Xianyang Zhang

Testing the structural stability of temporally dependent functional observations and application to climate projections

Abstract: We develop a self-normalization (SN) based test to test the structural stability of temporally dependent functional observations. Testing for a change point in the mean of functional data has been studied in Berkes, Gabrys, Horvath and Kokoszka [4], but their test was developed under the independence assumption. In many applications, functional observations are expected to be dependent, especially when the data is collected over time. Building on the SN-based change point test proposed in Shao and Zhang [23] for a univariate time series, we extend the SNbased test to the functional setup by testing the constant mean of the finite dimensional eigenvectors after performing functional principal component analysis. Asymptotic theories are derived under both the null and local alternatives. Through theory and extensive simulations, our SN-based test statistic proposed in the functional setting is shown to inherit some useful properties in the univariate setup: the test is asymptotically distribution free and its power is monotonic. Furthermore, we extend the SN-based test to identify potential change points in the dependence structure of functional observations. The method is then applied to central England temperature series to detect the warming trend and to gridded temperature fields generated by global climate models to test for changes in spatial bias structure over time.

Simultaneous Inference for High-dimensional Linear Models

ABSTRACT This article proposes a bootstrap-assisted procedure to conduct simultaneous inference for high-dimensional sparse linear models based on the recent desparsifying Lasso estimator. Our procedure allows the dimension of the parameter vector of interest to be exponentially larger than sample size, and it automatically accounts for the dependence within the desparsifying Lasso estimator. Moreover, our simultaneous testing method can be naturally coupled with the margin screening to enhance its power in sparse testing with a reduced computational cost, or with the step-down method to provide a strong control for the family-wise error rate. In theory, we prove that our simultaneous testing procedure asymptotically achieves the prespecified significance level, and enjoys certain optimality in terms of its power even when the model errors are non-Gaussian. Our general theory is also useful in studying the support recovery problem. To broaden the applicability, we further extend our main results to generalized linear models with convex loss functions. The effectiveness of our methods is demonstrated via simulation studies. Supplementary materials for this article are available online.

Two Sample Inference for the Second-order Property of Temporally Dependent Functional Data

Motivated by the need to statistically quantify the difference between two spatio-temporal datasets that arise in climate downscaling studies, we propose new tests to detect the differences of the covariance operators and their associated characteristics of two functional time series. Our two sample tests are constructed on the basis of functional principal component analysis and self-normalization, the latter of which is a new studentization technique recently developed for the inference of a univariate time series. Compared to the existing tests, our SN-based tests allow for weak dependence within each sample and it is robust to the dependence between the two samples in the case of equal sample sizes. Asymptotic properties of the SN-based test statistics are derived under both the null and local alternatives. Through extensive simulations, our SN-based tests are shown to outperform existing alternatives in size and their powers are found to be respectable. The tests are then applied to the gridded climate model outputs and interpolated observations to detect the difference in their spatial dynamics.

Gaussian approximation for high dimensional vector under physical dependence

We develop a Gaussian approximation result for the maximum of a sum of weakly dependent vectors, where the data dimension is allowed to be exponentially larger than sample size. Our result is established under the physical/functional dependence framework. This work can be viewed as a substantive extension of Chernozhukov et al. (2013) to time series based on a variant of Stein’s method developed therein.

Conditional mean and quantile dependence testing in high dimension

Motivated by applications in biological science, we propose a novel test to assess the conditional mean dependence of a response variable on a large number of covariates. Our procedure is built on the martingale difference divergence recently proposed in Shao and Zhang (2014), and it is able to detect a certain type of departure from the null hypothesis of conditional mean independence without making any specific model assumptions. Theoretically, we establish the asymptotic normality of the proposed test statistic under suitable assumption on the eigenvalues of a Hermitian operator, which is constructed based on the characteristic function of the covariates. These conditions can be simplified under banded dependence structure on the covariates or Gaussian design. To account for heterogeneity within the data, we further develop a testing procedure for conditional quantile independence at a given quantile level and provide an asymptotic justification. Empirically, our test of conditional mean independence delivers comparable results to the competitor, which was constructed under the linear model framework, when the underlying model is linear. It significantly outperforms the competitor when the conditional mean admits a nonlinear form.

Distance Metrics for Measuring Joint Dependence with Application to Causal Inference

Abstract Many statistical applications require the quantification of joint dependence among more than two random vectors. In this work, we generalize the notion of distance covariance to quantify joint dependence among random vectors. We introduce the high-order distance covariance to measure the so-called Lancaster interaction dependence. The joint distance covariance is then defined as a linear combination of pairwise distance covariances and their higher-order counterparts which together completely characterize mutual independence. We further introduce some related concepts including the distance cumulant, distance characteristic function, and rank-based distance covariance. Empirical estimators are constructed based on certain Euclidean distances between sample elements. We study the large-sample properties of the estimators and propose a bootstrap procedure to approximate their sampling distributions. The asymptotic validity of the bootstrap procedure is justified under both the null and alternative hypotheses. The new metrics are employed to perform model selection in causal inference, which is based on the joint independence testing of the residuals from the fitted structural equation models. The effectiveness of the method is illustrated via both simulated and real datasets. Supplementary materials for this article are available online.

Predictive modeling of microbiome data using a phylogeny-regularized generalized linear mixed model

Recent human microbiome studies have revealed an essential role of the human microbiome in health and disease, opening up the possibility of building microbiome-based predictive models for individualized medicine. One unique characteristic of microbiome data is the existence of a phylogenetic tree that relates all the microbial species. It has frequently been observed that a cluster or clusters of bacteria at varying phylogenetic depths are associated with some clinical or biological outcome due to shared biological function (clustered signal). Moreover, in many cases, we observe a community-level change, where a large number of functionally interdependent species are associated with the outcome (dense signal). We thus develop “glmmTree,” a prediction method based on a generalized linear mixed model framework, for capturing clustered and dense microbiome signals. glmmTree uses the similarity between microbiomes, which is defined based on the microbiome composition and the phylogenetic tree, to predict the outcome. The effects of other predictive variables (e.g., age, sex) can be incorporated readily in the regression framework. Additional tuning parameters enable a data-adaptive approach to capture signals at different phylogenetic depth and abundance level. Simulation studies and real data applications demonstrated that “glmmTree” outperformed existing methods in the dense and clustered signal scenarios.