive effect of QTL using the new method is significantly different from the estimate obtained using ordinary interval mapping. The above results were further confirmed by the detection of QTL for dried soymilk in 222 F2:4 families in soybean.

Point process data have become increasingly popular these days. For example, many of the data captured in electronic health records (EHR) are in the format of point process data. It is of great interest to study the association between a point process predictor and a scalar response using generalized functional linear regression models. Various generalized functional linear regression models have been developed under different settings in the past decades. However, existing methods can only deal with functional or longitudinal predictors, not point process predictors. In this article, we propose a novel generalized functional linear regression model for a point process predictor. Our proposed model is based on the joint modeling framework, where we adopt a log-Gaussian Cox process model for the point process predictor and a generalized linear regression model for the outcome. We also develop a new algorithm for fast model estimation based on the Gaussian variational approximation method. We conduct extensive simulation studies to evaluate the performance of our proposed method and compare it to competing methods. The performance of our proposed method is further demonstrated on an EHR dataset of patients admitted into the intensive care units of the Beth Israel Deaconess Medical Center between 2001 and 2008.

For tting penalized Fourier regression, Wu (2013) provide some R-functions in the settings of univariate nonparametric regression, partial linear models, additive models, and nonparametric logistic and Poisson regression. We improve these functions to follow the R-function format so that users could apply generic R-functions summary, plot, lines, fitted, predict, and residuals. In addition, we write some R-function to select penalty parameters by AICc or BIC. Some data examples are used to illustrate our R-functions. A small simulation study is conducted to compare the penalized Fourier to spline approaches.