Rank method for partial functional linear regression models

Parameter regimes in partial functional panel regression

<p>In modern manufacturing, the efficiency and accuracy of quality inspection significantly influence production costs and product quality. With the development of sensing technology, signals of process variables can be collected in high resolution, which can be regarded as multichannel profile data. They have abundant information to characterize the multistage manufacturing process and help with quality inspection tasks. Motivated by the urge to take the place of manual inspection, we target at modeling based on multichannel profile data for data-driven quality inspection. However, quality inspection modeling based on multichannel profile data is challenging due to the complexity of data structures and manufacturing processes. Specifically, in the pipe tightening process, accurate diagnosis of defect types is required, but the available samples for each defect type are limited and imbalanced. Moreover, the profile data is incomplete since the pre-tightening process before the pipe tightening process is unobserved. In the ceramic firing process, the identification of crucial phases and stages is required. Nevertheless, the process data streams from different stages are unsynchronized and high-dimensional, and a hierarchical structure exists between phases and stages. To tackle the challenges, we propose models based on functional data analysis, machine learning, and deep learning. For the pipe tightening process, we propose an innovative classification framework to train on imbalanced datasets based on deep metric learning. A neural network and padding mechanism specially crafted for processing profile data are also proposed to handle incomplete profile data. For the ceramic firing process, we propose a real-time diagnostic system, facilitating the real-time synchronization of unsynchronized and high-dimensional process data streams. We also develop a hierarchical sparse partial functional linear regression model (HSPFLR) and corresponding parameter estimation algorithm, enabling the simultaneous identification of crucial phases and stages. The effectiveness of our frameworks is demonstrated through simulation studies and real-world case studies. The proposed methods can replace traditional manual inspection methods, improving product quality and reducing inspection costs.</p>

In this paper, estimation of the linear regression\nmodel is made by ordinary least squares method and the\npartially linear regression model is estimated by penalized\nleast squares method using smoothing spline. Then, it is\ninvestigated that differences and similarity in the sum of\nsquares related for linear regression and partial linear\nregression models (semi-parametric regression models). It is\ndenoted that the sum of squares in linear regression is reduced\nto sum of squares in partial linear regression models.\nFurthermore, we indicated that various sums of squares in the\nlinear regression are similar to different deviance statements in\npartial linear regression. In addition to, coefficient of the\ndetermination derived in linear regression model is easily\ngeneralized to coefficient of the determination of the partial\nlinear regression model. For this aim, it is made two different\napplications. A simulated and a real data set are considered to\nprove the claim mentioned here. In this way, this study is\nsupported with a simulation and a real data example.

Checking the adequacy of functional linear quantile regression model

This study examines the optimal selections of bandwidth and semi-metric for a functional partial linear model. Our proposed method begins by estimating the unknown error density using a kernel density estimator of residuals, where the regression function, consisting of parametric and nonparametric components, can be estimated by functional principal component and functional Nadayara-Watson estimators. The estimation accuracy of the regression function and error density crucially depends on the optimal estimations of bandwidth and semi-metric. A Bayesian method is utilized to simultaneously estimate the bandwidths in the regression function and kernel error density by minimizing the Kullback-Leibler divergence. For estimating the regression function and error density, a series of simulation studies demonstrate that the functional partial linear model gives improved estimation and forecast accuracies compared with the functional principal component regression and functional nonparametric regression. Using a spectroscopy dataset, the functional partial linear model yields better forecast accuracy than some commonly used functional regression models. As a by-product of the Bayesian method, a pointwise prediction interval can be obtained, and marginal likelihood can be used to select the optimal semi-metric.

We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time. Predictor and response processes have smooth random trajectories, and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects. In longitudinal studies, the number of repeated measurements per subject is often small and may be modeled as a discrete random number and, accordingly, only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit. We propose a functional regression approach for this situation, using functional principal component analysis, where we estimate the functional principal component scores through conditional expectations. This allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory. The resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories. Asymptotic properties for a sample of n subjects are investigated under mild conditions, as n→∞, and we obtain consistent estimation for the regression function. Besides convergence results for the components of functional linear regression, such as the regression parameter function, we construct asymptotic pointwise confidence bands for the predicted trajectories. A functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced, extending the standard R2 to the functional case. The proposed methods are illustrated with a simulation study, longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index.

Local polynomial estimation in partial linear regression models under dependence

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.

This paper considers partial linear single-index regression models when all the variables are measured with multiplicative distortion measurement errors. To eliminate the effect caused by the distortion, we propose the conditional absolute mean calibration. This method avoids to use the nonzero expectation conditions imposed on the variables in the literature. Using the calibrated variables, a profile least squares estimator is obtained. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. A smoothly clipped absolute deviation penalty is employed to select the relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.

Inference in functional linear quantile regression

Partially functional linear regression in reproducing kernel Hilbert spaces

This article considers the analysis of clustered data via partial linear regression models. Adopting the idea of modeling the within-cluster correlation from the method of generalized estimating equations, a least squares type estimate of the slope parameter is obtained through piecewise local polynomial approximation of the nonparametric component. This slope estimate has several advantages: (a) It attains n1/2-consistency without undersmoothing; (b) it is efficient when correct within-cluster correlation is used, assuming multivariate normality of the error; (c) the preceding properties hold regardless of whether or not the nonparametric component is of cluster level; and (d) this estimation method naturally extends to deal with generalized partial linear models. Simulation studies and a real example are presented in support of the theory.

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.

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