Jin-Ting Zhang

Robust principal component analysis for functional data

A method for exploring the structure of populations of complex objects, such as images, is considered. The objects are summarized by feature vectors. The statistical backbone is Principal Component Analysis in the space of feature vectors. Visual insights come from representing the results in the original data space. In an ophthalmological example, endemic outliers motivate the development of a bounded influence approach to PCA.

Local Polynomial Mixed-Effects Models for Longitudinal Data

We consider a nonparametric mixed-effects model yi(tij)=η(tij)+vi(tij)+ϵi(tij),j=1,2,…,ni;i=1,2,…,n for longitudinal data. We propose combining local polynomial kernel regression and linear mixed-effects (LME) model techniques to estimate both fixedeffects (population) curve η(t) and random-effects curves vi(t). The resulting estimator, called the local polynomial LME (LLME) estimator, takes the local correlation structure of the longitudinal data into account naturally. We also propose new bandwidth selection strategies for estimating η(t) and vi(t). Simulation studies show that our estimator for η(t) is superior to the existing estimators in the sense of mean squared errors. The asymptotic bias, variance, mean squared errors, and asymptotic normality are established for the LLME estimators of η(t). When ni is bounded and n tends to infinity, our LLME estimator converges in a standard nonparametric rate, and the asymptotic bias and variance are essentially the same as those of the kernel generalized estimating equation estimator proposed by Lin and Carroll. But when both ni and n tend to infinity, the LLME estimator is consistent with a slower rate of n1/2 compared to the standard nonparametric rate, due to the existence of within-subject correlations of longitudinal data. We illustrate our methods with an application to a longitudinal dataset.

Analysis of variance for functional data

Introduction Functional Data Motivating Functional Data Why Is Functional Data Analysis Needed? Overview of the Book Implementation of Methodologies Options for Reading This Book Nonparametric Smoothers for a Single Curve Introduction Local Polynomial Kernel Smoothing Regression Splines Smoothing Splines P-Splines Reconstruction of Functional Data Introduction Reconstruction Methods Accuracy of LPK Reconstructions Accuracy of LPK Reconstruction in FLMs Stochastic Processes Introduction Stochastic Processes x2-Type Mixtures F-Type Mixtures One-Sample Problem for Functional Data ANOVA for Functional Data Introduction Two-Sample Problem One-Way ANOVA Two-Way ANOVA Linear Models with Functional Responses Introduction Linear Models with Time-Independent Covariates Linear Models with Time-Dependent Covariates Ill-Conditioned Functional Linear Models Introduction Generalized Inverse Method Reparameterization Method Side-Condition Method Diagnostics of Functional Observations Introduction Residual Functions Functional Outlier Detection Influential Case Detection Robust Estimation of Coefficient Functions Outlier Detection for a Sample of Functions Heteroscedastic ANOVA for Functional Data Introduction Two-Sample Behrens-Fisher Problems Heteroscedastic One-Way ANOVA Heteroscedastic Two-Way ANOVA Test of Equality of Covariance Functions Introduction Two-Sample Case Multi-Sample Case Bibliography Index Technical Proofs, Concluding Remarks, Bibliographical Notes, and Exercises appear at the end of most chapters.

Approximate and Asymptotic Distributions of Chi-Squared–Type Mixtures With Applications

In this article we study how to approximate a random variable T of general chi-squared–type mixtures by a random variable of the form via matching the first three cumulants. The approximation error bounds for the density functions of the chi-squared approximation and the normal approximation are established. Applications of the results to some nonparametric goodness-of-fit tests, including those tests based on orthogonal series, smoothing splines, and local polynomial smoothers, are investigated. Two simulation studies are conducted to compare the chi-squared approximation and the normal approximation numerically. The chi-squared approximation is illustrated using a real data example for polynomial goodness-of-fit tests.

SiZer for smoothing splines

Smoothing splines are an attractive method for scatterplot smoothing. The SiZer approach to statistical inference is adapted to this smoothing method, named SiZerSS. This allows quick and sure inference as to “which features in the smooth are really there” as opposed to “which are due to sampling artifacts”, when using smoothing splines for data analysis. Applications of SiZerSS to mode, linearity, quadraticity and monotonicity tests are illustrated using a real data example. Some small scale simulations are presented to demonstrate that the SiZerSS and the SiZerLL (the original local linear version of SiZer) often give similar performance in exploring data structure but they can not replace each other completely.

Forward Variable Selection for Sparse Ultra-High Dimensional Varying Coefficient Models

ABSTRACT Varying coefficient models have numerous applications in a wide scope of scientific areas. While enjoying nice interpretability, they also allow for flexibility in modeling dynamic impacts of the covariates. But, in the new era of big data, it is challenging to select the relevant variables when the dimensionality is very large. Recently, several works are focused on this important problem based on sparsity assumptions; they are subject to some limitations, however. We introduce an appealing forward selection procedure. It selects important variables sequentially according to a reduction in sum of squares criterion and it employs a Bayesian information criterion (BIC)-based stopping rule. Clearly, it is simple to implement and fast to compute, and possesses many other desirable properties from theoretical and numerical viewpoints. The BIC is a special case of the extended BIC (EBIC) when an extra tuning parameter in the latter vanishes. We establish rigorous screening consistency results when either BIC or EBIC is used as the stopping criterion. The theoretical results depend on some conditions on the eigenvalues related to the design matrices, which can be relaxed in some situations. Results of an extensive simulation study and a real data example are also presented to show the efficacy and usefulness of our procedure. Supplementary materials for this article are available online.

Modeling HIV Dynamics Using Unified Mixed-Effects Models

SYNOPTIC ABSTRACT Studies of HIV dynamics play a crucial role in understanding the pathogenesis of HIV infection and in evaluating antiviral therapies. To accelerate AIDS clinical trials, viral load (HIV-1 RNA copies) is now used as a surrogate marker. The time range of collecting the viral load data may be divided into three successive stages. There are several existing models for the first and second stage viral load data. Recently a semiparametric nonlinear mixed-effects (NLME) model has been proposed for the complete viral load data which include the third stage viral load data, i.e., the data of those patients who fail the therapies. An important and challenging problem is if the existing models that are good only for the first one or two-stage viral load data can be generalized so that they are applicable for the complete viral load data. Another question is which model is the most preferred. In this paper, we propose a unified mixed-effects model, which models population characteristics and individual variations semiparametrically, so that all existing and recently proposed models are its special cases. We employ a basis-based approach to solve this model. We also discuss and generalize the existing and recently proposed models, including uniexponential, biexponential, multiexponential and biexponential semiparametric models. We employed AIC and BIC to compare these models and found that for the complete viral load data, the recently proposed biexponential semiparametric NLME model is the best.

Testing the equality of several covariance functions for functional data: A supremum-norm based test

Abstract Testing the equality of covariance functions is crucial for solving functional ANOVA problems. Available methods, such as the recently proposed L 2 -norm based tests work well when functional data are less correlated but are less powerful when functional data are highly correlated or with some local spikes, which are often the cases in real functional data analysis. To overcome this difficulty, a new test for the equality of several covariance functions is proposed. Its test statistic is taken as the supremum value of the sum of the squared differences between the estimated individual covariance functions and the pooled sample covariance function. The asymptotic random expressions of the test statistic under the null hypothesis and under a local alternative are derived and a non-parametric bootstrap method is suggested. The root- n consistency of the proposed test is also obtained. Intensive simulation studies are conducted to demonstrate the finite sample performance of the proposed test. The simulation results show that the proposed test is indeed more powerful than several existing L 2 -norm based competitors when functional data are highly correlated or with some local spikes. The proposed test is illustrated with three real data examples collected in a wide scope of scientific fields.

Discretization approach and nonparametric modeling for long-term HIV dynamic model

Modeling viral load response in long-term HIV dynamics is important for AIDS clinical study. It allows one to examine how a patient recuperates for an antiviral treatment. This paper investigates a discretization approach to the viral load based on a long-term HIV dynamic model. We propose a two-stage nonparametric procedure for estimating the viral load and the time-varying new productive virus. Simulation study shows that the proposed estimated methods are efficient and powerful for fitting long-term viral load measurements.

New Tests for Equality of Several Covariance Functions for Functional Data

ABSTRACT In this article, we propose two new tests for the equality of the covariance functions of several functional populations, namely, a quasi-GPF test and a quasi-Fmax  test whose test statistics are obtained via globalizing a pointwise quasi-F-test statistic with integration and taking its supremum over some time interval of interest, respectively. Unlike several existing tests, they are scale-invariant in the sense that their test statistics will not change if we multiply each of the observed functions by any nonzero function of time. We derive the asymptotic random expressions of the two tests under the null hypothesis and show that under some mild conditions, the asymptotic null distribution of the quasi-GPF test is a chi-squared-type mixture whose distribution can be well approximated by a simple-scaled chi-squared distribution. We also propose a random permutation method for approximating the null distributions of the quasi-GPF and Fmax  tests. The asymptotic distributions of the two tests under a local alternative are also investigated and the two tests are shown to be root-n consistent. A theoretical power comparison between the quasi-GPF test and the L2-norm-based test proposed in the literature is also given. Simulation studies are presented to demonstrate the finite-sample performance of the new tests against five existing tests. An illustrative example is also presented. Supplementary materials for this article are available online.